Localized modes in chiral photonic structures

Anisotropy and chirality of photonic structures give rise to altered polarization of light propagating in them. A notable specific feature related to polarization in chiral structures is the geometric (topological) phase [61]. The geometric phase is defined as the phase acquired when an oscillatory system travels along a closed trajectory in the parameter space. The notion of a geometric phase, first introduced in optics [62, 63] and mechanics [64], presently includes, in addition, various phenomena of quantum and relativistic physics [65, 66].

In recent years, photonics has experienced another spike of interest in topological ideas [67, 68] due to the popularity of the graphene model and the topological insulator concept [69], on the one hand, and progress in optical technologies, on the other hand. For example, recently in the polarization optics the 3D structure of the light polarization field with nontrivial topology was measured directly [70].

The geometric phase in a twisted nematic not only allows explanation of polarization phenomena [71] but is also employed to design wave fronts using so-called Pancharatnam–Berry-phase optical elements [45, 72]. The geometric phase can be manipulated regardless of the total phase, which allows it to be used for shifting frequencies in modulators by adding the phase plate mechanical rotation frequency to the field frequency [73–75]; also, switching with the use of a ferroelectric LC is possible [76]. The authors of Ref. [77] proposed a waveguide in which a geometric phase is used instead of the refractive index gradient for total internal reflection.

A simple example of a geometric phase in polarization optics is provided by the QHQ-device depicted in Fig. 5. A full-wave phase plate (b, c, d) having a vertical axis is placed between parallel polarizers (a, e) turned 45° with respect to the vertical plane. The plate is split into two quarter-wave parts (b, d) and the half-wave part at the center (c). The incident radiation passing through the polarizer (a) becomes linearly polarized. The quarter-wave plate (b) makes polarization circular. The half-wave plate (c) changes the sign of circular polarization. Then, inverse transformation takes place. The quarter-wave plate (d) makes light linearly polarized again, after which it passes unobstructed through the polarizer (e) and leaves it. The central half-wave plate (c) can rotate in its plane, and the phase of the passing circular radiation is essentially determined by optical axis rotation. Evidently, for vertical and horizontal positions of the optical axis (0 and π/2) the phase of the outgoing radiation differs by π. Generally speaking, the change in the radiation phase corresponds to the double angle of rotation. The above qualitative considerations can be justified by the rigorous quantitative description in the algebraic language of Jones matrices and geometric language of the Poincaré sphere [61, 75].

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If the position of the optical axis is neither vertical nor horizontal, the QHQ-device becomes chiral. Mirror reflection of the QHQ-device with polarizers corresponds to the right-left circular polarization conversion. The chirality of the structure and the optical field changes while the sign of the geometric phase remains unaltered. In other words, the geometric phase in this case is even with respect to chirality sign, i.e., it has the same sign for right- and left-hand structures. A more complicated case of the nonadiabatic geometric phase also even with respect to chirality sign is described in Section 3.2.2. On the other hand, chirality of the optical field can be altered without changing structure chirality by turning both polarizes through 90° and changing right- to left-handedness of circular polarization. In this case, the geometric phase is odd with respect to the structure chirality sign, i.e., it has different signs for right- and left-hand turns of the half-wave plate.

For polar holography and a polarization diffraction grating (Fig. 4b), phase control is more common than light transmission control (what is meant here is the geometric phase determined by a change in polarization in an anisotropic medium rather than the dynamic phase depending on the medium thickness).

Let us turn to a mathematical description of light propagation in a helical structure confining ourselves to the case of normal incidence of the light propagating along the z axis of the helicoid. The Maxwell's equation in optics is usually written at permeability μ = 1:

The wave is described by vector E of complex amplitudes for electric field components in orthogonal directions x and y. Projection of the permittivity tensor onto the xy plane in the cholesteric depth z has the form

Here, the optical axis coincident with the cholesteric director is described by the twist angle measured from the x axis toward the y axis; p is the helix pitch. Positive and negative pitches correspond to right- and left-handed helicoids, respectively.

To find the simplest solution, the symmetry between electric and magnetic field intensities needs to be restored and permeability taken into consideration. The explicitly written magnetic field strength H increases the field vector dimensionality from 2 to 4:

Accordingly, the order of the differential equation is reduced from second to first. Suppose that the principal axes of the magnetic and electric permeability tensors coincide. This assumption makes possible a transition to the orthonormal basis u, ν, z uniformly rotating together with the cholesteric director, so that the u axis always goes along the director:

In the Berreman method [78], Maxwell's equations in a stationary case take a form more general than Eqn (1):

Matrix for the rotating basis formulated in [79, 80] can be reduced as follows:

where is the dimensionless wavelength, is the dimensionless coordinate, and λ₀ is the wavelength in a vacuum. There are different units for electric and magnetic strengths in the SI system; therefore, they have to be normalized via the vacuum impedance . The differential transfer matrix can be just as well written in the nonrotating basis [78, 81].

