We compare the nonlinear and dispersive properties of the recently discovered mid-infrared nonlinear crystal CdSiP2 with other chalcopyrite materials to establish its potential for super-continuum generation through a second-order nonlinear process.
© 2011 OSA
The nonlinear optical crystal CdSiP2 (CSP) belongs to the tetragonal point group , with lattice constants a = 5.68Å, c = 10.431 Å, and Z = 4 for the unit cell parameters . It was grown in the past in small sizes that did not allow measurement of essential physical properties [2–7].
CSP is optically negative uniaxial chalcopyrite so that non-critical type-I (oo-e) phase-matching is possible in contrast to the positive ZnGeP2 (ZGP) which is the only commercially available II-IV-V2 type chalcopyrite. The birefringence, (ne-no), of CSP was found to be - 0.06 approaching a wavelength of 6 µm , - 0.051 near 2 µm , - 0.05 at 1 µm , and - 0.045 at 840 nm  in earlier work. An isotropic point, i.e. ne = no, was also observed close to the band-edge [5–7]: at room temperature this point occurs at 2.41 eV (514.5 nm) [6,7] and near this point, optical activity can be observed .
Recently, high optical quality crystals of CSP with sizes reaching 70 × 25 × 8 mm3 were grown successfully from the melt by using high purity starting materials via the horizontal gradient freeze technique [8–10]. Important physical characteristics, including transparency, birefringence, and thermo-mechanical properties have already been measured and can be found in the literature .
In the present work we analyze the nonlinear and dispersive properties of CSP predicting broadband continuum generation phase-matching at a pump wavelength of 2.43 µm. A comparison with other chalcopyrite materials in the context of super-continuum generation through a second-order nonlinear process is also established.
2. Nonlinearity of CSP
The value of the d36 nonlinear optical coefficient of CSP was estimated relative to ZGP by second-harmonic generation (SHG) near 4.6 µm . The fundamental beam that we used was produced by a femtosecond KNbO3-based optical parametric amplifier (OPA) pumped near 800 nm at a repetition rate of 1 kHz. The OPA used a 6-mm-long KNbO3 crystal cut at θ = 41.9° for type I phase matching. As an OPA seed we used the frequency doubled idler output of a BBO type-II OPA near 1 μm. The idler pulses of the KNbO3 OPA were temporally broadened but simultaneously spectrally narrowed (typical FWHM of 90 nm). The uncoated CSP sample was 0.53 mm thick with an aperture of 7 × 7 mm2. It was cut at (θ = 43°, φ = 45°) for type-I SHG. An uncoated sample of ZGP, cut at (θ = 50.5°, φ = 0°) with identical size and thickness, was used as a reference.
For the chosen wavelength and crystal thickness, the spectral acceptance was much larger than the spectral extent of the pump pulses (~12 times for CSP and ~7 times for ZGP). The angular acceptance was also much larger (>11 times for CSP and >16 times for ZGP) than the angular extent of the incident beam focused by a 25 cm BaF2 lens. Finally, according to the beam diameter at the position of the crystals, the birefringence walk-off (tanρ = 0.009 for CSP and tanρ = 0.005 for ZGP) could be neglected. The incident pulse energy near 4.6 µm was limited to less than 2 µJ and the internal conversion efficiency was below 10%. This justifies the plane-wave approximation. The effective nonlinear coefficient was estimated by correcting the relative SHG efficiency only for the different Fresnel losses and index of refraction although both corrections did not exceed 5%.
An average of 20 measurements for each crystal was taken, in which the results in terms of SHG output did not deviate by more than ± 5%. The experimentally measured internal phase-matching angles were used then to derive the ratio for the d36 coefficients: we found d36(CSP) = 1.07d36(ZGP) with a relative error estimate of ± 5%. Assuming d36 = 75 pm/V for of ZGP at a fundamental wavelength of λF = 9.6 µm , which gives 79 pm/V at λF = 4.56 µm when using Miller’s rule, it is found d36 = 84.5 pm/V for CSP at λF = 4.56 µm.
As a test for the reliability of the measurement, we also measured d36 of HgGa2S4 relative to ZGP using a sample of 0.48 mm thickness cut at (θ = 38.6°, φ = 45°) for type-I SHG. The result was d36(HgGa2S4) = 0.328d36(ZGP) at λF = 4.58 µm, leading to d36(HgGa2S4) = 25.9 pm/V near 4.6 µm, which is in very good agreement with previous estimates using the Maker fringe technique at 1064 nm or SHG at 3.5 µm, i.e. d36(HgGa2S4) ≈1.8d36(AgGaS2) ·.
