Tunable Terahertz Wavelength Conversion Based on Optofluidic Infiltrated Rib Silicon Waveguides

Terahertz (THz) sources have attracted special attention for various applications. Compared to the standard silica-based optical fibers, silicon waveguides have more advantages such as higher refractive index and lower absorption loss over the THz region. In this paper, for the first time, a rib silicon waveguide based on the photonic crystal (PC) idea is designed and by using the scalar modulation instability (SMI) phenomenon, tunable wavelength conversion for THz wave generation is simulated. By changing the structural parameters such as the air-hole diameter of the PC and infiltration of optical fluids into the air holes, linear and nonlinear characteristics of the waveguide are controlled and hence the generated THz radiation is considered as a tunable source. Simulation presents that the maximum converted wavelength, 326.17 μm, is obtained when the air-hole diameter is set at d = 0.86 μm and the pump wavelength is in the normal dispersion regime. Also, we have infiltrated optical fluids into PC air holes in order to change the dispersion properties of the waveguide while keeping the geometrical parameters unchanged. This led to converted wavelengths of 70.5, 76.8 and 107.5 μm, all located in the THz region. The loss of this waveguide is less than 0.8dBcm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \raisebox{1ex}{$ dB$}\!\left/ \!\raisebox{-1ex}{$ cm$}\right. $$\end{document}, which is less than that of previously reported in similar silicon waveguides.


Introduction
THz radiation as an electromagnetic radiation in the frequency range of 0.1 to 10 THz (wavelength range of 30 μm to 3 mm) has attracted a lot of interest in both science and technology [1]. The THz band which is located between the infrared and microwave region is a low energy and non-ionizing radiation that may penetrate in non-polar and non-metallic materials such as cloth, paper, stone and plastics [2].
The development of efficient THz wave sources is important for measurement applications and biology [3]. There are many methods for generating THz radiation such as: photoconductive antennas [4], large particle accelerators [5,6], quantum cascade lasers [7], and optical rectification in nonlinear crystals [8,9]. The important ones that have been recently used is four-wave mixing (FWM) in silicon waveguides [10].
Compared to the conventional silica-based optical fibers, a rib silicon waveguide is a more consistent structure for generating THz waves through the FWM. In fact silicon has more inherent advantages than silica so that the absorption loss of the silicon is less than 0.23 cm −1 in range of 1.2-6.9 μm and 25-200 μm [11], while absorption coefficient of optical fiber is 5 cm −1 in the THz region. Moreover, silicon nonlinear refractive index is 200 times greater than the silica [12] and also the silicon refractive index (about 3.5) is much larger than the air refractive index (about 1), which leads to a very strong light confinement. Additionally, the crystalline nature of silicon is in such a way that if the transverse magnetic (TM) mode is launched, the stimulated Raman scattering (SRS) does not occur, because this scattering strongly depends on the waveguide geometry and the mode polarization, so SRS cannot occur when an input pulse excites the TM mode [10]. Finally, the rib silicon waveguide is compatible with the CMOS technology which enables large-scale integration with a low cost [10]. Low cost, abundant availability as well as good linear and nonlinear optical properties have made silicon as an ideal material for fabricating nanoscale photonic complex devices [13].
Owing to the interplay of nonlinear effects and dispersion, many nonlinear systems can exhibit an unstable phenomenon known as the modulation instability (MI) which leads to generation of new side bands [14]. MI is a third-order nonlinear phenomenon that can be utilized for wavelength conversion in optical fibers/waveguides [14]. The simplest form of MI is considered as scalar modulation instability (SMI) which has no dependency on the polarization state of the lightwave [14]. The underlying physics of SMI is the production of new photons which is known as Stokes and anti-Stokes. In the case of SMI, in order to satisfy the phase-matching condition, the pump wavelength (λ p ) must be adjusted very close to the zero-dispersion wavelength (ZDW), λ 0 , of the waveguide in either normal (λ p < λ 0 ) or anomalous (λ p > λ 0 ) dispersion regimes [15].
In this paper, for the first time, we design a photonic crystalbased rib silicon waveguide to achive tunable wavelength conversion in the THz region. We examine wavelength conversion via the SMI phenomenon in both the anomalous and the normal dispersion regimes. To this end, we change the structural parameters (both geometry and optofluidic infiltration) of the waveguide to engineer the dispersion and nonlinear characteristics and hence to control the phase-matching condition and finally tune the converted wavelengths in THz region. As the wavelength conversion strongly depends on group-velocity dispersion (GVD) parameter, β 2 , the merit of the optofluidic infiltration is to obtain tunable wavelength conversion while keeping the waveguide geometry unchanged.
The optical properties of fluids are generally very wide. An optical fluid can be applied to any photonic device which has an air hole in its structure [16,17]. The fluid interacts with the confined field and changes the dispersion properties of device [17]. Fluid can be successfully infiltrated in photonic crystal (PC) and microstructures and change the waveguide properties. Most recently, the optofluidic infiltration in photonic crystal fibers (PCFs) has been used for wavelength conversion in the telecommunication region [15]. Therefore, in this work, we intend to investigate the wavelength conversion in the PCbased rib silicon waveguide to access the THz region not only with the help of geometrical changes but also with optofluidic infiltration possessing various refractive indices. In fact, by infiltrating the optofluidic into the PC air holes of the waveguide and changing the optical fluid's refractive index, n F , the dispersion properties can be changed and thereby the converted wavelengths are tuned [18].
The paper structure is as below. In Section 2, the theoretical framework of the wavelength conversion in both anomalous and tnormal dispersion regimes based on the SMI is presented. In Section 3, we design a PC-based rib silicon waveguide and determine its linear and nonlinear characteristics. In Section 4, the wavelength conversion is simulated when geometrical parameters are changed or when the waveguide is infiltrated with various optical fluids. Finally, the paper is concluded in Section 5.

