Abstract

Multiple internal reflection effects on the optical modulation of a commercial reflective parallel-aligned liquid-crystal on silicon (PAL-LCoS) spatial light modulator (SLM) are analyzed. The display is illuminated with different wavelengths and different angles of incidence. Non-negligible Fabry-Perot (FP) effect is observed due to the sandwiched LC layer structure. A simplified physical model that quantitatively accounts for the observed phenomena is proposed. It is shown how the expected pure phase modulation response is substantially modified in the following aspects: 1) a coupled amplitude modulation, 2) a non-linear behavior of the phase modulation, 3) some amount of unmodulated light, and 4) a reduction of the effective phase modulation as the angle of incidence increases. Finally, it is shown that multiple reflections can be useful since the effect of a displayed diffraction grating is doubled on a beam that is reflected twice through the LC layer, thus rendering gratings with doubled phase modulation depth.

© 2014 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  25. I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. 43(34), 6278–6284 (2004).
    [Crossref] [PubMed]
  26. J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51(2), 111–115 (2013).
    [Crossref]

2014 (3)

2013 (5)

2010 (1)

L. Lobato, A. Márquez, A. Lizana, I. Moreno, C. Iemmi, and J. Campos, “Characterization of a parallel aligned liquid cristal on silicon and its application on a Shack-Hartmann sensor,” Proc. SPIE 7797, 77970Q (2010).
[Crossref]

2009 (1)

2008 (2)

2007 (2)

M. L. Álvarez, A. Márquez, L. A. Puerta, R. Estévez, E. Fernández, A. Beléndez, and I. Pascual, “Characterization of a low resolution liquid crystal display to be used as an optical modulator,” J. Opt. A.: Pure Appl. Opt. 40, 95–103 (2007).

Y.-K. Jang and P. Bos, “Analysis of the multireflection effects in compensated liquid crystal devices,” J. Appl. Phys. 101(3), 033131 (2007).
[Crossref]

2004 (2)

2001 (1)

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40(11), 2558–2564 (2001).
[Crossref]

1999 (1)

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38(6), 1051–1057 (1999).
[Crossref]

1996 (2)

Y. Bao, A. Sneh, K. Hsu, K. M. Johnson, J.-Y. Liu, C. M. Miller, Y. Monta, and M. McLain, “High-speed liquid crystal fiber Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(9), 1190–1192 (1996).
[Crossref]

R. Carey, P. A. Gago-Sandoval, D. M. Newman, and B. W. J. Thomas, “Programmable liquid crystal waveplates in ellipsometric measurements,” Meas. Sci. Technol. 7(4), 505–510 (1996).
[Crossref]

1990 (2)

1964 (1)

F. Gires and P. Tournois, “Interféromètre utilisable pour la compression d’impulsions lumineuses modulées en fréquence,” C. R. Acad. Sci. Paris 258, 6112–6115 (1964).

Albero, J.

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51(2), 111–115 (2013).
[Crossref]

V. Calero, P. García-Martínez, J. Albero, M. M. Sánchez-López, and I. Moreno, “Liquid crystal spatial light modulator with very large phase modulation operating in high harmonic orders,” Opt. Lett. 38(22), 4663–4666 (2013).
[Crossref] [PubMed]

Álvarez, M. L.

M. L. Álvarez, A. Márquez, L. A. Puerta, R. Estévez, E. Fernández, A. Beléndez, and I. Pascual, “Characterization of a low resolution liquid crystal display to be used as an optical modulator,” J. Opt. A.: Pure Appl. Opt. 40, 95–103 (2007).

Amako, J.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38(6), 1051–1057 (1999).
[Crossref]

Arias, J.

A. Vargas, M. M. Sánchez-López, P. García-Martínez, J. Arias, and I. Moreno, “Highly accurate spectral retardance characterization of a liquid crystal retarder including Fabry-Perot interference effects,” J. Appl. Phys. 115(3), 033101 (2014).
[Crossref]

Bao, Y.

Y. Bao, A. Sneh, K. Hsu, K. M. Johnson, J.-Y. Liu, C. M. Miller, Y. Monta, and M. McLain, “High-speed liquid crystal fiber Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(9), 1190–1192 (1996).
[Crossref]

Beléndez, A.

