PHYSICAL REVIEW B 74, 115110 共2006兲
Properties of Faraday chiral media: Green dyadics and negative refraction Cheng-Wei Qiu,1,2 Le-Wei Li,1,* Hai-Ying Yao,1 and Saïd Zouhdi2 1Department 2Laboratoire
of Electrical and Computer Engineering, National University of Singapore, Kent Ridge, Singapore 119260 de Génie Electrique de Paris, CNRS, Ecole Supérieure D’Électricité, Plateau de Moulon 91192, Gif-Sur-Yvette Cedex, France 共Received 17 April 2006; published 18 September 2006兲
Selected properties of generalized Faraday chiral media are thoroughly studied in this paper where Green’s dyadics are formulated for unbounded and layered structures, and the possibility of negative refractive index, the backward eigenwaves, and quantum vacuum are also investigated. After a general representation of the Green’s dyadics is obtained, the scattering coefficients of the Green’s dyadics are determined from the boundary conditions at each interface and are expressed in a greatly compact form of recurrence matrices. In the formulation of the Green’s dyadics and their scattering coefficients, three cases are considered, i.e., the current source is immersed in 共i兲 the intermediate, 共ii兲 the first, and 共iii兲 the last regions, respectively. We present here layered dyadic Green’s functions for generalized Faraday chiral media. This kind of Faraday chiral media can also be manipulated to achieve negative refraction and possible backward wave propagation is presented as well. As compared to the existing results, the present work mainly contributes: 共1兲 the exact representation of the dyadic Green’s functions, with irrotational part extracted out, for the gyrotropic Faraday chiral medium in multilayered geometry; 共2兲 the general DGFs and scattering coefficients which can be reduced to either layered chiroferrite, chiroplasma or other simpler cases; and 共3兲 negative refractive index and backward waves achieved with less restriction and more advantages compared to chiral media. DOI: 10.1103/PhysRevB.74.115110
PACS number共s兲: 42.70.Qs, 41.20.Jb, 42.25.Bs
¯ −1 · B, H = i cE +
I. INTRODUCTION
Recently, composite materials have attracted considerable attention in various areas.1–4 Among these materials, a double negative material 共DNG兲5 exhibits a left-handedness ruling the polarizations of electric and magnetic fields which is referred to as left-handed materials.6–10 Those materials can possess negative refraction of the waves and thus are considered to be negative-index media 共NIM兲 which open new avenues to achieving unprecedented physical properties and functionality unattainable with natural materials.11–14 Metamaterials with negative refraction in microwave region and related applications have been studied, including metamaterial waveguides,15 SRR and spiral resonators,16,17 leaky wave antennas,18 and subwavelength cavity resonators.19 Since the negative refraction by the artificial NIM was experimentally verified by Shelby,20 more studies on metamaterials have been carried out such as tensorparameter retrieval using quasistatic Lorentz theory,21 S-parameter retrieval using the plane wave incidence,22 and constitutive relation retrieval using the transmission line method.23,24 In this paper, we propose a different way to achieve negative refraction and backward waves by using Faraday chiral media. Although some recent works25,26 have shown that chiral media can also exhibit negative refraction, this chiral route still has many limitations 共e.g., pure chirality, both permittivity and permeability tending zero and chiral nihility only valid at or near a resonant frequency兲. Herein, in particular, we examine the Faraday chiral medium with gyrotropy, which is characterized in a rectangular coordinate sys¯ as follows: tem by chirality c, ¯⑀, and D = ¯⑀ · E + icB, 1098-0121/2006/74共11兲/115110共10兲
共1兲
共2兲
where
冤 冤
冥 冥
⑀ − ig 0 ¯⑀ = ig ⑀ 0 , 0 0 ⑀z
共3兲
− iw 0 ¯ = iw 0 . 0 0 z
共4兲
It is found that these gyrotropic Faraday chiral media have advantages over normal chiral media: 共i兲 negative index of refraction in a gyrotropic chiral medium can be realized with less restrictions, since the refractive indices are greatly reduced by those gyrotropic parameters; 共ii兲 two backward eigenwaves are found in certain frequency bands; and 共iii兲 the parameters in permittivity and permeability tensors and chirality admittance can be positive even when negative refraction occurs. The negative-index effects and eigenwaves propagating in a backward direction can be realized from gyrotropic Faraday chiral media as shown in this paper. Further, we will consider potential applications in NIM, phase conservation, and quantum fields. Since the interaction between materials and electromagnetic waves is another important aspect in material characterization, Green dyadics are of particular interest for Faraday chiral media, which can describe the wave interaction in a macroscopic view. Dyadic Green’s functions,27 which relate directly the radiated electromagnetic fields and the source distribution, provide a good way to characterize the macroscopic performance of artificial complex media including metamaterials. DGFs play an important role in solving
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QIU et al.
