## Abstract

We propose and investigate a ribbon waveguide for difference-frequency generation of terahertz (THz) wave from infrared light sources. The proposed ribbon waveguide is composed of a nonlinear optic crystal and has a thickness less than the wavelength of the THz wave to support the surface-wave mode in the THz region. By utilizing the waveguide dispersion of the surface-wave mode, the phase matching condition between infrared pump, idler and THz waves can be realized in the collinear configuration. Owing to the weak mode confinement of the THz wave, the absorption coefficient can also be reduced. We design the ribbon waveguide which uses LiNbO_{3} crystal and discuss the phase-matching condition for DFG of THz wave. Highly efficient THz-wave generation is confirmed by numerical simulations.

© 2007 Optical Society of America

## 1. Introduction

Applications of terahertz (THz) wave have been exploited in various fields such as applied physics, communications, sensing, and life sciences. Hence, efficient and compact sources of THz wave are of great importance. The difference-frequency generation (DFG) and the parametric generation in nonlinear optic (NLO) crystals is one of the most promising techniques, and several demonstrations have been made by using lithium niobate (LiNbO_{3}, LN) [1–6]. In these experiments, the pump light in the infrared region was launched into the NLO crystal (with the idler light beam), and the THz wave was generated through the second-order nonlinearity.

Although the THz-wave generation by the DFG has a great advantage that the coherent THz wave is obtained, it is not straightforward to increase in the conversion efficiency. The major difficulties are threefold. The first is the phase matching among the infrared pump, idler and THz waves. This is due to the fact that the refractive index varies significantly in the infrared and THz regions. For example, the refractive indices of LN for extraordinary light are 2.16 and 5.17 at wavelengths of 1.06 μm and 299 μm (=1 THz), respectively [7]. Therefore, the previous experiments have been made only in the non-collinear configuration, in which the THz wave was radiated at an angle of >65 degree with respect to the pump beam [1–6]. The second is the low output-coupling efficiency originating from the high refractive index in THz region. In the case of LN, the reflectivity at the crystal surface (open to the air) is more than 46 %. To improve the output-coupling efficiency, several efforts have been made so far. In refs. [2] and [3], a silicon prism array was used to improve the coupling efficiency, though the wavefront could be non-uniform due to the use of multiple prisms. In other work, the surface emission configuration has been examined by using three pieces of LN crystals [5] and by using the slanted periodically polled LN [6,8]. However, these methods suffer from the complex experimental setup and the short length of the effective interaction. The third is the high attenuation in the THz region. In most of NLO crystals, the absorption coefficient in the THz region is extremely large, and hence, the THz wave generated in the NLO crystal can be subsequently absorbed. This results in the severe degradation of the conversion efficiency. At the present, the only way to cope with this problem is to shorten the propagation distance in the NLO crystal by aligning the pump and the signal beams near the crystal surface [2,3,5].

In order to cope with these difficulties, we propose a ribbon waveguide for DFG in collinear configuration. The key idea is to make the size of the waveguide much less than the wavelength of the THz wave such that the THz wave is guided in a surface-wave mode [9]. Unlike in the previous designs of THz waveguide [4,8,10], the field intensity of the THz-wave is distributed outside of the waveguide, and the effective refractive index for the guided THz wave has a value between the material and the surrounding air, depending on the confinement of the THz wave. By designing the size of the waveguide properly, we can adjust the effective refractive index of the guided mode in such a way that the phase matching condition is satisfied in collinear configuration. Also, since the effective refractive index becomes much smaller than that of the NLO crystal, the reflectivity at the crystal surface is reduced, resulting in the efficient output coupling. Furthermore, since the electric field outside the NLO does not suffer from the material absorption, the propagation loss can be reduced significantly [11–13].

So far, there have been several discussions of the waveguide structure for DFG. For example, refs. [4], [8] and [10] introduced the waveguide structure to enhance the DFG by utilizing the strong confinements of both the infrared and the THz waves, and the efficient DFG was demonstrated in non-collinear configuration [4]. On the contrary, in our proposal, the confinement of the THz wave is rather weak in order to achieve the phase matching in the collinear configuration, and the purpose of using the waveguide is entirely different.

In this paper, we focus on the use of LN crystal and investigate the DFG of THz waves in the LN ribbon waveguide in the collinear configuration. The organization of the paper is as follows. In Section 2, we derive the effective refractive index of the THz guided wave required for the phase matching condition in collinear configuration. In Section 3, we calculate the dispersion relation of the LN ribbon waveguide in THz region and show how the effective refractive index and the propagation loss are determined by the size of the waveguide. In Section 4, we numerically investigate the DFG of the THz wave in the ribbon waveguide. Before concluding this paper, we briefly discuss the effect of the size of the waveguide in Section 5. Finally, we summarize this paper in Section 6.

