# A duality between metamaterials and conventional materials in multilayered anisotropic planar structures.

1. INTRODUCTION

Metamaterials are materials in which the real parts of the components of permittivity or/and permeability tensors can be negative. They may be classified as epsilon negative (ENG), mu negative (MNG) and double negative (DNG). Conventional materials are called double positive (DPS).

The structure, taken into consideration here, is a multilayered anisotropic planar structure composed of conventional materials and metamaterials and situated between two half spaces. As shown in Figure 1, a plane wave is obliquely incident on the structure. In recent years, numerous methods have been employed for analyzing this structure [1-12].

Initial works on the subject were based on 4 x 4 characteristic matrix of a single anisotropic slab [1,2]. Later efforts include generalization of the problem for stratified structures by different methods [3-5]. Morgan et al. paid attention to a numerical solution, and introduced an efficient and simple algorithm for this case [6]. Others proposed various techniques based on eigenvalue computation, Ricatti differential equation, and transmission line method [7-11], which are more complex. The characteristic matrix algorithm [1, 2] had a serious drawback and showed instability for thick layers compared to the wavelength. To avoid this instability which was due to the numerical finite difference algorithm, the use of hybrid matrix of the structure is suggested [12].

In this paper, we employ a full-wave matrix method, whereby the domain is decomposed into forward and backward travelling waves [13,14]. We prove three theorems for the reflection and transmission coefficients of incident plane waves onto Figure 1.

In [18] these duality theorems are proved in the special case of isotropic structures, limiting application since most metamaterials are anisotropic. In this paper we expand the application scope of these theorems to include anisotropic media as well as isotropic ones.

The purposed theorems are given a mathematical proof and indicate a duality for the propagation and reflection of electromagnetic waves in dual media of DPS [left right arrow] DNG and ENG [left right arrow] MNG. Independent of wave frequency, the number of layers, their thickness, and the type of polarization hold true in case of any change in any of these conditions.

At last, the purposed theorems are validated through some examples solved using state space method.

2. PROBLEM CONFIGURATION

Consider a multilayered anisotropic planar structure in which layer l is a layer with thickness [d.sub.l] and the anisotropic constitutive parameters below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Supposing [e.sup.+j[omega]t], the solution of the wave equation in layer l, which would be some of two waves with orthogonal TE and TM polarizations, is thus [13, 14]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where l= 0,1,.. .N+1, and superscripts I and II relate to TE and TM polarizations, respectively. [A.sub.l] and [C.sub.l] are the amplitudes of forward and backward traveling TE waves; Bl and [D.sub.l] are the amplitudes of forward and backward traveling TM waves; in the above relations, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is obtained from phases matching and is identical for all layers.

From the dispersion relation [13] for each layer, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

In the region l = 0, we have the following expressions:

[A.sub.0] = [E.sub.0] (5)

[B.sub.0] = [H.sub.0] (6)

[C.sub.0] = [R.sub.TE] * [E.sub.0] (7)

[D.sub.0] = [R.sub.TM] * [H.sub.0] (8)

In which [R.sub.TE] and [R.sub.TM] are reflection coefficients for TE and TM polarizations in the left most half space, respectively. And in the region l = N + 1, we have:

[A.sub.N+1] = [T.sub.TE] * [E.sub.0] (9)

[B.sub.N+1] = [T.sub.TM] * [H.sub.0] (10)

[C.sub.N+1] = 0 (11)

[D.sub.N+1] = 0 (12)

In which [T.sub.TE] and [T.sub.TM] are transmission coefficients for TE and TM polarizations in the right most half-space, respectively.

The boundary conditions at the boundaries of the layers may be written as a matrix equation:

MX=Y (13)

In which the M, X and Y are matrix of coefficients, unknowns' vector and sources' vector, respectively and defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Y = [[-[E.sub.0] -[E.sub.0] -[H.sub.0] -[H.sub.0] 00 ... 00000000 ... 00000 0].sup.T] (16)

where [[phi]] is the null matrix and:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

in which:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

3. PROOF OF THE THEOREMS

3.1. Theorem 1

Consider a multilayered anisotropic planar structure made of a combination of common materials and metamaterials situated between two half spaces composed of lossless media. Now all layers including two half spaces are filled by their dual media according to the interchanges DPS [left right arrow] DNG and ENG [left right arrow] MNG. Then, the reflection (R) and transmission (T) coefficients from the structure become the complex conjugates of their counterparts. Consequently, the reflected power and transmitted power from the structure are the same for the two dual cases.

