Abstract

We show that phase aberrations in an imaging system can be mitigated using binary-amplitude masks that reduce destructive interference in the image spatial frequency domain. Appropriately designed masks increase the magnitude of the optical transfer function and prevent nulls. This offers a low-cost, transmission-mode alternative to phase correction as used in active and adaptive optics, without a restriction on the waveband of operation.

© 2011 Optical Society of America

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2010 (3)

2009 (2)

J. Osborn, R. M. Myers, and G. D. Love, “PSF halo reduction in adaptive optics using dynamic pupil masking,” Opt. Express 17, 17279–17292 (2009).
[CrossRef] [PubMed]

J. W. Stayman, N. Subotic, and W. Buller, “An analysis of coded aperture acquisition and reconstruction using multi-frame code sequences for relaxed optical design constraints,” Proc. SPIE 7468, 74680D (2009).
[CrossRef]

2008 (3)

S. Goodwin, B. R. Stoner, J. Carlson, and S. Rogers, “Artificial eyelid dynamic aperture optical arrays for large scale coding elements with application in the visible to MWIR,” Proc. SPIE 7096, 70960E (2008).
[CrossRef]

M. E. McNie, D. O. King, N. Price, D. J. Combes, G. W. Smith, A. G. Brown, N. T. Gordon, S. M. Stone, K. M. Brunson, K. L. Lewis, C. W. Slinger, and S. Rogers, “A large area reconfigurable MOEMS microshutter array for coded aperture imaging systems,” Proc. SPIE 7096, 70960D (2008).
[CrossRef]

C. Torti, S. Gruppetta, and L. Diaz-santana, “Wavefront curvature sensing for the human eye,” J. Modern Opt. 55, 691–702 (2008).
[CrossRef]

2007 (1)

2005 (1)

2004 (1)

A. S. Kutyrev, R. Arendt, S. H. Moseley, R. A. Boucarut, T. Hadjimichael, M. Jhabvala, T. King, M. J. Li, J. Loughlin, D. Rapchun, D. S. Schwinger, and R. F. Silverberg, “Programmable microshutter arrays for the JWST NIRSpec: optical performance,” IEEE J. Sel. Top. Quantum Electron. 10, 652–661(2004).
[CrossRef]

2001 (3)

2000 (2)

1998 (1)

1997 (1)

R. Storn and K. Price, “Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

1995 (1)

1992 (1)

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, and G. G. Yang, “Programmable binary phase-only optical device based on ferroelectric liquid crystal SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

1977 (1)

1974 (1)

Andrews, N.

Arendt, R.

A. S. Kutyrev, R. Arendt, S. H. Moseley, R. A. Boucarut, T. Hadjimichael, M. Jhabvala, T. King, M. J. Li, J. Loughlin, D. Rapchun, D. S. Schwinger, and R. F. Silverberg, “Programmable microshutter arrays for the JWST NIRSpec: optical performance,” IEEE J. Sel. Top. Quantum Electron. 10, 652–661(2004).
[CrossRef]

Artal, P.

Birch, P.

Birch, P. M.

Booth, M. J.

Boucarut, R. A.

A. S. Kutyrev, R. Arendt, S. H. Moseley, R. A. Boucarut, T. Hadjimichael, M. Jhabvala, T. King, M. J. Li, J. Loughlin, D. Rapchun, D. S. Schwinger, and R. F. Silverberg, “Programmable microshutter arrays for the JWST NIRSpec: optical performance,” IEEE J. Sel. Top. Quantum Electron. 10, 652–661(2004).
[CrossRef]

Broomfield, S. E.

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, and G. G. Yang, “Programmable binary phase-only optical device based on ferroelectric liquid crystal SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

Brown, A. G.

M. E. McNie, D. O. King, N. Price, D. J. Combes, G. W. Smith, A. G. Brown, N. T. Gordon, S. M. Stone, K. M. Brunson, K. L. Lewis, C. W. Slinger, and S. Rogers, “A large area reconfigurable MOEMS microshutter array for coded aperture imaging systems,” Proc. SPIE 7096, 70960D (2008).
[CrossRef]

Brunson, K. M.

