Numerical Investigation and Design of Optical On-Chip Waveguide with Engineered Dispersion for Generation of Supercontinuum-Based Frequency Combs

In this paper, with dispersion engineering, two waveguides with silicon core and SiO2 cladding are proposed to generate the supercontinuum spectrum and optical frequency combs. By injecting a pulse with a peak power of 800 W and a pulse duration of 50 fs, the output supercontinuum spectrum is obtained from a wavelength of 1100 nm to 4000 nm. Also, broadband optical frequency combs based on a supercontinuum have been obtained by applying a maximum power of 1 kW and a pulse width of 100 fs. Our proposed structure shows promise for achieving flat dispersion and can be useful in engineering applications. The structure exhibits two zero dispersion wavelengths at 1890 nm and 2850 nm, respectively. This flat dispersion can be very useful to achieve the desired output spectrum. Due to the materials used and the flat structure of the proposed waveguides, these can be used for integrated optical circuits as well as applications in optical communications, spectroscopy, and sensors.


Introduction
This article discusses the increasing demand for high-capacity programs, including high-quality videos, artificial intelligence, and 5G technology, which has led to the development of high-volume transmission networks.Optical networks have emerged as a viable solution for meeting these communication needs due to their high bandwidth and rapid capacity expansion.Optical signal processing (OSP) techniques serve as a suitable platform for processing high volume data in various optical fields, such as analogy and digital signal processing, nonlinear applications, and a new type of data modulation [1][2][3].OSP technology is crucial for processing high-speed fiber optic communication signals.OSP systems possess significant potential to meet current and future demands for high-volume data programs [4].Most recently, there is an urgent need for compression and device miniaturization in optical circuits, which can be achieved by using the nonlinear effects of structures containing silicon [5], Hydex [6], SiN [7,8], AlN [9], diamond [10], III-V semiconductor [11][12][13], and chalcogenide [14][15][16][17].In 2012 Zhang et al. designed a waveguide with four ZDWs to produce a super continuous spectrum with a width of 670 nm [18].In 2015 Kuyken et al. designed silicon nanophotonic waveguides to produce frequency combs in the range of 1500 nm-3300 nm [19].In 2017 Ciret et al. reported supercontinuum in the range of 1200 nm to 2870 nm at the − 20 dB level in tapered and dispersion-managed (DM) silicon nanophotonic waveguides [20].In 2018 Kou et al. presented a supercontinuum spectrum in the range of 2-5 μm in a suspended silicon waveguide [21].Silicon-on-insulator (SOI) waveguide is considered an important substrate in compact circuits due to its high nonlinear properties and compatibility with CMOS integrated circuits.There are several ways to increase the nonlinear coefficient of the structure.One of the methods is a strong optical field confinement in the structure, which increases with a high index contrast between the core and the cladding, which results in a highly efficient four-wave mixing (FWM) process that is important in many applications [22,23].The FWM phenomenon is especially important for phase-sensitive processing where signal and idler wave interference is desired [24].One of the most important applications of nonlinear effects is supercontinuum spectrum generation (SCG) as well as optical frequency comb generation (FCG).
The supercontinuum is the broadening of a very short pulse that enters the input of a waveguide or fiber with a high nonlinear coefficient.The interaction of various parameters such as dispersion and nonlinear effects such as self-phase modulation, Raman effects, and four-wave mixing, etc., cause this expansion.Typically, the broad SCG is formed from compressing of solitons and soliton breaking effect in the anomalous dispersion region, and then the propagation of the dispersive wave [25][26][27][28].SCG sources include wide and flat output spectra that have been used extensively in spectroscopy [29,30], microscopy [31], imaging [32], metrology, tomography, pulse compression, fingerprinting, and sensors [29,30].Especially the infrared spectral range is very important because the molecules of the most of materials oscillate in this region [31].The SCG has been extensively studied in nonlinear media such as fibers and photonic crystal fibers.However, the waveguides that can be implemented on chips have recently attracted much attention [32,33].In addition to optimizing the optical waveguide structure, the appropriate pumping wavelength and maximum input pulse power can be used to broaden the pulse.However, it should be noted that in practice, each has limitations.The dispersion profile also plays an important role in the width and flattening of the SCG, especially if the pumping wavelength is close to the zero dispersion wavelengths (ZDW), which can be adjusted based on the waveguide structure in an urgent regime.To increase the width of the produced spectrum, pumping is injected into the anomalous dispersion zone, which produces solitons and four-wave mixing [34,35].The width of the supercontinuum spectrum in fibers is usually greater than in waveguides and fibers have lower losses.However, to produce a wide spectrum, the effective length of the waveguides is shorter than that of the fibers, and the flat nature of the waveguides allows them to be implemented on chips [36][37][38].
