Alfredo Dubra's PhD thesis - Imperial College London

A Shearing Interferometer for
the Evaluation of Human
Tear Film Topography
Alfredo Dubra Suárez
Submitted in partial fulfilment of the requirements for the degree of Doctor of
Philosophy

Abstract
The evaluation of the optical quality of the human eye beyond spectacle prescription
has recently become a very active field, given the number and importance of
applications that can benefit from it: refractive surgery, retinal imaging, monitoring
of intraocular lens performance, psychophysical experiments and adaptive optics
spectacles for improving visual acuity and increasing accommodation range in presbyopic
subjects. Several ophthalmic wavefront sensors have been recently developed
to measure the imaging properties of the eye, most of them based on the use of a
Shack-Hartmann configuration.
Despite the current widespread use of wavefront sensing based refractive surgery, there
are a number of related issues not yet well understood, one of the most important
being the variability of the wavefront sensor measurements due to eye movements,
heart beat, fluctuations of accommodation and tear film dynamics.
The work in this thesis focuses on the study of tear film dynamics and its effects on
the optical quality of the eye. In order to do so, a lateral shearing interferometer to
measure the pre-corneal tear film topography was designed, built and used to study
20 subjects. The collected data was processed with a novel reconstruction technique,
which is applicable to any lateral shearing interferometry experiment, and the results
are discussed in terms of the tear film influence of the optical quality of the eye, which
can be used to understand the effects on vision, wavefront sensing and adaptive optics.
Finally, the feasibility of using interferometry to evaluate the wavefront error of the
eye by modifying the lateral shearing interferometer was also tested.
1

Acknowledgements
Pursuing the PhD degree at Imperial has been a life changing experience both inside
and outside the lab, and for this, I have to thank a few people.
To begin with, thanks Paula for your support in the mad idea of coming to London
to start a PhD with no funding. I then have to say that I am greatly indebted to
my supervisor Chris Dainty for getting this funding (and Luis Diaz-Santana) and
providing the academic freedom and confidence in the project. I am also grateful
to Chris for the work environment he created both in the Applied Optics groups at
Imperial and NUI Galway.
Much of the research undertaken for this thesis was the result of several discussions
with Carl Paterson, even before he was my supervisor.
The experiments performed during the PhD were only possible thanks to the patience
of and quality of work from, the guys in the mechanical and optics workshop: Paul
Brown, Martin Kehoe, Simon Johnson and Martin Dowman.
Thanks to all the people that volunteered to put their eyes in front of a laser beam and
have their mouths stuck in a wax block for nothing in return. That is true generosity.
It is said that one should not mix business with pleasure, but the most valuable thing
I will take from my PhD is the friends I made at work; thanks to Fred Reavell for his
immense patience with my poor engineering skills, the explanation of English culture
and motorcycle tuition. Being in a lab with no windows and having to deal with the
same project for three years (and a series of unsuccessful experiments) could have
driven me insane, had it not been for the company of David Lara, Steve Gruppetta,
Jonathan Brooks, Karen Hampson, Gordon Kennedy, James McIlroy, Ian Munro,
David Catlin, Simon Clay and Allison Craig.
There have been some seriously hard times on a personal level during the PhD, which
I only came through with the company and support from Mo and Si. Thanks to them
I came closer to the true English culture (and slang), sometimes difficult to find in
such a cosmopolitan place and more importantly, I found two friends for life.
There was a time when working long hours in front of the computer and a lonely lab
made me lose all perspective of life. Fortunately, Chris Dainty took me to Galway for
2

a few months, where I found it (maybe for the first time). This only happened thanks
to the friendship of David Lara, Andrew Cronin, Larry Murray and David Merino.
Thanks to them for repeatedly telling me what is really important over and over again
(yes, sometimes I can be a bit stubborn).
I also have to thank to all my family for making me feel supported all the way through,
despite the physical distance.
Finally, thanks to Cecilia, for reminding me that there is life outside the lab.
Alf
London, June 2004
3

Contents
Contents 4
List of Figures 8
1 Introduction 12
1.1 Wavefront sensing in the human eye . . . . . . . . . . . . . . . . . . . 14
1.1.1 Applications of wavefront sensing in the human eye . . . . . . . 17
1.1.2 Repeatability in wavefront sensing measurements . . . . . . . . 18
1.2 Tear film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3 Thesis synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Lateral shearing interferometer design 24
2.1 System requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Lateral shearing interferometry . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Interferogram intensity modulation . . . . . . . . . . . . . . . . . . . . 26
2.4 Wedge design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Wedge manufacture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Illumination branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6.1 Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6.2 Spatial filter and telescope . . . . . . . . . . . . . . . . . . . . . 32
2.6.3 Polarizing beam splitter and quarter waveplate . . . . . . . . . 33
2.6.4 Focusing lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7 Imaging branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.7.1 Shear and tilt estimation . . . . . . . . . . . . . . . . . . . . . 39
2.7.2 Polarisation control for optimization of interferogram contrast . 39
3 Data processing 42
3.1 Carrier frequency estimation . . . . . . . . . . . . . . . . . . . . . . . 42
4

CONTENTS
3.1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.3 Implementation details . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.4 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Shear estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.3 Implementation details . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.4 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Pupil position and size estimation . . . . . . . . . . . . . . . . . . . . 53
3.3.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.2 Implementation details . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.3 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Phase recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.2 Implementation details . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.3 Noise introduced by undesired reflections . . . . . . . . . . . . 58
3.4.4 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 Phase unwrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5.2 Implementation details . . . . . . . . . . . . . . . . . . . . . . . 61
3.5.3 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 Wavefront reconstruction from shear phase maps 67
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.1 The transfer function of the reconstruction problem . . . . . . 69
4.2.2 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.3 Data extension and extension method . . . . . . . . . . . . . . 70
4.2.4 Filter poles and subdivision method . . . . . . . . . . . . . . . 72
4.3 Computing performance . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4 Noise performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 Preliminary experiments 79
5

CONTENTS
5.1 Prydal type experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1.1 Normal illumination experiment . . . . . . . . . . . . . . . . . 80
5.1.2 Focused illumination and estimation of undesired eye reflections 82
5.2 Estimation of errors for tear topography measurements . . . . . . . . . 84
5.2.1 Repeatability test and associated error . . . . . . . . . . . . . . 85
5.2.2 Alignment and shear error propagation . . . . . . . . . . . . . 86
5.2.3 Simulated interferograms and non-linear errors . . . . . . . . . 88
5.3 Validation experiment with a deformable mirror . . . . . . . . . . . . . 88
5.4 Small detail test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6 Tear topography dynamics experiment 90
6.1 Data collection protocol . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.2 Eye movements and effect on tear topography measurements . . . . . 92
6.2.1 Effect of eyeball rotation on tear topography experiment . . . . 92
6.2.2 Effect of head movement on tear topography . . . . . . . . . . 95
6.2.3 Asymmetries in the optical system . . . . . . . . . . . . . . . . 95
6.3 Global error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.4 Wavefront error RMS introduced by the tear topography . . . . . . . . 97
6.4.1 Wavefront error RMS evolution for individual series . . . . . . 98
6.4.2 RMS mean values . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.4.3 RMS mean evolution . . . . . . . . . . . . . . . . . . . . . . . . 101
6.5 Second moment of the AC term . . . . . . . . . . . . . . . . . . . . . . 103
7 Summary and discussion 107
7.1 Future work: Radial shearing interferometry . . . . . . . . . . . . . . . 110
7.2 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
A Preliminary study of feasibility of interferometric wavefront sensing
in the eye 112
A.1 Experimental setups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
A.1.1 Experimental setup for 632.8 nm . . . . . . . . . . . . . . . . . 114
A.1.2 Experimental setup for 780 nm . . . . . . . . . . . . . . . . . . 115
A.2 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.2.1 Data acquired with 632.8 nm source . . . . . . . . . . . . . . . 117
A.2.2 Data acquired with 780 nm source . . . . . . . . . . . . . . . . 118
A.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6

CONTENTS
B Laser safety calculations 124
B.1 MPE for He-Ne laser in tear topography experiment . . . . . . . . . . 125
B.2 MPE for He-Ne laser in full ocular wavefront sensor . . . . . . . . . . 125
B.3 MPE for infrared LEDs . . . . . . . . . . . . . . . . . . . . . . . . . . 125
B.4 MPE for infrared laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Bibliography 127
7

List of Figures
1.1 Young’s experiment for demonstrating that the capability of changing
the fixating distance of the human eye is due to the lens. . . . . . . . . 13
1.2 Schematic representation of Scheiner’s principle, used to prescribe spectacles.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Gullstrand’s method for estimating the aberrations introduced by the
cornea. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Viewing chart proposed by Helmholtz to verify the existence of aberrations
in the human eye. . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Glass wedge geometry and angle definition for reflected rays. . . . . . 27
2.2 Lateral shearing interferogram spectra produced by a glass wedge. . . 29
2.3 Illumination system of the lateral shearing interferometer. . . . . . . . 32
2.4 Relative intensities of reflections from a 50-50 non- polarising beam
splitter (NPBS) and a polarising beam splitter (PBS). . . . . . . . . . 33
2.5 Interference patterns that result from using different lenses in the illumination
branch of the shear interferometer and an artificial cornea. . 36
2.6 Imaging branches of the double shearing interferometer used for tear
topography estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.7 Sketch of the 3 dimensional optical setup for tear topography estimation
with double lateral shearing interferometry. . . . . . . . . . . . . . . . 38
2.8 Effect of the orientation of the polarisation state incident on the first
glass wedge on the intensity and contrast of the interferograms produced
by the glass wedges in the optical system illustrated in figure 2.7. . . . 41
3.1 Estimation of the interferogram modulation carrier frequency intensity
modulation from spectra. . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 As figure 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
8

LIST OF FIGURES
3.3 Geometry of autocorrelation of an image consisting of two sheared
copies of an object with finite dimensions. . . . . . . . . . . . . . . . . 48
3.4 Improvement on shear estimation algorithm by filtering in Fourier domain. 49
3.5 Shear estimation algorithm applied to tear film interferograms with
different features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6 As figure 3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.7 Center and radius estimation algorithm applied to tear film interferograms
with different features. . . . . . . . . . . . . . . . . . . . . . . . 55
3.8 As figure 3.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.9 Outputs of Takeda’s phase recovery method applied to tear interferograms.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.10 As figure 3.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.11 Phase unwrapping applied to different tear topographies. . . . . . . . 65
3.12 As figure 3.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1 Illustration of the method for extending the difference data outside the
pupil, by copying the difference values in the boundary of the pupil
outside the pupil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Example of the interlaced subgrids for a grid of dimensions N x = 6 by
N y = 7 and shears s x = 3 and s y = 2. . . . . . . . . . . . . . . . . . . 73
4.3 a) Plot of the number of real multiplications required to perform the
wavefront reconstruction over a square grid with a total of N points and
shear s

LIST OF FIGURES
5.2 Sequence of reflections from the front surface of the tear illustrating
bubble movements on the tear. . . . . . . . . . . . . . . . . . . . . . . 83
5.3 Images obtained by illuminating the eye with a converging beam and
placing the front surface of the eye roughly at the beam focus, and
simulated interferograms. . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Reconstructed topography of two simulated small bubbles. . . . . . . . 89
6.1 Light scattered, as opposed to the specular reflections, by the cornea
and the crystalline lens when the tear topography experiment was being
performed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Eyeball modelled as two spherical surfaces. . . . . . . . . . . . . . . . 93
6.3 Estimated movement of the center of the illuminated area of the tear. 94
6.4 Correlation between the relative change in the radius of the interferograms
and the estimated change in defocus. . . . . . . . . . . . . . . . 96
6.5 Typical wavefront RMS, defocus and astigmatism evolution with respect
to the diffraction limit. . . . . . . . . . . . . . . . . . . . . . . . 99
6.6 Mean RMS and residual RMS over the data series with respect to the
diffraction limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.7 Temporal evolution of the estimated wavefront error RMS and residual
wavefront error RMS for 30 data series corresponding to 19 different
subjects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.8 Normalized second order moment of the AC term of tear topography
lateral shearing interferograms (M N ) illustrating some of the different
tear topography features. . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.1 Modified imaging branch to convert the lateral shearing interferometer
into a radial shearing interferometer. . . . . . . . . . . . . . . . . . . . 110
7.2 Typical radial shearing interferograms from tear film topography with
no tilt between the interfering wavefronts. Recorded with the experimental
setup described in figure 7.1. . . . . . . . . . . . . . . . . . . . 111
A.1 Illumination branch of the wavefront sensing experiment using a 632.8 nm
He-Ne laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
A.2 Imaging branch of the wavefront sensing experiment using a 632.8 nm
He-Ne laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
10

LIST OF FIGURES
A.3 Illumination branch of the wavefront sensing experiment using a 780 nm
diode laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.4 Imaging branch of the wavefront sensing experiment using a 780 nm
diode laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.5 Interferograms and SH spot patterns that resulted from using 3.5, 1.0
and 0.35 µW at λ = 632.8 nm. . . . . . . . . . . . . . . . . . . . . . . . 119
A.6 Interferograms and SH patterns that resulted from the 4 µW at 780 nm
setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.7 Interferograms and SH patterns that resulted from the 4 µW at 780 nm
setup with accommodation paralyzed with cyclopentolate hydrochloride
1%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
B.1 Definition of subtended angle α by an apparent source at a distance r. 124
11

Chapter 1
Introduction
Assessing and improving the optical quality of the human eye can be traced back as
far as the 13 th century, when defocus was initially corrected by the use of spectacles.
However, it was not until the 18 th century that monochromatic aberrations in the
optics of the human eye other than defocus were studied by Thomas Young [1]. In his
work, Young describes a series of fundamental experiments related to the imaging and
focusing properties of the eye, like the one shown in figure 1.1, where it is demonstrated
that the focusing capability of the eye is purely due to the change of shape of the
crystalline lens. Young also reported noticing that his own eyes were astigmatic and
consulting with a colleague to find this was not unusual and that some people used
to hold a positive lens obliquely to compensate for it. He investigated this topic
further by modifying a method proposed by Scheiner in 1619 to prescribe spectacles
[2]. The principle of this method is illustrated in figure 1.2: in an refractive error-free
(emmetropic) eye an incoming parallel pencil of rays of monochromatic light will be
focused at the retina as shown in figure 1.2 (a) and therefore if a mask with two
small holes is placed in front of the eye as in (b) the subject will see one spot. If
on the other hand, the eye is ametropic, then the light from the incoming beam will
be focused either in front (myopia) or behind the retina (hyperopia), producing two
spots on the retina instead of one as shown in (c) and (d). Young’s modification to
this experiment consisted of using 4 slits instead of 2 holes, and found that when the
eye of the subject had aberrations other than defocus or astigmatism, there was no
lens sphero-cylindrical correction that could make the image of the four slits coincide
onto the retina, showing the presence of other aberrations in the eye.
12

1. Introduction
Figure 1.1: Young’s experiment for demonstrating that the capability of changing the
fixating distance of the human eye is due to the lens. With the face looking down,
the eye is placed on top of a a small cuvette filled with water so that the cornea
is submerged. At the bottom of the cuvette there is a lens with the same refractive
power as the cornea. Then by looking through this lens, focusing at different distances,
Young observed that the accommodation range was the same as without the cuvette,
confirming then that the accommodation of the eye is due entirely to the lens and
not to the cornea. This image was reproduced from Young’s Bakerian lecture on the
mechanism of the eye [1].
(a)
(b)
(c)
(d)
Figure 1.2: Schematic representation of Scheiner’s principle, used to prescribe spectacles.
In an emmetropic eye, an incoming parallel pencil of rays will be focused at
the retina as shown in (a). Therefore, by placing a mask with two small holes the
subject will see one spot when the eye is emmetropic (b) and two when the eye is
either myopic (c) or hyperopic (d).
13

1. Introduction
(a)
(b)
Figure 1.3: Gullstrand’s method for estimating the aberrations introduced by the
cornea. The method is a rudimentary corneal topography, consisting of imaging a set
of squares like the one in (a) after reflection from the first corneal surface and measuring
its deformation as shown in (b). This images were reproduced from Helmholtz’s
Handbuch der Physiologischen Optik [3].
One century later, Gullstrand [3] studied extensively the monochromatic aberrations
of the eye, the influence of the concavity of the tear in vision quality, the scattering
introduced by the fibres in the lens and the cornea and the lack of rotational symmetry
shown by the aberrations in the eye. In his work, Gullstrand tried to characterize the
aberrations of the eye in terms of the caustic, and studied the aberrations introduced
by the cornea by performing a very elementary corneal topography, consisting of
imaging a set of distorted squares on the front corneal surface and measuring their
deformation, as shown in figure 1.3.
1.1 Wavefront sensing in the human eye
We can think of wavefront sensing in the eye as the evaluation of the monochromatic
aberrations of the eye other than piston, tip and tilt, as these do not degrade the
imaging properties of the eye. We could also say that Young is the pioneer of pupilplane
wavefront sensing in the eye, while Gullstrand gave the first steps in image-plane
wavefront sensing in the human eye. However, we can consider that the modern era
of wavefront sensing in the eye did not start until 1961, with the work by Smirnov [4]
where a subjective experiment was set up to sample the wavefront slope across the
pupil and then to estimate the point spread function (PSF) of the eye. Smirnov also
suggested the tear film as a possible source for the noticeable inter-subject variability
14

1. Introduction
in the measurements and noticed that the aberrations change with the accommodating
distance. In the same work, Smirnov suggested the use of contact lenses for correcting
high order aberrations. Many of Smirnov’s ideas and observations are the basis of
modern wavefront sensing techniques, both subjective and objective and some of his
ideas, such as the compensation of the aberrations have been put into practice.
Following Smirnov’s work, at least two more subjective or psychophysical wavefront
sensing methods (i.e., those in which the aberrations of the eye are estimated from
responses provided by the subject being examined) have been developed, based on
wavefront slope evaluation in the pupil plane. Firstly, the cross-cylinder aberroscope
proposed by H. and B. Howland [5], in which a grid is sandwiched between two crossed
cylindrical lenses and projected onto the retina. The retinal image of the grid is
distorted by the aberrations of the eye. The subject is asked to draw this grid so
that the aberrations can be estimated the drawings. This technique did not become
popular given the strong reliance on the subject’s ability to replicate the image seen.
Later, in 1992 Webb et al.[6] and then He et al.[7] built more automated versions
of Smirnov’s experiment, where the wavefront slope at different positions within the
pupil is measured by feedback provided by the vernier acuity of the subject.
In 1969 Berny [8] used a completely different approach by performing a Foucault
knife-edge test experiment to measure the wavefront aberration itself, rather than the
wavefront slope.
With the development of more sensitive films and cameras, a number of objective
techniques for evaluating the aberrations of the eye were developed in recent years.
In 1984 Walsh et al. [9, 10, 11] transformed the subjective Howland and Howland
cross-cylinder aberroscope into an objective method by recording the image of a grid
projected onto the retina with a camera.
In 1997, Navarro et al.[12, 13, 14] developed an instrument to evaluate the slope of the
wavefront, sampling the pupil at different locations with a narrow collimated incoming
beam and recording its displacement on the retinal surface with respect to a reference
spot.
Recently, in 1996 Ragazzoni [15] proposed a new slope wavefront sensor, in which the
beam to analyze is moved in a circular pattern over a glass pyramid producing four
images that are combined to estimate local wavefront slopes. This wavefront sensor
15

1. Introduction
was then modified and tested in the eye by Iglesias et al. [16].
Iglesias et al. also proposed an image-plane wavefront sensing method based on applying
phase retrieval algorithms to an experimentally estimated point spread function
(PSF) for the eye [17, 18, 19].
In 1994 Liang et al. [20, 21] introduced the use of the Shack-Hartmann sensor in the
eye, which was then to become the preferred instrument for wavefront sensing in the
eye. In this wavefront sensor, an array of lenslets placed in a plane conjugated to the
pupil of the eye forms an array of spots on a camera. The displacement of these spots
with respect to their reference position is proportional to the average wavefront slope
at the plane of the lenslet. Since the work by Liang et al., an increasing amount of
research has been done in the field of wavefront sensing in the eye, leading to a better
understanding of the field. Artal et al. [22] recognized the double pass problem as an
autocorrelation. Therefore, if equal entrance and exit pupils are used in a wavefront
sensing instrument, the odd aberrations of the eye would be underestimated in the
measurements. Artal then proposed using unequal entrance and exit pupil sizes to
minimize this problem. To tackle the same problem, Díaz and Dainty [23] tested an
original approach, setting up a single pass experiment by using the autofluorescence
of the retinal lipofuscin as a light source for wavefront sensing. Salmon et al. [24]
explored the problem further by performing a comparison between a Smirnov-type
single-pass experiment and a Shack-Hartmann sensor (double-pass) with unequal entrance
pupils, concluding that the results were similar to within experimental errors.
López and Artal [25] and then Llorente et al. [26] explored the difference in the
estimated aberrations for different wavelengths, which is a vital point for clinical applications.
Ideally, one would like to use visible wavelengths, but this causes the iris of
the eye to contract, and therefore pupil dilation drugs would be needed to measure the
aberrations over large pupils. On the other hand, if infrared wavelengths are used, a
large pupil can be used without the need for drugs, but then the presence of chromatic
aberration in the eye and different scattering characteristics need to be accounted for.
Thibos et al. [27] noticed that the Shack-Hartmann measurements can be affected by
tear film break-up and cataracts, which suggests that the measurements from wavefront
sensors should not be used alone without other studies such as retro-illumination
images. More recently, some experimental studies have been performed to characterize
the dynamics of the aberrations [28] and their statistics [29], but theoretical modelling
is still pending.
16

