1. Introduction

Chalcogenide glasses based on the chalcogen elements S, Se, and Te are very promising materials for use in fiber optics and integrated optics since they have many unique optical properties and exhibit a good transparency in the infrared region [1–3]. In particular, As-Se and closely related glass systems possess a high third-order Kerr nonlinearity with an ultrafast time response [4–8], a large Raman-gain coefficient [9], and a large Brillouin-gain coefficient [10,11]. These three kinds of third-order nonlinearities can be applied to all-optical switching, optical bistability, regeneration, Raman and Brillouin amplification and lasing, slow light, and so on [3]. The Kerr nonlinearity of As₂Se₃ glass, which is representative of the As-Se glass system, is estimated to be 500-1000 times larger than that of fused silica [4–8]. Especially Ag₂₀As₃₂Se₄₈ glass has a high Kerr nonlinearity greater than 3000 times that of fused silica [8]. However the mechanisms producing the nonlinear refraction and the nonlinear absorption in chalcogenide glasses still are not well understood.

There are generally many physical mechanisms of optical nonlinearities in materials and they depend on the wavelength, pulse duration, and peak power of incident optical pulses. To evaluate the optical nonlinearities with different pump wavelengths and pulse widths brings us useful information in studying the mechanisms of their nonlinearities and in considering applications of their nonlinearities. However, to change the pulse width is not easy especially in high power lasers. Moreover, there is often a lack of information about the nonlinear refraction and nonlinear absorption in the nanosecond regime because shorter (picosecond or femtosecond) pulses are usually used in their characterization. Most recently, we successfully measured the pulse-width dependence of free-carrier nonlinearities in crystalline silicon at 1.06 µm by using pulse compression based on stimulated Brillouin scattering (SBS) and the Z-scan technique [12], where compressed Stokes pulses were used as pump pulses for the conventional Z-scans [13,14]. The proposed method can separate a fast (instantaneous) Kerr nonlinearity and a slow (cumulative) nonlinearity such as a thermal nonlinearity with respect to the pulse width because the magnitude of the cumulative nonlinearity should be proportional to the pulse width.

In this paper we measure the dependence of optical nonlinearities in As₂Se₃ glass on the incident pulse-width in the nanosecond regime using our proposed method [12] and discuss the mechanisms of the optical nonlinearities. The pump wavelength is 1.064 µm and the pulse width is changed from 11.5 ns to 1.6 ns. We also present the experimental results done with 1.052-µm and 60-ps optical pulses separately. The measured results clearly show that both the nonlinear refractive index n _2eff and the nonlinear absorption coefficient β_eff increase linearly with pulse width. The linearity of n _2eff and β_eff against the pulse width comes from a cumulative nonlinearity generated by linear absorption in the weak-absorption region (socalled mid-gap absorption). The nonlinearity is presumably attributed to photostructual changes since it cannot be explained by free-carrier effects and thermal effects.

2. Theoretical background

We have already derived the effective nonlinear refractive index n_2eff and the nonlinear absorption coefficient β_eff for free-carriers generated by linear (one-photon) absorption (OPA) and two-photon absorption (TPA) in the previous paper [12]. We shall derive the effective nonlinear refractive index due to thermal contributions and further formulate a general cumulative nonlinearity, which will be needed for coming discussions.

First, we derive n _2eff when a fast Kerr nonlinearity and a slow thermal nonlinearity exist together. The rate equation for a temperature increase ΔT generated by OPA may be written as

where τ is the dissipation time due to thermal diffusion, ρ is the density, and C is the specific heat. α is the linear absorption of the medium and αI(t) is the heat source. We consider an incident Gaussian pulse with the following optical intensity profile:

where I₀ is the peak intensity and t₀ is the pulse width defined by the half width at e ^-1 of the maximum. Refer to discussion on the free-carrier nonlinearities done in [12], since Eq. (1) is analogous to the rate equation governing the temporal variation of the density of photoinduced free-carriers. Substituting Eq. (2) into Eq. (1) and assuming τ≫t₀, we have the following solution:

where π^1/2t₀I₀ is the fluence (energy density). The change in refractive index due to the temperature increase is given by

where dn/dT is the thermo-optic coefficient of the material. In the closed-aperture Z-scan to determine the nonlinear refractive index [13], the time-averaged nonlinear phase shift ΔΦ₀ of

