Abstract

We use aberration theory to derive a generalized pupil function of the Golay3 imaging system when astigmatisms exist in its submirrors. Theoretical analysis and numerical simulation using ZEMAX show that the point spread function (PSF) and the modulation transfer function (MTF) of the Golay3 sparse aperture system have a periodic change when there are piston errors. When the peak–valley value of the wavefront (PVtilt) due to the tilt error increases from zero to λ, the PSF and the MTF change significantly, and the change direction is determined by the location of the submirror with the tilt error. When PVtilt becomes larger than λ, the PSF and the MTF remain unvaried. We calculate the peaks of the signal-to-noise ratio (PSNR) resulting from the piston and tilt errors according to the Strehl ratio, and show that the PSNR decreases when the errors increase.

© 2013 Optical Society of America

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References

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2012

2011

F. J. Liu, W. Q. Ying, W. F. Liang, and S. T. Ting, “Design of Golay3 telescope system, ” Opt. Precision Eng. 19, 2677–2683 (2011).

2009

2007

2004

S.-J. Chung, D. W. Miller, and O. L. de Weck, “ARGOS testbed: study of multidisciplinary challenges of future spaceborne interferometric arrays,” Opt. Eng. 43, 2156–2167 (2004).
[CrossRef]

J. L. Flores, M. Strojnik, G. Paez, and G. Garcia-Torales, “Effects of misalignment errors on the optical transfer functions of synthetic aperture telescopes,” Appl. Opt. 43, 5926–5932 (2004).
[CrossRef]

2003

A. Meinel and M. Meinel, “Extremely large sparse aperture telescope,” Opt. Photon. News 14(10), 26–29 (2003).
[CrossRef]

2002

2001

G. Rousset, L. M. Mugenier, F. Cassaing, and B. Sorrente, “Imaging with multi-aperture optical telescopes and an application,” C. R. Acad. Sci. Ser. IV Phys. Astrophys. 2, 1957–1968 (2001).
[CrossRef]

1998

D. B. DeYoung, J. D. Dillow, S. P. Corcoran, E. V. Andrews, J. Yellowhair, and K. DeVries, “Ground demonstration of an optical control system for a space-based sparse aperture telescope,” Proc. SPIE 3356, 1156–1167 (1998).
[CrossRef]

1994

1992

1983

L. C. Chan and P. Whiteman, “Hardware-constrained hybrid coding of video imagery,” IEEE Trans. Aerosp. Electron. Syst. 19, 71–84 (1983).
[CrossRef]

1970

Andrews, E. V.

D. B. DeYoung, J. D. Dillow, S. P. Corcoran, E. V. Andrews, J. Yellowhair, and K. DeVries, “Ground demonstration of an optical control system for a space-based sparse aperture telescope,” Proc. SPIE 3356, 1156–1167 (1998).
[CrossRef]

Angel, J. R. P.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Buchroeder, R. A.

Cassaing, F.

G. Rousset, L. M. Mugenier, F. Cassaing, and B. Sorrente, “Imaging with multi-aperture optical telescopes and an application,” C. R. Acad. Sci. Ser. IV Phys. Astrophys. 2, 1957–1968 (2001).
[CrossRef]

Chan, L. C.

L. C. Chan and P. Whiteman, “Hardware-constrained hybrid coding of video imagery,” IEEE Trans. Aerosp. Electron. Syst. 19, 71–84 (1983).
[CrossRef]

Chung, S.-J.

S.-J. Chung, D. W. Miller, and O. L. de Weck, “ARGOS testbed: study of multidisciplinary challenges of future spaceborne interferometric arrays,” Opt. Eng. 43, 2156–2167 (2004).
[CrossRef]

Colucci, D.

Corcoran, S. P.

D. B. DeYoung, J. D. Dillow, S. P. Corcoran, E. V. Andrews, J. Yellowhair, and K. DeVries, “Ground demonstration of an optical control system for a space-based sparse aperture telescope,” Proc. SPIE 3356, 1156–1167 (1998).
[CrossRef]

de Weck, O. L.

