1 Introduction

Silicon photonics has recently received considerable attention and the most promising photonic integration platforms, because they can be mainly attributed to the combination of a high index contrast and the availability of the devices fabrication technology [1]. Recently, the silicon waveguides structures have been used and demonstrated a record decline in the footsteps of the waveguide, and in particular, wavelength selection device, which can be used for application in supercontinuum generation [2, 3], and ring resonator optical modulator [4–6]. The supercontinuum (SC) is an important spectral range for a variety of applications, such as in optical coherence tomography, telecommunication, and broadband sources [7–10]. The highly nonlinear waveguide in combination with femtosecond pulse polarized (soliton) can be used to provide such a source through the process that induces broad spectrum call SC. The silicon-on-insulator (SOI) waveguide has also been investigated in both theory and experiment, which SOI can support solitons in the wavelength region near 1.55 µm and this has been used for several nonlinear effects, such as solitonic effects, Raman scattering (SRS), four-wave mixing (FWM), and self-phase modulation (SPM) [11, 12]. The SC generation depends on both nonlinear medium and anomalous dispersion when they are designed suitably to provide the anomalous dispersion [13–17]. Yin et al. [2] have proposed the SC in the silicon ridge waveguides length L of 1.2 cm long, which can be used to create a supercontinuum extending over 470 nm, from which the coverage wavelength range is around 1.28–1.75 µm with a −30 dB using the pulses with a width of 50 fs and the power peak of 25 W.

Apparently, ring resonators play an important role in the integrated optics and have received considerable attention [18, 19] because of there are several physical mechanisms that contribute to the nonlinear refractive index and nonlinear absorption of materials, which depend on the wavelength, a peak intensity of an incident light, and pulse width [20, 21, 24]. In addition, the nonlinear ring resonators depend on the intensity-dependent phase shift accumulated over a ring resonator, which was done under the assumption of an ideal Kerr nonlinearity, where the refractive index of material is relatively changed with the optical intensity, where they have been theoretically and experimentally demonstrated by making use of a nonlinear ring resonator in several applications in full-width at half maximum (FWM), modulators, optical filters, sensors, and optical communications. From which all these analyses of these, nonlinear ring waveguide of steady-state and dynamic transmission properties have extensively been investigated using an iterative method [20–27]. Till date, the silicon racetrack ring-based creates the combination of both the bus waveguide and ring resonator. Such a device is formed by a racetrack resonator coupled to a bus waveguide, where the ring radius R and straight arm’s length L, which can be used for optical modulator [5, 6], coherent perfect absorption in a silicon photonics [28], a series-coupled silicon racetrack resonators, and the Vernier effect [29].

In this paper, the SC generation in SOI racetrack waveguide can be generated, from which the broadband SC in the mid-infrared (MIR) regime is obtained. In the proposed model, the racetrack waveguide is coupled to a bus waveguide of length L b in the racetrack waveguide structure, which can be modeled by the combination of the ring-half waveguide with the ring radius R and bus ridge waveguide with lengths L 1 and L 2. Therefore, the dynamic propagation properties of light within the racetrack waveguide can be calculated using the ridge waveguide evolution, which is governed by the generalized nonlinear Schrödinger equation (GNLSE) with the split-step Fourier method (SSFM) [10, 16, 30] and iterative method for ring waveguide [20–27]. The SC generation process uses the ultra-shot polarized pulses (solitons), which depend strongly on the material dispersion. Principally, a soliton pulse can be formed when the SOI exhibits the anomalous dispersion at the operation wavelength close to the pump wavelength, which can be tailored to fall in this regime with a suitable design, which can be confirmed in both theoretical and experimental studies [11, 14, 15, 29]. Therefore, in this study, the SOI racetrack waveguide is designed by a straight arm’s length L, which is fabricated along the direction on the (001) surface. This racetrack waveguide is the anomalous dispersion, where the effective mode index (n eff) for the fundamental TE mode and effective mode area (A eff) are calculated. The simulation results are obtained using the finite difference eigenmode (FDE) of MODE solutions, which can predict that the TE mode has a dispersion reduction at 1550 nm, the bus ridge waveguides are calculated using the GNLSE with SSFM solver, and the iterative method is used for the ring waveguide, which can be used to calculate the dynamic properties when an optical pulse with an arbitrary temporal profile, which is incident into the nonlinear single racetrack ridge waveguide and double-coupler racetrack waveguide. More details are given in the following sections.

2 Theoretical background

The SC generation process employs the ultra-short pulse, which depends strongly on the material dispersion, where the zero dispersion wavelength (ZDW) of such waveguides can be tailored to fall in this regime with a suitable design and material properties. The racetrack ridge waveguide consists of a bus ridge waveguide of length L b and racetrack ridge waveguide of length L (=L R + L 1 + L 2, L R = 2πR, where R being the ring radius), from which the straight arms are the ridge waveguide of length L 1 and L 2, which is shown in Fig. 1. The SC generation in the straight arms is the ridge waveguide of lengths L 1 and L 2, which exhibits the anomalous dispersion at the operating wavelength near 1550 nm. Therefore, the silicon (n si) and silica refractive indices are specified for the required channel waveguide and substrates, over the interval of a wavelength of interest.

