## Abstract

A recent study [J. Opt. Soc. Am. A **15**, 2720 (1998)] showed that the eikonal equation can be generalized to include higher-order terms that can describe the dispersion of a pulse in a linear medium. In this companion paper, the formalism of this generalized eikonal approximation is investigated for stationary waves in both linear and nonlinear media. A local refractive index can be defined that includes higher-order terms in the form of the second derivatives of the amplitude of the wave. It is shown that wave phenomena such as diffraction, self-focusing, and self-trapping of a finite beam in linear and nonlinear media can be derived naturally as a result of the differences in the local refractive indices that arise from the spatial variation of the wave amplitude and the nonlinearity of the medium. This formalism can be readily extended to the study of other complex electromagnetic wave-propagation phenomena in both linear and nonlinear media.

© 1998 Optical Society of America

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### Equations (39)

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(1)
$${\nabla}^{2}\mathbf{E}+k_{0}{}^{2}{n}^{2}\mathbf{E}=0,$$
(2)
$$\mathbf{E}=\varphi (\mathbf{r})exp[{\mathit{ik}}_{0}L(\mathbf{r})]\stackrel{\u02c6}{e},$$
(3)
$$(\nabla L{)}^{2}={n}^{2}+\frac{1}{k_{0}{}^{2}\varphi}{\nabla}^{2}\varphi ,$$
(4)
$$\nabla \xb7({\varphi}^{2}\nabla L)=0.$$
(5)
$$(\nabla L{)}^{2}={n}^{2},$$
(6)
$$(\nabla L{)}^{2}=n_{G}{}^{2},$$
(7)
$$n_{G}{}^{2}={n}^{2}+\frac{1}{k_{0}{}^{2}\varphi}{\nabla}^{2}\varphi .$$
(8)
$$\mathbf{E}=\stackrel{\u02c6}{z}\varphi (\mathbf{r})exp[{\mathit{ik}}_{0}L(\mathbf{r})].$$
(9)
$$\mathbf{H}=({\mathit{ik}}_{0}\sqrt{\mu /\u220a}{)}^{-1}\nabla \times \mathbf{E}.$$
(10)
$$\mathbf{P}=\mathrm{Re}(\mathbf{E}\times {\mathbf{H}}^{*})=\frac{n}{c\mu}({\varphi}^{2}\nabla L).$$
(11)
$${\mathit{\upsilon}}_{g}=\frac{c}{n_{0}{}^{2}}\nabla L.$$
(12)
$$\nabla (n_{G}{}^{2})=2{n}_{G}\frac{\partial}{\partial \xi}({n}_{G}\mathit{\xi})=\frac{\partial n_{G}{}^{2}}{\partial \xi}\mathit{\xi}+2n_{G}{}^{2}\frac{\partial \mathit{\xi}}{\partial \xi}.$$
(13)
$$\nabla (n_{G}{}^{2})=\stackrel{\u02c6}{\xi}\frac{\partial}{\partial \xi}n_{G}{}^{2}+\stackrel{\u02c6}{\tau}\frac{\partial}{\partial \tau}n_{G}{}^{2}.$$
(14)
$$\frac{\partial \stackrel{\u02c6}{\xi}}{\partial \xi}=\stackrel{\u02c6}{\tau}K,$$
(15)
$$K=\frac{1}{2n_{G}{}^{2}}\left(\frac{\partial n_{G}{}^{2}}{\partial \tau}\right).$$
