Chalcogenide glasses as kind of diamagnetic magneto-optical materials have promising applications in the field of integrated optics and optical communication systems due to their excellent properties, such as easy to be processed into waveguide and temperature independence of the Verdet constants. For clarifying the influence factors following the compositional variation on Faraday effect and finding a glass with a large Verdet constant, novel pseudo-ternary chalcogenide glass system, GeS2 - In2S3 - PbI2, was prepared and investigated. The composition, wavelength and temperature dependences on the Verdet constants were systematically investigated at the wavelengths of 635, 808, 980 and 1319 nm. PbI2 was confirmed to have positive contribution to the Verdet constant and the Becquerel rule was proved to be an effective guidance for predicting the Verdet constant in chalcogenide glasses. The 60GeS2·15In2S3·25PbI2 glass was found to possess the largest Verdet constant (V = 0.215 min·G−1·cm−1, @808nm), which is great larger than that of commercial diamagnetic glasses. These glasses also possess good glass-forming ability and VIS-IR transmittance, therefore be a good candidate for next-generation integrated optical isolator and other magneto-optical devices.
© 2017 Optical Society of America
The rapid development of integrated optics and optical communication systems requires novel magneto-optical materials with large Verdet constant and low optical absorption for fabricating nonreciprocal photonic devices such as optical isolators and optical circulators [1–9]. Materials with large Verdet constant are benefit for highly compact magneto-optical devices, thus ferromagnetic materials such as yttrium iron garnet (YIG: Y3Fe5O12) [10,11] and paramagnetic materials such as terbium boroaluminosilicate (Kigre M-32)  have been widely used in telecommunication and high-power laser systems as Faraday rotators, due to their large Verdet constant at the operation wavelength. However, to pursue a large Verdet constant, these materials require incorporating massive amount of rare earth ions such as Ce3+ or Tb3+ [1,8,13], which may seriously deteriorate its chemical stability and optical transmittance. Meanwhile, magneto-optical effects of these materials also show strong temperature dependences , which is harmful to the stability of the optical systems operating at a varied temperature condition.
On the contrary, the Verdet constants of diamagnetic materials generally have low intrinsic temperature coefficients according to the classical electromagnetism theory [15,16]. Thus these materials are expected to be used as optical isolator in rough occasions with large temperature variation. Among the developing diamagnetic materials, glass, as a kind of amorphous material, has many distinctive properties, for example, easy to be processed into waveguide on a semiconductor substrate by using thermal evaporation, pulsed laser deposition or fibration technique, suggesting that it is a good candidate for integrated devices. As of now, several series diamagnetic glass systems have been studied including silicate, borate and tellurite glasses. Unfortunately, these materials remain defectiveness by their small Verdet constant compared to ferromagnetic materials and paramagnetic materials.
Chalcogenide glasses are compounds primarily from chalcogen elements, and because of their low photon energies inducing prominent wide transparent region from visible to infrared region [17–19], they were widely researched as passive and active infrared materials [20,21]. Recently, chalcogenide glasses attract much attention in the integrated optics, and they are proved as potential candidate for harnessing on-chip signal processing using their large stimulated Brillouin scattering and Kerr effect . In fact, Chalcogenide glasses also possess another unique advantage i.e. large Verdet constant due to their high refractive index dispersion, making them attractive as diamagnetic magneto-optical materials used in integrated optical devices working in the visible to mid-infrared region.
However, one problem being existence at present is that only few chalcogenide glasses have been studied on the Faraday effect, thus the main contributions to large Verdet constant in chalcogenide glasses have not been cognized clearly, which makes a confusion in the design of novel diamagnetic materials. For clarifying the influence factors of the glass compositional variation on Faraday effect and pursuing large Verdet constant, GeS2-In2S3-PbI2 pseudo-ternary chalcogenide glasses were selected as a start. The glass-forming region and some basic properties such as thermal, optical properties were investigated. Their magneto-optical properties were also studied, and the relationship between Verdet constant and glass composition were discussed in detail. The result shows the Verdet constants of these glasses are great larger than that of commercial diamagnetic glasses, and roughly 22 times larger than that of SiO2. Our work was aimed at affirming an effective theoretical guidance for estimating the Verdet constant and further optimizing the compositions in chalcogenide glasses.
