Abstract

We propose an efficient approximation to the nonlinear phase diversity (PD) method for wavefront reconstruction and correction from intensity measurements with potential of being used in real-time applications. The new iterative linear phase diversity (ILPD) method assumes that the residual phase aberration is small and makes use of a first-order Taylor expansion of the point spread function (PSF), which allows for arbitrary (large) diversities in order to optimize the phase retrieval. For static disturbances, at each step, the residual phase aberration is estimated based on one defocused image by solving a linear least squares problem, and compensated for with a deformable mirror. Due to the fact that the linear approximation does not have to be updated with each correction step, the computational complexity of the method is reduced to that of a matrix-vector multiplication. The convergence of the ILPD correction steps has been investigated and numerically verified. The comparative study that we make demonstrates the improved performance in computational time with no decrease in accuracy with respect to existing methods that also linearize the PSF.

© 2013 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  3. D. R. Gerwe, M. M. Johnson, and B. Calef, “Local minima analysis of phase diverse phase retrieval using maximum likelihood,” The Advanced Maui Optical and Space Surveillance Technical Conference, Kihei, Hawaii (2008).
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    [CrossRef]
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    [CrossRef]
  7. F. Martinache, “Kernel phase in Fizeau interferometry,” Astrophys. J. 724, 464–469 (2010).
    [CrossRef]
  8. S. Meimon, T. Fusco, and L. M. Mugnier, “Lift: a focal-plane wavefront sensor for real-time low-order sensing on faint sources,” Opt. Lett. 35, 3036–3038 (2010).
    [CrossRef]
  9. C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. J. Antonello, M. Verhaegen, R. Fraanje, T. van Werkhoven, H. C. Gerritsen, and C. U. Keller, “Semidefinite programming for model-based sensorless adaptive optics,” J. Opt. Soc. Am. A 29, 2428–2438 (2012).
    [CrossRef]
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    [CrossRef]
  14. A. Tokovinin and S. Heathcote, “Donut: measuring optical aberrations from a single extrafocal image,” Publ. Astron. Soc. Pac. 118, 1165–1175 (2006).
    [CrossRef]
  15. A. Polo, S. F. Pereira, and P. H. Urbach, “Theoretical analysis for best defocus measurement plane for robust phase retrieval,” Opt. Lett. 38, 812–814 (2013).
    [CrossRef]
  16. ANSI, “Methods for reporting optical aberrations of eyes,” (2004).
  17. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).
  18. F. Gustafsson, Statistical Sensor Fusion (Holmbergs i Malmö AB, 2010).
  19. G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University, 1996).
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    [CrossRef]

2013 (1)

2012 (2)

J. Antonello, M. Verhaegen, R. Fraanje, T. van Werkhoven, H. C. Gerritsen, and C. U. Keller, “Semidefinite programming for model-based sensorless adaptive optics,” J. Opt. Soc. Am. A 29, 2428–2438 (2012).
[CrossRef]

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

2010 (2)

2009 (1)

2006 (2)

L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” Adv. Imaging Electron Phys. 141, 1–76 (2006).
[CrossRef]

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

2005 (1)

C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

2001 (1)

1999 (1)

1998 (1)

1992 (1)

1976 (1)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “Phase retrieval by iterated projections,” Optik 35, 237–246 (1972).

Andrei, R.

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Antonello, J.

Blanc, A.

L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” Adv. Imaging Electron Phys. 141, 1–76 (2006).
[CrossRef]

Braat, J. J. M.

C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

Calef, B.

D. R. Gerwe, M. M. Johnson, and B. Calef, “Local minima analysis of phase diverse phase retrieval using maximum likelihood,” The Advanced Maui Optical and Space Surveillance Technical Conference, Kihei, Hawaii (2008).

Cassaing, F.

Dirksen, P.

C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

Doelman, N.

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Fienup, J. R.

Fraanje, R.

