Abstract

A method for determining the pupil phase distribution of an optical system is demonstrated. Coefficients in a wavefront expansion were estimated using likelihood methods, where the data consisted of multiple irradiance patterns near focus. Proof-of-principle results were obtained in both simulation and experiment. Large-aberration wavefronts were handled in the numerical study. Experimentally, we discuss the handling of nuisance parameters. Fisher information matrices, Cramér-Rao bounds, and likelihood surfaces are examined. ML estimates were obtained by simulated annealing to deal with numerous local extrema in the likelihood function. Rapid processing techniques were employed to reduce the computational time.

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. I. S. Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26(9), 2141–2160 (1985).
    [CrossRef]
  2. J. H. Seldin and J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7(3), 412–427 (1990).
    [CrossRef]
  3. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983).
    [CrossRef]
  4. G. R. Brady and J. R. Fienup, “Improved optical metrology using phase retrieval,” 2004 Optical Fabrication & Testing Topical Meeting, OSA, Rochester, NY, paper OTuB3 (2004).
  5. G. R. Brady and J. Fienup, “Phase retrieval as an optical metrology tool,” Optifab: Technical Digest, SPIE Technical Digest TD03, 139–141 (2005).
  6. G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express 17(2), 624–639 (2009).
    [CrossRef] [PubMed]
  7. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).
  8. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2005).
  9. A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variable with the ‘simulated annealing’ algorithm,” ACM Trans. Math. Softw. 13(3), 262–280 (1987).
    [CrossRef]
  10. 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).
    [CrossRef] [PubMed]
  11. C. R. Rao, “Information and accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).
  12. H. Cramér, Mathematical Methods of Statistics (Princeton University Press, 1946).

2009 (1)

2007 (1)

2005 (1)

G. R. Brady and J. Fienup, “Phase retrieval as an optical metrology tool,” Optifab: Technical Digest, SPIE Technical Digest TD03, 139–141 (2005).

1990 (1)

1987 (1)

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variable with the ‘simulated annealing’ algorithm,” ACM Trans. Math. Softw. 13(3), 262–280 (1987).
[CrossRef]

1985 (1)

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26(9), 2141–2160 (1985).
[CrossRef]

1983 (1)

1945 (1)

C. R. Rao, “Information and accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).

Barrett, H. H.

Brady, G. R.

G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express 17(2), 624–639 (2009).
[CrossRef] [PubMed]

G. R. Brady and J. Fienup, “Phase retrieval as an optical metrology tool,” Optifab: Technical Digest, SPIE Technical Digest TD03, 139–141 (2005).

Corana, A.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variable with the ‘simulated annealing’ algorithm,” ACM Trans. Math. Softw. 13(3), 262–280 (1987).
[CrossRef]

Dainty, C.

Fienup, J.

G. R. Brady and J. Fienup, “Phase retrieval as an optical metrology tool,” Optifab: Technical Digest, SPIE Technical Digest TD03, 139–141 (2005).

Fienup, J. R.

Guizar-Sicairos, M.

Lara, D.

Marchesi, M.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variable with the ‘simulated annealing’ algorithm,” ACM Trans. Math. Softw. 13(3), 262–280 (1987).
[CrossRef]

Martini, C.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variable with the ‘simulated annealing’ algorithm,” ACM Trans. Math. Softw. 13(3), 262–280 (1987).
[CrossRef]

Rao, C. R.

C. R. Rao, “Information and accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).

Ridella, S.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variable with the ‘simulated annealing’ algorithm,” ACM Trans. Math. Softw. 13(3), 262–280 (1987).
[CrossRef]

Seldin, J. H.

Stefanescu, I. S.

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26(9), 2141–2160 (1985).
[CrossRef]

Teague, M. R.

ACM Trans. Math. Softw. (1)

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variable with the ‘simulated annealing’ algorithm,” ACM Trans. Math. Softw. 13(3), 262–280 (1987).
[CrossRef]

Bull. Calcutta Math. Soc. (1)

C. R. Rao, “Information and accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–91 (1945).

J. Math. Phys. (1)

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26(9), 2141–2160 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Optifab: Technical Digest, SPIE Technical Digest (1)

G. R. Brady and J. Fienup, “Phase retrieval as an optical metrology tool,” Optifab: Technical Digest, SPIE Technical Digest TD03, 139–141 (2005).

Other (4)

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

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2005).

G. R. Brady and J. R. Fienup, “Improved optical metrology using phase retrieval,” 2004 Optical Fabrication & Testing Topical Meeting, OSA, Rochester, NY, paper OTuB3 (2004).

H. Cramér, Mathematical Methods of Statistics (Princeton University Press, 1946).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Data-acquisition system for collecting multiple irradiance patterns near the focus of an optical element.

Fig. 2
Fig. 2

Detector data for the highly aberrated test lens using a pupil sampling of P = 1024 at image plane: (a) z = z1 and (b) z = z2.

Fig. 3
Fig. 3

(a) FIM and (b) its inverse, for Fringe Zernike coefficients {αn, n = 2,…, 9, 16} in the exit pupil of the highly aberrated test lens. Shown on a logarithmic scale.

Fig. 4
Fig. 4

Likelihood surface plotted along two axes at a time for pairs of parameters: (a) α9 and α16, (b) α4 and α9, (c) α3 and α8, (d) α7 and α9, (e) α7 and α16, and (f) α2 and α3.

