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Aluminum nitride photonic integrated circuits: from piezo-optomechanics to nonlinear optics

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Abstract

The commercial success of radio-frequency acoustic filters in wireless communication systems has launched aluminum nitride (AlN) as one of the most widely used semiconductors across the globe. Over recent years, AlN has also been investigated as an attractive photonic integrated platform due to its excellent characteristics, such as enormous bandgaps (∼6.2 eV), quadratic and cubic optical nonlinearities, Pockels electro-optic effects, and compatibility with the complementary metal-oxide semiconductor technology. In parallel, AlN possesses outstanding piezoelectric and mechanical performances, which can provide new aspects for controlling phonons and photons at the wavelength scale using nanophotonic architectures. These characteristics pose AlN as a promising candidate to address the drawbacks in conventional silicon and silicon nitride platforms. In this review, we aim to present recent advances achieved in AlN photonic integrated circuits ranging from material processing and passive optical routing to active functionality implementation such as electro-optics, piezo-optomechanics, and all-optical nonlinear frequency conversion. Finally, we highlight the challenges and future prospects existing in AlN nanophotonic chips.

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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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Figures (39)

Figure 1.
Figure 1. Illustration of wurtzite AlN materials prepared in different crystalline forms and their key deployments for diverse photonic integrated applications. The unit cell of wurtzite AlN is presented in the center, where aluminum (Al) and nitride (N) atoms are denoted. Insets: (a) Reprinted from [26] under a CC BY license. (b) Reprinted with permission from Xiong et al., Nano Lett. 7, 3562 (2012) [27]. Copyright 2012 American Chemical Society. https://doi.org/10.1021/nl3011885. (c) Reprinted by permission from Nature Publishing Group: Guo et al., Light: Sci. & Appl. 6, e16249 (2017) [28]. Copyright 2017. (d) Reprinted with permission from Feng et al., ACS Photonics 5, 699 (2018) [29]. Copyright 2018 American Chemical Society. https://doi.org/10.1021/acsphotonics.7b01215. (e) Reprinted by permission from Nature Publishing Group: Bruch et al., Nat. Photonics 15, 21 (2021) [30]. Copyright 2021.
Figure 2.
Figure 2. Comparison of the bandgap (horizontal) and nonlinearity (vertical) for common photonic integrated platforms. The black triangles and colored circles denote materials possessing ${\rm \chi }^{(3)}$ and ${\rm \chi }^{(2)}$${\rm \chi }^{(3)}$ effects, respectively, whereas the magnitude of the ${\rm \chi }^{(2)}$ susceptibility is reflected by the right-hand side colorbar. a-Si, amorphous silicon; c-Si, crystalline silicon; Chalc., chalcogonides. Reprinted from Semiconductors and Semimetals, Vol. 107, Liu et al., “AlN nonlinear optics and integrated photonics,” pp. 223–281, Copyright 2021 with permission from Elsevier [22].
Figure 3.
Figure 3. (a) Schematic of $c$-oriented poly-crystalline AlN films grown on SiO$_2$/Si substrates, highlighting the domain boundaries (purple lines). Reprinted with permission from Xiong et al., Nano Lett. 7, 3562 (2012) [27]. Copyright 2012 American Chemical Society. https://doi.org/10.1021/nl3011885. (b) Transmission electron microscopy image of poly-crystalline AlN waveguide facets, showing columnar structures. Reprinted from [21] under a CC BY-NC 4.0 license.
Figure 4.
Figure 4. High-temperature annealing of sputtered AlN-on-sapphire thin films. (a) Mechanisms for improved AlN crystal qualities after annealing at different temperatures. Reprinted from J. Cryst. Growth 502, Xiao et al., “Improvement mechanism of sputtered AlN films by high-temperature annealing,” pp. 41–44, Copyright 2018 with permission from Elsevier [91]. (b) Illustration of a two-layer AlN structure after sputtering. (c) Overlapped AlN thin films in a “face-to-face” fashion. (b),(c) Reprinted from [91] under a CC-BY license.