the beam transmitted through the sample is evaluated on the axis at the focus. The average of on-axis phase shift over the laser pulse shape (2) is given by

where

k₀ is the free-space wave number, L_eff=[1-exp(-αL)]/α is the effective sample length, γ is the Kerr coefficient due to bound electrons, and I₀ is the peak on-axis intensity. The coefficients 1/2^1/2 and 1/2 appearing in the third equation of Eq. (5) are an averaging factor for the instantaneous Kerr nonlinearity and the cumulative nonlinearity, respectively. In the conventional Z-scan experiment where fast nonlinearities and a temporally Gaussian pulse are implicitly assumed, we can define the effective nonlinear refractive index n_2eff as shown in Eq. (6). This agrees with Eq. (20) in [13]. Since the refractive index change Δn_T is proportional to the fluence, the thermal contribution to n_2eff is accordingly proportional to the pulse width t₀. Using the pulse width t _FWHM (=2(ln2)^1/2 t₀) defined by full width at half maximum, Eq. (6) is rewritten as

Thus a plot of n _2eff versus t _FWHM should be a straight line as in the case of plasma effects of freecarriers generated by OPA [12].

Next, we shall extend the formulation for the free-carrier nonlinearity developed in the previous paper [12] to a general cumulative nonlinearity where the refractive index and absorption coefficient are slowly changed with incident intensity through other mechanism. In fact, a very large nonlinearity that can be observed with a cw He-Ne laser has been found in As₂S₃ and As₂Se₃ glasses [15,16], which is presumably attributed to photostructural changes typically found in chalcogenide glasses [17]. We here assume that the mechanism generating the slow nonlinearity is induced by OPA, resulting in an effective third-order nonlinearity. In [12], we can replace the density N of the photoexcited charge-carriers by the density N of the absorbed photons. The phase change Δϕ and optical intensity I of the optical wave per unit propagation distance within the nonlinear material can be expressed as

where

γ is the Kerr coefficient, α is the linear absorption coefficient, β is the TPA coefficient, ħω (=hc/λ) is the photon energy, σ_r is the change in the refractive index per unit of the absorbed photon density, and σ_ab is the change in the absorption coefficient per unit photon density. The mechanism generating a change in the refraction and absorption is included in the two coefficients σ_r and σ_ab. In the case of free-carrier nonlinearities, σ_r and σ_ab are called as the FCD (free-carrier dispersion) coefficient and the FCA (free-carrier absorption) coefficient, respectively, and σ_r <0 and σ_ab >0. The effective nonlinear refractive index and nonlinear absorption coefficient when the Kerr nonlinearity and the cumulative nonlinearity coexist in the sample are given by

By plotting n _2eff and β _eff as a function of t _FWHM, we can deduce σ_r and σ _ab from the slope of the straight lines, and γ and β from the intercept, respectively. If the mechanism generating the slow nonlinearity is induced by TPA, it becomes an effective fifth-order process [12,14]. In this case, αI/ħω in Eq. (10) should be replaced by βI²/(2ħω) and eventually the last term in Eqs. (11) and (12) should be replaced by a term proportional to the fluence as follows [12]:

$\frac{\alpha }{2\hslash \omega }\frac{\sqrt{\pi }{t}_{\mathrm{FWHM}}}{\sqrt{2\ln 2}}\to \frac{\beta }{4\hslash \omega }\frac{\sqrt{\pi }{t}_{\mathrm{FWHM}}{I}_0}{\sqrt{2}\sqrt{2\ln 2}}$

If the incident fluence is kept constant, the TPA’s contribution to n _2eff and β _eff is constant independently of the pulse width.

3. Experiment

The As₂Se₃ chalcogenide glasses used in this work were prepared by a conventional meltquenching method. The mixture of As and Se elements of 99.9999 % purity in an evacuated fused-quartz ample was melted in a rocking furnace at ~1000 °C for 25 h and then was rapidly cooled by being put into water. We obtained the sample used for Z-scan measurements by cutting the glass into a parallel plate and polishing both surfaces of it. The thickness of the sample was 1.3 mm. Although a 1.064-µm pump wavelength is longer than a band gap (Tauc gap) wavelength of 0.705 µm, there is still considerable absorption loss because of weak-absorption tail (mid-gap absorption) [17]. The linear refractive index and absorption coefficient of glasses prepared in this way have already been measured to be 2.808 and 0.549 cm^-1 (2.38 dB/cm) at 1.047 µm using Brewster’s angle [18].