Four normal waves correspond to Eqn (3). They are determined by the eigenvalues of matrix having the sense of refractive indices n. The respective eigenvectors of matrix have the sense of polarizations J₀. Based on the helicoid axis reversal symmetry, these four normal waves can be classified as two pairs of counter-directed waves. In each pair, the wave with a larger refractive index has a lower phase velocity. This wave will be called slow and the other wave fast:

Substituting the solution for into Eqn (3) yields

where is the unit matrix and superscript '+' in is omitted. The refractive indices are as follows:

where d_μ = (_eμ_o – _oμ_e)/2 is the antisymmetry of permittivities, and the overbar defines the arithmetic mean over ordinary and extraordinary permittivities:

The scale invariance of Maxwell's equations (3) and normalization of material parameters (see Supplement in [80]) reduce the structure to two crucial parameters: electric and magnetic anisotropies

Consider the case of equal anisotropies δ = δ_μ. The clarity and elegance of this single-parametric set of structures should compensate the difficulty of their physical realization in the optical range for the reader [82]. Dispersion equation (7) is simplified due to the symmetry of permittivities d_μ = 0:

Without further loss of generality, we assume the normalization . Then, . In other words, the permittivity is normalized to the geometric mean of _o and _e: , the arithmetic mean being not less than unity. The second term in the right-hand part of the dispersion equation becomes squared anisotropy . Anisotropy here is akin to the standard deviation of permittivities:

The dispersion curve of Eqn (11) is a hyperbola, and pure imaginary values give rise to an exact circle in the stop-band. Dispersion equation (11) can be written for both the refractive index and the wave vector:

This symmetry of and dispersions appears to have been noticed for the first time in Ref. [21]. It indicates the symmetry of longwave and shortwave limits (Fig. 6). In the longwave limit, the medium is homogeneous, and a manifestation of anisotropy reduces to polarization rotation opposite to the helix twist. The negative optical activity ceases as the situation comes close to the static field case . It is not identical with natural optical activity, since normal waves are not purely circular [83]. In the shortwave limit , the symmetric positive optical activity is supported by the Mauguin waveguide regime. Violation of the equal anisotropies condition, δ ≠ δ_μ, breaks symmetry, as shown by the dashed lines in Fig. 6. Ordinary and extraordinary waves can then be distinguished in the high-frequency limit equivalent to helix untwisting. Both positive and negative optical activities of a helicoidal medium can be described either in the algebraic language of Jones matrices or in the geometric language of Mauguin's rolling cone [83–85].

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Thus, it is possible to classify effects of a helical photonic structure on optical fields by grouping phenomena into three classes according to the ratio of light wavelength λ inside the structure and helix pitch p:

• λ < p — negative optical activity, polarization rotation opposite to the helix twist;
• λ > p — positive optical activity, Mauguin's effect, linearly polarized light is adiabatically controlled by the helix twist;
• λ = p — equality is conserved within the frequency range for an incident circularly polarized wave with the sign coincident with the helix twist sign; selective volume reflection (Bragg circular diffraction) takes place.

Let us focus on the circular Bragg diffraction at λ = p. The refractive index n_f for a fast wave acquires purely imaginary values. This is the case when the phase velocity becomes infinite and the group velocity meaningless. It would be reasonable here to write dispersion equation (11) in the form of a trigonometric identity where a certain angle χ ∈ [0,π/2] acts instead of the wavelength:

This means that at the wavelength

the refractive index of the fast wave n_f = iδ sin (2χ) describes full reflection in the bulk cholesteric. This wave is conventionally called a diffracting wave [1]. The spatial form of the diffracting wave shown in Fig. 1 can be derived from the nontrivial solvability of Eqn (6). It can be shown as in [21] that both the incident and the reflected waves have circular polarization with the sign coinciding with the helix twist sign. At a fixed point in space, intensity vectors of the incident and reflected waves rotate in opposite directions. The phase difference between them is 2χ. For a superposition of these waves, the electric and magnetic fields are oriented at the same angle -χ to the optical axis of the helical structure. The problem of polarization is closely related to that of the absence of nodes and antinodes in such a standing wave (see Fig. 9 below).

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The selectivity of reflection of only one of the two circular polarizations is compensated by the enhanced reflective power compared with that of a scalar (nonchiral) Bragg reflector consisting of a few layers of isotropic materials. The measure of reflectivity is radius r of the semicircle in Fig. 6. This dimensionless radius with the sense of the extinction coefficient determines not only the depth of the stop-band but also its width. It makes an exponential contribution to the Q-value and the laser oscillation threshold of resonators. In the case of equal anisotropies, this radius (in accordance with Eqn (10)) is equal to the anisotropy: r_μ = δ. In the absence of magnetic anisotropy, e.g., in a cholesteric, the circle undergoes deformation proportional to electric anisotropy, and its radius decreases by roughly half, r ≈ δ/2. The reflective power of a scalar Bragg reflector is optimal when optical paths in the layers are of equal length, and each alternating layer holds a quarter-wavelength. In this case, the stop-band width is given by expression [86]:

This means that at equal periods and contrasts of the refraction index, the frequency interval of reflection of a scalar structure is π/2 times narrower than in a cholesteric, while the thickness of the entire structure needed for a fixed light reflection value at the frequency in the stop-band center becomes greater. Two factors in this comparison are of importance. First, a helical structure in the sense of permittivity tensor (2) considered in two orthogonal projections on planes xz and yz represents two sinusoidal profiles displaced by the quarter-pitch of a helicoid relative to each other. Both profiles independently contribute to the reflection of the diffracting wave. It doubles the reflective power and quantitatively compensates for reflection of only half of the polarized light by the helical structure. For a scalar reflector with the sinusoidal profile, . Second, the reflectivity of the stepwise profile of the alternating layers at equal anisotropies is stronger than that of the sinusoidal profile, , because the first term of Fourier expansion of the stepwise profile has an anisotropy amplitude 4/π times that of the stepwise profile itself. A reflector consisting of two similarly thick helical structures with opposite chirality signs reflects arbitrarily polarized light as efficiently as the beetle integument shown in Fig. 2. Its reflective power with respect to the thickness is similar to that of the scalar reflector with the sinusoidal profile and twice that with respect to the width.

