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Dispersion of Linear and Nonlinear Refractive Index in Chalcogenide Glass Yulia Kuzutkina, Alexander Melnikov, Elena Romanova, Viacheslav Kochubey, Nabil S. Abdel-Moneim*, David Furniss*, Angela Seddon* Saratov State University, Astrakhanskaya 83, 410012 Saratov, Russia * University of Nottingham, University Park, NG7 2RD Nottingham, UK Tel: (7845) 210728, Fax: (7845) 278529, e-mail:
[email protected] ABSTRACT Fundamental mechanisms of the intensity-dependent refractivity in optical glasses have been less understood than those in two-level atoms and crystalline semiconductors. In the work, the concept based on the strong dependence of nonlinear refractive index on the bandgap energy is developed. Spectral dependence of the nonlinear refractive index near the bandgap frequency in chalcogenide glasses is considered and compared with direct- and indirect gap crystalline semiconductors. Keywords: optical glass, nonlinear optics, dispersion of refractive index. 1. INTRODUCTION Values of linear and nonlinear refractive index (Kerr coefficient) in optical glass are basic parameters, in addition to linear and nonlinear losses, in design of non-linear optical devices. These parameters are extremely important when dealing with propagation of the ultra-short laser pulses. They are also responsible for some photo-induced phenomena. Among different optical glasses, chalcogenide glasses exhibit the highest third-order nonlinear response. In general, they have Kerr coefficient (n2) 2-3 orders of magnitude greater than that of fused silica. Wide range of transparency in infrared, high values of refractive index, good mechanical properties, high chemical and thermal stability are other features, which enabled prospective applications of chalcogenide glasses. Chalcogenide glasses are the best candidates for all-optical signal processing at above 40 Gb/s rates. Fundamental mechanisms of the intensity-dependent refractivity in glasses have been less understood than those in simple systems such as two-level atoms and crystalline semiconductors. A concept based on using the Kramers–Kronig relation was developed in [1] to derive dispersion of n2 in direct-gap crystalline semiconductors. Dispersion curves for the third-order nonlinear optical coefficients in indirect-gap crystalline semiconductors were obtained in [2]. The nonlinear refractive index was shown to be positive-valued below the indirect gap frequency and the spectral variation of the nonlinear optical coefficients was greatly reduced here in comparison with direct semiconductors (Fig.1). Similar spectral dependencies can be expected for amorphous semiconductors [3]. Static disorders in amorphous semiconductors and phonons in indirect-gap semiconductors appear to exert similar influences upon photo-electronic excitation. In crystalline semiconductors, nonlinear refraction and absorption coefficients are strongly dependent on the bandgap energy Eg. Far from the bandgap frequency in direct semiconductors n2~1/Eg4 , ȕ2~1/Eg3. Linear coefficients (Į and n0) do not depend so strongly on Eg: n0~1/Eg1/4. It has been shown in [3] that similarly Eg is a decisive parameter in determining nonlinear optical properties of amorphous semiconductors.
Figure.1. Coefficients of linear (Į) and nonlinear (ȕ) absorption, linear (n0) and nonlinear (n2) refraction in direct-gap (solid) and indirect-gap (dashed) crystalline semiconductors versus the photon energy [3].
