Temperature-Dependent Sellmeier Equations of IR Nonlinear Optical Crystal BaGa₄Se₇

Abstract

The thermal dependent principal refractive indices of a new promising IR nonlinear optical crystal BaGa₄Se₇ at wavelengths of 0.546, 0.5806, 0.644, 0.7065, 1.530, 1.970, and 2.325μm were measured by using the vertical incidence method within the temperature range from 25 to 150 °C. We derived equations of thermal refractive index coefficients as a function of wavelength that could be used to calculate the principal thermal refractive indices at different wavelengths. The temperature-dependent Sellmeier equations were also obtained and used to calculate the phase matching angles for the optical parametric process of BaGa₄Se₇ crystal at different temperatures.

1. Introduction

At present, there has been an intensive trend to explore novel mid-far IR nonlinear optical (NLO) crystals for generating mid-far IR laser sources, which have many important applications including atmospheric monitoring, laser radar, and laser guidance [1,2,3,4]. In IR nonlinear optics, the chalcopyrite-type AgGaQ₂ (Q = S, Se) and ZnGeP₂ crystals have been practically used since the 1970s. However, AgGaQ₂ (Q = S, Se) has a low laser-damage threshold, and ZnGeP₂ exhibits strong two-photon absorption of the conventional 1064 nm pumping laser sources, severely limiting their applications[5,6,7].

Recently, BaGa₄Se₇(BGSe) was reported as a promising NLO crystal for practical applications in the mid-IR spectral range. The crystal belongs to the monoclinic space group Pc. It possesses intriguing overall properties for IR NLO applications including wide transparent range (0.47 to 18 µm), suitable birefringence, large nonlinear optical coefficients, and high laser damage threshold [8,9,10,11,12]. The outstanding properties of BaGa₄Se₇have been exemplified by some recent laser experiments [13,14,15,16,17]. Given the outstanding NLO properties and the preliminary laser experiment results, BGSe may have attractive applications in three aspects: (1) realizing mid IR output using the conventional 1 µm laser as the pumping source; (2) the generation of IR laser in the long wavelength, especially the 8–15 μm range; (3) the second-harmonic generation the CO₂ laser.

During high-power laser output process, the effect of self-heating in NLO crystal leads to variation of refractive index, and thus the phase-matching (PM) direction will be distorted, which could dramatically influence the stability of output power and the output mode of the laser cavity. To solve this problem, NLO crystal can be attached to some cooling device to obtain a stable temperature. To predict the PM angles more accurately at elevated temperature, it is of significance to determine the principal refractive indices at different temperatures and to obtain the thermal refractive index coefficients and the temperature-dependent Sellmeier equations, which are important parameters for designing a frequency-conversion device at the actual operating temperature.

In this work, we determine the principal refractive indices at different temperatures of BGSe crystal, and present its thermal refractive index coefficients and temperature-dependent Sellmeier equations for the first time.

2. Results

2.1. Thermal Refractive Index Coefficients

As shown in

Table 1. Principal refractive indices of BaGa₄Se₇crystal.

λ (μm)/T (°C)		25	50	80	100	120	150
0.546	nx	2.66645	2.66960	2.67344	2.67597	2.67877	2.68300
	ny	2.68304	2.68676	2.69131	2.69419	2.69726	2.70200
	nz	2.77179	2.77628	2.78191	2.78551	2.78945	2.79520
0.5875	nx	2.62699	2.6296	2.63275	2.6348	2.6371	2.64033
	ny	2.64356	2.64675	2.65052	2.65282	2.65552	2.65961
	nz	2.72365	2.72722	2.73171	2.73443	2.73755	2.74210
0.644	nx	2.58907	2.59125	2.59385	2.59558	2.59744	2.60005
	ny	2.60532	2.60812	2.61143	2.61328	2.61574	2.61882
	nz	2.67846	2.68136	2.68502	2.68711	2.68956	2.69329
0.7065	nx	2.55958	2.56149	2.56376	2.56512	2.56688	2.56905
	ny	2.57556	2.57800	2.58081	2.58253	2.58465	2.58754
	nz	2.6439	2.64641	2.64945	2.65118	2.65333	2.65643
1.530	nx	2.46278	2.46442	2.46615	2.46721	2.46829	2.46947
	ny	2.47979	2.48122	2.48295	2.48356	2.48503	2.48688
	nz	2.53538	2.5368	2.53808	2.53914	2.53991	2.54222
1.970	nx	2.45312	2.45536	2.45634	2.45717	2.45821	2.45955
	ny	2.46986	2.47120	2.47316	2.47437	2.47499	2.47719
	nz	2.52443	2.52577	2.52757	2.52808	2.52903	2.53065
2.325	nx	2.44903	2.45101	2.45229	2.45303	2.45402	2.45508
	ny	2.46581	2.46738	2.4692	2.46984	2.47082	2.4727
	nz	2.51973	2.52116	2.52268	2.5233	2.52415	2.52584

