Investigation of Terahertz Generation from Bulk and Periodically Poled LiTaO_{3} Crystal with a Cherenkov Phase Matching Scheme
 Author: Li Zhongyang, Bing Pibin, Yuan Sheng, Xu Degang, Yao Jianquan
 Publish: Current Optics and Photonics Volume 19, Issue3, p297~302, 25 June 2015

ABSTRACT
Terahertz (THz) wave generation from bulk and periodically poled LiTaO_{3} (PPLT) with a Cherenkov phase matching scheme is numerically investigated. It is shown that by using the crystal birefringence of bulk LiTaO_{3} and a grating vector of PPLT, THz waves can be efficiently generated by difference frequency generation (DFG) with a Cherenkov phase matching scheme. The frequency tuning characteristics of the THz wave via varying wavelength of difference frequency waves, phase matching angle, poling period of PPLT and working temperature are theoretically analyzed. The parametric gain coefficient in the lowloss limit and the absorption coefficient of the THz wave during the DFG process in the vicinity of polariton resonances are numerically analyzed. A THz wave can be efficiently generated by utilizing the giant second order nonlinearities of LiTaO_{3} in the vicinity of polariton resonances.

KEYWORD
Terahertz wave , Difference frequency generation , Cherenkov phase matching

I. INTRODUCTION
In the study of modern terahertz (THz) optoelectronics, monochromatic THz sources play an important role in high resolution THz applications, such as environmental gas monitoring and highdensity and highspeed wireless communications [14]. Difference frequency generation (DFG) with two closely spaced laser frequencies
ω _{1} andω _{2} in secondorder nonlinear optics crystals is one of the promising processes for efficient monochromatic THz wave output in the wide frequency tuning range and for room temperature operation [5, 6]. Recently, highefficiency THz wave generation based on DFG employing Cherenkov phase matching has been intensively researched [79]. THz wave generation with Cherenkov phase matching is an attractive scheme because strong THz absorption originated from lattice vibration modes of the nonlinear optical crystal can be overcome. Unfortunately, the maximum frequencies generated by DFG with a Cherenkov phase matching scheme are limited to 7.2 THz [10]. Moreover, the output power of the THz wave within the highfrequency band is extremely low. Such limitations on output frequencies and power are caused by the dramatically increased absorption of the nonlinear materials in the vicinity of polariton resonances. However, at the same time, polariton resonances can induce giant second order nonlinearities [1113]. The giant second order nonlinearities in the vicinity of polariton resonances can be exploited to extend the output frequencies and enhance the output power of the THz wave. In this paper, we explore THz wave generation based on DFG in the vicinity of polariton resonances with a Cherenkov phase matching scheme. A surfaceemitted configuration is employed to overcome the strong absorption problem in the vicinity of polariton resonances. The nonlinear crystals utilized in this paper are bulk and periodically poled LiTaO_{3} (PPLT). To the best of our knowledge, THz wave generation based on DFG utilizing bulk LiTaO_{3} and PPLT with a Cherenkov phase matching scheme has not been investigated. The advantage of utilizing LiTaO_{3} over LiNbO_{3} lies in the fact that LiTaO_{3} has a significantly reduced rate of opticallyinduced index change damage due to the photorefractive effect [14]. According to Ref. [14], the optically induced indexchange damage increases at rates of 2.8×10^{−3} cm^{2}/μW and 4.9×10^{−4} cm^{2}/μW in LiNbO_{3} and LiTaO_{3}, respectively. The second advantage is that the poling electric field required for LiTaO_{3} is one order of