Fig. 1
figure 1

Structure of the racetrack waveguide resonator and the definition of electric fields, where a a single-coupler racetrack waveguide, and b a double-coupler racetrack waveguide

Full size image

The Sellmeier equation is given by Eq. (1), where the material dispersion of silicon can be calculated accurately [30]:

$$n_{\text{Si}}^{2}(\lambda )=1+\frac{{{B}_{1}}{{\lambda }^{2}}}{{{\lambda }^{2}}-\lambda _{1}^{2}}+\frac{{{B}_{2}}{{\lambda }^{2}}}{{{\lambda }^{2}}-\lambda _{2}^{2}},$$
(1)

where λ j and B j are the jth resonant wavelength and strength, respectively. The used silicon parameters are λ 1 = 0.2904 µm, λ 2 = 0.3669 µm, B 1 = 9.733, and B 2 = 0.936. In a simulation, the effective mode index n eff = λβ(ω)/2π and the propagation constant is β(ω), where the fundamental TE mode over a range of wavelength can be obtained using the finite difference Eigen mode solver with MODE solutions, which is shown in Fig. 2. Subsequently, the used parameter for numerical calculation of a group-velocity dispersion is D(λ) = −(λ/c) (d2 n eff/dλ 2) (ps nm−1 km−1), and the effective mode is\({{A}_{\text{eff}}}=|\int{(E\times {{H}^{*}})\cdot \hat{z}\text{d}A}{{|}^{2}}/|\int{(E\times {{H}^{*}})\cdot \hat{z}{{|}^{2}}\text{d}A}\) at the pump wavelength 1550 nm as well as all other higher order dispersion parameters.

Fig. 2
figure 2

Simulation result, where a (color online) dispersion and effective index (long-dashed curve) and n si (dashed curve). The vertical dotted line denotes wavelength = 1550 nm, b the ridge waveguide geometry, and c the mode field of the SOI waveguide with W = 760 nm, H = 220 nm for the fundamental TE mode

Full size image

2.1 Single-coupler racetrack waveguide

Figure 1a shows the structure of a nonlinear single-coupler racetrack ridge waveguide resonator, which consists of a bus ridge waveguide of length L b and racetrack ridge waveguide of length L = L R + L 1 + L 2 = n R λ, L R = 2πR, where R is the ring radius, the straight arms are the bus ridge waveguide lengths L 1 and L 2, n R is an integer, and λ is the pumped wavelength. The coupling coefficient is κ 1. The conventional iterative method for this configuration and the bus ridge waveguide of racetrack ridge waveguide using a generalized nonlinear Schrödinger equation with the split-step Fourier method solves [2, 11]. The nonlinearity of the racetrack ridge waveguide resonator is assumed to satisfy the Kerr-type, i.e., the refractive index is given by the following:

$$n\text{ }=\text{ }{{n}_{0}}+{{n}_{2}}I\text{ }=\text{ }{{n}_{0}}\text{ }+\text{ }\frac{{{n}_{0}}{{n}_{2}}|E{{|}^{2}}}{2{{\eta }_{0}}}=\text{ }{{n}_{0}}+\frac{{{n}_{2}}P}{{{A}_{\text{eff}}}},$$
(2)

where n 0 and n 2 are the linear and nonlinear refractive indices, respectively, and η 0 is the wave impedance in the vacuum. I, E, and P are the optical intensity, the optical electric field, and the optical power, respectively. A eff is the effective mode area of the bus ridge and the racetrack ridge waveguides. The nonlinearity of the racetrack ridge waveguide takes place instantaneously. Since the length of the ring ridge waveguide (L R) in this paper is short, therefore, the group-velocity dispersion effect is neglected in the following formulation. Furthermore, the slowly varying complex electric field at each position is defined, as shown in Fig. 1. The field amplitudes of incident wave at the coupling point between the bus ridge waveguide and the racetrack ridge waveguide are given by E in(t), from which the transmitted wave is E through(t), where E c1(z, t) and E c2 (z, t) are the fields inside the right-quarter cavity, E c3(y, t) is the field inside the right-bus cavity, E c4 (z, t) is the field inside the below-half cavity, E c5(y, t) is the field inside the left-bus cavity, and E c6 (z, t) is the field inside the left-quarter cavity satisfy the following equations:

$${{E}_{\text{c}1}}({{L}_{\text{R}}}=0,t)=\sqrt{1-\gamma }\left[ -i\sqrt{{{\kappa }_{1}}}{{E}_{\text{in}}}(z=0,t)-\sqrt{1-{{\kappa }_{1}}}{{E}_{\text{c}6}}(L,t) \right],$$
(3)
$${{E}_{\text{th}}}(z=0,t)=\sqrt{1-\gamma }\left[ \sqrt{1-{{\kappa }_{1}}}{{E}_{\text{in}}}(z=0,t)-i\sqrt{{{\kappa }_{1}}}{{E}_{\text{c}6}}(L,t) \right],$$
(4)

where κ and γ are the intensity coupling coefficient of the coupler and the fractional intensity loss, respectively. E c3(y, t) and E c5(y, t) are the field amplitudes inside the bus ridge waveguide of racetrack ridge waveguide model of the SC use GNLSE with SSFM. Using the slowly varying envelope of the pulse, the following equations can be obtained:

