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Abstract
In this report we review the literature concerning the theory of scattering in transparent glass ceramics (TGCs). The parameters dictating scattering are all relational and include the size of the scatterers relative to the wavelength of propagating light, the volume fraction of crystals relative to the vitreous matrix and the refractive index difference between crystal and glass. We extend the theory by including material dispersion for the case of low, i.e., <5% volume fraction of nanocrystals relative to the vitreous matrix where Rayleigh scattering is appropriate. We examine five crystal-glass sets and calculate the scattering coefficient from the visible to the mid-infrared. We show that for all combinations, the scattering coefficient departs from the fixed refractive index case by as much as 48% at 2.5 µm for BaF2 nanocrystals in an aluminosilictate glass. For every case studied, the addition of material dispersion reduces the dependence on wavelength from the standard λ−p where p = 4 to as much as p = 4.8 for the case of PbF2 nanocrystals in an aluminosilictate glass. For the accurate estimation of the Rayleigh scattering coefficient in a TGC, knowledge of material dispersion is therefore essential.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Glass ceramics in which a glass matrix contains small, usually nanoscale, crystals were first discovered by Stookey [1]. There are several methods to fabricate TGCs. Precisely controlled heating and cooling of a precursor multicomponent glass beyond the crystallisation temperature can produce nucleation and precipitation of nanocrystals into the vitreous matrix [2]. Alternatively, nucleating agents such as TiO$_{2}$ or ZrO$_{2}$ [3,4] can be added, but care must be taken so as to not introduce additional optical loss from impurities. They are of interest to the optics community because they can display an unusually high degree of transparency despite the typically large volume fraction of nanocrystals relative to the vitreous matrix. As such, they are now referred to as transparent glass ceramics (TGCs). One of the key reasons for this interest relates to the fact that a TGC can allow a separation between the optical properties of the crystal from the optical properties of the glass.
For the creation of optical gain, rare earth ions are doped into either a crystal or glass. For TGCs, it has been shown for several oxyfluoride glass systems that the rare earth ion partitions almost entirely into crystal because of the strong affinity between the rare earth ion and fluorine ion. In this case, the fluorescence properties of the rare earth ion are entirely governed by the crystal. For other systems, e.g., the common oxides, partitioning to the crystal phase is less complete. Crystal properties such as increased rare earth doping without clustering and reduced maximum phonon energies can be accessed whilst maintaining the processing flexibility of the vitreous matrix. As such, many TGCs have been explored to improve florescence whilst maintaining the ease of fabrication of a glass [5] for the creation of new gain media for lasers and optical amplifiers.
The presence of crystals within a glass creates scattering. When large and efficiently generated optical gain is needed, an understanding of the degree of scattering is critical. The scattering is dependent on the volume fraction of crystals relative to the glass, the size of the nanocrystal relative to the wavelength of the propagating light, and the refractive index difference between the crystal and glass. It has been shown that the measured degree of scattering is lower than that calculated using Rayleigh scattering alone, suggesting that coherence effects such as interparticle interference increase transmission.
In this report, we briefly review the literature that is concerned with theoretically modelling scattering in TGCs in order to highlight the major concepts underpinning scattering by both particles in a vitreous matrix and spatial fluctuations of the refractive index of the matrix. As a starting point, we restrict our investigation to the conditions in which Rayleigh scattering is appropriate. To do this, we confine the problem to particle scattering in which the volume fraction of particles is <5% and scattering is incoherent, meaning no interparticle interference is present. We investigate the effects on the scattering coefficient when literature-sourced material dispersion parameters [6] are added to the model for Rayleigh scattering. We show that for all TGCs that we examined the material dispersion alters the degree of scattering relative to Rayleigh scattering for fixed set of refractive indices. For the case of a BaF$_2$ nanocrystal in silica TGC, we show for wavelengths around 833 nm that scattering is removed because of index matching. Whilst this study is confined to silicate and fluoride glasses, material dispersion will potentially play a greater role in chalcogenide (and oxide chalcogenides) TGCs because the crystal and glass can be composed of the same anion making index matching more likely.
