Abstract

Switchable, continuous, complex-amplitude modulation is demonstrated with two cascaded, twisted nematic liquid-crystal televisions (LCTV’s), both operating in phase- and amplitude-coupled modulation modes. The condition for full-range complex modulation is that one of the LCTV’s must provide a 2π-range phase modulation. A look-up table encoding method is proposed that permits the compensation of phase–amplitude coupling and nonlinearity in the two individual LCTV modulations. Experimental techniques for determining the LCTV-device parameters, for maximizing the phase-mostly modulation range and the amplitude-mostly modulation contrast, and for testing the complex-amplitude modulation are developed. Optical complex-amplitude Fresnel holograms are shown.

© 1996 Optical Society of America

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References

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  5. J. M. Florence, R. D. Juday, “Full complex spatial filtering with a phase-mostly DMD,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE 1558, 505–516 (1991).
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  7. N. Konforti, E. Marom, S. T. Wu, “Phase-only modulation with twisted liquid-crystal spatial light modulators,” Opt. Lett. 13, 251–253 (1988).
    [CrossRef] [PubMed]
  8. K. Lu, B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial light modulator,” Opt. Eng. 29, 240–245 (1990).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  17. C. H. Gooch, H. A. Tarry, “The optical properties of twisted nematic liquid-crystal structures with twist angles ≤90°,” J. Phys. D. 8, 1575–1584 (1975).
    [CrossRef]
  18. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 5.
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1995 (2)

1994 (1)

C. Soutar, K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid-crystal cell,” Opt. Eng. 33, 2704–2712 (1994).
[CrossRef]

1993 (4)

1992 (1)

1991 (1)

1990 (1)

K. Lu, B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial light modulator,” Opt. Eng. 29, 240–245 (1990).
[CrossRef]

1988 (1)

1986 (1)

1984 (1)

1975 (1)

C. H. Gooch, H. A. Tarry, “The optical properties of twisted nematic liquid-crystal structures with twist angles ≤90°,” J. Phys. D. 8, 1575–1584 (1975).
[CrossRef]

Amako, J.

Au, A.

Bartelt, H.

Bates, T. D.

Chao, T. H.

T. H. Chao, A. Yacoubian, B. Lau, W. J. Miceli, “Optical wavelet processor for target detection,” in Optical Computing, Vol. 10 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 242–245.

Chipman, R. A.

Efron, U.

Florence, J. M.

R. D. Juday, J. M. Florence, “Full complex modulation with two one-parameter SLM’s,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE 1558, 499–504 (1991).

J. M. Florence, R. D. Juday, “Full complex spatial filtering with a phase-mostly DMD,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE 1558, 505–516 (1991).

Gonçalves, L.

Gooch, C. H.

C. H. Gooch, H. A. Tarry, “The optical properties of twisted nematic liquid-crystal structures with twist angles ≤90°,” J. Phys. D. 8, 1575–1584 (1975).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1968), Chap. 3.

Gregory, D. A.

Juday, R. D.

R. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
[CrossRef] [PubMed]

R. D. Juday, J. M. Florence, “Full complex modulation with two one-parameter SLM’s,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE 1558, 499–504 (1991).

J. M. Florence, R. D. Juday, “Full complex spatial filtering with a phase-mostly DMD,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE 1558, 505–516 (1991).

Kirsch, J. C.

Konforti, N.

Lau, B.

T. H. Chao, A. Yacoubian, B. Lau, W. J. Miceli, “Optical wavelet processor for target detection,” in Optical Computing, Vol. 10 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 242–245.

Lu, K.

C. Soutar, K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid-crystal cell,” Opt. Eng. 33, 2704–2712 (1994).
[CrossRef]

K. Lu, B. E. A. Saleh, “Complex amplitude reflectance of the liquid crystal light valve,” Appl. Opt. 30, 2354–2362 (1991).
[CrossRef] [PubMed]

K. Lu, B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial light modulator,” Opt. Eng. 29, 240–245 (1990).
[CrossRef]

Marom, E.

