5.4 Theoretical signal for piston steps
5.4.2 Piston error without atmospheric disturbances
The results presented in this Section have been published by Surdej et al. [75]. From Eq. (5.4), the intensity on the detector is given by
S(x) = |U2(x)|2
= |U1(x)|2+ 2[1−cos(ψ0)]Uc∗(x)Uc(x)
+2Re{U1∗(x)Uc(x)(exp(iψ0)−1)} (5.26)
where Uc(x) is the convolution of U1(x) and the Fourier transform of the transmission
function T(x) of the phase mask:
Uc(x) =
1
λ
Z
U1(x0)T(x−x0)dx0. (5.27)
A normalization of the intensity in the plane of the entrance pupil, such that |U1(x)|2 = 1,
leads to the following expression for the intensity in the plane of the exit pupil:
S(x) = 1 +F(x) (5.28)
and the signal F(x) is given by
F(x) = 2[1−cos(ψ0)]Uc∗(x)Uc(x) + 2Re{U1∗(x)Uc(x)(exp(iψ0)−1)}. (5.29)
Note that the structure of the signal of the Zernike Phase Contrast Sensor is the same as for the curvature signal, expressed by Eq. (3.4).
The signal is a periodic function of ∆ϕ, and the range of the measurable steps ∆p, expressed in wavefront, is therefore limited to the range [−λ/2, λ/2].
Except for ψ0 = 0, the signal of the Zernike Phase Contrast Sensor is always the sum
of two terms, an anti-symmetric and a symmetric one, shown in Figure 5.4. The anti-symmetric term is given by:
Fas(x) = [1−f(b|x|)][sign(x) sin(ψ0) sin(∆ϕ)], (5.31)
and the symmetric term by:
Fs(x) = −[1−f(b|x|)] {f(b|x|)[1−cos(ψ0)][1−cos(∆ϕ)]}. (5.32)
The lobes, which are present in the signal with the sharp-edge mask, disappear with the apodized one. −0.4 −0.2 0 0.2 0.4 −1 −0.5 0 0.5 1 x [mm] anti−symmetric term symmetric term sharp edge phase mask −0.4 −0.2 0 0.2 0.4 −1 −0.5 0 0.5 1 x [mm] anti−symmetric term symmetric term apodized phase mask
Figure 5.4: Symmetric and anti-symmetric parts of the signal for (left) the sharp edge phase mask and (right) the phase mask with the apodized profile.
The anti-symmetric term of the Zernike Phase Contrast Sensor is the same as the signal obtained with the Mach-Zehnder interferometer for an optical path difference of
δ0 =λ/4 [86].
The weights of both terms depend on the values of ∆ϕ and ψ0. They are plotted in
Figure 5.5. The highest sensitivity to phase steps is, for the symmetric as well as for the anti-symmetric term, obtained if ψ0 =π/2.
5.4 Theoretical signal for piston steps 57
−pi −3pi/4 −pi/2 −pi/4 0 pi/4 pi/2 3pi/4 pi −1 −0.5 0 0.5 1 1.5 2 x sin(x) 1−cos(x)
Figure 5.5: Weights of the symmetric and anti-symmetric term.
When the phase step is in the range ∆ϕ ∈ [0. . . π/2[ (Fig. 5.6), the anti-symmetric part of the signal dominates, whereas in the range ∆ϕ ∈ [π/2. . . π[, the symmetric part dominates. The same applies to the phase of the optical path differenceψ0.
The phase mask used in the experiment is always the one with sharp edges. When the terms ”sharp edge phase mask” or ”gaussian phase mask” are used, they refer to the analytical expressions of functions (3.25) or (3.24) used inside the signal equation (5.30).
−0.4 −0.2 0 0.2 0.4 −1 −0.5 0 0.5 1 x [mm] ∆p=0 ∆p=λ/8 ∆p=λ/4 sharp edge phase mask −0.4 −0.2 0 0.2 0.4 −1 −0.5 0 0.5 1 x [mm] ∆p=3*λ/8 ∆p=λ/2 sharp edge phase mask −0.4 −0.2 0 0.2 0.4 −1 −0.5 0 0.5 1 x [mm] ∆p=0 ∆p=λ/8 ∆p=λ/4 apodized phase mask −0.4 −0.2 0 0.2 0.4 −1 −0.5 0 0.5 1 x [mm] ∆p=3*λ/8 ∆p=λ/2 apodized phase mask
Figure 5.6: Theoretical signal profiles for different piston steps ∆p and phase masks. (top) ∆p = 0, ∆p = λ/8, ∆p = λ/4. (bottom) ∆p = 3λ/8, ∆p = λ/2. Both mask shapes are represented: the side lobes of the signal are a signature of the sharp edge phase mask.
Figure 5.7: Left: Smooth function, its square of the module of its derivative and Laplacian. Right: Step function, [1−f(b|x|)]f(b|x|) and [1−f(b|x|)]sign(x).
A comparison of the exact signal due to a discontinuous step, described by Eq. (5.30), with the signal for a smooth aberration, described by Eq. (5.17), shows
2 αb2|∇ϕ(r)| 2 ∝ [1−f(b|x|)] {f(b|x|)[1−cos(∆ϕ)]}=F1(b|x|), (5.33) 2 αb2∆ϕ(r)∝[1−f(b|x|)] sign(x)[sin(∆ϕ)]}=F2(b|x|). (5.34)
The symmetrical part F1(b|x|) of the signal is proportional to the modulus square of the
gradient of the phase andF2(b|x|), or the anti-symmetrical part of the signal, is proportional
to the Laplacian of the phase function. This comparison is illustrated in Figure 5.7. The plot on the left shows the phase as a step function approximated by a smooth function, the square of the module of its derivative and its Laplacian. The plot on the right shows the discontinuous phase, and the termsF1(b|x|) andF2(b|x|) of the signal, calculated with
Fourier transformations. The anti-symmetrical part has a similar shape in both plots, which means that the approximation of the discontinuous step by a smoothed step is justified. This is, however, not the case for the symmetrical part, where the simple peak in the approximated signal is replaced by two peaks in the exact signal.
It is interesting to note that the signal of the Mach-Zehnder interferometer for the choice δ0 =λ/4 can be retrieved with a proper combination of two images of the Zernike
Phase Contrast Sensor, acquired with different optical path differences:
F(x)|ψ0=π/2−F(x)|ψ0=−π/2 = 2×S(x)|MZ = 2×[1−f(b|x|)][sign(x) sin(∆ϕ)] (5.35) Signal width
The parameter b appearing in Eq. (5.30) is proportional to the pinhole diameter a, and inversely proportional to the wavelength λ:
b= πa
5.4 Theoretical signal for piston steps 59
Figure 5.8: Illustration of the definition of the signal width for the sharp edge phase mask and gaussian profile phase mask.
This parameter determines the width of the signal, ∆x, illustrated in Fig. 5.8. As for the signal obtained with the Mach-Zehnder interferometer, the width ∆x of the signal is defined for the gaussian phase mask as the distance between two points where the intensity of the signal has dropped to 10% of its maximum value and for the sharp edge phase mask as the distance between the two first zeroes of the curve. It can be expressed in mm as [86]:
∆x [mm] = 253.5λ
c a (5.37)
where c=(2√ln 2)−1 for a gaussian type mask and c=1 for a sharp edge mask. The sig-
nal width is proportional to the wavelength and is inversely proportional to the pinhole size. The signal width is an important parameter to optimize because this is where the information on the phase discontinuity is contained (see Section 5.5).