polarization properties of symmetrical and asymmetrical nonreciprocal chiral photonic bandgap...

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Vladimir R. Tuz Vol. 26, No. 9 /September 2009 /J. Opt. Soc. Am. B 1693

Polarization properties of symmetrical andasymmetrical nonreciprocal chiral photonic

bandgap structure with defect

Vladimir R. Tuz

Department of Theoretical Radio Physics, Kharkov National University, Svobody Square 4, Ukraine([email protected])

Received March 18, 2009; revised July 1, 2009; accepted July 17, 2009;posted July 23, 2009 (Doc. ID 108949); published August 14, 2009

The polarization properties of perfectly periodical and defective one-dimensional photonic bandgap structureswith nonreciprocal chiral (bi-isotropic) layers are studied. The method of solution is based on the 2�2 block-representation transfer-matrix formulation. Numerical simulations are carried out for different types of struc-tures (symmetrical or asymmetrical) in order to reveal the dependence of the reflection and transmission co-efficients on frequency, chirality, nonreciprocity parameters, and angle of wave incidence. © 2009 OpticalSociety of America

OCIS codes: 160.1585, 230.4170, 260.5430.

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. INTRODUCTIONesearch on artificial chiral media has been especially in-

ense regarding microwaves in the past two decades1–4], and chirality is an example of novel material effectst its best. In general, nonreciprocal chiral (bi-isotropic)edia are defined via the following constitutive equationshere the electric and the magnetic inductions are re-

ated to the electromagnetic field intensity via the permit-ivity, permeability, and magnetoelectric interaction pa-ameters (the gyration parameters in terms ofrystalloptics):

D = �E + �H, B = �H + �E. �1�

ere �=�+ i� and �=�− i�; � describes the degree of inher-nt nonreciprocity in the medium (Tellegen parameter);nd � is a measure for handedness of the material (chiral-ty parameter). It is well known that the eigenwaves inomogeneous chiral media are two circularly polarized

right- and left-handed) waves that exhibit some interest-ng behaviors [4]. Thus, the real part of chirality param-ter � defines the optical rotatory dispersion, whichauses a rotation of polarization due to different phase ve-ocities of the right- and left-handed circularly polarizedave. Then, the imaginary part of � and Tellegen param-

ter � define the circular dichroism, which modifies theature of polarization of the propagating wave by causingn ellipticity (due to a difference between the absorptionoefficients of the right- and left-handed circularly polar-zed waves and nonorthogonality of electric and magneticeld vectors). The effect of the optical activity in chiraledia can be so strong that the refractive index becomes

egative, even if permittivity � and permeability � areoth positive. Specifically, this negative refraction may oc-ur for one circularly polarized wave (backward wave),hile for the other circular polarization the refractive in-ex remains positive. These properties have attracted

0740-3224/09/091693-9/$15.00 © 2

onsiderable attention to chiral media and may open newotential applications in optics [5–9].In the present paper, we consider chiral media in an ap-

lication to construct photonic bandgap crystals [10,11].here are a lot of publications where both periodic anduasi-periodic [12,13] photonic crystals are studied, in-luding a group of chiral photonic crystals [14–18]. Iniew of the above-mentioned behaviors of chiral media,hiral photonic crystals possess rich optical propertiesnd are characterized by selective reflection in a specificaveband, in addition to circular polarization of reflectednd transmitted light. Such artificial structures areidely used in modern integrated optics and optoelectron-

cs, laser techniques, millimeter and submillimeter waveevices, and optical communications.Nonreciprocal photonic structures are also of special in-

