Systematic z-scan measurements of the third order ... - OSA Publishing

Systematic z-scan measurements of the third order nonlinearity of chalcogenide glasses Ting Wang,1 Xin Gai,1* Wenhou Wei,1 Rongping Wang,1 Zhiyong Yang,1 Xiang Shen,2 Steve Madden,1 and Barry Luther-Davies1 1

Centre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS), Laser Physics Centre, Research School of Physics and Engineering, Australian National University, Canberra 0200, Australia 2 Laboratory of Infrared Material and Devices, the Advanced Technology Research Institute, Ningbo University, Ningbo 315211, China *[email protected]

Abstract: We report measurements of the third order optical nonlinearity of 51 chalcogenide glasses in the near infrared. Substituting more polarizable elements (Se for S, Sb for As) into the glasses increased their nonlinearity but also reduced the optical bandgap increasing two-photon absorption. Overall the measured values are an extremely good fit to the semi-empirical Miller’s rule whilst the normalized real and imaginary parts are in satisfactory agreement with the scaling for indirect gap semiconductors reported by Dinu. At 1550nm we find that there is an upper limit to the nonlinearity of ≈10−13cm2/W above which two-photon absorption becomes significant. © 2014 Optical Society of America OCIS codes: (160.2750) Glass and other amorphous materials; (190.4400) Nonlinear optics, materials.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

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Received 21 Mar 2014; revised 5 Apr 2014; accepted 9 Apr 2014; published 22 Apr 2014

1 May 2014 | Vol. 4, No. 5 | DOI:10.1364/OME.4.001011 | OPTICAL MATERIALS EXPRESS 1011

14. M. D. Pelusi, F. Luan, D. Y. Choi, S. J. Madden, D. A. P. Bulla, B. Luther-Davies, and B. J. Eggleton, “Optical phase conjugation by an As(2)S(3) glass planar waveguide for dispersion-free transmission of WDM-DPSK signals over fiber,” Opt. Express 18(25), 26686–26694 (2010). 15. T. D. Vo, H. Hu, M. Galili, E. Palushani, J. Xu, L. K. Oxenløwe, S. J. Madden, D. Y. Choi, D. A. P. Bulla, M. D. Pelusi, J. Schröder, B. Luther-Davies, and B. J. Eggleton, “Photonic chip based transmitter optimization and receiver demultiplexing of a 1.28 Tbit/s OTDM signal,” Opt. Express 18(16), 17252–17261 (2010). 16. X. Gai, D. Y. Choi, S. Madden, and B. Luther-Davies, “Interplay between Raman scattering and four-wave mixing in As2S3 chalcogenide glass waveguides,” J. Opt. Soc. Am. B 28(11), 2777–2784 (2011). 17. M. R. E. Lamont, B. Luther-Davies, D. Y. Choi, S. Madden, X. Gai, and B. J. Eggleton, “Net-gain from a parametric amplifier on a chalcogenide optical chip,” Opt. Express 16(25), 20374–20381 (2008). 18. X. Q. Su, R. P. Wang, B. Luther-Davies, and L. Wang, “The dependence of photosensitivity on composition for thin films of Ge (x) As (y) Se1-x-y chalcogenide glasses,” Appl. Phys. A Mater. 