Synthetic phase-shifting for optical testing : Point-diffraction interferometry without null optics or phase shifters

An innovative iterative search method called the synthetic phaseshifting (SPS) algorithm is proposed. This search algorithm is used for maximum-likelihood (ML) estimation of a wavefront that is described by a finite set of Zernike Fringe polynomials. In this paper, we estimate the coefficient, or parameter, values of the wavefront using a single interferogram obtained from a point-diffraction interferometer (PDI). In order to find the estimates, we first calculate the squared-difference between the measured and simulated interferograms. Under certain assumptions, this squared-difference image can be treated as an interferogram showing the phase difference between the true wavefront deviation and simulated wavefront deviation. The wavefront deviation is the difference between the reference and the test wavefronts. We calculate the phase difference using a traditional phase-shifting technique without physical phase-shifters. We present a detailed forward model for the PDI interferogram, including the effect of the finite size of a detector pixel. The algorithm was validated with computational studies and its performance and constraints are discussed. A prototype PDI was built and the algorithm was also experimentally validated. A large wavefront deviation was successfully estimated without using null optics or physical phase-shifters. The experimental result shows that the proposed algorithm has great potential to provide an accurate tool for non-null testing. ©2013 Optical Society of America OCIS codes: (100.2650) Fringe analysis; (110.3175) Interferometric imaging; (120.5050) Phase measurement. References and links 1. V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE 4771, 276–286 (2002). 2. D. Malacara, Optical Shop Testing (Wiley, 1978). 3. J. E. Greivenkamp and R. O. Gappinger, “Design of a nonnull interferometer for aspheric wave fronts,” Appl. Opt. 43(27), 5143–5151 (2004). 4. J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26(24), 5245–5258 (1987). 5. J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub‐Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35(10), 2962–2969 (1996). 6. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004). 7. H. H. Barrett, C. Dainty, and D. Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A 24(2), 391–414 (2007). 8. J. A. Sakamoto and H. H. Barrett, “Maximum-likelihood estimation of parameterized wavefronts from multifocal data,” Opt. Express 20(14), 15928–15944 (2012). 9. J. A. Sakamoto, H. H. Barrett, and A. V. Goncharov, “Inverse optical design of the human eye using likelihood methods and wavefront sensing,” Opt. Express 16(1), 304–314 (2008). 10. W. P. Linnik, “A simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. URSS 5, 208– 210 (1933). #193124 $15.00 USD Received 1 Jul 2013; revised 13 Oct 2013; accepted 21 Oct 2013; published 28 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.026398 | OPTICS EXPRESS 26398 11. R. N. Smartt and W. H. Steel, “Theory and Application of Point-Diffraction Interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975). 12. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988). 13. D. Ruijters, B. M. ter Harr Romeny, and P. Suetens, ”Efficient GPU-accelerated elastic image registration,” in Proceedings Sixth IASTED international conference on biomedical engineering (BioMed), pp. 419–424 (2008). 14. G. E. Sommargren, “Phase shifting diffraction interferometry for measuring extreme ultraviolet optics,” No. UCRL-JC–123549, CONF-9604150–1, Lawrence Livermore National Lab., CA (1996). 15. C. R. Mercer and K. Creath, “Liquid-crystal point-diffraction interferometer for wave-front measurements,” Appl. Opt. 35(10), 1633–1642 (1996). 16. R. M. Neal and J. C. Wyant, “Polarization phase-shifting point-diffraction interferometer,” Appl. Opt. 45(15), 3463–3476 (2006). 17. M. Paturzo, F. Pignatiello, S. Grilli, S. De Nicola, and P. Ferraro, “Phase-shifting point-diffraction interferometer developed by using the electro-optic effect in ferroelectric crystals,” Opt. Lett. 31(24), 3597–3599 (2006). 18. J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE 5531, 264–272 (2004). 19. P. Su, J. Burge, R. A. Sprowl, and J. Sasian,”Maximum likelihood estimation as a general method of combining subaperture data for interferometric testing,” Proc. SPIE 6342, 1X–1X-6 (2006).


