Stimulated polariton scattering (SPS) has proved to be an efficient scheme to generate terahertz (THz) wave [1-7]. A polariton is a coupled quantum between a pump laser and the infrared- and Raman-active transverse optical (TO) modes in a crystal; it behaves like a phonon near the resonant frequency associated with the TO mode, and exhibits photon-like behavior for lower non-resonant frequencies [8]. SPS is composed of second- and third-order nonlinear frequency-conversion processes in which a pump photon stimulates a Stokes photon at the difference frequency between the pump photon and the polariton. At the same time, a THz wave is generated by the parametric process due to the nonlinearity arising from both electronic and vibrational contributions of the crystal. The TO phonon resonances can contribute substantially to the magnitudes of second- and third-order nonlinearities, which are beneficial to THz generation via SPS. A THz wave parametric oscillator (TPO) based on SPS processes exhibits many advantages, such as narrow linewidth, coherence, a wide range of tunability, high-power output, and room-temperature operation [1-5].

Ferroelectric crystals such as MgO:LiNbO₃, LiTaO₃, KTiOPO₄, RbTiOPO₄, and KTiOAsO₄ are characterized by strong infrared- and Raman-active TO phonon resonances, which are employed to generate a THz wave via SPS [9-13]. Ferroelectric crystals have a wide transmission range, a relatively high second-order nonlinear coefficient, and a high optical damage threshold. However, the quantum conversion efficiency of SPS in the above mentioned ferroelectric crystals is relatively low. The reasons are as follows: First, the wave vectors of the pump, Stokes, and THz waves are noncollinear, which intensively restricts the interaction volume among the three mixing waves. Second, the effective nonlinear optical coefficients involving both second- and third-order nonlinearities are not large enough. Third, the THz absorption coefficients of ferroelectric crystals are extremely large, particularly in high THz frequency range.

Similar to the above mentioned ferroelectric crystals, BaTiO₃ has infrared- and Raman-active TO phonon modes, which can be employed to generate THz wave via SPS [14]. BaTiO₃ with its perovskite structure has five atoms per unit cell and four sets of optical branches. At room temperature it has a tetragonal unit cell with C_4v point-group symmetry, and the optical branches are split into three A₁ and four E infrared- and Raman-active TO modes, and one B₁ Raman-active mode [15]. The four E TO modes in the tetragonal phase, with frequencies of 34, 180, 308 and 498 cm^-1, are doubly degenerate, having their polarization along the x- and y- axes [16]. The three A₁ TO modes in the tetragonal phase, with frequencies of 178, 260 and 515 cm^-1, are polarized along the z- axis [14]. BaTiO₃ is an attractive material for the nonlinear optical interaction between optical and THz waves, due to its wide transmission range (0.4~5 μm) [17], a relatively high nonlinear coefficient (d₁₅ = 13.7 pm/V at 1064 nm) [18], and a high optical damage threshold (83 GW/cm² at 596 nm, with 1 picosecond pulsewidth) [19]. It should be noted that BaTiO₃ suffers from severe photorefractive effect [20], so in the process of THz generation by SPS we should avoid the photorefractive effect.

In this Letter, we theoretically study SPS from the A₁ TO modes of BaTiO₃ bulk crystal with the noncollinear phase-matching (NPM) condition, for the first time. We analyze the phase-matching (PM) characteristics in the NPM configuration. The effective parametric gain length and effective parametric gain coefficient for the THz wave in the NPM configuration are calculated. THz phonon flux densities in BaTiO₃ are theoretically simulated by solving coupled wave equations under NPM conditions.

A surface-emitted TPO with a NPM configuration comprises a single-resonant optical parametric oscillator with a Fabry-Perot cavity, as shown in Fig. 1. The configuration was first reported by T. Ikari et al [2]. The nonlinear optical crystal is a BaTiO₃ crystal. The resonant cavity for the Stokes wave consists of two plane-parallel mirrors M₁ and M₂ with high reflectance. The pump wave passes through the cavity at its edges M₁ and M₂, and the Stokes wave propagates along the x axis of the BaTiO₃.A THz wave vector perpendicular to the output surface is achieved by setting the pump wave’s angle of incidence to the crystal surface. θ is the angle between the vectors of the pump and Stokes waves within the crystal, and φ is the angle between the vectors of the pump and THz waves within the crystal. To take advantage of the largest nonlinear optical coefficient d₁₅ of BaTiO₃, we consider that the pump, Stokes and THz waves are an o-wave, o-wave and e-wave respectively. The cavity mirrors and BaTiO₃ crystal are mounted on a rotating stage. The wave-lengths of the Stokes, and THz waves can be tuned by rotating the stage continuously, since the angle θ changes continuously. Since THz waves are severely absorbed by BaTiO₃, it is important to couple the THz wave out of the BaTiO₃ before the pump, Stokes and THz waves get absorbed [21].