, refractive indices n_x,n_y, and n_zall increase slowly with the increased temperature at different wavelengths. For a specific wavelength, the index can be assumed as a linear variation with increased temperature, which could be expressed as

n = n₀ + dn/dT(T − T₀)

(1)

where n is the refractive index at temperature T, n₀ is the refractive index at T₀, and dn/dT is the thermal refractive index coefficient.

According to the measured principal indices, the relationship between dn/dT of the principal refractive indices and the wavelength λ could be fitted, shown in Equation(2):

dn_x/dT = (0.6837/λ³ − 1.7607/λ² + 1.6316/λ + 0.0318) ×10⁻⁴
dn_y/dT = (0.2692/λ³ − 0.3112/λ² + 0.2201/λ + 0.4867) × 10⁻⁴
dn_z/dT = (0.7223/λ³ − 1.5170/λ² + 1.2953/λ + 0.1296) × 10⁻⁴

(2)

where λ is in micrometer and 0.254 μm ≤ λ ≤ 2.325 μm. The thermal refractive index coefficients for n_x, n_y, and n_z were calculated by monadic linear regression method and are given in

Table 2. Thermal refractive index coefficients (×10⁻⁵) of BaGa₄Se₇ crystal at different temperatures.

λ (μm)	dnx/dT		dny/dT		dnz/dT
	Experimental	Calculated	Experimental	Calculated	Experimental	Calculated
0.546	13.20	13.14	15.10	15.00	18.70	18.51
0.5875	10.70	10.79	12.70	12.87	14.70	15.01
0.644	8.79	8.80	10.80	10.86	11.80	11.88
0.7065	7.58	7.52	9.54	9.38	9.96	9.72
1.53	5.39	5.37	5.56	5.73	5.23	5.30
1.97	4.86	4.96	5.78	5.53	4.86	4.91
2.325	4.69	4.62	5.34	5.45	4.71	4.64

. Table 2 shows that these constants are all positive with magnitude level of 10⁻⁴. Refractive indices corresponding to different wavelengths at different temperatures were calculated by using Equations (1) and (2) (T₀ is 25 °C, n₀ is the experimental value of refractive index at 25 °C). Figure 1 shows the theoretical and measured values of n_x, n_y, and n_z. The difference between the measured and calculated values was less than 2 × 10⁻⁴. This result verifies the reliability of Equation (2). The coefficients of Equation (2) are listed in Table 2.

2.2. Sellmeier Equationsof BaGa₄Se₇ Crystal

The Sellmeier equation of given temperature can be fit as Equation (3) and the Sellmeier coefficients are listed in

Table 3. The Sellmeier coefficients (×10⁻⁵) of BaGa₄Se₇ crystal at different temperatures.

T (°C)	Sellmeier	A	B	C	D
25	nx	5.96012	0.24529	0.08482	0.00163
	ny	6.0459	0.24249	0.08794	0.00218
	nz	6.31414	0.27001	0.10096	0.00305
50	nx	5.96791	0.24482	0.08687	0.00114
	ny	6.0502	0.24582	0.08787	0.00169
	nz	6.32003	0.27192	0.10225	0.0029
80	nx	5.97698	0.24475	0.08903	0.0018
	ny	6.05737	0.24815	0.08893	0.00138
	nz	6.3241	0.27579	0.10323	0.00225
100	nx	5.98331	0.24424	0.09079	0.0023
	ny	6.06074	0.24965	0.08977	0.00131
	nz	6.33053	0.27559	0.10525	0.00295
120	nx	5.98866	0.24496	0.09186	0.00242
	ny	6.06751	0.25147	0.08998	0.0019
	nz	6.33286	0.27849	0.10584	0.00265
150	nx	5.99506	0.24542	0.09425	0.0026
	ny	6.07652	0.25243	0.09205	0.00178
	nz	6.34616	0.27808	0.10862	0.00363

. The difference between the measured and calculated values was less than 3.6 × 10⁻⁴.

n²(i, T) = A + B/(λ² − C) − Dλ² (I = x, y, z; T/°C = 25, 50, 80, 100, 120, 150)

(3)

2.3. Temperature-Dependent Sellmeier Equations

The temperature-dependent Sellmeier equations of n_x, n_y, and n_z were obtained by using the least-square-fit method, as shown in Equation (4) and

Table 4. Constants in Equation (4).

