# Characteristics of Collimators Based on the Large-Mode-Area CMCF for Coupling Laser Beam
## 2 Theoretical Analysis of the Coupling Loss Mechanism  
The fiber collimator is a basic optical device in fiber optic communication systems and optical fiber sensing systems, which consists of 1/4 pitch self-focusing lens and single-mode fiber. Its purpose is to collimate the Gaussian beam so as to improve the coupling efficiency between the fibers. The injected light becomes a parallel beam after passing through the self-focusing lens, which is then coupled into the other self-focusing lens. Through the self-focusing lens, the parallel beam is coupled into the output fiber. In this way, the coupling efficiency between the fibers is greatly improved. However, the self-focusing lens does not eliminate the coupling loss caused by the assembly errors. There is a long space between the two fiber collimators, which allows the insertion of optical elements. The fiber collimator consists of a tail fiber and a self-focusing lens, and its structure and coupling are shown in Figure 1.  
Figure 1: Structure of fiber collimator and coupling in an ideal case. As shown in Figure 1, \(W_{t}\) is the beam waist radius of the Gaussian beam exiting from the fiber collimator under the case of ideal coupling; \(L_{5}/L_{1}\) are the gap distance between the incident/exiting fibers and the self-focusing lens; \(L_{2}/L_{4}\) are the self-focusing lens lengths, which are less than 1/4 pitch; \(L_{3}\) is the coupling distance, which is ideally twice the distance from the collimator end-face to the beam waist position of the exiting beam. Three types of alignment tolerance are introduced: off-axis deviation X, angular deviation \(\theta\), and axial deviation Z, as shown in Figure 2. The system coupling losses introduced by these deviations are not negligible. Therefore, it is important to reduce the coupling loss introduced by the assembly error to improve the stability of the coupling system. The coupling process between the two collimators was first analyzed and then expressed with the transfer matrix [25]. According to the characteristic of Gaussian beam, the transfer matrix of the collimator coupling is established, which consists of five parts, as follows: the matrix of the gap between the tail fiber in Collimator 1 and the self-focusing lens, \(\mathbf{T}_{1}\); the matrix of the self-focusing lens in Collimator 1, \(\mathbf{T}_{2}\); the matrix of the working distance between the two collimators, \(\mathbf{T}_{3}\); the matrix of the self-focusing lens in Collimator 2, \(\mathbf{T}_{4}\); and the matrix of the gap between the tail fiber and the self-focusing lens in Collimator 2, \(\mathbf{T}_{5}\). Therefore, the total transfer matrix \(\mathbf{T}\) of the collimator-based coupling can be obtained as shown in Equation (1). \[\mathbf{T}=\mathbf{T}_{5}\mathbf{T}_{4}\mathbf{T}_{3}\mathbf{T}_{2}\mathbf{T}_{1}= \tag{1}\] \[\left[ \begin{array}{cc}1&L_{5}\\ 0&1\end{array} \right]\left[ \begin{array}{cc}\cos(\sqrt{A}L_{4})&\frac{\sin( \sqrt{A}L_{4})}{n_{0}\sqrt{A}}\\ -n_{0}\sqrt{A}\sin(\sqrt{A}L_{4})&\cos(\sqrt{A}L_{4})\end{array} \right]\left[ \begin{array}{cc}1&L_{3}\\ 0&1\end{array} \right]\]  
where \(\sqrt{A}\) is the focusing constant of the self-focusing lens, and \(n_{0}\) is the refractive index in the center of the self-focusing lens. The process of optical coupling is clearly described in Equation (1), in which the beam at any position in the collimator can be found easily. Usually, the incident beam can be represented by a Gaussian beam. According to the mode coupling theory, the energy-coupling efficiency \(\eta\) can be expressed by Equation (2), which is caused by the mode-field mismatching. \[\eta=-10\lg\frac{|\iint E_{1}\cdot E_{2}^{*}dxdy|}{\iint|E_{1}^{2}|dxdy\cdot \iint|E_{2}^{2}|dxdy} \tag{2}\]  
where \(E_{1}\) and \(E_{2}\), respectively, represent the electric field of the exiting and incident beams, and \(x\) and \(y\) are horizontal and vertical coordinates anywhere on the cross section of collimators, respectively. The coupling of the beams in the collimator can be calculated by (2).