Low-temperature terahertz absorption measurements and polariton laser gain modelling for 5&#x2009;mol&#x0025; MgO:LiNbO<sub>3</sub>

Ameera A. Jose; Ondrej Kitzler; Helen M. Pask; David J. Spence

doi:10.1364/OME.474141

1. Introduction

Terahertz (THz) wave generation via stimulated polariton scattering (SPS) has been extensively studied in the pursuit of compact and tunable THz lasers [1,2]. Lasers based on SPS deliver reasonably high output power and wide frequency tunability, ideal for use in spectral fingerprinting and imaging applications involving many organic and inorganic compounds [3,4]. The scattering process involves a fundamental laser field that excites transverse optical (TO) phonon modes in polar crystals and scatters parametrically into two lower-energy Stokes and polariton (THz) fields. The fundamental field couples to the TO modes to create phonon-polaritons which can be tuned at the THz frequency by noncollinear angle phase-matching of the fundamental and Stokes fields.

MgO:LiNbO$_3$ (MLN) has been the most popular SPS crystal due to its large nonlinear coefficient [5]. It has several active TO phonon modes; of particular interest is the lowest A$_1$-symmetry polariton mode at $\sim$252 cm$^{-1}$ (7.6 THz) which has been widely utilized in modern SPS systems for THz generation from 1 to 3 THz [4]. While frequency tuning is still possible beyond this range, THz generation above 3 THz has been a challenge due to the dramatic increase in absorption, which severely limits the usefulness of the crystal for generating higher frequencies.

Recent studies on SPS lasers demonstrate experimental techniques to address the key issue of absorption, most notably, introducing more complex geometries such as surface-emitted configurations [6,7] where the generated THz field is directly out-coupled into the air to eliminate absorption losses. Meanwhile, cooling the crystal down to cryogenic temperatures has been demonstrated [2,8] as this will decrease the linewidth of the polaritons and, therefore, reduce the crystal’s absorption coefficient.

Accurate knowledge of the THz absorption coefficient is crucial as it forms the basis for accurately modelling SPS gain. Schwarz and Maier [9–11] investigated the polariton damping for MLN at lower frequencies and observed that even in interesting regions where the absorption is expected to be relatively low, there exist low-frequency modes associated with defect centers in the crystal that couple to polaritons and create unexpected increased absorption. These resonances are not extended phonon modes of the C$_3v$ crystal symmetry and appear as dips in gain measurements. Due to the presence of these unexpected modes, it is crucial to consider the overall polariton damping in the crystal to correctly determine the SPS gain.

The temperature-dependent absorption coefficient has been previously reported [12] for both stoichiometric and congruent LN crystals with a range of doping concentrations from 0.9 to 5.4 THz. For 5 mol% congruent MLN, the experimental data available [13,14] is only up to 2 THz and the results of temperature dependence are still lacking. The most common approach to determining the absorption coefficient at high frequencies is by fitting a mode-based model to experimental reflectivity spectra to recover the complex dielectric function [15]. Fitting reflectivity data does not necessarily predict well the losses, as additional absorbing modes can significantly alter the absorption while having a relatively minor effect on the reflectivity.

In this study, we performed transmission measurements at the Australian Synchrotron to provide a more complete temperature-dependent absorption coefficient spectra for a 5 mol% congruent MLN crystal. Spectra were collected at temperatures from 37 to 300 K and a frequency range from 0.75 to 6 THz covering almost the entire low-frequency region of the lowest A$_1$-symmetry polariton mode. The results were used to accurately model the SPS gain curves, taking into account the effect of low-frequency polariton damping. The SPS gain was described using the Gaussian-mode theory reported in [16] which is valid for narrow fundamental beam sizes, typical in SPS intracavity lasers. Our results are especially useful for the design and performance-enhancement of MLN-based SPS intracavity lasers [17,18] operating at room or cryogenic temperatures, but are generally useful for a broader range of studies involving MLN.