Figure 5.
Figure 5. Illustration of “top-down” masking approaches for patterning low-loss AlN-on-sapphire photonic circuits. Relative film thicknesses are enlarged for clarity and are not to scale. Here, FOx-16 and ZEP520A are common negative and positive e-beam resists, respectively, whereas SPR955 is an i-line photoresist.
Figure 6.
Figure 6. Smooth dry etching of AlN with an optimized process. (a) SEM image of as-grown and etched AlN surfaces with nickel as the etching mask. (b),(c) AFM images of as-grown and etched AlN surface with the RMS roughness of 0.68 and 0.77 nm, respectively. (a)–(c) Reprinted from Vacuum 116, Liu et al., “Smooth etching of epitaxially grown AlN film by Cl$_2$/BCl$_3$/Ar-based inductively coupled plasma,” 1pp. 58–162, Copyright 2015 with permission from Elsevier [105].
Figure 7.
Figure 7. Ridge waveguides based on poly-crystalline AlN films. (a),(b) Sketched cross sections of buried and suspended AlN waveguide structures, respectively. (c) Simulated fundamental TE$_{00}$ mode profiles of single-mode AlN ridge waveguides at wavelengths (${\rm \lambda}$) of 0.4, 1.55, and 3.8 $\mathrm {\mu }$m, respectively. The right color bar indicates the normalized electric field intensity. (d) SEM image of a practical AlN waveguide. Reprinted with permission from Xiong et al., Nano Lett. 7, 3562 (2012) [27]. Copyright 2012 American Chemical Society. https://doi.org/10.1021/nl3011885. (e) Reported propagation losses of poly-crystalline AlN waveguides at various spectral windows. KIT, Karlsruhe Institute of Technology; NUS, National University of Singapore.
Figure 8.
Figure 8. Periodically poled AlN ridge waveguides. (a) Top: AFM image of re-grown AlN thin films with alternating stripes of Al- and N-polar domains; bottom: SEM image of fabricated AlN waveguides with alternating polarities. Reprinted with permission from Alden et al., Appl. Phys. Lett. 108, 261106 (2016) [116]. Copyright 2016, AIP Publishing LLC. (b) Illustration of transverse polarity-inverted AlN structure via a direct wafer bonding process. Reprinted from [117] under a CC BY 4.0 license.
Figure 9.
Figure 9. Fundamentals of optical MRRs. (a) Schematic of a MRR. The white arrows indicate the direction of light propagation, whereas intrinsic and coupling loss rates are denoted as $\kappa _0$ and $\kappa _\mathrm {ext}$, respectively, with $\kappa = \kappa _0 + \kappa _\mathrm {ext}$. (b) Lorentzian resonance profiles of a MRR at the output port. $\Delta {\rm \lambda }$, full width at the half maximum; FSR, free spectral range; ${\rm \lambda }_\mathrm {res}$, resonant wavelength.
Figure 10.
Figure 10. Optical microresonators in AlN platforms. (a)–(c) SEM images of suspended AlN microwheels, microdisks, and photonic crystal cavities. (d),(e) SEM images of AlN MRRs patterned from positive and negative tone resists, respectively. (f) SEM image of a dually resonant AlN MRR. (g) State-of-the-art $Q_\mathrm {int}$ achieved from different research groups at various spectral windows. The orange, green, and blue dashed lines indicate microresonators made of single-, nano-, and poly-crystalline AlN thin films, respectively. Yale, Yale University ($\circ$); MIT, Massachusetts Institute of Technology ($\pentagon$); U-M, University of Michigan ($\Delta$); THU, Tsinghua University ($\bigstar$); HUST, Huazhong University of Science and Technology ($\rhd$); NUS, National University of Singapore ($\Diamond$). (a) Reprinted with permission Xiong et al., Appl. Phys. Lett. 100, 171111 (2012) [124]. Copyright 2012, AIP Publishing LLC. (b) Reprinted with permission from Han et al., Appl. Phys. Lett. 106, 161108 (2015) [125]. Copyright 2015, AIP Publishing LLC. (c) Reprinted with permission from Pernice et al., Appl. Phys. Lett. 100, 091105 (2012) [126]. Copyright 2012, AIP Publishing LLC. (d) Reprinted with permission from [95] © The Optical Society. (e) Reprinted with permission from [39] © The Optical Society. (f) Reprinted with permission from Bruch et al., Appl. Phys. Lett. 113, 131102 (2018) [57]. Copyright 2018, AIP Publishing LLC.
