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The nonlinear refractive index n2 of SiO2-Al2O3-La2O3 (SAL) glasses of 10 to 24 mol% La2O3 is determined via Z-scan technique at 800 nm in the sub-100 fs time regime. n2 (5.8 to 9.3 × 10−16 cm2/W) correlates linearly with the La2O3 concentration scaling by the factor (0.264 ± 0.007)×10−16 cm2/W per mol% of La2O3. The relation between n2 and the linear refractive index n0 of the SAL systems is successfully described by the theoretical model of Boling, Glass and Owyoung (BGO theory). Increasing concentrations of heavy La3+ ions and non-bridging oxygen accompanied by a rising volume of the O2− ions are considered to be responsible for the n0 and n2 increase as well as a slight Urbach energy decrease. The upper limit of the La2/3O hyperpolarizability is estimated to amount to (2.2 ± 0.2)×10−36 esu.
© 2014 Optical Society of America
1. Introduction
During the past SiO2-Al2O3-La2O3 (SAL) glasses have been proposed as promising material for new generations of high power optical fibers and as host material for bulk laser media [1,2]. These glasses show a large solubility for various rare earth dopants such as Sm3+ [2], Yb3+ [3], Tm3+ [4], Ho3+ [4] or Eu3+ [5], which is crucial for efficient lasing. Furthermore, compared to high refractive index glasses (e.g. tellurites or chalcogenides), SAL systems possess high glass transition temperatures Tg and low thermal expansion coefficients αth [6–8] and allow a combination with conventional silica fiber based components [9]. Other glasses with comparable thermochemical properties, such as ZnO-Al2O3-SiO2 [10] or MgO-Al2O3-SiO2 [11] for instance, suffer from strong phase separation, which in SAL systems can be minimized by optimizing the Al2O3/La2O3 ratio [1] and by rapid cooling. Thus SAL systems have been used as core material for silica cladded fibers with numerical apertures above 0.4 [1, 9].
The structural properties of SAL systems have been investigated comprehensively during the past [12–19]. Some few papers only, on the other hand, are dealing with the nonlinear optical properties of these glasses. Kiryanov et al. measured n2 of Yb3+ doped SAL systems in the dopant resonance and the thermal regime [3]. Litzkendorf et al. estimated the range of n2 by supercontinuum generation around 1500 nm [9]. The latter work indicates a significantly increased n2 of undoped SAL systems compared to that of pure Al2O3 or SiO2. For both materials n2 was measured to be around 3×10−16 cm2/W [20–26]. Hence SAL systems may also be a feasible material for nonlinear optical applications in fibers and in bulk materials used for white light generation or all optical switching, including optical Kerr gating, for which a huge n2 value is desired [27, 28].
In the present work, n2 values of selected SAL systems of varying La2O3 content were measured by the Z-scan technique at 800 nm in the sub-100 fs time regime and compared to those of pure SiO2 (Heraeus F300). The Urbach energy was determined as system order parameter and the linear refractive index n0 measured as a function of the La2O3 content. The influence of the La2O3 content onto n0 and thus the linear polarizability is discussed. The relation between n2 and n0 of the SAL systems is successfully described by BGO theory [29].
2. Experimental
2.1. Glass sample preparation
Four SAL glass mixtures of high-purity powder (SiO2, Al(OH)3, La2O3) containing 10 – 24 mol% La2O3 were melted conventionally in 500 g batches in a discontinuous two-step process. High alumina concentrations (20 mol%) served to increase the lanthanum solubility in silica glass and to avoid phase separation and crystallization. The powder pre-melt (first step) in a covered platinum crucible in air atmosphere at 1400–1650 °C was quenched after 8 hours in ultra-pure water to obtain fritted glass particles. The re-melt (second step) of the dried particles at 1650 °C for refining and homogenizing was partly stirred (platinum stirrer) to avoid striae and bubbles and finally cast into a stainless steel mold. The obtained glass blocks (25 × 25 × 120 mm3) were slowly cooled in a furnace at 100 K/h.
