Investigation of Stimulated Polariton Scattering from the B_{1}symmetry Modes of the KNbO_{3} Crystal
 Author: Li Zhongyang, Wang Mengtao, Wang Silei, Yuan Bin, Bing Pibin, Xu Degang, Yao Jianquan
 Publish: Current Optics and Photonics Volume 2, Issue1, p90~95, 25 Feb 2018

ABSTRACT
Stimulated polariton scattering from the B_{1}symmetry modes of a KNbO_{3} crystal to generate a terahertz wave (THzwave) with a noncollinear phasematching scheme is investigated. The frequencytuning characteristics of the THzwave by varying the phasematching angle and pump wavelength are analyzed. The expression for the effective parametric gain length under the noncollinear phasematching condition is deduced. Parametric gain and absorption characteristics of the THzwave in KNbO_{3} are theoretically simulated. The characteristics of KNbO_{3} for a terahertz parametric oscillator (TPO) are compared to those of MgO:LiNbO_{3}. The analysis indicates that KNbO_{3} is an excellent optical crystal for a TPO, to enhance the THzwave output.

KEYWORD
Stimulated polariton scattering , KNbO3 , Terahertz parametric oscillator

I. INTRODUCTION
Over the past two decades, with the everincreasing number of applications for terahertz (THz) radiation, such as imaging, biology, medicine, communications, security technologies, and quality control [16], there is growing demand for THz sources with excellent performance. Among many electronic and optical methods for terahertzwave (THzwave) generation, the terahertz parametric oscillator (TPO) [7] based on stimulated polariton scattering (SPS) processes exhibits many advantages, such as narrow linewidth, coherence, a wide tunable range, high power output, and roomtemperature operation. A polariton is a coupled photonphonon transversewave field, and polariton scattering is a nonlinear effect that occurs in crystals with both infrared and Ramanactive transverse optical (TO) modes [7]. In SPS, the interaction of a fundamental laser field with a polariton mode of a crystal generates a THzwave and a Stokes wave. The wavelengths of the generated THzwave and Stokes wave depend on the phasematching condition, giving rise to tunability. Typically the refractive index of the THzwave is substantially larger than that for the opticalpump wave, and phase matching is impossible for collinear interactions. Noncollinear phase matching can perform well for THz generation [811]. One bonus of noncollinear phase matching is convenient frequency tuning of the THzwave. However, the noncollinear phasematching configuration, in which the pump, Stokes, and THzwaves are all not parallel to each other, significantly reduces parametric gain. Thus it is vitally important to increase the effective parametric gain length in the noncollinear phasematching configuration.
A frequently employed material for TPOs is the nonlinear optical crystal MgO:LiNbO_{3}, because of its relatively large secondorder optical nonlinearity and its wide transparency range [12]. Unfortunately, the quantum conversion efficiency of such a TPO is extremely low, as the THzwave is intensely absorbed by the MgO:LiNbO_{3} crystal. At a frequency of 1.5 THz, the absorption coefficient is about 45 cm^{1} [13]. KNbO_{3} crystal is an attractive material for the nonlinear optical interaction between optical and THzwaves, due to its wide transmission range (0.44.5 μm) [14], high nonlinear coefficient (
d _{33} = 27.4 pm/V at 1064 nm) [15], and relatively high opticaldamage threshold of 350 MW/cm^{2} [16]. KNbO_{3} has four infrared and Ramanactive TO phonon modes, called B_{1}symmetry modes, located at 187, 243, 270, and 534 cm^{1} [17]. When pump excitation is sufficiently strong, a THzwave can be generated from the efficient parametric scattering of laser light via SPS.In this Letter, we theoretically study the characteristics of KNbO_{3} for a TPO with a noncollinear phasematching scheme. We analyze the frequencytuning characteristics of the THzwave by varying the phasematching angle and pump wavelength. The expression of the effective parametric gain length under the noncollinear phasematching condition is deduced. The gain and absorption characteristics of THzwaves in KNbO_{3} and MgO:LiNbO_{3} are investigated.
II. THEORETICAL MODEL
A surfaceemitting TPO with a noncollinear phasematching scheme comprises a singleresonant optical parametric oscillator with a FabryPerot cavity, as shown in Fig. 1. The configuration was first reported by T. Ikari
