Abstract

The generalized phase contrast method (GPC) is a versatile tool for efficiently rerouting photon energy into desired spatial distributions. We demonstrate that GPC-based patterned projection shows robustness to shift in wavelength and can maintain the projection length scale and high efficiency over a range [0.75λ 0;1.5λ 0] with λ 0 as the characteristic design wavelength. The GPC has the capacity to combine multiple wavelengths along the same optical path and to efficiently redirect them into desired distributions such as for array illumination, beam shaping, or grayscale image projection. This opens the possibility for creatively incorporating various multi-wavelength approaches into patterned illumination that can enable new broad-band optical applications.

©2008 Optical Society of America

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2007 (4)

S. E. Chung, W. Park, H. Park, K. Yu, N. Park, and S. Kwon, “Optofluidic maskless lithography system for real-time synthesis of photopolymerized microstructures in microfluidic channels,” Appl. Phys. Lett. 91, 041106 (2007).
[Crossref]

C. A. Alonzo, P. J. Rodrigo, and J. Gluckstad, “Photon-efficient grey-level image projection by the generalized phase contrast method,” New J. Phys. 9, 132 (2007).
[Crossref]

D. Palima, C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Generalized phase contrast matched to Gaussian illumination,” Opt. Express 15, 11971–11977 (2007).
[Crossref] [PubMed]

J. Glückstad, D. Palima, P. J. Rodrigo, and C. A. Alonzo, “Laser projection using generalized phase contrast,” Opt. Lett. 32, 3281–3283 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (3)

2004 (3)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

V. R. Daria, P. J. Rodrigo, and J. Gluckstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. 84, 323–325 (2004).
[Crossref]

P. J. Rodrigo, V.R. Daria, and J. Glückstad, “Real-time three-dimensional optical micromanipulation of multiple particles and living cells,” Opt. Lett. 292270–2272 (2004).
[Crossref] [PubMed]

2003 (2)

N. Kitamura and F. Kitagawa, “Optical trapping - chemical analysis of single microparticles in solution,” J. Photochem. Photobiol. C 4, 227–247 (2003).
[Crossref]

S. Shoji, H. B. Sun, and S. Kawata, “Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference,” Appl. Phys. Lett. 83, 608–610 (2003).
[Crossref]

2002 (3)

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt. 7, 60–71 (2002).
[Crossref] [PubMed]

Y. C. Chen, N. R. Raravikar, L. S. Schadler, P. M. Ajayan, Y. P. Zhao, T. M. Lu, G. C. Wang, and X. C. Zhang, “Ultrafast optical switching properties of single-wall carbon nanotube polymer composites at 1.55 µm,” Appl. Phys. Lett. 81, 975–977 (2002).
[Crossref]

E. L. Heffer and S. Fantini, “Quantitative oximetry of breast tumors: a near-infrared method that identifies two optimal wavelengths for each tumor,” Appl. Opt. 41, 3827–3839 (2002).
[Crossref] [PubMed]

2001 (1)

2000 (1)

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[Crossref]

1999 (1)

S. Singh-Gasson, R. D. Green, Y. J. Yue, C. Nelson, F. Blattner, M. R. Sussman, and F. Cerrina, “Maskless fabrication of light-directed oligonucleotide microarrays using a digital micromirror array,” Nat. Biotechnol. 17, 974–978 (1999).
[Crossref] [PubMed]

1997 (2)

1996 (1)

J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. 130, 225–230 (1996).
[Crossref]

1995 (1)

J. Glückstad, “Adaptive array illumination and structured light generated by spatial zero-order self-phase modulation in a Kerr medium,” Opt. Commun. 120, 194–203 (1995).
[Crossref]

1990 (2)

1986 (1)

1985 (1)

1965 (1)

Ajayan, P. M.

