## Abstract

A new zoom mechanism was proposed for the realization of a freeform varifocal panoramic annular lens (PAL) with a specified annular center of the field of view (FOV). The zooming effect was achieved through a rotation of the varifoal PAL around an optical axis, which is different from a conventional zooming method by moving lenses back and forth. This method solves the problem of FOV deviation from the target scope during the zooming process, since the optical axis was not taken as the zooming center of the FOV. The conical surface corresponding to a certain acceptance angle was specified as the annular center of the FOV, and it was adopted as the reference surface of zooming for the FOV. As an example, the design principle and optimization process of a freeform varifocal PAL was discussed in detail. The annular center of the FOV was specified at the acceptance angle of 90°. The absolute FOV in the direction of acceptance angles is relative to the specified annular center, with cosine deviation from ± 20° at 0° rotational angle to ± 10° at ± 180° rotational angle on both sides around optical axis. An X–Y polynomial (XYP) was used for the representation of freeform surfaces for its simple form and convergence efficiency. The correction for irregular astigmatism and distortion and the position offset of an entrance pupil caused by an irregular aperture spherical aberration are also discussed. The results from the analysis of the modulus of the optical transfer function (MTF) and f-theta distortion show that the zooming method by a rotation of the varifocal freeform PAL is feasible.

© 2011 OSA

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### Equations (22)

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(1)
$${\mathit{r}}_{j+1}={\mathit{r}}_{j}+{d}_{sj}\cdot {\mathit{s}}_{j},$$
(2)
$${\mathit{s}}_{j+1}=n{\mathit{s}}_{j}+{\mathit{e}}_{j}\cdot \left[\sqrt{1-{n}^{2}+{(n{\mathit{e}}_{j}\cdot {\mathit{s}}_{j})}^{2}}-n{\mathit{e}}_{j}\cdot {\mathit{s}}_{j}\right],$$
(3)
$${\text{\Phi}}_{1,2}\text{=}{\text{\Phi}}_{1}\text{+}{\text{\Phi}}_{2}\text{-}{\text{D}}_{1}{\text{\Phi}}_{1}{\text{\Phi}}_{2},$$
(4)
$${D}_{1}=\frac{{d}_{{}_{1}}}{{n}_{1}},$$
(5)
$${\text{\Phi}}_{1,3}={\text{\Phi}}_{1,2}+{\text{\Phi}}_{3}-{D}_{2}{\text{\Phi}}_{1,2}{\text{\Phi}}_{3},$$
(6)
$${D}_{2}=\frac{{d}_{2}}{{n}_{2}}+\frac{{d}_{1}{\text{\Phi}}_{1}}{{n}_{1}\left({\text{\Phi}}_{1}+{\text{\Phi}}_{2}-{D}_{1}{\text{\Phi}}_{1}{\text{\Phi}}_{2}\right)}.$$
(7)
$${\text{\Phi}}_{1,k}={\text{\Phi}}_{1,k-1}+{\text{\Phi}}_{k}-{D}_{k-1}{\text{\Phi}}_{1,k-1}{\text{\Phi}}_{k},$$
(8)
$${D}_{k-1}=\frac{{d}_{k-1}}{{n}_{k-1}}+\frac{{d}_{k-2}{\text{\Phi}}_{k-2}}{{n}_{k-2}\left({\text{\Phi}}_{k-2}+{\text{\Phi}}_{k-1}-{D}_{k-2}{\text{\Phi}}_{k-2}{\text{\Phi}}_{k-1}\right)}.$$
(9)
$${\text{\Phi}}_{1,k}=f({\text{\Phi}}_{1},{\text{\Phi}}_{2}\mathrm{...}{\text{\Phi}}_{k}).$$
(10)
$${\text{\Phi}}_{j}=({n}_{j}-{n}_{j-1}){c}_{j}.$$
(11)
$${\text{\Phi}}_{1,k}=F({c}_{1},{c}_{2}\mathrm{...}{c}_{k}).$$
(12)
$${c}_{j,\parallel}={c}_{h1j}{\mathrm{cos}}^{2}\theta +{c}_{h2j}{\mathrm{sin}}^{2}\theta ,$$
(13)
$${c}_{j,\perp}={c}_{h1j}{\mathrm{sin}}^{2}\theta +{c}_{h2j}{\mathrm{cos}}^{2}\theta .$$
(14)
$${\text{\Phi}}_{1,k,\parallel}=F({c}_{1,\parallel},{c}_{2,\parallel}\mathrm{...}{c}_{k,\parallel}),$$
(15)
$${\text{\Phi}}_{1,k,\perp}=F({c}_{1,\perp},{c}_{2,\perp}\mathrm{...}{c}_{k,\perp}).$$
(16)
$$Astig\triangleq \left|{\text{\Phi}}_{1,k,\parallel}-{\text{\Phi}}_{1,k,\perp}\right|.$$
(17)
$${\mathit{e}}_{j}={\mathit{e}}_{j,T}+{\mathit{e}}_{j,R}.$$
(18)
$${\text{s}}_{j}=({\text{s}}_{j}\cdot {\text{e}}_{j,T}){\text{e}}_{j,T}+({\text{s}}_{j}\cdot {\text{e}}_{j,R}){\text{e}}_{j,R},$$
(19)
$${\mathit{s}}_{j+1}=({\mathit{s}}_{j+1}\cdot {\mathit{e}}_{j,T}){\mathit{e}}_{j,T}+({\mathit{s}}_{j+1}\cdot {\mathit{e}}_{j,R}){\mathit{e}}_{j,R}.$$
(20)
$$({\text{s}}_{1}\cdot {\text{e}}_{1,R}){\text{e}}_{1,R}=0.$$
(21)
$$({\mathit{s}}_{k+1}\cdot {\mathit{e}}_{k,R}){\mathit{e}}_{k,R}=0.$$
(22)
$$z(x,y)=\frac{c{r}^{2}}{1+\sqrt{1-(1+k){c}^{2}{r}^{2}}}+{\displaystyle \sum _{i=1}^{N}{A}_{i}{x}^{2m}{y}^{n}},$$