Quadratic nonlinearity (${\chi }^{(2)}$) is the basis of many important nonlinear optical processes,1 such as second-harmonic generation (SHG), sum-frequency generation, difference-frequency generation, optical parametric oscillation, and nonclassical spontaneous parametric downconversion, making it very promising in a broad range of applications, including wavelength conversion, frequency metrology, and quantum information processing.2^–5 Among the materials possessing ${\chi }^{(2)}$ nonlinearity, lithium niobate (LN), a multifunctional ferroelectric crystal with excellent properties including low absorption loss, a wide transparent window ($0.35-5\mu \mathrm{m}$), a high electro-optic coefficient ($r_{33}\sim 34\mathrm{pm}/\mathrm{V}$), and strong second-order nonlinearity ($d_{33}\sim 25\mathrm{pm}/\mathrm{V}$), plays an important role in nonlinear and quantum optics.6 Efficient wavelength conversion in LN waveguides typically relies on periodically poled lithium niobate (PPLN) to achieve quasi-phase matching (QPM).7 However, these nonlinear photonic devices based on bulk LN suffer from large footprints and low conversion efficiencies due to the relatively large optical mode areas in proton-exchanged and titanium-diffused waveguides.8^,9

Recently, LN-on-insulator (LNOI) technology has attracted considerable attention because the high refractive index contrast between the core and cladding leads to strong optical confinement, making it possible to achieve high-density photonic integration and efficient nonlinear interaction.10^–12 To date, the LNOI platform has hosted a wide variety of high-performance photonic devices, including high-speed electro-optic modulators, efficient acousto-optic frequency shifters, and high-gain chip-scale amplifiers.13^–15 In addition, the enhanced second-order nonlinear interaction could considerably improve the wave-mixing efficiencies in various nonlinear optical processes.16^–20

As a mainstream approach for SHG using LN, the QPM scheme has been successfully implemented in a hybrid silicon nitride–LNOI waveguide with a normalized conversion efficiency (${\eta }_{\text{norm}}$) of $160\%{\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$ in 2016.21 It is significantly improved to higher than $2000\%{\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$ in periodically poled LNOI (PPLNOI) waveguides, although it is still lower than the theoretical value, possibly due to imperfect domain poling.22^–25 Moreover, achieving uniform domain inversion faces great challenges for on-chip LNOI devices that require small domain poling periods, especially for generating short-wavelength light (e.g., blue and ultraviolet lights), since the random domain growth and leakage currents can result in poor poling quality.26 Many efforts were then made in improving the poling process,27 such as actively monitoring the poling process and adapted poling approach, achieving a ${\eta }_{\text{norm}}$ of up to $4600\%{\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$ for SHG in a 0.6-mm-long waveguide28 and an absolute conversion efficiency (${\eta }_{\mathrm{abs}}$) of $82.5\%$ for a 21-mm-long PPLNOI waveguide.29 It is important to note that there is an upper limit of conversion efficiency in theory for PPLNOI devices pumping at telecommunication wavelength, which is around $5000\%{\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$, depending on the specific configurations in PPLNs.28 In addition, birefringent phase-matching (BPM) in an $x$-cut LNOI has been demonstrated, but ${\eta }_{\text{norm}}$ ($2.7\%{\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$) is constrained by the low nonlinear susceptibility ($d_{31 }\sim 4.7\mathrm{pm}/\mathrm{V}$) inevitably used when both polarization states interact with each other.30 Another competent approach that keeps making progress is modal phase matching (MPM).31 Initially, the poor modal overlap between modes of different orders in the MPM generally results in low conversion efficiency.32^,33 To solve this problem, researchers have modified the ${\chi }^{(2)}$ profile in the waveguide cross section to improve the modal overlap, including chemically changing ${\chi }^{(2)}$ by ion exchange34^,35 or coating the waveguide with a material with ${\chi }^{(2)}=0$,36^,37 exhibiting enhanced conversion efficiency compared to the conventional MPM. However, in these methods, ${\chi }^{(2)}$ in part of the waveguide core region is zero and has no contribution to the SHG process. To solve this issue, a polarization-reversed double-layer LNOI waveguide with reconfigured ${\chi }^{(2)}$ profile38^,39 is proposed to fully leverage the high nonlinear coefficient of LN and achieve ${\eta }_{\text{norm}}$ of $5540\%{\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$, which is already comparable to the highest achievable ${\eta }_{\text{norm}}$ of PPLNOI waveguides.28 Nevertheless, it is still highly desirable to enable significantly higher conversion efficiency toward its theoretical limit for MPM-based LNOI SHG devices.

