## 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.

Assume the functionality of transparent is

The w0 is called minimal beam width, which describes the width of narrowest part of laser beam. If we perform fourier transform on transparent, we could get that light intensity of different spatial frequencies.

And then we substitute it back to the result of fresnel diffraction that we previously introduced:

If we let

Then the above formula could be simplified into

And again we let

Then we will get our final expression of gaussian beam:

Express the above formula in terms of light intensity would be

## The meaning of each term in gaussian beam

The expression of gaussian beam in terms of electric field is

There are 5 terms in it. What do they mean?

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.

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!

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