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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.

既然本系列的目標是要講解光學相干斷層掃描，當然就需要先說明甚麼是光的相干性/同調性是甚麼東西。各位在高中或大一普物可能做過干涉的實驗，通常會需要用到雷射光。各位有沒有想過，為什麼拿兩支手電筒照在同一個點上，不會產生干涉呢？？？這就是相干性的問題。

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+i φ). 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+i φ)，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。

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.

完全相干的光源，一階同調性的絕對值為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：

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:

既然本系列的目標是要講解光學相干斷層掃描，當然就需要先說明甚麼是光的相干性/同調性是甚麼東西。各位在高中或大一普物可能做過干涉的實驗，通常會需要用到雷射光。各位有沒有想過，為什麼拿兩支手電筒照在同一個點上，不會產生干涉呢？？？這就是相干性的問題。

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

## 兩支手電筒照在同一個點上為什麼沒有干涉圖形？

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

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

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.

完全相干的光源，一階同調性的絕對值為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.

表示他是完全同調的。

表示他是完全同調的。

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.

可以得到這個結果是因為，如果在時間間隔τ之內沒有發生碰撞，則<exp(iφ(t+τ)-iφ(t))>=1，但如果有發生碰撞的話<exp(iφ(t+τ)-iφ(t))>=0，因此最後的平均值就取決於兩事件發生的機率。我們可以看到，當光源每隔一段時間就會跳掉相位的話，他的同調性會隨時間遞減，時間常數為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.

也就是說，如果我們把這個光束分成兩道光，進行干涉實驗的話，如果兩條光路的時間差小於τ0，則可以視為是同調的，可以產生穩定的干涉條紋。但如果兩條光路時間差大於τ0，則兩道光就不再同調，無法形成干涉條紋。

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

本篇採用非常非常簡單的方式來說明光的同調性，更深入的內容還請讀者再自行深究囉。

可以得到這個結果是因為，如果在時間間隔τ之內沒有發生碰撞，則<exp(iφ(t+τ)-iφ(t))>=1，但如果有發生碰撞的話<exp(iφ(t+τ)-iφ(t))>=0，因此最後的平均值就取決於兩事件發生的機率。我們可以看到，當光源每隔一段時間就會跳掉相位的話，他的同調性會隨時間遞減，時間常數為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.

也就是說，如果我們把這個光束分成兩道光，進行干涉實驗的話，如果兩條光路的時間差小於τ0，則可以視為是同調的，可以產生穩定的干涉條紋。但如果兩條光路時間差大於τ0，則兩道光就不再同調，無法形成干涉條紋。

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

本篇採用非常非常簡單的方式來說明光的同調性，更深入的內容還請讀者再自行深究囉。

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