# 高斯光束传播

This is Section 5.1 of the Laser Optics Resource Guide.

(1)$$I \! \left( r \right) = I_0 \exp{\left( \frac{-2 r^2}{w \! \left( z \right)^2} \right)} = \frac{2P}{\pi w \! \left( z \right)^2} \exp{\left( \frac{-2 r^2}{w \! \left( z \right)^2} \right)}$$

##### 图 1: 高斯光束的束腰定义为辐照度为其最大值 1/e2 (13.5%) 的位置

(2)$$w_0 = \frac{\lambda}{\pi \theta}$$
(3)$$\theta = \frac{\lambda}{\pi w_0}$$

##### 图 2: 高斯光束通过其束腰 (w0)、瑞利长度 (zR) 和发散角 (θ) 定义

(4)$$w \! \left( z \right) ^2 = w^2 _0 + \theta ^2 z^2 = w^2 _0 + \left( \frac{\lambda}{\pi w_0 ^2} \right)^2 z^2$$
(5)$$w \! \left( z \right) = w_0 \sqrt{1 + \left( \frac{\lambda z}{\pi w_0 ^2} \right)^2}$$

(6)$$z_R = \frac{\pi w_0 ^2}{\lambda }$$

(7)$$w \! \left( z \right) = w_0 \sqrt{1 + \left( \frac{\lambda z}{\pi w_0 ^2} \right)^2} = w_0 \sqrt{1 + \left( \frac{z}{z_R} \right)^2}$$

## 高斯光束处理

### 高斯光束的薄透镜公式

(8)$$\frac{1}{s'} = \frac{1}{s} + \frac{1}{f}$$

(9)$$\frac{f}{s'} = \frac{f}{s} + 1 \text{ or } \frac{1}{\left( \frac{s'}{f} \right)} = \frac{1}{\left( \frac{s}{f} \right)} + 1$$

##### 图 4: 薄透镜公式能够在已知透镜到物体的距离 (s) 和透镜的焦距(f) 时确定图像的位置 (s’ )

(10)$$\boxed{ \frac{1}{s'} = \frac{1}{ s + \frac{z_R ^2}{\left( s + f \right)}} + \frac{1}{f} }$$

(11)$$\frac{f}{s'} = \frac{1}{\left( \frac{s}{f} \right) + \frac{\left( \frac{z_R}{f} \right)^2}{\left( \frac{s}{f} + 1 \right)} } + 1$$

##### 图 6: zR/f=0 的曲线对应于传统的薄透镜公式。zR/f>0 的曲线表 明，高斯成像具有瑞利长度所定义的最小和最大成像距离

(12)$$\alpha = \frac{w'_0}{w_0} = \frac{1}{\sqrt{1 - \left( \frac{s}{f} \right)^2 + \left( \frac{z_R}{f} \right)^2}}$$

(13)$$\boxed{ \frac{1}{ s' + \frac{{z'}_R ^2}{\left( s' - f \right)} } = \frac{1}{s} + \frac{1}{f} }$$

(14)$$\frac{1}{\alpha ^2} = \frac{ \left| s \right| - f}{s' -f} = \left( \frac{w_0}{w'_0} \right)^2$$

### 将高斯光束聚焦到一个点上

In many applications, such as laser materials processing or surgery, it is highly important to focus a laser beam down to the smallest spot possible to maximize intensity and minimize the heated area. In cases such as these, the goal is to minimize w0' (Figure 7). A modified version of Equation 14 may be used to identify how to minimize the output beam waist3:

(15)$$\frac{\left( \left| s \right| -f \right)^2}{f^2 - \left( \frac{{w'}_0}{\theta} \right)^2 } = \left( \frac{w_{0}}{{w'}_0} \right)^2$$

##### Figure 7: 将激光束聚焦到尽可能小的尺寸对于包括这种激光切割装置在内的广泛应用至关重要

(16)$$\left( \left| s\right| -f \right)^2 = f^2 \left( \frac{w_0}{w'_0} \right)^2 - \left( \frac{w_0}{\theta} \right)^2 = f^2 \left( \frac{w_0}{w'_0} \right)^2 - z_R ^2$$
(17)$$\left( w'_0 \right) ^2 \left[ \left( \left| s \right| - f \right)^2 + z_R ^2 \right] = f^2 w_0 ^2$$

Solving for w0' results in:

(18)$$\boxed{ w'_0 = w_0 \frac{f}{\sqrt{\left( \left| s \right| - f \right) ^2 + z_R ^2}} = \alpha w_0 }$$
(19)$$\boxed{ \alpha = \frac{f}{\sqrt{\left( \left| s \right| -f \right)^2} + z_R ^2} }$$

