Abstract

The ability to retrieve the complex-valued, generalized pupil function of an imaging system from undersampled measurements of the defocused system point spread function (PSF) is examined through numerical simulations. The ability to do so degrades as the detector pixel pitch increases when using a fixed number of PSF measurements. Two strategies for obtaining better results with undersampled data are demonstrated using additional PSF measurements with (i) random shifts due to system pointing fluctuations and (ii) intermediate amounts of defocus.

© 2009 Optical Society of America

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References

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2009 (2)

2006 (2)

2004 (1)

J. J. Green, B. H. Dean, C. M. Ohara, D. C. Redding, and Y. Zhang, “Target selection and imaging requirements for JWST fine phasing,” Proc. SPIE 5487, 944-953 (2004).
[CrossRef]

2003 (1)

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

2002 (1)

1999 (2)

J. R. Fienup, “Phase retrieval for undersampled broadband images,” J. Opt. Soc. Am. A 16, 1831-1837 (1999).
[CrossRef]

R. D. Fiete, “Image quality and λFN/p for remote sensing systems,” Opt. Eng. 38, 1229-1240 (1999).
[CrossRef]

1997 (1)

1993 (1)

1977 (1)

A. Papoulis, “Generalized sampling expansion,” IEEE Trans. Circuits Syst. CAS-24, 652-654 (1977).
[CrossRef]

1928 (1)

H. Nyquist, “Certain topics in telegraph transmission theory,” Trans. Am. Inst. Electr. Eng. 47, 617-644 (1928).
[CrossRef]

1915 (1)

E. T. Whittaker, “On the functions which are represented by the expansions of the interpolation theory,” Proc. - R. Soc. Edinburgh, Sect. A: Math. 35, 181-194 (1915).

Almoro, P.

Brady, G. R.

Dean, B. H.

J. J. Green, B. H. Dean, C. M. Ohara, D. C. Redding, and Y. Zhang, “Target selection and imaging requirements for JWST fine phasing,” Proc. SPIE 5487, 944-953 (2004).
[CrossRef]

DeRosa, R. T.

Fienup, J. R.

Fiete, R. D.

R. D. Fiete, “Image quality and λFN/p for remote sensing systems,” Opt. Eng. 38, 1229-1240 (1999).
[CrossRef]

Georges, J.

Goodman, J. W.

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

Green, J. J.

J. J. Green, B. H. Dean, C. M. Ohara, D. C. Redding, and Y. Zhang, “Target selection and imaging requirements for JWST fine phasing,” Proc. SPIE 5487, 944-953 (2004).
[CrossRef]

Hege, E. K.

Jefferies, S. M.

Kang, M. G.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

Lloyd-Hart, M.

Nyquist, H.

H. Nyquist, “Certain topics in telegraph transmission theory,” Trans. Am. Inst. Electr. Eng. 47, 617-644 (1928).
[CrossRef]

Ohara, C. M.

J. J. Green, B. H. Dean, C. M. Ohara, D. C. Redding, and Y. Zhang, “Target selection and imaging requirements for JWST fine phasing,” Proc. SPIE 5487, 944-953 (2004).
[CrossRef]

Osten, W.

Papoulis, A.

A. Papoulis, “Generalized sampling expansion,” IEEE Trans. Circuits Syst. CAS-24, 652-654 (1977).
[CrossRef]

Park, M. K.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

Park, S. C.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

Pedrini, G.

Redding, D. C.

J. J. Green, B. H. Dean, C. M. Ohara, D. C. Redding, and Y. Zhang, “Target selection and imaging requirements for JWST fine phasing,” Proc. SPIE 5487, 944-953 (2004).
[CrossRef]

Thurman, S. T.

Whittaker, E. T.

E. T. Whittaker, “On the functions which are represented by the expansions of the interpolation theory,” Proc. - R. Soc. Edinburgh, Sect. A: Math. 35, 181-194 (1915).

Zhang, Y.

J. J. Green, B. H. Dean, C. M. Ohara, D. C. Redding, and Y. Zhang, “Target selection and imaging requirements for JWST fine phasing,” Proc. SPIE 5487, 944-953 (2004).
[CrossRef]

Appl. Opt. (4)

IEEE Signal Process. Mag. (1)

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

A. Papoulis, “Generalized sampling expansion,” IEEE Trans. Circuits Syst. CAS-24, 652-654 (1977).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

R. D. Fiete, “Image quality and λFN/p for remote sensing systems,” Opt. Eng. 38, 1229-1240 (1999).
[CrossRef]

Opt. Express (1)

Proc. - R. Soc. Edinburgh, Sect. A: Math. (1)

E. T. Whittaker, “On the functions which are represented by the expansions of the interpolation theory,” Proc. - R. Soc. Edinburgh, Sect. A: Math. 35, 181-194 (1915).