both source-free and source-incorporated boundary value problems for electromagnetic scattering, radiation, and propagation.28 However, DGFs in complex media like gyrotropic or metamaterials have never been studied especially in multilayered structures, though the DGFs for some isotropic,29 chiral,30 anisotropic,31 chiroplasma,32 and bianisotropic33–35 media have been formulated over the last three decades. The technique of eigenfunctional expansion provides a systematic approach in electromagnetic theory for interpreting various electromagnetic representations;36 most importantly, it is applicable in almost all the fundamental coordinates. Even in the cylindrical structure considered in detail in this paper, the eigenfunctional expansion technique can provide an explicit form of the dyadic Green’s functions, so that it becomes easy and convenient when the source distribution is independent from the azimuth directions or when the far-zone fields are computed. Different from the existing work, this paper aims at two important points, namely, Green dyadics and negative refraction index. Physical properties of materials are often described by using these two concepts. For the part of DGFs, we achieve 共i兲 the direct development of the unbounded dyadic Green’s functions with the irrotational part extracted out in an unbounded gyrotropic Faraday chiral medium where the eigenfunctional expansion technique is employed, and 共ii兲 the formulations of the scattering dyadic Green’s functions and their coefficients in layered gyrotropic Faraday chiral media where each layer can be stratified with arbitrary thickness and material parameters. For the negative-index part, after DGFs have been obtained, we also realize some characteristics of metamaterials 共e.g., backward waves and negative refraction兲 by using this kind of material with a number of advantages and less limitations. Some potential applications are suggested. Throughout the paper, a time dependence e−it is assumed but always suppressed. II. DGFS FOR UNBOUNDED GYROTROPIC FARADAY CHIRAL MEDIA
A homogeneous gyrotropic Faraday chiral medium has been characterized as in Eq. 共1兲. Experimentally, there might be some problems of effectively controlling the gyroelectric 共g兲 and the gyromagnetic 共w兲 parameters simultaneously. However, gyrotropic Faraday chiral materials may occur unintentionally in the fabrication process of chiroplasma and chiroferrite, and theoretical physics goes ahead often. Due to the generality of the material discussed in this paper, it is valuable to investigate the DGFs for this material in its multilayered structure as well as its potential ways to achieve negative refraction and characteristics of left-handed materials. Substituting Eq. 共1兲 into the source incorporated Maxwell’s equations, we have ¯ −1 · ⫻ E兴 − 2c ⫻ E − 2¯⑀ · E = iJ. 共5兲 ⫻ 关 The electric field can thus be expressed in terms of the DGF and electric source distribution as follows:
E共r兲 = i
冕
¯ 共r,r⬘兲 · J共r⬘兲dV⬘ , G e
V⬘
共6兲
where V⬘ denotes the volume occupied by the source. Substituting Eq. 共6兲 into Eq. 共5兲 leads to ¯ = ¯I␦共r − r⬘兲, ¯ 兴 − 2 ⫻ G ¯ − 2¯⑀ · G ¯ −1 · ⫻ G ⫻ 关 e c e e 共7兲 where ¯I and ␦共r − r⬘兲 denotes the identity dyadic and Dirac delta function, respectively. According to the well-known Ohm-Rayleigh method, the source term in Eq. 共7兲 can be expanded in terms of the solenoidal and irrotational cylindrical vector wave functions in cylindrical coordinates. Thus, we obtain ¯I␦共r − r⬘兲 =
冕 冕 ⬁
⬁
⬁
d
dh
兺
关Mn共h,兲An共h,兲
n=−⬁
−⬁
0
+ Nn共h,兲Bn共h,兲 + Ln共h,兲Cn共h,兲兴,
共8兲
where the vector wave functions M, N, and L in a cylindrical coordinate system are defined Mn共h,兲 = ⫻ 关⌿n共h,兲zˆ 兴,
共9兲
1 ⫻ Mn共h,兲, k
共10兲
Nn共h,兲 =
Ln共h,兲 = 关⌿n共h,兲兴,
共11兲
with k = 冑2 + h2, and the generating function given by ⌿n共h , 兲 = Jn共兲ei共n+hz兲. The coefficients An共h , 兲, Bn共h , 兲, and Cn共h , 兲 in Eq. 共8兲 are to be determined from the orthogonality relations among the cylindrical vector wave functions. Therefore, scalar-dot multiplying both sides of Eq. 共8兲 with M−n⬘共−h⬘ , −⬘兲, N−n⬘共−h⬘ , −⬘兲, and L−n⬘共−h⬘ , −⬘兲 each at a time and integrating them over the entire source volume, we obtain from the orthogonality that An共h,兲 =
1 M⬘ 共− h,− 兲, 42 −n
共12兲
Bn共h,兲 =
1 N⬘ 共− h,− 兲, 42 −n
共13兲
L⬘ 共− h,− 兲. 4 共2 + h2兲 −n
共14兲
Cn共h,兲 =
2
The unbounded dyadic Green’s function can thus be expanded as follows: ¯ 共r,r⬘兲 = G 0
冕 冕 ⬁
d
0
⬁
⬁
−⬁
dh
兺
关Mn共h,兲an共h,兲
n=−⬁
+ Nn共h,兲bn共h,兲 + Ln共h,兲cn共h,兲兴,
共15兲
where the vector expansion coefficients an共h , 兲, bn共h , 兲, and cn共h , 兲 are unknown vector coefficients to be determined from the orthogonality and permittivity and perme-
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ability tensors’ properties. To obtain these unknown vectors, we substitute Eq. 共15兲 and Eq. 共8兲 into Eq. 共7兲, noting the instinct properties of the vector wave functions of ⫻ Nn共h,兲 = kMn共h,兲,
共16兲
⫻ Mn共h,兲 = kNn共h,兲,
共17兲
⫻ Ln共h,兲 = 0.
