## Abstract

The optical properties of the α/β-BaTeMo_{2}O_{9} (α/β-BTM) crystal in the terahertz range were characterized by the terahertz time domain spectroscopy (TDS) system. Frequency-dependent refractive indices and absorption coefficients of two crystals were measured from 0.2 to 2 THz, and discussions and comparisons were made on birefringence, absorption and phonon modes by referring to their structures. The Sellmeier equations for both crystals were also fitted in their transparent ranges. Based on the mode properties and parameters, a feasible scheme for terahertz generation via stimulated polariton scattering (SPS) with β-BTM was proposed and the angle tuning characteristics were calculated. Simulations show that β-BTM has great potential and should be more advantageous than LiNbO_{3} in generating high-frequency terahertz waves.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

BaTeMo_{2}O_{9} (BTM) crystals, which were firstly grown in Shandong University, have anion groups with asymmetric structure and demonstrate outstanding performance as both second- and third-order nonlinear materials [1]. There are two transformable polymorphous phases of BTM crystal, α-BTM with orthorhombic structure and β-BTM with monoclinic structure, depending on the reaction temperature during the crystal growth process [2]. Both polymorphs of BTM crystals are good candidates for nonlinear frequency conversion. On one hand, both crystals are excellent Raman media with high Raman gain and appropriate Raman shift, and remarkable achievements have been made in the wavelength range from 1178 nm to 1531 nm [3,4]. On the other hand, the β-BTM crystal has a larger second-order nonlinearity (*d*_{31}=-9.88 pm/V) than α-BTM, and it has been successfully used to achieve high-efficiency yellow lasers by self-frequency-doubled Raman conversion [5].

The superior performance of β-BTM crystals on Raman and second-order nonlinear frequency conversion promotes another potential application—terahertz generation based on stimulated polariton scattering (SPS), which requires the material to possess transverse optical phonon modes that are both Raman and infrared active, as well as good physical characteristics for high conversion efficiency. To date, only few practical materials are available for terahertz generation via SPS, and LiNbO_{3}, KTP and their isomorphs are typical representatives [6–11]. The lack of material limits the terahertz power scaling and wavelength extension, therefore, seeking new materials have always been an urgent and challenging work. It has been shown that the BTM crystals have diversified vibration modes that are both Raman and infrared active [12,13]; Specially, the good physical properties of β-BTM, such as large nonlinear figure of merit (the $d_{\textrm{eff}}^2$/${\textrm{n}^3}$ value of β-BTM is 3.17 times of LiNbO_{3} for the second harmonic generation process from 1064 nm to 532 nm [1]), high Raman gain, wide transmission range (0.4–5.5 µm) and high damage threshold (>500 MW/cm^{2}), make it a promising candidate for terahertz generation through the SPS process.

In this paper, the optical properties of BTM crystals in the terahertz range were studied by the terahertz time domain spectroscopy (TDS). The refractive indices and absorption coefficients of α- and β-BTM crystals in the terahertz range were measured, and the Sellmeier equations in the 0.2–2 THz region were fitted. Significant difference on dispersion and birefringence between two polymorphous phases was revealed from the comparison. Based on the characterizations, the scheme of terahertz generation via SPS in β-BTM was proposed and the tuning curves were given, laying the basis for realizing a practical terahertz source.

## 2. Measurement scheme and sample preparation

A commercial terahertz TDS system (TAS 7500TS, Advantest Corp.) was employed to measure the optical properties of the BTM samples, which consisted of two channels of ultrashort 1550-nm fiber lasers for terahertz generation and detection. Phase-modulated dual-laser-synchronized control technology without a mechanical optical delay line enabled extremely high-speed terahertz spectroscopy (8 ms per scan) and high frequency resolution (3.8 GHz). The whole optical path for terahertz wave, including the sample chamber, was purged by flowing dry air to eliminate the water vapor absorption. The maximum signal-to-noise ratio (SNR) reached over 40 dB averaged by 2048 times per measurement.

Both the α- and β-BTM crystals were grown by the top-seeded solution growth method [2]. The dielectric axes *x*, *y* and *z* of the orthorhombic α-BTM crystal coincide with the crystallographic axes *b*, *c* and *a*, respectively, but only one dielectric axes of the monoclinic β-BTM crystal (*z*) overlaps the crystallographic axis (*b*), and the other two orthorhombic refractive axes (*x* and *y*) of β-BTM locates in the *a*-*c* plane. The angle is 25.5° between *y*- and *a*-axis while it is 24.603° between *x*- and *c*-axis [12]. To characterize their optical properties in the terahertz range, thin wafers (∼ 1 mm for α-BTM and ∼ 0.6 mm for β-BTM) were cut from bulk single crystals and polished, with clear apertures of 7 mm × 9 mm to facilitate the measurements. Since the investigated crystals are biaxial, sample transmission directions along three dielectric principle axes (*x*, *y* and *z*) were prepared with detailed specifications given in Table 1.

