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Instruments Insight Trusit Dave PhD, BSc, MCOptom, FAAO Figure 7 The wavefront error map shows the difference between the aberrated and reference (plane) wavefront. Once this information is processed for all points over the pupil, a 2D wavefront error map can be displayed (right) Figure 8 Society of America 4 . This nomenclature groups each term according to the radial order (n) and angular frequency (m), thus each term is written in the form Z m n. The radial order (n) groups Zernike modes in terms ρ (rho), whereas the angular frequency (m) groups the modes in terms of θ (theta). The coefficient, C, in each term varies according to the number of modes in the Zernike polynomial. C defines the level of a particular mode of aberration in microns and can have a positive or negative value. A list of the Zernike terms and their optical equivalent up to the fourth order is shown in Table 1. There are, in fact, an infinite number of Zernike terms which can be used to fit an individual wavefront. In practice, however, Zernike terms up to the 4th radial order are measured. It is worth understanding the actual terms in the equation. Let us take the example of secondary astigmatism or Z 2 4. −1 Z1 = 4ρ sin( θ ) 1 Z1 = 4ρ cos( θ ) − Z 2 6 2 2 = ρ sin(2θ ) 0 2 Z2 = 3(2ρ −1) Z 2 = 6ρ 2 cos(2 θ ) 2 Individual modes of Zernike polynomials up to fourth order. ‘n’ represents the radial order and ‘m’ the angular frequency Table 1 The double index notation of Zernike term, the equation of each mode in the Zernike polynomial and the appearance of each mode in a colour-coded wavefront error map Zernike term WFE Map Optical equivalent Vertical prism Horizontal prism Astigmatism Defocus Astigmatism The Zernike equation is Z 2 4 = √10(4ρ 4 −3ρ 2 )cos(2θ) What does this mean? As you can see, there are only two variables in all the modes in Table 1. Broadly speaking, you can categorise each equation into three areas. Figure 9 Wavefront error map defining ρ and θ − Z 3 8 3 3 = ρ sin(3θ ) −1 3 Z3 = 8(3ρ − 2ρ)sin( θ ) 1 3 Z3 = 8(3ρ − 2ρ) cos( θ ) Z 3 = 8ρ 3 cos(3 θ ) 3 − Z 4 10 4 4 = ρ sin(4θ ) −2 4 2 Z4 = 10(4ρ −3ρ ) sin(2θ ) 0 4 2 Z4 = 5(6ρ − 6ρ + 1) 2 4 2 Z4 = 10(4ρ − 3ρ )cos(2θ ) Z 4 = 10ρ 4 cos(4 θ ) 4 Trefoil Coma Coma Trefoil Tetrafoil Secondary astigmatism Spherical aberration Secondary astigmatism Tetrafoil 44 | November 19 | 2004 OT
Instruments insight Z 2 4 = √10(4ρ 4 −3ρ 2) cos(2θ) Normalisation term Radial term Angular term Normalisation of each mode means that observation of the coefficients immediately gives an indication of the level of influence each type of aberration has on the total aberration 5 . The radial term, in this case of the fourth order, describes the variation of the wavefront error with distance from pupil centre. The largest power ρ is raised to is the radial order, i.e. to the power 4. The angular frequency describes the number of repeat cycles which are made over 360 degrees. For example, in secondary astigmatism, there are two repeat cycles over the circular pupil, thus the radial order is 2. Also, notice that where the radial order is assigned a negative value, the equation uses the sine of θ to describe the angular variation. The first order term, prism, is not relevant to the wavefront as they represent tilt and are corrected using a prism. The second order terms represent low order aberrations, namely defocus and astigmatism. Defocus represents the spherical component of the wavefront. The astigmatic terms conversely describe the cylinder. Using these three terms, any sphero-cylindrical lens can be described. Wavefront maps with defocus and astigmatism only can be perfectly corrected using spectacles and contact lenses. Caution, however, needs to be exercised when interpreting these terms as in practice, patient wavefront error maps have high order terms. As a result, defocus and astigmatism terms do not represent the sphere and cylinder terms found in subjective refraction. An estimate of subjective refraction can be made if only these three terms are fit to the wavefront data. Second order terms are referred to as low order aberrations. Every mode after second order is a high order aberration. These terms cannot be measured in conventional refraction or autorefraction. The only method of quantifying these terms is by using an