2.2. SAL glass characterization
The chemical glass composition was determined by quantitative electron probe microanalysis (EPMA) using energy dispersive X-ray (EDX) spectrometry on an electron microprobe (JEOL Ltd., JXA-8800L). Glass transition temperatures (Tg) and thermal expansion coefficients (αth at 600 °C) were ascertained by a vertical dilatometer (LINSEIS Messgeraete GmbH, L75V) using a 5 K/min heating rate. The glass densities ρ were determined by the Archimedes’ principle in ethanol. The molar volumes Vm were calculated from ρ. All obtained values are summarized in Table 1.
Table 1. Chemical composition, glass transition temperature Tg, thermal expansion coefficient αth, density ρ and molar volume Vm of the SAL glasses and the F300 quartz glass sample (Heraeus Quarzglas GmbH & Co. KG)
All samples were polished to a thickness of 1 mm and cleaned by methanol. Transmission as well as reflection spectra were acquired by a commercial spectrometer (PerkinElmer, Lambda 900) between 210 and 850 nm. The wavelength dependent absorption and reflection coefficients α(λ) and r(λ) were retrieved numerically from the obtained spectra.
2.3. Z-scan measurements
The nonlinear refractive index n2 of the glasses was determined by a Z-scan technique (Fig. 1, in [31], Sheik-Bahae et al.) using laser pulses of 50 fs duration at 1 kHz repetition rate and 800 nm central wavelength generated by a commercial CPA (chirped pulse amplification) system (Mantis Oscillator and Legend Elite amplifier, both from Coherent Inc.). After passing through a home-built variable attenuator (half wave plate in combination with a thin film polarizer) the beam was focused by a 500 mm BK7 lens to a 1/e2 spot radius w0 ≈ 40 μm. In the far field behind the focus the laser beam passed an aperture of 20 % pulse energy transmission onto a photo detector (PD 10, read out with an Pulsar 2 system, both from Ophir Inc.). The back reflection from a 0.2 mm BK7 cover slip, recorded by a photo diode of the same type served as reference. The sample was moved within the focal region along the laser propagation direction and the transmitted pulse energy recorded for each position. Z-scan signals at different laser pulse energies served to improve the accuracy and to investigate higher order nonlinearities. The accuracy was improved additionally by recording Z-scan signals of a reference sample, as proposed in [32]. The reference consisted of a 0.7 mm thick slab of fused silica with 1200 ppm OH and (n2)r = (2.7 ± 0.3)×10−16 cm2/W [21–24, 26].
3. Results and data analysis
3.1. Linear refractive index and Urbach energy
With the reflection and absorption coefficients r(λ) and α(λ) retrieved from the spectroscopic results and by applying the Fresnel equations the linear refractive index n0 has been determined for each SAL glass (Fig. 2).
Fig. 2 Wavelength dependent linear refractive index n0 of SAL glass samples of various La2O3 concentrations.
In addition the α(λ) values in the low photon energy region (E < 6 eV) lie considerably below 103 cm−1 and enable fitting by the function [33, 34]
to determine the Urbach energy EU as depicted in Fig. 3. Here E is the photon energy and α0 the band tailing parameter [35]. The EU values range from 0.53 eV to 0.76 eV in agreement with comparable literature data [36]. The shape of the absorption tail and consequently EU thus yield information on the glass structural order [36].Fig. 3 Absorption coefficients α(E) of SAL glass samples as function of photon energy E and La2O3 concentration fitted exponentially to determine the Urbach energy EU [eV].
3.2. Z-scan measurements
A representative stack of Z-scan traces recorded at different pulse energies (pulse peak intensities in the focal plane) is shown in Fig. 4. These curves were corrected by subtracting a back-ground trace measured at very low pulse energy in order to eliminate transmission fluctuations due to surface effects (e.g. small scratches).
Fig. 4 Intensity dependent Z-scan traces of SAL glass with 10.5 mol% La2O3 (sample 1) recorded at various pulse peak intensities in the focal plane.
As pointed out in [31] and [37], the ”Peak-to-Valley” transmission ratio ΔTPV of a Z-scan can be approximated linearly to the maximum phase shift induced by the laser pulse in the focal plane. Hence the correlation between ΔTPV and n2 and accordingly the pulse peak intensity is linear as well, if higher order nonlinearities can be neglected. According to Fig. 5 at least up to peak intensities of 100 GW/cm2, the latter is the case.
