Mid-Infrared Highly Efficient, Broadband, and Flattened Dispersive Wave Generation via Dual-Coupled Thin-Film Lithium-Niobate-on-Insulator Waveguide

Abstract

We designed a structure of dual-coupled ridge waveguide in thin-film lithium-niobate-on-insulator (LNOI) and numerically studied the highly efficient, broadband, and flattened dispersive wave-enhanced supercontinuum generation in the mid-infrared region. By leveraging the mode coupling of the proposed dual-coupled waveguide structure, one of the supermodes, namely the anti-symmetric mode, can produce additional zero-dispersion wavelengths in the mid-infrared region, and consequently multiple normal dispersion regions for dispersive wave emission. Given the rich geometrical degrees of freedom powered by this dual-coupled LNOI waveguide structure, we can tailor the dispersion profile so that a well-established mode-locked fiber laser in the telecommunication band can serve as the pump. Thus, the whole system can potentially be fiber-to-chip integrated and packaged, enabling a compact, cost-effective, and low system-complexity platform. We numerically show that the broadband dispersive wave covering the wavelength range of 1.92~3.55 $\mathsf{\mu }$m (−20 dB level, near octave-spanning) with spectral flatness of 6.31 dB can be achieved using a 1550 nm, 190 pJ femtosecond pump seed. When the dual hump-shaped spectrum is obtained, the conversion efficiency of the mid-infrared dispersive wave can be up to 19.31%. The influence of the pumping conditions on the performance of mid-infrared dispersive wave generation was also studied. This work provides a competitive candidate for efficient, broadband, and flattened mid-infrared spectrum generation, which can find important applications in spectroscopy, metrology, and communication.

1. Introduction

Mid-infrared (MIR) supercontinuum generation (SCG) plays an essential role in molecular spectroscopy [1,2,3], medical diagnostics [4], infrared countermeasures [5], etc. When high-order dispersion is present, the dispersive wave (DW) on the longer wavelength side can be generated during the process of SCG if the phase-matching condition is satisfied [2]. The DW can maintain the coherence of the pump source and help extend the spectral range significantly into the MIR regime [6]. Due to the rich applications in the MIR regime, high-performance spectrum generation in this regime has attracted significant interest. To evaluate the performance of the MIR SCG and DW, the conversion efficiency, bandwidth, and flatness of the spectrum are the crucial metrics, especially the spectrum covering 3~5 $\mathsf{\mu }$m band, which is one of the atmospheric transparency windows [1]. High efficiency is an important research direction. Grassani et al. designed a large cross-section silicon nitride (Si₃N₄) channel waveguide and engineered the dispersion. By pumping the structure at around 2 $\mathsf{\mu }$m, conversion efficiency of the MIR DW up to 35% was obtained [7]. Similar conversion efficiency of 36.26% was achieved on the LNOI platform when pumping at around 2 $\mathsf{\mu }$m [8]. Broadband and flattened SCG and DW are preferred in practical applications because they enable parallel operation, improve the accuracy of the measurement, and reduce the degree of the difficulty in power equalization [9]. For example, in trace gas detection, a broadband and flattened supercontinuum and DW source without many cluttered dips can maintain a good spectral uniformity and distinguish the absorption spectrum of a detected gas molecular accurately, thus improving the robustness of the gas detection system. However, there is typically a tradeoff between a large bandwidth and high degree of flatness in conventional SCG. A promising approach for MIR SCG and DW generation is to use cheap and commercially matured mode-locked femtosecond fiber lasers at 1550 nm as the pump seed. For example, Guo et al. leveraged such a fiber-to-chip system to pump Si₃N₄ waveguides for MIR-DW generation [6]. The dispersion engineering of the Si₃N₄ platform relies on the thick films fabricated with the photonic Damascene process [10] to compensate for the material dispersion, which is a non-commercial and complicated technique. For a single Si₃N₄ waveguide structure, it is difficult to obtain a broadband and flattened DW at a longer wavelength phase-matched point. The conversion efficiency is limited to 1% [6]. In order to acquire a large bandwidth and a high flatness of the SCG and DW, Guo et al. proposed parallel dual-coupled Si₃N₄ waveguide with different widths [11]. Based on this structure, they produce a four zero-dispersion-wavelength (ZDW) dispersion profile. The flattened MIR-DW generation spanning from 2.7 to 3.7 $\mathsf{\mu }$m has been achieved, but the conversion efficiency is still limited to 1~5% [11] because of the primary-stage dispersion engineering. Although a four-ZDW dispersion profile can also be produced by a vertical stacked slot waveguide structure [12,13,14], it is difficult to grow or deposit the different materials layer by layer.

