Abstract

A method for characterizing the phase response of spatial light modulators (SLMs) by using a Sagnac interferometer is proposed and demonstrated. The method represents an improvement over conventional diffraction-based or interferometric techniques by providing a simple and accurate phase measurement while taking advantage of the inherent phase stability of a Sagnac interferometer. As a demonstration, the phase response of a commercial liquid crystal on a silicon SLM is characterized and then linearized by using a programmable lookup table. The transverse phase profile over the SLM surface is also measured.

© 2009 Optical Society of America

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    [CrossRef]
  23. H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003).
    [CrossRef]
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    [CrossRef]
  25. V. Delaubert, N. Treps, C. C. Harb, P. K. Lam, and H.-A. Bachor, “Quantum measurements of spatial conjugate variables: displacement and tilt of a Gaussian beam,” Opt. Lett. 31, 1537-1539 (2006).
    [CrossRef] [PubMed]
  26. S. Serati, X. Xia, O. Mughal, and A. Linnenberger, “High-resolution phase-only spatial light modulators with sub-millisecond response,” Proc. SPIE 5106, 138-145 (2003).
    [CrossRef]

2008

2007

2006

V. Delaubert, N. Treps, C. C. Harb, P. K. Lam, and H.-A. Bachor, “Quantum measurements of spatial conjugate variables: displacement and tilt of a Gaussian beam,” Opt. Lett. 31, 1537-1539 (2006).
[CrossRef] [PubMed]

M. S. Millán, J. Otón, and E. Pérez-Cabré, “Chromatic compensation of programmable Fresnel lenses,” Opt. Express 14, 6226-6242 (2006).
[CrossRef] [PubMed]

D. Engström, G. Milewski, J. Bengtsson, and S. Galt, “Diffraction-based determination of the phase modulation for general spatial light modulators,” Appl. Opt. 45, 7195-7204 (2006).
[CrossRef] [PubMed]

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre Gaussian light beams formed by spatial light modulation,” J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast Sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[CrossRef]

2005

E. J. Fernández, B. Považay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45, 3432-3444(2005).
[CrossRef] [PubMed]

E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. Wulff, J. Courtial, and M. Padgett, “3D interferometric optical tweezers using a single spatial light modulator,” Opt. Express 13, 3777-3786 (2005).
[CrossRef] [PubMed]

2004

X. Xun and R. W. Cohn, “Phase calibration of spatially nonuniform spatial light modulators,” Appl. Opt. 43, 6400-6406 (2004).
[CrossRef] [PubMed]

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, “Nano-displacement measurements using spatially multimode squeezed light,” J. Opt. B 6, S664-S674 (2004).
[CrossRef]

2003

2001

1999

1996

1995

Ahnelt, P.

E. J. Fernández, B. Považay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45, 3432-3444(2005).
[CrossRef] [PubMed]

Ambs, P.

Arsenault, H. H.

Artal, P.

E. J. Fernández, B. Považay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45, 3432-3444(2005).
[CrossRef] [PubMed]

Bachor, H.-A.

V. Delaubert, N. Treps, C. C. Harb, P. K. Lam, and H.-A. Bachor, “Quantum measurements of spatial conjugate variables: displacement and tilt of a Gaussian beam,” Opt. Lett. 31, 1537-1539 (2006).
[CrossRef] [PubMed]

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, “Nano-displacement measurements using spatially multimode squeezed light,” J. Opt. B 6, S664-S674 (2004).
[CrossRef]

Bengtsson, J.

Bergeron, A.

Bowen, W. P.

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, “Nano-displacement measurements using spatially multimode squeezed light,” J. Opt. B 6, S664-S674 (2004).
[CrossRef]

Cai, L. Z.

Campos, J.

Carcole, E.

Cohn, R. W.

Cooper, J.

Cottrell, D. M.

Courtial, J.

Davis, J.

Davis, J. A.

Delaubert, V.

Dholakia, K.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre Gaussian light beams formed by spatial light modulation,” J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light modulators in atom optics,” Opt. Express 11, 158-166 (2003).
[CrossRef] [PubMed]

Domján, L.

Doucet, M.

Drexler, W.

