Formulation

Rigorous coupled wave analysis formulation

This documentation provides a reference to important formula definitions and helps people understand the implementation in the code. Some variables will have (Variable Name) to denote the short-form variable naming convention.

Rigorous coupled wave analysis (RCWA) is a method that solves diffraction problem in a periodic structure, such that we can leverage Fourier decomposition to help us formulate the problem into semi-analytical matrix equation. Using the boundary condition, we can solve the equation and find the diffraction efficiency of the device we are interested in.

Defining the problem

RCWA aims to solve diffraction efficiency problems inside periodic structures, by finding eigen electric-magnetic field modes existing in the structure that obeys Maxwell Equation under the assumption of linear material, source-free, and monochromatic.

Maxwell Equation

$$\nabla\cdot\vec D=\rho$$

$$\nabla\cdot\vec B=0$$

$$\nabla\times\vec E=-{\partial\vec B\over\partial t}$$

$$\nabla\times\vec H=\vec J+{\partial\vec D\over\partial t}$$

• $\vec D$ : Electric displacement
• $\vec B$ : Magnetic flux density
• $\vec H$ : Magnetic field vector (H)
• $\vec E$ : Electric field vector (E)
• $\rho$ : Charge density
• $J$ : Current density
• $t$ : time (t)

Linear material assumption

$$\vec D=\varepsilon\vec E$$

$$\vec B=\mu\vec H$$

• $\varepsilon$ : Permittivity
• $\mu$ : Permeability

Source-free assumption

$$\rho=0$$

$$J=0$$

Monochromatic assumption

$$\vec E\propto e^{-i\omega t}$$

$$\vec H\propto e^{-i\omega t}$$

• $\omega$ : Angular frequency

Normalization of magnetic field

$$\tilde{\vec H}=i\sqrt{\mu_0\over\varepsilon_0}\vec H$$

• $\varepsilon_0$ Free-space permittivity
• $\mu_0$ : Free-space permeability
• $k_0$ : Free-space wave number $2\pi\over\lambda_0$
  • $\lambda_0$ : Free-space wavelength (wl)
• $\varepsilon_r$ : Relative permittivity $\varepsilon\over\varepsilon_0$ (er)
• $\mu_r$ : Relative permeability $\mu\over\mu_0$ (ur)

See derivation for the normalization term

Normalized Maxwell equation

Linear, source-free, monochromatic.

$$\nabla\times\vec E=k_0\mu_r\tilde{\vec H}$$

$$\nabla\times\tilde{\vec H}=k_0\varepsilon_r\vec E$$

Fourier decomposition

The entire problem is formulated in Fourier space with respect to the periodic structure.

Plane-wave decomposition

$$\vec E(x,y;z)=\sum_{mn}\vec E(m,n;z)e^{i(k_x(m,n)x+k_y(m,n)y)}$$

$$\tilde{\vec H}(x,y;z)=\sum_{mn}\vec H(m,n;z)e^{i(k_x(m,n)x+k_y(m,n)y)}$$

• $\vec E(m,n;z)$ : Electric field coefficients
• $\vec H(m,n;z)$ : Magnetic field coefficients
• $k_x(m,n), k_y(m,n)$ : x, y component of the wave vector
• $m,n$ : Mode number, integer

Fourier expansion of the material

$$\varepsilon_r(x, y)=\sum_{mn}\varepsilon_r(m,n)e^{i(mG_xx+nG_yy)}$$

$$\mu_r(x,y)=\sum_{mn}\mu_r(m,n)e^{i(G_x(m, n)x+G_y(m, n)y)}$$

• $\varepsilon_r(m,n)$ : Permittivity coefficients
• $\mu_r(m,n)$ : Permeability coefficients
• $G_x, G_y$ : x, y component of the grating lattice vector

Wave and grating lattice vector

$$k_x(m,n)=k_{0x}+G_x(m, n)$$

$$k_y(m,n)=k_{0y}+G_y(m, n)$$

• $k_{0x}, k_{0y}$ : x, y component of the incidence wave vector

For rectangular lattice:

$$G_x(m, n) = mG_{0x}$$

$$G_y(m, n) = nG_{0y}$$

• $G_{0x}, G_{0y}$ : x, y component of the grating lattice vector

$$G_{0x}={2\pi\over d_x}, G_{0y}={2\pi\over d_y}$$

• $d_x$ : Period in x
• $d_y$ : Period in y

Convolution

For any field $U$ (ex. $E$ or $H$) interact with material coefficient $u$ (ex. $\varepsilon_r$ or $\mu_r$):

$$U(x,y;z)=\sum_{mn}U(m,n;z)e^{i(k_x(m,n)x+k_y(m,n)y)}$$

$$u(x, y)=\sum_{mn}u(m,n)e^{i(G_x(m, n)x+G_y(m, n)y)}$$

The effect of $u$ acts on $U$ is:

$$u(x,y;z)\cdot U(x,y;z)=\sum_{mn}U'(m,n;z)e^{i(k_x(m,n)x+k_y(m,n)y)}$$

$$U'(m,n;z)=\sum_{m_1n_1}U(m_1,n_1;z)u(m-m_1,n-n_1)$$

See derivation of Fourier convolution

Matrix equation

Maxwell equation in matrix form

$$ {d\over dz} \begin{bmatrix} E_x \\ E_y \end{bmatrix}= \left( \begin{bmatrix} K_x\\ K_y \end{bmatrix} \varepsilon_r^{-1} \begin{bmatrix} K_y & -K_x \end{bmatrix} + \begin{bmatrix} 0&\mu_r\\ -\mu_r&0 \end{bmatrix} \right) \begin{bmatrix} \tilde{H_x} \\ \tilde{H_y} \end{bmatrix} $$

$$ {d\over dz} \begin{bmatrix} \tilde{H_x} \\ \tilde{H_y} \end{bmatrix}= \left( \begin{bmatrix} K_x\\ K_y \end{bmatrix} \mu_r^{-1} \begin{bmatrix} K_y & -K_x \end{bmatrix} + \begin{bmatrix} 0&\varepsilon_r\\ -\varepsilon_r&0 \end{bmatrix} \right) \begin{bmatrix} E_x \\ E_y \end{bmatrix} $$

See derivation of matrix equation

Solving Boundary condition

(WIP)

Scatter Matrix

(WIP)

Finding Diffraction Efficiency

(WIP)