Figure 11.
Figure 11. Cleavage and coupling of AlN waveguides. (a),(b) “Standard” and “dice-and-cleave” methods for addressing Si and sapphire substrates, respectively. Insets of (a) and (b): SEM images of cleaved AlN waveguide facets w/ and w/o polishing treatment. (c) Adiabatic evanescent coupling between a tapered optical fiber and AlN nanophotonic waveguide with relevant mode distributions in the bottom. Reprinted from [137] © The Optical Society.
Figure 12.
Figure 12. (a) Sketched index ellipsoid of $c$-oriented AlN crystals with $X$, $Y$, and $Z$ being the principal axes. (b) Schematic of an AlN crystal after applying a vertical voltage $V$ (i.e., electric field $E_z$), which induces phase retardation of optical beams with a $z$-axis polarization (indicated by vertical arrows).
Figure 13.
Figure 13. Electrode design for integrated AlN EOMs. (a) Common placement of signal (S) and ground (G) electrodes (from left to right) for exciting horizon ($E_x$) or vertical ($E_z$) electric fields in AlN waveguides. (b) Numerical simulation of the optical and electric fields in an AlN nanophotonic waveguide with GSG electrodes on top.
Figure 14.
Figure 14. Nanophotonic EOMs made of poly-crystalline AlN thin films. (a) Micrograph of a microring-based AlN EOM with GSG electrodes on top. (b) Frequency response in telecom-C (top) and near-VIS (bottom) band with Q-factors in the insets. (a),(b) Reprinted with permission from Xiong et al., Nano Lett. 7, 3562 (2012) [27]. Copyright 2012 American Chemical Society. https://doi.org/10.1021/nl3011885. (c) Micrograph of asymmetric AlN MZI modulators with parallel Al/TiN plate electrodes highlighted in the bottom. Reprinted from [150] © The Optical Society.
Figure 15.
Figure 15. Principle of electro-optic transducers. (a) Diagram of microwave ($\mathrm {\mu }$W)-optical photon conversion via cavity-enhanced DFG and SFG parametric processes. (b) Top: schematic of cavity electro-optic systems containing an inductor-capacitor (LC) microwave resonator and Pockels ${\rm \chi }^{(2)}$ optical resonator; bottom: implementation of cavity electro-optic transduction with co-integrated planar superconducting and optical resonators. Reprinted from [127] under a CC BY-NC 4.0 license.
Figure 16.
Figure 16. Electro-optic transducers in AlN. (a) Sketched cross section of integrated photonic-superconducting systems with poly-crystalline AlN and NbTiN thin films. (b) Optical micrograph of the fabricated device. Reprinted from [127] under a CC BY-NC 4.0 license. (c) Sketched cross section of flip-chip bonded single-crystalline AlN photonics and NbN superconducting chips. (d) Optical micrograph of the fabricated device, showing coupled MRRs, concentric GSG RF electrodes, and “Ouroboros” LC resonators. Figure 1(c) reprinted with permission from Fu et al., Phys. Rev. A 103, 053504 (2021) [170]. Copyright 2021 by the American Physical Society.
Figure 17.
Figure 17. Integrated phononic circuits in AlN. (a) Poly-crystalline AlN-on-SiO$_2$/Si racetrack resonators with a laterally integrated IDT for launching high-frequency Raleigh SAWs. Reprinted with permission from Nature Publishing Group: Tadesse and Li, Nat. Commun. 5, 5402 (2014) [191]. Copyright 2014. (b) Suspended AlN optomechanical waveguides with an integrated curve-shaped IDT at one end for acoustic excitation. Reprinted from [192] © The Optical Society. (c) Comparison of the $f_\mathrm {m} \cdot Q_\mathrm {m}$ reported in various types of poly-crystalline AlN phononic resonators at ambient air conditions.
Figure 18.
