Evaluation of deformation tolerance of the hologram medium

M. Matsumura

doi:10.1364/JOSA.64.000928

Journal of the Optical Society of America
Vol. 64,
Issue 7,
pp. 928-933
(1974)
•https://doi.org/10.1364/JOSA.64.000928

Evaluation of deformation tolerance of the hologram medium

M. Matsumura

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M. Matsumura, "Evaluation of deformation tolerance of the hologram medium," J. Opt. Soc. Am. 64, 928-933 (1974)

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Abstract

The deformation tolerance of the hologram recording medium is evaluated analytically for small holograms. The deformation shape is supposed to be inclination or curvature. This simple supposition is justified for small holograms, which are in common use for high-density information recording. The relation is derived between the aberration types and the deformation types. It is shown that the only aberrations that need to be considered are distortion and astigmatism, which are caused by inclination and curvature of the medium, respectively. Numerical examples of the deformation tolerance for these aberrations are given and the dependence of the tolerance upon the recording arrangement is shown. For maximum deformation tolerance, there is an optimum arrangement, which corresponds to symmetrical or nearly symmetrical arrangement of the object and the reference beam.

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Deformations	Distortion First order	First order	Astigmatism Second order	Third order	Coma Third order
Δa	D_x⁽¹⁾		A_x⁽²⁾		A_x⁽³⁾
Δa	D_x⁽¹⁾		A_xy⁽²⁾		A_x⁽³⁾
Δb	D_y⁽¹⁾		A_y⁽²⁾		A_y⁽³⁾
Δb	D_y⁽¹⁾		A_xy⁽²⁾		A_y⁽³⁾
Δc		A_x⁽¹⁾	A_xx⁽⁴⁾, A_xy⁽⁴⁾	A_x⁽⁵⁾
Δc		A_x⁽¹⁾	A_yx⁽⁴⁾	A_x⁽⁵⁾
Δd		A_xy⁽¹⁾	A_xy⁽⁴⁾, A_yx⁽⁴⁾	A_xy⁽⁵⁾
Δe		A_y⁽¹⁾	A_yy⁽⁴⁾, A_yx⁽⁴⁾,	A_y⁽⁵⁾
Δe		A_y⁽¹⁾	A_xy⁽⁴⁾	A_y⁽⁵⁾

Deformation	Related aberrations		Deformation tolerance $E^{″} ≦ \frac{1}{4} λ$
Deformation	Orders	Symbols	Deformation tolerance $E^{″} ≦ \frac{1}{4} λ$
$Δ a [\frac{d_{H}}{2}]$	First	D_x⁽¹⁾	$\frac{λ}{4 ∣ Z_{1} ∣}$	Eq. (3.1a)
	Second	A_x⁽²⁾	$\frac{λ}{2 d_{H} ∣ X_{2} ∣}$	Eq. (3.1b)
	Third	A_x⁽³⁾	$\frac{λ}{{d_{H}}^{2} ∣ Z_{3} ∣}$	Eq. (3.1c)
$Δ c {[\frac{d_{H}}{2}]}^{2}$	First	A_x⁽¹⁾	$\frac{λ}{4 ∣ Z_{1} ∣}$	Eq. (3.1d)
	Second	A_xx⁽⁴⁾	$\frac{λ}{2 d_{H} ∣ X_{2} ∣}$	Eq. (3.1e)
	Third	A_x⁽⁵⁾	$\frac{λ}{{d_{H}}^{2} ∣ Z_{3} ∣}$	Eq. (3.1f)