113(3), 575–581 (2013). 19. G. Yang, H. S. Jain, A. T. Ganjoo, D. H. Zhao, Y. S. Xu, H. D. Zeng, and G. R. Chen, “A photo-stable chalcogenide glass,” Opt. Express 16(14), 10565–10571 (2008). 20. D. A. P. Bulla, R. P. Wang, A. Prasad, A. V. Rode, S. J. Madden, and B. Luther-Davies, “On the properties and stability of thermally evaporated Ge-As-Se thin films,” Appl. Phys. A Mater. 96(3), 615–625 (2009). 21. W. H. Wei, R. P. Wang, X. Shen, L. Fang, and B. Luther-Davies, “Correlation between Structural and Physical Properties in Ge-Sb-Se Glasses,” J. Phys. Chem. C 117(32), 16571–16576 (2013). 22. R. P. Wang, A. Smith, B. Luther-Davies, H. Kokkonen, and I. Jackson, “Observation of two elastic thresholds in GexAsySe1-x-y glasses,” J. Appl. Phys. 105, 056109 (2009). 23. P. Němec, S. Zhang, V. Nazabal, K. Fedus, G. Boudebs, A. Moreac, M. Cathelinaud, and X. H. Zhang, “Photostability of pulsed laser deposited GexAsySe100-x-y amorphous thin films,” Opt. Express 18(22), 22944–22957 (2010). 24. M. Sheik Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Vanstryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990). 25. J. S. Sanghera, L. B. Shaw, P. Pureza, V. Q. Nguyen, D. Gibson, L. Busse, I. D. Aggarwal, C. M. Florea, and F. H. Kung, “Nonlinear properties of chalcogenide glass fibers,” Int. J. Appl. Glass Sci. 1(3), 296–308 (2010). 26. H. C. Nguyen, K. Finsterbusch, D. J. Moss, and B. J. Eggleton, “Dispersion in nonlinear figure of merit of As2Se3 chalcogenide fibre,” Electron. Lett. 42(10), 571–572 (2006). 27. X. Gai, B. Luther-Davies, and T. P. White, “Photonic crystal nanocavities fabricated from chalcogenide glass fully embedded in an index-matched cladding with a high Q-factor (>750,000),” Opt. Express 20(14), 15503– 15515 (2012). 28. T. Wang, O. Gulbiten, R. P. Wang, Z. Y. Yang, A. Smith, B. Luther-Davies, and P. Lucas, “Relative contribution of stoichiometry and mean coordination to the fragility of Ge-As-Se glass forming liquids,” J. Phys. Chem. B 118(5), 1436–1442 (2014). 29. M. A. Popescu, Non-Crystalline Chalcogenide (Kluwer Academic, 2001). 30. W. Boyd, Nonlinear Optics (Academic Press Inc., 2003). 31. C. C. Wang, “Empirical relation between the linear and the third-order nonlinear optical susceptibilities,” Phys. Rev. B 2(6), 2045–2048 (1970). 32. H. Ticha and L. Tichy, “Semiempirical relation between non-linear susceptibility (refractive index), linear refractive index and optical gap and its application to amorphous chalcogenides,” J. Optoelectron. Adv. Mater. 4, 381–386 (2002). 33. V. G. Ta’eed, N. J. Baker, L. B. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007). 34. M. Sheik-Bahae, D. J. Hagan, and E. W. Vanstryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65(1), 96–99 (1990). 35. M. Dinu, “Dispersion of phonon-assisted nonresonant third-order nonlinearities,” IEEE J. Quantum Electron. 39(11), 1498–1503 (2003).

