Resolution distribution and information capacity of a hologram*

Stephen S. Hsiao

doi:10.1364/JOSA.63.001108

Journal of the Optical Society of America
Vol. 63,
Issue 9,
pp. 1108-1119
(1973)
•https://doi.org/10.1364/JOSA.63.001108

Resolution distribution and information capacity of a hologram

Stephen S. Hsiao

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Stephen S. Hsiao, "Resolution distribution and information capacity of a hologram*," J. Opt. Soc. Am. 63, 1108-1119 (1973)

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Abstract

The spatial variation of image resolution is analyzed and the number of resolvable image points in an object space is determined, when a hologram of finite recording capacity is used. The object space is divided into different sections by the maximum spatial frequency of the hologram, the hologram size, and the reference beam. The expressions for the image resolution at different planes are derived. The spatial variation of image resolution changes its structure at different planes, whereas the number of resolvable points at any plane remains constant and is equal to n², the recording capability of the hologram. The number of resolvable points in object space is also calculated for each region of object space, for both the point-source reference and the plane-wave reference. The total number of resolvable points in object space varies between (1/2)n³ and (2/3)n³ for a point-source reference, whereas the total number for the plane-wave reference is (1/3)n³. When the image is separable from its convolution term by an offset reference beam, the number of resolvable points is reduced to less than 1/20 the value when no separation is required.

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Space section	Number of resolvable points
	$N_{υ} (IA) = \frac{1}{6} \cdot \frac{1}{2} n^{3}$
I	$\begin{array}{l} N_{υ} (IB) & = \frac{1}{6} \cdot \frac{1}{2} \cdot \frac{1}{8} n^{3} \\ N_{υ} (IC) & = \frac{1}{6} \cdot \frac{1}{2} \cdot \frac{1}{8} n^{3} \\ N_{υ} (I) & = N_{υ} (IA) + 4 N_{υ} (IB) + 4 N_{υ} (IC) = \frac{1}{6} n^{3} \end{array}$
	$N_{υ} (IIA) = \frac{1}{6} \cdot \frac{1}{2} n^{3}$
II	$\begin{array}{l} N_{υ} (IIB) & = \frac{1}{6} \cdot \frac{1}{2} \cdot \frac{1}{8} n^{3} \\ N_{υ} (IIC) & = \frac{1}{6} \cdot \frac{1}{2} \cdot \frac{1}{8} n^{3} \\ N_{υ} (II) & = N_{υ} (IIA) + 4 N_{υ} (IIB) + 4 N_{υ} (IIC) = \frac{1}{6} n^{3} \end{array}$
	$N_{υ} (IIIA) = \frac{1}{6} \cdot \frac{1}{2} n^{3}$
III	$\begin{array}{l} N_{υ} (IIIB) & = \frac{1}{6} \cdot \frac{1}{2} \cdot \frac{1}{8} n^{3} \\ N_{υ} (IIIC) & = \frac{1}{6} \cdot \frac{1}{2} \cdot \frac{1}{8} n^{3} \\ N_{υ} (III) & = N_{υ} (IIIA) + 4 N_{υ} (IIIB) + 4 N_{υ} (IIIC) = \frac{1}{6} n^{3} \\ N_{υ} (IVA) & = \frac{1}{6} \cdot \frac{1}{2} \cdot {(\frac{(W - 2 X_{0}) 2 X_{0}}{λ z_{0}})}^{3} \end{array}$
	$\begin{array}{l} N_{υ} (IVB) = & \frac{1}{16} \cdot \frac{1}{2} {(\frac{2 X_{0}}{z_{0}})}^{3} {(W - 2 X_{0})}^{2} W \\ - \frac{1}{6} \cdot \frac{1}{8} {(\frac{(W - 2 X_{0}) 2 X_{0}}{λ z_{0}})}^{3} \end{array}$
IV	$\begin{array}{l} N_{υ} (IVC) = & \frac{1}{16} \cdot \frac{1}{2} {(\frac{2 X_{0}}{λ z_{0}})}^{3} (2 X_{0} W) (W - 2 X_{0}) \\ + \frac{1}{6} \cdot \frac{1}{16} {(\frac{(W - 2 X_{0}) 2 X_{0}}{λ z_{0}})}^{3} \end{array}$
	$\begin{array}{l} N_{υ} (IV) & = N_{υ} (IVA) + 4 N_{υ} (IVB) + 4 N_{υ} (IVC) \\ = \frac{1}{24} {(n - \frac{{(2 X_{0})}^{2}}{λ z_{0}})}^{3} + \frac{1}{8} n^{2} (n - \frac{{(2 X_{0})}^{2}}{λ z_{0}}) \\ N_{υ} (VA) & = \frac{1}{12} {(\frac{W^{2}}{λ z_{0}} - n)}^{3} + \frac{1}{12} n^{3} \\ N_{υ} (VB) & = \frac{W^{5} X_{0}}{16 {(λ z_{0})}^{3}} - \frac{1}{6} \cdot \frac{1}{8} {(\frac{W^{2}}{λ z_{0}})}^{3} \end{array}$
V	$\begin{array}{l} N_{υ} (VC) & = \frac{1}{6} \cdot \frac{1}{16} {(\frac{W^{2}}{λ z_{0}})}^{3} \\ N_{υ} (V) & = N_{υ} (VA) + 4 N_{υ} (VB) + 4 N_{υ} (VC) \\ = \frac{1}{24} {(\frac{W^{2}}{λ z_{0}}) \cdot [{(\frac{W^{2}}{λ z_{0}})}^{2} + 3 n^{3}] \\ + 3 n (n - \frac{W^{2}}{λ z_{0}}) \cdot (\frac{W^{2}}{λ z_{0}})} \end{array}$

