1. Introduction

A high brightness source of radiation is widely desired for terahertz (30–1000 μm) applications in imaging, nondestructive evaluation, security, and bio-medical technologies [1]. ε-polytype gallium selenide (hereinafter GaSe) promises efficient laser down-conversion almost over the entire THz range [2, 3], except for the phonon absorption gap around 44 μm [2, 4]. Owing to the large birefringence of GaSe (0.35 at 10 μm [5] and up to 0.79 [6] at 1 THz), phase matching can be achieved in multiple geometries: the common forward k ₁ − k ₂ = k ₃ type II (oe → e) [2, 3, 7–11 ]; (eo → e) [2] and (oe → o); [12], type I (ee → o) [10, 13]; (oo → e) [14]; the original (ee → e) [15, 16]; and backward type II (oe → e) [17]. The potential of GaSe for THz generation rests also on its other outstanding physical properties. It has the widest transparency range of 0.62–20 μm that continues at wavelengths ≥50 μm [18]; the second-highest optical damage threshold among IR anisotropic crystals [5, 19]; and high second order nonlinear susceptibility (d ₂₂=54 pm/V at 10 μm; 24.3 pm/V at THz frequencies) [7]). GaSe also possesses high thermal conductivity (0.162 W/(m·K)) in the plane of growth layers [5], and low two-photon absorption (≈0.5 cm/GW at 0.755–0.875 μm) [19].

Unfortunately, layered GaSe is difficult to grow as large, high optical quality monocrystalline samples (i.e. with absorption coefficient of 0.1–0.2 cm⁻¹ in the maximum transparency range and 1 cm⁻¹ in the THz range), due to the poor technological control of point- and micro-defects [20, 21]. In fact, the available information on the minimal optical loss in GaSe in the THz band [22–24 ] is based on incorrectly interpreted data from [25] (see for comparison FTIR data in [4]). The weak electrostatic bonding between layers in crystal structure makes it difficult to fabricate optical quality surfaces in arbitrary crystallographic directions. Modified growth technology with heat field rotation, designed to obtain highly uniform melt mixing and low temperature gradient in the plane of the thin crystallization front, accomplished a reduction by a factor of 2–3 in the optical loss coefficients in the mid-IR and THz ranges; but lattice strengthening was not achieved [18,26–28 ]. As a result, poor mechanical properties restrict the use of GaSe to in-lab applications.

However, GaSe is an excellent matrix material and can be doped by different elements. GaSe that is heavily doped with isovalent sulfur (S) is also referred to as solid solution GaSe:GaS or GaSe_1−xS_x, where x is the mixing ratio of the parent materials. Such crystals appear to be the most effective approach to controlling the physical properties of GaSe by varying sulfur concentration [18, 21, 28–32 ]. Due to the substitution of gallium vacancies by sulfur atoms that in turn reduce the stacking faults, the original layered structure of GaSe is strengthened sufficiently to be cut and polished to optical quality in arbitrary directions, provided it is first prepared by encapsulation in a polymer matrix that shrinks slightly during polymerization [18, 28]. This technology has allowed us to manufacture layered GaSe_1−xS_x samples [18,28] where absorption properties for o- and e-wave in the THz band can be studied directly. It was verified that optimal doping leads to additional decrease of optical loss by a factor of 2–3 (as was previously established for the mid-IR [29,30]), and that the optical damage threshold increases fivefold [33, 34]. Moreover, GaSe_1−xS_x crystals grown by modified technology with heat field rotation have demonstrated the lowest optical losses in the THz range among known nonlinear crystals, indicating the possibility of raising the frequency conversion efficiency by up to 15 times. The combination of increased hardness, reduced optical losses and improved frequency conversion efficiency make GaSe_1−xS_x suitable for field applications [1, 18, 28–30 ].