1. Introduction For more than a decade, chalcogenide glasses have been used as platform for ultra-fast alloptical processing due to their high linear refractive indices (2.0-3.0 at 1.55μm), high third order nonlinear refractive index, n2 (≈100-1000 × silica), and femtosecond response time [1– 7]. Furthermore, in many chalcogenide glasses two-photon absorption (2PA) at telecommunications wavelengths can be negligible and no free carrier effects are present. These characteristics provide a distinct advantage for third order nonlinear optics compared with materials such as crystalline silicon (c-Si) and III-V semiconductors [8–10]. One of the best-known chalcogenide glasses, As2S3, was first used in fibre form for all-optical switching by Asobe et al in 1992 [11]. Subsequently As2S3 has been used to make rib waveguides that demonstrated many ultra-fast nonlinear effects including signal regeneration [12]; radiofrequency (RF) spectral analysis [13]; mid-span spectral inversion for dispersion

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compensation [14]; time-division demultipexing [15]; and parametric amplification [16, 17], etc. In spite of this, As2S3 has the disadvantages of having only a moderate nonlinearity (≈90 × fused silica) and also suffering from residual photosensitivity even at infrared (IR) wavelengths and this means that the waveguides can be unstable when operated at high average power. As a result, better chalcogenide glasses are needed which are more nonlinear and display no IR photosensitivity. There have been many measurements reporting that larger nonlinearities are available from other chalcogenide glasses [1–6] as well as some results suggesting that low photosensitivity can be also obtained [18, 19]. It is known, for example, that substituting more polarizable Se atoms for S reduces the bandgap and increases the refractive index with the result that As2Se3 is about 4 times more nonlinear than As2S3. However, this comes at the expense of a reduced glass transition temperature, Tg, increasing two-photon absorption and large photosensitivity. The addition of four-fold coordinated Ge helps raise Tg and creates a glass system that has an exceptionally wide glass-forming range. This has allowed many physical properties, such as the linear and nonlinear refractive index, bandgap, glass transition temperature, etc, to be tuned via the composition and hence the Ge-As-S(Se) system has become one of the most widely characterized of the chalcogenides [5, 6, 18–23]. Combining all the existing data on nonlinearity from this and closely related glass systems, there is a general trend that n2 increases with the normalized photon energy, hυ/Eg, where Eg is the optical bandgap of the glass, although that there are significant ( × 2-3) differences between n2 values measured by different authors for similar glasses. Whilst this trend suggests that narrow bandgap glasses should have best nonlinearity, as the normalised photon energy increases above 0.5, two-photon absorption (2PA) appears and, hence, the figure of merit, FOM2PA = n2 /β2λ, where β2 is the two photon absorption coefficient, generally decreases. FOM2PA characterizes the nonlinear phase shift obtainable in the materials over a distance limited by two-photon absorption and values > 10 are desirable for efficient all-optical devices. However, in many papers discouraging values of FOM2PA have been reported with values 100 [25]; for Ge11.5As24Se64.5, ≈60 [5] whilst values for As2Se3 remained poor at ≈1.9 [26]. Some papers have reported reduced photosensitivity for specific ternary glass compositions in the Ge-As-Se system of glasses. Yang et al [19] reported a photo-stable glass with composition Ge10As35Se55, whilst Su et al [18] showed that compositions characterized by a mean coordination number (MCN – the sum of the products of the valency times the abundance of the individual atomic constituents) around 2.5 were the most photostable. An

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Burning Threshold (mW)

example of one of these glasses is Ge11.5As24Se64.5 and this has been used to fabricate very high-Q photonic crystal resonators. The resonant frequency of these structures is extremely sensitive to any change in refractive index, but no changes due to photosensitivity were apparent at the < 10ppm level when pumped on resonance with ≈1550nm light [27]. Hence these results suggest that photostable compositions exist at least in the Ge-As-Se ternary system. Whether such photostable glasses exist in other system remains an open question. Ge11.5As24Se64.5 is one of the more nonlinear glasses from the Ge-As-Se system with about 2.5 × the nonlinearity of As2S3. It also has the somewhat unusual property for a chalcogenide glass that thermally evaporated thin films have the same bond structure as the bulk glass and it is, therefore, thermally stable [20]. Our recent studies of the thermal properties of glasses in the Ge-As-Se system suggested that Ge11.5As24Se64.5 corresponds to one of the strongest glass formers, which may explain the apparent higher stability of its chemical bonds [28]. However, its weakness is a lower resistance to laser damage than As2S3 and this restricts applications that require the highest intensity. We quantified the relative damage resistance of several samples of chalcogenide glasses using tightly-focussed CW radiation at 830nm (from a micro-Raman system). In fact the issue of optical damage of Se-containing chalcogenides initially became apparent during Raman measurements. Se-based glasses underwent catastrophic damage once a threshold was exceeded and the glass surface “burnt” and evaporated creating gross craters far bigger than the irradiated spot. This was generally not observed with S-based glasses. Furthermore, as the Se was progressively replaced by S, the damage threshold rose (see Fig. 1) and the tendency for the surface to “burn” disappeared. In our tests Ge11.5As24S64.5 glass ended up with a better damage resistance than As2S3. 10

As2S3

8 6 4 2 0 0.00

0.25

0.50

0.75

1.00

Se/(Se+S) Fig. 1. Variation of optical damage threshold measured for CW radiation at 830nm for glasses in the Ge11.5As24(SexS1-x)64.5 for x = 0, 25%, 50%, 75% and 100%.

Substituting S for Se should not change the glass network but the nonlinear index would be expected to decrease because substitution of S widens the bandgap decreasing the normalised photon energy which, as pointed out above, correlates with a reducing value of n2. An increase in the bandgap may partly explain why the damage resistance rises with increasing S-content since absorption by the exponential tail extending from the band-edge will drop. However, we also observed a smaller damage threshold in waveguide experiments at 1550nm, well away from the band edge of both glasses. In this case Ge11.5As24Se64.5 rib waveguides with air as an upper cladding were significantly more prone to fiber “fuse” damage originating from defects at the waveguide surface than As2S3 waveguides when operated at high continuous powers. Thus, substituting Se for S may be limited as a way of increasing optical nonlinearity once optical damage is taken into account. Thus it is interesting to explore other tactics for increasing the nonlinearity.

