Weakly deformed optical microdisks: A third-order perturbation theory for transverse-magnetic modes

In this section we explain the general setup of the perturbation theory for weakly deformed microdisks with a mirror-reflection symmetry and TM polarized fields. Furthermore, we summarize the known results of the first- and second-order corrections. The detailed derivation of these corrections can be found in [33].

In the perturbation theory the cavity's boundary is expressed in polar coordinates as

$$\begin{eqnarray}&&r(\phi )=R+\lambda f(\phi )\end{eqnarray} \tag{ 5 }$$

where λ is a formally small deformation parameter and f(ϕ) is the deformation function. In contrast to the case of the circular cavity the boundary conditions here explicitly depend on the polar angle ϕ as [ψ_±,in  − ψ_±,out ](r(ϕ), ϕ) = 0 and ∂_r [ψ_±,in  − ψ_±,out ](r(ϕ), ϕ) = 0 for TM polarization. Using a Taylor series of these boundary conditions in λ yields

$$\begin{eqnarray}&&[{\psi }_{\pm ,{\mathsf{in}}}-{\psi }_{\pm ,{\mathsf{out}}}](R,\phi )=-\displaystyle \sum _{p=1}\displaystyle \frac{{\lambda }^{p}}{p!}{f}^{p}(\phi ){\partial }_{r}^{p}\left[{\psi }_{\pm ,{\mathsf{in}}}-{\psi }_{\pm ,{\mathsf{out}}}\right](R,\phi ),\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\partial }_{r}[{\psi }_{\pm ,{\mathsf{in}}}-{\psi }_{\pm ,{\mathsf{out}}}](R,\phi )=-\displaystyle \sum _{p=1}\displaystyle \frac{{\lambda }^{p}}{p!}{f}^{p}(\phi ){\partial }_{r}^{p+1}\left[{\psi }_{\pm ,{\mathsf{in}}}-{\psi }_{\pm ,{\mathsf{out}}}\right](R,\phi ).\end{eqnarray} \tag{ 7 }$$

Both equations need to be evaluated in the orders of λ to derive the perturbation theory corrections. To do so, the wave function is expressed with the ansatz

$$\begin{eqnarray}&&{\psi }_{\pm ,{\mathsf{in}}}(r,\phi )=\displaystyle \frac{{J}_{m}({nkr})}{{J}_{m}({nx})}{\chi }_{\pm ,m}(\phi )+\displaystyle \sum _{p\ne m}{a}_{p}\displaystyle \frac{{J}_{p}({nkr})}{{J}_{p}({nx})}{\chi }_{\pm ,p}(\phi ),\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{\psi }_{\pm ,{\mathsf{out}}}(r,\phi )=(1+{b}_{m})\displaystyle \frac{{H}_{m}({kr})}{{H}_{m}(x)}{\chi }_{\pm ,m}(\phi )+\displaystyle \sum _{p\ne m}({a}_{p}+{b}_{p})\displaystyle \frac{{H}_{p}({kr})}{{H}_{p}(x)}{\chi }_{\pm ,p}(\phi )\end{eqnarray} \tag{ 9 }$$

where the (dimensionless) frequency x of the mode in the deformed cavity is the quantity of interest (k = x/R) as well as the coefficients a_p and b_p . These quantities are again expanded in a series of the perturbation parameter λ up to the third order as

$$\begin{eqnarray}\begin{array}{rcl}x & = & {x}_{0}+{x}_{1}\lambda +{x}_{2}{\lambda }^{2}+{x}_{3}{\lambda }^{3}\\ {a}_{p} & = & {a}_{p}^{(1)}\lambda +{a}_{p}^{(2)}{\lambda }^{2}+{a}_{p}^{(3)}{\lambda }^{3}\\ {b}_{p} & = & {b}_{p}^{(2)}{\lambda }^{2}+{b}_{p}^{(3)}{\lambda }^{3}\end{array}\end{eqnarray} \tag{ 10 }$$

where x₀ is the frequency of the considered (unperturbed) mode in the circular cavity with S_m (x₀) = 0. Putting both, the expansion (10) and the ansatz (8)–(9), into equations (6)–(7) and comparing the coefficients in order λ¹ provides the first-order corrections

$$\begin{eqnarray}&&{x}_{1}=-{x}_{0}{A}_{{mm}},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{a}_{p}^{(1)}=({n}^{2}-1)\displaystyle \frac{{x}_{0}}{{S}_{p}}{A}_{{pm}}.\end{eqnarray} \tag{ 12 }$$

