Enhanced Third-Harmonic Generation in Diamond Photonic Crystal Slabs via Doubly Resonant Quasi-Bound States in the Continuum
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 $\Gamma$-point quasi-bound-state-in-the-continuum (quasi-BIC) TH mode satisfy the frequency-tripling condition $3\omega_1\approx\omega_3$. 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 $\eta=2.7\times10^{-7}~\mathrm{W}^{-2}$ under moderate quality factors, which is projected to reach ~$0.034~\mathrm{W}^{-2}$ 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.