In order to exploit fully the potential of GaSe_1−xS_x in THz applications, it is necessary to know accurately the refractive index dispersion for both the ordinary (n_o) and extraordinary (n_e) waves across the entirety of the transparency range. Dispersion properties for the pumping wavelength in the visible region are well described by Sellmeier equations recommended in a widely used handbook [5]. Additional information can be found in recently published papers [35, 36]. Dispersions may be estimated from the available dispersion data for the parent GaS and GaSe crystals by using the following second order relation for the refractive indices [36]:

Alternatively, they may be measured directly. Dispersion equations for the entire transparency range of GaS were recently reported in [37]. However, due to the insufficient technological control of crystal quality, limited availability and poor mechanical properties, only three sets [4, 36, 38] among the known sets of dispersion equations [4, 36, 38–45 ] for GaSe are suitable for use in the THz range. In fact, the dispersion equations claimed in [38] as valid for the 0.65–18 μm range and recommended in the handbook [5] are also in widespread use for THz frequencies (see, for example, [2, 3]).

Numerous dispersion curves for n_o, determined experimentally by THz TDS, are available for GaSe. Limited data exists on n_e, principally for unprocessed [2–4 ,6–9 ,11,12,15,17,46,47] samples, but very little published data is available for processed (cut and polished at θ=90°) [15,16] samples. Recent data [48], reporting birefringence of B=0.79 in the THz range, matches well with the previous experimental data of 0.78–0.80 [4, 6, 15, 16], but differs unacceptably from the estimated values of 0.39 and 0.49 obtained by using the dispersion equations from [38] and [36] respectively.

Most studies of dispersion properties were carried out using GaSe crystals of suitable optical quality (absorption coefficient 0.2 cm⁻¹ in the maximum transparency range) grown by the common vertical Bridgman technique. In the last decade [8, 9, 46] it was shown that the measured magnitude of the refractive indices is heavily dependent on the optical quality of the GaSe crystal. For low quality samples (absorption coefficient α=75–250 cm⁻¹ in the THz range) the value of n_o exceeds 3.5 [46]. According to calculations in [4], for GaSe with α=10–150 cm⁻¹ reported in [8, 9], it is close to 3.6. In addition, it was also shown [49] that refractive index values may vary strongly (by at least ±0.05) among measurements produced even by the same THz TDS facility.

Experimental data for the phase matching (PM) angles for o-wave generation by down-conversion in GaSe [10, 13] are in good agreement throughout the THz range with PM angles estimated using the data in [38], but show mismatch for the generation of forward [2–4 ] and backward [17] e-waves. For the long wavelengths (≥300 μm) external PM angles differ by up to 2–3° [2, 3, 17]. However, due to non-critical spectral PM, this difference leads to great (from few hundreds up to thousands of μm) difference in the generated wavelength. Therefore selection of the best set of dispersion equations for the entire transparency range of GaSe remains one of the key issues in obtaining accurate estimates of GaSe_1−xS_x dispersion properties for frequency conversion, polarization optics and other applications. To achieve this, reference measurements should be performed on well-established materials; or alternatively, a criterion should be established and employed for selecting acceptable measurement data.

Very recently, dispersion properties of GaSe_1−xS_x at THz frequencies were approximated in the form of Sellmeier equations in [36]. New approximated dispersion equations for GaSe (slightly modified from [38]) were used together with published experimental data on phase matching in GaSe_1−xS_x. Dispersion properties for a limited range of crystal compositions, x=0.01, 0.14, 0.26, 0.37 [47], 0.26 [15], and 0.29 [16], were also measured by THz TDS. A number of GaSe_1−xS_x compositions were studied and reported in brief for x=0.05, 0.11, 0.22, 0.29, 0.44 [48]. Results in [48] were found to be in disagreement with the previously published data as to the gradient of the refractive indices and the change in birefringence with S content, as well as the spectral features of generated emission. A detailed comparative analysis of available data on dispersion properties of GaSe_1−xS_x in the THz range has not yet been carried out.

In order to exploit fully the potential of GaSe_1−xS_x and to expand its applications, it is necessary to know accurately its frequency-dependent refractive indices. In the present work we report, for the first time to the best of our knowledge, detailed THz TDS measurements of o- and e-wave dispersions in the 0.3–4.0 THz range for a wide range of solid solutions of GaSe_1−xS_x (x=0, 0.05, 0.11, 0.22, 0.29, 0.44, 1). Criteria are proposed for selecting data sets measured by THz TDS that are of acceptable quality for the determination of dispersion properties. Finally, it has been established that several different available equations for o- or e-waves are most suitable for different spectral ranges. The dispersion equations most commonly used in practice were found to be unsuitable for calculating n_e dispersion and birefringence in the THz range and for estimating large PM angles for e-wave generation.

