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Polarization properties of a symmetrical and asymmetrical nonreciprocal chiral photonic bandgap structure with defect

The polarization properties of perfectly periodical and defective one-dimensional photonic bandgap structures with nonreciprocal chiral (bi-isotropic) layers are studied. The method of solution is based on the 2x2-block-representation transfer-matrix formulation. Numerical simulations are carried out for different type of structures (symmetrical or asymmetrical) in order to reveal the dependence of the reflection and transmission coefficients on frequency, chirality, nonreciprocity parameters, and the angle of wave incidence.


INTRODUCTION
The research on artificial chiral media has been especially intense within microwaves in the past two decades [1][2][3][4], and chirality is an example of novel material effects at best. In general, the nonreciprocal chiral (bi-isotropic) media are defined via the following constitutive equations where the electric and the magnetic inductions are related to the electromagnetic field intensity via the permittivity, permeability and magnetoelectric interaction parameters (the gyration parameters in terms of crystalloptics): Here i ξ = χ + ρ and i ζ = χ − ρ ; χ describes the degree of inherent nonreciprocity in the medium (Tellegen parameter), ρ is a measure for handedness of the material (chirality parameter). As known, the eigenwaves in homogeneous chiral media are two circularly polarized (right-and left-handed) waves which exhibit some interesting behaviors [4]. Thus, the real part of the chirality parameter ρ defines the optical rotatory dispersion, which causes a rotation of polarization due to different phase velocities of the right-and left-handed circularly polarized wave. Then, the imaginary part of ρ and Tellegen parameter χ define the circular dichroism, which modifies the nature of polarization of the propagating wave by causing an ellipticity (due to a difference between the absorption coefficients of the right-and left-handed circularly polarized waves and nonorthogonality of electric and magnetic field vectors). The effect of the optical activity in chiral media can by so strong, that the refractive index becomes negative, even if the permittivity ε and permeability μ are both positive. Specifically, this negative refraction may occur for one circularly polarized wave (backward wave), while for the other circular polarization the refractive index remains positive. These properties have attracted considerable attention to chiral media and may open new potential applications in optics [5][6][7][8].
In the present paper, we consider the chiral media in their application to constructing photonic bandgap crystals [9][10][11]. There are a lot of publications where both ideally periodic and quasi-periodic [12][13][14] photonic crystals are studied, including the group of chiral photonic crystals [15][16][17][18][19]. In view of the mentioned above behaviors of chiral media, the chiral photonic crystals possess rich optical properties and are characterized by the selective reflection in a specific waveband in addition to circular polarization of the reflected and transmitted light. Such artificial structures are widely used in the modern integrated optics and optoelectronics, laser techniques, millimeter and submillimeter wave devices, and optical communications.
Nonreciprocal photonic structures are also of special interest. The nonreciprocity is a comparable effect to the chirality from theoretical point of view, but in terms of manufacturing real-life samples, nonreciprocal materials are not as advanced [3]. It is known that, externally, the nonreciprocity can be created by a static magnetic field in ferrites or plasmas, but in those cases the medium is anisotropic. In recent years, some new mechanisms of obtaining nonreciprocity have been proposed [20,21] in addition to natural nonreciprocal materials [22,23]. In particular, the importance of nonreciprocal photonic crystals is related, to possibility of creating omnidirectional reflectors, optical diodes, and etc. [24][25][26]. The optical diode is a device transmitting light in one direction and blocking it in the reverse direction (i.e., characterized by a nonreciprocal transmission). Various types of optical diodes have been proposed. The wellknown optical insulator based on linear polarizers and Faraday rotator is characterized by a low level of losses and a high ratio of the direct transmission to the reverse attenuation. Optical diodes can also be based on combinations of some other effects [7,27].
The present paper is devoted to study of both one-dimensional perfectly periodic chiral photonic structures and those containing a defect, with taking into account the media nonreciprocity. We also consider an asymmetric defective structure which has two multilayer sections with the mirror handedness of chirality and nonreciprocity parameters on either sides of the defective element for the purpose of the optical diodes constructing. The method of solution is based on the 2 2 × -block-representation transfer-matrix formulation proposed earlier in [18].
As the excitation fields, the plane monochromatic waves ( ) Making use of the transfer-matrix formalism [28], the equation coupling the field amplitudes at the structure input and output for the incident fields of E type ( s e = ) and H type determined from solving the boundary-value problem and are presented in [18,29].
The algorithm from the matrix polynomial theory [30] for raising the matrix T to the power N was introduced in [31] to study the structure with a large number of periods ( 1 N >> ) Here n λ are the eigenvalues of the transfer-matrix T , Thus, there appears a possibility for the diffraction control of the polarization rotation.