Fig. 5 Peak-to-Valley ratio ΔTPV as function of the incident pulse intensity I0. The percentage values represent the La2O3 mol% of each sample.
As proposed in [32] (n2)s of each sample was calculated from the Z-scan data using
with (n2)r being the nonlinear refractive index of a reference, Lr and Ls the effective thicknesses (c.f. [32]), Rr and Rs the reflectivities, (ΔTPV)r and (ΔTPV)s the Peak-to-Valley values, and Er and Es the pulse energies applied to the reference (r) material and the samples (s), respectively. The spectroscopic results show, that the linear absorption of the samples around 800 nm (1.6 eV) can be neglected (Fig. 3). Therefore the effective thickness is the geometric one [37]. By this reference method, the accuracy of the measurement is improved relative to a ΔTPV analysis based on a single Z-scan measurement. Furthermore, the reference method eliminates possible deviations due to an aberrant laser beam.The values of the linear (n0) and nonlinear (n2) refractive indices, the Urbach energies EU as well as the Abbe number νd of the SAL glasses are summarized in Table 2. The obtained n2 values are furthermore depicted in Fig. 6 with the error bars representing the standard deviation of repeated measurements. n2 increases linearly with the molar La2O3 concentration in the SAL glass. Its increment is (0.264 ± 0.007)×10−16 cm2/W per mol% of La2O3.
Table 2. Linear refractive index n0 at 800 nm and nd at 587.6 nm, Abbe number νd, nonlinear refractive index n2 at 800 nm and Urbach energy EU of the SAL glasses and fused silica (F300)
Fig. 6 Nonlinear refractive index n2 of SAL glass (black squares) as a function of the La2O3 concentration as well as the linear refractive indices n0 measured in the present work (blue stars), by Iftekhar et al. (green triangles) [17] and Dejneka et al. (red circles) [1].
4. Discussion
The prepared SAL glass samples are homogeneous and of very similar Al2O3 content [Al2O3] ≈ 21 ± 1 mol%, i.e. in this study the glass maker SiO2 is replaced by the glass modifier La2O3. The glass densities ρ, molar volumes Vm and the refractive indices n0 (linear) and n2 (nonlinear) are found to depend linearly on the La2O3 content (Tables 1 and 2, Fig. 6). The macroscopic properties of the SAL glasses are compared to previous findings in the literature [1, 17] and rationalized within the framework of empirical macroscopic models (sect. 4.1). Further insight is obtained from the application of empirical microscopic models (sect. 4.2).
4.1. Macroscopic properties of SAL glasses
The composition of all prepared SAL glass samples lies within that region of the ternary La2O3-Al2O3-SiO2 system which was previously identified to yield homogeneous glass without phase separation and/or crystallization (cf. e.g. [17]). The La2O3 dependent increase of the glass density ρ(Δρ/Δ[La2O3] = 7.9 × 10−2 g cm−3 mol%−1) and the linear refractive index n0 (Δn0/Δ[La2O3] = 9.4 × 10−3 mol%−1) in this work are, however, found to be stronger than those recently reported to amount to 6×10−2 g cm−3 mol%−1 and to 6×10−3 mol%−1, respectively [17]. The relatively steep gradients and the small deviations from the linear n0([La2O3]) and n2([La2O3]) dependences in Fig. 6 are possibly a result of the two-step sample preparation in the present work including a re-melt of a relatively large glass sample (cf. sect. 2.1). This assumption is supported by the n0 values obtained at 777 nm from re-melted SAL glasses in ref. [1] which fit the linear regression of the present n0 values very well in spite of their broad range of Al2O3 content from 6 to 20 mol% (cf. Fig. 6).