Y. C. Chen, N. R. Raravikar, L. S. Schadler, P. M. Ajayan, Y. P. Zhao, T. M. Lu, G. C. Wang, and X. C. Zhang, “Ultrafast optical switching properties of single-wall carbon nanotube polymer composites at 1.55 µm,” Appl. Phys. Lett. 81, 975–977 (2002).
[Crossref]

Alonzo, C. A.

Berger, A. J.

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt. 7, 60–71 (2002).
[Crossref] [PubMed]

Bevilacqua, F.

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt. 7, 60–71 (2002).
[Crossref] [PubMed]

Blattner, F.

S. Singh-Gasson, R. D. Green, Y. J. Yue, C. Nelson, F. Blattner, M. R. Sussman, and F. Cerrina, “Maskless fabrication of light-directed oligonucleotide microarrays using a digital micromirror array,” Nat. Biotechnol. 17, 974–978 (1999).
[Crossref] [PubMed]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

Brown, G. M.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[Crossref]

Butler, J.

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt. 7, 60–71 (2002).
[Crossref] [PubMed]

Cerrina, F.

S. Singh-Gasson, R. D. Green, Y. J. Yue, C. Nelson, F. Blattner, M. R. Sussman, and F. Cerrina, “Maskless fabrication of light-directed oligonucleotide microarrays using a digital micromirror array,” Nat. Biotechnol. 17, 974–978 (1999).
[Crossref] [PubMed]

Cerussi, A. E.

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt. 7, 60–71 (2002).
[Crossref] [PubMed]

Chen, F.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[Crossref]

Chen, Y. C.

Y. C. Chen, N. R. Raravikar, L. S. Schadler, P. M. Ajayan, Y. P. Zhao, T. M. Lu, G. C. Wang, and X. C. Zhang, “Ultrafast optical switching properties of single-wall carbon nanotube polymer composites at 1.55 µm,” Appl. Phys. Lett. 81, 975–977 (2002).
[Crossref]

Chen, Z. Z.

Cheng, Y. Y.

Chung, S. E.

S. E. Chung, W. Park, H. Park, K. Yu, N. Park, and S. Kwon, “Optofluidic maskless lithography system for real-time synthesis of photopolymerized microstructures in microfluidic channels,” Appl. Phys. Lett. 91, 041106 (2007).
[Crossref]

Daria, V. R.

Daria, V.R.

Deng, X. M.

Fantini, S.

Frieden, B. R.

Gluckstad, J.

C. A. Alonzo, P. J. Rodrigo, and J. Gluckstad, “Photon-efficient grey-level image projection by the generalized phase contrast method,” New J. Phys. 9, 132 (2007).
[Crossref]

P. J. Rodrigo, V. R. Daria, and J. Gluckstad, “Dynamically reconfigurable optical lattices,” Opt. Express 13, 1384–1394 (2005).
[Crossref] [PubMed]

V. R. Daria, P. J. Rodrigo, and J. Gluckstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. 84, 323–325 (2004).
[Crossref]

Glückstad, J.

Green, R. D.

S. Singh-Gasson, R. D. Green, Y. J. Yue, C. Nelson, F. Blattner, M. R. Sussman, and F. Cerrina, “Maskless fabrication of light-directed oligonucleotide microarrays using a digital micromirror array,” Nat. Biotechnol. 17, 974–978 (1999).
[Crossref] [PubMed]

Grier, D. G.

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. USA 102, 13081–13086 (2005).
[Crossref] [PubMed]

Hara, T.

Heffer, E. L.

Holcombe, R. F.

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt. 7, 60–71 (2002).
[Crossref] [PubMed]

Hsiang, D.

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt. 7, 60–71 (2002).
[Crossref] [PubMed]

Jakubowski, D.

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt. 7, 60–71 (2002).
[Crossref] [PubMed]

Juskaitis, R.

Kawata, S.

S. Shoji, H. B. Sun, and S. Kawata, “Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference,” Appl. Phys. Lett. 83, 608–610 (2003).
[Crossref]

Kitagawa, F.