E. J. Fernández, B. Považay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45, 3432-3444(2005).
[CrossRef] [PubMed]

Engström, D.

Escalera, J. C.

Fabre, C.

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, “Nano-displacement measurements using spatially multimode squeezed light,” J. Opt. B 6, S664-S674 (2004).
[CrossRef]

Fernández, E. J.

E. J. Fernández, B. Považay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45, 3432-3444(2005).
[CrossRef] [PubMed]

Freegarde, T.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre Gaussian light beams formed by spatial light modulation,” J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

Frumker, E.

Gagnon, F.

Galt, S.

Gauvin, J.

Gherardi, D. M.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre Gaussian light beams formed by spatial light modulation,” J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

Gingras, D.

Grosse, N.

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, “Nano-displacement measurements using spatially multimode squeezed light,” J. Opt. B 6, S664-S674 (2004).
[CrossRef]

Guo, C.-S.

Haist, T.

H. J. Tiziani, T. Haist, J. Liesener, M. Reicherter, and L. Seifert, “Applications of SLMs for optical metrology,” Proc. SPIE 4457, 72-81 (2001).
[CrossRef]

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608-610 (1999).
[CrossRef]

Harb, C. C.

Hermann, B.

E. J. Fernández, B. Považay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45, 3432-3444(2005).
[CrossRef] [PubMed]

Hsu, M. T. L.

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, “Nano-displacement measurements using spatially multimode squeezed light,” J. Opt. B 6, S664-S674 (2004).
[CrossRef]

Hurley, D. H.

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast Sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[CrossRef]

D. H. Hurley and O. B. Wright, “Detection of ultrafast phenomena by use of a modified Sagnac interferometer,” Opt. Lett. 24, 1305-1307 (1999).
[CrossRef]

Iemmi, C.

Jordan, P.

Kiire, T.

Koppa, P.

Lam, P. K.

V. Delaubert, N. Treps, C. C. Harb, P. K. Lam, and H.-A. Bachor, “Quantum measurements of spatial conjugate variables: displacement and tilt of a Gaussian beam,” Opt. Lett. 31, 1537-1539 (2006).
[CrossRef] [PubMed]

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, “Nano-displacement measurements using spatially multimode squeezed light,” J. Opt. B 6, S664-S674 (2004).
[CrossRef]

Ledesma, S.

Leitgeb, R.

E. J. Fernández, B. Považay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45, 3432-3444(2005).
[CrossRef] [PubMed]

Liesener, J.

H. J. Tiziani, T. Haist, J. Liesener, M. Reicherter, and L. Seifert, “Applications of SLMs for optical metrology,” Proc. SPIE 4457, 72-81 (2001).
[CrossRef]

Linnenberger, A.

S. Serati, X. Xia, O. Mughal, and A. Linnenberger, “High-resolution phase-only spatial light modulators with sub-millisecond response,” Proc. SPIE 5106, 138-145 (2003).
[CrossRef]

Livesey, J.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre Gaussian light beams formed by spatial light modulation,” J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

Lõrincz, E.

Maître, A.

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, “Nano-displacement measurements using spatially multimode squeezed light,” J. Opt. B 6, S664-S674 (2004).
[CrossRef]

Márquez, A.

Matsuda, O.

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast Sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[CrossRef]

McGloin, D.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre Gaussian light beams formed by spatial light modulation,” J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light modulators in atom optics,” Opt. Express 11, 158-166 (2003).
[CrossRef] [PubMed]

Melville, H.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre Gaussian light beams formed by spatial light modulation,” J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light modulators in atom optics,” Opt. Express 11, 158-166 (2003).
[CrossRef] [PubMed]

Milewski, G.

Millán, M. S.

Mosk, A. P.

Mughal, O.

S. Serati, X. Xia, O. Mughal, and A. Linnenberger, “High-resolution phase-only spatial light modulators with sub-millisecond response,” Proc. SPIE 5106, 138-145 (2003).
[CrossRef]

Muroya, T.

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast Sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[CrossRef]

Nakadate, S.

Ohzu, H.