共18兲
For the compactness of the subsequent manipulation, we define
冤
冥
␣t − ␣a 0 ¯ = = ␣a ␣t 0 , ␣ 0 0 ␣z ¯ −1
␣t =
, − w2
␣a =
2
− iw , 2 − w2
␣z =
1 . z
共20兲
By substituting Eq. 共15兲 into Eq. 共7兲, taking, respectively, the anterior scalar product with the vector wave equations, and performing the integration over the entire source volume, we can formulate the equations satisfied by the unknown vectors and the known scalar and vector parameters in a matrix form as given by
关⌽兴关X兴 = 关⌰兴,
共19兲
共21兲
where
where
冤
关⌽兴 = −
h 2␣ t + 2␣ z − 2⑀
冉
2hg + ikh␣a + 2ck k − i2
−
冊
冉
2 g k2
2hg + ikh␣a + 2ck k h 2⑀ + 2⑀ z k2 ␣t − 2 k2 2
The quantities 关X兴 and 关⌰兴 are known and parameter column vectors are given, respectively, by
ih2 共⑀z − ⑀兲 k3
p =
关X兴 = 关an共h,兲,bn共h,兲,cn共h,兲兴T ,
Solving Eq. 共21兲, we have the solutions to an共h , 兲, bn共h , 兲, and cn共h , 兲 as follows:
q =
cn共h,兲 =
1 关␣3An共h,兲 + 3Bn共h,兲 + ␥3Cn共h,兲兴, ⌫
␣z2
− 2 − 2
冥
ih 共⑀z − ⑀兲 . k
h 2⑀ z + 2⑀ k2
1
␣z2
共22兲
兵共␣2t + ␣2a兲h2⑀ − 4ih␣a⑀c − 共4⑀2c + ⑀␣z⑀z兲2 共25兲
兵− 共␣2t + ␣2a兲h4⑀z + 4ih2␣a共2hc + g兲⑀z
+ ⑀z关4h22c + 4ghc + 共g2 − ⑀2兲2 + 2␣th2⑀⑀z兴2其.
1 an共h,兲 = 关␣1An共h,兲 + 1Bn共h,兲 + ␥1Cn共h,兲兴, ⌫ 1 关␣2An共h,兲 + 2Bn共h,兲 + ␥2Cn共h,兲兴, ⌫
1
i 2g
+ ␣t共g2 − ⑀2兲2 − ␣t␣zh2⑀z其,
关⌰兴 = 关An共h,兲,Bn共h,兲,Cn共h,兲兴T .
bn共h,兲 =
冊
共26兲 It should be noted that the coupling coefficients 1, ␥1, ␣2, ␥2, ␣3, and 3 were assumed to be zero in Ref. 32. Here it is proved that those coupling coefficients must be considered in the formulation since they are not always zero, and the coupling coefficients ␣1,2,3, 1,2,3, and ␥1,2,3 are given in detail below
where ⌫ = ⑀z␣t共k2 − k21兲共k2 − k22兲/␣z
共23兲
1 关− p ± 冑p2 + 4⑀z␣t/␣zq兴 2 ⑀ z␣ t/ ␣ z
共24兲
␣1 =
1 2 ␣t 2 2 2 共h ⑀z + ⑀兲 − 2 ⑀⑀z , ␣z ␣z
共27兲
and 2 = k1,2
␣2 = 1 =
1 k␣z2
关ih␣a共h2⑀z + 2⑀兲
+ 2c共h2⑀z + ⑀2兲 + hg⑀z2兴,
with p and q given, respectively, below: 115110-3
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QIU et al.
␥1 = −
k2 i 2 2 2 ␣3 = 2 关gk␣t + ih ␣a共⑀ − ⑀z兲 ␣z
+ 2hc共⑀ − ⑀z兲 − g⑀z2兴,
␥2 = −
冋冉
冊
k2 ␣t i h h2 + 2 共⑀ − ⑀z兲/␣z + ighk2 ␣a/␣z2 2 3 = k ␣z
2 =
1 k2
冋冉
h2
Ln共h,兲 = Lnt共h,兲 + Lnz共h,兲,
共33兲
⬘ 共− h,− 兲 = L−nt ⬘ 共− h,− 兲 + L−nz ⬘ 共− h,− 兲, L−n
共34兲
Nn共h,兲 = Nnt共h,兲 + Nnz共h,兲,
共35兲
⬘ 共− h,− 兲 = N−nt ⬘ 共− h,− 兲 + N−nz ⬘ 共− h,− 兲, N−n
共36兲
共29兲
册
+ 2gck2 /␣z2 − h共⑀2 − ⑀⑀z − g2兲2/␣z2 ,
冊
共30兲
␣t − 2 共h2⑀z + 2⑀兲/␣z ␣z
册
− 共h2⑀⑀z + 2共⑀2 − g2兲兲2/␣z2 ,
␥3 =
Note that there are some special relations between vectors L and N as shown below:
再 冉
共31兲
冊
冋
+ k2 ⑀␣t +
h ␣t + ␣z
+ 2ih g␣a + 2
+
1 k2
2
k2 4k2 2c
册
共h2⑀ + 2⑀z兲
⬘ 共− h,− 兲 = L−nz
2
/␣z2
+ 4ghc
冎
3
/␣z2
关h2共g2 − ⑀2兲 − 2⑀⑀z兴4/␣z2 .