The optical band gap Egap of the SAL glasses is expected to lie far beyond the measured transmission and reflection spectral range limit (≥ 210 nm; ≤ 5.9 eV): the absorption coefficients α (210 nm) in Fig. 3 are much lower than α(Egap) ≈ 103 to 104 cm−1 considered as an indicator for the Egap position (cf. e.g. [38]). Thus two- or three-photon absorption can be neglected. The latter is also confirmed by the absence of asymmetries in the Z-scan signals, which are typical for nonlinear refractive index measurements, accompanied by nonlinear absorption [37]. The Urbach energy values of the SAL glasses in the range of 0.5 to 0.8 eV are much larger than those of fused silica below 0.1 eV [38] reflecting the glass modifying effect of La2O3 and an increased concentration of non-bridging oxygen ions O2−.
Based on the assumption of one major polarizable constituent of the glass mixture (La2O3) in combination with the Lorentz-Lorenz-oscillator [39,40] approach, Boling et al. [29] formulated a well accepted [41, 42] semi-empirical relationship (BGO theory)
where nd is the linear refractive index at 587.6 nm and νd is the Abbe number. is the nonlinear refractive index with respect to the electric field of the laser beam in cgs units, whereas n2 is the index related to the intensity in SI units. The conversion of n2 into and vice versa can be found e.g. in [43]. In Fig. 7 the values indicated as ”BGO” were calculated by applying (3) and the measured linear refractive indices. From that figure, the good agreement between BGO theory and the measured values is evident.Fig. 7 Nonlinear refractive index with respect to the laser electric field in cgs units as a function of the La2O3 content c(La2O3): black squares: converted from the measured n2 and n0 values, blue triangles: calculated according to the BGO theory (3), using the measured n0 values, red circles: calculated according to (6) solved for using only the oxygen hyperpolarizabilities of SiO2, Al2O3 and La2O3 given by Adair et al. [44] (c.f. section 4.2).
Here it should be pointed out, that the nonlinear refractive indices n2 and of F300 and the reference differ significantly (10 %), as shown in Figs. 6 and 7. The Heraeus F300 sample possesses a tiny content of OH (< 1 ppm), which was achieved by adding chlorine to the glass mixture during manufacturing. It is assumed that the increased n2 value may be due to residual chlorine.
Additional empirical correlations between nonlinear optical properties and the band gap energy Egap, as for instance proposed in [42], [45] and [46] will become amenable as soon as the Egap values of the SAL glasses are determined.
4.2. Microscopic models
On the microscopic level, the physical origin of both refractive indices lies in the deformation of the electron cloud of the glass atoms by the incident light wave, i.e. n0 and n2 are directly related to the polarizability α and the hyperpolarizability γ, respectively. These electronic glass properties are most important in the fs time regime [47], irrespective of counteracting slow processes contributing to n2 such as the nuclear motional response of the ions or thermal and electrostrictive effects.
The polarizability of La2O3 and of selected SAL glasses were investigated by Zhao et al. [48] and Duffy et al. [49]. Applying the Clausius-Mosotti equation to the measured n0 values (Table 2) reveals the average polarizability of the SAL-systems (αmeasured in Table 3) representing the weighted average of the molecular polarizabilities of all ions in the glass system. The polarizability of the whole system is, however, considered to be mostly affected by the anions (O2−) [47, 50] due to their large volume whereas the polarizability of the cations appears to stay less important. The oxygen polarizability follows the relation [35, 50]
with Vm being the molar volume, ∑αi = x2αLa3+ + yαSi4+ +(1 − x−y)2αAl3+ the molar cation polarizability, and NO2− = 3x + 2y + 3(1 − x − y) the number of oxide ions. The cation polarizabilities are taken from Duffy et al. [49]. The anion polarizabilities calculated in the present work are very close to those reported from similar systems in [49].Table 3. Optical parameters n2, , αO2−, αcation, αmeasured, 〈γmeasured〉, and the r value for each SAL and the fused silica (F300) sample
The average cation polarizability αcation in Table 3 increases with the La2O3 content by more than a factor of two whereas the O2− polarizability αO2− increases due to the O2− volume change by less than 6.5 %. The anion polarizability is, however, generally higher at least by a factor of three. Nevertheless, the sum of both, αcation + αanion, is still smaller than the average αmeasured value (Table 3). Replacing SiO2 in the SAL system by Al2O3 increases the average polarizability by reducing the density of covalent bonds and the generation of a high concentration of non-bridging oxygen (NBO) [17,19]. As a measure of the NBO concentration a r-value is assigned to the SAL system according to Iftekhar et al. [17]
