Enhanced Third-Harmonic Generation in Diamond Photonic Crystal Slabs via Doubly Resonant Quasi-Bound States in the Continuum

The gist: The authors propose and numerically demonstrate doubly resonant third-harmonic generation in a diamond photonic crystal (PhC) slab, in which the fundamental harmonic (FH) and the third harmonic (TH) modes are simultaneously resonant within the same membrane. This design establishes a robust route toward efficient, monolithic on-chip frequency conversion in an all-diamond platform for quantum and nonlinear photonics.

Design Strategy for Doubly Resonant THG

The research focuses on overcoming the challenge of vanishing nonlinear coupling by performing band engineering that tailors the nonlinear overlap between FH and TH modes via the χ(3) tensor The design utilizes a hexagonal-lattice PhC slab where a K-point band-edge FH mode is used, which is readily confined in a PhC heterostructure cavity owing to the relatively large TE bandgap of the hexagonal lattice Candidates for the TH mode (ω3) are identified as nondegenerate modes at the Γ point, which are symmetry-protected BICs because their mode symmetry is incompatible with the radiation continuum The authors introduce asymmetry by continuously deforming the circular hole into an equilateral triangular hole while keeping the hole area constant, which transforms Mode II into Mode II(t), which acquires a nonzero β3 with the FH mode owing to the broken symmetry.

Coupled-Mode Theory and Figure of Merit

The THG efficiency is formulated using temporal coupled-mode theory (CMT) The normalized THG efficiency η is defined by the expression η = 16ω3/2β32τ1/6τ3/2τex1,3·L1(ω)3L3(ω). Under the doubly resonant condition (ω = ω1 and 3ω = ω3), the line shapes equal unity, so η ∝ Q13Q3β32. This expression, Equation (8), is defined as the figure of merit (FOM) that guides the optimization below. The nonlinear overlap β3 is defined by the integral involving the nonlinear polarization vector P(3)(3ω) and the field distributions.

Optimization of Unit Cell Geometry

The optimization process maximizes Q13Q3β32 over the three-dimensional unit-cell geometry. The material dispersion of single-crystal diamond is used, and the third-order susceptibility tensor is set to χ1111 = 3χ1122 = 2.1 × 10−21 m2V−2. The FOM Q3β32 is maximized at (d/√3)/a = 0.4375, 0.455, 0.4625, for t/a = 0.50, 0.55, and 0.60, respectively. The corresponding three-dimensional band diagram for the designed structure (t/a, d/a) = (0.55, 0.451√3) is shown in Fig. 3.

Cavity Design and Performance Verification

Three-dimensional finite difference time domain (3D-FDTD) simulations were performed to verify the design in realistic PhC heterostructure cavities The designed cavity comprises core, transition, and cladding layers with parameters (t, dcore, dtrans, dclad) = (0.55a, 0.46√3a, 0.44√3a, 0.4√3a). The FH mode quality factor Q1 increases exponentially with the number of cladding layers nclad. The calculated nonlinear overlap reaches β32 = 5.8 × 106 J−2. Inserting these values into Eq. (8) gives a normalized THG efficiency η = 2.7 × 10−7 W−2. This value of η corresponds to Q1 = 4000, obtained with nclad = 5.

Benchmarking and Scalability

The proposed design outperforms most previously reported structures, exceeding the record-high THG performance reported to date [22] by two orders of magnitude. The PhC cavity can be excited directly from free space, enabling much higher-power pumping and improved efficiency. The design is scalable via the lattice constant a, and the broad transparency of diamond allows a single geometry to be mapped onto fabricable wavelengths for a range of applications. The results establish a robust route toward efficient, monolithic on-chip frequency conversion in an all-diamond platform for quantum and nonlinear photonics.

Conclusion

In conclusion, we have proposed and numerically demonstrated doubly resonant third-harmonic generation in a centrosymmetric diamond photonic crystal slab Combining a K-point band-edge mode at the fundamental frequency with a Γ-point BIC at the third harmonic, and deforming the circular holes into equilateral triangles that break the lattice symmetry, we simultaneously activate a symmetry-forbidden nonlinear overlap for a realistic slab thickness. A closed-form expression for the coupled-mode-theory THG efficiency provided a material-independent figure of merit that guided both the unit-cell and the heterostructure-cavity design. Three-dimensional FDTD simulations of the designed cavity, which co-confines the fundamental and third-harmonic modes, yield η = 2.7 × 10−7 W−2, projected to reach 0.034 W−2 at the fabrication-limited quality factor. Because the scheme relies only on lattice symmetry and a material-independent figure of merit, it offers a general and scalable route to monolithic frequency conversion, which is readily transferable to other wavelengths, χ(3) processes, and material platforms for nonlinear-optical and colorcenter quantum applications in diamond.

