# 高斯光束

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Wave Eq.

$$\boldsymbol{\nabla}^2 \boldsymbol{\mathbf{E}} - \frac{1}{c^2} \frac{\partial^{2}{ \boldsymbol{\mathbf{E}} }}{\partial{t}^{2}} = 0$$
assume propagation within small angle of $z$ axis
$$\boldsymbol{\mathbf{E}} = \hat{\boldsymbol{\mathbf{\epsilon}}} E( \boldsymbol{\mathbf{r}} , t) = 2 \hat{\boldsymbol{\mathbf{\epsilon}}} U(x, y, z) \mathrm{e} ^{ \mathrm{i} (kz - \omega t)}$$
$U(x, y, z)$ is the envelope. Plug in, use ‘slowly varying envelope approximation’
$$2 \mathrm{i} k \frac{\partial U}{\partial z} = \frac{\partial^{2}{U}}{\partial{x}^{2}} + \frac{\partial^{2}{U}}{\partial{y}^{2}}$$
The general solution is a linear combination of the following basis
$$U_{mn}(x, y, z) = \frac{C}{w(z)} \exp\left[-\frac{r^2}{w^2(z)}\right] \exp\left[ \mathrm{i} k\frac{r^2}{2R(z)}\right] H_m \left[\frac{\sqrt{2}x}{w(z)} \right] H_n \left[\frac{\sqrt{2} y}{w(z)} \right] \mathrm{e} ^{- \mathrm{i} \phi_{mn}(z)}$$
$$\phi_{mn}(z) = (m+n+1)\tan^{-1}(z/z_R)$$
$$w(z) = w_0\sqrt{1 + z^2/z_R^2} \qquad R(z) = z + z_R^2 / z \qquad z_R = \pi w_0^2 / \lambda$$
$$H_0 = 1 \qquad H_1 = 2x \qquad H_2 = 4x^2 - 1$$
This is called the Hermite-Gauss mode, denoted $TEM_{mn}$. $H$ are Hermite polynomials and $\phi$ is the Gouy phase-shift, $z_R$ is the Rayleigh length. The second exp factor makes the wave front a spherical wave with curvature $R(z)$, because
$$l - R = \sqrt{R^2 + z^2} - R \approx \frac{r^2}{2R}$$

$TEM_{00}$ is the fundamental Gaussian mode.

In cylindrical coordinates, the basis change to Laguerre-Gauss modes $TEM_{lm}^*$

$$U_{lm}(r, \theta, z) = \frac{C'}{w(z)} \left[\frac{\sqrt{2}r}{w(z)} \right] ^{ \left\lvert m \right\rvert } \exp\left[-\frac{r^2}{w^2(z)}\right] \exp\left[ \mathrm{i} \frac{r^2}{2R(z)}\right] L_l^{ \left\lvert m \right\rvert } \left[\frac{2r^2}{w^2(z)} \right] \mathrm{e} ^{ \mathrm{i} m\theta} \mathrm{e} ^{- \mathrm{i} \phi_{lm}(z)}$$
$$\phi_{lm}(z) = (2l + \left\lvert m \right\rvert + 1) \tan^{-1} (z/z_R)$$
This is analogous to solving SHO in polar coordiantes while Hermite-Gauss modes are in Cartesian coordinates.