Nonlinear one-way edge-mode interactions for frequency mixing in topological photonic crystals

Topological photonics aims to utilize topological photonic bands and corresponding edge modes to implement robust light manipulation, which can be readily achieved in the linear regime of light-matter interaction. Importantly, unlike solid-state physics, the common test bed for new ideas in topological physics, topological photonics provides an ideal platform to study wave mixing and other nonlinear interactions. These are well-known topics in classical nonlinear optics but largely unexplored in the context of topological photonics. Here, we investigate nonlinear interactions of one-way edge modes in frequency mixing processes in topological photonic crystals. We present a detailed analysis of the band topology of two-dimensional photonic crystals with hexagonal symmetry and demonstrate that nonlinear optical processes, such as second- and third-harmonic generation, can be conveniently implemented via one-way edge modes in this setup. Moreover, we demonstrate that more exotic phenomena, such as slow-light enhancement of nonlinear interactions and harmonic generation upon interaction of backward-propagating (left-handed) edge modes, can also be realized. Our work opens up new avenues towards topology-protected frequency mixing processes in photonics.

Figure 1

(a) Schematic band structure showing the emergent edge modes due to the nontrivial topology of the bulk frequency bands. The edge modes can couple via SHG and THG frequency mixing processes. (b) Real-space illustration showing the unidirectional propagation of coupled edge modes along the system edge. The red arrow indicates the excitation source of the fundamental wave, whereas the green and blue waves are generated as a result of nonlinear wave mixing.

Figure 2

(a) Unit cell with lattice constant $a$ of the PhC, where $r$ and ${\epsilon }_1$ are the radius and relative permittivity of the cylinders, respectively, and ${\epsilon }_2$ and ${\mu }_i$ are the relative permittivity and off-diagonal component of the relative permeability of the background magnetic material, respectively. (b) The first Brillouin zone with symmetry points $\mathrm{\Gamma }$, $K$, $K^{\prime }$, and $M$. (c) Photonic band structure of the PhC, computed for $r=0.4a$, ${\epsilon }_1=3$, ${\epsilon }_2=18$, and ${\mu }_i=0$. (d) Nontrivial topological bands for ${\mu }_i=0.8$ [other parameters are the same as in (c)], where the Chern number of each band and the gap Chern numbers are provided (except for the last two bands, which touch each other and thus have the same Chern number). (e) Chern-number-graded gap phase diagrams when varying $r$, ${\epsilon }_1$, ${\epsilon }_2$, and ${\mu }_i$, determined for $r=0.4a$, ${\epsilon }_1=3$, ${\epsilon }_2=18$, and ${\mu }_i=0.8$.

Figure 3

(a) Photonic band structure of a 1D PhC strip that is periodic along the $x$ axis and has a finite size of 30 unit cells along the $y$ axis (top and bottom edges are terminated by a perfect electric conductor at $r=0.42a$). Other simulation parameters are ${\epsilon }_1=3$, ${\epsilon }_2=20$, and ${\mu }_i=0.8$. Edge modes in the three gaps around $\overline{\omega }=0.2$, 0.4, and 0.6 are depicted by red and blue lines and are formed at the top and bottom edges of the PhC, respectively. (b) Field profiles of the three one-way edge modes at $\overline{\omega }=0.2$, 0.4, and 0.6 of the top edge. Exponential decay of the field around the PhC edge can be observed (integers indicate the number of unit cells). (c) Dispersion curves of edge modes can be tailored by changing the edge termination, as indicated in the sketch.

