Abstract

We present a novel method of producing arbitrarily valued binary phase-only modulation from a commercially available ferroelectric liquid-crystal spatial light modulator that is used in conjunction with simple polarization components. By cascading of such stages, modulators with four and eight equally spaced phase levels are constructed with 128 × 128 pixels. Near-diffraction-limited performance, when stopped down to 64 × 64 pixels, is reported in producing simple diffraction patterns and when used to generate asymmetric spot arrays in the Fourier plane of a lens.

© 1995 Optical Society of America

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References

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  1. D. C. O’Brien, R. J. Mears, T. D. Wilkinson, W. A. Crossland, “Dynamic holographic interconnects that use ferroelectric liquid-crystal spatial light modulators,” Appl. Opt. 33, 2795–2803 (1994).
    [CrossRef]
  2. I. D. Thomas, “Programmable correction for optical aberrations using a phase-only spatial light modulator,” M.S. thesis (Oxford University, Oxford, UK, 1993).
  3. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  4. D. R. Pape, L. J. Hornbeck, “Characteristics of the deform-able mirror device for optical information processing,” Opt. Eng. 22, 675–681 (1983).
  5. J. Amako, T. Sonehara, “Kinoform using an electrically controlled birefringent liquid crystal spatial light modulator,” Appl. Opt. 30, 4622–4628 (1991).
    [CrossRef] [PubMed]
  6. M. O. Freeman, T. A. Brown, D. M. Walba, “Quantized complex ferroelectric liquid crystal spatial light modulators,” Appl. Opt. 31, 3917–3929 (1992).
    [CrossRef] [PubMed]
  7. J. A. Davis, S. W. Connely, G. W. Bach, R. A. Lilly, D. M. Cottrell, “Programmable optical interconnections with large fan-out capability using the magneto-optic spatial light modulator,” Opt. Lett. 14, 102–104 (1989).
    [CrossRef] [PubMed]
  8. S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, G. G. Yang, “Programmable binary phase-only optical device based on ferroelectric liquid crystal SLM,” Electron. Lett. 28, 26–28 (1992).
    [CrossRef]
  9. Model 2DX × 128, supplied by Thorn Electrical Music Industries, Central Research Laboratories, Smectic Technology, Dawley Road, Hayes, Middlesex UB3 1HH, UK.
  10. S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, “4-level, phase-only, spatial light modulator,” Electron. Lett. 29, 1661–1663 (1993).
    [CrossRef]
  11. T. M. Kreis, “Computer-aided evaluation of fringe patterns,” Opt. Lasers Eng. 19(4), 221–240 (1993).
    [CrossRef]
  12. M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [CrossRef] [PubMed]
  13. M. A. A. Neil, E. G. S. Paige, “Breaking of inversion symmetry in 2-level, binary, Fourier holograms,” IEE Conf. Publ. London 379, 85–90 (1993).
  14. M. A. A. Neil, E. G. S. Paige, “Improved Transmission in a 2-level, phase-only, spatial light modulator,” Electron. Lett. 30, 445–446 (1994).
    [CrossRef]

1994 (2)

D. C. O’Brien, R. J. Mears, T. D. Wilkinson, W. A. Crossland, “Dynamic holographic interconnects that use ferroelectric liquid-crystal spatial light modulators,” Appl. Opt. 33, 2795–2803 (1994).
[CrossRef]

M. A. A. Neil, E. G. S. Paige, “Improved Transmission in a 2-level, phase-only, spatial light modulator,” Electron. Lett. 30, 445–446 (1994).
[CrossRef]

1993 (3)

M. A. A. Neil, E. G. S. Paige, “Breaking of inversion symmetry in 2-level, binary, Fourier holograms,” IEE Conf. Publ. London 379, 85–90 (1993).

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, “4-level, phase-only, spatial light modulator,” Electron. Lett. 29, 1661–1663 (1993).
[CrossRef]

T. M. Kreis, “Computer-aided evaluation of fringe patterns,” Opt. Lasers Eng. 19(4), 221–240 (1993).
[CrossRef]

1992 (2)

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, G. G. Yang, “Programmable binary phase-only optical device based on ferroelectric liquid crystal SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

M. O. Freeman, T. A. Brown, D. M. Walba, “Quantized complex ferroelectric liquid crystal spatial light modulators,” Appl. Opt. 31, 3917–3929 (1992).
[CrossRef] [PubMed]

1991 (1)

1989 (1)

1987 (1)

1984 (1)

1983 (1)

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

Allebach, J. P.

