Broadband Second Harmonic Generation in a z-Cut Lithium Niobate on Insulator Waveguide Based on Type-I Modal Phase Matching

Abstract

We numerically investigate a second harmonic generation (SHG) in a z-cut lithium niobate on insulator (LNOI) waveguide based on type-I mode phase matching (MPM) between two fundamental modes. A mode overlap factor that is close to unity is achieved and the normalized SHG efficiency reaches up to 72.1% W⁻¹cm⁻² at the telecommunication band, together with a large spectral tunability of 2.5 nm/K. Moreover, a bandwidth of about 100 nm for the broad SHG in a 5 mm-long LNOI ridge waveguide is demonstrated with this scheme. This stratagem will inspire new integrated nonlinear frequency conversion methods for versatile applications.

1. Introduction

Lithium niobate (LiNbO₃ or LN) is a well-known multifunctional crystal with a wide optical transparency window and excellent piezoelectric and electro-optic properties [1,2,3,4,5]. With the birth of lithium niobate on insulator (LNOI), it has sparked significant interest in exploring new on-chip optical phenomena and novel functionalities, such as high-speed electro-optic modulators [6,7,8], lasers and amplifiers [9,10,11], broadband frequency combs [12,13,14], and single photon generation [15,16,17]. In particular, the large quadratic nonlinear susceptibility (d_eff = 27 pm·V⁻¹) [18,19,20] and the high refractive index contrast between LN and the silica substrate, dramatically enhancing the light–matter interaction within a subwavelength scale, render it a unique chip-scale platform for nonlinear optical processes.

To achieve efficient wavelength conversion, the phase matching among interaction waves should be strictly satisfied [21,22,23], which ensures that generated nonlinear optical signals are accumulated constructively. Generally, the current state of the art of phase matching in LN comprises three approaches: birefringent phase matching (BPM) [24,25,26], quasi-phase matching (QPM) [27,28,29], and modal phase matching (MPM) [30,31,32,33,34,35,36]. BPM achieves precise phase matching based on the birefringence effect of LN, under special light propagation direction and polarization configuration. However, it is relatively difficult to implement in waveguides and has limited conversion efficiency. QPM is realized by periodic poling to achieve domain inversion of the crystal to compensate the phase mismatching between different optical fields. It takes advantage of high conversion efficiency, but is subject to the complex engineering technology of small poling periods and uniform domain poling with an appropriate duty cycle, along with limited wavelength tunability. Meanwhile, MPM is fulfilled by carefully designing the waveguide geometries and engineering its dispersion to match phases of the fundamental mode with a higher-order mode, while the mismatch of their mode profiles usually degrades the conversion efficiency significantly. On the other hand, remarkable achievements have been implemented in x-cut LNOI waveguides. For example, Wang et al. employed a periodically poled x-cut LNOI waveguide with QPM [26], Lu et al. proposed novel angle-cut ridge waveguides [30], and Luo et al. developed a semi-nonlinear waveguide by MPM [36] with x-cut LNOI to demonstrate second-harmonic generation with high normalized efficiency. However, the wave is restricted to propagating in a specific direction to fulfill the phase-matching condition in x-cut LNOI, which imposes restrictions on a wide range applications, like micro-rings and micro-disks. In contrast, z-cut LNOI may be worthy of attention since it is homogeneous in the waveguide plane, and has unique applications for micro-ring-based high-Q nonlinear photonics.

In this work, we propose and simulatively demonstrate a design of a z-cut few-micron-thick LNOI waveguide for broadband and efficient SHG, which accesses type-I mode phase matching between two fundamental modes, enabling a spatial mode overlap close to unity. A normalized conversion efficiency of 72.1% W⁻¹cm⁻² in the telecommunication band is theoretically demonstrated. By employing dispersion engineering of the waveguide, a broad bandwidth of 100 nm for SHG generation is also realized in a 5 mm-long waveguide, almost one order of magnitude larger than that achieved by traditional QPM or MPM. Furthermore, the strong thermo-optic effect in such a type-I phase-matching scheme makes it possible to obtain a large wavelength tunability. The simulated thermal tuning slope of the phase-matching wavelength is 2.5 nm/K, which is two or three times larger than that of similar work. Our work provides a new idea for tunable ultrabroadband nonlinear frequency conversion and prospects for various applications.

2. Theoretical Design

Figure 1a illustrates the schematic structure of the z-cut LN strip waveguide for SHG. The optical axis is collinear with the z-axis. The FW pump light propagates along the y-axis and generates a second harmonic wave (SHW) in the waveguide. Instead of matching phases of FW with a higher-order mode of SHW, the phase-matching condition between the fundamental quasi-transverse-electric mode (TE₀₀) of FW and the fundamental quasi-transverse-magnetic mode (TM₀₀) of SHW is perfectly achieved, by carefully designing the geometry of the waveguide. There is a large mode overlap factor between these two fundamental modes, benefiting high nonlinear conversion efficiency. For details, the geometry of the waveguide is optimized by finite difference eigenmode solver (MODE solutions, Lumerical), which can be experimentally fabricated by using standard electron beam lithography and inductively coupled plasma etching processes [26], and the excitation of the fundamental mode can be achieved by well designed spot-size converters (SSC) or grating couplers. Figure 1b shows the simulated phase-matched modal profiles of the TE₀₀ mode of FW and TM₀₀ mode of SHW for SHG at the telecommunication band, where the height (h) and width (w) of the waveguide are 2 μm and 2.71 μm, respectively. We plot the effective refractive indices of different modes in the LN strip waveguide as functions of the pump wavelength in Figure 1c. A zoomed-in plot of the black-dashed-line box is shown in Figure 1d, where a cross between the TM₀₀ at 0.775 μm and the TE₀₀ at 1.55 μm is present, indicating a suitable phase-matching wavelength around 1.55 μm at a room temperature of T = 20 °C.