Space section	Number of resolvable points
	$\begin{array}{l} N_{υ} (I^{'} B^{'}) & = \frac{1}{6} \cdot \frac{1}{16} \cdot \frac{5}{32} n^{3} \\ N_{υ} (I^{'} C^{'}) & = \frac{1}{6} \cdot \frac{1}{16} \cdot \frac{1}{64} n^{3} \\ N_{υ} (I^{'} D^{'}) & = \frac{1}{6} \cdot \frac{1}{16} \cdot \frac{11}{16} n^{3} \\ N_{υ} (I^{'} E^{'}) & = \frac{1}{6} \cdot \frac{1}{16} \cdot \frac{1}{32} n^{3} \end{array}$
I^′	$\begin{array}{l} N_{υ} (I^{'}) & = N_{υ} (I^{'} B^{'}) + 2 N_{υ} (I^{'} E^{'}) \\ + 2 N_{υ} (I^{'} C^{'}) + N_{υ} (I^{'} D^{'}) \\ = \frac{1}{6} \cdot \frac{1}{16} \cdot \frac{30}{32} n^{3} ≑ \frac{1}{6} \cdot \frac{1}{16} n^{3} \end{array}$
	$\begin{array}{l} N_{υ} ({II}^{'} B^{'}) & = \frac{1}{6} \cdot \frac{1}{16} \cdot \frac{5}{32} n^{3} \\ N_{υ} ({II}^{'} C^{'}) & = \frac{1}{6} \cdot \frac{1}{16} \cdot \frac{1}{64} n^{3} \\ N_{υ} ({II}^{'} D^{'}) & = \frac{1}{6} \cdot \frac{1}{16} \cdot \frac{11}{16} n^{3} \\ N_{υ} ({II}^{'} E^{'}) & = \frac{1}{6} \cdot \frac{1}{16} \cdot \frac{1}{32} n^{3} \end{array}$
II^′	$\begin{array}{l} N_{υ} ({II}^{'}) & = N_{υ} ({II}^{'} B^{'}) + 2 N_{υ} ({II}^{'} C^{'}) \\ + N_{υ} ({II}^{'} D^{'}) + 2 N_{υ} ({II}^{'} E^{'}) \\ = \frac{1}{6} \cdot \frac{1}{16} \cdot \frac{30}{32} n^{3} ≑ \frac{1}{6} \cdot \frac{1}{16} n^{3} \\ N_{υ} ({III}^{'} B^{'}) & = \frac{1}{2} \cdot \frac{9}{16^{3}} n^{3} \end{array}$
III^′	$\begin{array}{l} N_{υ} ({III}^{'} B^{'}) & = \frac{1}{2} \cdot \frac{3}{16^{3}} n^{3} \\ N_{υ} ({III}^{'}) & = N_{υ} ({III}^{'} B^{'}) + 2 N_{υ} ({III}^{'} C^{'}) = \frac{1}{2} \cdot \frac{15}{16^{3}} \\ N_{υ} ({IV}^{'} B^{'}) & = \frac{1}{16} \cdot \frac{1}{16^{3}} n^{3} \end{array}$
IV^′	$\begin{array}{l} N_{υ} ({IV}^{'} C^{'}) & = \frac{1}{6} \cdot \frac{1}{16^{3}} n^{3} \\ N_{υ} ({IV}^{'}) & = N_{υ} ({IV}^{'} B^{'}) + 2 N_{υ} ({IV}^{'} C^{'}) = \frac{1}{2} \cdot \frac{1}{16^{3}} n^{3} \\ N_{υ} (V^{'} B^{'}) & = \frac{1}{2} \cdot \frac{9}{16^{3}} n^{3} \end{array}$
V^′	$\begin{array}{l} N_{υ} (V^{'} C^{'}) & = \frac{1}{2} \cdot \frac{3}{16^{3}} n^{3} \\ N_{υ} (V^{'}) & = N_{υ} (V^{'} B^{'}) + 2 N_{υ} (V^{'} C^{'}) = \frac{1}{2} \cdot \frac{15}{16^{3}} n^{3} \\ N_{υ} ({IV}^{'} B^{'}) & = \frac{1}{16^{3}} [\frac{1}{16} \cdot \frac{1}{2} {(\frac{2 X_{0}}{λ z_{0}})}^{3} W {(W - 2 X_{0})}^{2} \\ - \frac{1}{6} \cdot \frac{1}{8} {(\frac{(W - 2 X_{0}) 2 X_{0}}{λ z_{0}})}^{3}] \end{array}$
VI′	$\begin{array}{l} N_{υ} ({VI}^{'} C^{'}) & = \frac{1}{16^{2}} [\frac{{(2 X_{0})}^{4} W (W - 2 X_{0})}{16 \cdot 2 \cdot λ^{3} {z_{0}}^{3}} \\ + \frac{1}{6} \cdot \frac{1}{16} {(\frac{(W - 2 X_{0}) 2 X_{0}}{λ z_{0}})}^{3}] \\ N_{υ} ({VI}^{'}) & = N_{υ} ({VI}^{'} B^{'}) + 2 N_{υ} ({VI}^{'} C^{'}) \\ = \frac{1}{16^{3}} \cdot \frac{1}{2} n [n^{2} - {(\frac{{(2 X_{0})}^{2}}{λ z_{0}})}^{2}] \\ N_{υ} ({VII}^{'} B^{'}) & = \frac{1}{16^{2}} {(\frac{X_{0}}{λ z_{0}})}^{3} W (2 W - X_{0}) (X_{0} + W) \end{array}$
VII^′	$\begin{array}{l} N_{υ} ({VII}^{'} C^{'}) & = \frac{1}{16^{2}} {(\frac{X_{0}}{λ z_{0}})}^{3} \frac{W^{2} (2 W - X_{0})}{2} \\ N_{υ} ({VII}^{'}) & = N_{υ} ({VII}^{'} B^{'}) + 2 N_{υ} ({VII}^{'} C^{'}) \\ = \frac{1}{2} \cdot \frac{1}{16^{2}} n [n^{2} - {(\frac{{X_{0}}^{2}}{λ z_{0}})}^{2}] \\ N_{υ} ({VIII}^{'} B^{'}) & = \frac{W^{5} (3 X_{0} - W)}{16 {(λ z_{0})}^{3} \cdot 3} \\ N_{υ} ({VIII}^{'} C^{'}) & = \frac{1}{6} \cdot \frac{1}{16} \cdot \frac{W^{6}}{{(λ z_{0})}^{3}} \end{array}$
VIII^′	$\begin{array}{l} N_{υ} ({VIII}^{'} D^{'}) & = \frac{1}{4 \cdot 12} (\frac{W^{4}}{λ^{3} {z_{0}}^{3}}) (4 W^{2} - 15 X_{0} W + {12 X}_{0}^{2}) \\ N_{υ} ({VIII}^{'} E^{'}) & = \frac{W^{5}}{8 \cdot 12 {(z_{0} λ)}^{3}} (3 X_{0} - 4 W) \\ N_{υ} ({VIII}^{'}) & = N_{υ} ({VIII}^{'} B^{'}) + 2 N_{υ} ({VIII}^{'} C^{'}) \\ + N_{υ} ({VIII}^{'} D^{'}) + 2 N_{υ} ({VVV}^{'} E^{'}) \\ = \frac{1}{16} \cdot \frac{1}{2} n (\frac{W^{2}}{λ z_{0}}) [2 n - 3 (\frac{W^{2}}{λ z_{0}})] \end{array}$