Figure 18. Acousto-optic phase modulation in suspended AlN waveguides. (a) Top: schematic of the waveguide structure for co-located optical and acoustic modes; bottom: optical micrograph of a serpentine AlN waveguide with deposited RF electrodes highlighted in the red-dashed region. Reprinted by permission from Nature Publishing Group: Nat. Photonics 10, 766 (2016) [110], Copyright 2016. (b) Top: AlN spiral waveguide arrays for QPM between mechanical and optical modes and the packaged device for microwave excitation; bottom: optical spectra of modulated frequency combs at various input RF powers. Reprinted by permission from Nature Publishing Group: Nat. Photonics 13, 323 (2019) [210], Copyright 2019.
Figure 19.
Figure 19. Large-angle acousto-optic deflectors realized in AlN photonic platforms. Left: overall view of the device configuration with shallow-etched areas colorized blue. Right: enlarged view of finger-shaped IDTs (period $\Lambda$ of $\sim$2.7 $\mathrm {\mu }$m) in the dashed square regions with contact pads (top) and back-end trench reflectors (bottom). Reprinted with permission Li et al., APL Photonics 4, 080802 (2019) [213]. Copyright 2019, AIP Publishing LLC.
Figure 20.
Figure 20. Optical non-reciprocity via STM in AlN-based photonic chips. (a) Optical micrograph of an acoustically pumped AlN racetrack resonator with phonon–photon interaction regions, optical (TE$_{10}$ and TE$_{00}$) and acoustic (Lamb S$_0$) mode profiles in the right. Reprinted with permission from Nature Publishing Group: Sohn et al., Nat. Photonics 12, 91 (2018) [220], Copyright 2018. (b) Schematic of hybrid AlN-on-Si$_3$N$_4$ photonic chips with an unreleased Si substrate (cross sections in the right bottom). Reprinted from [221] under a CC BY license. (c) Magnetic-free optical isolators from a hybrid device consisting of AlN actuators (yellow, green) and Si$_3$N$_4$ MRRs (blue) with a released Si substrate. Reprinted with permission from Nature Publishing Group: Tian et al., Nat. Photonics 15, 828 (2021) [222], Copyright 2021.
Figure 21.
Figure 21. Principle of piezo-optomechanical transduction. (a) Illustration of mechanical phonon-mediated interactions between microwave and optical photons in a piezoelectric platform. The piezoelectric and optomechanical coupling are indicated by $g_\mathrm {em}$ and $g_\mathrm {om}$, respectively. (b) Schematic of cavity piezo-optomechanical transduction with $\omega _\mathrm {e}$, $\Omega _\mathrm {m}$, and $\omega _\mathrm {o}$ being the resonance frequencies of the microwave, mechanical, and optical cavities, respectively. A red-detuned optical pump (blue arrow) with a frequency $\omega _\mathrm {p}$ is employed for providing linearized optomechanical interaction.
Figure 22.
Figure 22. Piezo-optomechanical crystal transducers. (a) Bidirectional microwave–optical transducer comprising a pair of radially symmetric IDTs coupled to an AlN optomechanical crystal. Reprinted with permission from Vainsencher et al., Appl. Phys. Lett. 109, 033107 (2016) [205]. Copyright 2016, AIP Publishing LLC. (b) Qubit-to-optical transducer in a hybrid AlN-on-Si platform with the simulated resonant modes (top) and fabricated devices (bottom). Reprinted with permission from Nature Publishing Group: Mirhosseini et al., Nature 588, 599 (2020) [239], Copyright 2020.
Figure 23.
Figure 23. Triply resonant piezo-optomechanical transducer in an AlN platform. (a) Schematic of the device architecture composed of a frequency-tunable superconducting “Ouroboros” microwave resonator (yellow) over an AlN optomechanical microdisk. (b) Images of the fabricated AlN optomechanical microdisk (left) and NbN superconducting “Ouroboros” resonators (right). The cross-section view of the integrated device is sketched at the bottom. (c) Amplitude spectra at 900 mK for the optical, mechanical, and microwave resonances from the left to right with $Q$-factors of $5.4 \times 10^5$, $1.1 \times 10^4$, and $2.6 \times 10^3$, respectively. (a)–(c) Reprinted from [26] under a CC BY license.