A variety of complicated problems can be addressed based on the simplest models. Dispersion equation (12) for a helical structure gives dispersion curves in the form of conical sections. Such an exact solution can be compared with another possible analytical solution for the case of a step-index profile. Here, the dispersion curve of the Rytov equation [87] only roughly corresponds to a parabola (see Section 6 in [86]). The solution proposed by Abeles [88, 89] comes from joining the Rayleigh waves at the interfaces and raising the respective matrix to the power through the Chebyshev identity. The simplicity of helical symmetry is attributable to the smoothness, as opposed to discreteness of crystal translational symmetry. The solution of dispersion equation (7) for cholesterics is in excellent agreement with the measured transmission [90] and luminescence [91] spectra. The discovery of this solution has a remarkable history [92–95]. The author of [96] treats it as a paradigmatic case. It is poorly known and as a rule rarely mentioned in classical studies on wave physics in one-dimensional periodic and layered media [97, 98]. Many authors confine themselves to the generality of the Hill scalar equation

the simplest case of which is the Mathieu equation for harmonic potential f(z) = cos (kz) that has no analytical solution [99].

Let us consider a Fabry–Perot resonator consisting of two flat mirrors (see Fig. 7) with their reflecting surfaces facing each other and oriented in the xy plane. The surfaces cross the z axis at points 0 and L. A nematic LC (nematic) is placed between the mirrors. The unit vector of the predominant direction of LC molecules is called the LC director. In the twisted state, the nematic layer is split into plane-parallel sublayers. The director in each sublayer is constant but turns in passing from one sublayer to another. It is supposed that there are no external orienting fields and the twist is uniform, i.e., the director uniformly rotates clockwise in the layer plane. In Fig. 7, the full angle of twist is 80°, but further consideration holds equally well for an arbitrary angle.

The field of the LC director determines the local permittivity tensor at all points of the medium. The axis of extraordinary permittivity coincides with the LC director. Let us consider a nematic with positive uniaxial anisotropy. The ordinary and extraordinary refractive indices correspond (in terms of phase velocity) to slow and fast waves, respectively; they equal n_{e, o} = n ± δn.

Experimental and theoretical studies of defect modes in nematic-filled nonchiral resonators in electric and magnetic fields are reported in [140]. Two series of transmission bands are distinguished in the spectrum. They correspond to the modes in which light polarization is either transverse or parallel to the predominant direction of LC molecules; these are ordinary and extraordinary modes, o- and e-modes, respectively. Application of the external field leads to the inclination of liquid crystal molecules localized far from the boundaries. E-modes undergo essential displacement in fields exceeding the Fréedericksz threshold, while o-modes do not change their position. Twisting of the predominant direction of nematic LC molecules in the PC layer plane creates coupling between defective o- and e-modes. Normal modes can be referred to as L-modes, virtually longitudinal with respect to the predominant direction of the molecules and T-modes almost transverse to it. The situation is complicated by the fact that normal modes change polarization from linear to elliptical, and o-modes become also displaced under effect of the external field.

To calculate the LC structure in a twist-cell under voltage, the variational free energy minimization method was employed. It proved difficult to come to the analytical solution for deformation under the influence of electric voltage. Therefore, the numerical method of minimizing free energy through gradient descent was used. To simulate a spectrum, the following parameters of the structure were chosen: ITO film (n_ITO = 1.88858 + 0.006 i, d_ITO = 140 nm), glass substrate (n_sub = 1.45), orientant (n_PVA = 1.515, d_PVA = 1000 nm), defective layer filled with 5CB LC (n_|| = 1.701 + 3.9 × 10⁻⁴ i, n_⊥ = 1.536 + 3.9 × 10⁻⁴ i, d = 10815 nm). The thickness of the defective layer does not strictly correspond to the spacer thickness and is found from the distance between spectral peaks (intermode spacing). Materials of the alternating layers were zirconium dioxide (ZrO₂) and silicon dioxide (SiO₂).

Figure 8A presents an experimental light transmission spectrum for the case of a polarizer oriented across the rubbing direction of the nearest substrate. It shows a second series of minor peaks, besides the main one. There is good agreement between theoretical and observed spectra. Although extinction of materials was considered in the calculations, the spectral peaks turned out to be higher than in experiment due to the weak light scattering by structural inhomogeneities and other imperfections inherent in experimental procedures, including tolerance limits for manufacturing multilayer structures.

Figure 8B presents a calculated transmission spectrum of nonpolarized radiation at the long-wavelength wing of the photonic band gap under increasing cell voltage. The spectrum gives evidence of four stages in the displacement of transmission peaks.