Here, we demonstrate SHG with an unprecedented conversion efficiency in a double-layer $x$-cut LNOI waveguide by MPM. Our model reveals that, with the realistic waveguide properties enabled in advanced nanofabrication, an achievable conversion efficiency can be close to the theoretical limit. A record-high ${\eta }_{\text{norm}}$ of $9600\%{\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$ is obtained, 2 times higher than that given by PPLNOIs, while ${\eta }_{\mathrm{abs}}$ is as high as $85\%$. Such a highly efficient frequency converter also exhibits tunability of $0.071\mathrm{nm}/\mathrm{K}$ by the thermal-optic effect of LN. These results pave the way for developing a variety of on-chip frequency converters with ultrahigh efficiency in frequency metrology, quantum optics, and ultrafast optics.

To achieve an efficient SHG, the phase matching (PM) condition should be satisfied,1 which implies $$k_{2\omega }=2k_{\omega },$$(1)where $k_{\omega }$ and $k_{2\omega }$ are the wave vectors at fundamental and second-harmonic (SH) frequencies, respectively. Due to the relation of $k=n\omega /c$, Eq. (1) indicates that one needs to find fundamental and SH modes with identical effective refractive indices ($n_{\omega }=n_{2\omega }$). Considering the material and waveguide dispersions, this condition can be fulfilled by selecting the fundamental mode at $\omega$ and higher-order mode at $2\omega$ in MPM.

To utilize the largest component of the second-order nonlinear tensor ($d_{zzz}=d_{33}=25\mathrm{pm}/\mathrm{V}$), ${\mathrm{TE}}_{00}$ mode at the fundamental wavelength and ${\mathrm{TE}}_{01}$ mode at SH wavelength in an $x$-cut LNOI waveguide are chosen. The cross section of the double-layer LNOI waveguide is shown in Fig. 1(a). The waveguide has a device-layer thickness, $H$, of 500 nm ($h_1=h_2=250\mathrm{nm}$), a top width, $w$, of 850 nm, an etching depth, $h$, of 450 nm, and a sidewall angle, $\theta$, of 66.5 deg.

The finite element method is used for calculating the properties of waveguide eigenmodes, including their effective refractive indices and mode profiles, based on which we can access the PM condition and modal overlap. To evaluate SHG power and conversion efficiency, especially in a high-power regime where an analytical solution is unavailable, we use numerical calculation based on the formula below21$$\frac{\partial A_{\mathrm{SH}}}{\partial y}=-\frac{{\alpha }_{\mathrm{SH}}}{2}{A}_{\mathrm{SH}}-i{\kappa }_{\mathrm{SH}}{A}_f^2\exp [i\mathrm{\Delta }ky],$$(2)$$\frac{\partial A_f}{\partial y}=-\frac{{\alpha }_f}{2}{A}_f-i{\kappa }_f{A}_{\mathrm{SH}}{A}_f^{*}\exp [-i\mathrm{\Delta }ky],$$(3)where ${\alpha }_{\mathrm{SH}}$ and ${\alpha }_f$ are the waveguide propagation losses at fundamental and SH wavelengths, respectively. $A$ is proportional to the amplitude of the interacting light. The coupling coefficients are $${\kappa }_{\mathrm{SH}}=\frac{d_{33}{\omega }_{\mathrm{SH}}}{\pi c{n}_{\mathrm{SH}}}•\frac{\iint E_f^2(x,z)E_{\mathrm{SH}}(x,z)\mathrm{d}x\mathrm{d}z}{\iint E_{\mathrm{SH}}^2(x,z)\mathrm{d}x\mathrm{d}z},$$(4)$${\kappa }_f=\frac{{2d}_{33}{\omega }_f}{\pi c{n}_f}•\frac{\iint E_f^2(x,z)E_{\mathrm{SH}}(x,z)\mathrm{d}x\mathrm{d}z}{\iint E_f^2(x,z)\mathrm{d}x\mathrm{d}z},$$(5)the phase mismatch $\mathrm{\Delta }k$ is $$\mathrm{\Delta }k=k_{\mathrm{SH}}-2k_f.$$(6)