N. Kitamura and F. Kitagawa, “Optical trapping - chemical analysis of single microparticles in solution,” J. Photochem. Photobiol. C 4, 227–247 (2003).
[Crossref]

Kitamura, N.

N. Kitamura and F. Kitagawa, “Optical trapping - chemical analysis of single microparticles in solution,” J. Photochem. Photobiol. C 4, 227–247 (2003).
[Crossref]

Kwon, S.

S. E. Chung, W. Park, H. Park, K. Yu, N. Park, and S. Kwon, “Optofluidic maskless lithography system for real-time synthesis of photopolymerized microstructures in microfluidic channels,” Appl. Phys. Lett. 91, 041106 (2007).
[Crossref]

Lading, L.

Lanning, R.

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt. 7, 60–71 (2002).
[Crossref] [PubMed]

Lee, S. H.

Liang, X. C.

Lohmann, A. W.

Lu, T. M.

Y. C. Chen, N. R. Raravikar, L. S. Schadler, P. M. Ajayan, Y. P. Zhao, T. M. Lu, G. C. Wang, and X. C. Zhang, “Ultrafast optical switching properties of single-wall carbon nanotube polymer composites at 1.55 µm,” Appl. Phys. Lett. 81, 975–977 (2002).
[Crossref]

Ma, R. Y.

Mogensen, P.C.

Neil, M. A. A.

Nelson, C.

S. Singh-Gasson, R. D. Green, Y. J. Yue, C. Nelson, F. Blattner, M. R. Sussman, and F. Cerrina, “Maskless fabrication of light-directed oligonucleotide microarrays using a digital micromirror array,” Nat. Biotechnol. 17, 974–978 (1999).
[Crossref] [PubMed]

Palima, D.

Park, H.

S. E. Chung, W. Park, H. Park, K. Yu, N. Park, and S. Kwon, “Optofluidic maskless lithography system for real-time synthesis of photopolymerized microstructures in microfluidic channels,” Appl. Phys. Lett. 91, 041106 (2007).
[Crossref]

Park, N.

S. E. Chung, W. Park, H. Park, K. Yu, N. Park, and S. Kwon, “Optofluidic maskless lithography system for real-time synthesis of photopolymerized microstructures in microfluidic channels,” Appl. Phys. Lett. 91, 041106 (2007).
[Crossref]

Park, W.

S. E. Chung, W. Park, H. Park, K. Yu, N. Park, and S. Kwon, “Optofluidic maskless lithography system for real-time synthesis of photopolymerized microstructures in microfluidic channels,” Appl. Phys. Lett. 91, 041106 (2007).
[Crossref]

Raravikar, N. R.

Y. C. Chen, N. R. Raravikar, L. S. Schadler, P. M. Ajayan, Y. P. Zhao, T. M. Lu, G. C. Wang, and X. C. Zhang, “Ultrafast optical switching properties of single-wall carbon nanotube polymer composites at 1.55 µm,” Appl. Phys. Lett. 81, 975–977 (2002).
[Crossref]

Rodrigo, P. J.

Schadler, L. S.

Y. C. Chen, N. R. Raravikar, L. S. Schadler, P. M. Ajayan, Y. P. Zhao, T. M. Lu, G. C. Wang, and X. C. Zhang, “Ultrafast optical switching properties of single-wall carbon nanotube polymer composites at 1.55 µm,” Appl. Phys. Lett. 81, 975–977 (2002).
[Crossref]

Shah, N.

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt. 7, 60–71 (2002).
[Crossref] [PubMed]

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

Shoji, S.

S. Shoji, H. B. Sun, and S. Kawata, “Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference,” Appl. Phys. Lett. 83, 608–610 (2003).
[Crossref]

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

Singh-Gasson, S.