Here, S_p  = S_p (x₀) is shorten and A_mp are the Fourier harmonics of the deformation function

$$\begin{eqnarray}&&{A}_{{pm}}=\displaystyle \frac{{\varepsilon }_{p}}{2\pi R}{\int }_{0}^{2\pi }f(\phi ){\chi }_{\pm ,p}(\phi ){\chi }_{\pm ,m}(\phi )\ {\mathsf{d}}\phi \end{eqnarray} \tag{ 13 }$$

with ε_p  = 2 (ε_p  = 1) for p ≠ 0 (p = 0).

In the same way the second-order corrections are derived by evaluating equations (6)–(7) with the ansatz (8)–(9) and the expansion (10) and comparing the coefficients of the order λ². With the second-order Fourier harmonics of the deformation function

$$\begin{eqnarray}&&\quad {B}_{{pm}}=\displaystyle \frac{{\varepsilon }_{p}}{2\pi {R}^{2}}{\int }_{0}^{2\pi }{f}^{2}(\phi ){\chi }_{\pm ,p}(\phi ){\chi }_{\pm ,m}(\phi )\ {\mathsf{d}}\phi \end{eqnarray} \tag{ 14 }$$

the second-order corrections read

$$\begin{eqnarray}&&{x}_{2}=\displaystyle \frac{1}{2}{x}_{0}\left(3{A}_{{mm}}^{2}-{B}_{{mm}}\right)+{x}_{0}^{2}\left({A}_{{mm}}^{2}-{B}_{{mm}}\right)\displaystyle \frac{{H}_{m}^{{\prime} }}{{H}_{m}}-({n}^{2}-1){x}_{0}^{2}\displaystyle \sum _{p\ne m}\displaystyle \frac{{A}_{{mp}}{A}_{{pm}}}{{S}_{p}},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{a}_{p}^{(2)}=({n}^{2}-1)\displaystyle \frac{{x}_{0}}{{S}_{p}}\left\{{A}_{{pm}}{A}_{{mm}}\left(\displaystyle \frac{{x}_{0}}{{S}_{p}}{S}_{p}^{{\prime} }-1\right)+\displaystyle \frac{1}{2}{B}_{{pm}}\left[1+{x}_{0}\left(\displaystyle \frac{{H}_{m}^{{\prime} }}{{H}_{m}}+\displaystyle \frac{{H}_{p}^{{\prime} }}{{H}_{p}}\right)\right]\right.\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\left.+\,{x}_{0}({n}^{2}-1)\displaystyle \sum _{k\ne m}\displaystyle \frac{{A}_{{pk}}{A}_{{km}}}{{S}_{k}}\right\},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{b}_{p}^{(2)}=\displaystyle \frac{1}{2}({n}^{2}-1){x}_{0}^{2}{B}_{{pm}}.\end{eqnarray} \tag{ 18 }$$

Again the functions are evaluated at x₀.

In general, the perturbation theory is expected to give accurate results for sufficiently small λ. In [33] the range of λ has been estimated based on the area where the refractive index is changed via the deformation. However, even if this criterion is violated the perturbation theory can give reasonable results [43]. Contrary, pairs of quasi-degenerate modes within a symmetry class may spoil the accuracy of the perturbation theory which then needs to be adapted [33]. However, this specific case is omitted in the following.

