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Abstract
Nonlinear sum frequency generation (SFG) and difference frequency generation (DFG) are fundamental methods to obtain new light sources for various applications. However, most of the on-chip SFG and DFG are based on conventional resonators, lacking robustness against fabrication defects. Here, we demonstrate topologically protected SFG and DFG in a second-order topological photonic system. The mechanism is based on the nonlinear interaction between three high-Q corner modes inside dual topological band gaps. The frequency matching condition for SFG and DFG is precisely satisfied by designing a valley-photonic-crystal-like topological system, which provides more freedoms to tune the corner modes. The topological SFG and DFG are achieved with high conversion efficiency, and the underlying topological physics is revealed. This work opens up avenues toward topologically protected nonlinear frequency conversion, and can find applications in the fields of on-chip single-photon detections and optical quantum memories with robustness against defects.
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1. Introduction
Inspired by condensed matter physics, Haldane and Raghu introduced the concept of topology into photonic crystals (PCs) for the first time, offering a novel method for manipulating light against structural disorder [1]. Recently, a series of higher-order topological insulators have been proposed and experimentally verified on various platforms, including optical and acoustic systems [2–6]. Differing from the traditional first-order topological PCs with bulk-boundary correspondence, the second-order photonic topological system support topological states with dimensionality two lower than the system. For the two-dimensional (2D) case, a second-order topological insulator may possess 0D corner states, whose mode fields are highly localized on a subwavelength scale [4–7]. Taking advantage of these tightly localized corner states, researchers have successfully realized high-quality nanoscale topological cavities [8–10] and low-threshold topological lasers [10,11].
The fusion of material nonlinearity and topological photonics has provoked some novel findings, such as topological edge solitons in Kerr nonlinear coupled helical waveguides [12], nonlinear control of topological edge states [13], and nonlinearity-induced topological transitions [14,15,16]. Wavelength conversion is one of the fundamental nonlinear optical processes. Recently, some interesting works have been reported to achieve third harmonic generation (THG) based on topological edge states [17] and second harmonic generation (SHG) based on topological corner states [18–20]. By utilizing the nonlinear interaction between doubly-resonant corner states with high quality (Q) factors, the conversion efficiency of SHG can be significantly improved than that based on conventional methods [18–20]. Sum and difference frequency generation (SFG/DFG) are the other two crucial nonlinear parametric conversion processes, which can be used in single-photon detections and optical quantum memories, etc [21–23]. Compared with SHG, SFG and DFG are more flexible, and allow a broader conversion bandwidth. To achieve efficient on-chip conversion process, most of the reported SFG and DFG are based on ultrahigh-Q whispering-gallery-mode (WGM) microcavities, e.g., microrings and microdisks [22,24–26], which can provide abundant traveling-wave modes to satisfy the desired momentum and frequency matching conditions. However, the nonlinear conversion processes in these WGM cavities may yield many unexpected signals (besides SHG and SFG signals), resulting in nonlinear noise and consume the pump energy [27]. In addition, these traditional WGM cavities are not topologically protected, which means that even if slight perturbation will significantly decrease the Q factor and the mode overlap factor, and the conversion efficiency of SFG or DFG will be inevitably affected. Therefore, if we can realize SFG or DFG in a standing-wave topological nanocavity with finite resonances inside the band gaps, the above-mentioned limits may be broken. Furthermore, for the localized modes inside the standing-wave topological nanocavity, their average k vectors are zero, so that the condition of momentum matching could be naturally satisfied. However, to achieve efficient SFG or DFG, we have to create high-Q topological corner states with three different resonant frequencies and make them rigorously satisfy the frequency matching condition, which is more challenging than that of SHG and has not yet been realized.
In this paper, we demonstrate topologically protected SFG and DFG in a second-order topological PC with dual band gaps for the first time, to the best of our knowledge. The corner states in the second band gap possess extremely high value of Q factor over 3.5 × 107, which is comparable to that of WGMs [22,24–26], but is immune to the defects and disorders. This characteristic greatly facilitates the efficient generation of sum and difference frequencies. More importantly, we design a valley-photonic-crystal-like topological system, which provides extra freedom to make it possible to meet the frequency matching condition for SFG and DFG. Additionally, we also investigate the robustness of SFG and DFG against structural defects. These results may open up a novel avenue for achieving high-efficiency on-chip SFG and DFG with topological protection.
