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Optical Activity of a Nonideal 1D Photonic Crystal

Vladimir Rumyantsev*, Stanislav Fedorov, Marina Proskurenko

A.A. Galkin Donetsk Institute for Physics and Engineering of NASU, Donetsk, Ukraine

Abstract: Natural optical activity of a nonideal 1D multilayer system is considered phenomenologically and the dependence of its specific rotation angle on concentration of impurity layers is simulated numerically. Specific features caused by the corresponding disordering types of the gyrotropic photonic crystal were revealed. Based on the developed phenomenological theory, a microscopic theory of dispersion of optical activity is constructed for the case of a molecular crystalline superlattice, whose layers includes point defects.

Keywords: Light propagation, nonideal 1D gyrotropic photonic crystal, specific angle of optical rotation, exciton region of the spectrum

Received October 26, 2013; Revised November 28, 2013; Accepted December 20, 2013

Citation: Vladimir Rumyantsev, Stanislav Fedorov, Marina Proskurenko, Optical Activity of a Nonideal 1D Photonic Crystal. Journal of Optoelectronics Engineering, 1 (1). 19-27. January 2013. doi:10.12691/joe-1-1-4

1. Introduction

Investigations of the optical properties of dielectric superlattices occupy a prominent place in condensed matter physics (e.g. [1,2,3,4,5]). These studies are mainly stimulated by the needs of electrical engineering and electronics in layer structures and, therefore, by the need to simulate composite materials with the given properties. The study of the spatial dispersion effects is of considerable interest, since they are a valuable and often unique tool for revealing subtle structural features of spatially dispersive media. In connection with this, the development of the theory of the aforementioned effects that reveal of their specificity in multilayer systems and finding corresponding frequency characteristics based on model representations are of current interest. The urgency of these investigations is evident because, at present, there are many complicated organic complexes and polymers that are optically active due to the features of their structure or the optical activity of molecules that enter into their composition (it follows, for example, from the work of Kizel' [3] and Mohrig et al. [6]). The problem of finding normal electromagnetic waves (which are necessary to calculate gyrotropy characteristics) in spatially dispersive superlattices is unsolved. Nevertheless, it is evident that, in the case when the thickness of the layers of a multilayer system considerably exceeds the characteristic scales of spatial dispersion, the corresponding values can be calculated approximately if the contribution of each layer to the gyrotropy is considered to be independent. With regard to natural optical activity (NOA), this means that the knowledge of only the specific rotation angles of the layer (ω is the light frequency, n is the number of a unit cell of the one-dimensional (1D) lattice, α is the number of a layer in the cell) and the concentration of impurity layers (if they are present) is sufficient to find the specific rotation angle of the polarization plane. In this work, the aforementioned approximation is used to calculate of a 1D superlattice that contains randomly distributed extraneous layers, which differ from the corresponding layers of an ideal system in the physico-chemical composition and/or in the thickness. The expression for obtained below makes it possible to numerically simulate the concentration dependence of the optical activity, which was done for a SiO₂-liquid crystal (LC) multilayer system and for 1D superlattice with optically active sublattice of potassium dithionate . With the known microscopic expressions for , the approach described in this work makes it possible to reveal and identify the predominating gyrotropy mechanisms, which are important for experiment, to establish the relation of the above mentioned function with microcharacteristics of the medium (such as the dipole, quadrupole, and magnetic dipole moments of structural units), and to find the corresponding frequency characteristics. Superlattices that consist of macroscopically homogeneous systems with pointlike defects are of special interest. In this case, is a function of not only the concentration of defects in extraneous layers, but also of the concentration of pointlike defects.

In this work, numerical calculations were performed for the frequency dispersion of the optical activity of a nonideal superlattice made up of layers of impurity molecular crystals. The latter condition widens the possibilities of simulating composite materials because it permits one to continuously vary their gyrotropy properties and the disordering parameters (with respect to the composition or/and thickness). For the gyrotropic potassium dithionate sublattice we take as defect layers a model molecular crystal with vacancies. of such a system is obviously a function not only of foreign layer concentration but of vacancy concentration as well.

