New Sellmeier and Thermo-Optic Dispersion Formulas for AgGaS 2

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.


Introduction
In previous papers [1,2] we have reported the Sellmeier equations for AgGaS 2 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 LiNbO 3 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 λ 1 = 1.7718 µm at 20 • C.

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 θ 1 = 37.2 • (θ 2 = 52.8• ) and φ = 45 • , and θ 1 = 39.7 • (θ 2 = 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 mm 3 .
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.

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 mm 3 .
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 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 (Δθext/ΔT) and the angular acceptance angles (Δθext•l) calculated with the following new Sellmeier equations, are tabulated in Table 1.We next measured the temperature variation of the phase-matching angles for SHG and sum-frequency generation (SFG) of a CO 2 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 (∆T•l) at full width at half-maximum (FWHM) that were determined from the temperature variation of the phase-matching angles (∆θ ext /∆T) and the angular acceptance angles (∆θ ext •l) calculated with the following new Sellmeier equations, are tabulated in Table 1.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 CO 2 laser because the polarization directions of λ 1 and λ 2 were mutually orthogonal in the experimental setup, in which the 5.2955 µm input (λ 2 ) 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 λ 1 = 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 = 661.183nm and λ s = 754.752nm 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 LiNbO 3 OPO in the 5-12 µm range [4] described above.  The superscripts of the interacting wavelengths denote the polarization directions.
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 4.9-5.0µm 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 (∆T•l) for SHG of a CO 2 laser and its harmonics that are tabulated in Table 1 and those listed in Table 1 of [2].For instance, ∆T•l = 35.8• C cm observed in the present experiment at λ 2 = 5.2955 µm is a factor of ~4 smaller than the ∆T•l = 139 • C cm value observed in previous experiments [2].This may account for the systematic error in previous measurements of ∆θ ext /∆T.
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 AgGaS 2 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 (∆T•l) tabulated in Table 1.The newly constructed thermo-optic dispersion formula is expressed as  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 = 0.68162-0.68290µm).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 = 1.846−2.353µm [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.of [6]).They give the phase-matching angles of θpm = 28.90°,30.38°, and 33.24° to generate λi = 5.0, 8.5, and 12.0 μm at 20 °C, respectively.Hence, we did not use his index formulas for the present calculations.The dashed line (T/K) is the theoretical curve at θpm = 34.30°calculated with the Sellmeier and thermo-optic dispersion formulas presented in [2].Closed circles are the data points taken from [6].

Conclusions
In this study, we have reported the new Sellmeier and thermo-optic dispersion formulas for AgGaS 2 that reproduce well the present experimental results as well as the temperature-dependent Crystals 2019, 9, 129 6 of 6 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.
1 • C. By using a Nd:YAG laser-pumped KTiOPO 4 (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 dλ 1 /dT = +0.16nm/ • C and λ 1 = 1.7718 µm at 20 • C. Since dλ 1 /dT is defined as

Figure 1 .
Figure 1.Temperature tuning curve for type-1 90° phase-matched second-harmonic generation (SHG) in AgGaS2.The solid line is the theoretical curve calculated with the Sellmeier and thermo-optic dispersion formulas presented in this text.Open circles are our experimental points.The 90° phase-matching wavelength at 20 °C is λ1 = 1.7718 μm with ΔT•l = 7.3 °C cm.

Figure 1 .
Figure 1.Temperature tuning curve for type-1 90 • phase-matched second-harmonic generation (SHG) in AgGaS 2 .The solid line is the theoretical curve calculated with the Sellmeier and thermo-optic dispersion formulas presented in this text.Open circles are our experimental points.The 90 • phase-matching wavelength at 20 • C is λ 1 = 1.7718 µm with ∆T•l = 7.3 • C cm.

Figure 2 .
Figure 2. Thermo-optic dispersion curves of AgGaS 2 .Closed circles are the values at 20°C estimated from the data points of Aggarwal and Fan at 308K (35 • C) and 97 K (−176 • C) [8].Open circles are our experimental points.

Figure 3 .
Figure 3. Temperature-tuning curves for type-1 difference-frequency generation (DFG) between the two laser diodes in AgGaS2.The signal wavelength is 0.791116 μm.The solid line (K) is the theoretical curve calculated with the Sellmeier and thermo-optic dispersion formulas presented in this text.The dashed lines (Z/T), (H/K), and (R) are the theoretical curve calculated with the Sellmeier equations of Zondy and Touahri[11], Harasaki and Kato[1], and Roberts[13] coupled with our thermo-optic dispersion formula presented in this text.Closed circles are the data points taken from[5].

Figure 4 .
Figure 4. Temperature-tuning curves for type-1 DFG between two PPLN optical parametric oscillators (OPOs) pumped by a Nd:YAG laser in AgGaS2.The solid line (K) is the theoretical curve at θpm = 34.20°calculated with the Sellmeier and thermo-optic dispersion formulas presented in this text.The dashed line (T/K) is the theoretical curve at θpm = 34.30°calculated with the Sellmeier and thermo-optic dispersion formulas presented in[2].Closed circles are the data points taken from[6].

Figure 3 .
Figure 3. Temperature-tuning curves for type-1 difference-frequency generation (DFG) between the two laser diodes in AgGaS 2 .The signal wavelength is 0.791116 µm.The solid line (K) is the theoretical curve calculated with the Sellmeier and thermo-optic dispersion formulas presented in this text.The dashed lines (Z/T), (H/K), and (R) are the theoretical curve calculated with the Sellmeier equations of Zondy and Touahri[11], Harasaki and Kato[1], and Roberts[13] coupled with our thermo-optic dispersion formula presented in this text.Closed circles are the data points taken from[5].

Figure 3 .
Figure 3. Temperature-tuning curves for type-1 difference-frequency generation (DFG) between the two laser diodes in AgGaS2.The signal wavelength is 0.791116 μm.The solid line (K) is the theoretical curve calculated with the Sellmeier and thermo-optic dispersion formulas presented in this text.The dashed lines (Z/T), (H/K), and (R) are the theoretical curve calculated with the Sellmeier equations of Zondy and Touahri[11], Harasaki and Kato[1], and Roberts[13] coupled with our thermo-optic dispersion formula presented in this text.Closed circles are the data points taken from[5].

Figure 4 .
Figure 4. Temperature-tuning curves for type-1 DFG between two PPLN optical parametric oscillators (OPOs) pumped by a Nd:YAG laser in AgGaS2.The solid line (K) is the theoretical curve at θpm = 34.20°calculated with the Sellmeier and thermo-optic dispersion formulas presented in this text.The dashed line (T/K) is the theoretical curve at θpm = 34.30°calculated with the Sellmeier and thermo-optic dispersion formulas presented in[2].Closed circles are the data points taken from[6].

Figure 4 .
Figure 4. Temperature-tuning curves for type-1 DFG between two PPLN optical parametric oscillators (OPOs) pumped by a Nd:YAG laser in AgGaS 2 .The solid line (K) is the theoretical curve at θ pm = 34.20 • calculated with the Sellmeier and thermo-optic dispersion formulas presented in this text.The dashed line (T/K) is the theoretical curve at θ pm = 34.30• calculated with the Sellmeier and thermo-optic dispersion formulas presented in[2].Closed circles are the data points taken from[6].

Table 1 .
Temperature phase-matching properties of AgGaS 2 for harmonic generation of a CO 2 laser at 10.5910 µm.