Radiophysics and Quantum Electronics, Vol. 56, No. 10, March, 2014 (Russian Original Vol. 56, Nos. 10, October, 2013)

SOME FEATURES OF LIGHT SCATTERING IN OPTICALLY ANISOTROPIC MEDIA E. L. Bubis, 1 ∗ M. A. Novikov, 2 V. V. Lozhkarev, 1 V. M. Gelikonov, 1 G. V. Gelikonov, 1 and V. I. Rubakha 1

UDC 535.43+535.5

Experimental results on observation of the light scattering in optically anisotropic transparent media are presented. During observation at angles close to the normal to the light beam in the crystal, a periodic trace line, which reflects the light-scattering dependence on the polarization of the laser light beam passing through the medium, is seen. Generalization of the well-known Umov experiment to more complex optically anisotropic media is presented. A simple theoretical model for a qualitative interpretation of the experimental data is proposed.

1.

INTRODUCTION

Although the current main concern in optics is in the study of the optical properties of nano objects and optical materials based on the latter, growing of optically perfect large-size crystals for quantum optics is of keen interest. The study of various inclusions in crystals, which define their optical quality and ultimate optical strength is still an important problem. In this respect, Umov’s experiment [1–3] is of interest. In Umov’s experiment, color spirals related to the dependence of the rotational capability of the chiral liquid on the wavelength are seen during lateral observation when white light passes through a layer of turbid optically active (chiral) liquid. In [4], it is shown that a similar phenomenon is also observed in chiral crystals during the light propagation along the optical axis. In this work, the method capabilities are extended to studying the light scattering not only in chiral media, but also in anisotropic crystals and polymers [5, 6]. In addition, our studies show that such a method for studying the optical properties of anisotropic media can also be used for a luminescent analysis of inclusions in crystals. It is assumed that after updating this method can be used for studying the crystal qualities, e.g., in laser high-power optics. A simple theoretical model is proposed for a qualitative interpretation of the observed optical effects. 2.

EXPERIMENTAL DETAILS

In the scattering-study experiments, the samples were illuminated by the linearly polarized radiation of a helium-neon laser (wavelength λ = 0.63 μm), which is single-mode by its transverse structure, or singlemode radiation of the laser pointer (λ = 0.53 μm). Power of both sources did not exceed 10 μW. To observe the scattering traces in the spar and lithium-iodate crystals, a laser beam was launched in the direction close to the optical axis (Z cut). The LiIO3 crystal, which was grown and worked at the Institute of Applied Physics of the Russian Academy of Sciences, was used. The photograph of the scattering in a 4.2 cm-long spar crystal is shown in Fig. 1a. Since the crystal-birefringence value (the difference between the Δn indices ∗

[email protected] 1

Institute of Applied Physics of the Russian Academy of Sciences; 2 Institute of Microstructure Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 56, Nos. 10, pp. 726–732, October 2013. Original article submitted July 16, 2013; accepted October 31, 2013. c 2014 Springer Science+Business Media New York 0033-8443/14/5610-0651  651

a)

b)

c)

d)