2. Experimental

2.1. Material synthesis and crystal growth

The modified vertical Bridgman method was employed to grow single solid solution crystals of GaSe_1−xS_x, x=0, 0.05, 0.11, 0.22, 0.29, 0.44, 1). The starting materials for the synthesis were Ga 99.9997, Se 99.99 and S 99.95, which were additionally purified by remelting in a continuously evacuated ampoule. The weighing of a stoichiometric charge of Ga and Se, and the nominal 0, 1, 2.5, 5, 7, 11, and 31.5 mass% S was performed with the accuracy of ±0.1 mg. Synthesis ampoules were loaded to >60% in volume to minimize the quantity of rest gases. All growth and synthesis ampoules were evacuated to a residual pressure of ≈10⁻⁴ Torr. After several hours of melt homogenization, the temperature was slowly decreased to 40 K below the melting point of GaSe at 1238 K, at the rate of approximately 10 K/h, before switching off the oven.

For the growth process, the polycrystalline material was loaded into a single wall ampoule. The internal wall of the container was coated with a layer of pyrolytic carbon, which protected the melt from reaction with ampoule wall. The unseeded crystal growth was performed by the vertical Bridgman method, with heat field symmetry change and rotation as described in detail in [26]. The sealed growth ampoule was loaded into a furnace having a temperature gradient of ≈15 K/cm at the estimated level of the crystallization front. After homogenization of the melt at the temperature 30 K above the melting point, the ampoule was mechanically lowered at a speed of 10 mm/day. Other details on the synthesis process were reported elsewhere [18,28,37].

2.2. Sample fabrication

Two types of GaSe and GaSe_1−xS_x samples were fabricated for this study. The first type was cleaved from as-grown boules, i.e. it had faces orthogonal to the c-axis, so that a beam traversing the sample traveled parallel to the c-axis. The second type of sample was cut so that a traversing beam traveled orthogonally to the c-axis. To make this type of sample, a section of the grown boule was first immersed in methyl acrylate monomer mixed with a thermoinitiator and placed in an oven for polymerization. Once set, the boule section was cut perpendicularly to the growth layers (⊥ c̄) and then polished with fine 0.8–1.2 μm fraction separated from POLYRIT powder (ErmakChem Co., Russia) [18, 26, 28]. The produced samples of both types were free of precipitates, broken layers, dislocations, micro-bubbles, or other visual defects. Their thicknesses were from <1 mm to 1–2 mm.

2.3. Crystal characterization

The properties of the GaSe_1−xS_x samples were characterized at room temperature. Scanning electron microscopy (SEM) with a SEM Quanta 200 3D (FEI, Netherlands) microscope was employed to study the surface morphology of the samples. This microscope, provided with an EDAX ECON VI micro analyzer, was also used to measure element composition. An X-ray diffractometer Shimadzu XRD 6000 (Japan), and a transmission electron microscope (TEM) CM12 (Philips, Netherlands) were used in the SAED method to analyze the structure. UV-visible-near-IR transmission spectra were recorded by a Cary 100 Scan (Varian Inc.) spectrophotometer over the spectral range of 190–900 nm with a spectral resolution of 0.2–4 nm.

The measurements of linear optical properties at 0.2–4 THz were performed using a THz TDS. The THz TDS used a standard configuration incorporating a Ti:Sapphire (λ=800 nm) femtosecond laser, four off-axis parabolic mirrors, a biased GaAs emitter, and electro-optic detection with a 〈110〉 ZnTe crystal and balanced Si photodiodes. Samples under test were placed in the polarized THz beam such that they interacted with the incident radiation in either Ē ⊥ c̄ (o-wave) or Ē ‖ c̄ (e-wave) configuration, where Ē is the electric field vector of the incident THz wave. The spectral resolution was varied to test for resolution effects, and was 3.75, 7.5, 15, and 30 GHz. Test spectra were also recorded using GaAs as a standard reference material whose properties are well known. Other details can be found elsewhere [28, 50, 51].