1. Introduction
1.1.1 Applications of wavefront sensing in the human eye
There are a number of applications for wavefront sensing in the eye of which the
most widespread in clinical applications is refractive surgery, in its different techniques:
photorefractive keratectomy (PRK) and laser-assisted in situ keratomileusis
(LASIK), where only sphere and cylinder are corrected, and the more recent wavefrontguided
LASIK, also referred to as custom ablation, where higher order aberrations
are accounted for [30, 31]. These operations are currently being performed on more
than a million people per year. The importance of good understanding of the wavefront
sensing measurements cannot be overemphasized, given the potential permanent
degradation to vision in such large number of individuals. There are currently several
patents related to ophthalmic wavefront sensors [32, 33, 34, 35, 36] and also commercial
instruments available in the market, like the BriteEye Aberrometer from Suzhou
Medical General Factory, the Zywave wavefront aberrometer from Bausch and Lomb,
the WASCA Analyser from Carl Zeiss Meditec, the LADARWave from Alcon, the KR-
9000PW Wavefront Analyzer from Topcon and the COAS from Wavefront Sciences.
Wavefront sensing in the eye in combination with corneal topography, allows the
study of the aberrations of the crystalline lens, which could lead to better design
and evaluation of intraocular lenses. These lenses are currently being implanted in
numbers comparable to or larger than those involved in refractive surgery [37].
Adaptive optics has proven to be the most challenging application of wavefront sensing
in the eye to implement in a reliable manner. Adaptive optics was first proposed by
Babcock in 1953, for compensating the aberrations introduced by the atmosphere in
telescope images [38], and based on the same principles and technology it has recently
been applied to the correction of the aberrations of the human eye [21, 39, 40, 41, 42].
In principle, all ophthalmic imaging instruments such as fundus cameras [43, 44, 45,
46, 47], confocal laser scanning ophthalmoscopes (CLSOs) [48, 49, 50, 41] and optical
coherence tomography systems (OCTs) [51, 52] can benefit by improving the optical
quality of the eye with an adaptive optics system to a resolution in which individual
photoreceptors can be resolved within small retinal patches. If high-resolution imaging
is required over a retinal patch larger than the isoplanatic angle [53, 54, 55], then a
more complex multi-conjugate adaptive optics system would be needed.
Improving the resolution of retinal images down to almost the diffraction limit has also
17

1. Introduction
been achieved by using deconvolution techniques on series of retinal images using the
information on the aberrations of the eye obtained with wavefront sensors [56, 57, 58].
These deconvolution techniques might be cheaper and easier to implement alternatives
to adaptive optics.
A less viable application for wavefront sensing in the eye is the design of phase plates
that can be used as spectacles for superior vision improvement in comparison with
the usual sphero-cylindrical correction [59, 60]. The drawback of this is that the best
optical quality would only be achieved when looking in a particular direction. When
the eye turns to look in a slightly different direction, vision will almost certainly be
worse than with a normal pair of spectacles.
One of the applications that attracts the most attention to the field of wavefront
sensing in the eye is the possibility of achieving what has been called supernormal
vision, through the use of adaptive optics. This has so far only been achieved in
laboratory experimental setups [43, 61, 62], with instruments of sizes and weights
far from the size and weight of a normal pair of spectacles (one and three order of
magnitudes larger respectively). A similarly challenging goal is the use intraocular
adaptive optics either to compensate presbyopia or to improve visual acuity. This
possibility has started to be explored by Vdovin et al. [63] who built and tested a
small liquid crystal correction element of size comparable to that of a crystalline lens.
Adaptive optics in the eye might also allow, in the near future, psychophysical experiments
in which light is focused onto single photoreceptors, to gain knowledge of their
response to different wavelengths. So far this has not been achieved due to technical
difficulties and problems not well understood yet. In the same way, the theoretical
imaging diffraction limit has not yet been achieved experimentally.
1.1.2 Repeatability in wavefront sensing measurements
The variability of wavefront sensing measurements in the eye was identified in the very
early stages of the field by Smirnov, and a number of sources for this variability have
been suggested: fluctuations of accommodation, change in axial length and corneal
shape due to the heart beat, eye and head movement and tear film dynamics. We
believe that the understanding of the effect of these sources in wavefront sensing
measurements could lead to better use of the measurements, for example in the context
18

1. Introduction
Figure 1.4: Viewing chart proposed by Helmholtz to verify the existence of aberrations
in the human eye. When this target is viewed with one eye and a steady fixation,
some rotating sectors can be seen. The time course of the movements correspond to
that of the fluctuations of the accommodation.
of refractive surgery. Reported values for the variability of the root-mean-squared
(RMS) of the wavefront error in the eye are typically of the order of 0.1 µm [4, 24,
64, 17, 21]. These values are not an issue for spectacle prescription. 0.1 µm of sphere
(defocus) for an 8 mm pupil corresponds only to a twentieth of a diopter (D), which is
less than half the minimum step between consecutive ophthalmic lenses. On the other
hand, the variability observed is of concern when considering a diffraction limited
performance imaging system for the eye (e.g. fundus camera), as the wavefront RMS
should be kept below one fourteenth of a wavelength according to Marechal’s criterion,
that is 0.03 − 0.05 µm for visible wavelengths [10].
Is is very likely that the largest source of variability of the optical quality of the eye
is the fluctuations of accommodation, whose amplitude and frequency were studied
by Campbell et al. [65] and Gray et al. [66]. The measured fluctuations depend
on several factors like target luminance, pupil diameter, target form, vergence and
contrast, with amplitudes as large as 0.5 D. If we consider that a typical fluctuation
amplitude is around 0.2 D, which seems reasonable from Charman’s review [67], this
is equivalent to a wavefront error RMS of approximately 0.5 µm for an 8 mm diameter
pupil. This is one order of magnitude larger than the wavefront aberration that is
needed to take a perfect optical system out of the diffraction limit regime. For this
reason, it is common practice when using both wavefront sensors and adaptive optics
in research environments, to paralyze or reduce these fluctuations by the use of drugs
(usually cyclopentolate) [37, 28, 21, 17, 25, 14, 68]. A simple experiment to verify the
existence of these microfluctuations can be performed with the aid of figure 1.4.
19

1. Introduction
A less significant but not negligible source of variability seems to be the axial length
change of the eye when the heart beats. Hofer [28] quotes that typical changes in the
axial length of 4 µm would produce defocus changes of 0.01 D which for an 8 mm pupil
corresponds to a wavefront RMS of 0.02 µm.
The role of eye movement in wavefront sensing measurements has, to our knowledge,
not yet been studied on its own, although the movements of the eye themselves are
well studied and their characteristics appear in current ophthalmology textbooks such
as Adler’s [69]. For example, it has been proved that when the eye is fixating it has
three characteristic movements, namely microsaccades of ∼ 5 arcmin of amplitude at
a frequency of 2 − 3 Hz; drift ∼ 1 arcmin every 2 − 5 min and high frequency tremor
∼ 40 arcsec at 90 Hz.
There seems to be agreement in the ophthalmic wavefront sensing community about
the tear film playing a role in the variability of measurements [4, 70, 39, 46, 71, 28,
58, 72, 21, 24, 68, 60], although to our knowledge before the research described in
this thesis there was no quantitative study on the tear topography dynamics and its
effects on wavefront sensing measurements. Most of the research related to this topic
so far had been done for the extreme situations of tear break-up, like that by Tutt et
al. [68] showing that the tear break-up affects the contrast of retinal images. Some
efforts by Himebaugh et al. [71] to provide a technique to study the changes on the
tear topography made in parallel to and independently of our work, led to a retroillumination
instrument, which is yet to produce definite results. A more suitable
technique to study the tear topography, was proposed by Licznerski et al. [73, 74,
75, 76], where information about the tear topography can be obtained using shearing
interferometry, and some semi-quantitative measurements were performed. However,
this experimental setup had an important limitation in that the tear topography
estimation was made using a gradient projection along only one direction.
There are other sources of variability in wavefront sensing measurements that might
not be noticeable if measurements are taken within a few seconds, but are nevertheless
non-negligible. They can significantly bias the wavefront aberration estimation,leading
to, for example, an incorrect corneal ablation profile. They include instrument myopia
[77] which can be as large as 5 D, the change in refraction due to changes in ambient
light [78] and the corneal topography changes that can follow reading due to eyelid
pressure, around 0.25 D sphere and 0.25 D cylinder.
20

1. Introduction
1.2 Tear film
It is accepted that the tear film consists of three layers [76, 79], the outermost being the
lipid layer of around 50 nm thickness, followed by the aqueous layer with a thickness
of around 4 − 8 µm and finally a mucous layer of around 20 − 50 nm. Each of the
layers has a different function, but from the optical point of view we should only be
concerned with their thickness and refractive indices. Actually, from the refractive
index point of view, we will assume, to a first approximation, a mean refractive index
n tear = 1.337 [80], so that we can concentrate on the tear thickness only. What is
more, in the rest of this thesis, we will estimate the topography of the front surface
of the film, assuming an unchanged back surface, which is reasonable given the time
scales involved in our experiments (a few seconds).
The first question we should ask ourselves is whether the tear film is thick enough
to produce optical path length differences that are significant in terms of the other
sources of variability of wavefront sensing in the eye measurements. Let us begin by
assuming that the difference between two tear topography maps at two given instants
is a parabolic surface that can be described by the defocus Zernike polynomial. We can
then relate the sag s of this parabolic surface to the wavefront error RMS amplitude
(a 2,0 ) it would produce in the eye through the formula
(
2 √ )
3
s =
a 2,0 (1.1)
n tear − 1
≈ 10 a 2,0 (1.2)
Therefore, if we assume even the most conservative reported values of tear thickness
[81, 82, 83, 84, 85, 86, 87, 88, 89], that is around 3 µm, then a sag of only 15 % of the
tear thickness (which seems plausible) would be enough to take a perfect eye out of the
diffraction limit. To see, under the same assumptions, the sag required to introduce
a defocus amplitude comparable to the intervals in spectacle prescription, we can use
the following
s =
d 2 Φ s
8(n tear − 1)
(1.3)
where d is the pupil diameter and Φ is the prescription in diopters. Therefore, for the
tear to play a noticeable role in vision (let us say a quarter of a diopter) in daylight
conditions with a typical value for d of 3 mm, the sag would have to be 0.8 µm –a
value physically plausible.
21

1. Introduction
It should also be mentioned that decrease in spatial-contrast sensitivity has been
reported by Rolando et al. to be associated with tear disfunction [90], although no
quantitative study of the tear topography was performed for this study.
1.3 Thesis synopsis
Chapter 2 describes the design of and theory behind the double-shearing interferometer
built for the study of the dynamics of human tear topography. The specifications
of the interferometer and role of all the optical elements are explained in sufficient
detail so that anyone trying to replicate or improve the experiment can do so without
great difficulty.
In Chapter 3 the theory, implementation of and elementary testing of the algorithms
used in the data processing from the raw shearing interferograms to the unwrapped
topography difference maps are discussed.
Chapter 4 is devoted to the discrete fourier transform (DFT) based integration methods
used to estimate the tear topography maps from the difference maps. We present
the theory of both methods in detail, and evaluate the computing and noise performance
in comparison with the conventional least-squares pseudoinverse reconstruction
methods.
Chapter 5 presents some preliminary experiments, where the feasibility of using
the interference of the reflections on the front and back surface of the tear to estimate
the tear topography was tested. The repeatability of and some errors in our
double-shearing interferometer are estimated. Finally a simple validation experiment
was performed and the ability of the experiment to detect small details on the tear
topography was tested.
Chapter 6 begins by describing the data collection protocol, follows with a discussion
of the error that can be expected in our data from eye movements and then with
an estimation of the global errors on the data. After this, the estimated wavefront
RMS from the processed tear topography data is analyzed with regards to the optical
quality of the eye. Finally, it is shown based on experimental data, that the method
proposed by Licznerski et al. [74] to evaluate the tear break-up is not adequate either
as originally proposed or after some suggested improvements.
22

1. Introduction
Chapter 7 presents a short summary and suggests the use of radial shearing interferometry
for the study of tear topography instead or double lateral shearing interferometry.
Appendix A explores experimentally the possibility of using lateral shearing interferometry
for wavefront sensing in the eye.
Appendix B shows the safety calculations performed to comply with the British and
European standard BS EN 60825-1:1994 with amendments 1, 2 and 3.
The DVDs attached to this thesis contain the following:
• Documents in HTML format that link to MPG movies showing the sequences
of raw tear lateral shearing interferograms and estimated tear topography maps
when these could be obtained.
• Compressed files in ZIP format with the same movies in AVI format (higher
resolution).
• Compressed files in ZIP format with the raw images recorded (TIF format).
• Matlab scripts (M) files containing the experimental configuration for each individual
data series for the tear topography experiments and some notes.
• Matlab binary .MAT files containing the input parameters for the tear topography
data processing and the outcomes (e.g. interferograms center and diameter).
• Raw images for all the other experiments described in this thesis in TIF format.
23

Chapter 2
Lateral shearing interferometer
design
Designing and building an instrument to measure the dynamics of the topography of
the pre-corneal tear film has proven a task far more difficult than initially thought. In
what follows we describe the specifications, theory and technical details regarding the
design of the double-shearing interferometer that was built and tested on 20 subjects.
2.1 System requirements
We aimed at producing an instrument able to measure changes in the topography of
the tear film with sub-micron depth resolution and lateral resolution higher than 20
to 40 samples across a pupil diameter of about 7 mm diameter, at a minimum rate of
2 Hz. In this way, we would be able to resolve temporally and spatially tear topography
features such as the rough surface, bubbles, lines and break-ups, not visually obvious
in current SH wavefront sensor images, and study their effect both in the optical
quality of the eye and SH wavefront measurements. The only specification we could
not meet is the area under study in the pupil, given the large numerical aperture
required and the fact that for comfort the instrument was to be placed at least 15 mm
away from the front surface of the cornea. The diameter of the illuminated area over
the tear film was around 3 mm.
The range of tear depth changes to estimate seem appropriate for an interferometric
technique, although not all interferometric methods would be suitable due to the
24

2. Lateral shearing interferometer design
movements of the eye. Self referencing interferometers are able to cope better with
eye movement. If we now look at the techniques to extract the phase from an interferogram,
we find ourselves again restricted by eye movement to single frame techniques
(or synchronized multi-frame acquisition). Based on this and some previous semiquantitative
experiments by Licznerski et al. [73] it was decided to implement a
lateral shearing interferometer as described below.
2.2 Lateral shearing interferometry
In a lateral shearing interferometer (LSI), the beam containing the phase information
to be measured is amplitude divided into two beams, laterally shifted and then recombined.
The lack of a static coherent reference beam makes the resulting interferogram
less sensitive with respect to eye movement. If we assume that the amplitude division
produces two beams with equal intensity and phase profiles, then the equation that
describes the intensity pattern resulting from the interference of the sheared copies of
the electric field E 0 (⃗r) and E 0 (⃗r + ⃗s) is
I(⃗r) = I 0 (⃗r) + I 0 (⃗r + ⃗s) + 2 √ I 0 (⃗r) I 0 (⃗r + ⃗s) cos [φ(⃗r + ⃗s) − φ(⃗r)] , (2.1)
where ⃗r is the position vector in the interferogram plane, ⃗s is the shear between
the two beams, I 0 (⃗r) = |E(⃗r)| 2 , φ is the phase of the complex electric field E(⃗r),
the function of interest. Here we have assumed scalar electric fields, but provided
that when performing the amplitude division the polarisation state of the two sheared
beams remains unchanged with respect to each other, then equation 2.1 remains valid.
It is also implicitly assumed that the spatial and temporal coherence of the beams is
such that the modulus of the complex degree of coherence [91] is virtually one for the
shear and wavefront under consideration.
In order to reconstruct the two-dimensional phase map φ, at least 2 interferograms
with different shear directions (not necessarily orthogonal) are needed, because the
phase difference in equation 2.1 is insensitive to phase maps constant along the direction
of shear.
25

2. Lateral shearing interferometer design
2.3 Interferogram intensity modulation
The fundamental problem in interferometry is to extract the phase from interferograms,
despite fluctuations in the intensity profile across the interferogram.
non-uniform intensity in real interferograms can be due to a number of factors, such
as intensity profile of the source, non-uniform absorption/reflection of the phase object
to be analyzed or diffraction. We will avoid phase-shifting techniques [92] because
they imply the acquisition of multiple interferograms for every phase map to be reconstructed
[93, 94], requiring either non-simultaneous acquisition, which is not desirable
as we do not know how fast the tear topography changes, or a complex optical setup
for the simultaneous acquisition of the interferograms needed. Instead, we modulate
the intensity of the interferogram spatially, by introducing a controlled amount of tilt
between the two interfering beams, and then use the phase recovery algorithm proposed
by Takeda [95]. The spatial frequency of the intensity modulation ⃗ f c acts as a
phase carrier frequency and is related to the tip (ε x ) and tilt (ε y ) angles between the
interfering wavefronts by
The
( tan εx
⃗f c =
λ
, tan ε )
y
. (2.2)
λ
The intensity of the interferogram once the tilt associated to ⃗ f c is introduced is given
by
I(⃗r) = I 0 (⃗r) + I 0 (⃗r + ⃗s) + 2 √ [
I 0 (⃗r) I 0 (⃗r + ⃗s) cos φ(⃗r + ⃗s) − φ(⃗r) + 2πf ⃗ ]
c · ⃗r (2.3) .
The first two terms on the right, are referred to as the DC term and the third one as
the AC term.
2.4 Wedge design
The amplitude division for the shear interferometer can be achieved in different ways,
typically by the use of beam-splitters [96], diffraction gratings [97, 98, 99] and wedged
plates [73]. The use of wedged plates was preferred for their robustness and ease
of production in our optical workshop. Even though a wedge plate can be defined
by only two parameters, namely the angle between the surfaces and the thickness at
one given point, the selection of these two is not trivial in our case. The choice of
tilt to introduce between the interfering wavefronts depends on the optical system,
26

2. Lateral shearing interferometer design
the density of elements in the detector that records the interferograms, the expected
spatial spectra of the tear and source profile and the signal-to-noise ratio.
In the rest of this section the geometry and angle nomenclature is as shown in figure
2.1, and only the reflections on the front surface and first reflection on the back
surface of the wedge will be considered. The consequences of this consideration will
be discussed later in section 3.4.3.
a 1
a 2
a 3
O d
A
a 1
a 5
C
a 4
a 3
B
n w n m
Figure 2.1: Glass wedge geometry and angle definition for reflected rays.
From figure 2.1 and the use of Snell’s law it can be shown that for a ray incident on
the wedge at the point A, the tilt t = |α 5 − α 1 | between the reflection on the front
and back surface is given by
[ ( )]} ∣ t =
1
∣∣∣
∣
{n arcsin w sin 2δ + arcsin sin α 1 − α 1 . (2.4)
n w
where n w is the refractive index of the wedge and we assume that it is in air. This
equation shows that the tilt between each pair of rays resulting from the same incident
ray, depends on the angle of incidence α 1 . Therefore, unless all the rays of the incident
beam at the wedge are parallel, the reflected wavefronts will be distorted. Hence, the
wedges must be placed in a region of the optical system where the incident beam is
parallel.
If we now calculate the shear s = AC introduced by the wedge using basic trigonometry,
we have,
s =
[
sin δ sin
[ ( )]
cos 2δ + arcsin sin α1
n w
( )]
2δ + 2 arcsin sin α1
n w
cos
[
δ + arcsin
(
sin α1
n w
)]AO (2.5)
27

2. Lateral shearing interferometer design
which for δ ≪ 1 and α 1 ≫ δ can be approximated by
[ ( )]
sin 2 arcsin sin α1
n w
s ≈
( )] δ AO (2.6)
cos
[arcsin 2 sin α1
n w
From either of these two equations we can see that the shear is non-uniform across
the wedge, and therefore, the wavefront reflected on the back surface of the wedge will
be non-uniformly stretched in the direction of the shear. The relative change in shear
D s across the beam width w at the wedge, can be expressed as
D s = 2 s max − s min
s max + s min
(2.7)
where s min and s max are the shear of the rays incident on the wedge closest and furthest
from the wedge apex respectively. By using equation 2.5 assuming a collimated
incident beam, the relative change in shear across the beam can be rewritten as
D s =
w
AO mean
(2.8)
where w is the beam width at the wedge. This suggests that the shear non-uniformity
can be kept below a given tolerance T by making the distance AO mean large in comparison
to w or equivalently, making the wedged plate thicker,
w
AO mean
< T (2.9)
2.5 Wedge manufacture
For manufacturing purposes it will be more convenient to invert and combine formulas
2.4, 2.6 and the condition 2.9, so that we have
{
arcsin
AO mean
δ = 1 2
≈
[ ]
sin(α1 ± t)
− arcsin
n w
[ sin(α1 )
n w
]}
( )]
2 s mean cos
[arcsin 2 sin α1
n w
[ ( )]
sin 2 arcsin sin α1
n w
{arcsin
[
sin(α1 )±t
n w
]
− arcsin
(2.10)
[
sin(α1 )
n w
]}(2.11)
AO mean > w T . (2.12)
It should be noticed that if the two equations for AO mean can not be simultaneously
satisfied, then the experiment will need to be modified to change the input parameters
until the three equations are satisfied to within the accepted tolerances. In practice,
28

2. Lateral shearing interferometer design
f AC
max
f DC
max
f c
f sampling
Figure 2.2: Lateral shearing interferogram spectra produced by a glass wedge. The
red circle in the center contains the DC term, while the other two circles contain the
AC terms that carry the phase information. The indicated spatial frequencies are: the
sampling frequency f sampling , the carrier frequency f c and the maximum DC and AC
term frequencies fmax DC and fmax AC respectively.
this is not an important limitation because we have an extra degree of freedom in the
shear, that can be tuned later by sliding the wedge position along the optical axis of
the system, as will be explained in section 2.7.1.
Two conditions must be met to avoid aliasing when spatially sampling the interferogram.
Firstly, the spectra of the AC and DC terms of the interferogram intensity
must not overlap, as it is the case in the example shown in figure 2.2. This is ensured
by choosing a carrier frequency f c (i.e. the wedges angle) larger than the sum of the
highest spatial frequencies of the DC and AC terms, that is
f DC
max + f AC
max < f c . (2.13)
Secondly, according to the sampling theorem, the spatial sampling frequency must be
equal to or larger than twice highest spatial frequency of the interferogram, that is
f c + f AC
max < f sampling /2. (2.14)
If information on the spatial content of the tear topography were available, one could
put figures into these conditions and estimate an adequate angle for the wedges and the
29

2. Lateral shearing interferometer design
number of camera pixels. However, we do not have an established model to describe
the dynamics of tear topography. Therefore, making no assumptions on the spatial
content of the tear topography, the angles of the glass wedges for the interferometer
were chosen so that the carrier frequencies of the interferograms were sampled at
around 8 camera pixels per fringe, with around 450 samples across each interferogram.
These values correspond to a tilt between the reflected wavefronts on the front and
back surfaces of the wedge of 12 mrad, and are a compromise between having high
spatial frequency modulation in the interferograms and reasonable sampling.
We now consider the shear introduced by the wedges, for which we recall equation 2.3
I(⃗r) = I 0 (⃗r) + I 0 (⃗r + ⃗s) + 2 √ [ ]
φ(⃗r + ⃗s) − φ(⃗r)
I 0 (⃗r) I 0 (⃗r + ⃗s) cos
s + 2πf s
⃗ c · ⃗r ,
where for convenience the phase difference has been multiplied and divided by the
shear amplitude s. We can now see that if the shear is small in comparison with the
spatial changes in the phase, the phase difference can be approximated by the phase
slope in the shear direction multiplied by the shear amplitude. We can further assume
that in terms of measured phase, the signal-to-noise ratio (SNR) is proportional to
the shear amplitude. On the other hand, the larger the shear the smaller the area
over which the pupils overlap, and so the poorer the phase estimation. Again, as with
the tilt, an optimum estimation of the shear needed for the experiment would need
some knowledge of the wavefronts to be measured, and therefore the choice of shear
is arbitrary, and in our case is approximately 5 % of the beam diameter for a beam
diameter of 3 mm. Finally, we will consider 2 % as an acceptable tolerance to the shear
variation across the beam.
Taking all the above in to consideration for a beam diameter of 3 mm, we reach the
following values for the wedge angle δ and thickness:
δ ≈ 10 arcmins, (2.15)
thickness ≈ 0.4 mm, (2.16)
The two glass wedges manufactured have angles 11 and 13 arcmins, and mean thicknesses
0.35 and 0.70 mm respectively. The extra thickness of the second wedge will
only affect the shear amplitude, but as will be shown later (see section 2.7.1), this
can be compensated for by displacing the wedge along the optical axis. Both glass
wedges were produced to be used at angle of incidence α 1 of 45 o and made of BK7,
30

