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Abstract
On-chip supercontinuum generators have emerged as an attractive optical source with small size, broad spectrum and high power efficiency. Nevertheless, there has long been a trade-off between spectral uniformity and bandwidth. We propose a novel silicon nitride waveguide with flat saddle-shaped all-normal dispersion, particularly for enhancing the nonlinear interactions over a wide band. By launching a 250-fs 30-kW input pulse, an ultra-flat (-6 dB) octave-spanning supercontinuum extending from 638 nm to 1477 nm can be generated. We analyze the performance of the supercontinuum generator in terms of spectral flatness and bandwidth under different input pulse conditions. Thanks to mature dispersion engineering, the pump wavelength can be flexibly selected within the flat dispersion region. The generated supercontinuum, therefore, can be applied to different spectral regions by shifting the center wavelength.
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1. Introduction
Over the past few decades, supercontinuum (SC) generation has been an active research field due to its potential applications in high-precision frequency metrology [1,2], optical coherence tomography [3–5], and spectroscopy [6]. Usually, supercontinua are generated in the anomalous dispersion regime, in which spectral broadening is caused by high-order soliton fission and dispersive wave generation. Although usually producing a SC with a large spectral bandwidth, soliton fission leads to complex temporal profiles and fine spectral structures, often associated with high sensitivity to phase noise and thus SC coherence degradation. More importantly, SC generation based on soliton fission may suffer from a poor spectral flatness, which significantly limits its applications in practice [7]. In contrast, a nonlinear fiber with all-normal dispersion (ANDi) can be used to obtain a SC source with good spectral coherence and flatness [8–13]. SC generation in the ANDi regime is typically produced over a smaller bandwidth, compared to that in the anomalous dispersion regime. However, by generating the ANDi SC, the spectral broadening is mainly dominated by self-phase modulation (SPM), which is much less sensitive to the input pulse noise. This mechanism usually helps achieve excellent spectral coherence in SC generation. Nevertheless, octave-spanning and flat supercontinua with good coherence have been difficult, requiring more sophisticated dispersion engineering in all-normal dispersion fibers, such as silica fibers [8], photonic crystal fibers (PCFs) [9–11], chalcogenide fibers [12,13], and so on. The success of fiber-based ANDi SC generation, especially based on PCFs, is partially attributed to mature dispersion engineering enabled by not only manipulation of the geometrical parameters but also matching the composition of glasses [14].
On-chip SC generation has already been proposed in various material platforms, including group IV materials [15–25], chalcogenide [26,27], AlN [28,29], LiNbO3 [30–32], and AlGaAs [33]. Silicon-based CMOS technologies may provide more advantages for on-chip SC generation due to mature fabrication and low cost. Particularly, silicon nitride (Si3N4) represents an attractive solution because of its broadband transparency window, relatively high Kerr nonlinearity, and negligible two-photon absorption [34].
Moreover, SC generation dramatically relies on dispersion characteristics. To obtain a flat, broadband, and highly coherent SC, dispersion engineering is critically important and is aimed at a low and flat all-normal dispersion profile. However, due to insufficient capacity to tailor dispersion, the reported dispersion profiles are usually parabolic, making it hard to obtain an ultra-flat, octave-spanning SC source on a silicon chip. Table 1 outlines recent results in SC generation, and one can see from it that the bandwidth and flatness of an on-chip SC source are typically in a trade-off. Although Ref. [21] shows both wide bandwidth and flat spectrum, the SC’s spectral coherence is not good. This particularly calls for a solution to flat, broadband, and coherent SC generation for practical applications.
Table 1. SC Generation in Silicon-Based Waveguidesa
Here, we propose a Si3N4 waveguide with a low and flat all-normal dispersion profile. It is shown that an ultra-flat SC extending from 638 nm to 1477 nm (∼1.21 octave) at the -6 dB level can be generated using a femtosecond pump pulse. We analyze the influence of waveguide dimensions on dispersion properties, showing that the sensitivity of dispersion to the dimension variations is relatively small. Moreover, we study the performance of the SC generation under different operating conditions, including varied pump wavelength, input pulse width, and peak power. The results show that, benefiting from the low and flat all-normal dispersion profile, the spectral flatness and bandwidth can be maintained as the input shifting from 1000 nm to 2600 nm, which means that the dispersion-flattened waveguides can be used for SC generation under quite flexible conditions.
