Abstract

A new direct search phase retrieval technique for determining the optical prescription of an imaging system in terms of Zernike coefficients is described. The technique provides coefficient estimates without the need to defocus point source images to generate phase diversity by using electric field (E-field) estimates in addition to intensity data. Numerical analysis shows that E-field patterns in the image plane produced by the Zernike polynomials are less correlated with each other than the intensity patterns. Therefore, the E-field pattern provides more information for Zernike coefficient estimation than the intensity pattern alone. The phase retrieval is accomplished through an iterative process that uses the measured point source data to estimate the E-field pattern in the image plane with the Gerchberg–Saxton (GS) algorithm. The estimated E-field is correlated with a modeled E-field to produce estimates of the Zernike coefficients. Then the coefficients that minimize the error between measured data and the intensity model are selected. By using E-field estimates rather than phase estimates from the GS algorithm, the limitations of phase unwrapping for Zernike decomposition are avoided. Simulated point source data shows the new phase retrieval algorithm avoids getting trapped in local minima over a wide range of random aberrations. Experimental point source data are used to demonstrate the phase retrieval effectiveness.

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References

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2013

2008

2006

A. Tokovinin and S. Heathcote, “Donut: measuring optical aberrations from a single extrafocal image,” Publ. Astron. Soc. Pac. 118, 1165–1175 (2006).
[CrossRef]

2003

T. G. Kolda, R. M. Lewis, and V. Torczon, “Optimization by direct search: new perspectives on some classical and modern methods,” SIAM Rev. 45, 385–482 (2003).
[CrossRef]

1995

1994

1993

1992

1988

1982

L. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[CrossRef]

1976

1972

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Blake, T.

J. C. Zingarelli, T. Blake, and S. Cain, “Improving ground based telescope focus through joint parameter estimation,” in Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, Hawaii, 2012.

Burrows, C. J.

Cain, S.

J. C. Zingarelli, T. Blake, and S. Cain, “Improving ground based telescope focus through joint parameter estimation,” in Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, Hawaii, 2012.

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping Theory, Algorithms and Software (Wiley-Interscience, 1998).

Goodman, J. W.

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

Heathcote, S.

A. Tokovinin and S. Heathcote, “Donut: measuring optical aberrations from a single extrafocal image,” Publ. Astron. Soc. Pac. 118, 1165–1175 (2006).
[CrossRef]

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 1997).

Hickson, P.

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice-Hall, 2011).

Kolda, T. G.

T. G. Kolda, R. M. Lewis, and V. Torczon, “Optimization by direct search: new perspectives on some classical and modern methods,” SIAM Rev. 45, 385–482 (2003).
[CrossRef]

Krist, J. E.

Kutner, M.

M. Kutner, C. Nachtsheim, J. Neter, and W. Li, Applied Linear Statistical Models (McGraw-Hill/Irwin, 2005).

Lewis, R. M.

T. G. Kolda, R. M. Lewis, and V. Torczon, “Optimization by direct search: new perspectives on some classical and modern methods,” SIAM Rev. 45, 385–482 (2003).
[CrossRef]

Li, W.

M. Kutner, C. Nachtsheim, J. Neter, and W. Li, Applied Linear Statistical Models (McGraw-Hill/Irwin, 2005).

Nachtsheim, C.

M. Kutner, C. Nachtsheim, J. Neter, and W. Li, Applied Linear Statistical Models (McGraw-Hill/Irwin, 2005).

Neter, J.

M. Kutner, C. Nachtsheim, J. Neter, and W. Li, Applied Linear Statistical Models (McGraw-Hill/Irwin, 2005).

Noll, R.

Paxman, R. G.

Pellizzari, C. J.

C. J. Pellizzari and J. D. Schmidt, “Phase unwrapping in the presence of strong turbulence,” in IEEE Aerospace Conference, Big Sky, Montana, 2010.

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping Theory, Algorithms and Software (Wiley-Interscience, 1998).

Roddier, F.

Rolland, J. P.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schmidt, J. D.

C. J. Pellizzari and J. D. Schmidt, “Phase unwrapping in the presence of strong turbulence,” in IEEE Aerospace Conference, Big Sky, Montana, 2010.

Schulz, T. J.

Shepp, L.

L. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[CrossRef]

Thompson, K. P.

Tobias, S.

Tokovinin, A.

A. Tokovinin and S. Heathcote, “Donut: measuring optical aberrations from a single extrafocal image,” Publ. Astron. Soc. Pac. 118, 1165–1175 (2006).
[CrossRef]

Torczon, V.

T. G. Kolda, R. M. Lewis, and V. Torczon, “Optimization by direct search: new perspectives on some classical and modern methods,” SIAM Rev. 45, 385–482 (2003).
[CrossRef]

Vardi, Y.

L. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[CrossRef]

Woods, D.

D. Woods, “The Space Surveillance Telescope: focus and alignment of a three mirror telescope,” in Advanced Maui Optical and Space Surveillence Technologies Conference, Maui, Hawaii, 2012.

Zingarelli, J. C.

J. C. Zingarelli, T. Blake, and S. Cain, “Improving ground based telescope focus through joint parameter estimation,” in Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, Hawaii, 2012.

Appl. Opt.

IEEE Trans. Med. Imaging

L. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Express

Optik

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Publ. Astron. Soc. Pac.

A. Tokovinin and S. Heathcote, “Donut: measuring optical aberrations from a single extrafocal image,” Publ. Astron. Soc. Pac. 118, 1165–1175 (2006).
[CrossRef]

SIAM Rev.

T. G. Kolda, R. M. Lewis, and V. Torczon, “Optimization by direct search: new perspectives on some classical and modern methods,” SIAM Rev. 45, 385–482 (2003).
[CrossRef]

Other

J. C. Zingarelli, T. Blake, and S. Cain, “Improving ground based telescope focus through joint parameter estimation,” in Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, Hawaii, 2012.

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

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping Theory, Algorithms and Software (Wiley-Interscience, 1998).

C. J. Pellizzari and J. D. Schmidt, “Phase unwrapping in the presence of strong turbulence,” in IEEE Aerospace Conference, Big Sky, Montana, 2010.

LSST Corporation, “Large Synoptic Survey Telescope,” (2011, accessed 3 June 2013). http://www.lsst.org/files/docs/overviewV2.0.pdf .

D. Woods, “The Space Surveillance Telescope: focus and alignment of a three mirror telescope,” in Advanced Maui Optical and Space Surveillence Technologies Conference, Maui, Hawaii, 2012.

M. Kutner, C. Nachtsheim, J. Neter, and W. Li, Applied Linear Statistical Models (McGraw-Hill/Irwin, 2005).

S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice-Hall, 2011).

E. Hecht, Optics (Addison-Wesley, 1997).

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Figures (9)

Fig. 1.
Fig. 1.

Image of pupil A(u1).

Fig. 2.
Fig. 2.

Pairwise correlation between two different (a) intensity patterns, ρh, and (b) E-field patterns, ρH, produced in the image plane by varying the Zernike coefficients for defocus, Z4. The pairwise correlation between two different (c) intensity patterns and (d) E-field patterns produced in the image plane by varying the Zernike coefficient for defocus and spherical error, Z11.

Fig. 3.
Fig. 3.

Block diagram of the E-field-based phase retrieval algorithm for estimating Zernike coefficients.

Fig. 4.
Fig. 4.

Phase retrieval simulation examples: (a) input wavefront in pupil, (b) simulated data in image plane, (c) estimated wavefront error in pupil using the E-field-based phase retrieval algorithm, (d) the real part of the E-field pattern in the image plane determined using the E-field-based phase retrieval algorithm, (e) the PSF in the image plane determined using the E-field-based phase retrieval algorithm, (f) estimated wavefront error in the pupil using only the intensity-based direct search LS algorithm, and (g) PSF in the image plane determined using only the intensity-based direct search LS algorithm.

Fig. 5.
Fig. 5.

Difference between the Zernike coefficients, Zi, simulated and phased retrieved using (a) only the intensity-based direct search LS algorithm versus the E-field-based phase retrieval algorithm and (b) zoomed-in plot of the E-field-based phase retrieval algorithm results. The error bars that correspond to the standard deviation of Δ are computed via Eq. (36).

Fig. 6.
Fig. 6.

Focus demonstration setup.

Fig. 7.
Fig. 7.

Results of the defocus demonstration. Phase retrieval results match theoretically predicted values of the focus aberration.

Fig. 8.
Fig. 8.

Astigmatism demonstration setup.

Fig. 9.
Fig. 9.

Astigmatism demonstration results.