Introduction
In optical metrology, interferometry is widely used in order to infer physical quantities of interest from an interferogram.Depending on the task, the physical quantities can be aberration coefficients, refractive-index gradient distribution, surface deformation, or alignment errors.The task of our work is to estimate the coefficient values of a test wavefront, expanded in a finite set of Zernike Fringe [ZF] polynomials [1], from a single interferogram.
The continuous irradiance pattern in an interferogram is given by the well-known expression, where 1 I and 2 I are the irradiance of the test and reference beam, and test ref φ φ − is the phase difference between the test and reference beams.The irradiance pattern is often detected by a digital detector which integrates the irradiance over each pixel in an array.It is usually assumed that the detector simply samples the interferogram at a discrete set of points.However, when measuring a strong asphericity or a large aberration of a test optic, the spatial fringe period is often significantly smaller than the area of one pixel.The integration over a pixel then blurs out the fine structure in the interferogram instead of merely sampling it.Both the blurring by the pixel and the sampling reduce the ability to perform the task for which the interferogram was intended.To avoid this issue, most interferometry is used in a null configuration [2].Null optics can compensate for a large aberration of a test optic such that the absolute value of the wavefront difference between any two adjacent pixels is less than half a wave, which is the well-known Nyquist condition [3].The main disadvantage of using null optics is that a particular null optic has to be designed for the specific test optic, which increases the cost of the system and the production time.In order to overcome these limitations, Sub-Nyquist interferometry (SNI) was developed [3][4][5].As a phase-shifting approach, SNI achieves a high dynamic range by recovering the phase information from the subsampled interferogram (by a sparse sensor) assuming the wavefront or surface slope is continuous.
Unlike SNI, we extract the phase information from a single interferogram obtained from a contiguous detector with a finite pixel size.For this approach, we estimate the ZF coefficients of the test wavefront, described by a finite set of ZF polynomials, via maximum-likelihood (ML) estimation.The advantages of the statistical approach in imaging are well understood [6].ML estimation in wavefront sensing has been rigorously discussed by Barrett et al. [7] and ML estimation of parameterized wavefronts from multifocal data has been developed using simulated annealing, a global optimization algorithm [8].Despite its advantages, the limitation of search algorithms for ML estimation in a high-dimensional parameter space, such as ZF coefficient space, is the computational time.A fast conjugate-gradient method can be applied if the likelihood is monotonic even in high-dimensional parameter space.However, if the likelihood surface has many local minima and strong coupling between parameters, time-consuming global optimization is often the only choice [8,9].To overcome this challenge, the innovative iterative search algorithm, called the synthetic phase-shifting (SPS) algorithm is developed.The SPS algorithm is far less computationally expensive than global search algorithms such as simulated annealing.
The SPS algorithm iteratively finds the coefficient values that minimize the squared difference between a single measured interferogram and an accurate computer-simulated interferogram.This approach becomes ML estimation if the random errors between the measured and simulated interferograms follow an independent and identically distributed normal distribution [6,7].Based on an accurate forward model, a simulated interferogram is generated with a test wavefront, expanded by the nominal coefficient values, and a known reference wavefront.We then calculate the squared-difference image between the measured and simulated interferograms.We show that, under certain constraints, this squareddifference image can be recognized as an interferogram which shows the phase difference between the true wavefront deviation and simulated wavefront deviation.The wavefront deviation is the difference between the reference and the test wavefronts.We apply a traditional phase-shifting method to extract the phase difference.By adding the result of the unwrapped phase, the coefficient values of the test wavefront are updated.These updated coefficient values are used as the nominal coefficient values for the next iteration.Note that the algorithm utilizes traditional phase-shifting techniques, but it does not require any physical phase-shifters.To our knowledge, ML estimation of parameterized wavefront using the SPS algorithm is the first static interferometric technique applied to data which do not satisfy the Nyquist condition.
In this paper, we established the theory of the SPS algorithm, validated it with computer simulations, and discussed its advantages and limitations.We have built a proof-of-concept prototype point-diffraction interferometer (PDI).An experimentally measured interferogram from the PDI was used to verify the practical applicability of the algorithm.The results show that the proposed algorithm has great potential for non-null testing and can achieve high measurement accuracy without using physical phase-shifters.

