Abstract

We show that computer generated holograms, implemented with amplitude-only liquid crystal spatial light modulators, allow the synthesis of fully complex fields with high accuracy. Our main discussion considers modified amplitude holograms whose transmittance is obtained by adding an appropriate bias function to the real cosine computer hologram of the encoded signal. We first propose a bias function, given by a soft envelope of the signal modulus, which is appropriate for perfect amplitude modulators. We also consider a second bias term, given by a constant function, which results appropriate for modulators whose amplitude transmittance is coupled with a linear phase modulation. The influence of the finite pixel size of the spatial light modulator is compensated by digital pre-filtering of the encoded complex signal. The performance of the discussed amplitude CGHs is illustrated by means of numerical simulations and the experimental synthesis of high order Bessel beams.

© 2005 Optical Society of America

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References

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Appl. Opt.

in Optical Pattern Recognition XI

R. D. Juday, J. M. Rollins, S. E. Monroe, Jr., and M. V. Morelli, �??Full-phase full-complex characterization of a reflective SLM,�?? in Optical Pattern Recognition XI, David P. Casasent, Tien-Hsin Chao; eds., Proc. SPIE 4043, 80-89 (2000).
[CrossRef]

Introduction to Fourier Optics

J. W. Goodman, �??Holography,�?? in Introduction to Fourier Optics (McGraw-Hill, 1996) pp. 295-392.

J. W. Goodman, �??Analog Optical Information Processing,�?? in Introduction to Fourier Optics (McGraw-Hill, 1996) pp. 217-294.

IRE Trans. Inform. Theory IT

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, �??Optical data processing and filtering systems,�?? IRE Trans. Inform. Theory IT-6, 386-400 (1960).
[CrossRef]

J. Opt. Soc. Am. A

J. Optics

I. Juvells, A. Carnicer, , S. Vallmitjana, and J. Campos, �??Implementation of real filters in a joint transform correlator using positive-only display,�?? J. Optics 25, 33 (1994).
[CrossRef]

JOSA A

Josep Nicolás, Juan Campos, María J. Yzuel, �??Phase and amplitude modulation of elliptic polarization states by nonabsorbing anisotropic elements: application to liquid-crystal devices,�?? JOSA A 19, 1013 (2002).
[CrossRef] [PubMed]

Lasers

A. E. Siegman, �??Wave Optics and Gaussian Beams�?? and �??Physical Properties of Gaussian Beams,�?? in Lasers (University Science Books, Mill Valley Ca., 1986) pp. 626-697.

Opt. Lett.

Optical Engineering

K. Lu, B. E. A. Saleh, "Theory and design of the liquid crystal TV as an optical spatial phase modulator," Optical Engineering 29, 240 (1990).
[CrossRef]

Proc. IEEE

J. Burch, �??A computer algorithm for the synthesis of spatial frequency filters,�?? Proc. IEEE 55, 599 (1967).
[CrossRef]

Prog. Opt.

W. H. Lee, �??Computer generated holograms,�?? Prog. Opt. 16, 121 (1978).

Other

L Allen, Stephen M Barnett, and Miles J Padgett, Optical Angular Momentum (Institute of Physics Publishing, 2003).
[CrossRef]

Supplementary Material (1)

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Figures (17)

Fig. 1.
Fig. 1.

(a) Modulus and (b) phase of a first order Bessel beam.

Fig. 2.
Fig. 2.

Profiles of the bias functions (of type 1 and 2) and the signal modulus a(x,y), for a first order Bessel beam, at the central row of the 2-D signal domain.

Fig. 3.
Fig. 3.

Signal spectra for the amplitude CGHs of types (a) 1 and (b) 2 encoding a first order Bessel beam, with carrier frequencies u0 =v0 =Δu/9.

Fig. 4.
Fig. 4.

Moduli of reconstructed signals for the CGHs of types (a) 1 and (b) 2, encoding the first order Bessel beam shown in Fig. 1.

Fig. 5.
Fig. 5.

SNR for non-compensated CGHs of types 1 (blue) and 2 (green) encoding a first order Bessel beam. (a) SNR versus u0 (with ρ0 =Δu/20), and (b) SNR versus ρ0 (with u0 =Δu/6).

Fig. 6.
Fig. 6.

Modulus of signal spectrum for a non-compensated type 2 CGH encoding a first order Bessel beam with radial frequency ρ0 =Δu/10 and carrier frequencies (a) u0 =v0 =Δu/6 and (b) u0 =v0 =Δu/3.

Fig. 7.
Fig. 7.