aberrometer. The third order terms describe coma and trefoil astigmatism. These terms cannot be corrected using spectacles, soft or RGP contact lenses. Attempts have been made to correct spherical aberration in contact lenses using aspheric designs. However, this correction only corrects for lens specific spherical aberration and not the combined effect of eye and lens. Therefore, the optical benefits of such a design may improve image quality for some patients but equally there may be some deterioration in image quality in other patients 6 . Root mean square error In order to summarise the wavefront error, scientists have tried to develop metrics to try and best describe the wavefront error as a numeric index. At the present time, the most widely used metric is the root mean square (RMS) error. All aberrometers display a RMS error of high and low order aberrations. This term describes the weighted mean of the individual Zernike modes. In other words, the RMS value describes the overall aberration and gives a ‘feel’ of how bad an individual’s aberrations are. The RMS error is simply: RMS=√(C2 -2 ) 2 +(C 0 2) 2 +(C 2 2) 2 +(C -3 3) 2 +(C -1 3) 2 ...etc There are number of disadvantages in using the RMS value. The most significant is that increases in RMS value do not necessarily result in decreases in image quality 5 . For example, if a subject has some defocus and spherical aberration, then it is possible that the spherical aberration will help reduce some of the image blur induced on the retina. However, because of the way RMS is calculated, the cancelling effects of one aberration over another are not considered 7 . As research continues, better predictors of visual function will develop 5,8 . Studies have evaluated normative values of ocular aberration. Porter et al 9 found a random variation in the wavefront aberration of 109 normal subjects with an artificial pupil diameter of 5.7mm. Chalita et al 10 performed aberrometry in 60 eyes and found mean values of 0.35±0.29 microns for coma; 0.36±0.31 for spherical aberration; and 0.31±0.14 microns for other high order terms for a pupil diameter of 7mm. It has also been documented that 91% of the RMS wavefront error can be accounted for by the second order and 99% if Zernike terms from the first four orders are used to fit the wavefront for a pupil diameter of 5mm 11 . Wang and Koch 12 measured ocular aberrations in subjects screened for refractive surgery. High order aberrations were found to have a mean RMS of 0.305±0.095 in 532 eyes across a 6mm pupil. Spherical aberration was found to be 0.128±0.074 and coma 0.170±0.089 microns. Any instrument involved in clinical measurement must be both accurate and repeatable. Cheng et al 13 measured the accuracy and repeatability of the COAS (Wavefront Sciences) Shack-Hartmann aberrometer on model eyes. They found the instrument to be both accurate and repeatable on model eyes. In another study, Cheng et al 14 evaluated the variation of monochromatic aberrometry measurements over a range of timescales (five measures in one second without realignment, five measures in one hour with realignment at the same time over five days and on five days at monthly intervals). The authors concluded that variations in aberrometry maps between days and months was due to biological changes to the eye, and that this fluctuation would prevent practitioners from achieving perfect optical correction, i.e. an aberration-free eye. Wavefront maps are also known to vary with accommodation 15 , tear film quality 16 and media opacities 17 . Part 2 of this article, to be published on December 3, will look at applications of wavefront aberrometry in clinical practice. About the author Dr Trusit Dave is Director of Optimed (Ophthalmic Medical Instruments), Clinical Consultant for Topcon GB and a partner in private practice in Coventry. References For a full set of references, email nicky@optometry.co.uk. Continuing Education and Training Systemic Pathology Orderline: 01252-810939 Fax: 01252-816176 NEW This book on ‘Systemic Pathology’ is designed to give practitioners and optometric students a better understanding of the principles and practice of clinical medicine and surgery, and in particular, of the further management of patients referred by optometrists. £39.50 45 | November 19 | 2004 OT
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