Besides the Si₃N₄ platform, lithium niobate (LN) is a competitive platform for wavelength conversion due to its strong ${\chi }^{\left(2\right)}$ and ${\chi }^{\left(3\right)}$ nonlinearities. For instance, SCG [15] and MIR DW generation [16] have been demonstrated in the bulk LN crystal. However, the LN crystal is bulky, thus presenting a limited dispersion engineering ability and high power consumption (sub-mJ level). Thin-film lithium-niobate-on-insulator (LNOI) is an emerging branch of LN, which presents a nano-scale size and high-index contrast compared with the bulk LN crystal. LNOI has been confirmed to be a promising candidate for nonlinear optics applications due to its appealing properties, such as strong ${\chi }^{\left(2\right)}$ (r₃₃ = 3 $\times$ 10⁻¹¹ m/V) and ${\chi }^{\left(3\right)}$ (n₂ = 1.8 $\times$ 10⁻¹⁹ m²/W) nonlinearities [17], high refractive index (n~2.21 @ 1550 nm) [18], ultralow loss (record @ 0.2 dB/m) [19], and wide transparency window (0.35~5 $\mathsf{\mu }$m) [18]. In addition to nonlinear effects, LNOI is also outstanding in the multi-physics field coupling and can realize a wide variety of piezo-optomechanical and thermo- and acoustic-optic devices [20]. The refractive index of LNOI is about 0.21 higher than that of Si₃N₄, which yields a stronger contribution of waveguide dispersion and better dispersion tailoring ability. Lithographic control of the waveguide geometry can precisely tailor the dispersion. Furthermore, compared with the Si₃N₄ platform, thick-film LNOI ranging from 600 to 900 nm is commercially available. In view of the aforementioned advantages of LNOI, tremendous advances have been demonstrated in this area in recent years, such as coherent and octave-spanning SCG [21,22], Kerr frequency comb generation [17,23,24], and an electro-optic micro-ring-based dual-comb interferometer [25]. An octave-spanning SCG on an LNOI waveguide was demonstrated by Yu et al. [21] and Lu et al. [22], which focused on the single-core waveguide and tailored the two-ZDW dispersion profiles for high performance SCG.

In this work, we leverage the four-ZDW dispersion profile produced by the anti-symmetric mode in the LNOI dual-coupled waveguide to obtain highly efficient and flat MIR DW-enhanced SCG. We designed the geometrical parameters of the dual-coupled waveguide to have a relatively flattened four-ZDW dispersion profile. We used the fourth-order Runge–Kutta method to solve the generalized nonlinear Schrödinger equation (GNLSE) to obtain the evolution in the frequency and time domain. The performances of the bandwidth, flatness, and conversion efficiency of the MIR-DW were numerically studied and are discussed. We also studied the influence on the MIR DW-enhanced SCG when the pumping conditions are changed.

2. Waveguide Design

We chose 0.85 $\mathsf{\mu }$m thick LN layer on an x-cut LNOI wafer to design the dual-coupled waveguide. Figure 1a shows the 3D schematic diagram and the cross-section (inset) of the dual-coupled ridge waveguide. The two parallel ridge waveguides are spaced with a certain gap. The whole structure is upper-clad with silicon dioxide (SiO₂) up to a 4 $\mathsf{\mu }$m height, which is illustrated in a translucent color. As illustrated in the inset, the c axis indicates the optical axis of the LN crystal which is aligned with the x-axis of the structure. The light propagates along the y-direction of the LN (coincident with the y-axis of the structure) resulting in an x-cut assembly. The top view of the structure is shown in Figure 1b. The top widths of the narrower and the wider waveguide are denoted by w₁ and w₂, respectively. The gap between the two parallel waveguides is defined as g, and the propagation length along the y-axis is defined as L. The height of the thin-film LN is fixed to H = 0.85 $\mathsf{\mu }$m. The etch depth and sidewall angle are denoted by d and $\theta$, respectively. When the two parallel ridge waveguides approach each other gradually, the interaction between the optical fields inside the two waveguides will affect the propagation constants of them due to the refractive index perturbation. Consequently, the group velocity dispersion (GVD) is changed when the two waveguides get close enough that the phase-matching condition of the two modes is satisfied and mode coupling occurs between them. According to supermode theory, mode coupling results in a pair of supermodes, namely, the symmetric and the anti-symmetric modes. The former always produces the normal GVD, while the latter always produces the anomalous GVD.

Mode coupling in this work is engineered for the MIR region. By leveraging the abrupt sign-changing of the dispersion that is produced by the anti-symmetric mode on the long wavelength side, the material dispersion of the waveguide is compensated for. In this way, a relative flattened dispersion profile with four ZDWs can be achieved, which is difficult for conventional single-core waveguides. Based on the above principle, Figure 2a illustrates the design of the dual-coupled waveguide. The geometries of the cross-sections of the two waveguides were selected to make the mode coupling occur in the MIR region by satisfying the phase-matching condition. The propagation constants of the two modes from each waveguide should be matched in the MIR region, namely, the effective refractive index should be the same, as ${\beta =n}_{\mathrm{eff}}\omega /c$, where $\mathsf{\omega }$ indicates the angular frequency of the light and c indicates the speed of light in vacuum. We simulate the effective refractive indices of the uncoupled and coupled modes by using the commercially available finite-element method (COMSOL, www.comsol.com, accessed on 6 June 2022). The Sellmeier equations of the material refractive index of LN and SiO₂ are chosen from [26,27], respectively. In Figure 2a, we can see the two uncoupled modes, and the effective indices of the TE₀₀ mode in the narrow waveguide and the TE₁₀ mode in the wide waveguide have an intersection at around 3.8 $\mathsf{\mu }$m for the top widths of w₁ = 0.9 $\mathsf{\mu }$m and w₂ = 2.8 $\mathsf{\mu }$m respectively with d = 0.8 $\mathsf{\mu }$m and $\theta$ = 80°. When the two waveguides approach each other and g = 0.3 $\mathsf{\mu }$m, the effective refractive indices and the mode profiles of the supermodes need to be re-calculated. It can be seen from Figure 2a that mode coupling occurs at the MIR region and there is no crossing point between the curves of the symmetric and anti-symmetric modes. By comparison, in the visible and near-infrared (NIR) region, the curves of the symmetric and anti-symmetric mode are degenerate to the uncoupled TE₁₀ mode in the wide waveguide and the uncoupled TE₀₀ mode in the narrow waveguide, respectively. The mode profiles of the symmetric and anti-symmetric modes at 1.55 $\mathsf{\mu }$m (pumping wavelength) and 3.6 $\mathsf{\mu }$m in the mode coupling region are also shown in Figure 2a. The symmetric mode evolves from the uncoupled TE₁₀ mode in the wide waveguide in the NIR, while the anti-symmetric mode evolves from the uncoupled TE₀₀ mode in the narrow waveguide in the NIR. Hereafter, in this study we focus on the anti-symmetric mode and its dispersion profile.

