Multilayer Optical Thin Film Simulation

Basic Usage Guide

To simulate a multilayer structure:

Set the wavelength range and number of spectral points.

Add layers between the semi-infinite media (air):
*Click "Add Layer" to add intermediate layers;
*For each layer, set refractive index (n), extinction coefficient (k) and thickness in nm.

View the results:
*Six plots show the optical response for both TE and TM polarizations;
*Use the angle slider to change the incident angle;
*Heatmaps show wavelength-angle dependence;

Key terms:
*TE: Electric field perpendicular to plane of incidence;
*TM: Magnetic field perpendicular to plane of incidence;
*R: Fraction of light reflected;
*T: Fraction of light transmitted;

How to use paired layers feature

To create periodic structures (e.g., DBR):

Add the layers you want to repeat
Check "Add to pair" for these layers
Enable "Enable Pairs"
Set the number of repetitions

Technical Details

Here, we use the Transfer Matrix Method (TMM) to analyze the propagation of electromagnetic waves through the multilayer structures. The following describes the implementation details.

Wave Propagation in Each Layer

In each layer, the electric field can be expressed as a superposition of forward and backward propagating waves: \(E(z) = A e^{ikz} + B e^{-ikz}\)
Where \(k = \frac{2\pi n}{\lambda}\) is the wave vector, \(n\) is the complex refractive index, and \(\lambda\) is the wavelength.

Boundary Conditions

At each interface between layers, the tangential components of the electric and magnetic fields must be continuous.
For TE polarization (s-polarization), the electric field is perpendicular to the plane of incidence.
For TM polarization (p-polarization), the magnetic field is perpendicular to the plane of incidence.

Transfer Matrix Formalism

The relationship between the fields in adjacent layers can be described by a transfer matrix: \(\begin{pmatrix} A_{j+1} \\ B_{j+1} \end{pmatrix} = M_j \begin{pmatrix} A_j \\ B_j \end{pmatrix}\)
Where \(M_j\) is the transfer matrix for the j-th layer, which accounts for both propagation through the layer and reflection/transmission at interfaces.

Matrix Components

For a layer with thickness \(d\) and refractive index \(n\), the transfer matrix is: \(M_j = \begin{pmatrix} e^{ik_jd_j} & 0 \\ 0 & e^{-ik_jd_j} \end{pmatrix} \begin{pmatrix} 1 & r_j \\ r_j & 1 \end{pmatrix}\)
Where \(r_j\) is the Fresnel reflection coefficient at the interface.

Fresnel Coefficients

For TE polarization: \(r_{TE} = \frac{n_1\cos\theta_1 - n_2\cos\theta_2}{n_1\cos\theta_1 + n_2\cos\theta_2}\)
For TM polarization: \(r_{TM} = \frac{n_2\cos\theta_1 - n_1\cos\theta_2}{n_2\cos\theta_1 + n_1\cos\theta_2}\)
Where \(\theta_1\) and \(\theta_2\) are the angles of incidence and refraction, related by Snell’s law: \(n_1\sin\theta_1 = n_2\sin\theta_2\).

Total Transfer Matrix

The total transfer matrix for a multilayer structure is the product of individual layer matrices: \(M_{total} = M_N \cdot M_{N-1} \cdots M_1\)

Reflection and Transmission Coefficients

The overall reflection and transmission coefficients can be extracted from the total transfer matrix: \(R = \left|\frac{B_0}{A_0}\right|^2, \quad T = \left|\frac{A_N}{A_0}\right|^2\)
Where \(A_0\) and \(B_0\) are the incident and reflected field amplitudes in the first layer, and \(A_N\) is the transmitted field amplitude in the last layer.