New Sellmeier and Thermo-Optic Dispersion Formulas for AgGaS₂

Abstract

This paper reports on the new Sellmeier and thermo-optic dispersion formulas that provide a good reproduction of the temperature-dependent phase-matching conditions for second-harmonic generation (SHG) and sum-frequency generation (SFG) of a CO₂ laser and a Nd:YAG laser-pumped KTiOPO₄ (KTP) optical parametric oscillator (OPO) in the 0.8859–10.5910 μm range as well as those for difference-frequency generation (DFG) between the two diode lasers in the 4.9–6.5 μm range and DFG between the two periodically poled LiNbO₃ (PPLN) OPOs in the 5–12 μm range thus far reported in the literature.

1. Introduction

In previous papers [1,2] we have reported the Sellmeier equations for AgGaS₂ that reproduce well the phase-matching angles of a Nd:YAG laser-pumped optical parametric oscillator (OPO) in the 2.6–5.3 μm range [3] and those of difference-frequency generation (DFG) between the signal and idler outputs of a Nd:YAG laser-pumped LiNbO₃ OPO in the 5–12 μm range [4] at room temperature. In addition, our Sellmeier and thermo-optic dispersion formulas [1,2] have also provided a good reproduction of the temperature-tuned phase-matching conditions for DFG between the two laser diodes in the 4.9–6.5 μm range [5] and those for DFG between two Nd:YAG laser-pumped PPLN OPOs in the 5–12 μm range [6] at elevated temperatures. However, this thermo-optic dispersion formula [2] constructed from the original work of Bhar et al. [7] has given dn_o/dT and dn_e/dT three times larger than the experimental values of Aggarwal and Fan [8] at 308 K (35 °C).

In order to clarify this large discrepancy, we measured dn_o/dT and dn_e/dT at 0.6328, 1.0642, 1.1523, 2.052, and 3.3913 μm by using a prism method and found that our experimental values of dn_o/dT = 6.048 × 10⁻⁵ °C⁻¹ and dn_e/dT = 6.549 × 10⁻⁵ °C⁻¹ at 1.0642 μm agree well with dn_o/dT ≃ 5.5 × 10⁻⁵ °C⁻¹ and dn_e/dT ≃ 6.0 × 10⁻⁵ °C⁻¹ at 1.06 μm measured by Aggarwal and Fan (Figure 1 of [8]). Thus, we have used our measured dn_o/dT and dn_e/dT and have constructed the new thermo-optic dispersion formula that provides a good reproduction of the abovementioned experimental results when coupled with our new Sellmeier equations that reproduce correctly the 90° phase-matched second-harmonic generation (SHG) wavelength of λ₁ = 1.7718 μm at 20 °C.

2. Experiments and Discussion

We used three different samples in the present experiments. One sample was cut at θ = 90° and φ = 45°. The other two samples were cut at θ₁ = 37.2° (θ₂ = 52.8°) and φ = 45°, and θ₁ = 39.7° (θ₂ = 50.3°) and φ = 0°. They were shaped as parallelepipeds with four polished surfaces corresponding to the two directions defined by the two values of the polar angle given. The dimensions of these samples are ~10 × 10 × 10 mm³.

These samples were mounted on a temperature-controlled copper oven, which was set on a Nikon stepmotor-driven rotation stage having an accuracy of ±0.02° to vary only the polar angle θ. The temperature stability of the oven was ±0.1 °C. By using a Nd:YAG laser-pumped KTiOPO₄ (KTP) OPO as a pump source, we first measured the 90° phase-matching wavelengths for type-1 SHG by heating a θ = 90° and φ = 45° cut crystal from 20 °C to 120 °C at 20 °C intervals. The resulting tuning points (open circles) are shown in Figure 1. As can be seen from this figure, these results give dλ₁/dT = +0.16 nm/°C and λ₁ = 1.7718 μm at 20 °C. Since dλ₁/dT is defined as

$$\frac{d{\lambda }_1}{dT}=\left(\frac{\partial n_2{}^e}{\partial T}-\frac{\partial n_1{}^o}{\partial T}\right)/\left(\frac{\partial n_1{}^o}{\partial {\lambda }_1}-\frac{1}{2}\frac{\partial n_2{}^e}{\partial {\lambda }_2}\right),$$

(1)

we obtain ∂n₂^e/∂T − ∂n₁^o/∂T = +1.0777 × 10⁻⁵ °C⁻¹ when using our Sellmeier equations presented in [1]. This is 27% lower than ∂n₂^e/∂T − ∂n₁^o/∂T = +1.4692 × 10⁻⁵ °C⁻¹ given by our thermo-optic dispersion formula (T/K) presented in [2].

We next measured the temperature variation of the phase-matching angles for SHG and sum-frequency generation (SFG) of a CO₂ laser (Coherent DEOS, Model EOM-10) and its SHG and third-harmonic generation as a pump source under the same experimental conditions as described in [9]. For reliable determination of a zero-angle of incidence, a reference He-Ne laser beam reflected from the entrance face of the crystal was aligned on a 0.2 mm slit located 2 m from the crystal.