Fig. 5
Fig. 5

12 simulated annealing trials for the estimation of wavefront parameters in the exit pupil of the highly aberrated test lens, plotting the optimal cost function versus iteration number on a log-log scale.

Fig. 6
Fig. 6

Comparison between the true and estimated irradiance patterns for the highly aberrated test lens.

Fig. 7
Fig. 7

Experimental data for the spherical test lens for image planes: (a) z = z1 and (b) z = z2. Scale bar corresponds to the intermediate image plane just before the imaging lens.

Fig. 8
Fig. 8

(a) FIM and (b) its inverse, for Fringe Zernike coefficients {αn, n = 2,…, 9, 16} in the exit pupil of the spherical test lens. Shown on a logarithmic scale.

Fig. 9
Fig. 9

Determining the nuisance parameters in the system for image plane z = z2 via a 2D grid search prior to the estimation of wavefront parameters.

Fig. 10
Fig. 10

12 simulated annealing trials for the estimation of wavefront parameters in the exit pupil of the highly aberrated test lens, plotting the optimal cost function versus iteration number on a log-log scale.

Tables (8)

Tables Icon

Table 1 Fringe Zernike coefficients {αn, n = 1,…, 37}, peak-to-valley, RMS, and variance, provided by ZEMAX for the highly aberrated test lens. Unlisted coefficients are zero.

Tables Icon

Table 2 Fringe Zernike Polynomials {Zn, n = 1,…, 9, 16, 25, 36, 37}.

Tables Icon

Table 3 Square-root of the CRB for Fringe Zernike coefficients {αn, n = 2,…, 9, 16} in the exit pupil of the highly aberrated test lens at λ = 0.6328 μm. Units are in waves λ.

Tables Icon

Table 4 Range in likelihood surface plots for Fringe Zernike coefficients {αn, n = 2,…, 9, 16} in the exit pupil of the highly aberrated test lens. Units are in waves.λ.

Tables Icon

Table 5 ML estimates of wavefront parameters for the highly aberrated test lens at λ = 0.6328 μm, including their standard deviations and the starting point in the search. Units are in waves λ.

Tables Icon

Table 6 Fringe Zernike coefficients {αn, n = 1,…, 37}, peak-to-valley, RMS, and variance, provided by ZEMAX for the highly aberrated test lens. Unlisted coefficients are zero.

Tables Icon

Table 7 Square-root of the CRB for Fringe Zernike coefficients {αn, n = 2,…, 9, 16} in the exit pupil of the highly aberrated test lens at λ = 0.6328 μm. Units are in waves λ.

Tables Icon

Table 8 ML estimates of wavefront parameters for the spherical test lens at λ = 0.6328 μm, including their standard deviations. Design values were used as a starting point in the search. Ranges in the search space were relative to the design values. Units are in waves λ.

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

u z (r)= 1 iλz d 2 r 0 u 0 ( r 0 )exp( ik |r r 0 | 2 + z 2 ) ,
u 0 ( r 0 )=exp[ikW( r 0 )] exp[ik R f ( r 0 )] R f ( r 0 ) ,
u z (r)= 1 iλz xp d 2 r 0 exp[ikW( r 0 )] exp[ik R f ( r 0 )] R f ( r 0 ) exp( ik |r r 0 | 2 + z 2 ) ,
u z (r)= 1 iλz xp d 2 r 0 exp[ikW( r 0 )] R f ( r 0 ) exp( ik r r 0 z ) ×exp[ ( ik |r r 0 | 2 + z 2 | r 0 | 2 + f 2 + r r 0 z ) ],
|r r 0 | 2 + z 2 | r 0 | 2 + f 2 + r r 0 z =zf+ r 2 2z + r 0 2 2 ( 1 z 1 f )+HOT,
HOT= |r r 0 | 4 8 z 3 + | r 0 | 4 8 f 3 + |r r 0 | 6 16 z 5 | r 0 | 6 16 f 5 +... .
u z (r)=A(r) xp d 2 r 0 exp[ikW( r 0 )] R f ( r 0 ) exp( ik r r 0 z )exp{ ik[ r 0 2 2 ( 1 z 1 f )+HOT ] },
A(r)= 1 iλz exp[ ik( zf+ r 2 2z ) ].
u z (r)A(r) F 2 { 1 R f ( r 0 ) exp[ikW( r 0 )]exp[ ik r 0 2 2 ( 1 z 1 f ) ] } ρ=r/λz ,
I(r)= | u z (r) | 2 1 λ 2 z 2 | F 2 { 1 R f ( r 0 ) exp[ikW( r 0 )]exp[ ik r 0 2 2 ( 1 z 1 f ) ] } ρ=r/λz | 2 .
W( r 0 ) n=1 N α n Z n ( r 0 ) ,
θ ^ ML argmax θ pr(g|θ).
θ ^ ML = argmax θ ln pr(g|θ).
θ ^ ML = argmin θ m=1 M [ g m g ¯ m (θ)] 2 .
F jk = [ θ j ln pr(g|θ) ] [ θ k ln pr(g|θ) ] g| θ ,
F jk = 1 σ 2 m=1 M g ¯ m (θ) θ j g ¯ m (θ) θ k ,
[ K θ ^ ] pp =Var{ θ ^ p } [ F 1 ] pp ,
units of F jk = 1 (units of θ j )(units of θ k ) .
z 1 = z f  +D z 1 = z f  0.6745 mm,    M 1 = 39.96,
z 2 = z f  +D z 2 = z f  0.7945 mm,    M 2 = 40.09,

Metrics