I. 0 < U < U_C = 0.78 V. Position of defect modes remains unaltered at voltages below the Fréedericksz threshold.

II. 0.78 V < U < 1.1 V. L-mode peaks move toward shorter wavelengths if the Fréedericksz threshold is slightly exceeded, but the position of T-modes remains virtually unaltered, because the refractive index for them does not change because of molecule inclination.

III. 1.1 V < U < 1.6 V. At high voltages, the middle layer of LC molecules inclines along the light propagation axis. It decreases the medium anisotropy, upsets the Mauguin regime, and enhances the ellipticity of polarization. The coupling of L- and T-modes increases upon reflection from the mirrors. Mode repulsion (avoided crossing) is augmented. Solid ovals in Fig. 8B outline areas where transmission peaks are pushed apart; dashed ovals mark regions of crossing between weakly coupled modes of the same parity. Each avoided crossing changes the mode number by one: +1 for T and −1 for L. In the general case, simple identification of the mode number as the number of standing wave antinodes is inapplicable to elliptically polarized resonant modes, because standing wave nodes are smeared out by the interference of counter-propagating elliptical waves of the same sign (Fig. 9).

IV. 1.6 V < U. At a voltage of twice the Fréedericksz threshold, the middle layer of LC molecules acquires an almost homeotropic orientation along the rotation axis. The molecules cannot create a torque. Therefore, the central layer eliminates the adhesion of near-surface LC layers. Each half of an LC layer returns to its orientation plane. Molecules in the left and right parts of the defect are oriented in horizontal and vertical planes, respectively. L- and T-modes merge and become polarization-degenerate. This regime is called polarization-independent. The refractive index goes down, as the voltage and LC inclination into the homeotropic state increase, while mode doublets move into the short-wavelength region toward the position of the untwisted homeotropic mode with linear polarization. It is shown in Section 3.2.2 that a mode with the same number has an even shorter wavelength at zero voltage.

Let us consider the field in an L-mode localized at the defect (see Fig. 9). Standing wave nodes and antinodes cancel each other out in the center of the defect due to ellipticity of counter-propagating traveling waves that form a standing wave. For example, two counterpropagating waves with identical circular polarization do not have nodes [137].

The phenomenon under study has a great potential for applications, such as controlling the structures with sharp optical properties and high-precision measurements in excellent agreement with numerical calculations. In addition to purely optical applications, it makes possible highly accurate characteristic of such LC parameters as viscosity and elasticity coefficients.

The geometric phase manifests itself in the transmission spectra of chiral structures via the phase matching condition that describes the total spectral shift taking into account various types of wave coupling. One of them of special interest in the present report is LC twisting and its respective spectral shift. Another is mode coupling associated with reflection from mirrors and the corresponding spectral shift of reflection. Wave couplings are responsible for the transition from simple to more complicated wave types listed in the table below and shown in Fig. 10.

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(1) The simplest class of eigenwaves comprises o- and e-waves propagating without a change in polarization in a homogeneous anisotropic medium with a constant direction of the optical axis. These waves are characterized by linear polarization either along or across the optical axis.

(2) Twisting of the optical axis causes coupling between o- and e-waves via nondiagonal components of the permittivity tensor. A new class of eigenwaves is represented by to- and te-waves [110]. These waves have elliptical polarization. Ellipticity of polarization is conserved; the principal axes of the ellipse are directed along and across the optical axis.

(3) Resonator mirrors create a coupling between to-, te-waves by changing the sign of ellipticity upon reflection. The right-hand polarized waves become left-hand polarized and vice versa. The third class of eigenwaves consists of ro-, re-waves. These are standing waves localized in the resonator. They have variable elliptical polarization. The principal axes of the ellipse deviate from directions along and across the optical axis. It is remarkable that ro-, re-waves on the mirrors have linear polarization [124, 125]. Their polarization is conserved during reflection. T-, L-modes in Fig. 8 can be regarded with a good accuracy as ro-, re-modes.

The table can be generalized to the case of anisotropic reflection from the mirrors by taking account of the phase difference between the reflected o- and e-components of the field. Anisotropy of reflection takes place even for mirrors from an isotropic material, because the medium inside the resonator is anisotropic. Therefore, polarization of the light leaving the resonator is no longer linear. However, the difference from linearity is as a rule insignificant, which accounts for the lack of its experimental observations [139]. The breaking of spatial symmetry of the twist-cell also contributes to the deviation from linear polarization [126].

Let us introduce the mean phase σ = nk₀L, anisotropy phase (angle) δ = δk L, and twist angle φ, where k₀ = ω/c is the wave vector modulus in a vacuum, δk = δnk₀. Then, the total traveling wave phase is

where υ is the twisted anisotropy phase. This expression is called the Mauguin formula [84]. Dividing both parts by k₀L yields effective refractive indices of the twisted medium:

The ellipticity parameter tan Θ = φ/δ reflects the smoothness of twisting relative to the anisotropy value. It corresponds to the adiabaticity parameter of the process known as Mauguin's waveguide regime. The smaller Θ is, the smoother the twisting and the closer the eigenwave polarization to the linear one.

