Angular distribution of light scattered by nanostructured glass predicted from an experimental wavelength dependence of turbidity

Michael P. Shepilov

doi:10.1364/OSAC.1.000939

1. Introduction

Nanostructured glasses (NGs)—phase-separated glasses (PSGs), nanoporous glasses, and glass-ceramics (GCs)—are light-scattering materials due to their inhomogeneous structure. In some wavelength range $(λ_{\min}, λ_{\max})$ comparable to that of visible light, experimental wavelength dependence of the scattering coefficient (SC, turbidity) of NG is often described by a power function [1,2]

α_{s} (λ) ≅ a λ^{- p}, (a, p = constant, λ_{\min} ​ < λ < λ_{\max}),

where

p > 4

(effect 1). It was found that effect 1 for PSGs is accompanied by effect 2: the scattered light is directed predominantly into the backward hemisphere [1]. The scattering demonstrating effects 1 and 2 had been called anomalous light scattering (ALS) [1] to distinguish it from the Rayleigh scattering, which is characterized by

p = 4

and forward-backward symmetry ([3], Chap. 5). For GCs, only effect 1 was noted [2] and experimental study of effect 2 was not carried out.

ALS is connected with interference of light scattered by different elements of the structure. These elements may be (a) phase-separated particles [1,4–7] or crystals [8–10] (interparticle interference), (b) elements of spinodal structure [11], and also (c) particle (crystal) and its diffusion zone [12–16]. It should be noted that in some cases the interference can significantly suppress the light scattering but does not lead to ALS (e.g., such a behavior was noted for spinodal structures [17] and in the system of hard particles [18]).

Recently [19] we assumed that SC of NG obeys Eq. (1) and used the simple model of NG structure to express the phase function for scattering of unpolarized light in terms of parameter $p$ (in other words, the relation between effects 1 and 2 was found). NG was modelled by a system of identical spherical Rayleigh scatterers (phase-separated particles or crystals) distributed in homogeneous glass matrix, and interparticle interference was taken into account in consideration of light scattering in the system. A generalization of this approach to the case of non-identical scatterers is difficult (if ever possible) [20]. The approach is inapplicable to NGs with spinodal-like structure (and especially, at early stages of transformation when the refractive index varies smoothly in space). And finally, the approach cannot be applied to the cases where SC of NG is determined by interference of light scattered by particle (crystal) and its diffusion zone.

In this paper, we use another approach to derive a relation between wavelength dependence of SC and angular distribution of light scattered by NG. The results are compared with results calculated for other models and with experimental data.

2. Relation between wavelength dependence of SC and the angular distribution of light scattered by NG

We assume that SC of NG obeys Eq. (1) with $p > 2$ and apply the approach of Debye and Bueche [21,22] to describe angular dependence of scattered light (i.e., here we consider the more general case than the case $p > 4$ considered earlier [19]).

The approach of Debye and Bueche describes the intensity of light scattered by inhomogeneous medium in the Rayleigh-Debye approximation [22] (otherwise known as the Rayleigh–Gans approximation [3,23,24]) and formulates the problem in terms of correlation function of dielectric constant $є (r)$ . The fluctuations of the dielectric constant are assumed to be small compared to its average value, so that the Rayleigh-Debye approximation is valid ([22], p. 459). Hendy [11] gives the quantitative condition for the validity of the approach, $k L Δ є ≪ 1$ , where $k = 2 π n / λ$ is the wave vector of the incident light propagating in a medium with effective refractive index $n$ , $λ$ is wavelength in vacuum, $Δ є$ is the difference in dielectric constant between scatterer and background media, and $L$ is the mean distance between phases. Bohren and Huffman ([3], Eqs. (6.2) and (6.3)), Mishchenko et al. ([23], Section 7.2), and Dombrovsky and Baillis ([24], Section 2.2.2) present two conditions for the validity of the Rayleigh-Debye approximation (the references were cited ([23], Section 7.2) which demonstrate that the Rayleigh-Debye approximation may often be useful outside its formally defined range of validity). These conditions may be rewritten in the form $Δ є / (2 n^{2}) = Δ є / (2 є) ≪ 1$ , $k d Δ є / (2 n^{2}) ≪ 1$ ( $d$ is characteristic size of inhomogeneity), and differ from the condition of Hendy. Anyway, we will assume that the approach of Debye and Bueche is applicable to calculation of light scattering by NG under consideration.