3.2. Proof of Theorem 1

If we apply the interchanges DPS - DNG and ENG - MNG in all

layers including two half spaces, we will have:

[[??].sup.lNew] = -[[??].sup.l] (22)

[[??].sup.lNew] = -[[??].sup.l] (23)

And according to relation (4) and the rules mentioned in [14-17] for choosing the correct sign of wave number, we will have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Consequently, these changes will be made in the components of the matrix of coefficients:

[p.sup.I.sub.lNew] = (-[[mu].sup.*.sub.lx])/(-[k.sup.I*.sub.zl]) * (-[k.sup.I.sub.z(l+1)]*)/(-[mu](l+1)[x.sup.*]) = [p.sup.I*.sub.l] (28)

[p.sup.II.sub.lNew] = (-[[epsilon].sup.*.sub.lx])/(-[k.sup.II*.sub.zl]) * (-[k.sup.I.sub.z(l+1)][I.sup.*])/(-[epsilon]](l+1)[x.sup.*]) = [p.sup.II*.sub.l] (29)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

It is observed that with the applied interchanges in constitutive parameters of the structure, the entire components of the matrix of coefficients become complex conjugates. As [E.sub.0] and [H.sub.0] are real, the components of vector of sources are equal with their complex conjugates. Consequently, the reflection and transmission coefficients of the new structure become complex conjugates of the old ones.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

Therefore:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

As the reflected and transmitted powers are equal to [P.sub.r] = RR* = [[absolute value of (R)].sup.2] and [P.sub.t] = TT* = [[absolute value of (T)].sup.2] respectively, they are identical for a multilayer anisotropic planar structure and its dual structure.

3.3. Theorem 2

If the interchanges DPS [left right arrow] DNG and ENG [left right arrow] MNG are made in all the layers except in the half spaces on the two sides of the multilayer structure (which is more realizable), then the reflection coefficients become complex conjugates and the reflected power remains the same.

3.4. Proof of Theorem 2

This theorem may be proved by consider the transmission line equivalent circuits of the multilayered anisotropic planar structure corresponding to TE and TM waves shown in Figure 2. Right and left half spaces are replaced by impedances equal to their characteristic impedances.

The ABCD matrixes of lth layer corresponding to TE and TM waves are given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

where [k.sup.I.sub.l] and [k.sup.II.sub.l] may be obtained from (4). [[eta].sup.I.sub.l] and [[eta].sup.II.sub.l] are characteristic impedances of lth layer corresponding to TE and TM waves and are given by [19]:

[[eta].sup.I.sub.l] = [square root of ([[mu].sup.lx]/[[epsilon].sub.lx][[epsilon].sub.lz]/ ([[epsilon].sub.lz][cos.sup.2][[theta].sub.0]+[[epsilon].sub.lx][sin.sup.2][[theta].sub.0])] (36)

[[eta].sup.II.sub.l] = [square root of ([[mu].sub.lx][[mu].sub.lz]/([[mu].sub.lz][cos.sup.2][[theta].sub.0] +[[mu].sub.lx][sin.sup.2][[theta].sub.0])/[[epsilon].sub.lx]] (37)

According to relations (22)-(27) and (34)-(37), if layer l is filled by its dual layer, its dual ABCD matrixes corresponding to TE and TM waves will become equal to the complex conjugates of the former ABCD matrixes.

Consequently, the overall ABCD matrixes of the dual multilayer planar structure (as the product of the individual line sections) corresponding to TE and TM waves will become equal to the complex conjugates of those of the original structure.

