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Terahertz (THz) wave generation via difference frequency mixing (DFM) process in strain silicon membrane waveguides by introducing the straining layer is theoretically investigated. The Si3N4 straining layer induces anisotropic compressive strain in the silicon core and results in the appearance of the bulk second order nonlinear susceptibility χ(2) by breaking the crystal symmetry. We have proposed waveguide structures for THz wave generation under the DFM process by .using the modal birefringence in the waveguide core. Our simulations show that an output power of up to 0.95 mW can be achieved at 9.09 THz. The strained silicon optical device may open a widow in the field of the silicon-based active THz photonic device applications
© 2014 Optical Society of America
1. Introduction
Nonlinear optics in silicon-based photonic device is a novel and quite attractive field [1]. The possibility to obtain the all optical data processing can be reinforced by its implementation on a CMOS-compatible, all silicon based platform. Up to now, the vigorous research has been focused on the exploitation of the optical nonlinear effect derived from the interaction of intense near infrared sources with the third-order nonlinear susceptibility χ(3) of silicon waveguides [2–4]. This is related to the inherent crystalline inversion symmetry of silicon crystal. Its centro-symmetry prohibits the bulk second order nonlinear susceptibility, leaving room only for surface contribution where the crystalline symmetry is naturally broken. However, recent experimental studies have indicated that the crystal symmetry can be broken by applying an anisotropic strain to the silicon membrane by depositing a Si3N4 straining layer on top of it. Therefore, the braking crystal symmetry creates a bulk second order nonlinear optical effect in strained silicon. Jacobsen et al have investigated linear electro-optic modulation in a silicon-on-insulator (SOI) channel waveguide coated with amorphous SiO2 (a-SiO2) and straining amorphous Si3N4 (a-Si3N4) layer [5,6]. The electro-optic modulation is associated with a second order nonlinear susceptibility estimated to be χ(2) = 15 pm/V. Cazzanelli et al have reported a second order susceptibility in the s-silicon by ab-initio simulation method and estimated to be about 200 pm/V in the wavelength region around 2 μm [7]. In the following, they have demonstrated second harmonic generation from the s-Si waveguide and estimated its second nonlinear susceptibility of 40 pm/V. The possibility to utilizing second-order nonlinear effect requires a low pump power to operate the active optical devices which are simple and reasonable.
The considerable development of efficient and robust THz sources is of great interest in material science, life science, chemistry, high-speed communication, nondestructive inspection, and environmental monitoring [8]. A plenty of THz sources have been developed based on quantum cascade lasers [9], photo conductive antennas [10], nonlinear optical process including terahertz parametric sources [11, 12], optical rectification [13], and difference frequency mixing (DFM) [14–21]. Especially, the DFM process can be used to produce tunable, narrow linewidth, high-power THz-waves at room temperature. THz wave generation under the DFM process is carried out by using second-order nonlinear optical materials such as LiNbO3, GaAs, GaP, and GaSe, respectively [14–21]. However, it is difficult to increase the conversion efficiency for the THz DFM because these nonlinear optical materials have a large absorption coefficient originated from their optical phonon absorptions. The exploitation of surface emitting configuration [22] and hybrid waveguide geometries [17] are effective ways to overcome the high absorption loss of the nonlinear optical materials. Another solution to avoid the material loss is to utilize the all silicon waveguide structure, which consists of the strained silicon waveguide for the optical pump source and high resistivity silicon waveguide for the produced THz wave. Although Waechter et al have demonstrated the wide band THz wave generation under the DFM process in a silicon waveguide [23], the monochromatic coherent THz wave generation from strained silicon waveguide under the DFM scheme has not ever been studied theoretically and experimentally.
In this paper we propose and analyze strained silicon waveguides that provide the modal phase matched difference frequency mixing (DFM) of mid-infrared sources. The appropriate pump wavelength and waveguide geometry enables us to achieve the efficient power conversion efficiency over the long interaction length. The designed waveguide structure described in this paper will help the construction of practical silicon-based THz photonic devices for DFM process.