$$\frac{\partial {{E}_{\text{c}3}}(y,T)}{\partial y}=-\frac{\alpha (\omega )}{2}{{E}_{\text{c}3}}(y,T)+\sum\limits_{m\ge 2}^{\infty }{\frac{{{i}^{m+1}}{{\beta }_{m}}}{m!}\frac{{{\partial }^{m}}{{E}_{\text{c}3}}(y,t)}{\partial {{T}^{m}}}}+i\left( \delta ({{\omega }_{0}})+i\frac{\delta }{\omega }\frac{\partial }{\partial T} \right){{E}_{\text{c}3}}(y,T)\times \int\limits_{-\infty }^{\infty }{R(T){{\left| {{E}_{\text{c}3}}(y,T-{T}') \right|}^{2}}}\text{d}{T}',$$
(5)
$$\frac{\partial {{E}_{\text{c}5}}(y,T)}{\partial y}=-\frac{\alpha (\omega )}{2}|{{E}_{\text{c}4}}(y,T)|+\sum\limits_{m\ge 2}^{\infty }{\frac{{{i}^{m+1}}{{\beta }_{m}}}{m!}\frac{{{\partial }^{m}}{{E}_{\text{c}5}}(y,T)}{\partial {{T}^{m}}}}+i\left( \delta ({{\omega }_{0}})+i\frac{\delta }{\omega }\frac{\partial }{\partial T} \right)|{{E}_{\text{c}5}}(y,T)|\times \int\limits_{-\infty }^{\infty }{R({t}'){{\left| {{E}_{c5}}(y,T-{T}') \right|}^{2}}}\text{d}{T}',$$
(6)

where E c is the electric field amplitude, α(ω) is the linear propagation loss, T = t − y/v g is the retarded time frame moving with the group velocity v g, which is equal to 1/β 1(ω 0) at the pumped frequency ω 0, R(t) = (1 − f R)δ(t) + f R h R(t) includes the instantaneous electronic response, and \({{h}_{\text{R}}}(t)=[(\tau _{1}^{2}+\tau _{2}^{2})/({{\tau }_{1}}\tau _{2}^{2})\exp (-t/{{\tau }_{2}})\sin (-t/{{\tau }_{1}})\) is the delayed Raman response, β m  = dm β/dωm is the mth dispersion order. The nonlinear parameter is δ = 2πn 2/λ 0 A eff + iβ TPA/2A eff, where β TPA is the two-photon absorption coefficient and n 2 is the nonlinear refractive index. The equation can be solved using the split-step Fourier method [31, 32], where the cavity fields at the end of a ring can be expressed by the cavity fields as follows:

$${{E}_{\text{c}2}}({{L}_{\text{R}}}/4,t)={{E}_{\text{c}1}}(L=0,t-{{\tau }_{\text{R}}})\exp (-\alpha {{L}_{\text{R}}}/4)\exp [-i({{\phi }_{0}}+{{\phi }_{\text{N}1}}(t-{{\tau }_{\text{R}}})]$$
(7)
$${{E}_{\text{c}4}}({{L}_{\text{R}}}/2,t)={{E}_{\text{c}3}}({{L}_{1}},t-2{{\tau }_{\text{R}}})\exp (-\alpha {{L}_{\text{R}}}/2)\exp [-i(2{{\phi }_{0}}+{{\phi }_{\text{N}2}}(t-2{{\tau }_{\text{R}}})]$$
(8)
$${{E}_{\text{c}6}}(L,t)={{E}_{\text{c}5}}({{L}_{2}},t-{{\tau }_{\text{R}}})\exp (-\alpha {{L}_{\text{R}}}/4)\exp [-i({{\phi }_{0}}+{{\phi }_{\text{N}3}}(t-{{\tau }_{\text{R}}})],$$
(9)

where τ R = n 0 L R/4c is the quarter round-trip time in the cavity, φ 0 = n 0 k 0 L/4 = n 0 πL/2λ 0 is the linear phase shift per quarter round-trip, and α is the amplitude attenuation coefficient of the ring waveguide. If the waveguide is lossless (α = 0), ϕ N1(t − τ R), ϕ N2(t − 2τ R), and ϕ N3(t − τ R) are nonlinear phase shifts due to propagation around the right-quarter, below-half, and left-quarter for ring waveguides, respectively, which are given by the following:

$${{\phi }_{\text{N}1}}(t-{{\tau }_{\text{R}}})]=\frac{\pi {{L}_{\text{R}}}{{n}_{2}}}{2{{\lambda }_{0}}{{A}_{\text{effr}}}}|{{E}_{\text{c}2}}({{L}_{\text{R}}}/4,t{{|}^{2}}=\frac{\pi {{L}_{\text{R}}}{{n}_{2}}}{2{{\lambda }_{0}}{{A}_{\text{effr}}}}|{{E}_{\text{c}1}}({{L}_{\text{R}}}=0,t-{{\tau }_{\text{R}}}){{|}^{2}}$$
(10)
$${{\phi }_{\text{N}2}}(t-2{{\tau }_{\text{R}}})]=\frac{\pi {{L}_{\text{R}}}{{n}_{2}}}{{{\lambda }_{0}}{{A}_{\text{effr}}}}|{{E}_{\text{c}4}}({{L}_{\text{R}}}/2,t{{|}^{2}}=\frac{\pi {{L}_{\text{R}}}{{n}_{2}}}{{{\lambda }_{0}}{{A}_{\text{effr}}}}|{{E}_{\text{c}3}}({{L}_{1}},t-2{{\tau }_{\text{R}}}){{|}^{2}}$$
(11)
$${{\phi }_{\text{N}3}}(t-{{\tau }_{\text{R}}})]=\frac{\pi {{L}_{\text{R}}}{{n}_{2}}}{2{{\lambda }_{0}}{{A}_{\text{effr}}}}|{{E}_{\text{c}6}}({{L}_{\text{R}}}/4,t{{|}^{2}}=\frac{\pi {{L}_{\text{R}}}{{n}_{2}}}{2{{\lambda }_{0}}{{A}_{\text{effr}}}}|{{E}_{\text{c}5}}({{L}_{\text{R}}}/4,t-{{\tau }_{\text{R}}}){{|}^{2}}.$$
(12)

From Eq. (4), the output field E th (z = 0, t) is a simple form of an iteration of the cavity field E c6 (L, t) with respect to the cavity round-trip time. Thus, the dynamic properties when an optical pulse with an arbitrary temporal profile is incident into the nonlinear single racetrack waveguide Eqs. (3)–(12) can be calculated.

