Theoretical analysis for best defocus measurement
plane for robust phase retrieval
Alessandro Polo,* Silvania F. Pereira, and Paul H. Urbach
Optics Research Group, Delft University of Technology, 2628 CJ Delft, The Netherlands *Corresponding author: a.polo@tudelft.nl
Received January 21, 2013; accepted January 26, 2013; posted January 29, 2013 (Doc. ID 183909); published March 5, 2013
We study the phase retrieval (PR) technique using through-focus intensity measurements and explain the depen-dence of PR on the defocus distance. An optimal measurement plane in the out-of-focus region is identified where the intensity distribution on the optical axis drops to the first minimum after focus. Experimental results confirm the theoretical predictions and are in good agreement with an independent phase measurement. © 2013 Optical Society of America
OCIS codes: 220.1080, 120.5050, 100.5070.
Phase retrieval (PR) concerns all the types of nonlinear algorithms for recovering phase information when only the intensity of a complex field is known [1–3]. PR has widely been applied in the last 20 years to determine the phase aberrations in the exit pupil of an optical sys-tem. PR methods that use through-focus (PRTF) intensity images show advantages over standard interferometric techniques [4,5], such as low demand on the temporal coherence requirement and system mechanical stability. Nevertheless, one of the main disadvantages of PRTF is the need for several through-focus measurements, which leads to a heavy computation load. With the recent devel-opments in high-performance computing systems and graphic processing units, PRTF has the potential to be used as a real-time phase measuring technique for application where fast phase measurements are needed, such as adaptive optics systems [6,7] or beam-shaping probing. However, when more than one measurement plane is used to retrieve the phase, uncertainties in the relative distance between two adjacent measurement planes are introduced into the system. In order to minimize the contribution of these uncertainties in the measurement results, it would be attractive to perform the PRTF using only one intensity measurement plane, added to some a priori knowledge of the optical system, such as numerical aperture (NA) and finite support of the pupil [8]. Lee et al. [9] introduced a statistical analysis in order to find the optimal defocus distance based on the Cramer–Rao lower bound. In this Letter, we show that in the limit of small aberrations [10] one can derive an analytical model that identifies the optimum measure-ment plane to achieve accurate PR. Simulations are car-ried out, and experimental results verify the predictions. The PR algorithm, considered in this Letter, is based on a nonlinear optimization with respect to the Zernike coef-ficients [11] that describes the phase-only distribution within the exit pupil of an optical system. A detailed de-scription of the algorithm can be found in [6,12,13]. As an example, we show the performance of this algorithm for four different random phase distributions consisting of 36 Zernike coefficients with the total RMS 0.04λ [Fig.1(a)] in an optical system with NA 0.1. The PRTF is performed using one single intensity measurement plane for different defocus distances u 2πz∕λNA2 [11]. Figure 1(b) shows the result of the performance
characterization in terms of RMS wavefront deviation as a function ofu for the considered cases.
Note that, as already pointed out in [9], the defocus distance plays an important role in the performance of the PRTF. In the following, we derive an analytical explanation of this behavior.
First, we consider the complex exit pupil field distribution [14]:
P⃗ξ; αm
n A⃗ξ expiϕ⃗ξ; αmn: (1)
Here, A⃗ξ is the amplitude distribution (for simplifica-tion we assume constant amplitude) at the exit pupil co-ordinate ⃗ξ, ϕ⃗ξ; αmn is the aberration function (in radians) of the optical path difference, andi is the imaginary unit. The phase distribution is described by a set of coeffi-cients αmn in the Zernike orthonormal base expansion up to the eighth order:
ϕ⃗ξ; αm n X8 n Xn m0 αm nZmn⃗ξ: (2)
The complex fieldUr⊥; f in the perpendicular plane at the geometrical focus of an optical system of focal lengthf can be evaluated by taking the Fourier transform F of the exit pupil field P at the frequency f r⊥∕λf [14]:
Ur⊥; f
expi2fkr⊥
iλf FP⃗ξ; αmn: (3) The through-focus field distribution at a distancez from the focus can be evaluated by computing the free-space
Fig. 1. (Color online) (a) Example of an arbitrary phase dis-tribution, (b) performance of the PR as a function ofu, with the intensity distribution along the axis (red curve) and the recon-structed RMS wavefront deviation (marked curves).
812 OPTICS LETTERS / Vol. 38, No. 6 / March 15, 2013
propagation of the angular spectrum of the quantity de-fined in Eq. (3): Uzr⊥ F−1fFUr⊥; f expikzzg; (4) wherekz ⃗k2− k2 x− k2y q
is thez component of the wave vector ⃗k.
To simplify the notation, we define the operatorGzP that maps the exit pupil field to an out-of-focus distance z as GzP F−1fFexpik∕2f r⊥FPexpikzzg∕iλf .
Hence, we can write Eq. (4) in a more compact way: Uzr⊥ GzP⃗ξ; αmn: (5)
Assuming thatϕ⃗ξ; αmn is small [10], we expand the expo-nential in Eq. (1) in a Taylor series up to the first order and use the linearity property ofF to obtain, from Eq. (5),
Uzr⊥ ≃ GzA⃗ξ iGzA⃗ξϕ⃗ξ; αmn: (6)
Since we are dealing with the measurements of only the intensity of a field, the square of the absolute value of the complex quantity in Eq. (6) should to be taken, obtaining
jUzr⊥j2≃ jGzA⃗ξj2 jGzA⃗ξϕ⃗ξ; αmnj2
2 RefiGzA⃗ξGzA⃗ξϕ⃗ξ; αmng: (7)
In this approximation the through-focus intensity distribu-tion can be expressed by a constant termjGzA⃗ξj2 that uniquely identifies the optical system (i.e., the Airy disk of the aberration-free pupil forz 0), a quadratic-phase-dependent term and a linear term that represents the inter-ference between these fields. Within the small aberration approximation, the quadratic-phase-dependent term can be neglected, so that the interference term is the only one that carries the phase information. The latter can be used to evaluate how a particular aberration function Zm
n⃗ξ contributes to the total intensity distribution
jUzr⊥j2for different through-focus planes.