J. Antonello, M. Verhaegen, R. Fraanje, T. van Werkhoven, H. C. Gerritsen, and C. U. Keller, “Semidefinite programming for model-based sensorless adaptive optics,” J. Opt. Soc. Am. A 29, 2428–2438 (2012).
[CrossRef]

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Fusco, T.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “Phase retrieval by iterated projections,” Optik 35, 237–246 (1972).

Gerritsen, H. C.

Gerwe, D. R.

D. R. Gerwe, M. M. Johnson, and B. Calef, “Local minima analysis of phase diverse phase retrieval using maximum likelihood,” The Advanced Maui Optical and Space Surveillance Technical Conference, Kihei, Hawaii (2008).

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University, 1996).

Gonsalves, R. A.

Goodman, J. W.

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

Gustafsson, F.

F. Gustafsson, Statistical Sensor Fusion (Holmbergs i Malmö AB, 2010).

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]

Idier, J.

L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” Adv. Imaging Electron Phys. 141, 1–76 (2006).
[CrossRef]

Janssen, A. J. E. M.

C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

Johnson, M. M.

D. R. Gerwe, M. M. Johnson, and B. Calef, “Local minima analysis of phase diverse phase retrieval using maximum likelihood,” The Advanced Maui Optical and Space Surveillance Technical Conference, Kihei, Hawaii (2008).

Keller, C. U.

J. Antonello, M. Verhaegen, R. Fraanje, T. van Werkhoven, H. C. Gerritsen, and C. U. Keller, “Semidefinite programming for model-based sensorless adaptive optics,” J. Opt. Soc. Am. A 29, 2428–2438 (2012).
[CrossRef]

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Korkiakoski, V.

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Lee, D. J.

Martinache, F.

F. Martinache, “Kernel phase in Fizeau interferometry,” Astrophys. J. 724, 464–469 (2010).
[CrossRef]

Meimon, S.

Mocœur, I.

Mugnier, L. M.

Noll, R. J.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976).
[CrossRef]

Paxman, R. G.

Pereira, S. F.

Polo, A.

Roggemann, M. C.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “Phase retrieval by iterated projections,” Optik 35, 237–246 (1972).

Schultz, T. J.

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]

Urbach, P. H.

van der Avoort, C.

C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University, 1996).

van Werkhoven, T.

Verhaegen, M.

J. Antonello, M. Verhaegen, R. Fraanje, T. van Werkhoven, H. C. Gerritsen, and C. U. Keller, “Semidefinite programming for model-based sensorless adaptive optics,” J. Opt. Soc. Am. A 29, 2428–2438 (2012).
[CrossRef]

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Welsh, B. M.

Wild, W. J.

Adv. Imaging Electron Phys. (1)

L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” Adv. Imaging Electron Phys. 141, 1–76 (2006).
[CrossRef]

Astrophys. J. (1)

F. Martinache, “Kernel phase in Fizeau interferometry,” Astrophys. J. 724, 464–469 (2010).
[CrossRef]

J. Mod. Opt. (1)

C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Lett. (5)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “Phase retrieval by iterated projections,” Optik 35, 237–246 (1972).

Proc. SPIE (1)

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

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

Other (5)

D. R. Gerwe, M. M. Johnson, and B. Calef, “Local minima analysis of phase diverse phase retrieval using maximum likelihood,” The Advanced Maui Optical and Space Surveillance Technical Conference, Kihei, Hawaii (2008).

ANSI, “Methods for reporting optical aberrations of eyes,” (2004).

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

F. Gustafsson, Statistical Sensor Fusion (Holmbergs i Malmö AB, 2010).

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University, 1996).

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Figures (8)

Fig. 1.
Fig. 1.

Optical system: focal plane (black), defocus plane (gray), unknown small aberration Zernike coefficients (α), known arbitrary diversity Zernike coefficients (βi).

Fig. 2.
Fig. 2.

Convergence in terms of wavefront error: 1 rad rms, no read-out/photon noise, 1000 photons per image. LIFT (top), ILPD (bottom).

Fig. 3.
Fig. 3.

