Progress In Electromagnetics Research, PIER 85, 381–391, 2008

PLANAR SLAB OF CHIRAL NIHILITYMETAMATERIAL BACKED BY FRACTIONALDUAL/PEMC INTERFACE

Q. A. Naqvi

Department of ElectronicsQuaid-i-Azam UniversityIslamabad, 45320, Pakistan

Abstract—Fields inside the chiral nihility slab which is backed byperfect electric conductor are determined. It is noted that both electricand magnetic fields exist inside the grounded chiral nihility slab whenit is excited by a plane wave. Electric field inside the slab disappears forexcitation due to an electric line source. Magnetic field inside the slabdisappears when geometry changes to corresponding dual geometry.Dual geometry means chiral nihility slab backed by perfect magneticconductor and excited by a magnetic line source. Using fractional curloperator, fields are determined for fractional order geometries whichmay be regarded as intermediate step between the two geometrieswhich are related through principle of duality. Discussion is extendedfor chiral nihility slab which is backed by perfect electromagneticconductor (PEMC).

1. INTRODUCTION

Chiral nihility is a special kind of chiral medium, for which the real partof permittivity and permeability are simultaneously zero or refractiveindex become zero at certain frequency known as nihility frequency [1–3]. In chiral nihility, the two eigenwaves are still circularly polarized butone of them is a backward wave. For backward waves phase velocityis antiparallel to the corresponding Poynting vector. Phenomena ofnegative refraction occurs when a plane wave enters from vacuumto chiral nihility medium. That is, when a plane wave obliquelyhits the interface due to vacuum and chiral nihility, one refractedeigenwave propagates on one side of normal at certain angle whileother eigenwave propagates at same angle on other side of the normalto the interface. Another interesting phenomena of negative reflection

382 Naqvi

of both eigenwaves occurs at the interface between chiral nihility andperfect electric conductor plane [2]. Due to these two phenomenon,both electric field and power flow disappear in particular regions ofplaner waveguide, composed of chiral nihility slabs backed by perfectelectric conductors, when it is excited by an electric line source [3]. Onthe other hand, magnetic field and power flow disappears in particularregions of planer waveguide, composed of chiral nihility slabs backedby perfect magnetic conductors, when it is excited by a magnetic linesource.

Our interest is to study behavior of fields inside and outside chiralnihility slab which is backed by perfect electric conductor. Anothergeometry which is dual to the first geometry has also been considered.Geometry containing chiral nihility slab backed by PEC and excitedby an electric line source and geometry containing chiral nihility slabbacked by PMC and excited by a magnetic line source are dual of eachother. For each geometry uniform plane wave or line source has beenconsidered as a source of excitation. Difference in behavior of fieldsinside grounded chiral nihility slab due to line source excitation andplane wave excitation is noted.

Discussion is further extended to two general geometries, firstdeals with chiral nihility slab backed by fractional dual interface whileother deals with chiral nihility slab backed by perfect electromagneticconductor (PEMC). PEC and PMC become special cases of eachgeneral geometry. Field corresponding to fractional or intermediategeometries between the two dual geometries are studied. Fractionalgeometries have been obtained using fractional curl operator [4].

2. GROUNDED CHIRAL NIHILITY SLAB

Consider a slab of chiral nihility metamaterial. The slab is of infinitelength and is backed by perfect electric conductor (PEC). Front faceof the chiral nihility slab is located at z = d1 while perfect electricconductor is located at location z = d2, where d2 > d1. The chiralnihility slab backed by PEC has been termed as grounded chiral nihilityslab.