Coupling of A₁ TO modes and the pump wave creates A₁-symmetry polaritons, with a dispersion curve. Figure 2 shows PM curves for a pump wavelength of 1064 nm, and an A₁-symmetry polariton dispersion curve in BaTiO₃. BaTiO₃ has a strong A₁ TO mode with a wavenumber of 178 cm^-1 and two medium-strength A₁ TO modes with wavenumbers of 260 and 515 cm^-1. The wavelengths of the Stokes waves generated by SPS depend on the PM angle θ between the pump wave and the Stokes wave, and the THz-wave frequencies are determined by the intersections of the PM curves and the A₁-symmetry polariton dispersion curve, as shown in Fig. 2. Polaritons generated by small-angle copropagating interactions between the pump and Stokes waves have terahertz frequencies for the 178 and 260 cm^-1 TO modes, and far-infrared frequencies associated with the 515 cm^-1 TO mode. For THz wave generation via SPS, two requirements have to be fulfilled: the energy conservation law ω_p = ω_s +ω_T , and the NPM condition k_p = k_s + k_T , as shown in the inset of Fig. 1. Here ω_p ,ω_s , and ω_T are the angular frequencies and k_p, k_s, and k_T the wave vectors of the pump, Stokes, and THz wave respectively. The PM condition can be rewritten as The theoretical values of the refractive index are calculated using a Sellmeier equation for BaTiO₃ in the infrared range [17] and THz range [14]. When the angle θ varies from 0° to 3.8°, the PM curves and the dispersion curve of the A₁-symmetry polariton modes intersect, which means a THz wave can be generated. It should be noted that the PM curve for θ = 0° intersects the polariton curve in the polariton-resonance region, which indicates that the collinear PM can be realized. At polariton-resonance region the refractive indices and nonlinear susceptibilities vary dramatically. A refractive index of 247 at 5.5 THz in the polariton-resonance region of KTiOPO₄ is obtained in backward SPS [7]. In the polariton-resonance region of 178 cm^-1 in BaTiO₃, the refractive indices of THz wave vary dramatically from 1.2 to 8.5. When the refractive index of the THz wave approximately equals those of the pump and Stokes waves, collinear PM can be realized. Owing to the strong coupling between TO mode and pump wave near polariton resonance, nonlinear susceptibilities can be dramatically enhanced, which can be employed to enhance the THz-wave output using a transverse-pumping geometry [22, 23].

The PM angles involved in the process of SPS will also vary with the THz frequency. Figure 3 shows the relation between the THz frequency and the PM angles θ and φ, in BaTiO₃ and MgO:LiNbO₃, with a pump wavelength of 1064 nm. When the frequency varies from 0 to 5 THz, θ varies from 0° to 1.9° in BaTiO₃ and from 0° to 2.8° in MgO:LiNbO₃, and φ varies from 44.5° to 60.3° in BaTiO₃ and from 63.4° to 67.7° in MgO:LiNbO₃. Compared to MgO:LiNbO₃, the PM angles θ and φ in BaTiO₃ are smaller, which means that the NPM configuration in BaTiO₃ can provide a larger effective interaction volume among the three mixing waves.

The PM angles θ and φ determine the magnitude of the effective parametric gain length of the Stokes and THz waves. Next we deduce the formula for the effective parametric gain length under the NPM condition, based on the theoretical model proposed in Ref. [24]. As shown in Fig. 3, the angle θ in the range of 0°-1.9° in BaTiO₃ is equivalent to a double-refraction walk-off angle in Ref. [24], since the magnitudes of both angles are approximately equal, and the effects of both are identical. Assuming that the pump, Stokes, and THz waves have Gaussian profiles, the transverse dimension of the Stokes is simultaneously narrowed by gain polarization and broadened by diffraction. The relation between the pump-wave radius w_p and the Stokes-wave radius w_s is given by

where λ_s is the wavelength of the Stokes wave and L is the optical cavity length, L = L' + (n_s −1) l, n_s is the refractive index of the Stokes wave, L' is the physical length of the Stokes cavity and l is the crystal length. The walk-off length l_ω is given by