Proc. SPIE (1)

J. J. Green, B. H. Dean, C. M. Ohara, D. C. Redding, and Y. Zhang, “Target selection and imaging requirements for JWST fine phasing,” Proc. SPIE 5487, 944-953 (2004).
[CrossRef]

Trans. Am. Inst. Electr. Eng. (1)

H. Nyquist, “Certain topics in telegraph transmission theory,” Trans. Am. Inst. Electr. Eng. 47, 617-644 (1928).
[CrossRef]

Other (2)

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

http://www.jwst.nasa.gov/

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Figures (9)

Fig. 1
Fig. 1

(a) Pupil function amplitude and (b) wavefront aberration function (in units of wavelengths) for the segmented-aperture telescope in the simulations.

Fig. 2
Fig. 2

(a) Pupil mask used in conjunction with the segmented aperture of Fig. 1, (b) overlay of (gray curve) the segmented aperture and (black curve) the misaligned pupil mask, and the net system (c) pupil function amplitude and (b) wavefront aberration function.

Fig. 3
Fig. 3

Simulated PSF measurements for d k = 1 λ of defocus and (a) Δ x = 20.8 μ m , (b) Δ x = 41.6 μ m , and (c) Δ x = 69.4 μ m . The corresponding detector sampling ratios for (a), (b), and (c) are Q = 2.0 , 1.0, and 0.6.

Fig. 4
Fig. 4

Initial pupil function estimate used for the retrieval algorithm.

Fig. 5
Fig. 5

Complex pupil retrieval results for the baseline scenario using three PSF measurements versus the detector sampling ratio after (open circles) 2,000 iterations and (x’s) 10,000 iterations. Phase retrieval results are also shown for comparison.

Fig. 6
Fig. 6

Retrieved (left column) pupil amplitude and (right column) wavefront aberration functions for (top row) Q = 2.0 , (middle row) Q = 1.0 , and (bottom row) Q = 0.6 for the baseline scenario using three PSF measurements.

Fig. 7
Fig. 7

Same as Fig. 5 except for the scenario using 12 PSF measurements (4 measurements with random pointing fluctuations at each of 3 defocus settings).

Fig. 8
Fig. 8

Retrieved (right column) pupil amplitude and (left column) wavefront aberration functions for (top row) Q = 2.0 , (middle row) Q = 1.0 , and (bottom row) Q = 0.6 using 12 PSF measurements (4 measurements with random pointing fluctuations at each of 3 defocus settings).

Fig. 9
Fig. 9

Same as Fig. 5 except for scenario using 12 PSF measurements at intermediate defocus values with no misregistrations.

Tables (1)

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Table 1 Numerical Simulation Parameters

Equations (18)