Fig. 1. Beam scattering during illumination of the Iceland-spar crystal (a), organic-glass bar (b), helical waveguide (c), and α-LiIO3 crystal in natural organics (d) by the helium-neon laser. for two normal waves) is a function of the light-propagation direction, using the crystal rotation with respect to its optical axis, one can change the scattering-trace period, which was observed in the experiment. The intermittent scattering traces can also be observed in organic glass, in which residual small birefringence is, as a rule, always present. Figure 1b shows a photograph of the scattering in a 50 cm-long organic-glass bar. Such a scattering pattern was also observed in a helical waveguide whose core has helical structure. The waveguide was manufactured at the Institute of Chemistry of High-Purity Materials of the Russian Academy of Sciences (Nizhny Novgorod). To ensure good visibility of traces, no high-quality optical polishing of the sample faces is required in all cases. In the absence of optical activity, the linear birefringence Δn value is related to the trace period Λ and the wavelength λ by a simple relationship Δn = λ/Λ and difference Δn for a polymer bar is equal to 2 · 10−5 . Similar traces related to chirality (circular birefringence) were observed in a lithium-iodate crystal during the light propagation along the optical axis. The spatial trace period agrees with the data presented in [7] in which the results of the study of chirality dispersion in this crystal are shown. It should be noted that linear light polarization in the traces, which was normal to the observation direction and the trace line, was observed in all experiments. In the studied sample of an Iceland-spar crystal from Turinskoye field (Krasnoyarsk Territory), an additional solid red-brown trace was observed at wavelength λ = 0.53 μm apart from the periodic scattering trace. Such a trace was not observed in other crystals. This trace seems to be related to photoluminescence of the crystal admixtures and microinclusions. Figure 2 shows the traces and the solid scattering line when the crystal is illuminated by a green laser pointer. The bottom strip corresponds to the photoluminescence trace. Note that in all cases, the scatteredlight traces are well observed by naked eye in slightly turbid samples located in a slightly darkened room.

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The proposed light-scattering experiments can be used for measuring the parameters of optical chirality and linear birefringence of the samples, including the largesize ones. 3.

THEORETICAL MODEL

In the general case, comprehensive theoretical description of the above-mentioned experiments in optical anisotropic media is rather difficult, since the amplitude and polarization of the observed scattered light at each crystal point during its lateral observation are functions Fig. 2. Illumination of the spar crystal by a green of not only the incident-light polarization, but also the laser pointer (λ = 0.53 μm). The red-brown lumicrystal-light scattering mechanism at this point. In the nescence trace is located in the bottom, while the general case, several physical mechanisms contribute to green traces are in the top. the light scattering [8]. Moreover, the medium-anisotropy influence on polarization of the observed scattered light should also be taken into account. The problem becomes still more complicated by the joint action of optical activity (chirality) and linear birefringence in the crystal. The above experiments showed that the studied scattering is rather high and always polarized normally to the scattering plane. Thus, it can be assumed that in this case we deal with scattering by macroscopic isotropic inclusions (particles) whose dimensions are small compared with the scattered wavelength, i.e., Tyndall scattering. To simplify the problem with allowance for the above-said, it is assumed that scattering is independent of the crystal anisotropy. Since it is also assumed that the scattering-particle dimensions are much smaller than the wavelength, the scattering is considered to be of dipole nature. In this case, the scattering features in the far zone can be described with allowance for the scattering coherence by the following formula: ω2 (1) e ∝ 2 M [N [NP]], c R where e is the scattered-radiation electric field at the observation point, ω is the incident-light frequency, R is the distance from the scattering point to the observation point, unit vector N determines the scatteringobservation direction, M is the scattering-particle density at the scattering point S, P = αE is the scatteringparticle polarization, α is the scattering-particle polarizability tensor, and E is the vector of the local incident field (in general case, elliptically polarized) at point S. To gain profound understanding of the dependence of intensity during the lateral observation of scattering on the incident-radiation polarization at the scattering point, we can restrict ourselves to the case of isotropic particles, i.e., the case where α is a scalar. Experimental configuration for the crystal is shown 0 X ´ X in Fig. 3. The incident light beam (in general, elliptically polarized) propagates along the Z axis. The observed scattering point S is located at a distance d from the crysEx tal boundary. Angle θ determines deviation of the wave P d Z vector from the crystal optical axis. In a general case, S from Fig. 3 and Eq. (1) the scattered-radiation intensity µ ex can be written as ez   E y ¾ J = |ex |2 + |ez |2 = const · |α|2 |Ex |2 + |Ey |2 sin2 (σ) , (2) Y N 0

Y where the angle σ is the difference of the scattering direction from normal to the Z axis in the observation plane Fig. 3. Umov’s experiment layout Y Z. For σ = 0, radiation is linearly polarized along the X axis. Obviously, with allowance for the scattering-particle anisotropy, contribution to the scattering com-

