Abstract

An innovative iterative search method called the synthetic phase-shifting (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

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References

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  1. V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE4771, 276–286 (2002).
    [CrossRef]
  2. D. Malacara, Optical Shop Testing (Wiley, 1978).
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    [CrossRef] [PubMed]
  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).
    [CrossRef]
  6. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).
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  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).
    [CrossRef]
  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).
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    [CrossRef] [PubMed]
  16. R. M. Neal and J. C. Wyant, “Polarization phase-shifting point-diffraction interferometer,” Appl. Opt.45(15), 3463–3476 (2006).
    [CrossRef] [PubMed]
  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).
    [CrossRef] [PubMed]
  18. J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004).
    [CrossRef]
  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).

2012

2008

2007

2006

2004

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004).
[CrossRef]

J. E. Greivenkamp and R. O. Gappinger, “Design of a nonnull interferometer for aspheric wave fronts,” Appl. Opt.43(27), 5143–5151 (2004).
[CrossRef] [PubMed]

2002

V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE4771, 276–286 (2002).
[CrossRef]

1996

J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub‐Nyquist interferometry: implementation and measurement capability,” Opt. Eng.35(10), 2962–2969 (1996).
[CrossRef]

C. R. Mercer and K. Creath, “Liquid-crystal point-diffraction interferometer for wave-front measurements,” Appl. Opt.35(10), 1633–1642 (1996).
[CrossRef] [PubMed]

1988

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

1987

1975

R. N. Smartt and W. H. Steel, “Theory and Application of point-diffraction interferometers,” Jpn. J. Appl. Phys.14, 351–356 (1975).

1933

W. P. Linnik, “A simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. URSS5, 208–210 (1933).

Barrett, H. H.

Brock, N. J.

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004).
[CrossRef]

Creath, K.

Dainty, C.

De Nicola, S.

Doyle, K. B.

V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE4771, 276–286 (2002).
[CrossRef]

Ferraro, P.

Gappinger, R. O.

Genberg, V.

V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE4771, 276–286 (2002).
[CrossRef]

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

Goncharov, A. V.

Greivenkamp, J. E.

Grilli, S.

Hayes, J. B.

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004).
[CrossRef]

Lara, D.

Linnik, W. P.

W. P. Linnik, “A simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. URSS5, 208–210 (1933).

Lowman, A. E.

J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub‐Nyquist interferometry: implementation and measurement capability,” Opt. Eng.35(10), 2962–2969 (1996).
[CrossRef]

Mercer, C. R.

Michels, G.

V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE4771, 276–286 (2002).
[CrossRef]

Millerd, J. E.

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004).
[CrossRef]

Neal, R. M.

Palum, R. J.

J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub‐Nyquist interferometry: implementation and measurement capability,” Opt. Eng.35(10), 2962–2969 (1996).
[CrossRef]

Paturzo, M.

Pignatiello, F.

Ruijters, D.

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).

Sakamoto, J. A.

Smartt, R. N.

R. N. Smartt and W. H. Steel, “Theory and Application of point-diffraction interferometers,” Jpn. J. Appl. Phys.14, 351–356 (1975).

Steel, W. H.

R. N. Smartt and W. H. Steel, “Theory and Application of point-diffraction interferometers,” Jpn. J. Appl. Phys.14, 351–356 (1975).

Suetens, P.

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).

ter Harr Romeny, B. M.

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).

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

Wyant, J. C.

R. M. Neal and J. C. Wyant, “Polarization phase-shifting point-diffraction interferometer,” Appl. Opt.45(15), 3463–3476 (2006).
[CrossRef] [PubMed]

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004).
[CrossRef]

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

Appl. Opt.

C. R. Acad. Sci. URSS

W. P. Linnik, “A simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. URSS5, 208–210 (1933).

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

R. N. Smartt and W. H. Steel, “Theory and Application of point-diffraction interferometers,” Jpn. J. Appl. Phys.14, 351–356 (1975).

Opt. Eng.

J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub‐Nyquist interferometry: implementation and measurement capability,” Opt. Eng.35(10), 2962–2969 (1996).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004).
[CrossRef]

V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE4771, 276–286 (2002).
[CrossRef]

Radio Sci.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

Other

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).

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).

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).

D. Malacara, Optical Shop Testing (Wiley, 1978).

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

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Figures (14)

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.

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,π/2 ,π, 3π /2 . (f)-(i) The squared-difference images between (a) and (b)-(e) respectively. The squared-difference images show the slowly varying ϕ diff (r). (Note: some spurious fringes may also be visible due to the aliasing by the monitor display sampling.)

Fig. 3
Fig. 3

The unwrapped-phase results after (a) 1, (b) 3, (c) 5, (d) 7, (e) 16, and (f) 25 iterations ofthe synthetic phase-shifting algorithm. The scale bars are in units of waves.

Fig. 4
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.

Fig. 5
Fig. 5

(a) The first simulated noisy interferogram using the true values in Table 2 and (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. 6

(a) The simulated noisy interferogram was generated using the true values in Table 4 (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. 7
Fig. 7

The absolute differences (in waves) between the true ZF coefficients of the test wavefront used in Subsection 3.2 and their estimates after 50 iterations of the SPS algorithm.