In this section the third-order corrections for the perturbation series are derived. To do so, the mode ansatz (8)–(9) and the expansion scheme (10) are used to expand the boundary conditions (6)–(7). Thus, the coefficients of the order λ³ can be compared. In order to simplify these calculations we first define the auxiliary function

$$\begin{eqnarray}&&{{ \mathcal D }}_{{lm}}(x)={x}^{l}\left[{n}^{l}\displaystyle \frac{{J}_{m}^{[l]}}{{J}_{m}}({nx})-\displaystyle \frac{{H}_{m}^{[l]}}{{H}_{m}}(x)\right]\end{eqnarray} \tag{ 19 }$$

where ^[l] denotes the l-th derivative with respect to the argument. Then, we express the final results with the help of this auxiliary function and its derivatives evaluated at the unperturbed frequency x₀; i.e. the third-order corrections are expressed with ${{ \mathcal D }}_{{lm}}={{ \mathcal D }}_{{lm}}({x}_{0})$ and ${\partial }_{x}^{u}{{ \mathcal D }}_{{lm}}={\partial }_{x}^{u}{{ \mathcal D }}_{{lm}}({x}_{0})$. The derivatives of ${{ \mathcal D }}_{{lm}}({x}_{0})$ are independent of the deformation and can be calculated explicitly with common computer algebra systems. However, these explicit lengthy expressions are omitted here.

With the help of the auxiliary function ${{ \mathcal D }}_{{lm}}(x)$ we can write

$$\begin{eqnarray}&&\begin{array}{l}{R}^{l}{\partial }_{r}^{l}\left[{\psi }_{\pm ,{\mathsf{in}}}-{\psi }_{\pm ,{\mathsf{out}}}\right](R,\phi )={{ \mathcal D }}_{{lm}}(x){\chi }_{\pm ,m}(\phi )\\ \quad +\,\sum _{p\ne m}{a}_{p}{{ \mathcal D }}_{{lp}}(x){\chi }_{\pm ,p}(\phi )-\displaystyle \sum _{p}{b}_{p}{x}^{l}\displaystyle \frac{{H}_{p}^{[l]}}{{H}_{p}}(x){\chi }_{\pm ,p}(\phi ).\end{array}\end{eqnarray} \tag{ 20 }$$

Here, ${{ \mathcal D }}_{{lm}}$ is evaluated at the frequency x given from the scheme (10). Hence, ${{ \mathcal D }}_{{lm}}(x)$ needs to be expanded as

$$\begin{eqnarray}&&{{ \mathcal D }}_{{lm}}(x)={{ \mathcal D }}_{{lm}}+\lambda {x}_{1}{\partial }_{x}{{ \mathcal D }}_{{lm}}+{\lambda }^{2}\left({x}_{2}{\partial }_{x}{{ \mathcal D }}_{{lm}}+\displaystyle \frac{1}{2}{x}_{1}^{2}{\partial }_{x}^{2}{{ \mathcal D }}_{{lm}}\right)+{\lambda }^{3}\left({x}_{3}{\partial }_{x}{{ \mathcal D }}_{{lm}}+{x}_{1}{x}_{2}{\partial }_{x}^{2}{{ \mathcal D }}_{{lm}}+\displaystyle \frac{1}{6}{x}_{1}^{3}{\partial }_{x}^{3}{{ \mathcal D }}_{{lm}}\right).\end{eqnarray} \tag{ 21 }$$

Inserting this expansion into equation (20) allows to write the boundary conditions (6) and (7) in a series of λ. Comparing the coefficients of the terms λ³ reads

