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# 高斯光束傳播

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

##### 圖 1: 高斯光束束腰定義：輻照度為其最大值 1/e2 (13.5%) 的位置

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

##### 圖 2: 高斯光束是以其束腰 (w0)、雷利範圍 (zR), 及發散角 (θ) 定義

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

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

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

## 高斯光束操控

### 適合高斯光束的薄透鏡方程式

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

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

##### 圖 4: 薄透鏡公式在已知透鏡與物件距離 (s) 及透鏡焦距 (f ) 的情況下，可判定影像位置 (s')

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

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

##### 圖 5: 重新對焦高斯光束時的「物體」為輸入束腰，「影像」則為輸出束腰

A plot of the normalized image distance (s’/f) versus the normalized object distance (s/f) shows the possible output waist locations at a given normalized Rayleigh range (zR/f) (圖 6). This plot shows that Gaussian beams focused through a lens have a few key differences when compared to conventional thin lens imaging. Gaussian beam imaging has both minimum and maximum possible image distances, while conventional thin lens imaging does not. The maximum image distance of a refocused Gaussian beam occurs at an object distance of -(f + zR), as opposed to –f. The point on the plot where s/f is equal to -1 and s’/f is equal to 1 indicates that the output waist will be at the back focal point of the lens if the input is at the front focal point of a positive lens.

##### 圖 6: The curve where zR/f=0 corresponds to the conventional thin lens equation. The curves where zR/f>0 show that Gaussian imaging has minimum and maximum image distances defined by the Rayleigh range

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

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

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

### 高斯光束聚焦至一點

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

##### 圖 7: 將雷射光束聚焦至最小尺寸，對各式各樣的應用相當關鍵，包括此項雷射切削配置

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

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

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

##### 圖 8: 在放大倍率為 2 的情況下，輸出束腰將是輸入束腰的兩倍，而輸出發散則為輸入光束發散的一半

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

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

s >> zR時，透鏡至聚焦點的距離等於透鏡焦距。

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

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

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

s >> zR, 時，透鏡至聚焦點的距離等於透鏡焦距。

### 高斯焦點位移

##### 圖 10: 聚焦光束束腰在目標前的特定位置時，才會在目標產生最小光束半徑，聚焦束腰位在目標時則非如此

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

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

### Collimating a Gaussian Beam

Achieving a truly collimated beam where the divergence is 0 is not possible, but achieving an approximately collimated beam by either minimizing the divergence or maximizing the distance between the point of observation and the nearest beam waist is possible. Since the output divergence is inversely proportional to the magnification constant α, the output divergence reaches a minimum value when |s| = f (圖 11).

### Off-the-Shelf Lasers

1. Paschotta, Rüdiger. Encyclopedia of Laser Physics and Technology（雷射物理與技術百科全書）, RP Photonics, October 2017 www.rp-photonics.com/encyclopedia.html.
2. Self, Sidney A. “Focusing of Spherical Gaussian Beams.” （球面高斯光束聚焦）Applied Optics（應用光學）第 22 卷第 5 期，1983 年 1 月。
3. O'Shea, Donald C. Elements of Modern Optical Design（現代光學設計要素）。 Wiley,1985 年。
4. Self, Sidney A. “Focusing of Spherical Gaussian Beams.” Applied Optics, vol. 22, no. 5, Jan. 1983.
5. Katz、Joseph 及 Yajun Li。“Optimum Focusing of Gaussian Laser Beams:Beam WaistShift in Spot Size Minimization” （高斯雷射光束最佳聚焦：光斑大小最小化的束腰偏移）。 Optical Engineering（光學工程）第 33 卷第 4 期，1994 年 4 月，第 1152–1155頁，doi：10.1117/12.158232。

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