2. Lateral shearing interferometer design
He-Ne
laser
633nm
ND filter
microscope objective
spatial filter
f = 250
1
stop aperture
s-polarisation
l/4 f 2 = 40
PBS
eye
p-polarisation
circular-polarisation
Figure 2.3: Illumination system of the lateral shearing interferometer.
which at the wavelength of the He-Ne laser used (632.8 nm) has a refractive index
n w = 1.51509.
2.6 Illumination branch
The illumination system of the shearing interferometer, shown in figure 2.3, consists
of a light source, a neutral density absorption filter, a microscope objective, a pinhole,
a collimating lens, a polarising beam-splitter, a quarter waveplate and a lens. The
purpose of this part of the optical system is to produce a smooth (ideally perfectly uniform)
illumination over a circular region of the front surface of a subject’s precorneal
tear and a fixation target.
2.6.1 Source
The source to be used needs to be quasi-monochromatic because multiple wavelengths
would produce different phase differences and, therefore, multiple overlapping interferograms.
31

2. Lateral shearing interferometer design
A polarized source is desirable to keep the number of optical elements to a minimum
thus reducing undesired reflections, which as we will show in section 3.4.3 can be a
non-negligible source of error in the data processing.
Finally, it would also be desirable to use a source with an intensity profile as uniform
and symmetrical as possible to produce uniform illumination over the cornea and thus
a uniform signal-to-noise ratio across the interferograms. With this in mind we used
a single-mode linearly polarized He-Ne laser with an output of 2.5 mW at 632.8 nm
(Spectra Physics S/N 47075/1919), with a Gaussian intensity profile.
The intensity of the source output is adjusted by using absorption neutral density
filters to keep the power reaching the eye within the safety limits described in Appendix
B.
2.6.2 Spatial filter and telescope
The microscope objective, pinhole, collimating lens and stop aperture shown in figure
2.3 form a telescope to magnify the collimated output of the source and a spatial
filter to produce a smooth intensity profile. The microscope objective is ×40 with a
numerical aperture (NA) 0.65 and a focal length of approximately 3 mm at 632.8 nm,
the pinhole diameter is 10 µm and the collimating lens has a focal length of 250 mm.
In a normal telescope, one would like to match the NAs to make use of all the incoming
light, but in our case, we have chosen the combination of microscope objective and
collimating lens so that the NA of the first one is about 10 times larger than the
second, so that the beam at output of the telescope only takes the central part of the
intensity profile of the laser, to produce a flatter intensity profile. The stop aperture
narrows the beam so that less than the whole diameter of the rest of the optics is
used, providing some tolerance for eye movement.
2.6.3 Polarizing beam splitter and quarter waveplate
The block formed by the polarising beam-splitter (PBS) and the quarter waveplate
serves two purposes, first, to maximize the light reflected back from the eye towards
the imaging branch of the optical setup and second, to minimize the reflections from
the beam-splitter itself.
32

2. Lateral shearing interferometer design
NPBS 50-50
PBS
1
1
2
0.5 x ARC
0.5
1
2
0.5 x ARC
2
0.5 x TR
0.5 x ARC
0.5 x TR
p-polarisation
s-polarisation
ER x ARC
TR
ARC
TR
0.5 x ARC
0.5
ER
Figure 2.4: Relative intensities of reflections from a 50-50 non- polarising beam splitter
(NPBS) and a polarising beam splitter (PBS). ARC is the intensity reflection coefficient
for the antireflection coating, TR is the reflection coefficient of the tear film, and
ER is related to the extinction ratio of the PBS and the polarisation of the incident
beam.
The laser tube was rotated (and thus the polarization) to maximize the intensity
reflected from the PBS towards the eye (s-polarisation). Then, the quarter waveplate
was oriented with its axis at 45 o with respect to the s-polarisation, changing the
polarisation state of the light transmitted towards the eye to circular, so that the
light reflected at the front surface of the tear is fully transmitted through the PBS.
As the reflection on the tear surface occurs at normal incidence, the polarization state
is not altered except for the change in rotation direction.
The minimization of undesired reflections from the beam splitter surfaces is a crucial
point in the experimental setup. The intensity of the light reflected from the tear film
is about 2 % of the incident light (assuming a tear mean refractive index of 1.337 [81].
With reference to figure 2.4 we can see that if we used a non-polarising beam splitter
(NPBS) to combine the illumination and the imaging branches, the relative intensity of
the undesired reflections on the NPBS surfaces would be approximately 2×0.5 2 ×ARC
where ARC is the intensity reflection coefficient of the anti-reflection coating on the
surfaces, which in the best case scenario (laser-line matched) is 0.001. The ratio of
the intensities reflected from the eye to the undesired reflections from the NPBS is
about 20, which is very poor. If on the other hand we use the combination of a PBS
and a quarter waveplate the intensity of the reflections is a bit more complicated to
calculate because it depends on both the extinction ratio of the beam-splitting surface
and the polarisation of the incident beam, which in figure 2.4 we have represented as
33

2. Lateral shearing interferometer design
ER. The experimentally measured value for the ratio of the light reflected from the
tear to that reflected from the PBS is about 2400 that represents an improvement
of two orders of magnitude in intensity and one in SNR. This ratio of intensities is
a good value from the point of view of the detection, because with the camera that
we used the contrast of the undesired interference pattern that would result from the
light reflected from the eye and the undesired reflections from the PBS, is comparable
(a factor of 2 larger) to the readout noise of the camera. Ideally one would make the
contrast between the beam of interest and the undesired reflections equal to or smaller
than the noise of the camera to record the interferogram.
2.6.4 Focusing lens
The final part of the illumination branch of the experimental setup to consider is the
positive lens that makes the beam directed towards the eye converging, focusing at
the center of curvature of the cornea. The two main issues to consider here are the
eye clearance for comfort and the size of the illuminated area.
We considered that the minimum clearance between the closest surface of the lens to
the cornea should be no less than 15 mm; that is the accepted distance for spectacles.
If we bear in mind that the curvature of the wavefront incident on the tear film should
match the curvature of the cornea and the beam before the lens is collimated, then
we have a condition to be met by the focal length f of the lens,
f > 15 mm + R tc (2.17)
where R tc = 8 mm is the radius of a typical cornea, and therefore, f > 23 mm. The
diameter of the illuminated area D illum for a typical cornea is
( ) D
D illum = 2R tc arctan
(2.18)
2f
where D is the minimum of the lens and incoming beam diameters, and f is the
focal length of the lens. This tell us that D illum depends on the F-number of the
lens, assuming the diameter of the incident beam is adjusted accordingly. In the end
we compromised between illuminating the largest possible area over the pupil and
relatively small optics (17 mm clear diameter) in the rest of the experimental setup to
keep costs down, choosing to look at the focal length range 30 − 40 mm. Given the
low F-number, a doublet is preferable to a singlet to keep aberrations low, although
34

2. Lateral shearing interferometer design
this is not crucial because the aberrations of the optical system and corneal surface
are static and will be removed in the data processing. We tried two doublets: an
air-spaced one formed by a 50 mm meniscus and an 80 mm plano-convex lens of the
same material, and an cemented achromat. Also, for the sake of comparison, we
tested a 30 mm biconvex and a 40 mm plano-convex. The shear interferograms they
produce are shown in figure 2.5 and clearly indicate that the preferable lens (in terms
of straighter fringes and hence a smaller spread of the AC and DC spectra)is the 40 mm
achromat, as one would expect. The diameter of the illuminated area for a typical
cornea would then be 3.4 mm if we used the whole clear aperture of the achromat
(17 mm). However, when using the full aperture of the lens, there is no tolerance
of eye misalignment or movement and therefore we traded off some tolerance to eye
misalignment by using only a fraction of the aperture. We have arbitrarily chosen
15 mm of usable aperture which has proven to be reasonable for the tested subjects,
providing ±0.1 mm sideways and ±0.3 mm depth tolerance, without vignetting.
2.7 Imaging branch
The imaging branch consists of: a focusing lens, shared with the illumination branch,
a telescope to demagnify the image of the tear film to a size suitable for the camera;
the purpose-built glass wedges; and two 4f systems to allow the placement of the
wedges. Given a corneal radius of curvature R c , we can calculate the position d of
the planes T ′ and T ′′ conjugated to the front surface of the tear, assuming that to
first order the image is flat (the sag is only 2 % for a typical corneal radius and the
illuminated area in our experiment), by using the formulas
(
d T ′ = f 4 1 − f 2 2 f )
4
f3 2 R
(
d T ′′ = f 6 1 − f 2 2 f 4 2 f )
6
f3 2 f 5 2 R
(2.19)
(2.20)
where the notation is that indicated in figure 2.6. The magnifications M at those
planes are given by
M T ′ = f 2 f 4
f 3 R
(2.21)
M T ′′ = f 2 f 4 f 6
f 3 f 5 R . (2.22)
35

2. Lateral shearing interferometer design
(a)
(b)
(c)
(d)
Figure 2.5: Interference patterns that result from using different lenses in the illumination
branch of the shear interferometer and an artificial cornea: (a) is from a 30 mm
biconvex lens, (b) from a 40 mm plano-convex lens, (c) from a 30 mm air-spaced doublet
and(d) from a 40 mm achromat.
In theory, we should adjust the camera position along the optical axis for it to be
conjugated to the tear, according to the radius of curvature of the cornea of every
subject, but in practice, we set the experiment assuming a typical corneal radius of
curvature of 8 mm. Under this assumption we choose the focal lengths of the lenses
for the magnification of the imaging system to be unity.
The glass wedges are introduced in the imaging system to produce sheared and tilted
copies of the incident beams by being used in reflection at 45 o . This produces reflections
from the front and back surface of the wedges with almost equal intensity and
thus results in very high contrast interferograms. The first of the wedges is also used
36

2. Lateral shearing interferometer design
T''
CCD
camera
f 6 = 150
CCD
camera
T''
f 6 = 150
f 5 = 150
T'
f 4 = 30 f 3 = 150
f = 40 2
T
wedge 2
wedge 1
300 180
180 190
eye
Figure 2.6: Imaging branches of the double shearing interferometer used for tear
topography estimation. For ease of understanding we have drawn both branches with
their optical axis in the paper plane, but in reality the beam after wedge number 2
and the CCD camera are in the perpendicular plane.
in transmission and we will neglect the undesired reflections in both transmission and
reflection modes, which are around 0.5 % for linear polarisation incident at 45 o with
respect to the reflection plane.
From the imaging point of view, the glass wedges can be thought of as mirrors, and
being so thin (less than a millimeter) we will neglect the optical path difference between
the reflections in their front and back surface.
As explained previously, we need to produce two lateral shearing interferograms with
different shear directions. Given the symmetry of the camera pixels and for ease of
data processing we have chosen orthogonal directions. In the diagram 2.6 we show
both of the branches after the wedges in the same plane for ease of visualization, but
in the experimental setup one of the two is in the plane perpendicular to the paper,
so that the shears introduced are orthogonal. If the branches were left as shown,
two synchronized cameras would be needed. Instead both branches were folded using
mirrors at 45 o making the optical axis go along the edges of a rectangular prism, thus
superimposing the image planes T ′′ (figure 2.7).
37

P
P
2. Lateral shearing interferometer design
CCD camera
wedge
P
P
P
PP
P
He-Ne (632.8nm)
ND filter
microscope objective
spatial filter
P
l/2 plate
wedge
aperture stop
PBS
l/4 plate
eye
Figure 2.7: Sketch of the 3 dimensional optical setup for tear topography estimation
with double lateral shearing interferometry. After the light is reflected back from the
front surface of the tear, the first glass wedge produces two horizontally sheared and
tilted copies of the incident beam by reflection and a third copy by transmission. The
second wedge allows the first pair of copies of the beam to go through unchanged,
while on reflection produces a second pair of copies of the beam carrying information
from the eye, sheared and tilted in the perpendicular direction.
Even though so far the imaging system has been considered in a conventional way, it
is very important to notice that there is only one ray to be traced from each point at
the tear surface because the front surface of the tear behaves like a mirror and the
illumination is such that there is only one ray reaching each point of it. Geometrically
this would give an infinite depth of focus, however, diffraction makes the depth of
focus finite. Without entering into the problem of how to define depth of focus in this
experiment, we will considered that in our experimental setup it is large enough to
allow for all the radii of the corneas imaged to be in focus.
2.7.1 Shear and tilt estimation
Using the paraxial equations, we will now estimate the tilts t 1f and t 2f and shears
s 1f and s 2f introduced by the wedges in the camera plane from the tilts t 1 and t 2 and
shears s 1 and s 2 introduced by the wedges. It can be shown that the tilts between the
38

2. Lateral shearing interferometer design
front and back reflections on the wedge surfaces at the camera plane are given by,
( )
f5
t 1f = t 1 (2.23)
f 6
and that the shears are
s 1f = f 5 t 1 + f 6 t 1 +
t 2f = t 2 (2.24)
( ) ( ) ( )
f5 t 1
f6 f6 t
d T ′′ − √ 1
s 1 − d 5 (2.25)
f 6 f 5 2 f 5
s 2f = s 2
√
2
+ t 2 d 6 (2.26)
where d 5 is the distance from the first wedge to the lens labelled f 5 and d 6 is the
distance from the second wedge to the camera. These equations show that by sliding
the optical wedges along the optical axis, the shear amplitude can be tuned, and this
was used to achieve the desired value of 5 % of the beam diameter.
2.7.2 Polarisation control for optimization of interferogram contrast
When observing the interferograms produced by the experimental setup described
above, one can observe a clear difference in their intensities.
This is due to the
dependence of the reflection coefficients of the glass wedges on the incident state of
polarisation. It is therefore necessary to calculate the interferogram intensities and
contrasts as a function of the incident state of polarisation. The Fresnel reflection r
and transmission t coefficients related to air-wedge transitions will be denoted with
the subindex a while the inverse transitions will be denoted with the subindex w. We
can calculate the relative amplitudes of the reflected beams from the front and back
surface of the first glass wedge E 1F and E 1B respectively. Assuming an incident beam
with unit amplitude and elliptical polarization described by the angles θ (azimuth)
and β (ellipticity) and ignoring multiple reflections, the coefficients are given by
⃗ E 1F = (t ‖a t ‖w r ⊥a cos θ, t ⊥a t ⊥w r ‖a e iβ sin θ) (2.27)
⃗ E 1B = (t ‖a t ‖w t ⊥a r ⊥w t ⊥w cos θ, t ⊥a t ⊥w t ‖a r ‖w t ‖w e iβ sin θ) (2.28)
Similarly for E 2F and E 2B ,
⃗ E 2F = (t ‖a t ‖w r ⊥a e iβ sin θ, t ⊥a t ⊥w r ‖a cos θ) (2.29)
⃗ E 2B = (t ‖a t ‖w t ⊥a r ⊥w t ⊥w e iβ sin θ, t ⊥a t ⊥w t ‖a r ‖w t ‖w cos θ ) (2.30)
39

2. Lateral shearing interferometer design
1.001
C 12
0.16
I 12
1
C 34
0.14
I 34
inteferogram contrast
0.999
0.998
0.997
inteferogram intensity
0.12
0.1
0.08
0.06
0.04
0.996
0.02
0.995
0 50 100 150
polarization orientation (deg)
0
0 50 100 150
polarization orientation (deg)
(a)
(b)
Figure 2.8: Effect of the orientation of the polarisation state incident on the first glass
wedge on the intensity and contrast of the interferograms produced by the glass wedges
in the optical system illustrated in figure 2.7. Notice there is virtually no dependence
for the visibility but there is a very significant dependence for the intensity.
where it should be noted that due to the way the interferometer is set, the axes
of the system are swapped between the wedges by the reflections from two mirrors
between the first and second wedge. The first thing to notice from examination of
equations 2.27 to 2.30 is that neither the contrast nor the intensities will depend on the
ellipticity β. Thus, the only effect of polarisation on both the contrast and intensity
of the interferograms will arise from the azimuth of the polarisation state, suggesting
that we only need to rotate the polarisation state incident on the first wedge with, for
example, a linear retarder of half a wavelength to tune these parameters. Figure 2.8
(a) shows that the contrast is almost maximum for any polarisation state, while the
situation is very different for the interferogram intensities shown in (b), where there
are two orthogonal directions in which the intensities will be equalized, although not
maximized. A half-wave plate was placed in the optical system just before the first
glass wedge as shown in figure 2.7, so that by rotating this plate the intensity of both
lateral shearing interferograms were equalized.
40

Chapter 3
Data processing
The methods used to process the recorded data, from the raw interferograms to the
topography slope maps are now presented and discussed. The algorithms described
here might not be optimum either mathematically or computationally, but they keep
the complexity to a minimum and, within reasonable levels, minimize the user input
(automation). The quantitative use of an interferometric technique to study the tear
film topography is already a novel field, and we think that by keeping the data processing
simple, not only will this help in its understanding, but it will also keep artifacts
to a minimum. We have sometimes, therefore, traded performance and accuracy for
simplicity.
Before processing, a visual inspection of all the raw interferograms was made, and
those with vignetting, a blink, or where the eye was too far (or too close) to the
experimental setup were tagged as non-usable. From now on we will refer to all the
other interferograms as usable, and these are the only ones to be processed as described
below.
3.1 Carrier frequency estimation
3.1.1 Theory
The first step of the data processing is the estimation of the carrier frequency ⃗ f c of
the interferograms described by equation 2.3 that we recall here once more
I(⃗r) = I o (⃗r) + I o (⃗r + ⃗s) + 2 √ [
I o (⃗r) I o (⃗r + ⃗s) cos φ(⃗r + ⃗s) − φ(⃗r) + 2πf ⃗ ]
c · ⃗r .
41

3. Data processing
If we look at the square modulus of the Fourier transform of the intensity of the
interferograms (spectra), under certain conditions (smooth tear topography) we should
be able to identify three peaks. The major central peak corresponds to the DC term,
and the two other peaks, distributed symmetrically with respect to the zero frequency
correspond to the AC term.
Ideally, we would like to estimate the frequency of the intensity modulation ⃗ f c , produced
by the tilt between the two interfering wavefronts. However, tilt between the
two interfering wavefronts is not the only source of linear phase terms in lateral shearing
interferograms. Both defocus and astigmatism would produce linear terms as well
with an associated frequency ⃗ f lin . This can be understood by looking back at equation
2.15 where the phase difference term is to first order a derivative, hence, as defocus
and astigmatism are second order polynomials, their derivative will produce linear
terms. Actually, second order polynomials will not be the only ones leading to linear
terms in the phase difference term, but we assume these are the dominant ones,
as is corroborated later with the experimental data. To estimate absolute values of
defocus and astigmatism in the tear topography would require calibrating the carrier
frequency. In our case this is not important, because we are not interested in measuring
the absolute topography of the tear film, only its changes with respect to an
initial topography map. We will therefore ignore the offset in the carrier frequency estimation
due to the presence of defocus and astigmatism in our optical setup, corneal
topography and eye misalignment.
3.1.2 Algorithm
The algorithm for the estimation of the carrier frequency has two steps: calculating
the modulus of the spectrum of the raw interferogram and then looking for the spectra
maximum in a region of the Fourier space that excludes the DC term. The coordinates
of this maximum with respect to the origin of coordinates in the Fourier space
is what from now on we will refer to as the estimated carrier frequency. As we have
just explained, we should be aware that the algorithm will not retrieve the frequency
associated with the tilt between the wavefronts, but the vectorial sum of the carrier
frequency and the linear term in the difference of the sheared wavefronts. This algorithm
relies on the implicit assumption that there is no other spatial frequency in the
AC term that has more energy than that corresponding to the sum f ⃗ c + f ⃗ lin .
42

3. Data processing
3.1.3 Implementation details
In practice the algorithm calculates the square modulus of the discrete Fourier transform
(DFT) of the raw interferograms, and then performs the search of the maximum
value over a semi-annular region centered at the zero spatial frequency, with the inner
and outer radii corresponding to the limits of the range we expect the carrier
frequency to have. This search range was set to spatial frequencies with wavelengths
between 5 and 17 image pixels. The maximum of the AC term is only found within
pixel accuracy, sub-pixel accuracy by spectra interpolation was not pursued. The only
consequence of this coarse estimation is to leave a residual tip/tilt on the retrieved
phase maps that is removed later.
3.1.4 Tests
To illustrate the performance and limitations of the algorithms described in this chapter,
we will show the results of their application to 8 interferograms that display the
different features we have found in the almost 5000 usable interferograms. These results
are by no means an exhaustive test, although they provide a good sample of all
the possible scenarios that could be found.
Figure 3.1 shows a series of raw interferograms and the corresponding frontal view
of the 3 dimensional plot of the calculated spectra. The region of the spectra within
the red vertical lines was excluded from the search for the maxima (inner diameter
of the maximum search area), and the DC term can be clearly recognized in it for
all figures. However, the AC terms in the far left and right of the spectra plots
can be difficult to identify in some of the interferograms, because when the phase or
intensity of the interferogram contains rapid variations, the spectra of the DC and AC
term broaden and overlap, as can be seen in figures 3.2 b), f) and h). These figures
illustrate two very important problems that are critical for automated processing of
the data. First, the algorithm will always find a maximum which might not be related
to the modulation frequency, but to other features on the interferograms, such as
the speckle grain size in figures 3.2 f) and h). Second and more important is the
issue of the spectra overlapping, since the method we use to retrieve the phase of
the interferogram is based on the assumption that the DC and AC spectral terms do
not overlap. When this is not the case there will be a non-linear influence of the DC
43