erest. Nonreciprocity is a comparable effect to chiralityrom a theoretical point of view, but bi-isotropic media ofhe nonreciprocal kind appear not to have been made ar-ificially as yet [19,20]. Actually the nonreciprocal materi-ls are anisotropic; as example, nonreciprocity can be cre-ted by an external static magnetic field in ferrites orlasmas. In recent years, some new mechanisms of ob-aining nonreciprocity have been proposed [21,22] in ad-ition to natural nonreciprocal materials [23,24]. In par-icular, the importance of nonreciprocal photonic crystalss related to the possibility of creating omnidirectional re-ectors, optical diodes, etc. [25–27]. The optical diode is aevice transmitting light in one direction and blocking itn the reverse direction (i.e., characterized by a nonrecip-ocal transmission). Various types of optical diodes haveeen proposed. A well-known optical insulator based oninear polarizers and a Faraday rotator is characterizedy a low level of losses and a high ratio of direct transmis-ion to the reverse attenuation. Optical diodes can also beased on combinations of some other effects [7,28].The present paper is devoted to the study of both one-

009 Optical Society of America

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1694 J. Opt. Soc. Am. B/Vol. 26, No. 9 /September 2009 Vladimir R. Tuz

imensional perfectly periodic chiral photonic structuresnd those containing a defect, while taking into accounthe media nonreciprocity. Although the nonreciprocal bi-sotropic medium is an idealization, the investigation ofptical properties of a photonic crystal in its present forman reveal fundamental distinctions of nonreciprocaltructures in contrast to reciprocal ones. We also considern asymmetric defective structure that has twoultilayer sections with the mirror handedness of chiral-

ty and nonreciprocity parameters on either side of the de-ective element for the purpose of the optical diodes con-tructing. The method of solution is based on the 2�2lock-representation transfer-matrix formulation pro-osed earlier in [17].

. PERFECTLY PERIODIC NONRECIPROCALHIRAL PHOTONIC STRUCTUREirst, we consider a finite one-dimensional periodic struc-ure of N basic elements (periods) (Fig. 1). Each of the pe-iods includes a homogeneous isotropic (with permittivi-ies �1, �1) and a bi-isotropic (with �2, �2, �, �) layer withhicknesses d1 and d2, respectively. The total length of thetructure period is L=d1+d2. In general, the material pa-ameters �1, �1, �2, �2, �, � are frequency dependent andomplex for lossy media. In the particular case of ��0,=0 the structure layers are chiral and reciprocal (theasteur layers); when �=0, ��0 they are achiral andonreciprocal (the Tellegen layers). The outer half spaces0 and zNL are homogeneous, isotropic, and haveermittivities �0, �0 and �3, �3, respectively.For the excitation fields, the plane monochromatic

aves �exp�−i�t�� with perpendicular (Ee �x0, Hxe =0, s=e)

r parallel (Hh �x0, Exh=0, s=h) polarization are selected.

hey are obliquely incident from the region z0 at anngle �0 to the z axis.Making use of transfer-matrix formalism [29], the

quation coupling the field amplitudes at the structure in-ut and output for the incident fields of E type �s=e� and

type �s=h� is obtained in [17] as

V0 = T VN+1 = �T01TN−1T̃�VN+1 = �T01TNT13�VN+1, �2�

here V0= �A0s B0

s 0 B0s��T and VN+1= �AN+1

s 0 AN+1s� 0�T are

ectors containing the field amplitudes at the structurenput and output; T is the matrix transpose operator; and

01, T, T̃ are the transfer-matrices of the illuminatedoundary, the repeated heterogeneity, and the last ele-

y

z

0 d L mL mL+d ( +1)m L NL

� �1 1 � �2 2

� �

� �3 3� �0 0

T01 T T

Ae

0 Ah

0

k

�0

Be

0 Bh

0

Ae

m Ah

m

Be

m Bh

mA

e

N+1Ah

N+1

1 1

ig. 1. (Color online) Finite photonic bandgap structure of iso-ropic and bi-isotropic layers.

ent, which is loaded on the waveguide channel havinghe admittance Y3

s . We denote here As, As� and Bs, Bs� ashe amplitudes of copolarized �s� and cross-polarized �s��omponents of the transmitted and reflected fields, re-pectively. The elements of the transfer matrices in Eq.2) are determined from solving the boundary value prob-em and are presented in [17,30].

