1 Introduction

The nonlinear refraction of second-order nonlinear materials is of interest for a variety of nonlinear optical applications [1]. The nonlinear index n 2 plays an important role in the spatial, spectral, and temporal pulse evolution in many χ (3) mixing experiments involving ultrashort and high-energy optical pulses [2, 3]. Accurate knowledge of n 2 is particularly useful in assessing spectral and temporal pulse broadening due to self-phase modulation, chirp reversal, and self-compression as well as the minimum pulse width attainable in ultrashort pulse harmonic generators, optical parametric amplifiers, and oscillators [4–6]. It can also be used to control such pulse shaping and nonlinear phase compensation schemes [7]. The nonlinear index can also give rise to self-focusing, spatial beam distortion, and catastrophic optical damage in high-energy frequency-conversion processes. Knowledge about n2 can provide a useful estimate of the material damage limits in such experiments [8, 9].

The AgGaS2 (AGS) and ZnGeP2 (ZGP) crystals measured in our paper play an important role in the generation of ultrafast mid-infrared laser [10]. The AGS crystal is always used to generate the widely tunable ultrashort mid-infrared laser [11], and the ZGP crystal can be used to generate long wavelength mid-infrared lasers [12]. Then, determining the nonlinear parameters as nonlinear refractive index of these crystals is important considering their potential applications. However, there are no measurements of the nonlinear refraction of these crystals in mid-infrared, the third-order nonlinear susceptibilities were determined only for 1260 nm and only for ZGP crystal [13].

In this paper, we present the measurements and analyses of the third-order nonlinear refraction indices of AGS and ZGP in the mid-infrared region performed using our Z-scan platform. The probe light is produced by an OPA-DFG system. Our measurements showed that there is a small difference between the values of n 2 of o-polarization and e-polarization. We compared the measured n 2 value of the direct band gap material AGS with the theoretical values and found that they were approximately equal. The detected value of n 2 for the indirect semiconductor ZGP at ~1300 nm agrees with the results obtained under similar conditions in a previous study [14]. Besides, we calculated the dispersion distances according to the experimental conditions by considering the pulse broadening effect for ultrashort probe pulses, and the results demonstrate that the broadening due to group velocity dispersion (GVD) is negligible in our experiments. We believe these measured values can be used to analyze and design the optical system in the mid-infrared region because the AGS and ZGP are meaningful for nonlinear optics in the mid-infrared region [15–18].

2 Z-scan platform in the mid-infrared region

In the experiments, a laser pulse from the OPA-DFG system similar to [11] is used to produce the probe light. The nonlinear optical properties are studied in the mid-infrared region. An intensity-autocorrelation device is used to determine the pulse duration accurately. The laser beam is focused (Fig. 1) by a lens with a focal length of 300 mm in the Z-scanning process. The sample is mounted on an adjustable stage, which can translate along the optical Z-axis through the focusing zone. The energy of the laser pulse is measured using the pyroelectric detector (3 A, Ophir). An iris diaphragm, whose diameter can be varied from 1 to 8 mm, is located at 750 mm behind the lens, and the pyroelectric power probe is placed close behind the iris diaphragm (shown in Fig. 1). The scheme with a confining iris diaphragm makes it possible to determine the sign and magnitude of n2 of the materials under study. The sample is held on a motorized precision translation stage (FS-3150XY, Sigma Tech). The signal of the photosensor is fed into the header processor (Nova II, Ophir) to measure the dependence of the sample transmittance on the laser power. The header is connected to a PC through a serial port. The general control of the measuring system is implemented using the program LabVIEW. The translation step per cycle in the scanning process is approximately 100 µm.