2. Background and related works
2.1 Historical overview of the concepts developed for light scattering in TGCs
It is widely known that for Rayleigh scattering to be relevant, the following condition must be met, $x<<1$ whereby $x = \frac {2\pi r}{\lambda }$, where r is the radius of the scattering particle and $\lambda$ is the wavelength of the incident light. Another important aspect of Rayleigh scattering is the assumption that particles are randomly spaced, resulting in the scattered light being incoherent. Debye and Bueche presented their theory for scattering by an inhomogeneous solid [7] in 1949 using a correlation function to determine the amount of light scattered by an inhomogeneous solid, such as a TGC. The correlation function gives the relationship between distance and local variations that are superimposed onto an average dielectric constant. Debye was able to simplify the problem by utilising a simple correlation function reducing to an equation for the absolute value of scattered intensity that closely aligns with the theory proposed by Rayleigh fifty years earlier.
Following Debye, Hopper released a series of papers on the stochastic theory of scattering from idealised and spinodal structures in the mid-80s [8,9]. In [8], Hopper presents a general theory relating to the use of the autocorrelation function (ACF) as a means of analytically calculating scattering. The ACF is a time domain measure of the stochastic process memory and is able to reveal how any two values of a signal change as their distance changes [10]. Edgar et al. [11] applied derivations based on Rayleigh and Debye’s work and compared the results with experimental results based on fluorozirconate glass ceramics involving barium chloride ($BaCl_2$) crystals. The results indicated that the glass ceramic was six times more transparent than could be predicted by the Rayleigh and Debye’s models, aligning with previously developed conclusions that Rayleigh and Debye’s models were vastly overestimating the amount of scattering in TGCs.
Shortly after, Edgar proposed another explanation for the observed high transparencies of glass ceramics [12]. Continuing on from Rayleigh and Debye’s work, the equation
The work done by Edgar et al. in [11] was continued by Shepilov in [13], focusing on fluorozirconate glass ceramics containing barium chloride crystals. Similar to the results of Edgar et al., Shepilov calculated that the turbidity had been reduced by a factor of five. The calculations carried out by Shepilov were done on the basis of mondisperse nano-crystals with a diameter of 11 nm. The crystals were present in the host glass at a volume fraction of 20%. Calculations conducted by Shepilov in this work directly contradicted the results of Edgar et al., who concluded that the high transparency of glass ceramics was independent of interference effects. Shepilov instead proposed that the discrepancy is caused entirely by interference effects, at least in the wavelength region between 400 and 700 nm.
Many more papers have been presented in the last decade and a half, however, they have been more focused on experimental work and applying the existing theory. No attempt was made to unify or even explain all the theories together until Borrelli et al. in 2018 [14]. Borrelli et al. categorised the theories into two distinct approaches: Single Particle Scattering (such as the work done by Rayleigh) and Correlation Function Scattering (such as the work done by Hopper). The requirements for the applicability of Rayleigh scattering, i.e. that the particle size is significantly smaller than the wavelength of incident light and that the density of crystals is low, allows for the assumption that the field at a crystal is uniform and thus the effects of interference between scattered waves are negligible. It is heavily emphasised that if the volume fraction of the glass ceramic is significantly large it is no longer feasible to treat the crystals as independent sources. Interference effects must be taken into account. Borrelli et al. use Rayleigh scattering for very limited, low density cases with small particles and preferentially use equations derived from Hopper’s theory for more dense systems.
One of the key arguments that has been used to justify Rayleigh scattering is the presence of a dependence on wavelength to the power of negative four that was initially proposed by Rayleigh and has been found experimentally numerous times, although often at notably different magnitudes. However, Borrelli et al. were able to find this dependence working from Hopper’s ACF theory, and thus concluded that using the often found proportionality as evidence to the accuracy of Rayleigh scattering is a fallacy, and note that it breaks down regardless in the regime of Mie scattering.