2.2 Double-coupler racetrack waveguide

Figure 1b shows the configuration of a nonlinear double-coupler racetrack ridge waveguide resonator and the definition of the electric fields for analysis, where the system consists of a bus ridge waveguide of length L b and racetrack ridge waveguide resonator of length L (=L R + L 1 + L 2, L R = 2πR, where R is the ring radius). The bus ridge waveguide of racetrack ridge waveguide resonator lengths is L 1 and L 2, and the coupling coefficients κ i are the coupling factor. The conventional iterative method for this configuration and the bus ridge waveguide of racetrack ridge waveguide is the GNLSE with SSFM solved [2, 11]. The nonlinearity in the double-coupler racetrack ridge waveguide resonator is assumed to be the Kerr-type, which is followed by Eq. (2). The field amplitudes of incident wave at the coupling point between the bus ridge waveguide and the racetrack ridge waveguide are E in(t), transmitted wave is E through(t), E c1(z, t), E c2 (z, t) and E c4 (z, t) are the fields inside the right-quarter cavity, E c3(y, t) is the field inside the right-bus cavity, E c5 (z, t), E c7 (z, t), and E c8 (z, t) are the fields inside the right-quarter cavity, and E c6(y, t) is the field inside the left-bus cavity. Mathematical relations of these fields are given by the following equations:

$${{E}_{\text{c}1}}({{L}_{\text{R}}}=0,t)=\sqrt{1-\gamma }\left[ -i\sqrt{{{\kappa }_{1}}}{{E}_{\text{in}}}(z=0,t)-\sqrt{1-{{\kappa }_{1}}}{{E}_{\text{c}8}}(L,t) \right]$$
(13)
$${{E}_{\text{th}}}(z=0,t)=\sqrt{1-\gamma }\left[ \sqrt{1-{{\kappa }_{1}}}{{E}_{\text{in}}}(z=0,t)-i\sqrt{{{\kappa }_{1}}}{{E}_{\text{c}8}}(L,t) \right]$$
(14)
$${{E}_{\text{c}5}}({{L}_{\text{R}}}/4,t)=\sqrt{1-\gamma }[\sqrt{1-{{\kappa }_{2}}}{{E}_{\text{c}4}}({{L}_{\text{R}}}/4,t)-i\sqrt{{{\kappa }_{2}}}{{E}_{\text{add}}}(z=0,t)]$$
(15)
$${{E}_{\text{drop}}}(z=0,t)=\sqrt{1-\gamma }\left[ \sqrt{1-{{\kappa }_{2}}}{{E}_{\text{add}}}(z=0,t)-i\sqrt{{{\kappa }_{2}}}{{E}_{\text{c}4}}({{L}_{\text{R}}}/4,t) \right].$$
(16)

The cavity fields at the end of a ring can be expressed by the cavity fields as follows:

$${{E}_{\text{c}2}}({{L}_{\text{R}}}/4,t)={{E}_{\text{c}1}}(L=0,t-{{\tau }_{\text{R}}})\exp (-\alpha {{L}_{\text{R}}}/4)\exp [-i({{\phi }_{0}}+{{\phi }_{\text{N}1}}(t-{{\tau }_{\text{R}}})]$$
(17)
$${{E}_{\text{c}4}}({{L}_{\text{R}}}/4,t)={{E}_{\text{c}3}}({{L}_{1}},t-{{\tau }_{\text{R}}})\exp (-\alpha {{L}_{\text{R}}}/4)\exp [-i({{\phi }_{0}}+{{\phi }_{\text{N}2}}(t-{{\tau }_{\text{R}}})]$$
(18)
$${{E}_{\text{c}6}}({{L}_{\text{R}}}/4,t)={{E}_{\text{c}5}}({{L}_{\text{R}}}/4,t-{{\tau }_{\text{R}}})\exp (-\alpha {{L}_{\text{R}}}/4)\exp [-i({{\phi }_{0}}+{{\phi }_{\text{N}3}}(t-{{\tau }_{\text{R}}})]$$
(19)
$${{E}_{\text{c}8}}({{L}_{4}}/4,t)={{E}_{\text{c}7}}({{L}_{2}},t-{{\tau }_{\text{R}}})\exp (-\alpha {{L}_{\text{R}}}/4)\exp [-i({{\phi }_{0}}+{{\phi }_{\text{N}4}}(t-{{\tau }_{\text{R}}})].$$
(20)

The cavity fields amplitudes of E c3(y, t) and E c7(y, t) are the fields inside the bus ridge waveguide of racetrack ridge waveguide. The model of the SC generation with the GNLSE for the slowly varying envelope of the pulse can be expressed by the following equations:

$$\frac{\partial {{E}_{\text{c}3}}(y,T)}{\partial y}=-\frac{\alpha (\omega )}{2}{{E}_{\text{c}3}}(y,T)+\sum\limits_{m\ge 2}^{\infty }{\frac{{{i}^{m+1}}{{\beta }_{m}}}{m!}\frac{{{\partial }^{m}}{{E}_{\text{c}3}}(y,T)}{\partial {{T}^{m}}}}+i\left( \delta ({{\omega }_{0}})+i\frac{\delta }{\omega }\frac{\partial }{\partial T} \right){{E}_{\text{c}3}}(y,T)\times \int\limits_{-\infty }^{\infty }{R({T}'){{\left| {{E}_{\text{c}3}}(y,T-{T}') \right|}^{2}}}\text{d}{T}'$$
(21)
$$\frac{\partial {{E}_{\text{c}7}}(y,T)}{\partial y}=-\frac{\alpha (\omega )}{2}|{{E}_{\text{c}7}}(y,T)|+\sum\limits_{m\ge 2}^{\infty }{\frac{{{i}^{m+1}}{{\beta }_{m}}}{m!}\frac{{{\partial }^{m}}{{E}_{\text{c}7}}(y,T)}{\partial {{T}^{m}}}}+i\left( \delta ({{\omega }_{0}})+i\frac{\delta }{\omega }\frac{\partial }{\partial t} \right)|{{E}_{\text{c}7}}(y,T)|\times \int\limits_{-\infty }^{\infty }{R({T}'){{\left| {{E}_{\text{c}7}}(y,T-{T}') \right|}^{2}}}\text{d}{T}',$$
(22)

where τ R = n 0 L R/4c is the quarter round-trip time of the cavity, φ 0 = n 0 k 0 L/4 = n 0 πL/2λ 0 is the linear phase shift per quarter round-trip, and α is the amplitude attenuation coefficient of the ring waveguide. When the waveguide is lossless, (α = 0), φ N1(t − τ R), φ N2(t − τ R), φ N3(t − τ R), and φ N4(t − τ R) are nonlinear phase shifts due to propagation around the right-quarter, below-half, and left-quarter rings, respectively, which are given by the following:

$${{\phi }_{\text{N}1}}(t-{{\tau }_{\text{R}}})]=\frac{\pi {{L}_{\text{R}}}{{n}_{2}}}{2{{\lambda }_{0}}{{A}_{\text{eff}}}}|{{E}_{\text{c}2}}({{L}_{\text{R}}}/4,t{{|}^{2}}=\frac{\pi {{L}_{\text{R}}}{{n}_{2}}}{2{{\lambda }_{0}}{{A}_{\text{eff}}}}|{{E}_{\text{c}1}}({{L}_{\text{R}}}=0,t-{{\tau }_{\text{R}}}){{|}^{2}}$$
(23)
$${{\phi }_{\text{N}2}}(t-{{\tau }_{\text{R}}})]=\frac{\pi {{L}_{R}}{{n}_{2}}}{2{{\lambda }_{0}}{{A}_{\text{effr}}}}|{{E}_{\text{c}4}}({{L}_{\text{R}}}/4,t{{|}^{2}}=\frac{\pi {{L}_{\text{R}}}{{n}_{2}}}{2{{\lambda }_{0}}{{A}_{\text{effr}}}}|{{E}_{\text{c}3}}({{L}_{1}},t-{{\tau }_{\text{R}}}){{|}^{2}}$$
(24)
$${{\phi }_{\text{N}3}}(t-{{\tau }_{\text{R}}})]=\frac{\pi {{L}_{\text{R}}}{{n}_{2}}}{2{{\lambda }_{0}}{{A}_{\text{effr}}}}|{{E}_{\text{c}6}}({{L}_{\text{R}}}/4,t{{|}^{2}}=\frac{\pi {{L}_{\text{R}}}{{n}_{2}}}{2{{\lambda }_{0}}{{A}_{\text{effr}}}}|{{E}_{\text{c}5}}({{L}_{\text{R}}}/4,t-{{\tau }_{\text{R}}}){{|}^{2}}$$
(25)
$${{\phi }_{\text{N}4}}(t-{{\tau }_{\text{R}}})]=\frac{\pi {{L}_{\text{R}}}{{n}_{2}}}{2{{\lambda }_{0}}{{A}_{\text{effr}}}}|{{E}_{\text{c}8}}({{L}_{\text{R}}}/4,t{{|}^{2}}=\frac{\pi {{L}_{\text{R}}}{{n}_{2}}}{2{{\lambda }_{0}}{{A}_{\text{effr}}}}|{{E}_{\text{c}7}}({{L}_{2}},t-{{\tau }_{\text{R}}}){{|}^{2}}.$$
(26)

From Eqs. (14) and (16), the output fields E th (z = 0, t) and E drop (z = 0, t) are simple for an iteration of the cavity fields E c4(L/2, t) and E c8(L, t), respectively, with respect to the cavity round-trip time. Thus, we can calculate the dynamic properties when an optical pulse with an arbitrary temporal profile incidences into the nonlinear ring ridge waveguide, from Eqs. (13)–(26).

