## Abstract

Terahertz (THz) wave generation via four-wave mixing (FWM) in silicon membrane waveguides is theoretically investigated with mid-infrared laser pulses. Compared with the conventional parametric amplification or wavelength conversion based on FWM in silicon waveguides, which needs a pump wavelength located in the anomalous group-velocity dispersion (GVD) regime to realize broad phase matching, the pump wavelength located in the normal GVD regime is required to realize collinear phase matching for the THz-wave generation via FWM. The pump wavelength and rib height of the silicon membrane waveguide can be tuned to obtain a broadband phase matching. Moreover, the conversion efficiency of the THz-wave generation is studied with different pump wavelengths and rib heights of the silicon membrane waveguides, and broadband THz-wave can be obtained with high efficiency exceeding 1%.

©2012 Optical Society of America

## 1. Introduction

The development of efficient and compact sources of THz-wave is of great interest for applications in various fields such as applied physics, communications, sensing, and life sciences [1]. The difference-frequency generation (DFG) in nonlinear optical crystals is an important technique for coherent THz-wave generation [2–7]. However, it is difficult to increase the conversion efficiency for DFG based THz-wave generation, because most of nonlinear optical crystals have a large absorption in the THz–wave region. Surface-emitting THz-wave generation can be used to overcome the high absorption loss [8–11]. Unfortunately, this method requires a specially designed crystal, and the interaction length is limited by the size of the base material [12].

To cope with these difficulties, Suizu *et al.* proposed a way to generate THz-waves in an optical fiber via FWM process, which is a promising method for realizing a reasonable THz-wave source [12]. FWM in silicon waveguide had been studied not only theoretically but also experimentally [13, 14]. Compared with conventional fiber, the silicon rib membrane waveguide will be a more viable structure for THz-wave generation via FWM. There are five major inherent advantages. First, the silicon membrane has an absorption loss below 0.23 cm^{−1} over 1.2-6.9 μm and 25-200 μm [15], while the absorption coefficient of the optical fiber in the THz-wave region is about 5 cm^{−1}. Second, the nonlinear refractive index n_{2} of silicon is about 200 times larger than that of silica [16]. Third, the refractive index of silicon (around 3.5) is much larger than that of air, which implies a much stronger light confinement. Fourth, the crystalline nature of silicon that makes stimulated Raman scattering (SRS) depend strongly on the waveguide geometry and mode polarization, and SRS cannot occur when an input pulse excites the TM mode [16]. Fifth, the silicon membrane waveguide is also CMOS compatible and enable low-cost large-scale integration [15]. Moreover, the silicon waveguide also have been modified to show second-order nonlinearity at technically relevant levels [17, 18]. Even the THz-wave generation based on DFG has already been experimentally demonstrated in a silicon waveguide by Waechter *et al* [19]. Despite this progress, there is still a strong motivation to investigate the THz-wave generation based on FWM due to the high third-order nonlinearity of silicon waveguide.

In this paper, we investigate efficient THz-wave generation via FWM in silicon membrane waveguides using Mid-infrared pump and signal waves. The organization of the paper is as follows. In Section 2, we analyze the collinear phase matching condition and phase matching bandwidth with the dispersion relation of silicon membrane waveguides. In section 3, we numerically investigate the conversion efficiency of the THz-wave generation for different pump wavelengths and rib heights of the waveguides. Finally, we summarize this paper in Section 4.

## 2. Phase matching condition and phase matching bandwidth for THz-wave generation

We use degenerate FWM to generate THz-wave, which typically involves two pump photons at angular frequency *ω*_{p} passing their energy to a signal wave at angular frequency *ω*_{s} and a THz-wave at angular frequency *ω*_{THz}. Figure 1
shows the energy conservation diagrams and phase-matching condition for collinear configuration, which ensures that the THz-wave is generated through FWM and grows while copropagating with the pump and signal beam. These relationships can be written as the following equations [20]:

*k*

_{p},

*k*

_{s}and

*k*

_{THz}represent the propagation wave number of pump, signal and THz-wave, respectively.