2. Experimental procedure
All samples were prepared by using the conventional melt-quenching technique from the high purity raw materials of Ge (grains, 6N, Nanjing Germanium Co., Ltd. China), In (grains, 5N, SCRC Co., Ltd. China), S (powders, 6N, SCRC Co., Ltd. China) and PbI2(powders, 98%, SCRC Co., Ltd. China). The raw materials were encapsulated in a silica ampoule with a 10−4 Pa vacuum and heated at 1000°C for 18 h. The melt was then quenched in cold water and annealed near glass transition temperature for 2 h. Samples were finally obtained after the glass rods were cut and polished to mirror smoothness with a thickness of 3.0 mm. These samples have fine performances such as striae-free, and the parallelism with 3 minutes of arc.
All samples were first characterized by X-ray diffraction (XRD) (Rigaku D/max-RB with Cu Kα radiation and a power of 40 kV, 50 mA) at room temperature. Samples with no diffraction peaks were termed glass, and samples with narrow, strong peaks were labeled crystalline.
Differential scanning calorimetry (DSC) curves of the prepared specimens were obtained using a comprehensive thermal analysis apparatus (NET-ZSCH STA 449C). The weight of each specimen was ~10mg and the heating rate was 10 K/min with a precision of ± 1K. The powdered glass sample was encapsulated in an aluminum crucible and pure α-Al2O3 was used as a standard.
Optical measurements were carried out at room temperature. The UV-VIS-NIR transmission spectra of samples were measured in the range of 400-1500 nm using the JASCO V570 spectrophotometer. Refractive indices were measured at 632.8, 934.7 and 1549.2nm using Prism Coupler (Model 2010 M, Mctricon, USA) which can provide index accuracies of ± 0.0003.
The Verdet constants were tested at 635, 808, 980 and 1319nm by using a home-made optical bench (as seen in Fig. 1) consisting of a light source, two polarizers, a photo detector and a permanent magnet with hollow circular cylinder shape providing an axial magnetic field of about 10000 gauss at the center. One of polarizer was seated at a motorized high precision rotation mount (PRM1-Z7, THORLABS, USA) controlled by a computer. The light beam provided by semiconductor laser passed through a polarizer to produce a linear-input polarization state. After passing through the sample, the angle of the polarization rotation was detected by the polarimeter with an angular resolution of less than 0.1°, and the Verdet constant is determined using the following Eq. (1) 
The glass-forming region of this novel GeS2-In2S3-PbI2 chalcogenide glass system is shown in Fig. 2. This system has a relatively narrow glass-forming region which is mainly located at the GeS2-rich domain. Binary samples of the present system are formed by combining GeS2 with In2S3 or PbI2. The maximum contents of In2S3 and PbI2 are 30 mol% and 10 mol%, respectively. In the pseudo-ternary system, the amount of dissolved PbI2 is more than 25 mol% when the ratio of GeS2 to In2S3 remains 8:2 (in Series A). The glass-forming region extends to over 25.5 mol% amount of In2S3 with a constant PbI2 content of 15 mol% (in Series B), however it is not glassy when In2S3 content is less than 8.5 mol%.
The glass transition temperature (Tg) and thermal stability were obtained according to the DSC measurement, and the characteristic DSC-TG curve of the 72GeS2·18In2S3·10PbI2 glass is shown in Fig. 3 for example. The shape of the curve is flat, and it is difficult to find the characteristic temperatures by eyes. Actually, these temperatures are determined by the software of TA Instruments Universal Analysis 2000 which identified temperatures by the sudden change of the first order derivative. Tg is defined as the mid-temperature of an endothermic step shown in the curve. The crystallization temperature (Tx) corresponds to theonset temperature of crystallization. Thus the characteristic temperatures of Tg and Tx are equal to 577.6 K and 705.9 K, respectively, as shown in Fig. 3. The characteristic temperatures and glass-forming criteria of (100-x) (0.8GeS2·0.2In2S3)·xPbI2 (x = 0, 5, 10, 15, 20, 25 mol%) glasses are summarized in Table 1. It can be seen that Tg decreases from 630 K to 505.7 K and Tx decreases from 726 K to 615.5 K when PbI2 content increases from 0 to 25 mol%. A criterion ΔT defined as ΔT = Tx-Tg, is used for evaluating the stability of a glass against crystallization . The large value of ΔT favors the stability of a glass. In this study, all ΔT value of PbI2 contained glasses are large than 100K, indicating that these glasses are easily obtained in bulk forms and have adequate stabilities for fiberization. Furthermore, the values of ΔT increase first and then decrease with the addition of PbI2 in GeS2-In2S3 glass, and the highest value is obtained when PbI2 is up to 10 mol%.