(16)
$$\varphi =Aexp(-{r}^{2}/\sigma _{0}{}^{2}),$$
(17)
$$n_{G}{}^{2}={n}^{2}+\frac{1}{k_{0}{}^{2}}\frac{4}{\sigma _{0}{}^{2}}\left(\frac{{r}^{2}}{\sigma _{0}{}^{2}}-1\right),$$
(18)
$$K\approx \frac{1}{{n}^{2}}\frac{4}{k_{0}{}^{2}\sigma _{0}{}^{4}}r,$$
(19)
$${\omega}^{2}(z)=\omega _{0}{}^{2}\left(1+\frac{4{z}^{2}}{k_{0}{}^{2}{n}^{2}\sigma _{0}{}^{4}}\right).$$
(20)
$${z}_{0}=\frac{{k}_{0}n\sigma _{0}{}^{2}}{2},$$
(21)
$$\nabla L=\stackrel{\u02c6}{z}\frac{\partial L}{\partial z}=\stackrel{\u02c6}{z}{n}_{G},$$
(22)
$$\frac{\partial}{\partial z}({\phi}^{2}{n}_{G})=0.$$
(23)
$$\varphi =\frac{f(r,\theta )}{n_{G}{}^{1/2}},$$
(24)
$$\frac{{\partial}^{2}f}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial f}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial}^{2}f}{\partial {\theta}^{2}}-{a}^{2}f=0,$$
(25)
$$n={n}_{0}+{n}_{2}{\varphi}^{2}+{n}_{4}{\varphi}^{4}+\dots .$$
(26)
$${n}_{0}\gg {n}_{2}{\varphi}^{2},\hspace{0.5em}{n}_{4}{\varphi}^{4},\dots ;$$
(27)
$${n}^{2}\approx n_{0}{}^{2}+2{n}_{0}{n}_{2}{\varphi}^{2},$$
(28)
$$n_{G}{}^{2}\approx n_{0}{}^{2}+2{n}_{0}{n}_{2}{\varphi}^{2}+\frac{1}{k_{0}{}^{2}\varphi}{\nabla}^{2}\varphi ,$$
(29)
$$n_{G}{}^{2}=n_{0}{}^{2}+2{n}_{0}{n}_{2}{\varphi}^{2}+\frac{1}{k_{0}{}^{2}}\frac{4}{\sigma _{0}{}^{2}}\left(\frac{{r}^{2}}{\sigma _{0}{}^{2}}-1\right),$$
(30)
$$K\propto \frac{1}{k_{0}{}^{2}}\frac{8}{\sigma _{0}{}^{4}}r-\frac{8{n}_{0}{n}_{2}}{\sigma _{0}{}^{2}}{\varphi}^{2}r.$$
(31)
$$\nabla L=\stackrel{\u02c6}{z}\frac{\partial L}{\partial z}=\stackrel{\u02c6}{z}{n}_{G}.$$
(32)
$$\frac{{\partial}^{2}f}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial f}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial}^{2}f}{\partial {\theta}^{2}}={a}^{2}f-{b}^{2}{f}^{3},$$
(33)
$$\frac{{\partial}^{2}f}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial f}{\partial r}-{a}^{2}f+{b}^{2}{f}^{3}=0,$$
(34)
$$\frac{{\partial}^{2}f}{\partial {y}^{2}}-{a}^{2}f+{b}^{2}{f}^{3}=0.$$
(35)
$$n_{0}{}^{2}<n_{G}{}^{2}<n_{0}{}^{2}+2{n}_{0}{n}_{2}{\varphi}_{max}^{2},$$
(36)
$$f(y)=\frac{\sqrt{2}a}{b}\mathrm{sech}a(y-{y}_{0})$$
(37)
$$\varphi (y)={\left[\frac{n_{G}{}^{2}-n_{0}{}^{2}}{{n}_{0}{n}_{2}}\right]}^{1/2}\mathrm{sech}a(y-{y}_{0}),$$
(38)
$$n_{G}{}^{2}=n_{0}{}^{2}+{n}_{0}{n}_{2}{\varphi}^{2}(y={y}_{0}).$$
(39)
$$\frac{1}{k_{0}{}^{2}}\frac{{\nabla}^{2}\varphi}{\varphi}+2{n}_{0}{n}_{2}{\varphi}^{2}(y)=n_{G}{}^{2}-n_{0}{}^{2}={n}_{0}{n}_{2}{\varphi}^{2}({y}_{0}).$$