M. E. McNie, D. O. King, N. Price, D. J. Combes, G. W. Smith, A. G. Brown, N. T. Gordon, S. M. Stone, K. M. Brunson, K. L. Lewis, C. W. Slinger, and S. Rogers, “A large area reconfigurable MOEMS microshutter array for coded aperture imaging systems,” Proc. SPIE 7096, 70960D (2008).
[CrossRef]

Buffington, A.

Buller, W.

J. W. Stayman, N. Subotic, and W. Buller, “An analysis of coded aperture acquisition and reconstruction using multi-frame code sequences for relaxed optical design constraints,” Proc. SPIE 7468, 74680D (2009).
[CrossRef]

Buscher, D.

Bustin, N.

Carlson, J.

S. Goodwin, B. R. Stoner, J. Carlson, and S. Rogers, “Artificial eyelid dynamic aperture optical arrays for large scale coding elements with application in the visible to MWIR,” Proc. SPIE 7096, 70960E (2008).
[CrossRef]

Chang, J.

K. H. Lee, J. Chang, and J. B. Yoon, “High performance microshutter device with space-division modulation,” J. Micromech. Microeng. 20, 075030 (2010).
[CrossRef]

Chen, L.

Combes, D. J.

M. E. McNie, D. O. King, N. Price, D. J. Combes, G. W. Smith, A. G. Brown, N. T. Gordon, S. M. Stone, K. M. Brunson, K. L. Lewis, C. W. Slinger, and S. Rogers, “A large area reconfigurable MOEMS microshutter array for coded aperture imaging systems,” Proc. SPIE 7096, 70960D (2008).
[CrossRef]

Cox, I. G.

Crawford, F. S.

Dainty, J.

Débarre, D.

Diaz-santana, L.

C. Torti, S. Gruppetta, and L. Diaz-santana, “Wavefront curvature sensing for the human eye,” J. Modern Opt. 55, 691–702 (2008).
[CrossRef]

Doel, P.

Dunlop, C.

Fernández, E. J.

Fletcher-Holmes, D. W.

Fomin, S. V.

A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis (Dover, 1975).

Glindemann, A.

Goodman, J. W.

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

Goodwin, S.

S. Goodwin, B. R. Stoner, J. Carlson, and S. Rogers, “Artificial eyelid dynamic aperture optical arrays for large scale coding elements with application in the visible to MWIR,” Proc. SPIE 7096, 70960E (2008).
[CrossRef]

Gordon, N. T.

M. E. McNie, D. O. King, N. Price, D. J. Combes, G. W. Smith, A. G. Brown, N. T. Gordon, S. M. Stone, K. M. Brunson, K. L. Lewis, C. W. Slinger, and S. Rogers, “A large area reconfigurable MOEMS microshutter array for coded aperture imaging systems,” Proc. SPIE 7096, 70960D (2008).
[CrossRef]

Gorman, A.

Gourlay, J.

Gruppetta, S.

C. Torti, S. Gruppetta, and L. Diaz-santana, “Wavefront curvature sensing for the human eye,” J. Modern Opt. 55, 691–702 (2008).
[CrossRef]

Guirao, A.

Hadjimichael, T.

A. S. Kutyrev, R. Arendt, S. H. Moseley, R. A. Boucarut, T. Hadjimichael, M. Jhabvala, T. King, M. J. Li, J. Loughlin, D. Rapchun, D. S. Schwinger, and R. F. Silverberg, “Programmable microshutter arrays for the JWST NIRSpec: optical performance,” IEEE J. Sel. Top. Quantum Electron. 10, 652–661(2004).
[CrossRef]

Harvey, A. R.

Hofer, H.

Iglesias, I.

Jhabvala, M.

A. S. Kutyrev, R. Arendt, S. H. Moseley, R. A. Boucarut, T. Hadjimichael, M. Jhabvala, T. King, M. J. Li, J. Loughlin, D. Rapchun, D. S. Schwinger, and R. F. Silverberg, “Programmable microshutter arrays for the JWST NIRSpec: optical performance,” IEEE J. Sel. Top. Quantum Electron. 10, 652–661(2004).
[CrossRef]

King, D. O.

