In its simplest form, a **Fabry-Pérot cavity** is a region of light-transmitting medium surrounded by two mirrors, which may transmit some of the incoming light. Such a setup can be used as e.g. an interferometer or a laser cavity.

Below, we calculate its quasinormal modes in 1D. We divide the \(x\)-axis into three domains: left \(L\), center \(C\), and right \(R\). The cavity \(C\) has length \(\ell\) and is centered on \(x = 0\). Let \(n_L\), \(n_C\) and \(n_R\) be the respective domains’ refractive indices:

In its simplest “microscopic” form, the reflection at the boundaries is simply caused by the index differences there. Consider this ansatz for the electric field \(E_m(x)\), where \(m\) is the mode:

\[\begin{aligned} E_m(x) = \begin{cases} A_1 e^{- i k_m n_L x} & \mathrm{for}\; x < -\ell/2 \\ A_2 e^{- i k_m n_C x} + A_3 e^{i k_m n_C x} & \mathrm{for}\; \!-\!\ell/2 < x < \ell/2 \\ A_4 e^{i k_m n_R x} & \mathrm{for}\; x > \ell/2 \end{cases} \end{aligned}\]

The goal is to find the modes’ wavenumbers \(k_m\). First, we demand that \(E_m\) and its derivative \(\dv*{E_m}{x}\) are continuous at the boundaries \(x = \pm \ell/2\):

\[\begin{aligned} A_1 e^{i k_m n_L \ell/2} &= A_2 e^{i k_m n_C \ell/2} + A_3 e^{- i k_m n_C \ell/2} \\ A_4 e^{i k_m n_R \ell/2} &= A_2 e^{- i k_m n_C \ell/2} + A_3 e^{i k_m n_C \ell/2} \end{aligned}\] \[\begin{aligned} - i k_m n_L A_1 e^{i k_m n_L \ell/2} &= - i k_m n_C A_2 e^{i k_m n_C \ell/2} + i k_m n_C A_3 e^{- i k_m n_C \ell/2} \\ i k_m n_R A_4 e^{i k_m n_R \ell/2} &= - i k_m n_C A_2 e^{- i k_m n_C \ell/2} + i k_m n_C A_3 e^{i k_m n_C \ell/2} \end{aligned}\]

Rearranging the four equations above yields the following linear system:

\[\begin{aligned} 0 &= A_1 - A_2 e^{i k_m (n_C - n_L) \ell/2} - A_3 e^{- i k_m (n_C + n_L) \ell/2} \\ 0 &= A_2 e^{- i k_m (n_C + n_R) \ell/2} + A_3 e^{i k_m (n_C - n_R) \ell/2} - A_4 \\ 0 &= n_L A_1 + n_C \big( A_3 e^{- i k_m (n_C + n_L) \ell/2} - A_2 e^{i k_m (n_C - n_L) \ell/2} \big) \\ 0 &= n_C \big( A_3 e^{i k_m (n_C - n_R) \ell/2} - A_2 e^{- i k_m (n_C + n_R) \ell/2} \big) - n_R A_4 \end{aligned}\]

Which can be rewritten in matrix form as follows, with the system matrix on the left:

\[\begin{aligned} \begin{bmatrix} 1 & -e^{i k_m (n_C - n_L) \ell/2} & -e^{- i k_m (n_C + n_L) \ell/2} & 0 \\ 0 & e^{- i k_m (n_C + n_R) \ell/2} & e^{i k_m (n_C - n_R) \ell/2} & -1 \\ n_L & -n_C e^{i k_m (n_C - n_L) \ell/2} & n_C e^{- i k_m (n_C + n_L) \ell/2} & 0 \\ 0 & -n_C e^{- i k_m (n_C + n_R) \ell/2} & n_C e^{i k_m (n_C - n_R) \ell/2} & -n_R \end{bmatrix} \cdot \begin{bmatrix} A_1 \\ A_2 \\ A_3 \\ A_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \end{aligned}\]

We want non-trivial solutions, where we cannot simply satisfy the system by setting \(A_1\), \(A_2\), \(A_3\) and \(A_4\); this constraint will give us an equation for \(k_m\). Therefore, we demand that the system matrix is singular, i.e. its determinant is zero:

\[\begin{aligned} 0 = &- n_C (n_L + n_R) \big( e^{i k_m (2 n_C - n_L - n_R) \ell/2} + e^{- i k_m (2 n_C + n_L + n_R) \ell/2} \big) \\ &+ (n_C^2 + n_L n_R) \big( e^{i k_m (2 n_C - n_L - n_R) \ell/2} - e^{- i k_m (2 n_C + n_L + n_R) \ell/2} \big) \end{aligned}\]

