# Laser Lens Size Measurement Using Swept-Source Optical Coherence Tomography
## 2 Principle and Systems
### K-trigger  
According to the principle of SS-OCT, the depth information \(\Delta z\) of the sample can be obtained by the inverse Fourier transform of the wave number \(k\). However, the interference spectrum is sampled in linear wavelength domain. If the inverse Fourier transform is directly applied on the interference spectrum, the axial resolution degrades because of the nonlinear relationship between wavelength and wave number [24; 25; 26]. In order to improve axial resolution of the system, the resampling procedure in linear frequency domain is essential for the OCT images. Figure 2 shows a schematic diagram of k-trigger signal. The optical signal is divided into two optical paths with different path lengths by a 50/50 coupler 1. The interference signal occurs in the 50/50 coupler 2 and recorded by the dual balanced detector, then is transmitted into data acquisition card. The signals from the balanced detection can be expressed as follows [27; 28]. \[P_{MZI}(t_{i})=a\cos(k[t_{i}]\cdot n\cdot\Delta l)\;\;i=1,\;2,\;\ldots N \tag{2}\]  
where \(a\) is the spectral envelope of the incident light, \(k[t_{i}]\) is wave number corresponding the time instant of the \(i\)-_th_ sample point, \(n\) is the refractive index of the fiber, \(\Delta L\) is the optical path difference (OPD) of two optical paths. Figure 1: Experimental setup of swept-source optical coherence tomography (SS-OCT). PC—polarization controller; DAQ—data acquisition card; MZI—Mach–Zehnder interferometer. Figure 2: Schematic diagram of k-trigger signal. The spectrum \(X(k)\) can be obtained by performing the discrete Fourier transforms (DFT) of the signal \(P_{M\text{Z}I}(t_{i})\), then spectrum \(Z(k)\) can be expressed as  
\[Z(k)=\left\{ \begin{array}{ll}X(k)&k=0;\\ 2X(k)&k=1,2,\ldots,\frac{N}{2}-1;\\ 0&k=\frac{N}{2},\frac{N}{2}+1,\ldots,N;\end{array} \right. \tag{3}\]  
The analytic signal \(z(t)\) is obtained by inverse DFT of \(Z(k)\), which contains the phase information of the wave numbers. The analytic signal \(i\)-\(th\) and \((i\)+\(1)\)-\(th\) can be expressed as  
\[z_{i}(t)=x_{i}(t)+jH_{i}(x(t))=Ae^{j(\theta_{i})} \tag{4}\]  
\[z_{i+1}(t)=x_{i+1}(t)+jH_{i+1}(x(t))=Be^{j(\theta_{i+1})} \tag{5}\]  
In addition,  
\[z=z_{i}^{*}(t)\cdot z_{i+1}(t)=Ae^{-j(\theta_{i})}\cdot Be^{j(\theta_{i+1})}= ABe^{j(\theta_{i+1}-\theta_{i})} \tag{6}\]  
Then the adjacent phase difference \(\Delta\varphi_{i}\) can be obtained as  
\[\Delta\varphi_{i}=\theta_{i+1}-\theta_{i}=\arctan[\text{Im}(z)/\text{Re}(z)] \tag{7}\]  
The average phase difference \(\Phi\) can be expressed as  
\[\Phi=\frac{\sum\Delta\varphi_{i}}{N-1}\ \ \ i=0,1,2,3,\ldots N-2; \tag{8}\]  
The phase of the equalization in the wave number can be obtained as  
\[\varphi_{i}=\theta_{0}+i\cdot\Phi\ \ \ i=1,2,3,\ldots N-1; \tag{9}\]  
The interference spectrum and the k-rigger calibration signal are acquired synchronously by the DAQ card. The sampling points of k-trigger corresponding to the sampling points of interference spectrum are obtained simultaneously. Then, the equal phase calibration about OCT interference spectrum in wave number domain can be obtained. The k-trigger signal of the sampling point can be expressed as a function of phase. The interference signal can also be expressed as a function of corresponding phase approximately because of the same sampling time. Assuming the interference spectrum signal can be expressed by a function \(f(\theta)\) which is corresponding to the phase of the sampling point, where \(\theta\) is the demodulation phase of the k-trigger sampling point, the linear interpolation of the interference spectrum at equal-phase point \(f(\varphi_{i})\) can be expressed as  
\[f(\varphi_{i})=f(\theta_{i})+\frac{f(\theta_{i+1})-f(\theta_{i})}{(\theta_{i +1}-\theta_{i})}(\varphi_{i}-\theta_{i}) \tag{10}\]  
where, \(\varphi_{i}\) is the equalization phase, \(\theta_{i}\leq\varphi_{i}\leq\theta_{i+1}\). With the above proposed method, the interference spectrum in linear wave number domain can be obtained.