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Surface wave modes in PEMC backed chiral slab
Optics Communications 283 (2010) 2536–2546
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Surface wave modes in PEMC backed chiral slab Husnul Maab a,b, Qaisar Abbas Naqvi a,* a b
Department of Electronics, Quaid-i-Azam University, 45320 Islamabad, Pakistan GIK Institute of Engineering Sciences and Technology, 23640 Topi, Pakistan
a r t i c l e
i n f o
Article history: Received 22 August 2009 Received in revised form 16 February 2010 Accepted 16 February 2010
a b s t r a c t The characteristics of surface wave modes in a PEMC backed chiral slab are studied theoretically. First, the analytical solution of electromagnetic fields and dispersion relations are carried out. Then, the fractional field solutions are found using the fractional curl operator. The numerical results are given by assuming that wave numbers k and k are either real or imaginary. These results are also evaluated at real and imaginary values of fractional parameter describing the order of curl operator. The discussion contains fractional dispersion curves at various cut-off frequencies and the fractional surface waves in chiralPEMC and achiral-PMC slabs respectively. For numerical analysis it is assumed that the fractional order of the curl operator is related to chiral admittance, thickness of the slab, and PEMC admittance. For the values of the fractional order equal to 0, 1, and 2 geometry corresponds to PMC backed ordinary dielectric slab, PEMC backed chiral slab, and PEC backed chiral slab respectively. Consequently TE, HE (even), and HE (odd) modes are produced in the respective geometries. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Phenomenons of optical activity and circular dichroism distinguish chiral material from ordinary dielectric material [1,2]. The optical activity and circular dichroism respectively refer to the rotation of the plane of polarization and change in polarization ellipticity of optical waves by a medium. By definition, an object is said to be chiral if it cannot be superimposed on its mirror image by translatory motion or rotation. A chiral medium is characterized by either a left-handedness or a right-handedness in its microstructure. Consequently, in a chiral medium left- and right-circularly polarized (LCP and RCP) waves propagate with different phase velocities [2]. Chiral medium is a special form of bi-isotropic media which has been extensively studied in literature. The reflection and transmission of electromagnetic phenomenon for semiinfinite chiral media and for finite chiral slabs have been analyzed by Bassari et al. [3]. Lakhtakia et al. addressed various issues related to chiral media [2,4–6]. The theory of electromagnetic wave propagation in cylindrical waveguides filled with chiral material, which is also known as chirowaveguides, has been presented and discussed in [7–11]. Lakhtakia [12] analyzed propagation in a parallel-plate waveguide wholly filled with a chiral medium. Mariotte et al. presented an excellent overview of guided waves in chiral medium [13]. Some of canonical problems whose solution eventually useful in chiral devices were reviewed by Cory [14]. Lacava and Lumini proposed an alternative formulation for guided electromagnetic fields in grounded chiral slab [15]. * Corresponding author. E-mail address: [email protected] (Q.A. Naqvi). 0030-4018/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.02.039
The study of chirowaveguides is focused by many researchers due to the existence of hybrid propagation modes [7–11], which is the resultant of coupling effect between electric and magnetic fields in the presence of chirality. Moreover, there exit bifurcated modes with the same cut-off frequencies and unequal propagation characteristics. These phenomena produce several potential applications such as directional couplers, mode converters, electronic switches, and dipole and infinite arrays printed on chiral slab related to chirowaveguides. In literature the chiral slab backed by perfect electric conducting boundary, is analyzed by various researchers [1,2,6]. In the present paper a chiral slab backed by a perfect electromagnetic conducting (PEMC) boundary is analyzed. The problem is further investigated a for fractional solutions using fractional curl operator i.e., curl where a is the order of fractional parameter. The PEMC and chiral admittances are also linked with order of fractional curl operator. In this way changing the order of fractional curl operator effects both PEMC and chiral admittance. It is noted that for a ¼ 0 situation reduces to PMC backed achiral slab while for a ¼ 1 the situation corresponds to PEMC backed chiral slab. The numerical results of fractional surface waves and fractional modes are presented. In electromagnetic theory, PEMC is the generalization of PEC and PMC boundaries [16]. The boundary conditions of PEMC boundary are defined by