*k*

_{NL}is the nonlinear phase mismatch, which induced by self phase modulation (SPM) and cross phase modulation (XPM) [21]. We can also define

*k*

_{L}=

*k*

_{s}+

*k*

_{THz}-2

*k*

_{p}as the linear phase mismatch due to dispersion [22]. Since the signal and THz-wave are located symmetrically around the pump frequency, the linear phase mismatch only depends on even-order dispersion parameters as [20]

*β*

_{2p}is the group-velocity dispersion, and

*β*

_{2mp}is the even-order dispersion at the pump frequency. Ω

_{sp}=

*ω*

_{s}-

*ω*

_{p}=

*ω*

_{p}-

*ω*

_{THz}is the signal-pump (or pump-THz) frequency detuning, which is a large value due to

*ω*

_{THz}<<

*ω*

_{p}. Since Ω

_{sp}is so large, the higher-order dispersions become important for the phase-matching [20]. However, the higher-order dispersions cannot be accurately calculated using numerical method. Therefore, Eq. (3) can’t be used to calculate the linear phase mismatch, and we use it only to explain the influence of the higher-order dispersions on the linear phase mismatch in the following part of the paper. The linear phase mismatch

*k*

_{L}can be calculated using the Equation:

*k*

_{L}=

*k*

_{s}+

*k*

_{THz}-2

*k*

_{p}= (

*n*+

_{s}ω_{s}*n*-2

_{THz}ω_{THz}*n*)/

_{p}ω_{p}*c,*where

*n*,

_{s}*n*, and

_{THz}*n*represent the fundamental TM mode effective indices of the signal, THz-wave, and pump.

_{p}*c*is the speed of light in vacuum.

Figure 2 shows the dimension of silicon membrane waveguide, in which the rib is suspended over an air filled cavity comprising the lower cladding, whilst the upper cladding is also air. This waveguide is constructed by etching away the buried-oxide insulating material in silicon on insulator (SOI) locally under the rib [15]. The waveguide should be designed not only confining the THz-wave, but also satisfying the single-mode condition [23]:

where W/H is the ratio of the rib width to overall rib height, and*r*is the ratio of the slab height (H-h) to overall rib height. Here we set h = H/2 and r = 0.5.

The dimension of the waveguide is optimally designed to realize the collinear phase matching. The rib waveguides with width of 12 μm and rib heights varied from 14 μm up to 17 μm can satisfy the single-mode condition. To determine the performance of these waveguides, we need to simulate the mode profiles at the THz-band using a finite-difference mode solver [24]. There are two assumptions before simulation. The first is that the pump and signal waves are both fundamental TM mode, thus the SRS can be neglected [16]. The second is that the pump and signal waves are Mid-infrared waves and exceed 2.2 μm, leading to a negligible of two-photon absorption (TPA) and free-carrier absorption (FCA) and thus enabling efficient parametric generation [25, 26]. The fundamental TM mode profiles of the silicon membrane waveguide at the wavelength of 35 μm for several rib heights are shown in Fig. 2. It is clear that the designed waveguides can confine THz-wave.

To realize phase matching, we first simulate the dispersion of the waveguides. For the silicon membrane waveguides mentioned above, the fundamental TM mode effective indices *n _{eff}* as a function of pump wavelength are numerically determined using the finite-difference mode solver. Note that in the calculations, the material dispersion of the silicon is determined by a Sellmeier equation mentioned in [27].The dispersion relation is then calculated from

*β(ω) = n*. Higher-order dispersion is finally calculated via numerical differentiation from

_{eff}(ω)ω/c*β*, and the results of the

_{n}= d^{n}β/dω^{n}*n*and even-order dispersion are shown in Fig. 3 . The zero dispersion wavelengths of the four waveguides are 6 μm, 6.1 μm, 6.2 μm, and 6.3 μm, respectively. The fourth-order dispersion

_{eff}*β*

_{4}and sixth-order dispersion

*β*

_{6}are all negative for the pump wavelength ranging from 4 μm to 7 μm. In the calculation process, the initial errors would be amplified when taking so many derivatives of discrete date and the curves of higher-order dispersion will be distorted for a high spectral resolution. As we only concern the sign of higher-order dispersions other than precision, we can decrease spectral resolution to obtain smooth curves as shown in Fig. 3. Despite the value of higher-order dispersions can’t be accurately calculated, we can distinguish the sign of higher-order dispersions, which will be used to explain the change of linear phase mismatch with different pump wavelengths for a large signal-pump frequency detuning Ω

_{sp}in the following part.

In order to determine the effective mode area *A*_{eff}, we calculate the mode profiles at wavelength from 2 μm to 7 μm. Figure 4
shows the fundamental TM mode profiles of the waveguides with different heights. Because of the little variation of the mode profiles for the Mid-infrared waves, we use 4.3 μm as the operation wavelength to simulate the mode profiles and calculate the effective mode areas. The effective mode areas *A*_{eff} of the waveguides are 42 μm^{2}, 48 μm^{2}, 54 μm^{2} and 58 μm^{2}, respectively.