Figure 4

(a–c) Simulated field profile intensities of ${\mathbf{E}}_1$, ${\mathbf{E}}_2$, and ${\mathbf{E}}_3$ at $\overline{\omega }=0.2$, ${\overline{\mathrm{\Omega }}}_2=2\overline{\omega }=0.4$, and ${\overline{\mathrm{\Omega }}}_3=3\overline{\omega }=0.6$, respectively, where ${\mathbf{E}}_1$ is induced by an external source indicated by an arrow, whereas ${\mathbf{E}}_2$ and ${\mathbf{E}}_3$ are generated by the corresponding nonlinear polarizations. SHG and THG are simulated separately with ${\chi }^{\left(2\right)}={10}^{-21}$ C ${\mathrm{V}}^{-2}$ and ${\chi }^{\left(3\right)}={10}^{-30}$ C m ${\mathrm{V}}^{-3}$. In the simulations, the absorbing boundary condition (ABC) is used for the left edge and PEC for the other edges. (d–f) Detailed analysis of SHG, where (d) shows the field intensity profile and power ${\overline{P}}_2$ (normalized by its peak value) at SH in a much larger simulation domain so as to resolve the oscillations of ${\overline{P}}_2$ due to the small phase mismatch $\mathrm{\Delta }\overline{k}$ of the edge modes at FF $\overline{\omega }=0.2$ and SH ${\overline{\mathrm{\Omega }}}_2=0.4$ (dotted and solid lines correspond to full numerical simulations and CMT, respectively); (e) shows the edge modes at FF ${\omega }_0$ (plotted in terms of $2{\omega }_0$) and, for SHG, ${\mathrm{\Omega }}_2$; and (f) shows the effect of phase matching, where the theoretically calculated ${\mathrm{\Lambda }}_2=2\pi /\mathrm{\Delta }\overline{k}$ and numerically extracted oscillation period of ${\mathbf{E}}_2$ are in excellent agreement. The $\overline{\omega }\left(\mathrm{\Delta }\overline{k}\right)$ curve is calculated according to the edge modes presented in (e). (g–i) Results corresponding to (d)–(f), respectively, but calculated for the THG. (j) Illustration of the effect of a structural defect (simulated as the thin PEC rectangle) on the SHG interaction of topologically protected modes. Compared to the defect-free case shown in (d), one can see that the oscillation period of the amplitude of the second-harmonic mode after bypassing the defect is the same as that in (d).

Figure 5

(a) Dispersion of edge modes at ${\omega }_0$ and ${\mathrm{\Omega }}_2$ determined for $t=0.24a$, as in Fig. 3, whereas the other simulation parameters are the same as in Fig. 4. (b) The dispersion curve of the FF mode now shows a plateau leading to a peak of $n_g$ at ${\overline{\omega }}_g=0.1968$. The frequency of the phase-matching point ${\overline{\omega }}_k\equiv \overline{\omega }(\mathrm{\Delta }\overline{k}=0)$ is ${\overline{\omega }}_k=0.1972$. (c) Enhancement of the SH conversion efficiency due to the slow-light effect. Left panel: An example where the SH power $P_2$ at two frequencies, ${\overline{\omega }}_l=0.1960$ and ${\overline{\omega }}_h=0.1975$, shows the same oscillation period, yet its oscillation amplitude at ${\overline{\omega }}_l$ (closer to the maximum of $n_g$, located at ${\overline{\omega }}_g$) is enhanced compared to that at ${\overline{\omega }}_h$. Right panels: Schematics of the formal definition of the enhancement factor $\eta$ (top) and its frequency dependence in the slow-light regime (bottom).

Figure 6

SHG via interaction between forward- and backward-propagating edge modes. This setting exploits the existence of a gap with a negative Chern number at the SH for $r=0.41a$, ${\epsilon }_1=3$, ${\epsilon }_2=20$, ${\mu }_i=0.82$ [see Fig. 2] and edge termination at $t=0.82a$ [see Fig. 3]. (a, b) Field intensity profiles of ${\mathbf{E}}_1$ and ${\mathbf{E}}_2$ calculated for $2{\overline{\omega }}_0=0.74$, illustrating that whereas the mode at the FF propagates clockwise ($C=1$ for the ${\omega }_0$ gap), the edge mode at the SH is backward propagating ($C=-1$ for the ${\mathrm{\Omega }}_2$ gap). In the simulation, ABC is used for the bottom edge and PEC boundaries for the other edges. (c) Edge states of the two gaps, which show the hallmark of edge modes with a negative Chern number: the slope of the dispersion curve of the edge mode at the SH is negative. (d) Comparison between the theoretically calculated and the numerically extracted oscillation period ${\mathrm{\Lambda }}_2$, confirming that the phase-matching mechanism is involved in this unusual nonlinear interaction regime.