S. Singh-Gasson, R. D. Green, Y. J. Yue, C. Nelson, F. Blattner, M. R. Sussman, and F. Cerrina, “Maskless fabrication of light-directed oligonucleotide microarrays using a digital micromirror array,” Nat. Biotechnol. 17, 974–978 (1999).
[Crossref] [PubMed]

Song, M.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[Crossref]

Sun, H. B.

S. Shoji, H. B. Sun, and S. Kawata, “Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference,” Appl. Phys. Lett. 83, 608–610 (2003).
[Crossref]

Sussman, M. R.

S. Singh-Gasson, R. D. Green, Y. J. Yue, C. Nelson, F. Blattner, M. R. Sussman, and F. Cerrina, “Maskless fabrication of light-directed oligonucleotide microarrays using a digital micromirror array,” Nat. Biotechnol. 17, 974–978 (1999).
[Crossref] [PubMed]

Thomas, J. A.

Toyoda, H.

Tromberg, B. J.

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt. 7, 60–71 (2002).
[Crossref] [PubMed]

Wang, G. C.

Y. C. Chen, N. R. Raravikar, L. S. Schadler, P. M. Ajayan, Y. P. Zhao, T. M. Lu, G. C. Wang, and X. C. Zhang, “Ultrafast optical switching properties of single-wall carbon nanotube polymer composites at 1.55 µm,” Appl. Phys. Lett. 81, 975–977 (2002).
[Crossref]

Wang, Z.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

Wilson, T.

Wyant, J. C.

Wyrowski, F.

Yu, K.

S. E. Chung, W. Park, H. Park, K. Yu, N. Park, and S. Kwon, “Optofluidic maskless lithography system for real-time synthesis of photopolymerized microstructures in microfluidic channels,” Appl. Phys. Lett. 91, 041106 (2007).
[Crossref]

Yu, W. Y.

Yue, Y. J.

S. Singh-Gasson, R. D. Green, Y. J. Yue, C. Nelson, F. Blattner, M. R. Sussman, and F. Cerrina, “Maskless fabrication of light-directed oligonucleotide microarrays using a digital micromirror array,” Nat. Biotechnol. 17, 974–978 (1999).
[Crossref] [PubMed]

Zhang, X. C.

Y. C. Chen, N. R. Raravikar, L. S. Schadler, P. M. Ajayan, Y. P. Zhao, T. M. Lu, G. C. Wang, and X. C. Zhang, “Ultrafast optical switching properties of single-wall carbon nanotube polymer composites at 1.55 µm,” Appl. Phys. Lett. 81, 975–977 (2002).
[Crossref]

Zhao, Y. P.

Y. C. Chen, N. R. Raravikar, L. S. Schadler, P. M. Ajayan, Y. P. Zhao, T. M. Lu, G. C. Wang, and X. C. Zhang, “Ultrafast optical switching properties of single-wall carbon nanotube polymer composites at 1.55 µm,” Appl. Phys. Lett. 81, 975–977 (2002).
[Crossref]

Appl. Opt. (7)

Appl. Phys. Lett. (4)

V. R. Daria, P. J. Rodrigo, and J. Gluckstad, “Dynamic array of dark optical traps,” Appl. Phys. Lett. 84, 323–325 (2004).
[Crossref]

Y. C. Chen, N. R. Raravikar, L. S. Schadler, P. M. Ajayan, Y. P. Zhao, T. M. Lu, G. C. Wang, and X. C. Zhang, “Ultrafast optical switching properties of single-wall carbon nanotube polymer composites at 1.55 µm,” Appl. Phys. Lett. 81, 975–977 (2002).
[Crossref]

S. E. Chung, W. Park, H. Park, K. Yu, N. Park, and S. Kwon, “Optofluidic maskless lithography system for real-time synthesis of photopolymerized microstructures in microfluidic channels,” Appl. Phys. Lett. 91, 041106 (2007).
[Crossref]

S. Shoji, H. B. Sun, and S. Kawata, “Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference,” Appl. Phys. Lett. 83, 608–610 (2003).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. Optical setup for the generalized phase contrast method.