Maximum-likelihood estimation
In this study, a single interferogram is the raw CCD output from the PDI.The principle of the PDI is simple [10,11].Illumination light passes through a test lens and focuses onto a PDI plate, a neutral density filter with a clear pinhole in the center.When the focused beam transmits through the PDI plate, a reference beam is generated by the light diffraction from the pinhole.Simultaneously, the test beam is attenuated by the PDI plate and interferes with the reference beam (Fig. 1).This interference pattern forms an interferogram, g, in which the phase information of the test lens at the detector plane is encoded.The vector, g, is the 1 M × vector of random data from the CCD, and the total number of CCD pixels in the circular aperture is M. Our goal is to extract the test wavefront information from g using ML estimation of the test wavefront, expanded in a finite set of ZF polynomials.The CCD is assumed to be a discrete array of individual detector elements with no electronic coupling from one element to another, and each element is identical.We assume that the PDI noise follows Gaussian statistics, so the data vector, , g follows a probability density function (PDF) denoted by where arg min operator returns the parameter value at which the function is minimized.Therefore, analysis of an interferogram reduces to nonlinear least-squares regression between the interferogram and simulated interferogram.

The synthetic phase-shifting algorithm
Conventional search algorithms for ˆML θ are computationally expensive, not only because of the high-dimensional parameter space, but also because of the complicated likelihood function.To overcome this challenge, we propose an innovative iterative search algorithm called the SPS algorithm.For this algorithm, we focus on the squared-difference image between a single measured interferogram and an accurate computer simulated interferogram, which is an where the difference of the wavefront deviations is denoted by The true wavefront deviation is ( ) ( ) ( ),  The first assumption is that a CCD pixel is small compared to the spatial fringe period of ( ), diff φ r so the irradiance fluctuation due to ( ) diff φ r is constant over a pixel.This is a much less stringent assumption than requiring that the actual interferogram irradiance be constant over a pixel, which is the usual implicit assumption in interferometry.The squared-difference images in Figs.π From a single measured interferogram, we subtract each phase-shifted simulation result to get the four phase-shifted squared-difference images.After obtaining the wrapped phase map of ( ), diff m φ r Goldstein's branch-cut phase-unwrapping algorithm [12] is applied.In this study, the residual modeling error for the reference wavefront, is integrated into the simulated test wavefront.Therefore, the simulated test wavefront is rewritten as For an ideal case, we know the reference wavefront exactly, so ˆ( | ) 0.
The true test wavefront at the detector is Note that the algorithm utilizes the traditional phase-shifting technique, but entirely in the computer; it does not require any physical phase-shifters.

The iterative process of the synthetic phase-shifting algorithm
Since the SPS algorithm is a search method for the nonlinear least squares regression, we propose an iterative process in order to increase the accuracy of the estimation.Because of the first assumption in Subsection 2.2, we assume that the resulting sampled unwrapped-phase map is continuous and is well approximated by 37 ZF polynomials.At each iteration, j, the ZF coefficients of the unwrapped-phase, , j diff θ are obtained by their orthonormality.At the th j iteration of the SPS algorithm, the nominal ZF coefficient vector of the test wavefront is ˆ.j θ This coefficient vector is updated by adding :  process is optional for noise suppression, but it may improve the convergence rate to the true values.Future work is required to optimize the post-filtering algorithm.
Fig. 4. The flow chart of the iterative synthetic phase-shifting algorithm.The box-averaging filtering process is optional for noise suppression, but it may improve the convergence rate.