with x and y being again the molar concentrations of La2O3 and SiO2, respectively. The r-values in Table 3 show an increase of about 23 % indicating a significant NBO increase with the La2O3 enrichment. The exact NBO number is, however, hard to estimate without knowing the fraction of 3-coordinated anions. This is a further necessary parameter to identify the NBO number. Another indicator of a NBO enrichment is the measured increase in the Urbach energy EU (Table 2), representing the systems order [36]. In a less ordered system, less oxygen can bind to the cations, leading to more NBO species.On the microscopic scale the rise of the nonlinear refractive index n2 of the SAL systems in Fig. 6 can in a simplified consideration of one single polarizable constituent only (La2O3 in this case) be discussed by the relationship γ = Qα2 [47] with Q being an empirically determined constant. reveals the average hyperpolarizability 〈γ〉 of the SAL systems by applying the formula
where f is the Lorentz local field correction factor defined as and N is the number density of the glass calculated from the glass density and stoichiometry [44]. Thus the correlation constant Q between the α2 and γ for the present SAL systems is calculated to Q = (7.8 ± 0.6)×1010esu/cm6 (Fig. 8).Fig. 8 Correlation γ = Qα2 between the hyperpolarizability γ and the polarizability α. For the investigated SAL systems Q = (7.8 ± 0.6)×1010esu/cm6.
As discussed previously, the linear polarizability α is not only affected by the oxygen (due to its larger volume and NBO concentration), but also by the increasing content of La3+ ions. Thus the La3+ content of the glasses should as well impact the hyperpolarizability γ. This appears self-evident, when comparing the measured values with those calculated on the basis of the oxygen hyperpolarizabilities given by Adair et al. [44] (Fig. 7). Assuming an additive nature of the constituents’ hyperpolarizabilities [29] can be normalized by the function
revealing an upper limit for the La2/3O hyperpolarizability. Here N and γ are the number densities and hyperpolarizabilities of the SAL components. γSi1/2O and γAl2/3O were taken from [44].The molar fractions normalized to the oxygen were chosen for a direct comparison to the values of γLa2/3O given by Adair et al. [44]. Due to the impact of NBO onto γ an exact calculation is not possible, yet. The upper limit can be calculated by fitting n2 Norm and the number density of La2/3O linearly. It amounts to (2.2 ± 0.2)×10−36 esu (Fig. 9).
Fig. 9 Normalized n2 values according to (7) as function of the number density of La2/3O. The upper limit of γLa2O3 amounts to (2.2 ± 0.2)×10−36 esu.
Finally it should be pointed out, that the linear fits in Figs. 8 and 9 yield nonzero abscissa scales. This effect is attributed to the approximation of only one polarizable constituent of the glasses given by BGO theory. Such a constituent is given by La2O3, within the investigated parameter scale. With decreasing La2O3 content, however, this approximation looses its validity: the nonlinear refractive index and hence the hyperpolarizability will increasingly be affected by structural changes such as a decreasing NBO concentration as well as the neglected contributions of the other glass components.
5. Conclusion
The influence of La2O3 onto the linear n0 and nonlinear n2 refractive indices of SiO2-Al2O3-La2O3 (SAL) glasses was investigated. These systems were proposed to be suitable candidates as novel host material for high power fiber and bulk laser media, core material for optical fibers with a NA > 0.4 and as material for nonlinear optical applications.
The nonlinear increment Δn2/[La2O3] was determined to be (0.264 ± 0.007)×10−16 cm2/W per mol% of La2O3 by using the Z-scan method. Consequently, the hyperpolarizability γ could be calculated. Additionally, the linear refractive indices n0 (800 nm) and nd (587.6 nm) were determined via spectral measurements, revealing the polarizability α of the SAL systems. nd and n2 were correlated by the BGO theory. Thus the proportionality constant Q linking α2 and γ could be estimated to be Q = (7.8 ± 0.6)×1010 esu/cm6. Furthermore an upper limit of the = (2.2 ± 0.2)×10−36 La2/3O hyperpolarizability in the SAL systems was determined to γLa2/3O esu. An enhancement in the oxygen polarizability, an enrichment of non-bridging oxygen, and the increase of heavy La3+ cation concentrations are considered to be responsible for the raise of n2 with increasing La2O3 content.