The sample was placed at the confocal plane of two parabolic mirrors of the TDS system, so that the terahertz beam was focused into the sample perpendicularly. The transmitted terahertz wave that carried the sample information was collected by a spherical silicon lens and converted into current signal by the photoconductive switch, giving the time-domain curve via optical sampling and amplifying. After performing Fourier transforms, the refractive index *n* and the absorption coefficient *α* was obtained from the complex transmission coefficient by comparing the signal and reference electric fields. As the terahertz field was linearly polarized, it could be arranged parallel to all the three principle axes, so as to discuss the interactions between the terahertz wave and phonon modes.

## 3. Measurement results and discussions

The time-domain signals for all the samples in Table 1 and the corresponding references are shown in Fig. 1. Since the terahertz polarizations along two axes could be measured for each sample, e.g., it was possible to measure the transmitted signal for *x-* and *y*-polarizations for a *z*-cut sample, there were 6 time-domain signal curves in total for both the α- and β-BTM crystals. Identical curves were recorded when the terahertz polarization was along a certain axis even if the crystal orientation was different (neglecting the thickness difference), which meant that the crystal dielectric response depends on the field polarization rather than the propagation direction. Thereupon, the characterization could be accomplished with any two samples with different orientations for each crystal. The overall time delays of β-BTM samples were smaller because they were much thinner. It should also be noted that the time delays of the α-BTM samples in *x-* and *y-*polarization were similar, both much smaller than that in *z*-polarization, indicating group indices follow *n*_{αx}≈*n*_{αy}<*n*_{αz}. However, the time delays in *x-*, *y-* and *z-*polarizations were all significantly different for the β-BTM samples and the maximum delay occurred for *y*-polarization, which denote that *n*_{βx}<*n*_{βz}<*n*_{βy}. These conclusions provide proof that the orthorhombic α-BTM crystal is more symmetric than the monoclinic β-BTM crystal, and it can be verified by the refractive-index calculations in the following text.

The frequency-domain transmission spectra were obtained by Fourier transform to time-domain signals, as shown in Fig. 2. Although the TDS bandwidth exceeded 3 THz, the signal above 2 THz was seriously affected due to strong crystal absorption. Reliable transmission ranges for x-, y- and z-axes of the α-BTM crystal were 0.2–1.3 THz, 0.2–1.5 THz and 0.2–1.0 THz, respectively, while for the β-BTM crystal they were 0.2–1.8 THz, 0.2–1.2 THz and 0.2–1.5 THz, respectively. The distinct transmission difference along different axes came from the anisotropy of refractive-index and absorption. At the low-frequency end, the transmission is mainly decided by surface reflection related to refractive-index, while it depends on the absorption from the optical phonons at the high-frequency part. Two noticeable dips in the y-polarization transmission curve locating at 1.04 and 1.22 THz for α-BTM corresponds to two phonon modes in this range which cause strong absorption.

By dividing the complex electric fields and the reference, the refractive index was extracted from the phase [13]. The results are given in Fig. 3. Consistent with the predictions from the time-domain signals given in Fig. 1(a), the difference was minor between the *x-* and *y*-polarizations (*n*_{αy}-*n*_{αx}=0.13 at 0.6 THz) for α-BTM, while the birefringence was huge (*n*_{αz}-*n*_{αy}=1.75 at 0.6 THz) between *y*- and *z-*polarizations. In contrast, large birefringence existed among all three polarizations (*n*_{βy}-*n*_{βx}=1.93 and *n*_{βz}-*n*_{βy}=-0.97 at 0.6 THz) for β-BTM. The difference between *n*_{x} and *n*_{y} for two crystals is also illustrated in Fig. 4, where two *z*-cut samples were used and the terahertz polarization had a certain angle to the principle axes *x* or *y*. The apparent splitting (∼4.1 ps) of the time-domain signal after transmitting through the β-BTM sample accorded well with the phase difference calculated by the birefringence (*n*_{y}-*n*_{x}) and sample thickness (Fig. 4(b)), while the signal splitting was negligible for α-BTM with much smaller birefringence (Fig. 4(a)). Generally, the values of refractive indices and birefringence for both crystals are much larger than that in the visible to mid-infrared range, which favors their use as phase retarders and wave plates. A noticeable point here is that the relative value of *n*_{βz} and *n*_{βy} reverses across the reststrahlen band from mid-infrared to the terahertz band.