978-1-4673-2229-4/12/$31.00 ©2012 IEEE
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The aim of this work is to study dispersion of linear and nonlinear refractive index of AsíSíSe glass system. The experimental technique of measurement of the absorption spectra and dispersion of linear refractive index in the glass samples is validated via comparison with available dispersion curves for As2Se3 and As2S3 glass compositions. Evaluation of the nonlinear refractive index is based on the experimental results, as well as on the literature data and theoretical approaches discussed above. 2. DISPERSION OF LINEAR REFRACTIVE INDEX In experiment, spectra of transmittance and reflectance were measured by a spectrophotometer in the range of wavelengths 0.4 – 2.0 ȝm for chalcogenide glass samples shaped as thin plates. For evaluation of linear refraction and absorption coefficients, the two-dimensional model of plane monochromatic wave passing through a thin glass plate was used [4]. With account of multiple reflections at the sample interfaces, transmittance (T) and reflectance (R) are linked with the absorbance D = Įd of the sample with thickness d as follows: °T = (1 − R12 )2 exp(− D) / (1 − R122 exp(−2 D) ) ® R = R12 (1 + T exp(− D) ) °¯
(1)
Values of Į and n can be found by solution of the equations (1):
α = − ln ¨ ¨ T 2 − (1 − R ) + 4T 2 + (1 − R + 2 R 2 − T 2 ) ¸ / ( 2T ) ¸ / d §§
2
2
©©
(
)(
·
· ¹
n = 1 + R12 / 1 − R12
(2)
¹
)
(3)
where 2 · § R12 = ¨ T 2 + 1 + 2 R − R 2 − 4 ( R − 2 ) R + (1 + 2 R − R 2 + T 2 ) ¸ / ( 2 − R ) / 2 © ¹
(4)
The obtained values of linear refractive index were compared with the ones given by the Sellmeier formulas [5,6] for the compositions As2Se3 and As2S3 (Fig. 2a). Absorbance of the samples was compared with the results of measurements obtained for the different samples of the same compositions at the FTIR spectrometer (Fig. 2b). In the range of transparency, the obtained results of linear refractive index evaluation were in good agreement with the dispersion curves obtained by using Sellmeier formulas (Fig. 2a, dashed lines). These formulas are not feasible near the edge of absorption band where the major difference in the results shown in Fig.2 a is observed. In measurements of the absorption spectra, two different series of glass samples of the same compositions were used in measurements by spectrophotometer and by FTIR spectrometer. That was a reason of difference near the edge of absorption band in the absorption spectra shown in Fig. 2b. Dispersion curves of the linear refractive index of some other glass samples of the system As-S-Se are shown in Fig. 3a. The refractive index is increasing and the bandgap energy is decreasing (Fig. 3b) with substituting some amount of sulphur by selenium. 3.6 3.4
15
α , 1/cm
n
3.2
As2Se3
3.0 2.8 2.6
10
As2Se3
5
As2S3
As2S3
2.4 0
0.6
0.8
1.0
1.2
1.4
0.6
λ, µm
0.8
1.0
1.2
1.4
λ, µm
(a) (b) Figure 2. a) Spectral dependence of linear refractive index: measured (open circles) and calculated by Sellmeier formulas (solid lines); b) Absorption spectra measured by spectrophotometer (solid lines) and by FTIR spectrometer (dashed lines)
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1/2
3.6
-1/2
3.2
As40Se45S15
3.0
10
1/2
n
15
(αhν) , cm eV
3.4
2.8
As40Se15S45
2.6
5
As40Se45S15
As40Se15S45
2.4 0
0.6
0.8
1.0
1.2
1.4
1.4
1.6
1.8
2.0
hν, eV
λ, µm (a)
(b)
Figure 3. a) Spectral dependence of linear refractive index; b) Evaluation of the bandgap energy by Tauc method.
3. DISPERSION OF NONLINEAR REFRACTIVE INDEX It has been demonstrated in [3] that the optical gap energy Eg is a decisive parameter that determines nonlinear optical properties in optical glasses. In the low-frequency limit (Ȧ ĺ 0), value of Kerr coefficient can be evaluated by using this formula:
n20 = 1.7 × 10−14 (n 2 + 2)3 (n 2 − 1)(db / nES ) 2
(5)
(here n is the linear refractive index, db is the mean cation-anion bond length, ES is Sellmeier gap energy, ES § Eg, Eg is optical bandgap energy). Highly non-linear chalcogenide glasses have high density, large number of valence electrons, a tendency towards covalency and small Sellmeier gaps. Then loosely bound electrons can easy vibrate around their nuclei providing a strong linear and non-linear response when an electromagnetic field is applied to the glass. Near the bandgap frequency, nonlinear properties of amorphous semiconductors have not been yet well studied. In Fig.4, some results of measurements available in literature for As2Se3 and As2S3 glass compositions are shown together with the values of n2 given by (5) in the low-frequency limit (Ȧ ĺ 0). Dispersion curves are shown as an approximation of the sets of n2 values for each composition. The literature data for n2 values are positive near the bandgap frequencies. Positive-valued n2 were also obtained in experiments with chalcogenide glass samples in [7].
