Special Functions

cavity_mode_energy(θ::Real, p)

p = [E₀, n_eff]

Cavity mode energy as a function of incident angle.

\[\begin{aligned} E_\text{cavity}(\theta) = E₀ \left( 1 - \frac{\sin^2(\theta)}{n^2} \right)^{-1/2}, \end{aligned}\]

where $E₀$ is the energy of the cavity mode at zero degrees incidence angle.

cavity_transmittance(ν, p)

p = [n, α, L, R, ϕ]

Cavity transmittance as a function of frequency.

\[\begin{aligned} T(\nu) = \frac{T^2 e^{-\alpha L}}{1 + R^2 e^{-2 \alpha L} - 2 R e^{-\alpha L} \cos(4\pi n L \nu + 2\phi)}, \end{aligned}\]

where $\nu$ is the frequency, $\alpha$ and $n$ are the frequency-dependent absorption and refractive index, $L$ is the cavity length, and $\phi$ is the phase accumulated upon reflection against a mirror. $T$ and $R$ are the cavity transmittance and reflectance, respectively, where it is assumed that $T = 1 - R$.

coupled_energies(E_c::Vector{Real}, E_v::Real, V::Real, branch=0)

Coupled energies found by diagonalizing the coupled-oscillator model Hamiltonian with principle energies E_c (cavity photon) and E_v (material excitation), and interaction energy V. The lower and upper branches are called with branch=0 and branch=1, respectively.

\[\begin{aligned} H = \begin{pmatrix} E_1(\theta) & \Omega_R/2 \\\Omega_R/2 & E_2 \end{pmatrix} \end{aligned}\]

Diagonalizing $H$ gives the interaction energies as a function of $\theta$. These exhibit avoided crossing behavior near where the cavity mode energy equals the material excitation energy.

\[\begin{aligned} E_{\pm}(\theta) = \frac{1}{2}(E_1(\theta) + E_2) \pm \frac{1}{2} \sqrt{(E_1(\theta) - E_2)^2 + \Omega_R^2} \end{aligned}\]

https://en.wikipedia.org/wiki/Avoided_crossing

dielectric_imag(ν, p)

p = [A, ν₀, Γ]

Frequency-dependent imaginary part of the dielectric function.

\[\begin{aligned} \varepsilon_2(\nu) = \frac{A \Gamma \nu}{(\nu^2 - \nu_0^2)^2 + (\Gamma\nu)^2} \end{aligned}\]

dielectric_real(ν, p)

p = [A, ν₀, Γ]

Frequency-dependent real part of the dielectric function in terms of the background index and Lorentzian oscillator,

\[\begin{aligned} \varepsilon_1(\nu) = n^2 + \frac{A (\nu_0^2 - \nu^2)}{(\nu^2 - \nu_0^2)^2 + (\Gamma\nu)^2}, \end{aligned}\]

where $n$ is the background real index, $\nu_0$ is the excitation frequency, $\Gamma$ is the line width of the oscillator, and $A$ is the oscillator strength. In this model, $n$ is omitted to allow for a sum of oscillators. The background index must be added later.

fsr(peak_positions::Vector{Real)

Find the free spectral range for all adjacent peaks. Return the list of FSRs, the average FSR for the range, and the standard deviation.

fsr(x1::Real, x2::Real)

Solve for one of three variables in the equation for free spectral range in terms of the other two.

\[\begin{aligned} \Delta \nu = \frac{1}{2 L n}, \end{aligned}\]

where $\Delta\nu = |\nu_2 - \nu_1|$, $L$ is the intracavity length, and $n$ is intracavity index of refraction.