3. Data processing
term on the retrieved phase. We should therefore either visually inspect the spectra
of all the interferograms to be processed, or find an automated method to decide on
whether or not we consider the DC and AC spectra do overlap, to discard them as
not usable 1 .
3.2 Shear estimation
3.2.1 Theory
Let us assume that we have an image that consists of the sum of the intensities of
two copies of the same object A (represented by a triangle in figure 3.3) shifted with
respect to each other by a vector ⃗s. The normalised autocorrelation C AA of this image
can be defined as
∫ ∫ [ A( r ⃗′ ) + A( ⃗ [
r ′ + ⃗s)]
A( r ⃗′ − ⃗r) + A( ⃗ ]
r ′ + ⃗s − ⃗r) d 2 r ′
C AA (⃗r; ⃗s) =
∫ ∫ [ A( r ⃗′ ) + A( r ⃗ ] 2
(3.1)
′ + ⃗s) d 2 r ′
where we assume that A is a real function, and in practice will only take non-zero
values over a finite range. For an autocorrelation, we expect to get a maximum value
when ⃗r is zero, but in this particular case, we will also find that C AA has two other
peaks at ⃗r = −⃗s and ⃗r = ⃗s. These peaks correspond to the situation illustrated in
figure 3.3 b) and c) respectively. Thus, the position of the side peaks are at ± the
shear. This is the idea behind the algorithm used to estimate the shear between the
pupils in the shearing interferograms.
3.2.2 Algorithm
The shearing interferograms are more complex than the illustration of figure 3.3, they
also contain the interference (AC) term. The autocorrelation of such interferograms,
has multiple side fringes with the same spacing as the fringes in the interferogram as
illustrated by figure 3.4 (a) and (b). If we low-pass filter the raw interferogram (c) to
remove the AC term, one could resolve central peak and two side peaks (f). Although
1 If we were to automate the evaluation of the spectra overlapping, we could use as a first step
the second order moment of the area around the estimated maximum, as defined in section 6.5 and
then setting a threshold value to determine when the AC term corresponds to a relatively smooth
topography.
44

3. Data processing
mk1/46
(a)
(b)
st3/15
(c)
(d)
im4/17
(e)
(f)
pb2/92
(g)
(h)
Figure 3.1: Estimation of the interferogram modulation carrier frequency intensity
modulation (left) from spectra (right). The frequencies between the red lines indicate
the area where we expect to have only the DC term. In the top row, we see a normal
smooth front tear surface, the second row shows a rough surface typical after blink, in
the third small bubbles on the tear surface can be noticed and at the bottom a clear
bump in the tear formed by the one of the eyelids is shown.
45

3. Data processing
mk1/33
(a)
(b)
kh3/47
(c)
(d)
te2/49
(e)
(f)
te2/13
(g)
(h)
Figure 3.2: As figure 3.1. The top row shows a very undulated tear surface due
to blink prevention, the following row corresponds to a tear topography with a hole
(break-up) and the two bottom ones are interferograms produced by extremely rough
tear surfaces in front of contact lenses.
46

3. Data processing
y
y
y
x
x
s
x
-s
(a)
(b)
(c)
Figure 3.3: Geometry of autocorrelation of an image consisting of two sheared copies
of an object with finite dimensions: a) the image to autocorrelate, b) and c) the
situations in which the autocorrelation has local maxima other than at the origin.
the peaks are visible, they are not strong compared with the background, and in order
to improve this and therefore the sensitivity of the algorithm, we also remove the very
low spatial frequencies of the object. We therefore virtually retain only the edges of
the figure as illustrated in figure 3.4 (e). This stage requires the non-overlapping AC
and DC spectra.
3.2.3 Implementation details
The interferogram band-pass filtering and the autocorrelation calculation were performed
using the DFT and the autocorrelation theorem. There are two advantages
of using the DFT rather than the straightforward calculation of the autocorrelation
here: firstly a significant speed increase by making use of the fast implementations of
the DFT and secondly, the bandpass filtering in the Fourier domain becomes simply
a multiplication by a binary mask.
Before using the DFT we pad the data with zeros so that the area of the padded data
is no less than twice the size of the interferogram plus the shear to avoid overlapping
of the periodized autocorrelation (resulting from using the DFT).
The mask that performs the band-pass filtering is simply a binary annular mask. The
inner radius was chosen so that the spatial frequencies within the shear search range
[s min , s max ] would be preserved and all the lower frequencies removed. The outer
radius was chosen to keep all the frequencies within the carrier frequency search range
and remove those above it.
Finally, once the correlation of the filtered interferograms has been evaluated, the
47

3. Data processing
mk1/46
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3.4: Improvement on shear estimation algorithm by filtering in Fourier domain:
(a) raw shearing interferogram, (c) and (e) the same interferogram after low- and bandpass
filtering respectively, (b), (d) and (f) show frontal views of the corresponding
autocorrelation plots. The position of the two side peaks with respect to the central
peak in the autocorrelation plot in (d) is the shear.
48

3. Data processing
maximum value within the annular section s min < r < s max is found. The position
of the maximum was only determined with 1 pixel accuracy. Sub-pixel search of
the maximum by interpolation would not provide any advantage in our case because
the algorithm that reconstructs the wavefront from the slope of the phase maps only
uses integer values of shear amplitude. If we assume that the error in the maximum
position determination is half a pixel, then the relative error for a typical shear value
(30 pixels) is about 2%. In practice the error of the algorithm is higher, due to noise
in the image acquisition and pixelation.
3.2.4 Tests
The value of the autocorrelation at the side peaks can not be determined theoretically
unless one knows the image function A and the shear ⃗s. However, we can estimate it
roughly if we use the fact that the actual overlapping of the images over areas with
non-zero values is small in comparison with the non-overlapping areas. Evaluating
formula 3.1 for ⃗r = −⃗s (and expanding the denominator) gives,
∫ ∫ [
C AA (−⃗s; ⃗s) = 1 A( r ⃗′ )A( r ⃗′ + ⃗s) + A( r ⃗′ )A( ⃗ ]
r ′ + 2⃗s) d 2 r ′
2 + 2 ∫ ∫ [ A 2 ( r ⃗′ ) + A( r ⃗′ )A( r ⃗ ] , (3.2)
′ + ⃗s) d 2 r ′
which shows that if the overlap between the two sheared images is small, then the value
of the normalized autocorrelation at the side peaks is around 0.5. This is the case
for our filtered interferograms that only have non-negligible values at the edges of the
pupil and tear features. This can be clearly seen in figures 3.5 and 3.6, where the first
columns show raw interferograms, the images in the second columns are the band-pass
filtered interferograms and the three dimensional plots are the central region of the
autocorrelation. By comparing the values of the autocorrelation side peaks indicated
in the figures, and the filtered interferograms, it can be seen that other than when the
tear surface is so rough that the fringes can not be recognized, the values at the side
peaks are close to 0.5. More interesting is that even in the situation when the fringes
can not be identified at all, the side peaks seem clearly identifiable, illustrating the
robustness of the algorithm.
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3. Data processing
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Figure 3.5: Shear estimation algorithm applied to tear film interferograms with different
features. The first column contains the input images (raw interferograms), the
second shows the band-pass filtered images and the third one is the central region of
the autocorrelation of the images on the second column.
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3. Data processing
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3. Data processing
3.3 Pupil position and size estimation
There are three parameters to estimate from the interferograms that are crucial for
the phase map estimation and wavefront integration algorithms: the position, size and
shape of our region of interest (pupil) on the raw interferograms. Ideally, the pupils at
the detector plane would be circular. However, in practice the recorded interferograms
have shapes which are not exactly circular, sometimes clearly elliptical, due to corneal
astigmatism, and some times very irregular due to very rough tear surface and tear
break-ups. Despite this asymmetry, to first approximation, the pupils can in general
be modelled as circular and the algorithm proposed below is based on this assumption.
3.3.1 Algorithm
To estimate the pupil position and radius, a normalised correlation between a binarised
low pass-filtered version of the raw interferograms and a model of a pupil was
maximized, by varying the parameters of the model.
The raw interferograms are low-pass filtered to remove the AC term and then binarised
to eliminate the non-uniformities in illumination, then they are correlated with a series
of models consisting of binarised sums of pairs of circles shifted with respect to each
other by the estimated shear, with different radii (r model ). Finally, we take the value
of r model that maximizes the normalised correlation as the pupil size.
3.3.2 Implementation details
Again, as with the shear estimation, it is convenient for speed and ease of filtering to
use the DFT and the correlation theorem, again padding as before. The low-pass filter
is a circular binary mask in the Fourier space, with a radius equal to the minimum
modulation frequency we considered for all the algorithms.
The intensity profiles of the interferograms change from one series of data to the next,
due to fluctuations of intensity of the source, eye position, etc., requiring different
threshold values to binarize the low-pass filtered images. Despite several attempts
to find a satisfactory automated method for determining the threshold value (e.g.
histogram based) that was able to cope with diffraction rings, tear film features,
52

3. Data processing
etc., we ended up having to set this value manually for each individual series of
interferograms.
The algorithm calculates the peak of the correlation between the binarised low-pass
filtered interferogram and the pupil model by changing the radius of the model r model
in steps of one pixel. In order to reduce the number of computations performed,
the algorithm does not actually calculate the correlation for each value in the range
passed to the algorithm. Instead, two series of calculations are performed, first a coarse
estimation in which the step size between consecutive values of r model is set to √ N,
where N is the number of integers in the radii search range, and a second, in which
the step is set to one, and a smaller range of width 2 √ N centered in the value of the
coarse estimation that yielded the maximum normalised correlation. This works on
the assumption of the correlation being a slow-varying function of the value r model , and
reduces the number of correlation calculations from N to 2 √ N. In a typical situation,
the size of the search range needed to be around 100 pixels, and thus the reduction
on the number of calculations achieved by this two-step calculation was 5. As in the
shear estimation algorithm, we make no attempt to achieve sub-pixel position or size
resolution because the integration algorithm is only applicable to integer values.
3.3.3 Tests
In order to illustrate the performance of the method, we show in figures 3.7 and
3.8 raw interferograms with the edge of the pupils that give the highest normalised
correlation indicated by two red circles, the corresponding binarized low-pass filtered
interferograms used for the correlation calculation and the calculated correlation peak
values for the two-step algorithm.
It can be seen from the figures, that the model seems adequate, both visually on the
left column and numerically on the correlation peak value on the right column, but
it is also clear that there is room for improvement both in the pupil model and the
thresholds election. A further step to improve the algorithm would be to consider
fitting elliptical pupils, though the dimension of the search parameter space would
then increase from one to three (ellipse major and minor radii and orientation).
53

3. Data processing
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(f)
1.2 1.3 1.4 1.5 1.6 1.7 1.8
pupil radius (m m)
(i)
1.2 1.3 1.4 1.5 1.6 1.7 1.8
pupil radius (m m)
(l)
Figure 3.7: Center and radius estimation algorithm applied to tear film interferograms
with different features. The first and second columns show the estimated pupils (red
circles) superimposed on the raw interferograms and corresponding binarised low-pass
filtered interferograms respectively. On the right we can see the calculated peak values
of the normalised correlation within the r model search range. The values from the first
step of the search algorithm are plotted in blue and those on the second step in red.
54

3. Data processing
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Figure 3.8: As figure 3.7.
55

3. Data processing
3.4 Phase recovery
The phase recovery algorithm we now describe is that proposed by Takeda [95], to
extract the phase map of an interferogram with tilt fringes.
3.4.1 Algorithm
Let us begin by rewriting the equation describing the intensity of the interferograms
(Eq. 2.3) in terms of complex exponentials for convenience,
I(⃗r) = I o (⃗r) + I o (⃗r + ⃗s)
+ e +2πi f ⃗ √ c·⃗r +i [φ(⃗r+⃗s)−φ(⃗r)]
I o (⃗r) I o (⃗r + ⃗s) e
+ e −2πi ⃗ f c·⃗r √ I o (⃗r) I o (⃗r + ⃗s) e −i [φ(⃗r+⃗s)−φ(⃗r)] (3.3)
Recalling the Fourier shift theorem and assuming that the carrier frequency is such
that it separates the DC term from the AC term in the Fourier domain, we filter one
of the AC terms in the Fourier domain simply by masking out the other terms, and
then remove the tilt associated with the carrier frequency (using the shift theorem
again) by shifting the AC term to the origin of coordinates of the Fourier domain.
The resulting intensity I Takeda is now the complex magnitude
I Takeda (⃗r) = √ I o (⃗r) I o (⃗r + ⃗s) e ± i [φ(⃗r+⃗s)−φ(⃗r)] (3.4)
were the plus or minus sign depends on which of the AC terms is masked out. If we
now take the complex logarithm of both sides of the equation, and then retain the
imaginary part, we can separate the phase information,
W {± [φ(⃗r + ⃗s) − φ(⃗r)]} = I [ln I Takeda (⃗r)] (3.5)
from the magnitude information, obtained by taking the modulus of equation 3.4
√
Io (⃗r) I o (⃗r + ⃗s) = |I Takeda (⃗r)| (3.6)
Note that the recovered estimated phase is wrapped (W) in the interval [−π, π].
3.4.2 Implementation details
The algorithm calculates the DFT of the raw interferograms, padded as before, then
everything except for one of the AC terms is masked out with a circular binary mask,
56

3. Data processing
centered at the carrier frequency (or its negative) and with radius half of the carrier
frequency. The remaining AC term is translated to the center of the Fourier domain
and then the inverse DFT is computed to obtain I Takeda .
Finally the phase and
magnitude maps are obtained by taking the imaginary part of the complex logarithm
and the modulus respectively.
Note that is has been assumed that the DC and AC terms do not overlap. Overlapping
would give rise to aliasing both in the phase and intensity maps, thus producing
misleading results. The algorithm should not be used blindly, but accompanied with
some estimation of the overlapping of the DC and AC terms in the Fourier domain.
3.4.3 Noise introduced by undesired reflections
Two sources of noise in the experiment are the unwanted reflections from the PBS and
the glass wedges. We now estimate the magnitude of undesired reflections in Takeda’s
phase recovery method.
We can describe the electric field at the output of the tear film experiment as the
superposition of two sheared and tilted copies of the electric field to be studied (E T )
and the undesired reflections (E R ). The observed intensity at the output of the interferometer
will then be
I T+R (⃗r) =
∣ E T (⃗r) + E T (⃗r + ⃗s) e 2πi ⃗ f·⃗s + E R (⃗r) + E R (⃗r + ⃗s) e 2πi ⃗ f·⃗s
∣ 2 (3.7)
where we assume that the source has a coherence length greater than the length of
the optical system. If we now apply Takeda’s algorithm, we get the recovered phase
ψ rec
ψ rec (⃗r) = arctan
{ }
IT,T sin ψ T,T + I R,R sin ψ R,R + I R,T sin ψ R,T + I R,T sin ψ T,R
I T,T cos ψ T,T + I R,R cos ψ R,R + I R,T cos ψ R,T + I R,T cos ψ T,R
(3.8)
where I a,b (⃗r) = √ I a (⃗r)I b (⃗r + ⃗s) and φ a,b = φ a (⃗r) − φ b (⃗r + ⃗s). If the amplitude of
the undesired reflections is much smaller than the amplitude of the light reflected
from the tear film, i.e. I R /I T ≪ 1 and it is assumed that the intensity profiles are
approximately constant over the pupil, then, to first order, we can neglect the terms
57

3. Data processing
in I R,R and approximate the recovered phase by
ψ rec (⃗r) ≈ ψ T,T (3.9)
( )
IR,T
+ [(sin ψ R,T + sin ψ T,R ) cos ψ T,T − (cos ψ R,T + cos ψ T,R ) sin ψ T,T ]
I T,T
If we now define the phase error ψ error as the difference between the recovered phase
and ψ T,T , it is possible to show that this error is bounded by
( )
IR,T
max {|ψ error |} ≈ 2 . (3.10)
I T,T
If we now assume that the scale of the spatial intensity fluctuations is smaller than
the shear amplitude (i.e. I a (⃗r) ≈ I a (⃗r + ⃗s)), then to first order
√
I R,R
max {|ψ error |} ≈ 2
(3.11)
I T,T
We can now use this approximate formula and the measured value for (I R,R /I T,T ) of
about 1/2000 to see that the reflections introduced by the polarizing beam-splitter in
our experimental setup would introduce a maximum error in the reconstructed phase
of around 0.05 radians. Of more concern though are the unwanted reflections from the
glass wedges which have values I R /I T ≈ 1/210, producing a maximum phase error
of about 0.14 radians, equivalent to a wavefront error of λ/50.
It is important to
remember that this error is not on the tear topography but on the phase difference
maps, that to first order are gradients of the tear topography.
3.4.4 Tests
Takeda’s algorithm produces two outputs from each interferogram, a phase map
wrapped between −π and π and a map containing information on the amplitude
of the electric fields. In principle, one would think that the information of interest
is contained purely within the phase map, but the information in the amplitude map
should not be overlooked, as is now explained. Figures 3.9 and 3.10 show a series
of raw interferograms, and the corresponding normalised amplitude maps (values between
0 and 1), and wrapped phase maps retrieved by the algorithm. Even though
we designed the optical system to achieve almost uniform illumination across the interferograms,
it is clear that the amplitude maps are not so. There are a number of
reasons for this:
58

3. Data processing
• Diffraction rings at the pupil edge (seen on the right side of figure 3.9 b). This
could be reduced by conjugating the aperture stop to the tear film for each
subject, but we believe this is not an important issue. In practice, to reduce the
influence of these rings, the pupil radius used for data processing was 95 % of
the estimated pupil radius.
• Dependence of the tear front surface reflection coefficient on the angle of incidence,
that results in changes in the amplitude when the topography departs
from a smooth spherical surface (3.9 e). This has no implications in the phase
map estimation.
• The steepness of the tear film with respect to a spherical shell can reflect some
of the incident rays out of the optical system (vignetting) producing completely
dark areas in the amplitude map, such as the ones in figures 3.9 h and k. There
is little we can about this, because we can only reduce it by increasing the
numerical aperture of the lens closest to the eye. It is a problem that will always
be present, and will lead to areas with undefined phase.
• When the tear breaks up, the optically rough mucus layer of the tear film is
exposed, and most of the light is scattered out of the optical system, again
producing dark areas (3.10 e).
• Finally, when the optical surface is rough the AC and DC terms overlap considerably,
giving non-linear aliasing of phase and amplitude (3.10 b, h and k). This
effect can only be minimized by the use of a higher intensity modulation frequency,
accompanied by a larger sampling density, but we are currently limited
by the number of pixels in our camera.
If we now look at wrapped phase maps in figures 3.9 and 3.10, we can see that there
are two types of discontinuities, the ones that are a closed line or begin and end at
the pupil edge, and the ones that are lines that begin and/or end within the pupil.
The latter correspond to discontinuities in the phase, and to see how they are related
to vignetting and areas with almost zero or zero intensity, we placed a red spot on
each pixel of the intensity map where the amplitude is less than one per cent of the
maximum value. As we can see in figures 3.9 and 3.10, whenever we have a red spot
indicating low amplitude, we have a matching phase discontinuity in the wrapped
phase map.
59

3. Data processing
Phase discontinuities are physically produced in some situations like atmospheric propagation
or with diffractive elements, but because of surface tension true phase discontinuities
cannot be produced at the tear film surface.
3.5 Phase unwrapping
Phase unwrapping is a subject in itself, not only in optical interferometry, but also
synthetic aperture radar, tomography and nuclear magnetic resonance among other
fields [100]. The basic idea behind many algorithms consists of calculating the wrapped
gradients from the wrapped phase, and then integrating these. The main difficulty in
practise is performing the integration when there are certain types of discontinuities
in the initial data that result in residues in the calculated gradient maps.
3.5.1 Principle
For the sake of simplicity, we have chosen one of the simplest to implement and, we
believe, most intuitive unwrapping algorithm, that is a least-square estimation. The
algorithm finds the phase values ψ unwrapped that minimize the sum of the squared
differences between the data gradients and the gradients of the estimated phase map.
In practise, we do not calculate the exact gradients of the data ψ rec or the estimated
wavefront ψ estimated , instead we calculate the differences between neighboring points,
and minimize the error ɛ 2 defined as
ɛ 2 = ∑ i,j
{[∆ x ψ estimated (i, j) − ∆ x ψ rec (i, j)] 2 + [∆ y ψ estimated (i, j) − ∆ y ψ rec (i, j)] 2}
where ∆ x ψ(i, j) = ψ(i, j) − ψ(i − 1, j) and ∆ y ψ(i, j) = ψ(i, j) − ψ(i, j − 1).
(3.12)
3.5.2 Implementation details
The algorithm performs three steps. First, it calculates the differences, then wraps
the differences in the interval [−π, π], and finally performs the integration using an
optimized least-square integrator described in the next chapter. All that should be
pointed out here is that not all the gradients are given the same weight, the implications
of which is that the solution will deviate slightly from the least square solution
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3. Data processing
0 0.2 0.4 0.6 0.8 1 -3 -2 -1 0 1 2 3
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Figure 3.9: Outputs of Takeda’s phase recovery method applied to tear interferograms.
The first column is a series of raw interferograms, the central column shows
the normalised amplitude maps, and on the right we show the estimated phase maps
wrapped in the interval [−π, π]. The red spots on the amplitude maps indicate the areas
where the amplitude value is less than one percent of the maximum value, and can
be compared with the phase discontinuities in the same locations in the corresponding
wrapped phase maps.
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3. Data processing
in the presence of noise.
3.5.3 Tests
The algorithm was tested on the wrapped phase maps from figures 3.9 and 3.10, and
the results are shown in figures 3.11 and 3.12. Comparing with figures 3.9 and 3.10, we
see that the phase discontinuities have been smoothed out in the unwrapping process.
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3. Data processing
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Figure 3.11: Phase unwrapping applied to different tear topographies. The central
column contains the corresponding unwrapped phase maps corresponding to the interferograms
on the left. Note the difference in the colorbar scales.
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3. Data processing
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65

