Abstract

This paper presents what is to the author’s knowledge a new theory for phase-only holograms. It explains many phenomena observed in the reconstruction of phase-only Fourier holograms and in particular the existence and nature of noise. Using this theory it was demonstrated that any reconstruction is the convolution of the target reconstruction with a two-dimensional set of impulses. The nature of this convolution function depends on the target reconstruction and in particular the spacing between the spots. In the simplest case of a two-spot generating hologram the convolution function was calculated analytically. The theory is also verified with examples from spot generating holograms, symbology, and image projection holograms.

© 2010 Optical Society of America

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2008 (2)

2005 (2)

A. G. Georgiou, M. Komarcevic, T. D. Wilkinson, and W. A. Crossland, “Hologram optimisation using liquid crystal modelling,” Mol. Cryst. Liq. Cryst. 343, 511–526 (2005).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Robert, 2005).

2004 (2)

2001 (1)

2000 (2)

1995 (1)

1994 (1)

1986 (2)

1983 (1)

S. Kirkpatrick, C. Gelatt, and J. M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

1978 (1)

1972 (1)

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

1971 (1)

H. Dammann and K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Abushagur, M. A. G.

Akahori, H.

Ashkin, A.

Bjorkholm, J. E.

Bonas, I. G.

Christmas, J.

Chu, H. H.

Chu, S.

Collings, N.

Cooper, J.

Courtial, J.

Crossland, A. W.

Crossland, W.

Crossland, W. A.

A. Georgiou, J. Christmas, J. Moore, A. Jeziorska, A. Davey, N. Collings, and W. A. Crossland, “Liquid crystal on silicon device characteristics for holographic projection of high-definition television images,” Appl. Opt. 47, 4793–4803 (2008).
[CrossRef] [PubMed]

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10, 035302 (2008).
[CrossRef]

A. G. Georgiou, M. Komarcevic, T. D. Wilkinson, and W. A. Crossland, “Hologram optimisation using liquid crystal modelling,” Mol. Cryst. Liq. Cryst. 343, 511–526 (2005).

A. Georgiou and W. A. Crossland, “Image projection using phase-only holograms,” in Photon04 Conference Proceedings (2004).

W. A. Crossland, I. G. Manolis, M. M. Redmond, K. L. Tan, T. D. Wilkinson, M. J. Holmes, T. R. Parker, H. H. Chu, J. Croucher, V. A. Handerek, S. T. Warr, B. Robertson, I. G. Bonas, R. Franklin, C. Stace, H. J. White, R. A. Woolley, and H. G., “Holographic optical switching: the “ROSES” demonstrator,” J. Lightwave Technol. 18, 1845–1854 (2000).
[CrossRef]

Croucher, J.

Dammann, H.

H. Dammann and K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Davey, A.

Dziedzic, J. M.

Fienup, J. R.

Franklin, R.

G., H.

Gelatt, C.

S. Kirkpatrick, C. Gelatt, and J. M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Georgiou, A.

A. Georgiou, J. Christmas, J. Moore, A. Jeziorska, A. Davey, N. Collings, and W. A. Crossland, “Liquid crystal on silicon device characteristics for holographic projection of high-definition television images,” Appl. Opt. 47, 4793–4803 (2008).
[CrossRef] [PubMed]

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10, 035302 (2008).
[CrossRef]

A. Georgiou and W. A. Crossland, “Image projection using phase-only holograms,” in Photon04 Conference Proceedings (2004).

Georgiou, A. G.

A. G. Georgiou, M. Komarcevic, T. D. Wilkinson, and W. A. Crossland, “Hologram optimisation using liquid crystal modelling,” Mol. Cryst. Liq. Cryst. 343, 511–526 (2005).

Gerchberg, R. W.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Robert, 2005).

J. W. Goodman, Statistical Optics (Wiley, 2000).

Gortler, K.

H. Dammann and K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Handerek, V. A.

Holmes, M. J.

Ilias, M. G.

Jeziorska, A.

Johnson, E. G.

Jordan, P.

Kirkpatrick, S.

S. Kirkpatrick, C. Gelatt, and J. M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Komarcevic, M.

A. G. Georgiou, M. Komarcevic, T. D. Wilkinson, and W. A. Crossland, “Hologram optimisation using liquid crystal modelling,” Mol. Cryst. Liq. Cryst. 343, 511–526 (2005).