~ þ M~ ^ ðH n EÞ ¼ 0; ~ ~ ^ ðD M BÞ ¼ 0; n
ð1Þ ð2Þ
^ denotes the unit normal vector to the boundary and M is where n the PEMC admittance. The value of M can be set as M ¼ tan h [16]
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which leads to M ¼ 0 ðh ¼ 0Þ and M ¼ 1 ðh ¼ p=2Þ for the corresponding PMC and PEC boundaries respectively. Idea of fractional a curl operator ðcurl Þ is introduced by Engheta [17]. The application of fractional curl operator yields new solutions to Maxwell equations and are termed intermediate solutions. Fractional solutions of Maxwell equations may be obtained as
~a Þ ¼ ðikÞa curla ð~ ~ ð~ Ea ; gH E; gHÞ;
ð3Þ
pffiffiffiffiffiffiffiffiffi where k ¼ x l is the propagation constant, g ¼ l= is the ~a Þ are the new solutions to some known intrinsic impedance, ð~ Ea ; gH ~ and the fractional order (FO) a may possess both real fields ð~ E; gHÞ, ~a Þ represents the and complex values. The fractional solution ð~ Ea ; gH intermediate solutions between initial and dual to initial solutions. For a ¼ 0 and a ¼ 1 the corresponding field solutions become pffiffiffiffiffiffi
~ ðE ; gH Þ ¼ ð~ E; gHÞ; 1 1 ~ ~ ~ EÞ: ðE ; gH Þ ¼ ðgH; ~ ~0
~0
ð6Þ
~ ¼ 0; ~ þ k2 H r H þ 2pr H
ð7Þ
2~
pffiffiffiffiffiffi
where k ¼ x l and p ¼ xln with x being the angular frequency of the time harmonic fields [11]. It is identified that in an unbounded chiral medium, a right-circularly polarized and left-circularly polarized plane waves propagate with wave numbers.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 l x2 l2 n2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ xln þ x2 l x2 l2 n2 : kþ ¼ xln þ
ð8Þ ð9Þ
@ In the chiral slab, for no variation along y-axis @y ¼ 0 , the electromagnetic field distribution inside the chiral slab can be described as
Ef ðx; zÞ ¼
( 4 X
) ics z
Hf ðx; zÞ ¼
eibx ;
efs e
s¼1
The fractional operators are very useful because they make simple the description of a transitional and non-ideal situation in the electromagnetic problems. These problems are the extensions of some known and thoroughly studied canonical cases [17]. Naqvi et al. apply the fractional curl operator to: propagation in an infinite chiral medium [19,20,27], the fractional solutions and the corresponding sources [21], fractional surface waveguide, and fractional rectangular cavity resonator [22,23]. Lakhtakia [24] derived a theorem which shows that a dyadic operator commutes with curl operator, that can be used to find new solutions of Faraday and Ampere–Maxwell equations.
( 4 X
ð10Þ
) hfs eics z eibx ;
ð11Þ
s¼1
where f ¼ x; y or z; s ¼ 1; 2; 3 or 4; efs ; hfs are unknown coefficients, cs is the wave number along z-axis, and b ¼ b0 ib00 is the propagation constant along x direction. As the chiral medium supports two bulk plane waves; therefore, the four different wave numbers along z-axis are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kþ b2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ¼ c4 ¼ k2 b2 :
c1 ¼ c3 ¼
ð12Þ ð13Þ
Using Eqs. (10) and (11) in source free Maxwell equations, the relations of unknown coefficients of electromagnetic field may be
2. Formulation Consider a homogeneous, reciprocal, and isotropic chiral slab of uniform thickness d which is placed on a perfect electromagnetic conducting (PEMC) plate of infinite length as shown in Fig. 1. The chiral medium inside the slab is described by complex permittivity , complex permeability l, and complex chiral admittance n. The geometry is divided into two regions. Region 1 ð0 < z < dÞ contains chiral slab whereas region 0 ðz > dÞ is a free space having permittivity 0 and permeability l0 . To assume, the time harmonic dependency eixt , the constitutive relations for isotropic reciprocal chiral medium can be written as [1].