(a) SNR versus u0 for compensated CGHs of types 1 (blue) and 2 (green) encoding a first order Bessel beam. (b) SNR versus u0 for compensated CGHs of type 2 encoding Bessel beams of orders 2 (green), 3 (pink), 4 (black), and 5 (blue).

Fig. 8.
Fig. 8.

(a) Phase modulation of first order Bessel beams synthesized with compensated type 3 CGHs (a) without and (b) with coupled phase.

Fig. 9.
Fig. 9.

SNR versus u0 for compensated CGHs of type 3 (a) without and (b) with coupled phase. The signal spatial frequency is ρ0 =Δu/20. The orders of encoded Bessel beams are 1(blue), 2 (green), 3 (red), 4 (cyan), and 5 (magenta).

Fig. 10.
Fig. 10.

Normalized efficiency versus u 0 for (a) non-compensated and (b) compensated type 2 CGHs encoding Bessel beams of orders 2 (green), 3 (pink), 4 (black), and 5 (blue).

Fig. 11.
Fig. 11.

Arrangement of a TNLC-SLM with two linear polarizers to obtain amplitude modulation.

Fig. 12.
Fig. 12.

Experimental (a) amplitude and (b) phase modulations versus gray level g, and (c) phase versus amplitude modulation, measured for the TNLC-SLM with polarizers oriented at angles θ1=90°, θ2=0°.

Fig. 13.
Fig. 13.

Spatial filtering setup for the synthesis of arbitrary complex fields with amplitude CGHs.

Fig. 14.
Fig. 14.

Experimentally recorded first order Bessel beams generated with type 3 compensated amplitude CGHs, employing carrier frequencies u0 =v0 =Δu/4. The beam spatial frequencies are (a) ρ0 =Δu/12, (b) ρ0 =Δu/16, (c) ρ0 =Δu/20, and (d) ρ0 =Δu/24.

Fig. 15.
Fig. 15.

Experimentally recorded Bessel beams of orders (a) 2, (b) 3, and (c) 4, generated with type 3 compensated amplitude CGHs employing carrier frequencies u0 =v0 =Δu/4. The beam spatial frequency is ρ0 =Δu/20.

Fig. 16.
Fig. 16.

Interferograms of the experimentally generated Bessel beams of order (a) 1, (b) 2, (b) 3, and (b) 4, with spatial frequency ρ0 =Δu/20, employing a on-axis reference plane wave.

Fig. 17.
Fig. 17.

(393 KB) Rotating ring shaped field with azimuthal intensity modulation, generated by a sequence of type 3 compensated amplitude CGHs. The CGHs encode the complex field J4(2πρ0r) cos(4θ-Δϕ) in a finite circular support, with increasing values of the phase shift Δϕ.

Equations (21)

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s x y = a x y exp [ x y ] ,
h x y = c 0 { b x y + a x y cos [ ϕ x y 2 π ( u 0 x + v 0 y ) ] } ,
b x y = [ 1 + a 2 x y ] 2 .
H u v = c 0 [ B u v + 1 2 S u + u 0 v + v 0 + 1 2 S * u + u 0 v + v 0 ] ,
B u v = A 0 A u v exp ( ρ 2 α 2 ) ,
h x y = c 0 [ b x y + t s x y + t s * x y ] ,
h m x y = h x y exp [ i ϕ c x y ] ,
exp [ i ϕ c x y ] = n = 0 i n n ! ϕ c n x y .
h m x y = n = 0 i n γ n n ! h n + 1 x y .
h n + 1 x y = ( c 0 ) n + 1 q = 0 n + 1 ( n + 1 ) ! ( n + 1 q ) ! q ! ( b + t s ) n + 1 q t s * q ,
s m x y = { n = 0 ( n + 1 ) i n γ n c 0 n + 1 b n x y n ! } t s x y .
b x y = 1 .
S m u v = c 0 2 ( Δ u ) 2 [ W u v S u + u 0 v + v 0 ] P u v ,
W u v = a b sinc ( au ) sinc ( bv ) ,
S m u v = c 0 2 ( Δ u ) 2 [ W u v S u + u 0 v + v 0 ] P u v .
T u v S u + u 0 v + v 0 = S u + u 0 v + v 0 W 1 u v .
SNR = Ds s x y 2 dxdy Ds s x y β s t x y 2 dxdy ,
β = D s Re { s x y s t * x y } dxdy D s s t x y 2 dxdy ,
η norm = η i 1 D s s m x y 2 dxdy D s s x y 2 dxdy .
s r θ = a 0 J w ( 2 π ρ 0 r ) exp ( iwθ ) circ ( r R ) ,
s r θ = a 0 J 4 ( 2 π ρ 0 r ) cos ( 4 θ Δ ϕ ) circ ( r R ) ,

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