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. 9

The simulated noisy interferograms using the true values in Table 2 with a pixel size of (a) 48μ× 48μm, and (c) 96μ× 96μm. (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.)

Fig. 10
Fig. 10

The effect of different ranges of deviation between the initial values and the true values on the convergence to a threshold value.

Fig. 11
Fig. 11

The experimental setup of the prototype point-diffraction interferometer.

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.)

Fig. 14
Fig. 14

A single line profile, through the x and y axes, comparison between the measured data [(a) and (c)] and the estimated results [(b) and (d)].

Tables (8)

Tables Icon

Table 1 The system construction parameter values for the reference beam at the detector plane used in Section 3.1. The values were chosen such that the departure of the test wavefront from the reference wavefront changes by less than half a wave over two adjacent pixels.

Tables Icon

Table 2 The true ZF coefficients, the initial values and the final estimates of the test wavefront at the detector plane used in Subsection 3.1. The values of terms that are not listed are zero. All ZF coefficient values are in waves.

Tables Icon

Table 3 The system construction parameters for the reference beam used in Subsection 3.2.

Tables Icon

Table 4 The true ZF coefficients and the initial values of the test wavefront used in Subsection 3.2. The estimation results are after 50 iterations of the SPS algorithm. All ZF coefficient values are in waves. The corresponding aberration types for some indices are shown in Table 2.

Tables Icon

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.

Tables Icon

Table 6 ZF coefficients, peak-to-valley and RMS of the exit pupil aberration of the aspheric lens (Edmund 49104) obtained from Zemax. The center of the reference sphere is located at the paraxial image plane. The exit pupil diameter is 20.01 mm. The values of non-symmetric terms are zero.

Tables Icon

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 in Table 2.

Tables Icon

Table 8 All system construction parameters were experimentally measured using a micrometer, except Cref which was calculated by a local Michelson contrast measurement. The measurement error of dPDI was large due to the limited access to the CCD plane.

Equations (27)

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I(x,y)= I 1 + I 2 +2 I 1 I 2 cos[ ϕ test ϕ ref ],
pr(g|θ)= m=1 M 1 2π σ m 2 exp{ [ g m g ¯ m (θ)] 2 2 σ m 2 },
θ ^ ML argmax θ pr(g|θ),
θ ^ ML argmin θ m=1 M [ g m g ¯ m (θ)] 2 ,
Δ g m 2 [ g m g ¯ m (θ)] 2 .
Δ g m 2 cos[ 2π  ϕ diff ( r m ) ],
ϕ diff (r)=[ ϕ test (r) ϕ ref (r) ][ ϕ ^ test (r|θ) ϕ ^ ref (r) ] = ϕ exp (r) ϕ sim (r).
ϕ test ( r m ) ϕ ^ test ( r m |θ)+ ϕ diff ( r m ).
θ ^ j+1   = θ ^ j + θ diff j .
σ RMSD = m=1 M [ ϕ test ( r m ) ϕ ^ test ( r m ) ] 2 M ,
σ RMS = m=1 M [ ϕ diff ( r m ) c diff ] 2 M ,
σ RSS =(n1) ( σ test 2 + σ test 2 ) 1/2 ,
[ p=1 P θ p Z p (r) ϕ ref (r) ]=[ p=1 P θ p Z p (r) ϕ ref (r) ],
E(r|θ)= | U test (r)+ U ref (r)  | 2 ,
U test (r)=α(r)exp[ 2πi ϕ ^ test (r) ],
ϕ ^ test (r)= p=1 P θ p Z p (r) ,
U ref (r)=β(r)exp( ik | r r c | 2 + d PDI 2 )=β(r)exp[ 2πi ϕ ^ ref (r) ],
g ¯ m ( θ )R N ¯ m =Rη T fr A m E ¯ ( r|θ )dr =Rη T fr ( α ( r ) 2 +β ( r ) 2 +2α( r )β( r )cos{ 2π[ ϕ ^ test ( r|θ ) ϕ ^ ref ( r ) ] } )dr,
g ¯ m (θ)= A m { B ^ (r)+ γ ^ (r)cos[ 2π  ϕ sim (r|θ) ] }dr ,
g m = A m { B(r)+γ(r)cos[ 2π  ϕ exp (r) ] }dr + n m ,
Δ g m 2 [ g m g ¯ m (θ)] 2 .
Δ g m 2 [ A m ( γ(r){ cos[ 2π  ϕ exp (r) ]cos[ 2π  ϕ sim (r) ] }+B(r) B ^ (r) )dr + n m ] 2 .
Δ g m 2 ( A m 2γ(r)sin{ π[ ϕ exp (r) ϕ sim (r) ] }sin{ π[ ϕ exp (r)+ ϕ sim (r) ] }dr + A m [ B(r) B ^ (r) ] dr+ n m ) 2 .
Δ g m 2 ( B m + γ m sin[ π  ϕ diff ( r m ) ] ) 2 ,
B m A m { B(r) B ^ (r) }dr + n m .
γ m A m { 2γ(r)sin[ π  ϕ sum (r) ] }dr .
Δ g m 2 B m + γ m cos[ 2π  ϕ diff ( r m ) ],

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