magnitude lower than that for LiNbO_{3} [15]. The coercive fields are 1.7 kV/mm and 21 kV/mm for stoichiometric LiTaO_{3} and congruent LiNbO_{3}, respectively [15]. In this paper, a Cherenkov phase matching scheme is realized by using the crystal birefringence of bulk LiTaO_{3} and a grating vector of PPLT. We numerically simulate the frequency tuning characteristics via varying pump wavelengths, phase matching angle, poling period of PPLT and working temperature. Parametric gain coefficient in the lowloss limit of the THz wave during the DFG process in the vicinity of polariton resonances is numerically analyzed.II. THEORETICAL MODEL
The schematic drawing of surfaceemitted DFG in bulk LiTaO_{3} and PPLT is presented in Fig. 1. As shown in Fig. 1(a), two input optical waves,
λ _{1} andλ _{2}, at the frequencies ofω _{1} andω _{2} in the infrared (IR) domain are propagating collinearly along theX axis in a bulk LiTaO_{3} crystal and are polarized along theZ andY axes, respectively. The two optical waves are located close to the lateral surface (XY ) of the crystal. A THz wave is emitted due to the oscillating nonlinear polarization at the difference frequency between the two input waves. The radiation angleα between directions of the optical and THz wave propagation is determined by the refractive index of input optical waves in the crystal, and the refractive index of the THz wave,where
K _{THz} =ω _{THz}n _{THz}/c is the wave vector,n _{THz} is the refractive index atω _{THz} frequency;K _{IR} =n _{1}^{e}(θ )ω _{1}/c n _{2}^{o}ω _{2}/c ,n _{1}^{e} (θ ) andn _{2}^{o} are the extraordinary and ordinary refractive indices of optical wavesλ _{1} andλ _{2} atω _{1} andω _{2} frequencies, respectively.θ is the angle between the directions of the optical wave propagation and the optical axis of LiTaO_{3}. For the THz wave generation based on DFG, the energy conservation condition has to be fulfilledwhere
λ _{THz} is the wavelength of the THz wave. From Eq. (1), it follows that the THz wave is emitted perpendicular to the directions of the optical wave propagation ifK _{IR}=0. The effective nonlinear coefficientd_{eff} in the DFG process is given bywhere
d _{15} andd _{22} are the nonlinear coefficients, the angleφ is an azimuthal angle of the optical wave vector with respect to theX axis.As shown in Fig. 1(b), two input optical waves,
λ _{1} andλ _{2}, at the frequencies ofω _{1} andω _{2} in the infrared (IR) domain are propagating collinearly along theX axis in the PPLT crystal. Bothλ _{1} andλ _{2} are polarized along theZ axis. The two optical waves are located close to the lateral surface (XZ ) of the crystal. The optical axis of the LiTaO_{3} crystal parallels theZ axis. A THz wave can be generated in a direction perpendicular to the optical wave propagation if poling period Λ of the PPLT satisfiesThe parametric gain coefficient
g _{0} in the lowloss limit during DFG processes in cgs units can be determined by the following expression [16]:where
ω _{THz} is the absorption coefficient in the THz region,ω _{0j},S_{j} and Γ_{j} denote eigenfrequency, oscillator strength of the polariton modes and the bandwidth of thej th A_{1}symmetry phonon mode in the LiTaO_{3} crystal, respectively.I _{λ1} is the power density of the optical waveλ _{1}.d' _{E} andd'_{Q} are nonlinear coefficients related to pure parametric (secondorder) and Raman (thirdorder) scattering processes, respectively. In this letter, the data for LiTaO_{3} is taken from reference [17].The theoretical values of the optical wavelengths are calculated using a wavelength and temperatureindependent Sellmeier equation for 0.5% MgOdoped stoichiometric LiTaO_{3} (MgO: LiTaO_{3})in the IR [18]. The Sellmeier equation for MgO:LiTaO_{3} of Dolev et al. [18] in the IR range in a temperature range from room temperature to 200℃ can be written as
