## Abstract

We compare the effects of spherical aberration on the penetration
depth of single-photon and two-photon excitation for instances in which
the aberration is caused by the refractive-index mismatch when a beam
is focused through an interface. It is shown both theoretically and
experimentally that two-photon fluorescence imaging experiences less
spherical aberration and can thus propagate to a deeper depth within a
thick medium.

© 2000 Optical Society of America

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### Equations (5)

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(1)
$${k}_{0}\mathrm{\Phi}\left({\mathrm{\theta}}_{1},{\mathrm{\theta}}_{2},-d\right)=-{k}_{0}d\left({n}_{1}cos{\mathrm{\theta}}_{1}-{n}_{2}cos{\mathrm{\theta}}_{2}\right),$$
(2)
$$I\left(r,\mathrm{\varphi},z\right)=|{I}_{0}{|}^{2}+4|{I}_{1}{|}^{2}{cos}^{2}\mathrm{\varphi}+|{I}_{2}{|}^{2}+2cos\left(2\mathrm{\varphi}\right)\mathrm{Re}\left({I}_{0}{I}_{2}*\right).$$
(3)
$${I}_{1-\mathrm{p}}\left(r,\mathrm{\varphi},z\right)={I}_{\mathrm{ex}}\left(r,\mathrm{\varphi},z\right){I}_{f}\left(r,\mathrm{\varphi},z\right),$$
(4)
$${I}_{2-\mathrm{p}}\left(r,\mathrm{\varphi},z\right)=I_{\mathrm{ex}}{}^{2}\left(r,\mathrm{\varphi},z\right).$$
(5)
$$I\left(d\right)={\int}_{-d}^{\infty}{\int}_{0}^{2\mathrm{\pi}}{\int}_{0}^{\infty}{I}_{1-\mathrm{p}}\left(r,\mathrm{\varphi},z\right)r\mathrm{d}r\mathrm{d}\mathrm{\varphi}\mathrm{d}z.$$