Let us make two remarks. First, the approach of Debye and Bueche was used for calculation of light scattering by NGs [11,17]. Second, if the amplitude of change in dielectric constant of the medium and the sizes of inhomogeneity regions are sufficiently small, the approach can be applied regardless of the type of structure. This may be a system of monodisperse or polydisperse particles surrounded by diffusional field or distributed in homogeneous matrix, or spinodal structure at any stage of transformation. All information on the light-scattering properties of such a system is contained in the correlation function of the dielectric constant.

Let us consider NG with spatially varying dielectric constant $є (r) = є + η (r)$ where $є$ is the average dielectric constant and $η (r)$ are superimposed local variations ( $〈 η 〉 = 0$ ). The correlation function $γ (r)$ [21] is defined from

{〈 η (r') η (r^{'} + r) 〉}_{r'} = γ (r) {〈 {(η (r'))}^{2} 〉}_{r'} : = γ (r) \bar{η^{2}}

and does not depend on the direction in space for isotropic NG. If the conditions for validity of the Debye and Bueche approach are satisfied (see above), the scattered intensity

I (θ)

coming from a volume

V

under an angle

θ

at a large enough distance

R

is related to the intensity

I_{0}

of the incident light as [21]

\frac{I (θ)}{I_{0}} = \frac{\bar{η^{2}}}{є^{2}} \frac{π^{2} V}{λ_{m}^{4} R^{2}} ω (4 π \sin (θ / 2) / λ_{m}) i (θ) .

Here

λ_{m}

is the wavelength of light in the NG,

q = 4 π sin (θ / 2) / λ_{m}

is the magnitude of the scattering vector

q

, and

ω (q)

is the Fourier transform of the correlation function

ω (q) = \int γ (r) e^{i q r} d^{3} r = 4 π \int_{0}^{\infty} \frac{\sin q r}{q r} γ (r) r^{2} d r

called the correlation-volume by Debye and Bueche [21] and the structure factor by Hendy [11]. The factor

i (θ)

depends on the polarization of the incident light:

i (θ) = i_{⊥} = 1, (incident light polarized perpendicular to the scattering plane, V_{V});

i (θ) = i_{∥} = {cos}^{2} θ, (incident light polarized parallel to the scattering plane, H_{H});

i (θ) = i_{u} = (i_{⊥} + i_{∥}) / 2 = (1 + \cos^{2} θ) / 2, (unpolarized incident light) .

This assumes no depolarization of the scattered radiation. Debye and Bueche [21] considered unpolarized incident light. Some authors [12–14] applied the Rayleigh-Debye approximation to incident light polarized perpendicular to the scattering plane. Kerker [22] used both polarizations (5a) and (5b) in his consideration of the Rayleigh-Debye approximation. In Eqs. (5a) and (5b), we also give the designations

V_{V}

and

H_{H}

which are used in discussion of experiments [25].

Below we use the vacuum wavelength $λ = λ_{m} / n$ and consider that $n^{2} ≅ є$ . Integrating the scattering intensity (3) over all directions in space, one can derive the expression for the SC in the form

α_{s} (λ) = \frac{π^{3} \bar{η^{2}}}{λ^{4}} \int_{0}^{π} \sin θ (1 + \cos^{2} θ) ω (4 π n \sin (θ / 2) / λ) d θ .

The assumption that the wavelength dependence of the SC (6) obeys Eq. (1) with $p > 2$ leads to relation

ω (4 π n \sin (θ / 2) / λ) ≅ b {(4 π n \sin (θ / 2) / λ)}^{p - 4}, (b = constant, p > 2) .

This relation must be satisfied at least for those values of

[4 π n \sin (θ / 2) / λ]

that make the main contribution to the integral in Eq. (6). Then the angular dependence of the scattered intensity (3) is given by

I (θ) \propto P_{p} (θ) ≅ i (θ) {(\sin (θ / 2))}^{p - 4}, (p > 2),

where the phase function

P_{p} (θ)

is independent of wavelength and normalized to unity at

θ = π

. The condition

p > 2

provides integrability of the scattered light intensity over angles.

For unpolarized incident light [Eq. (5c)] the phase function (8) takes the following form

P_{u, p} (θ) = (1 + \cos^{2} ​ θ) {(\sin (θ / 2))}^{p - 4} / 2 .

