. INTRODUCTIONnumber of publications have discussed the propagation
f electromagnetic waves in one-dimensional birefringenttratified media [1,2]. Of particular interest is the formu-ation pioneered by P. Yeh that employs a translation ma-rix approach to discuss periodic linearly birefringenttructures. The propagation of light across a single periods analyzed in conjunction with Floquet’s theorem to de-ermine the dispersion relation and Bloch waves of theystem. This approach has been used to study the proper-ies of layered media with misaligned optic axes from oneayer to the next [2]. Numerical solutions have been foundor normal incidence of an optical beam for identical bire-ringent plates with alternating azimuth angles and layerhicknesses [1].
More recently the present authors have used this tech-ique to study elliptically birefringent nonreciprocal me-ia [3], where elliptically polarized normal modes charac-erize the system locally. Elliptical birefringence refers tohe difference in refractive index between the ellipticallyolarized normal modes of any given layer in the photonicrystal stack. Analytic solutions were found for light inci-ent perpendicularly into stratified media consisting oflternating magneto-optic (MO) layers with different gy-ation vectors [4] and birefringence levels [3]. Adjacentayers were assumed to have their anisotropy axesligned to each other. This model captures some impor-ant features of one-dimensional magnetophotonic crystalaveguides, particularly the presence of MO gyrotropynd locally alternating birefringence levels. Such systems
re currently being studied for use in integrated fast op-ical switches and ultrasmall optical isolators [5–7]. Workn these systems extends prior theoretical and experi-ental efforts on magnetophotonic crystals in order to en-
ompass the elliptical birefringence that often character-zes planar magnetophotonic structures [8–12].
A particularly interesting feature of periodic ellipticallyirefringent gyrotropic media concerns the character ofhe dispersion branch solutions to the Floquet theorem.he work of Pochi Yeh cited before [1] discusses the for-ation of a bandgap away from the Brillouin zone bound-
ry in linearly birefringent media with misaligned aniso-ropy axes. Merzlikin et al. have pointed out thatagnetically tunable bandgaps can arise in stratified me-
ia that combine circularly and linearly birefringent lay-rs [13]. In the present work we note that magneticallyunable bandgaps can also exist in elliptically birefrin-ent stratified MO media. A range of frequency splits isossible since local normal modes span a wide spectrumf polarization states. The work presented here traces therigin of these band structural features to normal-modeoupling arising from the simultaneous presence of gyrot-opy and linear birefringence in the periodic system, ashe Bloch states of the system propagate across layeroundaries. The dispersion relation for this type of sys-em is found to contain additional terms that describe theormation of magnetically tunable bandgaps away fromhe Brillouin zone boundary and that determine the mag-itude of the gap. A new kind of parameter is identifiederewith that characterizes the coupling of these layer-
008 Optical Society of America
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120 J. Opt. Soc. Am. B/Vol. 25, No. 1 /January 2008 A. A. Jalali and M. Levy
ependent normal modes. The underlying phenomenon ishe continuity of transverse electric and magnetic fieldomponents across interlayer boundaries. Such a couplingan yield magnetization-dependent optical bandgaps in-ide the Brillouin zone as discussed below. The formula-ion presented herein can serve a tool for the design ofagnetically tunable bandgaps in nonreciprocal magne-
ophotonic structures.After introducing the formalism to be employed in this
ork (Section 2), it is shown that the transfer matrix forlliptically birefringent magnetophotonic crystals can bearameterized in terms of an intermodal coupling param-ter (Section 3). The conditions for the frequency splittingf degenerate Bloch states is then discussed (Section 4),nd the dispersion relation for MO layered structures inhe presence of elliptical birefringence is derived and dis-ussed (Section 5). Section 6 examines the character ofloch states for these systems.
