Potential of Sub-THz-Wave Generation in Li 2 B 4 O 7 Nonlinear Crystal at Room and Cryogenic Temperatures

: Due to their high optical damage threshold, borate crystals can be used for the efﬁcient nonlinear down-conversion of terawatt laser radiation into the terahertz (THz) frequency range of the electromagnetic spectrum. In this work, we carried out a thorough study of the terahertz optical properties of the lithium tetraborate crystal (Li 2 B 4 O 7 ; LB4) at 295 and 77 K. Approximating the terahertz refractive index in the form of Sellmeier’s equations, we assessed the possibility of converting the radiation of widespread high-power laser sources with wavelengths of 1064 and 800 nm, as well as their second and third harmonics, into the THz range. It was found that four out of eight types of three-wave mixing processes are possible. The conditions for collinear phase matching were fulﬁlled only for the o − e → o type of interaction, while cooling the crystal to 77 K did not practically affect the phase-matching curves. However, a noticeable increase of birefringence in the THz range with cooling (from 0.12 to 0.16) led to an increase in the coherence length for o − o → e and e − e → e types of interaction, which are potentially attractive for the down-conversion of ultrashort laser pulses.


Introduction
A significant development in high-power visible and near-infrared (IR) laser sources has encouraged research into nonlinear media suitable for the efficient conversion of radiation into other spectral ranges [1,2]. One unique case of such a source is the THL-100-a terawatt laser complex generating femtosecond pulses in the visible range. This complex consists of a high-power titanium-sapphire laser, a second harmonic generator, and a gas amplifier operating at the C-A transition in XeF. The THL-100 generates~30 fs pulses at a wavelength of 475 nm with an energy of up to 1.2 J (corresponding to a peak power of 40 TW) [3]. One promising application of such a laser system is the nonlinear optical generation of intense terahertz pulses (down-conversion) [4]. Several applications of the intense THz waves are intensively developing nowadays. Compact charged particle accelerators could use the benefits of THz radiation (their dimensions limited by the wavelength of the accelerating electromagnetic field [5,6]). Advances of intense THz generation led to the development of nonlinear optical approaches into new spectral ranges [7]. Moreover, there are a lot of studies on THz waves' selective effect on living organisms [8]. An additionally unsolved problem is the designing and building THz Light Detection and Ranging (LiDAR) system that demands a high-power terahertz source. Since many hazardous industrial and greenhouse gases have absorption lines at sub-terahertz frequencies (where the atmospheric transparency windows are located), such a LiDAR would allow for the monitoring of their small components in the surface layer at kilometer distances for environmental and climatic purposes [9,10].
To achieve the effective down-conversion of intense laser radiation, it is crucial to find nonlinear crystals with a sufficiently large aperture, high optical quality, high damage threshold, and high transparency at pump wavelengths. Nonlinear optical crystals of the borate family satisfy these criteria [11]. Among them, the crystal of lithium tetraborate, Li 2 B 4 O 7 (LB4), has the shortest wavelength of ultraviolet (UV) absorption edge at 0.16 µm, low optical losses (α ≤ 10 −4 cm −1 ), and a high damage threshold (up to 40 GW/cm for 10 ns pulses). However, it has lower nonlinear coefficients at a wavelength of 1064 nm (d 31 = 0.12 pm/V, d 33 = 0.47 pm/V) compared to other borate crystals (β-BBO: |d 22 | = 2.2 pm/V; LBO: |d 31 | = 0.67 pm/V, |d 32 | = 0.85 pm/V) [12]. Nevertheless, these properties make LB4 particularly effective for harmonic generation down to wavelengths near the UV absorption edge [13,14]. These properties also make LB4 a promising converter of UV and visible laser radiation into the THz range using pump intensities near the crystal laser-induced damage threshold. Additionally, the damage threshold and conversion efficiency of the crystal can be increased via cryogenic cooling due to decreases in the absorption of both pump radiation and terahertz waves [15].
