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Abstract
A temperature-dependent Sellmeier equation for GaP, valid for wavelengths between 0.7 and 12.5 μm over a temperature range of 78 to 450 K, is presented. The temperature dependence values of the generated wavelengths in nonlinear frequency conversion calculated using this equation match well the experimentally measured values.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Orientation patterned GaP (OP-GaP) has been of increasing recent interest for quasi-phase-matched (QPM) nonlinear optical frequency mixing due to several attractive properties of GaP, including high thermal conductivity, wide transparency extending into the visible wavelength range and low linear and nonlinear absorption at various wavelengths of interest due to its large direct bandgap energy (2.87 eV) [1]. For accurate determination of the requisite grating spacing for QPM interactions, the refractive index values at the interacting wavelengths need to be known with high accuracy. Since different wavelengths can be potentially generated from the same crystal by changing the temperature of the crystal, it is also important to know the refractive index values at different temperatures.
Measurements of the refractive index of GaP at room temperature over a wide wavelength range spanning the visible to microwave wavelengths has been reported [2]. From the frequency dependent expression for the dielectric constant given in Ref. [2], a Sellmeier equation with four poles has been derived for GaP:
where the wavelength λ is in microns, and A1 = 1.39, A2 = 0.172, B1 = 4.131, B2 = 0.234, C1 = 2.57, C2 = 0.345, D1 = 2.056, and D2 = 27.5. Three of the poles are in the ultraviolet region and the pole D2 is in the long wave infrared.Experimental results are available in the literature describing the frequency conversion of pump wavelengths of 1.064 µm, 1.559 µm, and 2.09 µm in orientation patterned GaP (OP-GaP) crystals fabricated with grating spacings of 20.8 µm, 61.1 µm, and 92.7 µm, respectively, based on the index values given in Eq. (1) [3–5]. The signal and idler wavelength pairs (in µm) predicted by Eq. (1) for these grating spacings and pump wavelengths are (1.388, 4.752), (1.928, 8.137), and (3.550, 5.076), respectively, and their experimentally measured values are respectively (1.385, 4.591), (1.923, 8.236), and (3.54, 5.1). The predicted and experimentally observed values are close, indicating that the room temperature index values obtained from Eq. (1) are reliable as a starting point.
The dispersion formula for refractive index of GaP was also given in Ref. [6]. (at 297 K and 105 K) in the ‘Pikhtin form’. The grating spacings for quasi-phase matched three-wave mixing predicted by this formula were far from the experimentally measured values. By contrast, the grating spacings Λ predicted by Eq. (1) were different from the experimental measurements by less than 1%. (See Table 1.)
Table 1. Experimental wavelengths and grating spacings, and predicted grating spacings
A temperature dependent Sellmeier equation for GaP was previously obtained from refractive index values measured with a prism [7]. However these measurements were performed over a limited temperature and wavelength range (295 to 395 K, 1 to 5.5 µm).
2. Measurement technique
To obtain the refractive indices of GaP over a wide range of temperature and wavelength, the method outlined by Moss [8] and used more recently for accurate measurement of refractive indices of GaAs [9] was used. A single crystal, (100)-oriented wafer of GaP grown by the Bridgman technique was polished on both sides and thinned to 158 ± 2 µm. Transmission spectra of the samples were taken on a Perkin Elmer FTIR spectrometer run with a step size of 0.125 cm−1 over the wavenumber range of 16,667 to 400 cm−1 (i.e., wavelength range of 0.67 to 25 µm) and over a temperature range of 78 K to 450 K.
A representative spectrum taken at room temperature is shown in Fig. 1, over a limited spectral range for clarity. The spectra showed a series of fringes corresponding to constructive and destructive interference. If d denotes the sample thickness, then constructive interference at a wavelength λm indicates the relationship:
where the fringe number m is an integer and n(λm) is the value of the refractive index at λm. Assuming the 295 K value of the refractive index to be known from literature [2] at a long wavelength, say around 10 µm where the fringes are relatively sparse, the fringe number m at that wavelength can be accurately determined from Eq. (1), even with the uncertainty of 2 µm in the wafer thickness. From the value of m at a given fringe wavelength, the fringe numbers at all the shorter and longer wavelengths at which the fringes appeared are obtained. For discrete values of d within its measured limits ( ± 2 µm), the values of n(λm) are obtained from Eq. (2) at the values of the fringe maxima, and then at arbitrary wavelengths through spline interpolation.Several GaP samples were studied. Ideally, if the samples were identical in all but thickness, the n(λ)d product determined from the measured fringe spectra (using Eq. (2), with known m) for any one sample would match the same product (to within a multiplicative constant) for any other sample of the same material, across the spectrum. This was not found to be the case, indicating that the refractive index is very sensitive to the concentration of sample impurities and/or crystal defects. The correct refractive index values for intrinsic GaP at room temperature were taken to be those given by Eq. (1), and these were compared to the n(λ)d product from the GaP sample having the best spectral fringe contrast. If the refractive index of the GaP sample reported here matched that of the samples studied by Parsons and Coleman [2] and Bond [cited in Ref. [2]], then the ratio (n(λ)d)FTIR / n(λ)Eq. (1) would simply equal a constant d, representing the sample thickness. As it was, the dispersion curve deviated somewhat from Parsons and Coleman’s result, so that the ratio was equal to a slightly wavelength-dependent function d(λ) (see Fig. 2). The variation however was not large: its maximum extent over the wavelength range of 0.7 to 12 µm was ~200 nm, or ~0.1%.