Amako, J.

Bach, G. W.

Broomfield, S. E.

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, “4-level, phase-only, spatial light modulator,” Electron. Lett. 29, 1661–1663 (1993).
[CrossRef]

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, G. G. Yang, “Programmable binary phase-only optical device based on ferroelectric liquid crystal SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

Brown, T. A.

Connely, S. W.

Cottrell, D. M.

Crossland, W. A.

Davis, J. A.

Freeman, M. O.

Gianino, P. D.

Hornbeck, L. J.

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

Horner, J. L.

Kreis, T. M.

T. M. Kreis, “Computer-aided evaluation of fringe patterns,” Opt. Lasers Eng. 19(4), 221–240 (1993).
[CrossRef]

Lilly, R. A.

Mears, R. J.

Neil, M. A. A.

M. A. A. Neil, E. G. S. Paige, “Improved Transmission in a 2-level, phase-only, spatial light modulator,” Electron. Lett. 30, 445–446 (1994).
[CrossRef]

M. A. A. Neil, E. G. S. Paige, “Breaking of inversion symmetry in 2-level, binary, Fourier holograms,” IEE Conf. Publ. London 379, 85–90 (1993).

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, “4-level, phase-only, spatial light modulator,” Electron. Lett. 29, 1661–1663 (1993).
[CrossRef]

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, G. G. Yang, “Programmable binary phase-only optical device based on ferroelectric liquid crystal SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

O’Brien, D. C.

Paige, E. G. S.

M. A. A. Neil, E. G. S. Paige, “Improved Transmission in a 2-level, phase-only, spatial light modulator,” Electron. Lett. 30, 445–446 (1994).
[CrossRef]

M. A. A. Neil, E. G. S. Paige, “Breaking of inversion symmetry in 2-level, binary, Fourier holograms,” IEE Conf. Publ. London 379, 85–90 (1993).

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, “4-level, phase-only, spatial light modulator,” Electron. Lett. 29, 1661–1663 (1993).
[CrossRef]

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, G. G. Yang, “Programmable binary phase-only optical device based on ferroelectric liquid crystal SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

Pape, D. R.

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

Seldowitz, M. A.

Sonehara, T.

Sweeney, D. W.

Thomas, I. D.

I. D. Thomas, “Programmable correction for optical aberrations using a phase-only spatial light modulator,” M.S. thesis (Oxford University, Oxford, UK, 1993).

Walba, D. M.

Wilkinson, T. D.

Yang, G. G.

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, G. G. Yang, “Programmable binary phase-only optical device based on ferroelectric liquid crystal SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

Appl. Opt. (5)

Electron. Lett. (3)

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, “4-level, phase-only, spatial light modulator,” Electron. Lett. 29, 1661–1663 (1993).
[CrossRef]

M. A. A. Neil, E. G. S. Paige, “Improved Transmission in a 2-level, phase-only, spatial light modulator,” Electron. Lett. 30, 445–446 (1994).
[CrossRef]

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, G. G. Yang, “Programmable binary phase-only optical device based on ferroelectric liquid crystal SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

IEE Conf. Publ. London (1)

M. A. A. Neil, E. G. S. Paige, “Breaking of inversion symmetry in 2-level, binary, Fourier holograms,” IEE Conf. Publ. London 379, 85–90 (1993).

Opt. Eng. (1)

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

Opt. Lasers Eng. (1)

T. M. Kreis, “Computer-aided evaluation of fringe patterns,” Opt. Lasers Eng. 19(4), 221–240 (1993).
[CrossRef]

Opt. Lett. (1)

Other (2)

Model 2DX × 128, supplied by Thorn Electrical Music Industries, Central Research Laboratories, Smectic Technology, Dawley Road, Hayes, Middlesex UB3 1HH, UK.

I. D. Thomas, “Programmable correction for optical aberrations using a phase-only spatial light modulator,” M.S. thesis (Oxford University, Oxford, UK, 1993).

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Figures (16)

Fig. 1
Fig. 1

Scheme for producing phase-only modulation from a single FLCSLM pixel showing coordinate axes and angles (see text).

Fig. 2
Fig. 2

Intensity transmission plotted versus device parameters Δ and Φ. Contours range from <10% (black) to >90% (white) in steps of 10%: (a) full scheme, {0, π/2} or {0, 3π/2}; (b) reduced scheme, {0, π/2} or {0, 3π/2}; (c) full scheme, {0, π/4}, {0, 3π/4}, {0, 5π/4}, or {0, 7π/4}; (d) reduced scheme, {0, π/4}, {0, 3π/4}, {0, 5π/4}, or {0, 7π/4}.