At the same time, Shepilov et al. published an article revising the current understanding of the field of light scattering in glass ceramics [15]. The particular focus of this paper was to review the published literature to analyse whether traditional models of light scattering, such as Rayleigh scattering and Mie scattering, were sufficient to model most TGCs with the exception of some notable edge cases, or whether these particular cases made up the bulk of scattering examples in TGCs. In all cases, the amount of light scattered is dependent on the wavelength to the inverse power of some value. For Rayleigh scattering, this value $=4$, whereas for Mie scattering it is $\le 4$, and in the presented edge cases it is $>4$. The scattering that was displayed by these edge cases was given the name Anomalous Light Scattering (ALS). There are, however, few papers that have considered the ALS problem in detail. What is known is the shape, orientation, polydispersity, and their distinct refractive indices influence the degree of light scattered due to ALS.
ALS relies on the relationship between wavelength and the extinction coefficient of scattering. The ALS model has not yet been fully developed into an equation that can be directly applied, however a general form has been highlighted in multiple papers [15,16].
where the extinction coefficient $\alpha _s(\lambda )$ is dependent on some arbitrary constant $a$ multiplied by wavelength $\lambda$ to the inverse power of some constant $p$. In order for this relationship to be considered ALS, p must be greater than 4. If p = 4, this would indicate that Rayleigh scattering was dominant and p < 4 would indicate Mie scattering is dominant [15].Recently [16], it was concluded that the wavelength dependency of light scattering is to the inverse power of 7.1. This is highly notable, as it provides a further indication that even in the limiting case of low volume fractions of crystals, light scattering is categorised by ALS instead of pure Rayleigh scattering. In other words, even when the nanocrystals are sparsely distributed within the glass ceramic, interference effects are key to predicting the amount of light scattered and the particles cannot be considered independent scatterers. Shepilov posits that this is being caused by the relationship between the size of the particles and the wavelength of light. If the distance between particles is much less than the wavelength of the incoming light, the material will be treated as an almost homogeneous material, resulting in far less scattering than would be predicted by a model of independent scatterers.
Despite the growing consensus that interparticle scattering is most likely the cause of the observed increased transparency, in this investigation we re-examine Rayleigh scattering relevant to TGCs with the inclusion of material dispersion as an additional wavelength-dependent parameter to the problem. We note that for volume fractions of the nanocrystals of <5% that less interparticle scattering occurs and therefore interference can be neglected. In this case, the total scattering cross section is the sum of the scattering from individual incoherent scatterers. There are some recent indications that interference effects impact the scattering significantly even at small volume fractions where it is expected that Rayleigh would apply [16], however, more work is needed to confirm this.
Low density particle scattering as examined here is based on the work done by Rayleigh [17] and Debye [7]. This model relies on two key assumptions. The first is that the volume fraction of crystals is sufficiently low such that interference effects become negligible. The exact number at which the interference effects become significant has not been agreed on, but it is at least less than 5% [13]. The second key assumption is that the scattering particles are much smaller than the wavelength of the incoming light. This assumption is governed by $\frac {2\pi r}{\lambda } << 1$. It should also be noted that the work done by Edgar et al. [11,12] also falls under this method as it is an application of Rayleigh scattering.
Also important to note is that almost all of the presented works have made the assumptions that the scattering particles are monodisperse, hard spheres, with the exception of the most recent works done by Shepilov [15,16] which considered polydisperse hard spheres. This an extremely important consideration, as almost all TGCs that are physically created will be a system of polydisperse spheres. For the sake of simplicity, however, a monodisperse system is usually considered when doing any theoretical calculations. The assumption of hard spheres is also important. By assuming hard spheres, it is consequently assumed that the effects of absorption are negligible. This is key in simplifying the math used to model the transparency of the glass, negating the need for the introduction of complex numbers into the calculation.
3. Theory
3.1 Light scattering theory
3.1.1 Rayleigh
The equations presented here are derivations based on the theory presented in [17], but do not come strictly from that paper. Equation (3) [18] and Eq. (4) [19] are used to calculate the intensity of scattered light by any single particle due to Rayleigh scattering in a vacuum and the Rayleigh scattering cross-section in a vacuum, respectively.