Fully independent, the result for d36(CSP) was confirmed in frequency doubling of 100 ns long pulses at the second harmonic (λF = 4.78 µm) of a TEA CO2 laser operating at 4 Hz repetition rate .
3. Sellmeier equations and phase-matching properties
The different Sellmeier equations that have been proposed in previous works for CSP are listed in Table 1 . The first Sellmeier equations that appeared in the literature on CSP were in fact derived from first-principles band structure calculations . Two versions of Sellmeier equations appeared more recently based on ordinary and extraordinary principal refractive index measurement by the minimum deviation method with a 30° prism in the 0.66 µm - 5 µm wavelength range [8,9]. The latter one predicts an isotropic point at 508.8 nm . In addition, the birefringence was independently derived from polarized light interference spectra in the entire transparency range of CSP .
Note that, based on older data of the temperature dependence of the birefringence , thermo-optic coefficients had also been fitted for CSP . The Sellmeier coefficients of Schunemann et al.  were also extended with temperature dependence, with validity between 10°C and 70°C, which was derived fitting optical parametric oscillator (OPO) tuning data recorded for a pump wavelength of 1.99 µm emitted by a Tm:YALO laser .
These Sellmeier equations were then modified by Kato et al.  by taking into account the experimental OPO wavelengths for non-critical room-temperature phase-matching at a pump wavelength of 1064.2 nm , as well as the introduction of the isotropic point at room temperature. These equations were also extended with thermo-optic coefficients, as it will be shown in the next section, based on the same 1.99 µm pumped OPO tuning data  and the non-critical OPO tuning data recorded in the 25°C - 150°C temperature range . The validity of the Sellmeier and thermo-optic coefficients  was specified from the isotropic point up to the long-wave clear transparency limit of CSP around 6.5 µm . Finally, the Sellmeier equations of Kemlin et al.  were determined from independent phase-matching angle measurements between 3 µm and 9.5 µm using the sphere method.
The physical validity of these Sellmeier equations can be assessed from the corresponding infrared pole wavelengths λIR. These values were taken as initial parameters at λIRo,e = 20.4 µm for both the ordinary and the extraordinary indices by Lambrecht et al. in their calculations . The Sellmeier equations of Schunemann et al. [8,9,17] cannot provide λIRo,e since the fits were based on expressions using infrared correction terms. In the case of Kemlin et al. , the values can be deduced from the ordinary and extraordinary refractive index equations that were experimentally obtained, which leads to λIRo = 45.5 µm and λIRe = 22.2 µm respectively. The procedure is the same for Kato et al. , but the two infrared poles obtained are closer, i.e. λIRo = 24.8 µm and λIRe = 25.6 µm. Note that all these values are compatible with several optically active phonons measurements previously reported [21,22].
The DFG phase-matching curves predicted by the equations of Lambrecht et al.  are not shown in Fig. 2 because they noticeably deviate from the experimental points. Nevertheless, the discrepancy is not so large in the case of SHG, which is a good result for dispersion equations coming from a computational method. The best agreement is obtained with the equations of Kemlin et al. .
The temperature dependence of the Sellmeier equations is an important issue for applications. We then combined the room temperature equations of Kemlin et al.  with the thermo-optic coefficients derived by Kato et al. , which provides the following expressions for the ordinary and extraordinary refractive indices of CSP:Equations (1) are the temperature-dependent Sellmeier equations of highest accuracy for CSP to the best of our knowledge.
4. Broadband infrared continuum generation
Continuum generation based on a second-order nonlinear process is attracting recently increasing attention for various applications and the potential to generate such continuum using ultrashort laser pulses has recently been evaluated for a number of oxide materials operating at shorter wavelengths . An OPO where the idler and signal are in the same polarization state is of particular interest for the generation of a broadband continuum. However, in such OPOs, or for the case of ultrashort pump pulses in the so-called synchronously pumped OPOs, normally only the signal wave is resonated. Thus, to obtain smooth mid-infrared continuum, having in mind that seeding in this spectral range will require additional complexity in the case of OPA, most straightforward of the travelling-wave approaches seems to be the optical parametric generator (OPG), pumped by amplified picosecond or femtosecond pulses that possess sufficient intensity to initiate the parametric process from fluorescence noise .
In all the above cases, the nonlinear processes are equivalent to type-I DFG. The spectral acceptance for such DFG is very broad when the first derivative of the angular tuning curve with respect to the wavelength is infinite at degeneracy, i.e. at λs = λi = λp/2, which leads to a spectrally non-critical phase-matching at the corresponding angle and higher derivatives have to be considered . But phase-matching at degeneracy is not necessarily given at any pump wavelength, as can be seen e.g. in Fig. 2 for the case of CSP which is a crystal possessing moderate birefringence at λp = 1.064 µm. It is then important to define the pump wavelength range over which an OPO or OPG can be phase-matched at degeneracy.