Figure 4 shows the wavelength dispersion of the refractive index for (100-x) (0.8GeS2·0.2In2S3) · xPbI2 (x = 5, 10, 15, 20, 25) series glasses. The curves are fitted by the Prism Coupler automatically using the equation described below Table 2, which are used for theoretical calculation of the Verdet constant. The inset shows the relationship between the refractive index and the content of PbI2 at wavelength 808nm. These glasses have relatively high refractive index and the value increases linearly with the additionof the PbI2.Fig. 5) . In this series glasses, the λ* increases monotonously with the addition of PbI2.
The Verdet constants of (100-x) (0.8GeS2·0.2In2S3) · xPbI2 (x = 5, 10, 15, 20, 25) series glasses at 635, 808, 980 and 1319 nm are shown in Fig. 6. Verdet constant of SiO2 glass was measured as a reference, and its value is 0.013 min·G−1·cm−1 at the wavelength 635 nm, which agrees well with previous reported in , demonstrating that the Verdet constant test bench works well. From Fig. 6, it can be seen that Verdet constants have a linear variation with the content of PbI2 and values increase gradually with the addition of PbI2 in this system glasses. However they decrease monotonously with the red shift of the testing wavelength. The highest value is as large as 0.286 min·G−1·cm−1 for 60GeS2·15In2S3·25PbI2 glass at the wavelength of 635nm, which is great larger than that of commercial diamagnetic glasses (Schott, SF6, V = 0.069 min·G−1·cm−1 @ 633 nm for example) , and roughly 22 times larger than that of SiO2.
According to the previous work [29,30], the main basic structural units in GeS2-In2S3 chalcogenide glasses are [GeS4] and [InS4] tetrahedral, and also a part of ethane-like [S3Ge-GeS3] units. When PbI2 was added into glass, the amount of ethane-like [S3Ge-GeS3] units and [GeS4] tetrahedral diminishes, whereas the [PbIn], [GeS3I] and [GeS2I2] s.u. were formed in network gradually. The behavior of the glass-forming region of GeS2-In2S3-PbI2 series glasses observed in the present work appears to support this proposed structural variation. In GeS2-PbI2 glasses, the amount of PbI2 is limited to 10%, while it is up to over 25% in GeS2-In2S3-PbI2 glasses. That is due to the metallic bonds such as Ge-Ge would be disrupted when PbI2 is added into the glasses, and this process induces the high solubility of PbI2.
Tg dependence can be explained by the average bond energy. Addition of PbI2 into GeS2-In2S3 glasses results in a decrease in the average bond energy due to the number of weaker Pb-I (197 ± 38 kJ/mol ,) bonds increase while that of relatively stronger Ge-S (551 ± 25 kJ/mol), In-S (289 ± 17 kJ/mol) and Ge-Ge (274 ± 21 kJ/mol) bonds decrease. Hence, Tg in GeS2-In2S3-PbI2 glasses is lower than that of GeS2-In2S3 glasses. PbI2 as a network modifier is benefit for modifying the rigidity of the network and enhancing the glass-forming ability, leading to the increasing of the ΔT. However the breakage of network becomes distinct due to the decrease of the [GeS4] and [SbS3] s.u. when the amount of PbI2 is more than 10 mol%, and this structural degradation causes the decrease of the ΔT.
The refractive index of the glass is generally determined by the ionic polarization and the stacking density of structural units in network . Larger ionic polarization and the stacking density of structural units lead to larger refractive index. In this work, the polarizability of I- and Pb2+ ions is larger than that of S2- and Ge4+. Thus the refractive index increases linearly with the addition of PbI2.
The absorption edge, λ*, of a glass is related to the optical band gap which is defined as the excitation energy of the electron from the top of the valence band to the bottom of the conduction band. And λ* will shift to long wavelength when anionic radius increases and electronegativity decreases. In GeS2-In2S3-PbI2 glasses, I- ion has comparable electronegativity but larger ionic radius than S2- ion, meanwhile the Pb2+ ion also has loose electronic shell and 6s2 outermost electron. So, when PbI2 content increases, the value of the optical band gap decreases, which corresponds to a red-shift of the visible absorption edge .