Tables (2)

Tables Icon

Table 1. Defocus Demonstration Parameters

Tables Icon

Table 2. Astigmatism Demonstration Zernike Coefficient Estimates

Equations (46)

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W(u1)=Z1·ϕ1(u1)++ZN·ϕN(u1),
P(u1)=A(u1)exp[j·W(u1)],
H(m,W)=u1P(u1)exp(j2πmu1)=F[P(u1)],
h(m,W)=|H(m,W)|2=H(m,W)·H(m,W)*.
I(m,W)=θ^·h(m,W)+B^,
B^=m=1Md(d(m))Md,
θ^=m=1Md(d(m)B^),
d(m)=d˜1(m)+d˜2(m),
E[d˜1(m)]=θ·h(m),
E[d˜2(m)]=B.
P(d˜1,d˜2m1:md)=x=1Mdθ·h(m)d˜1(m)eθ·h(m)d˜1(m)!Bd˜2(m)eBd˜2(m)!,
L(θ,h(m),B)=m=1Mdd˜1(m)ln(θ·h(m))θ·h(m)d˜2(m)ln(B)B.
Θ=E[L(θ^,h(m),B^)|d(m),θ^,hold(m),B^]λm=1Mdh(m)=m=1Mdθ^·hold(m)·d(m)θ^·hold(m)+B^ln[θ^·h(m)]θ^·h(m)+B^·d(m)θ^·hold+B^ln(B^)B^λm=1Mdh(m),
m=1Mdh(m)=1.
dΘdh(m)=m=1Mdθ^hold(m)d(m)(θ^·hold(m)+B^)h(m)θ^λ.
hnew(m0)=θ^·hold(m0)·d(m0)(θ^·hold(m0)+B^)(θ^+λ)
λnew=m=1Mdθ^·hold(m)·d(m)(θ^·hold(m)+B^)θ^.
H^(m)=[h(m)]1/2exp[j·ϖ(m)],
Hd(m)=[h^(m)]1/2exp[j·ϖ(m)].
W^(u1)=angle(F1[Hd(m)]).
H(m,Zi,j)=F{A(u1)exp[j·Zi,j·ϕi,j(u1)]},
ρH(Zi,Zj)=RE[m=1MdH(m,Zi)·H(m,Zj)*](m=1MdH(m,Zi)·H(m,Zi)*)1/2(m=1MdH(m,Zj)·H(m,Zj)*)1/2
ρh(Zi,Zj)=m=1Mdh(m,Zi)·h(m,Zj)(m=1Mdh(m,Zi)2)1/2·(m=1Mdh(m,Zj)2)1/2.
Q{h^(m),h[m,W(u1)]}=m=1Md{h^(m)h[m,W(u1)]}2.
Wold(u1)=Z2·ϕ2(u1)++Z11·ϕ11(u1),
W2+New(u1)=Wold(u1)+ΔZ·ϕ2(u1)W2New(u1)=Wold(u1)ΔZ·ϕ2(u1)W11+New(u1)=Wold(u1)+ΔZ·ϕ11(u1)W11New(u1)=Wold(u1)ΔZ·ϕ11(u1),
h[m,W2+New(u1)]=|F{A(u1)exp[j·W2+New(u1)]}|2h[m,W2New(u1)]=|F{A(u1)exp[j·W2New(u1)]}|2h[m,W11+New(u1)]=|F{A(u1)exp[j·W11+New(u1)]}|2h[m,W11New(u1)]=|F{A(u1)exp[j·W11New(u1)]}|2.
Q=[Q{h^(m),h[m,W2+New(u1)]}Q{h^(m),h[m,W2New(u1)]}Q{h^(m),h[m,W11+New(u1)]}Q{h^(m),h[m,W11New(u1)]}].
ρ(H^(m),H(m,W))=RE[m=1MdH^(m)·H(m,W)*](m=1MdH^(m)·H^(m)*)1/2(m=1MdH(m,W)·H(m,W)*)1/2.
H[m,W2+New(u1)]=F{A(u1)exp[j·W2+New(u1)]}H[m,W2New(u1)]=F{A(u1)exp[j·W2New(u1)]}H[m,W11+New(u1)]=F{A(u1)exp[j·W11+New(u1)]}H[m,W11New(u1)]=F{A(u1)exp[j·W11New(u1)]}.
ρ=[ρ{H^(m),H[m,W2+New(u1)]}ρ{H^(m),H[m,W2New(u1)]}ρ{H^(m),H[m,W11+New(u1)]}ρ{H^(m),H[m,W11New(u1)]}].
d(m,W)=θ·h(m,W)+B+n(m),
[Z2Z3Z11]=unif(1,1).
Δ=|Zi||T=1100Z^i100|wherei(4,11).
ΔMean=C=1100Δ100,
ΔSTD={C=1100[(ΔΔMean)2]/100}1/2.
1f=1SI+1SO,
ΔFocus=SISD,
M=SISO.
dx=D1024.
W1̲̲=2·π·((Δy)2+SO2)1/2/λ.
=(x2̲̲+y2̲̲)1/2,
Λ̳=π·2λ·f.
W2̲̲=2·π·((M·Δy)2+(SIΔz)2)1/2/λ.
WT̲̲=W1̲̲+Λ̳+W2̲̲.
Zi=(ϕi̲̲·WT̲̲).

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