Ȝ, ȝm Figure 4. Experimental values of Kerr coefficient and approximating curves. In the spectral range Ȝ 2Ȝg (Ȝg is the bandgap wavelength), the model of anharmonic oscillator [8] can be used to evaluate the values of n2:
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n2 (λ ) = n20 / (1 −
λ 3.5 ± 0.5 ) 2λg
(6)
The results of calculation of the linear and nonlinear refractive index are summarized in Table 1. Table 1. Glass composition As2S3 As40 Se15S45 As40 Se45S15 As2Se3
Bandgap energy Eg (eV) 2.3 1.82 1.65 1.6
Linear refractive index n, Ȧ ĺ 0 2.4 2.6 2.8 2.9
Nonlinear refractive index n20·10-14 (cm2/W), Ȧ ĺ 0 1.24 2.9 5.2 6.6
Nonlinear refractive index n2·10-14 (cm2/W), Ȝ=2Ȝg 4.6 10.9 19.1 24.4
4. CONCLUSIONS Dispersion of Kerr coefficient in chalcogenide glasses of the system As-S-Se has been studied near the bandgap wavelength (Ȝg < Ȝ 2Ȝg). In the spectral range, Kerr coefficients are positive-valued that is similar to the indirect-gap crystalline semiconductors. Far from the bandgap wavelength (Ȝ > 2Ȝg), dispersion of n2 is well described by the model of anharmonic oscillator. Maximum of a dispersion curve is located near the two-photon bandgap Ȝ = 2Ȝg that is more similar to the direct-gap crystalline semiconductors. In summary, the similarity between nonlinear spectral features of amorphous semiconductors and indirect-gap crystalline semiconductors assumed in [3] reveals near the bandgap wavelength. However the shapes of dispersion curves are different. Far from the bandgap, the shapes of dispersion curves fit the features of direct-gap crystalline semiconductors. REFERENCES [1] M. Sheik-Bahae, et al.: Dispersion of bound electronic nonlinear refraction in solids, IEEE J. of Quantum Electron., vol. 27, 6, pp. 1296-1309, 1991. [2] M. Dinu, Dispersion of phonon-assisted nonresonant third-order nonlinearities, IEEE J. of Quantum Electron., vol. 39, 11, pp 1498-1503, 2003. [3] K. Tanaka: Nonlinear optics in glasses: How can we analyze?, J. of Physics and Chemistry of Solids, vol. 68, pp. 896-900, 2007. [4] J. Tauc, in Amorphous and Liquid Semiconductors, J. Tauc, ed. (Plenum, London, 1974), pp. 171-206. [5] W. S. Rodney, I. H. Malitson, T.A. King , JOSA, vol. 48, p. 633.,1958. [6] W. Burckhardt, Z. Wiss, Friendrich_Schuller – Univ. Jena. Math. – Naturwiss. R., vol. 32, 2–3, p. 205, 1983. [7] E. Romanova, et al.: Experimental studies of non-linear properties of chalcogenide glasses, in Proc. ICTON 2009, Azores, Portugal, June 28 - July 2, 2009, We.B5.1. [8] M.E. Lines: Oxide glasses for fast photonic switching: A comparative study, J. of Applied Phys., vol. 69, pp. 6876-6884, 1991.
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