Thus, in this work, we studied the terahertz refractive index and absorption coefficient of the lithium tetraborate nonlinear crystal (Li 2 B 4 O 7 ; LB4) at room (295 K) and liquid nitrogen (77 K) temperatures in the range up to 2 THz. Additionally, we measured and the refractive index and transmission of the crystal in the main transparency window, as well as comparing them with previously published data. The terahertz refractive index was approximated in the form of Sellmeier's equations. The possibility of converting the radiation of widespread high-power laser sources with wavelengths of 1064 and 800 nm, as well as their second (532 and 400 nm) and third (355 and 266 nm) harmonics, into the THz range was assessed. To the best of our knowledge, such thorough studies of terahertz optical properties and possible nonlinear interactions have not previously been carried out.

Materials and Methods
The LB4 single crystal was grown in the Institute of Geology and Mineralogy SB RAS, Russia [16]. The samples were plane-parallel plates with a thickness of 350 µm. The plates were polished to optical quality. A dielectric antireflection coating was not applied. The optical z-axis of the samples was placed parallel to the polished surfaces to study optical properties for ordinary and extraordinary waves. The optical properties were measured in both the main transparency window and the terahertz range.
The refractive indices in the main transparency window were measured using a Metricon 2010/M prism coupler system (Metricon Corporation, Pennington, NJ, USA). The operating principle of the system is based on determining the critical angle of laser radiation out-coupling from the measuring prism in the total internal reflection mode. The force of pressing the crystal samples to the prism was 30-40 psi. The system was equipped with five semiconductor lasers with wavelengths of 517.1, 636, 846, 1309.6, and 1547.2 nm, which made it possible to estimate the dispersion of the refractive index of the samples.
The transmission of samples in the visible and IR spectral ranges was investigated by a Cary 5000 spectrophotometer (Varian, Belrose, Australia) with a spectral range of 190-3300 nm and a resolution of 1 nm and by an FT-801 IR Fourier spectrometer (Simex, Novosibirsk, Russia) with a spectral range of 400-4000 cm −1 (2.5-25 µm) and a resolution of 2 cm −1 . The terahertz properties were measured using a conventional terahertz time-domain spectrometer (THz-TDS) with a liquid nitrogen bath cryostat. A description of the experimental setup can be found elsewhere [17,18]. In this study, the experimental setup included two smaller diaphragms with apertures of 5 mm due to the small dimensions of the samples under investigation. Additionally, a newly developed measurement procedure described in [19] was applied. This allowed us to measure the absorption coefficient and the refractive index of the crystals for both ordinary and extraordinary waves during one cryostat cooling cycle (without warming up and changing the position of the sample) under nearly the same conditions. THz signals were acquired with a time step of 125 fs in the 60 ps time range, which corresponded to a spectral resolution of about 20 GHz. The temperature expansion coefficient of the LB4 crystal was taken into account when extracting its THz optical properties. Figure 1a shows the transmission of the LB4 samples in the main transparency window. The crystal was practically transparent up to the wavelength of 3 µm. Then, vibrational absorption bands that are characteristic of the borate structure appeared in the IR range [20,21]. These results are in good agreement with those of prior works [22,23]. The estimated absorption coefficient (see the inset in Figure 1a) was below 0.1 cm −1 . Its measurement accuracy was limited by the small thickness of the samples and the noise of the spectrophotometer. The obtained results confirmed the low absorption losses and high optical quality of the samples under study. and a resolution of 1 nm and by an FT-801 IR Fourier spectrometer (Simex, Russia) with a spectral range of 400-4000 cm −1 (2.5-25 µ m) and a resolution of 2 cm −1 . The terahertz properties were measured using a conventional terahertz time-domain spectrometer (THz-TDS) with a liquid nitrogen bath cryostat. A description of the experimental setup can be found elsewhere [17,18]. In this study, the experimental setup included two smaller diaphragms with apertures of 5 mm due to the small dimensions of the samples under investigation. Additionally, a newly developed measurement procedure described in [19] was applied. This allowed us to measure the absorption coefficient and the refractive index of the crystals for both ordinary and extraordinary waves during one cryostat cooling cycle (without warming up and changing the position of the sample) under nearly