The algorithm from the matrix polynomial theory [31]or raising the matrix T to the power N was introduced in32] to study the structure with a large number of periodsN�1�:

V0 = �T01n=1

4

�nNFn�T13�VN+1. �3�

ere �n are the eigenvalues of the transfer matrix T, FnPInP−1, P is the matrix where columns are the set of in-ependent eigenvectors of T, and In is the unit dyad in nimensions.From Eq. (3), the required reflection and transmission

oefficients are determined by the expressions Rss

B0s /A0

s , �ss=AN+1s /A0

s , and Rss�=B0s� /A0

s , �ss�=AN+1s� /A0

s forhe copolarized and cross-polarized waves, respectively.

Parametrical dependences of the reflection and trans-ission spectra of plane monochromatic waves of the fi-ite structure with bi-isotropic layers have interleaved ar-as with high (stop bands) and low (passbands) averageevels of reflection (Figs. 2 and 3). Due to an interferencef the reflected waves from outside boundaries of layers−1, small-scale oscillations appear in the passbands.It is well known that the eigenwaves of an infinite con-

enient isotropic medium have transverse magnetic andransverse electric polarizations or their linear combina-ions, and, in general, they have elliptical polarization.n the other hand, in a bi-isotropic medium, fields can be

plit into left- and right-hand circularly polarized eigen-aves, which have different propagation constants �±,nd each of these eigenwaves sees the medium as if itere an isotropic medium with equivalent parameters �±

nd �± [2,17,30]. Right- and left-hand circular polariza-ions are also the eigenwave polarizations in cholestericsfor the difference between the constitutive equations ofholesteric liquid crystals and isotropic chiral materials,ee [33]). However, while the photonic bandgap in choles-erics exists only for one circular polarization that coin-ides with the chirality sign of the medium, in chiral pe-iodic media both circular polarizations are diffractingolarizations. From this characteristic of chiral periodictructures follows the independence of their reflectionpectra from chirality parameter � under normal inci-ence; a chiral periodic structure and an achiral one withhe same parameters are characterized by identical re-ection and transmission spectra (the stop band andassband positions). Medium chirality causes change onlyn the rotation of the polarization plane of the transmittedave. The theoretical angle of the polarization rotation inchiral medium is given by �=−Re��k0z [2,4], where

e�� is the real part of the chirality parameter, k0 is theavenumber in free space, and z is the distance passed by

he wave through chiral layers. The complexity of � modi-es the nature of the propagating wave by introducing el-

ipticity. This is due to the different absorption coeffi-

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ients for the right- and left-handed circularly polarizedaves. Theoretical ellipticity depends on the imaginaryart of the chirality as follows [4]: �= �exp�2k0z Im��1� / �exp�2k0z Im��−1� , where � is the ellipticity de-ned as the ratio between the major and minor axes of thellipse, and Im�� is the imaginary part of the chiralityarameter. As follows from our numerical calculation, theross-polarized components of reflected and transmittedaves are equal to each other Reh = Rhe , �eh = �he for

ig. 2. (Color online) (a) Frequency and (b) angular dependencesf isotropic and bi-isotropic layers. �j=�j=1, j�2, �2=2, �2=1, �

he structure with reciprocal chiral layers (the Pasteurayers). The nonzero value of the nonreciprocal (Tellegen)arameter ��0� causes a disturbance of the amplitudef small-scale oscillations in the passbands for both theopolarized and cross-polarized waves (Fig. 2). The cross-olarized components appear in the reflected field evenor normal incidence ��0=0� of the exciting wave withReh = Rhe when �0=0 and Reh � Rhe when �0�0. The

agnitudes of the cross-polarized components of the

reflection and transmission spectra of a finite photonic structure=0.05, d1 /L=d2 /L=0.5, N=5. (a) �0=25°. (b) k0L=10.

of the=0.1, �

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ransmitted field remain equal to each other indepen-ently from the angle of incidence �eh = �he . The addi-ional effect by the media nonreciprocality is the trans-ission spectra symmetry breakdown related to �=0

Fig. 3).

ig. 3. (Color online) Transmission coefficient magnitude of (a)nd chirality parameter � for a finite photonic structure of is1 /L=d2 /L=0.5, N=5.

