高斯光束

贡献者： addis

本文处于草稿阶段。
本文缺少预备知识，初学者可能会遇到困难。
翻译成中文

预备知识　电场波动方程

　　 Wave Eq.

\begin{matrix} (1) & \nabla^{2} E - \frac{1}{c^{2}} \frac{\partial^{2} E}{\partial t^{2}} = 0 . \end{matrix}

assume propagation within small angle of

z

axis

\begin{matrix} (2) & E = \hat{ϵ} E (r, t) = 2 \hat{ϵ} U (x, y, z) e^{i (k z - ω t)}, \end{matrix}

U (x, y, z)

is the envelope. Plug in, use ‘slowly varying envelope approximation’

\begin{matrix} (3) & 2 i k \frac{\partial U}{\partial z} = \frac{\partial^{2} U}{\partial x^{2}} + \frac{\partial^{2} U}{\partial y^{2}} . \end{matrix}

The general solution is a linear combination of the following basis

\begin{matrix} (4) & U_{m n} (x, y, z) = \frac{C}{w (z)} \exp [- \frac{r^{2}}{w^{2} (z)}] \exp [i k \frac{r^{2}}{2 R (z)}] H_{m} [\frac{\sqrt{2} x}{w (z)}] H_{n} [\frac{\sqrt{2} y}{w (z)}] e^{- i ϕ_{m n} (z)} . \end{matrix}

\begin{matrix} (5) & ϕ_{m n} (z) = (m + n + 1) \tan^{- 1} (z / z_{R}), \end{matrix}

\begin{matrix} (6) & w (z) = w_{0} \sqrt{1 + z^{2} / z_{R}^{2}}, R (z) = z + z_{R}^{2} / z, z_{R} = π w_{0}^{2} / λ, \end{matrix}

\begin{matrix} (7) & H_{0} = 1, H_{1} = 2 x, H_{2} = 4 x^{2} - 1 . \end{matrix}

This is called the Hermite-Gauss mode, denoted

T E M_{m n}

H

are Hermite polynomials and

ϕ

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

\begin{matrix} (8) & l - R = \sqrt{R^{2} + z^{2}} - R \approx \frac{r^{2}}{2 R} . \end{matrix}

图 1：triangle

　　 $T E M_{00}$ is the fundamental Gaussian mode.

　　 In cylindrical coordinates, the basis change to Laguerre-Gauss modes $T E M_{l m}^{*}$

\begin{matrix} (9) & U_{l m} (r, θ, z) = \frac{C^{'}}{w (z)} {[\frac{\sqrt{2} r}{w (z)}]}^{| m |} \exp [- \frac{r^{2}}{w^{2} (z)}] \exp [i \frac{r^{2}}{2 R (z)}] L_{l}^{| m |} [\frac{2 r^{2}}{w^{2} (z)}] e^{i m θ} e^{- i ϕ_{l m} (z)}, \end{matrix}

\begin{matrix} (10) & ϕ_{l m} (z) = (2 l + | m | + 1) \tan^{- 1} (z / z_{R}) . \end{matrix}

This is analogous to solving SHO in polar coordiantes while Hermite-Gauss modes are in Cartesian coordinates.

致读者：小时百科一直以来坚持所有内容免费无广告，这导致我们处于严重的亏损状态。长此以往很可能会最终导致我们不得不选择大量广告以及内容付费等。因此，我们请求广大读者热心打赏 ，使网站得以健康发展。如果看到这条信息的每位读者能慷慨打赏 20 元，我们一周就能脱离亏损，并在接下来的一年里向所有读者继续免费提供优质内容。但遗憾的是只有不到 1% 的读者愿意捐款，他们的付出帮助了 99% 的读者免费获取知识，我们在此表示感谢。