Abstract

A spatial light modulator design consisting of cascaded or sandwiched layers of ferroelectric liquid crystals (FLC’s) is investigated. The interrelation between the FLC material, the polarization of the incident illumination, and the achievable modulation states is characterized. Magnitude modulation is accomplished by standard methods by addressing the FLC layer with linearly polarized light and following it with a properly oriented analyzer. When the FLC is addressed with circularly polarized light, lossless phase modulation results with the phase states separated by twice the angle of rotation of the optical axes. A continuum of elliptical polarization states ties together the lossless phase states achievable by using circular polarization with the more well-known 0°–180° phase states obtainable with linearly polarized light. Layers of various bistable FLC materials can be cascaded, possibly with polarization control layers between some of the layers, to yield a spatial light modulator that produces multiple quantized bits of complex-valued modulation and with independent control of magnitude and phase states. Four-state phase modulation, ternary amplitude–phase modulation, and four-state magnitude modulation are demonstrated experimentally by using two layers of FLC.

© 1992 Optical Society of America

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References

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  1. W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), pp. 121–232.
  2. G. Moddel, K. M. Johnson, W. Li, R. A. Rice, L. A. Pagano-Stauffer, M. A. Handschy, “High speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).
  3. C. Warde, A. D. Fisher, “Spatial light modulators: application and functional capabilities.” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987).
  4. R. D. Juday, “Optical correlation with a cross-coupled spatial light modulator,” in Spatial Light Modulators and Applications, Vol. 8 of OSA 1988 Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 238.
  5. K. Lu, B. Saleh, “Complex amplitude reflectance of the liquid crystal light valve,” J. Opt. Soc. Am. A (to be published).
  6. D. L. Flannery, J. S. Loomis, M. E. Milkovich, “Transform-ratio ternary phase-amplitude filter formulation for improved correlation discrimination,” Appl. Opt. 27, 4079–4083 (1988).
  7. D. R. Pape, L. J. Hornbeck, “Characteristics of the deformable mirror device for optical information processing,” Opt. Eng. 22, 675–681 (1983).
  8. S. Voran, L. Scharf, “Memoryless scalar quantizers for noisy environments,” OCS Tech. Rep. 89-43 (Optoelectronic Computing Systems Center, University of Colorado, Boulder, Colo., 1989).
  9. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 121–126.
  10. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 17.

1989

G. Moddel, K. M. Johnson, W. Li, R. A. Rice, L. A. Pagano-Stauffer, M. A. Handschy, “High speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).

1988

1983

D. R. Pape, L. J. Hornbeck, “Characteristics of the deformable mirror device for optical information processing,” Opt. Eng. 22, 675–681 (1983).

Fisher, A. D.

C. Warde, A. D. Fisher, “Spatial light modulators: application and functional capabilities.” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987).

Flannery, D. L.

Handschy, M. A.

G. Moddel, K. M. Johnson, W. Li, R. A. Rice, L. A. Pagano-Stauffer, M. A. Handschy, “High speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).

Hornbeck, L. J.

D. R. Pape, L. J. Hornbeck, “Characteristics of the deformable mirror device for optical information processing,” Opt. Eng. 22, 675–681 (1983).

Johnson, K. M.

G. Moddel, K. M. Johnson, W. Li, R. A. Rice, L. A. Pagano-Stauffer, M. A. Handschy, “High speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).

Juday, R. D.

R. D. Juday, “Optical correlation with a cross-coupled spatial light modulator,” in Spatial Light Modulators and Applications, Vol. 8 of OSA 1988 Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 238.

Lee, W. H.

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), pp. 121–232.

Li, W.

G. Moddel, K. M. Johnson, W. Li, R. A. Rice, L. A. Pagano-Stauffer, M. A. Handschy, “High speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).

Loomis, J. S.

Lu, K.

K. Lu, B. Saleh, “Complex amplitude reflectance of the liquid crystal light valve,” J. Opt. Soc. Am. A (to be published).

Milkovich, M. E.

Moddel, G.

G. Moddel, K. M. Johnson, W. Li, R. A. Rice, L. A. Pagano-Stauffer, M. A. Handschy, “High speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).