In some cases, another normalization of this phase function is useful

P_{u, p} (θ) = (1 + \cos^{2} ​ θ) {(\sin (θ / 2))}^{p - 4} / {8 [{(p - 2)}^{- 1} - 2 p^{- 1} + 2 {(p + 2)}^{- 1}]} .

for which

\int_{0}^{π} \sin θ P_{u, p} (θ) d θ = 1

Using phase function $P_{u, p} (θ)$ in Eq. (10), one can calculate the asymmetry parameter $g_{p}$ for $unpolarized incident light$ [3,23]

\begin{array}{l} g_{p} = 〈 \cos θ 〉 = \int_{0}^{π} \cos θ \sin θ P_{u, p} (θ) d θ / \int_{0}^{π} \sin θ P_{u, p} (θ) d θ \\ = [{(p - 2)}^{- 1} - 4 p^{- 1} + 6 {(p + 2)}^{- 1} - 4 {(p + 4)}^{- 1}] / [{(p - 2)}^{- 1} - 2 p^{- 1} + 2 {(p + 2)}^{- 1}] . \end{array}

It is easy to show that Eq. (12) holds for

p > 2

.

Examples of phase functions $P_{u, p} (θ)$ in Eq. (10) for unpolarized incident light are shown in Fig. 1

Fig. 1 Phase functions $P_{u, p} (θ)$ in Eq. (10) for scattering of unpolarized incident light in (a) Cartesian and (b) polar presentation. The results are presented for several values of the parameter $p$ , which determines the wavelength dependence of SC (1).

Download Full Size | PDF

. Asymmetry parameter

g_{p}

in Eq. (12) is presented in Fig. 2

Fig. 2 Asymmetry parameter $g_{p}$ in Eq. (12) as a function of absolute value $p (p > 2)$ of exponent in wavelength dependence of SC (1).

Download Full Size | PDF

as a function of

p

. It should be noted that asymmetry factor

g_{p}

is independent of wavelength, positive at p < 4 and negative at p > 4. Earlier the wavelength-dependent negative asymmetry factors

g

were predicted for systems of particles [26–28].

3. Comparison with results obtained for other models of light scattering in NG

It was noted in Section 1 that the similar problem was considered earlier in terms of interparticle interference [19]. The equations obtained here for phase function and asymmetry parameter in the case of unpolarized incident light are equivalent to those derived in Ref. [19]. The only difference consists in the region of applicability of the results: the limitation $p \geq 4$ was considered in [19], whereas more general condition $p > 2$ is used in the present paper. Thus, the results do not depend on the approach (at least for $p \geq 4$ ) and are universal in some degree.

Let us compare the phase function (8) with phase functions calculated in the Rayleigh-Debye approximation for some models of NGs.

Goldstein [12] proposed a theory of the light scattering from Rayleigh particle and its diffusion zone for the incident light polarized perpendicular to the scattering plane (5a) ( $V_{V}$ geometry of experiment). The dependence of intensity of the scattered light on wavelength and scattering angle was presented in the form $I (λ, θ) \propto λ^{- 4} {[f (C \sin (θ / 2))]}^{2}$ where $f (u)$ is the function introduced by Goldstein and the dimensionless parameter $C = (4 π / λ) {(D t)}^{1 / 2}$ is expressed in terms of diffusion coefficient $D$ and time $t$ of growth of the particle. The non-normalized phase function is determined by the square of $f (C \sin (θ / 2))$ . The examples of phase functions calculated for different values of $C (C = 0.1 - 10)$ were presented by Hammel et al. [29].

We calculated the Goldstein phase functions $P_{G} (C, θ) = {[f (C \sin (θ / 2)) / f (C)]}^{2}$ normalized to unity at $θ = π$ and found that this function is practically independent of $C$ for $C ≲ 0.5$ (maximum difference between the phase functions calculated for $C = 0.5$ and $C = 0$ is $≅ 0.02$ at $2 θ ≅ 110 °$ . In the limit of small values of $C$ , the approximation $f (u) ≅ (4 / 15) u^{2}$ [12] gives: $P_{G} (C \to 0, θ) = {[\sin (θ / 2)]}^{4}$ . This approximation for $f (u)$ leads to the relation $I (λ, θ) \propto λ^{- 8} {[\sin (θ / 2)]}^{4}$ for the intensity of scattered light. Multiplying this relation by $i (θ)$ (5c) for non-polarized light and integrating over all directions in space, we obtain the inverse eighth-power dependence of SC $α_{s,G} (λ)$ on wavelength. Thus, the Goldstein theory gives for $C ≲ 0.5$ (or $λ ≳ λ_{\min} : = 8 π {(D t)}^{1 / 2}$ )

α_{s,G} (λ) \propto λ^{- 8}, P_{G} (C, θ) ≅ {[\sin (θ / 2)]}^{4}, (for i_{⊥} and C ≲ 0.5) .