Figure 24.
Figure 24. Energy conservation diagrams of various nonlinear optical processes. (a) Second-harmonic generation ($\omega _1$ being the incident photon frequency). (b) Optical parametric downconversion ($\omega _\mathrm {p}$, $\omega _\mathrm {s}$, and $\omega _\mathrm {i}$ being the pump, signal, and idler photon frequency, respectively). A zero frequency detuning $\Delta$ indicates a degenerate process. (c) Degenerate four-wave mixing with two identical pump photons ($\omega _\mathrm {p}$). (d) Non-degenerate four-wave mixing with two different pump photons ($\omega _1$ and $\omega _2$).
Figure 25.
Figure 25. Example of geometrical optimization for phase-matched SHG. The simulation adopts 1.0-$\mathrm {\mu }$m-thick AlN-on-sapphire thin films with anisotropic refractive indices. (a) Effective refractive indices of various visible modes at 780 nm (solid lines) compared to fundamental TM and TE infrared modes at 1560 nm (dashed black and red lines, respectively). (b) Enlarged view of mode crossing between TM$_{00}$ IR mode and TM$_{20}$ visible modes. Insets: electric profiles of each mode.
Figure 26.
Figure 26. SHG in AlN nanophotonic platforms. (a) Schematic of a dually coupled MRR with intracavity VIS (TM$_{20}$, top) and IR (TM$_{00}$, bottom) optical modes for fulfilling modal phase matching. Reprinted by permission from Nature Publishing Group: Guo et al., Light: Sci. & Appl. 6, e16249 (2017) [28]. Copyright 2017. (b) Colored SEM micrograph of cascaded MRRs fabricated from AlN-on-sapphire thin films. (c) State-of-the-art SHG efficiency in a single-crystalline AlN MRR at an optimal temperature. (b),(c) Reprinted with permission from Bruch et al., Appl. Phys. Lett. 113, 131102 (2018) [57]. Copyright 2018, AIP Publishing LLC.
Figure 27.
Figure 27. Optical parametric downconversion in a poly-crystalline AlN MRR. (a) Schematic illustration of the experimental scheme for on-chip photon-pair source generation and detection. (b) Effective refractive indices of phase-matched IR (TM$_{00}$) and near-VIS (TM$_{20}$) modes (profiles in the insets) when scanning the ring width. (c) Self-correlation of degenerate SPDC photon pairs with power-dependent generation rate in the right inset. (d) Self-correlation of the idler photon in the nearest non-degenerate downconversion. (a)–(d) Reprinted by permission from Nature Publishing Group: Guo et al., Light: Sci. & Appl. 6, e16249 (2017) [28]. Copyright 2017.
Figure 28.
Figure 28. Quadratic OPOs in a single-crystalline AlN chip. (a) Colored SEM image of the fabricated AlN chip with cascaded MRRs. (b) Degenerate and non-degenerate OPO spectra collected when varying the temperatures. (c) Recorded signal (blue) and idler (red) wavelengths versus the temperature, with a numerical simulation displayed as a black line. (a)–(c) Reprinted from [88] © The Optical Society.
Figure 29.
Figure 29. Quadratic soliton combs in an AlN MRR. (a) Illustration of quadratic comb generation from a continuous-wave VIS pump in a coupled microring system possessing the ${\rm \chi }^{(2)}$ nonlinearity. (b) Near-VIS (left) and IR (right) spectra in four distinct comb states labeled by (i)–(iv). A sech$^2$ fit is applied to the infrared comb spectrum in (iv). (a),(b) Reprinted by permission from Nature Publishing Group: Bruch et al., Nat. Photonics 15, 21 (2021) [30]. Copyright 2021.
Figure 30.
Figure 30. Diagram of the experimental apparatus for DTHG in a composite AlN/Si$_3$N$_4$ chip. The ring resonator is pumped by an IR continuous-wave laser, whereas the output is monitored by VIS and IR photodetectors. An Andor system is used for collecting scattered light from the top of the device. Reprinted from [78] © The Optical Society.
Figure 31.