According to the evolution of the Verdet constant, Pb2+ ion has greater contribution to Verdet constant than Ge4+ and In3+ ones. This can be ascribed to the unique electronic structure of Pb2+ ion, viz., s2-sp electron jumps involving 1S0→1P1, 3P0,1,2 transitions, which has been demonstrated to give rise to large Verdet constant in some kinds of oxide glass systems . Faraday effect of the magneto-optical materials originates from the magnetic-field-induced Larmor precession of electron orbits . According to the classical electromagnetism theory, Becquerel deduced the relationship between the Verdet constant and the dispersion of the refractive index in diamagnetic materials . The equation is described asEq. (3) for each glass. Figure 7 shows the theoretical and experimental values of the Verdet constant dispersions typically for 60GeS2·15In2S3·25PbI2 glass. The theoretical values are in a good agreement with the experimental data and the error is within 11% at long wavelength. However, the deviation is as large as 33% at wavelength 635nm. We believe that it is mainly caused by the error of the refraction index fitting formula which does not work well at the short wavelength near the absorption edge, and the error will be cut down while a more precision formula is used for refractive index fitting.
In order to clarify the temperature dependence on the Verdet constant, the values at 635 nm for 60GeS2·15In2S3·25PbI2 and terbium aluminosilicate glass (Tb glass) are measured at different temperatures and the results are shown in Fig. 8. Tb glass as paramagnetic materials is chosen for contrast and the absolute value of the Verdet constant is used here for comparing clearly. From Fig. 8, the Verdet constant of Tb glass decreases linearly when the temperature rises from 278 K to 368 K. And the slope coefficient is about 8.9 × 10−4 K−1. While the Verdet constant of chalcogenide glass changes slightly and can be regarded as a constant. In particular, when the temperature exceeds 350 K, the Verdet constant of Tb glass will be smaller than that of chalcogenide glass. The results show that chalcogenide glass is a good candidate for the sensing element in high stability Faraday effect sensors.
As a high-performance magneto-optical material, it should provide a relatively large Verdet constant and low optical absorption at the operation wavelength. A suitable figure of merit (FOM) thus can be defined as Eq. (3). The values of FOM for all examined glasses are listed in Table. 3. Figure 9(a) shows the evolution of the FOM as the wavelength for 68GeS2·17In2S3·15PbI2 glass. The FOM increases first and then decreased, and the maximum values were obtained within the wavelength range from 700 nm to 800 nm. It indicates that this magneto-optical glass is most suitable for applications in the devices working at 700 nm - 800 nm wavelength region.
In Fig. 9(b), the relationship between the FOM and the molar content of PbI2 is shown at wavelength 635, 808, 980 and 1319 nm, respectively. It should be noted that the absolute values obtained in this paper are smaller than the real values, because all samples were prepared from as-received raw materials, thus the absorption coefficient is great larger than their intrinsic absorption due to the high concentration of the impurities. However, some significant information can still be obtained from the evolution of the FOM. From Fig. 9(b), the value of the FOM increases monotonously with the addition of PbI2 at the wavelength of 980 and 1319 nm, whereas it increases first and then decreases at 635nm. From Eq. (5), it can be known that the FOM depends on both the Verdet constant and the absorption coefficient of the glass. When PbI2 is added into the glass, the Verdet constant and the absorption coefficient of the glass are both changed, which lead to a relatively complex relationship between FOM and the content of PbI2. On the one hand, the absorption coefficient changes little at far away from the absorption edge wavelength region when the content of PbI2 increases, thus the variation tendency of the FOM is similar with that of the Verdet constant, i.e. the values increase monotonously with the addition of the PbI2 at infrared wavelength region. On the other hand, the absorption coefficient increases significantly at 635 nm due to the red-shift of the absorption edge when PbI2 is added into the glass, leading to a negative variation of FOM at 635 nm when PbI2 reaches to 25 mol%. The evolution of the FOM at wavelength 808nm is anomaly. It is mainly caused by the fluctuation of the absorption spectrum which introduced by the changing lamb of the spectrophotometer.