2. Design and analysis of topological corner states
We start from a valley PC shown in Fig. 1(a), which is composed of two kinds of dielectric circular rods with different radii to form a honeycomb lattice in air. The radii of the red and blue rods are r1 and r2, respectively, and the dielectric constant of these rods is 12.25. The side length of the primitive cell is $l = a/\sqrt 3 $, where a = 1 µm is the lattice constant. In this work, we use finite-element method to make the first-principles calculations, and focus on the transverse-magnetic (TM) modes. When r1 = r2, the PC lattice possesses C6-symmetry, and the typical band diagram has two Dirac points at K, as shown by the blue dots in Fig. 1(c). To open up the Dirac points and create two complete band gaps, we replace the red dielectric rods in Fig. 1(a) with three identical pillars forming an equilateral triangle, as shown in Fig. 1(b). Thus, the system’s inversion symmetry is broken, and the symmetry is correspondingly changed from C6 to C3. As expected, the Dirac points are gapped out and form two band gaps, as shown by the blue-shadowed areas of the band diagram in Fig. 1(d). It is known that distinct topological characteristics can be achieved by choosing different unit cells. In Fig. 1(b), we select the green and red hexagonal areas as two different unit cells UC1 and UC2, respectively, and the enlarged ones are also depicted in Figs. 1(e) and (f). One can see that UC2 is a translation of UC1 by l along the vertical direction. UC1 and UC2 share the same band structure shown by Fig. 1(d), as they are formed in actually the same PC structure. Since the band diagram shown in Fig. 1(d) is similar to that of a valley PC, we thus refer to the structure shown in Fig. 1(b) as a valley-PC-like lattice. But, different from the common valley PC, each red rod shown in Fig. 1(a) is now substituted by three identical rods in Fig. 1(b). The center-to-center distance (as denoted by d in Fig. 1(f), where d is the distance between the center of a red rod and the center of a triangle composed of them) of these three rods can be freely tuned, and the pointing direction of the triangle composed of them can be rotated, which provide extra freedoms to engineer the bands to meet the frequency matching condition for SFG and DFG.
Fig. 1. (a) Schematic of a 2D valley PC system. ${{\mathbf a}_{\mathbf 1}} = (1/2,\sqrt 3 /2)a\textrm{ }$ and ${{\mathbf a}_{\mathbf 2}} = ( - 1/2,\sqrt 3 /2)a$ are the lattice vectors, and a is the lattice constant. (b) A valley-like PC system transformed from (a), where each red rod in (a) is replaced by three identical ones. (c) Band diagram for (a) at r1 = r2 = 0.167a. The two blue dots at K are Dirac points. (d) Band diagram for (b). The Dirac points are gapped out and form two band gaps, as shown by the blue-shadowed areas. Here r1 = 0.135a, r2 = 0.167a and d = 0.27 l are taken. (e) and (f) UC1 and UC2 represent two different unit cells as marked by green and red hexagonal frames in (b), respectively.
To show this more clearly, we study the evolution of the band structure at K point for the valley-PC-like lattice shown in Fig. 1(b) as d varies. From Fig. 2(b), we can see that the whole band evolutions are symmetric with respect to d = l/2, and lattice at an arbitrary d and its partner “l-d” have the same band structure. When d varies from 0 to l, the first band gap undergoes a process of opening, closing, and reopening, implying a possible topological phase transition. While for the second band gap, it keeps open during this evolution process, which indicates that the topological property of the lattice in the second band gap are the same for different d.