3 Simulation results and discussion

In a simulation, an SOI racetrack waveguide consists of a bus ridge waveguide of length L b and racetrack ridge waveguide of length L, with the straight arms are the ridge waveguide of lengths L 1 and L 2, which is shown in Fig. 1. The soliton propagation behaviors inside the coupling SOI racetrack ridge waveguide are simulated and studied. The dispersion is the anomalous dispersion, which is obtained from the straight arms of the ridge waveguide of lengths L 1 and L 2. The propagating mode dispersion is a combination of both the material properties and geometry of waveguide. The numerical result of the effective mode index (n eff) is shown in Fig. 2a, which shows the wavelength dependence of n si and the dispersion curve fitting of the dispersion data, in which the Taylor series expansion of up to tenth dispersion is achieved, where the fundamental TE mode is obtained using a FDE-mode solver. The dispersion of the ridge waveguide geometry when the width W is 760 nm and the height H is 220 nm is shown in Fig. 2b. From which the zero dispersion point of the fundamental TE mode is found to be 1536 nm, which is shown in Fig. 2a, while the anomalous dispersion D = +28.89 ps nm−1 km−1 at 1550 nm. Figure 2c is confirmed that these waveguides support the propagation of fundamental TE mode. This result clearly demonstrates that the zero dispersion point of the fundamental TE mode of 1536 nm remains in anomalous dispersion with a ridge waveguide geometry, when both the width W = 760 nm and height H = 220 nm are employed. In the simulation, a 50 fs pulse at a wavelength of 1550 nm is input into the system, where the input peak power is 25 W (≈1 pJ) and the coupling coefficients ĸ i = 0.5. Regarding the SOI racetrack waveguide design is a straight arm’s and relatively short waveguide (L = 0.5 cm), which is sufficient for SC generation that can be used to create a SC with a −30 dB spectra extending to be 2100 nm. It covers the wavelength range from 1043 to 3152 nm with maximum output peak power of 9.82 W, where the extended spectrum of over 1613 nm is obtained within the wavelength range from 1030 to 2643 nm, where the maximum output peak power is 21.19 W for single-couple racetrack waveguide and double-couple racetrack waveguide, respectively. By launching the femtosecond pulses, higher order solitons are formed. The physical processes underlying the SC formation in both the couple racetrack ridge waveguides are related to the soliton fission, Cherenkov radiation, self-phase modulation, and accumulations, which occur inside the racetrack waveguide resonator until the amplitude and spectrum width are stabled and given in the proposed scheme, where the attractive platform is that the on-chip SC broadband sources can be useful for practical applications. The SC generation results are formed by the GNLSE in Eqs. (5)–(6) and Eqs. (21)–(22) with SSFM solves [31, 32] for single-coupler waveguide and double-coupler waveguide, respectively.

An iteration of each cavity field with respect to the cavity round-trip time is performed and calculated, where the dynamic properties of an optical pulse with an arbitrary temporal profile are obtained, which is incidence into the nonlinear ring waveguide using Eqs. (3), (4) and Eqs. (7)–(12) for single-coupler ring ridge waveguide and Eqs. (13)–(20) and Eqs. (23)–(26) for double-coupler ring ridge waveguide are obtained using the parameters in Table 1, where \({{E}_{\text{input}}}(0,t)=\sqrt{{{P}_{0}}}\text{sech}(t/{{T}_{0}})\) is an initial soliton order N, P 0 is the peak power of “sech” pulse of width T p ≈ 1.76T 0, where T p = 50 fs, P 0 = 50 W, the nonlinear length L NL = 1/Re(δ)P 0, β 2 = −0.0381 ps2 m−1, and β 3 = 3.7368 × 10−3 ps3 m−1, and the pump wavelength is 1550 nm, which is the estimated value from Fig. 2, where β 2 and β 3 are the second-order dispersion and third-order dispersion values, respectively. The input parameters satisfy the condition N = (L D/L NL)1/2 order soliton, where the dispersion length is \({{L}_{\text{D}}}=T_{\text{p}}^{2}/\left| {{\beta }_{2}} \right|\) and soliton fission length, L fiss ≈ L D/N [2, 11]. Using the FDE-mode, we obtain A eff = 0.1832 µm2 and Re(δ) = 2πn 2/λ 0 A eff = 132.7729 W−1 m−1, where the dispersion length L D is 21.20983 mm, the nonlinear length L NL is 0.30127 mm, the order soliton N is 8, and the soliton fission length L fiss ≈ 0.25 cm. In Fig. 3, the soliton fission of eighth-order (N = 8) occurs within 0.25 cm after the pulse is launched and the SC is formed soon after. The Cherenkov radiation wavelength for the dominant spectral peak is located at 1.36 µm, which corresponds to the dispersion that produces the long tail, which is shown Fig. 3b. It can be verified by setting f R = 0, which is mostly shifted by the SPM. It is not introduced by the Raman scattering because of its much narrower Raman-gain spectrum, and also the power-clamping effect of TPA [2]. Thus, the primary nonlinear process of the SC generation is the soliton fission and Cherenkov radiation rather than FWM, which is supported by the appearance of idler terms, which can be seen after 0.25 cm of propagation. The apparent asymmetry occurs, because FWM is the balanced in frequency [11], from which the SC extension of over 440 nm at a −30 dB with the coverage wavelength range from 1.33 to 1.77 µm can be observed.

Table 1 Parameters of a racetrack waveguide system used in the simulation
Full size table
Fig. 3
figure 3

Simulation result of SC generation along the ridge waveguide length L = 0.6 cm, where a spectral profiles (color online) and spectral evolution, b temporal profiles (color online) and temporal evolution when a 50 fs pulse propagates as an eighth-order soliton at the pump wavelength of 1550 nm and input power is 25 W of the SOI waveguide with W = 760 nm, H = 220 nm. The dotted curves show, for comparison, the corresponding input profiles