F. J. Martínez, A. Márquez, S. Gallego, M. Ortuño, J. Francés, A. Beléndez, and I. Pascual, “Averaged Stokes polarimetry applied to evaluate retardance and flicker in PA-LCoS devices,” Opt. Express 22(12), 15064–15074 (2014).
[Crossref] [PubMed]

M. L. Álvarez, A. Márquez, L. A. Puerta, R. Estévez, E. Fernández, A. Beléndez, and I. Pascual, “Characterization of a low resolution liquid crystal display to be used as an optical modulator,” J. Opt. A.: Pure Appl. Opt. 40, 95–103 (2007).

Bengtsson, J.

Bernet, S.

Bos, P.

Y.-K. Jang and P. Bos, “Analysis of the multireflection effects in compensated liquid crystal devices,” J. Appl. Phys. 101(3), 033131 (2007).
[Crossref]

Calero, V.

Campos, J.

Carey, R.

R. Carey, P. A. Gago-Sandoval, D. M. Newman, and B. W. J. Thomas, “Programmable liquid crystal waveplates in ellipsometric measurements,” Meas. Sci. Technol. 7(4), 505–510 (1996).
[Crossref]

Chambers, J. B.

Cottrell, D. M.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38(6), 1051–1057 (1999).
[Crossref]

Coulston, S.

G. Lester, A. Strudwick, and S. Coulston, “Electronically switchable diffractive optical elements,” Opto-Electron. Rev. 12, 313–316 (2004).

Davis, J. A.

J. A. Davis, J. B. Chambers, B. A. Slovick, and I. Moreno, “Wavelength-dependent diffraction patterns from a liquid crystal display,” Appl. Opt. 47(24), 4375–4380 (2008).
[Crossref] [PubMed]

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40(11), 2558–2564 (2001).
[Crossref]

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38(6), 1051–1057 (1999).
[Crossref]

Engström, D.

Estapé, M.

Estévez, R.

M. L. Álvarez, A. Márquez, L. A. Puerta, R. Estévez, E. Fernández, A. Beléndez, and I. Pascual, “Characterization of a low resolution liquid crystal display to be used as an optical modulator,” J. Opt. A.: Pure Appl. Opt. 40, 95–103 (2007).

Fernández, E.

Francés, J.

Gago-Sandoval, P. A.

R. Carey, P. A. Gago-Sandoval, D. M. Newman, and B. W. J. Thomas, “Programmable liquid crystal waveplates in ellipsometric measurements,” Meas. Sci. Technol. 7(4), 505–510 (1996).
[Crossref]

Gallego, S.

García-Martínez, P.

A. Vargas, M. M. Sánchez-López, P. García-Martínez, J. Arias, and I. Moreno, “Highly accurate spectral retardance characterization of a liquid crystal retarder including Fabry-Perot interference effects,” J. Appl. Phys. 115(3), 033101 (2014).
[Crossref]

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51(2), 111–115 (2013).
[Crossref]

V. Calero, P. García-Martínez, J. Albero, M. M. Sánchez-López, and I. Moreno, “Liquid crystal spatial light modulator with very large phase modulation operating in high harmonic orders,” Opt. Lett. 38(22), 4663–4666 (2013).
[Crossref] [PubMed]

Gires, F.

F. Gires and P. Tournois, “Interféromètre utilisable pour la compression d’impulsions lumineuses modulées en fréquence,” C. R. Acad. Sci. Paris 258, 6112–6115 (1964).

Goksör, M.

Haist, T.

Hsu, K.

Y. Bao, A. Sneh, K. Hsu, K. M. Johnson, J.-Y. Liu, C. M. Miller, Y. Monta, and M. McLain, “High-speed liquid crystal fiber Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(9), 1190–1192 (1996).
[Crossref]

Iemmi, C.

Jang, Y.-K.

Y.-K. Jang and P. Bos, “Analysis of the multireflection effects in compensated liquid crystal devices,” J. Appl. Phys. 101(3), 033131 (2007).
[Crossref]

Jesacher, A.

Johnson, K. M.

Y. Bao, A. Sneh, K. Hsu, K. M. Johnson, J.-Y. Liu, C. M. Miller, Y. Monta, and M. McLain, “High-speed liquid crystal fiber Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(9), 1190–1192 (1996).
[Crossref]

Leaird, D. E.