Fig. 2. (a) Illustration of k1, k2 and the Brillouin zone used for calculating bulk polarization (the blue-shadowed area). (b) Evolution of the band structure at K point for the lattice shown in Fig. 1(b) as d varies. The pale and dark blue areas are the first and second band gaps, respectively, and the corresponding bulk polarizations are also marked. The insets show the unit cells at d = 0.27 l and d = 0.73 l. Here r1 = r2 = 0.135a. (c) A vertically flipped case. The used parameters are the same to that of Fig. 2(b).
The higher-order topological properties of a system can be characterized by bulk polarization, which is associated with the Wannier center (the center of the maximally localized Wannier function) [2]. In 2D systems, bulk polarization can be expressed by the following formula [28]:
Using Eqs. (2) and (3), we calculate the bulk polarization (P1, P2) for the first and second band gaps when d changes, as labeled in Fig. 2(b). As predicted, for the first band gap, the bulk polarization changes abruptly from (1/3, 1/3) to (0, 0) at the “closing” point d = 0.5 l; while for the second band gap, the bulk polarization keeps (−1/3, −1/3) invariant for varying d. Commonly, higher-order topological phenomena are expected to appear at the boundaries between two lattices with topologically distinct higher-order phases. If we choose the unit cells at d = 0.27 l and d = 0.73 l as the partner unit cells (see the inserts in Fig. 2(b)), the width of the first band gap can reach the maximum, and the two unit cells are in different topological phases for the first band gap. However, their bulk polarizations are identical for the second band gap, so that no higher-order corner states will occur in the second band gap at the interface between these two unit cells. To make the partner unit cells have different topological phases in both band gaps, we vertically flip the lattice depicted in Fig. 1(b). In this case, the band gap diagram keeps invariant, but the corresponding bulk polarizations become opposite, as shown in Fig. 2(c). As expected, the bulk polarizations for both band gaps at d = 0.27 l in Fig. 2(b) are now completely different from that at d = 0.73 l in Fig. 2(c). Noticing that the unit cells at d = 0.27 l in Fig. 2(b) and (d) = 0.73 l in Fig. 2(c) (see the inserts) are exactly the same with the ones shown in Figs. 1(e) and (f), respectively – that is why we choose UC1 and UC2 plotted in Fig. 1 as the partner unit cells for topological SFG and DFG.
In the following investigation, we keep d = 0.27 l unchanged to ensure the two band gaps broad enough, and just tune r1 or r2 to meet the frequency matching condition for SFG/DFG, simultaneously keeping the topological characteristics for UC1 and UC2 distinct. As an example, Fig. 3 shows the calculated bulk polarizations at r1 = 0.135a, r2 = 0.167a, and d = 0.27 l. The bulk polarizations (P1, P2) for the first band gap of UC1 and UC2 are (1/3, 1/3) (Fig. 3(a)) and (0, 0) (Fig. 3(b)), respectively. While for the second band gap, the bulk polarizations of UC1 and UC2 are (-1/3, -1/3) (Fig. 3(c)) and (1/3, 1/3) (Fig. 3(d)), respectively. We can see that the bulk polarizations for UC1 and UC2 are quite different, either in the first band gap or in the second band gap. By using the data of bulk polarizations, the corresponding Wannier centers can be further calculated by P1a1 + P2a2, where ${{\mathbf a}_{\mathbf 1}} = (1/2,\sqrt 3 /2)a\textrm{ }$ and ${{\mathbf a}_{\mathbf 2}} = ( - 1/2,\sqrt 3 /2)a$ are the lattice vectors as shown in Fig. 1(a). The positions of the Wannier centers (the red dots) for UC1 and UC2 in different band gaps are marked in Fig. 3.
Fig. 3. (a) and (b) Bulk polarizations for UC1 and UC2 below the first band gap, respectively. (c) and (d) Bulk polarizations for UC1 and UC2 below the second band gap, respectively. The inserts show the corresponding locations of the Wannier centers (the red dots).