At the oblique incidence of the exciting wave, the pat-ern of the structure reflection and transmission becomesore complicated. The rotation spectra acquire a pro-ounced diffraction character: the rotation and ellipticityre heavily suppressed in the photonic bandgap and

rized and (b) cross-polarized waves as function of frequency k0Lc and bi-isotropic layers. �j=�j=1, j�2, �2=2, �2=1, �=0.05,

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trongly oscillate near the bandgap boundaries. Here, thengle of rotation increases in one wavelength range andecreases in another. At large angles of incidence begin-ing from some cutoff angle, the magnitudes of the cross-olarized components of the reflected field and the copo-arized and cross-polarized components of the transmittedeld decrease to zero � Rss� , �ss , �ss� →0�. Thus, there ap-ears a possibility for the diffraction control of the polar-zation rotation.

. SYMMETRICAL AND ASYMMETRICALONRECIPROCAL CHIRAL PHOTONICTRUCTURE WITH DEFECT

n this section we consider a nonreciprocal chiral photonictructure with a periodicity defect that is created via re-lacing the mth period of the sequence (Fig. 4). In gen-ral, material parameters of periods of both structure sec-ions, coming before and after the defective element, areifferent. Their difference is defined with some subscriptubstitution above the parameter. Thus, for the first andast structure sections and the defect element we have:�2�, �2�, ��=��+ i��, ��=��− i��}, ��2� ,�2� ,�� ,�� and�2� ,�2� ,�� ,��, respectively. Without loss of generality,he material parameters of isotropic slabs and the layerhicknesses are left unchanged, i.e., �1�=�1�=�1�=�1, �1��1�=�1�=�1 and dj�=dj�=dj�=dj, j=1,2. It is defined that

he structure is symmetrical when Re��=Re��, Re��Re�� and is asymmetrical when Re��=−Re�� ore��=−Re��.Instead of Eq. (2), the equation coupling the field am-

litudes at the defective structure input and output is ob-ained as

V0 = �T01�T��m−1T��T��N−mT13�VN+1

=�T01n=1

4

��n��m−1Fn��T�n=1

4

��n��N−mFn��T13�VN+1,

�4�

here T�, T� and T� are the transfer matrices of the pe-iod of the first and the last structure sections and the de-

y

0 d L 2L mL mL+

� �1 1 � �2 2

� �

� �1 1� �0 0

T01 T

Ae

0 Ah

0

k

�0

Be

0 Bh

0

Ae

m Ah

m

Be

m Bh

m

T

1

Fig. 4. (Color online) Finite photonic bandgap structu

ective element, respectively. Obviously, due to the non-ommutativity of the matrix product, the spectra of theeflected and transmitted fields depend on the defectivelement position within the structure.

It is well known that the defects inside a layered iso-ropic one-dimensional sample produce additional reso-ance modes (defective modes) in the stop bands. Such de-ects are widely used to produce high-Q laser cavities inertical-cavity surface-emitting lasers. In an analogy tohe isotropic periodic structures, a defect can be producedn a chiral structure by adding an isotropic layer in

iddle of a sample (see [32] and its bibliography). In thisase, the defective modes also exhibit a number of impor-ant polarization-related features.