Acknowledgments
The authors would like to thank the Free State of Thuringia and the European Regional Development Fund (EFRE) for their support within the framework of the LASIL Project (contract number 2012 VF 0020) and the NEODIN Project (contract number TNA I-1/2010). We wish to thank P. Dittmann and A. Ludwig for their help in preparing glass melts and for glass characterization, J. Dellith and A. Scheffel for EPMA analysis, and M. Arnz for technical support.
References and links
1. M. J. Dejneka, B. Z. Hanson, S. G. Crigler, L. A. Zenteno, J. D. Minelly, D. C. Allan, C. Douglas, W. J. Miller, and D. Kuksenkov, “La2O3-Al2O3-SiO2 Glasses for High-Power, Yb3+-Doped, 980-nm Fiber Lasers,” J. Am. Ceram. Soc. 85(5), 1100–1106 (2002). [CrossRef]
2. S. Kuhn, A. Herrmann, J. Hein, M. C. Kaluza, and C. Rüssel, “Sm3+-doped La2O3Al2O3SiO2-glasses: structure, fluorescence and thermal expansion,” J. Mater. Sci. 48(22), 8014–8022 (2013). [CrossRef]
3. A. V. Kiryanov, M. C. Paul, Y. O. Barmenkov, S. Das, M. Pal, and L. Escalante-Zarate, “Yb3+ Concentration Effects in Novel Yb Doped Lanthano-Alumino-Silicate Fibers: Experimental Study,” IEEE J. Quantum Electron. 49(6), 528–544 (2013). [CrossRef]
4. Q. Zhang, J. Ding, Y. Shen, G. Zhang, G. Lin, J. Qiu, and D. Chen, “Infrared emission properties and energy transfer between Tm3+ and Ho3+ in lanthanum aluminum germanate glasses,” J. Opt. Soc. Am. B 27(5), 975–980 (2010). [CrossRef]
5. H. Yang, G. Lakshminarayana, S. Zhou, Y. Teng, and J. Qiu, “Cyan-white-red luminescence from europium doped Al2O3-La2O3-SiO2 glasses,” Opt. Express 16(9), 6731–6735 (2008). [CrossRef] [PubMed]
6. K. Richardson, D. Kroll, and K. Hirao, “Glasses for Photonic Applications,” Int. J. Appl. Glass Sci. 1(1), 74–86 (2010). [CrossRef]
7. J. Kirchhof and A. Funke, “Reactor Problems in Modified Chemical Vapour Deposition (II). The mean viscosity of quartz glass reactor tubes,” Cryst. Res. Technol. 21(6), 763–770 (1986). [CrossRef]
8. S. Manning, H. Ebendorff-Heidepriem, and T. M. Monro, “Ternary tellurite glasses for the fabrication of nonlinear optical fibres,” Opt. Mater. Express 2(2), 140–152 (2012). [CrossRef]
9. D. Litzkendorf, S. Grimm, K. Schuster, J. Kobelke, A. Schwuchow, A. Ludwig, J. Kirchhof, M. Leich, S. Jetschke, J. Dellith, J.-L. Auguste, and G. Humbert, “Study of Lanthanum Aluminum Silicate Glasses for Passive and Active Optical Fibers,” Int. J. Appl. Glass Sci. 3(4), 321–331 (2012). [CrossRef]
10. D. Ehrt, H. T. Vu, A. Herrmann, and G. Völksch, “Luminescent ZnO-Al2O3-SiO2 Glasses and Glass Ceramics,” Adv. Mater. Res. 39–40, 231–236 (2008). [CrossRef]
11. M. Tiegel, A. Herrmann, C. Rüssel, J. Körner, D. Klöpfel, J. Hein, and M. C. Kaluza, “Magnesium aluminosilicate glasses as potential laser host material for ultrahigh power laser systems,” J. Mater. Chem. C 1, 5031–5039 (2013). [CrossRef]
12. R. Weber, S. Sen, R. E. Youngman, R. T. Hart, and C. J. Benmore, “Structure of High Alumina Content Al2O3SiO2 Composition Glasses,” J. Phys. Chem. B 112(51), 16726–16733 (2008). [CrossRef] [PubMed]
13. P. Florian, N. Sadiki, D. Massiot, and J. P. Coutures, “27Al NMR Study of the Structure of Lanthanum- and Yttrium-Based Aluminosilicate Glasses and Melts,” J. Phys. Chem. B 111(1), 9747–9757 (2007). [CrossRef] [PubMed]