$$\begin{eqnarray}&&\begin{array}{l}\left({x}_{3}{\partial }_{x}{{ \mathcal D }}_{1m}+{x}_{1}{x}_{2}{\partial }_{x}^{2}{{ \mathcal D }}_{1m}+\displaystyle \frac{1}{6}{x}_{1}^{3}{\partial }_{x}^{3}{{ \mathcal D }}_{1m}\right){\chi }_{\pm ,m}(\phi )\\ \quad -\,\displaystyle \sum _{p}{\chi }_{\pm ,p}(\phi )\left\{{b}_{p}^{(3)}{x}_{0}\displaystyle \frac{{H}_{p}^{{\prime} }}{{H}_{p}}+{b}_{p}^{(2)}{x}_{1}\left[{x}_{0}\left(\displaystyle \frac{{H}_{m}^{{\prime\prime} }}{{H}_{m}}-\displaystyle \frac{{H}_{m}^{{\prime} 2}}{{H}_{m}^{2}}\right)-\displaystyle \frac{{H}_{m}^{{\prime} }}{{H}_{m}}\right]\right\}\\ \quad +\sum _{p\ne m}\left[{a}_{p}^{(3)}{{ \mathcal D }}_{1p}+{a}_{p}^{(2)}{x}_{1}{\partial }_{x}{{ \mathcal D }}_{1p}+{a}_{p}^{(1)}\left({x}_{2}{\partial }_{x}{{ \mathcal D }}_{1p}+\displaystyle \frac{1}{2}{x}_{1}^{2}{\partial }_{x}^{2}{{ \mathcal D }}_{1p}\right)\right]{\chi }_{\pm ,p}(\phi )\\ \ =\,\left[-\displaystyle \frac{f(\phi )}{R}\left({x}_{2}{\partial }_{x}{{ \mathcal D }}_{2m}+\displaystyle \frac{1}{2}{x}_{1}^{2}{\partial }_{x}^{2}{{ \mathcal D }}_{2m}\right)-\displaystyle \frac{{f}^{2}(\phi )}{2{R}^{2}}{x}_{1}{\partial }_{x}{{ \mathcal D }}_{3m}-\displaystyle \frac{{f}^{3}(\phi )}{6{R}^{3}}{{ \mathcal D }}_{4m}\right]{\chi }_{\pm ,m}(\phi )\\ \quad +\,\displaystyle \sum _{p}\displaystyle \frac{f(\phi )}{R}{b}_{p}^{(2)}{x}_{0}^{2}\displaystyle \frac{{H}_{p}^{{\prime\prime} }}{{H}_{p}}{\chi }_{\pm ,p}(\phi )\\ \quad +\,\sum _{p\ne m}\left[-\displaystyle \frac{f(\phi )}{R}\left({a}_{p}^{(2)}{{ \mathcal D }}_{2p}+{a}_{p}^{(1)}{x}_{1}{\partial }_{x}{{ \mathcal D }}_{2p}\right)-\displaystyle \frac{{f}^{2}(\phi )}{2{R}^{2}}{a}_{p}^{(1)}{{ \mathcal D }}_{3p}\right]{\chi }_{\pm ,p}(\phi )\end{array}\end{eqnarray} \tag{ 22 }$$

and

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \sum _{p}{b}_{p}^{(3)}{\chi }_{\pm ,p}(\phi )=\left[\displaystyle \frac{f(\phi )}{R}\left({x}_{2}{\partial }_{x}{{ \mathcal D }}_{1m}+\displaystyle \frac{1}{2}{x}_{1}^{2}{\partial }_{x}^{2}{{ \mathcal D }}_{1m}\right)+\displaystyle \frac{{f}^{2}(\phi )}{2{R}^{2}}{x}_{1}{\partial }_{x}{{ \mathcal D }}_{2m}+\displaystyle \frac{{f}^{3}(\phi )}{6{R}^{3}}{{ \mathcal D }}_{3m}\right]{\chi }_{\pm ,m}(\phi )\\ \quad +\sum _{p\ne m}\left[\displaystyle \frac{f(\phi )}{R}\left({a}_{p}^{(2)}{{ \mathcal D }}_{1p}+{a}_{p}^{(1)}{x}_{1}{\partial }_{x}{{ \mathcal D }}_{1p}\right)+\displaystyle \frac{{f}^{2}(\phi )}{2{R}^{2}}{a}_{p}^{(1)}{{ \mathcal D }}_{2p}\right]{\chi }_{\pm ,p}(\phi )-\displaystyle \sum _{p}\displaystyle \frac{f(\phi )}{R}{b}_{p}^{(2)}{x}_{0}\displaystyle \frac{{H}_{p}^{{\prime} }}{{H}_{p}}\,{\chi }_{\pm ,p}(\phi ).\end{array}\end{eqnarray} \tag{ 23 }$$