. WAVES IN A BIREFRINGENTAGNETOPHOTONIC MEDIUM
n the optical wavelength regime, the permeability of a bi-efringent uniaxial MO medium is very close to the per-eability of vacuum �0; its relative permeability is close
o unity. The relative permittivity tensor � of the mediumor magnetization along the z axis has the form
� = ��xx i�xy 0
− i�xy �yy 0
0 0 �zz� , �1�
here we assume no absorption of the light in the me-ium. This implies that all components of the relative per-ittivity ��i,j , i , j=x ,y ,z� are real, and it is not assumed
hat �xx=�yy. By solving the wave equation upon normalncidence of a monochromatic plane wave [with time de-endence exp �i�t�] propagating parallel to the z axis on airefringent MO medium, one obtains eigenmodes
e± =1
�2�cos � ± sin �
±i cos � − i sin �
0� , �2�
orresponding to the refractive indices
n±2 = � ± ��2 + �xy
2 . �3�
ere �= ��yy+�xx� /2, �= ��yy−�xx� /2, and tan�2��=� /�xy.he propagation constant in the medium is defined as
onsider a plane wave normally incident on a periodictack structure of alternating elliptically birefringentagnetophotonic layers (Fig. 1), where the elliptical bire-
ringence parameters of adjacent layers may differ, buthe anisotropy axis are aligned. The Bloch states for thisystem can be expressed in terms of local normal modes.hus in the nth layer the optical electric field can be writ-
en as
E�n��z� = �E01�n�ei�+
�n��z−zn� + E02�n�e−i�+
�n��z−zn��e+�n� + �E03
�n�ei�−�n��z−zn�
+ E04�n�e−i�−
�n��z−zn��e−�n�. �5�
ere E0i�n� �i=1, . . . ,4� are the complex amplitudes of the
artial waves corresponding to each normal mode. Theloch states for this system satisfy the Floquet–Bloch
heorem through the following eigenvalue equation:
T�n−1,n+1�E = exp�iK��E, �6�
here the transfer matrix T�n−1,n+1� relates the four eigen-ode amplitudes E in the second layer of a unit cell [the
egion between z= �n−2�� and z= �n−1��] to the corre-ponding amplitudes in the second layer of the adjacentnit cell [the region between z= �n−1�� and z=n�]. K ishe Bloch wave vector, and � is the period of the periodictructure. With this functional basis, the transfer matrix�n−1,n+1� acquires the following form:
�f1,1 + g1,1 sin2 ��n,n+1� f1,2 + g1,2 sin2 ��n,n+1� g1,3 sin 2��n,n+1� g1,4 sin 2��n,n+1�
f2,1 + g2,1 sin2 ��n,n+1� f2,2 + g2,2 sin2 ��n,n+1� g2,3 sin 2��n,n+1� g2,4 sin 2��n,n+1�
g3,1 sin 2��n,n+1� g3,2 sin 2��n,n+1� f3,3 + g3,3 sin2 ��n,n+1� f3,4 + g3,4 sin2 ��n,n+1�
g4,1 sin 2��n,n+1� g4,2 sin 2��n,n+1� f4,3 + g4,3 sin2 ��n,n+1� f4,4 + g4,4 sin2 ��n,n+1�� , �7�
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A. A. Jalali and M. Levy Vol. 25, No. 1 /January 2008 /J. Opt. Soc. Am. B 121
here
f1,1 = exp�− i�+�n+1�d�n+1���cos��+
�n�d�n��
−i
2� n+�n�
n+�n+1�
+n+
�n+1�
n+�n� �sin��+
�n�d�n�� , �8�
g1,1 = exp�− i�+�n+1�d�n+1���cos��−
�n�d�n�� − cos��+�n�d�n��
−i
2� n−�n�
n+�n+1�
+n+
�n+1�
n−�n� �sin��−
�n�d�n��
+i
2� n+�n�
n+�n+1�
+n+
�n+1�
n+�n� �sin��+
�n�d�n�� , �9�
f1,2 =i
2exp�i�+
�n+1�d�n+1���n+�n+1�
n+�n�
−n+
�n�
n+�n+1�sin��+
�n�d�n��,
�10�
g1,2 =i
2exp�i�+
�n+1�d�n+1�� �� n+�n�
n+�n+1�
−n+
�n+1�
n+�n� �sin��+
�n�d�n��
+ �n+�n+1�
n−�n�
−n−
�n�
n+�n+1��sin��−
�n�d�n�� , �11�
g1,3 = exp�− i�−�n+1�d�n+1����n−
�n+1� + n+�n+1�
4n+�n+1� ��cos��+
�n�d�n��
− cos��−�n�d�n��� +
i
4�n−�n+1�
n−�n�
+n−
�n�
n+�n+1��sin��−
�n�d�n��
−i
4�n−�n+1�
n+�n�
+n+
�n�
n+�n+1��sin��+
�n�d�n�� , �12�
ig. 1. Schematic diagram of a one-dimensional birefringentagnetophotonic crystal with period of �. The magnetophotonic
rystal extends indefinitely in the x and y directions. A planeave is incident normally to the layered structure. A unit cell
pans the region between zn−1 and zn+1.
