Second-Harmonic Generation at a Fourth-Order Exceptional Point Degeneracy

Abstract

An anomalous flat-band dispersion provided by a degenerate band edge (DBE) of longitudinal optical modes in a double-grating waveguide is used to enhance second-harmonic generation (SHG). The DBE is a fourth-order exceptional point degeneracy (EPD) in a lossless and gainless waveguide, characterized by the coalescence of four eigenmodes that establish a frozen mode in a cavity. At a DBE resonance, the cavity quality factor scales $Q\propto N^5$, where $N$ is the number of unit cells of the grating waveguide. In our numerical experiments, we observe the peak intensity of the fundamental field in the edge-excited cavity scaling as $I_1\propto N^{3.6}$. This leads to a highly efficient SHG process that is radiated vertically from the cavity (i.e., normal to the grating) without requiring collinear phase matching, with a conversion efficiency scaling as $η\propto N^{8.27}$. These results establish DBE-based waveguides as promising platforms for miniaturized efficient nonlinear photonic devices.

Figures (6)

• Figure 1: (a) Schematic of the three-dimensional double-grating waveguide supporting a DBE composed of $N$ unit cells along the $x$ direction and width $w$ in the transverse $z$ direction. The upper section is partially removed for visual clarity. A pump wave at the DBE resonance frequency $f_{r,D}$ propagates in the positive $x$ direction and generates a second harmonic at $2f_{r,D}$, which radiates in the $y$ direction, i.e., vertically. Collinear phase matching at $2f_{r,D}$ is not required, whereas the phase condition for vertically-emitted second-harmonic is satisfied when the pump is tuned at the DBE. (b) Dispersion diagram of $z$-polarized modes near the fundamental frequency, with the flat band at the Brillouin zone edge indicating the DBE at $f_D$ (black dashed). The bracket indicates the difference between the resonant frequency $f_{r,D}$ (blue dashed-line), and the DBE frequency $f_D$ (black-dashed line). The inset shows the dispersion diagram around $2f_D$ (black-dashed line). The SHG at $2f_{r,D}$ would not excite a mode near $\operatorname{Re}(kd/\pi)=2$.
• Figure 2: Transverse section of a double grating supporting a DBE of guided longitudinal modes in the $x$ direction, polarized along $z$ (out-of-plane). The two oppositely-facing gratings are shifted by a distance $s$ along $x$ to break the mirror symmetry, enabling the occurrence of the DBE that leads to a very flat band in the dispersion diagram provided in the first equation of \ref{['eq:DBERBEDisp']}. Orange and gray areas indicate high-index (AlGaAs) and low-index (AlGaO) regions of the structure, respectively.
• Figure 3: (a) Difference between the DBE resonance frequency $f_{r,D}$ and the ideal DBE frequency $f_D$ as a function of the number $N$ of unit cells. Black dots are the simulation data; the red line is the fit from Eq. \ref{['eq:DBEresonFreqAsymp']}. (b) Normalized intensity profiles $I_1(x)/I_{1,{\text{max}}}$ of the $z$-polarized field versus longitudinal position $x/(Nd)$ for various waveguide lengths $N$, evaluated at the center of the top waveguide core and at their respective DBE resonances. All normalized profiles collapse onto a single curve, indicating that the spatial distribution of the field is preserved as $N$ increases. This confirms that increasing the number of unit cells enhances the field amplitude without altering the shape of the intensity envelope. The intensity peak approaches $x=Nd/2$ for increasing $N$. (c) Field intensity values $I_1(N)$ normalized by the incident intensity $I_0 = \qty{3.86e8}{\V\squared/\m\squared}$, evaluated at the cavity center $x = Nd/2$ (orange) and at one-third of the length from the center $x = Nd/3$ (blue) as a function of $N$. Solid lines represent power-law fittings of the form $I_{\text{fit}}(N) = aN^b$, yielding exponents of $b = 3.6$ and $b = 3.35$, respectively. (d) Loaded quality factors of the DBE-supporting double grating (represented by black dots and evaluated at the DBE resonance for each $N$) for varying grating length. The red line is a fitting function that shows the asymptotic $\propto N^5$ scaling of the DBE quality factor $Q_D$ with waveguide length.
• Figure 4: Fundamental-field intensity in the cavity at the DBE resonance, for $N=200$. The Bloch model (green dashed curve, retrieved from Eq. (\ref{['pump_field']})), shows excellent agreement with the numerical simulation (blue curve). The red-dashed line is the intensity of the propagating portion of the field, $|A_0e^{ik^{(0)}x}+A_2e^{ik^{(2)}x}|^2$, while the black-dashed line is the intensity of the evanescent portion of the field, $|A_1e^{ik^{(1)}x}+A_3e^{ik^{(3)}x}|^2$. All the quantities are normalized with respect to the value of the peak intensity in the center of the cavity.
• Figure 5: (a) Normalized intensity profiles along the longitudinal $x$ axis of a double grating with $N=330$ unit cells at the center of the top waveguide core. The solid blue line shows the intensity $I_1$ at the fundamental frequency $f_1 = f_{r,D}$, while the solid red line represents its squared value, $I_1^2$. The dashed yellow line denotes the normalized intensity $I_2$ at the second harmonic ($f_2 = 2f_{r,D}$). The near-perfect overlap between $I_1^2$ and $I_2$ confirms that the second-harmonic field distribution follows the nonlinear polarization generated by the DBE-enhanced field, and that no guided Bloch modes are substantially excited at $f_2$. (b) Conversion efficiency of double grating (black dots) cavities of different lengths computed at their respective DBE resonances. The solid red line depicts the fitting function that shows the asymptotic $N^{8.27}$ scaling of the nonlinear conversion efficiency $\eta_D$ with waveguide length. The inset depicts the field at the fundamental frequency $f_1=f_{r,D}$ being excited from the left and the field at the second-harmonic frequency leaking out of the structure.