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Fig. 2.
Fig. 2. (a) Magnitude and (b) phase of the normalized zero-order, α ¯ oe-16-2-1331-i001 , filter parameter, [exp(iθ)-1] oe-16-2-1331-i002 , and the SRW oe-16-2-1331-i003 at different illumination wavelengths.

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Fig. 3.
Fig. 3. Performance metrics at different wavelengths (a) Contrast ratio (logarithmic scale); (b) Efficiency.

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Fig. 4.
Fig. 4. Efficiency of GPC array illuminator as the wavelength λ is tuned away from the design wavelength λ 0. Periodic array - oe-16-2-1331-i004 ; aperiodic array - oe-16-2-1331-i005 ; solid line - efficiency for matched illumination and SRW profiles. Inset pictures: outputs at different wavelengths (rendered with false color, assuming λ 0=550 nm). Inset plot: illumination profile (magenta) and SRW profile (black) at λ 0.

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Fig. 5.
Fig. 5. Wavelength dependence of the efficiency when generating shapes using GPC illuminated with a Gaussian beam. Insets show the incident Gaussian illumination and the generated patterns at λ=450 nm, 550 nm, and 650 nm (rendered with false color).

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Fig. 6.
Fig. 6. Wavelength dependence of the efficiency, normalized root mean square error (nrmse) and structural similarity (mssim) of grayscale images projected using GPC. Insets show the generated images at λ=450 nm, 550 nm, and 650 nm (rendered with false color).

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Equations (30)