Miceli, W. J.

T. H. Chao, A. Yacoubian, B. Lau, W. J. Miceli, “Optical wavelet processor for target detection,” in Optical Computing, Vol. 10 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 242–245.

Miura, H.

Paul-Hus, G.

Pezzaniti, J. L.

Roberge, D.

Saleh, B. E. A.

K. Lu, B. E. A. Saleh, “Complex amplitude reflectance of the liquid crystal light valve,” Appl. Opt. 30, 2354–2362 (1991).
[CrossRef] [PubMed]

K. Lu, B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial light modulator,” Opt. Eng. 29, 240–245 (1990).
[CrossRef]

Sheng, Y.

Sonehara, T.

Soutar, C.

C. Soutar, K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid-crystal cell,” Opt. Eng. 33, 2704–2712 (1994).
[CrossRef]

Tam, E. C.

Tarry, H. A.

C. H. Gooch, H. A. Tarry, “The optical properties of twisted nematic liquid-crystal structures with twist angles ≤90°,” J. Phys. D. 8, 1575–1584 (1975).
[CrossRef]

Wu, C. S.

Wu, S. T.

Yacoubian, A.

T. H. Chao, A. Yacoubian, B. Lau, W. J. Miceli, “Optical wavelet processor for target detection,” in Optical Computing, Vol. 10 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 242–245.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 5.

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 5.

Appl. Opt. (8)

J. Opt. Soc. Am. B (1)

J. Phys. D. (1)

C. H. Gooch, H. A. Tarry, “The optical properties of twisted nematic liquid-crystal structures with twist angles ≤90°,” J. Phys. D. 8, 1575–1584 (1975).
[CrossRef]

Opt. Eng. (2)

K. Lu, B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial light modulator,” Opt. Eng. 29, 240–245 (1990).
[CrossRef]

C. Soutar, K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid-crystal cell,” Opt. Eng. 33, 2704–2712 (1994).
[CrossRef]

Opt. Lett. (2)

Other (5)

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 5.

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1968), Chap. 3.

T. H. Chao, A. Yacoubian, B. Lau, W. J. Miceli, “Optical wavelet processor for target detection,” in Optical Computing, Vol. 10 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 242–245.

R. D. Juday, J. M. Florence, “Full complex modulation with two one-parameter SLM’s,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE 1558, 499–504 (1991).

J. M. Florence, R. D. Juday, “Full complex spatial filtering with a phase-mostly DMD,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE 1558, 505–516 (1991).

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Figures (13)

Fig. 1
Fig. 1

Schematic diagram of the coordinate systems for the polarizer–LCTV–analyzer sandwich.

Fig. 2
Fig. 2

Intensity transmittance T as a function of the orientation ψ of the parallel polarizer and the analyzer at five different gray-level values: 0, 50, 75, 100, and 150.

Fig. 3
Fig. 3

Ratio of the maximum and minimum intensity transmittances, T max and T min, respectively, for the simultaneously rotated parallel polarizer and analyzer ψ as a function of gray-level values.

Fig. 4
Fig. 4

Linear relation between cos - 1 ( T ) and ψ2, when γ = nπ and ψ1 = 0.

Fig. 5
Fig. 5

Intensity transmittance T and phase shift δ of the twisted nematic LCD as functions of β for four configurations of the polarizer ψ1 and the analyzer ψ2 described in Eqs. (5) and (6): α is the twist angle, and ψ D is the orientation of the front LC molecules.

Fig. 6
Fig. 6

Schematic diagram of an experimental cascade of two LCTV’s for complex-amplitude modulations: Phase mod., phase modulator; Amp. mod., amplitude modulator.

Fig. 7
Fig. 7

Coupled phase δ and amplitude A modulations: (a) δ from LCTV1, (b) δ from LCTV2, (c) A from LCTV1, and (d) A from LCTV2.

