Abstract

An analysis of dynamic phase-only holograms, described by fractional notation and recorded onto a pixelated spatial light modulator (SLM) in a reconfigurable optical beam-steering switch, is presented. The phase quantization and arrangement of the phase states and the SLM pixelation and dead-space effects are decoupled, expressed analytically, and simulated numerically. The phase analysis with a skip–rotate rule reveals the location and intensity of each diffraction order at the digital replay stage. The optical reconstruction of the holograms recorded onto SLM’s with rectangular pixel apertures entails sinc-squared scaling, which further reduces the intensity of each diffraction order. With these two factors taken into account, the highest values of the nonuniform first-order diffraction efficiencies are expected to be 33%, 66%, and 77% for two-, four-, and and eight-level one-dimensional holograms with a 90% linear pixel fill factor. The variation of the first-order diffraction efficiency and the relative replay intensities were verified to within 1 dB by performing the optical reconstruction of binary phase-only holograms recorded onto a ferroelectric liquid crystal on a silicon SLM.

© 2001 Optical Society of America

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References

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  1. K. L. Tan, W. A. Crossland, R. J. Mears, “Dynamic holography for optical interconnections. I. Noise floor of low-cross-talk holographic switches,” J. Opt. Soc. Am. A 18, 195–204 (2001).
    [CrossRef]
  2. H. Dammann, “Blazed synthetic phase-only holograms,” Optik (Stuttgart) 31, 95–104 (1970).
  3. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  4. J. A. Cox, “Diffractive efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
    [CrossRef]
  5. P. D. Gianino, C. L. Woods, “General treatment of spatial light modulator dead-zone effects on optical correlation. I. Computer simulations.” “II. Mathematical analysis,”Appl. Opt. 32, 6527–6535, 6536–6541 (1993).
    [CrossRef] [PubMed]
  6. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  7. 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]
  8. M. R. Feldman, C. C. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
    [CrossRef] [PubMed]
  9. S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
    [CrossRef]
  10. 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]
  11. C. Dragone, “An N×N optical multiplexer using a planar arrangement of two star couplers,” IEEE Photonics Technol. Lett. 3, 812–815 (1991).
    [CrossRef]
  12. M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process 2, 221–242 (1992).
  13. E. G. Steward, Fourier Optics: An Introduction, 2nd ed. (Wiley, New York, 1987), pp. 106–115.
  14. Reconfigurable Optical Switches for Aerospace and Telecommunications Systems (ROSES), , was a collaborative Department of Trade and Industry–Engineering and Physical Sciences Research Council Link Photonics project of British Aerospace, Central Research Laboratories, the Cambridge University Engineering Department, King’s College London, Northern Telecom, and Thomas Swan & Co. Ltd. (TS) and was managed by TS (Cambridge University Engineering Department, Cambridge). The binary SLM’s developed under this program contain a 6-mm-tall linear array of 540×1pixels at 20-μm pixel pitch.
  15. CDRR materials were developed at Hull University in the .
  16. M. M. Redmond, “Alignment and preliminary LC assessment of binary SLM W2/D22,” ROSES-CUED 8th quarterly rep. (Cambridge U. Engineering Department, Cambridge, UK, 1998), pp. 6–7.
  17. P. Berthele, B. Fracasso, J. L. de Bougrenet de la Tocnaye, “Design and characterization of a liquid-crystal spatial light modulator for a polarization-insensitive optical space switch,” Appl. Opt. 37, 5461–5468 (1998).
    [CrossRef]

2001 (1)

1998 (1)

1993 (1)

P. D. Gianino, C. L. Woods, “General treatment of spatial light modulator dead-zone effects on optical correlation. I. Computer simulations.” “II. Mathematical analysis,”Appl. Opt. 32, 6527–6535, 6536–6541 (1993).
[CrossRef] [PubMed]

1992 (1)

M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process 2, 221–242 (1992).

1991 (1)

C. Dragone, “An N×N optical multiplexer using a planar arrangement of two star couplers,” IEEE Photonics Technol. Lett. 3, 812–815 (1991).
[CrossRef]

1989 (3)

1987 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

1970 (1)

H. Dammann, “Blazed synthetic phase-only holograms,” Optik (Stuttgart) 31, 95–104 (1970).

Allebach, J. P.

Bach, G. W.

Berthele, P.

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Bryngdahl, O.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

Connely, S. W.

Cottrell, D. M.

Cox, J. A.

J. A. Cox, “Diffractive efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

Crossland, W. A.

Dammann, H.