~ D ¼ ~ E in~ B; ~ ~ B ¼ lH þ in~ E:
2 E þ 2pr ~ E ¼ 0; r2~ Eþk ~
ð4Þ ð5Þ
Upon substituting the constitutive relations in source free Maxwell equations, the wave equations for electromagnetic fields inside the chiral slab take the form [15].
written
in
terms
of
ezs
as
exs ¼ cbs ezs ; eys ¼ ð1Þsþ1 ikbs ezs ;
ezs ; hzs ¼ ð1Þsþ1 gi ezs ; k1 ¼ k3 ¼ kþ ; c pffiffiffiffiffiffiffiffiffiffi k2 ¼ k4 ¼ k . Quantity gc ¼ l=c ; is the intrinsic impedance of
hxs ¼
ð1Þsþ1 bicgs ezs ; hys c
¼
bkgs c
chiral medium and c ¼ þ ln2 , the permittivity of chiral slab. Introducing n ¼ 0; l ¼ l0 and ¼ 0 in (6) and (7) reduced to wave equations of free space 2 E0 þ k0~ E0 ¼ 0; r2~ 2~
r H0 þ
2~ k0 H 0
ð14Þ
¼ 0;
ð15Þ
pffiffiffiffiffiffiffiffiffiffi
where k0 ¼ x l0 0 . The electromagnetic field expressions in free space (region 0) may be expressed as
Ef0 ðx; zÞ ¼ ef0 eðdzibxÞ ; Hf0 ðx; zÞ ¼ hf0 e
ðdzibxÞ
ð16Þ ;
ð17Þ
where ef0 ; hf0 are unknown coefficients and d is a positive valued decay rate guided mode quantity in transversal z-direction and is given by
d¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 k0 :
ð18Þ
Using the same procedure as discussed above for the unknown coefficient in chiral slab, the relations of the unknown coefficients of electromagnetic field outside (in region 0) the chiral slab in terms of electric and magnetic z-coefficients are ex0 ¼ idb ez0 ; ey0 ¼ xbl0 hz0 ; hx0 ¼ idb hz0 ; hy0 ¼ xb0 ez0 , where ez0 and hz0 are the un-
Fig. 1. Geometry of PEMC backed chiral slab.
known coefficients in free space region outside the chiral slab, which will be determined using boundary conditions at z ¼ d. Application of the appropriate tangential boundary conditions at z ¼ 0 (a planar interface between PEMC and chiral medium) [16] and z ¼ d (a planar interface between chiral medium and free space) yield the following unknown coefficients inside and outside the PEMC backed chiral slab.
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h
2
2
ez1 ¼ jaj2 fka c1 c2 k þ idk c2 gc2 þ jaj2 fika c22 kþ dk c1 gs2 i Þ2 k c2 B1 eic1 d ez4 ; þ ða ð19Þ h 2 2 ez2 ¼ jaj2 fka c1 c2 kþ idk c1 gc1 þ jaj2 fika c21 k þ dkþ c2 gs1 i ðaÞ2 kþ c1 B2 eic2 d ez4 ; ð20Þ ez0 ¼
2edd
Y 1 e z4 ;
ð21Þ
2edd Y 2 ez4 ; ixl0 D
ð22Þ
ix0 gc D
hz0 ¼
¼ g1 ðMgc iÞ; ka ¼ ez3 ¼ ez1 ; D ¼ ez2 =ez4 ; a ¼ g1 ðMgc þ iÞ; a
where
c
c
2 idk ; Y 1 ¼ ðkþ 22 2 2 2
2 k 21 Þjaj2 s1 s2 þ 2 c1 c2 dk jaj2 ½ 1
x0 gc ; B1 ¼ ka c1 þ idkþ ; B2 ¼ ka c2 þ c c c1 c2 k2 ½ðaÞ2 ðaÞ2 ; Y 2 ¼ ka c1 c2 k2 ½ðaÞ ðaÞ jaj c 2 2 2 2 2 k c1 s2 þ c2 kþ s1 c2 þ ka jaj ½kþ c2 þ k c1 s1 s2 ; c1 ¼ cosðc1 dÞ; c2 ¼ cosðc2 dÞ; s1 ¼ sinðc1 dÞ; s2 ¼ sinðc2 dÞ. Note that in the above relations the bar over quantities represent the complex conjugate. The dispersion relation for the surface wave modes [for detail see, [15]], guided in chiral slab is obtained as follow
h gc c2 k ðk20 c21 d2 k2þ Þs1 c2 dxk2 c1 c2 ð0 g2c þ l0 Þc1 c2 2aa þ 0:5dx 2
2 ðk 21
2
2 2 2 kþ Þð 0
c c
2 2 c 1 kþ ðk0 2