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ponent |ez |2 can also take place for σ = 0. The scattering-field e components can be observed separately by varying the polarization-analyzer orientation at the observation point. For crystals with linear birefringence and chirality, the situation is more complicated than that in the Umov experiment, since the effect in the latter case generally depends on the orientation of the main anisotropy axes with respect to the light-propagation direction. In such media, the light-polarization transversality is generally known to be violated. However, for weak linear anisotropy, which is characteristic of the above-considered experiments, this can be neglected and we can restrict ourselves to the case where the main axes X  and Y  of this anisotropy are rotated by an angle η with respect to the X and Y axes. In this case, the lateral-scattering intensity depends only on the light-beam electric-field components at the scattering point, which are normal to the Z axis. In a chiral crystal, with allowance for both linear birefringence and chirality, polarization of eigenwaves is elliptical if light propagates in the direction other than that of the optical axis and, moreover, the chirality anisotropy should be taken into consideration. However, if we consider the case where the light-propagation direction slightly deviates from the Z axis, then the chirality constant can be assumed to be independent of the η and θ angles. To take the light-polarization variation into account during light propagation in the crystal with allowance for the light-polarization transversality, it is expedient to use the method of the Jones matrices [9]. Using this method, one can rather simply determine the light polarization as it propagates along the Z axis with the help of the Jones matrix M, which relates the field at point S with the incident-radiation field at the crystal input     Ex in Ex out =M . (3) Ey out Ey in With allowance for Eq. (2), Eq. (3) shows that the scattered-light intensity can be written as   J = const · |α|2 |Mxx Ex in |2 + |Mxy Ey in |2 sin2 (σ) ,

(4)

where Mxx and Mxy are the elements of the matrix   Mxx Mxy . M= Myx Myy It can be shown that the Jones matrix can be written in the form    A −B   S(η)  M = S(−η)  B A∗ 

(5)

with allowance for both linear birefringence and chirality. Here,    cos η sin η   S(η) =  − sin η cos η  is the matrix of the coordinate-system rotation by an angle η, which determines orientation of the main axes of the crystal linear anisotropy with respect to the X axis. The matrix elements are A = cos(ΔΦ) + i sin(ΔΦ)× × cos(η) and “∗ ” denotes complex conjugation. The resulting phase delay ΔΦ has the form (ΔΦ)2 = (Δϕ)2 + ρ2 .

(6)

Equation (6) represents the superposition principle for the phase delay in an anisotropic crystal with allowance for chirality and linear birefringence [10]. The Faraday rotation is ρ = πd (nl − nr )/λ, where nl and 654

nr are the refractive indices of the left-hand and right-hand polarized waves and d is the distance covered by light in the crystal. Phase delay for the linear birefringence of an uniaxial crystal at small angles θ of the light-direction deviation from the optical axis is  2  1 no − n2e 2 θ ≈ (no − ne ) θ 2 , (7) 2 Δϕ = 2π δn d/λ, δn = no − ne (θ) = no 2 n2e where it is assumed that |no − ne |  1 and θ  1. Here, no and ne are the refractive indices for the ordinary and extraordinary waves, respectively, ne (θ) represents the well-known refractive-index dependence [10] on the angle θ, and ne = ne (θ  = π/2). (nl − nr )/(no − ne ), one can neglect the linear birefringence If θ  (Δϕ ≈ 0), and the Jones matrix takes the form   cos ρ sin ρ   . M= (8) sin ρ cos ρ In this case, the difference between the chiral crystal and chiral liquid is absent. For the incident light polarized along the X axis, using Eqs. (4) and (7) for σ = π/2 (see Fig. 3), we write

and

J = const · |α|2 |Ex |2 cos2 [π (nl − nr ) d/λ]

(9)

J = const · |α|2 |Ey |2 sin2 [π (nl − nr ) d/λ]