Theory of the Study
Modulation instablity (MI) is a general feature of the lightwave propagating in an optical medium where dispersion and nonlinear effects are simultaneously present. Based on MI, when a pump wave with the angular frequency of ω p is input into the optical waveguide, the generated sidebands (Stokes and anti Stokes) are symmetrically located with resect to the pump frequency (zero detuning); so that the energy conservation condition will be met, i.e. 2ω p = ω a + ω s . The angular frequencies of the sidebands, Stokes and anti Stokes, are defined respectively by ω s and ω a [15].
The frequency detuning of the sidebands with respect to the pump frequency is given by Ω = ω p − ω s = ω a − ω p where ω s ¼ 2πc λ s and ω a ¼ 2πc λ a , and the Stokes and anti -Stokes wavelengths are presented by λ s and λ a , respectively [14,15].
Linear stability analysis, phase-matching condition and FWM can be used to describe the SMI in an optical medium [14]. From the phase-matching condition, one can readily determine the peak frequency detunings Ω max at which the maximum gain of SMI occurs, thus Δk = β 2 Ω 2 + 2γp 0 = 0. Therefore, the position of peak frequency detunings in the anomalous dispersion regime are easily determined as [14,15]: where p 0 is the peak power of the input pump and β 2 is the second-order dispersion coefficient of the waveguide which is calculated at pump wavelength, λ p . Also, γ denotes the nonlinear parameter of the waveguide. In the normal dispersion regime, the mode propagation constant, β, should be expanded up to fourth order β 4 to determine Ω max as [14,15]: where the phase-matching criterion, It should be noted that in the anomalous dispersion regime where β 2 is negative, the first term of Δk = β 2 Ω 2 + 2γp 0 can be cancelled out by the second term (which is always positive), so the phase-matching condition is satisfied. On the other hand, in the normal dispersion regime, since β 2 is positive, the negative term corresponds to β 4 so that the phase-matching condition The intensity gain coefficient of the SMI in the anomalous dispersion regime can be calculated as [14,15]: Both theoretical and experimental investigations show that the SMI may occurs in the normal dispersion regime as well where the higher-order dispersion coefficients are taken into account. Therefore, considering dispersion terms up to the fourth-order coefficient β 4 , the intensity gain coefficient in the normal dispersion regime becomes [14,15]: It will be shown in Sectoin 4.2 that the converted wavelengths (generated sidebands) in the normal dispersion regime can be located in the THz region.