Laczik, Z.

Leach, J.

Manolis, I. G.

Mears, R.

Moore, J.

O’Brien, D.

Padgett, M.

Parker, T. R.

Redmond, M. M.

Robertson, B.

Saxton, W. O.

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

Sinclair, G.

Stace, C.

Tan, K. L.

Vecchi, J. M. P.

S. Kirkpatrick, C. Gelatt, and J. M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Warr, S. T.

White, H. J.

Wilkinson, T.

Wilkinson, T. D.

Woolley, R. A.

Appl. Opt. (3)

J. Lightwave Technol. (1)

J. Opt. A, Pure Appl. Opt. (1)

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10, 035302 (2008).
[CrossRef]

J. Opt. Soc. Am. A (2)

Mol. Cryst. Liq. Cryst. (1)

A. G. Georgiou, M. Komarcevic, T. D. Wilkinson, and W. A. Crossland, “Hologram optimisation using liquid crystal modelling,” Mol. Cryst. Liq. Cryst. 343, 511–526 (2005).

Opt. Commun. (1)

H. Dammann and K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Optik (Stuttgart) (1)

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

Science (1)

S. Kirkpatrick, C. Gelatt, and J. M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Other (3)

A. Georgiou and W. A. Crossland, “Image projection using phase-only holograms,” in Photon04 Conference Proceedings (2004).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Robert, 2005).

J. W. Goodman, Statistical Optics (Wiley, 2000).

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Figures (8)

Fig. 1
Fig. 1

A phase-only hologram will distort the phase of a flat wavefront. The lens focuses the distorted wavefront into its focal plane forming the reconstruction plane, which is given by the Fourier transform of the phase-only hologram.

Fig. 2
Fig. 2

Target function E T is equal to function E T . However, the function E T consists of impulses (δ functions) that have amplitude of ε. This manipulation may be performed because E T is a known variable and can be re-arranged in any way as long as its value for each u and v does not change.

Fig. 3
Fig. 3

Phase-only hologram can be represented as the product of a real and positive function τ ( x , y ) multiplied with the ideal hologram H ( x , y ) . On the reconstruction plane this is equivalent to convolving the target plane with a convolution function T ( u , v ) .

Fig. 4
Fig. 4

(a) Target reconstruction consisting of four spots. (b) The function T ( u , v ) consisting of a single central M 0 order, 12 orders with amplitude M 1 , and many more with coefficients of M 2 , M 3 , etc. (c) When the function T ( u , v ) is convolved with the target plane, each target spot is replaced with the function T ( u , v ) resulting into the four target orders and 24 noise orders due to the M 1 orders of the function T ( u , v ) .

Fig. 5
Fig. 5

Reconstruction of a four-spot generating hologram. The four thick circles in the center denote the target spots. All other circles show the predicted position of the noise spots with coefficients of M 1 , M 2 , and M 3 . The image is plotted on a logarithmic scale.

Fig. 6
Fig. 6

Symbology reconstruction on (a) linear and (b) logarithmic scales. Note the halo around the symbol caused by the convolution of the function T ( u , v ) with the target symbol. (c) The convolution function T ( u , v ) . A central order with amplitude M 0 is surrounded by smaller orders of M 1 , M 2 , and so on. (d) A cross section of the reconstruction showing clearly the extent of the halo around the symbol.

Fig. 7
Fig. 7

(a) If the symbol consists of a single spot the efficiency is unity (blazed grating). For symbols with more spots efficiency reduces down to around π / 4 (shown in detail in Fig. 8) and then increases slower until it reaches unity when the symbol occupies the entire reconstruction plane. (b) The RMS noise increases, reaching a peak value when about a quarter of the reconstruction is used.

Fig. 8
Fig. 8

Efficiency of a spot generating hologram drops initially as the number of spot increases because more and more noise spots are created. After a certain number of spots, there are so many noise orders that they coincide with target orders and efficiency increases.