2. Device design and modeling
The proposed Si3N4 waveguide is shown in Fig. 1, which is partially etched to have a slab to support two same-sized Si3N4 strips. We choose low-stress Si3N4 as the material, which allows us to obtain the waveguide thickness. Due primarily to the high material absorption of the buried silica layer (> 10 dB/cm at around 2700 nm, from Ref. [35]), it is challenging to use Si3N4-on-insulator in the mid-infrared region [35,36], and thus we design a suspended structure, which can be realized by selectively removing the buried SiO2 layer [37]. The waveguide parameters are: Si3N4 stripe width W = 550 nm, slot width G = 200 nm, Si3N4 stripe height H1 = 1350 nm, Si3N4 slab thickness H2 = 400 nm, and sidewall angle α = 85° [18,38–40]. We use the quasi-TM mode.
A suggested fabrication process for the suspended waveguide is given as follows. First, using low-pressure chemical vapor deposition (LPCVD), silicon nitride is deposited on top of a silicon dioxide layer on a silicon wafer. Then, the waveguide pattern is created in the photoresist (ZEP520A) using electron beam lithography (EBL). Following the development of the resist, a chromium film is evaporated and lifted off to form a hard mask. The pattern is then transferred to the silicon nitride layer by reactive ion etching (RIE). The chromium hard mask is then removed using chrome etching [41]. After that, holes far away from the waveguide are made through EBL and RIE, which can be used to allow hydrofluoric (HF) acid to reach the silica layer [42]. Then, the air-clad waveguide is dipped into a diluted HF solution. Due to the high etching selectivity of HF between silicon oxide and silicon nitride (200:1), the silicon oxide beneath the waveguide can be locally removed [43].
A flat saddle-shaped dispersion profile is obtained in Fig. 2(a). We modify the waveguide to obtain an all-normal dispersion profile, with the dispersion coefficient β2 lower than 0.22 ps2/m over a 2160-nm-wide range from 600 nm to 2760 nm, which results from a novel dispersion flattening approach [44]. According to the study in [44], using two materials with a relatively low index contrast to form a bilayer structure, we reported recently how to achieve flattened dispersion. Here, we design an equivalent bilayer waveguide using a single material, as shown in Fig. 1, to obtain a flattened dispersion profile, in which the high-index layer is a slab, and the low-index layer consists of an air slot and two high-index strips, which has an average index that is needed in [44]. In this waveguide, the index contrast will not be affected due to the use of a single material. It means that we can arbitrarily modify the refractive index of the equivalent upper layer by adding an air slot and ultimately realize an index contrast that we need. As shown in Ref. [44], the confinement factor of the equivalent bilayer waveguide changes almost linearly with wavelength over a wide bandwidth, which means that the effective refractive index also varies linearly with wavelength. In this way, we obtain a saddle-shaped all-normal dispersion profile, which can keep flat over a 2160-nm bandwidth. The optical field distributions at different wavelengths across the flat-dispersion region are shown in Fig. 2(c). Moreover, we note that the next higher-order mode supported by this waveguide is an anti-symmetric mode, which is hard to be excited, and the next higher-order symmetric mode does not exist. Thus, we can ignore the dispersion variation caused by coupling from the higher-order mode to the fundamental mode. In addition, we calculate the overlap factors between the mode located at the pump wavelength used in this work (960 nm) and other fundamental modes at the spectral broadening region. As shown in Fig. 2(d), the normalized optical field overlaps are higher than 0.95, which means that the influence of modal field mismatch is negligible. Figure 3 shows the dispersion profiles of the fundamental TM mode with different waveguide dimensions. We change the W, H1, H2, and G within a range of ±2% around the values given above. The results indicate that the sensitivity of dispersion to the dimension variations is relatively small, which means that the requirements for precision in fabrication can be reduced.
Fig. 2. (a) Flattened all-normal dispersion for SC generation in the proposed Si3N4 waveguide. (b) The nonlinear coefficient γ of the Si3N4 waveguide decreases quickly with wavelength. (c) The mode profiles for different wavelengths across the flat-dispersion region. (d) Normalized optical field overlaps of a fixed mode located at the pump wavelength with other modes at the spectral broadening region.