The optical frequency comb (OFC) consists of a spectrum with equal frequency lines, each of which has a relatively constant phase so that the spectrum appears as a comb.This OFC spectrum can be employed in many applications such as astronomy [39], optical communications [40], microwave photonics [41], and other fields.Several methods have been proposed for creating frequency combs with a flat spectrum for use in high-capacity optical communications [42].The first method is to use combs based on electro-optical modulators.However, this method usually involves bulky components and has limited comb lines spanning over a few nanometers or exhibiting the triangular spectrum [43,44].The second method is to use integrated mode-locked lasers.In this method, the optical linewidth of the mode-locked laser is usually too large for coherent optical communication [45,46].The third method is to use the nonlinear micro-resonator-based Kerr frequency combs.In this method, the spacing of the frequency comb depends on the free spectral range (FSR) of the used micro-resonator, where its tunability is limited [47].The fourth method is based on SCG by injecting a laser pulse into a fiber or optical waveguide suitable for practical application because of the long time stability of the comb [48,49].Highly nonlinear fiber with normal dispersion is a desirable method for the generation of broadband OFC spectrum [50].However, such a fiber must be hundreds of meters long and should exhibit a normal, almost near zero-dispersion along this length [51].The production of long fiber with the mentioned characteristics has serious challenges in manufacturing technology.One method is to use high nonlinear optical integrated circuits based on the waveguides which not only use materials with high nonlinear coefficients but also the high optical limitation that can be achieved in these waveguides; thus, the desired length in these structures is significantly reduced [52][53][54][55].In this paper, by dispersion engineering and investigating the effect of dimensions on the dispersion profile, two silicon waveguides are proposed for the generation of supercontinuum spectrum and frequency combs based on SCG.Complex circuits are typically required to maintain a consistent spacing between the frequency comb lines.These circuits often involve the use of electro-optic modulators and other components, resulting in increased complexity of optical circuits.Consequently, integrating these circuits becomes more challenging.In the context of supercontinuum-based frequency comb generation, this challenge can be overcome with phase-matching techniques that include dispersion engineering [56][57][58][59][60][61][62].If the dispersion is effectively managed, stable frequency combs can be generated within a waveguide, eliminating the necessity for complex circuits.This paper aims to investigate dispersion engineering techniques by optimizing the structure and enabling the generation of frequency combs in a waveguide, which eliminates the need for complex circuits [54].As a result, the essential foundation for integrated optical circuits is provided.Due to the materials used and the flat structure of the proposed waveguides, these can be used for integrated optical circuits as well as applications in optical communications, spectroscopy, and sensors.

Theory
The nonlinear Schrödinger equation can be solved numerically using the split-step Fourier method to study the generation of supercontinuum spectrum and frequency combs.The simplified form of this equation is as follows [56]: where, D represents the linear part and N represents the nonlinear part of this equation.The value of D is obtained from the following equation: In Eq. ( 2), α represents the waveguide loss coefficient, which can be neglected due to the low waveguide length.β n (n ≥ 2) Indicates the various orders of dispersion that can be obtained from the following equation [55][56][57][58]: In this equation, 0 represents the pumping wavelength.In addition, the nonlinear part of Eq. ( 1) is obtained from [59]: where A is the input pump envelope and e = γ + i TPA where: In addition, TPA represents the losses related to the twophoton absorption coefficient and TPA represents the two-photon absorption coefficient, with a value of TPA = 5 × 10 −12 at a wavelength of 1555 nm [60].The effective mode area is obtained from the following equation: where, |F(x.y)| is the modal distribution function.In Eq. ( 5), n 2 is the nonlinear refractive index related to the Kerr effect, c is the speed of light in a vacuum, A ef f is the effective mode area at the central wavelength for the propagation mode, and λ is the pumping wavelength.Also, in Eq. ( 4), R ( T′ ) is a function of the Raman response and is obtained from the following equation [61]: where the value of f R = 0.1 and h R (T) is obtained from: (1) where the value of 1 = 10 fs and 2 = 3 ps for Si [61].
The group velocity dispersion D (λ) is calculated using the propagation mode wavelength-dependent effective refractive index: where Re n eff is the real part of the effective refractive index and determined by using the Sellmeier equation [62].