To find the optimized and feasible phase-matching condition, we numerically simulate and calculate the effective refractive index difference (Δn) between TE₀₀ mode at 1.55 μm and TM₀₀ mode at 0.775 μm in z-cut LN strip waveguides with different widths and heights. The results are presented in Figure 2a. The phase-matching condition is nearly satisfied in the yellow area where Δn is close to zero. It should be noted that the variation of the height is smaller than that of the width of the waveguide to the same variation of Δn, so that the phase matching is more sensitive to the height than width of the waveguide, which may bring challenges to the fabrication. However, this problem can be solved by temperature tuning as shown in Figure 2b, where the width of the waveguide is set as w = 2.71 μm. For a particular height of waveguide, we can always find the exact phase-matching condition by tuning the temperature. Therefore, the numerical simulation results show the proposed phase-matching stratagem is practical for the nonlinear frequency conversion. The theoretically calculated relationship between the amplitude and propagation length of the SHW of the white star in Figure 2a is also illustrated in Figure 2c, where h = 2 μm, w = 2.71 μm, and Δn = 0. The calculated SH intensity follows a quadratic response with respect to the propagation distance in the low-conversion limit when the phase-matching condition is satisfied between the designed modes.

Next, the conversion efficiency is estimated with the geometry designed above. When the phase-matching condition is satisfied, the normalized SHG efficiency is given by the following expression under small signal conditions [34,35]:

$${\eta }_{norm}=\frac{8{\pi }^2}{{\epsilon }_{0}c{n}_1^2{n}_2{\lambda }^2}\frac{{\xi }^2{d}_{eff}^2}{A_{eff}},$$

(1)

where c is the speed of light in vacuum, ε₀ is the vacuum permittivity, deff is the effective nonlinear coefficient of lithium niobite (d_eff = d₃₁ = 4.3 pm/V), and a type-I configuration is employed. Equation (1) shows that the efficiency of SHG is mainly determined by the effective nonlinear coefficient d_eff, the effective mode area A_eff, and the spatial mode overlap factor of the fundamental and second-harmonic modes. Here the effective mode area is A_eff = (A₁²A₂)^1/3, where

$$A_i=\frac{{\left({\displaystyle {\int }_{all}{\left|E_i\right|}^{2}dxdz}\right)}^3}{{\left|{\displaystyle {\int }_{{\chi }^{\left(2\right)}}{\left|E_i\right|}^2{E}_{i}dxdz}\right|}^2},i=1,2$$

(2)

and the spatial mode overlap factor

$$\zeta =\frac{{\displaystyle {\int }_{{\chi }^{\left(2\right)}}{\left(E_{1x}^{\ast }\right)}^2{E}_{2z}dxdz}}{{\left|{\displaystyle {\int }_{{\chi }^{\left(2\right)}}{\left|E_1\right|}^2{E}_{1}dxdz}\right|}^{2/3}{\left|{\displaystyle {\int }_{{\chi }^{\left(2\right)}}{\left|E_2\right|}^2{E}_{2}dxdz}\right|}^{1/3}},$$

(3)

where ∫_χ(2) and ∫_all denote two-dimensional integration over the LN material and whole space, respectively. E_1x is the x-component of E₁(x, z), the electric field of the fundamental mode TE₀₀, and E_2z is the z-component of E₂(x, z), the electric field of the second-harmonic mode TM₀₀. It can be seen from Equation (1) that the efficiency of SHG conversion in waveguide mainly depends on three elements: effective nonlinear coefficient, spatial mode overlap factor, and effective mode area. In order to improve the efficiency of frequency conversion, we can increase the overlap integral of spatial mode field. Due to the high refractive index contrast between the LN film waveguide layer and the substrate layer (SiO₂), which can achieve strong field confinement, the frequency conversion efficiency in the waveguide will naturally be higher than the bulk. Equation (3) indicates that the magnitude of the overlapping integral of the mode field depends strictly on the symmetry of the two modes involved in the nonlinear interaction. Unfortunately, different modes have their own unique spatial symmetry in the waveguide, which makes the mode overlap factor greatly reduced. The calculated mode overlap factor is ζ = 0.9890 and the effective mode area A_eff = 3.05 μm². Near-unity mode overlap factor is achieved since the phase matching is utilized between the fundamental mode of FW and SHW, which is much larger than traditional modal phase matching between different order modes [35,36]. The calculated normalized conversion efficiency is 72.1% W⁻¹cm⁻², which is also higher than previous MPM works [30,37].