H. Dammann, “Blazed synthetic phase-only holograms,” Optik (Stuttgart) 31, 95–104 (1970).

Davis, J. A.

de Bougrenet de la Tocnaye, J. L.

Dragone, C.

C. Dragone, “An N×N optical multiplexer using a planar arrangement of two star couplers,” IEEE Photonics Technol. Lett. 3, 812–815 (1991).
[CrossRef]

Feldman, M. R.

Fracasso, B.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Gianino, P. D.

P. D. Gianino, C. L. Woods, “General treatment of spatial light modulator dead-zone effects on optical correlation. I. Computer simulations.” “II. Mathematical analysis,”Appl. Opt. 32, 6527–6535, 6536–6541 (1993).
[CrossRef] [PubMed]

Guest, C. C.

Lilly, R. A.

Mears, R. J.

Redmond, M. M.

M. M. Redmond, “Alignment and preliminary LC assessment of binary SLM W2/D22,” ROSES-CUED 8th quarterly rep. (Cambridge U. Engineering Department, Cambridge, UK, 1998), pp. 6–7.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Seldowitz, M. A.

Steward, E. G.

E. G. Steward, Fourier Optics: An Introduction, 2nd ed. (Wiley, New York, 1987), pp. 106–115.

Sweeney, D. W.

Taghizadeh, M. R.

M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process 2, 221–242 (1992).

Tan, K. L.

Turunen, J.

M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process 2, 221–242 (1992).

Weissbach, S.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

Woods, C. L.

P. D. Gianino, C. L. Woods, “General treatment of spatial light modulator dead-zone effects on optical correlation. I. Computer simulations.” “II. Mathematical analysis,”Appl. Opt. 32, 6527–6535, 6536–6541 (1993).
[CrossRef] [PubMed]

Wyrowski, F.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

Appl. Opt. (3)

IEEE Photonics Technol. Lett. (1)

C. Dragone, “An N×N optical multiplexer using a planar arrangement of two star couplers,” IEEE Photonics Technol. Lett. 3, 812–815 (1991).
[CrossRef]

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

Opt. Commun. (1)

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

Opt. Comput. Process (1)

M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process 2, 221–242 (1992).

Opt. Lett. (2)

Optik (Stuttgart) (2)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

H. Dammann, “Blazed synthetic phase-only holograms,” Optik (Stuttgart) 31, 95–104 (1970).

Other (6)

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

J. A. Cox, “Diffractive efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

E. G. Steward, Fourier Optics: An Introduction, 2nd ed. (Wiley, New York, 1987), pp. 106–115.

Reconfigurable Optical Switches for Aerospace and Telecommunications Systems (ROSES), , was a collaborative Department of Trade and Industry–Engineering and Physical Sciences Research Council Link Photonics project of British Aerospace, Central Research Laboratories, the Cambridge University Engineering Department, King’s College London, Northern Telecom, and Thomas Swan & Co. Ltd. (TS) and was managed by TS (Cambridge University Engineering Department, Cambridge). The binary SLM’s developed under this program contain a 6-mm-tall linear array of 540×1pixels at 20-μm pixel pitch.

CDRR materials were developed at Hull University in the .

M. M. Redmond, “Alignment and preliminary LC assessment of binary SLM W2/D22,” ROSES-CUED 8th quarterly rep. (Cambridge U. Engineering Department, Cambridge, UK, 1998), pp. 6–7.

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Figures (11)

Fig. 1
Fig. 1

Free-space 1×N optical switch with a coherent 4f setup. The size of each replay replication, ΔR, is fλ/d, a consequence of pixel periodicity.

Fig. 2
Fig. 2

Beam steering with periodic diffractive optical elements with a full period shown: (a) gratings composed of a microprism array, (b) step-phase gratings blazed for the desired main replay order, (c)–(e) thin optical elements that have only limited numbers of phase levels m, and up to 2π phase depth. With each beam-steering technique the phase profile and the associated replay plane image within the two central replications are depicted for three routing fractions (σ=1/8, 2/8, 3/8).

Fig. 3
Fig. 3

Overlap of higher replay orders in the numerical replay grid of binary gratings. FFT, fast Fourier transform.

Fig. 4
Fig. 4

Modulo-1 skip–rotate rule used to locate higher replay orders of a σ=1/10 quaternary hologram replay fraction.

Fig. 5
Fig. 5

First ten orders of a (1/10, 3/8) quaternary base hologram. Diamonds, locations of the orders present, with the number of contour lines proportional to the intensity values; circles, absent orders. These diamonds do not represent the beam profile of the replay peaks.