Cases	Space regions
	VI^′	V^′	I^′	II^′	III^′	IV^′
2X₀ < W	$\frac{1}{16^{3}} \cdot \frac{1}{2} \cdot n [n^{2} - {(\frac{{(2 X_{0})}^{2}}{λ z_{0}})}^{2}]$	$\frac{1}{16^{3}} \cdot \frac{15}{2} n^{3}$	$\frac{1}{16^{3}} \cdot \frac{5}{2} n^{3}$	$\frac{1}{16^{2}} \cdot \frac{5}{2} n^{3}$	$\frac{1}{16^{3}} \cdot \frac{15}{2} n^{3}$	$\frac{1}{16^{3}} \cdot \frac{1}{2} n^{3}$
	VII^′ ≡ V^′ + VI^′		I^′	II^′	III^′	IV^′
4W > 2X₀ > W	$\frac{1}{2} \cdot \frac{1}{16} n [n^{2} - {(\frac{{X_{0}}^{2}}{z_{0}})}^{2}]$		$\frac{1}{16^{3}} \cdot \frac{5}{2} n^{3}$	$\frac{1}{16^{2}} \cdot \frac{5}{2} n^{3}$	$\frac{1}{16^{3}} \cdot \frac{15}{2} n^{3}$	$\frac{1}{16^{3}} \cdot \frac{1}{2} n^{3}$
	VIII^′ ≡ V^′ + VI^′ + I^′			II^′	III^′	IV^′
X₀ > 2W or 2X₀ > 4W	$\frac{1}{16} \cdot \frac{1}{2} n (\frac{W^{2}}{λ z_{0}}) [2 n - 3 (\frac{W^{2}}{λ z_{0}})]$			$\frac{1}{16^{2}} \cdot \frac{5}{2} n^{3}$	$\frac{1}{16^{3}} \cdot \frac{15}{2} n^{3}$	$\frac{1}{16^{3}} \cdot \frac{1}{2} n^{3}$