CRediT authorship contribution statement Sangmin Ji: Conceptualization, Methodology, Software, Investigation, Formal analysis, Visualization, Writing – original draft. Satoshi Iwamoto: Supervision, Writing – review & editing. Hideki Gotoh: Supervision, Project administration, Writing – review & editing. Acknowledgements This work was supported by Iketani Science and Technology Foundation (0361158-A); Hirose Foundation (12th Research Grant); JSPS KAKENHI (23K19197, 24K17579). Disclosures The authors declare no conflicts of interest. Data availability Data underlying the results may be obtained from the authors upon reasonable request. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the preparation of this work the authors used Claude AI and Gemini AI in order to review the grammar and the correspondence between figures and main text. After using this tool/service, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article. References 1 M. D. Levenson and N. Bloembergen, Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media, Phys. Rev. B 10, 4447 (1974). https://doi.org/10.1103/PhysRevB.10.4447 2 R. P. Mildren and J. R. Rabeau, Optical engineering of diamond, Wiley (2013). DOI:10.1002/9783527648603 3 B. J. M. Hausmann, I. Bulu, V. Venkataraman, P. Deotare & M. Lončar, Diamond nonlinear photonics, Nat. Photon.

1. The AI system can perform automated band engineering optimization by using a closed-form expression for THG efficiency derived from coupled-mode theory as a figure of merit to optimize the unit cell geometry, ensuring scalability across various wavelengths.

2. The system can predict the optimal slab thickness by combining the hole size d that maximizes the FOM increases with t, while d satisfying the tripling condition ω3 = 3ω1 decreases with t, allowing it to select the best physical configuration for a target frequency conversion wavelength.

3. The AI can design and simulate complex heterostructure cavities using three-dimensional finite difference time domain (3D-FDTD simulations to determine mode properties, including eigenfrequencies and quality factors, for simultaneous FH and TH mode confinement.

4. The system can calculate the expected performance benchmark by evaluating the normalized THG efficiency η = 2.7 × 10−7 W−2 under fabrication-limited conditions (Q ≈ 200,000), providing a quantitative comparison against literature where Diamond PhC cavity [This work, Q1 = 200,000] 3.4 × 10−2 W-2.

5. The system can identify the critical structural parameters for maximizing efficiency by locating the optimum ncore = 6 as determined by simulations where "β32 peaks near ncore = 3–4 and decreases for ncore > 4."

We propose and numerically demonstrate doubly resonant third-harmonic generation (THG) in a diamond photonic crystal (PhC) slab, in which the fundamental harmonic (FH) and the third harmonic (TH) modes are simultaneously resonant within the same membrane. A hexagonal-lattice slab with triangular air holes is designed so that a K-point band-edge FH mode and a Γ-point quasi-bound-state-in-the-continuum (quasi-BIC) TH mode satisfy the frequency-tripling condition 3ω 1 aboutω 3. Modifying the hole shape from circular to equilateral triangular breaks the in-plane symmetry that otherwise forces the nonlinear coupling to vanish, thereby converting a TH mode with negligible overlap into one with finite while simultaneously reducing the required slab thickness. Guided by a closed-form expression for THG efficiency derived from coupled-mode theory, we design the unit cell and a PhC heterostructure cavity. Three-dimensional simulations of the designed cavity yield a normalized THG efficiency η=2.7 times10-7 W-2 under moderate quality factors, which is projected to reach 0.034 W-2 at the fabrication-limited quality factor (Q = 200,000). Because the operating wavelength is set by the lattice constant, this design, combined with the ultra-wide transparency window of diamond, can map a single geometry across various fabricable wavelengths, spanning from telecommunication bands to color-center-resonant visible and deep-UV outputs. These results establish a robust route toward efficient, monolithic on-chip frequency conversion in an all-diamond platform for quantum and nonlinear photonics.