Fig. 2 Calculated sample thickness at 295 K, dividing n(λ)d product (from spectral data) by refractive index values from Eq. (1).
From the values of d(λ), thicknesses at other temperatures were then calculated using the known temperature-dependent thermal expansion coefficient [10, 11]. Spectra at these other temperatures yield fringe orders m and the wavelengths λm corresponding to the transmittance maxima, and from these and Eq. (2), the refractive indices at other temperatures are obtained, referenced to the samples and measurements of Parsons and Coleman and Bond at 300 K. (The nominal “300 K” mentioned in Ref. [2] was assumed here to be “room temperature,” and the spectral data taken here at 295 K was matched to Eq. (1).)
3. Results and discussion
From the FTIR spectra of the GaP sample, the refractive index values were obtained over a temperature range of 78 K to 450 K and wavelength range of 0.7 to 12.5 µm, and the dispersion curves are shown in Fig. 3(a). Fitting the values of to Eq. (1), the Sellmeier coefficients were found for each measured temperature. However, the temperature dependence of these coefficients could not be fit using simple polynomial expressions. Using instead an equivalent form of the Sellmeier equation (as used in Ref. [12] and discussed in Ref. [13] on p. 99, Eq. (38)):
the parameters through in Eq. (3) were found for each temperature, and in this case, the values of , , and could be fit to quadratic expressions in with good accuracy. These expressions are provided in Table 2 below. We note that the poles (at nm and 27.53 µm) are close to the poles of 345 nm and 27.5 µm in the Parsons and Coleman Sellmeier equation (Eq. (1)). The difference between the experimental results shown in Fig. 3(a) and the values obtained from Eq. (3) is the fit error (shown in Figs. 3(b)–3(d)).Fig. 3 (a) Temperature and wavelength dependent refractive indices of GaP, in 50 K steps; n(λ,T) increases with temperature. (b, c, d) Fit errors between the Sellmeier expression and the experimental refractive index at different temperatures.
Table 2. Temperature-dependent Sellmeier fit coefficientsa
4. Comparison with nonlinear experiments
The temperature dependent Sellmeier equation for GaP, given by Eq. (3) and Table 2, was used to calculate the signal and idler wavelengths generated by frequency conversion of three different pump wavelengths in OP-GaP crystals with three different grating spacings. Figure 4 shows the predicted values (solid lines) and the experimentally measured values (symbols). The predicted and measured values were in good correspondence with one another, indicating that the Sellmeier equation is reliable. (This comparison to experiment validates the wavelength- and temperature-dependence of Eq. (3), although it should be pointed out that adding a wavelength-independent offset in the refractive index will yield the same phase matching curves.) The deviation from experiment was small in all cases, but it was most pronounced in the case of the 1.559 µm pump. The predicted wavelengths are indeed sensitive to refractive index, the pump wavelength, and the grating spacing, and a <0.3% error in any combination of these would account for the slight discrepancy.
Fig. 4 Comparison between predicted signal and idler wavelengths (solid line) and the experimentally measured values (symbols) for (a) 1.064 µm pump OPO, with Λ = 20.8 µm, (b) 1.559 µm pump OPO, with Λ = 61.1 µm, and (c) 2.09 µm pump OPO, with Λ = 92.7 µm.
5. Summary
The temperature dependent Sellmeier equation for GaP was obtained over the wavelength range 0.7 to 12.5 μm, and over the temperature range 78 K to 450 K for the first time, significantly extending the room temperature results of Parsons and Coleman [2]. A good match was obtained between the temperature dependent signal and idler wavelengths predicted by the equation and the experimentally measured values for three pump wavelengths and grating spacings.
Acknowledgements
The authors thank Dr. Paulina Kuo, NIST, for bringing Ref. [6] and related work to their attention.
References and links
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