Modal Purcell factor in $\mathcal{PT}$-symmetric waveguides

We study the spontaneous emission rate of a dipole emitter in $\mathcal{PT}$-symmetric environment of two coupled waveguides using the reciprocity approach generalized to nonorthogonal eigenmodes of non-Hermitian systems. Considering emission to the guided modes, we define and calculate the modal Purcell factor composed of contributions of independent and interfering nonorthogonal modes leading to the emergence of cross-mode terms in the Purcell factor. We reveal that the closed-form expression for the modal Purcell factor within the coupled mode theory slightly alters for the non-Hermitian coupled waveguide compared to the Hermitian case. It is true even near the exceptional point, where the eigenmodes coalesce and the Petermann factor goes to infinity. This result is fully confirmed by the numerical simulations of active and passive $\mathcal{PT}$-symmetric systems being the consequence of the mode nonorthogonality.

Figure 1

Schematics of the $\mathcal{PT}$-symmetric system: Gain (refractive index $n_r=n_{\mathrm{co}}+i\gamma$) and loss ($n_l=n_{\mathrm{co}}-i\gamma$) waveguides of the width $w$ and height $h$ are embedded in the dielectric medium with index of $n_{cl}$. Waveguides are separated by the distance $g$. Modes propagate in the $\widehat{z}$ direction.

Figure 2

Distribution of the electric field component $E_x$ in the $\mathcal{PT}$-symmetric regime ($\gamma =3.5\times {10}^{-4}$). Distributions of $|E_x|$ for (a) “even” and (b) “odd” supermodes.

Figure 3

Distribution of the electric field component $E_x$ in the broken-$\mathcal{PT}$-symmetric regime ($\gamma =8\times {10}^{-4}$). Distributions of $|E_x|$ for (a) “loss” and (b) “gain” supermodes.

Figure 4

Effective mode indices versus the non-Hermiticity parameter $\gamma$. Black curves correspond to the real parts of the effective indices. Red curves correspond to the imaginary parts of the effective indices. Dashed (black and red) curves are related to the first supermode whereas dot-dashed ones related to the second supermode.

Figure 5

Petermann factors $K_1$ and $K_2$, respectively, for the first and second supermodes of the $\mathcal{PT}$-symmetric coupled waveguides as functions of the non-Hermiticity parameter $\gamma$. Parameters of the waveguide coupler: $g=0.8 \mu \mathrm{m}, h=0.2 \mu \mathrm{m}, n_{\mathrm{cl}}=1.444$, and $n_{\mathrm{co}}=3.478$.

Figure 6

Purcell factor distribution in the plane ($x, y$) (a) for the Hermitian system characterized by $\gamma =0$, (b) in the $\mathcal{PT}$-symmetric phase ($\gamma =3.5\times {10}^{-4}$), (c) in the broken-$\mathcal{PT}$-symmetric state ($\gamma =8\times {10}^{-4}$). Parameters of the waveguide coupler: $g=0.8 \mu \mathrm{m}, h=0.2 \mu \mathrm{m}, n_{\mathrm{cl}}=1.444$, and $n_{\mathrm{co}}=3.478$.

Figure 7

Distribution of the Purcell factor (a) diagonal and (b) off-diagonal terms depending on the emitter position $x$ at $y=0$ for different values of $\gamma$. Parameters of the coupled waveguide are given in the caption of Fig. 6.

Figure 8

Distribution of the Purcell factor at the line $y=0$ as function of the emitter position $x$ and non-Hermiticity parameter $\gamma$. Parameters of the coupled waveguide are given in the caption of Fig. 6.

Figure 9

Effective mode indices for the passive coupler versus the non-Hermiticity parameter $\gamma$. Black curves correspond to real parts of the effective indices. Red curves correspond to imaginary parts. Dashed (black and red) curves are related to the first supermode whereas dot-dashed ones related to the second supermode. Parameters of the passive waveguide coupler: $g=0.8 \mu \mathrm{m}, h=0.2 \mu \mathrm{m}, n_{\mathrm{cl}}=1.444$, and $n_{\mathrm{co}}=3.478$.

Figure 10

Petermann factors $K_1$ and $K_2$, respectively, for the first and second supermodes of the passive coupler as functions of the non-Hermiticity parameter $\gamma$. Parameters of the guiding system are given in the caption of Fig. 9.

Figure 11

Purcell factor distribution in the plane ($x, y$) (a) below phase transition ($\gamma =3.5\times {10}^{-4}$), (b) above the phase transition ($\gamma =8\times {10}^{-4}$). Parameters of the guiding system are given in the caption of Fig. 9.

Figure 12

Distribution of the Purcell factor at the line $y=0$ as function of emitter position $x$ and non-Hermiticity parameter $\gamma$. Parameters of the guiding system are given in the caption of Fig. 9.