## Abstract

The synthesis of spherical computer-generated holograms is investigated. To deal with the staggering calculation times required to synthesize the hologram, a fast calculation method for approximating the hologram distribution is proposed. In this method, the diffraction integral is approximated as a convolution integral, allowing computation using the fast-Fourier-transform algorithm. The principles of the fast calculation method, the error in the approximation, and results from simulations are presented.

© 2006 Optical Society of America

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### Figures (9)

Fig. 1

Setup of the calculation method.

Fig. 2

(a) Geometry of the approximation and (b) geometry to solve for α.

Fig. 3

Relative distance error distributions for various object point locations.

Fig. 4

(Color online) Distance error profiles for (a) ro = 1 cm, (b) ro = 3 cm, and (c) ro = 9 cm.

Fig. 5

Computer-generated holograms for an object at (a) θ o = 0, (b) θ o = −π∕4, and (c) θ o = −π∕2.

Fig. 6

(a) Amplitude distribution and (b) Fourier spectrum amplitude distribution of the point spread function.

Fig. 7

Hologram of two point sources.

Fig. 8

Reconstruction images of a small object at (a) θ o = 0, (b) θ o = −π∕4, and (c) θ o = −π∕2.

Fig. 9

(a) Sample object and (b) reconstruction image.

### Equations (16)

$f h ( ϕ h , θ h ) = C ∫ ∫ f o ( ϕ o , θ o ) exp ⁡ ( i k d ) d d ϕ o d θ o ,$
$d = { r o 2 + r h 2 − 2 r o r h [ sin ⁡ ( θ h ) sin ⁡ ( θ o ) + cos ( θ h ) cos ⁡ ( θ o ) cos ⁡ ( ϕ h − ϕ o ) ] } 1 / 2 .$
$d = [ r o 2 + r h 2 − 2 r o r h cos ⁡ ( ϕ h − ϕ o ) cos ⁡ ( θ h − θ o ) ] 1 / 2 .$
$h ( ϕ , θ ) = exp ⁡ { i k [ r o 2 + r h 2 − 2 r o r h cos ⁡ ( ϕ ) cos ⁡ ( θ ) ] 1 / 2 } [ r o 2 + r h 2 − 2 r o r h cos ⁡ ( ϕ ) cos ⁡ ( θ ) ] 1 / 2 .$
$f h ( ϕ h , θ h ) = C ∫ ∫ f o ( ϕ o , θ o ) h ( ϕ h − ϕ o , θ h − θ o ) d ϕ o d θ o ,$
$f h = C f o ∗ h ,$
$ν ϕ ( ϕ , θ ) = 1 2 π ∂ ∂ ϕ ∠ [ h ( ϕ , θ ) ] ,$
$ν ϕ ( ϕ , θ ) = 1 2 π ∂ ∂ ϕ { k [ r o 2 + r h 2 − 2 r o r h cos ⁡ ( ϕ ) cos ⁡ ( θ ) ] 1 / 2 }$
$= 1 2 π k r o r h sin ⁡ ( ϕ ) cos ⁡ ( θ ) [ r o 2 + r h 2 − 2 r o r h cos ⁡ ( ϕ ) cos ⁡ ( θ ) ] 1 / 2 .$
$| ν ϕ | max ⁡ = k × min ⁡ ( r o , r h ) 2 π ,$
$B W ϕ = 2 | ν ϕ | max ⁡ = k × min ⁡ ( r o , r h ) π ,$
$N ϕ > 4 π × min ⁡ ( r o , r h ) λ ,$
$N θ > 2 π × min ⁡ ( r o , r h ) λ ,$
$d = { r o 2 + r h 2 − 2 r o r h [ sin ⁡ ( θ h ) sin ⁡ ( θ o ) cos ⁡ ( ϕ h − ϕ o ) + cos ⁡ ( θ h ) cos ⁡ ( θ o ) cos ⁡ ( ϕ h − ϕ o ) ] } 1 / 2 ,$
$d = [ r o 2 + r h 2 − 2 r o r h cos ⁡ ( ϕ h − ϕ o ) cos ⁡ ( θ h − θ o ) ] 1 / 2 .$
$d = [ r o 2 + r h 2 − 2 r o r h cos ⁡ ( θ h ) cos ⁡ ( ϕ h − ϕ o ) ] 1 / 2 ,$