### #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."

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.

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.)

It is obvious that the exp(i~) + exp(-i~) could be written in terms of cosine. So let's do that:

S(k) is usually assumed to be Gaussian. That is $S(k)=\frac{1}{\Delta&space;k&space;\sqrt{\pi}}e^{-[\frac{(k-k_0)}{\Delta&space;k}]^2}$. 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:
S(k)通常我們會假設成Gaussian，即$S(k)=\frac{1}{\Delta&space;k&space;\sqrt{\pi}}e^{-[\frac{(k-k_0)}{\Delta&space;k}]^2}$。所以說，如果我們有一個波長選擇器，可以只偵測單一波長的光強度，然後我們對不同波長來掃強度的話，他就會像是一個Gaussian被好多個cosine調節。長得就像這樣子：
The period of each modulation cosine equals $\pi/(z_R-z_{S_n})$, 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).

## 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 $S_0&space;=&space;\int_{0}^{\infty}S(k)\textup{d}k$.  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:

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.
TDOCT看起來很不錯，但他其實還是會有一些困難，首先我們的解析度會受限於我們調整參考光徑的解析度，所以如果我們想要得到1微米的解析度，我們就必須能夠精確調整參考光徑的長度1微米。而且重點是想想看如果我們是要去掃一整片影像的時候，或是這個儀器掃描長度1000次等等，都必須要能維持同樣的精確度。而且如果我們要掃的厚度比較厚，比方說1cm好了，就表示我們要精確調整1微米1微米1微米直到1cm為止，這電腦模擬模擬一下很快，真實世界可是非常耗時的呢！

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