Dispersion Engineering and Waveguide Design
We have designed two waveguides with silicon core and SiO 2 cladding to produce supercontinuum spectrum and frequency combs.To achieve the final structure, we have used the effect of waveguide dimensional changes on dispersion and dispersion engineering.Figure 1 shows the structure of the two proposed waveguides.The first structure shown in Fig. 1a is very familiar, while the second structure is suitable for flat dispersion applications (Fig. 1c) [63].
The dispersion profile plays a critical role in the development of the supercontinuum.If the input pulse is introduced to the normal dispersion regime, the self-phase modulation phenomena cause the spectrum to symmetrically broaden, resulting in a flat spectrum.Conversely, if the input pulse is introduced to the anomalous dispersion regime, soliton breaking becomes the crucial factor leading to pulse broadening, ultimately causing the output spectrum to widen while containing many harmonics in the wavelengths related to the anomalous part of the spectrum.In situations where the dispersion is within the anomalous dispersion regime and near the zero dispersion wavelength (ZDW), spectral content initially broadens due to self-phase modulation (SPM), displacing it to the vicinity of the ZDW and throughout the anomalous dispersion regime.This occurrence can significantly influence the produced supercontinuum's temporal and spectral properties through soliton dynamics.
Additionally, other nonlinear effects' presence directs the spectrum toward normal dispersion, resulting in a smoother output spectrum in this region.Figure 1e shows the dispersion profile of the first structure.In this structure, the height h adjusts 300 nm and the width w is equal to w = 1000 nm.The curve has two zero-dispersion wavelengths around the wavelengths of 1980 nm and 2800 nm. Figure 2 shows the dispersion profile of Structure 2, in which Fig. 2a, b, c, and d correspond to the heights h 1 , h 2 , h 3 , and h 4 , respectively, and Fig. 2e is related to width changes of the waveguide.Based on Fig. 2a, a flat dispersion profile is obtained when h 1 = 180 nm, but the amount of dispersion is high in the anomalous dispersion regime, which is unfavourable for supercontinuum generation [29,30].If h 1 is increased, the dispersion profile becomes non-flat, as shown in Fig. 2b, where increasing h 2 from 380 nm to higher values results in a flatter dispersion profile, but the amount of dispersion increases in the anomalous dispersion regime.Similarly, changing the h 3 parameter in Fig. 2c produces a similar effect on the dispersion profile.However, changes in h 4 do not have a significant effect on the dispersion profile, as shown in Fig. 2d.
When w is increased from 950 nm to higher values, the dispersion profile becomes flatter according to Fig. 5e, but it shifts towards the normal dispersion regime.In the normal dispersion regime, although the output supercontinuum spectrum becomes flatter and smoother, the broadening of the spectrum is not significant.The selected dimensions for Structure 2 are shown in Table 1.This structure has two zero-dispersion wavelengths in 1890 nm and 2850 nm in the selected dimensions.According to the obtained dispersion index for two structures, it is observed that the wavelength distance between two ZDWs in the second structure is greater, while its maximum profile value in the anomalous dispersion regime is less.This indicates that the second structure is more suitable for SCG and especially FCG.