One can conclude that the Goldstein theory provides at

λ ≳ λ_{\min}

the description of wavelength dependence of SC and angular distribution of scattered light in Eq. (13), which are the particular case of Eqs. (1) and (8) for

p = 8

and polarization

i_{⊥}

.

In the case of $C > 0.5 (λ < λ_{\min})$ , the Goldstein theory leads to the phase function which depends on $C$ , i.e., on $λ$ . Calculations show that wavelength dependence of SC cannot be described by Eq. (1) in this case, and hence, our approach is inapplicable.

To calculate SC of GCs, Edgar [15] considered light scattered from crystal and its diffusion zone and obtained the same dependences as those given by Eqs. (1) and (8) for $p = 8$ and unpolarized incident light. The similar conclusion may be drawn for the polarized light [16].

The scattering of unpolarized light by spinodal structure at late stage of decomposition was examined theoretically by Hendy [11]. He used the results of numerical simulation of the structure and concluded that the wavelength dependence of SC obeys Eq. (1) with the value of $p = 8$ . The angular distribution of scattered light does not depend of wavelength in certain spectral range and is described by Eq. (8) with $p = 8$ and the factor $i_{u}$ (5c) of unpolarized light. Hendy also argued that the results can be applied to light scattering by GCs.

Thus, in all examples of calculations considered above [11,12,15,16], wavelength dependence of SC and phase function obey Eqs. (1) and (8), respectively, with the value of $p = 8$ . This phase function is backward-directed and leads to the asymmetry parameter $g_{p} = - 0.57$ [see Fig. 2 or Eq. (12)]. We could not find other examples of calculations carried out in the Rayleigh-Debye approximation for NGs.

4. Comparison with experimental data

4.1 Three examples in which phase function and wavelength dependence of SC were measured

Let us consider three examples in which wavelength dependence of SC (1) was experimentally observed and angular distribution of scattered light was also studied.

The scattering of visible light in inhomogeneous sodium borosilicate glass characterized by a linear dependence of SC on the inverse eighth power of the wavelength (Eq. (1), $p = 8$ ) has been studied experimentally by Kolyadin [30,31]. He studied the angular distribution of scattered light for two polarization (5a) and (5b) of incident beam and found that normalized angular distribution is independent of wavelength. This conclusion is consistent with the prediction of our approach [see text after Eq. (8)]. Figure 3

Fig. 3 Comparison of the experimental data on angular distribution of light scattered by sodium borosilicate glass obtained by Kolyadin [30,31] (crosses and circles) with the phase function calculated by Eq. (8) for $p = 8$ and two polarizations (5a) and (5b) (curves). The value $p = 8$ was observed in this glass experimentally [30,31].

Download Full Size | PDF

shows the experimental angular distribution obtained by Kolyadin (points taken from Fig. 1(A) in Ref. [30] or Fig. 3(a) in [31]) in comparison with the phase function

P_{p} (θ)

in Eq. (8) calculated for

p = 8

and two polarizations (5a) and (5b) (curves). As noted above [see Eq. (13)], the Goldstein theory gives for

C ≲ 0.5

the same phase function as Eq. (8) for

p = 8

and

V_{V}

polarization. The experimental data were normalized to unity at

θ = 180 °

using the extrapolation to

θ = 180 °

made by Kolyadin. With the exception of small angles, the experimental and theoretical phase functions are in good agreement. A similar discrepancy at small angles was noted by Goldstein [12] which compared the phase function calculated by his theory (polarization

V_{V}

,

C = 1

) with the experimental data obtained by Kolyadin for another glass sample [32] (see below, Section 4.2). To explain the discrepancy, Goldstein proposed three possibilities for experimental errors. From our point of view, the phase function

P_{p} (θ)

cannot tend to zero at

θ \to 0

. For the case of interparticle interference, we presented arguments in support of this statement [19].