We studied the influence of etch depth, sidewall angle, and gap variation on the dispersion curve. Figure 2b–d show the calculated dispersion as a function of wavelength when d, $\theta$, and g vary, respectively. It can be seen in Figure 2b that the variation in the etch depth has a more significant influence on the ZDW at longer wavelengths than the one at shorter wavelengths. For the ZDW at shorter wavelengths, it redshifts when d increases from 600 to 650 nm, then it blueshifts as d increases further from 650 to 800 nm. For the ZDW at longer wavelengths, when d increases from 600 to 800 nm, it redshifts monotonously. Since the MIR-DW is generated in the normal dispersion region where the value of the dispersion is below zero, a larger value of the ZDW at longer wavelengths is preferred in order to generate a stronger MIR-DW at longer wavelengths. Only when d is equal or greater than 650 nm can the dispersion profile possess the normal dispersion in the NIR region; thus, a four-ZDW profile. As we can see from Figure 2c, the variation in the sidewall angle which can be tuned from 40° to 80° according to the advanced fabrication technology [20] has little influence on the ZDW at shorter wavelengths. When $\theta$ increases from 60° to 65°, the ZDW at shorter wavelengths blueshifts, then it redshifts to the initial location as $\theta$ further increases from 65° to 80°. By comparison, for the ZDW at longer wavelengths, it blueshifts as $\theta$ increases from 60° to 80°. When $\theta$ is equal to or larger than 70°, the dispersion profile is a four-ZDW shape; otherwise, it has two ZDWs only and keeps the anomalous dispersion between them. Since we pump the dual-coupled waveguide with a 1.55 $\mathsf{\mu }$m source, it is beneficial to locate the pump at the falling edge of the dispersion profile in the anomalous dispersion region. In this way, we can obtain both negative GVD and the third-order dispersion (TOD) parameters, which is preferred for MIR-DW enhancement. For the reasons described above, we fixed $\theta$ to be 80° in this study. Consequently, d = 0.8 $\mathsf{\mu }$m and $\theta$ = 80° were chosen to fulfill the desired dispersion profile of the dual-coupled waveguide. Figure 2d shows that the ZDWs at shorter wavelength are almost superimposed when g varies from 200 to 400 nm. For the ZDW at longer wavelengths, it blueshifts as g increases from 200 to 400 nm. As for the shape of the dispersion profile, when g is equal to 200 nm, it becomes a two-ZDW curve. In other cases, it is a four-ZDW profile. A smaller g does not yield a stronger change in the dispersion profile. Since the mode coupling can occur over a larger spectral bandwidth when g decreases, the variation in the phase constant of the anti-symmetric mode with frequency dependency is reduced, which results in a reduced depth and spectral range of the normal dispersion region. Finally, we chose g = 0.3 $\mathsf{\mu }$m for a relatively flattened dispersion profile and an appropriate mode coupling distance. To summarize, the geometric parameters of the coupled waveguides are chosen as H = 0.85 $\mathsf{\mu }$m, d = 0.8 $\mathsf{\mu }$m, $\theta$ = 80°, w₁ = 0.9 $\mathsf{\mu }$m, w₂ = 2.8 $\mathsf{\mu }$m, and g = 0.3 $\mathsf{\mu }$m.