A series of GeS2-In2S3-PbI2 pseudo-ternary chalcogenide glasses have been systematically explored and stable glasses have been prepared. The glass-forming region is mainly located at the Ge-rich domains and the amount of PbI2 dissolved is up to 25 mol%. The parameter ΔT, which is defined as ΔT = Tx-Tg, is larger than 100K in all of the PbI2 contained glasses, indicating that these glasses have adequate stabilities for fiberization. The magneto-optical properties of (100-x)(0.8GeS2·0.2In2S3)·xPbI2 (x = 5, 10, 15, 20, 25) series glasses were studied. The results indicate that high polarizability Pb2+ ions are responsible for large Verdet constant, and it can be regarded as a constant between the temperatures of 275 K and 370 K. Through the comparison with the experiment data, the Becquerel rule is proved to be an effective theoretical guidance for estimating the Verdet constant and further optimizing the compositions in chalcogenide glasses. The performance of these glasses was also evaluated and the result shows they are a good candidate for the next generation of integrated optical isolator and other magneto-optical equipment.
National Nature Science Foundation of China (61405241, 61475189); the West Light Foundation from Chinese Academy of Science of China (CAS) and Natural Science Basic Research Project in Shaanxi Province (2014JQ8345, 2015JQ5141); Project of Key Laboratory of Optoelectronic Detection Materials and Devices of Zhejiang Province (2017001).
References and links
1. L. Sun, S. Jiang, and J. R. Marciante, “Compact all-fiber optical Faraday components using 65-wt%-terbium-doped fiber with a record Verdet constant of -32 rad/(Tm),” Opt. Express 18(12), 12191–12196 (2010). [CrossRef] [PubMed]
2. C. J. Firby and A. Y. Elezzabi, “High-speed nonreciprocal magnetoplasmonic waveguide phase shifter,” Optica 2(7), 598–606 (2015). [CrossRef]
3. L. Bi, J. Hu, G. F. Dionne, L. Kimerling, and C. A. Ross, “Monolithic integration of chalcogenide glass/iron garnet waveguides and resonators for on-chip nonreciprocal photonic devices,” Proc. SPIE 7941, 794105 (2011). [CrossRef]
4. H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22(1), 240–253 (2005). [CrossRef]
6. A. Haddadpour, V. F. Nezhad, Z. Yu, and G. Veronis, “Highly compact magneto-optical switches for metal-dielectric-metal plasmonic waveguides,” Opt. Lett. 41(18), 4340–4343 (2016). [CrossRef] [PubMed]
9. T. Yoshino, S. Torihata, M. Yokota, and N. Tsukada, “Faraday-effect optical current sensor with a garnet film/ring core in a transverse configuration,” Appl. Opt. 42(10), 1769–1772 (2003). [CrossRef] [PubMed]
10. K. Taki, Y. Miyazaki, and Y. Akao, “Optical propagation properties in gyromagnetic waveguides using faraday effects of YIG thin films on GGG substrates,” Jpn. J. Appl. Phys. 19(5), 925–938 (1980). [CrossRef]
11. B. Geist, R. Ronningen, A. Stolz, G. Bollen, and V. Kochergin, “Radiation stability of visible and near-infrared optical and magneto-optical properties of terbium gallium garnet crystals,” Appl. Opt. 54(10), 2866–2869 (2015). [CrossRef] [PubMed]