The curves of the reflection and transmission spectra ofsymmetrical and asymmetrical bi-isotropic photonic

tructure with a defective isotropic layer are presented inigs. 5 and 6, respectively. For clarity, we consider sepa-ately the spectrum properties of the structure of twoypes. The structure of the first type consists of achiral,onreciprocal layers (the Tellegen layers) [Figs. 5(a), 5(b),(a), and 6(b)], and the second type are chiral, reciprocalnes (the Pasteur layers) [Figs. 5(c), 5(d), 6(c), and 6(d)].s can be seen in Figs. 5 and 6, in addition to the above-entioned defective modes in the stop bands, there are

ome changes in the magnitude of the high-frequencysmall-scale) oscillations in the passbands for both typesf structures. Those effects are determined by the compo-ition of the eigenmodes of the structure sections placedefore and after the defective element and by additionaligenmodes that appear as a result of the wave polariza-ion transformation [32]. Especially interesting behaviorf the perturbation of the mentioned small-scale oscilla-ions shows for the structure with the Tellegen layers. Inhis case, the frequency bands �4.0�k0L�4.5,9.5�k0L10.0� of the high reflection of the cross-polarized field

omponent appear in the passbands. At the same time,he copolarized component dominates in the transmittedelds, and eventually in these bands we have the cross-olarized reflection and the copolarized transmission

Rss �0, Rss� �0 and �ss �0, �ss� �0.The spectrum properties of the structure with the Tel-

z

+1)m L NL

� �1 1 � �2 2

� �

� �3 3

T

Ae

N+1Ah

N+1

T

sotropic and bi-isotropic layers with periodicity defect.

d (

� �2 2

� �

1

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egen layers in the stop bands are practically the same ashe properties of the convenient defective isotropic peri-dic structure. Note only that there are defective modes inhe stop bands of both copolarized and cross-polarizedomponents of the reflected and transmitted fields. Theeflection (transmission) level of cross-polarized defectiveodes is much less than the copolarized ones for the sym-etrical structure [Figs. 5(a) and 5(b)], and these cross-

olarized modes are practically absent for the asymmetri-al structure, i.e., �ss �1 [Figs. 6(a) and 6(b)].

ig. 5. (Color online) (a), (c) Reflection and (b), (d) transmiss=19 periods with the isotropic defect in the middle �m=10� of t

0=25°, d1 /L=d2 /L=0.5. (a), (b) Tellegen layers. ��=��=0, ��=��

The defective mode behavior is more interesting in thetructure with the Pasteur layers. As can be seen in Figs.(c) and 5(d), there is a possibility of obtaining differentombinations of the polarization of the reflected andransmitted fields using the symmetrical structure. Forxample, for the chosen structure parameters we have:Reh �0, Ree �0, �ee � �eh at k0L�2.6 and Ree � Reh ,�ee � �eh at k0L�11.6. The particular properties of thesymmetrical structure is that the cross-polarized compo-ent of the transmitted field is practically absent in the

ectra of a finite symmetrical bi-isotropic photonic structure ofucture. �j=�j=1, j�2, �2�=�2�=2, �2�=1, �2�=�2�=�2�=1, ��=��=0,(c), (d) Pasteur layers. ��=��=0.1, ��=��=0.

ion sphe str=0.1.

ptTtswTfb

umiitdti

FN� �


assbands and stop bands, but the significant polariza-ion transformation occurs on the bandgap boundaries.he first effect is determined by the mutual discharge ofhe polarization rotation, which provides two structuredections placed before and after the defective elementith equal material parameters and mirror handedness.he oblique incidence of the wave and the mentioned dif-

raction character of the rotation near the bandgapoundaries explain the second feature.

ig. 6. (Color online) (a), (c) Reflection and (b), (d) transmissi=19 periods with the isotropic defect in the middle �m=10� of t

0=25°, d1 /L=d2 /L=0.5. (a), (b) Tellegen layers. ��=��=0, ��=0.