Furthermore, degenerate phase-matching can be allowed at a pump wavelength λpOpt for which the first derivative of the group velocity of the signal and idler waves is vanishing, leading to the broadest wavelength range that can be generated for a given crystal cut at the proper phase-matching angle θOpt . Experimentally this situation has been realized for the first time in the visible and near-infrared with a picosecond OPG based on KDP .
According to Eqs. (1) at 21°C, the above condition is fulfilled in a CSP crystal cut at θOpt = 42.8° and pumped at λpOpt = 2.43 µm. This deviates from an earlier prediction of 2.55 µm [8,9], but it confirms that such wavelengths can be provided from Cr2+:ZnSe or similar laser systems. We performed the same kind of calculation for the commercially available nonlinear chalcopyrite crystals AgGaS2, AgGaSe2 and ZGP using the corresponding Sellmeier equations [27–29], and we found for their respective optimum pump wavelengths λpOpt : 2.04 µm, 2.86 µm and 2.63 µm, slightly deviating from the values that can be found in the literature . The corresponding calculated angular tuning curves are plotted in Fig. 3 , which shows the complementary of these materials for the generation of mid-infrared super-continuum.
The full wavelength range of the super-continuum emitted by a crystal cut at the phase-matching angle θOpt can be then estimated from the third derivative of the group velocity [23,25]. While Fig. 3 provides a nice illustration basically valid for DFG, in the case of high parametric gain in an OPG this factor also enters the expression for the bandwidth and the parameter that can be quantitatively compared is the so-called parametric gain bandwidth. Expressed using the FWHM convention, this parametric gain bandwidth is written in the degenerate case :
The values calculated from Eq. (2) are listed in Table 2 for CSP, AgGaS2, AgGaSe2 and ZGP. For comparison, we have assumed a crystal length of L = 1 cm and Ip = 1 GW/cm2. In order to complete the comparison, we calculated the broadest continuum that can be generated using a quasi-phase-matched (QPM) orientation-patterned GaAs (OP-GaAs) crystal . Instead of optimum phase-matching angle, there is optimum grating period in this case which amounts to 173 µm. Note that there is a fundamental difference with respect to super-continuum generation when dealing with QPM materials. It is related to the group walk-off of the generated waves with respect to the pump which limits the effective interaction length and hence the conversion efficiency . This fact reflects the plane-wave approximation used to derive Eq. (2), i.e. the limited validity of this formalism for ultrashort laser pulses. In order to enable a fair comparison, we selected crystal length and pump intensity that correspond to picosecond rather than femtosecond pump sources, so that the full crystal length can be utilized in the process of parametric amplification. In Table 2 are also given the effective nonlinear coefficients deff of the different materials in the direction corresponding to θOpt. For their calculation, we used d36 values from the cited literature corrected by Miller’s rule for the actual three-wave process. Note that the effective nonlinear coefficient of AgGaSe2, CSP and AgGaS2 is maximum at an azimuthal angle of φ = 45°, whereas it is maximum at φ = 0° for ZGP. Both cases correspond to negative and positive chalcopyrite crystals in type-I phase-matching configuration. The parametric gain bandwidths are not very different for the materials considered due to the weak dependence on the dispersion and even weaker on the gain in Eq. (2). OP-GaAs shows somewhat narrower bandwidth due to the higher βi,s value.
The capability to generate a stable broadband super-continuum is closely related to the thermal properties of the nonlinear material. On the basis of the thermo-optic coefficients that are available for the four chalcopyrites [29,32,33] and GaAs  under investigation, we calculated the temperature dependence of the sinc2(∆k(T)L/2) function as shown in Fig. 4 , and the corresponding thermal acceptances L.∆T (FWHM) are included in Table 2.
The temperature acceptance of CSP is very large but lower than in ZGP. It is known that temperature tuning in ZGP is impractical, contrary to the case of CSP for which this still seems a feasible approach to tune the wavelength , also under non-critical phase-matching conditions .
In conclusion, we compared existing dispersion relations for the CSP nonlinear crystal and extended the most reliable of them with temperature dependence. With the measured nonlinear coefficient, it was possible to estimate the parametric gain bandwidth for the special phase-matching configuration ensuring ultra-broad parametric amplification bandwidth in an OPG pumped by ultrashort pulses. CSP can be pumped by Cr2+:ZnSe ultrafast laser systems or tandem OPGs to generate super-continuum in the mid-infrared extending up to its upper transparency limit.
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