The phase incursion for a cycle in the resonator is equal to the 2ρ angle,

Figure 11a shows dispersion curves of te- and to-waves for φ = π/2 and φ = 0. The o-mode number N_o = (σ – δ})/2π = 2L/λ_o plotted along the y-axis is proportional to light field frequency. The phase incursion on the x-axis is the wave vector multiplied by the resonator length. Figure 11a, unlike Fig. 6, shows no splitting of the to-wave branch corresponding to the stop-band of a cholesteric LC for φ = π/2 and σ = υ at point B. The splitting stays beyond the framework of the model because Eqn (15) disregards reflection in the bulk of the LC.

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The dotted lines corresponding to the phases of o- and e-waves of an untwisted structure, υ(φ = 0) = δ, are straight, since frequency dispersion of the materials is not taken into consideration. Points O and T mark the third mode frequency for o- and to-waves. In such a numeration, the o-mode with the zero number corresponds to point B.

Figure 11b presents the dispersion curves of re- and ro-waves obtained by matching phase (17):

Let us find the spectral shift of the twisted structure relative to the untwisted one and consider the o-wave without loss of generality:

The second term can be neglected far from Gooch–Terry maxima [85, 111], which yields a corollary of the Mauguin formula (15):

Let us proceed from eigenwave frequency dispersion to the wave shape, phase, and polarization changing with variation of the twist-layer depth [113, 125]. Figure 12 shows smooth shift trajectories of various polarizations of the waves penetrating deep into the twisted layer. The space, the points of which unambiguously correspond to light polarization ellipses, is usually mapped on a sphere of unit radius called a Poincaré sphere. To study right-hand twisted to- and ro-waves, it is convenient to map right-hand polarization on the upper hemisphere of the Poincaré sphere as in Refs [10, 112] but not on the lower one as in [61, 110, 141]. Figure 12b presents a cylindrical projection of polarization trajectories. The ro-wave polarization trajectory RR' is a spherical trochoid. It describes the trajectory of the point rigidly bound to a cone rolling over the plane. Mauguin described such a cone rolling slip-free along the 'equator' of the Poincaré sphere [84]. Geometric calculations related to this cone are termed the Mauguin–Poincaré rolling cone method. It allows the Mauguin formula (15) to be treated as the Pythagorean theorem for the addition of orthogonal components of the cone angular velocity.

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Parallel displacement of a geometric object over a curved surface results in its rotation about its own axis. A classical example is Foucault's pendulum, whose plane of swing is rotated by the diurnal rotation of the Earth. Similarly, the parallel transfer of polarization over the curved surface of a Poincaré sphere causes state phase incursion, called the geometric phase. This phase originates from a global geometric characteristic (transfer trajectory) and is independent of local characteristics, such as velocity of the state traveling along the trajectory. In the context of polarization optics, the following 'geometric' formula holds for closed trajectories:

where β is the geometric phase, and Ω is the area enclosed by the trajectory on the Poincaré sphere. A rigorous proof with the use of the Stokes theorem is provided by D N Klyshko in Ref. [61]. To recall, the 'geometric' formula corresponds to the Gauss–Bonnet theorem and can be understood as the addition of spherical excesses of triangles making up the enclosed area.

Reference [85] demonstrates the direct relationship between geometric formula (21) and Mauguin's formula (15). Here, only the Aharonov–Anandan nonadiabatic phase corresponding to eigenwave ellipticity works [142, 143]. This minor contribution from the geometric phase should be distinguished from the geometric phase in the zeroth-order adiabatic approximation responsible for the phase jump by π in a cell twisted through π radians.

To simulate experimental spectra (Fig. 13a), zirconium dioxide (ZrO₂) and silicon dioxide (SiO₂) were used as materials for alternating layers (just as in Fig. 8). The thickness of layers making up dielectric mirrors was 83 nm for SiO₂ and 66 nm for ZrO₂. ITO thickness was 117 nm and n_ITO = 1.88858 + 0.006 i, taking into account absorption; for the glass substrate, n_sub = 1.45. Polyvinyl alcohol with a diffraction index of 1.515 served as the orientant with a layer thickness of 300 and 600 nm, the difference being due to different LC orientations at the mirrors. For MBBA crystals, n_|| = 1.737 + 3.9 × 10⁻⁴ i and n_⊥ = 1.549 + 3.9 × 10⁻⁴ i. The thickness of the nematic LC layer was 7980 nm, and the angle of twist φ = 80°.

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Figure 13b demonstrates experimental and theoretical values of spectral shifts of the transmission peaks. Good agreement is observed for all 14 peaks. The measured spectral shift is proportional to the third power of the wavelength, in conformity with the Mauguin formula without regard for the resonator (20).

The shift related to the optical axis twist considered in this section can be observed directly without polarizers, and the desired measuring accuracy is ensured by the use of a resonator. The proposed experimental design ensures the absence of serious disturbing interferences during transition to the twisted structure. In this scheme, however, the untwisted structure retains a constant refractive index for the o-wave alone. The respective shift of the e-wave can be measured under the experimental conditions with a transition from a twisted to a planar structure that could be observed with the use of a photo-orientant with reversible intermolecular bonds.