g2c þ l0 Þs1 s2 þ g c
c
2 d2 k Þc1 s2
2
þ ða þ a Þdxk c1 c2 ð0 gc l0 Þ ¼ 0: 0
i
ð23aÞ
00
When b ¼ k0 and b ¼ 0, the attenuation propagation constant outside the chiral slab becomes equal to zero (i.e., d ¼ 0) and (23a) takes the following form
k c1c sinðc1c dÞ cosðc2c dÞ þ kþ c2c sinðc2c dÞ cosðc1c dÞ ¼ 0 where c1c
ð23bÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 ¼ kþ k0 ; c2c ¼ k k0 . Eq. (23b) is similar to the one
reported in [15] and the references therein. Eqs. (23a) and (23b) can be numerically evaluated to find the propagation constant b and cut-off frequencies of the guided surface wave modes inside the chiral slab. 3. Expressions for guided electromagnetic fields The field expressions for region 1 are obtained by inserting (19) and (20) in the expressions containing the unknown coefficients defined for the inside of the chiral slab. Introducing the resultant expressions in Eqs. (10) and (11) , the following set of expressions is obtained
o 2iez4 n c1 c2 ½W 1 C 1 W 11 C 2 þ ½k c1 W 2 S1 þ W 12 kþ c2 S2 eibx ; bD ð24Þ o 2iez4 n 2 k ½W 2 C 1 W 22 C 2 ½kþ c2 W 1 S1 þ W 21 k c1 S2 eibx ; Ey ¼ bD ð25Þ o 2ez4 n ½k W 2 C 1 þ kþ W 12 C 2 ½c2 W 1 S1 c1 W 11 S2 eibx ; ð26Þ Ez ¼ D n o 2e gc Hx ¼ z4 c1 c2 ½W 1 C 1 þ W 11 C 2 þ ½k c1 W 2 S1 W 12 kþ c2 S2 eibx ; bD ð27Þ o 2ez4 n 2 1 1 ibx gc Hy ¼ k ½W 2 C 1 þ W 2 C 2 ½kþ c2 W 1 S1 W 1 k c1 S2 e ; bD ð28Þ Ex ¼
gc Hz ¼
2iez4 f½k W 2 C 1 kþ W 12 C 2 ½c2 W 1 S1 þ c1 W 11 S2 geibx ; D ð29Þ
where 2
Þ2 c1 þ jaj2 c2 ka ½ða Þ2 k c1 s1 þ jaj2 kþ c2 c2 ; W 1 ¼ dk ½ða Þ2 c1 jaj2 c2 þ d½ða Þ2 kþ c2 s1 jaj2 k c1 c2 ; W 2 ¼ ka c1 c2 ½ða 2
W 11 ¼ dk ½jaj2 c1 þ ðaÞ2 c2 ka ½jaj2 k c1 s1 þ ðaÞ2 kþ c2 c2 ; W 12
¼ ka c1 c2 ½jaj2 c1 ðaÞ2 c2 þ d½jaj2 kþ c2 s1 ðaÞ2 k c1 c1 ; 2
Þ2 c2 ka ½jaj2 k c1 s1 þ ða Þ2 kþ c2 c2 ; W 21 ¼ dk ½jaj2 c1 þ ða Þ2 c2 þ d½jaj2 kþ c2 s1 ða Þ2 k c1 c2 ; W 22 ¼ ka c1 c2 ½jaj2 c1 ða C 1 ¼ cosðc1 zÞ; C 2 ¼ cosðc2 zÞ; S1 ¼ sinðc1 zÞ; S2 ¼ sinðc2 zÞ: The electromagnetic field expressions in free space are obtained by substituting (21) and (22) into the unknown coefficient relations defined for free space, which are then, plugged-in into (16) and (17), yielding
2idez4 Y 1 edd1 ibx ; Db 2iez4 Y 2 edd1 ibx ; Ey0 ¼ Db 2ez4 Ez0 ¼ Y 1 edd1 ibx ; D 2de g0 Hx0 ¼ z4 Y 2 edd1 ibx ; Dbk0 2k0 ez4 g0 Hy0 ¼ Y 1 edd1 ibx ; Db 2ie g0 Hz0 ¼ z4 Y 2 edd1 ibx ; Dk0 Ex0 ¼
ð30Þ ð31Þ ð32Þ ð33Þ ð34Þ ð35Þ
where d1 ¼ d z. It is interesting to note that the expressions (24)– (35) and the dispersion relation given in Eq. (23a) along with cut-off frequency expression (23b) reduce to standard TE even modes, when n ¼ 0 and M ¼ 0 which is a planar waveguide composed of PMC backed achiral slab. However, when n ¼ 0 and M ¼ 1 the geometry of the problem reduces to PEC backed achiral slab. For these values, the field expressions (24)–(35) including Eqs. (23a) and (23b) convert to standard TE odd modes that compromise with the results shown in reference [15].