The cross-section averages the intensity calculated in Eq. (3) over all angles of scattering. $d$ and $n$ represent the diameter and refractive index of the spherical scattering particles, $\lambda$ and $I_0$ are the wavelength and intensity of the incident light, $R$ is the distance to the particle, $\theta$ is the angle of scattering, and $I$ is the intensity of the scattered light. $\sigma _S$ is the Rayleigh scattering cross-section. To calculate the fraction of light scattered by a medium, the scattering cross-section is multiplied by the density of particles per unit volume.
In order to calculate the amount of light scattered by a transparent dielectric, as is the case for TGCs, a slightly different equation is needed, i.e.,
Equation (5) [13] can be used to calculate the extinction coefficient, $\alpha _R(\lambda )$ for light scattered by a particle in any medium, and is thus applied to calculate the amount of light that is expected to be scattered by a TGC. In this equation $\nu$ represents the volume fraction of crystals in the medium, $D$ represents the diameter of the crystals, $n_m$ represents the refractive index of the glass medium, and $m$ is the relative refractive index ratio $\frac {n_c}{n_m}$ where $n_c$ and $n_m$ is the refractive index of the nanocrystals and matrix, respectively.
3.2 Material dispersion theory
3.2.1 Sellmeier
Sellmeier’s equation is an empirically derived equation based on the Cauchy Equation [20] that relies on constants determined for each material using a least-square fitting procedure over a wide range of wavelengths [21]. Of the two equations, Sellmeier’s equation is preferred to calculate the refractive index of a material because, while the Cauchy equation is simpler and highly accurate within the visible spectrum, it quickly breaks down for longer wavelengths. Comparatively, Sellmeier’s equation whilst slightly more complicated, is able to accurately predict the refractive index over a wide range of wavelengths and therefore is our preferred equation.
The Sellmeier equation presented in the most general form is an infinite series [22]:
where $n$ is the refractive index of the material, $A$, $B_i$, and $C_i$ are the experimentally derived Sellmeier coefficients of the material. It should be noted that in many versions of the equation $A \equiv 1$, however there are some cases where this was not the case for the experimentally derived coefficients and as such it has been treated as a constant variable. The constant values are typically determined for a wavelength $\lambda$ measured in microns. In this equation, each term typically represents an absorption resonance of the material that has a strength of $B_i$ and occurs at a wavelength of $\sqrt {C_i}$ microns [23].In most cases, the equation is limited to three terms:
While the three term equation is the most commonly used, a two term version also appears on occasion,. Further, another variation of the equation is seen below [24].
4. Methodology
4.1 Model development
Table 1 lists the experimentally determined Sellmeier coefficients for a series of glasses and crystals sourced from the literature. There were several materials where the experimentally derived coefficients were calculated for the two-term Sellmeier equation. In those cases, the coefficients of the third term were set to zero, which will result in the third term equating to zero and thus cancelling the third term out of the sum, creating an effective two term equation without having to define a separate function. There was only one material for which the standard three term equation could not be applied, ZnS.
Table 1. Database of Sellmeier coefficient for various materials commonly found in TGCs.
The Rayleigh function of the model was benchmarked against calculations regarding independent scattering as determined by Rayleigh scattering done by Shepilov in 2008 [13]. The theoretical model developed here matched the results of the calculations done by Shepilov. Unfortunately, benchmarking the Rayleigh function against experimental data proved difficult because most work done in the field to date has indicated non-negligible interparticle interference effects.
5. Results
The model was applied to fused silica glass, aluminosilicate glass, and ZBLAN glass. For each type of glass, the material dispersion was calculated and the refractive index was plotted with respect to wavelength, along with the difference in refractive indexes of the composite material. The material dispersion calculation was then fed into the calculation of Rayleigh scattering, with a plot of the extinction coefficient being returned.