Lester, G.

G. Lester, A. Strudwick, and S. Coulston, “Electronically switchable diffractive optical elements,” Opto-Electron. Rev. 12, 313–316 (2004).

Lingel, C.

Liu, J.-Y.

Y. Bao, A. Sneh, K. Hsu, K. M. Johnson, J.-Y. Liu, C. M. Miller, Y. Monta, and M. McLain, “High-speed liquid crystal fiber Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(9), 1190–1192 (1996).
[Crossref]

Lizana, A.

Lobato, L.

L. Lobato, A. Márquez, A. Lizana, I. Moreno, C. Iemmi, and J. Campos, “Characterization of a parallel aligned liquid cristal on silicon and its application on a Shack-Hartmann sensor,” Proc. SPIE 7797, 77970Q (2010).
[Crossref]

Lu, K.

K. Lu and B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29(3), 240–246 (1990).
[Crossref]

Márquez, A.

F. J. Martínez, A. Márquez, S. Gallego, M. Ortuño, J. Francés, A. Beléndez, and I. Pascual, “Averaged Stokes polarimetry applied to evaluate retardance and flicker in PA-LCoS devices,” Opt. Express 22(12), 15064–15074 (2014).
[Crossref] [PubMed]

L. Lobato, A. Márquez, A. Lizana, I. Moreno, C. Iemmi, and J. Campos, “Characterization of a parallel aligned liquid cristal on silicon and its application on a Shack-Hartmann sensor,” Proc. SPIE 7797, 77970Q (2010).
[Crossref]

A. Lizana, N. Martín, M. Estapé, E. Fernández, I. Moreno, A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Influence of the incident angle in the performance of Liquid Crystal on Silicon displays,” Opt. Express 17(10), 8491–8505 (2009).
[Crossref] [PubMed]

I. Moreno, A. Lizana, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express 16(21), 16711–16722 (2008).
[Crossref] [PubMed]

M. L. Álvarez, A. Márquez, L. A. Puerta, R. Estévez, E. Fernández, A. Beléndez, and I. Pascual, “Characterization of a low resolution liquid crystal display to be used as an optical modulator,” J. Opt. A.: Pure Appl. Opt. 40, 95–103 (2007).

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. 43(34), 6278–6284 (2004).
[Crossref] [PubMed]

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40(11), 2558–2564 (2001).
[Crossref]

Martín, N.

Martínez, F. J.

Martínez, J. L.

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51(2), 111–115 (2013).
[Crossref]

McLain, M.

Y. Bao, A. Sneh, K. Hsu, K. M. Johnson, J.-Y. Liu, C. M. Miller, Y. Monta, and M. McLain, “High-speed liquid crystal fiber Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(9), 1190–1192 (1996).
[Crossref]

Miller, C. M.

Y. Bao, A. Sneh, K. Hsu, K. M. Johnson, J.-Y. Liu, C. M. Miller, Y. Monta, and M. McLain, “High-speed liquid crystal fiber Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(9), 1190–1192 (1996).
[Crossref]

Monta, Y.

Y. Bao, A. Sneh, K. Hsu, K. M. Johnson, J.-Y. Liu, C. M. Miller, Y. Monta, and M. McLain, “High-speed liquid crystal fiber Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(9), 1190–1192 (1996).
[Crossref]

Moreno, I.

A. Vargas, M. M. Sánchez-López, P. García-Martínez, J. Arias, and I. Moreno, “Highly accurate spectral retardance characterization of a liquid crystal retarder including Fabry-Perot interference effects,” J. Appl. Phys. 115(3), 033101 (2014).
[Crossref]

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51(2), 111–115 (2013).
[Crossref]

V. Calero, P. García-Martínez, J. Albero, M. M. Sánchez-López, and I. Moreno, “Liquid crystal spatial light modulator with very large phase modulation operating in high harmonic orders,” Opt. Lett. 38(22), 4663–4666 (2013).
[Crossref] [PubMed]

L. Lobato, A. Márquez, A. Lizana, I. Moreno, C. Iemmi, and J. Campos, “Characterization of a parallel aligned liquid cristal on silicon and its application on a Shack-Hartmann sensor,” Proc. SPIE 7797, 77970Q (2010).
[Crossref]