## 2. Phase matching condition for DFG of THz wave in collinear configuration

In this section, we discuss the phase matching condition for DFG in a waveguide in collinear configuration and describe it by using the effective refractive index. We consider a dielectric waveguide, which is composed of the NLO crystal core and the air cladding (Fig. 1), and launch the pump and idler beams in the infrared region for the DFG. We denote the angular frequencies of the pump and the idler beams as *ω*
_{p} and *ω*
_{i}, respectively, and assume *ω*
_{p}>*ω*
_{i}. The waveguide can support guided modes in both the infrared and the THz regions. The THz wave is generated through DFG and grows while copropagating with pump and idler beam. In this section, we assume that the infrared waves propagate in the fundamental mode for simplicity.

The phase matching condition for the DFG in collinear configuration is expressed by,

where *β*
_{p}, *β*
_{i} and *β*
_{THz}, are the propagation constants of guided modes for pump, idler and THz waves, respectively. Here, we introduce the effective refractive index, which is defined by *n̅*
_{x} =*β*
_{x}/*k*
_{x} (suffix x=p, i, or THz), where *k* is the angular wavenumber (2*π*/λ). In order to satisfy the phase matching condition of Eq. (1), the waveguide must have the effective refractive index *n̅*
_{THz,required} at the THz frequency as follows,

Equation (2) can be further simplified if the pump and idler waves propagate in the same transverse mode. When we denote the effective refractive index of the guided mode in the infrared region by *n*
_{IR}(*ω*), *n̅*
_{p} and *n̅*
_{i} are expressed by *n̅*
_{IR} (*ω*
_{p}) and *n̅*
_{IR}(*ω*
_{i}), respectively. By using the Taylor expansion of *n̅*
_{IR}(*ω*) around *ω*_{p}, Eq. (2) is transformed as:

Here, we ignored the second and higher-order terms of Taylor expansion. On the right-hand side of Eq. (3), the third term is negligibly small compared with other terms because *ω*
_{i} ≫ *ω*
_{THz}. Eventually, the effective reflective index *n̅*
_{THz,required} required for the phase
matching becomes almost constant for a fixed pump wavelength and does not depends on the frequency of the THz wave to be generated.

We calculated *n̅*
_{THz,required} when we used the LN as NLO crystal. Here, we assumed all pump, idler and THz waves have the same polarization parallel to the optical axis, in order to utilize the largest second-order nonlinear tensor d_{33}. Figure 1(b) shows the required effective refractive index of the THz wave *n̅*
_{THz,required} as a function of the pump wavelength. We evaluated Eq. (2) by replacing *n̅*
_{IR} with the refractive index of the LN for the extraordinary light *n*
^{(e)}. (As will be shown in the next section [Fig. 4(b)], the electric field of the fundamental mode is strongly confined in the core in the near-infrared region because of the high index-contrast. Hence, *n̅*
_{IR} ≈ *n*
^{(e)} is a good approximation.) When the pump wavelength is 1.06 μm (*n*
^{(e)}=2.16), *n̅*
_{THz,required} =2.21. The refractive index of the LN is about 5.17 in THz region, and hence, the phase-mismatching becomes too large in collinear configuration in a bulk crystal. When the mode confinement of the THz wave is strong, as in the refs. [4,13], the effective refractive index is near that of the LN bulk crystal and the phase matching in collinear configuration is almost impossible. (In fact, in these references, the non-collinear configuration was used.)

## 3. Dispersion relation of THz ribbon waveguides and phase-matching condition

In the case of using bulk LN, the refractive index in the THz region is much larger than *n̅*
_{THz,required}, and hence, the phase matching in the collinear configuration is impossible. However, if we use the waveguide structure and reduce the effective refractive index down to *n̅*
_{THz,required} by utilizing the waveguide dispersion, the phase matching condition can be satisfied in the collinear configuration. Although many types of waveguides can be used for this purpose, we focus on the ribbon (thin-film) waveguide as shown in Fig. 2 [11–13]. The thickness of the ribbon waveguide is much less than the wavelength of the THz wave, and this waveguide can support surface-wave modes in the THz regime. The reasons why we chose the ribbon waveguide are mainly twofold. The first is that the ribbon waveguide has a low propagation loss compared to those of other waveguide structures which can support surface-wave modes [11–13]. The second is that the structure is simple and the mode confinement can be easily controlled by changing the thickness of the waveguide.