The SH power $P_{2\omega }$ generated by a device with waveguide length $L$ can be expressed as $$P_{2\omega }={\eta }_{\text{norm}}{L}^2{P}_{\omega }^{2}g(L),$$(7)where $P_{\omega }$ is the pump power and $${\eta }_{\text{norm}}=\frac{8d_{33}^2}{{\epsilon }_{0}c{n}_{\mathrm{SH}}{n}_f^2{\lambda }_{\mathrm{SH}}^2{S}_{\mathrm{eff}}}{\mathrm{sinc}}^2\left(\frac{\mathrm{\Delta }kL}{2}\right)$$(8)is the normalized efficiency expressed in ${\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$, where $d_{33}$ is the nonlinear coefficient of the material, ${\epsilon }_0$ is the vacuum permittivity, $c$ is the speed of light in vacuum, and $n_{\omega }$ and $n_{2\omega }$ are the effective refractive indices for the fundamental mode and the SH mode, respectively. $S_{\text{eff}}$ can be expressed as $$S_{\mathrm{eff}}=\frac{{[\iint E_f^2(x,z)\mathrm{d}x\mathrm{d}z]}^2\iint E_{\mathrm{SH}}^2(x,z)\mathrm{d}x\mathrm{d}z}{{[\iint E_f^2(x,z)E_{\mathrm{SH}}(x,z)\mathrm{d}x\mathrm{d}z]}^2},$$(9)where $E(x,z)$ is the spatial mode profile of transverse waveguide modes. $S_{\mathrm{eff}}$ mainly depends on the mode size at the different wavelengths and the overlap between them. Finally, the loss factor $g(L)$ is expressed as $$g(L)=e^{-2{\alpha }_{f}L}\frac{(e^{\mathrm{\Delta }\alpha L}+1{)}^2-4{\cos }^2(\frac{\mathrm{\Delta }kL}{2})e^{\mathrm{\Delta }\alpha L}}{\mathrm{\Delta }{\alpha }^2+\mathrm{\Delta }{k}^2},$$(10)where $$\mathrm{\Delta }\alpha ={\alpha }_f-\frac{{\alpha }_{\mathrm{SH}}}{2}.$$(11)

The effective refractive indices of these two modes as a function of wavelength are calculated and shown in Fig. 1(d). The PM is achieved at the wavelength of 1552.2 nm. Figures 1(b) and 1(c) show the dominant electric field component ($E_z$) of the ${\mathrm{TE}}_{00}$ mode at 1552.2 nm and ${\mathrm{TE}}_{01}$ mode at 776.1 nm, respectively. The conversion efficiency of SHG highly depends on the modal overlap ($\mathrm{\Gamma }$) $$\mathrm{\Gamma }={\iint }_{\mathrm{LN}}{d}_{33}(x,z)E_{z,\omega }^2(x,z)E_{z,2\omega }(x,z)\mathrm{d}x\mathrm{d}z.$$(12)

Since the electric field of the ${\mathrm{TE}}_{01}$ mode changes its sign across the waveguide core, as shown in Fig. 1(c), the positive and negative contributions to the integral almost cancel out, and the resulting $\mathrm{\Gamma }$ is generally small. Methods of enhancing $\mathrm{\Gamma }$ reported in previous works include introducing a double-ridge waveguide31 and modifying ${\chi }^{(2)}$ profile.34^–37 Here, by introducing a polarization-reversed double-layer structure [see $d_{33}$ distribution in Fig. 1(a)], the nonlinear coefficient $d_{33}$ has the opposite signs as well across the waveguide, leading to a significant enhancement of $\mathrm{\Gamma }$ and resolving the fundamental issue in the MPM scheme. Figure 1(e) shows that ${\eta }_{\text{norm}}$ reaches a maximum value when the two LN layers have approximately equal thicknesses.

Previous works investigated the SHG process in double-layer LNOI for device-layer thicknesses larger than 550 nm and conducted experiments with a thickness of 570 nm.39 In fact, ${\eta }_{\text{norm}}$ depends on waveguide geometry, and we find that smaller thickness leads to higher ${\eta }_{\text{norm}}$ [see Fig. 1(f)] because of more compact mode sizes, but mode hybridization makes PM difficult when the thickness is smaller than $\sim 500\mathrm{nm}$ [see a case for $H=450\mathrm{nm}$ in Fig. 1(g), in which two SH modes hybridize and PM condition is not satisfied for any widths]. Therefore, an LN thickness of 500 nm is chosen in this work to reach a relatively high ${\eta }_{\text{norm}}$ and ensure that the PM condition can be accessed.