Coeffecient	A	B	C	D	E	F	G
nx	5.9622	0.2375	0.090768	−0.002209	0.000205	8.0 × 10−8	1027.6
ny	6.03935	0.240995	0.08975	−0.00175	0.0002358	8.0 × 10−8	1263.6
nz	6.31176	0.26518	0.10476	−0.00299	0.000191	1.1 × 10−7	1228.7

, where iis x, y,or z, λ is the wavelength in micrometer, and A, B, C, D, E, F, and G are constants.

n²(i, λ, T) = A + ET + (B + F(T − 25) × (T + G))/(λ² − C) + Dλ² (i = x, y, z).

(4)

Average differences between calculated and experimental indices are found to be in the order of 2.8 × 10⁻⁴, 2.4 × 10⁻⁴ and 2.7 × 10⁻⁴ for n_x, n_y and n_z, respectively. Fitting quality is also demonstrated by the fact that maximum difference (accounting for all principal indices) between theoretical and measured values is found to be 5 × 10⁻⁴, that is still within our experimental accuracy.

Table 5. Calculated phase matching angles for 1064 nm → 3900nm + 1463 nm OPO.

Temperature (°C)	25	50	80	100	120	150
PM angle (θ, φ)	(54.9, 0)	(55, 0)	(55.2, 0)	(55.6,0)	(56.1,0)	(56.7, 0)

lists the calculated phase matching angles in the xz plane for 3900 nm output OPO pumped by 1064 nm laser (1064 nm → 3900 nm + 1463 nm), based on the above Sellmeier equations. It is shown the influence of temperature on the PM angle is not obvious below 80 °C.

3. Measurement of Refractive Indices

BaGa₄Se₇ is a biaxial crystal and belongs to the monoclinic space group Pc [8]. To measure its three independent principal refractive indices n_x, n_y, and n_z, one BaGa₄Se₇ crystal was cut into two right-angle prisms with the sides of the right angle aligned with different directions, as shown in Figure 2. The apex angles of two prisms were approximately 14.63° and 18.16°, respectively. In our experiment, the refractive index could be calculated by using n = sin(α + β)/sin(α), where n is the principal refractive index, α is the apex angle of the prism, and β is the deviation angle of refractive light. When the incident light travels perpendicular to the b face of Prism 1, refractive indices n_x and n_z could be measured. Refractive indices n_y as well as n_x could also be measured when the light travels perpendicular to the c face of Prism 2.

Our experimental system is comprised ofa refractive index measurement instrument (SpectroMaster UV-VIS-IR, Trioptics, Wedel, Germany) with a high accuracy of 1 × 10⁻⁵ and a homemade temperature-stabilized heating furnace with an accuracy of ±0.1 °C, as shown in Figure 3.

In the measurement, mercury lamp at wavelengths of 0.546, 1.530, 1.970, and 2.325 μm, helium lamp at wavelengths of 0.5875 and 0.7065μm, and Chromium lamp at wavelength of 0.644 μm were used. The prisms were placed in a temperature-stabilized furnace, and heated to the target temperature (25, 50, 80, 100, 120, and 150 °C, respectively) with 30 °C/h and allowed to reach thermal equilibrium. During the measurement process, the incident angle of the central collimator beam was defined as 0°.The sample table was first adjusted to be horizontal and the accurate apex angle of the prism was measured, after which the sample was aligned square to the collimator, and the goniometer was rotated manually to find the refractive signal and measure the deviation angle using the measurement program of the instrument. At each temperature, the deviation angle was usually measured four times at different wavelengths, and the results were averaged to obtain a single determination of deviation. The measured results are listed in Table 1.

4. Conclusions

In summary, the thermal refractive index coefficients of BGSe crystal were measured for the first time. The temperature-dependent Sellmeier equations were fitted, and were then used to calculate PM angles for BGSe OPO crystal at different temperatures. The thermal refractive index coefficients and the temperature-dependent Sellmeier equations of BGSe crystal will be highly useful for designing a mid IR frequency conversion system based on BGSe crystal.