(20)$$\theta ' = \frac{\lambda}{\pi w'_0} = \frac{\lambda}{\pi \alpha w_0} = \frac{\theta}{\alpha}$$
(21)$$z'_R = \frac{w'_0}{\theta '} = \frac{\alpha w_0}{\left( \frac{\theta}{\alpha} \right)} = \alpha ^2 z_R$$
(22)$$\boxed{ s' = f + \alpha ^2 \left( \left| s \right| - f \right)}$$

##### 图 8: 对于放大倍率 2，输出束腰将是输入束腰的两倍，输出发散将是输入光束发散的一半

There are two limiting cases which further simplify the calculations of the output beam waist size and location: when s is much less than zR or much greater than zR.3 When the lens is well within the laser’s Rayleigh range, then s << zR and (|s|-f)2<〖z_R〗2. Equation 19 simplifies to:

(23)$$\alpha = \frac{f}{z_R} = \frac{f \theta}{w_0}$$

(24)$$\theta ' = \frac{\theta}{\alpha} = \frac{ \theta w_0}{f \theta} = \frac{w_0}{f}$$
(25)$$z_R ' = \alpha ^2 z_R = \frac{f^2}{z_R ^2} z_R = \frac{f^2}{z_R}$$
(26)$$\boxed{ w'_0 = \alpha w_0 = f \theta }$$
(27)$$\boxed{ s' = f + \alpha^2 \left( \left| s \right| - f \right) = f + \frac{ \left| s \right| f^2}{z_R ^2} \approx f }$$

(28)$$\alpha = \frac{f}{\left| s \right|}$$

(29)$$w'_0 = \frac{f w}{ \left| s \right|}$$

(30)$$\theta ' = \frac{\theta}{\alpha} = \frac{\left| s \right| \theta }{f}$$
(31)$$z'_R = \alpha^2 z_R = \frac{f^2 z_R}{s^2}$$
(32)$$\boxed{ w'_0 = \alpha w_0 = \frac{fw_0}{\left| s \right|} }$$
(33)$$\boxed{ s' = f + \alpha ^2 \left( \left| s \right| - f \right) = f + \frac{\left( \left| s \right| - f \right) f^2}{s^2} = f + \frac{f^2}{ \left| s \right|} \approx f }$$

s >> zR, 时，从透镜到聚焦点的距离等于透镜的焦距。

### 高斯焦距变换

Counterintuitively, the intensity of a focused beam in a target at a fixed distance (L) away from the lens is not maximized when the waist is located at the target. The intensity on the target is actually maximized when the waist occurs at a location before the target (Figure 10). This phenomenon is known as Gaussian focal shift.

##### 图 10: 目标处的光束半径在聚焦光束的束腰出现在目标前的特定位置，而不是目标处时达到最小值

(34)$$w_L = \frac{w_0}{z_R} \left[ f^2 - 2 \left( \left| s \right| - f \right) \left( L - f \right) + \left( \frac{L-f}{\alpha} \right)^2 \right]$$

(35)$$\frac{ \text{d} }{ \text{d}f } \left[ w_L \! \left( f \right) \right] = \frac{w_0}{z_R} \left[ 2f + 2 \left( \left| s \right| -f \right) + \left( L - f \right) - \frac{2 \left( L - f \right)}{\alpha ^2} \right]$$
(36)$$\frac{ \text{d} }{ \text{d}f } \left[ w_L \! \left( f \right) \right] = 0 \text{ when } f = \left( \frac{1}{\left| s \right| + \frac{z_R ^2}{\left| s \right|}} + \frac{1}{L} \right) ^{-1}$$

### 准直高斯光束

##### 图 11: 要对高斯光束进行准直，束腰到准直透镜的距离应该等于透镜的焦距

1. Paschotta, Rüdiger. Encyclopedia of Laser Physics and Technology, RP Photonics, October 2017, www.rp-photonics.com/encyclopedia.html.
2. “Gaussian Beam Optics.” CVI Laser Optics, IDEX Optics & Photonics.
3. O'Shea, Donald C. Elements of Modern Optical Design. Wiley, 1985.
4. Self, Sidney A. “Focusing of Spherical Gaussian Beams.” Applied Optics, vol. 22, no. 5, Jan. 1983.
5. Katz, Joseph, and Yajun Li. “Optimum Focusing of Gaussian Laser Beams: Beam Waist Shift in Spot Size Minimization.” Optical Engineering, vol. 33, no. 4, Apr. 1994, pp. 1152–1155., doi:10.1117/12.158232.

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