Equations (44)

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E R ( u , v ) = y = y = + x = x = + h ( x , y ) e j ( u x + v y ) d x d y .
h ( x , y ) = e j h ̃ ( x , y ) ,     where   h ̃ ( x , y ) = h ( x , y ) .
h k = h ( x k , y k ) ,     h ̃ k k = h ( x k , y k ) .
I out = n = 1 S E R n E R n ,
E R n = E R ( u n , v n ) ,
ϑ I out ϑ h ̃ k = n = 1 S { E R n ϑ ϑ h ̃ k [ x y e j h ̃ e + j ( u n x + v n y ) d x d y ] + E R n ϑ ϑ h ̃ k [ x y e + j h ̃ e j ( u n x + v n y ) d x d y ] } = 0 ,
ϑ I out ϑ h ̃ k = n = 1 S { E R n e j h ̃ k e + j ( u n x k + v n y k ) + E R n e + j h ̃ k e j ( u n x k + v n y k ) }
= n = 1 S 2 I [ | E R n | e j ψ n e j h ̃ k e + j ( u n x k + v n y k ) ] ,
ϑ I out ϑ h ̃ k = n = 1 S 2 | E R n | sin ( u n x k + v n y k + ψ n h ̃ k )
tan ( h ̃ k ) = n = 1 S | E R n | cos ( u n x k + v n y k + ψ n ) n = 1 S | E R n | sin ( u n x k + v n y k + ψ n ) .
h ̃ k = arctan ( n = 1 S | E T n | sin ( u n x k + v n y k + ψ n ) n = 1 S | E T n | cos ( u n x k + v n y k + ψ n ) ) + N π ,
ϑ 2 I out ϑ h ̃ k 2 = 2 n = 1 S | E T n | cos ( u n x k + v n y k + ψ n h ̃ k ) < 0.
F = n = 1 S | E T n | sin ( u n x + v n y + ψ n ) ,
G = n = 1 S | E T n | cos ( u n x + v n y + ψ n ) ,
h ̃ = arctan ( F G ) ,
h = G + j F G 2 + F 2 = n = 1 S | E T n | e j ( u n x + v n y + ψ n ) | n = 1 S | E T n | e j ( u n x + v n y + ψ n ) | .
h ( x , y ) = τ ( x , y ) n = 1 S | E T n | e j ( u n x + v n y + ψ n ) ,
τ ( x , y ) = 1 | n = 1 S | E T n | e j ( u n x + v n y + ψ n ) | .
H ( x , y ) = n = 1 S | E T n | e j ( u n x + v n y + ψ n ) ,
α n = u n x + v n y + ψ n ,
τ ( x , y ) = 1 | n = 1 S | E T n | e j α n | = [ ( n = 1 S | E T n | e j α n ) ( m = 1 S | E T m | e j α m ) ] 1 / 2 = [ n = 1 S m = 1 S | E T n | | E T m | e j ( α n α m ) ] 1 / 2 .
τ ( x , y ) = ε 2 S [ 1 + 1 S n = 1 S m = 1 S , m n e j ( α n α m ) ] 1 / 2 = ε 2 S [ 1 + W S ] 1 / 2 ,
W = n = 1 S m = 1 S , m n e j ( α n α m ) .
τ ( x , y ) = ε 2 S [ 1 + ( 1 2 ) 1 ! ( 1 S n = 1 S m = 1 S , m n e j ( α n α m ) ) + ( 1 2 ) ( 3 2 ) 2 ! ( 1 S n = 1 S m = 1 S , m n e j ( α n α m ) ) 2 ] = ε 2 S [ 1 + ( 1 2 ) 1 ! W + ( 1 2 ) ( 3 2 ) 2 ! W 2 ] .
τ ( x , y ) = 1 S [ c = 0 M c d = 0 m c e j n = 1 S μ c , d , n α n ] ,
n = 1 S μ c , d , n = 0.
M c = n = c ξ n , c S n .
I N = ( 2 2 π ) σ 2 ,
I S = π 2 σ 2 .
η = π 4 + ( 1 π 4 ) S S max ,
M 0 + M 1 = 1 ,
M 1 + M 2 = 1 3 ,
M 2 + M 3 = 1 5 ,
M n 1 + M n = 1 2 ( n 1 ) + 1 ,
M n + M n + 1 = 1 2 n + 1 .
M 0 + M 1 = 1 ,
M 1 M 2 = 1 3 ,
M 2 + M 3 = 1 5 ,
M n 1 M n = 1 2 ( n 1 ) + 1 ,
M n + M n + 1 = 1 2 n + 1
M 0 = 1 1 3 + 1 5 + = arctan   1 = π 4
M 1 = 1 π 4 .

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