Figure 31. Raman lasing in single-crystalline AlN MRRs. (a) Illustration of the energy level in SRS and the device structure with simulated mode profiles in the pump and Stokes wavelengths. Right inset: cleaved AlN waveguide facet in the output. (b) Sketch of on-resonance Stokes light with a Raman shift $\Omega _\mathrm {R}$ from the pump frequency $f_\mathrm {pump}$. The cavity resonance is indicated by pink color, whereas the Raman gain linewidth is denoted by $\Gamma _\mathrm {R}$. Reprinted with permission from Liu et al., ACS Photonics 5, 1943–1950 (2018) [60]. Copyright 2018 American Chemical Society, https://doi.org/10.1021/acsphotonics.7b01254. (c),(d) Cascaded Raman lasing spectra with distinct $\Omega _\mathrm {R}$ when the pump is TM- and TE-polarized, respectively. (a),(c),(d) Reprinted from [87] © The Optical Society.
Figure 32.
Figure 32. SCG in a single-crystalline AlN waveguide. Top: observed (solid line) and simulated (dotted line) power spectral density (PSD) from a 2.6-$\mu$m-wide AlN waveguide. Inset: the calculated first-order coherence. Bottom: SCG spectra from a set of AlN waveguides with varied widths, showing shifted short-wavelength dispersive waves (SWDWs) and long-wavelength dispersive waves (LWDWs). Reprinted from [346] © The Optical Society.
Figure 33.
Figure 33. Dispersion engineering of AlN MRRs. Left: example of $D_\mathrm {int}$ curves from 50-$\mathrm {\mu }$m-radius AlN MRRs (fixed width of 2.3 $\mathrm {\mu }$m), showing height-dependent behavior. Right: wWafer-scale thickness mapping of a 2-inch AlN wafer using a spectroscopic ellipsometer. Reprinted from [131] under a CC BY license.
Figure 34.
Figure 34. DKS combs in a single-crystalline AlN chip. By adjusting the resonance-pump detuning, comb spectra in different states (i)–(iv) are recorded (left), corresponding to distinct RF beatnotes in the low-frequency side (right). Reprinted from [369] © The Optical Society.
Figure 35.
Figure 35. Octave-spanning DKS for on-chip self-referencing in AlN. (a) Schematics of implementing DKS comb-based $f$$2f$ interferometry from an AlN nanophotonic chip, including a ${\rm \chi }^{(3)}$ octave-spanning DKS and ${\rm \chi }^{(2)}$ SHG. (b) SEM image of fabricated AlN chip composed of MRR-based Kerr comb generators and spiral SHG waveguides. (c) Octave-spanning DKS spectra in dispersion-engineered AlN MRRs, featuring a sub-terahertz $f_\mathrm {rep}$ down to 220 GHz. (a)–(c) Reprinted from [131] under a CC BY license.
Figure 36.
Figure 36. Conceptual architecture of a fully integrated, self-locked frequency comb in a single AlN photonic chip by implementing functionalities including octave-spanning DKSs, microresonator-enhanced SHG, and electro-optical phase modulation.
Figure 37.
Figure 37. Principle of harmonic frequency comb generation. The fundamental Kerr comb in the near-IR band is formed via a cascaded FWM process and is converted into red and green comb lines via SHG/SFG and third-order SFG processes, respectively. Reprinted from [309] © The Optical Society.
Figure 38.
Figure 38. High-efficiency visible harmonic combs in AlN. (a) Schematic of the device configuration with a strong ${\rm \chi }^{(2)}$ coupling. (b) SHG phase matching between TM00 and TM20 modes at different AlN ring widths. Insets: simulated mode profiles and device optical image. (c) Near-IR Kerr comb (left) and visible harmonic comb (right) spectra of AlN MRRs at different phase-matched wavelengths. The dotted lines indicate the spectral envelope. (a)–(c) Reprinted from [404] © The Optical Society.
Figure 39.
Figure 39. Broadband UV harmonic combs in AlN. (a) Schematic of frequency upconverting a normal-GVD supercontinuum (red) around 780 nm into the UV regime (purple) around 390 nm. (b) Phase matching design of near-VIS (TM$_{00}$) and UV (TM$_{20}$) modes in a uniform AlN waveguide. (c) Sketch of chirp-tapered AlN waveguides. (d) Near-VIS and shifted UV comb spectra from straight AlN waveguides at varied widths. (e),(f) Near-VIS and UV comb spectra collected from linear- and chirp-tapered AlN waveguides, respectively. (a)–(f) Reprinted from [40] under a CC BY license.

