## Abstract

Distortion can be corrected in an image by placing a fourth-order aspheric optical element near the image plane. Moving the aspheric surface longitudinally changes the amount of distortion that is added by the aspheric surface without changing the paraxial image. Third-order astigmatism limits the performance of distortion correctors and may be eliminated by adding another fourth-order aspheric surface. Example elements were fabricated by diamond turning and were shown to introduce distortion without significantly degrading image quality. Three arrangements of distortion correctors are discussed: a single-element planoaspheric arrangement, an antisymmetric two-element arrangement, and a bi-aspheric arrangement in which distortion is not adjustable.

© 1992 Optical Society of America

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### Equations (21)

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(2)
$$\overline{u}=4a(n-1){h}^{3}$$
(3)
$$\mathrm{\Delta}\mathrm{\eta}=4a(n-1){\mathrm{\eta}}^{3}d,$$
(4)
$$\text{SPH}=-4a(n-1){u}^{3}{d}^{4},$$
(5)
$$\text{CMA}=12a(n-1){u}^{2}(\mathrm{\eta}-\text{d}\overline{u}){d}^{3},$$
(6)
$$\text{AST}=-12a(n-1)u{(\mathrm{\eta}-\text{d}\overline{u})}^{2}{d}^{2},$$
(7)
$$\text{DST}=4a(n-1)\hspace{0.17em}{(\mathrm{\eta}-\text{d}\overline{u})}^{3}d.$$
(8)
$$\text{FFC}=a(n-1)u{(\mathrm{\eta}-\text{d}\overline{u})}^{4}.$$
(9)
$$\frac{-(\mathrm{\eta}-\text{d}\overline{u})}{3u}<d<\frac{(\mathrm{\eta}-\text{d}\overline{u})}{3u}.$$
(10)
$$\overline{u}=4a(n-1){\mathrm{\eta}}^{3}=\text{DST}/d.$$
(11)
$$M(d)=\text{AST}+\text{FFC}+\overline{u}=3\text{DST}\left(\frac{d}{\mathrm{\eta}}\right)+\frac{\text{DST}}{4}\frac{\mathrm{\eta}}{d}+\frac{\text{DST}}{d},$$
(12)
$$\text{AST}={\text{AST}}_{1}+{\text{AST}}_{2},$$
(13)
$$\text{DST}={\text{DST}}_{1}+{\text{DST}}_{2},$$
(14)
$$\text{AST}=-12(n-1)u{\mathrm{\eta}}^{2}(-{a}_{1}{{d}_{1}}^{2}+{a}_{2}{{d}_{2}}^{2})$$
(15)
$$\text{DST}=4(n-1){\mathrm{\eta}}^{3}{a}_{1}{d}_{1}({d}_{1}/{d}_{2}-1).$$
(16)
$$\text{SPH}=-4n(n-1){u}^{3}\left[{a}_{1}{\left({d}_{2}+\frac{t}{n}\right)}^{4}-{a}_{2}{{d}_{2}}^{4}\right],$$
(17)
$$\text{CMA}=12(n-1){u}^{2}\mathrm{\eta}\left[{a}_{2}{{d}_{2}}^{3}-{a}_{1}{\left({d}_{2}+\frac{t}{n}\right)}^{3}\right],$$
(18)
$$\text{AST}=-12(n-1)u{\mathrm{\eta}}^{2}\left[{{d}_{2}}^{2}({a}_{1}-{a}_{2})+2{d}_{2}{a}_{1}\frac{t}{n}+{a}_{1}{\left(\frac{t}{n}\right)}^{2}\right],$$
(19)
$$\text{DST}=4(n-1){\mathrm{\eta}}^{3}\left[{d}_{2}({a}_{1}-{a}_{2})+{a}_{1}\left(\frac{t}{n}\right)\right],$$
(20)
$$\text{FFC}=(n-1){\mathrm{\eta}}^{4}({a}_{1}-{a}_{2}),$$
(21)
$${d}_{2}=\left(\frac{t}{n}\right)\frac{{a}_{1}+\sqrt{{a}_{1}{a}_{2}}}{{a}_{1}-{a}_{2}}.$$