5.1 Fused silica matrix
Both sets of coefficients were modelled in a theoretical glass ceramic containing barium fluoride crystals over a range of wavelengths from 0.4 $\mu$m to 2.5 $\mu$m, seen in Fig. 1, which demonstrates the drastic effect that a discrepancy in the Sellmeier coefficient can cause. Using the 1994 Sellmeier coefficients, the difference in refractive index falls within 0.02, compared to the 0.7 seen with the 2020 Sellmeier coefficients.
Fig. 1. Comparison of theoretical chromatic dispersion of fused silica glass and BaF$_2$ using two different sets of Sellmeier coefficients.
Fig. 2. Extinction coefficient of a TGC with BaF$_2$ crystals in a fused silica glass matrix using two different sets of Sellmeier coefficients. The solid line represents the case with dispersion considered, while the dotted line represents the case where the refractive index of both materials was taken to be equivalent to the 550 nm values for all wavelengths. For this case, the volume fraction was taken to be 0.01 and the diameter of the crystals was taken to be 11 nm.
It is clear that material dispersion significantly alters the Rayleigh scattering coefficient creating large departures both above and well below the scattering coefficient calculated for fixed refractive index. The 2020 values indicate that there will be no Rayleigh scattering at $\approx$ 0.833 $\mu$m, evidenced by the sharp downwards peak, before plateauing as it approaches longer wavelengths. This result is both mathematically and theoretically sound. The peak matches the point at which the difference in the expected refractive indices of the two materials is equal to zero. Mathematically, the Rayleigh function applies Eq. (5), which is expected to go to zero when the refractive indices of the two materials are equal.
5.2 Aluminosilicate matrix
For the aluminosilicate glass, two theoretical TGCs were investigated, both containing fluoride crystals. Material dispersion was investigated for both theoretical glass ceramics, with the refractive index of the composite materials being compared. The effects of differing volume fractions and crystal diameters were then investigated.
5.2.1 Barium fluoride
Figure 3 shows the comparative effects of material dispersion on the two materials for wavelengths between 0.4 $\mu$m and 2.5 $\mu$m. It is clear that the refractive index of aluminosilicate decreases at a faster rate than that of barium fluoride for the entire spectrum, resulting in the curve in Fig. 3 decreasing steadily throughout the range. At no point do the refractive indices intersect, and the difference in the refractive indices changes by less than 0.025 across the spectral band.
Accounting for the effects of material dispersion, the extinction coefficient generally decreases faster as the wavelength increases compared to the fixed refractive index case. Using a regression analysis, we calculate that p = 4.4 and the maximum departure of 47% from the fixed refractive index case occurs at 2.5 $\mu$m. We note that as the the wavelength increases, the p parameter becomes larger as seen in Fig. 4 but calculating beyond 2.5 $\mu$m is meaningless because of absorption by aluminosilicate glass.
Fig. 4. Comparison of extinction coefficients of a TGC with BaF$_2$ crystals in a aluminosilicate glass matrix with (a) a variety of volume fractions from 1% to 5% with a 11nm crystal diameter and (b) a variety of crystal diameters from 50nm to 200nm with a volume fraction 0.01. The solid line represents the case with dispersion considered, while the dotted line represents the case where the refractive index of both materials was taken to be equivalent to the 550 nm values for all wavelengths.
Figure 4(a) and (b) also highlight that the extinction coefficient will increase as the volume fraction and crystal diameter increase respectively. The effects of this increase are more significant for an increased diameter. Mathematically, this aligns with Eq. (5) where the extinction coefficient is proportional to the volume fraction, and to the crystal diameter cubed.
5.2.2 Lead fluoride nanocrystals
Applying the Sellmeier formula at shorter wavelengths the refractive index of lead fluoride decreases significantly quicker than that of aluminosilicate glass, however, at longer wavelengths aluminosilicate’s refractive index decreases slightly faster. This is confirmed in Fig. 5, where it can clearly be seen that the difference in the two refractive indices swiftly decreases until $\approx$ 1.263 $\mu$m, at which point it begins to increase. Throughout the band, the difference in refractive index only changes by <0.06.