Tables (9)

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Table 1. Comparison of Physical Properties in AlN and Other Photonic platformsa,b

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Table 2. Fiber-to-Chip Interface of AlN Photonic Circuits

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Table 3. Elastic Properties of Phononic Platforms and Substrate Materials

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Table 4. Chip-Scale Optical Non-reciprocal Devices Realized Using Different Schemes

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Table 5. Comparison of Piezo-Optomechanical Crystal Transducers in Different Platforms

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Table 6. Comparison of State-of-the-Art SHG in Planar PICsa

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Table 7. Comparison of On-Chip THG Efficiency with ${\rm \chi }^{(3)}$- or ${\rm \chi }^{(2)}$-Based Processes

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Table 8. Characteristics of Raman-Active Phonons in Poly- and Single-Crystalline AlNa

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Table 9. Octave-Spanning Kerr Combs Generated in Different Photonic Platforms

Equations (26)

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$$\begin{pmatrix} a_\mathrm{B} \\ b_\mathrm{A} \end{pmatrix} = \begin{pmatrix} t & \kappa \\ -\kappa & t \end{pmatrix} \begin{pmatrix} a_\mathrm{A} \\ b_\mathrm{B} \end{pmatrix},$$
$$\triangle\left(\frac{1}{n^{2}}\right)_{i}=\left(\begin{array}{ccc} 0 & 0 & r_{13}\\ 0 & 0 & r_{13}\\ 0 & 0 & r_{33}\\ 0 & r_{51} & 0\\ r_{51} & 0 & 0\\ 0 & 0 & 0 \end{array}\right)\left(\begin{array}{c} E_{x}\\ E_{y}\\ E_{z} \end{array}\right).$$
$$\begin{array}{c} n_{x,y}=n_\mathrm{o}-\frac{1}{2}r_{13}n_\mathrm{o}^{3}E_{z},\\ n_{z}=n_\mathrm{e}-\frac{1}{2}r_{33}n_\mathrm{e}^{3}E_{z}. \end{array}$$
$$\Gamma=\frac{g}{V}\frac{\iint_\mathrm{AlN}E_{x,\mathrm{o}}^{2}E_{z}dxdz}{\iint_\mathrm{AlN}E_{x,\mathrm{o}}^{2}dxdz}.$$
$$V_{\pi}=\frac{\lambda n_\mathrm{eff}}{r_{33}n_\mathrm{e}^{4}\Gamma}\frac{g}{L}.$$
$$H_\mathrm{inter}=\hbar g_\mathrm{eo,0}(ab^{{\dagger}}c+a^{{\dagger}}bc^{{\dagger}}).$$
$$\eta=\eta_\mathrm{o}\eta_\mathrm{e}\frac{4C_\mathrm{eo}}{(1+C_\mathrm{eo})^{2}},$$
$$\bf{d^{T}}=\left(\begin{array}{cccccc} 0 & 0 & 0 & 0 & d_{15} & 0\\ 0 & 0 & 0 & d_{15} & 0 & 0\\ d_{31} & d_{31} & d_{33} & 0 & 0 & 0 \end{array}\right),$$
$$\triangle\left(\frac{1}{n^{2}}\right)_{i}=\left(\begin{array}{cccccc} p_{11} & p_{12} & p_{13} & 0 & 0 & 0\\ p_{12} & p_{11} & p_{13} & 0 & 0 & 0\\ p_{13} & p_{13} & p_{33} & 0 & 0 & 0\\ 0 & 0 & 0 & p_{44} & 0 & 0\\ 0 & 0 & 0 & 0 & p_{44} & 0\\ 0 & 0 & 0 & 0 & 0 & p_{66} \end{array}\right)\left(\begin{array}{c} \sigma_{xx}\\ \sigma_{yy}\\ \sigma_{zz}\\ \sigma_{yz}\\ \sigma_{zy}\\ \sigma_{xy} \end{array}\right).$$
$$H_\mathrm{int}=\hbar g_\mathrm{om}\left(\hat{a}\hat{c}^{{\dagger}}+\hat{a}^{{\dagger}}\hat{c}\right)+\hbar g_\mathrm{em}\left(\hat{b}\hat{c}^{{\dagger}}+\hat{b}^{{\dagger}}\hat{c}\right).$$