The input impedance and the reflection coefficient at the input port of TE and TM transmission lines are:

[Z.sup.I.sub.in] = [A.sup.I][Z.sub.L]+[B.sup.I]/[C.sup.I][Z.sub.L]+[D.sup.I], [R.sup.TE] = [Z.sup.I.sub.in]-[[eta].sup.0]/[Z.sup.I.sub.in]+[[eta].sub.0] (38)

[Z.sup.II.sub.in] = [A.sup.II][Z.sub.L]+[B.sup.II]/[C.sup.II][Z.sub.L]+[D.sup.II], [R.sup.TM] = [Z.sup.II.sub.in]-[[eta].sup.0]/[Z.sup.II.sub.in]+[[eta].sub.0] (39)

where [[eta].sub.0] is the intrinsic impedance of the left half space. Considering the fact that the elements of ABCD matrixes of the two dual structures are complex conjugates of each other and that [[eta].sub.0] is real, it can be deduced:

[([R.sub.TE]).sub.new] = [([R.sub.TE]).sup.*.sub.old], [([R.sub.TM]).sub.new] = [([R.sub.TM]).sup.*.sub.old], (40)

3.5. Theorem 3

If the structure is backed by a perfect electric conductor and the media interchanges DPS [left right arrow] DNG, and ENG [left right arrow] MNG are made in the layers, then the reflection coefficients of the two dual structures become complex conjugates of each other, and the reflected powers are equal.

3.6. Proof of Theorem 3

This theorem may also be deduced in a similar way with theorem 2. In this case, ZL = 0. Thus (38) and (39) reduce to:

[Z.sup.I.sub.in] = [B.sup.I]/[D.sup.I], [R.sup.TE] = [[Z.sup.I.sub.in]-[[eta].sub.0]]/[[Z.sup.I.sub.in]+[[eta].sub.0]] (41)

[Z.sup.II.sub.in] = [B.sup.II]/[D.sup.II], [R.sup.TM] = [[Z.sup.II.sub.in]-[[eta].sub.0]]/[[Z.sup.II.sub.in]+[[eta].sub.0]] (42)

and again with the same explanations (40) is deduced.

4. NUMERICAL EXAMPLES

4.1. Example 1

We consider a slab with the parameters shown in the Table 1. The reflection and transmission coefficients of this slab and its dual medium are presented in Figures 3 and 4 versus frequency for a normal incidence. These curves are drawn using Matlab code based on the state space method [20, 21].

4.2. Example 2

Now consider a bilayer structure with the parameters shown in Table 2 backed by a PEC. The reflection coefficients of the structure and its dual are represented in Figure 5 versus frequency for a normal incidence.

Excellent agreement is observed between the curves, which confirm the validity of our theorems.

5. CONCLUSIONS

Several theorems have been proved in this paper, for an incidence of plane waves on a multilayered anisotropic planar structure. These theorems indicate a kind of duality for the reflection and transmission between two dual multilayered anisotropic structures with interchanges DPS [left right arrow] DNG or ENG [left right arrow] MNG. Moreover, independent of wave frequency, the number of layers, their thickness, and the type of polarization the theorems hold true in case of any change in any of these conditions.

Each theorem was given a mathematical proof and at last, the validity of the theorems has been verified by some examples solved using the state space method.

Received 11 April 2012, Accepted 6 July 2013, Scheduled 15 July 2013

REFERENCES

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[3.] Barkovskii, L. M. and G. N. Borzdov, "Electromagnetic waves in absorbing plane-layered anisotropic and gyrotropic media," J. Appl. Spectrosc., Vol. 23, 985-991, Sept. 1976.

[4.] Barkovskii, L. M. and G. N. Borzdov, "Reflection of electromagnetic waves from layered continuously inhomogeneous anisotropic media: Multiple reflection method," Opt. Spectrosc. (USSR), Vol. 45, 701-705, Oct. 1978.

[5.] Graglia, R. D. and P. L. E. Uslenghi, "Electromagnetic scattering from anisotropic materials, part I: General theory," IEEE Trans. Antennas Propagat., Vol. 32, 867-869, Aug. 1984.

[6.] Morgan, M. A., et al., "Electromagnetic scattering by stratified inhomogeneous anisotropic media," IEEE Trans. Antennas Propagat. , Vol. 35, 191-198, Feb. 1987.

[7.] Graglia, R. D., et al., "Reflection and transmission for planar structures of bianisotropic media," Electromagnetics, Vol. 11, 193-208, 1991.