To better understand the advantage of our structure, we plot the dispersion profiles of the anti-symmetric mode of Si₃N₄-coupled waveguides in [11], an uncoupled mode in a single-core waveguide on LNOI, and the symmetric and anti-symmetric modes of our structure together to have a clear comparison (Figure 2e). The width of the single-core waveguide on LNOI was selected to have a similar phase-matching wavelength as that of the anti-symmetric mode of our structure. As we can see in Figure 2e, the dispersion profiles of the uncoupled mode in a single-core waveguide on LNOI and the symmetric mode of our structure have a similar shape, which has two ZDWs with anomalous dispersion between them and normal dispersion on the longer wavelength side. By comparison, for the anti-symmetric mode of Si₃N₄ and LN dual-coupled waveguides, similar dispersion profiles with four ZDWs due to the mode coupling in the MIR region can be obtained. Because of the larger refractive index of LN, the ZDW at shorter wavelengths redshifts compared to that of the Si₃N₄ waveguide. When referring to the MIR region between 2 and 3.25 $\mathsf{\mu }$m, the dispersion curve yielded by our structure via LNOI waveguide has a more flattened profile than that of the Si₃N₄ waveguide, which is preferred to the broadband and flat MIR-DW generation. Figure 2f shows the TOD curves of the symmetric and anti-symmetric modes of the proposed dual-coupled waveguide. We can see that the TODs of the symmetric and the anti-symmetric modes have the opposite sign at the pumping wavelength of 1.55 $\mathsf{\mu }$m. The former is positive while the latter is negative. We prefer the latter because a negative TOD means a negative dispersion slope of GVD, which is helpful for enhancing the DW generation on the longer wavelength side (MIR).

When high-order dispersion is not negligible, the soliton pump will transfer energy to the DW, which is generated in the normal dispersion regime at the phase-matching wavelength, during the soliton-fission based SCG. The DW occurs when the phase-matching condition is satisfied [7]:

(1)$\beta (\omega )=\beta ({\omega }_{\mathrm{s}})+(\omega -{\omega }_{\mathrm{s}}){\nu }_{\mathrm{g}}^{-1}+\frac{\gamma P_0}{2}$

where $\beta \left(\omega \right)$ is the propagation constant of the anti-symmetric mode, ${\omega }_{\mathrm{s}}$ is the central angular frequency of soliton, $v_{\mathrm{g}}$ is the group velocity of soliton, $\gamma$ is the nonlinear coefficient, and $P_0$ is the peak power of the pump. According to Equation (1), the integrated dispersion ${\beta }_{\mathrm{int}}$ (or phase mismatching parameter) is given by:

(2)$\begin{array}{l}{\beta }_{\mathrm{int}}=\beta (\omega )-\beta ({\omega }_{\mathrm{s}})-(\omega -{\omega }_{\mathrm{s}}){\nu }_{\mathrm{g}}^{-1}-\frac{\gamma P_0}{2}\\ \approx {\displaystyle \sum _{n\ge 2}\frac{{(\omega -{\omega }_s)}^n}{n!}\frac{{\partial }^n\beta ({\omega }_{\mathrm{s}})}{\partial {\omega }^n}}\end{array}$

Since ${\gamma P}_0$ is relatively small compared to the former Taylor expansion, it is usually neglected. Figure 2g shows the calculated ${\beta }_{\mathrm{int}}$ as a function of the wavelength at a 1.55 $\mathsf{\mu }$m pump wavelength. The anti-symmetric mode of Si₃N₄-coupled waveguides in [11], an uncoupled mode in a single-core waveguide on LNOI, and the anti-symmetric mode of the LNOI dual-coupled waveguide are shown together as a comparison. When compared with the single-core waveguide on LN, our structure has a lower MIR dispersion barrier and a higher visible one, which is beneficial for transferring power to the MIR-DW. By comparison, for the anti-symmetric mode of Si₃N₄-coupled waveguides, it features a shorter phase-matching wavelength range at the MIR region compared with our structure. We find that there is an upward trend towards zero in the ${\beta }_{\mathrm{int}}$ curve of our structure around 2.5 $\mathsf{\mu }$m, which will produce another MIR DW, as mentioned later. However, the upward trend towards zero is between 2.5 and 3 $\mathsf{\mu }$m in the ${\beta }_{\mathrm{int}}$ curve of the anti-symmetric mode of Si₃N₄-coupled waveguides. This part of the ${\beta }_{\mathrm{int}}$ curve is closer to the MIR phase-matching wavelength point compared with our structure. That results in a limited bandwidth and flatness of the MIR-DW generation.

3. Results and Discussion

The SCG in the proposed dual-coupled LNOI waveguide is governed by the following GNLSE [28]:

(3)$\begin{array}{cc}\hfill \frac{\partial \tilde{A}}{\partial \mathrm{z}}& =i[\beta (\omega )-\beta ({\omega }_0)-{\beta }_1({\omega }_0)(\omega -{\omega }_0)]\tilde{A}(\mathrm{z},\omega )-\frac{\alpha (\omega )}{2}\tilde{A}(\mathrm{z},\omega )\hfill \\ & +i\gamma (\omega )(1+\frac{\omega -{\omega }_0}{{\omega }_0})\mathcal{F}\left\{{\left|A(\mathrm{z},t)\right|}^{2}A(\mathrm{z},t)\right\}\hfill \end{array}$

(4)$\tilde{A}(\mathrm{z},\omega )=\mathcal{F}\{A(\mathrm{z},t)\}={(\frac{A_{\mathrm{eff}}(\omega )}{A_{\mathrm{eff}}({\omega }_0)})}^{-1/4}\tilde{E}(\mathrm{z},\omega )$

(5)$\gamma (\omega )=\frac{n_2{n}_0{\omega }_0}{c{n}_{\mathrm{eff}}(\omega )\sqrt{A_{\mathrm{eff}}(\omega )A_{\mathrm{eff}}({\omega }_0)}}$

