## Abstract

Numerical values are shown for the aberrations that are induced by a wavelength shift from construction to reconstruction. The holograms discussed in this paper may be constructed and reconstructed from a plane wave or point source. Four types of hologram geometries have been studied: in-line, off-axis, near-image plane, and lensless-fourier transform. The aberrations are first considered where the reconstruction beam geometry is the same as that needed to reconstruct an ideal image if no wavelength shift is present. The change in wavelength, however, causes aberrations and these are discussed as a function of the various hologram geometries and parameters. A considerable reduction in the aberrations can be realized in off-axis holograms by a technique of aberration balancing. In one example the total aberrations were reduced to approximately a Rayleigh limit of λ/4 in a hologram that was recorded at 4880 Å and reconstructed at 6328 Å. The minimum aberrations possible using the balancing process are discussed with respect to various hologram parameters.

© 1971 Optical Society of America

Full Article |

PDF Article
### Equations (14)

Equations on this page are rendered with MathJax. Learn more.

(1)
$$\text{MAX}\mid {\mathrm{\Delta}}_{S}\mid =-\text{MAX}\mid {\mathrm{\Delta}}_{A}\mid ,$$
(2)
$${\mathrm{\Delta}}_{C}=0,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{all}\hspace{0.17em}x,y\u220a\hspace{0.17em}\text{hologram}\hspace{0.17em}\text{surface}.$$
(3)
$${\mathrm{\Delta}}_{\text{MAX}}=\frac{1}{{\mathrm{\lambda}}_{C}}[-{\scriptstyle \frac{1}{8}}({{\rho}_{m}}^{4}S)+{\scriptstyle \frac{1}{2}}({{\rho}_{m}}^{3}{C}_{X})-{\scriptstyle \frac{1}{2}}({{\rho}_{m}}^{2}{A}_{X})].$$
(4)
$$\frac{1}{{R}_{1}}=\frac{1}{{R}_{IR}}-\frac{1}{{R}_{C}},$$
(5)
$${S}_{C}=\text{sin}{\alpha}_{IR}-\text{sin}{\alpha}_{C},$$
(6)
$$\frac{1}{{R}_{2}}=\frac{1}{{{R}_{C}}^{3}}-\frac{1}{{{R}_{IR}}^{3}},$$
(7)
$$\frac{1}{{R}_{3}}=\frac{\text{sin}{\alpha}_{C}}{{{R}_{C}}^{2}}-\frac{\text{sin}{\alpha}_{IR}}{{{R}_{IR}}^{2}},$$
(8)
$$\frac{1}{{R}_{4}}=\frac{{\text{sin}}^{2}{\alpha}_{C}}{{R}_{C}}-\frac{{\text{sin}}^{2}{\alpha}_{IR}}{{R}_{IR}}.$$
(9)
$$-\frac{1}{4}{{\rho}_{m}}^{2}\left[\frac{1}{{R}_{2}}-\mu \left(\frac{1}{{{R}_{O}}^{3}}-\frac{1}{{{R}_{R}}^{3}}\right)\right]=\frac{1}{{R}_{4}}-\mu \left(\frac{{\text{sin}}^{2}{\alpha}_{O}}{{R}_{O}}-\frac{{\text{sin}}^{2}{\alpha}_{R}}{{R}_{R}}\right),$$
(10)
$$(1/{R}_{3})-\mu \left(\frac{\text{sin}{\alpha}_{O}}{{{R}_{O}}^{2}}-\frac{\text{sin}{\alpha}_{R}}{{{R}_{R}}^{2}}\right)=0,$$
(11)
$$(1/{R}_{O})=(1/{R}_{R})-1/(\mu {R}_{1}),$$
(12)
$$\text{sin}{\alpha}_{O}=\text{sin}{\alpha}_{R}-({S}_{C}/\mu ).$$
(13)
$$\text{sin}{\alpha}_{O}=\frac{({{R}_{O}}^{2}{\mu}^{2}{{R}_{1}}^{2})/{R}_{C}+{S}_{C}{\mu}^{2}{{R}_{1}}^{2}+2{S}_{C}{R}_{1}\mu {R}_{O}+{S}_{C}{{R}_{O}}^{2}}{-(2{\mu}^{2}{R}_{1}{R}_{O}-{{R}_{O}}^{2}\mu )};$$
(14)
$$\begin{array}{l}\left(-\frac{{{\rho}_{m}}^{2}{\mu}^{2}}{4{R}_{2}}-\frac{{{\rho}_{m}}^{2}}{4{{R}_{1}}^{3}}-\frac{{\mu}^{2}}{{R}_{4}}-\frac{{\mu}^{4}{{R}_{1}}^{3}}{{{R}_{3}}^{2}}\right){{R}_{O}}^{4}\\ +\hspace{0.17em}\left\{-{{\rho}_{m}}^{2}\left[\left(\frac{7}{4}\right)\frac{\mu}{{{R}_{1}}^{2}}+\frac{{\mu}^{3}{R}_{1}}{{R}_{2}}\right]-\frac{4{\mu}^{3}{R}_{1}}{{R}_{4}}+\frac{2{\mu}^{3}{{R}_{1}}^{2}{S}_{C}}{{R}_{3}}+{{S}_{C}}^{2}\mu \right\}\\ \times \hspace{0.17em}{{R}_{O}}^{3}+[-\left(\frac{19}{4}\right)\frac{{{\rho}_{m}}^{2}{\mu}^{2}}{{R}_{1}}-\frac{{{\rho}_{m}}^{2}{\mu}^{4}{{R}_{1}}^{2}}{{R}_{2}}-\frac{4{\mu}^{4}{{R}_{1}}^{2}}{{R}_{4}}\\ +\hspace{0.17em}\frac{2{S}_{C}{\mu}^{4}{{R}_{1}}^{3}}{{R}_{3}}+4{{S}_{C}}^{2}{R}_{1}{\mu}^{2}]\hspace{0.17em}{{R}_{O}}^{2}+(-6{{\rho}_{m}}^{2}{\mu}^{3}+6{{S}_{C}}^{2}{\mu}^{3}{{R}_{1}}^{2}){R}_{O}\\ -\hspace{0.17em}3{{\rho}_{m}}^{2}{\mu}^{4}{R}_{1}+3{\mu}^{4}{{S}_{C}}^{2}{{R}_{1}}^{3}=0.\end{array}$$