optical properties of a quasi-periodic generalized fibonacci structure of chiral and material layers
TRANSCRIPT
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Vladimir R. Tuz Vol. 26, No. 4 /April 2009 /J. Opt. Soc. Am. B 627
Optical properties of a quasi-periodic generalizedFibonacci structure of chiral and material
layers
Vladimir R. Tuz
Department of Theoretical Radio Physics, Kharkov National University, Svobody Square 4, Ukraine([email protected])
Received November 5, 2008; revised January 14, 2009; accepted January 20, 2009;posted January 26, 2009 (Doc. ID 103691); published March 6, 2009
The reflection and transmission coefficients of the perpendicular and parallel polarization plane electromag-netic waves of a finite quasi-periodic Fibonacci sequence of chiral and convenient isotropic magnetodielectriclayers are obtained using the 2�2-block-representation transfer-matrix formulation. A correlation has beenestablished between geometrical and spectral properties of the structure under consideration. Numerical simu-lations are carried out for different structures to reveal the dependence of the reflection and transmission co-efficients on the frequency, chirality parameter, and the angle of wave incidence. © 2009 Optical Society ofAmerica
OCIS codes: 160.1585, 230.4170, 260.5430.
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. INTRODUCTIONn recent years, much attention has been given to so-alled photonic crystals [1–3], which constitute a speciallass of artificial structures with periodic variation of theielectric properties over a spatial scale on the order of aight wavelength. Such structures represent artificial ma-erials of a new type possessing physical characteristicsnachievable in the natural dielectric media (as well as inemiconductors and metals). Indeed, the properties ofhotonic crystals depend both on the physical parametersf a base material and on the geometry of a particularayer and the period of its structure. Such artificial struc-ures are widely used in the modern integrated optics andptoelectronics, laser and X-ray techniques, and opticalommunications. Using photonic crystals, it is possible toompletely control light wave propagation and employ thehenomenon of light wave localization. Among the mostmportant properties of photonic crystals is a wide rangef variation of their eigenwaves phase speed, with the dis-ributed feedback determining the conditions of waveropagations within the bounded passbands.Nonperiodic but deterministic (quasi-periodic) struc-
ures constitute a separate field of research. There arearious groups of these structures, namely, substitutionalFibonacci, Thue–Morse, Rudin–Shapiro, and double-eriodic) sequences [4–10] or model fractal structuresCantor sets or Koch fractals) [11–13]. Quasi-periodic sys-ems do not have translation symmetry and were mainlyonsidered as suitable theoretical models to describe theonceptual transition from a perfect periodic structure toome random version [14,15].
Enhanced functionality of the mentioned structures isossible by incorporating materials with spatial disper-ion [bi-anisotropic, reciprocal (chiral), and nonreciprocali-isotropic media] [16–19]. For example, the chiral pho-onic crystals can include cholesteric liquid crystals, chi-
0740-3224/09/040627-6/$15.00 © 2
al smectics, artificial chiral sculptured thin films andrystals, etc. Such structures are characterized by selec-ive reflection in a specific wavelength region and circularolarization of the reflected and transmitted light. Theseroperties are attractive in the design of cascaded high-Qnd stopband frequency filters, polarizers, high-precisionatched loads, etc. In [20] it has been proposed to intro-
uce Fibonaccian defects into the chiral photonic crystalso create reflective color-displays without the need forack-lighting, polarizers, or color filters.In the present work, the optical properties of a gener-
lized Fibonacci sequence of chiral and convenient isotro-ic layers are investigated. The method of solution isased on the 2�2-block-representation transfer-matrixormulation proposed in [21]. This method generalizes the�2-matrix approach applied to right-hand and left-handircular wave polarization [18,19] and the 4�4-matrixpproach of Berreman [22], and it allows the investiga-ion of reflection and transmission in terms of linear-waveolarization.