4. Fractional dual expressions for guided electromagnetic fields a
In this section fractional curl operator, curl , is used to fraction~a Þ alize the principle of duality that is, new set of solutions ð~ Ea ; gH to the Maxwell equations are obtained. These new solutions are known as intermediate or fractional solutions between the original solution and dual to original solution. According to [17], the fractional electromagnetic fields may be obtained as
~a Þ ¼ ðikÞa curla ð~ ~ ð~ Ea ; gH E; gHÞ;
ð36Þ
where g and k are the intrinsic impedance and wave number of unbounded dielectric medium respectively. In PEMC backed chiral ~ take the folslab, fractional solutions with original solution ð~ E; gc HÞ lowing form
~a Þ ¼ ðik Þa curla ð~ ~ ð~ Ea ; gc H E; gc HÞ;
ð37Þ
where gc is the intrinsic impedance of chiral material and k represent the right and left circular wave numbers in chiral slab. Eq. (37) can easily be reduce to Eq. (36) by setting n ¼ 0 (i.e., non-chiral material). Using above relations, the fractional electromagnetic fields both inside and outside the PEMC backed chiral slab are obtained as: In region 1 ð0 < z < dÞ
H. Maab, Q.A. Naqvi / Optics Communications 283 (2010) 2536–2546
4iez4 a ~ ^ Eax3 sðaÞ^zeibx ; Eax ¼ ½E CðaÞ^x Eax2 SðaÞy bD x1 4iez4 a ~ ^ þ Eay3 SðaÞ^zeibx ; ½E SðaÞ^x þ Eay2 CðaÞy Eay ¼ bD y1 4ez4 a ~ ^ þ Eaz3 CðaÞ^zeibx ; Eaz ¼ ½E sðaÞ^x Eaz2 SðaÞy D z1
4ez4 a ^ Hax3 sðaÞ^zeibx ; ½H CðaÞ^x Hax2 SðaÞy bD x1 4ez4 a ^ þ Hay3 SðaÞ^zeibx ; ¼ ½H SðaÞ^x þ Hay2 CðaÞy bD y1 4iez4 sðaÞ ^x Haz2 SðaÞy ^ þ Haz3 CðaÞ^z eibx ; ¼ Haz1 b D
2539
ð38Þ
~a ¼ gc H x
ð41Þ
ð39Þ
~a gc H y
ð42Þ
ð40Þ
~a gc H z
ð43Þ
Fig. 2. Fractional dispersion diagrams for modes in chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 . In the diagrams b0 is plotted versus frequency, for real and imaginary values of k; k separately. The dispersion diagrams corresponding to modes of achiral-PMC and chiral-PEMC waveguides where chiral and PEMC admittances are interrelated to fractional order parameter given by (53).