g1,4 = exp�i�−�n+1�d�n+1����n+
�n+1� − n−�n+1�
4n+�n+1� ��cos��+
�n�d�n��
− cos��−�n�d�n��� −
i
4�n−�n+1�
n−�n�
−n−
�n�
n+�n+1��sin��−
�n�d�n��
+i
4�n−�n+1�
n+�n�
−n+
�n�
n+�n+1��sin��+
�n�d�n�� . �13�
ere d�n� is the thickness of layer n in the stack and�n,n+1�=��n�−��n+1�. The other elements of the T�n−1,n+1�
atrix can be obtained by the following symmetry rela-ions:
f2,1 = f 1,2* , g2,1 = g1,2
* ;
f2,2 = f 1,1* , g2,2 = g1,1
* ;
g2,3 = g1,4* ;
g2,4 = g1,3* ;
g3,1 = g1,3 �n+�n+1� ↔ n−
�n+1��;
g3,2 = g1,4 �n+�n+1� ↔ n−
�n+1��;
f3,3 = f1,1, g3,3 = g1,1 �n+�n+1� ↔ n−
�n+1� and n+�n� ↔ n−
�n��;
f3,4 = f1,2, g3,4 = g1,2 �n+�n+1� ↔ n−
�n+1� and n+�n� ↔ n−
�n��;
g4,1 = g3,2* ;
g4,2 = g3,1* ;
f4,3 = f 3,4* , g4,3 = g3,4
* ;
f4,4 = f 3,3* , g4,4 = g3,3
* . �14�
he ↔ sign means the exchange of parameters n+�n,n+1�
ith n−�n,n+1�.
. NORMAL-MODE COUPLING AND THEPLITTING OF DEGENERATE BLOCHTATESeh has discussed the propagation of electromagneticaves in alternating linearly birefringent layers where
he normal modes differ in adjacent layers owing to an-sotropy axes misalignment [1]. He points out that in thisase a new type of constructive interference in the scat-ered waves arises from the coupling between slow andast waves. Forbidden frequency zones or bandgaps canow appear away from the Brillouin zone boundaries. Healls this an exchange Bragg condition because forwardropagating fast (slow) Bloch states couple to backwardropagating slow (fast) Bloch states. A similar type of con-tructive interference exists in elliptically birefringentO layered systems with different local modes in adja-
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122 J. Opt. Soc. Am. B/Vol. 25, No. 1 /January 2008 A. A. Jalali and M. Levy
ent layers. It is shown here that such local normal-modeariations lead to the opening up of a bandgap in the Bril-ouin zone. This effect can be traced to the presence oformal-mode coupling between adjacent layers.Notice that the unit cell transformation matrix
�n−1,n+1�, Eq. (7), depends on the relative elliptical bire-ringence parameter ��n,n+1� and the individual normal-ode propagation constants for each layer. When the pa-
ameter ��n,n+1� equals zero, the normal modes in adjacentayers are the same, and they remain uncoupled. This cane seen explicitly from the form of the T�n−1,n+1� matrixormulated in terms of normal modes. It is clear from thexpression that if ��n,n+1�=0 then the off-block-diagonalomponents of the T�n−1,n+1� matrix are zero and there iso admixture of the local normal modes. This situation isimilar to the case of a periodic layered medium consist-ng of isotropic layers, where transverse electric (TE) andransverse magnetic (TM) waves, or right- and left-ircularly polarized waves, remain uncoupled [10].