Space section	Number of resolvable points
I	$\begin{array}{l} N_{υ} (IA) & = \frac{1}{6} \cdot \frac{1}{2} \cdot n^{3} \\ N_{υ} (IB) & = \frac{1}{6} \cdot \frac{1}{16} \cdot n^{3} \\ N_{υ} (IC) & = \frac{1}{6} \cdot \frac{1}{16} \cdot n^{3} \\ N_{υ} (I) & = N_{υ} (IA) + 4 N_{υ} (IB) + 4 N_{υ} (IC) = \frac{1}{6} n^{3} \end{array}$
II	$\begin{array}{l} N_{υ} (IIA) & = \frac{1}{6} \cdot \frac{1}{2} \cdot n^{3} \\ N_{υ} (IIB) & = \frac{1}{6} \cdot \frac{1}{16} \cdot n^{3} \\ N_{υ} (IIC) & = \frac{1}{2} \cdot \frac{1}{16} \cdot n^{3} \\ N_{υ} (II) & = N_{υ} (IIA) + 4 N_{υ} (IIB) + 4 N_{υ} (IIC) = \frac{1}{6} n^{3} \end{array}$

Space section	Number of resolvable points
	$\begin{array}{l} N_{υ} (I^{'} B^{'}) & = \frac{1}{64} \cdot \frac{5}{48} n^{3} \\ N_{υ} (I^{'} C^{'}) & = \frac{1}{64} \cdot \frac{1}{48} \cdot \frac{1}{2} n^{3} \end{array}$
I^′	$\begin{array}{l} N_{υ} (I^{'} D^{'}) & = \frac{1}{64} \cdot \frac{11}{24} n^{3} \\ N_{υ} (I^{'} E^{'}) & = \frac{1}{64} \cdot \frac{1}{48} n^{3} \\ N_{υ} (I^{'}) & = 2 N_{υ} (I^{'} B^{'}) + 2 N_{υ} (I^{'} C^{'}) + N_{υ} (I^{'} D^{'}) \\ + 2 N_{υ} (I^{'} E^{'}) \\ = \frac{1}{16^{2}} \cdot \frac{5}{2} n^{3} \\ N_{υ} ({II}^{'} B^{'}) & = \frac{1}{16^{2}} \cdot \frac{9}{32} n^{3} \end{array}$
II^′	$\begin{array}{l} N_{υ} ({II}^{'} C^{'}) & = \frac{1}{16^{2}} \cdot \frac{3}{32} n^{3} \\ N_{υ} ({II}^{'}) & = N_{υ} ({II}^{'} B^{'}) + 2 N_{υ} ({II}^{'} C^{'}) = \frac{1}{16^{2}} \cdot \frac{15}{32} n^{3} \\ N_{υ} ({III}^{'} B^{'}) & = \frac{1}{16^{3}} \cdot \frac{2}{3} n^{3} \end{array}$
III^′	$\begin{array}{l} N_{υ} ({III}^{'} C^{'}) & = \frac{1}{16^{3}} \cdot \frac{1}{6} n^{3} \\ N_{υ} ({III}^{'}) & = N_{υ} ({III}^{'} B^{'}) + 2 N_{υ} ({III}^{'} C^{'}) = \frac{1}{16^{3}} n^{3} \end{array}$

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Equations (53)

Journal of the Optical Society of America