et al . [18]. The nonlinear optical crystal is KNbO_{3}. The resonant cavity for the Stokes wave consists of two planeparallel mirrors M_{1} and M_{2} of high reflectance. The pump wave passes through the cavity at the edges of M_{1} and M_{2}, and the Stokes wave propagates along the xaxis of the KNbO_{3}. A THzwave vector perpendicular to the output surface is achieved by setting the angle of incidence of the pump wave to the crystal surface. The polarizations of the pump, Stokes, and THzwaves are all along thez axis of the KNbO_{3} crystal.θ is the angle between the vectors of the pump and Stokes waves within the crystal, andφ is the angle between the vectors of the pump and THzwaves within the crystal. The cavity mirrors and KNbO_{3} crystal are mounted on a rotating stage. The wavelength of the Stokes wave, and hence the wavelength of the THzwave, can be tuned by rotating the stage continuously, since that changes the angleθ continuously.The theoretical values of refractive index are calculated using the Sellmeier equation for KNbO_{3} in the infrared range at 22℃ [14] and in the THz range [17], respectively. In this Letter, the theoretical parameters for KNbO_{3} are taken from reference [17].
III. TUNING CHARACTERISTICS
For tunable THzwave generation, two requirements must be fulfilled: the energy conservation law
ω _{p} =ω _{s} +ω _{T}, and the noncollinear phasematching condition _{p} =k _{s} +k _{T}, as shown in the inset of Fig. 1. Here,k ω _{p},ω _{s} andω _{T} are the angular frequencies, while _{p},k _{s} andk _{T} are the wavevectors of the pump, Stokes and THzwave, respectively. The phase matching condition can be rewritten as = + cosk θ . By varying one parameter of the noncollinear phasematching condition, such as the angleθ or the pump wavelengthλ _{p}, we can obtain a family of phasematching curves. Figure 2 shows the dispersion curve of the B_{1}symmetry polariton modes in KNbO_{3} and the phasematching curves for the 1064nm laser pump. When the phasematching curves are superimposed on the dispersion curve of the B_{1}symmetry polariton modes, the points of intersection of these curves are expected to determine the allowed frequencies and wave vectors of the THz wave. As the angleθ is changed continuously, the frequency tuning of the THz wave is realized simultaneously, which is the basic principle of the socalled angletuning method for a TPO. When the angleθ varies from 0° to 7.3°, the phasematching curves and the dispersion curve of the B_{1}symmetry polariton modes intersect, which means a THzwave can be generated.According to the noncollinear phasematching condition, tuning of a THzwave can be realized by varying the pump wavelength
λ _{p}. Figure 3 shows the dispersion curve of the B_{1}symmetry polariton modes in KNbO_{3} and the phasematching curves at a fixed phasematching angleθ of 1°, when the pump wavelengths are 400, 633, 1064, 1550, and 1064 nm respectively. From the figure we find that when the pump wavelengthλ _{p} changes, the intersection points of the phasematching curves and the dispersion curve of the B_{1}symmetry polariton modes change, which means that frequency tuning of the THz wave is realized simultaneously. The shorter the pump wavelength, the higher the frequency shift of the intersection point; that is, a higherfrequency THzwave will be achieved.IV. EFFECTIVE PARAMETRIC GAIN LENGTH CHARACTERISTICS
The effective parametric gain length is of vital importance for THzwave output, as the noncollinear phasematching scheme is employed in the TPO. Next we deduce the expression for the effective parametric gain length under the noncollinear phasematching condition, based on the theoretical model proposed in Ref. [19]. In this Letter we regard the phasematching angle θ between the vectors of the pump and Stokes waves as a double refraction walkoff angle, since the magnitudes of both angles are approximately equal, and the effect of both is identical. Assuming the three mixing waves have Gaussian profiles, the Stokes spot size is simultaneously narrowed by gain polarization and broadened by diffraction. The balance determines the final Stokeswave spot size. The relationship between the pumpwave radius
w _{p} and the Stokeswave radiusw _{s} is given bywhere
λ _{s} is the wavelength of the Stokes wave andL is the optical cavity’s length. The walkoff lengthl_{ω} is given bywhere
θ is used as a substitute for the double refraction walkoff angle. The effective parametric gain lengthL _{eff} is given bywhere
l is the crystal’s length, which is the propagation length of the Stokes wave within the KNbO_{3} crystal. The effective parametric gain lengthL _{eff} versus the pump wavelengthλ _{p} is shown in Fig. 4, when the frequency of the THzwave equals 1, 2, 3, 4, and 5 THz respectively. From the figure we find that as the pump wavelength increases, the effective parametric gain length gradually decreases. The reason is that as the pump wavelength increases, the phasematching angleθ is enlarged. The pump wave in the shortwavelength region can effectively lengthen the effective parametric gain length. Compared to frequencies of 2, 3, 4, and 5 THz, the effective parametric gain length is maximal at a THzwave frequency of 1 THz, because the phasematching angleθ is minimalat 1 THz.Figure 5 shows the effective parametric gain length versus the radius of the pump wave, when the frequency of the THzwave is 1, 2, 3, 4, and 5 THz respectively. From the figure we find that the effective parametric gain length increases rapidly and smoothly with increasing pumpwave radius. A pump wave with a large beam radius can generate a Stokes wave and a THzwave with a large beam radius simultaneously, resulting in a long effective parametric gain length. Actually, for maximum conversion efficiency the pump beam diameter must be increased until the effective parametric gain length is equal to the crystal’s length.