When ��n,n+1� differs from zero the Bloch states requiren admixture of local normal modes, since the off-block-iagonal components of T�n−1,n+1� differ from zero. More-ver, the strength of the coupling, parameterized as theeight of the off-block-diagonal terms, can be seen to in-
rease as sin 2��n,n+1�. We thus see that ��n,n+1� parameter-zes the degree of admixture of the normal modes. In theext section we shall discuss the form taken by this cou-ling in the wave-vector dispersion of the system.As ��n,n+1� changes away from zero the polarization
tate of the normal modes changes according to Eq. (2).ocal normal modes in adjacent layers acquire differentolarization states. These normal modes in adjacent lay-rs are now coupled by the continuity of tangential com-onents of the magnetic and electric fields across theoundary. Changes in normal-mode polarization thus af-ect the solution to the Floquet–Bloch theorem throughode coupling across the boundary.This effect can be seen in Fig. 2 below. The new band-
ap that develops away from the Brillouin zone edge wille denoted the gyrotropic degenerate bandgap. The casee are considering differs from Yeh’s treatment because of
he elliptical birefringence and nonreciprocity of the MO
ig. 2. Energy band diagram for BiIG periodic structure (fifthranch) with d�n�=0.4 and d�n+1�=0.6. The dashed lines corre-pond to the case where ��n,n+1�=0 in the transfer matrix. Theolid curves correspond to a realistic case with �=0.14.
ystems under consideration. In this case it is the gyrot-opy of the system that plays a central role in the forma-ion of this new type of gap.
. DISPERSION RELATION AS A FUNCTIONF INTERMODAL COUPLING PARAMETER
n this section we explicitly find the dependence of theispersion relation on the intermodal coupling parameter�n,n+1�. The derivation and final form of the dispersion re-ation highlight the ��n,n+1� dependence. We will considerhe polarization state dependence on ��n,n+1� in Section 6.
If one maintains the same index contrast in the photo-ic crystal and the same normal-mode propagation con-tants in each layer but allows the polarization state ofhe normal modes to vary, the only parameter thathanges in the T�n−1,n+1� matrix is ��n,n+1�, and the bandplitting that emerges is a result of normal-mode cou-ling.The characteristic equation of the T�n−1,n+1� matrix in
q. (7) is a polynomial function of of order four. de-otes the eigenvalue exp �iK�� of the Floquet–Bloch ex-ression, Eq. (6). It does not denote wavelength as is oth-rwise customary. For MO materials the gij’s are muchmaller than fi,j’s in the T�n−1,n+1� matrix. The character-stic equation can be written as
g�� = f�� + h��sin2 ��n,n+1� + j��sin2 2��n,n+1�
+ k��sin2 3��n,n+1� + l��sin2 4��n,n+1� = 0, �15�
here f�� is the characteristic equation of the T�n−1,n+1�
atrix when ��n,n+1�=0; h, j, k, and l are functions of
±�n,n+1�, �±
�n,n+1�, and d�n,n+1�. We note that for a typical MOeriodic structure, f�h� j�k� l. The lambdas corre-ponding to the zeros of f�� are denoted 0� (which corre-ponds to the case where the off-block-diagonal elementsf T�n−1,n+1� equal zero) and those corresponding to the ze-os of g�� by 0. On first-order expansion of g�� around
rom this expression one can obtain 0 in terms of 0�,
0 − 0� = −h�0��sin2 ��n,n+1�
f��0�� + h��0��sin2 ��n,n+1�,
0 0� −h�0��
f��0��sin2 ��n,n+1�, �17�
here a prime on the functions indicates the first deriva-ive with respect to their argument. In Eq. (17) use hasade of the fact that g�0�=0 and f�0��=0. Terms propor-
ional to j, k, and l and their derivatives have been ne-lected, as these are all small polynomial expressions in. From this simple analysis we can see that the differ-nce in eigenvalues is directly proportional to sin2 ��n,n+1�.