Numerical studies
The SPS algorithm was verified with a series of computer simulation studies, and its application and performance was demonstrated.Since a monochromatic full-frame CCD (Kodak KAF-09000) was used for the prototype PDI, the fill factor of the CCD in the forward model was assumed to be 100% with a pixel size of 12 m 12 m.
μ μ × In order to represent the contiguous detector adequately, the simulation sub-pixelized the detector grid so that the resulting simulated interferogram did not suffer from aliasing.This super-sampled simulation was then down-sampled by averaging the sub-pixel values of each sub-pixelized block.The final sampling size of the simulated interferogram was equal to the detector size, 1024 1024 × pixels.As shown in Appendix 7.1, the forward model is a mixture of the measured test beam amplitude and an analytical function with 37 ZF coefficients.Due to the super-sampling requirement for the simulation, the test beam amplitude data were interpolated and a large matrix had to be calculated at each iteration.Therefore, the simulation was implemented using a parallel algorithm in the graphics processing unit (GPU) programming language, CUDA (Computer Unified Device Architecture).Particularly, we utilized the texture memory of the GPU to minimize the execution time for the test beam interpolation.A CUDA Cubic B-Spline Interpolation library implemented cubic interpolation using the texture memory [13].The calculation time of the simulation (32768 × 32768 pixels) in double precision took 3.5 seconds using an NVIDIA Tesla C2075 card with 6 GB of on-board memory.The SPS algorithm was written in a combination of MATLAB ® , C and CUDA, and a quad-core Intel Xeon 2.13 GHz processor was used.The model did not account for the effect of the micro lenses of the CCD.
To emulate the real experimental data, we first generated the simulated interferogram using the forward model and added zero-mean Gaussian noise with the measured noise variance of the CCD.After generating the data, we disregarded our knowledge of the true ZF coefficients, and estimated them by ML estimation using the SPS algorithm.The system construction parameters for the reference beam are , , is the center position of the reference beam (Fig. 1 and Appendix 7.1).They were assumed to be known exactly in the numerical studies and the same experimentally measured test-beam amplitude was used in each of the simulations.The piston-corrected root-mean-square deviation (RMSD) between the measured and the simulated test wavefronts was chosen as a figure of merit for the numerical studies.It is denoted as where m is the index of the CCD pixel and M is the total number of pixels within a circular aperture.In this study, the SPS algorithm was applied to interferograms whose spatial frequencies were lower and higher than the Nyquist condition.

An interferogram with a spatial frequency lower than the Nyquist condition
The first simulated noisy interferogram [Fig.5(a)] was generated using the system construction parameter values (Table 1) and the true ZF coefficient values of the test wavefront (Table 2).The distance from the PDI plate to the CCD, , PDI d was chosen such that the interferogram would satisfy the Nyquist condition.We then disregarded our knowledge of the true coefficient values, and estimated them by ML estimation using the iterative SPS algorithm.The initial values, 1  ˆ, θ and the final estimated results are shown in Table 2.The piston-corrected RMSD converged to ~0.008 waves after 10 iterations of the SPS algorithm.The final estimated result is shown in Fig. 5(b).

An interferogram with a spatial frequency higher than the Nyquist condition
For the second numerical study, the noisy data were simulated using the system construction values in Table 3 and true parameter values in Table 4.All 37 true ZF coefficient values were chosen randomly from the initial values within a 10 waves ± range.The local spatial frequency of the interferogram was higher than the Nyquist condition.After generating the noisy interferogram [Fig.6(a)], we disregarded our knowledge of true ZF coefficient values and estimated them with the SPS algorithm.After 50 iterations, the estimated result [Fig.6(b)] is compared to the noisy interferogram.A quantitative comparison between the true parameter values and their estimates is shown in Table 4 and the absolute differences between two are shown in Fig. 7. Figure 8 shows a single profile of the true test wavefront and its estimated wavefront.The piston-corrected RMSD of the second simulated experimental data converged to ~0.009 waves after 50 iterations.