(6)$A_{\mathrm{eff}}(\omega )=\frac{(\int {\int }_{-\infty }^{+\infty }{\left|E(x,\mathrm{z},\omega )\right|}^{2}dxd\mathrm{z}{)}^2}{\int {\int }_{-\infty }^{+\infty }{\left|E(x,\mathrm{z},\omega )\right|}^{4}dxd\mathrm{z}}$

where $\tilde{A}(\mathrm{z}, \omega )$ is the Fourier transform of $A(\mathrm{z}, t)$ given by Equation (4), $\beta$ is the propagation constant of the selected mode, ${\beta }_1$ is the first order dispersion parameter, ${\omega }_0$ is the angular frequency of the pump, $\alpha$ is the linear propagation loss of the nonlinear waveguide, and γ is the nonlinear coefficient. $A_{\mathrm{eff}}$ is the effective mode area given by Equation (6), $\tilde{E}(\mathrm{z}, \omega )$ is the Fourier transform of the electric field. $n_2$ is the nonlinear refractive index, $n_0$ is the linear refractive index, c is the speed of light in the vacuum, and $n_{\mathrm{eff}}$ is the effective index of the selected mode. E (x, z, ω) is the frequency-dependent intensity of electric field distribution. We safely neglected the stimulated Raman scattering in the model for two reasons. First, the contribution of the Raman effect typically becomes significant after the soliton fission length [29]. The propagation lengths we considered to evaluate the performance of SCG and MIR-DW generation are just around the soliton fission length, where the Raman contribution is still weak. Second, the Raman effect was neglected in previously reported experimental results that aimed at SCG and Kerr frequency comb generation in LNOI platforms [21,22,23] because its contribution is weak. The ${\chi }^{\left(2\right)}$-based second harmonic generation (SHG) is also neglected in this study because it makes a substantial contribution to the frequency generation on the shorter wavelength side, not the MIR region. The dispersion parameters considered in our study are up to the 11-th orders in order to fulfill the dispersion characteristics within a broadband spectrum. Here, we chose the anti-symmetric mode of our structure as the selected mode. Because of the weak mode coupling between the anti-symmetric mode and other high-order modes [11], we can assume that there is no mode coupling. We simulate Equation (3) with the fourth-order Runge–Kutta method [30].

We chose the femtosecond pulse centered at 1.55 μm as the pump source. The pulse envelope is a hyperbolic secant shape as:

(7)$A\left(\mathrm{z}=0,\tau \right)=\sqrt{P_0}\mathrm{sech}\left(\frac{\tau }{{\tau }_0}\right)+{\sigma }_{\mathrm{Quantum}}$

where τ₀ = τ_FWHM/2ln(1 + 2^1/2) ≈ τ_FWHM/1.7632 is the pulse width, τ_FWHM is the full-width at half maxima, and σ_Quantum is the quantum noise.

The full-width at half maxima of the pump τ_FWHM is 75 fs and the peak power P₀ is 4.5 kW, corresponding to a pulse energy of 190 pJ. Assuming a commercially available repetition rate of 100 MHz for the mode-locked fiber laser at 1.55 $\mathsf{\mu }$m, the average power consumption of this work is only 19 mW. The length of the dual-coupled waveguide is 1.2 mm, and the propagation loss was chosen to be 3 dB/m [21]. As shown in Figure 3a,d, the spectral and temporal evolution profiles are presented along the propagation length. We note that the evolution dynamic after the soliton fission length is not accurate because the stimulated Raman scattering is neglected in this work. For the case that the evolution dynamic after the soliton fission length is focused, the stimulated Raman scattering effect of LN [15,31] should be considered for more accurate results. We can see from Figure 3a that the spectrum initially broadens symmetrically around the pump wavelength in the frequency domain due to the self-phase modulation (SPM). Accordingly, in the time domain, the soliton goes through a temporal compression process as shown in Figure 3d. When the propagation length reaches the soliton fission length, i.e., l_fission = l_D/N_soliton, around 0.83 mm, the spectrum starts to explode into a fairly wide range. Here, N_soliton = (l_D/l_NL)^1/2 is the soliton number, and l_D = τ₀²/|β₂| and l_NL = 1/γP₀ are the dispersion and nonlinear lengths, respectively. Almost at the same time of soliton fission, the MIR-DW at around 2.5 μm and NIR-DW at around 0.68 μm are generated. As a result of the upward trend towards zero in the ${\beta }_{\mathrm{int}}$ curve around 2.5 $\mathsf{\mu }$m, the MIR-DW at this band is obtained. Correspondingly, in the time domain, Figure 3d shows the fundamental sub-solitons and the DW pulses are observed temporally emitted from the compressed high order soliton due to their different group velocity. As both the MIR-DW and NIR-DW are generated in the normal dispersion region, the pulse of MIR-DW on the longer wavelength side exhibits acceleration relative to the sub-solitons, while the pulse of NIR-DW at shorter wavelength side shows a delay. When the propagation length reaches around 0.9 mm, the power of the MIR-DW increases and the spectrum is flattened. Figure 3b shows the spectrum at L = 0.9 mm extracted from the spectral evolution profile in Figure 3a. We find that the spectrum is very smooth with a high spectral flatness of 6.31 dB achieved in the wavelength range of 1.92~3.55 $\mathsf{\mu }$m, calculated by Equation (8):