In order to solve this set of equations for the third-order corrections we introduce the third Fourier Harmonics of the deformation

$$\begin{eqnarray}&&\quad {C}_{{pm}}=\displaystyle \frac{{\epsilon }_{p}}{2\pi {R}^{3}}{\int }_{0}^{2\pi }{f}^{3}(\phi ){\chi }_{\pm ,m}(\phi ){\chi }_{\pm ,p}(\phi )\ {\mathsf{d}}\phi .\end{eqnarray} \tag{ 24 }$$

Using orthogonality of the χ_q (ϕ) we get from equations (22) and (23) the third-order corrections as

$$\begin{eqnarray}\begin{array}{rcl}{b}_{p}^{(3)} & = & \left({x}_{2}{\partial }_{x}{{ \mathcal D }}_{1m}+\displaystyle \frac{1}{2}{x}_{1}^{2}{\partial }_{x}^{2}{{ \mathcal D }}_{1m}\right){A}_{{pm}}+\displaystyle \frac{1}{2}{x}_{1}{\partial }_{x}{{ \mathcal D }}_{2m}{B}_{{pm}}+\displaystyle \frac{1}{6}{{ \mathcal D }}_{3m}{C}_{{pm}}\\ & & +\displaystyle \sum _{k\ne m}\left[\left({a}_{k}^{(2)}{{ \mathcal D }}_{1k}+{a}_{k}^{(1)}{x}_{1}{\partial }_{x}{{ \mathcal D }}_{1k}\right){A}_{{pk}}+\displaystyle \frac{1}{2}{a}_{k}^{(1)}{{ \mathcal D }}_{2k}{B}_{{pk}}\right]-\displaystyle \sum _{k}{b}_{k}^{(2)}{x}_{0}\displaystyle \frac{{H}_{k}^{{\prime} }}{{H}_{k}}{A}_{{pk}},\end{array}\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}\begin{array}{rcl}{x}_{3}\,{\partial }_{x}{{ \mathcal D }}_{1m} & = & -{x}_{1}{x}_{2}{\partial }_{x}^{2}{{ \mathcal D }}_{1m}-\displaystyle \frac{1}{6}{x}_{1}^{3}{\partial }_{x}^{3}{{ \mathcal D }}_{1m}+{b}_{m}^{(3)}{x}_{0}\displaystyle \frac{{H}_{m}^{{\prime} }}{{H}_{m}}+{b}_{m}^{(2)}{x}_{1}\left[{x}_{0}\left(\displaystyle \frac{{H}_{m}^{{\prime\prime} }}{{H}_{m}}-\displaystyle \frac{{H}_{m}^{{\prime} 2}}{{H}_{m}^{2}}\right)+\displaystyle \frac{{H}_{m}^{{\prime} }}{{H}_{m}}\right]\\ & & -\left({x}_{2}{\partial }_{x}{{ \mathcal D }}_{2m}+\displaystyle \frac{1}{2}{x}_{1}^{2}{\partial }_{x}^{2}{{ \mathcal D }}_{2m}\right){A}_{{mm}}-\displaystyle \frac{1}{2}{x}_{1}{\partial }_{x}{{ \mathcal D }}_{3m}{B}_{{mm}}-\displaystyle \frac{1}{6}{{ \mathcal D }}_{4m}{C}_{{mm}}+{b}_{m}^{(2)}{x}_{0}^{2}\displaystyle \frac{{H}_{m}^{{\prime\prime} }}{{H}_{m}}{A}_{{mm}}\\ & & -\sum _{p\ne m}\left[{A}_{{mp}}\left({a}_{p}^{(2)}{{ \mathcal D }}_{2p}+{a}_{p}^{(1)}{x}_{1}{\partial }_{x}{{ \mathcal D }}_{2p}\right)+\displaystyle \frac{1}{2}{B}_{{mp}}{a}_{p}^{(1)}{{ \mathcal D }}_{3p}-{A}_{{mp}}{b}_{p}^{(2)}{x}_{0}^{2}\displaystyle \frac{{H}_{p}^{{\prime\prime} }}{{H}_{p}}\right],\end{array}\end{eqnarray} \tag{ 26 }$$