This approach takes into account the spectral shift due to mode coupling associated with twisting and reflection from mirrors. The shift is described analytically and detected in experiment in twist-structures free of deformation by the external field. Generalization of the Mauguin–Poincaré rolling cone method made it possible to solve the problem geometrically; it was independently confirmed using Jones's [107] and Berreman's [78] matrix formalisms.

The method shows that the twist-related spectral shift in question characterizes the twisted layer rather than the whole resonator, which serves only to facilitate measurement. Indeed, the twisted layer produces no peaks. What, then, actually does shift as the layer twists? Evidently, the phase of the eigenwave leaving the twisted layer does, which suggests a change in the effective refraction index. The optical length of the resonator is not the sole characteristic from which it can be deduced. For example, polarization diffraction gratings are highly susceptible to the magnitude of the effective refractive index and make experimental verification of the described phenomenon possible.

The assumption of intermediate optical response determining the medium response interval n_|| > n_eff > n_⊥ has been formulated. However, despite its self-evidence, this assumption can be invalid. In the case of a twisted nematic LC, this fact is explained as the contribution from the geometric phase. It is a nonlocal characteristic; therefore, n_eff must not be treated as , where ε_eff is a certain local response of the material. On the contrary, n_eff should be understood in the sense of phase incursion n_eff 2πL/λ. In addition, the pictorial representation of the geometric phase is given in the form of the region enclosed by the trajectory on the Poincaré sphere, with spectral shifts related to the areas of a spherical rectangle and triangle. The observed spectral shift exists not only in the twist-nematic LC but also in any other twisted anisotropic medium.

Finally, it should be noted that in the case in question,as in the example of the QHQ-device discussed in Section 3.1.1, the nonadiabatic correction for the geometric phase is unrelated to the chirality sign of the structure, i.e., it has the same sign for right- and left-hand twisted structures.

The introduction of different types of defects into the structure of an ideal CLC gives rise to narrow transmission bands in the PC band gaps corresponding to localized defect modes. There are a few methods to induce photonic defect modes in CLCs, e.g., using a thin layer of a homogeneous isotropic [149] or homogeneous anisotropic substance inserted between two CLC layers [185, 186], a twist defect (jump in the rotation angle of the cholesteric helix) [133, 153, 154], or a defect caused by a local change in the CLC helix pitch [134, 150, 187]. Reference [149] was the first to report the use of numerical analysis for the study of defect modes in CLCs as materials with photonic band gaps. One of the elucidated optical effects associated with a defect in the form of a thin layer of an isotropic dielectric placed between cholesteric layers is the induction of defect modes in the CLC stop-band for both circular polarizations of normally incident light. An analytical approach to the theory of optical defect modes in CLCs with an isotropic defective layer was developed in Ref. [152] in the framework of a model allowing the exclusion of polarization mixing and deriving the equation for light with only diffracting polarization. Defect modes induced in CLCs by a twist defect, i.e., a sharp turn of the director around the cholesteric axis at the interface between two CLCs, were analyzed analytically in [155, 188]. Light propagation in CLCs containing a combined defect formed by the dielectric layer and a twist defect at the interface between dielectric and CLC layers was investigated in [156, 189]. The localized modes in CLCs, like defect modes in scalar periodically layered media, can be used to produce narrow-band tunable filters [189], amplifiers of polarization plane rotation [190], polarization azimuthal stabilizers [191], and other devices. Low-threshold lasing in CLCs is a subject of extensive research [192]. There are two possibilities for such oscillation, one at the edge of the band gap [193] and the other in the defect mode [185], the lasing threshold in the latter case usually being lower, because the mode frequency is closer to the center of the stop-band, reflection from the CLC layers is stronger, and photonic state density is higher [194].

Nowadays, limitations imposed on the optical properties of natural materials are being successfully overcome due to replacing them with metamaterials. Of special interest are composite media containing metallic nanoparticles used to design nanostructured metal-dielectric photonic crystals and their application for the development of new approaches to light control. Resonance of effective permittivity has been predicted for nanocomposites made of spherical metal nanoparticles dispersed in a transparent matrix [195, 196]. Both the position and the width of the resonance region in the visible wavelength range depend on nanoparticle size and concentration. Effective characteristics of a nanocomposite consisting of metallic nanoparticles suspended in a dielectric matrix are formed by virtue of plasmon resonance of nanoparticles and can acquire unique values in the optical range that are lacking in natural materials. An example is provided by the real part of the effective refractive index, which can be much higher or lower than unity and can even vanish [195, 197–199].

A combination of resonant medium dispersion and intrinsic dispersion of a photonic crystal opens up promising prospects for controlling the spectral and optical properties of photonic crystals [200–210].

To describe nanocomposite materials containing various inclusions, effective medium theories, such as the Maxwell-Garnett and Bruggeman theories, were developed together with approaches to the derivation of formulas for these models [211–216]. Let us consider the simplest case in which the defective layer is a dielectric with metallic inclusions spherical in shape. The effective permittivity of the nanocomposite layer ε_mix is defined by the Maxwell-Garnett equation finding wide application for considering matrix media when the matrix material contains dispersed isolated inclusions of a minor volume fraction,

Here, f is the filling factor, i.e., a nanoparticle fraction in the matrix, ε_d and ε_m (ω) are permittivities of the matrix and the material of the nanoparticles, respectively, and ω is the radiation frequency. The effective medium models make use of the quasi-static approximation with the applicability condition given by the smallness of the size of nanoparticles and the distance between them in comparison with the optical wavelength in the medium. The nanoparticle size is significantly smaller than the field penetration depth in the material. The permittivity of the metal from which nanoparticles are made is found using the Drude approximation:

where ε₀ is a constant taking account of contributions from inter-band transitions, ω_p is the plasma frequency, and γ is the quantity inverse to electron relaxation time. For metal nanoballs suspended in a transparent optical glass, ε₀ = 5, ω_p = 9.6 eV, γ = 0.02 eV [217], ε_d = 2.56.