M. E. McNie, D. O. King, N. Price, D. J. Combes, G. W. Smith, A. G. Brown, N. T. Gordon, S. M. Stone, K. M. Brunson, K. L. Lewis, C. W. Slinger, and S. Rogers, “A large area reconfigurable MOEMS microshutter array for coded aperture imaging systems,” Proc. SPIE 7096, 70960D (2008).
[CrossRef]

King, T.

A. S. Kutyrev, R. Arendt, S. H. Moseley, R. A. Boucarut, T. Hadjimichael, M. Jhabvala, T. King, M. J. Li, J. Loughlin, D. Rapchun, D. S. Schwinger, and R. F. Silverberg, “Programmable microshutter arrays for the JWST NIRSpec: optical performance,” IEEE J. Sel. Top. Quantum Electron. 10, 652–661(2004).
[CrossRef]

Kolmogorov, A. N.

A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis (Dover, 1975).

Kutyrev, A. S.

A. S. Kutyrev, R. Arendt, S. H. Moseley, R. A. Boucarut, T. Hadjimichael, M. Jhabvala, T. King, M. J. Li, J. Loughlin, D. Rapchun, D. S. Schwinger, and R. F. Silverberg, “Programmable microshutter arrays for the JWST NIRSpec: optical performance,” IEEE J. Sel. Top. Quantum Electron. 10, 652–661(2004).
[CrossRef]

Lee, K. H.

K. H. Lee, J. Chang, and J. B. Yoon, “High performance microshutter device with space-division modulation,” J. Micromech. Microeng. 20, 075030 (2010).
[CrossRef]

Lewis, K. L.

M. E. McNie, D. O. King, N. Price, D. J. Combes, G. W. Smith, A. G. Brown, N. T. Gordon, S. M. Stone, K. M. Brunson, K. L. Lewis, C. W. Slinger, and S. Rogers, “A large area reconfigurable MOEMS microshutter array for coded aperture imaging systems,” Proc. SPIE 7096, 70960D (2008).
[CrossRef]

Li, M. J.

A. S. Kutyrev, R. Arendt, S. H. Moseley, R. A. Boucarut, T. Hadjimichael, M. Jhabvala, T. King, M. J. Li, J. Loughlin, D. Rapchun, D. S. Schwinger, and R. F. Silverberg, “Programmable microshutter arrays for the JWST NIRSpec: optical performance,” IEEE J. Sel. Top. Quantum Electron. 10, 652–661(2004).
[CrossRef]

Loughlin, J.

A. S. Kutyrev, R. Arendt, S. H. Moseley, R. A. Boucarut, T. Hadjimichael, M. Jhabvala, T. King, M. J. Li, J. Loughlin, D. Rapchun, D. S. Schwinger, and R. F. Silverberg, “Programmable microshutter arrays for the JWST NIRSpec: optical performance,” IEEE J. Sel. Top. Quantum Electron. 10, 652–661(2004).
[CrossRef]

Love, G. D.

Major, J.

McNie, M. E.

M. E. McNie, D. O. King, N. Price, D. J. Combes, G. W. Smith, A. G. Brown, N. T. Gordon, S. M. Stone, K. M. Brunson, K. L. Lewis, C. W. Slinger, and S. Rogers, “A large area reconfigurable MOEMS microshutter array for coded aperture imaging systems,” Proc. SPIE 7096, 70960D (2008).
[CrossRef]

Moseley, S. H.

A. S. Kutyrev, R. Arendt, S. H. Moseley, R. A. Boucarut, T. Hadjimichael, M. Jhabvala, T. King, M. J. Li, J. Loughlin, D. Rapchun, D. S. Schwinger, and R. F. Silverberg, “Programmable microshutter arrays for the JWST NIRSpec: optical performance,” IEEE J. Sel. Top. Quantum Electron. 10, 652–661(2004).
[CrossRef]

Muller, R. A.

Munro, I.

Muyo, G.

Myers, R.

Myers, R. M.