The observed phase-matching angles and the temperature phase-matching bandwidths (ΔT·l) at full width at half-maximum (FWHM) that were determined from the temperature variation of the phase-matching angles (Δθ_ext/ΔT) and the angular acceptance angles (Δθ_ext·l) calculated with the following new Sellmeier equations, are tabulated in

Table 1. Temperature phase-matching properties of AgGaS₂ for harmonic generation of a CO₂ laser at 10.5910 μm.

Wavelength (μm) 1			Phase-Matching Angle at 20 °C θpm (deg)	Angular Acceptance Δθext·l (deg·cm)	ΔT·l (°C·cm)
λ1	λ2	λ3			Measured	Calculated
10.5910o	10.5910o	5.2955e	68.5	1.77	35.8	35.5
5.2955o	5.2955o	2.6478e	32.6	0.66 (0.656)	20.6	20.8
5.2955e	5.2955o	2.6478e	50.1	1.23	21.2	21.4
3.5303o	3.5303o	1.7652e	33.4	0.44 (0.438)	15.2	15.2
3.5303e	3.5303o	1.7652e	51.5	0.83	15.8	15.7
10.5910o	5.2955o	3.5303e	43.3	0.80		28.5
10.5910e	5.2955o	3.5303e	58.5	1.38	29.0	29.4

¹ 1/λ₁ + 1/λ₂ = 1/λ₃. ² The superscripts of the interacting wavelengths denote the polarization directions.

.

$$\begin{array}{c}{n}_o{}^2=5.79495+\frac{0.22799}{{\lambda }^2-0.07963}-2.4534\times {10}^{-3}{\lambda }^2+3.1814\times {10}^{-7}{\lambda }^4-9.7051\times {10}^{-9}{\lambda }^6,\\ n_e{}^2=5.54077+\frac{0.22223}{{\lambda }^2-0.09670}-2.5240\times {10}^{-3}{\lambda }^2+2.0930\times {10}^{-6}{\lambda }^4-2.2984\times {10}^{-8}{\lambda }^6,\\ (0.661\le \lambda \le 10.5910),\end{array}$$

(2)

where λ is in micrometers. The phase-matching angles tabulated in Table 1 agree well with the values presented in [1,2]. We did not measure the temperature phase-matching bandwidth for type-1 THG of a CO₂ laser because the polarization directions of λ₁ and λ₂ were mutually orthogonal in the experimental setup, in which the 5.2955 μm input (λ₂) was generated by a type-1 SHG process. This index formula was obtained by adjusting the Sellmeier constants of Equations (1) and (2) presented in [1] to give the best fit to the type-1 90° phase-matched SHG wavelength of λ₁ = 1.7718 μm at 20 °C and the type-1 90° phase-matched DFG wavelength of λ_i = 5.333 μm generated by mixing λ_p = 661.183 nm and λ_s = 754.752 nm at 20 °C [10]. It reproduces correctly the phase-matching angles of a Nd:YAG laser-pumped OPO in the 2.6–5.3 μm range [3] and those of DFG between the signal and idler outputs of a Nd:YAG laser-pumped LiNbO₃ OPO in the 5–12 μm range [4] described above.

Although our Sellmeier and thermo-optic dispersion formulas presented in [2] reproduce well the temperature-dependent phase-matching conditions for the 90° phase-matched DFG between the two laser diodes in the 4.9–5.0 μm range [5] and those for the critically phase-matched DFG between the two PPLN OPOs pumped by a Nd:YAG laser in the 5–12 μm range [6], we found large differences between the temperature phase-matching bandwidths (ΔT·l) for SHG of a CO₂ laser and its harmonics that are tabulated in Table 1 and those listed in Table 1 of [2]. For instance, ΔT·l = 35.8 °C cm observed in the present experiment at λ₂ = 5.2955 μm is a factor of ~4 smaller than the ΔT·l = 139 °C cm value observed in previous experiments [2]. This may account for the systematic error in previous measurements of Δθ_ext/ΔT.

Aggarwal and Fan [8] have reported that dn_o/dT and dn_e/dT at 308 K (35 °C) measured by using the temperature-induced shift in the frequency of the interference fringes in the Fourier transform infrared spectroscopy (FTIR) transmittance spectrum of AgGaS₂ etalon are ~1/3 of our calculated values and those of Bhar et al. [7] and Zondy and Touahri [11]. Hence, we attempted to measure dn_o/dT and dn_e/dT at 0.6328, 1.0642, 1.1523, 2.052, and 3.3913 μm by using a minimum deviation method with a prism cut at an apex angle of 20°48’.