2. Specific Rotation Angle of Light Polarization Plane in Nonideal 1D Superlattice

In keeping with the above mentioned approximate approach, the rotation angle of the polarization plane of light has the following form:

[picture] | (1)

In Equation (1) and are the configuration dependent thickness of the α-th layer of the n-th unit cell and the specific rotation angle of the plane of polarization of light caused by this layer, respectively, σ is the number of layers of the unit cell. It is evident that Equation (1) is valid for the case of light propagation along layer optical axes perpendicular to the plane of the layers of a nonideal, topologically ordered, 1D superlattice consisting of N unit cells. We suppose that the number of cells N is so large that it is possible to form a configuration that averages in correspondence with the general principles of physics of disordered systems.

According to Ziman [7], the experimentally measured rotation angle is defined as , where is the configuration averaging operator that acts on the configuration-dependent function . There are two types of disorders in the considered nonideal 1D superlattice; therefore, there are two types of configuration dependences. The disordering of the first type is caused by the fact that the superlattice contains extraneous (defective) layers that differ from the corresponding layers of an ideal system in the physico-chemical composition (the corresponding configuration_dependent quantity is ). The second disordering is caused by the presence of layer defects that differ from an ideal superlattice in thickness (the corresponding configuration_dependent quantity is ). It is evident that these factors do not depend on each other. The connection of and with the configuration-dependent stochastic variables and has the following form:

[picture] | (2)

and

[picture] | (3)

Here , if the-th layer of the n-th unit cell is the layer of -type ( and - in any other case; , if thickness of the - th layer of the n-th unit cell equals ( and - in any other case. is the specific rotation angle of the - th layer of type. Now and then index enumerates layers of variable composition, - of variable thickness.

Using Equations (1)–(3) and the averaging rules for , we obtain the following expression:

[picture] | (4)

where

are the concentrations of defects of layers that differ from the layers of the basic compound (for which the layers of the first sort are conventionally taken) in the composition and/or thickness, respectively. The first summand in Equation (4) corresponds to the rotation angle of the light polarization plane of an ideal 1D superlattice made up of layers of the first sort. The second summand is caused by the disordering of the superlattice with respect to the composition; it vanishes in the absence of variations in the composition. The third summand reflects disordering in the thickness (in the absence of this disordering, this summand vanishes). The last summand is caused by the simultaneous disordering of the superlattice with respect to both the composition and the thickness of the layers. The absence of at least one of the disorderings leads to the vanishing of the fourth summand in Equation (4). Each of the four summands of (4) in the sum over α has the meaning of the rotation angle per one unit cell. Unlike (which is measured in degrees/unit of length), these angles are measured in degrees.

To illustrate the obtained results, let us consider NOA in a nonperfect 1D superlattice with two elementary layers in a unit cell (see Figure 1) the first layer of which is SiO₂ (α-quartz) (= 780 degrees/mm, λ =152.3 nm) and the second layer is a model liquid crystal (= 2000 degrees/mm). Let the first sublattice contain impurity layers that only differ from the basic (ideal) sublattice in thickness and let both the thickness and the composition vary in the second liquid crystal sublattice ( = 2500 degrees/mm). Let the concentration and thickness of the layer of the basic compound in the first and the second sublattices be denoted as , and , , respectively, and those of the impurity as , and , (index C(T) denotes the variations in the impurity layers in the composition or in the thickness). Based on Equation (4), simple transformations permit one to obtain the following concentration dependence of the specific rotation angle of the light polarization plane in the considered two-sublattice nonideal 1D superlattice, , as follows:

Here, is the averaged period of the cell of the 1D superlattice,

The diagram of the concentration dependence of the specific rotation angle of the light polarization plane in the studied nonideal superlattice is presented in Figs 2 and 3. Figure 2 reflects the variation of the superlattice layers only in thickness, and Figure 3 presents the function . Moreover, Figure 3 shows a considerable variation of the specific angle with variation of the relative thicknesses ,

Figure 1. Problem scheme: imperfect two-sublattice photonic crystal with randomly included defect layers of variable composition and thickness

Figure 2. Concentration dependence of the specific rotation angle of the light polarization plane in the studied nonideal superlattice: ; and are, respectively, (1) 0.3 and 0.5, (2) 0.3 and 0.1, and (3) 3 and 0.1

Figure 3. Concentration dependence of the specific rotation angle of the light polarization plane in studied nonideal superlattice: , , and ; concentration is 0.1 for (1), 0.3 for (2), and 0.9 for (3)

of the layers at constant , and permits one to conclude that, for particular values of the relative thicknesses of the layers, the value of depends rather weakly on the variation of the superlattice composition.

The study of these composite polymer and LC superlattices is important due to the variety of their functional properties in a wide application area (as [9,10,11]). Below, by the example of systems with only one type of disorder (in thickness), we study the NOA of two-layer superlattices made up of layers of binary mixed (the first layer) and orientation-disordered (the second layer) molecular crystals. In the following section of this work, using the microscopic expressions for , features of the frequency–concentration dependence related to the relative positions of excitonic energies of both sublattices and with layer thicknesses are studied for the excitonic region of the spectrum.

3. Microscopic Theory of the Optical Activity of Imperfect Superlattices

3.1. Theoretical Fundamentals

The expression for the specific rotation angle of the polarization plane of linearly polarized light transmitted through a nonideal two-layer molecular superlattice (the first layer is a mixed molecular crystal and the second layer is orientationally disordered), as follows from Equation (4), has the form:

[picture] | (5)

Where - specific rotation angles for mixed and orientationally disordered molecular crystals; - thicknesses of first and second layers of corresponding perfect superlattice; - concentrations of isotopic impurities and orientationally disordered molecules, - concentrations of foreign layers in relevant sublattices with thickness differing from those of the perfect system. Here each of the summands has the sense of the rotation angle (by a corresponding sublattice) per one unit cell.

In the most general case, the specific rotation angle for a multicomponent topologically ordered impurity (including both mixed and orientationally disordered) molecular crystal with a primitive lattice in the exciton region of the spectrum is, within the framework of single-level model, of the form:

[picture] | (6)

Here - volume of the unit cell of molecular crystal, r – number of molecular groups, each relating to a definite -th or - th type of molecules; . Quantities and are:

[picture] | (7)

[picture] | (8)

In Equations (7), (8) - Levy - Civita tensor, , - molecular excitation energies, - matrix elements of the dipole-moment operators, - matrix elements of operators derived from operators of molecular currents of the -th and-th molecular types, respectively,

[picture] | (9)

Matrix elements of locator function and matrix (proportional to matrix of the resonance intermolecular interaction) are of the form:

[picture] | (10)

Matrix elements of locator function in the nodal representation are expressed through matrix as follows:

[picture] | (11)

Equation (11) is written in terms of projection operator , - unit operator. , configuration-dependent random unit equals 1, if in node n there is an-type molecules, and it is zero in any other case. Matrix elements relate to the matrix of resonance intermolecular interaction (figuring in exciton Hamiltonian [8]) as:

[picture] | (12)

3.2. Results and Discussion

3.2.1. Specific Rotation Angles for Mixed and Orientationally Disordered Molecular Crystals

For binary systems, values in mononodal approximation are:

[picture] | (13)

In Equation (13) - molecular concentration for the basic substance of a mixed or orientationally disordered molecular crystal, and . Zeros of function define renormalized exciton energy levels of a molecular crystal, each lying in a definite interval of values (see Figure 3).