Fig. 3. Dispersion profiles of the fundamental TM mode with different structural parameters. We change the (a) W, (b) H1, (c) H2, and (d) G within a range of ±2% around the optimum values. We also change the (e) α within a range of ±2° around the optimum values.
We also investigate the influence of the waveguide sidewall angle on dispersion properties. As shown in Fig. 3(e), the variation of the sidewall angle can result in changes in the dispersion profile. However, the dispersion profile can be recovered by tailoring other waveguide parameters in a new design iteration, once the sidewall angle is found to be different from its designed value after fabrication. Figure 2(b) shows the nonlinear coefficient, γ, of the waveguide, which decreases with wavelength rapidly.
We use the generalized nonlinear Schrödinger equation (GNLSE) to investigate SC generation [36], taking the factors of wavelength-dependent loss, all-order dispersion, SPM, Raman scattering, and self-steepening into account. The GNLSE can be given as follows:
In (1), $A = A (z, t)$ is the complex amplitude of the pump pulse and α is the propagation loss. βm represents the mth-order dispersion coefficient, which is associated with the Taylor expansion of the propagation constant around the center frequency. All-order dispersion is taken into consideration. γn is the nth-order nonlinear coefficient, which is defined as $\gamma_n=\omega_0 \partial^n[\gamma(\omega) / \omega] / \partial \omega^n$, where ω0 is the angular frequency of the carrier. The shock time τshockR for Raman nonlinearity is calculated with consideration on wavelength-dependent effective mode area. hR(t) represents the Raman response function.
A chirp-free hyperbolic secant pulse centered at 960-nm wavelength, with a full width at half-maximum (FWHM) of 250 fs and a peak power of 30 kW, is launched into a 4 cm-long Si3N4 waveguide. The pump used in this work can be obtained from commercially available solid-state femtosecond light sources, from such as Light Conversion, Coherent, etc. We set the time window length to be 50 ps (i.e., frequency resolution Δf = 20 GHz), and the whole bandwidth in the frequency domain to be 3000 THz (i.e., time resolution Δt = 0.33 fs). The nonlinear refractive index n2 of silicon nitride is 3 × 10−19 m2/W [45]. Moreover, the nonlinear coefficient γ and the dispersion coefficient β2 at the pump wavelength are 1.64 /m/W and 0.09 ps2/m, respectively. The Raman shock time is 1.05 fs. More Raman scattering properties are given in Ref. [46]. The propagation loss is set to be 0.5 dB/cm, and since Si3N4 exhibits negligible two-photon absorption for wavelength beyond 500 nm, the nonlinear loss can be ignored.
3. Results and discussion
Figure 4(a) shows the spectral evolution of the input optical pulse along the dispersion-flattened nonlinear waveguide. As a result of SPM, the pulse initially experiences nearly symmetric spectral broadening, which becomes much more significant then, with blue-shifted and red-shifted spectral components due to optical wave breaking (OWB) at near 2 mm and 8 mm, respectively. After around 3 cm, the spectrum does not change much, except becoming smoother by transporting energy evenly from the pump wavelength to the whole spectral range. The temporal evolution of the pump pulse along the waveguide is shown in Fig. 4(b). As shown in Fig. 4, for a longer propagation distance, there is a slight improvement in spectral flatness. However, due to the accumulated all-normal dispersion and propagation loss, the pulse gradually increases in width and decreases in power, so it is necessary to determine an appropriate waveguide length.
Fig. 4. (a) Spectral and (b) temporal evolution along the Si3N4 waveguide. (c) Generated spectra at different propagation distances.
Figure 4(c) shows the pulse spectra at different propagation distances. We note that the spectrum becomes significantly broadened and extends from 638 nm to 1477 nm at 4 cm, covering 1.21 octave at the -6 dB level. The coherence of the generated SC is examined by calculating the modulus $|{{\textrm{g}_{\textrm{12}}}^{\textrm{(1)}}} |$ of the complex degree of first-order coherence from 100 simulations with random quantum noise [7]. At a propagation distance of 4 cm, the generated SC is perfectly coherent over the entire bandwidth.