(8)$F=\max ({\log }_{10}(spectrum({\lambda }_1))-{\log }_{10}(spectrum({\lambda }_2)))$

where F is the spectral flatness, spectrum is the normalized output spectrum, and ${\lambda }_1$ and ${\lambda }_2$ are the wavelengths in a certain wavelength region. The conversion efficiency of the MIR-DW is as high as 18.48%. This is calculated by the ratio between the energy of the MIR-DW and the soliton pump, which corresponds to on-chip energy in experiments. However, the intensity of the MIR-DW at the phase-matching point of 3.63 μm is weak due to the insufficient nonlinear accumulation at this short propagation length. When L = 0.95 mm, there are dual hump-shaped MIR-DWs in the output spectrum in Figure 3c. One is centered at around 2.46 $\mathsf{\mu }$m, and the other is located at about 3.65 $\mathsf{\mu }$m, which corresponds to the normal dispersion region in Figure 2e. Since the intensity of the MIR-DW at around 3.65 $\mathsf{\mu }$m is accumulated to a relatively high level, these dual hump-shaped MIR-DWs are obtained at this propagation length. The conversion efficiency of the MIR-DW is calculated as 19.31%. Figure 3e,f show that the normalized temporal pulse envelopes at the dual-coupled waveguide output exhibit the soliton fission effect which splits into two components.

We further studied the influence on the MIR-DW generation when the pump condition changes, i.e., the pulse width τ₀ and the coupled pulse energy. We can see the spectral and temporal evolution with the propagation length in Figure 4a,d when the full-width at half maxima τ_FWHM is reduced to 50 fs. As the peak power P₀ is kept at 4.5 kW, the pulse energy is reduced to around 130 pJ. Figure 4a shows the similar soliton evolution process as mentioned above. However, the soliton fission length is different from the one with τ_FWHM = 75 fs due to the reduced pulse width. The dual hump shape is more apparent for MIR-DWs generated in the two normal dispersion regions. In the time domain, because of the lower N_soliton, the compressed high order soliton ejects fewer fundamental sub-solitons compared to the case of τ_FWHM = 75 fs. The DW pulses follow the same rules as previously mentioned with respect to the different group velocity. Here, we choose the propagation length of 0.8 mm and 0.86 mm to study the instantaneous spectral and temporal profiles of the MIR-DW generation. As we can see from Figure 4b, there are dual hump-shaped MIR-DWs centered at 2.45 and 3.6 $\mathsf{\mu }$m, respectively. The flatness of the SCG in the MIR region is reduced, of which the variation is 7.44 dB from 1.92 to 3.55 $\mathsf{\mu }$m. In regard to the bandwidth (at −10 dB level) of the MIR-DW, it is about 881 nm of the MIR-DW between 2 and 3 $\mathsf{\mu }$m. Moreover, the conversion efficiency of the MIR-DW is 17.50%. As for the output spectrum with the propagation length of 0.86 mm in Figure 4c, the phenomenon of the dual hump-shaped MIR-DWs generation is more obvious. The locations of the dual hump-shaped MIR-DWs are around 2.5 and 3.6 $\mathsf{\mu }$m. In addition, the conversion efficiency of the MIR-DW reaches as high as 22.53%. The −10 dB level of bandwidth is around 454 nm of the MIR-DW located between 2 and 3 $\mathsf{\mu }$m. Figure 4e,f show that the normalized temporal pulse envelopes at the dual-coupled waveguide output also present the soliton fission effect that splits into two components. For the above situations, the MIR-DW is much stronger than the NIR one because the dispersion characteristics at 1.55 $\mathsf{\mu }$m pump wavelength satisfy β₂ < 0 and β₃ < 0. Therefore, the pump energy is more inclined to be transferred to the DW at longer wavelength side.

In order to show the advantages of the proposed structure, we compare the performance of SCG and DW generation between the single-core, x-cut LNOI waveguide (dispersion in Figure 2e) and the proposed dual-coupled LNOI waveguide. Instantaneous spectra at two different propagation lengths are considered with the same pumping conditions, as shown in Figure 5. It can be seen from Figure 5a that when the propagation length is 0.9 mm, the instantaneous spectrum of the dual-coupled waveguide is apparently broadened with the relatively high degree of flatness. In contrast, for the case of the single-core waveguide, the soliton fission length has not been reached. The spectrum broadening determined by the SPM effect is still weak. We further chose a longer propagation length of 1.8 mm to compare the performance of the two structures. Figure 5b shows that there are intense dual hump-shaped MIR-DWs generated at 2~3 $\mathsf{\mu }$m and 3~4.5 $\mathsf{\mu }$m, respectively, for the dual-coupled waveguide. As for the single-core waveguide, the output spectrum has only one MIR-DW located at 3~4.5 $\mathsf{\mu }$m. The intensity of the MIR-DW is much lower than that obtained in the dual-coupled waveguide. Even if we only focus on the MIR-DW generated at 3~4.5 $\mathsf{\mu }$m, the conversion efficiency is about 1.48% for the single-core waveguide and 9.20% for the dual-coupled waveguide.