The character of the frequency-concentration dependence of rotator power is specified by the relative position of exciton energies and by values of parameters , . It follows from Equations (13) that for frequencies of light running over one of the regions of values (see Figure 4) the rotator power is divergent if damping is neglected.

Figure 4. Concentration dependence exciton energy levels of a molecular crystal: а); b) . is given in units of

For numerical calculations we limit ourselves with the two cases out of variety of types of the optical-activity concentration dependence for certain frequencies of electromagnetic wave. In the first case, the energy of electromagnetic excitation is in the interval , and the corresponding concentration dependence of the specific rotation angle of light passing through the layered structure under consideration is illustrated in Figure 5. In the second case (Figure .6), which is described by inequalities , for definite values of frequencies of light, the energy of electromagnetic excitation lies in the neighborhood of exciton resonance.

In particular calculations it is considered that for a mixed molecular crystal it can be, within a good accuracy, assumed , , (as a consequence, all ). For an orientationally disordered molecular crystal, as shown in [8], .

It is assumed that for a mixed molecular crystal , , , whereas for the orientationally disordered one - , , and , . Here, the numerical simulation was done for the following values of concentration of heterogeneous superlattice layers: , and for relative thicknesses ().The concentration dependences of the specific rotation angle of light are graphically shown in Figs. 5,6 in relative units ,

where

Figure 5. Concentration dependence of the specific rotation angle for

The frequency dependences of the specific rotation angle for specific values of concentration and is illustrated in Figs. 6 and 7. It is seen that changes in concentration with constant result in mutual removal of resonance frequencies (case а), whereas changes in with constant – in their approaching (case b). So the curves are determined by the specific dependence of exciton energy levels on the concentration of point defects considered superlattice (see Figure 4).

Figure 6. Concentration dependence of the specific rotation angle for

Figure 7. Frequency dependences of the specific rotation angle for values of concentration and equal, respectively, to: 1 – 0.2 and 0.17; 2 – 0.2 and 0.45; ω is given in units of

Figure 8. Frequency dependences of the specific rotation angle for values of concentration and equal, respectively, to: 1 – 0.2 and 0.2; 2 - 0.5 and 0.2; ω is given in units of )

3.2.2. Impurity Layers with Vacancies in 1D Superlattice

In the case of gyrotropic photonic crystals with impurity molecular crystal layers containing vacancies only, each of indices and assumes only two values: corresponding to the molecules of molecular crystal and corresponding to vacancies. Hence matrices and have only one non-zero element: ,. Taking this into account and using Equation (6) to write the one-node representation of the locator function we obtain the following expression for the rotatory power:

[picture] | (14)

Here is the exciton energy renormalized due to presence of vacancies:

[picture] | (15)

is the molecular excitation energy (see Figure 9), is the Fourier transform of the intermolecular interaction matrix, is the vacancy concentration.

For the nonideal 1D photonic crystal disordered only by the widths of layers Equation (4), which defines function takes the form:

[picture] | (16)

Where N is the number of elementary cells, is the number of sublattice of the crystalline inhomogeneous system, numbers the groups of layers of the same physico-chemical composition and thus having the same rotatory power ; is the number of groups of layers in the -th sublattice,

[picture] | (17)

denotes the concentrations of layers of the corresponding group, which differ by composition from the layers of the base substance of width (arbitrarily taken as the 1st type layers). The first term in Equation (16) is the angle of rotation of polarization plane of an ideal 1D-superlattice composed of layers of the base substance. The second term is due to compositional disordering of layers in the sublattice.

As applied to the considered gyrotropic photonic crystal Equation (16) becomes

[picture] | (18)

where is the specific angle of rotation of polarization plane in a layer of base substance, is the width of this layer, is the specific optical rotation of an admixture layer (a model molecular crystal) which in the context of exciton model is given by Equation (14), is the concentration of layers of molecular crystal. Since depends on vacancy concentration, the reduced angle of rotation is a function of both admixture layers and vacancies.