¯ 共r,r⬘兲 = G 0
冕 冕 ⬁
0
−⬁
⬁
⬁
dh
Lnz共h,兲 =
d
共32兲
共37兲
ik N⬘ 共− h,− 兲, h −nt
共38兲
ihk Nnz共h,兲, 2
共39兲
⬘ 共− h,− 兲 = L−nt
␣2 + ␣2a 1 ␣t 2 2 t − k h + 2 + 4ihk2 ␣ac/␣z2 2 2 ␣z ␣z 2
− ik Nnt共h,兲, h
Lnt共h,兲 =
− ihk ⬘ 共− h,− 兲, N−nz 2
共40兲
where the subscripts t and z denote, respectively, the transverse component and the longitude component and they apply similarly for the primed functions. Thus we can rewrite Eq. 共15兲 as follows:
1
⬘ 共− h,− 兲 + Nnt共h,兲M−n ⬘ 共− h,− 兲兴 兺 2 兵1Mn共h,兲M−n⬘ 共− h,− 兲 + 2关Mn共h,兲N−nt n=−⬁ 4 ⌫
⬘ 共− h,− 兲 + Nnz共h,兲M−n ⬘ 共− h,− 兲兴 + 4关Nnt共h,兲N−nz ⬘ 共− h,− 兲 + Nnz共h,兲N−nt ⬘ 共− h,− 兲兴 + 3关Mn共h,兲N−nz ⬘ 共− h,− 兲 + 6Nnz共h,兲N−nz ⬘ 共− h,− 兲其, + 5Nnt共h,兲N−nt
where the intermediates 1 to 6 are defined as follows:
1 = ␣1 , i2 2 = 1 + ␥1 , k h ih 3 = 1 − ␥1 , k ih ik 4 = 2 − ␥2 − 3 − ␥3 , k h
5 = 2 +
i2 ik 2 ␥2 − 3 + 2 ␥3 , k h h h
6 = 2 −
共42兲 共43兲
共44兲
共41兲
ih ihk h2 ␥2 + 2 3 + 2 ␥3 . k
共47兲
By applying the idea of Tai27 to obtain an exact expression of the irrotational term, we obtain from Eq. 共8兲 zˆ zˆ ␦共r − r⬘兲 =
冕 冕 ⬁
d
0
− 兲.
⬁
⬁
−⬁
dh
1
k2
⬘ 共− h, 兺 2 ⫻ 2 Nnz共h,兲N−nz 4 n=−⬁
共48兲
共45兲
共46兲
Apparently, the irrotational term of the unbounded DGF is contained in the Nnz共h , 兲N−nz ⬘ 共−h , −兲 dyadic hybrid modes. After some lengthy but careful algebraic manipulations, we rewrite Eq. 共41兲 in the following form
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PROPERTIES OF FARADAY CHIRAL MEDIA: GREEN¼
¯ 共r,r⬘兲 = − G 0
␣z zˆ zˆ ␦共r − r⬘兲 + 2 ⑀ z␣ t
冕 冕 ⬁
d ⫻
dh
−⬁
⬁
⬁
0
1
⬘ 共− h,− 兲 兺 2 兵1Mn共h,兲M−n⬘ 共− h,− 兲 + 2关Mn共h,兲N−nt n=−⬁ 4 ⌫
⬘ 共− h,− 兲兴 + 3关Mn共h,兲N−nz ⬘ 共− h,− 兲 + Nnz共h,兲M−n ⬘ 共− h,− 兲兴 + 4关Nnt共h,兲N−nz ⬘ 共− h,− 兲 + Nnt共h,兲M−n ⬘ 共− h,− 兲兴 + 5Nnt共h,兲N−nt ⬘ 共− h,− 兲 + 7Nnz共h,兲N−nz ⬘ 共− h,− 兲其, + Nnz共h,兲N−nt
where
7 = 2 +
1 2 2 2 2 k共k
− k21兲共k2 − k22兲 +
h ih ␥2 . 2 共ik3 + h␥3兲 − k 共50兲
The first term of Eq. 共49兲 is due to the contribution from the nonsolenoidal vector wave functions. The second integration term can be evaluated by making use of the residue theorem
¯ 共r,r⬘兲 = − G 0 ⫻
␣z i zˆ zˆ ␦共r − r⬘兲 + 2 ⑀ z␣ t 4
再
冕
⬁
共49兲
in the plane. This irrotational part of DGFs in a gyrotropic Faraday chiral medium is obtained for the first time when an eigenfunction expansion technique is applied. This irrotational part in specific cases agrees well with the previous solutions of a chiroplasma medium by letting ␣z = ␣t = 1 / or an isotropic medium by letting ⑀z = ⑀ further if we first set g = w = 0. The final expression of the unbounded DGFs is ⬎ ⬘, given after mathematical manipulations for ⬍
⬁
2
共− 1兲 j+1 ␣z dh 兺 兺 2 2 2j n=−⬁ ⑀z␣t共k1 − k2兲 j=1 −⬁
共1兲 共1兲 共1兲 共1兲 ⬘ 共− j兲 + Qn,h ⬘ 共− j兲 + Un,h ⬘ 共− j兲 + Vn,h ⬘ Mn,h 共 j兲P−n,−h 共 j兲M−n,−h 共 j兲N−nt,−h 共 j兲N−nz,−h 共− j兲;
⬘共1兲 共 j兲 + Qn,h共− j兲M−n,−h ⬘共1兲 共 j兲 + Un,h共− j兲N−nt,−h ⬘共1兲 共 j兲 + Vn,h共− j兲N−nz,−h ⬘共1兲 共 j兲. Mn,h共− j兲P−n,−h
The vector functions P−n,−h ⬘ 共− j兲, Qn,h共 j兲, Un,h共 j兲, and Vn,h共 j兲 in Eq. 共51兲 are given, respectively, by
⬘ 共− j兲 = 1M−n,−h ⬘ 共− j兲 + 2N−nt,−h ⬘ 共− j兲 P−n,−h
冎
共51兲
Green’s function and the scattering Green’s functions due to the presence of the dielectric boundaries. Such problems have never been studied to our knowledge and the generalized results of DGFs with the application in left-handed materials 共which will be discussed later兲 are reported.
⬘ + 3N−nz,−h 共− j兲,
共52兲
Qn,h共 j兲 = 2Nnt,h共 j兲 + 3Nnz,h共 j兲,
共53兲
III. FORMULATION OF SCATTERING DYADIC GREEN’S FUNCTIONS
Un,h共 j兲 = 5Nnt,h共 j兲 + 4Nnz,h共 j兲,
共54兲
Vn,h共 j兲 = 4Nnt,h共 j兲 + 7Nnz,h共 j兲.
