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#60 Optical coherence tomography-II 光學相干斷層掃描_中一

Today we will progress to the theoretical parts.

What are the 3 current terms? 電流的三個項是怎麼回事? 

Let's recall the current we mentioned in the previous episode:
The first term is simply the summation of the intensity of all reflected light without interference. This is just like a direct current and is therefore called "DC term."
第一個項其實蠻簡單的,其實就是每道反射光的光強直接加起來。如果只有這一項的話,表示每道反射光之間完全不相干涉的意思。這個項又被稱為"DC term",因為他就像直流電一樣。

The second term contains the effect of interference between the reference optical path and the reflected light from each histological layers. We call it "Cross-correlation terms" because it contains information across different optical path. Since we know R_R&Zsr already, if we could select out these terms we could easily know Rsn & zsn, which is just what we are craving for.
第二項當然就是干涉的項囉,他就是參考光徑的反射光與組織每層的反射光彼此干涉的項,研究者稱之為"Cross-correlation terms",因為他是參考光徑與樣本光徑的光之間的關係。因為我們想知道的東西是Rsn & zsn,而R_R&Zsr都是我們已知的東西,很容易除掉,所以這個項是我們最想要保存的東西。

The third term contains the interference between different reflective layers and is called "Auto-correlation terms". This term is troublesome because it contains too much unknown parameters. Fortunately the reflectivity between different layers of sample is "usually" small enough for us to simply neglect this term. (I mean "USUALLY" but not always.)
那第三項各位一定能依此類推了,他就是組織不同層的反射光之間彼此干涉的項,研究者稱之為"Auto-correlation terms",因為他是樣本光徑之內不同道光之間的關係。這個項會給我們帶來一些麻煩,但幸好樣本各層次的反射率並不大,遠遠小於參考光徑的反射率,所以通常這個項會很小(我說是「通常」喔)。

It is obvious that the exp(i~) + exp(-i~) could be written in terms of cosine. So let's do that:
眼尖的各位一定可以看出來,exp(i~)+ exp(-i~)根本就可以寫成cosine嘛。的確是這樣沒錯,所以我們就把它寫下來:

S(k) is usually assumed to be Gaussian. That is . Namely, if we have a spectrometer which could detect the intensity of single wavelength and we scanned it for the whole spectrum, the results would look like a Gaussian modulated by many cosine terms. Like this:
The period of each modulation cosine equals , and different terms have different period. We are going to delineate some information from this. There are 2 ways: Fourier domain OCT (abbreviated as FDOCT) & Time domain OCT (abbreviated as TDOCT).
每一個調節的cosine在k-domain的週期 = ,來自不同項的週期並不相同。我們就是要從這裏面隱藏的資訊解出不同層的反射率。基本上有兩種方法:Fourier domain OCT(以下簡稱為FDOCT)、Time domain OCT(以下簡稱TDOCT)。


We will talk about TDOCT first because it is much simpler. If we don't use a spectrometer and measure the intensity directly, we are measuring the sum of all intensity from different wavelengths. This is equivalent to integrate the current with respect to k. Assume that .  Recall we said previously S(k) is assumed to be a Gaussian, so its integration = 1. Assume k0>2Δk, the integration reads:
The strange R means "take real part." From above we know:
Now only the cross-correlation terms preserved its dependency on z_R. That is to say we could consider DC term and autocorrelation terms a constant as we changes the reference optical path length. The results would look like this:
上面的電流只剩下中間的項會隨z_R改變,也就是說只要我們紀錄z_R改變的情況下,所有波長的電流總讀值,就可以自動把DC & autocorrelation的項變成常數,所以畫出來的圖大概會長這樣:
In this figure there are 7 reflective surface in our scanning range.

We could understand why we should use a low-coherent light source. A low-coherent light source have broad spectrum (which means large Δk). The above formula shows that we will get sharper peaks if the Δk is large. We will get very broad peaks in our TDOCT scanning results if we use a laser as light source instead.

This is also instinctive. If the light source are low-coherent, only if we adjust the reference optical path length near the distance of certain reflective layer will they be able to interfere with each other. If the light source is highly coherent, the light from reference light path could simultaneously interfere with light reflected from layer 1, layer 2, ... and layer n and therefore becomes troublesome. That's the reason why we should use a low-coherent light source.

TDOCT looks simple enough and performs well. However, there are some difficulty utilizing it in every scenario. First, its resolution is limited by how precisely we could adjust the reference optical path length. We must be able to adjust the reference optical path length by 1 micrometer if we want a 1 micrometer resolution. It is also apparently time-consuming if we have a thick sample.

And that's why we need FDOCT. We will discuss it in next episode. Stay tuned!