The coordinate system is chosen as shown in Fig. 2 and the optical axis of the LN is set to y-axis. The ribbon waveguide has the thickness of *a* and the width of *b*. All guided modes are hybrid modes and they are categorized into TE-like and TM-like modes. Here, we assume that the pump and idler infrared waves have the same polarization parallel to the optic axis (extraordinary), and that the THz wave is generated in TE-like mode (E_{mny}) through the largest second-order nonlinear tensor element d_{33}. (In this paper, we follow the mode definition and the notation used by Marcatili *et al*. [14,15]. The subscripts *m* and *n* of E_{mny} start from 1.) The electric field of E_{mny} modes is expressed as:

where *ω*
_{THz} is the angular frequency of the THz wave, *u _{mn}*(

*x*) is the mode field distribution,

*β*is the propagation constant,

_{mn}*α*is the attenuation constant of E

_{mn}_{mny}mode,

*A*is a parameter expressing the intensity of the electric field, respectively. In order to obtain

*α*and

_{mn}*β*in both infrared and THz regions, we conducted numerical calculation rigidly by using the finite element method (FEM). For the calculation of guided modes in THz region, we use the values of the refractive index and the absorption coefficient obtained in ref [7] as a function of the frequency. For example, the refractive index for extraordinary wave and the (power) absorption constant at 1 THz are 5.17 and 2860 m

_{mn}^{-1}, respectively. The second-order nonlinear tensor d

_{33}was set to be 33×10

^{-12}m/W [16].

Figure 3(a) shows the effective refractive index *n̅*
_{THz} calculated for four lower-order E_{mny} modes as a function of the frequency in THz region. The thickness and width of the waveguide were *a*=9.4 7 μm and *b*=126 μm, respectively. The fundamental mode is E_{11}^{y}
, and it does not have a cut-off frequency [14]. The effective refractive indices for all guided modes have values between the refractive indices of the surrounding air and the core material. Figure 4(a) shows the mode profile of E_{11}^{y}
mode at 1 THz. Since the core thickness is much smaller than the wavelength (299.8 μm @ 1 THz), a part of the electric field exists outside of the core region as a surface wave. However, it is trapped in the vicinity of the core due to the high index-contrast. (The similar mechanism of wave-guiding can be seen in optical nano-wire fibers and super high-delta waveguides [16, 17].) As the frequency becomes high, the mode field is confined more tightly in the core and the effective refractive index approaches to that of the core material. Also, the waveguide can support higher-order modes at a frequency higher than the cutoff frequency of E_{21}^{y}
mode, which is about 0.85 THz in this calculation. For the pump wavelength of 1.06 μm, *n̅*
_{THz,required} required for the collinear phase matching condition is calculated to be around 2.21 (shown by the dotted line in Fig. 3(a)), and the phase-matching condition can be satisfied if the effective refractive indices of the guided modes are identical to *n̅*
_{THz,required} (ie., at the intersections of the dispersion curves and the dotted line in Fig. 3(a)). For example, the phase matching condition for E_{11}^{y}
mode is satisfied at around 1 THz.

Figure 3(b) shows the propagation loss 2*α*
_{11} (in power) of E_{11}^{y}
mode calculated by using the attenuation constant of LN crystal in ref. [7]. We also show the (power) absorption coefficient in LN crystal by the dotted curve. As can be seen, the propagation loss becomes much smaller than that of the core material (i.e., LN itself) because a significant part of the THz power is transmitted outside of the core. In this ribbon waveguide, the propagation loss of E_{11}^{y}
mode at 1 THz was reduced to be less than one third of that of the LN. The reduction of the propagation loss depends on the confinement of the electric field, and hence the attenuation constant increases as the frequency becomes high.

## 4. Numerical simulation of collinear DFG by using LN ribbon waveguides

To examine the feasibility of the proposed waveguide, we carried out the numerical simulation of the DFG of THz wave in collinear configuration. In this work, we developed a modified beam propagation method in which the second-order nonlinear interaction was taken into account. We assumed that the pump and idler infrared waves were launched at *z*=0 μm and only their fundamental modes were excited. The pump wavelength was 1.06 μm, and the difference frequency between the pump and idler waves was set to be the phase matching frequency (*f*
_{THz}= 1 THz) derived in the previous section.