2. Strained silicon based hybrid waveguide structure
In a non-strained silicon crystal which possesses a large third order nonlinear susceptibility χ(3), the bulk second order nonlinear dipolar polarization is forbidden by terms of crystal symmetry. By applying the strain in the silicon membrane, both χ(2) and χ(3) are simultaneously exhibit in it owing to the breaking of the crystal symmetry. In the strained silicon (s-Si) waveguide, we can compare the relative contributions of χ(2) and χ(3) to the nonlinear optical process. When a moderate level of pump light is injected the waveguide, three photon interaction process of χ(2) will generally dominate over the four-photon interaction process of χ(3). If we design an s-Si waveguide to be optimized for the χ(2) DFM process described in this paper, phase matching in the χ(3) process can be negligible.
We propose an s-Si based hybrid waveguide structure for THz DFM shown in Fig. 1.The waveguide is composed of two waveguide sections; a-Si3N4/Si/a-SiO2 ridge waveguide to confine infrared pump lights, Si quasi-channel waveguide for confinement of converted THz wave by the DFM process. The crystallographic structures of Si3N4 and SiO2 layers are amorphous, that of Si layers is high resistivity single crystal. The a-Si3N4 layer acts as a straining layer to apply inner stress in the cross section of the Si layer [5, 6]. Consequently, the s-Si layer becomes an active χ(2) medium. In order to avoid two-photon absorption in the DFM process, we chose the pump source with wavelength of λp = 2.3 μm, (2ħωp = 1.078 eV < Eg = 1.12 eV, where Eg is the bandgap of Si [24]). The mid-infrared sources such as laser diodes and optical parametric oscillators, and fiber lasers have been developed [25–27] and commercially available. The single mode operation of the light sources in the ridge waveguide is realized by satisfying the following condition [28],
where s, t, and w are slab thickness, ridge height from its bottom to top, and ridge width, respectively.
Fig. 1 Geometry of the strained silicon (s-Si) based hybrid waveguide structure. The s-Si ridge waveguide to confine infrared lights consist of amorphous Si3N4, high resistivity Si single crystal, and amorphous SiO2, layers with dimensions of tSi3N4, tSi-h, tSi-s, tSiO2, and wsSi, respectively. The quasi-channel waveguide for THz wave confinement is composed of high resistivity Si layers with dimensions, TSi-U, TSi-L, and WSi, respectively.
The pump and signal sources are confined to the s-Si ridge waveguide surrounded by the upper a-Si3N4 straining layer and lower a-SiO2 layer. The THz wave, generated through the DFM process, is guided by a quasi-channel Si waveguide, which is disposed so as to sandwich vertically on the a-Si3N4/s-Si/a-SiO2 ridge waveguide.
The modal properties of the optical pump and signal sources defined by the s-Si ridge waveguide in the right side of Fig. 1 are simulated. The waveguide dimensions, tSi3N4, tSi-h, tSi-s tSiO2, and wsSi are set to 0.5 μm, 0.15 μm, 0.85 μm, 1.0 μm, and 2.6 μm, respectively. The refractive indices of Si3N4, Si, and SiO2 are1.9741, 3.4452, and 1.4334 at λp = 2.3 μm, respectively [29]. Figure 2(a) shows modal profile of a fundamental TM mode ((a): E field polarized along the [001] crystallographic direction in Fig. 1 at the wavelength of 2.3 μm for the pump wave and TE mode ((b): E field polarized along the [10] crystallographic direction in Fig. 1) at the wavelength of 2.4723 μm for the signal wave. These mode profiles were calculated by using a finite-difference mode solver [30]. The full-width and half-maximum of horizontal and vertical of the TM-mode profile are about 3 μm and 0.8 μm, respectively, which are smaller than the 8 μm of mid-infrared chalcogenide optical fiber. Although the overlap between the channel waveguide mode and the fiber mode at 2.3 μm are small, the improvement could be made by increasing the channel mode size and using the appropriated coupling lens. The THz wave produced through the DFM process is guided by the quasi-channel Si waveguide which is consisted of Si slabs both below and above the a-Si3N4/s-Si/a-SiO2 ridge to provide lateral confinement, as shown in the left of Fig. 1. We note that the Si ridge is placed on top of the a-Si3N4 film to adjust the air gap of 1.0 μm in order to prevent from applying an unintentional stress to the ridge waveguide. We selected high resistivity Si as a THz waveguide material due to its lower absorption coefficient (~0.15 cm−1 at 3 THz [31]) than other nonlinear materials such as GaP (> 4 cm−1 at 3 THz) [32]. The s-Si based hybrid waveguide is constructed by photolithography and etching technique and wafer bonding applied for the buried-oxide insulating material in silicon on insulator (SOI).