To obtain higher-order topological corner states in both band gaps, we construct a large triangular supercell composed of UC1s (inner) and UC2s (outer), with a side length of 12a and 27a, respectively, as sketched in Fig. 4. The whole triangular supercell is surrounded by a perfect electric conductor layer. The bulk polarization values for the inner and outer parts are different in the two band gaps, and the positions of the Wannier centers are also different, as shown by the insets of Fig. 3. In the first band gap, the boundary between the inner and outer regions runs through the Wannier centers of the inner cells (see the enlarged images in Figs. 4(c) and (d)); while in the second band gap, the boundary crosses the Wannier centers of the outer cells (see the enlarged images in Figs. 4(e) and (f)). This leads to a separation of the charge associated with the Wannier centre [2]. As a result, the corners of the large triangular supercell are expected to host corner states in both band gaps.
Fig. 4. (a) Eigenspectrum of the triangular supercell at r1 = 0.135a, r2 = 0.167a, and d = 0.27 l. The corner states F1 and F2 are in the first band gap, while F3 and F4 are in the second band gap. (b) Frequency matching for SFG by varying r2. f1 + f2 = f3 is achieved at r2 = 0.167a. Besides, Q1, Q2 and Q3 (the Q factors of the corner modes F1, F2 and F3, respectively) versus r2 are also presented. (c)-(f) Ez field profiles of the corner modes F1, F2, F3 and F4. In the enlarged parts of the corner mode fields, the boundaries between UC1s (inner) and UC2s (outer) are marked by the zigzag dashed lines, and the Wannier centers are denoted by the red stars. We can see that the boundaries run through the Wannier centers of the inner cells (see Figs. (c) and (d)) or that of the outer cells (see Figs. (e) and (f)).
We then use finite-element method to perform calculations on this large triangular supercell for TM modes. Figure 4(a) exhibits the eigenspectrum of the triangular supercell at r1 = 0.135a, r2 = 0.167a, and d = 0.27 l. In the first band gap, two types of triply-degenerate corner states appear. The corner state F1 (at lower frequency f1 = 81.567 THz) has an electric field distribution as that shown by Fig. 4(c), which is localized on both sides of the corner, and has the typical feature of type II corner state mainly caused by next-nearest-neighbor coupling [30–32]. As for another type of corner state F2 (at higher frequency f2 = 83.203 THz), the electric field profile is shown by Fig. 4(d). One can see that the field is tightly concentrated at the corner sites, exhibiting the typical characteristic of Type I corner state, originating from nearest-neighbor coupling [4–6,30–32]. In addition, it is found that the second band gap also hosts two types of triply-degenerate corner states F3 and F4 at f3 = 164.77 THz and f4 = 171.0 THz, repectively, whose electric field distributions are depicted in Figs. 4(e) and (f), respectively. These two type of corner states are separated from the edge states (see Fig. 4(a)) due to next-nearest-neighbor coupling [30–32], so that their field profiles have the feature of type II corner state.
More importantly, these corner modes have high Q factors, especially for the corner states F3 and F4, possessing ultrahigh Q factors of ∼ 107, which is several orders of magnitude larger than that of other reported higher-order topological corner states [8–11]. That is because the gap width of the second band gap is broad (ranging from 148.04 THz to 192.9 THz), and the F3 and F4 corner modes locate near the center of the gap, obtaining better confinements. If we further engineer the band structure to make the sum of f1 and f2 exactly equal to f3 (or f4), the frequency matching condition for SFG/DFG will be satisfied in the topological system with dual band gaps. Thus, once the corner modes at f1 and f2 are excited under the pump of two appropriate excitation sources, the sum frequency signal at F3 (or F4) will be generated efficiently, due to the strong corner-corner nonlinear interactions.