. PROBLEM FORMULATIONhe structure under study consists of two basic elementsand � that are ordered according to the generalized Fi-
onacci sequence [7,8] (Fig. 1):
F1 = �, F2 = �,
Fv = Fv−1n Fv−2
m , for v � 3, �1�
here n and m are arbitrary integers and v is the itera-ion number. Note that, if n=m=1, the structure in Fig. 1s a classical Fibonacci sequence [4–7]. The basic ele-
ents � and � are made from isotropic magnetodielectricith permittivities ε , � , and chiral material with ε ,�
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628 J. Opt. Soc. Am. B/Vol. 26, No. 4 /April 2009 Vladimir R. Tuz
nd chirality parameter �, respectively. In general, theaterial parameters are frequency dependent and com-
lex for lossy media: εj���=εj/���+εj
//���, �j���=�j/���
�j//���, (j�1,2), ����=�/���+�//���. Here �/ is responsible
or optical rotatory power and �// produces circular dichro-sm. Optical thicknesses of both layers � and � are equalo each other and denoted as D. The thicknesses of thehole structure is L. The outer half-spaces (z0, z�L)re homogeneous, isotropic, and have permittivities ε0nd �0.As the excitation fields, the plane (in the XOZ plane)onochromatic waves with perpendicular (Ee �x0, Hx
e =0)r parallel (Hh �x0, Ex
h=0) polarization are selectedE- and H-waves). They are obliquely incident from theegion z0 at an angle 0 to the z axis.
�Ex0e
Ey0h � = ±� A0
e /�Y0e
iA0h/�Y0
h�exp�− i��t − kyy − kz0z��,
�Hy0e
Hx0h � =� A0
e�Y0e
iA0h�Y0
h�exp�− i��t − kyy − kz0z��, �2�
here kz0=k0 cos 0, ky0=k0 sin 0, Y0e =cos 0, Y0
h
�cos 0�−1.
. FIELDS IN A CHIRAL LAYERn the homogeneous along the x direction chiral layers,he electromagnetic field is characterized by the displace-ents D=ε2E− i�H and B=�2H+ i�E [16,17] and gov-
rned by the coupled differential equations
��Ex + k02�n2
2 + �2�Ex − 2ik02��2Hx = 0, ��Hx + k0
2�n22
+ �2�Hx + 2ik02�ε2Ex = 0, �3�
here n2=�ε2�2 is the medium refractive index and ��
�2 /�y2+�2 /�z2 is the two-dimensional Laplacian.The waves of the perpendicular �Ee �x0� and parallel
Hh �x0� linear polarizations can be presented as theuperposition of the waves of the right �Qs
+� and left �Qs−�
ircular polarizations [17]:
Exe = Qe
+ + Qe−, Hx
e = i�2−1�Qe
+ − Qe−�,
ig. 1. Quasi-periodic generalized Fibonacci structure of chiralayers.
Exh = − i�2�Qh
+ − Qh−�, Hx
h = Qh+ + Qh
−, �4�
here �2=��2 /ε2. Such a substitution transforms Eqs. (3)o two independent Helmholtz equations:
��Qs+ + � +�2Qs
+ = 0, ��Qs− + � −�2Qs
− = 0. �5�
ere s=e ,h, ±=k0�n±�� is the propagation constant ofhe right (RCP) and left (LCP) circularly polarized planeave in the unbounded chiral medium, respectively.Their general solutions for the RCP and LCP waves inbounded chiral layer can be written as
Qe± =
1
2�Ye±Ae± exp�i� y
±y + z±z�� + Be± exp�i� y
±y − z±z��,
Qh± =
�Yh±
2Ah± exp�i� y
±y + z±z�� + Bh± exp�i� y
±y − z±z��,
�6�
here As±,Bs± are the wave amplitudes, Y2e±=�2
−1 cos ±,
2h±= ��2 cos ±�−1 are the wave admittances, y
±
± sin ±, z±= ± cos ±, ± is the refraction angle in the
hiral medium, sin ±=sin 0n0 / �n2±�� and n0=�ε0�0.he substitution of Eqs. (6) into Eqs. (4) gives the fieldomponents of the e- and h-polarizations [21].