3. Broadband SHG Performance and the Spectral Tuning Capability

Noteworthily, such a phase-matching scheme possesses broadband performance. Figure 3a shows the conversion efficiency (normalized to the phase matched condition, Conv. effi. (norm.)) spectra with a waveguide length of L = 5 mm at T = 20 °C, which is dependent on the geometry of the strip waveguide and pump wavelength. Here, the height of the waveguide is set as h = 1.97 μm to obtain the optimized normalized conversion efficiency. The green curve illustrates that there exist two phase-matching points at 1.393 μm and 1.545 μm, respectively, with waveguide width w = 2.75 μm [see Figure 3a]. This is different from the traditional SHG methods, where only one phase-matching point could be realized. The corresponding modal dispersions of FW and SHW are shown in Figure 3b, accompanied by the effective index difference between the two modes. The two phase-matching points could be adjusted by the geometry of the waveguide and they are merged together at around 1.45 μm when the width of the waveguide is w = 2.698 μm. The nonlinear frequency conversion bandwidth of the merged phase-matching condition is approximately 35 nm, where the bandwidth is calculated at half of the maximum conversion efficiency. The nonlinear bandwidth could be further broadened by carefully varying the geometry of the waveguide. The optimized bandwidth is about 50 nm when w = 2.702 μm. It is much larger than the traditional methods for SHG, e.g., MPM. The bandwidth could be even larger by designing both the width and height to engineer the phase mismatch condition. The broadband characteristic is promising for quantum entanglement generation, quantum optical coherence tomography, Coherent Anti-Stoke Raman Spectroscopy (CARS), and so on.

Meanwhile, this scheme also has good spectral tuning capability by varying the temperature of the device. The SHG processes are simulated by FDTD with refractive indexes of LN at different temperatures from 0 °C to 20 °C. The normalized SHG conversion efficiencies are shown in Figure 3c with the waveguide geometry of w = 2.71 μm and h = 2 μm, and waveguide length of 5 mm. The SHG spectrum shifts to shorter wavelengths when the temperature increases. By mapping the phase-matched pump wavelength as a function of temperature, we obtained Figure 3d, showing a tuning slope of dλ_PM/dT = 2.5 nm/K, which is larger than the reported phase-matching works [30,35]. This is mainly caused by the dispersions of the relevant modes determined by the geometry of the waveguide, and the tuning slope could be further enlarged by carefully optimizing the parameters of the waveguide.

4. The Simulation Results of Ridge Waveguide

The design stratagem can also be applied to the ridge waveguide denoted in the schematic Figure 4a. Typical phase-matched modal profiles of TE₀₀ mode at FW and TM₀₀ mode of SHW are displayed in Figure 4b, where the waveguide dimensions are set as h₁ = 1.35 μm, h₂ = 0.3 μm, w = 2 μm, and θ = 75°. The mode overlap factor can reach up to ζ = 0.9808 and the effective mode area A_eff = 4.84 μm². The normalized conversion efficiency is evaluated as 44.86 % W⁻¹cm⁻², which is similar in magnitude to the strip waveguide. In addition, the ridge waveguide can also possess a broad SHG bandwidth. We fix the height of the waveguide and vary the width of it. Figure 4c shows the conversion efficiency spectra at T = 20 °C with a waveguide length of 5 mm. The bandwidth can reach up to 100 nm when the width is around 1.93 μm, which is double the optimized bandwidth of the strip waveguide. Meanwhile, the nonlinear process has better tolerance to the geometry of the waveguide compared with the strip waveguide, which will reduce the requirement of the fabrication process. As a result, it provides an alternative and effective way to realize the SHG process and benefit its applications.

5. Conclusions

We have numerically investigated a type-I MPM SHG process in a z-cut LNOI strip waveguide. The phase matching is achieved between two fundamental modes of FW and SHW, benefiting from a large mode overlap factor close to unity, and the normalized conversion efficiency is up to 72.1% W⁻¹ cm⁻². Such an SHG process has a broad bandwidth of about 50 nm (5 mm-long LNOI waveguide) in the telecommunication band and flexible tunability of the phase-matching wavelength is achieved with a tuning slope of 2.5 nm/K. Moreover, such an SHG process in the ridge waveguide is also investigated, indicating a larger bandwidth around 100 nm, accompanied by a better tolerance to the fabrication deviations. Compared with nano-LNOI (a few hundred nanometers in thickness), our few-micron-thick LNOI waveguide has advantages in several aspects, including handling more power, improving the coupling efficiency and having a moderate optical field confinement. Our approach also effectively achieves phase matching between two fundamental modes, enlarging the mode overlap for a high-efficiency nonlinear process, and simultaneously simplifying the fabrication processes. It is not only restricted to the LN straight waveguide, but can also be applied to other guided structures like micro-rings and micro-disks, since the waveguide direction is not restricted in a z-cut LN film. We believe our SHG stratagem with high efficiency and broad bandwidth will inspire new on-chip nonlinear frequency conversion methods and versatile applications.