Convergence in terms of wavefront error: 1 rad rms, read-out noise with SNR=3.16, no photon noise, 1000 photons per image. LIFT (top), ILPD (bottom).

Fig. 4.
Fig. 4.

Residual error in the aberration vector. On each box, the central mark is the median, the edges of the box are the 25th and 75th percentiles, the whiskers extend to the most extreme data points not considered outliers, and outliers are plotted individually. The diamond signs represent mean values.

Fig. 5.
Fig. 5.

Relative residue.

Fig. 6.
Fig. 6.

Wavefront residual error versus increasing SNR.

Fig. 7.
Fig. 7.

Wavefront residual error versus increasing photon count.

Fig. 8.
Fig. 8.

Wavefront residual error versus increasing rms.

Tables (2)

Tables Icon

Table 1. Rms Values of the Corrected Wavefronts for No Read-Out/Photon Noise, 1000 Photons per Image, and 1 rad Initial rms

Tables Icon

Table 2. Rms Values of the Corrected Wavefronts for Read-Out Noise SNR=3.16, No Photon Noise, 1000 Photons per Image, and 1 rad Initial rms

Equations (43)

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ϕ(uj,vj)=Z(uj,vj)Tα,
ϕi(uj,vj)=Z(uj,vj)T(α+βi),
yi,j=μih(sj,tj,α,βi)+ni(sj,tj),
h(sj,tj,α,βi)=F(Π(u,v)exp(iϕi(u,v)))(sj,tj)×F(Π(u,v)exp(iϕi(u,v)))*(sj,tj)=F((exp(iϕi(u,v))Π(u,v)exp(iϕi(u,v))Π(u,v))(u,v))(sj,tj),
p(uj,vj,α,βi)=Π(uj,vj)exp(iϕi(uj,vj)).
W(uj,vj,α,βi)=(p(u,v,α,βi)p(u,v,α,βi)*)(uj,vj).
pj(α,βi)p(uj,vj,α+βi),pj(α,βi)p(uj,vj,α+βi),
Wj(α,βi)W(uj,vj,α,βi),hj(α,βi)h(sj,tj,α,βi).
hj(α,βi)=F(pj(α,βi)pj*(α,βi))(sj,tj)=F(Wj(α,βi))(sj,tj),
SNR=1m2j=1m2μihj(α,βi)1m2j=1m2σi2(=μim2σi),
pj(α,βi)=pj(α,βi)|α+βi=0+pj(α,βi)(α+βi)|α+βi=0(α+βi)+O(α+βi2),=Π(uj,vj)(1+iZT(uj,vj)(α+βi))+O(α+βi2).
hj(α,βi)=A0,j+A1,j(α+βi)+(α+βi)TA2,j(α+βi)+O(α+βi2),
A0,jF(pj(α,βi)pj*(α,βi))(sj,tj)|α+βi=0=hj(α,βi)|α+βi=0,A1,jF(pj(α,βi)(α+βi)pj*(α,βi)+pj(α,βi)pj*(α,βi)(α+βi)T)(sj,tj)|α+βi=0=hj(α,βi)(α+βi)|α+βi=0,A2,jF(pj(α,βi)(α+βi)pj*(α,βi)(α+βi)T+pj(α,βi)(α+βi)pj*(α,βi)(α+βi)T)(sj,tj)|α+βi=0.