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Although As is very good network-former, elemental arsenic is toxic and this makes it undesirable as a constituent used in the manufacturing of photonic devices. One interesting opportunity is to substitute antimony for arsenic and there has been a growing interest in GeSb-Se(S) glasses which are now commonly used in molded infrared optics. The glass-forming region of Ge-Sb-Se glasses is not as broad as in the case of Ge-As-Se, but nevertheless is wide enough to allow tuning of physical properties via composition [29]. In terms of nonlinearity, Sb has a larger polarizability than As and this should lead to larger linear and nonlinear indices. Some values of the third order nonlinearity of glasses in the Ge-Sb-Se(S) system have been measured using a 1064nm Nd laser although values of FOM2PA were quite poor, at least for the Se glasses, due to the proximity of the band-edge to the measurement wavelength [7]. So far the physical parameters of Ge-Sb-Se glasses over a wide compositional range have been less widely studied than for the Ge-As-Se system and the structure-property relations, therefore, remain less clear. Nevertheless, compared with the Ge-As-Se system, the Ge-Sb-Se system offers a major experimental advantage for research and, hence, should be a good system to further develop a fundamental understanding of chalcogenides. This relates to the relative atomic mass of Sb and (Ge, Se) which is large compared with ternaries containing As. In terms of elemental chemistry, it is interesting to understand how the elemental substitution of S by Se and As by Sb can tune the properties of the glasses. In this paper, we focus on systematic measurements of the linear and nonlinear refractive index of a wide range of chalcogenide glasses. By characterizing many materials from several glass systems using the same measurement system, we could eliminate differences that often exist between measurements of the same material undertaken in different laboratories. As a result we could be more certain of the trends that could reveal which glasses should be the target for nonlinear photonics. The nonlinearity of 51 chalcogenide glasses, including As2S(Se)3, GexAs(Sb)ySe100-x-y and Ge11.5As24SxSe64.5-x, were systematically studied using Z-scan method at the telecommunication wavelength of 1550nm. The spectral dispersive nonlinearities of five chalcogenide compositions (As2S3, Ge11.5As24Se64.5, Ge15Sb10Se75, Ge15Sb15Se70 and Ge12.5Sb20Se67.5) were also investigated from 1150nm to 1686nm. Our results provide database and insight for optimizing chalcogenide glass for third order nonlinear photonics. 2. Material preparation and experiments Bulk glasses were synthesized from high purity (5N) elements by the traditional meltquenching technique. The raw materials were weighed inside a dry nitrogen glove box and loaded into a pre-cleaned quartz ampoule. The loaded ampoule was dried under vacuum (10−6 Torr) at 110°C for 4 hours to remove surface moisture from the raw materials. The ampoule was then sealed under vacuum using an oxygen hydrogen torch, and introduced into a rocking furnace to melt the contents between 900 and 950°C, depending on the glass composition. The melt was homogenized for a period not less than 30 hours then the ampoule was removed from the rocking furnace at a predetermined temperature and air quenched. The resulting glass boule was subsequently annealed at a temperature 30°C below the glass transition temperature Tg, then slowly cooled to room temperature. After that, the bulk glasses were sectioned and parallel polished to about 2mm thickness for testing. The linear refractive indices at 1550 nm were measured using Metricon 2010 Prism Coupler with accuracy of ± 0.001. To measure the optical bandgap, Eg, it necessary to use relatively thin glass samples otherwise the Urbach tail dominates the measurement. We used a hot-pressing technique under an inert nitrogen atmosphere where a small amount (milligram) of bulk glass was first heated to a set point above the glass transition temperature Tg. After the glass became sufficiently soft, it was pressed onto a polished sapphire window substrate using weights to control the thickness of the resulting sample. Once a uniform layer had formed, the sample was slowly cooled down to ambient temperature. The absorption spectra of these layers which were typically 10-20µm thick were measured at room temperature using a dual beam Cary