where
a _{1e} = 4.5615,a _{2e} = 0.08488,a _{3e}= 0.1927,a _{4e} = 5.5832,a _{5e} = 8.3067,a _{6e} = 0.021696,b _{1e} = 4.782×10^{−7},b _{2e} = 3.0913 ×10^{−8},b _{3e} = 2.7326×10^{−8},b _{4e} = 1.4837×10^{−5},b _{5e} = 1.3647×10^{−7},a _{10} = 4.5082,a _{20} = 0.084888,a _{30} = 0.19552,a _{40} = 1.1570,a _{50} = 8.2517,a _{60} = 0.0237,b _{10} = 2.0704×10^{−8},b _{20} = 1.4449 ×10^{−8},b _{30} = 1.5978×10^{−8},b _{40} = 4.7686×10^{−6},b _{50} = 1.1127×10^{−5} andf = (T −24.5)×(T +570.82),T is the crystal temperature in ℃. All the figures in this work were numerically simulated according to the Eqs (18) using MATLAB software.III. FREQUENCY TUNING CHARACTERISTICS
3.1. THz Wave Generation with Bulk LiTaO_{3}
DFG processes are more versatile because of their wide tuning properties. According to Eq. (12), the frequencies of the THz wave can be tuned by varying phasematching angle
θ , wavelength of optical wavesλ _{1} andλ _{2} and working temperatureT . Figure 2 shows the tuning characteristics versus optical wavelengthλ _{1}. From the figure we find that asλ _{1} changes from 0.5 to 3 μm, a THz wave with frequency varying from 0.68 to 0.027 THz can be attained. The frequency of the THz wave is insensitive to the wavelengthλ _{1} because the refractive index difference betweenn _{1}^{e}(θ ) of optical waveλ _{1} andn _{2}^{o} of optical waveλ _{2} changes relatively little with optical wavelength in the infrared range.The frequency tuning can be realized by using the dependence of crystal birefringence on angle
θ between the directions of the optical wave propagation and optical axis. Figure 3 shows the tuning characteristics versus angleθ . From the figure we find that asθ changes from 0° to 90°, THz wave frequency varying from 0 to 0.3 THz can be attained. The frequency of the THz wave is insensitive to angleθ . Asθ changes from 0° to 90°, the effective nonlinear coefficientd_{eff} gradually increases. Whenθ equals 90°, thed_{eff} reaches the maximum value.Figure 4 shows the tuning characteristics versus working temperature
T . As working temperatureT varies from 20℃ to 77℃, THz wave frequency changes are relatively small. The tuning range is relatively limited because the difference betweenn _{1}^{e}(θ ) of optical waveλ _{1} andn _{2}^{o} of optical waveλ _{2} in the infrared range changes relatively little with working temperatureT . According to Eq. (6), with the increase of working temperatureT the absorption coefficientα _{THz} intensively increases due to the increase of the linewidth Γ_{j} of the A_{1}symmetry phonon modes.3.2. THz Wave Generation with PPLT
According to Eq. (4), the frequencies of the THz wave can be tuned by varying poling period Λ, wavelength of optical wave
λ _{1} andλ _{2}, and working temperatureT . Figure 5 shows the tuning characteristics versus optical wavelengthλ _{1} when the poling period Λ equals 13 μm. From the figure we find that asλ _{1} changes from 0.4 to 3 μm, frequencies of the THz wave varying from 8.5 to 10.7 THz can be attained. The frequency of the THz wave can be rapidly tuned by varying the wavelengthλ _{1} when the wavelengthλ _{1} is less than 1μm. Different from THz wave generation with bulk LiTaO_{3} where the frequencies of the generated THz wave are under 1 THz, the frequencies of THz wave generated from PPLT approach the polariton resonances of the LiTaO_{3} crystal.As
λ _{1} equals to 1.064 μm, the frequencies of THz wave versus the poling period Λ of PPLT are depicted in Fig. 6. From the figure we find that as Λ varies from 5 to 60 μm, a THz wave covering a tuning of 2.3 to 27.6 THz can be realized. The tuning range can cover the entire polariton resonances of the LiTaO_{3} crystal. The frequency of THz wave can be rapidly tuned by varying the poling period Λ for Λ less than 20 μm.Figure 7 shows the tuning characteristics versus working temperature