The further functionality expansion of the structurender study is connected with the use of an anisotropicaterial to construct the defect. As is well known, in an-

sotropic media uniform plane waves can be decomposedn two orthogonal polarization states (linear or circular)hat propagate with two different speeds. The two statesevelop a phase difference as they propagate, which al-ers the total polarization of the wave. In the subject be-ng discussed, the anisotropic defect can provide the dif-

ctra of a finite asymmetrical bi-isotropic photonic structure ofucture. �j=�j=1, j�2, �2�=�2�=2, �2�=1, �2�=�2�=�2�=1, ��=��=0,−0.1. (c), (d) Pasteur layers. ��=0.1, ��=−0.1��=��=0.

on spehe str1, � =

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tcra

tttoatm�dt

Fbd


erence of the defective mode configuration for theopolarized and cross-polarized waves. The other way toomplicate the spectrum features is an introduction of theonlinear defect. It is very important for the optical diodeonstruction, because, as usual, the amplitude-frequencyharacteristic of the diode is strongly nonlinear.

We also study a finite isotropic periodic structure withhe bi-isotropic defect. As above, we consider the nonre-iprocal and chiral defects separately. The curves of theeflection and transmission spectra of the both structures
re presented in Figs. 7(a)–7(d), respectively. One can see
1 2 � �

hat the low-level cross-polarized component appears inhe reflected and transmitted fields, except in the level ofhe transmission for the Pasteur defective layer. On thether hand, the significant polarization transformationppears within the defective modes both in reflection andransmission. Note the value of the defective layer per-ittivity is higher than those in the structure sections

�2��2� ,�2�=�2��. This condition provides the shift of theefective mode frequency from the right (left) boundary ofhe low-frequency (high-frequency) stop bands.

We do not discuss here the spectral properties of the de-

ig. 7. (Color online) (a), (c) Reflection and (b), (d) transmission spectra of a finite isotropic photonic structure of N=19 periods with thei-isotropic defect in the middle �m=10� of the structure. �j=�j=1, j�2, �2�=�2�=2, �2�=3, �2�=�2�=�2�=1, ��=��=��=��=0, �0=25°,

/L=d /L=0.5. (a), (b) Tellegen defective layer. � =0, � =0.1. (c), (d) Pasteur defective layer. � =0.1, � =0.
� �

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1

1

1

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1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

3

3

3

3


ective structure with simultaneous presence of the non-eciprocity and chirality of layers due to the limited sizef the publication. It is clear that the integrated spectraill contain all the mentioned features in their combina-

ions.

. CONCLUSIONhe polarization properties of both perfectly periodicalnd defective one-dimensional photonic bandgap struc-ures are studied. The structure period consists of nonre-iprocal, chiral (bi-isotropic), and isotropic layers. The pe-iodicity defect is created via replacing the mth period ofhe sequence with some element whose optical propertiesre different from the others. This defective element cane both isotropic and bi-isotropic. Two types of the struc-ure (symmetrical and asymmetrical) are considered.hey differ in the material parameters of periods of struc-ure sections coming before and after the defective ele-ent. This difference is defined via the handedness of theaterial of layers.The dependence of the reflection and transmission

pectra on the frequency, nonreciprocity, chirality param-ters, and angle of wave incidence are obtained. It ishown that the introduction of media nonreciprocity andhirality changes the pattern of structure reflection andransmission and causes the availability of the cross-olarized component in the reflected field for normal wavencidence, the perturbation of small-scale oscillations inhe passbands, and a change of the defective mode con-guration.The width of photonic bandgaps, as well as their spec-

ral position and the distance between them, depend onhe parameters of the problem and thus can be controlled.herefore, such systems can be used as controllable polar-

zation filters and mirrors, optical diodes, polarizationransformers, mode discriminators, multiplexers for cir-ularly or elliptically polarized waves, and sources of cir-ular (elliptical) polarization.

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polarization properties of symmetrical and asymmetrical nonreciprocal chiral photonic bandgap...

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