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p ( x , y ) = a ( x , y ) exp [ i ϕ 0 + i Δ ϕ ( x , y ) ]
= a ( x , y ) exp [ i 2 π d opt : s λ 0 + i 2 π Δ d opt : s ( x , y ) λ 0 ] ,
H ( f x , f y ) = { 1 + [ exp ( i θ ) − 1 ] S ( f x , f y ) } exp ( i ϕ f )
= { 1 + [ exp ( i 2 π Δ d opt : f λ 0 ) − 1 ] S ( f x , f y ) } exp ( i 2 π d opt : f λ 0 ) ,
α ¯ = ∣ α ¯ ∣ exp ( i ϕ α ¯ ) = 1 A 0 ∬ a ( x , y ) exp [ i Δ ϕ ( x , y ) ] d x d y ,
r ( x ′ , y ′ ) ≈ [ exp ( i 2 π Δ d opt : f λ 0 ) − 1 ] exp ( i 2 π d opt : f + d opt : s λ 0 ) ℑ − 1 { α ¯ S ( f x , f y ) ℑ { a ( x , y ) } } ,
I ( x ′ , y ′ ) ≈ ∣ p ( x ′ , y ′ ) + r ( x ′ , y ′ ) ∣ 2
  = ∣ a ( x ′ , y ′ ) exp [ i 2 π Δ d opt : s ( x ′ , y ′ ) λ 0 ] + α ¯ [ exp ( i 2 π Δ d opt : f λ 0 ) − 1 ] g ( x ′ , y ′ ) ∣ 2
= ∣ a ( x ′ , y ′ ) exp [ i Δ ϕ ( x ′ , y ′ ) ] + α ¯ [ exp ( i θ ) − 1 ] g ( x ′ , y ′ ) ∣ 2 ,
g ( x ′ , y ′ ) = ℑ − 1 { S ( f x , f y ) ℑ { a ( x , y ) } } .
g ( x ′ , y ′ ) ∝ 1 i λ 0 f ∬ S ( ξ , η ) A ( f x , f y ) exp [ − i 2 π f λ 0 ( x ′ ξ + y ′ η ) ] d ξ d η ,
A ( f x , f y ) = 1 i λ 0 f ∬ a ( x , y ) exp [ − i 2 π f λ 0 ( x ξ + y η ) ] d x d y .
I ( x ′ , y ′ ) = [ a ( x ′ , y ′ ) ] 2 ∣ exp [ i Δ ϕ ( x ′ , y ′ ) ] + K ∣ α ¯ ∣ sin ( θ 2 ) exp [ i ( θ + π 2 + ϕ α ¯ ) ] ∣ 2 .
I ( x ′ , y ′ ) = [ a ( x ′ , y ′ ) ] 2 ∣ exp [ i Δ ϕ ( x ′ , y ′ ) ] − 1 ∣ 2 .
α ¯ = α ¯ real + i α ¯ imag = 1 2 K + i 2 K cot ( θ 2 ) .
α ¯ = α ¯ real + i α ¯ imag = 1 2 + i 2 cot ( θ 2 ) ,
I target ( x ′ , y ′ ) = 2 [ a ( x ′ , y ′ ) ] 2 { 1 − cos [ Δ ϕ in ( x ′ , y ′ ) ] } .
cos [ Δ ϕ in ( x ′ , y ′ ) ] = 1 − I target ( x ′ , y ′ ) 2 [ a ( x ′ , y ′ ) ] 2 .
α ¯ real = 1 A 0 ∬ a ( x ′ , y ′ ) cos [ Δ ϕ ( x ′ , y ′ ) ] d x ′ d y ′ = 1 A 0 ∬ a ( x ′ , y ′ ) { 1 − I target ( x ′ , y ′ ) 2 [ a ( x ′ , y ′ ) ] 2 } d x ′ d y ′
= 1 − 1 2 A 0 ∬ I target ( x ′ , y ′ ) a ( x ′ , y ′ ) d x ′ d y ′ .
1 A 0 = ∬ I target ( x ′ , y ′ ) a ( x ′ , y ′ ) d x ′ d y ′ = 1 .
∬ ℜ ′ I target ( x ′ , y ′ ) d x ′ d y ′ ∬ ℜ ′ a 2 ( x ′ , y ′ ) d x ′ d y ′ = 1 ,
cos [ Δ ϕ in ( x , y ) ] = 1 − I target ( x , y ) 2 [ a ( x , y ) ] 2 .
cos [ Δ ϕ in ( x , y ) ] = 1 K − I target ( x , y ) 2 K [ a ( x , y ) ] 2 .
I ( x ′ , y ′ ) ≈ ∣ a ( x ′ , y ′ ) exp [ i π b ( x ′ , y ′ ) λ 0 λ ] + α ¯ [ exp ( iπ λ 0 λ ) − 1 ] g ( x ′ , y ′ ) ∣ 2 .
α ¯ = ∣ α ¯ ∣ exp ( i ϕ α ¯ ) = 1 A 0 ∬ a ( x , y ) exp [ i Δ ϕ ( x , y ) λ 0 λ ] d x d y .
A ( f x , f y ; λ ) = 1 i λ 0 f λ 0 λ ∬ a ( x , y ) exp [ − i 2 π f λ 0 ( x ξ λ 0 λ + y η λ 0 λ ) ] d x d y
= λ 0 λ A ( f x λ 0 λ , f y λ 0 λ , λ 0 ) .
g ( x ′ , y ′ ; λ ) ∝ 1 i λ 0 f λ 0 2 λ 2 ∬ S ( ξ , η ) A ( f x λ 0 λ , f y λ 0 λ ; λ 0 ) exp [ − i 2 π f λ 0 ( x ′ ξ λ 0 λ + y ′ η λ 0 λ ) ] d ξ d η
= 1 i λ 0 f ∬ S ( ξ λ λ λ 0 , η λ λ λ 0 ) A ( f x , λ , f y , λ ; λ 0 ) exp [ − i 2 π f λ 0 ( x ′ ξ λ + y ′ η λ ) ] d ξ λ d η λ ,

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