	Related aberrations		Tolerance of deformation
Deformation	Orders	Symbols	General formulas	Eq.	Normalized formulas	Eq.
$Δ a [\frac{d_{H}}{2}]$	First	D_x⁽¹⁾	$\frac{d_{H} Δ x_{I, limit}}{2 R_{I} ∣ Z_{1} ∣}$	(3.3a)	$\frac{Δ x_{I, limit}}{{(p_{0} d_{0})}_{0}} \times \frac{d_{H}}{d_{H_{0}}} \times \frac{λ}{∣ Z_{1} ∣}$	(3.5a)
	Second	A_x⁽²⁾	$\frac{Δ x_{I, limit}}{2 R_{I} ∣ X_{2} ∣}$	(3.3b)	$\frac{Δ x_{I, limit}}{{(p_{0} d_{0})}_{0}} \times \frac{λ}{d_{H_{0}} ∣ X_{2} ∣}$	(3.5b)
	Third	A_x⁽³⁾	$\frac{Δ x_{I, limit}}{\frac{3}{4} R_{I} d_{H} ∣ Z_{3} ∣}$	(3.3c)	$\frac{8}{3} \times \frac{Δ x_{I, limit}}{{(p_{0} d_{0})}_{0}} \times \frac{λ}{d_{H} d_{H_{0}} ∣ Z_{3} ∣}$	(3.5c)
$Δ c {[\frac{d_{H}}{2}]}^{2}$	First	A_x⁽¹⁾	$\frac{d_{H} Δ x_{I, limit}}{4 R_{I} ∣ Z_{1} ∣}$	(3.3d)	$\frac{1}{2} \times \frac{Δ x_{I, limit}}{{(p_{0} d_{0})}_{0}} \times \frac{d_{H}}{d_{H_{0}}} \times \frac{λ}{∣ Z_{1} ∣}$	(3.5d)
	Second	A_xx⁽⁴⁾	$\frac{Δ x_{I, limit}}{3 R_{I} ∣ X_{2} ∣}$	(3.3e)	$\frac{2}{3} \times \frac{Δ x_{I, limit}}{{(p_{0} d_{0})}_{0}} \times \frac{λ}{d_{H_{0}} ∣ X_{2} ∣}$	(3.5e)
	Third	A_x⁽⁵⁾	$\frac{Δ x_{I, limit}}{R_{I} d_{H} ∣ Z_{3} ∣}$	(3.3f)	$2 \times \frac{Δ x_{I, limit}}{{(p_{0} d_{0})}_{0}} \times \frac{λ}{d_{H} d_{H_{0}} ∣ Z_{3} ∣}$	(3.5f)

Deformations	Distortion First order	First order	Astigmatism Second order	Third order	Coma Third order
Δa	D_x⁽¹⁾		A_x⁽²⁾		A_x⁽³⁾
Δa	D_x⁽¹⁾		A_xy⁽²⁾		A_x⁽³⁾
Δb	D_y⁽¹⁾		A_y⁽²⁾		A_y⁽³⁾
Δb	D_y⁽¹⁾		A_xy⁽²⁾		A_y⁽³⁾
Δc		A_x⁽¹⁾	A_xx⁽⁴⁾, A_xy⁽⁴⁾	A_x⁽⁵⁾
Δc		A_x⁽¹⁾	A_yx⁽⁴⁾	A_x⁽⁵⁾
Δd		A_xy⁽¹⁾	A_xy⁽⁴⁾, A_yx⁽⁴⁾	A_xy⁽⁵⁾
Δe		A_y⁽¹⁾	A_yy⁽⁴⁾, A_yx⁽⁴⁾,	A_y⁽⁵⁾
Δe		A_y⁽¹⁾	A_xy⁽⁴⁾	A_y⁽⁵⁾

Deformation	Related aberrations		Deformation tolerance $E^{″} ≦ \frac{1}{4} λ$
Deformation	Orders	Symbols	Deformation tolerance $E^{″} ≦ \frac{1}{4} λ$
$Δ a [\frac{d_{H}}{2}]$	First	D_x⁽¹⁾	$\frac{λ}{4 ∣ Z_{1} ∣}$	Eq. (3.1a)
	Second	A_x⁽²⁾	$\frac{λ}{2 d_{H} ∣ X_{2} ∣}$	Eq. (3.1b)
	Third	A_x⁽³⁾	$\frac{λ}{{d_{H}}^{2} ∣ Z_{3} ∣}$	Eq. (3.1c)
$Δ c {[\frac{d_{H}}{2}]}^{2}$	First	A_x⁽¹⁾	$\frac{λ}{4 ∣ Z_{1} ∣}$	Eq. (3.1d)
	Second	A_xx⁽⁴⁾	$\frac{λ}{2 d_{H} ∣ X_{2} ∣}$	Eq. (3.1e)
	Third	A_x⁽⁵⁾	$\frac{λ}{{d_{H}}^{2} ∣ Z_{3} ∣}$	Eq. (3.1f)

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Journal of the Optical Society of America