Lumerical入门笔记
The FDTD(finite-difference time-domain) method（有限时域差分法）
- Time-domain simulation
- E and H fields are solved using a time-stepping algorithm
- E and H fields are solved at discrete positions in space
Advantages of the FDTD method
- 准确
- 普适
- 任意复杂形状
- 宽谱结果
Simulation time and memory requirements
当器件在一个方向上为无限长时，可以使用x,y坐标进行仿真
| 3D | 2D | |
|---|---|---|
| Memory Requirements | \(\sim V \cdot (\lambda/dx)^3\) | \(\sim A \cdot (\lambda/dx)^2\) |
| Simulation Time | \(\sim V \cdot (\lambda/dx)^4\) | \(\sim A \cdot (\lambda/dx)^3\) |
\(V\)：仿真体积
\(A\)：仿真面积
\((\lambda/dx)^3\)：仿真密度
Boundary conditions
PML
- Perfectly matched layer, absorbing boundary
Metal
- Perfectly reflecting boundary
Periodic
- Unit cell (structures and EM fields) contained in the simulation is repeated in the periodical direction (eg. source at normal incidence, along coordinate axes)
Bloch
- Periodic with a phase shift between each unit cell
Symmetric/ anti-symmetric
- Used to reduce the required stimulation volume when the structure and source have a plane of symmetry
Sources
- Dispole 偶极子光源
- Gaussian 高斯或大数值孔径（矢量）
- Plane wave 平面波光源
- Total-field scattered-field 全场散射场光源：主要用于散射、吸收、削光界面
- Mode 波导模式光源
- Import 自输入光源
Monitors
- Refractive index 折射率：用于查看器件设置是否正确
- Field time 时间：可以是点、线、面、立体的
- Movie 电影：可以查看相互作用
- Frequency-domain field profile/power 场分布/功率
- Mode expansion 模式分解
Running a simulation
一般情况下，进度表显示不超过95%结束可以得到准确的频域结果
正常结束运行有两种方法：
- 达到预先设置的Autoshutoff Min，强烈推荐
- 达到预先设置的仿真时间
FDTD Solutions Workflow
- 创建物理结构
- 添加仿真区+边界
- 添加光源
- 添加监视器
Using Eigen Solver
- 定义物理结构
- 定义仿真区域
- 频域分析
Basic Scripting
simple mathematics: plot some simple functions
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> x = linspace(-10,10,500);
> y = sin(x);
> plot(x, y, "x", "y", "sin(x)");
> y = exp(-x^2/9)*sin(10*x);
> plot(x, y, "x", "y", "exp(-x^2/9)*sin(10*x)");
> ?size(x);
Advanced Scripting
- 获得模式参数
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?getdata("mode1");
% 参数是 surface_normal dimension f neff loss TE polarization fraction waveguide TE/TM fraction x y z Ex Ey Ez Hx Hy Hz
- 获得频域曲线
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?getdata("frequencysweep");
% 参数是 neff loss vg D beta f f_vg f_D mode_number
例如
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f = getdata("frequencysweep", "f"); %得到频率
2.5D FDTD Propagator
How the FDTD method works
E and H are discrete in time \(\vec{E}(t) \rightarrow \vec{E}^{n\Delta t},\space \vec{H}(t) \rightarrow \vec{H}^{(n + \frac{1}{2})\Delta t}\) The basic FDTD time-stepping relation: \(\vec{E}^{n+1} = \vec{E}^{n} + \alpha \vec{\nabla} \times \vec{H}^{n+\frac{1}{2}}\)
\[\vec{H}^{n+\frac{3}{2}} = \vec{H}^{n+\frac{1}{2}} + \beta \vec{\nabla} \times \vec{E}^{n+1}\] \[\vec{H}^0 \rightarrow \vec{H}^\frac{1}{2} \rightarrow \vec{E}^1 \rightarrow \vec{H}^\frac{3}{2} \rightarrow ...\]不会产生人为的增益/衰减
Using the time-domain propagator
- 定义物理结构
- 定义仿真区域
- 添加光源
- 添加监视器
Dispersive Material
- 时域都是实数，频域是复数
- 频域关系：\(\vec{D}(\omega) = \varepsilon(\omega)\vec{E}(\omega)\)
- 时域关系：\(\vec{D}(t) = \varepsilon(t) \ast \vec{E}(t) = \int_0^t \vec{E}(t') \varepsilon(t - t')dt'\)
Sources
The propagator has a variety of sources available:
- Dipole
- Slab Gaussian beam （标量）
- Slab plane wave
- Mode
- Total field/scattered field
- Large Na source （矢量）
- User-defined
FDTD is a time domain technique
\[\vec{E}(\omega) = \int_0^{T_{Sim}} e^{i \omega t} \vec{E}(t) dt\]如果使用所谓“渐变”稳态场激励，就丧失了时域算法的优势
Tips
- What mesh size should I use?
- “Mesh accuracy” of 1 or 2 for initial setup (faster)
- Use “meh accuracy” of 2-4 for final simulations
- “Mesh accuracy” 5-8 is almost never necessary
- How long a simulation time should I use?
- Start with long simulations times and let the “auto-shutoff” feature find out when you can stop the simulation
- Check with point time monitors
- Avoid simulating homogeneous regions with no structure
- 与FDTD Solutions不同，光源的偏振特性不能在光源那里修改
Solvers
FDE Solver
Can be used for 1D(slab) or 2D structure cross sections
Look for solutions to Maxwell’s equations of the form \(\vec{E}(x,y,z,\omega) = exp(i\beta_m z) \vec{E}_m(x,y,\omega)\)
\[\vec{H}(x,y,z,\omega) = exp(i\beta_m z) \vec{H}_m(x,y,\omega)\]可计算的物理特性：
模式分布、群折射率、色散、弯曲损耗、耦合长度、重叠积分
varFDTD Solver
时域
Suitable for simulation of planar components including ring resonators, couplers, splitters, cross-overs, etc
3D planar waveguide geometry collapsed into 2D effective materials which capture material and waveguide dispersion.
应假设传播过程中在垂直方向几乎不改变
2D仿真速度但是接近3D仿真精度
EME Solver
频域
严格求解麦克斯韦方程
一次仿真可以获得多个模式输入和多个模式输出的结果
将波导器件沿一个方向分段，每段就是一个cell
When to use what solver
- 平面类波导
- 垂直方向上有耦合：EME
- 没有垂直方向上耦合
- 传播是全方向的：varFDTD
- 沿一个方向传播：EME
- 非平面类波导
- 光栅类：EME
- 光在垂直方向上有耦合的输入和输出：3D FDTD
Typical workflow
- 添加物理结构
- 添加仿真区域（EME中需要分段）
- 添加光源（EME中需要选输入输出端口、入射模式）
- 添加监视器
- 检查材料的拟合和内存需求
- 进行仿真
- 分析/输出
- 更改结构参数直到达到预期效果