Performing the appropriate transformations for and taking into account Equation (14), we arrive at the following expression for the reduced angle of rotation of polarization plane:

[picture] | (19)

Here

is a dimensionless combination obtained from Equation (5) under conditions that electromagnetic irradiation is directed along the optical axis (ОО), and the light frequency lies far from magnetodipole transitions, is the electron charge.

Figure 9. The dependence of the renormalized vacancies exciton energy of molecular crystals on the concentration of vacancies, where 1 corresponds to the excitation energy of the molecules, 2 corresponds to , and 3 for at

Figure 10. The concentration dependence of the angle of rotation of the plane of polarization of the light in the test imperfect photonic crystal for

In addition in the developed model of molecular crystal quadruple moments are approximately expressed as products of dipole moments (such replacement does not qualitatively alter the character of concentration dependence (19) of the model system and is necessary for numerical estimations). The critical vacancy concentration of the molecular crystal , which by definition satisfies the relation

[picture] | (20)

equals to

[picture] | (21)

Function related to this quantity has the form

It follows from Equation (19) that for the light frequencies falling into the codomain of the function , the rotatory power is divergent under the neglected attenuation.

To concretize the obtained results let us consider the propagation of electromagnetic irradiation through a gyrotropic photonic crystal of potassium dithionate – a 1D superlattice whose nonideality is due random substitution of layers by a model molecular crystal with vacancies. Concentration dependences of the reduced angle of rotation of polarization plane for the light frequency , and three different values of are plotted in Figs. 10, 11 and 12.

Figure 11. The concentration dependence of the angle of rotation of the plane of polarization of the light in the test imperfect photonic crystal for

The character of the dependence is essentially dependent upon the value of exciton energy as related to the given . Figure 10 depicts the case of the critical vacancy concentration for the molecular crystal parameters and . Figure 11 and 12 correspond to the cases of and and the same values of other parameters.

Figure 12. The concentration dependence of the angle of rotation of the plane of polarization of the light in the test imperfect photonic crystal for

4. Conclusion

Studies for optical properties of dielectric superlattices comprise a significant part of the modern-day condensed matter physics. Their importance is enhanced by the constantly growing demands of electronics and electrical engineering for the high-performance layered materials with prescribed optical characteristics. The related phenomena of spatial dispersion are of particular interest since they provide a valuable and very frequently the sole tool for revealing fine structural features in spatially dispersive media (e.g. [12]). This call for development of theoretical models, which would permit to gain a better understanding of the specified phenomena in multilayers and would hopefully allow to calculate the corresponding frequency characteristics. The problem appears even more topical in the light of the vast diversity of synthesized complex organic polymers, which happen to be optically active either due to their structural features or due to optical activeness of the comprising molecules [10,11]

In contrast to previous works, in which the optical activity of ideal crystals studied with a microscopic approach, present microscopic consideration is applied to dispersion of the optical activity of imperfect 1D-multilayer material. Our approach is helpful for a numerical simulation of the frequency-concentration dependence of the specific rotation angle for molecular superlattices in the exciton region of the spectrum.

The model was a two-sublattice system with parameters typical of orientationally disordered and mixed molecular crystals. Gyrotropy features due to disorder types have been studied. An 1D-superlattice with layers containing point defects has been investigated. This creates additional possibilities for simulation of optically active multilayer composite materials, which can find application in optoelectronics engineering. One of the ways to control the transmission of the signal through the optoelectronic system is the rotation of the polarization plane of the electromagnetic wave passing through this system.

Acknowledgement

The work was carried out in the frame of joint Ukrainian–Russian project of National Academy of Sciences of Ukraine and Russian Fund for Fundamental Studies (Grant Number 0112U004002) and supported by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme (project LIMACONA: "Light-Matter Coupling in Composite Nano-Structures", Grant Number: PIRSES-GA-2013-612600).

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