In the second example, we consider the light scattering by glass KF-7 [33]. It was found that Eq. (1) is fulfilled with $p = 4.8$ . The phase functions measured for wavelengths $λ = 365, 436, and 546 nm$ are somewhat different from each other. The experimental phase function for $λ = 546 nm$ and two polarizations (5a) and (5b) of incident light are shown in Fig. 4

Fig. 4 Experimental phase functions at $λ = 546 nm$ for glass KF-7 [33] in comparison with the phase functions calculated by Eq. (8) with the experimentally determined value of $p = 4.8$ [33]. Two polarizations (5a) and (5b) of incident light are used.

Download Full Size | PDF

in comparison with the phase function

P_{p} (θ)

calculated by Eq. (8) with

p = 4.8

. One can see the qualitative agreement between theory and experiment. The experimental phase function for

λ = 365 and 436 nm

are approximated by Eq. (8) with

p = 4.5 and 4.6

, respectively. However, it should be noted that, for these two wavelengths, the experimental curve

H_{H}

demonstrates the depolarization effect: the intensity of scattered light at

θ = 90 °

differs appreciably from zero.

In the third example, the light scattering by organic glass ST-1 [poly(methyl methacrylate) block] [33] is considered. It was found that Eq. (1) is fulfilled with $p = 3.0$ . The phase functions measured for wavelengths $λ = 405, 498, 546, and 691 nm$ were close to each other. The experimental phase function for $λ = 546 nm$ and two polarizations (5a) and (5b) of incident light are shown in Fig. 5

Fig. 5 Experimental phase functions at $λ = 546 nm$ for organic glass ST-1 [33] in comparison with the phase functions calculated by Eq. (8) with the experimentally determined value of $p = 3$ [33]. Two polarizations (5a) and (5b) of incident light are used.

Download Full Size | PDF

in comparison with the phase function

P_{p} (θ)

calculated by Eq. (8) with

p = 3

. One can see the satisfactory agreement between theory and experiment. Note that the experimental curve

H_{H}

demonstrates the appreciable depolarization effect.

We could not find other examples of such a type.

4.2 Application of theoretical phase functions obtained in the present paper to the approximation of the experimental angular distribution of light scattered by NG

There are a number of papers in which angular distribution of light scattered by NG had been measured at a single wavelength and wavelength dependence of SC was not studied. Here we show that Eq. (8) may be useful for approximation of experimental phase functions of scattered light.

One of such experiments was made by Kolyadin with inhomogeneous sodium borosilicate glass [32] different from that studied in [30,31]. The results of his measurements [32] are shown in Fig. 6

Fig. 6 Experimental data on angular distribution of light scattered by inhomogeneous sodium borosilicate glass [32] in comparison with the curves calculated by the Goldstein theory [12] and by Eq. (8).

Download Full Size | PDF

by points. The experimental data were normalized to unity at

θ = 180 °

using the extrapolation to

θ = 180 °

made by Kolyadin. Goldstein [12] compared these experimental data for

V_{V}

polarization with results calculated by his theory for

C = 1

. We made sure that this value of

C

is optimal for describing

V_{V}

experimental data and presented in Fig. 6 the curves of Goldstein for both polarizations (the polarization factor

i_{∥}

(5b) was used for

H_{H}

polarization). We also found that the best approximation of the experimental data by Eq. (8) takes place at

p = 7.24

and show the corresponding curves in Fig. 6. It is seen that curves calculated by the Goldstein theory and by Eq. (8) are close to each other and give a good approximation of the experimental data at

θ > 50 °

. At

θ ≲ 50 °

, Eq. (8) gives a slightly better approximation.

The curves of the phase functions for phase-separated borosilicate glass (Fig. 2(b) in [25]) are shown by points in Fig. 7

Fig. 7 Experimental data on angular distribution of light scattered by phase-separated borosilicate glass [25] in comparison with the curves calculated by the Goldstein theory [12] and by Eq. (8).

Download Full Size | PDF

to distinguish them from theoretical curves. In this case, the extrapolation of the experimental curves to

θ = 180 °

seems to be impossible. Therefore, we normalized the

V_{V}

curve to unity at

θ = 90 °

and used the same normalizing factor for the

H_{H}

curve. The same normalizing procedure was applied to theoretical curves. The best approximation of the experimental

V_{V}

curve by Eq. (8) was obtained for

p = 5.8

(Fig. 7). The Goldstein equation [12] was also used for the approximation, however, the best variant of this approximation calculated for

C = 2.45

differs significantly from the experimental

V_{V}

curve at

θ ≲ 60 °

. Using these values of

p

and

C

, and the polarization factor

i_{∥}

in Eq. (5b), we also obtained the theoretical

H_{H}

curves for our and Goldstein’s models. Figure 7 shows that the experimental

V_{V}

curve is well described by our model, whereas the experimental

H_{H}

curve demonstrates the depolarization effect and its description by our model is not so good. The Goldstein model cannot describe the experimental phase functions at angles

θ ≲ 60 °

. The analysis of Fig. 2(a) of [25] leads to the same conclusions [Fig. 2(b) was selected for the presentation in our Fig. 7 since the interval of scattering angle

θ

for the experimental

V_{V}

curve in this figure is wider than in Fig. 2(a)].