S.-J. Chung, D. W. Miller, and O. L. de Weck, “ARGOS testbed: study of multidisciplinary challenges of future spaceborne interferometric arrays,” Opt. Eng. 43, 2156–2167 (2004).
[CrossRef]

Dekany, R.

DeVries, K.

D. B. DeYoung, J. D. Dillow, S. P. Corcoran, E. V. Andrews, J. Yellowhair, and K. DeVries, “Ground demonstration of an optical control system for a space-based sparse aperture telescope,” Proc. SPIE 3356, 1156–1167 (1998).
[CrossRef]

DeYoung, D. B.

D. B. DeYoung, J. D. Dillow, S. P. Corcoran, E. V. Andrews, J. Yellowhair, and K. DeVries, “Ground demonstration of an optical control system for a space-based sparse aperture telescope,” Proc. SPIE 3356, 1156–1167 (1998).
[CrossRef]

Dierking, M. P.

Dillow, J. D.

D. B. DeYoung, J. D. Dillow, S. P. Corcoran, E. V. Andrews, J. Yellowhair, and K. DeVries, “Ground demonstration of an optical control system for a space-based sparse aperture telescope,” Proc. SPIE 3356, 1156–1167 (1998).
[CrossRef]

Dohlen, K.

Duncan, B. D.

Feng, S.

Feng, W.

Flores, J. L.

Garcia-Torales, G.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

Hua, W. X.

Hui, R. C.

Liang, W. F.

F. J. Liu, W. Q. Ying, W. F. Liang, and S. T. Ting, “Design of Golay3 telescope system, ” Opt. Precision Eng. 19, 2677–2683 (2011).

Lin, Q.

Liu, F. J.

F. J. Liu, W. Q. Ying, W. F. Liang, and S. T. Ting, “Design of Golay3 telescope system, ” Opt. Precision Eng. 19, 2677–2683 (2011).

Lloyd-Hart, M.

McCarthy, D.

Mcleod, B.

Meinel, A.

A. Meinel and M. Meinel, “Extremely large sparse aperture telescope,” Opt. Photon. News 14(10), 26–29 (2003).
[CrossRef]

Meinel, M.

A. Meinel and M. Meinel, “Extremely large sparse aperture telescope,” Opt. Photon. News 14(10), 26–29 (2003).
[CrossRef]

Miller, D. W.

S.-J. Chung, D. W. Miller, and O. L. de Weck, “ARGOS testbed: study of multidisciplinary challenges of future spaceborne interferometric arrays,” Opt. Eng. 43, 2156–2167 (2004).
[CrossRef]

Miller, N. J.

Mo, Z. Y.

Z. Y. Mo, Applied Optics, 3rd ed. (Publishing House of Electronics Industry, 2008).

Mugenier, L. M.

G. Rousset, L. M. Mugenier, F. Cassaing, and B. Sorrente, “Imaging with multi-aperture optical telescopes and an application,” C. R. Acad. Sci. Ser. IV Phys. Astrophys. 2, 1957–1968 (2001).
[CrossRef]

Paez, G.

Pauls, T. A.

T. A. Pauls, “Origins of sparse aperture imaging,” in Proceedings of IEEE Aerospace Conference (IEEE, 2001), pp. 1421–1426.

Qiang, F.

Rousset, G.

G. Rousset, L. M. Mugenier, F. Cassaing, and B. Sorrente, “Imaging with multi-aperture optical telescopes and an application,” C. R. Acad. Sci. Ser. IV Phys. Astrophys. 2, 1957–1968 (2001).
[CrossRef]

Scott-Fleming, I.

Sorrente, B.

G. Rousset, L. M. Mugenier, F. Cassaing, and B. Sorrente, “Imaging with multi-aperture optical telescopes and an application,” C. R. Acad. Sci. Ser. IV Phys. Astrophys. 2, 1957–1968 (2001).
[CrossRef]

Strojnik, M.

Ting, S. T.

F. J. Liu, W. Q. Ying, W. F. Liang, and S. T. Ting, “Design of Golay3 telescope system, ” Opt. Precision Eng. 19, 2677–2683 (2011).

Whiteman, P.