the same conditions. THz signals were acquired with a time step of 125 fs in the 60 ps time range, which corresponded to a spectral resolution of about 20 GHz. The temperature expansion coefficient of the LB4 crystal was taken into account when extracting its THz optical properties. Figure 1a shows the transmission of the LB4 samples in the main transparency window. The crystal was practically transparent up to the wavelength of 3 µ m. Then, vibrational absorption bands that are characteristic of the borate structure appeared in the IR range [20,21]. These results are in good agreement with those of prior works [22,23]. The estimated absorption coefficient (see the inset in Figure 1a) was below 0.1 cm −1 . Its measurement accuracy was limited by the small thickness of the samples and the noise of the spectrophotometer. The obtained results confirmed the low absorption losses and high optical quality of the samples under study. The refractive index of LB4 was measured at room and higher temperatures in [24,25]. As far as we know, cryogenic measurements have only been presented by Burak et al. [26], and the same data appeared in a review [27]. Burak et al. [25,26] only measured the refractive index in the range of 300-700 nm and approximated it with the Sellmeier equations [26]. This approximation significantly differs in the near-IR range from the Sellmeier equation developed by Sugawara et al. at room temperature [23] (orange and blue curves in Figure 1b). It should also be noted that the high-temperature dependence of the The refractive index of LB4 was measured at room and higher temperatures in [24,25]. As far as we know, cryogenic measurements have only been presented by Burak et al. [26], and the same data appeared in a review [27]. Burak et al. [25,26] only measured the refractive index in the range of 300-700 nm and approximated it with the Sellmeier equations [26]. This approximation significantly differs in the near-IR range from the Sellmeier equation developed by Sugawara et al. at room temperature [23] (orange and blue curves in Figure 1b). It should also be noted that the high-temperature dependence of the refractive index was investigated via effective nonlinear optical conversion by Umemura et al. [24], but their refined Sellmeier coefficients did not practically differ from those obtained by Sugawara et al. [23] and were not considered in the current work. The refractive indices we measured (see black squares in Figure 1b and Table 1) are in agreement with the works of Sugawara et al. [23] and Umemura et al. [24]. It was previously found that the optical and dielectric properties of the LB4 crystal behave discontinuously in the vicinity of 235 K [27,28]. However, we found that the refractive indices at 77 and 290 K did not practically differ [26,27] (as shown by blue and red triangles in Figure 1b, respectively). Thus, for further calculations, we neglected the temperature dependence of the refractive index in the main transparency window and used the Sellmeier equations developed by Umemura et al. [24] for calculations at room and liquid nitrogen temperature.  refractive index was investigated via effective nonlinear optical conversion by Umemura et al. [24], but their refined Sellmeier coefficients did not practically differ from those obtained by Sugawara et al. [23] and were not considered in the current work. The refractive indices we measured (see black squares in Figure 1b and Table 1) are in agreement with the works of Sugawara et al. [23] and Umemura et al. [24]. It was previously found that the optical and dielectric properties of the LB4 crystal behave discontinuously in the vicinity of 235 K [27,28]. However, we found that the refractive indices at 77 and 290 K did not practically differ [26,27] (as shown by blue and red triangles in Figure 1b, respectively). Thus, for further calculations, we neglected the temperature dependence of the refractive index in the main transparency window and used the Sellmeier equations developed by Umemura et al. [24] for calculations at room and liquid nitrogen temperature.  It should be noted that the change in the absorption coefficient with cooling ( Figure  2a) was not as radical as, for example, that of a lithium triborate (LBO) crystal [29]. Additionally, we considered the contribution of the hopping conductivity of lithium ions to It should be noted that the change in the absorption coefficient with cooling ( Figure 2a) was not as radical as, for example, that of a lithium triborate (LBO) crystal [29]. Additionally, we considered the contribution of the hopping conductivity of lithium ions to absorption. The dielectric susceptibility peak connected with lithium ion hopping was located at a frequency of about 2.8 GHz at 300 K [28]. Therefore, the absorption coefficient α e may not drop below 0.1 cm −1 even in sub-terahertz frequencies.