The effective parametric gain length of the Stokes wave L_eff-Stokes is given by

The relationship between the effective parametric gain length of the Stokes wave L_{eff -Stokes} and that of the THz wave L_{eff -THz} is given by

The effective parametric gain lengths L_{eff -Stokes} and L_{eff -THz} versus pump-wave radius w_p for THz frequencies of 1, 3, and 5 THz are shown in Fig. 4. The effective parametric gain lengths L_{eff -Stokes} and L_{eff -THz} monotonically increase with increasing pump-wave radius w_p. The maximum value of L_{eff -THz} is about 2 mm, and that of L_{eff -Stokes} is about 59.5 mm. The effective parametric gain lengths L_{eff -Stokes} and L_{eff -THz} rapidly increase when the pump wave radius w_p is in the range of 0~0.5 mm, and gradually increase when the pump wave radius w_p is in the range of 0.5~2 mm. When w_p is larger than 2 mm, L_{eff -Stokes} and L_{eff -THz} increase slowly. A pump wave with a larger radius can generate a Stokes wave and a THz wave with a large beam radius simultaneously, resulting in a long effective parametric gain length. Actually, for maximum conversion efficiency the pump beam’s diameter must be increased until the effective parametric gain length equals the crystal length.

The effective parametric gain lengths L_{eff -Stokes} and L_{eff -THz} versus the THz frequency for pump wavelengths of 532, 1550, and 3000 nm are shown in Fig. 5. The effective parametric gain length L_{eff -Stokes} monotonically decreases with increasing THz frequency. The separation between pump and Stokes waves enlarges as the PM angle θ increases with increasing THz frequency. The increase of θ induces the decrease of L_{eff -Stokes} . From Eq. (4) and Fig. 3, the effective parametric gain length L_{eff -THz} relates to L_{eff -Stokes} , the angle θ, and the angle φ. With increasing THz frequency, the angles θ and φ increase and the L_{eff -Stokes} decreases, so L_{eff -THz} does not vary monotonically.

The effective parametric gain L_{eff -Stokes} and L_{eff -THz} versus crystal length l for THz frequencies of 1, 3, and 5 THz are shown in Fig. 6. The effective parametric gain lengths L_{eff -Stokes} and L_{eff -THz} monotically enlarge with increasing crystal length l. L_{eff -Stokes} increases rapidly with crystal length l for the frequencies is 1 and 3 THz, and increases smoothly for a frequency of 5 THz. At lower frequencies of the THz wave, the pump and Stokes waves almost overlap as the PM θ is small. On the contrary, the pump and Stokes waves separate rapidly as the two beams partially overlap at higher THz frequencies.

For high-power THz wave generation, the magnitude of the parametric gain coefficient is important. We calculate the frequency dependence of THz-wave parametric gain g_T using the theoretical model proposed by Sussman [8, 25]. The analytic expression for g_T under the NPM condition in SI units can be written as [26]

where ω_0j , S_j , and Γ_j respectively denote eigenfrequency, oscillator strength of the polariton modes, and the bandwidth of the jth A₁-symmetry phonon mode in the BaTiO₃ crystal. I_p is the power density of the pump wave, g₀ is the low-loss parametric gain, and n_p and n_T are the refractive indices of the pump and THz waves, respectively. α_T is the material absorption coefficient in the THz region, while α_pr is the effective absorption coefficient arising from the propagation of the THz wave out of the interaction region among the pump, Stokes, and THz waves, as proposed by Schwarz and Maier [27]. a is a parameter that determines the ratio of the effective diameter of the interaction region to the pump radius w_p , and was estimated to be 0.2 [27]. d_E = 16πd₁₅ and d_Q are nonlinear coefficients related to pure parametric (second-order) and Raman (third-order) scattering processes, respectively.

When the THz frequency is far below that of the lowest TO mode of 178 cm^-1, Eq. (6) can be rewritten as [8]

The relationship between d_E and d_Q is given by [8, 28, 29]

where r₅₁ is the linear electro-optic coefficient of BaTiO₃. r₅₁ is 1640 pm/V at 546.1 nm in unclamped BaTiO₃ [30].