hj(α,βi)=B0,j(βi)+B1,j(βi)α+O(α2),
B0,j(βi)hj(α,βi)|α=0,B1,j(βi)hj(α,βi)α|α=0.
hj(α,βi)=C0,j+C1,j(α+βi)+(α+βi)TC2,j(α+βi)+O(α+βi3),
C0,jhj(α,βi)|α+βi=0(=A0,j),C1,jhj(α,βi)(α+βi)|α+βi=0(=A1,j),C2,j2hj(α,βi)(α+βi)(α+βi)T|α+βi=0=A2,j+F(2pj(α,βi)(α+βi)(α+βi)Tpj*(α,βi)+pj(α,βi)2pj*(α,βi)(α+βi)(α+βi)T)(sj,tj)|α+βi=0.
hj(α,βi)=D0,j(βi)+D1,j(βi)α+αTD2,j(βi)α+O(α3),
D0,j(βi)hj(α,βi)|α=0(=B0,j(βi)),D1,j(βi)hj(α,βi)α|α=0(=B1,j(βi)),D2,j(βi)2hj(α,βi)ααT|α=0.
Y1=bS+ASα+ΔbS(α)+n1,
ΔbS(α)O(α2),
αk=Δαk1=αk1α^k1.
b¯S,kΔbS(αk)=ASαk+nk,
ΔbS(αk)=O(αk2)=CSαk2,
minαkb¯S,kASαk2.
rLSαk+1αk.
α^kαkαkΔbS(αk)b¯S,k{2κ(AS)cos(θ)+tan(θ)κ(AS)2},
limαk0CSαk2b¯S,k{2κ(AS)cos(θ)+tan(θ)κ(AS)2}=0.
Π(uj,vj)={1Suj2+vj2r20uj2+vj2>r2,
pi,jpj(α,βi),Wi,jWj(α,βi).
Wi,j(pi,j+pi,jγiT(γiγ0))(pi,j*+pi,j*γiT(γiγ0))|γi=γ0=pi,jpi,j*|γi=γ0+[pi,jγiTpi,j*+pj(α,βi)pi,j*γiT]|γi=γ0(γiγ0)+(γiγ0)T[pi,jγiTpi,j*γi+pi,jγipi,j*γiT]|γi=γ0(γiγ0).
L[pi,jγiTpi,j*+pj(α,βi)pi,j*γiT]|γi=γ0.
L=iZjTΠjΠj*ΠjiZjTΠj*=i[Ze,jTZo,jT]ΠjΠj*Πji[Ze,jTZo,jT]Πj*,
L=i[Ze,jTZo,jT]ΠjΠjΠji[Ze,jTZo,jT]Πj=i[02Zo,jT]ΠjΠj.
Wi,jpi,jpi,j*|γi=γ0+[pi,jγiTpi,j*+pj(α,βi)pi,j*γiT]|γi=γ0(γiγ0)
L=iZjTΠjexp(iZjTγ0)Πj*exp(iZjTγ0)Πjexp(iZjTγ0)iZjTΠjexp(iZjTγ0)=i[Ze,jTZo,jT]Πjexp(iZjTγ0)Πj*exp(iZjTγ0)Πjexp(iZjTγ0)i[Ze,jTZo,jT]Πjexp(iZjTγ0)=i[Ze,jTZo,jT]Π˜jΠ˜j*Π˜ji[Ze,jTZo,jT]Π˜j*,
pi,jpi,j|γi=γ0+pi,jγi|γi=γ0(γiγ0)+12(γiγ0)T2pi,jγiγiT|γi=γ0(γiγ0).
Wi,jpi,jpi,j*|γi=γ0+[pi,jγiTpi,j*+pj(α,βi)pi,j*γiT]|γi=γ0(γiγ0)+(γiγ0)T[2pi,jγiγiTpi,j*+pj(α,βi)2pi,j*γiγiT+pi,jγiTpi,j*γi+pi,jγipi,j*γiT]|γi=γ0(γiγ0),
2pi,jγiγiTpi,j*+pj(α,βi)2pi,j*γiγiT.
μdiff=E[Δyj]=μ1h1,jμ2h2,j.
σdiff2=E[(Δyjμdiff)2]=σ1,j2+σ2,j2.
SNRdiff=μ1h1,jμ2h2,jσ1,j2+σ2,j2.
SNRdiff=1m2j=1m2μ1h1,jμ2h2,j1m2j=1m2σ1,j2+1m2jm2σ2,j2=1m21σ12+σ22j=1m2μ1(h1,jμ2h2,j)=1m2μ1μ2σ12+σ22.

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