Fig. 1
figure 1

Scheme of platform for automatic measurement of the nonlinear optical parameters for various media in the mid-infrared. f300 a lens with focal length of 300 mm

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A 1-mm-thick sample of CS2 is used to validate our measurement system (including the pulse duration measurement and beam waist measurement). The results are shown in Fig. 2. CS2 is frequently used as a reference material for nonlinear measurements [19, 20]. We measure the n 2 at 800 nm which is directly from Ti: sapphire laser system, and the sample CS2 is filled in the silica glass cell with a 1-mm path length. We employ the setup shown in Fig. 1 to gauge the transmittance when the translation stage is scanning. The pulse width measured by the intensity autocorrelation is about 50 fs, which agrees well with the value of 49 fs determined by autocorrelation (SSA, Coherent). In addition, the diameter of the beam waist detected by the knife-edge is approximately 79 µm, which agrees with the 74-µm beam waist measured by the beam profiler (SP620U, Ophir Optronics). The measured T PV is about 0.8, which makes the corresponding value of the nonlinear refractive index (n 2) 3.4714E-15 cm2/W. This result coincides with the previous value in [19], which was (3 ± 0.6)E-15 cm2/W.

Fig. 2
figure 2

Measurement results of CS2. a Dependences of the normalized transmission on the CS2 position with respect to the focus in the n 2 measurement. b Dependences of the normalized transmission on knife edge position with respect to the spot center in the knife-edge measurement. c Intensity autocorrelation trace corresponding to a 50-fs (FWHM) pulse

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3 Nonlinear refractive indices of AGS and ZGP

We measured the nonlinear refractions in the AGS and ZGP crystals based on our Z-scan platform. We determine the beam waist by knife-edge method and measure the pulse width by intensity autocorrelation in which a BBO or AGS crystal is used to produce the SHG signal. The characteristics of the nonlinear crystals and probe pulses used in our experiments are summarized in Table 1. In our experiment, we have adopted the incident energies which are not enough to invoke obvious nonlinear absorption. The AGS sample is cut with θ = 40° and ϕ = 45°, which can provide type-I phase matching for the difference-frequency processes of 2010 and 1330 nm. The ZGP crystal is cut with θ = 56.1° and ϕ = 0°, which can provide type-I phase matching for an OPA process with a 2050-nm pump and 5000-nm signal. We measured the nonlinear refraction of the ordinary and extraordinary polarizations for the crystals in the experiments. The results are shown in Fig. 3.

Table 1 Characteristics of the materials and the measured values of n 2 for samples with thicknesses of 1 mm, P is the polarizations, D 0 is the diameters of the beam waists, L R is the corresponding Rayleigh lengths
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Fig. 3
figure 3

Dependences of the normalized transmission on the crystal position with respect to the focus for extraordinary polarization on AGS at 4650 nm (a), ordinary polarization on AGS at 4650 nm (b), extraordinary polarization on ZGP at 2200 nm (c), ordinary polarization on ZGP at 2200 nm (d)

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The nonlinear refractive indices corresponding to the changes in the transmittancescan be calculated by the expression [21]

$${{n}_{2}}=\frac{{{T}_{\text{PV}}}\lambda \sqrt{2}}{0.404{{(1-S)}^{0.25}}2\pi {{I}_{0}}{{L}_{\text{eff}}}}$$
(1)

here, T PV is the normalized difference between the maximum and minimum transmissions in the limiting-aperture scheme; S is the aperture transmission, which is a fraction of the radiation incident on a power meter; L eff is the effective thickness of the sample, which is equal to the sample thickness because the thicknesses of the samples in our experiments are smaller than the Rayleigh lengths of the laser beams (shown in Table 1); is the radiation wavelength; and I 0 is the maximum radiation intensity in the waist plane. The value of I 0 for a Gaussian pulse can be calculated using the expression

$${{I}_{0}}=\frac{4E}{{{\tau }_{p}}\pi {{\omega }_{0}}^{2}}\sqrt{\frac{\ln 2}{\pi }}.$$
(2)

here, E is the energy of the incident laser pulse (considered the reflection of the surface), ω 0 is the beam radius at 1/e2 of the maximum intensity in the focal plane, and p is the duration (FWHM) of the laser pulse. The corresponding nonlinear refractive indices were calculated according to (1, 2), and the values are shown in Table 1 The results show that there is little difference between the values for o-polarization and e-polarization.