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5000 UV-Vis-NIR spectrophotometer between 200 to 1700 nm. From these data the optical bandgap of the material was extracted using a Tauc plot. The nonlinearities were measured using 1-2mm thick samples via the z-scan technique. The nonlinear refractive index was evaluating from the nonlinear phase change from “closed aperture” data and the nonlinear absorption coefficient from “open aperture” curves. The source used for these measurements consisted of ≈260fs pulses at 1.55μm generated by a Quantronix Palitra OPA pumped with a Ti: sapphire laser (Clark-MXR CPA 2001) at a repetition of 1 kHz. The beam from the Palitra was truncated using an aperture to improve its spatial coherence and then focused onto the sample using a 125mm focal length lens. The beam transmitted through the sample was intercepted in the far field by a rotating scattering screen (to reduce the speckle) and the light distribution on the screen was imaged onto a Xenics InGaAs camera. Around 300 frames were averaged and then stored for different positions of the sample as it was translated through the focus. These stored images could be reconstructed to create a “movie” of the z-scan signal and then post-processed to extract open and closed aperture signals. To ensure the accuracy of our z-scan method, which can suffer from errors due to uncertainties in the beam profile or pulse shape, we “calibrated” our measurements using an As2S3 bulk glass sample as a reference since its nonlinearity is known accurately at 1550nm from a wide range of experiments and has a value n2 of 2.9 ± 0.3 × 10−14 cm2/W [12]. For each sample a series of open and closed aperture z-scan measurements were recorded for different incident intensities with the traces normalised against a “zero” power trace which contained distortions introduced by the sample. After division of the closed aperture by the open aperture traces, the nonlinear phase change was extracted by fitting a numerical model to the data and the nonlinear phase change plotted against incident intensity. The slope of this curve was used to obtain the value of n2. If there was evidence of 2PA from the open aperture traces, the inverse transmission was plotted against intensity and its slope used to deduce the two-photon absorption coefficient. In most glasses at 1550nm nonlinear absorption was not observed. 3. Result and discussion Table 1 presents the linear and nonlinear indices at 1550nm for all the glass samples. Table 2 shows more detailed measurements of the dispersion of the nonlinearity and two-photon absorption coefficients as well as bandgap values for a sub-set of these glasses (As2S3, Ge11.5As24Se64.5, Ge15Sb10Se75, Ge15Sb15Se70 and Ge12.5Sb20Se67.5) from 1150nm to 1686nm. The range of glasses listed in Table 1 included the binaries As2S3 and As2Se3; a selection of compositions from the ternary GexAsySe100-x-y system with y = 10 and 20; a set of glasses in the GexSbySe100-x-y system with y = 10, 15, 20; and a series of mixed S-Se glasses Ge11.5As24(SexS1-x)64.5 with x = 0, 25%, 50%, 75% and 100%. The data from Table 1 is plotted in Fig. 2(a) in terms of the nonlinear susceptibility and in Fig. 2(b) as the nonlinear refractive index. From these results several trends are apparent. Firstly, the complete set of data conforms extremely well to the semi-empirical Miller’s rule relation between the linear and nonlinear susceptibility [30] written.  ( n02 − 1)  n n2 χ = 2 0 =α   0.0395 4π    ( 3)

4

(1)

Where χ(3) is the third order susceptibility in esu; α is the Miller’s coefficient for chalcogenide glass. Figure 2(a) plots the χ(3) as a function of [(n02 – 1) / 4π]4 and a good linear relation was achieved using a value of 2.7 × 10−10 for α. This value is consistent with that reported for ionic crystals [31], various oxide glasses, optical crystals, and some chalcogenide glasses [32].

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Table 1. The measured linear indices (n0), nonlinear indices (n2) at 1550nm for all the glass samples