The linear phase mismatch *k*_{L} as a function of THz wavelength in a waveguide with rib height of 15 μm is shown in Fig. 5
. It is found that the linear phase mismatch is larger for a pump wavelength located in the anomalous GVD regime (6.3 μm) than located in the normal GVD regime (5 μm, 4.5 μm). The Eq. (3) is used to explain this phenomenon. As Ω_{sp} = *ω*_{p}-*ω*_{THz} is large, the second term of Eq. (3) cannot be ignored and must be a large negative figure because *β*_{4p} and *β*_{6p} are negative from Fig. 3(c) and Fig. 3(d). If the pump wavelength located in the anomalous GVD regime (*β*_{2p}<0), the sum (*k*_{L}) of the first and second term of Eq. (3) must be a larger negative figure. If the pump wavelength located in the normal GVD regime (*β*_{2p}>0), the negative second term can be counteracted by the positive first term. Therefore, unlike the conventional parametric amplification or wavelength conversion using silicon waveguide, for which a pump wavelength in the anomalous GVD regime is needed to realize broad phase matching [28], proper pump wavelength located in the normal GVD regime can be used to realize a relative small linear phase mismatch for certain THz-waves, which is illustrated in the Fig. 6(a)
.

When the linear phase mismatch *k*_{L} is compensated by the nonlinear phase mismatch *k*_{NL}, the phase matching is realized according to Eq. (2). The nonlinear phase mismatch is defined as *k*_{NL} = 2*γP*_{P}, where *γ = ωn*_{2}*/cA*_{eff} is the effective nonlinearity coefficient of the waveguide, and *P*_{P} represents the pump peak power. If we assumed the pump wavelength is 4.3 μm and the rib height is 15 μm, the nonlinearity coefficient *γ =* 152.2 W^{−1}km^{−1} with *n*_{2} = 5 × 10^{−18} m^{2}W^{−1}, which is assumed according to [29–31]. The nonlinear phase mismatch *k*_{NL} = 6.08 cm^{−1} when the pump peak power is set to be 2000 W. Thus, the phase matching is realized when the linear phase mismatch is about −6 cm^{−1} as shown in Fig. 6(a), and the points of intersection are the phase matching point. The corresponding relationship of pump, signal and THz-wave wavelength for the phase matching point is described in Fig. 6(b). Therefore, phase matching for a widely tunable THz-wave ranging from 8.57 THz to 10 THz (or from 30 μm to 35 μm) can be realized by tuning the pump wavelength from 4.2 μm to 4.4 μm in the silicon membrane waveguide with rib height of 15 μm. If the pump wavelength is less than 4.2 μm, much broader THz-wave bandwidth can be achieved as the trend shown in Fig. 6. However, the corresponding signal wavelength satisfying the phase matching will be reduced less than 2.2 μm and the efficiency for THz-wave generation will be decreased due to TPA.

We can also tailor the rib height to tune the THz-wave bandwidth for a fixed pump, which is illustrated by Fig. 7(a)
. It is shown that the phase matching is changed for different rib heights, which means that THz-wave with different wavelength can be generated by changing rib heights. The corresponding relationship of rib height, signal and THz-wave wavelength for the phase matching point is described in Fig. 7(b). It is clear that the phase matching bandwidth of THz-wave ranging from 7.7 THz to 10 THz (or from 30 μm to 39 μm) can be achieved by tailoring the rib height from 14 μm to 17 μm when the pump wavelength is 4.3 μm. If the rib height H>17 μm, phase matching for THz-wave with much longer wavelength can be realized as the trend shown in Fig. 7. However, larger rib height means larger mode areas *A*_{eff}, which will lead to lower nonlinearity coefficient *γ* and reduce the efficiency of THz-wave generation.

