# 高斯光束

## 数学形式

${\displaystyle E(r,z)=E_{0}{\frac {w_{0}}{w(z)}}\exp \left({\frac {-r^{2}}{w^{2}(z)}}\right)\exp \left(-ikz-ik{\frac {r^{2}}{2R(z)}}+i\zeta (z)\right)\ ,}$

${\displaystyle r}$ 为径向坐标，以光轴中心为参考点
${\displaystyle z}$ 为轴向坐标，以光轴上光波最狭窄（束腰）位置为参考点
${\displaystyle i}$虚数单位（即 ${\displaystyle i^{2}=-1}$
${\displaystyle k={2\pi \over \lambda }}$波数（以“弧度/米”为单位）
${\displaystyle E_{0}=|E(0,0)|}$
${\displaystyle w(z)}$ 为当电磁场振幅降到轴向的1/e、强度降到轴向的1/e2的点的半径
${\displaystyle w_{0}=w(0)}$ 为激光的束腰宽度
${\displaystyle R(z)}$ 为光波波前的曲率半径
${\displaystyle \zeta (z)}$ 为轴对称光波的 Gouy 相移，对高斯光束的相位也有影响

${\displaystyle I(r,z)={|E(r,z)|^{2} \over 2\eta }=I_{0}\left({\frac {w_{0}}{w(z)}}\right)^{2}\exp \left({\frac {-2r^{2}}{w^{2}(z)}}\right)\ ,}$

## 波束参数

### 束腰

${\displaystyle w(z)=w_{0}\,{\sqrt {1+{\left({\frac {z}{z_{\mathrm {R} }}}\right)}^{2}}}\ .}$

${\displaystyle z_{\mathrm {R} }={\frac {\pi w_{0}^{2}}{\lambda }}}$

### 瑞利距离和共焦参数

${\displaystyle w(\pm z_{\mathrm {R} })=w_{0}{\sqrt {2}}.\,}$

${\displaystyle b=2z_{\mathrm {R} }={\frac {2\pi w_{0}^{2}}{\lambda }}\ .}$

### 曲率半径

${\displaystyle R(z)}$ 是光束波前的曲率半径，它是轴向距离的函数

${\displaystyle R(z)=z\left[{1+{\left({\frac {z_{\mathrm {R} }}{z}}\right)}^{2}}\right]\ .}$

### 光束偏移

${\displaystyle z\gg z_{\mathrm {R} }}$，参数 ${\displaystyle w(z)}$${\displaystyle z}$ 呈线性关系，趋近于一条直线。这条直线与中央光轴的夹角被称为光束的“偏移”，它等于

${\displaystyle \theta \simeq {\frac {\lambda }{\pi w_{0}}}\qquad (\theta \mathrm {\ in\ radians} ).}$

${\displaystyle \Theta =2\theta \ .}$

### Gouy 相位

${\displaystyle \zeta (z)=\arctan \left({\frac {z}{z_{\mathrm {R} }}}\right)\ .}$

### 复数形式的光束参数

${\displaystyle q(z)=z+q_{0}=z+iz_{\mathrm {R} }\ .}$

${\displaystyle {1 \over q(z)}={1 \over z+iz_{\mathrm {R} }}={z \over z^{2}+z_{\mathrm {R} }^{2}}-i{z_{\mathrm {R} } \over z^{2}+z_{\mathrm {R} }^{2}}={1 \over R(z)}-i{\lambda \over \pi w^{2}(z)}.}$

${\displaystyle {u}(x,z)={\frac {1}{\sqrt {{q}_{x}(z)}}}\exp \left(-ik{\frac {x^{2}}{2{q}_{x}(z)}}\right).}$

${\displaystyle {u}(x,y,z)={u}(x,z)\,{u}(y,z),}$

${\displaystyle {u}(r,z)={\frac {1}{{q}(z)}}\exp \left(-ik{\frac {r^{2}}{2{q}(z)}}\right).}$

## 功率和辐照度

### 流经孔隙的功率

${\displaystyle P(r,z)=P_{0}\left[1-e^{-2r^{2}/w^{2}(z)}\right]\ ,}$

${\displaystyle P_{0}={1 \over 2}\pi I_{0}w_{0}^{2}}$ 为电磁波传播的总能量

${\displaystyle {P(z) \over P_{0}}=1-e^{-2}\approx 0.865\ .}$

### 辐照度的峰值和平均值

${\displaystyle I(0,z)=\lim _{r\to 0}{\frac {P_{0}\left[1-e^{-2r^{2}/w^{2}(z)}\right]}{\pi r^{2}}}={\frac {P_{0}}{\pi }}\lim _{r\to 0}{\frac {\left[-(-2)(2r)e^{-2r^{2}/w^{2}(z)}\right]}{w^{2}(z)(2r)}}={2P_{0} \over \pi w^{2}(z)}.}$

## 参考文献

1. ^ Siegman (1986) p. 630.
2. ^ See Siegman (1986) p. 639. Eq. 29
• Saleh, Bahaa E. A. and Teich, Malvin Carl. Fundamentals of Photonics. New York: John Wiley & Sons. 1991. ISBN 0-471-83965-5. Chapter 3, "Beam Optics," pp. 80–107.
• Mandel, Leonard and Wolf, Emil. Optical Coherence and Quantum Optics. Cambridge: Cambridge University Press. 1995. ISBN 0-521-41711-2. Chapter 5, "Optical Beams," pp. 267.