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#58 Fundamentals--The coherence of light 常識集_光的同調性

Since we are going to talk about the optical coherence tomography in this series, we should of course briefly explain the coherence of light. Most of you should have done the experiment of interference in high school physics lab or in your freshman physics lab. We often use laser as the light source. Why can't we just take 2 flashlight for our interference experiment? That is the problem of coherence.

Why there is no interference pattern if we use 2 flashlights?


Let's recall the basic principle of interference. For simplicity, we only consider one dimension. Given 2 light source with electric field E1 & E2, E1 = E0 exp(iwt - ikz), E2 = E0 exp(iwt - ikz +iφ). The electric field at a point with distance z1 from E1 and z2 from E2 could be written as:
E =E1(z1, t)+E2(z2, t)
If we let t1 = t - z1/c, t2 = t - z2/c,τ = t2 - t1,then it could be re-written as:
E = E0(exp(iwt1)+exp(iwt2+iφ))) = E0exp(iwt1)[1+exp(iwτ+iφ)]
and the intensity of light would be:
I = 1/2εcE0^2[2 + 2cos(wτ+φ)].
我們回想一下干涉的原理是怎麼一回事,偷懶起見只考慮一維的情形。當有兩光源E1 & E2,
E1 = E0 exp(iwt - ikz), E2 = E0 exp(iwt - ikz +iφ),某點距離E1 z1, 距離E2 z2,則該處的電場為:E =E1(z1, t)+E2(z2, t)。如果我們令t1 = t - z1/c, t2 = t - z2/c,τ = t2 - t1,則E可以寫成
E = E0(exp(iwt1)+exp(iwt2+iφ))) = E0exp(iwt1)[1+exp(iwτ+iφ)]。光強度I則可寫成
I = 1/2εcE0^2[2 + 2cos(wτ+φ)]。

Now here is the main point. Because the thing we observed is the average of intensity. If we take the time average of intensity:
<I>= 1/2εcE0^2[2 + 2<cos(wτ+φ)>]
If the phase difference φ is a fixed value, then <cos> would not equal to 0. So we could observe interference pattern as we adjust the time delay τ. However, if the phase difference φ is not fixed and randomly changes with time, then <cos> = 0. And that's why we simply see the intensity add on and there is no interference with 2 flashlights. 現在重點來了,因為我們能看到的東西是光強度的平均值, 如果我們對上式取時間平均,
<I>= 1/2εcE0^2[2 + 2<cos(wτ+φ)>]。如果相位差φ是一個固定值,則<cos>會有值,我們調整時間差τ就可以看到干涉現象,光強會隨時間差的不同而有所變化。但如果相位差φ是一個隨時間變化的值,而且是隨機的話,那麼<cos> = 0,我們看到就會是單獨一支手電筒光強度的兩倍,也就不會有甚麼干涉現象。

定義同調性 Define "coherence"

So how could we define "coherence"? A naive way to do so is by autocorrelation function. autocorrelation function measures the correlation of electric field between time t and time t+τ. If the 2 light could form stable interference pattern, we had better completely know the electric field at time t+τ given the electric field at time t. So we defined the first order coherence as the following:
一個簡單的想法是,我們可以利用autocorrelation function來定義同調性。autocorrelation function觀察某一時刻t的電場強度和時刻t+τ的電場強度是否相關。如果兩束光要能形成穩定的干涉條紋,最佳情況就是:只要我知道你在t時刻的電場強度,我就一定知道你在t+τ時刻的電場強度,這樣就會有很高的同調性。因此我們定義一階同調性如下:
For a completely coherent light source, the absolute value of first order correlation = 1.

For example, if we substitute E = E0exp(iwt +iφ) with a fixed φ into the definition of first order coherence we will get:
舉例來說,如果電場E = E0exp(iwt +iφ), φ為固定值的話,帶入一階同調性的定義:
It means it is completely coherent.

Let's see another example. If the electric field reads E = E0exp(iwt +iφ(t)), in which φ would change every certain time. And the survival curve of fixed phase could be written as p(t) = e^(-t/τ0). Substitute that into our definition and we will get: 
再舉一個例子,如果電場 E = E0exp(iwt +iφ(t))φ每隔一段時間就會隨機跳掉變成另一個相位,每個固定相位的存活曲線為p(t) = e^(-t/τ0),則帶入定義式:
The result could be interpret like this. If the atom was not collided within τ, then <exp(iφ(t+τ)-iφ(t))>=1. However, if there is a collision within τ then <exp(iφ(t+τ)-iφ(t))>=0. So the final average would depend on the probability of collision. As you can see, if the light source would change its phase every certain time, the coherence would decrease with time with time constant τ0.

That is to say, if we divide the light source into 2 lightbeams for interference experiment, if the time difference between 2 paths is lesser than  τ0, then it could be interpreted as coherent and form stable interference pattern. However, is the time difference between 2 paths is greater than τ0, then they would lose their coherence and no interference pattern could be observed.

We try to introduce the concept of coherence. For more detail please refer to the textbook. Stay tuned~.