As the existing equations in the visible to mid-infrared range become inapplicable in the terahertz range [12,14], the dispersion equations for α- and β-BTM crystals were fitted in the terahertz transparent range, using the empirical Sellmeier model [15]

*i*denotes three principle axes

*x*,

*y*and

*z*,

*λ*is the wavelength in µm. The Sellmeier parameters

*A*,

*B*,

*C*,

*D*and

*E*, and the corresponding validity ranges are given in Table 2. The comparison of the fitted curves and measured results is shown in Fig. 3. The discrepancy for

*y*-polarization of α-BTM is resulted from two phonon modes at 1.04 and 1.22 THz, which lead to leaps in the dielectric curves.

The absorption coefficient was obtained from the terahertz field complex amplitude and refractive index [13], as shown in Fig. 5. Since the thickness of the α-BTM samples (∼ 1 mm) were obviously larger than that of the β-BTM samples (∼ 0.6 mm), the maximum measurable absorption coefficient for α-BTM was only 110 cm^{-1}, while it was 160 cm^{-1} for β-BTM [16]. Due to the far wings of the massive phonon modes in the high-frequency range, the absorption increases with frequency. Ostensibly it is random among different dielectric axes (generally *α*_{αz}>*α*_{αx}>*α*_{αy} for α-BTM, and *α*_{βy}>*α*_{βz}>*α*_{βx} for β-BTM), however, both crystals show good consistency from the aspect of crystallographic axis (*α*_{a}>*α*_{b}>*α*_{c} for both α- and β-BTM crystals), referring to the relationship between two coordinate systems given in Part 2. The terahertz polarization along the *c*-axis has the widest transmission range because both crystals are *c*-oriented two-dimensional layered structures [12,14], where the coupling between the electric field and the chemical bonds is the weakest.

The absorption spectrum shows the properties of molecular vibrational mode. On one hand, α-BTM crystallizes in the orthorhombic Pca_{21} ($\textrm{C}_{2\textrm{v}}^5$) structure, and group theory predicts 78*A*_{1}+78*B*_{1}+78*A*_{2}+78*B*_{2} Brillouin zone center modes, in which *A*_{1}, *B*_{1} and *B*_{2} mode are both Raman and infrared active [17]. Restricted by the bandwidth and SNR of the TDS system, most of the modes are not reflected in Fig. 5(a). However, two *A*_{1} modes at 1.04 THz and 1.22 THz, respectively, was observed with *y*-polarized terahertz field, which denote that the dipole generated by the vibration is along the *y*-axis (*c*-axis). It should be noted these two modes haven’t been reported or predicted by first-principle calculations to date. On the other hand, β-BTM crystallizes in the monoclinic P_{21} (C_{2}) structure and 39*A *+ 39*B* Brillouin zone center modes were predicted, where *A* and *B* modes are both Raman and infrared active [18]. When the terahertz field is along the *z*-axis *A* modes are excited. However, since the vibration of *B* modes locates in the *x-y* plane but not affiliated to a certain axis, these phonon states cannot be directly reflected in Fig. 5(b), even if absorption peaks are recorded for *x-* and *y*-polarizations.

## 4. Terahertz generation in β-BTM: a theoretical simulation

Compared with the uniaxial (e.g., LiNbO_{3}) and orthorhombic (e.g., KTP) crystals used for terahertz generation via SPS, the monoclinic β-BTM involves more dielectric tensor components and are more complicated [1]. Although both the *A* and *B* modes can be adopted in SPS, the *B*-modes are in the *x-y* plane and their polarization is unknown, as discussed in the former part. Thus, only the *A*-modes which coincide with a certain dielectric axis (*z*-axis) are considered to avoid the angle-dependent issues and facilitate the experiment. Figure 6 shows the classical model of SPS in β-BTM with a noncollinear phase-matching (PM) scheme. To guarantee a large effective nonlinear coefficient and sufficient Raman scattering efficiency, three interaction waves (pump *ω*_{P}, Stokes *ω*_{S} and terahertz *ω*_{T}) were all *z*-polarized. *θ* is the angle between the pump and Stokes wave vectors, and it plays an important role in wavelength tuning.