Waveguides are fabricated on an $x$-cut double-layer LNOI with a $2\text{-}\mu \mathrm{m}$-thick buried oxide.39 First, a layer of negative electron-beam resist is spin-coated as the etching mask, and device patterns are defined by electron-beam lithography. The acceleration voltage and beam current used in the electron-beam lithography are 100 kV and 1 nA, respectively. Then, the patterns are transferred to LN by argon plasma etching in an inductively coupled plasma reactive ion etching (ICP-RIE) tool (Oxford Instruments PlasmaPro 100 Cobra) with the following parameters: Ar flow of 15 sccm, DC bias of 190 V, ICP power of 1000 W, and pressure of 2 mTorr. The etching depth is measured to be 450 nm. We preserve a 50-nm-thick unetched LN layer to prevent the buried oxide from etching by the hydrofluoric acid in the following cleaning step. As shown in Fig. 1(h), the waveguide sidewall is smooth, indicating high fabrication quality and low scattering loss from the roughness of the sidewalls. The total waveguide length, $L$, is 10 mm.

The measurement setup is shown in Fig. 2. Pump light from a continuous-wave tunable laser (Santec TSL-570) is coupled into the waveguide using a lensed fiber. We use an in-line fiber polarization controller to ensure TE polarization at the input. When the pump laser is tuned to the PM wavelength, strong SH light is observed at the waveguide output facet. An SHG signal is collected using an aspheric lens (numerical aperture is 0.5) and detected by a photodetector (Thorlabs S130C). When measurement is performed under high-power pumping conditions, an erbium-doped fiber amplifier (EDFA) is used to boost the pump light.

By scanning the laser wavelength, we measure the dependence of the SHG signal on wavelength. Taking into account the measured fiber-to-chip coupling loss (around 8 dB per facet) and assuming that the SHG signal generated in the waveguide is completely collected by the lens, the pump power ($P_{\omega }$) at the input facet and the SHG power ($P_{2\omega }$) at the output facet of the waveguide can be determined. Then, ${\eta }_{\text{norm}}$ is calculated according to the formula: ${\eta }_{\text{norm}}=P_{2\omega }/(P_{\omega }L)$.2 We plot ${\eta }_{\text{norm}}$ as a function of wavelength in Fig. 3(a). Maximum ${\eta }_{\text{norm}}$ of $9600\%{\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$, $8845\%{\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$, and $9013\%{\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$ are obtained at PM wavelengths of 1485.7, 1518.3, and 1552.2 nm for three waveguides with different widths, respectively. To the best of our knowledge, these are the highest ${\eta }_{\text{norm}}$ reported in LN or LNOI waveguides pumped around the telecommunication band, indicating the advantage of MPM implemented in double-layer LNOI over QPM as well as the high fabrication quality of the SHG waveguides. The discrepancies of the maximum ${\eta }_{\text{norm}}$ between experiment and simulation with an ideal model can be attributed to the waveguide propagation loss. By fabricating a microring resonator with the same waveguide geometry as that used for SHG, and with a sufficiently large radius of $200\mu \mathrm{m}$ to avoid additional scattering and bend losses, the intrinsic quality factor and the corresponding waveguide loss are extracted to be $8.3\times {10}^5$ and $0.5\mathrm{dB}/\mathrm{cm}$ at 1550 nm, respectively [as shown in Fig. 3(b)]. Compared with the previous work of SHG in double-layer LNOI (${\eta }_{\text{norm}}=5540\%{\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$ with a propagation loss of $3.8\mathrm{dB}/\mathrm{cm}$),39 the lower loss achieved in this work can contribute to the improvement of the measured conversion efficiency. Taking into account the waveguide loss at the fundamental wavelength and the unequal thicknesses of the two LN layers ($h_1=220\mathrm{nm}$ and $h_2=280\mathrm{nm}$ measured by SEM) and assuming that the waveguide loss at SH wavelength is $0.8\mathrm{dB}/\mathrm{cm}$ in our simulation model, the measured spectra (black curves) are consistent with those of the corrected model (blue dashed curves). PM wavelengths extracted from Fig. 3(a) and the corresponding simulated results can be found in Fig. 3(c). The inset in Fig. 3(a) is the top-view optical micrograph of the scattered SH signal at the waveguide facet.