The Ga and Se content of the samples was determined by X-ray fluorescence analysis (XRF) using X-Ray Fluorescence Spectrometer (Shimadzu XRF 1800) with an accelerating voltage of 40 kV, a current of 95 mA, a scanning speed of 8°/min, a scanning step of 0.1°, and with the detection limit of ≈10⁻⁶. Samples with a known composition of Ga/Se/S powder were used as reference samples; the standard deviation was below 0.05%. Polytype structure was determined by using a nonlinear optical method (second harmonic generation of CO₂ laser) [52].

3. Results and discussions

3.1. Sample quality

The cleaved GaSe_1−xS_x 〈001〉 surface is atomically flat, so that the surface finish of these easily cleaved crystals is suitable for the study of optical properties [21, 31]. In contrast, cut and polished samples exhibit defects. The polished surfaces of samples of over 1 mm thickness had regions of local defects, although these samples were still of sufficient optical quality for THz measurements. Thinner samples, however, had high concentrations of defects, and as a result were unusable for this study.

Only Ga, Se and S peaks were seen in the EDAX spectrum, indicating the high purity of the product. EDAX spectra recorded while scanning the surface confirmed the uniform distribution of the sample composition. The analysis showed that the average atomic ratio of Ga:Se:S was close to the charge composition. The diffraction patterns clearly confirm the high quality of the ε-polytype crystalline structure (point group symmetry 6̄2m), which is similar to that of ε-GaSe widely used for THz applications.

Optical absorption coefficients of GaSe grown by modified technology are three times lower than those of crystals grown by the conventional Bridgman method. It was found that in unpolarised light the absorption coefficient for both GaS and GaSe crystals does not exceed 0.05 cm⁻¹ within their maximum transparency range, although it is significantly smaller for GaS than for GaSe. Absorption coefficients of GaSe:S in the IR and THz ranges decrease by a factor of at least 2 with increasing S content up to x=0.11; then rise again with further doping. Therefore the mixing ratio x=0.11 was identified in this study as the optimal doping level for the THz frequency band.

The narrow spectral width of the strongest phonon absorption peak at 0.59 THz (the E′ ⁽²⁾ rigid phonon mode) signals the high optical quality and structural homogeneity of the optimally doped crystal [53], since there is no spectral broadening due to site-to-site variation or defects. The o-wave absorption coefficient (α_o) of GaSe agrees well with the recently reported data in both spectral profile and magnitude. Absorption in GaSe is strongly anisotropic, with the e-wave absorption coefficient at least a factor of 2 to 3 lower than that for the o-wave [18]. The good match between the published and measured data for GaSe, together with the high optical quality of the grown crystals provide a firm basis for confidence in the accuracy of the measured dispersion properties of solid solution GaSe_1−xS_x. As an additional confirmation of the measurement methodology, the spectrum of the test-piece GaAs wafer was found to match well to the published data.

3.2. Dispersion properties studied by THz TDS

Examples of dispersion spectra as measured by THz TDS for GaSe_1−xS_x whose facets were orthogonal to the polarization of THz beam are depicted in Fig. 1.

 

Fig. 1 Measured n_o and n_e dispersion spectra for cleaved and polished (right superscript c and p respectively) GaSe_1−xS_x samples. The mixing ratio x is shown in brackets. As-measured and smoothed curves are indicated by the left superscript m and s respectively. Curves labeled as 1 or 2.5% are n_e(0.22) spectra calculated assuming thickness uncertainty of 1 or 2.5% respectively.

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In Fig. 1 it is seen that the ${}^m{n}_o^p(0.22)$ curve (our notation is described in the caption of Fig. 1) has a slope variation that is different from those of the other curves, and that its etalon pattern (sinusoidal oscillations around the mean) is strongly irregular. Numerous micro-defects of different nature, including admixture of second phases, were found in this sample. It is also seen that the etalon pattern on the ${}^m{n}_e^p(0.22)$ curve is more regular than that on the ${}^m{n}_o^p(0.22)$ curve, but is less regular than on ${}^m{n}_e^p(0.44)$. By examining different areas of processed (right superscript c) crystals, it was determined that irregularities arise from the presence of local surface defects caused by processing. Local surface irregularities are smaller (in density and size) for the harder sample (x=0.44), and therefore produce less distortion in the etalon pattern. Similar effects were observed in the presence of other types of local defects on the cleaved sample surface or within its bulk, such as an occasional precipitate or a local air gap between layers. Clearly, smaller size defects and lower defect density result in greater regularity of the etalon pattern and in better measurement reproducibility.