Pagano-Stauffer, L. A.

G. Moddel, K. M. Johnson, W. Li, R. A. Rice, L. A. Pagano-Stauffer, M. A. Handschy, “High speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).

Pape, D. R.

D. R. Pape, L. J. Hornbeck, “Characteristics of the deformable mirror device for optical information processing,” Opt. Eng. 22, 675–681 (1983).

Rice, R. A.

G. Moddel, K. M. Johnson, W. Li, R. A. Rice, L. A. Pagano-Stauffer, M. A. Handschy, “High speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).

Saleh, B.

K. Lu, B. Saleh, “Complex amplitude reflectance of the liquid crystal light valve,” J. Opt. Soc. Am. A (to be published).

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 17.

Scharf, L.

S. Voran, L. Scharf, “Memoryless scalar quantizers for noisy environments,” OCS Tech. Rep. 89-43 (Optoelectronic Computing Systems Center, University of Colorado, Boulder, Colo., 1989).

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 17.

Voran, S.

S. Voran, L. Scharf, “Memoryless scalar quantizers for noisy environments,” OCS Tech. Rep. 89-43 (Optoelectronic Computing Systems Center, University of Colorado, Boulder, Colo., 1989).

Warde, C.

C. Warde, A. D. Fisher, “Spatial light modulators: application and functional capabilities.” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987).

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 121–126.

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 121–126.

Appl. Opt.

Appl. Phys. Lett.

G. Moddel, K. M. Johnson, W. Li, R. A. Rice, L. A. Pagano-Stauffer, M. A. Handschy, “High speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).

Opt. Eng.

D. R. Pape, L. J. Hornbeck, “Characteristics of the deformable mirror device for optical information processing,” Opt. Eng. 22, 675–681 (1983).

Other

S. Voran, L. Scharf, “Memoryless scalar quantizers for noisy environments,” OCS Tech. Rep. 89-43 (Optoelectronic Computing Systems Center, University of Colorado, Boulder, Colo., 1989).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 121–126.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 17.

C. Warde, A. D. Fisher, “Spatial light modulators: application and functional capabilities.” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987).

R. D. Juday, “Optical correlation with a cross-coupled spatial light modulator,” in Spatial Light Modulators and Applications, Vol. 8 of OSA 1988 Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 238.

K. Lu, B. Saleh, “Complex amplitude reflectance of the liquid crystal light valve,” J. Opt. Soc. Am. A (to be published).

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), pp. 121–232.

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Figures (13)

Fig. 1
Fig. 1

Two orientations of the optical axes in a smectic-C* FLC modulator. The FLC has axes ne, no in state 0 and ne′, no′ in state 1.

Fig. 2
Fig. 2

Magnitude modulation and 0°–180° phase modulation by using a FLC output analyzer. The input polarization is along the x axis. The FLC axes in one state are aligned at some angle α with respect to the x axis and are rotated by an angle θo for the other state. The resulting two output polarization states are shown on the right. An analyzer oriented at some angle β produces output light states that have different magnitudes and, for the case shown, are out of phase by 180°

Fig. 3
Fig. 3

Phase modulation as a function of |α| for FLC material with a 45° rotation of the optical axes between its two states. The curves for quadrant 1 and quadrant 4 result from choosing the phase of α to lie in these quadrants of the complex plane.

Fig. 4
Fig. 4

Transmitted power (i.e., the power that is in the desired polarization state) associated with a given phase modulation for a 45° FLC material.

Fig. 5
Fig. 5

Two configurations of FLC magnitude modulators: (A) n FLC’s with one analyzer, (B) n FLC’s with n analyzers.

Fig. 6
Fig. 6

|E0| versus γ for a four-state magnitude modulator.

Fig. 7
Fig. 7

Transfer function of an exponential modulator consisting of three FLC’s.

Fig. 8
Fig. 8

Cascading of n FLC devices to perform n bits of phase modulation.

Fig. 9
Fig. 9

Relative phase shift between the two output states of a magnitude modulator caused by deviations in the thickness of the FLC layer from a pure half-wave retarder.