Disregarding the minor factor γ² yields the resonant frequency depending on characteristics of the starting materials and concentration of the dispersed phase f:

Figure 17 presents the dispersion dependence of nanocomposite permittivity for two filling factor values: f = 0.02, 0.1. It follows that frequency ω₀ corresponding to resonance in the defective layer shifts into the long wavelength spectral region. The half-width of the resonance curve changes insignificantly, but the curve undergoes a well apparent modification, and the frequency range within which the nanocomposite behaves as a metal expands at .

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The structure of interest consists of two identical layers of an ideal right-hand twisted CLC separated by a nanocomposite defect layer (Fig. 18). The cholesteric length is 10P, where P = 275 mm is the crystal helix pitch, and the thickness of the defective layer d = 5P/7. The medium outside the cholesteric is isotropic and has the refractive index n = (n_o + n_e)/2, where n_o = 1.4 and n_e = 1.6 are ordinary and extraordinary CLC refractive indices, respectively. Such a choice of an external medium results in weak Fresnel reflection from the cholesteric surface and interference bands from boundary surfaces. The numerical analysis of spectral properties and the field distribution in the sample was performed using the Berreman transfer matrix method [78], making possible the quantitative characteristic of light propagation in a CLC with a structural defect [218]. The equation describing propagation of light with frequency ω along the z-axis has the form

where Ψ(z) = (E_x,H_y,E_y,−H_x)^T, and Δ(z) is the Berreman matrix [78] depending on the dielectric function and the incident wave vector.

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Figure 19 shows the primary (f = 0) transmission spectrum under the normal incidence of a light beam on the cholesteric having a structural defect in the form of a dielectric plate. Similar to Ref. [149], the figure demonstrates peaks in the cholesteric stop-band corresponding to CLC defect modes induced for both circular polarizations of normally incident light. Moreover, the defect modes have the same wavelength and similar transmissivity characteristics at this wavelength.

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If the filling factor is nonzero and the nanocomposite resonance frequency ω₀ coincides with the defect mode frequency, the latter splits. Manifestation of the splitting effect in transmission, reflection, and absorption spectra is illustrated in Fig. 20. The difference from frequency splitting of two coupled oscillators consists in the presence of more than two modes in the reflection spectrum. It follows from Fig. 20a that due to the splitting defect modes possess equal wavelength for right-hand and left-hand circular polarizations but different transmission in the peak center. Calculations show that splitting increases with increasing volume fraction of nanoballs in the composite, as it does in a scalar PC with the resonant defect layer of nanocomposite [205]. The reflection and absorption spectra (Fig. 20b, c) are characterized by a strong dependence of reflection and absorption coefficients on the direction of incident light circular polarization. The appearance of the spectral region forbidden for both polarizations after splitting of the defect modes (Fig. 20a) is related in first and foremost to strong reflection and absorption of right-hand and left-hand circularly polarized waves (Fig. 20b,c).

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Figure 21 illustrates spatial distributions of the electric field in defect modes with the wavelength λ = 435.8 nm (Fig. 20a). Field localization most strikingly manifests itself in the region commensurate with the wavelength for the mode corresponding to right-hand diffracting polarization. New features in the transmission spectrum appear as a consequence of variation of the light incidence angle θ. The CLC band gap shifts to the short-wavelength region as θ increases, in accordance with the Bragg condition; simultaneously, the long-wavelength edge of the stop-band becomes displaced toward the resonance frequency ω₀ of the defective layer. At θ = 26° (Fig. 22a), the short-wavelength peak corresponding to the defect mode disappears from the stop-band, and only the long-wavelength peak of defect modes corresponding to right- and left-hand circular polarizations remains. Importantly, the resonance frequency ω₀ at this angle happens to be close to the long-wavelength boundary of the stop-band. Mixing of the resonant mode with the photonic modes leads to the splitting of the band gap, i.e., separation of the additional transparency band corresponding to diffracting polarization from the long-wavelength edge and the appearance of the frequency band gap in the vicinity of ω₀ for the waves of both polarizations due largely to field absorption in the nanocomposite layer. As the angle of incidence θ increases further, the resonance frequency ω₀ is detected in the continuous transmission spectrum, and the resulting resonance situation gives rise to an additional band gap in the transmission spectrum (Fig. 22b).

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Analogous effects can be realized in a different way, such as a change in the cholesteric helix pitch. Indeed, an increase in the helix pitch, e.g., by temperature variation, causes the displacement of the band gap into the long-wavelength region. In this case, mixing of the resonant mode with photonic modes results in the splitting of the stop-band (Fig. 23a) and the appearance of the band gap in the continuous spectrum (Fig. 23b).