14. S. Tanabe, K. Hirao, and N. Soga, “Elastic Properties and Molar Volume of Rare-Earth Aluminosilicate Glasses,” J. Am. Ceram. Soc. 75(3), 503–506 (1992). [CrossRef]
15. I. Pozdnyakova, N. Sadiki, L. Hennet, V. Cristiglio, A. Bytchkov, G. J. Coutures, and D. L. Price, “Structures of lanthanum and yttrium aluminosilicate glasses determined by X-ray and neutron diffraction,” J. Non-Cryst. Solids 354(18), 2038–2044 (2008). [CrossRef]
16. S. Iftekhar, E. Leonova, and M. Edén, “Structural characterization of lanthanum aluminosilicate glasses by 29Si solid-state NMR,” J. Non-Cryst. Solids 355(43–44), 2165–2174 (2009). [CrossRef]
17. S. Iftekhar, J. Grins, and M. Edén, “Composition-property relationships of the La2O3Al2O3SiO2 glass system,” J. Non-Cryst. Solids 356(20–22), 1043–1048 (2010). [CrossRef]
18. A. Aronne, S. Esposito, and P. Pernice, “FTIR and DTA study of lanthanum aluminosilicate glasses,” Mater. Chem. Phys. 51(2), 163–168 (1997). [CrossRef]
19. N. J. Clayden, S. Esposito, A. Aronne, and P. Pernice, “Solid state 27Al NMR and FTIR study of lanthanum aluminosilicate glasses,” J. Non-Cryst. Solids 258(1–3), 11–19 (1999). [CrossRef]
20. J. Marchi, D. S. Morais, J. Schneider, J. C. Bressiani, and A. H. A. Bressiani, “Dispersion of the nonlinear refractive index in sapphire,” J. Non-Cryst. Solids 351(10–11), 863–868 (2005). [CrossRef]
21. B. Schmidt, S. Laimgruber, W. Zinth, and P. Gilch, “A broadband Kerr shutter for femtosecond fluorescence spectroscopy,” Appl. Phys. B 76(8), 809–814 (2003). [CrossRef]
22. D. Blömer, A. Szameit, F. Dreisow, T. Schreiber, S. Nolte, and A. Tünnermann, “Nonlinear refractive index of fs-laser-written waveguides in fused silica,” Opt. Express 14(5), 2151–2157 (2006). [CrossRef] [PubMed]
23. S. Couris, M. Renard, O. Faucher, B. Lavorel, R. Chaux, E. Koudoumas, and X. Michaut, “An experimental investigation of the nonlinear refractive index (n2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369(3–4), 318–324 (2003). [CrossRef]
24. D. Milam, “Review and Assessment of Measured Values of the Nonlinear Refractive-Index Coefficient of Fused Silica,” Appl. Opt. 37(3), 546–550 (1998). [CrossRef]
25. A. Major, F. Yoshino, I. Nikolakakos, J. S. Aitchison, and P. W. E. Smith, “Dispersion of the nonlinear refractive index in sapphire,” Opt. Lett. 29(6), 602–604 (2004). [CrossRef] [PubMed]
26. D. Milam, J. M. Weber, and J. A. Glass, “Nonlinear refractive index of fluoride crystals,” Appl. Phys. Lett. 31(12), 822–825 (1977). [CrossRef]
27. Z. Yu, L. Gundlach, and P. Piotrowiak, “Efficiency and temporal response of crystalline Kerr media in collinear optical Kerr gating,” Opt. Lett. 36(15), 2904–2906 (2011). [CrossRef] [PubMed]
28. J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Copudec, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mat. 32(9), 1002–1006 (2010). [CrossRef]
29. N. Boling, A. Glass, and A. Owyoung, “Empirical relationships for predicting nonlinear refractive index changes in optical solids,” IEEE J. Quantum Electron. 14(8), 601–608 (1978). [CrossRef]
30. Heraeus Quarzglas GmbH & Co. KG, “Quartz Glass for Optics - Data and Properties” (Heraeus Quarzglas GmbH & Co. KG, 2014), optics.heraeus-quarzglas.com/media/webmedia_local/downloads/FusedsilicaandQuartzGlassforOpticsDataandProperties.pdf.