By fixing the pump wavelength at 1552.2 nm for the maximum ${\eta }_{\text{norm}}$, the SHG power is recorded as a function of the pump power, as shown in Fig. 4. As expected, in the low-power regime [$P_{\omega }<0.7\mathrm{mW}$ in the inset of Fig. 4(a)], their relation basically follows a quadratic dependence of an SHG process without pump depletion. We use an EDFA to further increase the pump power and investigate the device behavior under high-power pumping conditions. The measured SHG power is consistent with that simulated with pump depletion and waveguide propagation loss. We extract the ${\eta }_{\mathrm{abs}}$, defined as ${\eta }_{\mathrm{abs}}=P_{2\omega }/P_{\omega }$, from Fig. 4(a) and find that ${\eta }_{\mathrm{abs}}$ starts to saturate when $P_{\omega }$ is higher than 80 mW [Fig. 4(b)]. Notably, ${\eta }_{\mathrm{abs}}$ in our work reaches an unprecedented value of $85\%$ among the SHG waveguides reported in LNOI, simultaneously possessing the record-high ${\eta }_{\text{norm}}$, as illustrated in Fig. 3.

LN exhibits a significant thermo-optic effect for extraordinary light on the order of ${10}^{-5}$, while it is negligible for ordinary light, given by the following equations: $$\frac{\text{d}{n}_{e,\mathrm{FW}}}{\text{d}T}=-2.6+19.8\times {10}^{-3}T,$$(13)$$\frac{\text{d}{n}_{e,\mathrm{SH}}}{\text{d}T}=-2.6+22.4\times {10}^{-3}T.$$(14)

These equations are reliable in a temperature range from 300 to 515 K.40^,41 As a result, a temperature change would lead to an index change of LN and hence shift the PM wavelength of the SHG process. We leverage the thermo-optic effect in LN to achieve a tunable SHG device. The sample is mounted on a thermoelectric cooler and its temperature is changed by a temperature controller. Figure 5 shows the simulated and measured PM wavelengths at various temperatures. The PM condition moves toward longer wavelengths as temperature increases. The extracted tunability of our SHG device, $0.071\mathrm{nm}/\mathrm{K}$, agrees well with the simulated value.

Finally, we compare the LNOI waveguides for SHG at the telecommunication band, including MPM, QPM, and BPM schemes, in terms of the waveguide length, pump power, ${\eta }_{\mathrm{abs}}$ (in the unit of %), ${\eta }_{\mathrm{SHG}}$ (in the unit of $\%{\mathrm{W}}^{-1}$), ${\eta }_{\text{norm}}$ (in the unit of $\%{\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$), and SH power, as presented in Table 1.

By equalizing the thicknesses of the two LN layers and further reducing propagation loss, possibly down to $0.2\mathrm{dB}/\mathrm{cm}$, one can expect that the normalized conversion efficiency can be improved to $\mathrm{11,700}\%{\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$.

On the other hand, it is noteworthy that an optical microresonator can enhance the SHG efficiency in both QPM and MPM schemes,42 and an extremely high efficiency of 5,000,000% W^–1 has been reported.43 However, it is believed that the waveguide-based SHG process exhibits unique advantages, such as large bandwidth, high absolute conversion efficiency, wide tunability, and good tolerance of fabrication errors, which are highly desirable in practice.

In conclusion, highly efficient SHG in a double-layer LNOI waveguide is demonstrated. Leveraging the significantly enhanced modal overlap in the polarization-reversed LN layers and the high-quality nanofabrication, an on-chip normalized conversion efficiency as high as $9600\%{\mathrm{W}}^{-1}{\mathrm{cm}}^{-2}$ is achieved, associated with an absolute conversion efficiency of 85% when pumping around 1552.2 nm. Our results outperform other works on MPM31 and lend strong support to the competitiveness of the MPM scheme in such a new platform for a variety of nonlinear applications when compared to previously reported methods such as PPLN. The presented results pave the way for efficient nonlinear wave-mixing processes for classical and quantum applications.

Biographies of the authors are not available.