Non-linear optical response of bulk chalcogenide glasses near the fundamental absorption band edge

Magnitudes of the non-linear coefficients of absorption and refraction have been evaluated near the bandgap wavelengths of chalcogenide glasses of the system As-S-Se by using the interferometric pump-probe method and are compared with literature data. Photoexcited plasma dynamics and long-time scale variation of the dielectric constant have been studied by comparison with the results of numerical modelling of the behaviour of the glasses when heated by the ultra-short laser pulses.


Introduction
As other amorphous semiconductors, chalcogenide glasses have a disordered lattice and localised states of electrons in their bandgaps. Due to these specific structural and electronic properties, fundamental absorption band edge of amorphous semiconductors is not sharp. It has a spectral range of exponential decay of the single-photon absorption coefficient α that is usually referred to as the Urbach tail. Chalcogenide glasses have also weak absorption tails with α < 1 cm -1 [1].
Using of chalcogenide glasses, as highly non-linear materials transparent in the mid-infrared, can enhance non-linear optical effects and reduce the dimensions of integrated and fibre optical devices. However a theory of the non-linear optical response of chalcogenide glasses has not yet been fully developed. For direct-gap crystalline semiconductors, the values of non-linear refractive index n 2 in the low-frequency limit have been obtained by using the bond-orbital approximation [2]. The nonlinear Kramers-Kronig relation has been applied to obtain spectral dependencies of n 2 and of the twophoton absorption coefficient β 2 near the bandgap frequencies of the direct-gap and indirect-gap crystalline semiconductors in [3] and [4], respectively, by using two-band models. For amorphous semiconductors, only some experimental results of measurements of n 2 and β 2 are currently available. In particular, for chalcogenide glasses, these parameters have been measured mostly in the spectral range above their two-photon bandgap wavelengths [5][6][7][8].
In this paper, we study the non-linear optical response of chalcogenide glasses of the compositions As 40 Se 60-x S x (x = 0,10,15,20,30,40,45,60) (atomic %) by using the time-resolved interferometric pump-probe method [9]: evaluate n 2 and β 2 near their bandgap wavelengths and characterise plasma dynamics at the Urbach tail. Different compositions fit the pump pulse peak wavelength of 790 nm (the photon energy is shown in figure 1 as a dashed line) at different points of their Urbach tails. 1 To whom any correspondence should be addressed.

Non-linear optical response at low energies of the pump pulse
In the experiment, a glass sample (hot pressed between two precision aligned tungsten carbide plates with flatness of 0.08 µm and a surface finish of 0.009 µm at ~25 ˚C above its glass transition temperature in an in-house-built vacuum rig) was probed with two collinear 50 fs pulses separated by a fixed time delay. When a high intensity pump pulse was focused on the sample (by a lens with a focal distance of 300 mm after a 3 mm aperture) at some time between the two probe pulses, the second probe pulse experienced a disturbed dielectric constant. The induced phase shift φ and absorbance were measured by a spectrometer as a function of time by changing the delay Δt between the pump pulse and the second probe pulse. At relatively low pump energies E, the phase shift was induced by non-linear refraction and two-photon absorption (figure 2a) or single-and two-photon absorption (figure 2b) of the pump pulse. The effect of cross-phase modulation is described by the system of differential equations for the pump pulse intensity I pu and the probe pulse intensity I p :  The curves in figure 3b were obtained by fitting the values of n 2 calculated in the low-frequency limit [2]. Each curve ends at the bandgap wavelength λ g of a particular composition. In some range of wavelengths near λ g , these curves exhibit negative values of n 2 . In our measurements, only positivelyvalued n 2 have been obtained for all the glass compositions. In estimations of n 2 and β 2 , maximum magnitudes of the measured phase shift and absorbance at the probe beam axis have been used.

Non-linear optical response at high energies of the pump pulse
At higher energies, the effect of plasma formation due to the single-and two-photon absorption is to be taken into account. In [10], three basic scenarios of the plasma dynamics have been distinguished depending on the ratio P=hν/E g of the irradiating photon energy to the bandgap energy. In figure 4, the positive-valued maximum corresponding to the self-induced refraction of the pump pulse is followed by the fast phase shift decrease in As 40 S 60 sample (P=0.65) due to plasma formation and a subsequent slow phase shift increase due to the recombination of carriers. The effect of plasma formation is weakening when partially replacing S by Se. For the As 40 S 20 Se 40 sample, P=0.74 and for the As 40 S 15 Se 45 sample, P=0.77. In the latter sample, there were no free electrons after the pump pulse.
Permanent positive variation Δn p of the refractive index observed at the long-time scale and associated with photodarkening or the formation of excitons, has been compared with the results of numerical simulations of the glass heating [11]. A correlation with a sample surface heating above the glass transition temperature (T g ) and energy threshold of Δn p observation has been revealed (figure 5).

Conclusions
Study of the non-linear optical response of chalcogenide glass samples of the system As-S-Se has demonstrated that all the samples have the positive-valued n 2 unlike the direct-gap crystalline semiconductors having the negative-valued n 2 near their bandgap wavelengths. By comparison with the available literature data we have shown that the spectral curve obtained in [3] for the direct-gap crystalline semiconductors, agrees with the spectral dependence of the non-linear coefficients of chalcogenide glasses at wavelengths more than the two-photon bandgap wavelength.
A scenario of the photo-excited plasma dynamics depends on the ratio of the photon energy to the bandgap energy. If P > 0.75, fast trapping of carriers results in the lack of free electrons after the pump pulse. By numerical simulation of the glass sample heating in the irradiated zone, we have found a correlation between the permanent change of the dielectric constant at the long-time scale and glass heating above the glass transition temperature.