and

$$\begin{eqnarray}&&\begin{array}{l}{a}_{p}^{(3)}\,{{ \mathcal D }}_{1p}=-{a}_{p}^{(2)}{x}_{1}{\partial }_{x}{{ \mathcal D }}_{1p}-{a}_{p}^{(1)}\left[{x}_{2}{\partial }_{x}{{ \mathcal D }}_{1p}+\displaystyle \frac{1}{2}{x}_{1}^{2}{\partial }_{x}^{2}{{ \mathcal D }}_{1p}\right]+{b}_{p}^{(3)}{x}_{0}\displaystyle \frac{{H}_{p}^{{\prime} }}{{H}_{p}}\\ \quad +\,{b}_{p}^{(2)}{x}_{1}\left[{x}_{0}\left(\displaystyle \frac{{H}_{p}^{{\prime\prime} }}{{H}_{p}}-\displaystyle \frac{{H}_{p}^{{\prime} 2}}{{H}_{p}^{2}}\right)+\displaystyle \frac{{H}_{p}^{{\prime} }}{{H}_{p}}\right]-\left({x}_{2}{\partial }_{x}{{ \mathcal D }}_{2m}+\displaystyle \frac{1}{2}{x}_{1}^{2}{\partial }_{x}^{2}{{ \mathcal D }}_{2m}-{b}_{m}^{(2)}{x}_{0}^{2}\displaystyle \frac{{H}_{m}^{{\prime\prime} }}{{H}_{m}}\right){A}_{{pm}}\\ \quad -\,\displaystyle \frac{1}{2}{x}_{1}{\partial }_{x}{{ \mathcal D }}_{3m}{B}_{{pm}}-\displaystyle \frac{1}{6}{{ \mathcal D }}_{4m}{C}_{{pm}}-\displaystyle \sum _{k\ne m}\left[\left({a}_{k}^{(2)}{{ \mathcal D }}_{2k}+{a}_{k}^{(1)}{x}_{1}{\partial }_{x}{{ \mathcal D }}_{2k}-{b}_{k}^{(2)}{x}_{0}^{2}\displaystyle \frac{{H}_{k}^{{\prime\prime} }}{{H}_{k}}\right){A}_{{pk}}+\displaystyle \frac{1}{2}{a}_{k}^{(1)}{{ \mathcal D }}_{3k}{B}_{{pk}}\right].\end{array}\end{eqnarray} \tag{ 27 }$$

Note that the third-order corrections are written in terms of the first- and second-order corrections which can be found in section 3.1.

First, we validate our calculations via a proof of principle. Therefore, the perturbation theory is applied to a circular cavity with a diminished radius (see figure 1(a)). Such a shrunken circular cavity is described within the perturbation theory by the radius in polar coordinates as

$$\begin{eqnarray}&&\displaystyle \frac{r(\phi )}{R}=\left(1-\epsilon \right).\end{eqnarray} \tag{ 28 }$$

Here, the deformation parameter 0 <  < 1 directly corresponds to the perturbation parameter λ for a deformation function f(ϕ) = −R in equation (5).

On the other hand, the exact frequencies x _analyt in the shrunken circular cavity can be calculated analytically by rescaling the frequencies of the unperturbed circular cavity as

$$\begin{eqnarray}&&{x}_{{\mathsf{analyt}}}=\displaystyle \frac{{x}_{0}}{1-\epsilon }={x}_{0}\displaystyle \sum _{n=0}^{\infty }{\epsilon }^{n}.\end{eqnarray} \tag{ 29 }$$

Within this expansion of the exact frequencies the perturbation theory in first (second) [third] order is expected to capture the terms proportional to (²) [³]. Thus, the error of the perturbation theory Err x = ∣x _analyt  − x _PT ∣ scales with ² (³) [⁴] for ≪ 1. This scaling is verified in figure 2 for a mode (m, l) = (16, 1) in a cavity with refractive index n = 2.0. Here, the third-order corrections indeed provide a significant improvement of the perturbation theory. However, it should be kept in mind that the deformation does not lead to a coupling of modes with different azimuthal mode numbers and is therefore very special. Generic examples are discussed below.