[8.] Titchener, J. B. and J. R. Willis, "The reflection of electromagnetic waves from stratified anisotropic media," IEEE Trans. Antennas Propagat., Vol. 39, 35-40, Jan. 1991.

[9.] Tsalamengas, J. L., "Interaction of electromagnetic waves with general bianisotropic slabs," IEEE Trans. Microwave Therory Tech., Vol. 40, 1870-1879, Oct. 1992.

[10.] Lindell, I. V., et al., "Vector Transmission-line and circuit theory for bi-isotropic layered structures," Journal of Electromagnetic Waves and Applications, Vol. 7, 147-173, 1993.

[11.] Yang, H. D., "A spectral recursive transformation method for electromagnetic waves in generalized anisotropic layered media," IEEE Trans. Antennas Propagat., Vol. 45, 520-527, Mar. 1997.

[12.] Ning, J. and E. L. Tan, "Hybrid matrix method for stable analysis of electromagnetic waves in stratified bianisotropic media," IEEE Microwave Wireless Comp. Lett. , Vol. 18, 653-656, Oct. 2008.

[13.] Kong, J. A., "Theory of Electromagnetic Waves, EMW Publishing, 2005.

[14.] Oraizi, H. and A. Abdolali, "Design and optimization of planar multilayer antireflection meta-material coatings at Ku band under circularly polarized oblique plane wave incidence," Progress In Electromagnetics Research C, Vol. 3, 1-18, 2008.

[15.] Ziolkowski, R. W. and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E, tat. Phys. Plasmas Fluids Relat. , Vol. 64, No. 5, 617-625, 2001.

[16.] Kong, J. A., "Electromagnetic wave interaction with stratified negative isotropic media," Progress In Electromagnetics Research, Vol. 35, 1-52, 2002.

[17.] Oraizi, H. and A. Abdolali, "Mathematical formulation for zero reflection from multilayer metamaterial structures and their notable applications," IET Microwaves, Antennas & Propagation, Vol. 3, No. 6, 987-996, Sep. 2009.

[18.] Oraizi, H. and A. Abdolali, "Several theorems for reflection and transmission coefficients of plane wave incidence on planar multilayer metamaterial structures," IET Microwaves, Antennas & Propagation Journal, Vol. 4 , No. 11, 1870-1879, Nov. 2010.

[19.] Su, H. L. and K. H. Lin, "Design of an anisotropic quarter-wave polariser without insertion loss caused by mismatch," IEE Proc.Microw. Antennas Propag., Vol. 153, No. 3, 253-258, Jun. 2006.

[20.] Yang, H. D., "A spectral recursive transformation method for electromagnetic waves in generalized anisotropic layered media," IEEE Trans. Antennas Propagat., Vol. 45, 520-527, Mar. 1997.

[21.] Ning, J. and E. L. Tan, "Hybrid matrix method for stable analysis of electromagnetic waves in stratified bianisotropic media," IEEE Microwave Wireless Comp. Lett., Vol. 18, 653-656, Oct. 2008.

Maryam Heidary *, Ali Abdolali, Mohammad M. Salary, and Hossein Mirzaei

Electrical Engineering Department, Iran University of Science and Technology, Iran

* Corresponding author: Maryam Heidary (heidary_m@elec.iust.ac.ir).
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Table 1. Parameters of the slab under consideration.

Electric permittivity   Magnetic permeability   Layer thickness (m)

[MATHEMATICAL           [MATHEMATICAL
EXPRESSION NOT          EXPRESSION NOT          t = 0.001
REPRODUCIBLE            REPRODUCIBLE
IN ASCII]               IN ASCII]

Table 2. Parameters of the structure under consideration.

Layer       Electric         Magnetic              Layer
permittivity     permeability        Thickness (m)

1    [MATHEMATICAL       [MATHEMATICAL      [t.sub.1] = 0.001
EXPRESSION NOT      EXPRESSION NOT
REPRODUCIBLE        REPRODUCIBLE
IN ASCII]           IN ASCII]

2    [MATHEMATICAL       [MATHEMATICAL      [t.sub.2] = 0.001
EXPRESSION NOT      EXPRESSION NOT
REPRODUCIBLE        REPRODUCIBLE
IN ASCII]           IN ASCII]
```