Optical Properties in the THz Range
The dispersion of the THz refractive index (Figure 2b) also presented peculiarity upon cooling. For example, even though α o decreased by almost twice in the vicinity of a frequency of 1.8 THz, n o did not practically change; however, n o did change by 0.05 at a frequency of 0.5 THz. The refractive index for the extraordinary wave showed more noticeable changes of~0.08, which were frequency-independent within the measurement accuracy. These temperature changes were quite small in comparison to the LBO crystal, in which the refractive index of the x-axis n x changed by~0.24 upon cooling [19].
This behavior of the THz optical properties of the LB4 crystal may be associated with a complex phonon structure and its changes upon cooling [30][31][32]. For instance, the characteristic shape of the IR absorption spectrum in the region above 150 cm −1 shows a large number of absorption bands with a complex temperature behavior [30].

Assessment of THz-Wave Generation by Down-Conversion
The Li 2 B 4 O 7 compound forms a uniaxial ionic crystal, which belongs to the 4 mm point symmetry group [8]. Its tensor of the second-order nonlinear susceptibility coefficient d ij has the following form [33]: Within this study, when generating radiation in the THz frequency range, the Kleinman symmetry conditions were not valid because the resulting wave was located behind the infrared absorption band of the crystal. Table 2 provides effective nonlinear coefficients for eight types of three-wave interaction processes possible for THz-wave generation according to [34]. Among these, phase matching for selected wavelengths was only found to be possible for the o − e → o type of conversion. Table 2. Effective nonlinear coefficients for eight possible types of conversion in the LB4 crystal.

Conversion Type
In order to evaluate coherence lengths and phase-matching curves for these types of conversion, we approximated the dispersion of the refractive index for the THz range in the form of Sellmeier equations. It should be mentioned that a Sellmeier equation usually contains information about the frequency of the absorption peak (in the general case of the short-wavelength edge). However, due to the complex phonon structure of the crystal at frequencies above 2.5 THz, additional terms in the equation significantly complicated it and were omitted. Equations (2) and (3) The approximations made well fits the experimental data (white curves in Figure 2b). The refractive indices n o and n e for the pump radiation in the main transparency window were approximated using the expressions proposed by Umemura et al. [24] for both Crystals 2021, 11, 1321 6 of 9 295 and 77 K, since it was shown above that they only slightly differed for these two temperature points.
For the types of three-wave interaction with a collinear propagation, the efficiency of which is not identically equal to zero, we carried out a numerical simulation of the coherence length: where k(λ) = 2 π n(λ)/λ; 1/λ pump1 − 1/λ pump2 = 1/λ diff , λ pump1 , and λ pump2 are the pump radiation wavelengths; and λ diff is the resulting THz wavelength. A simulation was carried out for two temperatures of 295 and 77 K. Results are shown in Figure 3.
From the dependence of L c on the angle θ for fixed pump and difference wavelengths (Figure 3a), it can be seen that phase matching existed only for the o − e → o conversion type. However, the o − o → e and e − e → e types are also worth noting since their coherence lengths were found to weakly depend on θ. Figure 3a also demonstrates small differences in the coherence lengths for room and cryogenic temperatures. Figure 3b shows the phase-matching curves of the o − e → o conversion type when LB4 was pumped at the wavelengths of widespread high-power laser sources (1064 and 800 nm) and their second and third harmonics.
Crystals 2021, 11, x FOR PEER REVIEW 6 of 9 The approximations made well fits the experimental data (white curves in Figure 2b). The refractive indices no and ne for the pump radiation in the main transparency window were approximated using the expressions proposed by Umemura et al. [24] for both 295 and 77 K, since it was shown above that they only slightly differed for these two temperature points.