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A. A. Jalali and M. Levy Vol. 25, No. 1 /January 2008 /J. Opt. Soc. Am. B 123
Let us now consider the dispersion relation, Eq. (35) in3], for the case where ��n,n+1�=0, given by
cos K±� = cos��±�n+1�d�n+1��cos��±
�n�d�n��
−1
2N± sin��±
�n+1�d�n+1��sin��±�n�d�n��. �18�
n this case bandgaps appear only at the boundary of therillouin zone, displaying complex solutions for K�. On
he other hand, when ��n,n+1� differs from zero, the disper-ion relation acquires an additional term in the form of�h�0�� / f��0���sin2 ��n,n+1�, as follows:
cos K±� = �cos��±�n+1�d�n+1��cos��±
�n�d�n��
−1
2N± sin��±
�n+1�d�n+1��sin��±�n�d�n��
− R� h�0±� �
f��0±� ��sin2 ��n,n+1�. �19�
ere R denotes the real part of its argument and N±
n±�n� /n±
�n+1�+n±�n+1� /n±
�n�. For a general MO material N±2. This extra term in the dispersion relation is respon-
ible for bandgap formation away from the Brillouin zonedges. A complex solution for K+� (the same treatmentan be applied for K−�), and hence the existence of aandgap, occurs under the following conditions:
cos 2�� �
4 + 4u�0+� �sin2 ��n,n+1�
2 + N+,
cos 2�� �
− 4 + 4u�0+� �sin2 ��n,n+1�
2 + N+, �20�
here ��= �n+�n�d�n�+n+
�n+1�d�n+1��� / �2c� and u�0+� �
R�h�0+� � / f��0+
� ��.
ig. 3. Width of the gyrotropic degenerate bandgap versus�n,n+1�. The bandgap was calculated for the fifth branch of theand structure of the periodic structure of BiIG with d�n�=0.4nd d�n+1�=0.6. In one case the average in the refractive indices isept constant while ��n,n+1� is allowed to change (solid curve). Inhe other, the diagonal elements of the dielectric tensors of adja-ent layers are kept constant while the off-diagonal elements arellowed to change simultaneously (dashed curve).
If one maintains the same refractive index contrast andormal-mode propagation constants for each layer of theeriodic structure and allows the normal-mode polariza-ions to change, the gyrotropic degenerate bandgap widths a function of ��n,n+1� only. This bandwidth increases
onotonically with 0���n,n+1�� /2, as the range of ��xpands according to Eqs. (20). From Eq. (19) K acquirescomplex solution only for negative u�0+
� � for a typicalO periodic structure. The upper bound of �� will thus
ave larger values, and the lower bound lower values, as�n,n+1� increases. Notice that these bounds occur underquality in Eq. (20). This results in a wider gyrotropic de-enerate bandgap.
As an example let us consider a model system com-osed of bismuth iron garnet (BiIG) with typical valuesor the dielectric tensor in the near-infrared region:
��n� = �6.5411 i0.018 0
− i0.018 6.611 0
0 0 �zz� ,
��n+1� = �5.9859 i0.018 0
− i0.018 6.1699 0
0 0 �zz� . �21�
his structure simulates the effective index variation of aagnetophotonic crystal on a ridge waveguide [14]. Inig. 2, we show the band structure for the case where�n,n+1�=0 as dashed lines, corresponding to the solutionsor f��. There is no band splitting in the band structuret the crossover point. On the other hand, when ��n,n+1�
0, a bandgap opens up, as shown by the solid curves.In Fig. 3 we show the variation of the gyrotropic degen-
rate gap bandwidth for the same structure on variationf the off-diagonal components in the dielectric tensors,orresponding to the tuning of magnetization by an exter-al magnetic field applied to the photonic crystal struc-ure. We also show in the same figure the variation of theyrotropic degenerate gap bandwidth on the variation ofirefringence in the waveguide periodic structure. Weaintain the same refractive index contrast in the peri-
dic structure while the birefringence of the layers varies.n this case the condition of constancy of n± has beenlightly relaxed by �n± to obtain a wide range of variationn ��n,n+1�. However, we still require that �n± �n±. Inoth cases the gap bandwidth increases with ��n,n+1�.