共55兲
In this section, we extend our theoretical analysis to derive scattering DGFs for the fth region assuming that the current source is located in the sth layer. As such that the scattering DGFs have a similar form as the unbounded DGF as given in Eq. 共51兲, the expression for the scattering DGFs for each region of the layered gyrotropic Faraday chiral media can be constructed as follows:
Although the present results are more generalized and complicated, the present form of the DGFs expanded in terms of cylindrical vector wave functions is believed to be a correct one due to the rigorous formulation and multiple checks of the mathematical derivation processes. It can also be verified by reducing this generalized form to those forms for materials of simpler tensor forms such as gyroelectric and isotropic media. Also, this paper will contribute to both the unbounded
¯ , ¯ 共fs兲 = G ¯ +G G 1 2 es ¯ 共j = 1 , 2兲 are given by Eq. 共57兲, where the dyadics G j
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QIU et al.
¯ = i G j 4
冕
⬁
0
⬁
␣zs共2 − ␦0n兲共− 1兲 j+1 共1兲 f 1 fs 共1兲 共1兲 f N fs s N ⬘ 共− sj 兲 + 共1 − ␦sN兲BM jP⬘−n,−h共 j 兲兴 + 共1 − ␦ f 兲Qn,h共 j 兲 2 2 2 ⫻ 兵共1 − ␦ f 兲Mn,h共 j 兲关共1 − ␦s 兲A M jP−n,−h n=0 ⑀zs␣ts共k1s − k2s兲 js
dh 兺
共1兲 共1兲 f 共1兲 fs fs fs fs ⬘ 共− sj 兲 + 共1 − ␦sN兲BQj ⬘ 共− sj 兲 + 共1 − ␦sN兲BUj M−n,−h M⬘−n,−h 共sj 兲兴 + 共1 − ␦Nf 兲Un,h 共 j 兲关共1 − ␦s1兲AUj N−nt,−h N⬘−nt,−h ⫻关共1 − ␦s1兲AQj 共1兲 f 共1兲 fs fs fs ⬘ ⬘ 共− sj 兲 ⫻共sj 兲兴 + 共1 − ␦Nf 兲Vn,h 共 j 兲关共1 − ␦s1兲AVj N−nz,−h 共− sj 兲 + 共1 − ␦sN兲BVj N⬘−nz,−h 共sj 兲兴 + 共1 − ␦1f 兲Mn,h共− fj 兲关共1 − ␦s1兲C M jP−n,−h 共1兲 1 1 共1兲 fs s f fs fs ⬘ 共− sj 兲 + 共1 − ␦sN兲DQj + 共1 − ␦sN兲D M M⬘−n,−h 共sj 兲兴 + 共1 − ␦1f 兲Un,h共− fj 兲关共1 jP⬘−n,−h共 j 兲兴 + 共1 − ␦ f 兲Qn,h共− j 兲关共1 − ␦s 兲CQjM−n,−h 共1兲 共1兲 fs fs fs fs ⬘ 共− sj 兲 + 共1 − ␦sN兲DUj ⬘ N⬘−nz,−h 共sj 兲兴其, − ␦s1兲CUj N−nt,−h N⬘−nt,−h 共sj 兲兴 + 共1 − ␦1f 兲Vn,h共− fj 兲关共1 − ␦s1兲CVj N−nz,−h 共− sj 兲 + 共1 − ␦sN兲DVj
共57兲
where multiple transmissions and reflections have been taken into account, jf = 冑k2jf − h2 and the subscript f means the fth region. The ABCD coefficients where the superscripts and subscripts have been suppressed for compactness are scattering DGF coefficients to be determined from the boundary conditions. By considering the multiple transmissions and reflections, the scattering DGFs are thus constructed physically by inspecting Eq. 共57兲 and taking into account all the possible physical modes in the presence of the multiple interfaces as shown in Fig. 1. For instance, if the source is located in the first/last region 共i.e., 1 − ␦s1 = 0 / 1 − ␦sN = 0兲, the wavelets in the scattering DGFs are only excited by inward-coming/outward-going ⬘共1兲 共 js兲, M−n,−h ⬘共1兲 共 js兲, wavelets with excitation functions 关P−n,−h 共1兲 共1兲 ⬘ 共 js兲, and N−nz,−h ⬘ 共 js兲兴 / 关P−n,−h N−nt,−h ⬘ 共− js兲, M−n,−h ⬘ 共− js兲, N−nt,−h 共− js兲兴. When the source point is ⬘ 共− js兲, and N−nz,−h ⬘ located in any other layer, the excitation functions consist of both outward-going and inward-coming wavelets. If the observation point is in the first/last region 共i.e., 1 − ␦1f = 0 / 1 − ␦Nf = 0兲, the field terms consist of only outward-going/ inward-coming wavelets. The layered structure is shown in Fig. 1. Based on the principle of scattering superposition, we have ¯ 共fs兲共r,r⬘兲 = G ¯ 共r,r⬘兲␦s + G ¯ 共fs兲共r,r⬘兲, G 0 e f s
A. Recurrence matrix equation of DGFs’ scattering coefficients
To simplify the derivation of the general solution of these coefficients, we rewrite the boundary conditions in Eq. 共3兲 into the matrix form subsequently. Now, it is clear that the equations to be obtained here for the layered Faraday chiral medium are different from those in all the previous work. By using the boundary conditions, a set of linear equations satisfied by scattering coefficients can be obtained and then represented by a series of compact matrices as follows: 关Flj共f+1兲兴 · 兵关⌼lj共f+1兲s兴 + ␦sf+1关U共f+1兲兴其