Figures 5(a) and (b) show the power distribution of the generated THz wave when the powers of the pump and idler waves were set to be 100 W and 10 W, respectively. (We chose this input power in such a way that the power density was much lower than the damage threshold of LN for pulse pumping.) The nonlinear polarization at the difference frequency is induced in the core region through the second-order nonlinearity, and then it excites the guided mode(s). Although this ribbon waveguide can support two modes E_{11}^{y}
and E_{21}^{y}
at 1 THz, only the fundamental mode (E_{11}^{y}
) of the THz wave was rapidly formed and then its intensity increased drastically due to the parametric interaction. This is because E_{21}^{y}
mode is anti-symmetric with respect to x-axis, and hence, it can not be excited from the pump beam with a symmetric mode profile. (Even if we use asymmetric pumping, E_{21}^{y}
mode can not grow due to the phase mismatching.) For further understanding of the formation of the THz beam in the ribbon waveguide, the evolution of the electric field distribution is shown in a movie of Fig. 5(c). In order to observe the tail of the beam profile clearly, we showed the distribution of the absolute value of the electric field of the THz wave, which was normalized by the electric field at the center of the core. When the propagation distance was short (*z*: from 0 to ~5 μm), the size of the excited THz beam was as large as the launched infrared beams, because the source of the THz wave was only the nonlinear polarization caused by the second-order nonlinearity between the pump and the idler beams. Since this beam size was too small for the THz wave, a small amount of the THz beam was rapidly deflected in x-direction (*z*: from 5 to 20 μm), and eventually the beam size became large. However, the majority was trapped by the waveguide, and E^{11}^{y}
mode was formed after the propagation of about 100 μm with the help of the parametric gain. Once E_{11}^{y}
mode was formed, the beam profile was very stable.

We also investigated the dependence of the conversion efficiency on the THz frequency. Figure 6(a) shows the power of the generated THz wave as a function of the frequency when the idler wavelength was tuned while keeping the pump wavelength constant. The propagation distance was 1 mm. The conversion efficiency had two peaks at around 1 THz and 1.45 THz. These frequencies corresponded to the phase matching conditions for E_{11}^{y}
and E_{31}^{y}
modes, respectively. In Figs. 6(b) and (c), we show the power distributions of the THz wave in the y-z cross-section and the output beam profile when the difference frequency was set to be 1.0 and 1.45 THz, respectively. The result of Fig. 6(c) shows that the mode field of E_{31}^{y}
mode was formed dominantly, even though it was a higher-order mode. This is due to the fact that the other modes were not able to grow due to the phase mismatching and eventually E_{31}^{y}
mode was selectively generated. From Fig. 3(a), we found that the phase matching condition for E_{21}
^{y}
mode was satisfied at around 1.2 THz. However, E_{21}
^{y}
mode was anti-symmetric and could not be excited from the pump and idler waves with symmetric beam profile.

For the further investigation of the DFG for longer propagation distance, we numerically solved the coupled-mode equations for pump, idler, and THz waves, which were derived from the Maxwell equations by using the mode orthogonality. (This approach by using the coupled equations is valid because one guided mode of the THz wave can be selectively excited under the phase matching condition, and the guided mode is formed quickly in a short propagation distance as seen in Figs. 5 and 6.) Figure 7 shows the power of the generated THz wave calculated as a function of the propagation distance for several pump powers. For simplicity, we assume that the idler power was the same as the pump power, and the difference frequency was set to be 1 THz, at which the phase matching condition was satisfied for E_{11}
^{y}
mode. The power of the THz wave increased during the propagation of around ten to twenty millimeters, and then it decreased due to the propagation loss. In this way, the maximum propagation distance, i.e., the device length is limited by the propagation loss of the THz wave.

In order to deal with more realistic cases, we calculated the output power of the THz wave as a function of the pump power. First, we assumed the pulse pumping and calculate the output power of the THz wave for the relatively high input pump and idler power. Figure 8(a) shows the results when the device length was set to be 10 mm. Here, we used the ratio of the idler power to the pump power as a parameter. We noted that the intensity of the pump wave is much lower than the damage threshold of conventional LN crystals. The result in Fig. 8(a) shows that the power of the generated THz wave was around 40 to 280 mW for the pump power of 1 kW-peak. This relatively high conversion efficiency was attributed to the collinear configuration. It should be stressed that the output power was in the same order of the highest power derived by using LN crystal in the non-collinear configuration [2,3], even though the input (peak) power used in our simulation was three orders of magnitude lower than that used in those references. Although our calculations have been made by assuming the ideal conditions without experimental difficulties, we believe that the conversion efficiency can be improved dramatically by using the DFG in the proposed ribbon waveguide in the collinear configuration.

Also these results of high efficiencies suggest the possibility of the DFG of THz wave by using continuous-wave (CW) sources. Figure 8 (b) shows the output power of the THz wave calculated as a function of the pump power when CW sources were used for pump and idler waves. We noted that the average power used in the calculation can be obtained by using conventional tunable CW sources such Yb-doped fiber amplifiers/lasers, parametric oscillators and so on [20, 21]. The result of Fig. 8(b) shows that the output power of sub-micro watts could be obtained by using the ribbon waveguide.

