## Abstract

In this paper, we calculate the transverse spherical aberration *TA* of a thin lens and defines a normalized aberration *Y* equal to *TA* divided by the theoretical resolution limit. As a rule of thumb, (a) a thin lens that suffers only from spherical aberration may be considered effectively diffraction-limited as long as *Y* < 1.6. Similarly, (b) the coupling efficiency of a Gaussian beam to a single-mode fiber may be high even when Y > 1.6, and, specifically, (c) the lens need be diffraction-limited only over a radius approximately equal to the radius (to the 1/e-point) of the Gaussian beam.

© 2000 Optical Society of America

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### Equations (9)

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(1)
$$ta={l}^{\prime}h{L}_{\text{S}}$$
(2)
$$\begin{array}{c}{\text{LS}}_{}=\frac{{h}^{2}}{8{f}^{\prime 3}}\frac{1}{n\left(n-1\right)}\left[\frac{n+2}{n-1}{q}^{2}+4\left(n+1\right)pq\right.\\ \left.+\left(3n+2\right)\left(n-1\right){p}^{2}+\frac{{n}^{3}}{n-1}\right]\\ \equiv A{h}^{2}\end{array}$$
(3)
$$\begin{array}{l}a\left(h\right)=A{h}^{4}/4,\text{or}\\ a\left(\text{\theta}\right)=A{l}^{\prime 4}{\text{\theta}}^{4}/4,\end{array}$$
(4)
$$Y\propto {f}^{\prime}/\left(\text{\lambda}\xb7{\Phi}^{4}\right),$$
(5)
$$d\left(\text{\theta}\right)=\frac{TA{\text{\theta}}^{2}}{4NA},$$
(6)
$$U\left(\text{\theta}\right)=\mathrm{exp}\left(2\text{\pi}ip\left(\text{\theta}\right)/\text{\lambda}\right)\xb7\mathrm{exp}\left[-{\left(\text{\theta}/{\text{\theta}}_{0}\right)}^{2}\right],$$
(7)
$$T={\left\{2\text{\pi}{\displaystyle \underset{0}{\overset{NA}{\int}}\mathrm{exp}\left[-2{\left(\text{\theta}/{\text{\theta}}_{0}\right)}^{2}\right]\text{\theta}d\text{\theta}/\left({{\pi \theta}_{0}}^{2}/2\right)}\right\}}^{2}$$
(8)
$$u\left(r\right)={\displaystyle \underset{0}{\overset{NA}{\int}}2\text{\theta}U\left(\text{\theta}\right)}{\text{J}}_{0}\left(kr\text{\theta}\right)\text{d\theta}$$
(9)
$$c=T\frac{{\displaystyle \underset{0}{\overset{\infty}{\int}}u\left(r\right){g}^{*}\left(r\right)r\text{d}r}}{{\left[{\displaystyle \underset{0}{\overset{\infty}{\int}}{|u\left(r\right)|}^{2}r\text{d}r\xb7{\displaystyle \underset{0}{\overset{\infty}{\int}}{|g\left(r\right)|}^{2}r\text{d}r}}\right]}^{\xbd}}$$