共61兲
=关Fljf 兴 · 兵关⌼ljfs兴 + ␦sf 关D f 兴其. The intermediate matrices in 共61兲 are defined as follows: 关F M jf 兴 =
冋
ប j WM1 ប j WM2
册
T
,
共62兲
共58兲
¯ and G ¯ denote the total and unbounded electric where G e 0 DGFs, respectively, and superscripts f and s, respectively, denote the field point located in the fth region and source located in the sth region. The boundary conditions that must be satisfied by the dyadic Green’s function at the interface of regions f and f + 1 at = f 共f = 1 , 2 , . . . , N − 1兲 are shown as follows: ¯ 共fs兲共r,r⬘兲 = ˆ ⫻ G ¯ 关共f+1兲s兴共r,r⬘兲, ˆ ⫻ G e e ¯ 共fs兲 ¯ 共fs兲共r,r⬘兲 − G ˆ ⫻ 关␣ f · ⫻ G cf e 共r,r⬘兲兴 e
共59兲 共60兲
¯ 关共f+1兲s兴共r,r⬘兲 − ¯ 关共f+1兲s兴共r,r⬘兲兴. = ˆ ⫻ 关␣ f+1 · ⫻ G c共f+1兲Ge e
FIG. 1. geometry of cylindrically layered gyrotropic Faraday chiral media. 115110-6
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关FLjf 兴 =
where
冉 冉
冤
冉 冉
冊 冊
pjf nh បj WL1 + qjf 2jf f kjf pjf nh jj WL2 + qjf 2jf f kjf
冥
T
关U f 兴 = ,
冊 冊
共63兲 关D f 兴 =
共64兲
␣tf nh + ␣zf 2jf j j − 共cf + ih␣af 兲j j , f
共65兲
p ␣tf ប j + WL1 = ⌬qjf
− cf
冉
k2jf
冊 冊 冊 冊
冉
冉
ih2jf k2jf
hn pjf 2jf qjf + jj . kjf f kjf
h2共 pjf − qjf 兲 + k2jf qjf . kjf
d关H共1兲 n 共 jf 兲兴 d
冏
共76兲
关⌼lj,共f+1兲s兴 = 关Tljf 兴 · 兵关⌼lj,fs兴 + ␦sf 关D f 兴其 − ␦sf+1关U共f+1兲兴.
关TKlj 兴2⫻2 = 关Tlj,N−1兴关Tlj,N−2兴 ¯ 关Tlj,K+1兴关Tlj,K兴
共66兲
=
冋
TKlj,11 TKlj,12 TKlj,21 TKlj,22
册
共78兲
.
It should be noted that the coefficient matrices of the first and the last regions have the following specific forms 共67兲
共68兲
共69兲
冏
, = f
j j = Jn共 jf f 兲,
j j =
共75兲
冋 冋
关⌼lj,1s兴 =
关⌼lj,Ns兴 =
1s A1s lj Blj
0
0
0
0
CNs lj
DNs lj
册 册
共79兲
共80兲
.
Then we utilize the previously obtained recursive formula to obtain all the coefficients of Aljfs, Bljfs, Cljfs, and Dljfs. B. Specific applications: Three cases
ប j = H共1兲 n 共 jf f 兲,
冏
.
where 关Flj,共f+1兲f 兴−1 is the inverse matrix of 关Flj,共f+1兲f 兴, we rewrite the linear equation into
For simplicity, we define
ប j =
0 1
共74兲
关Tljf 兴 = 关Flj,共f+1兲f 兴−1 · 关Flj,f f 兴,
As in the matrices 关FLjf 兴, the subscript L denotes Q, U, or V, which come in pairs with ⌬23jf , ⌬45jf or ⌬47jf , respectively, with the definition of p ⌬qjf =
0 0
,
To shorten the expression, we also define
in p ⌬ ␣af j j f qjf
−
0 0
共77兲
in p ⌬ ␣af ប j f qjf
−
hn pjf 2jf qjf + បj , kjf f kjf
p ␣tf j j + WL2 = ⌬qjf
− cf
冉
ih2jf
1 0
Defining the following transmission T matrix
␣tf nh + ␣zf 2jf ប j − 共cf + ih␣af 兲ប j , W M1 = f W M2 =
冋 册 冋 册
d关Jn共 jf 兲兴 d
冏
共70兲 共71兲
. = f
关⌼lj,fs兴 =
冋
Bljfs Dljfs
册
,
1. Source in an intermediate layer
共72兲
The terms 2jf , 3jf , 4jf , 5jf , and 7jf are the weighting factors associated with the scattering coefficients Aljfs and Bljfs where l = M, Q, U, or V. They have the same forms as those in Eqs. 共2兲 and 共50兲 with the only change that each term relating to wave numbers 共e.g., 兲 will have a subscript of jf 共e.g., jf 兲 and each term relating to material parameters 共e.g., ⑀z兲 will have a subscript of f 共e.g., ⑀zf 兲 where j = 1, 2 and f represents the fth region. The following matrices are also used in the formulation: Aljfs Cljfs
To gain insight into the specific mathematical expressions of the physical quantities such as the transmission and reflection coefficient matrices, the following three cases are considered subsequently to demonstrate how these coefficients are determined by using the recursive algorithm when the source point is located in the first, the intermediate, and the last regions.
共73兲
关⌼lj,1s兴 =
关⌼lj,ms兴 =
关⌼lj,Ns兴 =
冋 冋
冋
册 册 册
1s A1s lj Blj
0
0
,
Bms Ams lj lj
Cms Dms lj lj 0
0
CNs lj
DNs lj
.