For the types of three-wave interaction with a collinear propagation, the efficiency of which is not identically equal to zero, we carried out a numerical simulation of the coherence length: where k(λ) = 2 π n(λ)/λ; 1/λpump1 −1/λpump2 = 1/λdiff, λpump1, and λpump2 are the pump radiation wavelengths; and λdiff is the resulting THz wavelength. A simulation was carried out for two temperatures of 295 and 77 K. Results are shown in Figure 3. From the dependence of Lc on the angle θ for fixed pump and difference wavelengths (Figure 3a), it can be seen that phase matching existed only for the o − e → o conversion type. However, the o − o → e and e − e → e types are also worth noting since their coherence lengths were found to weakly depend on θ. Figure 3a also demonstrates small differences in the coherence lengths for room and cryogenic temperatures. Figure 3b shows the phase-matching curves of the o − e → o conversion type when LB4 was pumped at the wavelengths of widespread high-power laser sources (1064 and 800 nm) and their second and third harmonics.
(a) (b) It follows from Figure 3b that cooling down the crystal from 295 to 77 K had practically no effect on the phase-matching curves. It should also be noted that for higher harmonics of pump radiation, a smaller phase-matching angle θpm is required. This is a fairly negative factor since it leads to a decrease in deff proportional to sin(θ). However, an increase in the d15 coefficient with a decrease of the wavelength can partially compensate for this. An estimate of the d15 in accordance with Miller's rule [35] provided an increase of its value by ~18% as the pump wavelength decreased from 1064 to 266 nm. As a result, according to our estimates, the resulting decrease in deff was only about 50%.
Let us consider other types of conversion without phase matching but with non-zero effective nonlinearity. Since a coherence length of several hundred microns is sufficient for efficient nonlinear frequency conversion, it is of interest to use this crystal with high-  It follows from Figure 3b that cooling down the crystal from 295 to 77 K had practically no effect on the phase-matching curves. It should also be noted that for higher harmonics of pump radiation, a smaller phase-matching angle θ pm is required. This is a fairly negative factor since it leads to a decrease in d eff proportional to sin(θ). However, an increase in the d 15 coefficient with a decrease of the wavelength can partially compensate for this. An estimate of the d 15 in accordance with Miller's rule [35] provided an increase of its value bỹ 18% as the pump wavelength decreased from 1064 to 266 nm. As a result, according to our estimates, the resulting decrease in d eff was only about 50%.
Let us consider other types of conversion without phase matching but with non-zero effective nonlinearity. Since a coherence length of several hundred microns is sufficient for efficient nonlinear frequency conversion, it is of interest to use this crystal with high-power femtosecond sources. The coherence lengths for the o − o → e and e − e → e types were found to be practically independent of the angle θ. In this case, it was advisable to take such an angle θ for which d eff was maximal. For the o − o → e conversion, the angles were θ = ±90 • . However, the picture was more complicated for the e − e → e type, as d e−e→e e f f consisted of two terms ( Table 2). The first one was proportional to sin 3 (θ), which reached the maximum value at the angle θ = ±90 • . The second term was proportional to cos 2 (θ) sin(θ), which was maximal at θ = ±35.26 • . Thus, the efficiency depended on the values of the coefficients d 15 , d 31 , and d 33 , which had to be experimentally determined in the THz region. However, for further estimates, we assumed that the ratio of these coefficients for the types of conversion that resulted in terahertz frequency generation approximately equaled their ratio for the main transparency window of the crystal, i.e., d 31 ≈ d 15 Figure 4 shows the dependence of the coherence length on the pump wavelength for the o − o → e and e − e → e conversion types at θ = ±90 • ; λ diff = 1000 µm and 3000 µm were used for the sake of estimation. power femtosecond sources. The coherence lengths for the o − o → e and e − e → e types were found to be practically independent of the angle θ. In this case, it was advisable to take such an angle θ for which deff was maximal. For the o − o → e conversion, the angles were θ = ±90°. However, the picture was more complicated for the e − e → e type, as − → consisted of two terms ( Table 2). The first one was proportional to sin 3 (θ), which reached the maximum value at the angle θ = ±90°. The second term was