Fig. 10
Fig. 10

Experimental setup that was used to measure phase modulation of the four-level FLC modulator.

Fig. 11
Fig. 11

Interferograms generated by the four-state FLC phase modulator.

Fig. 12
Fig. 12

Interferograms for ternary amplitude–phase modulator.

Fig. 13
Fig. 13

Output of the four-state FLC magnitude modulator.

Tables (2)

Tables Icon

Table I Results for the Four-state Phase Modulator

Tables Icon

Table II Results for the Four-State Magnitude Modulator

Equations (28)

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J r = exp [ - j π d / λ ( n e + n o ) ] [ exp ( - j Γ / 2 ) 0 0 exp ( j Γ / 2 ) ] ,
Γ = 2 π d λ ( n e - n o ) .
J π = exp ( j ϕ ) [ 1 0 0 - 1 ] .
R ( θ ) = [ cos θ sin θ - sin θ cos θ ] ,
J π ( θ ) = R ( - θ ) J π R ( θ ) = exp ( j ϕ ) [ cos 2 θ sin 2 θ sin 2 θ - cos 2 θ ] .
A ( β ) = R ( - β ) [ 1 0 0 0 ] R ( β ) = [ cos 2 β sin β cos β sin β cos β sin 2 β ] .
E o 0 = A ( β ) J π ( α ) [ 1 0 ] = exp ( j ϕ ) cos ( 2 α - β ) [ cos β sin β ]
E o 1 = A ( β ) J π ( α + θ o ) [ 1 0 ] = exp ( j ϕ ) cos ( 2 α + 2 θ o - β ) [ cos β sin β ] .
E o ( s ) = exp ( j ϕ ) cos ( 2 s θ o + β ^ ) [ cos β sin β ] .
E o ( s ) = J π ( s θ o ) [ 1 j ] = exp [ j ( ϕ + 2 s θ o ) ] [ 1 - j ] .
E i = 1 1 + α 2 [ 1 α ] ,
E o = C 1 ( s ) 1 + α 2 [ 1 α ] + C 2 ( s ) 1 + α 2 [ α * - 1 ] ,
C 1 ( 0 ) = 1 - α 2 1 + α 2 ,             C 1 ( 1 ) = 2 R e ( α ) 1 + α 2
C 2 ( 0 ) = 2 α 1 + α 2 ,             C 2 ( 1 ) = α 2 1 + α 2 .
cos 2 ϕ = 1 + α 4 2 α 2 - 2.
2 - 1 α 2 + 1.
ψ = arg [ C 2 ( 1 ) ] - arg [ C 2 ( 0 ) ] , = tan - 1 [ α 2 + 1 ( α 2 - 1 ) 2 6 α 2 - 1 - α 4 ] for quadrant 1 , = π - tan - 1 [ α 2 + 1 ( α 2 - 1 ) 2 6 α 2 - 1 - α 4 ] for quadrant 4.
P T = C 2 ( 0 ) 2 = 4 α 2 ( 1 + α 2 ) 2 .
E o ( s ) = A ( β ) i = 1 n [ J π ( s i θ i + α i ) ] [ 1 0 ] = cos γ [ cos β sin β ] ,
γ = β ^ + i = 1 n [ ( - 1 ) i + n 2 s i θ i ] .
E o ( s ) = i = 1 n [ cos ( 2 s i θ i + β ^ i ) ] [ cos β n sin β n ] .
cos β ^ i = 1
cos ( 2 θ i + β ^ i ) = k i
E o ( s ) = k m [ cos β n sin β n ] ,
m = i = 1 n s i i .
E o ( s ) = exp [ j ( 2 θ ^ + ϕ ^ ) ] [ 1 ( - 1 ) n j ] ,
θ ^ = i = 1 n s i θ i
E o ( s ) = R ( - s θ o ) J r R ( s θ o ) [ 1 j ] = exp [ j ( ϕ - π / 2 ) ] cos Γ 2 [ 1 j ] + exp [ j ( ϕ + 2 s θ o ) ] sin Γ 2 [ 1 - j ] .

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