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Compositions

n0

Ge5As10Se85 Ge7.5As10Se82.5 Ge10As10Se80 Ge12.5As10Se77.5 Ge15As10Se75 Ge17.5As10Se72.5 Ge20As10Se70 Ge22.5As10Se67.5 Ge25As10Se65 Ge27.5As10Se62.5 Ge30As10Se60 Ge32.5As10Se57.5 Ge35As10Se55 Ge10As20Se70 Ge15As20Se65 Ge16.67As20Se63.33 Ge20As20Se60 Ge22As20Se58 Ge22.5As20Se57.5 Ge25As20Se55 Ge30As20Se50 Ge7.5Sb10Se82.5 Ge10Sb10Se80 Ge12.5Sb10Se77.5 Ge15Sb10Se75 Ge17.5Sb10Se72.5 Ge20Sb10Se70 Ge22.5Sb10Se67.5 Ge25Sb10Se65 Ge27.5Sb10Se62.5 Ge30Sb10Se60 Ge15Sb15Se70 Ge17.5Sb15Se67.5 Ge20Sb15Se65 Ge20.83Sb15Se64.17 Ge22.5Sb15Se62.5 Ge25Sb15Se60 Ge10Sb20Se70 Ge12.5Sb20Se67.5 Ge15Sb20Se65 Ge16.67Sb20Se63.33 Ge17.5Sb20Se62.5 Ge20Sb20Se60 Ge22.5Sb20Se57.5 Ge11.5As24S64.5 Ge11.5As24S48.375Se16.125 Ge11.5As24S32.25Se32.25 Ge11.5As24S16.125Se48.375 Ge11.5As24Se64.5 As2S3 As2Se3

2.537 2.532 2.530 2.526 2.521 2.515 2.502 2.492 2.472 2.463 2.474 2.502 2.550 2.607 2.588 2.576 2.544 2.541 2.541 2.554 2.616 2.628 2.624 2.610 2.598 2.589 2.576 2.558 2.533 2.570 2.625 2.690 2.670 2.647 2.635 2.661 2.713 2.821 2.803 2.778 2.756 2.768 2.819 2.863 2.265 2.356 2.446 2.539 2.634 2.430 2.837

n2 (10−14cm2/W) 7.28 7.28 7.24 7.29 6.62 6.41 5.08 5.80 5.42 5.92 5.12 6.26 5.90 8.58 7.48 6.84 4.88 5.62 6.52 6.90 7.31 10.16 9.03 8.00 8.16 9.69 6.65 5.95 7.17 6.13 7.05 10.34 8.00 7.51 7.12 7.95 8.92 14.29 12.70 9.79 10.73 10.82 13.42 14.89 2.08 2.82 3.73 5.46 7.33 2.92 12.72



Table 2. Compositions, optical bandgap (Eg), nonlinear optical properties (n2, β2 and FOM) of five selected chalcogenide glasses. Compositions As2S3

Eg (eV) 2.22

Ge11.5As24Se64.5

1.75

Ge15Sb10Se75

1.72

Ge15Sb15Se70

1.62

Ge12.5Sb20Se67.5

1.57

Wavelength 1150nm 1250nm 1350nm 1450nm 1550nm 1686nm 1150nm 1250nm 1350nm 1450nm 1550nm 1686nm 1150nm 1250nm 1350nm 1450nm 1550nm 1686nm 1150nm 1250nm 1350nm 1450nm 1550nm 1686nm 1150nm 1250nm 1350nm 1450nm 1550nm 1686nm

n2 (10−14cm2/W) 4.33 3.67 3.50 3.23 2.85 2.79 11.8 10.4 8.83 7.67 7.90 6.83 12.5 9.00 7.67 8.30 7.50 7.33 15.5 14.9 13.7 12.2 10.0 10.0 20.3 17.5 13.5 12.0 11.4 9.40

β2 (10−9cm/W) < 0.01 < 0.01 < 0.01 < 0.01 < 0.01 < 0.01 1.20 0.35 0.11 < 0.01 < 0.01 0.10 1.27 0.35 0.12 0.05 < 0.01 < 0.01 5.94 2.78 0.81 0.49 0.35 0.27 7.44 3.05 0.94 0.45 0.37 0.22

FOM2PA > 30 > 25 > 24 > 22 > 20 > 19 1 2 6 > 57 > 59 4 1 2 5 11 > 52 > 51 0.2 0.4 1 2 2 2 0.2 0.5 1 2 2 3

Fig. 2. (a) Variation of the nonlinear susceptibility vs (n02-1)4 demonstrating that the data is a good fit to Miller’s rule 2(b) The nonlinear index n2 plotted against the linear index n0 along with the Miller’s rule according to Eq. (2).