A linearly polarized uniform plane wave, with time dependencytime harmonic exp(−jωt), is obliquely incident on the grounded chiralnihility slab. The electric and magnetic fields inside and outside thegrounded chiral nihility slab may be written in terms of unknowncoefficients as [5, 6]

E0 = exp(ikyy)[x exp (ik0zz) + A−N−

R exp(−ik0zz)

+B−N−L exp(−ik0zz)

], z<d1 (1)

Progress In Electromagnetics Research, PIER 85, 2008 383

E1 = exp(ikyy)[E+M+

R exp(ik+z z) + F+M+

L exp(ik−z z)

+E−M−Rexp(−ik+

z z)+F−M−Lexp(−ik−

z z)], d1 <z<d2 (2)

H0 = exp(ikyy)[

1k0η0

{yk0z exp (ik0zz) − zky exp (ik0zz)}

− i

η0

{A−N−

Rexp(−ik0zz)−B−N−L exp(−ik0zz)

}], z<d1 (3)

H1 = exp(ikyy)−i

η

[E+M+

R exp(ik+z z) − F+M+

L exp(ik−z z)

+E−M−R exp(−ik+

z z)−F−M−L exp(−ik−

z z)], d1 <z<d2 (4)

where

N±R = x ± ik0z

k0y − iky

k0z (5)

N±L = x ∓ ik0z

k0y +

iky

k0z (6)

M±R = x ± ik+

z

k+y − iky

k+z = x ± ik±

z

k± y − iky

k± z (7)

M±L = x ∓ ik+

z

k+y +

iky

k+z = x ∓ ik±

z

k± y +iky

k± z (8)

Superscript ± in Equations (5)–(8) represents the eigenwavespropagating in the ±z direction. The subscript R and L refer to theRCP and LCP eigenwaves satisfying the dispersion relations as

k2y + (k±

z )2 = (k±)2

where k± = ±ωκ at the nihility frequency. In above equations,k0 = ω

√µ0ε0, η0 =

√µ0/ε0, and η =

√µ/ε. k0z and ky satisfy the

dispersion relation

k2y + k2

0z = k20

It may be noted that relation k+z = −k−

z holds for all modespropagating inside the slab.

Unknown coefficients in field expressions (1)–(4) may be obtainedusing the boundary conditions. At z = d2, tangential components ofelectric field E1 must be zero. Application of boundary condition to xand y components of electric field at z = d2 and imposing restrictionk+

z = −k−z yields

E± = −F∓ (9)

384 Naqvi

Tangential components of electric and magnetic fields across thedielectric interface located at z = d1 must be continues. Continuity ofx-components and y-components of electric field yields

E− =exp (ik0zd1 + ik+

z d1)2Rf

(10)

E+ = −exp (ik0zd1 − ik+z d1)

2Rf(11)

where

Rf =k0η0k

+z

k+ηk0z

Substitution of unknowns coefficients in expressions (2) and (4) yieldselectric and magnetic fields inside the slab. On the other, if we excitethe grounded slab by a electric line source, it can be shown that electricfield inside the slab disappears. That is

∫ ∞

−∞E1(ky)dky = 0 (12)

Using duality principle, it can be shown that chiral nihility slab backedby PMC does not contain magnetic field at nihility frequency when itis excited by a magnetic line source. That is

∫ ∞

−∞H1(ky)dky = 0

In both geometries, for line source excitation, there is no power insidethe grounded chiral nihility slab.

Fractionalization of a given ordinary operators may be used toexplore the intermediate geometries between the two given geometries.The two given geometries must be connected through the givenordinary operator. Frac-tional curl operator has been used to studyvarious problems [7–16]. A linear operator may be fractionalized usingrecipe given in [4], which dictates that fractionalization of an operatormeans fractionalization of its eigenvalues. Our interest is to note thebehavior of electric and magnetic fields inside the chiral nihility slabfor different values of the order of curl operator. In other words, ourinterest is to see how fields in a geometry changes to fields in the dualgeometry.

Progress In Electromagnetics Research, PIER 85, 2008 385

3. CHIRAL NIHILITY SLAB BACKED BYFRACTIONAL DUAL INTERFACE

Using the concept of fractional curl operator (∇×)α [4], we can writeMaxwell equations for time harmonic fields as

(ki×)E0fd = (η0H0fd) (13)

(ki×)(η0H0fd) = −E0fd (14)

(k±i ×)E1fd = (ηH1fd) (15)

(k±i ×)(ηH1fd) = −E1fd (16)

Subscript fd stands for fractional dual. Fractional dual fields for regioninside and outside the slab may be obtained as