The convergence study
We performed two convergence studies in order to assess the robustness of the proposed algorithm.First, we simulated sensors with larger pixels to properly test the robustness of the SPS algorithm against a large wavefront deviation between two neighboring pixels.In other words, a different degree of pre-sampling low-pass filtering was introduced by increasing the size of a pixel by an integer multiple n.The output of a larger pixel was the average value of the corresponding n n × pixels of the finely-sampled detector ( 12 m 12 m μ μ × pixels).We used the noisy interferogram which satisfies the Nyquist condition (Subsection 3.1) as the outputs from the detector with the smallest pixel dimensions.Using the same true ZF coefficients in Table 2, we generated four different data sets depending on the different pixel sizes (Table 5 and Fig. 9).The corresponding piston-corrected RMSD of the test wavefront with different pixel sizes are shown in Table 5.It is important to note that the digitization error of a circular aperture contributes to the main piston-corrected RMSD errors for the larger pixel sizes.When the initial values were closer to the true values, the estimates converged to a threshold value with fewer iterations (Fig. 10).The threshold value (the piston-corrected RMSD) to terminate the iteration process for this study was 0.005 waves.Further optimization, such as post-filtering process, could improve the performance.

General design and layout
A proof-of-concept prototype PDI has been designed and built.The main configuration of our PDI was similar to a conventional PDI in a non-null configuration except that physical phaseshifters were not included (Fig. 11).An aspheric lens (Edmund optics 49104) with an entrance pupil diameter of 18 mm was tested for this study.According to Zemax, the exit pupil diameter and working f-number at the paraxial image location were xp D = 20.01 mm and f/# W = 2.537, respectively.According to the manufacturer, the aspheric lens has ~1.25 waves RMS surface accuracy.The lens is designed to achieve a minimum aberration when the flat side of lens is facing the detector plane.However, in order to introduce a large aberration, the lens was flipped such that the flat side was facing the light source (Fig. 1).The exit pupil aberration of the lens in the flipped configuration was obtained using Zemax (Table 6).The noisy data, g, consisted of the raw detector outputs from a CCD (Apogee Alta U9000) that measured the interference between the reference beam and the attenuated test beam at the detector plane.By translating the PDI plate in x, y and z directions, decenter and defocus were introduced in order to increase the fringe contrast.For the forward model, the amplitude function of the test beam was obtained experimentally via translation of the PDI plate such that the test beam does not pass through the pinhole but is attenuated by the PDI plate (Appendix 7.1).The scalar amplitude of the reference beam, ,

Application of the synthetic phase-shifting algorithm
We applied the SPS algorithm to a measured interferogram acquired using the prototype PDI.The design values at the detector plane were obtained from the Zemax model of the PDI in the on-axis configuration (Table 7).Without knowledge of the true values, the design values offered the best possible knowledge of the true values of the test lens.The measured system construction parameters (Table 8) were used for the reference beam.Unlike the numerical study, the true values were unknown for the physical data.Therefore, we set the termination criteria as the piston-corrected root-mean-square (RMS) of the ( ), where m is the index of the CCD pixel, M is the total number of pixels within a circular aperture and diff c is the piston term of ( ).
diff m φ r After 13 iterations, the piston-corrected RMS was ~0.006 waves, and the initial estimates of the test wavefront were obtained, including the alignment errors of the PDI system.Figure 13 shows the comparison between the measured and the estimated interferograms.The single line profiles through the measured and the estimated data in the x and y directions are compared in Fig. 14.Pearson's correlation coefficient between the measured data and the estimated data was ~0.95, indicating a high degree of correlation.In order to obtain the final estimates of , θ we estimated alignment errors and subtracted them from the resulting ML estimates of the SPS algorithm.Using Zemax, alignment errors were obtained from a sensitivity analysis of the tilt angle of the optical elements and the spacing between the elements.The range of the tilt angle was ± 0.5 ° and the range of the spacing between the elements was ± 0.5 mm.The ranges were within the alignment uncertainties of the prototype PDI.The final estimates, after subtracting the alignment errors, are compared with the design values in Table 7.The RMS difference error between the final estimated wavefront and the design wavefront was ~0.9 waves.
To compare this RMS difference error with the lens specification, we estimated root-sumsquare (RSS) wavefront error ( RSS σ ) of the test lens.The surface accuracy of the test lens ( test σ ) was given as ~1.25 waves RMS from the lens specification (Subsection 4.1).From this RMS surface accuracy, the RSS value from the two uncorrelated surface errors was calculated by where n = 1.58 was the index of refraction of the test lens.We obtained a RSS wavefront aberration of ~1.0 waves, which was comparable with the RMS difference error.An independent measurement of the test lens, using a null-testing setup, can be performed in future to validate our results.