Fig. 2 (a) The mode profile of TM mode pump at λp = 2.3 μm and (b) TE mode signal λs = 2.4723 μm of the s-Si waveguide. The waveguide dimensions are tSi3N4 = 0.5 μm, tSi-h = 0.15 μm, tSi-s = 0.75 μm, tSiO2 = 1 μm, and w = 2.6 μm, respectively.
3. Phase matching analysis
We utilize the DFM process to generate THz wave in strained silicon-based hybrid waveguide, which typically involves the pump and signal photons at angular frequency ωp passing their energy to a signal wave and a THz wave at angular frequency ωs and ωTHz, respectively. THz wave grows while copropagating with the pump and signal beam. The energy conservation and momentum conservation in the DFM process could be written in a form,
where kp, ks, and kTHz represent the propagation wave vector of the pump, signal, and THz wave, respectively. In the DFM process, the strained silicon channel waveguide act as a second order nonlinear optical waveguide. The second order nonlinear polarization P(2)(ωTHz = ωp-ωs) produced via the DFM process is expressed as,where Ep(ωp)and Es(ωs)are the vector of the electric field component of the pump and signal, respectively, χ(2)is the second order nonlinear optical tensor. Because of the non-uniformity of the strain distribution applied into the silicon core by the existing of the Si3N4 top layer on it, it is difficult to define the nonlinear optical tensor of the strained silicon crystal. Therefore, when we set the polarization of the pump and signal source to be TM- (E-field is parallel to [001] crystallographic direction in Fig. 1 and TE-mode (E-field is parallel to [10]), respectively, it is possible to produce a TE or TM polarized THz wave from the waveguide depend on the χ(2). In the following, we consider only TE-mode THz wave generation through the DFM process in the s-Si based hybrid waveguide.The dimension of the hybrid waveguide is designed to realize the collinear phase matched DFM. To determine the waveguide geometry, we need to simulate the modal indices and electric field profile of the pump, signal, and produced THz wave in the waveguide structure, respectively. To investigate collinear phase matching condition, we simulate the dispersion relation for three interacting waves in the waveguide. For the a-Si3N4/s-Si/a-SiO2 ridge waveguide, the fundamental TM- and TE- polarized of the pump and signal wave at the wavelength of around 2.3 μm are numerically determined in Fig. 3(a).The s-Si ridge waveguide possesses the modal birefringence between TE and TM mode owing to the mode confinement into the waveguide. This enables us to achieve the collinear phase matched THz wave generation through the modal phase matched DFM process.
Fig. 3 (a) Dispersion relation for the guided TM- and TE-like mode in the s-Si ridge waveguide in the wavelength range from 2.3 μm to 2.4723 μm, (b) that for the TE-like guided THz wave in the quasi-channel Si waveguide around 9 THz. The modal index required for phase matching npm represented in Fig. 3(b). The phase matching frequency position for TE-like THz wave is highlighted with open circle.
In the Fig. 1, we set the dimensions of the Si ridge waveguide of TSi-U = 8 μm, TSi-L = 8 μm, and WSi = 10 μm, respectively. And we used the refractive indices of the waveguide material of a-Si3N4, Si, and a-SiO2 are 2.7532, 3.4416, and 2.0843 at 9.09 THz, respectively. The calculated dispersion relationships for guided TM-like mode of produced THz wave are shown in Fig. 3(b). From the dispersion relation for guided TM-like and TE-like modes in Fig. 3(a), the refractive index required for the THz DFM npm which is calculated from the following equation, where np and ns are the effective indices for the TM-like mode pump and TE-like mode signal, respectively. In Fig. 3(b), calculated npm is shown, and the intersections of the npm and the guided THz wave highlighted with open circles at 9.09 THz (λTHz = 33.0033 μm) for TE mode THz wave indicate the perfect phase matching condition. These phase matching frequency positions could be varied by shifting the wavelength of the pump source or changing the dimension of the Si channel waveguide. The frequency conversion in the DFM process can be carried out by interacting among the fundamental modes of three guided waves in the s-Si based waveguide.
