利用傳輸矩陣法求解布拉格光柵的透射譜

采用傳輸矩陣法（TMM）計算具有任意折射率分布光柵結構的透射譜，TMM法描述如下：

能夠計算折射率呈階梯狀分布的波導的反射和透射率，以及波導的傳播常數。

在單模波導中，計算反射和透射率采用2×2的矩陣表示。

為了表示光柵（多個折射率突變界面），將矩陣乘成級聯網絡，

能夠計算光柵針對每個波長的透射值和反射值。

1、均勻波導的傳輸矩陣

? 傳輸矩陣定義如下：
$$\left[\begin{array}{c}{A}_1\\ B_1\end{array}\right]=\left[\begin{array}{cc}{T}_{11}& T_{12}\\ T_{11}& T_{12}\end{array}\right]\left[\begin{array}{c}{A}_2\\ B_2\end{array}\right]$$

傳遞矩陣的概念類似于散射參數矩陣，波導的均勻截面如圖（a）所示的傳輸矩陣如下：
$$T_{hw}=\left[\begin{array}{cc}{e}^{j\beta L}& 0\\ 0& e^{?j\beta L}\end{array}\right]$$
其中，β為場的復傳播常數，包括折射率和傳播損耗：
$$\beta =\frac{2\pi n_{eff}}{\lambda }?i\frac{\alpha }{2}$$
計算均勻波導的傳輸矩陣法其MATLAB代碼如下：

function T_hw=TMM_HomoWG_Matrix(wavelength,l,neff,loss) % Calculate the transfer matrix of a homogeneous waveguide. % Complex propagation constant beta=2*pi*neff./wavelength-1i*loss/2; T_hw=zeros(2,2,length(neff)); T_hw(1,1,:)=exp(1i*beta*l); T_hw(2,2,:)=exp(-1i*beta*l);

2、折射率呈階梯狀分布的波導

折射率呈階梯狀分布的波導的傳遞矩陣，如圖b所示，為
$$T_{is?12}=\left[\begin{array}{cc}1/t& r/t\\ r/t& 1/t\end{array}\right]=\left[\begin{array}{cc}\frac{n_1+n_2}{2\sqrt{n_1{n}_2}}& \frac{n_1?n_2}{2\sqrt{n_1{n}_2}}\\ \frac{n_1?n_2}{2\sqrt{n_1{n}_2}}& \frac{n_1+n_2}{2\sqrt{n_1{n}_2}}\end{array}\right]$$
其中r和t是基于菲涅耳系數的反射率和透射率。計算折射率呈階梯狀分布的波導界面的傳輸矩陣的MATLAB代碼如下：

function T_is=TMM_IndexStep_Matrix(n1,n2) % Calculate the transfer matrix for a index step from n1 to n2. T_is=zeros(2,2,length(n1)); a=(n1+n2)./(2*sqrt(n1.*n2)); b=(n1-n2)./(2*sqrt(n1.*n2)); %T_is=[a b; b a]; T_is(1,1,:)=a; T_is(1,2,:)=b; T_is(2,1,:)=b; T_is(2,2,:)=a;

3、布拉格光柵及反射和透射率

表述布拉格光柵的級聯矩陣如下：
$$T_p=T_{tw?2}{T}_{is?21}{T}_{hw?1}{T}_{is?12}$$
其中下標1和2表示低和高折射率的區域。然后構造有N個周期的均勻布拉格光柵：
$$T_p=(T_{tw?2}{T}_{is?21}{T}_{hw?1}{T}_{is?12}{)}^N$$
考慮了相移均勻布拉格光柵，即帶有兩個布拉格光柵反射器的一級FP腔，傳遞矩陣如下：
$$T=[(T_p{)}^N]T_{hw?2}[(T_p{)}^N]T_{hw?2}$$
計算由波導截面和材料界面組成的波導布拉格光柵腔的傳輸矩陣MATLAB代碼如下：

function T=TMM_Grating_Matrix(wavelength, Period, NG, n1, n2, loss) % Calculate the total transfer matrix of the gratings l=Period/2; T_hw1=TMM_HomoWG_Matrix(wavelength,l,n1,loss); T_is12=TMM_IndexStep_Matrix(n1,n2); T_hw2=TMM_HomoWG_Matrix(wavelength,l,n2,loss); T_is21=TMM_IndexStep_Matrix(n2,n1); q=length(wavelength); Tp=zeros(2,2,q); T=Tp; for i=1:length(wavelength)Tp(:,:,i)=T_hw2(:,:,i)*T_is21(:,:,i)*T_hw1(:,:,i)*T_is12(:,:,i);T(:,:,i)=Tp(:,:,i)^NG; % 1st order uniform Bragg grating% for an FP cavity, 1st order cavity, insert a high index region, n2.T(:,:,i)=Tp(:,:,i)^NG * (T_hw2(:,:,i))^1 * Tp(:,:,i)^NG * T_hw2(:,:,i); end