PSGs prepared by heat treatment of the same initial glass for different times $t$ were studied in [4]. The authors present the experimental phase functions for polarization $V_{V}$ . In Fig. 8, these phase function (points) are compared with the curves calculated by Eq. (8) with two values of $p$ : $p = 6.0 and 6.6$ . All the phase function are normalized to unity at $θ = 90 °$ . Excluding two points ( $θ = 40 and 50 °$ ) for the shortest heat treatment ( $t = 17 h$ ) and keeping in mind a possibility of experimental error, one can conclude that our model gives a good description of angular distribution of scattered light in the case under consideration. For example, the data for heat treatment $t = 115 h$ show ideal coincidence with the curve calculated for $p = 6$ .

Because the wavelength dependence of SC was not presented in Refs [4,25], one cannot conclude whether Eq. (1) is satisfied or not in these cases, and correspondingly, it is impossible to determine the $p$ value from wavelength dependence of SC. Nevertheless, the value of $p ≅ 6$ found from approximation of the experimental data (Figs. 7 and 8

Fig. 8 Experimental data on angular distribution of light scattered by phase-separated borosilicate glass heat-treated for different times $t$ [4] in comparison with the curves calculated by Eq. (8) with $p = 6 and 6.6$ .

Download Full Size | PDF

) seems to be reasonable. Indeed, it was shown [4] that the source of scattering in these glasses is interparticle interference, whereas typical values

p = 5 - 6.2

were demonstrated for wavelength dependence of SC of GCs [2] in which ALS is caused by interparticle interference.

Let us make two remarks relating the experimental data of Ref. [4] presented in Fig. 8. First, two points ( $θ = 40 and 50 °$ ) for the shortest heat treatment ( $t = 17 h$ ) differ significantly from other ones (Fig. 8). For this glass, the intensity of ALS is low and, in the range of small angles, becomes presumably less than the intensity of scattering, which manifests itself in this range and is assumed to have another nature. Second, the $p$ values for the best fit of the experiment (Fig. 8) decrease from $p = 6.6$ for short heat-treatment times ( $t = 17 and 42 h$ ) to $p = 6.0$ for the long heat treatment ( $t = 115 h$ ). This circumstance may be important for understanding the mechanism of phase separation.

Thus, in suitable cases, our model can be used for approximation of experimental phase functions.

4.3 On the asymmetry parameter

We demonstrated that Eq. (8) gives a satisfactory description of experimental angular distribution of light scattered by NGs. This means that if the value of $p$ has been determined from wavelength dependence of SC [Eq. (1)] or from direct approximation of phase function (as in Section 4.2), then it may be used for estimation of asymmetry parameter by Eq. (12) or Fig. 2.

Because values of $p = 6 - 8$ are often observed for NGs [1,2], asymmetry parameter calculated by Eq. (12) is estimated as $g_{p} = (- 0.4) - (- 0.6)$ for this materials. This conclusion is illustrated by backward-directed experimental phase functions shown in Figs. 3, and 6–8. The maximum absolute value of negative asymmetry parameter $g_{p} = - 0.57$ may be ascribed to inhomogeneous sodium borosilicate glass studied by Kolyadin [30,31] (Fig. 3).

It should be noted that the prediction of asymmetry parameter $g \sim - 0.5$ , which can be realized in the near infrared for system of dual-dipolar particles, is considered as an essential achievement [28], whereas such values are typical of NG in some wavelength range $(λ_{\min}, λ_{\max})$ comparable to that of visible light.

5. Summary

Light scattering in nanostructured glasses (phase-separated glasses and glass-ceramics) is considered.

It is assumed that wavelength dependence of light scattering coefficient of nanostructured glass is described by the power law with a constant exponent (−p). Such a behavior of scattering coefficient is often observed in these materials.