4. THz output characteristics and conversion efficiency
The DFM process can be expressed by the following coupling equations [33]:
where, Ai (i = p, s, and THz) are slowly varying amplitudes, αi (i = p, s, and THz) is the propagation loss coefficients, y is the propagation distance. If he mode field profiles of each mode are normalized, the modal Poynting vector for each mode estimates to 1. In the case of phase matching condition, propagation constant difference among three interacting waves () should be zero, and Eq. (4) is simplified. For instance, Eq. (4) for THz wave is expressed asThe coupling coefficient in the DFM process κ has units of second per meter and is referred aswhere χ(2) is the second order nonlinear susceptibility of the strained Si, Seff is the effective nonlinear interaction area and expressed bywhere Si (i = p, s, and THz) are the mode areas of the pump, signal, and produced THz waves; SNL is the area where all of the modes and the s-Si overlap .Figure 4 represents the electric field distribution of fundamental mode for the generated THz wave at frequency of 9.09 THz. The THz wave waveguide mode is confined in the Si quasi-channel waveguide structure. From Eq. (8), we estimated the effective nonlinear interaction area Seff ≈140.36 μm2. Since the pump wave powers are much higher than THz power, we assume that the depletion of the pump wave is ignored. Thus we obtain the following relation:
Fig. 4 The electric field distribution of the generated TE-like mode THz wave at 9.09 THz in the Si based quasi ridge waveguide
At the boundary of y = 0, THz power ATHz(0) = 0. We solved the Eq. (5) for the DFM process as,
The absorption loss of Si waveguide could be small in the 2 μm wavelength region compared with other IR region, and the propagation loss in the s-Si waveguide derived from cladding materials and scattering at interface roughness can be quite small by recent progresses of the Si optical waveguide fabrication technologies so that we assumed that αp, αs could be negligible. However, the material loss of s-Si crystal has not been reported. This concern can be important point for the practical use of s-Si based nonlinear optical devices. From Eq. (10), the corresponding THz power is obtained as,where Pi = |Ai|2 (i = p, s, and THz) correspond to the powers and L is the waveguide length. We assumed that the absorption loss of THz wave is mainly derived from the material loss of Si channel waveguide. We take the value of αTHz = 0.5 cm−1 at 9.09 THz [34] at room temperature. We vary the wavelength of the signal wave according to the fixed pump wavelength of 2.3 μm in order to calculate the conversion efficiency. The THz output power as a function of the THz wave frequency for various waveguide lengths is illustrated in Fig. 5(a).The optical power of pump and signal wave were Pp = Ps = 1W, respectively. The THz output power for both frequency increases monotonically as waveguide length increases. THz output power at 9.09 THz was obtained to be 0.95 mW when the waveguide length is L = 50 mm. The corresponding power conversion efficiency is η = PTHz /Pp Ps = 9.5 × 10−4 W−1. This conversion efficiency is higher than that obtained by using the LiNbO3 crystal (estimated from data in [17] to be ~1.8 x 10−7 W−1).The THz output power as a function of THz frequency is shown in. 5(b), with λp fixed at 2.3 μm. The phase matching bandwidth for THz wave as shown in this Fig. decreases from 250 GHz to 15 GHz as waveguide length increases from 5 mm to 50 mm. The s-Si based hybrid waveguide takes advantage of the lower propagation loss for THz wave and strong confinement in the Si channel waveguide, and leads to the high conversion efficiency owing to achievement of longer interaction length.Fig. 5 (a) Waveguide length dependence of the generated THz power and bandwidth. The input power of the pump and signal wave are each 1W. (b) THz output power as a function of THz frequency for TE-like mode with waveguide lengths of 10, 20, and 50 mm, respectively.
In this calculation, we used the second order nonlinear coefficient of χ(2) = 40 pm/V [7]. This value varies the strain distribution inside the Si core. The optimized fabrication process of strain silicon waveguide or the use of a photonic crystal structure will make it possible to utilize higher nonlinearity of the strained silicon waveguide. Especially, photonic crystal waveguide have shown the quite high nonlinear susceptibility of 1000 pm/V due to the unique slow light phenomena in the photonic crystal waveguide [6].
Our proposed strained Si based waveguide structure can be used as not only efficient THz source but also THz detector by nonlinear interaction between MIR pump and THz signal.
5. Conclusion
In summary, we have shown that a strained silicon (s-Si) based waveguide material can be designed to produce the collinear phase matched difference frequency generation of THz-wave. Owing to the presence of strain induced second-order nonlinearity χ(2), the proposed waveguide can be used in various active silicon-based photonic devices. This numerical result may predict a novel promising way of bridging the optical to THz frequency gap by the best use of the silicon nonlinear optical effects with low input signal power.
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