The dispersion and absorption of phonon modes in the β-BTM crystal were simulated using the Lorentz oscillator model [19,20]

*ɛ*(

*ν*) is the complex dielectric constant,

*ν*is the wavenumber,

*ν*

_{TO}is the eigen frequency of the corresponding polariton mode,

*ɛ*is the high-frequency dielectric constant,

_{∞}*ρ*is the oscillator strength, and

*γ*is the damping coefficient. According to the crystal lattice vibration parameters provided in 18, the dispersion and absorption of all the phonon modes were calculated and given in Fig. 7(a). For a certain resonant mode (e.g.,

*ν*

_{TO}=272 cm

^{-1}), the dispersion shows phonon characteristics at large wave vectors (

*ν*>

*ν*

_{TO}) with resonance enhanced absorption, while the absorption loss reduces rapidly at small wave vectors (

*ν*<

*ν*

_{TO}) and the dispersion shows electromagnetic radiation characteristics (parametric characteristics), resulting in terahertz radiation via the SPS process [21].

The PM condition should be fulfilled for the parametric process

*θ*is changed, thereby tunable terahertz waves are realized. The terahertz tuning range depends on the frequency range of the intersections. As to the vibrational mode

*ν*

_{TO}=272 cm

^{-1}, the intersection of the PM curve and the low-frequency branch of the dispersion curve can be changed from

*ν*=239.5 cm

^{-1}(7.25 THz) to

*ν*=272 cm

^{-1}(8.24 THz) when

*θ*is changed from 0° to 4.7°, yielding a continuous tuning range of 7.25–8.24 THz. Various vibrations modes at higher frequency, e.g., those centering at 320 cm

^{-1}, 599.4 cm

^{-1}, and 801 cm

^{-1}, enables the output frequency range extending to 9.36–10.23 THz, 12.06–18.31 THz and 23.69–25.05 THz, respectively, far beyond the universal definition of terahertz band (0.1–10 THz) and entering the infrared range.

To evaluate the performance of terahertz generation in β-BTM via SPS, a theoretical comparison was made on the absorption coefficients in the terahertz range between β-BTM and a commonly used material LiNbO_{3} [10], using the equation [19,20]

*α*

_{T}is the absorption coefficient,

*ν*

_{oj}and

*S*

_{j}are the eigen frequency and the oscillator strength of each polariton mode. As shown in Fig. 8, β-BTM generally has relatively weaker absorption than LiNbO

_{3}from 5 to 24 THz, especially within the tuning ranges mentioned above. Although the absorption coefficients from 0.2 to 1.5 THz (Fig. 5(b), y-pol) were higher than the calculated results given in Fig. 8, because of the inaccuracy of the low-frequency phonon-mode parameters during characterization [18], the calculations in the high-frequency part (above 50 cm

^{-1}or 1.5 THz) were reliable. It is known to all that absorption is the main factor that limits the available output frequency range to be lower than 4.9 THz in LiNbO

_{3}[22]. The advantage of β-BTM makes it a promising material for the generation of high-frequency terahertz and long-wave infrared radiation via SPS. Furthermore, compared with LiNbO

_{3}, there are intersections between the PM and dispersion curves when

*θ*=0° (Fig. 7(b)), indicating collinear terahertz generation is possible.

## 5. Conclusion

The optical properties of the α- and β-BTM crystals in the frequency range of 0.2–2 THz were studied by TDS. The refractive indices of both crystals were measured and the Sellmeier equations in their transparent frequency ranges were obtained by fitting the measurement data. It was found that the β-BTM crystal has much larger birefringence in the *x-y* plane than the α-BTM crystal. The characteristics accurately reflected the structural differences between the orthorhombic and monoclinic crystals. Two lattice vibration modes at 1.04 and 1.22 THz along the *y*-axis of α-BTM were discovered by analyzing the absorption coefficients. Moreover, the lowest absorption was obtained for both crystals along the *c*-axis, as predicted by their *c*-oriented two-dimensional layered structure. Based on the optical properties of β-BTM in the terahertz range, its potential performance in terahertz generation by SPS was investigated. The absorption and dispersion characteristics of its *A* vibration modes were calculated with the Lorentz oscillator model. The results show that by changing the angle *θ* between pump and Stokes waves, main tuning ranges of 7.25–8.24 THz, 9.36–10.23 THz, 12.06–18.31 THz, and 23.69–25.05 THz can be covered, extending from terahertz to infrared waves. Simulations also show that the absorption coefficients of β-BTM are lower than that of LiNbO_{3} in most of the tuning ranges. It is believed that the β-BTM crystal has great potential to enhance the output and extend the frequency range from terahertz to infrared.

## Funding

National Natural Science Foundation of China (61675146); Natural Science Foundation of Tianjin City (18JCYBJC16700).

## Disclosures

The authors declare no conflicts of interest.

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