Using this assumption, we apply the approach of Debye and Bueche to theoretical description of interference effects in such a light scattering and derive the expression for the angular distribution (the phase function) of the scattered light. The predicted phase function includes the value of p (p > 2) as parameter, does not depend on wavelength and is presented by the same expression as the phase function calculated earlier for p > 4 in the model of interparticle interference. The asymmetry parameter of the scattered light is a function of p which is positive at p < 4 and negative at p > 4.

For p = 8, the predicted phase function coincides with those calculated by other authors for some specific models of nanostructured glasses. This case is characterized by large absolute value of negative asymmetry parameter g = −0.57.

The predicted phase functions are in satisfactory agreement with the experimental phase functions presented in the literature. In particular, this agreement is good for p = 8, which demonstrates experimentally the large absolute value of negative asymmetry parameter g = −0.57.

Because the values p = 6−8 are often observed in nanostructured glasses, the negative asymmetry parameters g = (−0.4)−(−0.6) are predicted for these materials.

Thus, the method is proposed for evaluating phase function and asymmetry parameter of light scattered by nanostructured glass from measurement of wavelength dependence of the scattering coefficient.

Funding

Russian Foundation for Basic Research (RFBR) (16-03-01130).

Acknowledgment

The author expresses his gratitude to the RFBR for partial support of this work.

References

1. N. S. Andreev, “Scattering of visible light by glasses undergoing phase separation and homogenization,” J. Non-Cryst. Solids 30(2), 99–126 (1978). [CrossRef]

2. M. P. Shepilov, O. S. Dymshits, and A. A. Zhilin, “Light scattering in glass-ceramics: revision of the concept,” J. Opt. Soc. Am. B 35(7), 1717–1724 (2018). [CrossRef]

3. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2004).

4. J. P. Kerwawycz and M. Tomozawa, “Structural origin of the light scattering from a phase‐separated glass,” J. Appl. Phys. 51(8), 4391–4398 (1980). [CrossRef]

5. M. P. Shepilov, “The problem of theoretically describing anomalous light scattering by liquating glasses, caused by interparticle interference,” J. Opt. Technol. 70(12), 882–887 (2003). [CrossRef]

6. M. P. Shepilov, “On the problem of theoretical description of anomalous light scattering by phase separated glasses,” Phys. Chem. Glasses 46(2), 173–177 (2005).

7. M. P. Shepilov, O. S. Dymshits, A. A. Zhilin, A. E. Kalmykov, and G. A. Sycheva, “Features of the anomalous scattering of light in two-phase sodium borosilicate glass,” J. Opt. Technol. 80(11), 706–713 (2013). [CrossRef]

8. A. A. Zhilin, G. T. Petrovskii, O. S. Dymshits, V. V. Golubkov, and M. P. Shepilov, “Effect of phase transformations in Na₂O–Nb₂O₅–SiO₂ glasses on their transparency and Kerr coefficient variation,” Phys. Chem. Glasses 46(2), 232–235 (2005).

9. M. P. Shepilov, O. S. Dymshits, V. V. Golubkov, and A. A. Zhilin, “Anomalously low light scattering in the Na₂O-Nb₂O₅-SiO₂ glass-ceramics,” Adv. Mat. Res. 39–40, 270–273 (2008).

10. M. P. Shepilov, O. S. Dymshits, A. A. Zhilin, V. V. Golubkov, A. E. Kalmykov, I. P. Alekseeva, A. V. Myasoedov, A. A. Hubetsov, and S. S. Zapalova, “Effect of low NiO doping on anomalous light scattering in zinc aluminosilicate glass-ceramics,” J. Non-Cryst. Solids 473, 152–169 (2017). [CrossRef]

11. S. Hendy, “Light scattering in transparent glass ceramics,” Appl. Phys. Lett. 81(7), 1171–1173 (2002). [CrossRef]

12. M. Goldstein, “Theory of scattering for diffusion‐controlled phase separations,” J. Appl. Phys. 34(7), 1928–1934 (1963). [CrossRef]

13. J. J. Hammel and S. M. Ohlberg, “Light scattering from diffusion-controlled phase separations in glass,” J. Appl. Phys. 36(4), 1442–1447 (1965). [CrossRef]

14. D. M. Shetterly and R. J. Stubler, “Electromagnetic scattering from metastable immiscible droplets in glass,” J. Am. Ceram. Soc. 82(1), 5–16 (1999).