proportional to cos 2 (θ) sin(θ), which was maximal at θ = ±35.26°. Thus, the efficiency depended on the values of the coefficients d15, d31, and d33, which had to be experimentally determined in the THz region. However, for further estimates, we assumed that the ratio of these coefficients for the types of conversion that resulted in terahertz frequency generation approximately equaled their ratio for the main transparency window of the crystal, i.e., d31 ≈ d15 and d33/d15 ≈ 4. In this case, the term proportional to sin 3 (θ) dominated, and the maximum for − → occurred for the values of the angle θ = ±90°. Figure 4 shows the dependence of the coherence length on the pump wavelength for the o − o → e and e − e → e conversion types at θ = ±90°; λdiff = 1000 μm and 3000 μm were used for the sake of estimation. One can see that for a fixed generated wavelength λdiff, the coherence length Lc also weakly depended on the pump frequency, except for a small increase in the short-wavelength region. Thus, the use of short pump wavelengths for the generation of sub-THz radiation may be efficient because of both an increase in the quadratic susceptibility of the crystal and a small increase in the coherence length. Cooling the crystal from 295 to 77 K also led to a small growth of Lc. In general, a crystal with a thickness of at least 370 μm can be used to generate millimeter wavelengths λdiff > 1 mm (ν < 300 GHz) in the transparency window of the atmosphere.

Conclusions
Conducted analyses, experimental studies, and calculations showed that cooling of the LB4 crystal from room to liquid nitrogen temperature did not practically affect its refractive index in the main transparency window. At the same time, in the THz range, a significant decrease of absorption was observed for both ordinary and extraordinary waves. However, unlike LBO, absorption in LB4 did not decrease to a "zero" level at 77 K due to the peculiarities of the crystal structure. The THz refractive index ne demonstrated the highest temperature dependence that led to an increase in birefringence Δn = no − ne by more than 30% at 77 K.
The evaluation of the possibility of generating THz radiation from high-power sources of the visible and near-IR range (operating in the main transparency window of  One can see that for a fixed generated wavelength λ diff , the coherence length L c also weakly depended on the pump frequency, except for a small increase in the shortwavelength region. Thus, the use of short pump wavelengths for the generation of sub-THz radiation may be efficient because of both an increase in the quadratic susceptibility of the crystal and a small increase in the coherence length. Cooling the crystal from 295 to 77 K also led to a small growth of L c . In general, a crystal with a thickness of at least 370 µm can be used to generate millimeter wavelengths λ diff > 1 mm (ν < 300 GHz) in the transparency window of the atmosphere.

Conclusions
Conducted analyses, experimental studies, and calculations showed that cooling of the LB4 crystal from room to liquid nitrogen temperature did not practically affect its refractive index in the main transparency window. At the same time, in the THz range, a significant decrease of absorption was observed for both ordinary and extraordinary waves. However, unlike LBO, absorption in LB4 did not decrease to a "zero" level at 77 K due to the peculiarities of the crystal structure. The THz refractive index n e demonstrated the highest temperature dependence that led to an increase in birefringence ∆n = n o − n e by more than 30% at 77 K.
The evaluation of the possibility of generating THz radiation from high-power sources of the visible and near-IR range (operating in the main transparency window of LB4) via down-conversion showed that only four out of eight types of conversion are possible (they were found to have non-zero nonlinear coefficients). Phase-matching conditions were only fulfilled for the o − e → o conversion type. At the same time, the coherence length for the o − o → e and e − e → e types of interaction were estimated at several hundred microns, which is sufficient for efficient conversion when using high-power femtosecond lasers when taking the high damage threshold of LB4 into account. We suppose that cooling the crystal should increase conversion efficiency due to a decrease in the absorption losses of the pump and the THz waves, as well as a slight increase in the coherence length. Therefore, our study has shown the potential efficiency and provided the necessary conditions for the nonlinear generation of THz radiation in an LB4 crystal pumped by widespread high-power lasers with wavelengths of 1064 and 800 nm, as well as their harmonics.