共81兲
共82兲
共83兲
From Eq. 共78兲, the recurrence equation becomes 关⌼lj,fs兴 = 关Tlj,f−1兴 ¯ 关Tlj,s兴兵关Tlj,s−1兴 ¯ 关Tlj,1兴关⌼lj,1s兴 u共f − s − 1兲关Ds兴 − u共f − s兲关Us兴其,
共84兲
where u共x − x0兲 denotes the unit step function. When f = N,
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PHYSICAL REVIEW B 74, 115110 共2006兲
QIU et al.
the coefficients for the first region are given by: A1s lj
=
T共s兲 lj,11 T共1兲 lj,11
B1s lj = −
关⌼lj,1N兴 = 共85兲
,
关⌼lj,mN兴 =
T共s兲 lj,12 . T共1兲 lj,11
共86兲 关⌼lj,NN兴 =
For the last region, the coefficients are given by CNs lj
=
1s T共1兲 lj,21Alj
−
T共s兲 lj,21,
DNs lj
=
1s T共1兲 lj,21Blj
+
T共s兲 lj,22 .
共87兲
Substituting Eqs. 共85兲 to 共87兲 into Eq. 共84兲, the remaining coefficients can be obtained for the dyadic Green’s functions. If the source is located in the first or last region 共i.e., s = 1 or N兲, the formulation of coefficients can be tremendously simplified.
关⌼lj,11兴 =
关⌼lj,m1兴 =
关⌼lj,N1兴 =
冋 册 冋 册 冋 册 0
B11 lj
0
0
0 Bm1 lj
0 Dm1 lj
0
0
0 DN1 lj
共88兲
,
,
共89兲
,
共90兲
where m = 2 , 3 , . . . , N − 1. It can be seen that only four coefficients for the first region and the last region, but eight coefficients for each of the remaining regions or layers, need to be solved for. By following Eq. 共77兲, the recurrence relation for coeficients in the fth layer becomes 关⌼lj,f1兴 = 关Tlj,f−1兴 ¯ 关Tlj,1兴兵关⌼lj,11兴 + 关D1兴其.
共91兲
When f = N in Eq. 共91兲, a matrix equation satisfied by the coefficient matrices in Eqs. 共88兲–共90兲 can be obtained. The coefficients for the first region where the source is located 共i.e., s = 1兲 is given by: B11 lj = −
T共1兲 lj,12 T共1兲 lj,11
.
共92兲
The coefficients for the last region can be derived in terms of the coefficients for the first region given by: 共1兲 11 共1兲 DN1 lj = Tlj,21Blj + Tlj,22 .
共93兲
The coefficients for the intermediate layers can be then obtained by substituting the coefficients in Eqs. 共92兲 and 共93兲 to Eq. 共91兲. Thus, all the coefficients can be obtained by these procedures. 3. Source in the last region
When the current source is located in the first region 共i.e., s = N兲, the coefficients are:
0
0
0 AmN lj
0 CmN lj 0
0
CNN lj
0
,
共94兲
,
共95兲
.
共96兲
From the recurrence equation in Eq. 共77兲, similarly we have 关⌼lj,fN兴 = 关Tlj,f−1兴 ¯ 关Tlj,1兴关⌼lj,1N兴 − u共f − N兲关UN兴. 共97兲 By letting s = N, the coefficient for the first region is
2. Source in the first region
When the current source is located in the first region 共i.e., s = 1兲, the terms containing 共1 − ␦s1兲 in Eq. 共57兲 vanishes. The coefficient matrices in 共73兲 and 共79兲 will be further reduced to:
冋 册 冋 册 冋 册 0 A1N lj
A1N lj =
1
. T共1兲 lj,11
共98兲
And for the last region, it is found that 共1兲 1N CNN lj = Tlj,21Alj .
共99兲
Similarly, the rest of the coefficients can be obtained by substituting Eq. 共98兲 and Eq. 共99兲 into Eq. 共97兲. So far, for gyrotropic Faraday chiral media in layered structures, we have obtained a complete set of solutions to the DGFs in terms of the cylindrical vector wave functions and their scattering coefficients in terms of compact matrices. Reduction can be made for formulating the dyadic Green’s functions in less complex media, e.g, an anisotropic medium where c = 0, a bi-isotropic medium where g = w = 0, a gyroelectric medium where w = 0 and = z, a chiroferrite medium where p = 0 and ⑀ = ⑀z, or an isotropic medium where c = g = w = 0, ⑀z = ⑀, and = z. IV. NEGATIVE REFRACTION AND BACKWARD EIGENWAVES INSIDE GYROTROPIC FARADAY CHIRAL MEDIA
The permittivity and permeability considered here are defined in Eq. 共1兲. The generalized media studied in this paper can be reduced to: 共i兲 chiroplasma consisting of chiral objects embedded in a magnetically biased plasma, or 共ii兲 a chiroferrite by immersing chiral objects into magnetically biased ferrite. Assume that wave is given in form by ei共k·r−t兲, and it propagates along the z axis inside the gyrotropic Faraday chiral media. There are two approaches for obtaining the eigenmodes and wave numbers: 共i兲 starting from Eq. 共5兲 by setting J equal to zero and then obtaining the nontrivial solutions of wave numbers, and 共ii兲 starting from Eq. 共1兲 directly, listing all the relations of the field components and then solving the final equation consisting of wave numbers and parameters only. Here we choose the second method, which is less cumbersome and gives more insight to physical properties of the electromagnetic waves inside the media. Consider Eq. 共1兲 and
115110-8
⫻ E = iB,
共100兲
PHYSICAL REVIEW B 74, 115110 共2006兲
PROPERTIES OF FARADAY CHIRAL MEDIA: GREEN¼ TABLE I. Helicity and polarization states of k p− and ka− in three cases, assuming −␣t ⬍ ⬍ ␣t and c ⬎ 0. 关b兴 −⑀ ⬍ g ⬍ ⑀
g ⬍ −⑀ HEL k p− ka−
POL LCP RCPa
両 丣a
HEL 両 丣
POL LCP LCP
g⬎⑀ HEL 両a 丣
POL RCPa LCP
means that a backward wave can propagate in such a medium兲 if the conditions in Table I are maintained. It also shows that even when the chirality is small, a negative index of refraction may be easily achieved. By considering the Poynting’s vectors and the Maxwell equations, we will finally arrive at two impedances 1 共for ka−兲 and 2 共for k p−兲, which are given as follows:
1 =
a
Backward wave regimes.