PHOTONIC STRUCTURE WITH DEFECT
In this section we consider a nonreciprocal chiral photonic structure with a periodicity defect which is created via replacing the m-th period of the sequence (Fig. 4). In general, material parameters of periods of both structure sections, coming before and after of the defective element, are different. Their difference is defined with some prime substitution above the parameter. Thus where / T , // T and /// T are the transfer-matrices of the period of the first and the last structure sections and the defective element, respectively. Obviously, due to the non-commutativeness the matrix product, the spectra of the reflected and transmitted fields depend on the defective element position within the structure.
It is known that the defects inside a layered isotropic one-dimensional sample produce additional resonance modes (defective modes) in the stopbands. Such defects are widely used to produce high-Q laser cavities in vertical-cavity-surface-emitting lasers. In analogy to the isotropic periodic structures, a defect can be produced in a chiral structure by adding an isotropic layer in middle of a sample (see [31], and its bibliography). In this case, the defective modes also exhibit a number of important polarization-related features.
The curves of the reflection and transmission spectra of a symmetrical and asymmetrical bi-isotropic photonic structure with a defective isotropic layer are presented on Figs The further functionality expansion of the structure under study is connected with using of an anisotropic material to construct the defect. As is well known, in anisotropic media, uniform plane waves can be decomposed in two orthogonal polarization states (linear or circular) that propagate with two different speeds. The two states develop a phase difference as they propagate, which alters the total polarization of the wave. In the subject being discussed, the anisotropic defect can provide the difference of the defective mode configuration for the copolarized and cross-polarized waves. The other way to complicate the spectrum features is an introduction of the nonlinear defect. It is very important for the optical diode constructing, because, as usually, the amplitude-frequency characteristic of the diode is strongly nonlinear.
We also study a finite isotropic periodic structure with the bi-isotropic defect. As above, we We do not discuss here the spectral properties of the defective structure with simultaneous presence of the nonreciprocity and chirality of layers due to the limited size of the publication. It is clear that the integrated spectra will contain all the mentioned features in their combinations.

CONCLUSION
The polarization properties of both perfectly periodical and defective one-dimensional photonic bandgap structures are studied. The structure period consists of nonreciprocal, chiral (biisotropic) and isotropic layers. The periodicity defect is created via replacing the m-th period of the sequence with some element whose optical properties are different from the others. This defective element can be both isotropic and bi-isotropic. Two types of the structure (symmetrical and asymmetrical) are considered. They differ in the material parameters of periods of structure sections coming before and after of the defective element. This difference is defined via the handedness of the material of layers.
The dependences of the reflection and transmission spectra on the frequency, nonreciprocity, chirality parameters, and the angle of wave incidence are obtained. It is shown that the introduction of the media nonreciprocity and chirality changes the pattern of the structure reflection and transmission and causes the availability the cross-polarized component in the reflected field for the normal wave incidence, the perturbation of the small-scale oscillations in the passbands, a change of the defective mode configuration.
The width of photonic band gaps, as well as their spectral position and the distance between them, depend on the parameters of the problem and, thus, can be controlled. Therefore, such systems can be used as controllable polarization filters and mirrors, optical diodes, polarization transformers, mode discriminators, multiplexers for circularly or elliptically polarized waves, and sources of circular (elliptical) polarization.
List of Figure             (c) (d) Figure 7