A. Lizana, N. Martín, M. Estapé, E. Fernández, I. Moreno, A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Influence of the incident angle in the performance of Liquid Crystal on Silicon displays,” Opt. Express 17(10), 8491–8505 (2009).
[Crossref] [PubMed]

J. A. Davis, J. B. Chambers, B. A. Slovick, and I. Moreno, “Wavelength-dependent diffraction patterns from a liquid crystal display,” Appl. Opt. 47(24), 4375–4380 (2008).
[Crossref] [PubMed]

I. Moreno, A. Lizana, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express 16(21), 16711–16722 (2008).
[Crossref] [PubMed]

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. 43(34), 6278–6284 (2004).
[Crossref] [PubMed]

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40(11), 2558–2564 (2001).
[Crossref]

Newman, D. M.

R. Carey, P. A. Gago-Sandoval, D. M. Newman, and B. W. J. Thomas, “Programmable liquid crystal waveplates in ellipsometric measurements,” Meas. Sci. Technol. 7(4), 505–510 (1996).
[Crossref]

Ortuño, M.

Osten, W.

Pascual, I.

F. J. Martínez, A. Márquez, S. Gallego, M. Ortuño, J. Francés, A. Beléndez, and I. Pascual, “Averaged Stokes polarimetry applied to evaluate retardance and flicker in PA-LCoS devices,” Opt. Express 22(12), 15064–15074 (2014).
[Crossref] [PubMed]

M. L. Álvarez, A. Márquez, L. A. Puerta, R. Estévez, E. Fernández, A. Beléndez, and I. Pascual, “Characterization of a low resolution liquid crystal display to be used as an optical modulator,” J. Opt. A.: Pure Appl. Opt. 40, 95–103 (2007).

Patel, J. S.

Persson, M.

Puerta, L. A.

M. L. Álvarez, A. Márquez, L. A. Puerta, R. Estévez, E. Fernández, A. Beléndez, and I. Pascual, “Characterization of a low resolution liquid crystal display to be used as an optical modulator,” J. Opt. A.: Pure Appl. Opt. 40, 95–103 (2007).

Reichelt, S.

Ritsch-Marte, M.

Saleh, B. E. A.

K. Lu and B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29(3), 240–246 (1990).
[Crossref]

Sánchez-López, M. M.

A. Vargas, M. M. Sánchez-López, P. García-Martínez, J. Arias, and I. Moreno, “Highly accurate spectral retardance characterization of a liquid crystal retarder including Fabry-Perot interference effects,” J. Appl. Phys. 115(3), 033101 (2014).
[Crossref]

V. Calero, P. García-Martínez, J. Albero, M. M. Sánchez-López, and I. Moreno, “Liquid crystal spatial light modulator with very large phase modulation operating in high harmonic orders,” Opt. Lett. 38(22), 4663–4666 (2013).
[Crossref] [PubMed]

Slovick, B. A.

Sneh, A.

Y. Bao, A. Sneh, K. Hsu, K. M. Johnson, J.-Y. Liu, C. M. Miller, Y. Monta, and M. McLain, “High-speed liquid crystal fiber Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(9), 1190–1192 (1996).
[Crossref]

Sonehara, T.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38(6), 1051–1057 (1999).
[Crossref]

Strudwick, A.

G. Lester, A. Strudwick, and S. Coulston, “Electronically switchable diffractive optical elements,” Opto-Electron. Rev. 12, 313–316 (2004).

Thomas, B. W. J.

R. Carey, P. A. Gago-Sandoval, D. M. Newman, and B. W. J. Thomas, “Programmable liquid crystal waveplates in ellipsometric measurements,” Meas. Sci. Technol. 7(4), 505–510 (1996).
[Crossref]

Tournois, P.

F. Gires and P. Tournois, “Interféromètre utilisable pour la compression d’impulsions lumineuses modulées en fréquence,” C. R. Acad. Sci. Paris 258, 6112–6115 (1964).

Tsai, P.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38(6), 1051–1057 (1999).
[Crossref]

Vargas, A.

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Figures (9)

Fig. 1
Fig. 1

Layer structure of the LCoS display.