## 3. The efficiency of THz-wave generation

The FWM process can be described by the following coupling equations [29]:

*A*,

_{p}*A*and

_{s}*A*represent the slowly varying amplitude of the pump, signal and THz-waves, and

_{t}*z*is the propagation distance. The parameters α

*α*

_{p,}*and α*

_{s}*represent the linear propagation losses of the pump, signal and THz-wave, which are assumed as 0.138 cm*

_{t}^{−1}, 0.092 cm

^{−1}and 0.23 cm

^{−1}, respectively [15, 32]. The nonlinearity coefficient

*γ*(

_{xy}*xy*=

*ps*,

*st*,

*pt*) can be calculated with the averaged frequency,

*ω*(

_{xy}=*ω*)/2 [29]. Here, we assume

_{x}+ ω_{y}*n*

_{2}=5×10

^{−18}m

^{2}W

^{−1}for the pump and THz-wave [29–31]. Despite the n

_{2}for the signal wave is predicted to be about 8×10

^{−18}m

^{2}W

^{−1}[30], for simplicity, we also use

*n*

_{2}=5×10

^{−18}m

^{2}W

^{−1}for the signal in the simulation. Moreover, the value of n

_{2}used to calculate

*γ*is also assumed as 5×10

_{xy}^{−18}m

^{2}W

^{−1}.

The THz-wave generation via FWM is numerically studied by simultaneously injecting pump pulses and signal pulses in the Mid-infrared band [33]. The pump and signal pulses are taken to be hyperbolic-secant pulses with same pulse width of 12 ns and same repetition rate [33]. Here, we assume the signal peak power is half of the pump peak power. The multi-photon absorption can be neglected in this paper, which does not presents a significant obstacle for practical applications [26].

Figure 8(a)
depicts the peak power of the THz-wave along the waveguide for several pump peak powers, when the pump wavelength is 4.3 μm and the rib height is 15 μm. The input signal wavelength can be obtained from Fig. 6(b). The peak power of the THz-wave is increasing with the increase of the distance until the max. is attained, then the peak power of the THz wave drops to zero as a period of FWM completes and next period occurs. When the pump peak power is 2000 W, the maximum peak power (25 W) of THz-wave can be obtained in a 6-mm-long waveguide due to phase matching. The output spectra centered at 9.2304 THz for several pump peak powers are shown in Fig. 8(b). Furthermore, the maximum conversion efficiency *η* = 25/2000 = 1.25% in this case, which is calculated as the ratio of output THz-power with respect to the input pump peak:

For simplicity, we assume the waveguide length is 6 mm and the pump peak power is 2000 W in the following part.

THz-wave generation based on FWM is also a discrete wavelength conversion. Figure 9 shows the conversion efficiency of the generated THz-wave when the signal wavelength is tuned from 2299.8 nm to 2304.9 nm for a fixed pump located at 4.3 μm. It is shown that the maximum conversion efficiency occurs at the THz-wavelength of 32.5μm when the signal is tuned to satisfy the phase-matching condition. For this discrete wavelength conversion, the bandwidth of the generated THz-wave is about 1 μm, while the corresponding bandwidth of the signal is only 5.1 nm.

The output peak power and related wavelength of the THz-wave as a function of pump wavelength are shown in Fig. 10(a)
. With the increase of the pump wavelength, the frequency and peak power of the THz-wave increase, which is illustrated in Fig. 10(b). This can be explained that the THz-wave with relative high frequency can obtain more energy from the pump according to Eq. (1). The maximum conversion efficiency *η* is 1.39% at 9.8684 THz when the pump wavelength is 4.4 μm, and the minimal conversion efficiency *η* is 1.12% at 8.6455 THz when the pump wavelength is 4.2 μm.

When the pump wavelength is located at a fixed value such as 4.3 μm, we can also tailor the rib height to turn the bandwidth of the THz-wave. Figure 11(a)
shows the peak power and related wavelength of THz-wave as a function of rib height. It is clear that the peak power of THz-wave decreases as the rib height increases due to the increase of the effective mode area *A*_{eff}, which leads to a low nonlinearity coefficient *γ*. The frequency of the THz-wave decreases with the increase of the rib height, and the THz-wave with relative low frequency obtains less energy from the pump, which is also a reason for the low peak power of the low frequency THz-wave as shown in Fig. 11(b). The maximum conversion efficiency *η* is 1.71% at 10.1 THz when the rib height is 14 μm, and the minimal conversion efficiency *η* is 0.63% at 7.7 THz when the rib height is 17 μm.

## 4. Conclusion

We have presented a theoretical study of THz-wave generation using Mid-infrared pump and signal waves in silicon membrane waveguides. The simulation model allows us to show the importance of the pump wavelength to the collinear phase matching, which can be realized only when the pump wavelength locates in the normal GVD regime. Moreover, broadband phase matching can be achieved by tuning the pump wavelength and the rib height. Finally, we numerically discuss the conversion efficiency of the THz-wave generation in the silicon membrane waveguides. This method and results show a promising way to realize an efficient and compact THz-wave source.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61078029.

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