Neil, M. A. A.

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, and G. G. Yang, “Programmable binary phase-only optical device based on ferroelectric liquid crystal SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

Osborn, J.

Paige, E. G. S.

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, and G. G. Yang, “Programmable binary phase-only optical device based on ferroelectric liquid crystal SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

Paterson, C.

Porter, J.

Price, K.

R. Storn and K. Price, “Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

Price, N.

M. E. McNie, D. O. King, N. Price, D. J. Combes, G. W. Smith, A. G. Brown, N. T. Gordon, S. M. Stone, K. M. Brunson, K. L. Lewis, C. W. Slinger, and S. Rogers, “A large area reconfigurable MOEMS microshutter array for coded aperture imaging systems,” Proc. SPIE 7096, 70960D (2008).
[CrossRef]

Purvis, A.

Rapchun, D.

A. S. Kutyrev, R. Arendt, S. H. Moseley, R. A. Boucarut, T. Hadjimichael, M. Jhabvala, T. King, M. J. Li, J. Loughlin, D. Rapchun, D. S. Schwinger, and R. F. Silverberg, “Programmable microshutter arrays for the JWST NIRSpec: optical performance,” IEEE J. Sel. Top. Quantum Electron. 10, 652–661(2004).
[CrossRef]

Redondo, M.

Restaino, S. R.

Rogers, S.

M. E. McNie, D. O. King, N. Price, D. J. Combes, G. W. Smith, A. G. Brown, N. T. Gordon, S. M. Stone, K. M. Brunson, K. L. Lewis, C. W. Slinger, and S. Rogers, “A large area reconfigurable MOEMS microshutter array for coded aperture imaging systems,” Proc. SPIE 7096, 70960D (2008).
[CrossRef]

S. Goodwin, B. R. Stoner, J. Carlson, and S. Rogers, “Artificial eyelid dynamic aperture optical arrays for large scale coding elements with application in the visible to MWIR,” Proc. SPIE 7096, 70960E (2008).
[CrossRef]

Schwemin, A. J.

Schwinger, D. S.

A. S. Kutyrev, R. Arendt, S. H. Moseley, R. A. Boucarut, T. Hadjimichael, M. Jhabvala, T. King, M. J. Li, J. Loughlin, D. Rapchun, D. S. Schwinger, and R. F. Silverberg, “Programmable microshutter arrays for the JWST NIRSpec: optical performance,” IEEE J. Sel. Top. Quantum Electron. 10, 652–661(2004).
[CrossRef]

Sharples, R.

Silverberg, R. F.

A. S. Kutyrev, R. Arendt, S. H. Moseley, R. A. Boucarut, T. Hadjimichael, M. Jhabvala, T. King, M. J. Li, J. Loughlin, D. Rapchun, D. S. Schwinger, and R. F. Silverberg, “Programmable microshutter arrays for the JWST NIRSpec: optical performance,” IEEE J. Sel. Top. Quantum Electron. 10, 652–661(2004).
[CrossRef]

Singer, B.

Slinger, C. W.

M. E. McNie, D. O. King, N. Price, D. J. Combes, G. W. Smith, A. G. Brown, N. T. Gordon, S. M. Stone, K. M. Brunson, K. L. Lewis, C. W. Slinger, and S. Rogers, “A large area reconfigurable MOEMS microshutter array for coded aperture imaging systems,” Proc. SPIE 7096, 70960D (2008).
[CrossRef]

Smith, G. W.

M. E. McNie, D. O. King, N. Price, D. J. Combes, G. W. Smith, A. G. Brown, N. T. Gordon, S. M. Stone, K. M. Brunson, K. L. Lewis, C. W. Slinger, and S. Rogers, “A large area reconfigurable MOEMS microshutter array for coded aperture imaging systems,” Proc. SPIE 7096, 70960D (2008).
[CrossRef]

Smits, R. G.

Stayman, J. W.

J. W. Stayman, N. Subotic, and W. Buller, “An analysis of coded aperture acquisition and reconstruction using multi-frame code sequences for relaxed optical design constraints,” Proc. SPIE 7468, 74680D (2009).
[CrossRef]

Stone, S. M.