12. M. J. Weber, Handbook of Optical Materials (CRC, 2003), Chap. 2.
13. J. Qiu, K. Tanaka, N. Sugimoto, and K. Hirao, “Faraday effect in Tb3+-containing borate, fluoride and fluorophosphate glasses,” J. Non-Cryst. Solids 213, 193–198 (1997). [CrossRef]
14. A. Edgar, D. Giltrap, and D. R. MacFarlane, “Temperature dependence of Faraday rotation and magnetic susceptibility for Ce3+ and Pr3+ ions in fluorozirconate glass,” J. Non-Cryst. Solids 231(3), 257–267 (1998). [CrossRef]
15. M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. S. J. Russell, “Complex Faraday rotation in microstructured magneto-optical fiber waveguides,” Adv. Mater. 23(22), 2681–2688 (2011). [CrossRef] [PubMed]
16. J. Qiu and K. Hirao, “The Faraday effect in diamagnetic glasses,” J. Mater. Res. 13(5), 1358–1362 (1998). [CrossRef]
17. A. A. Wilhelm, C. Boussard-pledel, Q. Coulombier, J. Lucas, B. Bureau, and P. Lucas, “Development of far-infrared-transmitting Te based glasses suitable for carbon dioxide detection and space optics,” Adv. Mater. 19(22), 3796–3800 (2007). [CrossRef]
18. L. B. Shaw, B. Cole, P. A. Thielen, J. S. Sanghera, and I. D. Aggarwal, “Mid-wave IR and long-wave IR laser potential of rare-earth doped chalcogenide glass fiber,” IEEE J. Quantum Electron. 48(9), 1127–1137 (2001). [CrossRef]
19. A. Yang, M. Zhang, L. Li, Y. Wang, B. Zhang, Z. Yang, and D. Tang, “Ga-Sb-S chalcogenide glasses for mid-infrared applications,” J. Am. Ceram. Soc. 99(1), 12–15 (2016). [CrossRef]
20. B. J. Eggleton, B. Luther-Davies, and K. Richardson, “Chalcogenide photonics,” Nat. Photonics 5, 141–148 (2011).
21. H. Ou, S. Dai, P. Zhang, Z. Liu, X. Wang, F. Chen, H. Xu, B. Luo, Y. Huang, and R. Wang, “Ultrabroad supercontinuum generated from a highly nonlinear Ge-Sb-Se fiber,” Opt. Lett. 41(14), 3201–3204 (2016). [CrossRef] [PubMed]
22. B. J. Eggleton, T. D. Vo, R. Pant, J. Schr, M. D. Pelusi, D. Yong Choi, S. J. Madden, and B. Luther-Davies, “Photonic chip based ultrafast optical processing based on high nonlinearity dispersion engineered chalcogenide waveguides,” Laser Photonics Rev. 6(1), 97–114 (2012). [CrossRef]
23. H. Guo, H. Tao, Y. Gong, and X. Zhao, “Preparation and properties of chalcogenide glasses in the GeS2-Sb2S3-CdS system,” J. Non-Cryst. Solids 354(12-13), 1159–1163 (2008). [CrossRef]
24. F. Skuban, S. Lukic, D. Petrovic, and I. Guth, “Refractive-index dispersion of glassy semiconductors in the pseudo-binary As2Se3-SbSI system,” J. Non-Cryst. Solid 355, 2059–2062 (2009).
25. Test Methods of Colorless Optical Glass-Part 9: Coefficient of Optical Absorption, State Standard of the People’s Republic of China, GB/T 7962.9–2010.
26. J. Qiu, H. Kanbara, H. Nasu, and K. Hirao, “Wavelength dispersions of the Faraday effect in chalcogenide glasses,” J. Ceram. Soc. Jpn. 106(1230), 228–230 (1998). [CrossRef]
27. Y. Ruan, R. A. Jarvis, A. V. Rode, S. Madden, and B. Luther-Davies, “Wavelength dispersion of Verdet constants in chalcogenide glasses for magneto-optical waveguide devices,” Opt. Commun. 252(1-3), 39–45 (2005). [CrossRef]
28. J. M. Weber, Handbook of Optical Materials (CRC, 2003), Section 2.7.1.
29. H. Guo, H. Tao, Y. Zhai, S. Mao, and X. Zhao, “Raman spectroscopic analysis of GeS2-Ga2S3-PbI2 chalcohalide glasses,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 67(5), 1351–1356 (2007). [CrossRef] [PubMed]
30. T. Haizheng, Z. Xiujian, T. Wei, and M. Shun, “Micro-structural study of the GeS2-In2S3-KCl glassy system by Raman scattering,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 64(4), 1039–1045 (2006). [CrossRef] [PubMed]
31. J. A. Dean, Lange’s Handbook of Chemistry, 15th Ed., (McGraw-Hill Book Company, 1999), Section 4.
32. H. Guo, H. Tao, S. Gu, X. Zheng, Y. Zhai, S. Chu, X. Zhao, S. Wang, and Q. Gong, “Third- and second-order optical nonlinearity of Ge-Ga-S-PbI2 chalcogenide glasses,” J. Solid State Chem. 180(1), 240–248 (2007). [CrossRef]
33. N. F. Borrelli, “Faraday rotation in glasses,” J. Chem. Phys. 41(11), 3289–3293 (1964). [CrossRef]