Alfredo Dubra's PhD thesis - Imperial College London

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
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