Chapter 4
Wavefront reconstruction from
shear phase maps
In this chapter we present the algorithm used to reconstruct the tear topography
maps from the estimated finite difference maps extracted from the interferograms as
described in the previous chapter. The need for this algorithm arose due to the impossibility
of using conventional pseudoinverse reconstruction techniques and currently
available computing capabilities, given the large number of samples over which the
tear topography was estimated. The work described in this chapter has been published
in Applied Optics [101]
4.1 Introduction
Discrete zonal wavefront reconstruction from shear phase maps is, in principle, a simple
and well understood linear algebra problem [102, 103, 104]. The forward problem
is modelled as a set of linear equations and arranged in matrix form, where the forward
matrix contains the coefficients of the difference equations. Then, the pseudoinverse of
this matrix is calculated using iterative algorithms [105] and finally the reconstructed
wavefronts are obtained by multiplying the discrete difference data by the calculated
pseudoinverse. This method retrieves the best fit wavefronts in terms of the least
square solution with respect to the data.
In theory, there is no restriction to the problem size, that is, the number of samples over
which the wavefront is to be reconstructed. However, in practice, for large number
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4. Wavefront reconstruction from shear phase maps
of samples the computing and storing of the pseudoinverse, as well as the matrix
multiplication can be extremely computationally demanding. The need for faster and
less computer-demanding methods for when fast reconstruction is required or when
there are a large number of wavefront samples, led to the formulation of the problem as
a linear filter using fast implementations of the discrete Fourier transform (DFT) [106,
107, 108, 109, 110, 111], grouped under the name of FFTs [112]. The use of the DFT
for wavefront reconstruction from difference maps was first proposed by Freischlad and
Koliopoulos [106]. In their work, they calculate the transfer function corresponding to
problems with particular geometries related to Shack-Hartmann sensors (e.g. Hudgin
[102] and Fried [103]) for a rectangular pupil and a shear amplitude of one sample.
Elster and Weingärtner [108] treated the one dimensional (1D) case in which the
amplitude of the shear s is a divisor of the total number of samples N along the
direction of the shear. They also suggested a 1D data extension method for cases in
which N is not a multiple of s, but inconsistencies between the extended data and the
forward model introduce errors even in the absence of noise. The 2D data extension
for non-rectangular pupils has been approached by Roddier and Roddier [107], who
proposed an iterative solution that can in principle cope with any pupil shape, and
recently by Poyneer et al. [111], who suggested the extension of the data outside a
pupil so that it remains consistent with the data inside the pupil. In both works the
shear amplitude is one. However, in shearing interferometry, it is often desirable to
be able to adjust the shear to trade resolution for signal-to-noise ratio, and thus a
general reconstruction algorithm is desirable.
In this work, we treat the more general case of 2D wavefront reconstruction from
difference maps originated from sheared phase maps, for any integer shear amplitudes
and convex pupils using two different methods. In the first method we enlarge the
problem dimensions by extending the data to a larger domain while keeping it consistent
with the forward model of the problem. In the second, the original problem
is subdivided into a set of smaller problems with shear amplitude one. The solutions
retrieved by either method will have minimum norm, in the sense that all the modes
in the null space of the problem will have zero amplitude. The performance of the
proposed methods is evaluated by calculating their associated noise coefficient.
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4. Wavefront reconstruction from shear phase maps
4.2 Theory
4.2.1 The transfer function of the reconstruction problem
We will first generalize the linear filter proposed in reference [106] to any integer shear
and to allow for different weights for x- and y-difference data. We want to estimate
a wavefront φ, sampled over a regular grid of points labelled (n, m), inside a pupil
P , from an experimentally measured set of differences S x and S y defined over the
sub-pupils P x and P y contained within the pupil P . Under the forward model, the
phase differences are given by
D x (n, m) = φ(n + s x , m) − φ(n, m), (4.1)
D y (n, m) = φ(n, m + s y ) − φ(n, m). (4.2)
We can now try to estimate the wavefront by least-square fitting these phase differences
to the data over a rectangular grid of dimensions N x × N y , by minimizing the error
E 2 = ∑ n,m
{
|S x (n, m) − D x (n, m)| 2 + |S y (n, m) − D y (n, m)| 2} . (4.3)
If we now assume that the wavefront is periodic (with period N x × N y samples), and
band-limited, it can be proven [106] that minimizing E 2 leads to
[ ( ) [ ( )
exp −2πi k sx
N x
− 1] ˜Sx (k, l) + exp −2πi l sy
N y
− 1] ˜Sy (k, l)
˜φ(k, l) =
( ) ( )] , (4.4)
4
[sin 2 π k sx
N x
+ sin 2 π l sy
N y
where ˜g represents the DFT of g. There are three problems to address in applying
equation 4.4 to estimate φ from the experimental data S x , S y : the assumption of
periodicity on φ; data extension from the domains P x and P y to rectangular grids
that contain them; and the removal of the poles of the filter defined by equation 4.4.
4.2.2 Periodicity
It can be seen from equations 4.1 and 4.2 that for each value of m there are s x less
experimental differences S x than there are values of φ, and similarly for each value of
n there are s y less experimental differences S y than there are values of φ. Therefore,
for a rectangular pupil P with dimensions N x × N y , the dimensions of the pupils P x
and P y will be (N x − s x ) × N y and N x × (N y − s y ) respectively. Thus, we need to
68

4. Wavefront reconstruction from shear phase maps
add s x columns to the array of differences D x and s y rows to the array of differences
D y , while simultaneously being consistent with the forward model and meeting the
periodicity condition. Elster and Weingärtner [108] noticed that when N x is divisible
by s x and N y by s y (which we will henceforth refer to as the divisibility condition)
suitable values for these extra rows and columns can be calculated in terms of the
available data, thus,
S x (N x − s x + k x , m) = −
S y (n, N y − s y + k y ) = −
Nx
sx −1
∑
j=1
Ny
sy −1
∑
j=1
S x (s x (j − 1) + k y , m), (4.5)
S y (n, s y (j − 1) + k y ), (4.6)
for k x = 1, 2, . . . , s x and k y = 1, 2, . . . , s y . It is difficult to extend the wavefront
periodization by Elster and Weingärtner to the cases in which the divisibility condition
is not met, since this would require knowledge of differences not described by equations
4.1 and 4.2, and therefore not available in the data. It would also be undesirable to
use the periodization method described by Ghiglia and Pritt [100], based on mirror
reflections of the initial grid of data, as it quadruples the problem dimension and thus
requires significantly more operations.
4.2.3 Data extension and extension method
We now describe a data extension method for the more general case of a convex pupil
contained within a rectangular pupil, which also solves the cases where the divisibility
condition is not met. Our extension method, is a generalization of that described by
Poyneer et al.[111] for shears of amplitude one. It can be seen that for equations 4.1
and 4.2 to be valid, the following loop continuity condition has to be met
D x (n, m) + D y (n + s x , m) − D x (n, m + s y ) − D y (n, m) = 0. (4.7)
If we consider the difference data as a discrete gradient of the scalar field φ, we can
think of the loop continuity condition as the discrete equivalent of requiring the curl
of the gradient to be zero.
We can now use the loop continuity condition to extend the data outside the pupil, by
copying the difference values in the boundary of the pupil outside the pupil at intervals
69

4. Wavefront reconstruction from shear phase maps
d
e
d
e
b
d
b
e
b
b
d
c
e
b
c
a
d
a
P
a
d
a
P
(a)
(b)
Figure 4.1: Illustration of the method for extending the difference data outside the
pupil, by copying the difference values in the boundary of the pupil outside the pupil.
The black spots are the points over which the wavefront φ is to be evaluated, the gray
area indicates the pupil P and the curved arrows indicate the two values of the grid
that define the difference with coordinates at the origin of the arrow.
given by s x and s y . The method is illustrated with two examples in figures 4.1 (a) and
(b). The black spots are the points at which the wavefront φ is to be reconstructed,
the gray area indicates the pupil P and the curved arrows indicate phase differences
(solid lines measured data, dashed lines extended data). In figure 4.1 (a), the shears
are s x = s y = 1. The available difference data at the boundary of the pupil (indicated
a, b, c, d and e) are repeatedly copied outside the pupil boundary as indicated by the
dashed lines in the figure to the edge of the grid. It can be seen that for any of the
cells outside the pupil the loop continuity is satisfied. In figure 4.1 (b), the shears are
s x = 2 and s y = 1. In this case the x-differences are copied at intervals of s y = 1 as
before, but the y-differences are copied at intervals s x = 2. It can be seen that the
loop continuity condition is satisfied around loops of size s x × s y .
As there are no limits to extending the data using the loop continuity, we can choose
the dimensions of the rectangular grid to satisfy the divisibility condition. Then, the
data can be made consistent with a periodic wavefront using equations 4.5 and 4.6
and the reconstruction problem solved using equation 4.4.
It should be noted that whenever one extends the difference data, other than to meet
periodicity, one is no longer estimating the wavefront by the conventional least square
fit derived from equation 4.3. Instead, one has a weighted least square fit, in which the
data inside the pupil has weight one and the data in the boundary has a weight given
70

4. Wavefront reconstruction from shear phase maps
by the number of copies required for the extension. This suggests that unnecessary
data extension should be avoided.
4.2.4 Filter poles and subdivision method
In this section, addressing of filter poles problem in equation 4.4 will lead us to an
alternative method for approaching the reconstruction problem when the dimensions
of the original grid do not meet the divisibility condition.
From equations 4.1 and 4.2, it can be seen that there are some wavefronts, with period
s x × s y , which give rise to zero for all the differences. These define the null space and
correspond to the poles of the reconstruction filter (equation 4.4).
By setting the
values of the filter at the poles (which occur when ks x /N x and ls y /N y are integers) to
zero, one eliminates only these modes unsensed by the equations defining the forward
problem, thus, producing a solution with minimum norm. Therefore, we replace the
reconstruction filter of equation 4.4 by
⎧
⎪⎨
0 when k sx
[ ( )
[ ( )
N x
˜φ(k, l) = exp −2πi −1]
⎪⎩
k sx ˜Dx (k,l)+ exp −2πi −1] l sy ˜Dy (k,l)
Nx [
)
)]
Ny
elsewhere.
4
sin 2( π k sx
Nx
+sin 2( π l sy
Ny
and l sy
N y
are integers
(4.8)
Equivalently, we can see that sample points in the original problem are only linked
by difference data if they are separated by multiples of the shear distances s x , s y .
The original sample grid can by split into s x × s y interlaced subgrids with sample
spacing s x , s y which can be treated independently from each other, since points on
different subgrids are not linked by difference data (figure 4.2). Thus, the original
reconstruction problem can be subdivided into s x × s y independent problems each of
which can be treated as having shears s x = s y = 1, and which can therefore be solved
by direct application of equation 4.8. The resulting phase maps are re-interlaced onto
the original grid. Since there is no information about the relative piston term of these
interlaced grids in the original measurements, the piston term (i.e. the mean phase
over the pupil) for each of the reconstructed subgrids is set to zero. It is equivalent
to setting the null mode terms to zero, resulting in the minimum norm solution. We
shall refer to this method as subdivision.
71

4. Wavefront reconstruction from shear phase maps
x
y
subgrid (1,1) subgrid (2,1) subgrid (3,1)
subgrid (1,2) subgrid (2,2) subgrid (3,2)
Figure 4.2: Example of the interlaced subgrids for a grid of dimensions N x = 6 by
N y = 7 and shears s x = 3 and s y = 2. The relative phases between points in different
subgrids is undetermined by the data.
4.3 Computing performance
An important consideration when comparing alternative reconstruction methods is
their computational requirements. Figure 4.3 a) plots the number of real multiplications
required to perform the wavefront reconstruction over a square grid with a
total of N points and shear s

4. Wavefront reconstruction from shear phase maps
and subdivision methods (using Matlab 6.1 on a computer with a Pentium 4 processor
at 2.2 GHz and 500 MB RAM memory). Figure 4.3 b) shows the measured
time for an individual reconstruction against the number of reconstruction points N.
The trends of the plot are in agreement with those of the figure 4.3 a) for large N.
Differences between the methods are attributable to the exact details of the computational
implementations. If we compare the performance of the DFT methods
against reconstruction using the pseudoinverse for the case N = 3 × 10 3 , the speed
improvement factor is around 50 (which is comparable to that predicted by the ratio
of the number of multiplications, 2 N 2 /3 N log 2 N ≈ 170). If we try to put this in
the context of our tear topography estimation problem, we have that each slope map
is around 400 samples across, that is N ≈ 1 × 10 5 . This would imply that if we
were using a pseudoinverse reconstruction method, each topography map calculation
would take 3 orders of magnitude longer than with either of the DFT algorithms (i.e.
≈ 200 s/reconstruction). However, this was not the main reason for not using a pseudoinverse
method. The actual calculation of the pseudoinverse itself was not possible
at all with current technology in reasonable times (≈ 100 years with the available
computing power).
4.4 Noise performance
We now compare the DFT methods against the least square pseudoinverse in terms
of their associated noise coefficient[104], which quantifies the effect of white noise in
the data on the reconstructed phase,
C = 1 ∑
R ω R ω , (4.9)
N pupil
where N pupil is the number of wavefront samples inside the pupil of interest P , ω is an
index that runs through all the elements of the reconstruction matrix R. Figures 4.4
(a) and (b) show the noise coefficients for different pupil sizes and unit shear, without
any data extension. For unit shear no subdivision or data extension is necessary
for the DFT reconstruction. Note that the pseudoinverse has a lower (better) noise
coefficient for both pupil geometries.
We now investigate the effect of extending the data by enlarging the original grid
as previously explained. Figures 4.5 a) and b) show the noise coefficients versus
the number of extra rows/columns used for circular and square pupils respectively,
ω
73

4. Wavefront reconstruction from shear phase maps
Number of multiplications required for reconstruction
10 9
10 8
10 7
10 6
10 5
10 4
10 3
10 2
10 1 10 2 10 3 10 4 10 5
10 10 Number of samples
Reconstruction time (s)
10 2
10 1
10 0
10 1
10 2
10 3
10 4
10 5
10 1 10 2 10 3 10 4 10 5
10 3 Number of samples
(a)
(b)
Figure 4.3: a) Plot of the number of real multiplications required to perform the
wavefront reconstruction over a square grid with a total of N points and shear s

4. Wavefront reconstruction from shear phase maps
each of width 32 samples and with unit shear. There are two points to note from
these plots. Firstly, the noise coefficient actually decreases with small data extension.
This suggests that all the existing DFT methods could be improved by applying data
extension. Secondly, we can see that extending the pupil around all sides leads to
a smaller noise coefficient than keeping the pupil in a corner of the grid. This can
be understood by considering the noise variance of the various difference data points.
Recall that in order to use the DFT reconstruction methods, the gradient data need
to be made to meet the periodicity condition, using equations 4.5 and 4.6. That is, for
row (or column) of data in a subgrid, an extra data point given by the negative sum
of the data in the row is added to ensure that the gradients along the row sum to zero
(corresponding to a periodic phase function). The noise in the extra data is given by
the sum of the noise in the row of data. If we consider Gaussian noise of variance σ 2
in the original data, then these extra data points have noise variance N x /s x σ 2 (N x /s x
being the number of data points in the row of the subgrid). This suggests that for
large grids, there will be a considerable noise contribution around the edge due to
the extra periodization data. For the 2D reconstructor this will have greatest impact
on the reconstructed phase near the edges. Extending the data pushes these edges
further from the original pupil, reducing its effect in the reconstructed phase. This is
illustrated in figures 4.6 (a) and (b), which plot the spatial distribution of the noise
in the reconstructed phase maps (i.e. the noise variance) for square 32 × 32 sample
pupils with no data extension and with 4 rows and columns (two around each side)
data extension respectively. The plots correspond to the first and third circles of plot
4.5 (a).
The plots of spatial distribution of noise variance in figures 4.6 a) and b) correspond
to square pupils 32 samples across, unit shear, without data extension (first circle in
figure 4.5 a) and with 2 columns/rows at the pupil sides (third circle in figure 4.5 a).
4.5 Conclusions
We have presented two alternative methods to extend DFT-based wavefront reconstruction
in a convex pupil from difference maps to cope with arbitrary problem dimensions
and arbitrary integer shears, with performance comparable to that of the
pseudoinverse in terms of the noise propagation coefficient. Non-optimized algorithm
implementations showed significant speed increase with respect to the least squares
75

4. Wavefront reconstruction from shear phase maps
N ext/2 N ext/2 Next
N ext/2 N ext/2 Next
1.5
1.5
1.4
1.4
1.3
1.3
Noise coefficient
1.2
1.1
Noise coefficient
1.2
1.1
1
1
0.9
0.9
0.8
0 5 10 15
N ext
0.8
0 5 10 15
N ext
(a)
(b)
Figure 4.5: Effect on noise coefficient of extending the data by enlarging a square (a)
and a circular (b) pupil of 32 samples across and unit shear against N ext , the number
of columns/rows of padding for extension from two sides of the grid only (dots) and for
symmetric extension from all the four sides of the grid (circles). The noise coefficient
of the pseudoinverse is shown as a solid line for reference.
6
6
5
5
4
4
3
3
2
2
1
1
0
30
0
25
20
15
10
5
5
10
15
20
25
30
30
25
20
15
10
5
5
10
15
20
25
30
(a)
(b)
Figure 4.6: Spatial distribution of noise variance for DFT methods for a cornered (a)
and centered (b) square pupil of 32 samples across.
76

4. Wavefront reconstruction from shear phase maps
pseudoinverse reconstruction method. Our analysis shows that the noise performance
of the DFT reconstruction can actually be improved by judicious application of the
extension method.
77

Chapter 5
Preliminary experiments
5.1 Prydal type experiments
Prydal et al. [83] tried to measure the tear thickness by using the interference between
the front and back surfaces of the tear film using a coherent source, reproducing an experiment
by Green et al. [82] for the estimation of the corneal thickness. According to
the mean refractive index values for the tear (1.337) [81] and epithelium (1.401) [113],
if the boundary surfaces were optically smooth, then a large fringe visibility, around
0.44, was to be expected. However, this was not the case, and the fringes present
led to a tear thickness value an order of magnitude larger than previous estimations,
in clear disagreement with other mechanical and interferometric techniques. Prydal’s
model of the tear proved to be an oversimplified one, because the tear has a complex
three layer structure, each with a different refractive index. More importantly, the
deepest of these, the mucus layer, is very rough on the optical scale. It is believed
that what Prydal actually measured was the thickness of the tear and epithelium.
We want to study the tear topography and not its absolute thickness, and therefore
we want to obtain an interference pattern resulting from back reflections from the
front surface of the tear and some other surface within the eye. Provided that the
interferogram contrast is high enough, a modified version of Prydal’s experiment could
be used to serve this purpose. Two experiments performed to test the viability of this
approach are now described and their results discussed.
Both experiments use the same experimental setup: that described in sections 2.6 and
2.7 with the glass wedges removed.
78

5. Preliminary experiments
5.1.1 Normal illumination experiment
For this particular experiment, the eye was placed so that the spherically convergent
illumination was everywhere normal to the surface of the tear film. Two sequences
of 100 images at a 2.5 Hz rate were recorded from each of the two subjects (series
coded bb1, bb2, dm1 no wedges and dm2 no wedges). Figures 5.1 and 5.2 show two
sequences of 12 images each, where the illuminated area over the tear was around
3.1 mm in diameter. Note the diffraction rings at the pupil edge, which decrease in
thickness towards the center of the pupil, the bubbles and other tear features, and
finally the small sets of Newton rings with very low contrast (in comparison to the
diffraction rings) that are due to interference within optical elements and/or their
coatings.
These particular sets of images were selected because they illustrate some of the most
interesting things that can be seen using this technique. The first sequence (figure
5.1) starts with what is probably the most representative situation, a smooth tear
surface with some small bubbles distributed pretty much randomly over the pupil.
Before the next frame the top eyelid moved down and then up a fraction of the pupil
diameter, leaving a clearly delimited region full of bubbles which smooths out slowly in
the following 5 frames (approximately 2 seconds). The subject felt the need to blink
and tried to prevent it in the eighth frame of the sequence, where the tear surface
becomes very rough. Finally the subject blinked between the eighth and ninth frame,
producing again a smooth tear surface that remains so for the rest of the sequence
apart from the displacement of a small line of bubbles.
The second sequence (figure 5.2) shows a phenomenon that could be seen every time
this subject blinked. After the blink shown in the first image, a set of relatively
large bubbles are left near the top of the image pupil, and as time passes these set
of bubbles move downwards while their size is reduced until they almost disappear
before the next blink.
We failed to identify clearly an interference pattern on any of the 400 images recorded
with this experimental configuration that we could attribute with confidence to interference
between the reflections from the front surface and another surface within the
eye.
79

5. Preliminary experiments
bb2 (t = 0.000s)
bb2 (t = 0.390s)
bb2 (t = 0.781s)
bb2 (t = 1.172s)
bb2 (t = 1.562s)
bb2 (t = 1.953s)
bb2 (t = 2.344s)
bb2 (t = 2.734s)
bb2 (t = 3.125s)
bb2 (t = 3.515s)
bb2 (t = 3.906s)
bb2 (t = 4.297s)
Figure 5.1: Sequence of reflections from the front surface of the tear film (series coded
bb2).
80

5. Preliminary experiments
The recorded patterns look very similar, if not identical, to what the subject sees when
the experiment is being performed. We believe the image projected on the retina not
only shows information about the tear topography, but also about floaters within the
eye, and although this was not explored further, it should also show opacity in the
eye due to for example cataracts. Typically when the subject participates for the first
time in the experiment, asked to concentrate on one particular spot of these images
and to remain looking always in the same direction, they would choose a floater or
a bubble on the tear. Then because this feature is rigidly attached to the eye, the
subject would start moving the eye trying (unsuccessfully) to bring that object to the
fovea, resulting in a very frantic eye movement (one could push the analogy with a
dog chasing its own tail to its very limit). The subject would then be asked to try to
fixate on one of the set of small Newton rings that are due to the optics (and therefore
static with respect to the optical setup). In any case, it is an interesting experiment
to participate in as a subject, not only because it allows one to see what happens on
ones own tear film but also to see other structures (e.g. floaters) within ones own eye.
5.1.2 Focused illumination and estimation of undesired eye reflections
Despite the failure of the previous experiment to provide a quantitative tool for studying
the tear topography, a similar experiment with the same optical setup was performed,
but with the eye further away from the experiment, so that the focus of the
beam incident on the eye coincided with the front surface of the tear. In this way, if
there was a reflection from a surface within the eye other than that from the tear surface,
it would show up by producing an interferogram with concentric circular fringes.
These fringes would carry information on the optical path between the front surface
of the tear and the second reflecting surface. This is a similar experiment to that performed
by Prydal, the only difference being that in this case the axis of the incident
beam is perpendicular to the surface of the tear rather than at 45 o .
For this experiment, a series of 100 images on two subjects (series coded bb focused
and lm focused) were recorded. The two images shown at the top in figure 5.3
are representative of the recorded series of images, and show three types of ringing:
those at the edges, caused by diffraction, small sets due to interference from reflections
within optical elements in the setup and the very dim set almost centered covering the
81

5. Preliminary experiments
dm1_no_wedges (t = 0.000s)
dm1_no_wedges (t = 0.188s)
dm1_no_wedges (t = 0.391s)
dm1_no_wedges (t = 0.594s)
dm1_no_wedges (t = 0.781s)
dm1_no_wedges (t = 0.969s)
dm1_no_wedges (t = 1.172s)
dm1_no_wedges (t = 1.360s)
dm1_no_wedges (t = 1.563s)
dm1_no_wedges (t = 1.750s)
dm1_no_wedges (t = 1.9380s)
dm1_no_wedges (t = 2.156s)
Figure 5.2: Sequence of reflections from the front surface of the tear illustrating bubble
movements on the tear. (series coded dm1).
82