2. Experimental results and analysis

High-resolution transmission spectra were collected using the THz-far IR beamline at the Australian Synchrotron. The beamline provides intense beams of collimated and polarised light from the synchrotron. Here, we prepared two crystals of thicknesses 110 $\mu$m and 70 $\mu$m, respectively; the thinner MLN sample allowed accurate measurements for the more strongly absorbing higher frequencies. The exact thicknesses were determined by an optical profiler with sub-micron resolution. The samples were placed into a cryostat to investigate the spectra over temperatures from 37 to 300 K. All the experiments were carried out with light polarization oriented parallel to the crystal’s $c$-axis, thereby probing the A$_1$-symmetry modes. The spectra were measured using a Bruker IFS 125/HR Fourier Transform (FT) spectrometer with 2 cm$^{-1}$ resolution.

A complete set of measured temperature-dependent absorbance curves (inverse of transmittance) for the 110 $\mu$m MLN sample is plotted versus the THz wavenumber in Fig. 1. From the figure, two main results can be seen: (i) the absorbance decreases consistently upon cooling, (ii) there is a broad peak centered at 105 cm$^{-1}$(3.15 THz), found to be present at all temperatures. To retrieve the absorption coefficient from our transmission measurements $T(\omega )$, we must correct for surface reflections and interference effects. Modelling the crystal as an etalon [12], we use the relation

(1)$$\alpha_{i}(\omega) = \frac{1}{d}\; \ln \left[\left(\frac{1-R(\omega)^{2}+\sqrt{1-R(\omega)^{4}+4R(\omega)^{2}\bigl< T(\omega)\bigr>^{2}}}{2\bigl< T(\omega)\bigr>}\right)\right],$$

in which $R(\omega )$ is the surface reflectivity [15], $d$ is the crystal thickness, and $\bigl < T(\omega )\bigr >$ is the etalon transmission smoothed using Savitzky-Golay method to remove the interference fringes. We used the 110 $\mu$m crystal data to retrieve the absorption coefficient from 25 to 160 cm$^{-1}$, and data from the thinner 70 $\mu$m crystal to complete the spectral coverage up to 200 cm$^{-1}$. (Using the 70 $\mu$m crystal is not preferred at lower frequencies due to increased etalon effects, but gives better signal-to-noise for the frequencies with very high absorption coefficients.) The retrieved THz absorption coefficients are shown as the dotted lines in Fig. 2 for three different temperatures. To fit the measured data, we can use the well-established classical damped oscillator model [15,19]. The model calculates the dielectric function of the material from the knowledge of optically active modes with resonance frequency $\omega _{0}$, strength $S_{0}$, and linewidth $\Gamma _{0}$. For a single mode, the frequency-dependent complex dielectric function $\tilde {\varepsilon }(\omega )$ is given by:

(2)$$\tilde{\varepsilon}(\omega) = \varepsilon_{\infty} + \frac{S_0 \omega_{0}^{2}}{\omega_{0}^{2} - \omega^{2} - i\omega\Gamma_{0}} \;$$

Fig. 1. Measured temperature-dependent absorbance curves for a 5 mol% congruent MLN crystal.

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Fig. 2. THz absorption coefficient curves of the 5 mol% congruent MLN crystal for three different temperatures. The dashed curves are obtained from measurements, while the solid curves are fits from the oscillator model [15]. The inset figure emphasizes the curves in the low-frequency region.

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Higher frequency modes are here accounted for within the high-frequency dielectric constant term $\varepsilon _{\infty }$. The THz absorption coefficient relates to the dielectric function as:

(3)$$\alpha_{i}(\omega) = \frac{2\omega}{c}\;\operatorname{Im} \biggl[\sqrt{\tilde{\varepsilon}(\omega)}\biggr],$$

Equation (2) is appropriate for polaritons that have frequency-independent damping. However, the structure we observe is generally attributed to additional low-frequency modes that induce frequency-dependent damping of the polariton mode. Following the methods in [11], we replace the constant damping term $\Gamma _{0}$ with an effective damping function $\Gamma _{eff}(\omega )$:

(4)$$\Gamma_{eff}(\omega) = \Gamma_{0} + \frac{1}{i\omega}\biggl[\sum_{j=1}^{N}\frac{K_j}{\omega_{j}^{2} - \omega^{2} - i\omega\Gamma_{j}}\biggr],$$

where the summation in the square brackets represents the polariton coupling to $N$ low-frequency modes of frequency $\omega _{j}$, damping constant $\Gamma _{j}$, with coupling strength $K_{j}$ to the polariton mode. The calculated THz absorption coefficient curves for the three temperatures are shown in solid lines in Fig. 2. Contrasting with these is the absorption coefficient predicted for constant damping $\Gamma _{0}$, plotted as a grey line for 300 K: In this case, the absorption coefficient is a smooth monotonic curve that increases gradually as it approaches the polariton mode at 252 cm$^{-1}$. Clearly, the polariton mode alone in the dielectric function cannot explain the observed absorption feature at 105 cm$^{-1}$. With the addition of a single low-frequency mode, we achieve good agreement with the measured data at all temperatures (red, blue, and orange solid lines). It is notable that the model overshoots somewhat in the low-frequency region (inset figure). A more complex model with the inclusion of more than one low-frequency mode can give better fitting. In that low-loss region, the SPS gain does not depend critically on loss, and so for simplicity, we choose to use just one mode, sufficient to model the main peak in the frequency range where the losses start to determine the gain. LF modes can be mainly attributed to the vibrational defect modes or electronic traps in LN [11] and have been found [9,10] in both undoped and doped LN. The frequency of the LF mode we observed does not agree with these references. Differences may just reflect crystal-to-crystal variation, and indeed, our group has observed significant differences in laser performance for nominally identical crystals from different suppliers. It is possible that the strength of the LF modes could be reduced by improvements in crystal growth.

The fitted values of the low-frequency mode parameters are summarized in Table 1. The coupling strength and linewidth for both the low-frequency and main polariton modes decrease with cooling. At 300 K, the fitting parameter values for the polariton mode are consistent with our own measurement using Raman spectroscopy and Ref. [20]. The refractive index (determined from $\tilde {\varepsilon }(\omega )$) predicted using these values also matches the experimental data in Ref. [21]. We note that while the low-frequency modes have a very strong impact on the absorption coefficient, their effect on the reflectivity spectra is far smaller, suggesting that mode fitting [15] directly from the measured reflectivity is not the best approach for accurate determination of the absorption coefficient.

Table 1. Fitting parameters for the 5 mol% congruent MLN crystal at three different temperatures.

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3. Prediction of the SPS gain coefficient

Now that we have established parameters to describe the absorption characteristics of the crystal at a range of frequencies and temperatures, we can more accurately calculate the SPS gain at varying temperatures. The SPS gain has been commonly studied for plane wave excitation [19], with validity limited to single-pass generators with mm-scale beams. Here, we use the more recently developed model in [16] which is well-suited to describe systems with narrow Gaussian beams and (typically) low gain, such as laser oscillators. Following [16], we predict Gaussian Stokes gain coefficient, $g_{S}^{Gauss}$ as:

(5)$$\begin{array}{r} g_{S}^{Gauss}(\omega, T) = \alpha_{p}^{axis}(\omega, T)^2 \; \frac{1}{\sqrt{2}\sin\phi} \exp{\biggl(\frac{(\alpha_{i}(\omega,T)+g_{S}^{Gauss}\cos\phi)^{2}r_{x}^{2}}{32\sin^{2}\phi}\biggr)} \\ \times \int_{-\infty}^{\infty} \textrm{Erfc}{\biggl(\frac{(\alpha_{i}(\omega,T)+g_{S}^{Gauss}\cos\phi)r_{x}^{2}-8x\sin\phi}{4\sqrt{2}r_{x}\sin\phi}\biggr)}\\ \times \exp{\biggl(\frac{-\alpha_{i}(\omega,T) x}{2\sin\phi}\biggr)} \exp{\biggl(\frac{-2x^{2}}{r_{x}^{2}}\biggr)} \exp{\biggl(\frac{-g_{S}^{Gauss}x}{2\tan\phi}\biggr)}dx \end{array}$$