⫻ H = − iD.
共101兲
Since the electromagnetic fields, E and H, have only transverse components and the parameters are in gyrotropic form, Dz and Bz vanish. We will finally arrive at two equations where lengthy intermediate procedures have been suppressed
冋 册 冋 冋 册 冋册 冋
册 册
⑀Ex − igEy = , − kz共Hx + icEx兲 ⑀Ey + igEx
共102兲
Hx Ex k z − ␣ tE y − ␣ aE x = ic + . Hy Ey − ␣ aE y + ␣ tE x
共103兲
kz共Hy + icEy兲
Thus one set of equations can be obtained, and it contains only transverse electric components
冉 冉
冊 冉 冊 冉
冊 冊
kz kz ␣t − ⑀ Ey = 2ic + i g − ␣a Ex , kz kz
共104兲
kz kz 2ic + i g − ␣a Ey = − ␣t + ⑀ Ex . kz kz
共105兲
Hence, we can obtain four wave numbers for the eigenwaves propagating along the z axis as follows: ± c + 冑2c + 共␣t ± 兲共⑀ ± g兲 , ␣t ±
共106兲
⫿ c − 冑2c + 共␣t ⫿ 兲共⑀ ⫿ g兲 , ␣t ⫿
共107兲
k p± =
ka± =
where p and a denote the parallel and antiparallel directions of the real part of Poynting’s vector and = i␣a = w / 共2 − w2兲, while plus and minus signs refer to as the right- and left-circularly polarized 共RCP and LCP兲 forwarding waves, respectively. k p− and ka− are of particular interest since they will represent the properties of backward waves under specific cases as shown in Table I. The realization of negative refraction is of particular interest. Taking into account Eq. 共4兲 and respective polarization states, we can finally obtain two refraction indices for those backward eigenwaves: n± =
1
冑2c + 共␣t − 兲共⑀ + g兲 =
c0 关冑2c + 共␣t ± 兲共⑀ ± g兲 − c兴, 共␣t ± 兲
2 =
冑
2c
⑀+g + +w 1
2c
⑀−g + −w
,
共109兲
.
共110兲
Hence, when a Faraday chiral medium is employed to fabricate a perfect lens, impedance matching should be carefully carried out at the air-material interfaces. Within certain frequency bands, there are two backward waves. Therefore, each time only the one whose impedance is matched can be transmitted into the slab and double focusing effect can take place. Most of the other backward waves would be reflected due to the mismatch at the interface. Note that g ⬎ ⑀ can be realized with some modern technology in the future based on the theory of off-diagonal parameter amplification in gyrotropic media.37 Hence, negative refractive media can be realized even when all the parameters in material tensors are positive. It is possible to achieve a nearly zero-index material n− ⬇ 0 by requring ⑀ − g ⬇ 0 while the material still possesses a positive wave impedance 1 / c. Such a material has wide potential applications in airborne radome design, and highdirectivity antenna design since the phase of the propagating waves will keep unchanged inside this material. A zero-index or nearly zero-index medium provides potentials in quantum devices because the discrete quantized field will be greatly enhanced. The electric field strength with respect to nk photons of k mode can be expressed Ek =
冑
共nk + 1/2兲ប V, n d共n兲 d
共111兲
where ប is the Dirac constant and V stands for the volume of the medium. For a single photon, we have the critical field strength Ec =
共108兲
where plus and minus signs refer to as ka− and k p−, respectively. It can be seen that n+ will be negative when g ⬍ −⑀ and when n− will possess a minus sign when g ⬎ ⑀ 共which
1
冑2c + 共␣t + 兲共⑀ − g兲 =
冑
1
冑
3ប V. n d共n兲 2 d
共112兲
If the field strength has the same order of magnitude of Ec or less than Ec, the field can be viewed as a quantized one or a fluctuation of quantum vacuum. It is obvious that the critical field strength becomes very large when the refractive index
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is almost zero. Hence, the quantum vacuum fluctuation field becomes strong. In summary, a gyrotropic Faraday chiral medium provides us a very exciting opportunity to realize negative refraction, backward wave propagation, and related quantum effects.
V. CONCLUSIONS
In this paper, we studied some important electromagnetic properties of gyrotropic Faraday chiral media due to their significant potentials for metamaterials and interactions between electromagnetic waves and materials. Green dyadics, backward wave propagation, and negative refractive index associated with gyrotropic Faraday media have been well
*Email address:
[email protected]; http://www.ece.nus.edu.sg/lwli 1 J.
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investigated in this paper. The unbounded DGF for a gyrotropic Faraday chiral medium is formulated by use of the Ohm-Rayleigh method and cylindrical vector wave function expansion. Scattering DGFs for cylindrically stratified structures 共where each layer is an arbitrarily characterized gyrotropic Faraday chiral medium兲 are constructed and a recursive algorithm is proposed to obtain those scattering coefficients based on the boundary conditions matched at each interface. Those formulations can be reduced in form to the counterparts for anisotropic, chiroplasma, chiral, and isotropic media. In what follows, the eigenmodes and wave numbers in a gyrotropic Faraday chiral medium are studied. Finally, the properties of backward propagating eigenmodes and realization of negative refraction are well explored and discussed.
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