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In region 0 ðz > dÞ
Eax ¼ CðaÞEx þ SðaÞðgc Hx Þ; ðgc Hax Þ ¼ SðaÞEx þ CðaÞðgc Hx Þ;
o 4ez4 n 1þa a a 1þa ~ ^ þ ika0 bsðaÞ^z edd1 ibx ; ð44Þ Eax0 ¼ Y 1 di k0 CðaÞ^x i k0 SðaÞy bD ( ) a 1þa 4e di i b z4 1þa a a ~ ^ ^ ^ Ey0 ¼ Y 2 1a SðaÞx þ i k0 CðaÞy þ 1a SðaÞz edd1 ibx ; bD k0 k0
Eay ¼ CðaÞEy þ SðaÞðgc Hy Þ; ðgc Hay Þ ¼ CðaÞðgc Hx Þ SðaÞEy ;
(
ð45Þ
)
1þa
4ez4 d a ak ~ ^ þ ia ka0 CðaÞ^z edd1 ibx ; ð46Þ Eaz0 ¼ Y 1 ik0 sðaÞ^x i 0 SðaÞy b D b n o ~a ¼ 4ez4 Y 2 dia ka CðaÞ^x i1þa k1þa SðaÞy ^ bsðaÞ^z edd1 ibx ; g0 H x0 0 0 bDk0 ð47Þ ( ) 1þa a 4k e di i b a a ~ a ¼ 0 z4 Y 1 ^ 1a SðaÞ^z edd1 ibx ; g0 H SðaÞ^x i k0 CðaÞy y0 1a bD k0 k0 ð48Þ ( ) 1þa 1þa ~a ¼ 4ez4 Y 1 i d ka sðaÞ^x i k0 SðaÞy ^ þ i1þa ka0 CðaÞ^z edd1 ibx ; g0 H z0 b 0 k0 D b ð49Þ where n o a a a a a Eax1 ¼ i c1 c2 ðW 1 kþ C 1 W 11 k C 2 Þ þ ðkþ k c1 W 2 S1 þ W 22 k kþ c2 S2 Þ ; n o aþ1 aþ1 aþ1 aþ1 aþ1 Eax2 ¼ i ðkþ c2 W 1 S1 k c1 W 11 S2 Þ ðkþ k W 2 C 1 þ k kþ W 12 C 2 Þ ; n o a a a a Eax3 ¼ b ðkþ c2 W 1 S1 k c1 W 11 S2 Þ ðkþ k W 2 C 1 þ k kþ W 12 C 2 Þ ; ( !) Eay1 ¼ i
a
aþ1
a
a
c1 c2 ðkþ W 1 C 1 þ k W 21 C 2 Þ þ k a
2
a
c1
2
1a
kþ
1þa
W 2 S1 þ
c2
1a
k
W 22 S2
;
1þa
Eay2 ¼ i fk ðkþ W 2 C 1 k W 22 C 2 Þ ðkþ c2 W 1 S1 þ k c1 W 21 S2 Þg; ! ( ) W2 W 22 2 a a a 2 ðk Eay3 ¼ i b k C C c W S þ k c W S Þ ; 1 2 1 1 2 2 1 þ 1 1a 1a kþ k i a a a a Eaz1 ¼ fðkþ k c1 W 2 C 1 þ k kþ c2 W 12 C 2 Þ c1 c2 ðkþ W 2 S1 k W 12 S2 Þg; b a
i 1þa 1þa 1þa 1þa Eaz2 ¼ fðkþ k W 2 C 1 þ k kþ W 12 C 2 Þ ðkþ c2 W 1 S1 k c1 W 11 S2 Þg; b a
a
a
a
a
Eaz3 ¼ i fðkþ k W 2 C 1 þ k kþ W 12 C 2 Þ ðkþ c2 W 1 S1 k c1 W 11 S2 Þg; a
a
a
a
a
Hax1 ¼ fi c1 c2 ðW 1 kþ C 1 þ W 11 k C 2 Þ þ ðkþ k c1 W 2 S1 W 12 k kþ c2 S2 Þg; Hax2 ¼ i
1þa
1þa
1þa
1þa
1þa
fðkþ c2 W 1 S1 þ k c1 W 11 S2 Þ ðkþ k W 2 C 1 k kþ W 12 C 2 Þg; a
a
a
a
Hax3 ¼ bfðkþ c2 W 1 S1 þ k c1 W 11 S2 Þ ðkþ k W 2 C 1 k kþ W 12 C 2 Þg; ( Hay1 ¼ i
c1 c2 ðkaþ W 1 C 1 ka W 11 C 2 Þ þ k2
1þa
a
2
a
a
c1
W 2 S1 þ 1a
kþ
1þa
c2
!)