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A common example for a deformed cavity is the quadrupole which has been studied frequently in terms of ray-wave correspondence, dynamical tunneling and frequency splittings [9, 10, 44–46]. The boundary is given by the radius in polar coordinates as

$$\begin{eqnarray}&&\displaystyle \frac{r(\phi )}{R}=1+\epsilon \cos (2\phi )\end{eqnarray} \tag{ 30 }$$

where is a dimensionless deformation parameter which can be identified directly with the perturbation parameter λ for the deformation function $f(\phi )=R\cos (2\phi )$. In order to grade the perturbation theory the numerical reference value for the error Err x = ∣x _BEM  − x _PT ∣ of the frequency is calculated with the BEM. As shown in figure 3(a) for the mode with even parity and azimuthal mode number m = 3 in a quadrupole cavity with refractive index n = 1.5 the expected error scaling of the perturbation theory can be observed. In particular, the error of the second order scales with ³ and the error of the third-order scales with ⁴ until the overall error saturates for small due to the numerical tolerances in the BEM. For modes m > 3, however, the third-order corrections vanish and already the second-order corrections scale with ⁴ as shown exemplary for the mode (m, l) = (12, 1) in figure 3(b). The reason here is that in regard of the perturbation theory the quadrupole is a rather non-generic deformation due to the single cosine term in f(ϕ) which allows only for specific non-vanishing coupling coefficients in equations (13)–(14), (24). Therefore, A_pm , B_pm , and C_pm are rather sparse band matrices leading to the vanishing of the third-order corrections for m > 3.

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In order to avoid these non-generic behavior of the coupling coefficient A_pm , B_pm , and C_pm the cavity's boundary can be slightly modified to the so-called flatten quadrupole [10] defined by

$$\begin{eqnarray}&&\displaystyle \frac{r(\phi )}{R}=\sqrt{1+2\epsilon \cos (2\phi )}.\end{eqnarray} \tag{ 31 }$$

For small ∣∣ ≪ 1 the shape of the quadrupole and the flatten quadrupole cannot be distinguished by eye. However, due to the square root in equation (31) the flatten quadrupole exhibits more coupling terms between modes with different azimuthal mode numbers. Consequently, the third-order corrections are finite and lead to an improvement of the perturbation theory as shown in figure 4(a) for the mode (m, l) = (30, 1). Even though the improvement does not lead to a better scaling with the deformation parameter it yet can be advantageous, e.g., for the prediction of the mode coupling and Q-spoiling due to resonance-assisted tunneling [31, 47]. This can be seen in figure 4(b) where for the flatten quadrupole with  = 0.07 the complex frequencies of modes with l = 1 and different azimuthal mode numbers m are shown. The third-order perturbation theory describes the peak in the imaginary part of the frequency in a good agreement to the numerical BEM results whereas the second-order perturbation theory fails thereon. Note that the imaginary part of the frequency determines to a large amount the Q factor of the mode. Therefore, the peak shown in figure 4(b) refers to a spoiling of the Q factor.

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Generic examples with local deformations are notched cavities. They have been studied, e.g., for directional emission, mode selection and mode splitting [48–52]. In this section we focus on a doubly notched circular cavity (see figure 1(c)) described by the radius in polar coordinates as

$$\begin{eqnarray}&&\displaystyle \frac{r(\phi )}{R}=1-\epsilon \displaystyle \sum _{l=-\infty }^{\infty }\exp \left[-\displaystyle \frac{{\left(\phi -\pi l\right)}^{2}}{2{\sigma }^{2}}\right]\end{eqnarray} \tag{ 32 }$$

where the notches are at ϕ = 0 and ϕ = π. The notch depth  > 0 is the perturbation parameter whereas the angular notch width is fixed to σ = 0.1. In regard of the perturbation theory the notched circle is a generic deformation as there are no artificially vanishing coupling coefficients in A_pm , B_pm , and C_pm . Therefore, the expected error scaling of the perturbation theory is observed as shown in figure 5(a) for the mode (m, l) = (16, 1) with even parity in a cavity with refractive index n = 2. The third-order corrections again provide a significant improvement for small ≪ 1 in comparison to the second-order perturbation theory. On the other hand, for larger deformation, e.g.  > 0.07 in figure 5(a), the third-order corrections lead to more faulty result than the second order. Such a behavior is also expected as a high-order series approximation is accurate very close to the sampling point but then diverges faster than the lower-order approximation further away from the sampling point.