. REFLECTED AND TRANSMITTED FIELDSF A QUASIPERIODIC SEQUENCEue to the reflection and transmission of the given polar-
zation plane electromagnetic wave �s� by the chiral lay-rs, the cross-polarized components �s�� appear in the sec-ndary field. Denote their amplitudes as As� and Bs� forhe transmitted and reflected waves, respectively. Makingse of the transfer-matrix formalism [23,24], the equa-ions coupling the field amplitudes at the structure inputA0
s ,B0s ,B0
s�� and output �ALs ,AL
s�� for the incident fields ofhe E-type �A0
h=0� and H-type �A0e =0� are obtained as
�A0
s
B0s
0
B0s�� = Tv�
ALs
0
ALs�
0� = Tv−1
n Tv−2m �
ALs
0
ALs�
0�, T1 = T�
= T01P1T10, T2 = T� = T02P2T20, �7�
here T0j and Tj0 �j=1,2� are the transfer-matrices of theasic-element interfaces with outer half-spaces, and Pjre the propagation-matrices through the single layers �nd �. In the 2�2-block-representation [21] the transfer-atrices are
Tp� = �Tp�s � 0
0 �Tp�s���, T02 = �T02+
ss � �T02−ss �
�T02+ss� � �T02−
ss� ��,
T20 = �T20+ss � �T20+
ss� �
�T20−ss � �T20−
ss� �� ,
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Vladimir R. Tuz Vol. 26, No. 4 /April 2009 /J. Opt. Soc. Am. B 629
P1 = �E1� 0
0 �E1��, P2 = �E2+� 0
0 �E2−�� . �8�
ere Tp� corresponds to the matrices T01 and T10, and thelocks of these quasi-diagonal matrices are
Tp�s =
1
2�YpsY�
s Yps + Y�
s ±�Yps − Y�
s�
±�Yps − Y�
s� Yps + Y�
s � , �9�
here the upper sign relates to s=h, and the lower signelates to s=e in terms of the wave types.
The elements of the transfer-matrices T02 and T20 areetermined from solving the boundary-value problem re-ated to the field components (2) and (4). They are
0 0 L 0
Rp
5ATcdtvq[
T02±ee =
1
4�Y0eY2
e± Y0e + Y2
e± Y0e − Y2
e±
Y0e − Y2
e± Y0e + Y2
e±�, T02±eh
= ±1
4�Y0hY2
h± Y2h± + Y0
h Y2h± − Y0
h
Y2h± − Y0
h Y2h± + Y0
h� ,
T02±hh =
1
4�Y0hY2
h± Y2h± + Y0
h Y2h± − Y0
h
Y2h± − Y0
h Y2h± + Y0
h�, T02±he
= �1
� e e± Y0e + Y2
e± Y0e − Y2
e±
Ye − Ye± Ye + Ye±� , �10�
4 Y0Y2 0 2 0 2T20±ee =
1
4Y2e��Y0
eY2e±
� �Y2e� + Y0
e��Y2e� + Y2
e±� − �Y2e� − Y0
e��Y2e� − Y2
e±� �Y2e� − Y0
e��Y2e� + Y2
e±� − �Y2e� + Y0
e��Y2e� − Y2
e±�
�Y2e� − Y0
e��Y2e� + Y2
e±� − �Y2e� + Y0
e��Y2e� − Y2
e±� �Y2e� + Y0
e��Y2e� + Y2
e±� − �Y2e� − Y0
e��Y2e� − Y2
e±�� ,
�11�
T20±eh = �
1
4Y2h��Y0
hY2h±
� �Y2h� + Y0
h��Y2h� + Y2
h±� − �Y2h� − Y0
h��Y2h� − Y2
h±� �Y2h� + Y0
h��Y2h� − Y2
h±� − �Y2h� − Y0
h��Y2h� + Y2
h±�
�Y2h� + Y0
h��Y2h� − Y2
h±� − �Y2h� − Y0
h��Y2h� + Y2
h±� �Y2h� + Y0
h��Y2h� + Y2
h±� − �Y2h� − Y0
h��Y2h� − Y2
h±�� , �12�
T20±hh =
1
4Y2h��Y1
hY2h±
� �Y2h� + Y0
h��Y2h� + Y2
h±� − �Y2h� − Y0
h��Y2h� − Y2
h±� �Y2h� + Y0
h��Y2h� − Y2
h±� − �Y2h� − Y0
h��Y2h� + Y2
h±�
�Y2h� + Y0
h��Y2h� − Y2
h±� − �Y2h� − Y0
h��Y2h� + Y2
h±� �Y2h� + Y0
h��Y2h� + Y2
h±� − �Y2h� − Y0
h��Y2h� − Y2
h±�� , �13�
T20±he = �
1
4Y2e��Y0
eY2e±
� �Y2e� + Y0
e��Y2e� + Y2
e±� − �Y2e� − Y0
e��Y2e� − Y2
e±� �Y2e� − Y0
e��Y2e� + Y2
e±� − �Y2e� + Y0
e��Y2e� − Y2
e±�
�Y2e� − Y0
e��Y2e� + Y2