Similar to what was observed in Fig. 4 for BaF$_2$ crystals, when accounting for the effects of material dispersion, the extinction coefficient of the TGC decreases again faster relative to the fixed index case as the wavelength increases, although the rate of the decrease slows at longer wavelengths, see Fig. 6. Using regression analysis, we calculate that p = 4.8 and departs from the fixed index case by 32% at 400 nm. The shape of the curves are very similar for both of the crystals in aluminosilicate, although it should be noted that the PbF$_2$ crystals scattering coefficients are essentially an order of magnitude larger than that of the BaF$_2$ crystals. The relationship between volume fraction and extinction coefficient, and the crystal diameter and extinction coefficient discussed for the BaF$_2$ crystals also holds true here.
Fig. 6. Comparison of extinction coefficients of a TGC with PbF$_2$ crystals in a aluminosilicate glass matrix with (a) a variety of volume fractions from 1% to 5% with a 11nm crystal diameter and (b) a variety of crystal diameters from 50nm to 200nm with a volume fraction 0.01. The solid line represents the case with dispersion considered, while the dotted line represents the case where the refractive index of both materials was taken to be equivalent to the 550 nm values for all wavelengths.
5.3 ZBLAN matrix
For ZBLAN fluorozirconate glass, a further two theoretical TGCs were modelled containing the same two fluoride nanocrystals used for the aluminosilicate TGCs. Once again, the material dispersion of the materials was compared with respect to both crystals, and then Rayleigh scattering was applied for a variety of volume fractions and diameters of nanocrystals.
5.3.1 Barium fluoride nanocrystals
The calculations for BaF$_2$ crystals in ZBLAN glass matrix are shown in Fig. 7. The refractive indices are much closer for ZBLAN and BaF$_2$ and the difference in the rate of change is far less pronounced than it was for aluminosilicate glass seen in Fig. 3. In Fig. 7 it can be seen that the difference in refractive index only changes by approximately 0.006 over the entire spectral band.
When accounting for material dispersion, the extinction coefficient of this TGC, seen in Fig. 8 decreases faster relative to the fixed refractive index case with p = 4.3. The largest departure of 34% from the fixed index case occurs at 2.5 $\mu$m. Better index matching with this crystal-glass combination means the extinction coefficient is an order of magnitude smaller than what was seen for barium fluoride in an aluminosilicate glass matrix. The relationship between volume fraction and extinction coefficient, and the crystal diameter and extinction coefficient discussed for aluminosilicate TGCs continues to hold here.
Fig. 8. Comparison of extinction coefficients of a TGC with BaF$_2$ crystals in a ZBLAN glass matrix with (a) a variety of volume fractions from 1% to 5% with a 11nm crystal diameter and (b) a variety of crystal diameters from 50nm to 200nm with a volume fraction 0.01. The solid line represents the case with dispersion considered, while the dotted line represents the case where the refractive index of both materials was taken to be equivalent to the 550 nm values for all wavelengths.
5.3.2 Lead fluoride nanocrystals
The final theoretical TGC modelled in this paper utilised the lead fluoride crystal in an ZBLAN glass matrix. Material dispersion was calculated once more between 0.4 $\mu$m and 2.5 $\mu$m and the difference plotted in Fig. 9 where it is clear that the relationship between the two materials is similar to what was seen in Fig. 6 with a less rapid change in the difference between the refractive indices at longer wavelengths, resulting in the difference in the refractive indices changing by less than 0.07 over the spectral band.
Finally, the coefficient of extinction was calculated for the PbF$_2$ ZBLAN TGC accounting for material dispersion in Fig. 10. The results of this calculation were virtually identical to what was seen in Fig. 6 for PbF$_2$ in aluminosilicate, with a slightly higher coefficient. We calculate that p = 4.7 when material dispersion is included and the largest departure of 28% occurs at 400 nm.
Fig. 10. Comparison of extinction coefficients of a TGC with PbF$_2$ crystals in a ZBLAN glass matrix with (a) a variety of volume fractions from 1% to 5% with a 11nm crystal diameter and (b) a variety of crystal diameters from 50nm to 200nm with a volume fraction 0.01. The solid line represents the case with dispersion considered, while the dotted line represents the case where the refractive index of both materials was taken to be equivalent to the 550 nm values for all wavelengths.