Figure 6 shows the effective parametric gain length versus crystal length, when the frequency of THzwave is 1, 2, 3, 4, and 5 THz respectively. From the figure we find that the effective parametric gain length increases rapidly with increasing crystal length for frequencies of 1, 2, and 3 THz, and increases smoothly as the frequency climbs to 4 and 5 THz. At lower frequencies of the THzwave, the pump and Stokes waves almost overlap, as the phasematching
θ is small. On the contrary, the pump and Stokes waves separate rapidly, as the two beams only partially overlap at higher frequencies.V. GAIN AND ABSORPTION CHARACTERISTICS
For efficient generation of the THzwave, parametric gain is another crucial parameter. According to Ref. [20], the analytical expression for the THzwave parametric gain
g _{T} under the noncollinear phasematching condition can be written aswhere
α _{T} is the absorption coefficient in the THz region,ω _{0j},S_{j} and Γ_{j} respectively denote the eigenfrequency, oscillator strength of the polariton modes, and bandwidth of thej th B_{1}symmetry phonon mode in the KNbO_{3} crystal.I _{p} is the power density of the pump wave, andg _{0} is the lowloss parametric gain.n _{P} andn _{s} are the refractive indices of the pump and Stokes waves respectively. and are the nonlinear coefficients related to pure parametric (secondorder) and Raman (thirdorder) scattering processes respectively. Far below the lowest B_{1}symmetry polariton mode at 187 cm^{1}, Eq. (5) can be rewritten asThe nonlinear coefficients
d'_{E} andd'_{Q} relate to the electrooptic coefficientr :Thus the lowloss parametric gain coefficient
g _{0} is rewritten asAccording to Eqs. (4)~(9), we plot the parametric gain coefficient
g _{T} at a pump intensity of 100 MW/cm^{2} and the absorption coefficientα _{T} in KNbO_{3} and MgO:LiNbO_{3}, as shown in Figs. 7(a) and 7(b) respectively. From the figure we find that, as the frequency of the THzwave is far below the lowest B_{1}symmetry polariton mode at 187 cm^{1}, the values of the parametric gain coefficientg _{T} for KNbO_{3} and MgO:LiNbO_{3} are of the order of several cm^{1}. The gain coefficient of KNbO_{3} is larger than that of MgO:LiNbO_{3}. Meanwhile, the values of the absorption coefficientα _{T} of the THzwave in KNbO_{3} are much smaller than those of MgO:LiNbO_{3}. Compared to MgO:LiNbO_{3}, KNbO_{3} has a larger gain coefficient and smaller absorption coefficient, so KNbO_{3} is a more suitable selection for a TPO to enhance THzwave output.VI. CONCLUSION
SPS from the B_{1}symmetry modes of KNbO_{3} crystal to generate a THzwave with a noncollinear phasematching scheme is investigated. A widely tunable THzwave can be generated by varying the phasematching angle and pump wavelength. A pump wave with shorter wavelength and larger beam radius can significantly increase the effective parametric gain length under the noncollinear phasematching condition. Compared to MgO:LiNbO_{3}, KNbO_{3} has a larger gain coefficient and smaller absorption coefficient, so KNbO_{3} is an excellent optical crystal for a TPO.

[FIG. 1.] Schematic diagram of a surfaceemitting TPO using KNbO3 with a noncollinear phasematching scheme.

[FIG. 2.] Dispersion curve for the B1symmetry polariton modes in KNbO3 and the phasematching curves for the 1064nm pump laser.

[FIG. 3.] Dispersion curve for the B1symmetry polariton modes in KNbO3 and the phasematching curves, at room temperature and a fixed phasematching angle θ of 1°.

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[FIG. 4.] Effective parametric gain length versus pump wavelength, assuming the physical length of the resonant cavity is 80 mm, l is 60 mm, and wp is 1 mm.

[FIG. 5.] Effective parametric gain length versus the radius of the pump wave, assuming the physical length of the resonant cavity is 80 mm, l is 60 mm, and λp is 1064 nm.

[FIG. 6.] Effective parametric gain length versus crystal length, assuming the physical length of the resonant cavity is 80 mm, wp is 1 mm, and λp is 1064 nm.

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[FIG. 7.] THzwave parametric gain coefficient gT and absorption coefficient αT for KNbO3 and MgO:LiNbO3 at room temperature (λp = 633 nm, Ip = 100 MW/cm2): (a) gain coefficient gT, (b) absorption coefficient αT.