For all-optical processing, it is generally preferable to plot the relation between the nonlinear and linear refractive indices, therefore, Eq. (1) can be transformed into:

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n2 = 4.27 × 10

−16

(n

2 0

− 1)

n02

4

cm 2 W

(2)

and this is plotted in Fig. 2(b) as the solid line. It is worth noting that, in our measurements, errors in the value of n2 exist because of poor sample quality in some cases as it proved especially difficult to prepare homogeneous samples near the boundaries of the glass-forming region in the Ge-Sb-Se system. Thus, even after the normalization described above we estimate there was about ± 10% uncertainty in the absolute values that, in fact, may be the dominant source of the scatter in Fig. 2(b). Thus, we can conclude that the generalized Miller’s rule is an effective predictor of the relative nonlinear coefðcients of these chalcogenide glasses. Examining the different glass subgroups, we find that for the mixed Ge-As-(S-Se) quaternary glasses (open black squares) the nonlinearity increases steadily as Se substitutes for S. The larger polarizability of Se is also responsible for As2Se3 having about four times the nonlinearity of As2S3 which is consistent with previous reports [33]. On average, the glasses in the Ge-Sb-Se system (magenta stars, green inverted triangles, blue triangles) have higher nonlinearities than those in the Ge-As-Se system (black squares and red dots) for the same As(Sb)-concentration supporting the notion that substituting more polarizable Sb atoms for As should increase both the linear and nonlinear indices. Within the Ge-Sb-Se system itself, glasses with the highest Sb content of 20% also have the highest nonlinearity although there is overlap in the data for glasses with 10% Sb and 15% Sb. Within each sub-group with a fixed Sb concentration Table 1 shows that there is no monotonic change in indices with either the As or Se concentration. This last point is consistent with data on the linear index of refraction reported in [21] and reproduced in Fig. 3. In this case for each Sb level, the refractive index reached a minimum value for Ge and Se composition that corresponded to the stoichiometric glasses. Thus, on the basis of Miller’s rule we would anticipate a similar behaviour for n2 and this is reflected in Table 1.

Fig. 3. Variation of the linear refractive index of Ge-Sb-Se glasses for fixed Sb contentions of 10, 15 and 20 as a function of Ge concentration. The stoichiometric compositions correspond to the minima in these graphs [21].

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Fig. 4. (a) Typical open aperture traces and their corresponding fit to a numerical model. 4(b) Inverse transmission data for different wavelengths is plotted against irradiance. The slope of the lines is used to determine the 2PA absorption coefficient, β2.

Moving to the data of Table 2, for a subset of glasses a more complete characterization was undertaken to reveal the dispersion of the nonlinearity. A raw data set illustrating the technique used to extract the β2 is shown in Fig. 4. In Fig. 4(a) a series of open aperture traces have been fitted numerically to a theoretical model and using those fits the reciprocal transmission curves for different wavelengths and peak irradiance were extracted as shown in Fig. 4(b). From such curves the values of β2 could be extracted. Typical traces for the frequency dependence of β2 for four different glasses are shown in Fig. 5 plotted against normalized photon energy. Figure 5(b) show that β2 increases monotonically with hυ/Eg and that the threshold for observable 2PA is around hυ/Eg = 0.5 as would be anticipated. The values of n2 vs hυ/Eg are plotted in Fig. 5(a), which shows a monotonic increase with normalised photon energy. This trend is consistent with data reported by Petit et al for Ge-SbSe glasses [7] and Harbold et al for As-S-Se and Ge-As-S-Se glasses [6].

Fig. 5. (a) Plot of n2 vs normalised photon energy for the glasses listed in Table 2. 5(b) β2 vs normalised photon energy for the glasses listed in Table 2.

There are two models for the dispersion of the nonlinearity of bound electrons with which these results can be qualitatively compared. The first by Sheik-Bahae et al [34] was developed for direct-gap semiconductors and indicates that the proximity of a two-photon resonance gives rise to maximum value of n2 at normalised photon energy of about 0.53. At the same time β2 is predicted to reach a maximum value at hυ/Eg ≈0.7. A different model was provided by Dinu [35] for the case of indirect-gap semiconductors. In this case a maximum in the nonlinearity was predicted to occur when hυ/Eg ≈0.65 whilst β2 passed through a maximum at hυ/Eg ≈7/6.

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Fig. 6. (a) Normalised nonlinear refractive index, n2Eg4n02, plotted as a function of normalised photon energy. 6(b) Normalised two photon absorption, β2Eg3n02, plotted as a function of normalised photon energy. The solid line is fitting according to Dinu’s model [35] and the dashed line is Sheik Bahae’s model [34].