According to Eqs. (5)~(10), we plot the values of the THz wave parametric gain coefficient g_T and the absorption coefficient α_T in BaTiO₃ and MgO:LiNbO₃ in the range of 0~2 THz, at a pump intensity of 100 MW/cm², as shown in Fig. 7. In this Letter, the data for MgO:LiNbO₃ are taken from Ref. [8]. From the figure we find that the THz absorption coefficients α_T of BaTiO₃ and MgO:LiNbO₃ are approximately equal, whereas the gain coefficient g_T of BaTiO₃ is much larger than that of MgO:LiNbO₃. Below 1.0 THz, g_T in BaTiO₃ increases rapidly with frequency, while in the range of 1.0~2.0 THz, it varies slowly. The maximum value of g_T in BaTiO₃ is 23.4 cm^-1 at 1.7 THz. The large gain coefficient g_T in BaTiO₃ is derived from the third-order nonlinearity d_Q , which originates from ionic polarization. From Eq. (9) we find that g₀ is proportional to the pump intensity I_p. A pump wave with a picosecond or femtosecond pulse width, providing a pump intensity on the order of GW/cm², can enlarge the gain coefficient g_T by several orders of magnitude [31-33]. A recent experimental study of MgO:LiNbO₃ has shown that the conversion efficiency of a THz wave via SPS can be improved by three orders of magnitude through the use of subnanosecond pump pulses [34].

The coupled wave equations for THz-wave parametric mixing with THz-wave absorption can be found in Ref. [8]. The coupled wave equations describe the field-envelope variation of the pump, Stokes, and THz waves. With strong THz-wave absorption, and without phase mismatch or pump depletion, the coupled wave equations can be solved to give the THz-photon flux density ϕ_T , with a general solution [35] given by

where the initial THz-photon flux density is assumed to be zero, and ϕ_S(0) is the initial Stokes-photon flux density. According to Eq. (11), we plot the THz-photon flux density normalized to the initial Stokes-photon flux density ϕ_T/ϕ_S(0) for a pump wavelength of 546.1 nm, as shown in Fig. 8. ϕ_T/ϕ_S(0) monotonically rises with increasing pump-wave radius w_p , as shown in Fig. 8(a). The monotonic increase of the effective parametric gain length L_{eff -THz} with increasing pump-wave radius w_p , as shown in Fig. 4, leads to the monotonic increase of ϕ_T/ϕ_S(0) . ϕ_T/ϕ_S(0) slowly increases when w_p is in the range of 0~0.5 mm, then rapidly increases when w_p is larger than 0.5 mm. The maximum value of ϕ_T/ϕ_S(0) is 3.84. It should be noted that in the calculations we assume the pump intensity is invariant when the pump-wave radius is varying; in fact, the pump intensity decreases with the increase in pump-wave radius, which decreases the parametric gain. For this reason, the rate of increase of ϕ_T/ϕ_S(0) with the increase of pump-wave radius slows. Figure 8(b) shows the relation between ϕ_T/ϕ_S(0) and THz-wave frequency. ϕ_T/ϕ_S(0) increases in the range of 0~1.29 THz and decreases in the range of 1.29~2 THz, with a maximum value of ϕ_T/ϕ_S(0) 2.65 at 1.29 THz. Figure 8(c) shows the relation between ϕ_T/ϕ_S(0) and crystal length l. ϕ_T/ϕ_S(0) monotonically rises with the increase of crystal length l. The maximum value of ϕ_T/ϕ_S(0) is 2.13. L_{eff -THz} increases rapidly with increasing crystal length l, as shown in Fig. (6), which leads to the enhancement of THz-photon flux density along with crystal length l. From Fig. 8, the value of ϕ_T/ϕ_S(0) can be enhanced by selecting proper parameter values, so it is possible to generate a high-power THz wave by injecting a Stokes seed wave [34].

THz wave generation via SPS in BaTiO₃ bulk crystal has certain advantages over that in bulk MgO:LiNbO₃. First, the PM angles θ and φ in BaTiO₃ are smaller than those in MgO:LiNbO₃, indicating that the NPM configuration in BaTiO₃ can provide larger effective interaction volume among the three mixing waves and longer effective parametric gain length. Second, the effective parametric gain coefficients in BaTiO₃ are much larger than those in MgO:LiNbO₃.

SPS from the A₁ TO modes of BaTiO₃ bulk crystal under the NPM condition can efficiently generate a THz wave. The PM angles in BaTiO₃ are smaller than those in MgO:LiNbO₃. The effective parametric gain length of the Stokes and THz waves relate to the radius of the pump wave, the THz frequency, and the crystal length. The effective parametric gain coefficients in BaTiO₃ are much larger than those in MgO:LiNbO₃. Larger radius of the pump wave and longer crystal length can generate a THz wave with a larger phonon flux density. The high gain coefficients and large interaction volume among the three mixing waves indicate that BaTiO₃ is an attractive optical crystal for THz wave generation via SPS.