As illustrated by Fig. 1(b), the valley-PC-like lattice holds more degrees of freedom than that of a common valley PC shown in Fig. 1(a). This is critical and make it more readily for band design to meet the frequency matching condition for SFG (or DFG). As an example, we select to let the triangle composed of three red rods in UC2 (see Fig. 1(f)) downward-pointing, and set r1 = 0.135a and d = 0.156a. We then change the radius r2 gradually, while keep other parameters invariant. Seen from Fig. 4(b), with the increase of r2, all of the eigenfreqencies of the corner states will drop, but the drop rates of f1 and f2 are lower than that of f3 (here we select the eigenfrequency of the corner mode F3 as the target sum frequency f3). As a result, the curve of “f1 + f2” will inevitably intersect with another curve “f3”, and the cross point lies at r2 = 0.167a. The corresponding f3 = 164.77 THz, which is exactly the sum of f1 (81.567 THz) and f2 (83.203 THz), and thus the frequency matching condition is rigorously satisfied. Besides, we also find that the Q factors of all corner modes keep high when changing r2, as shown by Fig. 4(b). For example, when r2 = 0.167a, the Q factor of the corner mode F3 reaches Q3 = 3.54 × 107. This extremely high Q factor is key for efficient SFG and DFG, as will be shown below.
3. Topologically-protected SFG and DFG
We then investigate the nonlinear process of SFG. Theoretically, the generation of sum frequency is determined by the second-order polarization ${P^{\textrm{(}{\omega _1}\textrm{ + }{\omega _2})}} = 2{\varepsilon _0}{\chi ^{(2)}}{E^{({\omega _1})}}{E^{({\omega _2})}}$, where χ(2) is the second-order nonlinear susceptibility tensor. Since F1, F2, and F3 are all corner states of TM modes, they couple through $\chi _{zzz}^{(2)}$, so we only need to consider the z component of the second-order polarization $P_z^{({\omega _1} + {\omega _2})} = 2{\varepsilon _0}\chi _{zzz}^{(2)}E_z^{({\omega _1})}E_z^{({\omega _2})}$ with $\chi _{\textrm{zzz}}^{(2)} = 1 \times {10^{ - 21}}$ (C/V2 in MKS units, or 113 pm/V. The typical value of χ(2) ranges from 100 fm/V to 200 pm/V [33–36]). Then, we place two continuous-wave (CW) point pump sources (Source A and Source B) at the corner, as denoted in the enlarged part of Fig. 5(a). Both of the pump sources have the form of Ez = E0i sin(2πfit) (i = 1,2), where E0i denotes the amplitude, and fi is set to be same to the eigenfrequency of the corner mode Fi. Thus, the pump Source A at f1 = 81.567 THz will excite the corner mode F1, and Source B at f2 = 83.203 THz will excite the corner mode F2, as shown by the excited field distributions in Figs. 5(a) and (b), respectively. To reveal the features during the nonlinear frequency conversion process, we place a point detector inside the corner to record the temporal evolution information, and the corresponding spectrum can be obtained by using Fourier transformation. The second-order nonlinear interaction between the corner modes F1 and F2 is sensitive to the pump intensities. When the pump intensities of sources A and B are weak, no other spectral components occur. However, when the pump intensities are high enough, e.g., E01 = E02 = 107 V/m, we find except for the original components at f1 and f2, another new one at f3 = 164.77 THz is also generated, which is just the sum of f1 and f2, as shown by the blue solid line in Fig. 5(d). The Ez field profile for this new component is also presented in Fig. 5(c), which has the same feature as that of corner mode F3 (see Fig. 4(e)). This clearly implies that efficient SFG is successfully realized owing to the precise frequency matching between the corner modes with high Q factors.
Fig. 5. SFG in nonlinear topological system. (a-c) are the field distributions of the excited corner modes at the initial frequencies f1, f2, and the converted frequency f3 after SFG, respectively. (d) Spectrum of the SFG process (the blue curve). For comparison, the spectrum when χ(2) = 0 (i.e., a linear case) is also presented, as shown by the red dashed line.
To investigate the dependence of SFG on the pump intensities, we fix the power of Source A at p1 = 3.38 mW, and continuously vary the pump power of Source B (p2). The measured SFG power (pSF) versus p2 is illustrated in Fig. 6(a). One can see that the observed SFG signal is linearly enhanced with the increase of p2 when p1 keeps invariant. Similarly, if we keep p2 constant and only change p1, the same relationship will occur again. This indicates that PSF is actually proportional to the product of p1 and p2. Consequently, the normalized conversion efficiency of SFG can be defined as ηSFG = pSF/(p1 × p2). As shown in Fig. (6b), the conversion efficiency of SFG remains at a constant value, and is irrelevant to the change of p2 (or p1). For our topological SFG system, the conversion efficiency is calculated to be ηSFG =3.7 × 10−4 W-1, which is improved 2 to 3 orders of magnitude higher than that reported in other works [33,37].