We multiply by \(e^{i k_m (n_L + n_R) \ell / 2}\) and decompose the exponentials into sines and cosines:

\[\begin{aligned} 0 = i 2 (n_C^2 + n_L n_R) \sin\!(k_m n_C \ell) - 2 n_C (n_L + n_R) \cos\!(k_m n_C \ell) \end{aligned}\]

Finally, some further rearranging gives a convenient transcendental equation:

\[\begin{aligned} \boxed{ 0 = \tan\!(k_m n_C \ell) + i \frac{n_C (n_L + n_R)}{n_C^2 + n_L n_R} } \end{aligned}\]

Thanks to linearity, we can choose one of the amplitudes \(A_1\), \(A_2\), \(A_3\) or \(A_4\) freely, and then the others are determined by \(k_m\) and the field’s continuity.

Next, consider a “macroscopic” Fabry-Pérot cavity with complex mirror structures at boundaries, e.g. Bragg reflectors. If the cavity is large enough, we can neglect the mirrors’ thicknesses, and just use their reflection coefficients \(r_L\) and \(r_R\). We use the same ansatz:

\[\begin{aligned} E_m(x) = \begin{cases} A_1 e^{-i k_m n_L x} & \mathrm{for}\; x < -\ell/2 \\ A_2 e^{-i k_m n_C x} + A_3 e^{i k_m n_C x} & \mathrm{for}\; \!-\!\ell/2 < x < \ell/2 \\ A_4 e^{i k_m n_R x} & \mathrm{for}\; \ell/2 < x \end{cases} \end{aligned}\]

On the left, \(A_3\) is the reflection of \(A_2\), and on the right, \(A_2\) is the reflection of \(A_3\), where the reflected amplitudes are determined by the coefficients \(r_L\) and \(r_R\), respectively:

\[\begin{aligned} A_3 e^{- i k_m n_C \ell/2} &= r_L A_2 e^{i k_m n_C \ell/2} \\ A_2 e^{-i k_m n_C \ell/2} &= r_R A_3 e^{i k_m n_C \ell/2} \end{aligned}\]

These equations might seem to contradict each other. We recast them into matrix form:

\[\begin{aligned} \begin{bmatrix} 1 & - r_R e^{i k_m n_C \ell} \\ - r_L e^{i k_m n_C \ell} & 1 \end{bmatrix} \cdot \begin{bmatrix} A_2 \\ A_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \end{aligned}\]

Again, we demand that the determinant is zero, in order to get non-trivial solutions:

\[\begin{aligned} 0 &= 1 - r_L r_R e^{i 2 k_m n_C \ell} \end{aligned}\]

Isolating this for \(k_m\) yields the following modes, where \(m\) is an arbitrary integer:

\[\begin{aligned} \boxed{ k_m = - \frac{\ln\!(r_L r_R) + i 2 \pi m}{i 2 n_C \ell} } \end{aligned}\]

These \(k_m\) satisfy the matrix equation above. Thanks to linearity, we can choose one of \(A_2\) or \(A_3\), and then the other is determined by the corresponding reflection equation.

Finally, we look at the light transmitted through the mirrors, according to \(1 \!-\! r_L\) and \(1 \!-\! r_R\):

\[\begin{aligned} A_1 e^{i k_m n_L \ell/2} &= (1 - r_L) A_2 e^{i k_m n_C \ell/2} \\ A_4 e^{i k_m n_R \ell/2} &= (1 - r_R) A_3 e^{i k_m n_C \ell/2} \end{aligned}\]

We simply isolate for \(A_1\) and \(A_4\) respectively, yielding the following amplitudes:

\[\begin{aligned} A_1 &= (1 - r_L) A_2 e^{i k_m (n_C - n_L) \ell/2} \\ A_4 &= (1 - r_R) A_3 e^{i k_m (n_C - n_R) \ell/2} \end{aligned}\]

Note that we have not demanded continuity of the electric field. This is because the mirrors are infinitely thin “magic” planes; had we instead used the full mirror structure, then we would have demanded continuity, as you maybe expected.

- P.T. Kristensen, K. Herrmann, F. Intravaia, K. Busch, Modeling electromagnetic resonators using quasinormal modes, 2020, Optical Society of America.

© Marcus R.A. Newman, a.k.a. "Prefetch".
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