In the Z-scan experiments for determining the nonlinear refractive index and the nonlinear absorption coefficient of the sample, nanosecond and picosecond lasers are used. Since the details of SBS pulse compression of nanosecond pulses and the Z-scan experiments can be found in [12], we shall here describe them only briefly. The pump laser used in this work is an injection-seeded, Q-switched Nd:YAG laser, which provides 400-mJ, 11.5-ns Gaussian pulses at a 1.064-µm wavelength and a 10-Hz repetition rate. The optical pulses from this laser can be compressed by using SBS in heavy fluorocarbon liquids up to 1.6 ns. We utilize the compressed laser pulses to two kinds of Z-scan measurements, i.e. open-aperture and closed-aperture Z-scans for determining the nonlinear absorption coefficient and the nonlinear refractive index of the sample, respectively [13,14]. As another light source, we use a mode-locked and Q-switched YLF laser (Quantronix, 4216), which provides a train of pulses with a 60-ps duration and a 76-MHz repetition rate at 1.053 µm. The Q-switch is valid for single pulse to 400 Hz operation and its pulse width is about 400 ns. In our experiment, 60-ps pulses were extracted from Q-switch envelope with an electro-optic pulse selector at a repetition rate of 400 Hz. The sample is mounted on a translation table and is moved along the z-axis through the focal plane of a 20-cm focal length lens. The beam waist w₀ (defined by the half width at e^-2 of the maximum) at focus is 40 µm for 1.064-µm experiment and 56 µm for 1.053-µm experiment. We use the aperture’s linear transmission S=0.40 for our Z-scan experiments.

4. Results and discussion

Before presenting the detailed experimental results, we show variations of obtained Z-scan data, which give the accuracy of our experimental setup. The sensitivity and accuracy of the Z-scans depend on beam quality and stability of the laser system including the SBS pulse compressor as well as distortions and a tilt of the sample. Moreover the estimation of the beam waist w₀ is very important to calculate the peak on-axis intensity I₀ accurately. Figure 1 shows typical closed- and open-aperture Z-scan traces for an incident energy of 15 µJ and pulse widths of 11.5 ns and 1.6 ns. The closed-aperture data shown in the figures are ones which were already divided by the open-aperture data. Theoretical fits using w₀=40 µm are also shown in the figure, which are based on Eqs. (12) and (15) in [12]. We have the same degree of accuracy as one attained in the previous Z-scan experiment of crystalline silicon [12]. The only difference is the sign of the nonlinear refractive index. The closed-aperture Zscan traces indicate that the sign of n _2eff of As₂Se₃ glass is positive at the pulse width t _FWHM=11.5 s and 1.6 ns.

 

Fig. 1. Typical normalized open- and closed-aperture Z-scan data of As₂Se₃ glass at λ=1.064 µm with an incident energy of 15 µJ. The incident pulse-width is (a) t _FWHM=11.5 ns and (b) t _FWHM=1.6 ns. The solid lines are the best fits based on Eqs. (15) and (12) in [12].

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First, we examine the dependence of the effective nonlinear coefficients n _2eff and β _eff on the incident energy. Figures 2 and 3 show the experimental results for the pulse width t _FWHM=11.5 ns and 1.6 ns at 1.064 µm, respectively. The horizontal axis is shown in terms of the incident energy per pulse. An incident pulse energy of 10 µJ leads to the on-axis peak intensity (inside the sample) I₀=26.7 MW/cm² for t _FWHM=11.5 ns and I₀=192 MW/cm² for t _FWHM=1.6 ns. Note that the peak optical intensity at t _FWHM=1.6 ns is always about 7 times as much as one at t _FWHM=11.5 ns for a given value of the incident energy. In the case of 10-µJ illumination, the total number of photons absorbed in the interval from t=-∞ to t=0 is N(0)=4.8×10¹⁷ cm^-3. In the low energy region where data are not given in the figures, we could not obtain reliable results because of small Z-scan signals. It should be noted that, in the closed-aperture Z-scan, ΔΦ₀<π in the incident energy range shown in Figs. 2 and 3. At the low pulse energy <~10 µJ at t _FWHM=11.5 ns and < ~5 µJ at t _FWHM=1.6 ns, the values of n _2eff and β _eff increase drastically. In the case of t _FWHM=11.5 ns, the measured values of n _2eff and β _eff are approximately constant for the pulse energy >~10 µJ. On the other hand, in the case of t _FWHM=1.6 ns, the values of n _2eff and β _eff seem to decrease only slightly with incident energy withinthe measurement range. It is found that the values of n _2eff and β _eff at t _FWHM=11.5 ns are approximately 3 and 4 times as much as those at t _FWHM=1.6 ns, respectively. This indicates that a cumulative nonlinearity is present in the sample.