Fig. 2
Fig. 2

(a) Illumination scheme, where incident polarization is selected parallel to LC director. (b) Picture of the LCoS-SLM illuminated with θ0 = 45° and λ = 568 nm, where light scattered at the reflecting surfaces can be noticed. Incoming and emerging light are outlined by dashed lines. The reflected beams captured by a CCD camera are shown in the upper left inset box.

Fig. 3
Fig. 3

Reflected intensity versus gray level for (a) 488 nm, (b) 568 nm, (c) 647 nm for θ0 = 15°. Reflected intensities I1, I2, and I3 versus gray level for θ0 = 45° for (d) 488 nm, (e) 568 nm, and (f) 647 nm. Lines show the corresponding numerical curves after fitting the simplified physical model.

Fig. 4
Fig. 4

(a) Simplified layer structure and main transmitted/reflected beams. (b) Schematic of the interference condition at the LC layer. Voltage dependent LC director realignment in this layer is indicated

Fig. 5
Fig. 5

FP and air-glass interface reflectance corresponding to measurements (solid lines) and fitted data (crosses and asterisks) for wavelengths: (a) 488 nm, (b) 568 nm, and (c) 647 nm.

Fig. 6
Fig. 6

Estimated phase modulation (solid curve) and birefringence variation (dashes) for wavelengths: (a) 488nm, (b) 568 nm, and (c) 647nm.

Fig. 7
Fig. 7

Diffraction efficiency (η m ) at the m-th order diffraction efficiency versus gray level for (a) binary grating, and (b) blazed grating corresponding to the reflectance obtained for 568 nm.

Fig. 8
Fig. 8

CCD images of reflected beams (I1, I2, I3) with λ = 568 nm, corresponding to binary gratings with phase difference ranging from 0 to 9π/2 in steps of π/2. Effective phase modulations Δϕ2 for beam I2 and Δϕ3 = 2Δϕ2 for beam I3 are indicated on top of each image. Diffraction orders m = 0, ± 1 are indicated.

Fig. 9
Fig. 9

CCD images of reflected beams (I1, I2, I3) with λ = 568 nm, corresponding to a blazed grating with phase difference ranging from 0 to 9π/2 in steps of π/2. Effective phase modulations Δϕ2 for beam I2 and Δϕ3 = 2Δϕ2 for beam I3 are indicated on top of each image. Diffraction orders m = 0, + 1,… + 5 are indicated.

Tables (2)

Tables Icon

Table 1 Values obtained after the fitting process for the reflection coefficients and birefringence parameters at the three calibration wavelengths.

Tables Icon

Table 2 Mean square error between experimental data and model predictions for the data presented in Fig. 3.

Equations (15)

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2 Δ x = 2 d g sin θ 0 n 1 2 sin 2 θ 0 ,
Δ x > w 2 cos θ 0 ,
w < 2 d g sin θ 0 cos θ 0 n 1 2 sin 2 θ 0 ,
ρ F P = ρ 12 + ρ 23 exp ( i k 0 Δ L ) 1 + ρ 12 ρ 23 exp ( i k 0 Δ L ) ,
k 0 Δ L = 4 π λ 0 d n 2 t ,
I 1 = I i | ρ 01 | 2 ,
I 2 = I i | 1 ρ 01 2 | 2 | ρ F P | 2 ,
I 3 = I i | 1 ρ 01 2 | 2 | ρ 01 | 2 | ρ F P | 4 ,
ρ 01 2 = ( 1 + I 2 I 1 I 3 ) 1 ,
| ρ F P | 2 = I 2 I 1 ρ 01 2 ( 1 ρ 01 2 ) 2 .
k 0 Δ = β 0 t + Δ β ( g ) t ,
Δ β ( g ) = Δ β max ( 1 g 255 ) ,
Δ ϕ ( g ) = arg { ρ F P ( g ) } arg { ρ F P ( g = 0 ) } .
ρ F P ρ 23 exp ( i 2 k 0 d n 2 t ) ρ 23 exp ( i 2 Δ β max t ) .
c m = n = 0 N 1 ρ F P ( g n ) sin ( m π w P ) 1 m π w exp ( i 2 π m n w P ) ,

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