Perturbation theory for asymmetric deformed microdisk cavities

In an article by Dubertrand et al. [Phys. Rev. A 77, 013804 (2008)] the perturbation theory for slightly deformed optical microcavities with a mirror-reflection symmetry was developed. However, in real experiments such a mirror-reflection symmetry is often not present either intended or unintended via production tolerances. In this paper we therefore extended the perturbation theory to asymmetric boundary deformations. Consequently, we are able to describe interesting non-Hermitian phenomena like copropagation of optical modes in the (counter-)clockwise direction inside the cavity. The derived analytic formulas are demonstrated at two generic boundary shapes, the spiral and the double-notched circle where a good agreement to the numerical boundary element method is observed.

Figure 1

The boundary of (a),(b) symmetric and (c),(d) asymmetry deformed microcavities is shown as a solid curve. A dashed line illustrates the mirror-reflection symmetry. The cavities are (a) the quadrupole, (b) the Limaçon, (c) the spiral, and (d) the asymmetric double-notched circular cavity.

Figure 2

Second-order perturbation theory results for $x=kR$ values of optical modes with $l=1,2,3$ in the spiral cavity are shown as magenta triangles. Corresponding BEM results are shown as black open circles. The parameters of the spiral are $(\epsilon ,n)=(0.04,2.0)$.

Figure 3

Near-field intensity pattern of the optical modes with (upper panels) $(m,l)=(23,1)$ and (lower panels) $(m,l)=(19,2)$ in the spiral cavity with $(\epsilon ,n)=(0.04,2.0)$. The modes are calculated (a) with perturbation theory and (b) with BEM. The colormap ranges from no intensity (black) to high intensity (bright yellow).

Figure 4

The near-field intensity pattern of the mode pair with (top panels) $(m,l,\mu )=(14,1,0)$ and (bottom panels) $(m,l,\mu )=(14,1,1)$ in a spiral cavity with $(\epsilon ,n)=(0.04,2.0)$ is shown in (a). The corresponding BEM intensity pattern is shown in (b). The colormap ranges from no intensity (black) to high intensity (bright yellow).

Figure 5

The far-field intensity pattern ${\left|F\left(\varphi \right)\right|}^2$, see Eq. (16), of optical modes with (a) $(m,l,\mu )=(14,1,0)$ and (b) $(m,l,\mu )=(14,1,1)$ in a spiral cavity with $(\epsilon ,n)=(0.04,2.0)$ is shown as a solid magenta curve. The corresponding BEM result is shown as a dashed black curve. Both intensity patterns are normalized according to the area below the curves.

Figure 6

The (a) real and (b) imaginary part of $x=kR$ is shown as a function of notch size $\epsilon$ for the optical modes with $(m,l,\mu )=(14,1,0)$ and $(m,l,\mu )=(14,1,1)$. (c) shows the chirality, Eq. (15), of the modes vs notch size $\epsilon$. Black crosses and magenta triangles correspond to first- and second-order perturbation theory, respectively. BEM results are shown as black open circles. The refractive index of the spiral cavity is $n=2.0$. Insets show the boundary of the cavity for $\epsilon =0.0$ and $\epsilon =0.1$. A dashed curve illustrates the scaling with (a) $\epsilon$ and (b) ${\epsilon }^2$.

Figure 7

The error of second-order perturbation theory for the mode pair $(m,l)=(14,1)$ in a spiral cavity with refractive index $n=2.0$ is shown as function of notch size $\epsilon$ (black dots).

Figure 8

Black open circles (BEM) and magenta triangles (second-order perturbation theory) show the (a) real and (b) imaginary part of the splitting of the optical mode pair $(m,l)=(14,1)$ in a spiral cavity with $n=2.0$.

Figure 9

The (a) real and (b) imaginary part of $x=kR$ is shown as function of the angle ${\varphi }_2$ for the mode pair with $(m,l)=(8,1)$ in the double-notched cavity with $n=2.0$. (c) shows the chirality, Eq. (15), of the modes vs angle ${\varphi }_2$. Magenta triangles represent second-order perturbation theory and BEM results are shown as black open circles.

Figure 10

The (a) real and (b) imaginary part of $x=kR$ is shown as function of the angle ${\varphi }_2$ for the mode pair with $(m,l)=(10,1)$ in the double-notched cavity with $n=2.0$. (c) shows the chirality, Eq. (15), of the modes vs angle ${\varphi }_2$. Second-order perturbation theory results are shown as magenta triangles and BEM results are shown as black open circles. A dotted black line in the insets of (c) at ${\varphi }_2=1.0996$ is a guide to the eye.

Figure 11

The (a) real and (b) imaginary part of $x=kR$ for the mode pair $(m,l)=(10,1)$ is shown as a function of the size ${\epsilon }_2$ of the second notch. Both notches have the same width parameter ${\sigma }_1={\sigma }_2=0.035$. For ${\epsilon }_2=-0.06$ (two notches of equal size) and ${\epsilon }_2=0$ (a single notch), perturbation theory for symmetric deformations [19] can be applied (blue crosses). Second-order perturbation theory results for asymmetric cavities, Eqs. (23) and (31), are shown as magenta triangles and BEM results are shown as black open circles.

Figure 12

(a) Spiral cavity with $\epsilon =0.2$ and (b) rescaled spiral cavity with $\eta$ as in Eq. (A4) is shown as a black solid curve. The dashed curve is a circle of radius $R$, and the cyan filled region illustrates the area $\delta a$ where the deformation function changes the refractive index.

Figure 13

Area of the perturbation $\delta a$ depending on notch size $\epsilon$ of the spiral and the scaling factor $\eta$ is color encoded from dark (small perturbation area) to bright (large perturbation area). Black solid curves are contour lines. The white line is the optimal scaling to minimize the perturbation area; see Eq. (A4).

Figure 14

(a) Real and (b) imaginary part of complex wave number $x$ of the optical modes $(m,l)=(18,1)$ in a spiral cavity with $n=2.0$ is shown as function of notch size $\epsilon$. Dark blue crosses and magenta triangles represent perturbation theory with $[\eta$ as in Eq. (A4)] and without $\left(\eta =1\right)$ scaling. BEM results are shown as black open circles.