Consider the special case of geometrical focusz 0. In this particular case, the interference term will be the multiplication of two factors. The first factor,FA⃗ξ, is a real function (i.e., the Bessel function of the first kind). The second factor FA⃗ξϕ⃗ξ; αmn is, according to Eq. (2), a real and even/odd function according to the indexn. It is known that the Fourier transform of a real and even function is again real and even [15]. This implies that aberrations characterized by an even indexn in the Zernike expansion will not contribute to the first-order approximation of the intensity distribution in the geometrical focus, since the interference term will van-ish. Hence, the inverse problem of obtaining the phase coefficientαmn for these values ofn from such a measure-ment plane is strongly ill posed.
Consider now the interference term evaluated at an out-of-focus distancez ≠ 0. In this case, the interference term is always different from zero, but it is interesting to know in which plane the contribution to the intensity distribution is the maximum. To evaluate that, we
determine, for every z, the Zernike coefficients αmn for which the normalized interference term is minimum:
arg min
αm n
R
j2 RefiGzA⃗ξGzA⃗ξϕ⃗ξ; αmngjdr⊥
R jGzA⃗ξjdr⊥ 1 2RjiG zA⃗ξϕ⃗ξ; αmndr⊥ 1 2; (8) and we plot the value of the minimum (i.e.,L) with re-spect tou 2πz∕λNA2. With this normalizationL takes values between 0 and 1.
Figure2shows thatL has its minimum value in the geo-metrical focus of the optical system and it reaches its maximum value at a distance close tou 4π (i.e., where the intensity distribution along the optical axis has its first zero). This result demonstrates analytically that in
Fig. 2. (Color online) Intensity distribution along the optical axis for a system free of aberrations (red dashed curve); inte-gral of the minimum interference term achievable along the optical axis (blue solid curve). The maximum of this term is located in the region close tou 4π.
Fig. 3. (Color online) Through-focus intensity measurements. From top to bottom,u 0 (focal plane), u 2π, u 4.2π, and u 6π. From left to right, astigmatism, coma and second-order astigmatism.
the case of a complete arbitrary PR problem from one single measurement plane, the intensity distribution mea-sured in such a region has the biggest contribution of the exit pupil phase aberrations. Hence, this plane identifies the defocus position where the problem is least ill posed, and therefore the PR algorithm can be performed effi-ciently. We would like to emphasize that such an out-of-focus distance will not change in the presence of small aberrations, since it varies by aboutu 0.2π, giving a certain freedom in performing the measurement.
The theoretical predictions were validated with our experiment, performed using a deformable mirror to in-duce known aberrations in a spherical lens (0.1 NA) [16]. The wavelength of the experiment wasλ 638 nm. The through-focus intensity distributions at four different measurement planes (i.e., u 0, 2π, 4.2π, and 6π) were obtained by collecting the light with an NA 0.4 microscope objective and a CCD camera (Fig. 3). The PR results were compared with an independent phase measurement done by a Shack–Hartmann wavefront sensor (SH-WFS; Thorlabs WFS S300-14AR, λ∕50 accuracy). We generated three different Zernike polyno-mial distributions, namely astigmatism (α−22 0.05λ), coma (α−13 0.05λ), and second-order astigmatism (α−24 0.05λ). We performed the PR by optimizing for the first eight Zernike orders.
The RMS deviations of the retrieved phase distribu-tions with respect to the corresponding one measured by the SH-WFS for the considered out-of-focus measure-ment planes are listed in Table 1.
The retrieved phase aberrations from the intensity dis-tributions atu 4.2π defocus show the best agreement, and they are comparable with the accuracy of the SH-WFS used as a reference measurement. On the other hand, the intensity distributions at the geometrical focus (u 0) and at the intermediate planes u 2π and u 6π give less accurate results in the retrieved phase distri-butions, thus confirming the simulation predictions.
In conclusion, we have demonstrated analytically the identification of an optimal plane of measurement for
the focused field PR technique using one single intensity measurement in the linear approximation of small aberrations. Numerical simulation results were experi-mentally validated using 638 nm coherent visible light and a deformable mirror as a device to introduce known aberrations into the optical system. The PR was performed for one single intensity measurement in the geometrical focus and for three different out-of-focus planes. The results from the predicted optimal plane of measurement were in agreement with an independent measurement done with an SH-WFS. This demonstrates that PR can be performed in a more optimized way once the right plane is chosen. The use of one single intensity plane for PR is very attractive in a fast computation environment for applications like adaptive optics or beam-shaping probing.
This research is supported by the Dutch Ministry of the Economic Affairs and the Provinces of Noord-Brabant and Limburg in the frame of the “Pieken in de Delta” program. The authors are grateful to O. El Gawhary for helpful discussions. R. Horsten and R. Pols are acknowledged for their technical support.
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Table 1. PR RMS Wavefront Deviations for Different Aberrations and Different out-of-Focus Measurement
Planes
u 0 u 2π u 4.2π u 6π
RMSα−22 λ 0.26 0.09 0.01 0.04
RMSα−13 λ 0.14 0.07 0.01 0.04
RMSα−24 λ 0.21 0.08 0.02 0.06