A deeper understanding of the SC generation can be obtained by examining the spectrogram evolution of the pulse. As shown in Fig. 5, at the beginning of the pulse propagation, the spectral broadening is dominated by SPM, which induces new spectral components on the leading and trailing pulse edges. Moreover, due to the self-steepening effect, the pulse is asymmetric, with a sharp trailing edge in the time domain. Therefore, the SPM induces more blue-shifted spectral components on the steepened trailing edge, as we can see in Fig. 5 at 1.4 mm. Since the waveguide exhibits all-normal dispersion, the faster tail eventually catches up the slower blue-shifted components and thus OWB sets on [47,48]. At around 2.5 mm, the temporal overlap of different spectral components leads to the generation of new wavelength components at the short-wavelength region through the degenerate four-wave mixing process [48]. After around 8 mm of propagation, OWB also occurs on the leading pulse edge and generates new spectral components at the long-wavelength region. For a longer propagation, due to the accumulated dispersion, the walk-off among different spectral components becomes larger, and the spectrum eventually becomes flat and smooth at around 4 cm.
Fig. 5. Spectrogram of the pulse at (a) the input of the waveguide, (b) 0.5 mm, where the spectral broadening is dominated by SPM, (c) 1.4 mm, where more blue-shifted spectral components are induced due to the self-steepening, (d) 2.5 mm, where new wavelength components are generated at short wavelengths, (e) 8.0 mm, where new spectral components are generated at long wavelengths, and (f) 40 mm, where the spectrum becomes significantly broadened and flattened.
The generated SC varies with the peak power of the input pulse. Looking at Fig. 6(a), we note that the spectral width and flatness are improved by increasing the peak power from 10 kW to 30 kW, while keeping the pump wavelength at 960 nm and the FWHM of the input pulse to be 250 fs. The improvement in the spectral width and flatness is attributed to the increased SPM efficiency.
Fig. 6. (a) Spectral evolution over distance, when the peak power of the input pulse moves from 10 kW to 30 kW. (b) Generated supercontinua at a propagation distance of 4 cm, when the input pulse wavelength moves from 1000 nm to 2600 nm. The bandwidth parameters are at the -10 dB level.
The input pulse wavelength also plays an important role in SC generation. For waveguides with parabolic dispersion profiles, the second-order dispersion coefficients and group velocities vary greatly with wavelength, which limits the choice of the input pulse wavelength [24,49]. However, for the proposed Si3N4 waveguide with its saddle-shaped all-normal dispersion profile that is kept low and flat over a 2160-nm bandwidth, the influence of the input pulse wavelength on spectral flatness and bandwidth is relatively small, which means that the pump wavelength can be flexibly selected within the flat part of the dispersion profile to meet the demand for SC sources in different wavelength ranges. Figure 6(b) shows the generated supercontinua at a propagation distance of 4 cm. Keeping the FWHM of the input pulse to be 250 fs and the peak power at 30 kW, we change the pump wavelength from 1000 nm to 2600 nm. As shown in Fig. 2, we take all-order dispersion and Kerr nonlinearity into account. Therefore, while changing the pump wavelength, all other physical parameters related to the wavelength are changed accordingly. Under these conditions, the output spectra are significantly broadened and span more than 1 octave at the -10 dB level. This is mainly attributed to sophisticated dispersion engineering.
To further investigate the effect of the input pulse width on the SC generation, we change the FWHM of the input pulse from 200 fs to 400 fs, when keeping the pump wavelength and peak power to be 960 nm and 30 kW, respectively. As shown in Fig. 7, the spectral width and flatness are maintained with the FWHM of the input pulse increased from 200 fs to 400 fs. Different from the high-order soliton fission in the anomalous dispersion regime, ANDi SC generation is enabled by SPM and OWB [7], so the spectral width and flatness of the output SC are relatively insensitive to the input pulse width, which indicates that the SC generation is quite robust to the pump pulse width.
Fig. 7. Generated supercontinua at a propagation distance of 4 cm, as the FWHM of the input pulse changes from 200 fs to 400 fs. The bandwidth parameters are at the -10 dB level.
4. Conclusion
We have presented a Si3N4 waveguide with flat saddle-shaped all-normal dispersion, enabled by the newly proposed dispersion flattening scheme. With this unique dispersion profile, one can generate a flat SC extending from 638 nm to 1477 nm, covering 1.21 octave at the -6 dB level. The input pulse wavelength can be flexibly selected within the flat part of the dispersion profile, making it possible to meet the demand for SC sources in different spectral regions.
Acknowledgments
We acknowledge support by the Advanced Integrated Optoelectronics Facility at Tianjin University.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
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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