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The described resonance splitting may be helpful for explaining photosynthetic phenomena [17]. The chirality of photonic structures involved in photosynthesis is evidenced by the helical configuration of hydrogen bonds [219, 220]. An advantage of chiral structures for photosynthesis stems from their high reflectivity compared to that of a scalar Bragg reflector of the same thickness and refractive index contrast responsible for the higher Q-factor of the resonator. A detailed comparison is presented in Section 3.1.2. Another advantage is the self-organization and tunability of chiral structures.

The study system is presented in Fig. 24. It consists of a right-hand twisted cholesteric liquid crystal with a structural twist defect, a quarter-wave phase plate, and a metal film.

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The following CLC parameters were used for modeling: ordinary and extraordinary refractive indices n_e = 1.71 and n_o = 1.54, respectively, helix pitch p = 0.4 μm. These parameters correspond to the mixture of the chiral center (Merck, S-811) and the nematic liquid crystal (Merck, E44) [229]. For the given CLC, the band center is at the wavelength λ₀ = 650 nm. The CLC length is 4 μm. The center of the layer contains a twist defect with the same integer number of pitches on its right and left sides. The magnitude of the twist defect is given by angle α. It is assumed that rotation of the CLC layer occurs in a clockwise direction and observation is carried out in the direction of light propagation.

Let us assume in addition that CLC refractive indices n_e and n_o are equal to the respective indices of the phase plate. The thickness of the quarter-wave plate d = λ₀/4(n_e – n_o) = 0.96 μm. The structure is surrounded by a medium with the refractive index equaling the mean CLC refractive index n_ext = (n_e + n_o)/2. The thickness of the metallic film d_m = 50 nm; its permittivity is given in the form of the Drude approximation (31).

The simplest way to implement the proposed structure is to use polymer cholesteric elastomers instead of liquid ones [230]. Experiments [133, 231] revealed twist defect-induced photonic defect modes in such cholesteric materials. The twist defect can be caused by rotation of one part of the elastomer about the axis with respect to the other part, so that the helix undergoes a break in the continuous change of its director. The spectral properties of such materials, similarly to those of usual CLCs, can be modified by various impacts [176, 228]. The combination of a quarter-wave plate with a metallic film can be replaced with a polarization-preserving mirror, as proposed in Ref. [160].

Calculations of transmission spectra and field strength distribution in the structure were based on the transfer matrix method for calculating layered anisotropic structures [78].

Figure 25a presents a calculated transmission spectrum of the structure at the varied twist-defect angle α for the right-hand circular polarization of incident light. Calculations were performed for the case of rotation of the first CLC layer when the second layer bordering the phase plate remained motionless. In the absence of a twist defect (α = 0), the spectrum exhibited only one peak, corresponding to the localized mode at a wavelength of 643 nm. A rise in α caused a shift of the peak of the defect mode to the long-wavelength region. At α = 1.38 rad, the wavelengths of both peaks coincided, and the resonance frequency was split. The split peaks occurred at the wavelengths of 637 and 649 nm, respectively (see the inset in Fig. 25a). In the case of α variation, the spectrum demonstrated the avoided crossing characteristic of hybrid states.

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The white dashed line in Fig. 25a shows the position of the resonance wavelength for a CLC containing a twist defect [156],

where λ₂ and λ₁ are the longwave and shortwave boundaries of the CLC band gap. The thickness of the CLC being only 4 μm, its band gap is wider than pn_o < λ < pn_e; therefore, λ₂ – λ₁ ≠ p(n_e – n_o). This fact was used to plot the white dashed line in Fig. 25a.

The incident left-hand circularly polarized light produced a different spectrum (Fig. 25b). It did not contain the transmission peak corresponding to the CLC defect mode, the cause being the weak dip typical of the spectrum of light with nondiffracting circular polarization (see above). Splitting occurred when there was a coincidence of peak frequencies corresponding to the localized mode and the defect mode of the CLC.

Figure 26 demonstrates the spatial distribution of squared electric field modulus |E|²(z) for a wavelength of 649 nm at α = 1.38 rad and two circular polarizations of incident light. In both cases, the field is localized on the twist defect and the phase plate–metal interface. However, localization is much more pronounced for the right-hand circularly polarized light. A similar distribution pattern |E|²(z) is observed for the 637 nm wavelength.

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Variation in the helix pitch under the influence of external factors makes possible efficient tuning of the CLC transmission spectrum. The structure transmission spectrum (see Fig. 24) was calculated for different CLC helix pitches (Fig. 27).

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The condition for the existence of optical localized modes has the form

If CLC and phase plate parameters are substantially different and the ratio is not fulfilled, there are no optical states localized at the phase plate–metal interface. Therefore, the spectrum has a single peak corresponding to the CLC defect mode at a helix pitch significantly different from 400 nm.

Variation in the phase plate thickness changes the structure's transmission spectrum. The properties of a structure containing a phase plate of thickness d' = 5d = 4.8 μm were considered. The quarter-wave plate remained unaltered with respect to the wavelength λ₀ = 650 nm and birefringence of 0.17. The spectrum contained a few peaks corresponding to the localized modes of the CLC–phase plate–metal system. Variation in the twist defect allows the transmission spectrum of the structure of interest to be tuned (Fig. 28).

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