E0fd = (ki×)αη0H0

η0H0fd = (ki×)αE0

E1fd = (k±i ×)αηH1

ηHfd = (k±i ×)αE1

and these fields must satisfy the Maxwell equations.In order to deal with above equations, eigenvalues and eigenvector

of operators (ki×) and (k±i ×) are required. So first we calculate the

eigenvalues and eigenvectors of these cross product operators.Eigenvalues and eigenvectors of operator k1× =

(ky y+k0z z

k0

)×

A11 =1√2

[x + i

k0z

k0y − i

ky

k0z

]= N+

R, a11 = −i

A12 =1√2

[x − i

k0z

k0y + i

ky

k0z

]= N+

L , a12 = +i

A13 = iky

k0y + i

k0z

k0z, a13 = 0

Eigenvalues and eigenvectors of operator k2× =(

ky y−k0z zk0

)× are

A21 =1√2

[x + i

k0z

k0y + i

ky

k0z

]= N−

R, a21 = +i

A22 =1√2

[x − i

k0z

k0y − i

ky

k0z

]= N−

L , a22 = −i

A23 = iky

k0y − i

k0z

k0z, a23 = 0

386 Naqvi

Similarly eigenvalues and eigenvectors of operator k+1 × =

(ky y+k+

z zk0

)×

A+11 =

1√2

[x + i

k+z

k+y − i

ky

k+z

]= M+

R, a+11 = −i

A+12 =

1√2

[x − i

k+z

k+y + i

ky

k+z

]= M−

L , a+12 = +i

A+13 = i

ky

k+y + i

k+z

k+z, a+

13 = 0

Eigenvalues and eigenvector of operator k+2 × =

(ky y−k+

z zk0

)× are

A+21 =

1√2

[x + i

k+z

k+y + i

ky

k+z

]= M−

L , a+21 = i

A+22 =

1√2

[x − i

k+z

k+y − i

ky

k+z

]= M+

R, a+22 = −i

A+23 = i

ky

k+y − i

k+z

k+z, a+

23 = 0

Fractional dual fields are obtained by fractionalizing theeigenvalues of corresponding linear operator as given below

E0fd = exp(ikyy)[(−i)α 1√

2A11 + (+i)α 1√

2A12

]exp (ik0zz)

+[(−i)αA−N−

R + (i)αB−N−L

]exp(ikyy − ik0zz)

E1fd = exp(ikyy)[(−i)αE+M+

R exp(ik+z z) + (i)αF+M+

L exp(ik−z z)

+(−i)αE−M−R exp(−ik+

z z) + (i)αF−M−L exp(−ik−

z z)]

η0H0fd = exp(ikyy)[(−i)α+1 1√

2A11 + (+i)α+1 1√

2A12

]exp (ik0zz)

[+(−i)α+1A−N−

R + (i)α+1B−N−L

]exp(ikyy − ik0zz)

ηH1fd = exp(ikyy)[(−i)α+1E+M+

Rexp(ik+z z)+(i)α+1F+M+

L exp(ik−z z)

+(−i)α+1E−M−R exp(−ik+

z z) + (i)α+1F−M−L exp(−ik−

z z)]

Expressing k−z as (−k+

z ), above equations yields the following

E0fd =[cos

(απ

2

)x +

k0z

k0sin

(απ

2

)y

−ky

k0sin

(απ

2

)z

]exp(ikyy + ik0zz)

Progress In Electromagnetics Research, PIER 85, 2008 387

−12

[x2 cos

(απ

2

)− y2

k0z

k0sin

(απ

2

)

−z2ky

k0sin

(απ

2

)]exp(ikyy − ik0z(z − 2d1)) (17)

E1fd = E+

[−2i sin

(απ

2

)x − 2

k+z

k+sin

(απ

2

)y

−2iky

k+cos

(απ

2

)z

]exp(ikyy + ik+

z z)

+E−[−2i sin

(απ

2

)x − 2

k+z

k+sin

(απ

2

)y

−2iky

k+cos

(απ

2

)z

]exp(ikyy − ik+

z z) (18)

η0H0fd =[cos

((α + 1)π

2

)x +

k0z

k0sin

((α + 1)π

2

)y

−ky

k0sin

((α + 1)π

2

)z

]exp(ikyy + ik0zz)