 

Fig. 2. Dependence of (a) the effective nonlinear refractive index n _2eff and (b) the effective nonlinear absorption coefficient β _eff on the incident pulse energy for the pulse width t _FWHM=11.5 ns at 1.064 µm. A pulse energy of 10 µJ corresponds to the on-axis peak intensity (inside the sample) I₀=26.7 MW/cm².

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Fig. 3. Dependence of (a) the effective nonlinear refractive index n _2eff and (b) the effective nonlinear absorption coefficient β _eff on the incident pulse energy for the pulse width t _FWHM=1.6 ns at 1.064 µm. A pulse energy of 10 µJ corresponds to the on-axis peak intensity (inside the sample) I₀=192 MW/cm².

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For comparison, we also present the incident-intensity dependence of n _2eff and β _eff for picoseconds pulses. Figure 4 shows the experimental results for a 60-ps pulse width at 1.052 µm. The results at t _FWHM=300 ps in the previous work [8] are also reproduced in the figure. The horizontal axis is shown in terms of the optical intensity since the cumulative nonlinearity can be expected to diminish sufficiently. In the case of t _FWHM=60 ps, the optical intensity I₀=1.0 GW/cm² corresponds to an incident pulse energy of 3.80 µJ. On the other hand, in the case of t _FWHM=300 ps, the optical intensity I₀=1.0 GW/cm² corresponds to an incident pulse energy of 4.75 µJ, where it should be noted that the beam waist w₀=28 µm was used in [8]. In the low intensity region less than ~0.4 GW/cm², the n _2eff value rapidly increases with decreasing incident intensity. In the case of t _FWHM=300 ps, we have obtained n _2eff=4.03×10^-16 m²/W at the incident intensity I₀=0.074 GW/cm², but it is not shown in the figure. Such strong intensity dependence has been observed in the nanosecond regime as shown in Figs. 2 and 3 and in the picosecond regime in [4,7,8]. Moreover similar dependence has also been observed in As₂S₃ glass [4] and Ag₂₀As₃₂Se₄₈ glass [8]. Especially Ag₂₀As₃₂Se₄₈ glass has a large value of n _2eff=8.09×10^-16 m²/W, which is 27,000 times as large as that of fused silica (γ=3.0×10^-20 m²/W). As for n _2eff, the experimental results at t _FWHM=60 ps agree rather well with those at t _FWHM=300 ps and the value of n _2eff is estimated to be ~3.0×10^-17 m²/W, which is 1000 times larger than that of fused silica. On the other hand, the value of β _eff still reveals the pulse-width dependence. The reason why the β _eff value depends on the pulse width even at 300 ps compared to the case of n _2eff is that the relative magnitude of β, i.e. β/β _eff is smaller than that (γ/n _2eff) of γ at that pulse width (see Fig. 5). If the value of β decreases, the β _eff will depend on the pulse width to a lower value. The measured β _eff value at 60 ps is four times smaller than that at 300 ps, i.e. the β _eff value is ~5.0×10^-11 m/W at 60 ps and ~2.0×10^-10 m/W at 300 ps. Although the figure of merit F=2β _eff λ/n _2eff should be smaller than 1 for efficient operation of all-optical devices, we have F=3.5 for the glass used in this work at 60 ps. It is also confirmed that two values of n _2eff and β _eff are smaller than those at t _FWHM=1.6 ns as expected.

 

Fig. 4. Dependence of (a) the effective nonlinear refractive index n _2eff and (b) the effective nonlinear absorption coefficient β _eff on the incident optical intensity for the pulse width t _FWHM=60 ps at 1.052 µm. The results at t _FWHM=300 ps in [8] are also reproduced.