e±� − �Y2e� + Y0
e��Y2e� − Y2
e±� �Y2e� + Y0
e��Y2e� + Y2
e±� − �Y2e� − Y0
e��Y2e� − Y2
e±�� . �14�
The blocks of the quasi-diagonal propagation-matricesj �j=1,2� are
E1 = Diag�exp�− ikz1D� exp�ikz1D��, E2± = Diag�exp�
− i z±D� exp�i z
±D��. �15�
ere kzj=kj cos j, kyj=kj sin j, kj=k0nj, nj=�εj�j, Yje
�j−1 cos j, Yj
h= ��j cos j�−1, �j=��j /εj, sin jsin 0n0 /nj, and j=0,1.The reflection and transmission coefficients of the re-
ected �z0� and transmitted �z�L� fields are deter-ined by the expressions Rss=Bs /As , �ss=As /As and
ss�=B0s� /A0
s , �ss�=ALs� /A0
s for the co-polarized and cross-olarized waves, respectively.
. NUMERICAL RESULTS, SOLUTIONNALYSIS
he properties of a quasi-periodic multilayer system inonsistency with the generalized Fibonacci sequence areependent on both the electromagnetic, geometric charac-eristics of the basic element components and values,m ,n. It is known that the sequences with m=1 areuasi-periodic and those with m�2 are always aperiodic7].
m=a
emecfl
=tqdlspotsiipsa=rs[
ctom
Fso�
Faao
630 J. Opt. Soc. Am. B/Vol. 26, No. 4 /April 2009 Vladimir R. Tuz
The total number of the structure periods is deter-ined via the recursive formula Sv=nSv−1+mSv−2, S1S2=1, and the number of periods of each type �S� ,S��re determined in [8] as
S� =m
�n2 + 4m n + �n2 + 4m
2�v−2
−m
�n2 + 4m n − �n2 + 4m
2�v−2
,
S� =1
�n2 + 4m n + �n2 + 4m
2�v−1
−1
�n2 + 4m n − �n2 + 4m
2�v−1
.
In the case of a structure with nonchiral isotropic lay-rs ��=0�, the nondiagonal block-matrices of the transfer-atrix T2 equal to zero and the e and h modes of plane
lectromagnetic waves are independent of each other (un-oupled modes), and cross-polarized components in the re-ected and transmitted fields are absent ��Reh�= �Rhe�
ig. 2. Frequency dependences of the reflection and transmis-ion coefficient magnitudes for a generalized Fibonacci sequencef isotropic layers (without chirality): εj=�j=1, j�2, ε2=2+ iε2
//,2=1, �=0, 0=25°.
��eh�= ��he�=0�. The frequency dependences of the reflec-ion and transmission coefficient magnitudes for such se-uence are given in Fig. 2 for comparison. These depen-ences have interleaved areas with high and low averageevel of reflection and transmission. Let’s call them quasi-topbands and quasi-passbands (or pseudo stopbands andseudo passbands [9]), in analogy with the terms used inptics of the periodic structures. The interference withhe wave reflected from the outside boundaries givesmall-scale oscillations in the quasi-passbands. When thendex of iteration v rises, the number of structure periodsncreases and the reflection and transmission coefficienteaks get narrower with an increasing quantity of small-cale oscillations in quasi-passbands for each couple mnd n. In a presence of small real losses (εj
//�0, �j//�0, j
1,2) of magnetodielectric layers, the average level of theeflection and transmission reduces, and the amplitude ofmall-scale oscillations in the quasi-passbands decreases25] (Fig. 2).