The AGS is a direct band gap material at room temperature [22], and the theoretical value of n 2 can be calculated with the simple two-band model. The dispersion of bound-electronic nonlinear refraction for direct semiconductors is [23]:

$${{n}_{2}}(\lambda )\left[ {\text{c}{{\text{m}}^{2}}}/{\text{W}}\; \right]\cong B\frac{\lambda _{g}^{2}}{n_{0}^{2}}{{G}_{2}}\left( {{{\lambda }_{g}}}/{\lambda }\; \right)$$
(3)

where \({{\tilde{G}}_{2}}(x)\) is the normalized dispersion function and B is a constant with an approximate value of 0.65E-12 (in the units of (3) cm2/W) appropriate for wide-gap dielectrics [23, 24]. It was shown that for semiconductors, B = 1.22E-12 gives the best fit with the experimental results [23]. The band edge of AGS is 2.62 eV (g = 473 nm) [25] in our experiments, and x < 0.6, which allows the dispersion function to be approximated as

$${{\tilde{G}}_{2}}(x)=1+2.4{{x}^{2}}+\frac{2.6}{400{{(x-0.535)}^{2}}+1}.$$
(4)

Based on (3, 4), we calculate the theoretical values of n 2 for the two polarizations at 4650 nm in the AGS crystal, and the values are 1.12E-14 and 1.15E-14 cm2/W, respectively. These values are close to our experiment results. However, the ZGP crystal is an indirect band gap material [26], and (3, 4) cannot predict its nonlinear index precisely. We determined the nonlinear refractive index of ZGP at 4700 nm by considering that the ZGP is always used at long wavelength, and the measured value was 5.28E-14 cm2/W. Besides, we also determined the nonlinear refractive index of ZGP at 1300 nm, and the value was 4.09E-14 cm2/W, which was close to 5.37 E-14 cm2/W at 1260 nm in [13], the difference of n 2 may be caused by the distinct cut angles or the durations of the probe light.

It is important to account for a possible increase in the pulse width due to GVD in the samples because of the ultrashort duration of the probe pulses. Pulse broadening alters the peak intensity, which in turn alters the magnitude of the nonlinear phase shift. We evaluate the dispersion distance L D defined as the length over which a transform-limited Gaussian pulse broadens by a factor due to second-order dispersion [27] as follows:

$${{L}_{D}}=\frac{{{\tau }^{2}}}{4\ln 2}{{\left( \frac{{{\lambda }^{3}}}{2\pi {{c}^{2}}}\frac{{{\text{d}}^{2}}n}{\text{d}{{\lambda }^{2}}} \right)}^{-1}}$$
(5)

here, is the input pulse duration (FWHM). We list the calculated dispersion distances for each sample in Table 1. The calculations are based on the Sellmeier equations of each material [28]. The thicknesses of the samples in our experiments are 1 mm, as shown in Table 1. However, the dispersion distances are greater than 1 mm, so pulse broadening due to GVD is negligible.

4 Conclusions

In conclusion, we present accurate measurements of the nonlinear refractive index in two important mid-infrared nonlinear crystals using our femtosecond mid-infrared Z-scan platform. Our probe light is produced by an OPA-DFG system. The measured values of the AGS agree well with the theoretical values based on the two-band model for a direct band gap material. We also confirmed the validity of the measured values for the indirect semiconductor ZGP by comparing the results with those obtained at ~1300 nm in a previous study. Besides, we calculated the dispersion distances according to the experimental conditions by considering the pulse broadening effect for the ultrafast laser, and the results show that broadening due to GVD is negligible in our experiments. We believe these nonlinear measurements of AGS and ZGP are meaningful for the analysis and design of optical systems in the mid-infrared region.