Y. Takaki and H. Ohzu, “Liquid-crystal active lens: a reconfigurable lens employing a phase modulator,” Opt. Commun. 126, 123-134 (1996).
[CrossRef]

Okamoto, M.

H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003).
[CrossRef]

Otón, J.

Padgett, M.

Pérez-Cabré, E.

Piestun, R.

Považay, B.

E. J. Fernández, B. Považay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45, 3432-3444(2005).
[CrossRef] [PubMed]

Prieto, P. M.

E. J. Fernández, B. Považay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45, 3432-3444(2005).
[CrossRef] [PubMed]

Reicherter, M.

H. J. Tiziani, T. Haist, J. Liesener, M. Reicherter, and L. Seifert, “Applications of SLMs for optical metrology,” Proc. SPIE 4457, 72-81 (2001).
[CrossRef]

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608-610 (1999).
[CrossRef]

Reményi, J.

Rhodes, D. P.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. McGloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre Gaussian light beams formed by spatial light modulation,” J. Mod. Opt. 53, 547-556 (2006).
[CrossRef]

Rong, Z.-Y.

Sasada, H.

H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68, 012323 (2003).
[CrossRef]

Sattmann, H.

E. J. Fernández, B. Považay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45, 3432-3444(2005).
[CrossRef] [PubMed]

Schonbrun, E.

Seifert, L.

H. J. Tiziani, T. Haist, J. Liesener, M. Reicherter, and L. Seifert, “Applications of SLMs for optical metrology,” Proc. SPIE 4457, 72-81 (2001).
[CrossRef]

Serati, S.

S. Serati, X. Xia, O. Mughal, and A. Linnenberger, “High-resolution phase-only spatial light modulators with sub-millisecond response,” Proc. SPIE 5106, 138-145 (2003).
[CrossRef]

Shibuya, M.

Sibbett, W.

Silberberg, Y.

Spalding, G.

Sugawara, Y.

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast Sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[CrossRef]

Tachizaki, T.

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast Sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[CrossRef]

Takaki, Y.

Y. Takaki and H. Ohzu, “Liquid-crystal active lens: a reconfigurable lens employing a phase modulator,” Opt. Commun. 126, 123-134 (1996).
[CrossRef]

Tiziani, H. J.

H. J. Tiziani, T. Haist, J. Liesener, M. Reicherter, and L. Seifert, “Applications of SLMs for optical metrology,” Proc. SPIE 4457, 72-81 (2001).
[CrossRef]

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608-610 (1999).
[CrossRef]

Treps, N.

V. Delaubert, N. Treps, C. C. Harb, P. K. Lam, and H.-A. Bachor, “Quantum measurements of spatial conjugate variables: displacement and tilt of a Gaussian beam,” Opt. Lett. 31, 1537-1539 (2006).
[CrossRef] [PubMed]

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, “Nano-displacement measurements using spatially multimode squeezed light,” J. Opt. B 6, S664-S674 (2004).
[CrossRef]

Unterhuber, A.

E. J. Fernández, B. Považay, B. Hermann, A. Unterhuber, H. Sattmann, P. M. Prieto, R. Leitgeb, P. Ahnelt, P. Artal, and W. Drexler, “Three-dimensional adaptive optics ultrahigh-resolution optical coherence tomography using a liquid crystal spatial light modulator,” Vision Res. 45, 3432-3444(2005).
[CrossRef] [PubMed]

Valadéz, K. O.

van Putten, E. G.

Várhegyi, P.

Vellekoop, I. M.

Wagemann, E. U.

Wang, H.-T.

Wang, Y.

Wright, O. B.

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast Sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[CrossRef]

D. H. Hurley and O. B. Wright, “Detection of ultrafast phenomena by use of a modified Sagnac interferometer,” Opt. Lett. 24, 1305-1307 (1999).
[CrossRef]

Wulff, K.

Xia, X.

S. Serati, X. Xia, O. Mughal, and A. Linnenberger, “High-resolution phase-only spatial light modulators with sub-millisecond response,” Proc. SPIE 5106, 138-145 (2003).
[CrossRef]

Xun, X.

Yzuel, M. J.

Appl. Opt.