M. E. McNie, D. O. King, N. Price, D. J. Combes, G. W. Smith, A. G. Brown, N. T. Gordon, S. M. Stone, K. M. Brunson, K. L. Lewis, C. W. Slinger, and S. Rogers, “A large area reconfigurable MOEMS microshutter array for coded aperture imaging systems,” Proc. SPIE 7096, 70960D (2008).
[CrossRef]

Stoner, B. R.

S. Goodwin, B. R. Stoner, J. Carlson, and S. Rogers, “Artificial eyelid dynamic aperture optical arrays for large scale coding elements with application in the visible to MWIR,” Proc. SPIE 7096, 70960E (2008).
[CrossRef]

Storn, R.

R. Storn and K. Price, “Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

Subotic, N.

J. W. Stayman, N. Subotic, and W. Buller, “An analysis of coded aperture acquisition and reconstruction using multi-frame code sequences for relaxed optical design constraints,” Proc. SPIE 7468, 74680D (2009).
[CrossRef]

Torti, C.

C. Torti, S. Gruppetta, and L. Diaz-santana, “Wavefront curvature sensing for the human eye,” J. Modern Opt. 55, 691–702 (2008).
[CrossRef]

Vettenburg, T.

Vick, A.

Williams, D. R.

Wilson, T.

Yamauchi, Y.

Yang, G. G.

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, and G. G. Yang, “Programmable binary phase-only optical device based on ferroelectric liquid crystal SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

Yoon, G.-Y.

Yoon, J. B.

K. H. Lee, J. Chang, and J. B. Yoon, “High performance microshutter device with space-division modulation,” J. Micromech. Microeng. 20, 075030 (2010).
[CrossRef]

Zadrozny, A.

Appl. Opt. (2)

Electron. Lett. (1)

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, and G. G. Yang, “Programmable binary phase-only optical device based on ferroelectric liquid crystal SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. S. Kutyrev, R. Arendt, S. H. Moseley, R. A. Boucarut, T. Hadjimichael, M. Jhabvala, T. King, M. J. Li, J. Loughlin, D. Rapchun, D. S. Schwinger, and R. F. Silverberg, “Programmable microshutter arrays for the JWST NIRSpec: optical performance,” IEEE J. Sel. Top. Quantum Electron. 10, 652–661(2004).
[CrossRef]

J. Global Optim. (1)

R. Storn and K. Price, “Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

J. Micromech. Microeng. (1)

K. H. Lee, J. Chang, and J. B. Yoon, “High performance microshutter device with space-division modulation,” J. Micromech. Microeng. 20, 075030 (2010).
[CrossRef]

J. Modern Opt. (1)

C. Torti, S. Gruppetta, and L. Diaz-santana, “Wavefront curvature sensing for the human eye,” J. Modern Opt. 55, 691–702 (2008).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Express (6)

Opt. Lett. (2)

Proc. SPIE (3)

M. E. McNie, D. O. King, N. Price, D. J. Combes, G. W. Smith, A. G. Brown, N. T. Gordon, S. M. Stone, K. M. Brunson, K. L. Lewis, C. W. Slinger, and S. Rogers, “A large area reconfigurable MOEMS microshutter array for coded aperture imaging systems,” Proc. SPIE 7096, 70960D (2008).
[CrossRef]

J. W. Stayman, N. Subotic, and W. Buller, “An analysis of coded aperture acquisition and reconstruction using multi-frame code sequences for relaxed optical design constraints,” Proc. SPIE 7468, 74680D (2009).
[CrossRef]

S. Goodwin, B. R. Stoner, J. Carlson, and S. Rogers, “Artificial eyelid dynamic aperture optical arrays for large scale coding elements with application in the visible to MWIR,” Proc. SPIE 7096, 70960E (2008).
[CrossRef]

Other (2)

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

A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis (Dover, 1975).