The PC-Based Rib Silicon Waveguide Design
In this section, we design for the first time a PC-based rib silicon waveguide with suitable dispersion characteristics and investigate the wavelength conversion in this waveguide for a given input pump wave. If the circular air-hole diameter of PC changes and hence the waveguide dispersion characteristics vary, the converted wavelength range will be tuned. The waveguide which is studied in this work is made of silicon in which wavelength-dependent refractive index is given as [19]: The guidance mechanism of the waveguide is the same as the standard optical fibers, which is the total internal reflection [20].
Our proposal of the PC-based rib silicon waveguide is shown in Fig. 1 (a). The PC is arranged as a rectangular array of circular air holes inside the silicon waveguide background and is located at a distance of 14 μm from the top of the waveguide. The lattice pitch of the PC is fixed at Λ = 4.80 μm and the wavelength conversion process is controlled by changing the diameter of the air holes. In Fig. 1 (b), the mode which is well confined within the waveguide core, is located above the PC air holes. The air-hole diameter, d, can be changed which leads to change of waveguide dispersion. It should be noted that a similar waveguide has been recently reported by Pakarzadeh et al. for the pulse propagation in the THz region [21].
We have simulated the fundamental mode field distribution and mode propagation constant of the proposed waveguide [22,23]. A full-vectorial modal analysis based on the finite-difference eigenmode is used to obtain the field distribution of the fundamental mode and propagation constant of the waveguide [24,25]. The method discretizes the waveguide cross section using a 100 numbers of mesh cells and mesh step of 1 × 10 −6 μm in each direction and formulates Maxwell's equations in terms of a matrix eigenvalue problem. Perfectly matched layers with 18 μm width are used as absorbing boundaries placed on the boundary of the waveguide. Conformal meshing was also applied to all material interfaces to reduce meshing errors and increase the numerical accuracy of the simulations.  It should be noted that our proposal is inspired by a silicon membrane waveguide reported by Z. Wang et al. to generate THz radiation which makes it possible to create new wavelengths only by changing the waveguie geometry [10]. However, our proposed PC-based rib silicon waveguide has more advantages than the silicon membrane waveguide or other designs [11], with unique features such as freedom in design owing to the PC air holes and much lower loss (less than 0.8 dB cm ,) in operating wavelengths. In addition, our waveguide enables generating new wavelengths not only by changing the geometry, but also by infilterating optofluidic into the PC air holes. This makes our waveguide more suitable for industrial and commercial purposes in comparison to previous ones. Figure 2 shows dispersion curves, D, as a function of wavelength in the range of 4.0 to 7.0 μm for various air-hole diameters, d. As it is evident, by decreasing the diameter of the air holes, the zero-dispersion wavelength (ZDW) shifts toward larger wavelengths; therefore, by engineering the dispersion curves, it is possible to tune the frequency shift and also the gain of the converted wavelengths (see Eqs. (1)-(4)). The dependency of the ZDW on the air-hole diameter is given in Table 1.

Tunable Wavelength Conversion in the PC-Based Rib Silicon Waveguide
We simulate the tunable wavelength conversion by changing the waveguide geometry as well as by optofluidic infiltration in both normal and anomalous dispersion regimes. Then, the best situation for obtaining the wavelength conversion in the THz region is determined.