我們將光柵以折射率為n2的部分作為開始和結束。相移區域是采用高折射率材料n2來實現的。最后，生成透射T和反射R譜。通過對波長點的一維矩陣進行了計算。計算光柵的反射和透射率MATLAB代碼如下：

function [R,T]=TMM_Grating_RT(wavelength, Period, NG, n1, n2, loss) %Calculate the R and T versus wavelength M=TMM_Grating_Matrix(wavelength, Period, NG, n1, n2, loss); q=length(wavelength); T=abs(ones(q,1)./squeeze(M(1,1,:))).^2; R=abs(squeeze(M(2,1,:))./squeeze(M(1,1,:))).^2;

4、光柵物理結構設計

接下來將物理結構（波導幾何形狀）與有效折射率聯系起來。輸出波導部分為500×220 nm的條形波導和氧化物包層組成，其中波導寬度發生變化構成了光柵。

使用本征模計算光柵段的有效折射率，通過計算了波導與波長和波導寬度之間的有效折射率，然后對其進行參數化。數據可以通過曲線擬合為兩個函數：
$$\begin{gathered} n_{eff?\lambda }(\lambda )=a_0?a_1(\lambda ?{\lambda }_0)?a_2(\lambda ?{\lambda }_0{)}^2 \\ {n}_{eff?w}(w)=a_0?a_1(w?w_0)?a_2(w?w_0{)}^2 \end{gathered}$$
其中，lambda的單位值微米，w為μm中的波導的寬度，neff (w)為給定波導寬度w下的有效折射率相對于其在λ0處的值的偏差。

定義光柵的物理參數，并畫出頻譜的MATLAB代碼：

function Grating %This file is used to plot the reflection/transmission spectrum. % Grating Parameters Period=310e-9; % Bragg period NG=200; % Number of grating periods L=NG*Period; % Grating length width0=0.5; % mean waveguide width dwidth=0.01; % +/- waveguide width width1=width0 - dwidth; width2=width0 + dwidth; loss_dBcm=3; % waveguide loss, dB/cm loss=log(10)*loss_dBcm/10*100; % Simulation Parameters: span=30e-9; % Set the wavelength span for the simultion Npoints = 10000; % from MODE calculations switch 1 case 1 % Strip waveguide; 500x220 nm neff_wavelength = @(w) 2.4379 - 1.1193 * (w*1e6-1.554) - 0.0350 * (w*1e6-1.554).^2; % 500x220 oxide strip waveguide dneff_width = @(w) 10.4285*(w-0.5).^3 - 5.2487*(w-0.5).^2 + 1.6142*(w-0.5); end % Find Bragg wavelength using lambda_Bragg = Period * 2neff(lambda_bragg); % Assume neff is for the average waveguide width. f = @(lambda) lambda - Period*2*(neff_wavelength(lambda)+(dneff_width(width2)+dneff_width(width1))/2); wavelength0 = fzero(f,1550e-9); wavelengths=wavelength0 + linspace(-span/2, span/2, Npoints); n1=neff_wavelength(wavelengths)+dneff_width(width1); % low index n2=neff_wavelength(wavelengths)+dneff_width(width2); % high index [R,T]=TMM_Grating_RT(wavelengths, Period, NG, n1, n2, loss);figure; plot (wavelengths*1e6,[R, T],'LineWidth',3); hold all plot ([wavelength0, wavelength0]*1e6, [0,1],'--'); % calculated bragg wavelength xlabel('Wavelength [\mum]') ylabel('Response'); axis tight;