Full size image

3.1 Single-coupler racetrack waveguide

Figure 4 shows the temporal spectral profiles with 100 rounds, where spectral evolution of each round in the output E through port is obtained. The racetrack ridge waveguide consists of the input-through bus ridge waveguide of length L b = n bi λ 0 = 8 × 1.55 = 12.4 µm (without SC), and the racetrack ridge waveguide of length L = L R + L 1 + L 2 = L = 1.00155 cm, the ring radius R = n R λ 0/2π = 10 × 1.55 × 10−6/2π ≈ 2.4669 µm, L R = 2πR ≈ 0.1550 µm, and the straight arms of the ridge waveguide lengths L 1 = L 2 = n b λ 0 = 0.5 cm, where n R, n bi, and n b are the integer, which is shown in Fig. 1a. Figure 5 shows the temporal and spectral profiles inside a ring of the racetrack ridge waveguides, where the outputs E 1–E 6 and the output E through port for the first, second, and third round trips are obtained. The first round trip of propagation is as shown in Fig. 4a and d, where the input soliton peak power of the E input is 25 W, with the coupling coefficient κ = 0.5, and the induced E 1 and E 2 are 12.41 W each. The observed temporal spectrum of the propagation pulses within a short ring ridge waveguide (L R/4) by SMP, when E 2 propagates into the ridge waveguide length L 1 = 0.5 cm, the induced process of the SC generation is the soliton fission and Cherenkov radiation corresponding to Fig. 3, the spectrum is extended over 800 nm with −30 dB, where the maximum output peak power E 3 is 3.05 W. The induced E 4 is 3.05 W by SMP (L R/2) which propagates into E 5 within the ridge waveguide, and the length L 2 is 0.5 cm, where the observed spectra are broadened by the SPM due to low peak power of E 4. The obtained simulation result is verified by the ridge waveguide length L = 0.5 cm, with the maximum input is 3.05 W, in which the spectrum extends slightly. The E 5 spectrum is extended over 1800 nm with a −30 dB, where the maximum peak power is 4.55 W. The verified data are set by f R = 0, β TPA = 0 and n 2 < 1000 octaves that have the similar feature. The E through port outputs of both E 6 and E input are modulated to induce the SC spectrum extension, where the extension over 1800 nm with a −30 dB is obtained, where the maximum peak power is 12.41 W. The second round trip of propagation is as shown in Fig. 4b and e. The modulation of both input power peaks of the first round coupling propagation are the E input and E 6 are 25 and 4.55 W, respectively, where the induced E 1 can be observed, where the temporal peak power is located at 17.41 W, in which the many small peak power was seen. The maximum power peak of the coupling between E input and E 6 is 2.7 W, in which the E 1 of second round is induced. E 1 propagates to E 2 by SMP, which can be observed that the temporal spectrum is similar. The E 2 propagates to the ridge waveguide length L 1, which can be observed that the spectrum reduces when it is compared with the E 2 because of the spectra, it is extended by the peak power, which is located at 17.41 W corresponding to the result in Fig. 2 and on the SC for 2.7 W. The E 3 spectrum is extended over 880 nm with −30 dB and the maximum E 3 power peak is 4.99 W, when E 3 is propagated to E 4, the power peak is 4.99 W by SMP.

Fig. 4
figure 4

(Color online) a Temporal and b spectral profiles at the output E through port, c spectral evolution at the output E through port of each round trips for SOI single-coupler racetrack ridge waveguide at 100 rounds, when 50 fs pulse at pump wavelength of 1550 nm. The dotted curves show, for comparison, the corresponding input profiles

Full size image
Fig. 5
figure 5

(Color online) a–c Temporal, d–f spectral profiles inside ring of the racetrack ridge waveguide of E 1 to E 6 and the output E through port for the first round trip, the second round trip, and the third round trip, respectively, for SOI single-coupler racetrack ridge waveguide, when 50 fs pulse with the pump wavelength of 1550 nm. The dotted curves show, for comparison, the corresponding input profiles

Full size image

E 4 propagates to the ridge waveguide length L 2, where E 5 is induced, which can be observed that the spectra are broadened by the SPM and the SC of propagation. From which the spectrum is extended over 2470 nm with a −30 dB with the maximum power peak is 5.67 W, and E 6 is 4.55 W. The outputs at E through port of both E 6 and E input are modulated from which the SC spectrum can be extended over 2398 nm with a −30 dB, where the maximum peak power is 8.64 W, as shown in Fig. 4. In the third round trip of propagation, the nonlinear process drives the SC to generate a similar feature with the second round trip corresponding to the results in Fig. 4c and f but the maximum power peak increases within the racetrack waveguide, where at the 30, 50, and 100 rounds, the maximum power peaks of E 1–E 6 and E through are 15.56, 15,56, 11.01, 11.01, 10.6, 10.69, and 9.82 W, respectively, which demonstrates that the maximum peak power has the same values due to the accumulated E inside the racetrack waveguide resonator (L = (n R + n b)λ 0) until the amplitude is stabled [29]. For the 100 rounds of the stable E throgh port, the SC spectrum is extended over around 2100 nm with −30 dB, which covers a wavelength range from 1043 to 3152 nm, which is extended over 4.77 octaves with ridge waveguide length L of 0.5 cm, which is the same peak power corresponding to Fig. 2.

3.2 Double-coupler racetrack waveguide

Figure 6 shows the temporal spectral profiles with 100 rounds of spectral evolution, where each round of the E through and E drop port output is obtained. The add-drop racetrack ridge waveguide consists of both the input-through and add-drop bus ridge waveguide of length L b, the racetrack ridge waveguide length is L, the ring radius R, and the straight arms of the ridge waveguide length are L 1 = L 2, the initial condition and input parameter are used similarly to the single-coupler racetrack waveguide, which is shown in Fig. 1b. Figure 7 shows temporal and spectral profiles inside a ring of the double-coupler racetrack ridge waveguide of E 1 to E 8, from which the E through and E drop port outputs of the first, second, and third round trips. The first round trip of propagation is as shown in Fig. 7a and d, the input soliton power peak of the E input is 25 W, with the coupling coefficient κ = 0.5, the induced E 1 = 12.41 W, and E 2 = 12.41 W. The observed temporal and spectrum of propagation of a short ring ridge waveguide (L R/4) by SMP is seen, then E 2 propagates into the ridge waveguide length L 1 = 0.5 cm, where the SC generation is induced corresponding to result in Fig. 2, where the SC spectrum is extended over 790 nm at −30 dB, the output maximum power peak of E 3 = 4.54 W, then the induced E 4 = 4.54 W by SMP (L R/4), from which the nonlinear process of E 5 to E 8 is a similarly occurred within the E 1 to E 4 results. The E through and E drop port outputs of both E 8 and E input are similar to E 4 and E add outputs, respectively, which are modulated to induce both SC spectra extending over 780 nm with a −30 dB and the maximum power peak of both 19.10 W. The second round trip of propagation is as shown in Fig. 7b and e. Both the input power peaks between the E input and E 8 W for the first round of propagation coupling to the induced E 1 of second round are modulated, which can be observed that the temporal maximum power peak is located at 11.55 W and many small coupling power peaks are seen.