Fig. 6
Fig. 6

1-D dynamic hologram recording as depicted by the convolution of each calculated value of an infinitely repeated base hologram with the clear aperture transmittance of a single SLM pixel. A constant dead-space transmittance is appended to the pixel (inset), and the resultant spatial distribution is then multiplied by the finite hologram illumination. The reconstructed optical beam has its profile determined by this finite illumination, whereas the distribution of the diffractive power is dependent on both the hologram and the pixel functions.

Fig. 7
Fig. 7

Composite effects of spatial and phase quantization on the intensity of a phase-matched 3/8 quaternary replay fraction. Small open circles, the nonzero replay orders within the central replay replication along the intensity roll-off (dotted curves). ×, intensity of the central zero order. Dashed curve, intensities of the zero orders in other replications. (n, p), order and replication number.

Fig. 8
Fig. 8

Predicted first-order hologram replay efficiency at a given replay fraction in the central replay replication. The number of phase quantization levels are 2, 4 and 8. The sole effect of spatial quantization with fill factor ρ=0.9 is plotted (by a dotted curve).

Fig. 9
Fig. 9

Reflective experimental setup for measuring dynamic hologram replays.

Fig. 10
Fig. 10

Six replay orders of a σ=1/8 binary hologram: (a) 2-D scan of each replay order, (b) maximum relative coupled intensities, (c) predicted relative replay intensities.

Fig. 11
Fig. 11

Superimposed line scans of 60 successive replay orders (1/120 to 60/120): (a) experimental measurements with an FLC/Si SLM for hologram displays, collected automatically without adjustment of y and z axes, (b) simulated replay with oversampling of each hologram point function and zero-padded Gaussian illumination truncated at its spot diameter. Circles, highest intensities measured when the coupling condition for each first replay order was individually optimized.

Tables (1)

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Table 1 Losses in Use of a ROSES 2/21b SLM for Binary Hologram Encoding

Equations (21)

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|hˆ(xd, yd)|2=ns-t-ηm;nξ,ζδ[xd-(nξ+sN),yd-(nζ+tN)],
-N/2(nξ+sN, nζ+tN)<(N/2),
-12  (nσ+s, nτ+t) < 12,
h(nx, ny)=1x0y0kx=0x0-1ky=0y0-1H(kx, ky)×exp2πjnxkxx0+nykyy0×{sinc(πnx/x0)sinc(πny/y0)},
ηm;nσ=g=-sinc2nπm+g x0πm,
ηm;nσ=sinc2nπmg=-nn+gx02.
ηm;nσ=sinc2(nπ/m)sinc2(nπ/x0).
ηm=2;n=±1σ=1/2=sinc2(π/2)sinc2(π/2)=100%,
ηm=2;n=±1σ=1/4=sinc2(π/2)sinc2(π/4)=50%,
ηm=2;n=±1σ=1/8=sinc2(π/2)sinc2(π/8)=42.68%,
ηm=2;n=±1σ=1/16=sinc2(π/2)sinc2(π/16)=41.05%,
ηm;nσ,τ=sinc2(nπ/m)sinc2[nπ/lcm(m, x0, y0)],
H(xh, yh)
=p=-q=-δ(xh-p, yh-q)×Hˆ(p, q) * α rect xhρ, yhρ+p=-q=-δ(xh-p, yh-q)×β exp(jψ)*1-rect xhρ, yhρ×Hi(xh, yh),
h(xr, yr)
=p,qs,tnAm;nαρ2sinc[ρπ(p+s+nσ)]×sinc(ρπ[q+t+nτ)]+β(1-ρ2)exp(jψ)-β exp(jψ)pqρ2sinc(ρpπ)sinc(ρqπ)
 * hi(xr, yr),
|h(xr, yr)m;n0|2=np=-q=-ηm;ρ;n0σ,τ;p,q,
ηρ;n0σ,τ;p,q=s,tα2ρ4sinc2[ρπ(p+s+nσ)]×sinc2[ρπ(q+t+nτ)].
ηm;ρ;n0σ,τ;p=0,q=0=ns,tsinc2(nπ/m)sinc2nπlcm(m, x0, y0)×α2ρ4sinc2[ρπ(s+nσ)]×sinc2[ρπ(t+nτ)].
ηm;ρ;n=1σ,τ;p=0,q=0=sinc2(π/m)sinc2[π/lcm(m, x0, y0)]×α2ρ4sinc2(ρσπ)sinc2(ρτπ).

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