Abstract

We develop a method for subaperture piston phase retrieval in a telescope using a segmented primary mirror. We assume that the mirror subapertures are arranged on a two-dimensional lattice, and in addition, the separate subaperture point-spread functions are focused and overlapped on the focal plane. Therefore, the residual errors are the subaperture piston phase errors, represented as a phasor, a unit modulus complex number, for each subaperture. Under these conditions, we find considerable simplicity in the calculated optical transfer function (OTF) at special subaperture lattice spatial frequencies. We then construct a phasor-based error function based on the modulus squared of the difference between the measured OTF and the calculated OTF. The remaining steps in our piston phase retrieval algorithms are developed by calculating the error-function variation, with respect to each phasor element. The resulting equations for the error gradient are then used iteratively, in a phasor-based algorithm, to find the minimum of the error function. In the applications, we simulate photon-noise-limited piston retrieval for a segmented primary with 18 hexagonal subapertures. When we invoke phase diversity, the piston retrievals prove unique and accurate.

© 2012 Optical Society of America

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References

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  1. S. Bikkannavar, D. Redding, J. Green, S. Basinger, D. Cohen, J. Lou, C. Ohara, and S. Fang, “Phase retrieval methods for wavefront sensing,” Proc. SPIE 7739, 77392X (2010).
    [CrossRef]
  2. I. Surdej, N. Yaitskova, and F. Gonte, “On-sky performance of the Zernike phase contrast sensor for the phasing of segmented telescopes,” Appl. Opt. 49, 4052–4062 (2010).
    [CrossRef]
  3. M. R. Bolcar and J. R. Fienup, “Sub-aperture piston phase diversity for segmented and multi-aperture systems,” Appl. Opt. 48, A5–A12 (2009).
    [CrossRef]
  4. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
  5. R. W. Gerchberg and W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  6. W. H. Southwell, “Wave-front analyzer using a maximum likelihood algorithm,” J. Opt. Soc. Am. 67, 396–399(1977).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  8. S. T. Thurman and J. R. Fienup, “Complex pupil retrieval with undersampled data,” J. Opt. Soc. Am. A 26, 2640–2647 (2009).
    [CrossRef]
  9. G. C. Dente and M. L. Tilton, “New phasor reconstruction for speckle imaging,” Astron. Astrophys. 529, A58(2011).
    [CrossRef]
  10. O. Falconi, “Maximum sensitivities of optical direction and twist measuring instruments,” J. Opt. Soc. Am. 54, 1315–1320(1964).
    [CrossRef]

2011 (1)

G. C. Dente and M. L. Tilton, “New phasor reconstruction for speckle imaging,” Astron. Astrophys. 529, A58(2011).
[CrossRef]

2010 (2)

S. Bikkannavar, D. Redding, J. Green, S. Basinger, D. Cohen, J. Lou, C. Ohara, and S. Fang, “Phase retrieval methods for wavefront sensing,” Proc. SPIE 7739, 77392X (2010).
[CrossRef]

I. Surdej, N. Yaitskova, and F. Gonte, “On-sky performance of the Zernike phase contrast sensor for the phasing of segmented telescopes,” Appl. Opt. 49, 4052–4062 (2010).
[CrossRef]

2009 (2)

1982 (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

1977 (1)

1972 (1)

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

1964 (1)

Basinger, S.

S. Bikkannavar, D. Redding, J. Green, S. Basinger, D. Cohen, J. Lou, C. Ohara, and S. Fang, “Phase retrieval methods for wavefront sensing,” Proc. SPIE 7739, 77392X (2010).
[CrossRef]

Bikkannavar, S.

S. Bikkannavar, D. Redding, J. Green, S. Basinger, D. Cohen, J. Lou, C. Ohara, and S. Fang, “Phase retrieval methods for wavefront sensing,” Proc. SPIE 7739, 77392X (2010).
[CrossRef]

Bolcar, M. R.

Cohen, D.

S. Bikkannavar, D. Redding, J. Green, S. Basinger, D. Cohen, J. Lou, C. Ohara, and S. Fang, “Phase retrieval methods for wavefront sensing,” Proc. SPIE 7739, 77392X (2010).
[CrossRef]

Dente, G. C.

G. C. Dente and M. L. Tilton, “New phasor reconstruction for speckle imaging,” Astron. Astrophys. 529, A58(2011).
[CrossRef]

Falconi, O.

Fang, S.

S. Bikkannavar, D. Redding, J. Green, S. Basinger, D. Cohen, J. Lou, C. Ohara, and S. Fang, “Phase retrieval methods for wavefront sensing,” Proc. SPIE 7739, 77392X (2010).
[CrossRef]

Fienup, J. R.

Gerchberg, R. W.

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

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

Gonte, F.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Green, J.

S. Bikkannavar, D. Redding, J. Green, S. Basinger, D. Cohen, J. Lou, C. Ohara, and S. Fang, “Phase retrieval methods for wavefront sensing,” Proc. SPIE 7739, 77392X (2010).
[CrossRef]

Lou, J.

S. Bikkannavar, D. Redding, J. Green, S. Basinger, D. Cohen, J. Lou, C. Ohara, and S. Fang, “Phase retrieval methods for wavefront sensing,” Proc. SPIE 7739, 77392X (2010).
[CrossRef]

Ohara, C.