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An important quantity, e.g. for sensing applications, is the frequency splitting Δx = x₊ − x₋ between modes [53–56]. Especially, a local perturbation like a notch is known to cause a strong frequency splitting as modes with even parity undergo stronger shift in their frequencies as modes with odd parity. In figure 5(b) it is verified for the mode (m, l) = (16, 1) that the third-order perturbation theory is able to predict the frequency splitting accurately, especially for small notch depths  < 0.07.

An often considered example for a generic and non-local deformation is the limacon cavity [12–15, 39, 57, 58]. It is defined in polar coordinates (ρ, θ) by

$$\begin{eqnarray}&&\displaystyle \frac{\rho (\theta )}{R}=1+\epsilon \cos (\theta )\end{eqnarray} \tag{ 33 }$$

with deformation parameter . At first glance, the limacon seems again non-generic like the quadrupole cavity due to the single cosine term in the deformation function. However, for the perturbation it is crucial to reduce the area where the refractive index is changed via the deformation. Therefore, in [43] it was shown that a shift of the limacon's origin by R to the left along the x-axis minimizes this perturbation area (see figure 1(d)) and therefore leads to improved predictions of the second-order perturbation theory. Note that in the shifted limacon the radius r(ϕ) for the perturbation theory is defined implicitly and therefore does not contain only a single cosine term. In addition to further minimize the perturbation area we also take advantage of a rescaling of the cavity's radius [35]. More precisely, the perturbation theory is performed for the radius r_η (ϕ) = η r(ϕ) providing the complex frequency x_η . The final result from the perturbation theory is then given by x = η x_η . We select η dependent on the deformation parameter as $\eta ={\left(1+{\epsilon }^{2}/2\right)}^{-1}$. In figure 6 the results from the perturbation theory are compared to the BEM results for the mode (m, l) = (16, 1) in the limacon cavity with n = 2. For the real part of the complex frequencies the third-order corrections perform significantly better than the second-order results whereas in the imaginary part of the frequencies the second-order results are slightly more accurate for large deformation. In the second-order the imaginary part of x is even slightly more faulty if the rescaling scheme is applied. However, this is not a contradiction as the errors in the imaginary part are roughly a magnitude smaller compared to the errors in the real part which therefore dominate the overall error of the perturbation theory. Hence, the rescaling scheme improves the perturbation theory. It should also be noted that a deformation  ∼ 0.5 is rather large. For small deformations especially the third-order results again converge faster as seen in the magnification inset in figure 6(b).

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Within the perturbation theory also the far-field pattern of a mode can be calculated by [33]

$$\begin{eqnarray}&&{F}_{\pm }(\phi )=(1+{b}_{m})\displaystyle \frac{{e}^{-{im}\pi /2}}{{H}_{m}(x)}{\chi }_{\pm ,m}(\phi )+\displaystyle \sum _{p\ne m}({a}_{p}+{b}_{p})\displaystyle \frac{{e}^{-{ip}\pi /2}}{{H}_{p}(x)}{\chi }_{\pm ,p}(\phi ).\end{eqnarray} \tag{ 34 }$$

For the limacon cavity with a relatively strong deformation  = 0.43 the far-field pattern of the mode (m, l) = (16, 1) is shown in figure 7. Without the η-rescaling the far-field from the third-order corrections does not yield an obvious improvement in comparison to the second-order perturbation theory. However, by employing the η-rescaling significantly better predictions are obtained, see figure 7(b). Especially, the far-field from the third-order perturbation theory is in very well agreement the BEM results.

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