5. Preliminary experiments
whole pupil, which we believe results from the interference between the front surface
of the tear and some surface near the back of the cornea. This last set of fringes has
a contrast around 0.16 (estimated in the white rectangle on the left image away from
the edge to avoid confusion with the greater contrast diffraction rings. This means
that the undesired reflection from the surface near the back of the cornea is about 150
times less intense than the reflection from the front surface of the tear, assuming both
reflections are completely coherent. This is equivalent to a maximum error of λ/40 in
the phase recovery algorithm (see section 3.4.3).
Even though this matter will not be pursued in depth, it should be noticed that from
counting the fringes within the image (or at least in the central part, away from the
diffraction rings) one should be able to estimate the distance between the two reflecting
surfaces. To illustrate this point, the interferograms that would result if the reflecting
surface was 55 µ and 510 µm away from the front surface of the tear, typical for the
front and back surface of the cornea respectively, were simulated (bottom two images
of figure 5.3).
The simulation accounted for the focal length of the lens used, the refractive index of
the cornea (taken as 1.376), the double pass, the beam diameter, the wavelength and
the estimated contrast. Comparing the simulations with the data, it seems that one
of the reflections seem to come from a surface in between the epithelium and the back
of the stroma, or otherwise indicates thin corneas in both subject, although the fringe
visibility is quite poor for a good estimation of the number of fringes, not to mention
the difficulty in distinguishing the interference rings from the diffraction rings.
5.2 Estimation of errors for tear topography measurements
If the topography of the tear film was static multiple measurements could be used to
estimate a statistical error on the topography measurements. However, in principle
this cannot be done because the characteristic times involved in the tear topography
dynamics are not known.
83

5. Preliminary experiments
bb_focused
lm_focused
simulated thicknes = 55 mm
simulated thickness = 510 mm
Figure 5.3: The two top images were obtained by illuminating the eye with a converging
beam and placing the front surface of the eye roughly at the beam focus.
5.2.1 Repeatability test and associated error
In an attempt to estimate the errors introduced in the topography estimation by
the optical setup itself, the following experiment was performed. Three glass spheres
of different radii were placed one at a time in the position where the cornea of the
subject would be in a normal tear topography estimation experiment (see figure 2.7),
and a sequence of 50 interferograms was recorded for each of them. The spheres
were calibration accessories from a Humphrey-Zeiss Atlas Corneal topographer, having
radii 9.65, 8.00 and 6.15 mm. The data was then processed (series coded ae 965mm,
84

5. Preliminary experiments
ae 8mm and ae 615mm) and the corresponding topography maps reconstructed. The
resulting mean variation in the RMS of the topography is of the order of 10 nm. It
can therefore be considered that the error associated to the experimental setup, that
is, all the sources of noise related to the acquisition with a CCD camera, vibrations,
etc. in the estimated tear topography will be of the order 10 nm.
5.2.2 Alignment and shear error propagation
Although it is difficult to justify mathematically, experience would indicate that the
major sources of errors when the topography of the tear is relatively smooth (e.g. no
vignetting due to bubbles or tear film break-up) are the estimation of the position
of the interferograms and the shears not being perfectly horizontal or vertical. The
former problem is mainly due to the non-sphericity of the cornea of some subjects,
the slight asymmetries that exist between the two branches of the imaging system and
the non-uniform illumination of the cornea, while the latter is mostly related to eye
movement.
Estimation of the center of interferograms is not crucial for our experiment. The
crucial issue is the relative alignment of the two interferograms (horizontal and vertical
shears) corresponding to each tear topography map. Actually a systematic error in
estimating the position of the interferograms can be afforded, provided this error was
the same for each pair of interferograms and it was the same for the each series of
topography maps to be estimated.
A good estimation of the error introduce by this misalignment is difficult if not impossible,
because some idea of the statistics of the misalignment itself would be needed,
and this is not trivial to obtain. To estimate the distribution of misalignment errors six
series of interferograms were taken (series coded ad1, cp1, cp2, cp3, cw1 and cw3) with
a total of 550 pairs of interferograms. The centers estimated by the correlation-based
algorithm (see section 3.3) were compared with a visual estimation. These particular
series were chosen by alphabetical order and are not representative in the sense that
the subjects ad and cp have significant corneal astigmatism (about 2.5D) compared
to the other subjects, resulting in a poorer performance of the center estimation algorithm,
so one could argue that the typical misalignment will now be over-estimated.
We then assumed that the manually estimated position of the interferograms were
85

5. Preliminary experiments
the correct ones, and calculated the standard deviation of the misalignment for the
550 pairs of interferograms, and found values around 4 camera pixels. Some of the
histograms of these misalignments were also plotted to study their distributions and
different distribution shapes were found, with their mean typically non-zero, indicating
a systematic error within each of the series associated with the particular corneal
astigmatism of each subject.
The estimated shears were not perfectly horizontal or vertical. Typically when the
shear in the desired direction was about 35 camera pixels, the shear in the perpendicular
direction was 1 camera pixel (standard deviation). This is strongly related
to eye movement, because it changes from subject to subject and even within each
series of interferograms. Had this error in the orientation of the shear been due to a
misalignment of the wedges, the error would have been similar (i.e. same direction)
for all interferograms.
It is clear that the effect of misalignment and shear errors in the reconstructed topography
depends on the particular topography to be estimated. However, to get
an idea of how significant the resulting errors are, 7 topography maps (second and
third order Zernike polynomials) and their sheared phase maps were simulated, using
typical shear values (30 camera pixels) and pupil sizes (450 camera pixels). Then the
topography was estimated using the DFT wavefront reconstructor after introducing
all the possible combinations of misalignment error within the range −4 to 4 camera
pixels and the shear error along the perpendicular direction to the main shear in the
range −1 to 1 pixels. The relative RMS error for each of these were calculated and in
a somewhat conservative approach the maximum, 13 % was taken. Notice that this
error can be expressed as a percentage because the reconstructor is linear. It should
be taken into account that the topographies used for the simulation are relatively
smooth and therefore one should expect them to be less sensitive to small misalignments,
while in a real tear topography, small and abrupt features such as tear bubbles
and break-ups will be present. This features will be smoothed out in the wavefront
reconstruction if there is misalignment between the horizontal and vertical shearing
interferograms.
86

5. Preliminary experiments
5.2.3 Simulated interferograms and non-linear errors
We now describe a test of all the data processing software as a whole. Three series of
12 interferograms with similar characteristics to those in our data were simulated, i.e.
440 samples across the pupil, 20-sample shear amplitudes and similar carrier frequencies,
with the second, third and fourth radial order Zernike aberrations. The RMS
amplitude of the polynomials in each series was λ/2, λ and 5λ/2. From processing
the simulated interferograms topography maps were obtained, and two numbers were
extracted: the amplitude error, ɛ amplitude , that is the error in the amplitude of that
particular polynomial, and the residual error ɛ residual that is the RMS of the topography
with that particular polynomial subtracted. The mean values obtained for each
of the series are shown in the table 5.2.3.
Series RMS ɛ amplitude (%) ɛ residual (%)
λ/2 6 22
λ 4 7
5λ/2 18 80
5.3 Validation experiment with a deformable mirror
As a final check on the software and experimental setup as a whole, a simple experiment
with a 37-channel 15 mm diameter deformable membrane mirror (OKO technologies)
was performed to verify that the units and scaling in the data processing are
correct. For this experiment the experimental setup described in sections 2.6 and 2.7
was used after replacing the focusing lens that is closest to the eye by the surface of
the deformable mirror.
Five different voltage configurations were applied to the 37 hexagonal actuators ,
where the voltages were either the maximum (on) or minimum (off). The estimated
topographies from the different voltage configurations were: 1) all actuators off, 2)
all but one actuator off, 3) all but a horizontal line of actuators off, 4) one actuator
and its immediate neighbors on, while the rest remain off and 5) all the actuators
on. The mirror was tested using these 5 configurations with a commercial interferometer,
µPhase2 from Fisba Optik, and the peak-to-valley values of the estimated
mirror surface measured. Then the experiment was repeated with the lateral shearing
interferometer, and the peak-to-valley values obtained were compared with those from
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5. Preliminary experiments
1
10
1
20
15
0
5
0
10
5
-1
0
-1
0
-1 0 1
-1 0 1
Figure 5.4: Reconstructed topography of two simulated small bubbles, of around 0.13
and 0.26 mm in diameter, with heights (depths) of 11 and 22 nm respectively. The axis
units are millimeters and the units on the color scale are nanometers. The error in
the peak-to-valley value in each of the reconstructed topographies versus the original
simulated phase is 15 % and 10 % respectively.
the µPhase2, and it was found that the values differ by around 6%.
5.4 Small detail test
When developing the software the question of whether any small detail could be seen
in the final tear topography arose, because of the non-linear filtering in the Fourier
domain and the smoothing that occurs in the unwrapping and integration processes.
Therefore, some interferograms that would result from a flat tear topography and a
series of bubbles with increasing size were simulated. Figure 5.4 shows the reconstructed
topography of two small bubbles, of around 0.13 and 0.26 mm in diameter,
with heights (depths) of 11 and 22 nm respectively. The errors in the peak-to-valley
in each of the reconstructed topographies with respect to the original simulated topography
are 15% and 10% respectively, suggesting that the presence of small details
can in principle be detected.
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Chapter 6
Tear topography dynamics
experiment
6.1 Data collection protocol
The subjects for all the experiments described in this thesis were volunteers. They were
informed of the experimental protocol, approved by the St. Mary’s Local Research
Ethics Committee of the Kengsinton, Chelsea and Westminster Health Authority. All
the information was coded and strictly confidential. The laser radiation levels were for
all cases well below 2% of the maximum permissible exposures (MPEs) as indicated
on the British and European standard Safety of laser products (BS EN 60825-1:1994
with amendments 1, 2 and 3). The calculations of the MPEs are detailed in Appendix
B.
The measurements for each subject were taken in one session that lasted around one
hour, starting with a description of the experiment, followed by a verification of laser
radiation levels with a calibrated power meter and a demonstration of what the subject
was expected to do during the experiment. A bite registration with soft dental wax
was made and mounted on a x-y-z translation stage (from now on we will refer to
this block as the bite bar) for precise positioning of the subject with respect to the
experiment. The subject was then positioned in front of the experiment while using
the bite bar to align the subject’s dominant eye with the optical setup. This was
achieved by asking the subject to concentrate on one of the small features within the
big red circle projected by the laser onto the retina (only around 2 % of the light
89

6. Tear topography dynamics experiment
Figure 6.1: On the left we can see the light scattered, as opposed to the specular
reflections, by the cornea (large disc) and the crystalline lens (cone) when the tear
topography experiment was being performed (the exposure was 15 seconds). The
picture on the right is a superposition of the picture on the left on top of a picture of
the same subject with the room lights on. It should be noticed that when performing
the experiment, the intensity of the scattered light was extremely dim, and it was
enhanced in this picture for illustrative purposes only.
reaching the eye is reflected back by the front surface of the tear, most of the rest goes
through up to the retina producing a circular spot in which several structures can be
seen as described in section 5.1.1). Finally, when the eye was aligned with respect
to the optical system, the subject was asked to remain as steady as possible and to
blink normally, while sequences of 100 pairs of shearing interferograms (horizontal
and vertical shear) were recorded at a frame rate of 5 Hz. The radii of curvature of
the cornea was also measured with the Atlas corneal topographer (manufactured by
Humphrey-Zeiss).
The data analyzed in this chapter corresponds to 2450 pairs of usable interferograms,
each pair leading to one topography map. No attempt was made to process the data
in which there was vignetting (due to eye movement), the eyelids or eyelashes blocked
part of the illuminated area on the tear or the eye was either too close or too far
from the experiment so that the density of fringes in one of the interferograms was
too high to be sampled properly. The tolerance to alignment in depth is determined
by the fringes introduced by defocus. This can be understood by noticing that to
first order a shear interferometer is a slope sensor, so defocus will produce fringes that
either increase or decrease the number of fringes produced by the tilt introduced by
the wedges, depending on the orientation of the wedges. In our experiment the wedges
were oriented so that when the eye was too close, the number of fringes in one of the
90

6. Tear topography dynamics experiment
z
cornea
rotated cornea
Rc
h(x,y)
q
x
y
Rs
sclera
Figure 6.2: Eyeball modelled as two spherical surfaces.
interferograms increased while decreased in the other one and vice-versa when the eye
was too far, providing then a visual aid for alignment. The area illuminated on the
tear film was typically around 3 mm in diameter.
6.2 Eye movements and effect on tear topography measurements
Before presenting the estimated topography maps the effect that eye movements would
have on the tear topography estimation shall be discussed. For the sake of simplicity,
the rotation of the eyeball with respect to the center of the sclera and the movement
of the head will be considered independent.
6.2.1 Effect of eyeball rotation on tear topography experiment
Let us model the eyeball as a rigid body formed by two spherical surfaces, the sclera
with a typical radius (R s ) of approximately 11 mm and the cornea with a radius (R c )
of around 8 mm (typical values range from 7 to 9 mm). It will be also assumed that
the eyeball rotates around the center of the sclera which is a distance d ≈ 6 mm away
from the center of the corneal surface. Now, for the sake of simplicity but without
loss of generality only rotations in the y − z plane around the center of the sclera will
be considered, that as shown in figure 6.2 coincides with the origin of the system of
coordinates. Then, the change in corneal height ∆h with respect to the x − y plane
due to an eyeball rotation of angle θ is given by
91

6. Tear topography dynamics experiment
∆h(θ, x, y) = d (cos θ − 1) + √ R 2 c − x 2 − (y − d sin θ) 2 − √ R 2 c − x 2 − y 2
∀ (x, y) | x 2 + y 2 ≤ R 2 pupil < R2 c. (6.1)
In a real situation, when the eye is fixating the amplitude of the eyeball rotation is of
the order of 10 −3 rad [69], and thus we can approximate to first order in θ
∆h(|θ| ≪ 1, x, y)
≈
y d
√
R 2 c − x 2 − y 2 θ
∀ (x, y) | x2 + y 2 ≤ R 2 pupil < R2 c.(6.2)
The RMS of the change in topography ∆h for small angles over a pupil with diameter
R pupil < R c is,
RMS(θ) =
[∫∫
≈ d √
2
[
] 1/2
pupil ∆h2 (θ, x, y)dxdy
∫∫
pupil dxdy ( )
−
R2 c
R 2 pupil
ln
1 − R2 pupil
R 2 c
− 1] 1/2
θ. (6.3)
We now need to remove the RMS that is due to piston, tip and tilt, both because
these modes do not affect the optical quality of the eye, and because they cannot be
sensed by the experiment. Due to the angular symmetry, the piston and tip terms
cancel for the case where the rotation is in the y − z plane, leaving only the tilt which
is calculated as
RMS tilt (θ) =
≈
2
∣πRpupil
3 0
[
2 Rc
2 2 d
3 Rpupil
2 +
∫ 2π ∫ Rpupil
0
(
1
3 + 2 3
∆h(ρ, φ, θ) ρ sin φ ρ dρ dφ
∣
) √ ]
R 2 c
R 2 pupil
R 2 c
R 2 pupil
− 1
θ (6.4)
We can now subtract the tilt component from the overall change in topography, where
it should be noticed that even for large eye movement of one degree, around 99.5% of
the overall RMS is due to tilt when the typical corneal radius of curvature are used
and for a pupil radius (that is, the area imaged on our experiment as opposed to the
radius of the pupil of the eye) of 1.5 mm. Thus, the RMS of the change in topography
expected to be measured in our experiment purely due to eye movement, with the
assumptions made in our model of eyeball rotation can be estimated as
RMS(θ) ≈ (2.4 µm) × θ. (6.5)
To convert this topography RMS value to wavefront error RMS in full wavefront
sensing measurements in the eye, one only needs to multiply the topography RMS by
the difference in refractive indices between air (n air ≈ 1) and cornea (n cornea ≈ 1.37).
92

6. Tear topography dynamics experiment
0.2
0.1
0.1
vertical displacement (mm)
0.1
0
-0.1
-0.2
-0.2 0 0.2
horizontal displacement (mm)
0.08
0.06
0.04
0.02
0
-0.5 0 0.5
horizontal displacement (mm)
0.08
0.06
0.04
0.02
0
-0.5 0 0.5
vertical displacement(mm)
Figure 6.3: Estimated movement of the center of the illuminated area of the tear: on
the left a typical movement pattern over 30 seconds is shown (the radius of the red
circle corresponds to one standard deviation), and in the center and right we can see
the histograms of the x− and y−displacement of the interferograms center for all the
2450 usable frames. Both distributions have zero mean and 0.16 and 0.12 mm standard
deviations respectively, which would be equivalent to around 9 nm of wavefront error
RMS.
The movement of the eye was estimated from the recorded interferograms by estimating
the evolution of the center of each interferogram with respect to the first one of
the series. Figure 6.3 a) shows a typical eye movement pattern from the series coded
cp1 over a period of 30 seconds. From the standard deviation of the eye movement,
one can estimate the RMS of the wavefront error that it would be produced by eye
movement of those amplitudes. These can then be taken as an estimation of the
uncertainty in RMS due to eye movement, that is ≈ λ/70 for λ = 632.8 nm.
A completely different approach to estimate the effect of eye movement in the estimated
tear topography maps was also used, based on the correlation of the dominant
low-order modes (defocus and astigmatism) with the pupil displacement for each series
of data. In this way a set of correlation coefficients were generated, providing some
measure of how the changes in these mode, are linked to the movement of the eye.
These correlation coefficients provide a measure of how confidently one can say that
the estimated tear topography dynamics are due to eye movement or true change in
the tear topography. High correlation values would mean that the estimated changes
in the tear topography might be due more to eye movement than to true tear topography
changes.
When the correlation coefficients between eye movement and defocus and astigmatism
were calculated for all the series of data, it was found that in about half the series the
93

6. Tear topography dynamics experiment
values where higher than 0.7 and some were as high as 0.99.
6.2.2 Effect of head movement on tear topography
It was noticed in the course of the experiments that despite the use of the bite bar,
the head of the subject pivots a little bit around the biting point during the 20 to 30
seconds of the data acquisition in a plane vertical plane. This shows very clearly in
the raw data as noticeable fluctuations in the interferogram diameter of up to 10 %,
and on the number of fringes across the interferograms due to the changes in defocus
when the front tear surface changes its distance to the experimental setup along the
optical axis.
For this type of eyeball displacement no modelling was made, only the correlation
coefficient calculation used for eyeball rotation was performed. In this case, we look
for correlation between the defocus component of the measured topography maps and
the relative change in pupil diameter. Figure 6.4 shows a correlation value of 0.67
when all the 2450 usable pairs of interferograms are considered.
Again, as with the eye movement one could use the correlation coefficients as a measure
of how much of the changes in our measurements is due to head movement and how
much is due to true change in the tear topography.
6.2.3 Asymmetries in the optical system
The third and last issue with the experiment that was noticed through the movement
of the eye is the presence of small asymmetries between the two interferometer
branches, which should in principle be identical. However, after studying the data
small asymmetries of two types were noticed: a difference in magnification, which is
not constant throughout the different series and a difference in the movement of the
pair of interferograms on the interferometer output plane. If the pupil magnifications
are different, this will affect the wavefront integration, unless some pupil scaling (and
thus data interpolation) is done. In order to keep the data processing to a minimum,
such scaling was avoided, although it is shown later that the consequences of this do
not seem to have a large effect in the RMS estimation.
We should mention that if we were to repeat the experiments, the alignment of the
94

6. Tear topography dynamics experiment
1.2
1
0.8
(m m)
0.6
RMS measured
0.4
0.2
0
0.2
0.4
0.6
0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
relative pupil radii
Figure 6.4: Correlation between the relative change in the radius of the interferograms
and the estimated change in defocus. The correlation coefficient was calculated considering
all the interferograms from all the series together, and the obtained value was
0.67.
system should be improved until these asymmetries are negligible. The main practical
difficulty when trying to do this is not in focusing the optics, but in positioning the
mirrors that fold the optical path of the two branches so that the two interferograms
can be recorded by the same camera.
6.3 Global error estimation
Based on the experiments and simulations in section 5.2, and the data analysis in
section 6.2, a very rough estimation of the uncertainty in the wavefront error RMS
can be made.
To recap: the repeatability of the experimental setup plus the software was tested
to be around (n tear − n air )λ/60 (see section 5.2.1); it was shown that the effect of
reflections was lower than λ/50 in the slopes estimation, while the typical slope RMS
is of the order of a wavelength, so we can neglect this source of error with respect
to the others (see section 3.4.3); from the eyeball rotation estimation and model, the
estimated RMS error was around λ/70 (see section 6.2.1); from the pupil alignment
and shear estimation errors we estimated a 13 % (see section 5.2.2); and finally, the
95

6. Tear topography dynamics experiment
errors caused by the software and the hypothesis on which it is based (like correctly
sampled band-limited spatial spectra) which for the amplitudes measured (comparable
to or smaller λ/2) we estimate around 6 %. If we take a very conservative approach,
the total uncertainty in our tear topography RMS (σ RMS ) can be coarsely estimated
by combining the errors assuming their sources are independent, getting σ RMS ≈
[
(λ/50) 2 + (14 %) 2] 1/2 . This estimation does not consider the errors due to head
movement because we could not find a simple way to get an estimate.
6.4 Wavefront error RMS introduced by the tear topography
Once the tear topography maps have been estimated as described in chapters 3 and
4, they can be converted to wavefront error maps by a simple multiplication by the
difference in refractive index between the tear film and air (which is approximately
0.134). In this way, the effect that the changes in the tear topography have on the
optical quality of the eye, which is the main goal of the experiment, can be studied.
The results will be now be presented in various formats, that will help to address
different issues: data confidence, relevance of low order aberrations, dynamics of the
optical quality and inter-subject variability.
All the parameters calculated below to describe the optical quality of the eye, are
calculated under the hypothesis of the rest of the optics of the eye being perfect. This
is clearly an unreal situation, because we are already aware of a number of dynamic
factors in the eye, fluctuations of accommodation being dominant from the optical
quality point of view. However, it will serve for our purpose of studying the effects of
the tear film independently from the other changing features (though not completely
de-coupled from eye movement).
6.4.1 Wavefront error RMS evolution for individual series
A simple way to evaluate the imaging performance of a given optical system when
the amplitude of its aberrations are small is by looking at the wavefront error RMS.
If the RMS is smaller than λ/14, then the system is usually referred to as diffraction
limited, which means that it can be considered that the dominant effect in degrading
96

