Optical Activity of a Nonideal 1D Photonic Crystal

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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

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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)

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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:

(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:

(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:

(7)

(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,

(9)

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

(10)

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

(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:

(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:

(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

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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

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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

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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

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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

)

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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:

(14)

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

(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:

(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,

(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

(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:

(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

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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

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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

(20)

equals to

(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

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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

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