. BLOCH STATES IN PERIODICIREFRINGENT MEDIAet us define the nth unit cell as the combination of layersf n and �n+1�. The translation matrix from the secondayer of the �n−1�th unit cell to the second layer of theth unit cell is given by the T�n−1,n+1� matrix in Eq. (7).pon solving the eigenvalue equation (6), the eigenvec-
ors are given by
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ATSD
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124 J. Opt. Soc. Am. B/Vol. 25, No. 1 /January 2008 A. A. Jalali and M. Levy
EK = cK�A0
B0
C0
D0
�K
, �22�
here cK is an arbitrary constant. A0, B0, C0, and D0 areunctions of n±
�n,n+1�, �±�n,n+1�, and ��n,n+1�. According to Eq.
5), the Bloch wave solution in the second layer of the nthnit cell is then given (up to a constant factor) by
EK�n+1��z� = �A0ei�+
�n+1��z−n�� + B0e−i�+�n+1��z−n���eiKn�e+
�n+1�
+ �C0ei�−�n+1��z−n�� + D0e−i�−
�n+1��z−n���eiKn�e−�n+1�.
�23�
ote that this expression depends on the relative birefrin-ence parameter ��n,n+1� through the components of theatrix T�n−1,n+1�. Figure 4 shows the polarization states of
he Bloch eigenmode at the interface between layer �n1� and layer n and at the interface between layer n and
ayer �n+1� for artificially large values of the off-diagonalomponent of the dielectric tensors, in order to highlighthe polarization variation of the Bloch state. In this ex-mple, we have taken the material dielectric tensors as
��n� = �4 i0.1 0
− i0.1 2.5 0
0 0 �zz� ,
��n+1� = �6 i2 0
− i2 5 0
0 0 �zz� . �24�
n general, the polarization state of the Bloch mode willlso evolve as the wave traverses the unit cell, althoughor typical values of the dielectric tensors these changesill be small. Thus the character of the Bloch mode is af-
ected by the coupling.
ig. 4. Polarization of Bloch wave traveling through a MO bire-ringent periodic structure. The Bloch wave polarization is de-icted on the boundary of each layer in a unit cell (solid ellipses)
ust before the gyrotropic degenerate bandgap in the band struc-ure of the medium. Dashed ellipses show the local eigenmodesolarizations e for each layer.
+
Whereas Bloch states for nonelliptically birefringentyrotropic one-dimensional stacks are still circularly po-arized, elliptically birefringent stacks can have Bloch
odes whose polarization states differ from that of theormal modes in each layer and that depend on ��n,n+1� ac-ording to Eq. (23).
. CONCLUSIONSode coupling as a result of periodic variations in the po-
arization state of local normal modes in elliptically bire-ringent nonreciprocal periodic structures is reported andiscussed. Interlayer normal-mode coupling in such me-ia affects the polarization state of the Bloch waves andhe wave-vector frequency dispersion. This interlayer cou-ling is absent in periodic media possessing solely circu-ar birefringence throughout the crystal. As a conse-uence of local normal-mode coupling, extra terms appearn the dispersion relation characterizing the formation offrequency bandgap inside the Brillouin zone away from
he zone boundary. The bandwidth of this gap is found toe parameterized by a single characteristic coupling con-tant and is shown to increase monotonically with thisoupling parameter. An expression for the latter is pre-ented and shown to depend on the difference in diagonalomponents of the dielectric tensors and the gyrotropiesf adjacent layers in the periodic structure. Thus a ready-ade tool for designing wavelength dependent bandgaps
n nonreciprocal periodic magnetophotonic structures andalibrating their bandwidths is presented. Bloch-mode po-arization states are found to differ from those of the localormal modes and to evolve into different elliptical statess the wave propagates down the crystal. These Bloch-ode polarization states are found to depend on the
trength of the coupling between local normal modes inifferent layers.
CKNOWLEDGMENThis material is based on work supported by the Nationalcience Foundation under grants ECCS-0520814 andMR-0709669.
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Bertolotti, M. Scalora, M. Bloemer, and C. Bowden,“Birefringence in one-dimensional finite photonic bandgapstructure,” J. Opt. Soc. Am. B 20, 504–513 (2003).
3. M. Levy and A. A. Jalali, “Band structure and Bloch statesin birefringent one-dimensional magnetophotonic crystals:an analytical approach,” J. Opt. Soc. Am. B 24, 1603–1609(2007).
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