6. Tear topography dynamics experiment
the image quality is diffraction. This is a criterion proposed by Maréchal, and is the
one to be used as a reference for evaluating the optical quality of the tear. Let us now
look at the RMS evolution of a particular data series (code named cp3) to illustrate
some of the findings from the data analysis.
Figure 6.5 (a) shows the diffraction limit as a red continuous line and three different
plots. The blue continuous line is the RMS of the difference between each wavefront
and the mean wavefront of the series, thus providing information about how the optical
quality of the tear does evolve. From now on, whenever we refer to RMS without any
other reference, we will be referring to the RMS of the wavefront calculated in this
way. It was found that the fluctuations of the RMS are comparable to the diffraction
limit (i.e. 0.045µm for λ = 632.8nm), but this will be discussed further in the next
section. The dotted green and dashed black lines show the RMS of the wavefronts
reconstructed using only the horizontal and vertical shear interferograms respectively
with respect to the mean wavefront of the series (as in the blue continuous plot), after
a straightforward modification of the wavefront integration algorithm. These plots
are to be compared with the continuous blue plot for two reasons, first as a peace
of mind check that all three are of the same order of magnitude, to verify that the
pupil misalignments and difference in pupil magnification do not lead to catastrophic
results in the integration algorithm, and second, to have an idea of how different
would the results be if instead of performing a full wavefront estimation with two
shear interferograms, a partial estimation was performed by using only one shear and
thus using a simpler experimental setup and software. It was found that even though
the plots may look very dissimilar in some series, all three have comparable amplitudes,
and when defocus and astigmatism are removed, they are virtually identical.
Figure 6.5 (b) shows the same plots as (a) but now with the defocus and astigmatism
components of the wavefronts removed, and we will refer to these RMS values as
residual RMS. In this way, if we were not to have confidence in the defocus and
astigmatism estimation because of high correlation with eye movement, it can be seen
that the amplitude of the remaining aberrations would be in most cases below the
diffraction limit.
Finally, figure 6.5 (c) shows the evolution of the defocus and astigmatism components
of the wavefront difference. When studying these plots for all the series, it was found
that in most cases defocus is the dominant of the three aberrations, the two astigma-
97

6. Tear topography dynamics experiment
RMS evolution
Residual RMS
second order zernike evolution
0.08
diff limit ( l /14)
RMS
RMS hs
0.08
diff limit ( l /14)
diff limit ( l /14)
RMS vs astig 2
RMS
0.08
astig
1
RMS hs
defocus
amplitude (m m)
0.06
0.04
RMS vs
0 10 20
amplitude (m m)
0.06
0.04
amplitude (m m)
0.06
0.04
0.02
0.02
0.02
0
0
0
0 10 20
0 10 20
time (s)
time (s)
time (s)
(a) (b) (c)
Figure 6.5: Typical wavefront RMS evolution. The red horizontal line in the plots
indicates the diffraction limit for the wavelength used in the experiment. Plot (a)
shows the evolution of the RMS with respect to the initial wavefront estimated using
both interferograms (blue), and only one interferogram (green and black). Plot (b) is
similar to (a) but with defocus and astigmatism removed from the wavefronts, (c) plots
the evolution of the defocus and astigmatism components of the estimated wavefront.
tisms either being comparable to or below the diffraction limit. In order to analyze
these plots further, the correlation coefficients between these aberrations and pupil
position (associated to eyeball rotation) and pupil radius change (associated to head
movement) were used. For this particular case, it was found that the correlation of
defocus with head movement was 0.9 which is quite high, indicating that the large
fluctuations of the defocus component might be due to head movement rather than
true tear topography dynamics. Similarly, one of the astigmatisms correlates well
(again 0.9) with eyeball rotation, and again one should ask if the measured changes
are mostly due to eye movement rather than tear topography dynamics.
6.4.2 RMS mean values
Let us now look at the wavefront error results in a more global way, by plotting
the mean RMS and residual RMS over the data series with more than 75 usable
topography maps, corresponding to time intervals of around 30 seconds (see figure
6.6). Again, the red horizontal line shows the diffraction limit for the wavelength used
in the experiment for reference. The data is separated in three groups: on the left,
over the area with light gray background, we can see those series in which there were
no blinks during the data recording; on the center we have those series in which there
was at least one blink; and on the right over the dark gray area we show those series
98

6. Tear topography dynamics experiment
in which the subject was wearing contact lenses at the time of the experiment. The
numbers on top of the error bars are the maximum correlation coefficient between the
corresponding RMS and the eye and head movement. One can think of these numbers
as indicators of the likelihood of the RMS being an artifact due to the undesired eye
movements as opposed to true topography estimation.
The first thing to notice from the figure, is that in most cases the residual error
remains below the diffraction limit, while the RMS mean values including the defocus
and astigmatism components are comparable to the diffraction limit.
It is important to be aware that the results shown in figure 6.6 correspond only to a
sample of 14 subjects and within those, to the data series with no tear breakup, and
that all the high spatial frequencies higher than half that of the interferogram fringes
are lost, i.e. the smoothest tear surfaces. Thus, we are somehow underestimating
the RMS values, although not significantly, as the processed corresponds to the most
representative situation within the recorded data.
It seems from the data shown in the figure that the tear in front of the contact lenses is
no different from the wavefront RMS point of view to that in front of the cornea when
no contact lens is being worn. However, the processed data in this case represents a
small portion of the recorded data from contact lens users (less than 33 %). In the rest
of the data, the roughness of the tear surface is such that no fringes can be identified,
and therefore no quantitative analysis could be performed.
The conclusions from this section can therefore be summarized as follows:
1) The processed data for non contact lens users represents the typical tear topography
scenario within our set of recorded data, with a smooth tear topography that does
not degrade the eye beyond the diffraction limit.
2) The evolution of the residual wavefront error RMS in a normal situation seems
comparable to the diffraction limit, so we can say that the optical quality of the eye
is not significantly affected by the tear topography dynamics under normal conditions
(i.e. no tear break up or contact lenses).
3) In some cases the correlation coefficients between eye and/or head movement suggest
that the estimated values of defocus and astigmatism estimated from our experiment
might actually be partly due to eye and/or head movement and not only to tear
99

6. Tear topography dynamics experiment
topography changes. It is important to notice that even when these movements might
lead to an overestimation of the RMS, the estimated optical quality of the eye does
not seem to degrade far beyond the diffraction limit.
4) The estimated RMS mean amplitudes are consistent with the RMS variability
measurements reported for full wavefront sensors in the eye, ranging from 0.02 to
0.10 µm [4, 14, 28, 21], with and without paralyzed accommodation.
RMS residual
RMS
no blink blink contact lens on
0.1
0.1
0.2
amplitude ( m m)
0.05
0.5
0.2
0.2
0.2
0.2
0.1
0.4
0.2
0.3
0.3
0.4
0.5
0.5
0.2
0.1 0.7
0.6
0.3
0.3
0.2
0.2
0.6
0.1
0.3
0.3
0.1
0.4
0.3
0.1
0.2
0.3
0.2
0.1
0.3
0.4
0.4
0.3
0.2
0.2
0.4
0.5
0.2
0
cp1 cp2 cp3 cw1 jm1 jm2 jn1 kg2 kh4 pb3 sc3 sj1 cw2 cw3 jp2 sc2 st3 te4 im1 fr1 fr2 fr3
Figure 6.6: Mean RMS and residual RMS over the data series with more than 75
usable topography maps, corresponding to time intervals of 30 seconds. The red
horizontal line shows the diffraction limit (λ = 632.8 nm). The data is separated in
three groups: on the left, over the area with light gray background, we can see the
series in which there were no blinks during the data recording; in the center we have
the series in which there was at least one blink and on the right over the dark gray area
we show the series in which the subject was wearing contact lenses at the time of the
experiment. The numbers on top of the error bars are the the maximum correlation
coefficient between the corresponding RMS and the eye and head movement.
6.4.3 RMS mean evolution
We can now consider the temporal behavior of the RMS by looking at figure 6.7
where the evolution of estimated RMS (b) and residual RMS (a) (this time with
respect to the first topography map instead of the mean topography map as in the
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6. Tear topography dynamics experiment
previous sections) for 30 data series are plotted, using around 2300 topography maps
corresponding to 19 different subjects. The black lines are the mean RMS at the
corresponding time and the gray areas delimited by the blue lines are the areas within
one standard deviation from the mean. Again, the red horizontal line indicates the
diffraction limit at 632.8 nm.
The average blink rate is approximately 1 every 5 seconds [68], with considerable
variation between individuals and visual tasks. Having this in mind, it can be noticed
that the residual RMS stays below the diffraction limit even for periods up to 25
seconds. If we now look at the RMS including defocus and astigmatism, we find that
(ignoring the possibility of the measurements being biased to larger values due to eye
movement), it remains around twice the diffraction limit. This is interesting in the
context of wavefront sensing for refractive surgery, because it shows that the effect
of the tear topography dynamics is negligible. If the whole of the RMS was due to
defocus, for an 8 mm diameter pupil (and assuming that the RMS values measured
for a 3 mm pupil are similar to what they would be over a larger pupil), then an RMS
comparable to diffraction limit corresponds to 1/55 D and twice that to 1/25 D. These
values, from the refraction point of view are negligible if we bear in mind that the
graduation of spectacles go in steps of 1/8 D and in practice nobody prescribes with
more than 1/4 D accuracy.
If we now look at the same numbers in the context of adaptive optics for high resolution
retinal imaging or psychophysical experiments on individual photoreceptors,
even though the degradation of the optical quality of the eye is not very significant,
is by no means negligible if structures of the size of photoreceptors are to be resolved,
and some adaptive correction is needed, probably with a bandwidth of the order of
1 Hz. Again, the results obtained here are consistent with Roorda’s experience [114]
with static wavefront correction on a scanning laser ophthalmoscope (SLO), where
he reported some image resolution degradation when trying to resolve photoreceptors
after preventing the blink for a few seconds while having accommodation paralyzed.
6.5 Second moment of the AC term
Licznerski et al. [74] proposed to use the second moment of the AC term of lateral
shearing tear film interferograms as a tool to study the tear film break-up time. How-
101

6. Tear topography dynamics experiment
0.25
0.25
0.2
0.2
RMS residual
(mm)
0.15
0.1
RMS (mm)
0.15
0.1
0.05
0.05
0
0 5 10 15 20 25
time (s)
0
0 5 10 15 20 25
time (s)
Figure 6.7: Temporal evolution of the estimated wavefront error RMS (b) and residual
wavefront error RMS (a) for 30 data series (around 2300 topography maps) corresponding
to 19 different subjects. The blue dots correspond to the estimated RMS
(b) and residual RMS (a) wavefront aberration introduced by the tear for all the 30
data series. The black lines are the mean RMS at the corresponding time and the gray
areas delimited by the blue lines are the areas within one standard deviation from the
mean. Again, the red horizontal line indicates the diffraction limit at 632.8 nm.
ever, the second moment as defined by Licznerski depends not only on the surface of
the tear, but also on the intensity profile of the image, on the total intensity, on the
shear amplitude, spatial coherence of the source and pupil shape. Thus, monitoring
of the second order moment of the AC term (in the Fourier space) can only be used to
measure relative changes within a series of interferograms from a given subject with
unchanged conditions. The effect of shear amplitude, total intensity and coherence of
the source could be eliminated by defining a normalised second moment,
∫ ∫
f 2 |AC( f)|
M N =
⃗ 2 d 2 f
fR 2 ∫ ∫
|AC( f)| ⃗ 2 d 2 f
(6.6)
where the integral is performed over a circle with radius f R and AC is the part of the
spectrum of the interferogram centered on the carrier frequency. The moment defined
in this way ranges from 0 (when the AC term is a delta function) to 1 (when the AC
term is a ring of radius f R ). In practice, one has a discretised (pixelated) image, and
a discrete Fourier transform, and thus the need for the discrete version of equation
6.6, that is
M N =
∑ m 2 +n 2 ≤R 2
m,n
(
m 2 + n 2) |AC(m, n)| 2
R 2 ∑ m 2 +n 2 ≤R 2
m,n
|AC(m, n)| 2 (6.7)
102

6. Tear topography dynamics experiment
where (m, n) are the frequency coordinates in pixels or sample units with the carrier
frequency as origin of coordinates. Before performing any analysis of real data, one
can already see that the magnitude just defined will depend on the intensity profile
of the illumination, the pupil shape (this depends on the non-spherical corneal topography
of the subject under study) and the radius of the integration area. Thus, the
numbers obtained can only be compared with measurements from instruments with
equal illumination profiles and subjects with similar corneal topography. One also
requires that the spectra of the DC and AC terms do not overlap.
Figure 6.8 shows results for the data of figures 3.1 and 3.2 with the radius of the
integration area equal to half the estimated carrier frequency. Looking at the numbers
obtained and the corresponding figures, we notice firstly, that M N is not a good tool
to study tear break-up, as (f) shows a clear break-up without significant modification
of the topography other than in the surroundings of the break-up area, and N M is
comparable to that for (a) where the tear shows no break-up. Secondly, we can see
that N M seems to give an estimation of the roughness of the topography, roughness
being any departure of the tear topography from a sphere that cannot be described
as a second order polynomial. If the surface of the tear was so rough that the Fourier
transform could be thought of as white noise (equal amplitude for all frequencies), then
the corresponding value for N M (normalised moment of inertia of a disc) would be 0.5.
Figures 6.8 (e), (g) and (h) show values very close to this theoretical case, although
for different reasons, the first two because the spectrum of the DC term overlaps with
the spectrum of the AC term, and the second shows a more white noise appearance
(see figures 3.1 and 3.2). Although these few examples are not an exhaustive test, we
believe that the above defined normalised second moment is not a good number to
describe tear film break-up.
We can also test whether the normalised second moment of the shear interferograms
is related to the tear wavefront error RMS or RMS 2 (that if the RMS ≪ λ as it is
the case, is related to the Strehl ratio). This was put to the test by calculating the
correlation coefficients between M N and the total RMS (0.23), residual RMS (0.36),
total RMS squared (0.23) and residual RMS squared (0.32) for 2150 interferograms.
The low correlation values suggest that the normalised second order moment of the
interferograms on its own does not provide an idea of the optical quality of the tear
film. This was to be expected because to assess the optical quality of the tear we
should look at the tear topography and not at its derivatives, that to first order is
103

6. Tear topography dynamics experiment
what the shear interferograms provide.
104

6. Tear topography dynamics experiment
M = 0.134
N
M = 0.193
N
M = 0.254
N
mk1/46
st3/15
(a) (b) (c)
im4/17
M = 0.240
N
M = 0.484
N
M = 0.154
N
pb2/92
mk1/33
kh3/47
(d) (e) (f)
M = 0.412
N
M = 0.489
N
(g)
te2/49
(h)
te2/13
Figure 6.8: Normalized second order moment of the AC term of tear topography
lateral shearing interferograms (M N ) illustrating some of the different tear topography
features.
105

Chapter 7
Summary and discussion
The design of a double shearing interferometer for evaluating the dynamics of the
front surface of the pre-corneal tear film has been described. The instrument was
built and tested on 20 subjects, and the data was analyzed with purpose-developed
software. In order to achieve the topography integration from finite differences, we
generalized an existing reconstruction method based on the DFT, which allows the
processing of data volumes that can not be achieved with current computing powers
and conventional pseudoinverse reconstruction techniques.
The data processing software was tested in a set of recorded interferograms that
illustrate the different features found on the tear topography, to become familiar with
the limits of both the software algorithms and the theory on which these algorithms
are based.
We found that observing the back-reflection of the tear surface with normal coherent
illumination might be a qualitative tool for studying the tear topography evolution,
without the need for data processing and with relatively high resolution. The only
problem that might prevent this technique from being used clinically is the small
tolerance to eye misalignment.
A modified version of Prydal’s experiment was performed to estimate the tear film
thickness [83], with the purpose of estimating the amplitude of back-reflections from
surfaces within the eye other than the front surface of the tear. It was found, based
on fringe contrast estimation that the intensity of these reflections are around 150
times lower than those of the front surface of the tear, thus introducing very little
106

7. Summary and discussion
error in Takeda’s phase recovery algorithm [95]. It was also noticed on the concentric
ring pattern observed that counting the number of fringes within a certain radius
allows one to estimate the distance between the two back-reflecting surfaces. After
performing some computer simulations based on a simple model, we believe the deeper
reflecting surface is not always the same for all subjects. In the first subject, the
inner back-reflecting surface was likely to be the front surface of the cornea (around
50 − 60 µm deep) while on the second subject it was more likely to be the back surface
of the cornea (around 500 µm deep). This results may explain why Prydal estimated
a tear film thickness of around 40 µm, which is one order of magnitude larger than
the accepted values.
We tested the instrument repeatability and simulated the effect of pupil error alignments
and general data processing error for the current experimental setup, to get a
feeling of the amplitude of the errors or uncertainties to be expected in our measurements.
We also performed a simple validation experiment on the experiment against
a commercial interferometer.
Ethical approval and insurance cover was obtained for acquiring data from a number
of subjects, and 20 volunteers were recruited for the experiments. From the analyzed
data, the amplitude of the theoretical errors due to eye movement in the tear
topography maps was estimated. Then with this estimation and the results from previous
experiments and some simulations, a global error figure for the estimated tear
topography RMS was obtained.
We found that the contribution of the tear film to the degradation of the optical
quality of the eye even when the defocus and astigmatism are considered (not reliably
estimated from our measurements due to eye movement) is not significant, that is,
assuming the rest of the optics of the eye to be perfect (Strehl ratio = 1). It can be
concluded that the changes in the tear film topography are not large enough to bring
the optical quality of the eye below the diffraction limit. It should be kept in mind
that this conclusion has been drawn from the data that could be processed, which
corresponds to the smoothest tear surface, therefore leading to an underestimation of
the degradation of the optical quality of the eye by the tear film. Rough surfaces, can
degrade the optical quality of the eye more significantly, but these rough surfaces are
less common than the smoother surfaces found in the processed data.
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7. Summary and discussion
The values found for RMS variability are consistent with the reported variability in
full wavefront sensing in the eye measurements, with paralyzed accommodation.
It was noticed that the front surface of the tear in front of contact lenses is very rough
and although a quantitative analysis of the topography was not possible when the
typical roughness was present, we believe the subject should be studied further given
the high number of contact lens users.
From the data it can also be infered that the characteristic times associated with
change in the tear topography are of the order of seconds, so if one was thinking of
using an AO system in the eye just to correct the tear fluctuations, a bandwidth of
the order of 1 Hz or faster should be considered.
We put to the test the use of the amplitude of the second moment of the intensity
of a lateral shearing interferogram with tilt fringes as a tool to study tear break up
times. We first suggested a normalization of this moment to make it more robust
and independent of certain experimental parameters, and then showed, with a few
experimental examples, that this number is no good on its own for the estimation of
the break up time. It was also found that there is very little correlation of the second
moment with the actual optical quality of the tear, either in terms of wavefront error
RMS or Strehl ratio.
Even though the data processing could not be completely automated and despite
the inherent problems of using back-reflections from a moving eye, we believe that
interferometry (lateral shearing, radial shearing or else) could be useful as a tool
to study the tear film of subjects pre- and post-refractive surgery, and with tear
conditions, but only if the issue of eye movement is addressed in far more depth than
it was here.
After the tear topography experiments, the experimental setup was modified to be
used as a full wavefront sensor in the eye and a similar one was built for a different
wavelength. Each instrument recorded simultaneous lateral shearing interferograms
and SH spot patterns, for a qualitative study on the feasibility of using conventional
interferometry for wavefront sensing in the eye. The experiments performed suggest
that conventional interferometry is not as straightforward to use as other types of
wavefront sensing due to speckle. When recording interferograms with short exposure
times, the interferograms will be severely affected by speckle, while if one tries to
108

7. Summary and discussion
f6
f6
CCD
camera
T''
BS
f5
f5
BS
T'
f 4= 30 f 3= 150
f 2= 40
T
eye
180 190
Figure 7.1: Modified imaging branch to convert the lateral shearing interferometer
into a radial shearing interferometer.
increase the exposure times to reduce the speckle, the interferogram contrast will
decrease.
7.1 Future work: Radial shearing interferometry
Radial shearing interferometry is a self-referencing interferometric technique that can
allow the estimation of a phase map with only one interferogram, as opposed to lateral
shearing interferometry, that requires two interferograms [92]. In this technique, two
copies of the beam to study are made to interfere, with one of the beams expanded
with respect to the other. The intensity changes due to the beam profile spatial
fluctuations could be separated from the phase changes in the same way as it is done
in the lateral shearing interferometer described in Chapter 2 by introducing tilt fringes
and applying a similar phase recovery technique.
The imaging branch of the experimental setup of the lateral shearing interferometer
was modified in the way described by figure 7.1. The two copies of the beam are
created by the first beam splitter, and recombined through the second one. Each
branch of the interferometer would be a 4f system with different focal lengths, the
ratio of the focal lengths being the relative magnification of the pupils. This type of
configuration, with 50:50 beam splitters, would make better use of the light reflected
from the tear, because while in our lateral shearing interferometer only around 12 % of
the light is used, in this setup we make use of 50 % of it, a gain factor of more than 4.
More important would be the simplification of the data processing and elimination of
sources of error like the pupil alignment and asymmetries between the optical branches
109

7. Summary and discussion
ao2_radial_shear (t = 3.625s)
ao2_radial_shear (t = 4.203s)
Figure 7.2: Typical radial shearing interferograms from tear film topography with
no tilt between the interfering wavefronts. Recorded with the experimental setup
described in figure 7.1.
of the interferometer.
7.2 Publications
The following contributions related with this work have been published:
A. Dubra, C. Paterson, and J.C. Dainty. Wave-front reconstruction from shear phase
maps by use of the discrete Fourier transform. App. Opt. 43 (5) 1108-1113.
A. Dubra, J.C. Dainty, and C. Paterson. Measuring the Effect of the Tear Film on
the Optical Quality of the. Eye Invest. Ophthalmol. Vis. Sci. 2002 43: E-Abstract
2045.
A.Dubra, C.Paterson and J.C. Dainty. Tear film topography dynamics measurement
with a shear interferometer. Eye Invest. Ophthalmol. Vis. Sci. 2004 45: E-Abstract
2777.
A. Dubra, J.C. Dainty, and C. Paterson. Double-lateral shearing interferometer for
the quantitative measurement of the tear film topography (submitted to App. Opt.).
A. Dubra, J.C. Dainty, and C. Paterson. Study of the tear topography dynamics using
a lateral shearing interferometer (in preparation, to be submitted to Optics Express).
110