Tunable Wavelength Conversion Via Geometrical Change in Anomalous Dispersion Regime
As already mentioned, in the anomalous dispersion regime, β 2 becomes negative. By using the dispersion curves, D, shown in Fig. 2 and Table 1, if the pump wavelength λ p sets at 6.60 μm, the second-order dispersion coefficient β 2 ¼ − λ 2 2πc D, becomes negative. By taking the second derivative of the β 2 relative to the angular frequency, the fourth-order coefficient, β 4 , can be obtained which is reported in ref. [26].
The GVD parameter β 2 , effective mode area A eff (Or more precisely, the area of the intense spot seen in Fig. 1 (b)) and nonlinear parameter of the PC-based rib silicon waveguide, γ, are listed in Table 2 for various air-hole diameters calculated at  6.60 μm (because, according to Table 1, at this wavelength, for all air-hole diameters, the anomalous dispersion regime is fulfilled, i.e., λ p > ZDW), and γ ¼ By decreasing the air-hole diameters of the PC, the corecladding index contrast decreases soA eff γ decreaese as well. Core-cladding index contrast means the difference between the refractive index of the core and that of the cladding so when the air-hole diameter of the PC in the cladding decreases, the core-cladding index contrast reduces as well. This is because of the core purely consists of Si with the refractive index of~3.5 while the cladding consists of Si and/or air. When the air-hole diameter decreases, the contribution of the air in the cladding decreases and hence the refractive index of the cladding approaches to Si refractive index. This leads to the decrease in core-cladding index contrast.
Based on Eq. (1), in the anomalous dispersion regime, when the value of β 2 is smaller, the greater value of Ω max is obtained. Therefore, by decreasing the amount of β 2 , the frequency shift of the sidebands increases. This can be seen in Table 2 where the smallest value of β 2 corresponds to the smallest diameter of the air hole. Table 3 lists the Stokes and anti-Stokes frequencies (ω s and ω a ) as well as the wavelength shifts Δλ, where the pump wavelength is located at λ P = 6.60 μm and the pump peak power is fixed at 500 W. According to Eq. (1), for smaller β 2 , the larger Ω max is occurred, hence the larger wavelength shift of 1.22 μm is obtained. Figure 3 shows the gain coefficients as a function of the frequency detuning (Ω) simulated based on Eq. (3) for different air-hole diameters. According to Fig. 3, for d = 0.86 μm, the largest detuning corresponds to the smallest value of β 2 where the gain peak of the sidebands occurs. This has been already seen in Tables 2 and 3.
Based on Fig. 4, the gain coefficient increases. Additionally, increasing the pump power leads to shift the peak frequency detuning Ω max to the larger values. This is because of increased nonlinear interaction and phase matching between the interacting waves. It is worth noting that the practical values of λ p and P p can be readily obtained via mid infrared sources operating between 3 and 8 μm; for example, the second-harmonic generation of the intense CO laser using the ZnGeP 2 crystal or coherent radiation based on differencefrequency generation from tunable optical parametric oscillators (OPOs) [27,28].
As discussed above, in the PC-based rib silicon waveguide, it is possible to tune the converted wavelength by changing the waveguide geometry. In this case, the largest sideband peak was obtained for d = 0.86 μm at which β 2 has the smallest value, and correspondingly the largest wavelength shift of Δλ =1.22 μm is observed. Also, by increasing the pump peak  . 3 Gain coefficient versus the frequency detuning for various air-hole diameters in the anomalous dispersion regime. The pump wavelength is set up at λ P = 6.60 μm and the pump peak power is P p = 500 W Fig. 4 The gain coefficient versus the frequency detuning for different pump peak powers in the anomalous dispersion regime. The air-hole diameter is fixed at d = 0.86 μm power for this fixed air-hole diameter, the frequency shift increases. Therefore, it is possible to control the location of the converted wavelengths by changing the air-hole diameter as well as the input pump peak power. However, as it is seen from the simulation results, the largest wavelength shift occurred in the anomalous dispersion regime cannot reach the THz region; therefore, one should examine the wavelength conversion in the normal dispersion regime.

Tunable Wavelength Conversion Via Geometrical Change in Normal Dispersion Regime
In this section, we change the pump wavelength from λ P = 6.60 μm to λ P = 5.60 μm and repeat simulations but for normal dispersion regime. As it can be seen from Table 1, when the pump wavelength is set at 5.60 μm, for all air-hole diameters the normal dispersion regime is fulfilled, i.e., λ p < ZDW. In fact, in this regime, β 2 is positive and β 4 is negative [26]. The dispersion and nonlinear parameters of the PC-based rib silicon waveguide are listed in Table 4; while the wavelengths shift and Stokes and anti-stokes frequencies are given in Table 5.
As before, by decreasing the air-hole diameter, the corecladding index contrast decreases soA eff A eff increases and simultaneously γ decreases. According to Eq. (2), in the normal dispersion regime, the peak frequency detuning depends on both the second-and the fourth-order dispersion coefficients so that for the largest β 2 and the smallest β 4 , Ω max will become maximum. As it is seen in Table 4, the largest β 2 and simultaneously the smallest β 4 are related to the smallest value of air-hole diameter of d = 0.86 μm. This leads to the maximum wavelength shift of Δλ = 320.57 μm which is given in Table 5.
As a result, it is desired to work in the normal dispersion regime to access new wavelengths in the THz region.