Figure 1 also shows the uncertainty in the refractive index arising from the measurement uncertainty in the sample thickness. A difference of 1% in the estimated thickness (corresponding to ≈10 μm) produces a magnitude shift equivalent to a difference in the mixing ratio of about 0.2. Similar errors may be caused by a tilt of the sample away from zero incidence, which increases the effective thickness and affects measurement results [50].

The absorption spectra are plotted in Fig. 2 for cleaved and processed GaSe_1−xS_x, x=0.22, samples. Processed crystals were measured in both o-wave and e-wave transmission. Repeating the measurement at different points on the sample surface revealed local defects. This is seen in the spectra of ${}^s{\alpha }_o^p(0.22)$, one of which is anomalous, with features at 1.4 THz, indicating that they originate from a local defect. The inset in Fig. 2 shows anomalous dispersion caused by one of these defects.

 

Fig. 2 As-measured absorption spectra for cleaved and polished (right superscript c and p respectively) GaSe_1−xS_x samples. The mixing ratio x is shown in brackets. As-measured and smoothed curves are indicated by the left superscript m and s respectively. The inset shows anomalous dispersion caused by a local defect.

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Figure 3 shows examples of regular and irregular etalon patterns in the refractive index spectra. These etalon oscillations are caused by THz pulse echoes, which are reflected from the crystal-air interface and travel twice through the crystal, appearing in the time-domain trace as pulse features delayed from the main pulse peak. Applying Fourier Transform to a time-domain trace containing echoes results in a spectrum with etalon-like sinusoidal oscillations [51]. A plane-parallel, defect free sample with high THz transparency will produce a highly regular etalon oscillation pattern (Fig. 3(d)). A wedged sample will exhibit a reduced oscillation amplitude, because a part of the echo pulse will be deflected away from the detector (Fig. 3(c)). Sample wedging can be caused by the presence of packed or broken layers, or dislocations, or by material taken from a polycrystalline boule. In contrast to crystals of high optical quality, samples having surface defects and/or defects in the bulk give rise to highly irregular oscillations instead of regular etalon patterns (Fig. 3(a,b)), because the reflected echo pulses will suffer distortion and loss in traveling through the material.

 

Fig. 3 Etalon oscillation patterns from selected cleaved and polished (right superscript c and p respectively) samples with (a) processing defects on facets, (b) astoichiometric composition and micro precipitates, (c) wedged sample, (d) a near-perfect pattern from a crystal of high homogeneity and optical quality. The mixing ratio x is shown in brackets.

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Therefore the etalon oscillation pattern in the spectrum may be used to evaluate the optical quality of the crystal sample, and as a criterion for selecting the most reliable and consistent spectral data sets. Dispersion curves with the most regular etalon patterns that are also reproducible in multiple measurements on different areas of the crystal face are to be judged the most accurate and reliable. The best selected dispersion data for different compositions of GaSe_1−xS_x are presented in Fig. 4.

 

Fig. 4 Measured n_o and n_e dispersion spectra for cleaved and polished (right superscript c and p respectively) GaSe_1−xS_x samples selected for their reproducibility and the regularity of the etalon pattern. Also included is the dispersion curve for the GaAs wafer test piece.

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4. Dispersion equations

4.1. GaSe

The Sellmeier coefficients for GaSe that are suitable for application in the entire transparency range, as found in the literature [4, 36, 38], are presented in Table 1.

The refractive indices calculated using the parameters listed in Table 1 are shown in Fig. 5 together with the data measured in this work. In Fig. 5(a) it is seen that the estimated refractive indices n_o are very close to each other in the maximum transparency range and in the THz band. This is because the dispersion equations for o-wave in [4,36] are based on the n_o equations in [38] and are thus in close agreement. As a result, for all three equations n_o magnitudes at THz frequencies lie within the narrow range of 3.2–3.25. However, equations in [4] are significantly different at mid-IR wavelengths longer than 18 μm. The main purpose of those modifications was to reconcile the mismatch with their own experimental data. As a result, their validity range was extended up to 1500 μm. The modified equations also produce slightly lower magnitudes of n_o over the entire wavelength range.