A. Bergeron, J. Gauvin, F. Gagnon, D. Gingras, H. H. Arsenault, and M. Doucet, “Phase calibration and applications of a liquid-crystal spatial light modulator,” Appl. Opt. 34, 5133-5139 (1995).
[CrossRef] [PubMed]

J. Davis, E. Carcole, and D. M. Cottrell, “Intensity and phase measurements of nondiffracting beams generated with a magneto-optic spatial light modulator,” Appl. Opt. 35, 593-598 (1996).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Schematic of a three-mirror Sagnac interferometer. One of the mirrors has been replaced with a transversely asymmetric element that introduces a phase shift between its two halves. BS, beam splitter.

Fig. 2
Fig. 2

Schematic of experimental setup. SLM, spatial light modulator; BS, beam splitter; CCD, CCD camera.

Fig. 3
Fig. 3

Experimental results obtained with initial SLM configuration and photodiode based detection. I, Evolution of photocurrent as movie is played on SLM. Insets: A, spatial structure of movie encoded on SLM; B, spatial structure of light incident on photodiode; C, phase flicker observable at short time scales. II,  Phase response of SLM as a function of gray scale encoded; inset D, LUT utilized by the SLM to convert gray scale to applied voltage across the LCD.

Fig. 4
Fig. 4

Intensity at Sagnac dark port with fixed nonzero gray-scale value on the right-hand side of the SLM, and zero gray-scale value on the left. Orange (light) trace, unfiltered intensity with fluctuations dominated by fast flicker due to the SLM refresh process; black trace, intensity after low-pass filtering with a 60 Hz cutoff frequency. Insets: A, spatial structure of movie encoded on SLM; B, spatial structure of light incident on photodiode.

Fig. 5
Fig. 5

Experimental results obtained with linearized SLM configuration and photodiode based detection. I, Evolution of photocurrent as movie is played on SLM Insets: A, spatial structure of movie encoded on SLM; B, spatial structure of light incident on photodiode. II, Phase response of SLM as a function of gray scale encoded. Inset C, LUT utilized by SLM to convert gray scale to applied voltage across the LCD.

Fig. 6
Fig. 6

Experimental results obtained for a single pixel with linearized SLM configuration and CCD camera based detection. I, Evolution of incident intensity as movie is played on SLM. II, Phase response of SLM region corresponding to observed pixel as a function of gray scale encoded.

Fig. 7
Fig. 7

Transverse phase response profile of SLM based on CCD camera measurements. Color bar, change in gray scale needed to achieve a π phase shift. The greyed-out area on the periphery of the figure indicates the region over which light intensities were too low to accurately determine the phase.

Equations (10)

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E in ( x , y ) = A U ( x , y ) ,
E C ( x , y ) = 1 2 A U ( x , y ) e i k L e i ϕ ( x , y ) , E A ( x , y ) = 1 2 A U ( x , y ) e i k L e i ϕ ( x , y ) ,
E out ( x , y ) = 1 2 [ E C ( x , y ) E A ( x , y ) ] = 1 2 A U ( x , y ) e i k L [ e i ϕ ( x , y ) e i ϕ ( x , y ) ] = 1 2 E in ( x , y ) e i k L [ e i ϕ ( x , y ) e i ϕ ( x , y ) ] .
I out ( x , y ) = c ϵ 0 2 | E out ( x , y ) | 2 = 1 2 I in ( x , y ) [ 1 cos Δ ϕ ( x , y ) ] ,
Δ ϕ ( x , y ) = cos 1 [ 1 2 I out ( x , y ) I in ( x , y ) ] .
P out = A I out ( x , y ) d A = 1 2 A I in ( x , y ) [ 1 cos Δ ϕ ( x , y ) ] d A = 1 2 P in [ 1 cos Δ ϕ ¯ ] ,
Δ ϕ ¯ = cos 1 [ 1 2 P out P in ] .
Δ ϕ ¯ = cos 1 [ 1 VIS ( 1 2 P out P in ) ] ,
VIS = P max P min P max + P min ,
Δ ϕ ¯ ( θ ) = Δ ϕ ¯ ( θ exp ) cos ( θ exp ) cos ( θ ) ,

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