Discussion
A question always arises about the sign ambiguity of Eq. ( 1) when a single-frame interferogram analysis method is proposed.For ML estimation of parameterized wavefronts, the sign ambiguity problem can be addressed in the following way: a ZF coefficient vector, , ′ θ always exists such that Two limitations need to be addressed for this approach.First, we introduce a truncation error by truncating an infinite set of ZF polynomials to a finite set [6].Therefore, it is impossible to fully determine a test wavefront even in the absence of noise.However, the number of parameters can be increased at a cost of increased computational time.The other error source is the accuracy of prior knowledge of the reference wavefront.We can reduce the error if the PDI system is carefully calibrated.For the calibration, another ML estimation can be also carried out for estimation of the system construction parameters [19].If the exit pupil aberration of a test optic is the desired object wavefront, a reverse-ray tracing step can be added after estimating the wavefront at the detector plane.

Conclusion
In summary, using ML estimation with the novel SPS algorithm, a test wavefront, described by a finite set of ZF polynomials, was successfully estimated from a single interferogram from the PDI.The theory of the SPS algorithm and the forward model for the PDI have been established.The algorithm was verified with numerical and experimental studies using the PDI.The studies showed that the algorithm can recover the phase with a higher spatial frequency than the Nyquist frequency.The SPS algorithm can be used in a non-null configuration and does not require any physical phase shifters.
To conclude, the approach developed in this paper can provide an accurate tool for nonnull testing with a high dynamic range in terms of measurable (i.e.estimable) test wavefront deviation from the reference wavefront.This high dynamic range enables its non-null testing capability for various optics including highly aspheric lenses and free-form optics and eliminates the complexity to build customized null configurations such as null-lens or computer generated holograms.

Fig. 1 .
Fig. 1.The schematic of the PDI for non-null testing.Unlike a conventional PDI, physical phase-shifters are not required.The dashed line represents the test beam and its wavefront, and the solid line represents the reference wavefront generated by pinhole diffraction at the PDI plate.
2(f)-2(i) show the slowly varying ( ) diff φ r over a pixel.With our assumption, therefore, we can simply sample the irradiance fluctuation due to ( ) diff φ r at the center position of a CCD pixel instead of integrating over the pixel.The spatial frequency of ( ) diff φ r should at least satisfy the Nyquist condition to recover the phase information of ( ). diff φ r On the other hand, the second assumption is that the spatial frequency of ( well beyond the Nyquist frequency, so the fine structure of the irradiance fluctuation due to ( ) sum φ r is blurred out by the integration over the pixel (Appendix 7.2).The third assumption is that measured and the simulated mean irradiances are approximately equal.

Fig. 2 .
Fig. 2. Graphical demonstration of the synthetic four-step phase-shifting algorithm.(a) The experimental result was generated with the true parameter values based on the forward model, including noise.(b)-(e) The simulated interferograms, without noise, with nominal parameter values at phase-steps 0, 2 , , 3 2.π π π

Figure 3
Figure 3 shows the unwrapping results at different iterations.As iterations increase, the unwrapping map becomes null (i.e.showing only a constant piston-term difference), indicating the estimated test wavefront converges to the true test wavefront.The iterative process is summarized in the flow chart shown in Fig. 4. The box-averaging filtering (post-

Fig. 5 .
Fig. 5. (a) The first simulated noisy interferogram using the true values in Table2and (b) is the estimation result after 10 iterations of the SPS algorithm.Even though the spatial frequencies of the interferograms satisfy the Nyquist condition, some spurious fringes appear due to the aliasing by the monitor display sampling.