S. Bikkannavar, D. Redding, J. Green, S. Basinger, D. Cohen, J. Lou, C. Ohara, and S. Fang, “Phase retrieval methods for wavefront sensing,” Proc. SPIE 7739, 77392X (2010).
[CrossRef]

Redding, D.

S. Bikkannavar, D. Redding, J. Green, S. Basinger, D. Cohen, J. Lou, C. Ohara, and S. Fang, “Phase retrieval methods for wavefront sensing,” Proc. SPIE 7739, 77392X (2010).
[CrossRef]

Saxton, W. O.

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

Southwell, W. H.

Surdej, I.

Thurman, S. T.

Tilton, M. L.

G. C. Dente and M. L. Tilton, “New phasor reconstruction for speckle imaging,” Astron. Astrophys. 529, A58(2011).
[CrossRef]

Yaitskova, N.

Appl. Opt. (2)

Astron. Astrophys. (1)

G. C. Dente and M. L. Tilton, “New phasor reconstruction for speckle imaging,” Astron. Astrophys. 529, A58(2011).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

Optik (1)

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

Proc. SPIE (1)

S. Bikkannavar, D. Redding, J. Green, S. Basinger, D. Cohen, J. Lou, C. Ohara, and S. Fang, “Phase retrieval methods for wavefront sensing,” Proc. SPIE 7739, 77392X (2010).
[CrossRef]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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Figures (4)

Fig. 1.
Fig. 1.

Segmented primary aperture arrangement.

Fig. 2.
Fig. 2.

Color-shaded representation of randomly distributed subaperture piston errors in waves.

Fig. 3.
Fig. 3.

Noise-free aberrated PSF.

Fig. 4.
Fig. 4.

Plot of the residual phase errors.

Tables (2)

Tables Icon

Table 1. Simulation Parameters

Tables Icon

Table 2. Example of Randomly Selected Piston Phase Errors and Calculated Piston Phase Errors

Equations (18)

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I(x,y)=A2(AF)2|S˜(xλF,yλF)|2·K=1NK¯=1Nexp(i2πλF[(xc(K)xc(K¯))x+(yc(K)yc(K¯))y])·P(K)·P*(K¯),
Φ(fx,fy)dxdyexp(i·(2πfxx+2πfyy))I(x,y)=A2KK¯P(K)P*(K¯)dxdyS(x,y)·S*((xxc(K)+xc(K¯)+λFfx),(yyc(K)+yc(K¯)+λFfy)).
f¯x=(xc(K)xc(K¯))λFτxλF,f¯y=(yc(K)yc(K¯))λFτyλF,
Φ(f¯x,f¯y)=A2dxdy|S(x,y)|2KK¯P(K)P*(K¯)δrc(K)τ,rc(K¯),
H(τλF)=H(f¯x,f¯y)Φ(f¯x,f¯y)Φ(0,0)=1NKK¯P(K)P*(K¯)δrc(K)τ,rc(K¯).
I˜(l;m,n)=FFT{I(l;i,k)}.
m¯=Int[τxλFNxΔx],n¯=Int[τyλFNyΔx],
HM(l;m¯,n¯)I˜(l;m¯,n¯)I˜(l;0,0).
Elm¯,n¯[H(l;τ(m¯,n¯)λF)HM(l;m¯,n¯)][H(l;τ(m¯,n¯)λF)HM(l;m¯,n¯)]*.
Elτ[H(l;τλF)HM(l;τ)][H*(l;τλF)HM*(l;τ)]lτΓ(l;τ)·Γ*(l;τ).
H(l;τλF)=1NKK¯Pl(K)P(K)Pl(K¯)P(K¯)δrc(K)τ,rc(K¯).
Eθ(K)=EP(K)iP(K)=iN[P(K)Ω*(K)P1(K)Ω(K)],
Ω(K)l,τK¯[Γ(l;τ)Pl1(K)Pl(K¯)P(K¯)δrc(K)τ,rc(K¯)+Γ*(l;τ)Pl1(K)Pl(K¯)P(K¯)δrc(K)+τ,rc(K¯)].
P(K)Pold(K)·exp(iβEθ(K)),K=2,3,4,N,
rc(1)=(1,2b)sd,rc(2)=(0,2b)sd,rc(3)=(1,2b)sd,rc(4)=(1.5,1b)sd,rc(5)=(0.5,1b)sd,rc(6)=(0.5,1b)sd,rc(7)=(1.5,1b)sd,rc(8)=(2,0)sd,rc(9)=(1,0)sd,rc(10)=(1,0)sd,rc(11)=(2,0)sd,rc(12)=(1.5,1b)sd,rc(13)=(0.5,1b)sd,rc(14)=(0.5,1b)sd,rc(15)=(1.5,1b)sd,rc(16)=(1,2b)sd,rc(17)=(0,2b)sd,rc(18)=(1,2b)sd,
I(l;i,k)=dxdyp(iΔxx,kΔxy)·I(l;x,y)+η(i,k)I(l;iΔx,jΔx)+η(i,k),
Pl(5)=Pl(6)=Pl(13)=Pl(14)=1,l=1,Pl(5)=Pl(6)=Pl(13)=Pl(14)=i,l=2,Pl(5)=Pl(6)=Pl(13)=Pl(14)=1,l=3;
ΔθK=2N|arg(P^(K)/P(K))|2(N1).

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