31. M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14(17), 955–957 (1989). [CrossRef] [PubMed]
32. R. E. Bridges, G. L. Fischer, and R. W. Boyd, “Z-scan measurement technique for non-Gaussian beams and arbitrary sample thicknesses,” Opt. Lett. 20(17), 1821–1823 (1995). [CrossRef] [PubMed]
33. S. Zaynobidinov, R. G. Ikramov, and R. M. Jalalov, “Urbach energy and the tails of the density of states in amorphous semiconductors,” J. Appl. Spectrosc. 78(2), 223–227 (2011). [CrossRef]
34. I. T. Godmanis, A. N. Trukhin, and K. Hübner, “Exciton-phonon interaction in crystalline and vitreous SiO2,” Phys. Status Solidi B 116(1), 279–287 (1983). [CrossRef]
35. M. Abdel-Baki, F. A. Abdel-Wahab, A. Radi, and F. El-Diasty, “Factors affecting optical dispersion in borate glass systems,” J. Phys. Chem. Solids 68(8), 1457–1470 (2007). [CrossRef]
36. F. El-Diasty, F. A. Abdel-Wahab, and M. Abdel-Baki, “Optical band gap studies on lithium aluminum silicate glasses doped with Cr3+ ions,” J. Appl. Phys. 100, 093511 (2006). [CrossRef]
37. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990). [CrossRef]
38. K. Saito and A. J. Ikushima, “Absorption edge in silica glass,” Phys. Rev. B 62(13), 8584–8587 (2000). [CrossRef]
39. H. A. Lorentz, “Ueber die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der Körperdichte,” Ann. Phys. 245(4), 641–665 (1880). [CrossRef]
40. L. Lorenz, “Ueber die Refractionsconstante,” Ann. Phys. 247(9), 70–103 (1880). [CrossRef]
41. M. Bache, H. Guo, B. Zhou, and X. Zeng, “The anisotropic Kerr nonlinear refractive index of the beta-barium borate (β-BaB2O4) nonlinear crystal,” Opt. Mater. Express 3(3), 357–382 (2013). [CrossRef]
42. K. Tanaka, “Optical nonlinearity in photonic glasses,” J. Mater. Sci. - Mater. Electron. 16(10), 633–643 (2005). [CrossRef]
43. R. L. Sutherland, Handbook Of Nonlinear Optics (Marcel Dekker, 1996).
44. R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39(5), 3337–3350 (1989). [CrossRef]
45. M. E. Lines, “Oxide glasses for fast photonic switching: A comparative study,” J. Appl. Phys. 69(10), 6876–6884 (1991). [CrossRef]
46. K. Tanaka, “Nonlinear optics in glasses: How can we analyze?” J. Phys. Chem. Solids 68(5–6), 896–900 (2007). [CrossRef]
47. R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive-index measurements of glasses using three-wave frequency mixing,” J. Opt. Soc. Am. B 4(6), 875–881 (1987). [CrossRef]
48. X. Zhao, X. Wang, H. Lin, and Z. Wang, “Electronic polarizability and optical basicity of lanthanide oxides,” Physica B 392(1–2), 132–136 (2007). [CrossRef]
49. J. A. Duffy, “Polarisability and polarising power of rare earth ions in glass: an optical basicity assessment,” Phys. Chem. Glasses 46(1), 1–6 (2005).
50. V. Dimitrov and T. Komatsu, “An interpretation of optical properties of oxides and oxide glasses in terms of the electronic ion polarizability and average single bond strength (review),” J. Chem. Technol. Metall. 45(3), 219–250 (2010).