SCG and FCG
By numerically solving Eq. ( 1) by the split-step Fourier method, one can observe the spectrum of generated supercontinuum spectra in the structure and properties of the FCG.To investigate the supercontinuum spectrum, an input pulse with a power of 800 W and a pump pulse width of 50 fs is injected into the input of Waveguide 1.The value of the effective mode area in this structure is 0.33 μm 2 , which according to the nonlinear coefficient of silicon n 2 = 12 × 10 −14 cm 2 W −1 at about 1990 nm [64][65][66][67], the value of the nonlinear coefficient equal to γ = 115.93m −1 W −1 is obtained.The loss of propagation in the wave- length range of 1200 nm to 2400 nm has been determined to be 0.1 dB/cm [64,65].
The waveguide is designed to operate in a wide range of single-mode to not only create high confinement of light in the core, which increases the non-linear coefficient but also to facilitate phase matching for the generation of frequency combs.Here, a pumping pulse with a wavelength of 2000 nm is injected into the waveguide in the anomalous dispersion regime and near the first zero dispersion wavelength.By injecting this pulse, the interaction between linear and nonlinear effects with the input pulse results in the broadening of the output spectrum.In this case, due to the self-phase modulation effect, the pulse initially begins to broaden symmetrically around the pumping wavelength.When the propagation length equals the soliton splitting length, higher-order nonlinearities lead to the occurrence of soliton fission, causing the emergence of fundamental solitons with varying wavelengths and new frequency components are produced over a relatively wide range [68].The isolated optical pulses resulting from this process exhibit narrower temporal width and broader spectral width compared to the original pulses, thereby generating a broader bandwidth.Consequently, the self-frequency shift of the obtained solitons continuously shifts toward longer wavelengths, resulting in significant spectral broadening primarily on the red side.On the other hand, non-solitonic radiation leads to a broadening in the blue side of the spectrum [55].Simultaneously, dispersive waves (DW) are generated at wavelengths of approximately around 1400 nm and 2800 nm.Collectively, these mentioned phenomena contribute to the generation of a supercontinuum spectrum at the output of the waveguide.The supercontinuum spectrum and the corresponding spectrum evolution are shown in Fig. 3.This spectrum is prepared from wavelengths of 1100 nm to more than 4000 nm at a level of 20 dB.This bandwidth is very suitable for silicon waveguides considering the used waveguide length.
Numerous techniques have been proposed for generating frequency combs, which require complex circuits to extract the frequency shoulder lines, posing a challenge for integration.To generate frequency combs without the need for complex circuitry for phase matching, one can use FWM resulting from Eq. 1.For this purpose, the input power should be less than about 1kw.Under these conditions, Eq. 1 can be simplified because the FWM phenomenon dominates and the effects such as stimulated Raman scattering can be ignored due to the high power threshold [66].For use in optical applications such as communications, spectroscopy, and the production of supercontinuum-based frequency combs, the output spectrum is also simulated by numerically solving the Schrödinger equation at a pumping with the wavelengths of 1550 in the normal dispersion region.To create effective interactions between dispersion and nonlinear effects, the waveguide length is considered to be 0.8 mm in this case.These peaks in the spectrum occur at a pumping power of 500 watts.Increasing the input power leads to the occurrence of other nonlinear effects.These nonlinear effects generate harmonics and new frequencies, and further increasing the power results in the overlap of these frequency components and the production of a broadband SCG.Moreover, further increasing the waveguide length introduces new frequency components.These new components overlap with the FCG lines obtained from FWM and cause a frequency spacing disturbance.Additionally, dual-pumping can be used to adjust the frequency spacing between the frequency combs.[64,65].Figure 4 shows the spectrum corresponding to the observed ultra-broadband comb obtained via Structure 1.There are several sharp peaks in the spectrum, which indicates the ability to frequency combs generation in this structure [69][70][71].
To generate the supercontinuum spectrum in Structure 2, an input pulse with a peak power of 800 W and a pulse width of 50 fs at a wavelength of 1890 nm is injected into a 0.8 mm waveguide input.The effective area of mode in this structure is 0.4 μm 2 , and the nonlinear coefficient is calculated like the first structure, the value of which is γ = 99.68 m −1 W −1 .Figure 5 shows the mid-IR spectral broadening and pulse evolution in the proposed Waveguide 2.
As can be seen in this figure, the output spectrum is broadened at wavelengths between 1000 nm-1800 nm, and can be seen blue-and red-shifted solitons that cause spectral broadening from 1100 to 3800 nm. Figure 6 also shows the comb properties of Waveguide 2 by numerically solving Eq. ( 1) and injecting an input pulse at the wavelengths of 1550.The comb properties of the output spectrum of Waveguide 2 can be observed in this figure like Fig. 4. By comparing the two output spectra related to frequency combs, it is observed that Structure 2 is more suitable for the generation of flat frequency combs in the optical communication regime.
Table 2 presents a comparison of some features and output spectrum between the proposed structure and previous works in this field.Silicon's high refractive index allows for effective light confinement in the core of the waveguide, resulting in a high nonlinear coefficient and lower input power requirements.Pumping in the anomalous region leads to a wide bandwidth in the output spectrum, but it will have little coherence and smoothness.For example, in reference [72] a very flat spectrum is obtained.The objective of the proposed waveguide is to generate frequency combs within the telecommunication range.To achieve this, the design of the waveguide structure aimed to align the zero dispersion wavelength with this range.Injecting a pump near the ZDW resulted in the generation of the desired supercontinuum spectrum.Although the spectrum has an appropriate width, it exhibits lower flatness compared to some references.However, the spectrum of the output frequency comb demonstrates that it possesses the essential characteristics required for phase matching.
As mentioned before, the purpose of this study was to generate SCG based frequency combs in the telecommunication range and more, which can be used for applications such as spectroscopy, sensing, telecommunications, etc.According to the integration approach in modern systems, the generation of frequency combs in a waveguide without a complicated structure can be useful for today's systems.But, the spectral width of the output is restricted in this method.By employing dual pumping and using a parallel waveguide with a shifted ZDW, this problem can be resolved and the frequency comb lines can be extended to the required range.Additionally, through the implementation of dual pumping, there will also be more flexibility in regulating the distance  of the comb lines.The proposed waveguide can be easily fabricated based on existing fabrication techniques and the existence of researches as reported in [81][82][83], which has demonstrated the design of even more complex structures.