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Next, we examine the pulse-width dependence of n _2eff and β _eff. Figure 5 shows the experimental results when an incident energy per pulse was fixed at 15 µJ for the pulse width ranging from 11.5 ns to 1.6 ns. Our experimental data for a 60-ps pulse width at 1.052 µm are also shown in the figure. Although the data at 60 ps were obtained by using a pulse energy of about 4 µJ, the difference in incident pulse energy leads to no serious problem because it can be expected that the contribution of the cumulative nonlinearity is sufficiently small at 60 ps. If we use a 16-µJ pulse energy, higher-order nonlinearities will be generated and their contribution cannot be neglected [7]. It is found that the measured values of n _2eff and β _eff vary linearly with pulse width t _FWHM. This really indicates that a cumulative nonlinearity induced by OPA is present in As₂Se₃ glass. Although the cumulative nonlinearity can be increased by increasing the pulse width, the changes in the refractive index and the absorption coefficient never vary in proportion to the temporal variation of the incident pulse. It should be noted that the nonlinearity that can respond to the temporal change of the incident pulse is only the Kerr effect with an ultrafast response time. The cumulative nonlinearity may be detrimental to the operation of all-optical devices, for example, the normal operation may be spoiled if the operating point of devices is shifted by it. The two straight lines presented in the figure are the best fit to the experimental data of n _2eff and β _eff. From the slope of the fitted straight lines, we have σ_r=1.0×10^-22 cm³ and σ _ab=4.7×10^-18 cm², which are smaller than those of crystalline silicon at 1.064 µm (σ_r=-1.0×10^-21 cm³ and σ_ab=8.4×10^-18 cm²). Moreover, from the intercept (t _FWHM=0) of the straight line, we have the Kerr coefficient γ=3.0×10^-17 m²/W and the TPA coefficient β=5.0×10^-11m/W, which are of course the same values as those obtained at t _FWHM=60 ps.

In connection with the pulse-width dependence of n _2eff and β _eff, it is interesting to discuss one at lower incident pulse energy (or optical intensity) where the values of n _2eff and β _eff are enhanced. Although we examined their incident-intensity dependence at the pulse width t _FWHM=60 ps, 300 ps, 1.6 ns, and 11.5 ns, the observation of the enhanced nonlinearity became difficult as the pulse width was increased. In the examined pulse-width range, the attainable value of n _2eff never exhibits strong pulse-width dependence and it is approximately constant (3×10^-16 m²/W-5×10^-16 m²/W). The incident-intensity dependence of n _2eff at 300 ps agrees well with one at 60 ps as shown in Fig. 4. From these facts, it can be expected that the enhanced nonlinearity is not cumulative and can respond to the temporal change of the incident pulse. Although such an enhanced nonlinearity has no known cause, we can expect that it can be utilized for device applications and its response time is much shorter than 60 ps.

 

Fig. 5. Dependence of (a) the effective nonlinear refractive index n _2eff and (b) the effective nonlinear absorption coefficient β _eff on the incident pulse-width t _FWHM. The data for t _FWHM=1.6 -11.5 ns were obtained at an incident pulse energy of 15 µJ. The data at 60 ps are shown with a white circle.