(10)

for the intensity of scattered light arriving from the crystal point to the observation point and for the incident radiation polarized along the Y axis, respectively. Therefore, changing the light polarization at the crystal input, we can change the phase of the periodic visible trace. Obviously, the trace period corresponds to Faraday rotation by 90◦ , i.e., the circular half-wave phase plate and the spatial trace length L is equal to λ/(nl − nr ). Using the known L, one can determine the crystal chirality. The latter relationship and Eqs. (8) and (9) fully describe all features of the initial Umov experiment. If the medium was illuminated by white light, Umov observed various color spirals, which is explained by the length L dependence on the light wavelength. This dependence is also contributed by the frequency dispersion of the refractive indices of circular birefringence. If chirality is absent, the Jones matrix of the scattering medium given by Eq. (4) has the form   exp(i Δϕ)  0  S(η) M = S(−η)   0 exp(−i Δϕ)   2  cos (η) exp(i Δϕ) i sin(2η) sin(Δϕ)  =  i sin(2η) sin(Δϕ) cos2 (η) exp(−i Δϕ) + sin2 (η) exp(i Δϕ) . (11) For this case, Eqs. (4) and (11) yield the following scattered-light intensity:  2 J = const · |α|2 |Ex |2 1 + sin(2η) cos(2 Δϕ) .

(12)

Using Eq. (12), one can obtain the contrast of the periodic track of scattering: Jmax − Jmin = sin(2η). Jmax + Jmin

(13)

Therefore, for η = 0 or η = π/2, the contrast is zero, the intensity is J = const · |α|2 |Ex |2 , and the periodic trace of scattering is invisible. For η = π/4, Eq. (12) yields J = const · |α|2 |Ex |2 cos2 (Δϕ). In this case, the contrast is maximum and equal to unity. 655

With allowance for chirality and linear birefringence, we consider the case where η = 0. Using Eqs. (5) and (4), one can obtain the scattered-radiation intensity J = const · |α|2 |Ex |2 [1 − sin2 (γ) sin2 (ΔΦ)]2

(14)

for the incident-radiation field Ex component. If only the Ey component is present in incident light, we have J = const · |α|2 |Ey |2 sin4 (γ) sin4 (ΔΦ), where

 nl − nr 2 θ , γ = arcctg(ρ/Δϕ) = arcctg no − ne 

ΔΦ =

(15)

1/2 2π . d (nl − nr )2 + (no − ne )2 θ 2 λ

In both cases, the periodic-trace amplitude is the same and depends on the ratio between the chiral and linear-birefringence values. The above formulas can also be used for determining the parameters of the crystal chirality and linear birefringence. It is also expedient to consider the case where η = π/4. Equations (4) and (5) also yield the scatteredradiation intensity J = const · |α|2 |Ex |2 cos2 (ΔΦ) (16) for component Ex of the incident-radiation field. Obviously, in this case, periodic traces against a zero background with maximum amplitude (as in Eq. (10)) are formed and the trace period is governed by the joint chirality and linear-birefringence action according to the superposition principle given in Eq. (6) (see also [10]) and decreases with increasing angle θ. The above formulas allow us to explain the above-considered experimental observations of the light scattering in anisotropic media. 4.

CONCLUSIONS

To gain a more profound understanding of the features of such scattering and photoluminescence in crystals in such configuration and possibility to use them for the diagnostic purposes, additional studies of these phenomena in particular crystals, which are of applied interest, e.g., the KDP and DKDP crystals are required. In addition, it is also expedient to perform such studies with a more detailed analysis of the spatial microstructure of the scattering region. From our viewpoint, it is very useful to conduct further studies of photoluminescence in crystals, which can yield valuable information on the actual admixtures in crystals and their optical inhomogeneities related to the raw-material quality and their growth features. REFERENCES

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9. A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics, Wiley, London (1975). 10. J. F. Nye, Physical Properties of Crystals, Oxford Univ. Press, Oxford (1967).

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