Fig. 6 .Fig. 7 .
Fig. 6.(a) The simulated noisy interferogram was generated using the true values in Table4(Subsection 3.2) and (b) the estimated result after 50 iterations of the SPS algorithm.(Note: some spurious fringes may also be visible due to the aliasing by the monitor display sampling.)

Fig. 8 .
Fig. 8. (a) A single line profile of the true test wavefront used in Subsection 3.2 and its estimate after 50 iterations of the SPS algorithm.The true values and the estimates are almost overlapping each other.(b) The zoomed image of (a) at pixel 1022.

Fig. 9 .Fig. 10 .
Fig. 9.The simulated noisy interferograms using the true values in Table 2 with a pixel size of (a) 48 m 48 m, μ μ × and (c) 96 m 96 m. μ μ × (b) and (d) are the estimation results after the SPS algorithm was applied.(Note: some spurious fringes may also be visible due to the aliasing by the monitor display sampling).)

×
All 37 initial coefficient values of the test wavefront were randomly generated from the true values within1,

4. 2
Point-diffraction interferometer plateDifferent PDI schemes and PDI plate designs have previously been proposed, and much effort has been made to include phase-shifting techniques into PDI systems[14][15][16][17][18].Instead of undertaking complicated fabrication processes, we used a simple PDI plate.It consists of a chromium layer deposited on a fused silica plate with a 0through the chromium layer by a focused ion-beam (FIB).The irradiance transmittance of the PDI plate was measured as ~0.0001%.Images of the pinhole, taken with a scanning electron microscope, are shown in Fig.12.

Fig. 12 .
Fig. 12.The focused ion-beam etched pinhole on a chromium layer deposited on a fused silica plate.Images were taken with a scanning electron microscope.

Fig. 13 .
Fig. 13.(a) The measured interferogram obtained by using a f/# W = 2.537 aspheric lens.(b)The estimated interferogram after 13 iterations of the SPS algorithm.Pearson's correlation coefficient between the two images is ~0.95.(Note: some spurious fringes may also be visible due to the aliasing by the monitor display sampling.) the th p order ZF polynomial.Our approach can avoid this ambiguity if the distance between two parameter vectors, , prior knowledge of the search range of .θ In interferometry, we often have prior knowledge of the range of θ within a few tens of waves.For example, the distance between the two parameter vectors, , ′ − θ θ in Subsection 3.2 is ~135 waves.Since the initial guess values for the SPS algorithm is much closer to the true ZF coefficient, , θ than , ′ − θ θ the resulting estimates can avoid the sign ambiguity.

Table 5 . The piston-corrected RMSDs of the test wavefront at the detector plane with different pixel sizes. The true ZF coefficients of the test wavefront are shown in Table 2. Pixel size [µm × µm ] 12 × 12 a 24 × 24 48 × 48 96 × 96 Piston-corrected RMSD [waves]
aThe spatial frequency of the interferogram satisfies the Nyquist condition.

Table 7 . The design ZF coefficient values at the detector plane obtained from Zemax. The final estimates were obtained after subtracting the alignment errors (tip/tilt and distances between the elements). The corresponding aberration types for some indices are shown inTable 2 .
The design values offered the best possible knowledge of the true values of the test lens.

Table 8 . All system construction parameters were experimentally measured using a micrometer, except C ref which was calculated by a local Michelson contrast measurement. The measurement error of d PDI was large due to the limited access to the CCD plane. System Construction Parameter d
PDI [mm] x c [mm] y c [mm] C ref [ADU 1/2 ]