# 高斯光束

## 数学形式

$E(r,z) = E_0 \frac{w_0}{w(z)} \exp \left( \frac{-r^2}{w^2(z)}\right) \exp \left( -ikz -ik \frac{r^2}{2R(z)} +i \zeta(z) \right)\ ,$

$r$为场点距离光轴中心的径向距离
$z$为光轴上光波最狭窄位置束腰的位置坐标
$i$虚数单位（即$i^2 = -1$
$k = { 2 \pi \over \lambda }$波数（以弧度每米为单位）
$E_0 = |E(0,0)|$,
$w(z)$为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
$w_0 = w(0)$为激光的束腰宽度
$R(z)$为光波波前的曲率半径
$\zeta(z)$为轴对称光波的Gouy相位，对高斯光束的相位也有影响

$I(r,z) = { |E(r,z)|^2 \over 2 \eta } = I_0 \left( \frac{w_0}{w(z)} \right)^2 \exp \left( \frac{-2r^2}{w^2(z)} \right)\ ,$

## 波束参数

### 束宽

$w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 } \ .$

$z_\mathrm{R} = \frac{\pi w_0^2}{\lambda}$

### 瑞利距离和共焦参数

$w(\pm z_\mathrm{R}) = w_0 \sqrt{2}. \,$

$b = 2 z_\mathrm{R} = \frac{2 \pi w_0^2}{\lambda}\ .$

### 曲率半径

$R(z)$是光束波前的曲率半径，它是轴向距离的函数

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

### 光束偏移

$z \gg z_\mathrm{R}$，参数$w(z)$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

$\theta \simeq \frac{\lambda}{\pi w_0} \qquad (\theta \mathrm{\ in\ radians}).$

$\Theta = 2 \theta\ .$

### Gouy相位

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

### 复数形式的光束参数

$q(z) = z + q_0 = z + iz_\mathrm{R} \ .$

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

${u}(x,z) = \frac{1}{\sqrt{{q}_x(z)}} \exp\left(-i k \frac{x^2}{2 {q}_x(z)}\right).$

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

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

## 功率和辐照度

### 流经孔隙的功率

$P(r,z) = P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]\ ,$

$P_0 = { 1 \over 2 } \pi I_0 w_0^2$为电磁波传播的总能量

${ P(z) \over P_0 } = 1 - e^{-2} \approx 0.865\ .$

### 辐照度的峰值和平均值

$I(0,z) =\lim_{r\to 0} \frac {P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]} {\pi r^2} = \frac{P_0}{\pi} \lim_{r\to 0} \frac { \left[ -(-2)(2r) e^{-2r^2 / w^2(z)} \right]} {w^2(z)(2r)} = {2P_0 \over \pi w^2(z)}.$

## 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.
• Siegman, Anthony E. Lasers. University Science Books. 1986. ISBN 0-935702-11-3. Chapter 16.