where r$_x$ is the fundamental beam size, $\phi$ is the angle between the fundamental and THz fields (of order 65$^{\circ }$, and $\alpha _{p}^{axis}$ is a gain coefficient dependent on the intensity of the fundamental field, I$_f$ [2,19]). In comparison to the plane wave approach, the present theory assumes that both fundamental and Stokes beams are Gaussian beams with a specified 1/e$^2$ radius r$_x$ in the plane of THz generation. Fig. 3 depicts the calculated Stokes gain from 0 to 6 THz for the three temperatures. A significant broad dip at 3.15 THz is ascribed to the additional damping caused by the low-frequency mode, resulting in increased THz absorption and suppressed gain. At 300 K, the peak gain coefficient occurs around 1.8 THz, which is consistent with the experimental results in [18,22,23]. In contrast, the calculated gain at 300 K (grey curve in Fig. 3) using the constant damping model (also the grey curve in Fig. 2) incorrectly predicts the peak gain to be at a much higher frequency, reinforcing the importance of using high-quality spectral data.

Fig. 3. Calculated temperature-dependent Stokes gain coefficient for the 5 mol% congruent MLN crystal excited by a 1064 nm fundamental beam with intensity I$_f$ = 300 MW/cm${^2}$ and radius r$_x$ = 300 $\mu m$.

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Moreover, most experiments have only observed a single region of THz output [24,25] but Fig. 3 suggests a potential for a second band at higher frequencies. In practice, obtaining output at higher frequencies is complicated by two factors. First, the gain is relatively lower due to the large angle requirement between the phase-matched fundamental and Stokes beams for high THz frequency tuning. This is expected to reduce the Stokes gain coefficient below what is predicted in Fig. 3. Second, the large absorption at these frequencies makes it increasingly difficult to extract THz that may be generated. Nevertheless, there have been occasional reports of lasing in the second band [26,27], and we suggest that there should be more exploration in this region. Furthermore, cryogenic cooling of the crystal addresses this limitation by increasing the gain for the second peak and lowering the absorption of THz photons as they exit the crystal.

4. Summary

We carried out high-quality transmission measurements using synchrotron radiation and a high-resolution spectrometer to determine the THz absorption characteristics of a 5 mol% congruently grown MLN. We infer the presence of a broad 105 cm$^{-1}$ (3.15 THz) low-frequency mode that causes a significant increase in absorption. The effect of the low-frequency mode is to create a dip in the SPS gain around 3 THz and shift the gain maximum to lower frequencies, in agreement with many experiments. The prediction of a second gain maximum at higher frequencies is consistent with some recent experiments [26,27], and encourages more exploration of that region, particularly at lower temperatures.

Acknowledgments

The authors would like to thank Dominique Appadoo and the Australian Synchrotron for the generous support through beamtime access; Arnan Mitchell and Paul Jones of RMIT University for sample preparation assistance. Ameera Jose acknowledges receipt of the International Macquarie University Research Excellence Scholarship.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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	Symbol	300 K	225 K	77 K
	$ε_{\infty}$		8.31
Polariton mode properties	$ω_{0}$ (cm $^{- 1})$	252	254	258
	$Γ_{0}$ (cm $^{- 1})$	30	22	17
	$S_{0}$	15.5	14.3	14
	$ω_{1}$ (cm $^{- 1})$		105
Low-frequency mode properties	$Γ_{1}$ (cm $^{- 1})$	41	40	38
	$K_{1}$ (cm $^{- 4})$	26 $\times 10^{6}$	25 $\times 10^{6}$	22 $\times 10^{6}$

Low-temperature terahertz absorption measurements and polariton laser gain modelling for 5 mol% MgO:LiNbO₃

Abstract

1. Introduction

2. Experimental results and analysis

3. Prediction of the SPS gain coefficient

4. Summary

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (3)

Tables (1)

Equations (5)

Optical Materials Express