15. A. Edgar, “Core-shell particle model for optical transparency in glass ceramics,” Appl. Phys. Lett. 89(4), 041909 (2006). [CrossRef]

16. H. Li, Y. Zhang, D. Zhong, S. Xu, Q. Fei, and D. Wang, “Influence of blue-light polarization on light scattering in nanocrystal luminescent glass for white light-emitting-diodes,” Optoelectron. Lett. 6(3), 164–167 (2010). [CrossRef]

17. R. W. Hopper, “Stochastic theory of scattering from idealized spinodal structures: II. Scattering in general and for the basic late stage model,” J. Non-Cryst. Solids 70(1), 111–142 (1985). [CrossRef]

18. M. P. Shepilov, “On light scattering in fluorozirconate glass-ceramics containing BaCl₂ nano-crystals,” Opt. Mater. 30(6), 839–846 (2008). [CrossRef]

19. M. P. Shepilov, “Asymmetry parameter for anomalous scattering of light in nanostructured glasses,” Opt. Lett. 42(21), 4513–4516 (2017). [CrossRef] [PubMed]

20. M. P. Shepilov, “Light scattering in optical material containing polydisperse spherical nano-particles,” Opt. Mater. 31(2), 385–390 (2008). [CrossRef]

21. P. Debye and A. M. Bueche, “Scattering by an inhomogeneous solid,” J. Appl. Phys. 20(6), 518–525 (1949). [CrossRef]

22. M. Kerker, The Scattering of Light (Academic, 1969) (see Section 8.4).

23. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (NASA, Cambridge University Press, 2004).

24. L. A. Dombrovsky and D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach (Begell Haus, 2010), http://www.begellhouse.com/ebook_platform/6d17e856430c0b8d.html.

25. M. Tomozawa and J. P. Kerwawycz, “Depolarized light scattering from a phase‐separated glass,” J. Appl. Phys. 50(5), 3644–3647 (1979). [CrossRef]

26. M. I. Mishchenko, “Asymmetry parameters of the phase function for densely packed scattering grains,” J. Quant. Spectrosc. Radiat. Transf. 52(1), 95–110 (1994). [CrossRef]

27. R. Gómez-Medina, L. S. Froufe-Pérez, M. Yépez, F. Scheffold, M. Nieto-Vesperinas, and J. J. Sáenz, “Negative scattering asymmetry parameter for dipolar particles: Unusual reduction of the transport mean free path and radiation pressure,” Phys. Rev. A 85(3), 035802 (2012). [CrossRef]

28. B. X. Wang and C. Y. Zhao, “Achieving a strongly negative scattering asymmetry factor in random media composed of dual-dipolar particles,” Phys. Rev. A (Coll. Park) 97(2), 023836 (2018). [CrossRef]

29. J. J. Hammel, J. Mickey, and H. R. Golob, “Diffusion coefficients from light scattering in glass,” J. Coll. Int. Sci. 27(3), 329–335 (1968). [CrossRef]

30. A. I. Kolyadin, “Anomalous light scattering in glass,” Dokl. Akad. Nauk SSSR 109, 64 (1956) (in Russian) [Soviet Phys.‐Doklady 1, 406 (1956)].

31. A. I. Kolyadin, “Anomalous light scattering in glass,” Opt. Spektrosk. 1, 907 (1956) (in Russian).

32. A. I. Kolyadin, “Anomalous light scattering in glass,” in The Structure of Glass, Proceedings of the Third All‐Union Conference on the Glassy State, Leningrad, 1959, (Consultants Bureau, New York, 1960), p. 202.

33. A. I. Kolyadin, “Study of scattering and absorption of light in ultrapure glasses and optical ceramics,” Report of Vavilov State Optical Institute, NIR-XIY TT12–416–82/1F11–03–82 (Leningrad, 1984). (p. 36–39). (in Russian).

Angular distribution of light scattered by nanostructured glass predicted from an experimental wavelength dependence of turbidity

Abstract

1. Introduction

2. Relation between wavelength dependence of SC and the angular distribution of light scattered by NG

3. Comparison with results obtained for other models of light scattering in NG

4. Comparison with experimental data

4.1 Three examples in which phase function and wavelength dependence of SC were measured

4.2 Application of theoretical phase functions obtained in the present paper to the approximation of the experimental angular distribution of light scattered by NG

4.3 On the asymmetry parameter

5. Summary

Funding

Acknowledgment

References

Cited By

Figures (8)

Equations (15)

OSA Continuum