6. Discussion
As expected for Rayleigh scattering with a fixed m ratio that the scattering coefficient has the classic $p = 4$ relationship. With material dispersion included, for all cases we studied in which index matching is not possible, $p > 4$ with a positive or negative departure from the fixed refractive index case depending on the particular TGC. The largest departure can occur at either end of the spectral band, which was defined according to the transparency of silicate glass, again depending on the TGC. Overall, our calculations indicate that it is important to include material dispersion for a particular TGC if an accurate understanding of Rayleigh scattering is required because the departure can be as large as 48%.
Whilst the $p > 4$ relationship was found for the TGCs we studied, we see a sharp change in the scattering coefficient for the case $m = 1$ when the scattering coefficient becomes zero. Of course, this is expected given index matching but what is clear from Fig. 2 is the fact that value of the scattering coefficient is essentially maintained right up to the $m = 1$ condition, i.e., the coefficient does not gradually decay to zero. This is expected given the mathematical dependence of the scattering coefficient on the $m$ parameter. For higher volume fractions, i.e. $>5\%$, when it is expected that interparticle interference will become dominant, we will likely have a different scenario when index matching is possible. For wavelengths distant from the matched wavelength, the inference will be strong and ALS will be present. As the matched wavelength is approached, however, the degree of both scattering and the interference will decrease. What needs to be ascertained in future studies is whether ALS can be maintained right up to the matched condition.
Unfortunately, several difficulties were encountered when trying to compare the model developed here that considered material dispersion to prior models that did not. Ideally, one of the materials studied would have been a ZBLAN fluorozirconate glass with barium chloride crystals. This particular combination was the basis of several key papers [11–13,34–36]. Unfortunately, the Sellmeier coefficients of barium chloride do not appear to have ever been calculated, making it impossible to compare this model to the work done in these papers. Further, one of the benefits of TGCs is their ability to create stable crystals of which some are impossible, or near to, to create in other circumstances. While this is an overall benefit to the field of glass ceramics, it makes the determination of their Sellmeier coefficients extremely difficult and as such for many crystals comprising TGCs their Sellmeier coefficients are unknown.
Ultimately, the developed model is still fairly basic and is not applicable to many TGCs currently being developed. It assumes evenly spaced monodisperse hard spheres and that the volume fraction is sufficiently low that the effects of interparticle interference are negligible allowing Rayleigh scattering to dominate. By assuming hard spheres it is thus assumed that the effects of absorption are negligible, negating the need for complex numbers to be used in the calculations. While these assumptions are used to significantly simplify the model, they are also aligned with the vast majority of the work that has been done to date in the field of light scattering in TGCs. Recent work done by Shepilov [15,16] strongly indicates that for the vast majority of TGCs the dominant light scattering regime is ALS rather than Rayleigh scattering. However, unlike for Rayleigh scattering, no unified set of equations exists for ALS that can be applied to a computational model.
7. Conclusion
Despite the limitations, there is still a strong use case for this model. Most importantly, for the first time, our calculations indicate that material dispersion will have a significant effect on the amount of light scattered by a TGC. We show that the departure can be as large as 48% compared to the case in which a fixed m parameter is used. For all nanocrystal-glass sets that we studied where index matching was not possible, the p parameter was > 4. Whilst this was determined for the simple case of a low volume fraction i.e. Rayleigh scattering only, the degree of ALS is also expected to be highly dependent on material dispersion. Furthermore, while there are strong indications that in most TGCs the propagating light is subject to ALS, this has not yet been proven and is currently being determined for each TGC. Our model can be used to gain a more accurate understanding of Rayleigh scattering for cases in which the coherent properties of both the incoming and scattered light is negligible.
Funding
Asian Office of Aerospace Research and Development (FA2386-19-1-0043); Australian Research Council (DP210102442).
Acknowledgements
The authors would like to acknowledge the useful discussions had with Heike Ebendorff-Heidepriem, Yiqing Lu and Yunle Wei.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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