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Finally, we must discuss the origin of the cumulative nonlinearity. The first possibility is the plasma effect of free-carriers generated by OPA, which is the origin for crystalline silicon at 1.064 µm [12]. However this should be just rejected because the nonlinearity is self-defocusing, i.e. the sign of the nonlinear refractive index is minus. Moreover it has already been known that the quantum efficiency of photoconductivity related to the photoinduced charge-carriers is very low around 1 µm [19]. The second possibility is the thermal effect that was described in Section 2. To examine the possibility, we shall calculate the magnitude of the thermal contribution n _2eff-γ of Eq. (7) and compare it with the measured value. As an example, let us calculate the magnitude of n _2eff-γ at t _FWHM=11.5 ns. With the known values of α=0.549 cm^-1 [18], C=0.27 J/gK [20], ρ=4.64 gcm^-3, and dn/dT=3.5×10^-5 K^-1 [21] for As₂Se₃ glass, we have n _2eff-γ=7.8×10^-18 m²/W, which is about 1/30 of the measured value. The positive thermo-optic coefficient means that the absorption band edge will be red-shifted with temperature, resulting in an increase in absorption. Although the sign of both the nonlinear refraction and the nonlinear absorption due to the thermal effect is consistent with the present observation, we cannot explain the linearity of n _2eff against t _FWHM because of the large discrepancy in the magnitude of n _2eff-γ between prediction and experiment. A plausible origin is reversible photostructural changes accompanying photodarkening [17,19,22]. The positive sign of both the observed nonlinear refraction and nonlinear absorption is consistent with normal photodarkening, where the absorption increases as a result of a red-shift of optical absorption edge and hence the refractive index increases [23]. However, this normal photodarkening is not the origin because it is essentially induced by the interband absorption [19]. In our case, the sample was exposed to light with energy less than the bandgap where the absorption coefficient is α=0.55 cm^-1. In fact, Belykh et al. [24] have carried out an experiment of time-resolved transmission through As₂S₃ glass under conditions similar to our experiment (wavelength:1.064 µm, pulse width:10 ns, and irradiance:~100 MW/cm²), where nonlinear absorption with a very slow response time has been clearly observed. We believe that the observed linear-dependence of β _eff on the pulse width and the photoinduced transmission loss found in [24] are the same in essence although the examined substances are different. Belykh et al. have concluded that the mechanism of the photoinduced loss is connected with the nonlinear excitation of interband transition in the glass, which can be done by TPA or two-step absorption. However their conclusion is inconsistent with our experimental results. If the cumulative nonlinearity is induced by nonlinear absorption such as TPA and two-step absorption, the values of n _2eff and β _eff should be constant for a given incident fluence independently of the pulse width, as explained at the end of Section 2. The linearity of n _2eff and β 2 _eff against tFWHM is reliable evidence that the observed cumulative nonlinearity is induced by OPA. Therefore the present photostructural changes might be associated with defects which are responsible for weak-absorption tail. There are several possible sources for defects, for example, dangling bonds or valence alternation pairs [19] and wrong bonds, As-As and Se(S)-Se(S) [25]. The weak-absorption tail seems to be caused by wrong bonds because the density of wrong bonds is generally much greater than that of dangling bonds. It has also been found that intense mid-gap excitation increases the density of wrong bonds and further increases the refractive index via TPA, unaccompanied by photodarkening [26]. If our assumption is correct, wrong bonds are also created by OPA. In either case, we cannot understand how the wrong bonds relate to the observed slow nonlinearity at the present stage.

In connection with the cumulative nonlinearity, we must also discuss the origin of the enhanced nonlinearity in the low optical intensity region. The important points are as follows: the attainable value of n _2eff is approximately constant independently of the pulse width. The measured value of n _2eff decreases with incident intensity gradually. These points seem to reflect that the density of the sources (for example, defects such as dangling bonds) creating the nonlinearity is low compared with the density of the absorbed photons and its excitation is saturated for intense illumination.

5. Conclusion

We have investigated the pulse-width dependence of the effective nonlinear refractive index n _2eff and the effective nonlinear absorption coefficient β _eff in As₂Se₃ glass at 1.064 µm using the Z-scan method. In this experiment, the duration of incident pulses was changed from 11.5 ns to 1.6 ns by the pulse compression using stimulated Brillouin scattering in liquid Fluorinert. Moreover the experimental data for 60 ps at 1.052 µm have been added to those results. We measured the dependence of n _2eff and β _eff on the incident pulse energy (or optical intensity) at durations of 11.5 ns, 1.6 ns, and 60 ps. Moreover we measured the pulse-width dependence of n _2eff and β _eff at an incident pulse energy of 15 µJ. It has been found that the measured values of n _2eff and β _eff increase linearly with pulse width, which means that a cumulative nonlinearity is present in the glass. Our theoretical model developed in this paper shows that the cumulate nonlinearity is induced by OPA in the weak-absorption region. Moreover we determined the Kerr coefficient γ, the TPA coefficient β, and the changes in the refractive index and absorption coefficient per unit absorbed photon density, σ_r and σ _ab. The mechanism of the cumulative nonlinearity is presumably attributed to photostructural changes inherent in chalcogenide glasses. Moreover the origin of the enhanced nonlinearity found in the low incident intensity region was discussed.