−12

[x2 cos

((α + 1)π

2

)− y2

k0z

k0sin

((α + 1)π

2

)

−z2ky

k0sin

((α + 1)π

2

)]exp(ikyy − ik0z(z − 2d1)) (19)

η0H1fd = E+

[−2i sin

((α + 1)π

2

)x − 2

k+z

k+sin

((α + 1)π

2

)y

− 2iky

k+cos

((α + 1)π

2

)z

]exp(ikyy + ik+

z z)

+E−[−2i sin

((α + 1)π

2

)x − 2

k+z

k+sin

((α + 1)π

2

)y

− 2iky

k+cos

((α + 1)π

2

)z

]exp(ikyy − ik+

z z) (20)

Changing values of α between zero and one, we can find behavior offields inside intermediate geometries. α = 0 and α = 1 reproducesthe PEC and PMC cases respectively. In next section Chiral nihilityslab backed by PEMC characterized by admittance parameter M isconsidered. Mη → ±∞ and Mη → 0 reproduce PEC and PMC casesrespectively.

388 Naqvi

4. CHIRAL NIHILITY SLAB BACKED BY PEMC

Here it assumed that slab of chiral nihility metamaterial is backed byperfect electromagnetic conductor (PEMC). PEMC is generalization ofPEC and PMC and has been introduced by Lindell and Sihvola [17].Chiral material and PEMC has been studied by many authors [18–27]. Fields given in Equations (1)–(4) can be assumed in regions insideand outside the slab. Unknown coefficients in field expressions (1)–(4)may be obtained using the related boundary conditions. At z = d2,tangential components of field quantity (ME1 +H1) must be zero [17].Application of boundary condition to x and y components of electricfield yields

E± = −(

Mη + i

Mη − i

)F∓ (21)

Tangential components of electric and magnetic fields across thedielectric interface located at z = d1 must be continues. Continuity ofx-components and y-components of electric field yields

exp(ik0zd1) + (A− + B−) exp(−ik0zd1)

=(

2i

Mη + i

) [E+ exp(ik+

z d1) + E− exp(−ik+z d1)

]

(−A− + B−) exp(−ik0zd1)

=k+

z k0

k+k0z

(2i

Mη + i

) [E+ exp(ik+

z d1) − E− exp(−ik+z d1)

]

Continuity of x-components and y-components of magnetic fieldyields

(A− − B−) exp(−ik0zd1)

=η0

η

(2Mη

Mη + i

) [E+ exp(ik+

z d1) + E− exp(−ik+z d1)

]

exp(ik0zd1) − (A− + B−) exp(−ik0zd1)

=(

2Mη

Mη + i

)Rf

[E+ exp(ik+

z d1) − E− exp(−ik+z d1)

]

Solving above four equations simultaneously yields the unknowncoefficients

E− = −E+

(η22iRf + η2

02Mη

−η22iRf + η202Mη

)exp(2ik+

z d1) (22)

Progress In Electromagnetics Research, PIER 85, 2008 389

E+ =2 exp(ik0zd1 − ik+

z d1)P − QL

(23)

where

P =2i + 2MηRf

Mη + i

Q =(

η22iRf + η202Mη

−η22iRf + η202Mη

)

L =2i − 2MηRf

Mη + i

It may be noted that under the limit Mη → ±∞, results derived in thissection reduces to results derived in previous section for chiral nihilityslab backed by PEC.

5. CONCLUSIONS

Both electric and magnetic fields exist inside the grounded chiralnihility slab when it is excited by a plane wave. Electric field insidethe grounded chiral nihility slab disappears for excitation due toan electric line source. Magnetic field inside the slab disappearswhen geometry changes to corresponding dual geometry. Usingfractional curl operator, fields are determined for geometries whichmay be regarded as intermediate step between the two dual geometries.Neither electric fields nor magnetic field disappears for fractionalgeometries either for plane wave or line source excitation. Usingconcept of fractional geometries, one can select appropriate geometryrequired regarding distribution of field and power inside the nihilityslab.

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