$$\eta=\eta_{\mathrm{e}}\eta_{\mathrm{o}}\frac{4C_{\mathrm{em}}C_{\mathrm{om}}}{\left(C_{\mathrm{om}}+C_{\mathrm{em}}+1\right)^{2}}.$$
$$\Delta\omega\approx\gamma_{m}\left(1+C_\mathrm{em}+C_\mathrm{om}\right).$$
$$\eta=\eta_{\mathrm{em}}\eta_{\mathrm{o}}\frac{4C_{\mathrm{om}}}{\left(1+C_{\mathrm{om}}\right)^{2}}.$$
$$\begin{aligned} \vec{P}(t) & = \epsilon_0 ({\rm \chi}^{(1)}\vec{E}(t) + {\rm \chi}^{(2)}\vec{E}^2(t) + {\rm \chi}^{(3)}\vec{E}^3(t) + \cdots) \\ & = \vec{P}^{(1)}(t) + \vec{P}^{(2)}(t) + \vec{P}^{(3)}(t) + \cdots \\ & = \vec{P}_l(t) + \vec{P}_{nl}(t), \end{aligned}$$
$$\vec{E}(t) = \vec{E}_1(t) e^{{-}i \omega_1 t} + \vec{E}_2(t) e^{{-}i \omega_2 t} + \vec{E}_3(t) e^{{-}i \omega_3 t} + \mathrm{c.c.}$$
$$\eta_\mathrm{SHG}=\frac{P_\mathrm{SHG}}{P_\mathrm{p}^{2}}=g^{2}\frac{\hbar\omega_{b}}{(\hbar\omega_\mathrm{p})^{2}} \frac{2\kappa_{b,\mathrm{ext}}}{\delta_{b}^{2}+\kappa_{b}^{2}}\left(\frac{2\kappa_{a,\mathrm{ext}}}{\delta_{a}^{2}+\kappa_{a}^{2}}\right)^{2}.$$
$$\eta_\mathrm{SHG}^\mathrm{max} = \frac{g^2}{4 \hbar \omega_{a}} \frac{1}{\kappa_{a,0}^2 \kappa_{b,0}} = \frac{g^2}{8\hbar \omega_{a}^4} Q_{a,0}^2 Q_{b,0},$$
$$\eta_\mathrm{THG}=\frac{P_\mathrm{THG}}{P_\mathrm{p}^{3}}=g^{2}\frac{16Q_\mathrm{p,0}^{3}Q_\mathrm{TH,0}}{\hbar^{2}\omega_\mathrm{p}^{6}}.$$
$$\scalebox{0.94}{$\displaystyle \left(\frac{\partial}{\partial z}+\frac{\alpha}{2}-i\underset{k\geq2}{\sum}\frac{i^{k}\beta_{k}}{k!}\frac{\partial^{k}}{\partial t^{k}}\right)E\left(z,t\right)=i\gamma\left(1+\frac{i}{\omega_{0}}\frac{\partial}{\partial t}\right)\left[E\left(z,t\right)\int_{-\infty}^{+\infty}R\left(\tau\right)\left(\left|E\left(z,t-\tau\right)\right|^{2}d\tau\right)\right],$}$$
$$\beta\left(\omega_\mathrm{d}\right)=\beta\left(\omega_\mathrm{0}\right)+\left(\omega_\mathrm{d}-\omega_\mathrm{0}\right)\upsilon_\mathrm{g}^{{-}1}+\frac{\gamma P}{2}.$$
$$\beta_\mathrm{int}=\beta\left(\omega\right)-\beta\left(\omega_{0}\right)-\left(\omega-\omega_{0}\right)\upsilon_{g}^{{-}1} =\underset{k\geqslant2}{\sum}\frac{\left(\omega-\omega_{0}\right)^{k}}{k!}\frac{d^{k}}{d\omega^{k}}\beta\left(\omega_{0}\right).$$
$$\left(\frac{\partial}{\partial z}+\frac{\alpha}{2}-i\underset{k\geq2}{\sum}\frac{i^{k}\beta_{k}}{k!}\frac{\partial^{k}}{\partial t^{k}}\right)E\left(z,t\right)=i\frac{\omega_{0}}{2n_{0}c\varepsilon_{0}}\left(1+i\tau_\mathrm{sh}\frac{\partial}{\partial\tau}\right)P_\mathrm{NL}\left(z,t\right),$$
$$f_n = f_\mathrm{ceo} + n f_\mathrm{rep},$$
$$\omega_\mu = \omega_0 + \sum_\mathrm{m} \frac{D_\mathrm{m} \mu^\mathrm{m}}{m!} = \omega_0 + D_1 \mu + \frac{1}{2!} D_2 \mu^2 + \frac{1}{3!} D_3 \mu^3 + \ldots.$$
$$D_\mathrm{int} = \omega_\mu - \omega_0 - D_1 \mu = \frac{1}{2!} D_2 \mu^2 + \frac{1}{3!} D_3 \mu^3 + \underset{m\geqslant4}{\sum}\frac{1}{m!}D_{m}\mu^{m}.$$
$$f_\mathrm{ceo} = \delta_2-2\delta_1.$$

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