## 5. Discussion

In the aforementioned simulations, we used a ribbon waveguide with a fixed size. However this is not a unique solution, and the phase matching condition can be made for ribbon waveguides with different aspect ratios. Figure 9 shows the relationship of the thickness *a* and the width *b* to meet the phase matching condition for DGF at 1 THz. If we only consider the phase matching condition, we can choose any combination of *a* and *b* shown by the solid line in Fig. 9. However, we need to choose the size of the waveguide carefully, because it affects various characteristics.

The aspect ratio of the waveguide *b*/*a* affects the continuous tuning range and the propagation loss. As seen in Fig. 6(a), the continuous tuning is made possible in the vicinity of the phase matching frequency. This range can be estimated by the simple calculation of the phase mismatching. For simplicity, we assume that the phase matching condition of Eq. (4) is satisfied when the pump, idler and THz-wave angular frequencies are ω_{p0}, ω_{i0}, and ω_{Thz0}, respectively, and consider the case when the idler angular frequency is changed slightly by Δω while the pump wavelength is kept constant. The phase mismatch is expressed as:

We expand the right hand side of Eq. (5) by the Taylor expansion, and obtain the following relation after some calculation.

This result indicates that the phase mismatching is determined mainly by the slope (d*n̅*
_{THz}/dω∣_{ωTHz}) of the dispersion curve of the THz wave. That is, the continuous tuning range becomes wide as the slope of the dispersion curve becomes gentle. For rectangular waveguides, the dispersion relationship approaches a step-like function as the aspect ratio *b*/*a* approaches unity [14, 15]. Therefore, the waveguide having the large aspect ratio, i.e., the ribbon waveguide is suitable to improve the continuous tuning range.

As mentioned, the large aspect ratio is also preferable from the point of view of the propagation loss. In refs. [12] and [13], extensive studies on the propagation loss of THz waveguides have been made numerically and experimentally. They showed that the absorption in the dielectric core could substantially mitigated by using the ribbon waveguide structure, and the propagation loss became small as the aspect ratio became large under the condition of the fixed core area. On the other hand, in our case, we need to keep the effective refractive index at 1 THz the same (not the core area). However, the tendency is very similar to the results in ref [12]. The dotted curve in Fig. 9 shows the propagation loss calculated as a function of the width of the waveguide when the thickness is set to the value shown by the solid curve. As can be seen, the propagation loss decreases monotonically with the width. In this way, the large aspect ratio *b*/*a* is preferable.

On the contrary, the fabrication of the waveguide and the optical coupling would be difficult for the large aspect ratio. Therefore, we need to compromise it depending on the precision and/or the reliability of the fabrication technique. For the fabrication of the ribbon LN waveguide, several methods can be considered. One of the potential methods is to attach a thin LN crystal to a low-dielectric-constant substrate, polish the LN crystal to obtain the designed thickness, and adjust the width of the waveguide. In this mehtod, the LN ribbon waveguide is supported firmly by the substrate. Although the beam profile of the THz wave becomes slightly asymmetric, the change of *n̅*
_{THz} (as well as the phase-matching condition) is negligibly small if the refractive index of the substrate in the THz region is smaller than *n̅*
_{THz,required}. Also, several approaches to make thin-film LN waveguides might be applicable [21, 22].

In our discussion, we assumed the use of LN crystal. This is because the material constants of LN have been well known in both the infrared and THz regions. However, the discussions are not limited only to the LN, and the design approach would be applicable to the device using other NLO crystals such as Mg-doped LN and 4-dimethylamino-N-methyl-4-stilbazolium- tosylate (DAST) [23].

## 6. Summary

We proposed the LN ribbon waveguide for DFG of the THz wave in the collinear configuration. The proposed waveguide had a thickness less than the wavelength of the THz wave and supported the surface-wave mode in the THz regime. Owing to the reduced effective refractive index, the collinear phase matching was enabled in the THz regime. Also, since the effective refractive index became much smaller than that of the bulk LN, the reflectivity at the crystal surface was reduced, resulting in the efficient output coupling. Furthermore, since the electric field outside the waveguide did not suffer from the material absorption, the propagation loss could be reduced significantly. We designed the ribbon waveguide for DFG from infrared waves and numerically showed that the efficient THz-wave generation could be realized by using the proposed waveguide. Although our calculations have been made by assuming the ideal conditions without experimental difficulties, we believe that the conversion efficiency can be improved dramatically by using the DFG in the proposed ribbon waveguide in the collinear configuration.

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