Fig. 6. (a) Power of SFG signals changes with the pump power of Source B when the power of Source A is fixed at p1 = 3.38 mW. (b) Conversion efficiency of SFG versus the pump power p2. (c) Dependence of the conversion efficiency of SFG on the pump frequency of Source B when the frequency of Source A is fixed at f1 = 81.567 THz. (d-f) Field distributions of F1, F2, and the excited edge mode. (g) Spectrum of the SFG process when r2 = 0.1525a. (h-j) are the field profiles of the excited corner modes at the initial frequencies f3, f1, and the converted frequency f2 after DFG, respectively. (k) Spectrum of the DFG process.
To provide additional evidence of the efficiency of the SFG process, it is important to consider the enhanced nonlinear interactions between the higher-order corner modes. This can be demonstrated by comparing the nonlinear conversion efficiency with and without the involvement of the corner states. We keep the frequency of Source A invariant (namely, f1 = 81.567 THz), but change the frequency of Source B to make it detuned from the F2 corner mode. The frequency-dependent SFG efficiency is shown in Fig. 6(c). One can see the maximum conversion efficiency is achieved only when the pump frequency of Source B is exactly on-resonance with the corner mode F2. Conversely, when the pump frequency deviates from the resonance of F2, the conversion efficiency decreases quickly. This clearly indicates the corner-corner interaction plays crucial role in realizing efficient SFG in our topological system. We also note that there are other peaks with much lower conversion efficiency in Fig. 6(c). These SFG effects originate from the nonlinear interaction among corner mode, edge modes, and body modes. Obviously, this hybrid nonlinear interaction is far weaker than the corner-corner interaction, since the corner modes can better confine light within a tiny space to enhance the light-light nonlinear interaction.
A natural question is what will happen when the sum of the eigenfrequencies of F1 and F2 does not match that of F3, but match that of a nearby edge mode. To explore this, we reduce r2 from 0.167a to 0.1525a, and keep the other parameters invariant. Thus, the eigenfrequencies of F1, F2 and F3 are changed to f1 = 82.615 THz, f2 = 84.965 THz, and f3 = 169.15 THz, respectively. In this case, the sum of f1 and f2 exactly equals to the eigenfrequency of an edge mode at 167.58 THz, but is slightly lower than f3. As a result, when the frequencies of the two pump sources are set to be f1 and f2, respectively, one can see that the edge mode at the frequency of f1 + f2 is excited, while the F3 corner mode is now suppressed, as shown in Fig. 6(g). The corresponding electric field components of F1, F2, and the excited edge mode are depicted in Figs. 6(d-f), respectively. Although the Q factors of F1 and F2 increase slightly when r2 is reduced, and the Q factor of the edge mode is of the same order with that of F3, the conversion efficiency for this corner-edge interaction is only 1.4×10−5 W-1, which is much less than that of above-mentioned corner-corner interaction, due to the poor overlap of the electric fields of the corner modes and the edge mode. To show this, we calculate the mode overlap factor, which can be approximately expressed as [38]:
In addition to SFG, we also investigate topological DFG, which is a reversed nonlinear process of the former. Accordingly, DFG is expected to be realized in the same topological system. To verify this, we also place two CW point pump sources at the corner. Different from the case of SFG, now Source A operates at frequency of f3 = 164.77 THz to excite F3 corner mode, while Source B operates at a frequency of f1 = 81.567 THz to excite F1 corner mode. The nonlinear frequency conversion process is also recorded by a point detector inside the corner. Figure 6(k) exhibits the Fourier analysis spectrum of the DFG process. As expected, a new spectral peak at f2 = 83.203 THz is successfully generated, which is exactly the difference between the pump frequencies f1 and f3. Figures 6(h-j) are the excited field profiles of the corner modes during this DFG process.