When a sequence includes chiral layers ���0�, theross-polarized components in the reflected and transmit-ed fields appear (couple modes) (See Figs. 3–6). The levelf the cross-polarized reflection is not too big, and theaximum of the reflection coefficient magnitude for the
ig. 3. Transmission coefficient magnitude of (a) co-polarizednd (b) cross-polarized waves as function of the frequency k0Dnd the chirality parameter � for a classical Fibonacci sequencef chiral layers: εj=�j=1, j�2, ε2=2, �2=1.
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Fsoj
Fcl
Vladimir R. Tuz Vol. 26, No. 4 /April 2009 /J. Opt. Soc. Am. B 631
ross-polarized wave corresponds to the minimum of theeflection coefficient magnitude for the co-polarized wave.he reflection and transmission coefficient magnitudes ofhe cross-polarized components for both polarizations arequal to each other (�Reh�= �Rhe�, ��eh�= ��he�).
The bands become narrow and the number of oscilla-ions increases as m increases. The width of quasi-assbands decreases exponentially as m increases [8],nd then it tends to some constant when m is largenough (See Fig. 4). The obtained reflection and transmis-ion coefficient curves show that the optical properties ofhe structure depend on the parity of m and have a flip-ike characteristic, i.e., the quasi-passbands for even morrelate with the quasi-stopbands for odd m.
When m is fixed and n increases, the number of small-cale oscillations goes up with the quasi-stopband widthsecreasing (See Fig. 5). In this case the structure has andvantage as a filter with very sharp transparency win-ows, because quasi-stopbands start and terminate quitebruptly.An analysis of the angular dependences of the reflec-
ion and transmission coefficient magnitudes (Fig. 6)hows that the maximum wave transformation of the re-ected field is observed when the incidence angle 0 isearly 65° for the given structure parameters. In the case
ig. 4. Frequency dependences of the reflection and transmis-ion coefficient magnitudes for a generalized Fibonacci sequencef chiral layers with (a) even and (b) odd values of m: εj=�j=1,�2, ε2=2, �2=1, �=0.2, 0=25°.
f normal incidence 0=0, the cross-polarized componentn the reflected field is absent.
. CONCLUSIONhe electromagnetic properties of the finite quasi-periodiceneralized Fibonacci sequence of two basic elements are
ig. 5. Frequency dependences of the reflection and transmis-ion coefficient magnitudes for a generalized Fibonacci sequencef chiral layers with (a) odd and (b) even values of n: εj=�j=1,�2, ε2=2, �2=1, �=0.2, 0=25°.
ig. 6. Angular dependences of the reflection and transmissionoefficient magnitudes for a classical Fibonacci sequence of chiralayers: εj=�j=1, j�2, ε2=2, �2=1, �=0.1, k0D=6.
it�
tsnctfciiewad
sa
R
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
632 J. Opt. Soc. Am. B/Vol. 26, No. 4 /April 2009 Vladimir R. Tuz
nvestigated. The basic elements consist of chiral and iso-ropic layers. The method of solution is based on the 22-block-representation transfer-matrix formulation.The influence of the media chirality on optical proper-
ies of the structure is studied. The location of quasi-topbands and quasi-passbands and conditions of reso-ant structure transparency are determined. Theorrelation between the geometrical and spectral proper-ies of a generalized Fibonacci multilayer structure isound out. Two structure parameter combinations areonsidered. The first one is when n is constant and m var-es. In this case the structure has a flip-like characteristicn that the reflected field has the quasi-passbands forven m and quasi-stopbands for odd m. The second case ishen m is constant and n varies. The structure has andvantage as a filter with very sharp transparency win-ows.The obtained solution is suitable for the study other
ubstitutional and model fractal quasi-periodic structuresnd periodic structures with single and plural defects.
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