計算結果如下圖所示：

5、Lumerical求解

通過Lumerial的FDTD求解光柵透射率代碼如下：

############################################### # script file: Bragg_FDTD.lsf # # Create and simulate a basic Bragg grating # Copyright 2014 Lumerical Solutions ############################################### # DESIGN PARAMETERS ############################################### thick_Si = 0.22e-6; thick_BOX = 2e-6; width_ridge = 0.5e-6; # Waveguide width Delta_W = 50e-9; # Corrugation width L_pd = 324e-9; # Grating period N_gt = 280; # Number of grating periods L_gt = N_gt*L_pd;# Grating length W_ox = 3e-6; L_ex = 5e-6; # simulation size margins L_total = L_gt+2*L_ex; material_Si = ’Si (Silicon) - Dispersive & Lossless’; material_BOX = ’SiO2 (Glass) - Const’; # Constant index materials lead to more stable simulations # DRAW ############################################### newproject; switchtolayout; materials; # Oxide Substrate addrect; set(’x min’,-L_ex); set(’x max’,L_gt+L_ex); set(’y’,0e-6); set(’y span’,W_ox); set(’z min’,-thick_BOX); set(’z max’,-thick_Si/2); set(’material’,material_BOX); set(’name’,’oxide’); # Input Waveguide addrect; set(’x min’,-L_ex); set(’x max’,0); set(’y’,0); set(’y span’,width_ridge); set(’z’,0); set(’z span’,thick_Si); set(’material’,material_Si); set(’name’,’input_wg’); # Bragg Gratings addrect; set(’x min’,0); set(’x max’,L_pd/2); set(’y’,0); set(’y span’,width_ridge+Delta_W); set(’z’,0); set(’z span’,thick_Si); set(’material’,material_Si); set(’name’,’grt_big’);addrect; set(’x min’,L_pd/2); set(’x max’,L_pd); set(’y’,0); set(’y span’,width_ridge-Delta_W); set(’z’,0); set(’z span’,thick_Si); set(’material’,material_Si); set(’name’,’grt_small’);selectpartial(’grt’); addtogroup(’grt_cell’); select(’grt_cell’); redrawoff; for (i=1:N_gt-1) { copy(L_pd); } selectpartial(’grt_cell’); addtogroup(’bragg’); redrawon;# Output WG addrect; set(’x min’,L_gt); set(’x max’,L_gt+L_ex); set(’y’,0); set(’y span’,width_ridge); set(’z’,0); set(’z span’,thick_Si); set(’material’,material_Si); set(’name’,’output_wg’);# SIMULATION SETUP ############################################### lambda_min = 1.5e-6; lambda_max = 1.6e-6; freq_points = 101; sim_time = 6000e-15; Mesh_level = 2; mesh_override_dx = 40.5e-9; # needs to be an integer multiple of the period mesh_override_dy = 50e-9; mesh_override_dz = 20e-9;# FDTD addfdtd; set(’dimension’,’3D’); set(’simulation time’,sim_time); set(’x min’,-L_ex+1e-6); set(’x max’,L_gt+L_ex-1e-6); 158 Fundamental building blocks set(’y’, 0e-6); set(’y span’,2e-6); set(’z’,0); set(’z span’,1.8e-6); set(’mesh accuracy’,Mesh_level); set(’x min bc’,’PML’); set(’x max bc’,’PML’); set(’y min bc’,’PML’); set(’y max bc’,’PML’); set(’z min bc’,’PML’); set(’z max bc’,’PML’);#add symmetry planes to reduce the simulation time #set(’y min bc’,’Anti-Symmetric’); set(’force symmetric y mesh’, 1); # Mesh Override if (1){ addmesh; set(’x min’,0e-6); set(’x max’,L_gt); set(’y’,0); set(’y span’,width_ridge+Delta_W); set(’z’,0); set(’z span’,thick_Si+2*mesh_override_dz); set(’dx’,mesh_override_dx); set(’dy’,mesh_override_dy); set(’dz’,mesh_override_dz); }# MODE Source addmode; set(’injection axis’,’x-axis’); set(’mode selection’,’fundamental mode’); set(’x’,-2e-6); set(’y’,0); set(’y span’,2.5e-6); set(’z’,0); set(’z span’,2e-6); set(’wavelength start’,lambda_min); set(’wavelength stop’,lambda_max);# Time Monitors addtime; set(’name’,’tmonitor_r’); set(’monitor type’,’point’); set(’x’,-3e-6); set(’y’,0); set(’z’,0); addtime; set(’name’,’tmonitor_m’); set(’monitor type’,’point’); set(’x’,L_gt/2); set(’y’,0); set(’z’,0); addtime; set(’name’,’tmonitor_t’); set(’monitor type’,’point’); set(’x’,L_gt+3e-6); set(’y’,0); set(’z’,0);# Frequency Monitors addpower; set(’name’,’t’); set(’monitor type’,’2D X-normal’); set(’x’,L_gt+2.5e-6); set(’y’,0); set(’y span’,2.5e-6); set(’z’,0); set(’z span’,2e-6); set(’override global monitor settings’,1); set(’use source limits’,1); set(’use linear wavelength spacing’,1); set(’frequency points’,freq_points); addpower; set(’name’,’r’); set(’monitor type’,’2D X-normal’); set(’x’,-2.5e-6); set(’y’,0); set(’y span’,2.5e-6); set(’z’,0); set(’z span’,2e-6); set(’override global monitor settings’,1); set(’use source limits’,1); set(’use linear wavelength spacing’,1); set(’frequency points’,freq_points);#Top-view electric field profile if (0) {addprofile; References 159 set(’name’,’field’); set(’monitor type’,’2D Z-normal’); set(’x min’,-2e-6); set(’x max’,L_gt+2e-6); set(’y’, 0); set(’y span’,1.2e-6); set(’z’, 0); set(’override global monitor settings’,1); set(’use source limits’,1); set(’use linear wavelength spacing’,1); set(’frequency points’,21); }# SAVE AND RUN ############################################### save(’Bragg_FDTD’); run; # Analysis ############################################### transmission_sim=transmission(’t’); reflection_sim=transmission(’r’); wavelength_sim=3e8/getdata(’t’,’f’); plot(wavelength_sim*1e9, 10*log10(transmission_sim),10*log10(abs(reflection_sim)),’wavelength (nm)’, ’response’); legend(’T’,’R’); matlabsave(’Bragg_FDTD’);

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