Conclusion
Two silicon waveguides have been proposed to generate the supercontinuum spectrum and frequency combs based on the supercontinuum spectrum by dispersion engineering and investigating the effect of dimensions on the dispersion profile.For SCG, the pump pulse with a peak power of 800 W and pulse duration of 50 fs was injected into the input of both waveguides with a length of 0.8 mm.The first zero-dispersion wavelengths were located near the 1980 nm and 1890 nm in Waveguides 1 and 2, respectively.However, to investigate the frequency comb generation characteristics of the waveguides, an input pulse with a pulse duration of 100 fs was applied in the normal dispersion region of both waveguides at the wavelength of 1550 nm.The supercontinuum spectrum in Waveguides 1 and 2 has been achieved from the wavelength of 1100-4000 nm, and 1200-3800 nm, respectively.The generated optical frequency combs also have a wide bandwidth and can cover the optical communication area.The proposed structures can be used in optically integrated circuits, optical communications, and spectroscopy.

Fig. 1
Fig.1The waveguides studied in this research a and b the shape and mode profile of the first structure, c and d the structure and mode profile of the second proposed structure, and e dispersion index of Structure 1, in this structure, the size of height h is equal to 300 nm, and the size of width w is equal to w = 1000 nm.The curve has two zero-dispersion wavelengths around the wavelengths of 1980 nm and 2800 nm

Fig. 2
Fig. 2 The dispersion profiles of Structure 2, dispersion changes with respect to a h 1 , b h 2 , c h 3 , d h 4 , and e w

Fig. 3
Fig. 3 The supercontinuum spectrum generated in Structure 1, a supercontinuum spectrum with a pumping wavelength of 1890 nm, b spectrum evolution

Fig. 4 Fig. 5
Fig.4 Frequency comb spectrum generated in Structure 1 with pumping wavelength of 1550 nm

Fig. 6
Fig. 6 Frequency comb spectrum generated in Structure 2 with pumping wavelength of 1550 nm

Table 1
The design parameters of the proposed structure

Table 2
Comparing the characteristics of the output spectrum in the proposed structure with those in previous studies