Table 1 shows the comparison of the spectral bandwidth, flatness, and conversion efficiency with some previous works. We note that the listed LNOI single-core waveguide is z-cut [22], and not the same type used in Figure 2e. From Table 1, we can see that the bandwidth of the MIR-DW (at −20 dB level) of the proposed dual-coupled LNOI waveguide is about 1630 nm, which is the largest one. Meanwhile, our dual-coupled waveguide shows the highest degree of flatness within the largest spectral spanning in the mid-infrared, benefited from the well-tailored dispersion profile. Consequently, the conversion efficiency of our work shows significant improvement. We note that the comparison is somewhat unfair because our performances are determined by the numerical results rather than an experiment. However, considering the significantly improved level, the proposed waveguide has reasonable potential to yield a high-performance MIR-DW in the form of practically fabricated devices.

We further studied the influence of coupled pulse energy variation on MIR-DW generation, especially the conversion efficiency and the flatness. As the conversion efficiency of the MIR-DW is as high as 19.31% with τ_FWHM = 75 fs, P₀ = 4.5 kW and L = 0.95 mm, we just changed the peak power of the pump to study this effect, while the rest of the pump conditions and the propagation length remained constant. Figure 6a shows the output spectra with different coupled pulse energy ranging from 170 to 340 pJ, corresponding to the peak power of the pump varying from 4 to 8 kW. Figure 6b shows the corresponding conversion efficiency of the MIR-DW. When the coupled pulse energy is low, for example, 170 pJ, the power of the MIR-DW in the output spectrum is relatively low. This results in a low conversion efficiency of 10.79%. As the coupled pulse energy increases, we find a clear structure of dual hump-shaped MIR-DWs in the output spectra. Although there are many oscillating structures in the conversion efficiency curve, it is almost always above 10% and has a peak value of 19.31% at 190 pJ. The reason is that the pump energy transferred to the MIR-DW between 2 and 3 $\mathsf{\mu }$m is high at first, then it begins to go down, whereas the energy of the MIR-DW at around 3.65 $\mathsf{\mu }$m is continuously increasing. Accompanied by this phenomenon, the generated MIR-DWs can act as a sub-pump in the normal dispersion region when its energy reaches the threshold. Then, the MIR-DWs start to shed energy to other frequencies through nonlinear coupling, such as SPM and cross-phase modulation (XPM) effects. When the pump energy is further increased, the spectral components in the anomalous dispersion region refresh the energy of MIR-DWs through both DW generation and four-wave mixing (FWM) effects. Consequently, the conversion efficiency is lifted again. This reciprocating process is consistent with the experimental results in [7]. As for the flatness of these spectra in Figure 6c, it has a similar oscillating structure. When the coupled pulse energy is 190 pJ, the flatness of this spectrum is as high as 6.31 dB ranging from 1.92 to 3.55 $\mathsf{\mu }$m, whereas for the coupled pulse energy below 190 pJ, the flatness is low due to the relative low power of the MIR-DW on the longer wavelength side. When the coupled pulse energy increases to 210 pJ, the dual hump-shaped MIR-DWs generation is more obvious, so the flatness decreases. As the coupled pulse energy increases further, the part of the spectra between the dual hump-shaped MIR-DWs is lifted; thus, the spectra is flattened. When the coupled pulse energy reaches 340 pJ, the value of the flatness becomes very large because of the unbalanced power distribution between the dual hump-shaped MIR-DWs.

4. Conclusions

In summary, we designed and dispersion-engineered a structure of the dual-coupled waveguide on x-cut LNOI to achieve highly efficient and flat MIR-DW generation. We studied the mode characteristics and the dispersion engineering of the supermodes of this dual-coupled waveguide. By varying the geometric parameters of the proposed waveguide, i.e., the etch depth, the sidewall angle, and the gap between the dual-coupled waveguide, the influences on the dispersion profiles were revealed. As the dispersion profile of our structure is relatively flat, it effectively generates the MIR-DW having high efficiency and a high degree of flatness. A high spectral flatness of 6.31 dB is achieved in the wavelength range of 1.92~3.55 $\mathsf{\mu }$m by reasonable selection of the propagation length. The conversion efficiency of the MIR-DW is as high as 19.31% when the propagation length is longer. This conversion efficiency is relatively high and has the potential to achieve the MIR-DW of the highest conversion efficiency with pumping of 1.55 $\mathsf{\mu }$m sources in the near-infrared region. We note that the SHG effect, which exists in the LNOI platform, was not considered in the study. The frequency conversion by the SHG effect does grab energy from the pump source. Consequently, the conversion efficiency of MIR-DW generation is affected and reduced. However, since the conversion efficiency of the SHG effect is typically low, it will not significantly degrade the conversion efficiency of the MIR-DW. We further studied the influence of the pulse width and coupled pulse energy variations on the MIR-DW generation. We found that the dual hump-shaped MIR-DW generation is more obvious when the pulse width becomes shorter and the conversion efficiency of the MIR-DW is higher if we choose an appropriate propagation length. Although the values of the conversion efficiency oscillate when the coupled pulse energy is varied, it is almost always above 10% and has a peak value of 19.31% at 190 pJ. Moreover, we can obtain the spectral flatness as high as 6.31 dB when the coupled pulse energy is 190 pJ. Our results confirm that thin-film LNOI is a promising platform for efficient, broadband, and flat MIR-DW generation. In principle, mode coupling can be engineered at arbitrary wavelength regions on arbitrary material platforms. The results presented in this study can benefit a wide variety of technologies and applications.