T . As working temperatureT varies from 20℃ to 200℃, THz wave frequency changes relatively little. The rapid tuning of a THz wave cannot be realized by varying working temperature.IV. PARAMETRIC GAIN CHARACTERISTICS
LiTaO_{3} crystal has five infrared and Ramanactive transverse optical (TO) phonon modes at the frequencies of 200 cm^{−1}, 241 cm^{−1}, 357 cm^{−1}, 596 cm^{−1}, 657 cm^{−1}, which are called A_{1}symmetry modes. The five phonon modes which correspond to the THz wave frequencies of 6 THz, 7.23 THz, 10.71 THz, 17.88 THz and 19.71 THz, are useful for efficient THz wave generation because of the largest parametric gain in the vicinity of phonon polariton resonances, as shown in Fig. 8. From the figure we find that as the THz wave approaches polariton resonances, the parametric gain coefficient
g _{0} reaches sharply a maximum value and the absorption coefficientα_{THz} increases intensively to a great value. Such dramatic enhancements of parametric gain coefficients can be exploited for improving the output powers and extending the frequency bands of THz waves if a surfaceemitted configuration is employed to minimize the propagation path of the THz wave within the crystal.Compared with other works in which THz wave generations are far from polariton resonances to avoid intensive absorption, the theoretical model proposed in this letter can exploit dramatic enhancement of parametric gain in the vicinity of polariton resonances with a surfaceemitted configuration. The theoretical model proposed in this letter is useful to other materials with crystal birefringence properties, such as LiNbO_{3}, KTA, in which polaritons are both infraredactive and Ramanactive. The theoretical model is also useful to semiconductor optical waveguides with modal birefringence properties. By utilizing modal birefringence of fundamental TE and TM modes in a planar waveguide, the absorption at polariton resonances can be efficiently reduced [19].
V. CONCLUSION
THz wave generation at polariton resonance of LiTaO_{3} by DFG processes is investigated. It is shown that by using crystal birefringence of bulk LiTaO_{3} and grating vector of PPLT, a THz wave can be efficiently generated by DFG in the vicinity of polariton resonances with a Cherenkov phase matching scheme. In the case of bulk LiTaO_{3}, the frequency of the THz wave is insensitive to the wavelength
λ _{1}, phase matching angle and working temperature. In the case of PPLT, the tuning range of a THz wave is widened by varying the poling period Λ. Dramatic enhancement of parametric gain coefficients in the vicinity of polariton resonances can be exploited for improving the output power and extending the frequency bands of a THz wave by using a surfaceemitted configuration to minimize the propagation path of a THz wave within the crystal.

[FIG. 1.] Schematic diagram of the surfaceemitted DFG with a Cherenkov phase matching scheme. (a) LiTaO3, (b) PPLT.

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[FIG. 2.] THz wave frequency versus the wavelength of optical wave λ1. T=23℃, α = 90°, θ = 90°.

[FIG. 3.] THz wave frequency versus the angle θ . T=23℃, α = 90°, λ1=1.064 μm.

[FIG. 4.] THz wave frequency versus working temperature T. λ1=1.064 μm, α = 90°.

[FIG. 5.] THz wave frequency versus the wavelength of optical wave λl. T = 25℃, Λ=13 μm.

[FIG. 6.] THz wave frequency versus the poling period Λ of PPLT. T = 25℃, λ1=1.064 μm

[FIG. 7.] THz wave frequency versus working temperature T. λ1=1.064 μm, Λ=15 μm.

[FIG. 8.] Parametric gain coefficient in the lowloss limit g0 and absorption coefficient αTHz versus THz wave frequency at room temperature. λ1 = 1.064 μm, Iλl = 200 MW/cm2.