Fig. 6
figure 6

(Color online) a Temporal and b spectral profiles of the E through port output of 100 rounds, d temporal and e spectral profiles at the E drop port output of 100 rounds, e, f spectral evolution of both the E through and E drop port output of 1 round to 100 rounds, for SOI single-coupler racetrack ridge waveguide, when 50 fs pulse with pump wavelength of 1550 nm. The dotted curves show, for comparison, the corresponding input profiles

Full size image
Fig. 7
figure 7

(Color online) a–c Temporal, d–f spectral profiles inside ring of the racetrack ridge waveguide of E 1–E 8 and the E through and E drop port outputs for the first round trip, the second round trip, and the third round trip, respectively, for SOI single-coupler racetrack ridge waveguide, when 50 fs pulse of the pump wavelength of 1550 nm. The dotted curves show, for comparison, the corresponding input profiles

Full size image

E 1 propagates to E 2 by SMP, where the both maximum power peaks are 11.55 W, and the E 1 and E 2 spectra are 780 nm with −30 dB. Then, E 2 propagates to the ridge waveguide length L 1, the E 3 maximum power peak is 8.83 W, and the spectrum is extended over 1710 nm with −30 dB. The E 4 maximum power peak is 8.83 W by SMP (L R/4), where the nonlinear process similarly occurs to the propagation of solitons from E 5 to E 8, from which the maximum power peaks are 11.55, 11,55, 8.83, and 8.83 W, the output spectra are 780, 780, and 1711 nm with −30 dB, respectively. The E through and E drop port outputs of both E 8 and E input are a similar to the propagation of E 4 and E add, which are modulated to induce both SC spectra extending over 1656 nm with a −30 dB; the maximum power peak are 17.7 W. In the third round trip of propagation, the nonlinear process drives to obtain the SC generation, which is a similar feature with the second round trip, which is shown in Fig. 7c and f, but the maximum peak power increases within the racetrack waveguide the maximum power peaks of E 1–E 8, E through, and E drop are 10.19, 10.19, 10.16, 10.16, 10.19, 10.19, 10.16, 10.16, 20.90, and 20.90 W, respectively. For the 100 rounds, the maximum power peaks of E 1–E 8, E through and E drop are 10.76, 10.76, 12.13, 12.13, 10.76, 10.76, 12.13, 12.13, 21.19, and 21.19 W, respectively, which demonstrates that the maximum power peaks are similar between E 1–E 4 and E 5–E 8 because of E that are accumulated inside the racetrack waveguide resonator, the spectra extended of both E through and E drop port outputs are 1560 nm with −30 dB. For the 100 rounds of the stable E throgh and the same E drop port outputs, the SC spectrum is extended over around 1613 nm with −30 dB, covering a wavelength range from 1030 to 2643 nm, and extending over the 3.66 octaves with the ridge waveguide length L of 0.5 cm. The physical process of supercontinuum in the both coupled racetrack ridge waveguides is related to the soliton fission and Cherenkov radiation, where the self-phase modulation and accumulations occur inside the racetrack waveguide resonator until the amplitude and spectrum width are stable. However, there are many disadvantages for single and double couple SIO racetrack ridge waveguides owing to the noise in the spectrum. Recently, Li et al. [33] and Dudley et al. [34] have shown numerically that the noise properties of the SC can be significantly improved in terms of both of the coherence and intensity stability and effectively suppressed through the use of a sliding frequency filter. These studies may be used as a guide to improve the efficiency of the SC. The proposed scheme can be an attractive platform for on-chop SC broadband sources and proved useful in practical applications.

4 Conclusion

We have numerically demonstrated MIR SC generation by single-couple racetrack ridge waveguide and double-couple racetrack ridge waveguide with dispersion engineered to have anomalous dispersion silicon-on-insulator ridge waveguide, with a dispersion of D = + 28.89 ps nm−1 km−1 at near wavelength of 1550 nm. Using a 50 fs pulse with low peak power of 25 W, the broadband SC spectrum is extended over around 2100 nm with a −30 dB, covering a wavelength range from 1043 to 3152 nm, and extending over 4.77 octaves with ridge waveguide length for single-couple racetrack ridge waveguide. In the SC generation of the double-couple racetrack ridge waveguide at the output E throgh and the same E drop port, the SC spectrum extended over around 1613 nm at a −30 dB, covering a wavelength range from 1030 to 2643 nm, and extending over 3.66 octaves with ridge waveguide length. The physical process of supercontinuum generation in both the couple racetrack ridge waveguide is related to soliton fission and generation of Cherenkov radiation, self-phase modulation and accumulations are inside the racetrack waveguide resonator until the amplitude and spectrum width are stabled, the proposed scheme should prove useful in practical applications. The advantage of using the racetrack ridge waveguide for supercontinuum generation over the fiber optic system is that the smaller system can be arranged. However, the small supercontinuum fluctuation may be occurred, which can be neglected due to the high peak power output. Moreover, the compact racetrack system can be made by an on-chip concept, where the large system can be redundancy.