Tunable Wavelength Conversion Via Optofluidic Infiltration
As discussed above, one can control the waveguide dispersion curves by changing the geometrical parameters and hence tune the generated wavelength range. However, by infiltrating optical fluids with different refractive indices as the postfabrication method, one can easily engineer linear and nonlinear characteristics of the waveguide to access the desired wavelength range. From the practical point of view, it is a more appropriate and optimal tool that enables us to achieve significant results via small manipulations. The variation of waveguide geometry is not an optimal method for reaching different wavelength regions, because one cannot practically use a waveguide with different air-hole diameters. Therefore, we should think about changing the dispersion properties of the waveguide in a different way, such as infiltrating optical fluids to access different wavelength regions while keeping the waveguide geometry unchanged.
In this section, we set the waveguide dimensions at fixed parameters of Λ = 4.8 μm and d = 0.86 μm, to ensure the largest wavelength shift as discussed in the previous section; and then infiltrate different optical fluids with various refractive indices in the PC air holes. To this end, we infiltrate water  or other optical fluids into the PC air holes and then simulate β 2 and β 4 curves after that, compare them to the case of empty air holes (without infiltration). This is shown in Fig. 5 (a) whereby the refractive index of the holes increases, so the dispersion curve is changed and ZDW is shifted toward greater wavelengths. For instant, for the empty air hole, the ZDW is about 6.40 μm while for the optical fluid of C 8 H 18 this value is shifted to 6.56 μm.
As shown in Figs. 5 (b) and (c), β 2 and β 4 are plotted for both cases when the waveguide is uninfiltrated or infiltrated with various optical fluids. Evidently, by changing the refractive index of the fluid infiltrated into the PC air holes, the second-and fourth-order dispersion coefficients are changed and hence the wavelength shift is tuned according to Eq. (2).
Based on the obtained results in Sections 4-1 and 4-2, we obtain a larger wavelength shift for the normal dispersion regime. Therefore, in this section, simulations are only performed in the normal dispersion regime. Here, three different optical fluids including water, C 5 H 12 and C 8 H 18 are infiltrated into the PC air holes. The refractive indices of these fluids are as n water = 1.33400 [29], n C 5 H 12 ¼ 1:35377 [30] and n C 8 H 18 ¼ 1:39125 [30]. The pump wavelength is set at λ P = 5.9 μm, where for all cases the normal dispersion regime, λ P < ZDW, takes place and therefore the second-order dispersion coefficient is positive and simultaneously the fourthorder dispersion coefficient is negative.
The linear and nonlinear parameters for all cases of uninfiltrated and infiltrated PC holes are listed in Table 6 and corresponding wavelengths shifts, Stokes and anti-stokes frequencies are given in Table 7.
By increasing the refractive index of the air holes through the injection of optical fluid, the index contrast between the core and cladding decreases, soA eff A eff increases and γ decreases. As mentioned above, according to Eq. (2), in the normal dispersion regime, the peak frequency shift depends on both second-and fourth-order dispersion coefficients thus when β 2 is larger and β 4 is smaller, Ω max will be larger than before as well. As seen in Table 6, the largest value of β 2 and the smallest value of β 4 correspond to the case of C 8 H 18 where according to Table 7, this leads to the largest wavelength shift of Δλ = 101.58 μm. This largest wavelength shift corresponds to the maximum converted wavelength of λ = 107.48 μm which lies in the THz region. Figure 6 shows a gain coefficient versus the frequency detuning simulated based on Eq. (4) for both cases of C 8 H 18 optofluidic infiltration and empty air holes. As it is obvious, by infiltrating the optical fluids, the sideband peak frequency detuning is larger compared to that of the empty air hole. Therefore, the designed PC-based rib silicon waveguide can generate different wavelength shifts by changing the refractive index of the air holes via optofluidic infiltration. This provides Table 7 Stokes and anti-Stokes frequencies as well as the wavelength shifts of the PC-based rib silicon waveguide infiltrated by optical fluids with different refractive indices. The pump wavelength is set at λ P = 5.9 μm in the normal dispersion regime   Fig. 6 Gain coefficient versus the frequency detuning for thr empty air hole (blue line) and the optofluidic infiltrated air hole (red line). The pump wavelength is set at λ P = 5.9 μm in the normal dispersion regime with the pump peak power of P p = 2 kW an efficient method to access the THz region without changing the geometrical parameters.

Conclusion
In this paper, a rib silicon waveguide based on the photonic crystal (PC) idea was designed for the first time and tunable wavelength conversion for generating the THz radiation was simulated. As a result, the converted frequencies/wavelengths can easily lie in the THz region. When the pump wavelength is set at λ p = 5.60 μm in the normal dispersion regime and for the fixed lattice pitch of Λ = 4.80 μm, as the air-hole diameter is changed from d = 4.60 μm to d = 0.86 μm, the converted wavelength is tuned from λ = 2.15 μm to λ = 326.17 μm, respectively. This is while for the pump wavelength of λ p = 5.9 μm and the fixed lattice pitch of Λ = 4.80 μm and air-hole diameter of d = 0.86 μm, by infiltrating the optical fluids with different refractive indices in the PC air holes, the converted wavelength is tuned over the THz range from 70.5 μm to 107.5 μm.
Author's Contributions All authors contributed to the study conception and design. Material preparation, simulation results and analysis were performed by H. Pakarzadeh and S. Hosseinabadi. The first draft of the manuscript was written by S. Hosseinabadi and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Data Availability Data are available on request from the authors.
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