Fig. 3
Fig. 3

Simplified mode of operation in which the input and the output polarizations are fixed and only one rotatable wave plate is used.

Fig. 4
Fig. 4

Experimental arrangement used to measure modulator performance: P1–P3, polarizers; W1, λ/2 wave plate; W2, λ/4 wave plate; BS1, BS2, beam splitters.

Fig. 5
Fig. 5

Region of an interferogram used to measure the phase modulation (ϕ = π/2 in this case) at a step discontinuity in the state of the FLCSLM pixels.

Fig. 6
Fig. 6

(a) Measured phase modulation as a function of half-wave-plate rotation (θ h ) and calculated phase modulation ϕ assuming Φ = 30° and Δ = 223° (solid curve). (b) Measured intensity transmission as a function of half-wave-plate rotation (θ h ) and calculated intensity transmission assuming Φ = 30° and Δ = 223° (solid curve).

Fig. 7
Fig. 7

Experimental arrangement of the eight-level programmable phase modulator with Fourier-plane output: S1–S3, SLM’s; P1–P4, linear polarizers; W1, λ/2 retarder; W2, W3, λ/4 retarders; L1–L4, lenses with focal lengths of 200 mm; L5, lens with a focal length of 180 mm; FP, Fourier plane.

Fig. 8
Fig. 8

Experimental grating performance. Linear and logarithmic plots of line scans of the normalized intensity through Fourier-plane diffraction spots plotted versus normalized position X (equal to diffraction order): (a) clear aperture, (b) two-level grating, (c) four-level grating, and (d) eight-level grating.

Fig. 9
Fig. 9

Same as Fig. 8 but for theoretical grating performance.

Fig. 10
Fig. 10

Asymmetric spot-array generator pixel patterns: (a) four-level pattern and (b) eight-level pattern.

Fig. 11
Fig. 11

Fourier-plane asymmetric spot arrays: (a) four-level phase pattern and (b) eight-level phase pattern.

Fig. 12
Fig. 12

Poincaré sphere construction.

Fig. 13
Fig. 13

Poincaré sphere showing the effect of passing an arbitrary polarization through an FLCSLM to produce either polarization S + or S (see text).

Fig. 14
Fig. 14

T plotted on the complex plane as a function of θ a for ψ = 25°, also showing the unit circle.

Fig. 15
Fig. 15

Locus of F′ and the resulting elliptical transmission functions for a device with Φ = 30° and Δ = 223°.

Fig. 16
Fig. 16

Geometrical construction of the parabolic ellipse envelope.

Tables (2)

Tables Icon

Table 1 Predicted and Measured Peak Normalized Intensities of Various Diffraction Orders for Two-, Four-, and Eight-Level Gratings a

Tables Icon

Table 2 Predicted and Measured Fourier-Plane Percentage Powers of Features for Four- and Eight-Level Spot Array Generator Patterns Shown in Fig. 10

Equations (34)