Table 1. Sellmeier coefficients for ordinary and extraordinary waves for GaSe

View Table

 

Fig. 5 GaSe dispersions (a) and birefringence (b) as calculated and as measured by THz TDS in this study. Experimental points of the study: black; curves calculated by dispersion equations from [4]: olive, [36]: blue, [38]: red.

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It should be mentioned that the dispersion equations in [36], as well as the previous version of these equations for the mid-IR in [44], give the estimated phase matching (PM) angles at short wavelengths closer to the experimental values [54] than those estimated from the data in [38]. A careful fitting to the available data extends the validity range up to 1620 μm [36]. In fact, the n_o equation in [38] is adapted from [55, 56] (see also [25, 57] for details). By using this, together with the available experimental data for phase matching angles between 0.65–18 μm, the e-wave refractive indices were calculated and approximated in the form of Sellmeier equations. The resulting set of dispersion equations are recommended for the 0.65–18 μm range by the handbook [5]. However, it was found that the dispersion equations from [38] are in fact useful for PM estimation for down-conversion in the entire THz range (see, for example, [2, 3]). The experimental data for o-wave generation by down-conversion [10, 12, 13] likewise matched well with the estimated PM angles. This also originates from the data in [55, 56], based on the equation whose validity was claimed to be over a wider range of 0.4–300 μm or up to 1 THz.

It may be reasonably expected that PM angles calculated from the data in [38] for the generation of forward [3,4,12,58] and backward [17] e-waves at THz frequencies would differ significantly from those observed experimentally. As seen in Fig. 5(b), the differences in the calculated dispersion curves are so large that the estimated THz birefringence varies from 0.39 [36] to 0.49 [38] and further to 0.78 [4]. Only a limited amount of data is available on e-wave dispersion measured by THz TDS in unprocessed GaSe (see, for example, [4, 6, 9, 12, 45–47 ]), and even less for processed (cut and polished at θ=90°) samples [15, 16, 18]. Nevertheless, the measured birefringence is always found within the narrow range of 0.78–0.79 [4, 6, 15, 16, 18] that is significantly different from the values estimated in [36, 38].

On the other hand, in the case of z-cut GaSe, surprisingly, the difference between PM angles estimated using equations in [38] and those obtained from experimental data for tunable extraordinary THz e-wave generation for most commonly realized down-converters (see for example [2,3]) does not exceed 2° over the entire generation range of incidence angles (Fig. 6). The cause of such small variation can be understood by referring to Fig. 6, which shows that external PM angles for down-conversion are small.

 

Fig. 6 Phase matching (PM) angles in GaSe for THz e-wave generation by type II (oe → e) down-conversion of Nd:YAG laser for near-IR OPO emission (labeled as λ ₃=1.0642 μm), and for a hypothetical far-IR source at 23–28 μm (labeled as λ ₃=23 μm). Experimental points from [3]: cyan, [58]: olive; curves calculated by dispersion equations from [4]: black, [36]: blue, [38]: red.

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As a consequence, despite large differences in the values of n_e, at small PM angles, all n_e values are close to that of n_o, which is the same for all curves, according to the data in [4, 36, 38] (Fig. 5(a)). On the other hand, small differences in PM lead to large differences in the generated wavelength, either forward [3] or backward [17]). This ambiguity in wavelength and birefringence may be unacceptable for some applications.

At longer wavelengths, e.g. for a hypothetical source at 23 μm or longer, the PM angles are larger, leading to noticeable differences in PM curves calculated by using data in [4,36,38]. Clearly, therefore, any type of interaction involving THz e-waves that occurs at large PM angles will suffer from significant differences in the estimated angles. The dispersion equations in [4] are best suited for such PM angle calculations, because they are well matched to the best available data for birefringence and to the experimental results obtained in this study.