To compare our data with the normalised functions presented in these papers, the experimental values for n2 and β2 were multiplied by factors Eg4n02 and Eg3n02 and are replotted in Fig. 6(a) and 6(b) respectively. Clearly Fig. 6(a) does not show any sign of the maximum at hυ/Eg ≈0.53 as predicted by Sheik-Bahae (dashed line), but at the same time the data does not extend far enough to confirm the existence of the maximum around 0.65 as predicted by Dinu (solid line), although in most respects the fits to the Dinu model are good. The overall feature is, however, that n2Eg4n02 and β2Eg3n02 cluster suggesting that a universal relation exists as a function of normalised photon energy although this relation may not exactly correspond to either of the published models. The residual scatter in the data, especially in Fig. 6(a), could easily originate from uncertainty in the value of the bandgap since even using thin samples, the influence of tail states due to disorder and defects are difficult to eliminate from the Tauc plots. In addition, of course, neither model takes such states into account. 4. Conclusions We have provided an internally consistent set of measurement of the third order nonlinearity of a wide range of chalcogenide glasses in the telecommunications band. Using our full data set it becomes clear that the semi-empirical Miller’s rule is probably the best way of predicting the nonlinearity of chalcogenides and we find that a very good estimate of n2 can be obtained by simply measuring no, the linear refractive index. From our data the accuracy of such an estimate of n2 based on Eq. (2) is about ± 16% and in fact this is comparable to or smaller than the uncertainties in most experiments especially when comparing data from different laboratories. On the other hand our data does not seem to reproduce some of the more pronounced details of the direct-gap semiconductor model introduced by Sheik-Bahae but is quite consistent with Dinu’s model for indirect gap semiconductors (to within a standard deviation of 12%). However, the application of this model to predict the nonlinearity without performing direct measurements is of limited value because it requires two parameters to be determined, no and Eg. The latter can only be extracted accurately from a Tauc plot which requires thin samples to be prepared. Since it is extremely difficult to prepare thin polished samples of chalcogenide glasses, in many cases the optical bandgap has been estimated from transmission measurements on thick samples arbitrarily assigning the bandgap to the energy where the absorption is 1000cm−1. This significantly overestimates the value. Since the scalings in Fig. 6 are proportional to Eg4 and Eg3 whilst the abscissa scales with Eg−1 any error in the estimated values of Eg will translate to a large error in the estimate of n2 (and β2). Furthermore, there is no simple analytic expression that describes the functional form of n2(hν/Eg) which has to be obtained by a Kramers-Kronig transform from the relation β2(hν/Eg)

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and hence it is not straightforward to calculate a value for n2 even if no and Eg are known. The whilst the results do suggest that a universal relation exists between linear and nonlinear index, bandgap and photon energy, it is somewhat doubtful, therefore, whether this is more useful way of estimating n2 than a prediction based on Miller’s rule where it only necessary to measure no which can be done easily with good accuracy. To avoid 2PA it is important to operate any device at a normalized photon energy of ≤ 0.5. Figure 6(a) then suggests that there is a limit to the value of n2 that can be achieved of ≈10−13cm2/W achieved using glasses with a linear refractive index of about 2.65-2.7. Whilst larger values (by about a factor of >1.5) can be obtained by operating at larger normalised photon energy, inevitably this leads to strong 2PA and a rapid drop in FOM2PA which has serious consequences for nonlinear photonics. Thus, our data suggests that nonlinearities around 300-400 times those of fused silica are the largest that can be achieved from chalcogenide glasses at 1550nm. Our results present clear evidence that substituting Sb for As increases the nonlinearity. However, from preliminary tests it appears that the Ge-Sb-Se system of glasses are also prone to laser damage as is the case for the Ge-As-Se system. Hence, a reasonable approach may be to use high-Sb-content glasses to get the highest nonlinearity, but then substitute S for Se until the bandgap has increased to the point where the 2PA disappears. Hopefully this will provide a combination of the best nonlinearity and damage resistance. Exploring these possibilities will be the subject of further research. Acknowledgments This work was supported by the Australian Research Council (ARC) through its Centres of Excellence (Grant CE110001018); Discovery Projects (Grant DP110102753 and DP130100086); and Discovery Early Career Researcher Award (Grant DE120101036).

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