L. C. Chan and P. Whiteman, “Hardware-constrained hybrid coding of video imagery,” IEEE Trans. Aerosp. Electron. Syst. 19, 71–84 (1983).
[CrossRef]

Wittman, D.

Wizinowich, P.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Yaitskova, N.

Yellowhair, J.

D. B. DeYoung, J. D. Dillow, S. P. Corcoran, E. V. Andrews, J. Yellowhair, and K. DeVries, “Ground demonstration of an optical control system for a space-based sparse aperture telescope,” Proc. SPIE 3356, 1156–1167 (1998).
[CrossRef]

Ying, W. Q.

F. J. Liu, W. Q. Ying, W. F. Liang, and S. T. Ting, “Design of Golay3 telescope system, ” Opt. Precision Eng. 19, 2677–2683 (2011).

W. Feng, W. Q. Ying, and Q. Lin, “Analysis of the characteristics of the Golay3 multiple-mirror telescope,” Appl. Opt. 48, 643–652 (2009).
[CrossRef]

Appl. Opt.

C. R. Acad. Sci. Ser. IV Phys. Astrophys.

G. Rousset, L. M. Mugenier, F. Cassaing, and B. Sorrente, “Imaging with multi-aperture optical telescopes and an application,” C. R. Acad. Sci. Ser. IV Phys. Astrophys. 2, 1957–1968 (2001).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst.

L. C. Chan and P. Whiteman, “Hardware-constrained hybrid coding of video imagery,” IEEE Trans. Aerosp. Electron. Syst. 19, 71–84 (1983).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

S.-J. Chung, D. W. Miller, and O. L. de Weck, “ARGOS testbed: study of multidisciplinary challenges of future spaceborne interferometric arrays,” Opt. Eng. 43, 2156–2167 (2004).
[CrossRef]

Opt. Express

Opt. Photon. News

A. Meinel and M. Meinel, “Extremely large sparse aperture telescope,” Opt. Photon. News 14(10), 26–29 (2003).
[CrossRef]

Opt. Precision Eng.

F. J. Liu, W. Q. Ying, W. F. Liang, and S. T. Ting, “Design of Golay3 telescope system, ” Opt. Precision Eng. 19, 2677–2683 (2011).

Proc. SPIE

D. B. DeYoung, J. D. Dillow, S. P. Corcoran, E. V. Andrews, J. Yellowhair, and K. DeVries, “Ground demonstration of an optical control system for a space-based sparse aperture telescope,” Proc. SPIE 3356, 1156–1167 (1998).
[CrossRef]

Other

T. A. Pauls, “Origins of sparse aperture imaging,” in Proceedings of IEEE Aerospace Conference (IEEE, 2001), pp. 1421–1426.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

Z. Y. Mo, Applied Optics, 3rd ed. (Publishing House of Electronics Industry, 2008).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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Figures (14)

Fig. 1.
Fig. 1.

Submirror projection onto the entrance pupil.

Fig. 2.
Fig. 2.

Piston error of the submirror.

Fig. 3.
Fig. 3.

PSFs of the Golay3 sparse aperture system with piston errors: (a) piston=0λ, (b) piston=0.3λ, (c) piston=0.5λ, (d) piston=0.6λ, (e) piston=0.8λ, and (f) piston=1λ.

Fig. 4.
Fig. 4.

MTFs of the Golay3 sparse aperture system with piston errors: (a) piston=0λ, (b) piston=0.3λ, (c) piston=0.5λ, (d) piston=0.6λ, (e) piston=0.8λ, and (f) piston=1λ.

Fig. 5.
Fig. 5.

Tilt error of submirror.

Fig. 6.
Fig. 6.

PSFs of Golay3 sparse aperture imaging system with tilt errors: (a) PVtilt=0, (b) PVtilt=0.5λ, (c) PVtilt=1λ, and (d) PVtilt=2λ.

Fig. 7.
Fig. 7.

MTFs of the Golay3 sparse aperture system with tilt errors: (a) PVtilt=0, (b) PVtilt=0.5λ, (c) PVtilt=1λ, and (d) PVtilt=2λ.

