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The diffraction pattern of a gaussian transparent光強為高斯分布的繞射
In previous episodes, we did not specify the function of transparent, f(x,y). Today we will take gaussian distribution as an example to see how it propagates. The upcoming discussion is quite important for it underlies the laser propagation--the gaussian beam.
在前幾集裡面，我們並沒有去說明中間的透明片f(x,y)到底長甚麼樣子，今天我們就要以高斯分布作為例子，看看當f(x,y)為高斯分布時，光的傳播情形為何。這個結果非常重要，稱為Gaussian beam，是用來說明雷射傳播的重要理論。
Assume the functionality of transparent is
假設中央透明片的強度分布情形是
這裡w0被稱為minimal beam width，描述雷射最窄處的寬度。如果對這個透明片的強度分布做傅立葉轉換，我們可以得到通過他的平行光在不同空間頻率 的光強度：
And then we substitute it back to the result of fresnel diffraction that we previously introduced:
把不同空間頻率的光強度帶入Fresnel diffraction的結果：
=e^{-ikz}\iint_{-\infty}^{\infty}A(\nu_x,\nu_y)e^{i\pi\lambda&space;z(\nu_x^2+\nu_y^2)}e^{-i2\pi(\nu_x&space;x+\nu_y&space;y)}\textup{d}\nu_x\textup{d}\nu_y&space;\\&space;=&space;e^{-ikz}&space;\frac{\pi^2w_0^2}{\pi^2w_0^2-i\pi\lambda&space;z}e^{\frac{-\pi^2(x^2+y^2)}{(\pi^2&space;w_0^2&space;-&space;i\pi\lambda&space;z)}} "U(x,y,z)=e^{-ikz}\iint_{-\infty}^{\infty}A(\nu_x,\nu_y)e^{i\pi\lambda z(\nu_x^2+\nu_y^2)}e^{-i2\pi(\nu_x x+\nu_y y)}\textup{d}\nu_x\textup{d}\nu_y \\ = e^{-ikz} \frac{\pi^2w_0^2}{\pi^2w_0^2-i\pi\lambda z}e^{\frac{-\pi^2(x^2+y^2)}{(\pi^2 w_0^2 - i\pi\lambda z)}}")
If we let
如果我們令
=w_0^2[1+(\frac{z}{z_0})^2] "z_0 = \frac{\pi w_0^2}{\lambda}\\ w^2(z)=w_0^2[1+(\frac{z}{z_0})^2]")
Then the above formula could be simplified into
上式就可以化簡成
=e^{-ikz}\frac{1}{\sqrt{1+z^2/z_0^2}}\frac{1+iz/z_0}{\sqrt{1+z^2/z_0^2}}e^{\frac{-(x^2+y^2)}{w_0^2(1+z^2/z_0^2)}}e^{\frac{-ik(x^2+y^2)}{2z[1+z_0^2/z^2]}}\\ "U(x,y,z)=e^{-ikz}\frac{1}{\sqrt{1+z^2/z_0^2}}\frac{1+iz/z_0}{\sqrt{1+z^2/z_0^2}}e^{\frac{-(x^2+y^2)}{w_0^2(1+z^2/z_0^2)}}e^{\frac{-ik(x^2+y^2)}{2z[1+z_0^2/z^2]}}\\")
And again we let
再令
\\&space;R(z)=z[1+z_0^2/z^2] "\eta = \tan^{-1}(z/z_0)\\ R(z)=z[1+z_0^2/z^2]")
Then we will get our final expression of gaussian beam:
我們最後就可以得到Gaussian beam的表達式：
=e^{-ikz}\frac{w_0}{w(z)}e^{i\eta(z)}e^{\frac{-(x^2+y^2)}{w^2(z)}}e^{\frac{-ik(x^2+y^2)}{2R(z)}}\\ "U(x,y,z)=e^{-ikz}\frac{w_0}{w(z)}e^{i\eta(z)}e^{\frac{-(x^2+y^2)}{w^2(z)}}e^{\frac{-ik(x^2+y^2)}{2R(z)}}\\")
=\frac{w_0^2}{w^2(z)}e^{\frac{-2(x^2+y^2)}{w^2(z)}} "I(x,y,z)=\frac{w_0^2}{w^2(z)}e^{\frac{-2(x^2+y^2)}{w^2(z)}}")
我們得到Gaussian beam的電場表達式為
There are 5 terms in it. What do they mean?
這裡面有五個項，他們分別代表甚麼意思呢？
1. e^(-ikz)：the phase change comes from light propagation.光線往前進造成的相位變化。
2. w0/w(z)： due to the effect of diffraction, the beam width changes from w0 to w(z) and thus decreases its light intensity.
w0為在最窄處的光線寬度，因為繞射的關係，在遠方z處的beam width會變大成w(z)，因為光線變得分散，所以光的強度會減弱。
We will use gaussian beam in later episodes. Stay tuned!
在後面的文章裡面我們會再利用到Gaussian beam喔~~，敬請期待。
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把不同空間頻率的光強度帶入Fresnel diffraction的結果：
如果我們令
上式就可以化簡成
再令
我們最後就可以得到Gaussian beam的表達式：
Express the above formula in terms of light intensity would be
若換成光強度則為
若換成光強度則為
The meaning of each term in gaussian beam
The expression of gaussian beam in terms of electric field is我們得到Gaussian beam的電場表達式為
這裡面有五個項，他們分別代表甚麼意思呢？
1. e^(-ikz)：the phase change comes from light propagation.光線往前進造成的相位變化。
2. w0/w(z)： due to the effect of diffraction, the beam width changes from w0 to w(z) and thus decreases its light intensity.
w0為在最窄處的光線寬度，因為繞射的關係，在遠方z處的beam width會變大成w(z)，因為光線變得分散，所以光的強度會減弱。
3. e^(iη)：this term is called gout phase. This is the phase difference caused by the the fact that the diffracted light observed at each point is in fact the summation of light with various spatial frequencies.
這個項被稱為Gouy phase，是因為特定點的光來自不同spatial frequency的光互相加總造成的相位差。
這個項被稱為Gouy phase，是因為特定點的光來自不同spatial frequency的光互相加總造成的相位差。
4.e^(-(x^2+y^2)/w^2)：the light intensity is still a gaussian distribution in the same z plane. 同一z值的平面中，光強度為Gaussian distribution。
5. e^(-ik(x^2+y^2)/2R)：this means the wavefront of gaussian beam could be regarded as a spherical wave with radius R. 表示Gaussian beam的波前可以視為一個球面波，球面波的曲率半徑為R。
We will use gaussian beam in later episodes. Stay tuned!
在後面的文章裡面我們會再利用到Gaussian beam喔~~，敬請期待。
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本篇參考資料為：Optics by Ajoy Ghatak 的 Chapter 20, Fresnel Diffraction。
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