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s ( x , y , d ) = | P ( ξ , η ) exp [ i 2 π λ f ( ξ x + η y ) ] exp [ i 8 π d D 2 ( ξ 2 + η 2 ) ] d ξ d η | 2 ,
g m , n , k = ζ m , n , k + s ( x x k , y y k , d k ) h d ( n Δ x x , m Δ y y ) d x d y ,
G p , q , k = 1 N m = N 2 ( N 2 ) 2 n = N 2 ( N 2 ) 2 g m , n , k exp [ i 2 π N ( m p + n q ) ] ,
G p , q , k = Z p , q , k + N α = β = S ( u , v , d k ) exp [ i 2 π N ( x ̂ k u + y ̂ k v ) ] H d ( u , v ) sinc ( p Δ u α N Δ u u Δ u ) sinc ( q Δ u β N Δ u v Δ u ) d u d v ,
G p , q , k Z p , q , k + 1 N Δ x 2 α = β = S [ ( p α N ) Δ u , ( q β N ) Δ u , d k ] exp { i 2 π Δ u N [ x ̂ k ( p α N ) + y ̂ k ( q β N ) ] } H d [ ( p α N ) Δ u , ( q β N ) Δ u ] .
G p , q , k Z p , q , k + 1 N Δ x 2 S ( p Δ u , q Δ u , d k ) exp [ i 2 π Δ u N ( x ̂ k p + y ̂ k q ) ] H d ( p Δ u , q Δ u ) .
s ( m f Δ x f , n f Δ x f , d k ) = | 1 N p f = N 2 ( N 2 ) 2 q f = N 2 ( N 2 ) 2 P ̂ ( λ f p f Δ u , λ f q f Δ u ) × exp [ i 2 π N f ( p f m f + q f n f ) ] exp [ i 8 π d k Δ u 2 D 2 ( p f 2 + q f 2 ) ] | 2 ,
g ̂ m , n , k = 1 N p = N 2 ( N 2 ) 2 q = N 2 ( N 2 ) 2 G ̂ p , q , k exp [ i 2 π N ( m p + n q ) ] ,
G ̂ p , q , k = α = β = S ̂ [ ( p α N ) Δ u , ( q β N ) Δ u , d k ] × exp { i 2 π Δ u N [ x ̂ k ( p α N ) + y ̂ k ( q β N ) ] } × H d [ ( p α N ) Δ u , ( q β N ) Δ u ] ,
Φ = 1 1 K k = 1 K [ m = N 2 ( N 2 ) 2 n = N 2 ( N 2 ) 2 w m , n , k g m , n , k g ̂ m , n , k ] 2 [ m = N 2 ( N 2 ) 2 n = N 2 ( N 2 ) 2 w m , n , k g m , n , k 2 ] [ m = N 2 ( N 2 ) 2 n = N 2 ( N 2 ) 2 w m , n , k g ̂ m , n , k 2 ] ,
Φ g ̂ m , n , k = 2 K ( m , n ) w m , n , k g m , n , k g ̂ m , n , k [ ( m , n ) w m , n , k g m , n , k 2 ] [ ( m , n ) w m , n , k g ̂ m , n , k 2 ] w m , n , k [ g ̂ m , n , k ( m , n ) w m , n , k g m , n , k g ̂ m , n , k ( m , n ) w m , n , k g ̂ m , n , k 2 g m , n , k ] ,
G ̂ p , q , k = Φ Re ( G ̂ p , q , k ) + i Φ Im ( G ̂ p , q , k ) = 1 N m = N 2 ( N 2 ) 2 n = N 2 ( N 2 ) 2 Φ g ̂ m , n , k exp [ i 2 π N ( m p + n q ) ] .
Φ x ̂ k = ( p f , q f ) [ Φ Re ( G ̂ p f , q f , k ) Re ( G ̂ p f , q f , k ) x ̂ k + Φ Im ( G ̂ p f , q f , k ) Im ( G ̂ p f , q f , k ) x ̂ k ] = 2 π Δ u N ( p f , q f , α , β ) ( p f α N ) Im { G ̂ p f , q f , k S ̂ * [ ( p f α N ) Δ u , ( q f β N ) Δ u , d k ] × exp { i 2 π Δ u N [ x ̂ k ( p f α N ) + y ̂ k ( q f β N ) ] } H d * [ ( p f α N ) Δ u , ( q f β N ) Δ u ] } .
Φ y ̂ k = 2 π Δ u N ( p f , q f , α , β ) ( q f β N ) Im ( G ̂ p f , q f , k S ̂ * [ ( p f α N ) Δ u , ( q f β N ) Δ u , d k ] exp { i 2 π Δ u N [ x ̂ k ( p f α N ) + y ̂ k ( q f β N ) ] } × H d * [ ( p f α N ) Δ u , ( q f β N ) Δ u ] ) .
S ̂ ( p f Δ u , q f Δ u , d k ) = Φ Re [ S ̂ ( p f Δ u , q f Δ u , d k ) ] + i Φ Im [ S ̂ ( p f Δ u , q f Δ u , d k ) ] = G ̂ p f , q f , k exp { i 2 π Δ u N [ x ̂ k ( p α N ) + y ̂ k ( q β N ) ] } × H d * [ ( p α N ) Δ u , ( q β N ) Δ u ] .
W ( ξ , η ) = cos 2 [ 0.5 π ( ρ 0.5 D ) ( 0.025 D ) ] ,
E 2 = min ( c , φ 0 , t ξ , t η , ξ 0 , η 0 ) { ( ξ , η ) | c exp [ i φ 0 + i t ξ ( ξ ξ 0 ) + i t η ( η η 0 ) ] P ̂ ( ξ ξ 0 , η η 0 ) P ( ξ , η ) | 2 ( ξ , η ) | P ( ξ , η ) | 2 } ,
σ 2 = 1 N χ ( ξ , η ) χ arg 2 { exp [ i φ 0 + i t ξ ξ + i t η η ] P ̂ ( ξ ξ 0 , η η 0 ) P ( ξ , η ) } ,

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