Indirect link between resonant and guided modes on uniform and periodic slabs

A uniform or periodic dielectric slab can serve as an optical waveguide for which guided modes are important, and it can also be used as a diffraction structure for which resonant modes with complex frequencies are relevant. Guided modes are normally studied below the lightline where they exist continuously and emerge from points on the lightline, but isolated guided modes may exist above the lightline and they are the so-called bound states in the continuum. Resonant modes are usually studied above the lightline (defined using the real part of the complex frequency), but they are not connected to the guided modes on the lightline. In this work, through analytic and numerical calculations for uniform and periodic slabs, we establish an indirect link between the resonant and guided modes. It is shown that as the (Bloch) wave number is increased, a resonant mode continues its existence below the lightline, until it reaches its end at an exceptional point (EP) where a pair of improper modes emerge, and one branch of improper modes eventually approaches the lightline at the starting point of a guided mode. Leaky modes with a real frequency and a complex (Bloch) wave number (propagation constant) are also related to the improper modes. They emerge at EPs in eigenvalue formulations where the frequency is regarded as a parameter. Our study is based on a non-Hermitian eigenvalue formulation that includes resonant, improper, and leaky modes, and provides a complete picture for different kinds of eigenmodes on uniform and periodic slabs.

Figure 1

A uniform slab of thickness $2b$ and relative permittivity ${\varepsilon }_1$ surrounded by air.

Figure 2

Dispersion curves of odd (left panel) and even (right panel) TE modes for $1\le m\le 4$, showing guided (red), resonant (black), improper (blue), and leaky (cyan) modes. The dashed black line is the lightline. The orange and pink dots depict the locations of EPs for resonant and leaky modes, respectively.

Figure 3

Real parts [Figs. 3 and 3] and imaginary parts [Figs. 3 and 3] of $kh$ versus $\beta h$ for the odd and even TE modes with $m=1$, respectively.

Figure 4

Normalized free-space wave number $kh$ versus $\mathrm{Re}\left(\beta \right)h$ [Figs. 4 and 4] and $\mathrm{Im}\left(\beta \right)h$ [Figs. 4 and 4] for odd ($m=2$) and even ($m=1$) TE leaky modes, respectively.

Figure 5

Dispersion curves of a few odd (left panel) and even (right panel) TM modes, showing guided (red), resonant (black), improper (blue), and leaky (cyan) modes. The dashed black line is the lightline. The orange and pink dots depict the locations of EPs for resonant and leaky modes, respectively.

Figure 6

A dielectric slab that is invariant in $x$ and periodic in $y$ with period $d$. Each period consists of two uniform segments.

Figure 7

Dispersion curves of the first two TE odd modes on a periodic slab with ${\varepsilon }_1=11.56, {\varepsilon }_2=1, a=0.5d$, and $h=d$. The guided, resonant, and improper modes are shown in red, black, and blue, respectively. The insets highlight the local behavior of improper and resonant modes near the EPs marked as ${\mathrm{EP}}^{\left(r\right)}$.

Figure 8

Dispersion curves of two TE even modes on a periodic slab with $h=3d, {\varepsilon }_1=11.56, {\varepsilon }_2=1$, and $a=0.5d$. The guided, resonant, improper, and leaky modes are shown in red, black, blue, and cyan, respectively. The inset highlights the local behavior of improper and leaky modes near the EP marked as ${\mathrm{EP}}^{\left(l\right)}$.