Fig. 8.
Fig. 8.

MTFs of the Golay3 sparse aperture system with both piston and tilt errors: (a) PVtilt1=PVtilt2=PVtilt3=0.5λ, (b) PVtilt1=PVtilt2=PVtilt3=0.5λ, piston1=0.2λ, (c) piston1=piston2=piston3=0.5λ, and (d) piston1=piston2=piston3=0.5λ, PVtilt1=λ.

Fig. 9.
Fig. 9.

Relationship between S.D and piston error of the Golay3 sparse aperture system.

Fig. 10.
Fig. 10.

Relationship between S.D and tilt error of the Golay3 sparse aperture system.

Fig. 11.
Fig. 11.

Configuration of Golay3 sparse aperture imaging system.

Fig. 12.
Fig. 12.

Object.

Fig. 13.
Fig. 13.

Imaging results with different piston errors: (a) piston=0λ, (b) piston=0.1λ, (c) piston=0.2λ, (d) piston=0.3λ, (e) piston=0.4λ, and (f) piston=0.5λ.

Fig. 14.
Fig. 14.

Imaging results with different tilt errors: (a) PVtilt=0, (b) PVtilt=0.5λ, (c) PVtilt=1λ, and (d) PVtilt=2λ.

Tables (2)

Tables Icon

Table 1. Parameters of Two-Reflective System

Tables Icon

Table 2. PSNR of Imaging Results with Different Errors

Equations (21)

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P˜(ρ,θ)=P(ρ,θ)exp[jkW(ρ,θ)],
P(ρ,θ)={1in the aperture0outside the aperture},
W(ρ,θ)=W000+W200y2+W020ρ2+W111yρcosθ+W040ρ4+W131yρ3cosθ+W222y2ρ2cos2θ+W220y2ρ2+W311y3ρcosθ+W400y4,
W(ρ,θ)=W00+W20ρ2+W11ρcosθ+W40ρ4+W31ρ3cosθ+W22ρ2cos2θ,
{xn=ρcosθnyn=ρsinθn,
{ρ=R2d2/4sin(αβ)θn=(n1)2π3,n=1,2,3.
Pn=[(xxn)cosθn+(yyn)sinθn]2b2+[(xxn)sinθn+(yyn)cosθn]2a2,
P˜(x,y)=n=1NPn(xxn,yyn)×exp[jkWn(xxn,yyn)],
W00=Wp=2ABcos2θ,
AB=R2y2[x+Δhsin(αβ)]2Δhcos(αβ)R2x2y2.
PSFp=|F{P˜(x,y)}|2=|n=13F{Pn(x,y)exp[j2πλWp]}|2,
MTFp=|P(λzξ,λzη)P(λzξ,λzη)++|P(x,y)|dxdy|=|m=13n=13Pm(λzξ,λzη)exp[j2πλθxWp]Pn(λzξ,λzη)exp[j2πλθyWp]++|P(x,y)|dxdy|,
F{Pn(x,y)}=abJ1(2πl)lexp[j2π(ξxn+ηyn)],
l=a2(ξcosθn+ηsinφn)2+b2(ξsinθn+ηcosφn)2
Wt1(x,y)=xtanθ1y+ytanθ1x.
Wt1(x,y)=xθ1y+yθ1x.
P˜(x,y)=n=13Pn(x,y)exp[j2πλ[θy(xxn)+θx(yyn)]],n=1,2,3.
PSFt=|F{P(x,y)}|2=|n=13F{Pn(x,y)exp[j2πλ(θy(xxn)+θx(yyn))]}|2,
MTFt=|P(λzξ,λzη)P(λzξ,λzη)++|P(x,y)|dxdy|=|m=13n=13Pm(λzξ,λzη)exp[j2πλ(θy(xxn)+θx(yyn))]Pn(λzξ,λzη)exp[j2πλ(θy(xxn)+θx(yyn))]++|P(x,y)|dxdy|,
S.D=PSFaberration(0,0)PSF(0,0),
PSNR=10log10(2552×M×Ni=1Mj=1N|E(i,j)E0(i,j)|2),

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