More importantly, the SFG and DFG processes presented above are topologically protected, which means they are immune to the defects while keeping high conversion efficiency. To show this more clearly, we take the perturbed system sketched in Fig. 7 as an example, where several defects are randomly introduced into the bulk and edge of the topological system. The defects are created by removing some dielectric rods within the red dashed circles at different locations. We then reexamine the nonlinear features of SFG and DFG in this disturbed system. Figure 7(a-c) show the Ez field profiles of the corner modes under the excitation by CW point sources at frequencies f1 and f2. One can see except for the corner modes F1 and F2, the other corner mode F3 at f1 + f2 is also excited via nonlinear SFG effect. These mode profiles, as well as the corresponding spectrum shown in Fig. 7(d), have the same features as that plotted in Fig. 5. The same phenomenon can also be observed for DFG case, as shown by Fig. 7(e-h). This indicates that both SFG and DFG based on the nonlinear interaction between corner modes are robust against bulk and edge defects.
Fig. 7. (a-d) Robustness of SFG against defects. (e-h) Robustness of DFG against defects. The defects are randomly introduced into the topological system by removing several rods within the red dashed circles at different locations. (a-c) The Ez field profiles for the excited corner modes at f1, f2, and their sum via SFG. (e-g) The Ez field profiles for the excited corner modes at f1, f3, and their difference via DFG. (d) and (h) Detected SHG and DFG frequency spectra for the disturbed systems.
Of course, similar to other reported corner states, if the defects are introduced much closer to the corner site, the eigenfrequency of the corner mode will fluctuate, and the nonlinear conversion efficiency will be affected. Besides, in the real experiments, the Q factors of the corner modes will decrease due to the fabrication imperfections, e.g., unavoidable irregular scattering caused by the roughness of the sample’s sidewalls [39]. However, benefited from topological protection, the corner states will survive, and the field profiles of the corner modes are almost unaffected. As a result, the mode overlap factor, another key element for nonlinear conversion efficiency, will still keep high. This is a remarkable merit compared with other SFG/DFG methods based on traditional ultrahigh-Q resonators.
Finally, we discuss the experimental feasibility of the theoretical proposal. In the present work, to excite the corner modes F1 and F2, two CW point pump sources at the eigenfrequencies f1 and f2 are placed at the corner. Due to the scaling property of the PC system, the eigenfrequencies of the corner modes can be readily adjusted to match the pump sources. In realistic experiments, one can employ two polarized wavelength-tunable CW laser beams to directly excite the fundamental corner modes, via illuminating the top of the sample after focusing with a microscope objective, similar to the approach proposed in recent experimental works [17,40]. The laser beam size can even be larger than the total size of the sample, since only the on-resonance fundamental corner modes can be effectively excited, while the other off-resonance modes (e.g., edge modes and bulk modes) will be suppressed [17]. We also notice the progress of superlens, whose focal spot size is of subwavelength scale [41,42]. With the help of superlens, we can focus the pump laser beams on the corner site to excite the corner modes more efficiently. The generated SFG/DFG radiation can be imaged on a camera [17], with a dichroic mirror to filter the spectral components of the pump beams.
4. Conclusion
In summary, we design and investigate a valley-photonic-crystal-like topological system, which provides extra freedom to tune the corner modes to meet the frequency matching condition for SFG and DFG. Different from the conventional SFG and DFG methods based on high-Q microring, microdisk or other resonators, here the mechanism is based on the nonlinear interaction between three high-Q corner modes inside dual topological band gaps. In this way, topologically-protected SFG and DFG with high conversion efficiency are demonstrated. This work may open up a promising avenue for achieving efficient on-chip nonlinear frequency conversions with robustness against defects.
Funding
National Natural Science Foundation of China (11774098); Natural Science Foundation of Guangdong Province (2022A1515011950, 2023A1515010781); Science and Technology Program of Guangzhou (202002030500).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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