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Figures (4)

Fig. 1
Fig. 1

Decompositions of the OTF integral in the complex plane at spatial frequency ν = 1 / 2 . The black and dashed red curves depict, respectively, constructive and destructive contributions d u to the net OTF (blue phasor), for defocus of a circular pupil for (a) W 20 = λ / 4 , (b) W 20 = 0.642 λ (for which a null occurs), and (c) with a putative mask blocking the destructive interference components yielding a positive OTF.

Fig. 2
Fig. 2

Typical cost function shown for astigmatism as a function of tip-tilt ( t x , t y ) , and minimized for phase reference, φ 0 . Bright yellow and white regions indicate a high cost for large tip or tilt. The encircled dark red spots in the center indicate four modest tip-tilts that lead to equivalently performing masks, one of which is shown in Fig. 3a.

Fig. 3
Fig. 3

Aberration compensation of (a)–(c) astigmatism, (d)–(f) coma, and (g)–(i) an aberration representative for the human eye, of, respectively a root-mean-square optical path difference of 2 λ , 2 λ , and 3.5 λ . The aberration phases and contour masks with the relative transmission noted below are shown in (a), (d), and (g). The tangential (black), sagittal (blue), and diagonal (red) MTF without mask in (b), (e), and (h), and with mask in (c), (f), and (i).

Fig. 4
Fig. 4

(a) Off-axis aberration at λ 0 = 550 nm of a cemented doublet with contour mask optimized for panchromatic operation for 495 nm λ 605 nm ( Δ λ / λ 0 = 20 % ). (b) The tangential (black), sagittal (blue), and diagonal MTF (red) before and (c) after the introduction of the contour mask.

Equations (14)

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OTF ( ν ) = 1 supp ( P ) P * ( u ν ) P ( u + ν ) d u ,
OTF ( ν ) = 1 supp ( P ) Ω ( ν ) exp ( i Δ ( u , ν ) ) d u ,
OTF i ( ν ) = 1 supp ( P ) 1 T π 2 π 2 exp ( i Δ ) μ ( ν , Δ ) d Δ 1 supp ( P ) 1 T A Ω ( ν ) 2 π π 2 π 2 exp ( i Δ ) d Δ ,
MTF p ( ν ) 2 π MTF DL ( ν ) 0.64 MTF DL ( ν ) .
M ( u ) = 1 , u | ( l ( u ) φ 0 t · u + Δ φ 2 ) mod 2 π Δ φ ,
OTF ( ν ) MTF DL ( ν ) 4 π 2 π π exp ( i φ 1 ) π π exp ( i φ 2 ) d φ 2 d φ 1 ,
OTF c ( ν ) 2 π Δ φ MTF DL ( ν ) π 2 sin 2 ( Δ φ 2 ) 2 π 2 MTF DL ( ν ) .
OTF ( ν ) = 1 supp ( P ) Ω ( ν ) exp ( i Δ ( u , ν ) ) d u = 1 supp ( P ) exp ( i Δ ^ ) d μ ( Δ ^ ) ,
exp ( i Δ ^ ) d μ ( Δ ^ ) = π π exp ( i Δ ^ ) k = k min k max d μ ( Δ ^ + 2 π k ) ,
k = c · k min c · k max d μ c ( Δ ^ + 2 π k ) = k = c · k min c · k max d μ ( Δ ^ / c + 2 π k / c ) 1 / c = k = 2 π k min / δ 2 π k max / δ d μ ( Δ ^ δ / 2 π + k δ ) δ / 2 π ,
lim c k = c · k min c · k max d μ c ( Δ ^ + 2 π k ) = 1 2 π + d μ ( Δ ^ ) d Δ ^ = A Ω ( ν ) 2 π ,
OTF ( ν ) 1 supp ( P ) π π d φ 1 2 π Ω ( ν ) exp ( i φ 1 ) exp ( i l ( u + ν ) ) d u ,
OTF ( ν ) 1 supp ( P ) 1 2 π π π exp ( i φ 1 ) Ω ( ν ) exp ( i l ( u + ν ) ) d u d φ 1 .
OTF ( ν ) 1 supp ( P ) 1 2 π π π exp ( i φ 1 ) A Ω ( ν ) 2 π π + π exp ( i φ 2 ) d φ 2 d φ 1 ,

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