Fig. 8
Fig. 8

Example of operating curves for (a) LCTV1, (b) LCTV2, and (c) the realizable complex amplitude in the complex plane. When LCTV1 provides phase modulation that is lower than 2π, the realizable complex amplitude does not cover a complete circle.

Fig. 9
Fig. 9

Mapped plot of all realizable complex amplitudes A exp(jδ) as functions of gray levels (a) gl1 and (b) gl2.

Fig. 10
Fig. 10

Look-up tables used to generate gray levels for any required complex-transmittance value: (a) gray level gl1, and (b) gray level gl2.

Fig. 11
Fig. 11

Interferograms of the wedged shear-plate interferometer: (a) the gray levels of both LCTV1 and LCTV2 are uniform, (b) LCTV2 has two vertical bars of gray levels 255 and 125 and the coupled phase shift can be seen in the middle region, and (c) the complex amplitude encoded with the look-up table.

Fig. 12
Fig. 12

Fraunhofer diffraction patterns of a 2-D grating: (a) patterns encoded with the look-up table, and (b) the encoded phase in LCTV1 and amplitude in LCTV2.

Fig. 13
Fig. 13

Reconstructed images of switchable complex-valued Fresnel holograms.

Equations (20)

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J T = e j ϕ [ cos α - sin α sin α cos α ] × [ cos γ + j β sin γ γ α sin γ γ - α sin γ γ cos γ - j β sin γ γ ] ,
β = π d λ ( n e - n o ) , γ = α 2 + β 2 , ϕ = π d λ ( n e + n o ) = ϕ 0 + β
[ E e E o ] = [ cos ξ 1 sin ξ 1 ] ,
E out = [ E e E o ] = [ cos 2 ξ 2 sin ξ 2 cos ξ 2 sin ξ 2 cos ξ 2 sin 2 ξ 2 ] J T [ cos ξ 1 sin ξ 1 ] ,
T = E e 2 + E o 2 = { α γ sin γ sin [ α + ( ψ 1 - ψ 2 ) ] cos γ cos [ α + ( ψ 1 - ψ 2 ) ] } 2 + { β γ sin γ cos [ α - ( ψ 1 + ψ 2 ) + 2 ψ D ] } 2 .
δ = tan - 1 Im ( E e ) Re ( E e ) = tan - 1 Im ( E o ) Re ( E o ) = β + tan - 1 β γ sin γ cos [ α - ( ψ 1 + ψ 2 ) + 2 ψ D ] α γ sin γ sin [ α + ( ψ 1 - ψ 2 ) ] + cos γ cos [ α + ( ψ 1 - ψ 2 ) ]
T = [ cos γ cos α + α sin γ γ sin α ] 2 + [ β sin γ γ cos ( α - 2 ψ + 2 ψ D ) ] 2 ,
ψ max = α 2 + ψ D ± n π 2 ,
T max = [ cos γ cos α + α sin γ γ sin α ] 2 + [ β sin γ γ ] 2 .
ψ min = α 2 + ψ D ± ( 2 n + 1 ) π 4 ,
T min = [ cos γ cos α + α sin γ γ sin α ] 2 .
T = cos 2 [ α + ( ψ 1 - ψ 2 ) ] .
T = 1 - α 2 γ 2 sin 2 γ = 1 - ( α π ) 2 sinc 2 γ π .
δ = β + tan - 1 ( β γ tan γ ) .
δ = β - tan - 1 ( β γ tan γ ) .
T = ( α π ) 2 sinc 2 ( γ π ) .
δ = β ± m π ,             m = 0 , 1 , .
A exp ( j δ ) = A 1 A 2 exp [ j ( δ 1 + δ 2 ) ] ,
H ( u , v ) = sin ( 2 π u a ) exp ( j 2 π v b ) ,
F ( u , v , z ) = F ( u , v , 0 ) exp [ j 2 π z λ 1 - ( λ u ) 2 - ( λ ν ) 2 ]

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