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η max ( M ) = [ M π sin ( π M ) ] 2 ,
Δ = 2 π d λ ( n e n o ) .
S ± = A · W ± · S i .
T max = sin 2 ( Δ / 2 ) sin 2 ( Φ ) sin 2 ( ϕ / 2 )
T max = 1 sin 2 ( Δ / 2 ) sin 2 ( Φ ) cos 2 ( ϕ / 2 ) ,
S ± ( Φ , Δ , θ p , θ a ) = P ( θ a ) · W ( 45 , 90 ) · W ( ± Φ 2 , Δ ) · W ( 45 , 90 ) · S ( θ p , 0 ) .
4 T ( Φ , Δ , θ p , θ a ) = ( 1 + cos Δ ) ( 1 cos 2 θ p cos 2 θ a ) + cos 2 Φ ( 1 cos Δ ) ( 1 + cos 2 θ p cos 2 θ a ) + 2 cos Φ sin Δ ( cos 2 θ p sin 2 θ a + sin 2 θ p cos 2 θ a ) + 2 cos Δ sin 2 θ p sin 2 θ a + j [ sin 2 Φ ( cos Δ 1 ) ( cos 2 θ p + cos 2 θ a ) 2 sin Φ sin Δ ( sin 2 θ p + sin 2 θ a ) ] .
4 T ( Φ , Δ , θ h ) = 1 + cos Δ ( 1 + 2 sin 4 θ h ) + cos 2 Φ ( 1 cos Δ ) + 2 cos Φ sin Δ cos 4 θ h + j [ sin 2 Φ ( cos Δ 1 ) cos 4 θ h 2 sin Φ sin Δ ( sin 4 θ h + 1 ) ] .
I = { sin ( π X / M ) sin [ π N p ( X 1 ) ] π N p X sin [ π ( X 1 ) / M ] } 2 ,
S ( ψ ) = [ cos ψ j sin ψ ] .
S ( θ , ψ ) = R ( θ ) · S ( ψ ) ,
R ( θ ) = [ cos θ sin θ sin θ cos θ ] .
P 0 = [ 1 0 0 0 ] ,
W 0 ( Δ ) = [ exp ( j Δ 2 ) 0 0 exp ( j Δ 2 ) ] .
P ( θ ) = R ( θ ) · P 0 · R ( θ ) ,
W ( θ , Δ ) = R ( θ ) · W 0 ( Δ ) · R ( θ ) .
S 1 ( θ a , ψ ) = P ( θ a ) · S ( ψ ) ,
S 2 ( θ a , ψ ) = P ( θ a ) · S ( ψ )
T ( θ a , ψ ) = S 1 ( θ a , ψ ) · S 2 * ( θ a , ψ ) .
2 T ( θ a , ψ ) = cos ( 2 ψ ) + cos ( 2 θ a ) j sin ( 2 ψ ) sin ( 2 θ a ) .
2 T ( θ a , ψ , ϕ ) = exp ( j ϕ ) [ cos ( 2 ψ ) + cos ( 2 θ a ) j sin ( 2 ψ ) sin ( 2 θ a ) ] ,
F ( ψ , ϕ ) = exp ( j ϕ ) cos ( 2 ψ ) .
S ± ( Φ , Δ , θ i , ψ ) = W ( ± Φ 2 , Δ ) · S ( θ i , ψ ) .
2 F ( Φ , Δ , θ i , ψ ) = 1 + cos Δ + cos 2 Φ ( 1 cos Δ ) + j [ sin 2 Φ sin 2 ψ ( cos Δ 1 ) 2 sin Φ cos 2 ψ sin Δ sin 2 θ i ] .
P T max = 1 Re ( F ) 1 cos ( ϕ ) = sin 2 ( Φ ) sin 2 ( Δ / 2 ) sin 2 ( ϕ / 2 ) ,
P T max = 1 + Re ( F ) 1 + cos ( ϕ ) = 1 sin 2 ( Φ ) sin 2 ( Δ / 2 ) cos 2 ( ϕ / 2 ) .
F = 1 sin 2 ( Φ ) [ 1 cos ( Δ ) ] j sin ( Φ ) sin ( Δ ) .
( x 2 + y 2 ) 1 / 2 + [ ( x a ) 2 + ( y b ) 2 ] 1 / 2 = 1 ,
{ x = 0 , y = positive } T π / 2 = y = sin 2 ( 2 Φ ) sin 4 ( Δ / 2 ) 2 [ 1 + sin ( Φ ) sin ( Δ ) ] ,
{ x = 0 , y = negative } T 3 π / 2 = y = sin 2 ( 2 Φ ) sin 4 ( Δ / 2 ) 2 [ 1 sin ( Φ ) sin ( Δ ) ] ,
{ x = y = positive } T π / 4 = 2 x = sin 2 ( 2 Φ ) sin 4 ( Δ / 2 ) 2 2 [ 1 sin ( Φ ) sin ( Δ ) 2 sin 2 ( Φ ) sin 2 ( Δ / 2 ) ] ,
{ x = y = positive } T 3 π / 4 = 2 x = sin 2 ( 2 Φ ) sin 4 ( Δ / 2 ) 2 + 2 [ 1 + sin ( Φ ) sin ( Δ ) 2 sin 2 ( Φ ) sin 2 ( Δ / 2 ) ] ,
{ x = y = negative } T 5 π / 4 = 2 x = sin 2 ( 2 Φ ) sin 4 ( Δ / 2 ) 2 + 2 [ 1 sin ( Φ ) sin ( Δ ) 2 sin 2 ( Φ ) sin 2 ( Δ / 2 ) ] ,
{ x = y = positive } T 7 π / 4 = 2 x = sin 2 ( 2 Φ ) sin 4 ( Δ / 2 ) 2 2 [ 1 + sin ( Φ ) sin ( Δ ) 2 sin 2 ( Φ ) sin 2 ( Δ / 2 ) ] .

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