Author Contributions

Conceptualization, J.J. and Z.K.; methodology, J.J. and Z.K.; software, J.J.; validation, Z.K.; writing—original draft preparation, J.J.; writing—review and editing, Z.K., Q.H. and S.H.; visualization, J.J.; supervision, Z.K. and S.H. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

We thank Julian S. G. Evans for the helpful discussion and comments on this manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Table

Figure 1. (a) The 3D schematic diagram of the dual-coupled waveguide on x-cut LNOI. Inset: the cross-section view of the dual-coupled waveguide. The c-axis is the optic axis of the thin film LN and is parallel to the x-axis. (b) The 2D top view of the dual-coupled waveguide.

Figure 2. (a) The calculated effective refractive indices of the symmetric mode and the anti-symmetric mode of the dual-coupled waveguide, compared with the initial uncoupled modes separated in each core, which are the fundamental TE00 mode in the narrow waveguide and the TE10 mode in the wide waveguide. The mode coupling occurs at around 3.8 [Forumla omitted. See PDF.]m. Insets: the normalized electric field distribution of the supermodes at 1.55 and 3.6 [Forumla omitted. See PDF.]m respectively. The dispersion curves of the anti-symmetric mode in the dual-coupled waveguide as a function of the wavelength for (b) varied d, fixed [Forumla omitted. See PDF.] = 80°, and g = 300 nm, (c) varied [Forumla omitted. See PDF.], fixed d = 800 nm, and g = 300 nm, (d) varied g, fixed d = 800 nm, and [Forumla omitted. See PDF.] = 80°. (e) The dispersion curves of the anti-symmetric mode in Si3N4-coupled waveguides, the uncoupled mode of a selected single-core waveguide with H = 0.85 [Forumla omitted. See PDF.]m, d = 0.8 [Forumla omitted. See PDF.]m, [Forumla omitted. See PDF.] = 80°, w = 1.15 [Forumla omitted. See PDF.]m, and the symmetric and anti-symmetric modes in LN dual-coupled waveguide. (f) The TOD profiles of the symmetric and anti-symmetric modes of the dual-coupled LNOI waveguide. (g) The calculated integrated dispersion of the anti-symmetric mode in Si3N4-coupled waveguides, the uncoupled mode of a selected single-core waveguide and the anti-symmetric mode in the LN dual-coupled waveguide. The latter two show a similar phase-matching wavelength at around 3.6 [Forumla omitted. See PDF.]m.

Figure 3. (a) Spectral evolution along the propagation length with the full-width at half maxima τFWHM = 75 fs. (b) The spectra of the pump and the output at L = 0.9 mm. (c) The spectra of the pump and the output at L = 0.95 mm. (d) Temporal evolution along the propagation length with the full-width at half maxima τFWHM = 75 fs. (e) The normalized temporal pulse envelopes at input and output at L = 0.9 mm. (f) The normalized temporal pulse envelopes at input and output at L = 0.95 mm.

Figure 4. (a) Spectral evolution along the propagation length with the full-width at half maxima τFWHM = 50 fs. (b) The spectra of the pump and the output at L = 0.8 mm. (c) The spectra of the pump and the output at L = 0.86 mm. (d) Temporal evolution along the propagation length with the full-width at half maxima τFWHM = 50 fs. (e) The normalized temporal pulse envelopes at input and output at L = 0.8 mm. (f) The normalized temporal pulse envelopes at input and output at L = 0.86 mm.

Figure 5. (a) The instantaneous spectral profiles of the initial pump, and the single-core and the dual-coupled waveguides of x-cut LNOI at L = 0.9 mm. The full-width at half maxima of the pump is fixed to τFWHM = 75 fs. (b) The instantaneous spectral profiles of the initial pump, and the single-core and the dual-coupled waveguides of x-cut LNOI at L = 1.8 mm. The full-width at half maxima of the pump is fixed to τFWHM = 75 fs.

Figure 6. (a) The 3D waterfall of the output spectra at L = 0.95 mm with the full-width at half maxima τFWHM = 75 fs at different coupled pulse energies. (b) The conversion efficiency as a function of the coupled pulse energy. (c) The flatness as a function of the coupled pulse energy.

Table 1

The comparison of the spectral bandwidth, flatness, and conversion efficiency with some previous works.

Structure	MIR-DW Bandwidth (at −20 dB Level)	Flatness	Conversion Efficiency	Reference
Si3N4 single-core waveguide	0 nm	~15 dB(2.9–3.7 μm)	<1%	[6]
Si3N4 dual-coupled waveguide	~928 nm	~19 dB(2.56–3.65 μm)	~1–5%	[11]
LNOI single-core waveguide	N/A	~15 dB(1.84–2.17 μm)	N/A	[22]
LNOI dual-coupled waveguide	~1630 nm	~6.31 dB(1.92–3.55 μm)	18.48%	This work

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