4.2. GaSe_1−xS_x

Dispersion equations for GaSe_1−xS_x at THz frequencies can be designed using Eq. (1) together with those for GaSe from [4, 36, 38] and for GaS from [37], which are claimed to be valid between 0.8–20 μm and 0.75–1200 μm (0.25–4 THz). However, it should be mentioned that the authors of Eq. (1), first proposed in [59], do not claim it to be a priori evident. For example, dispersion properties of the solid solution compound AgGa_xIn_1−xSe₂ were correctly calculated in [60] by using a first order relation instead:

THz dispersion properties can be verified by THz TDS, and have been reported, but only for a limited range of GaSe_1−xS_x compositions: x=0.01, 0.14, 0.26, 0.37 [47], 0.26 [15], and 0.29 [16]. These results, however, disagree with each other and with the recently published data for x=0.05, 0.11, 0.22, 0.29, 0.44 [48] both in the magnitude of refractive indices and in their variation with S-doping. In particular, for the x=0.29 compound, n_o=3.03 in [16], 3.17 in [47] and 3.2 in [48], reflecting also the different slopes of n_o with S-doping. Furthermore, the decay and final disappearance of the rigid layer phonon peak E′ ⁽²⁾ as well as the appearance of new phonon modes with heavy S-doping was observed in [48], similar to the earlier reported changes with heavy Te-doping [53], but contrary to [16]. This phonon transformation should be accounted for by dispersion equations for GaSe_1−xS_x.

The published data and the measurements made in this work demonstrate significant variation in the magnitudes of n_o and n_e at THz frequencies with S-doping compared with those in the mid-IR [29]. Mid-IR dispersions arise from of the dielectric response of bonded electrons, which is different for GaSe and GaS; but similar bonding anisotropy results in the almost identical birefringence [29,41]. As a consequence, birefringence in solid solution GaSe_1−xS_x is almost constant with varying x, but the transparency range shifts towards shorter wavelengths with increasing x. The PM curve follows this shift, thus allowing PM optimization [29, 61].

In contrast, dispersions at THz frequencies are associated with phonon resonances via anomalous dispersion, as well as containing contribution from free charge carriers. The absorption strength of the fundamental phonon resonance at 46.8 μm in GaSe decreases with heavy S doping from its very high value of 4·10⁴ cm⁻¹ [4], whilst a GaS peak appears at the shorter wavelength of about 32 μm [37]. As a result, the total contribution of GaSe and GaS phonon peaks to the dielectric response in the THz range is decreased. However, due to persistently very high intensity of the 46.8 μm phonon peak, no shift in the transparency edge is observed. Only sub-micrometer thick samples show the barely observable indication of such shift. The common trend of refractive indices to decrease with rising S-content is clearly seen in Fig. 7.

 

Fig. 7 Variation of refractive index n_o of GaSe_1−xS_x at selected frequencies as a function of the mixing ratio x.

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In Fig. 7 it is seen that the variation of refractive index with S-content is characterized by a regular gradient, thus allowing PM optimization. Some deviations from this regular trend are observed, which may be attributed to measurement uncertainties, such as errors the determination of sample thickness, and to differences in the free charge carrier concentrations. The accuracy of the THz TDS measurements of GaSe_1−xS_x samples was confirmed by measurements of semi-insulating GaAs, which agreed with the published data for this material.

5. Conclusion

A range of high optical quality nonlinear crystals of solid solution GaSe_1−xS_x, x=0, 0.05, 0.11, 0.22, 0.29, 0.44, 1 was grown by modified Bridgman technique with heat field rotation. Ordinary and extraordinary wave dispersions were studied in detail by THz TDS at 0.3–4 THz as a function of sulfur concentration by using both cleaved and processed (θ=90°) crystals. A criterion is proposed for the selection of most accurate measurement data, based on the regularity and amplitude of the etalon oscillation pattern in the refractive index spectrum. Selected data show for the first time a clear regular trend in the dependence of the refractive index on the S-content. A comparative analysis of dispersion equations found in the literature and derived from the THz TDS measurements in this study was carried out. It was established that different dispersion equations represent most accurately the refractive indices for o- or e-waves of GaSe_1−xS_x in different wavelength regions of its transparency range. A set of dispersion equations for the THz band was identified which is different from the commonly used version. The reported results provide useful data with improved accuracy of the refractive index dispersions of GaSe_1−xS_x, and offer a basis for the design of dispersion equations for the entire transparency range.