Figure 7

(a, c) Real and (b, d) imaginary parts of ${\chi }_{\mathrm{eff},f}^{\left(2\right)}\left(x\right)$ and ${\chi }_{\mathrm{eff},s}^{\left(2\right)}\left(x\right)$ of Eq. (27) for SHG in one unit cell. While the real parts of ${\chi }_{\mathrm{eff},f}^{\left(2\right)}\left(x\right)$ and ${\chi }_{\mathrm{eff},s}^{\left(2\right)}\left(x\right)$ are even functions, with respect to the center of the unit cell, the imaginary parts of ${\chi }_{\mathrm{eff},f}^{\left(2\right)}\left(x\right)$ and ${\chi }_{\mathrm{eff},s}^{\left(2\right)}\left(x\right)$ are odd functions.

Figure 8

Evolution of the power of the second-harmonic wave as a function of the propagation distance. Red circles are from rigorous full-wave numerical simulations, while the black curve is obtained from solving the coupled-mode equations expressed as Eqs. (29).

Figure 9

(a, c) Real and (b, d) imaginary parts of ${\chi }_{\mathrm{eff},f}^{\left(3\right)}\left(x\right)$ and ${\chi }_{\mathrm{eff},t}^{\left(3\right)}\left(x\right)$ of Eqs. (B13) for THG in one unit cell. Similarly to the case of SHG, while the real parts of ${\chi }_{\mathrm{eff},f}^{\left(3\right)}\left(x\right)$ and ${\chi }_{\mathrm{eff},t}^{\left(3\right)}\left(x\right)$ are even functions, the imaginary parts of ${\chi }_{\mathrm{eff},f}^{\left(3\right)}\left(x\right)$ and ${\chi }_{\mathrm{eff},t}^{\left(3\right)}\left(x\right)$ are odd functions.

Figure 10

Evolution of the power of the third-harmonic wave vs. the propagation distance. Red circles are from numerical simulations, while the black curve is from solving Eqs. (B15).

Figure 11

(a) ${\chi }_{\mathrm{eff},f/s}^{\left(2\right)}\left(x\right)$ defined by Eqs. (27a) and (27b) for SHG in one unit cell at two frequencies, ${\overline{\omega }}_{\mathrm{slow}}=0.197$ and ${\overline{\omega }}_{\mathrm{fast}}=0.21$, where the group indices of the fundamental wave at the two frequencies are $n_g^f\left({\overline{\omega }}_{\mathrm{slow}}\right)\simeq 174$ and $n_g^f\left({\overline{\omega }}_{\mathrm{fast}}\right)\simeq 12$ [$n_g^s\left({\overline{\omega }}_{\mathrm{slow}}\right)\simeq 5$ and $n_g^s\left({\overline{\omega }}_{\mathrm{fast}}\right)\simeq 4$]. (b) The nonlinear coefficient ${\gamma }_{f/s}^{\left(2\right)}\left(x\right)$ of Eqs. (21) and (25) in one unit cell at the two frequencies ${\overline{\omega }}_{\mathrm{slow}}$ and ${\overline{\omega }}_{\mathrm{fast}}$. One can see the enhancement of the nonlinear coefficient due to the slow-light effect [$n_g^f\left({\overline{\omega }}_{\mathrm{slow}}\right)/n_g^f\left({\overline{\omega }}_{\mathrm{fast}}\right)\simeq 15$, whereas ${\overline{\gamma }}_{f,s}^{\left(2\right)}\left({\overline{\omega }}_{\mathrm{slow}}\right)/{\overline{\gamma }}_{f,s}^{\left(2\right)}\left({\overline{\omega }}_{\mathrm{fast}}\right)\simeq 18$ according to Eq. (28)].