Using the raw data obtained by heating the prism from 20 °C to 140 °C at 20 °C intervals, we constructed a tentative thermo-optic dispersion formula to extrapolate dn_o/dT and dn_e/dT at 5.2955 and 10.5910 μm. The interpolated and extrapolated values were then iteratively adjusted to give the best fit to the measured temperature phase-matching bandwidths (ΔT·l) tabulated in Table 1. The newly constructed thermo-optic dispersion formula is expressed as

$$\begin{array}{c}\frac{d{n}_o}{dT}=\left(\frac{5.4715}{{\lambda }^3}-\frac{13.1712}{{\lambda }^2}+\frac{11.1658}{\lambda }+2.6459\right)\times {10}^{-5}({{}^{°}\mathrm{C}}^{-1}),\\ \frac{d{n}_e}{dT}=\left(\frac{4.8348}{{\lambda }^3}-\frac{11.6335}{{\lambda }^2}+\frac{9.8950}{\lambda }+3.5135\right)\times {10}^{-5},\\ (0.6328\le \lambda \le 10.5910),\end{array}$$

(3)

where λ is in micrometers. Note that Equation (3) gives ∂B/∂T = ∂n_e/∂T − ∂n_o/∂T = 6.4991 × 10⁻⁵ − 5.9924 × 10⁻⁵ = 5.067 × 10⁻⁶ °C⁻¹ at 1.1523 μm, which is a factor of 1.7 larger than the value ∂B/∂T ≃ 3 × 10⁻⁶ °C⁻¹ which was reported by Suslikov et al. [12].

For reference, we have plotted dn_o/dT and dn_e/dT given by Equation (3) in Figure 2 together with the values at 20 °C (closed circles) estimated from the data points of Aggarwal and Fan at 308 K (35 °C) and 97 K (−176 °C) [8], and our experimental points (open circles). As can be seen from this figure, our calculated at 10.5910 μm are about 20% lower than the experimental values of Aggarwal and Fan [8].

In order to check the utility of Equations (2) and (3), we calculated the 90° phase-matching temperatures for type-1 DFG between the two diode lasers (λ_s = 0.791116 μm, λ_p = 0.68162–0.68290 μm). The resulting tuning curve (K) is shown in Figure 3 together with the experimental points (closed circles) of Willer et al. [5] and the tuning curves (Z/T), (H/K), and (R) that were calculated with the Sellmeier equations of Zondy and Touahri [11], Harasaki and Kato [1], and Roberts [13] coupled with our new thermo-optic dispersion formula (Equation (3)). Our tuning curve (K) reproduces the experimental points of Weller et al. [5] within an accuracy of ±5 °C except near 30 °C.

We next calculated the phase-matching temperatures at θ_pm = 34.30° for type-1 DFG between two Nd:YAG laser-pumped PPLN OPOs operating at λ_p = 1.60 μm and λ_s = 1.846−2.353 μm [6]. Since we found that our calculated values at θ_pm = 34.20° reproduce well the experimental points of Haidar et al. [6], we inserted our tuning curve (K) at θ_pm = 34.20° into Figure 4. It should be noted that our calculated values at θ_pm = 34.30° are ~12 °C larger than those of the tuning curve (K) because the phase-matching temperatures around the retracing point are strongly dependent on the phase-matching angle θ_pm. Also note that the dashed line (T/K) was formerly presented in Figure 3 of [6] by Haidar et al. and it agrees with fairly well with our tuning curve (K) at θ_pm = 34.20°. On the other hand, the Sellmeier equations of Roberts [13] give no retracing point for this process (Figure 1 of [6]). They give the phase-matching angles of θ_pm = 28.90°, 30.38°, and 33.24° to generate λ_i = 5.0, 8.5, and 12.0 μm at 20 °C, respectively. Hence, we did not use his index formulas for the present calculations.

In order to check further the utility of our Sellmeier and thermo-optic dispersion formulas, we measured the phase-matching angles for type-2 DFG at λ_i = 7.5190 μm between the signal (λ_p = 1.8645 μm) and the idler (λ_i = 2.4793 μm) of a Nd:YAG laser-pumped CsTiOAsO₄ OPO [9,14] in a θ₁ = 39.7° (θ₂ = 50.3°) and φ = 0° cut crystal. The measured phase-matching angles of θ_pm = 38.2 ± 0.2° at 20 °C and θ_pm = 39.3 ± 0.2° at 100 °C agree well the calculated values of θ_pm = 38.24° and 39.21°, respectively. Thus, we demonstrated the utility of Equations (2) and (3) in this wavelength range.

3. Conclusions

In this study, we have reported the new Sellmeier and thermo-optic dispersion formulas for AgGaS₂ that reproduce well the present experimental results as well as the temperature-dependent phase-matching conditions for type-1 DFG between the two laser diodes in the 4.9–6.5 μm range [5] and between two Nd:YAG laser-pumped PPLN OPOs in the 5–12 μm range [6]. We believe that these two formulas are highly useful for investigating the temperature-dependent phase-matching conditions for down conversion (OPO, OPA (optical parametric amplifier), and DFG) of near-IR solid state laser systems in different temporal regimes (pulse duration and pulse repetition rate) to the mid-IR spectral range.