Appendix A
Preliminary study of feasibility
of interferometric wavefront
sensing in the eye
As mentioned in the introduction, a number of wavefront sensors have been used to
assess the optical quality of the human eye. In this chapter the feasibility of using interferometric
techniques for evaluating the optical quality of the human eye is explored
through an experimental qualitative comparison of a lateral shearing interferometer
and a Shack-Hartmann wavefront sensor under similar conditions.
Because the retinal reflectance and scattering dependence on wavelength is known to
be very complex and subject-dependent [115], it is likely that different wavelengths
perform differently both for a SH wavefront sensor and an interferometer. It was
therefore decided to set up two similar experiments for two different wavelengths,
632.8 nm and 780 nm. These wavelengths were chosen because of their widespread
use in research laboratories [20, 24, 14, 28, 39] and commercially available wavefront
sensors (e.g. Zywave by Bausch & Lomb).
A.1 Experimental setups
There are a number of practical issues related to setting up a wavefront sensor for
the eye [116, 56], like the low light levels that can be used for safety reasons and the
double pass problem [22]. We do not intend to deal with these issues in depth here,
111

A. Preliminary study of feasibility of interferometric wavefront sensing in the
eye
but only to briefly describe how the experiments were setup for the data acquisition.
Most Shack-Hartmann wavefront sensors and laser ray tracers in the eye work by
delivering a narrow collimated beam (1 − 2 mm in diameter) into the eye, parallel to
the optical axis of the instrument and the eye, the latter assumed to be the line of
sight [117], and slightly off the pupil center to eliminate the corneal reflection. The
purpose of the narrow incoming beam is to try to reduce the loss of information on odd
aberrations due to symmetric double pass, by using an entrance pupil (beam diameter)
over which the optics of the eye is assumed to be almost diffraction limited. In this
way, the wavefront error of the outgoing beam will be mainly that of the second pass
through the optics of the eye after the reflection on the retina, when using a larger exit
pupil. It has also been shown that the phase information on the first pass through the
optics of the eye is almost completely eliminated after the non-specular reflection on
the retina, due to its roughness and movement. By delivering the incoming beam off
the optical axis but parallel to it, the corneal reflections are eliminated by vignetting.
After the incoming beam is reflected back from the retina, the wavefront emerging
from the eye is imaged by one or more pairs of lenses forming 4f systems, that produce
a copy (ignoring diffraction and lens aberrations) of both phase and intensity of the
beam exiting the eye on a plane we will denote output plane. In SH wavefront sensors
a lenslet array in the output plane focuses the light onto the surface of a detector
(typically a CCD camera) forming an array of spots. The displacement of these spots
from a reference position is proportional to the average slope (tilt) of the wavefront
over the corresponding lenslet, assuming the intensity over the lenslet is uniform.
This last hypothesis is not always true due to speckle, as we will see later, but which
can be reduced by time averaging using the random eye movements, or equivalently,
using a high frequency (a few KHz) scanning mirror [28]. In the lateral shearing
interferometers described below, either a glass wedge or plate with parallel sides is
placed after the last lens before the output plane (where the beam is collimated) at
45 o to the optical axis to produce the lateral shearing interferogram from the reflections
from the front and back surface of the wedge.
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A. Preliminary study of feasibility of interferometric wavefront sensing in the
eye
A.1.1
Experimental setup for 632.8 nm
This instrument is a modification of the interferometer described in sections 2.6 and
2.7, used for the tear topography estimation. The illumination branch shown in figure
A.1 uses a spatially filtered He-Ne single mode laser (632.8 nm) to deliver a 1 mm
diameter collimated beam to the eye, off-center by about two millimeters.
The polarising beam-splitter and the λ/4 waveplate are left as they were in the tear
film experiment to reduce the reflections from the beam-splitter, although this might
not be the best configuration to divert most of the light reflected back from the retina
down to the imaging branch due to the birefringent and depolarising properties of the
eye [118] which have great inter-subject variability.
He-Ne
laser
633nm
ND filter
microscope objective
spatial filter
f 1= 250
stop aperture
(off axis)
s-polarisation
PBS
l/4
eye
p-polarisation
circular-polarisation
Figure A.1: Illumination branch of the wavefront sensing experiment using a 632.8 nm
He-Ne laser, which delivers an off-center collimated 1 mm beam to the eye.
The imaging branch remains virtually unchanged from that described in 2.7 apart
from minor modifications that can be seen in figure A.2. The focusing lens that was
shared with the illumination branch, the first wedge and λ/2 waveplate were removed
113

A. Preliminary study of feasibility of interferometric wavefront sensing in the
eye
T''
T''
CCD
camera
f 6 = 150
f 5 = 150
T'
f 4= 30 f 3= 150
T
CCD
camera
eye
lenslet
array
wedge 2
300 180
180 170
Figure A.2: Imaging branch of the wavefront sensing experiment using a 632.8 nm
He-Ne laser. The exit pupil of the eye (T) is conjugated to a SH lenslet array (T”)
and a CCD camera that will record the laterally sheared interferograms (T”’).
and a second camera is placed after the second (and now only) glass wedge. The
resulting optical system produces a replica of the electric field at the pupil plane of
the eye demagnified by a factor of 5.
A.1.2
Experimental setup for 780 nm
The optical setup built for the wavefront sensing experiments at 780 nm (see figures A.3
and A.4) is quite similar to the one just described, although the illumination branch
is slightly simpler because the source, which in this case is a diode laser emitting at a
peak wavelength between 780 and 800 nm (depending on current and temperature) is
coupled to a single mode optical fiber, and thus the spatial filtering is produced within
the optical fiber by attenuation of all but the fundamental modes (gaussian intensity
profile and orthogonal polarisation states).
After the source there is a collimating lens, which we can now afford to choose with a
short focal length for compactness because the condition on uniform intensity over the
beam entering the eye is now not as demanding as it was for the tear topography experiment,
the aperture stop now being only 0.6 mm in diameter (as opposed to 15 mm
in the tear related experiments). Then the non-polarising beam splitter (NPBS) reflects
50 % of the light towards the 4f system in front of the eye with a magnification
of 2. This pair of lenses was not strictly needed but it was added thinking of using it
to compensate the sphere of the refractive error of the subjects. This is achieved by
displacing the subject and the lens together along the optical axis of the system, in
an arrangement that is known as the Badal optometer [77].
In this setup polarisation control was deliberately avoided, thus having the advantage
of collecting not only the light with the same polarisation state as the input beam, as
114

A. Preliminary study of feasibility of interferometric wavefront sensing in the
eye
a
diode laser
@ 790nm
single mode optical fiber
f 1= 60
stop aperture (off axis)
NPBS
f 2= 40
f 3= 80
eye
120 100
a/2
Figure A.3: Illumination branch of the wavefront sensing experiment using a 780 nm
diode laser, which delivers an off-center collimated 1.2 mm beam to the eye.
in the previous experiments, but also the light with the orthogonal polarisation state.
The price to pay for this gain is the tilting of the beam splitter and illumination
branch to eliminate the reflections from the beam splitter, which in this case are even
stronger in comparison with the signal in the tear interferometry experiment, because
the total power reflected from the retina is lower than that reflected by the tear. It
can be demonstrated by using the reflection law that if the optical axis of the part of
the illumination branch before the beam splitter is rotated an angle α and the beam
splitter is rotated half that angle, then the collimated beam after the beam splitter
will be parallel to the optical axis of the Badal optometer and that the undesired
back reflections from the beam splitter (leaving the beam splitter through the left
surface on the figure) will go off-axis at an angle α, allowing us to spatially filter these
reflections.
The imaging branch of the experiment (figure A.4) shares the Badal optometer and
beam-splitter with the illumination branch. The only purpose of the 4f system to the
left of the beam-splitter is to focus the light coming back from the eye so that a stop
aperture can be placed in the focal plane of this telescope to filter the off-axis beam
splitter reflections. Again, the final optical element before the planes conjugated to
the exit pupil of the eye is the wedge at 45 degrees, that produces the two reflections
on the camera that will register the interferograms and lets through the light that
goes to the SH sensor.
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A. Preliminary study of feasibility of interferometric wavefront sensing in the
eye
a/2
CCD
camera
CCD
camera
T'' T''
wedge
f 5= 40 f 4= 40
NPBS
f 2= 40
f 3= 80
T
eye
lenslet
array
40
80
80
stop aperture (BS backreflection filter)
120 100
Figure A.4: Imaging branch of the wavefront sensing experiment using a 780 nm diode
laser. The exit pupil of the eye (T) is conjugated to a SH lenslet array (T’) and a
CCD camera that will record the laterally sheared interferograms (T”).
A.2 Data acquisition
The interferograms and SH spot patterns were recorded with two CCD cameras, Retiga
1300 and Retiga EX (manufactured by QImaging), synchronized by an external TTL
signal. The exposures of the cameras were equal at all times, and the gain of the
electronics of each camera was adjusted to try to optimise the use of the dynamic
range and take into account the fact that the SH camera receives approximately 11
times more energy than the one recording the interferograms, and also the light is
focused into an array of spots as opposed to the interferogram recording camera in
which the energy is more evenly distributed over the pupil. The lenslet array used
to produce the SH spot patterns was a Fresnel microlens array on polycarbonate,
produced by WelchAllyn, with a focal lens of 7.5 mm and 200 µm pitch.
For the data acquisition, the subject was set in front of the experiment and the head
kept in position with the aid of a bite bar. In all the wavefront sensing experiments the
optical power entering the eye was kept below 1 % of the MPEs for 10 s, as calculated
in Appendix B.
A.2.1
Data acquired with 632.8 nm source
A subject (coded name ad) was tested with three different optical power levels entering
the eye and exposure times chosen so that the energy recorded by each camera on each
frame was always the same. The optical power entering the eye was adjusted by the
use of neutral density absorption filters placed between the laser and the spatial filter
to reduce the effect of undesired reflections. The experiment was repeated twice, with
116

A. Preliminary study of feasibility of interferometric wavefront sensing in the
eye
the wedge in opposite orientations (180 degrees rotation), producing in one a large
shear comparable to the pupil diameter, and in the other an almost negligible shear.
This could be used to get an idea of the spatial coherence of the light emerging from
the eye.
Figure A.5 shows the interferograms and SH patterns that resulted from using 3.5, 1.0
and 0.35 µW with camera exposures 0.2, 0.7 and 4.0 s respectively on each row. The
first and second columns display the lateral shearing interferograms with the wedge in
the orientations that produces large and small shear respectively. The third column
shows the associated SH spot pattern. Apart from the pupil size change in response
to the intensity of the light entering the eye, it can be clearly noticed that the speckle
both in the interferograms and the SH spots is reduced with increasing exposures. It
can also be seen that for the interferograms corresponding to a small lateral shear,
vertical fringes with very poor contrast seem to appear for the 4 second exposure but
these fringes are not of the spatial frequency produced by the wedge. We also looked
at the spectra of the interferograms looking for a peak at the frequency of the fringes
that the wedge would produce, and we found that there is indeed a peak, although
based on the experience gained from processing the tear film data for the previous
chapter, we believe no reliable quantitative data can be extracted.
A.2.2
Data acquired with 780 nm source
The same subject was tested for different exposures and this time with equal optical
power of 4 µW. The rows in figure A.6 show the interferograms and SH patterns
that resulted from the exposures of 0.5, 1.0 and 3.0 s respectively. Notice that the
optical magnification of the system is different from the previous experiment, hence
the increase in number of lenslets over the pupil in the SH spot patterns.
Again, as in the previous experiment, reduction of speckle can be noticed for longer
exposures. Fringes with poor contrast (≈ 0.1) can be observed for the interferograms
with small shear and no fringes at all can be noticed for the larger shearing interferograms.
If one could vary the shear in a continuous way like in the interferometer
proposed by Lohmann [97] and the spatial coherence characteristics of the source were
known, then the fringe contrast could be used to study the spatial coherence of the
retinal reflection. When looking at the amplitude of the spectra at the spatial fre-
117

A. Preliminary study of feasibility of interferometric wavefront sensing in the
eye
4
4
4
2
2
2
mm
0
mm
0
mm
0
-2
-2
-2
mm
mm
-4
4
3
2
1
0
-1
-2
-3
-4
4
3
2
1
0
-1
-2
-3
-4
ad / 3.20 mW / 0.2s -4
-4
-5 0 5
mm
(a) (b) (c)
-5 0 5
mm
(d) (e) (f)
-2
ad / 0.35 mW / 4.0s -4
-5 0 5
-4 -2 0 2 4
mm
mm
mm
mm
4
4
2
2
0
-2
-2
ad / 1.00 mW / 0.7s -4
-4
4
4
2
2
0
-4 -2 0 2 4
mm
-4 -2 0 2 4
mm
-4 -2 0 2 4
mm
(g) (h) (i)
mm
mm
0
0
-2
-4
-4 -2 0 2 4
mm
-4 -2 0 2 4
mm
Figure A.5: Interferograms and SH spot patterns that resulted from using 3.5, 1.0
and 0.35 µW at λ = 632.8 nm with camera exposures 0.2, 0.7 and 4.0 s respectively
on each row (subject ad). The first and second columns display the lateral shearing
interferograms with the wedge in the orientations that produces a large and small
shear respectively. The third column shows the associated SH spot pattern.
118

A. Preliminary study of feasibility of interferometric wavefront sensing in the
eye
quency of the fringes, we found a peak. Also a small decrease in amplitude, with
respect to the DC term, with increasing exposure was observed, we believe it was not
significant enough to assert that the fringe contrast increases with exposure times.
When comparing the SH spot patterns it can be noticed that at 780 nm there is a
uniform halo (not speckled) surrounding every spot, not present at 632.8 nm. The
presence of this scattering phenomenon at near infrared wavelengths is well known
[26, 25].
Finally, to rule out the possibility of the blur in the fringes being caused by fluctuations
of accommodation, the measurements were repeated under the same experimental
conditions, after paralyzing the accommodation (and pupil dilation as a side effect)
with cyclopentolate hydrochloride 1 % (see figure A.7). It was found that again as
the exposure increases the speckle is reduced, but no improvement in fringe contrast
was observed with respect to the measurements with the same light source and nonparalyzed
accommodation.
A.3 Conclusions
We have tested, we believe for the first time the use of a lateral shear interferometer for
wavefront sensing in the eye, using two different wavelengths and two shear amplitudes.
For the experiment at 632.8 nm we could not observe interference fringes for either
of the shear amplitudes (0.75 mm and 0.17 mm). In the experiment using 780 nm we
could not observe interference fringes for the larger shear amplitude (2.4 mm), but
fringes of contrast 0.1 could be seen for the smaller shear (0.17 mm) and all three
exposure times (0.5 s, 1 s and 3 s).
The fringe visibility of the interferograms depends on the complex degree of coherence
of the electric field at the pupil plane over the integration time [91]. The experiments
that were performed do not provide enough information to plot the fringe visibility
as a function of the shear amplitude. However, it can be said that the width of the
visibility function (in terms of the shear) is smaller than 0.17 mm at 632.8 nm and
of the order of 0.1 mm at 780 nm. This width tells us which is the maximum shear
amplitude for which interference fringes can be observed.
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A. Preliminary study of feasibility of interferometric wavefront sensing in the
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4
4
4
2
1
2
2
mm
0
mm
0
mm
0
-2
-2
-2
-4
4
ad / 4.00 mW / 0.5s
-4 -2 0 2 4
mm
-4
-4 -2 0 2 4
mm
4
4
-4
-4 -2 0 2 4
mm
(a) (b) (c)
2
2
2
mm
0
mm
0
mm
0
-2
-2
-2
-4
4
ad / 4.00 mW / 1.0s
-4 -2 0 2 4
mm
-4
-4 -2 0
mm
2 4
4
4
-4
-4 -2 0
mm
2 4
(d) (e) (f)
2
2
2
mm
0
mm
0
mm
0
-2
-2
-2
-4
ad / 4.00 mW / 3.0s
-4 -2 0 2 4
mm
-4
`-4 -2 0
mm
2 4
-4
-4 -2 0
mm
2 4
(g) (h) (i)
Figure A.6: Interferograms and SH patterns that resulted from the 4 µW at 780 nm
setup with exposures of 0.5, 1.0 and 3.0 s. The first and second columns display the
lateral shearing interferograms with the wedge in the orientations that produces a
large and small shear respectively. The third column shows the associated SH spot
pattern.
120

A. Preliminary study of feasibility of interferometric wavefront sensing in the
eye
From the experiments performed, it seems that conventional interferometry (e.g.
shearing interferometers or Twyman-Green) is not as straightforward to use as other
types of wavefront sensing (such as a Shack-Hartmann sensors), due to speckle. When
recording interferograms with short exposure times, the interferograms will be severely
affected by speckle, while if one tries to increase the exposure times (or scan the incoming
beam over the retina) to reduce the speckle, the interferogram contrast (and
thus the SNR) will decrease.
121

A. Preliminary study of feasibility of interferometric wavefront sensing in the
eye
5
5
5
3
3
3
1
1
1
mm
-1
mm
-1
mm
-1
-3
-3
-3
-5
ad / 4.00 mW / 0.5s
-6 -4 -2 0 2 4 6
mm
-5
-5 0 5
mm
5
5
5
-5
-5 0 5
mm
(a) (b) (c)
3
3
3
1
1
1
mm
-1
mm
-1
mm
-1
-3
-3
-3
-5
ad / 4.00 mW / 1.0s -5
-6 -4 -2 0 2 4 6 -5 0 5
mm
mm
5
5
5
-5
-5 0 5
mm
(d) (e) (f)
3
3
3
1
1
1
mm
-1
mm
-1
mm
-1
-3
-3
-3
-5
ad / 4.00 mW / 3.0s -5
-6 -4 -2 0 2 4 6 -5 0 5
mm
mm
-5
-5 0 5
mm
(g) (h) (i)
Figure A.7: Interferograms and SH patterns that resulted from the 4 µW at 780 nm
setup with exposures of 0.5, 1.0 and 3.0 s, with accommodation paralyzed with cyclopentolate
hydrochloride 1%.
122

Appendix B
Laser safety calculations
In all the experiments conducted for this thesis, the maximum permissible exposures
(MPEs) of the human eye to electromagnetic radiation were calculated based on the
British and European standard Safety of laser products (BS EN 60825-1:1994 with
amendments 1, 2 and 3). The MPE calculations below will only consider retinal
thermal damage as the cornea, aqueous humor, lens and vitreous humor are considered
transparent for all the wavelengths of the sources used in the experiments (i.e. within
the range 600 − 1400nm).
apparent
source
a
eye
r
Figure B.1: Definition of subtended angle α by an apparent source at a distance r.
The diameter of the pupil of the eye applicable to measuring laser irradiance exposure
for a wavelength between 400 and 1400 nm is 7 mm for t < 3 × 10 4 s, thus, the area of
the pupil to be considered is
A pupil ≈ 4 × 10 −5 m 2 .
(B.1)
123

B. Laser safety calculations
B.1 MPE for He-Ne laser in tear topography experiment
The wavelength λ of the He-Ne laser used for illuminating the precorneal tear film
in the shear interferometer is 632.8 nm, and the angular subtense α of the apparent
source for the tear topography experimental setup was greater than 300 mrad. Thus,
the MPE as a function of time is given by
{ 100
MP E(t; λ = 632.8 nm, α > 100 mrad) = 18 C 6 A pupil ×
−0.25 Wm −2 for t > 100 s
t 0.75 Jm −2 for t ≤ 100 s
(B.2)
where the factor C 6 depends only on the angular subtense of the apparent source. In
this case, α > α max , being α max = 100 mrad, therefore making C 6 take its maximum
value, i.e. α max /α min = 67 and the MPE becomes
{ 14.7 mW for t > 100 s
MP E(t; λ = 632.8 nm, α > 100 mrad) =
46 t 0.75 mJ for t ≤ 100 s
(B.3)
The exposure times for the experiments performed varied between 20 and 200 seconds
maximum, thus the most restrictive MPE is
MP E(t > 100 s; λ = 632.8 nm, α > 100 mrad) ≈ 15 mW
(B.4)
where this exposure is valid for up to 8.3 hours.
B.2 MPE for He-Ne laser in full ocular wavefront sensor
When the shear interferometer is modified to be a full ocular wavefront sensor, the
beam illuminating the eye is collimated, making the angular subtense to be used in
the calculations α min = 1.5 mrad. If under these conditions we consider the most
restrictive exposures, i.e. 10s < t < 8.3 hs, the MPE should be calculated as
MP E(t > 10 s; λ = 632.8 nm, α = 1.5 mrad) = A pupil × 10 Wm −2 (B.5)
which is
MP E(t > 10 s; λ = 632.8 nm, α = 1.5 mrad) = 0.4 mW
(B.6)
B.3 MPE for infrared LEDs
The sources used for pupil (iris) illumination were 4 LEDs with a central wavelength
(i.e. emission peak) λ c ≃ 880 nm. The following safety calculations are performed as
124

B. Laser safety calculations
if the LEDs were lasers, as indicated by the standard. The LEDs have a built-in lens
after the diode, making the apparent angular size of each LED the diameter of this
lens, that is 5mm, which then divided by the distance from the eye (50 mm), gives an
apparent subtended angle α ≈ 100 mrad. Then, the MPE for one LED is given by
MP E 1 (t > 100 s; λ = 880 nm, α > 100 mrad) = 18 C 4 C 6 C 7 A pupil × 100 −0.25 Wm −2
(B.7)
where C 4 and C 7 are wavelength dependent factors which for 880nm take the values
2.29 and 1, respectively. Again, we will consider the most restrictive case, i.e. exposures
greater than 100s and a value of 67 for C 6 , which gives the MPE for a single
LED
MP E 1 (t > 100 s; λ = 880 nm, α > 100 mrad) ≈ 33 mW (B.8)
If we have a very restrictive approach, we can assume that the power of the 4 LEDs
is coming from a subtended angle equal to that subtended by only one of them,
simplifying the calculation. Thus, we can safely say that the MPE for the combination
of the 4 LEDs at 50mm from the eye is
MP E 1 (t > 100 s; λ = 880 nm, α > 100 mrad) ≥ 33 mW
(B.9)
B.4 MPE for infrared laser
The source used for the Shack-Hartmann sensor was a laser diode emitting at λ =
780 nm. The apparent angular size of the source to be considered is 1.5 mrad as the
beam entering the eye is collimated, making the viewing distance infinite. For these
conditions and exposures times larger than 10s, the MPE is given by
MP E(t > 10 s; λ = 780 nm, α = 1.5 mrad) = 10 C 4 C 7 A pupil Wm −2
(B.10)
where C 4 and C 7 are wavelength dependent factors which for 780 nm take the values
1.45 and 1, respectively, yielding
MP E(t > 100 s; λ = 880 nm, α > 100 mrad) ≈ 0.6 mW
(B.11)
125

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