W 12 S2 1a
k
;
1þa
Hay2 ¼ i fk ðkþ W 2 C 1 þ k W 12 C 2 Þ ðkþ c2 W 1 S1 k c1 W 11 S2 Þg; ( ) ! W2 W 12 2 a a a 1 C þ C c W S k c W S Þ ; ðk Hay3 ¼ i b k 1 1 þ 2 1 1 2 1a 1 1a 2 kþ k a
a
a
a
Haz1 ¼ fðkþ k c1 W 2 S1 k kþ c2 W 12 S2 Þ þ c1 c2 ðkþ W 1 C 1 þ k W 11 C 2 Þg;
Eaz ¼ CðaÞEz þ SðaÞðgc Hz Þ; ðgc Haz Þ ¼ CðaÞðgc Hz Þ SðaÞEz :
ð50Þ
Similarly in region 0 ðz > dÞ
Eax0 ¼ CðaÞEx0 þ SðaÞðg0 Hx0 Þ; ðg0 Hax0 Þ ¼ SðaÞEx0 þ CðaÞðg0 Hx0 Þ; Eay0 ¼ CðaÞEy0 þ SðaÞðg0 Hy0 Þ; ðg0 Hay0 Þ ¼ CðaÞðg0 Hx0 Þ SðaÞEy0 ; Eaz0 ¼ CðaÞEz0 þ SðaÞðg0 Hz0 Þ; ðg0 Haz0 Þ ¼ CðaÞðg0 Hz0 Þ SðaÞEz0 :
ð51Þ
The discussion can be extended to show that how the characteristic parameters of chiral media and PEMC boundary are correlated with fractional order parameter a. Let assume an infinite chiral slab of thickness L. For normal incidence, the transmitted field in chiral b 31 , is defined by medium in term of transmission matrix T pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 31 b b 31 ¼ T 31 Rðjr k2 LÞ ¼ T 31 RðwÞ; jr ¼ j 0 l =l; ~ Et ¼ T E i where T 0 pffiffiffiffiffiffiffiffiffiffi k2 ¼ x l2 2 ; and RðwÞ denotes a matrix of rotation by an angle of w [1,18]. Using the fractional curl operator given in (36) for achiral medium, the fractional solution for field transmitted through an b a~ infinite achiral slab becomes ~ Ea;t ¼ T Ea;i where T a ¼ T 31 Rðap=2Þ [18]. Comparing the non-fractional transmitted field for chiral slab and fractional transmitted fields for achiral slab, we get the following relation.
pa=2 ¼ jr k2 L:
ð52Þ
It follows that the fractional field solution in chiral medium can be constructed as the field traverses through a non-chiral medium. From the above relation, it is suggested that for a wave transmission in chiral material, the fractional order a may be expressed in terms of chirality and the thickness of the chiral slab. In case of reflected field from an interface between PEC and bi-isotropic medium, the fractional order a, is associated with Tellegen parameter but independent of chirality parameter [18,25]. In view of the above statement, we assume that the fractional order (FO) a can be adjusted through chiral admittance and slab thickness for the guided electromagnetic fields in PEMC backed chiral slab and is given by
ap 2
¼ nd:
ð53Þ
The relationship between fractional order a and PEMC admittance is considered as M ¼ tanðap=4Þ [see for instance, [18]]. According to this relation, when a ¼ 0 the PEMC boundary reduces to PMC boundary whereas when a ¼ 1, and a ¼ 2, one can get the corresponding PEMC and PEC boundaries. Using the interrelationship among chiral admittance, PEMC admittance, and fractional order, we deduced the following observations. When a ¼ 0, the corresponding chiral and PEMC admittances become zero and the hybrid modes (HE) in PEMC backed chiral slab transform to standard TE even modes in PMC backed achiral slab. When 0 < a 6 1, the corresponding chiral and PEMC admittances reduce to some nonzero values. Therefore, the modes in the slab correspond to fractional intermediate solutions and the geometry of the problem behaves as PEMC backed chiral slab.
a
i 1þa 1þa 1þa 1þa Haz2 ¼ fðkþ k W 2 C 1 k kþ W 12 C 2 Þ ðkþ c2 W 1 S1 þ k c1 W 11 S2 Þg; b a
a
a
a
a
Haz3 ¼ i fðkþ k W 2 C 1 k kþ W 12 C 2 Þ ðkþ c2 W 1 S1 þ k c1 W 11 S2 Þg; ap a p ;SðaÞ ¼ sin ;sðaÞ ¼ sinðapÞ: CðaÞ ¼ cos 2 2
It may be noted that the fractional curl operator describes the polarization reversal of electromagnetic field [18]. Therefore, we can also represent the fractional field expressions, inside and outside the PEMC backed chiral slab, in terms of original electric and magnetic field solutions. In region 1 ð0 < z < dÞ
5. Numerical results The propagation constants c1 and c2 in chiral slab while the attenuation constant d in free space along the transversal direction (z-axis) depend on b ¼ b0 jb00 . The real and imaginary parts of propagation constant b represent phase constant and attenuation constant along the direction of propagation (x-axis). The values of these constants can be obtained numerically using the dispersion and critical expressions in (23a) and (23b) over a given frequency range of interest. It is to be noted that the critical condition for PEMC backed chiral slab geometry is obtained when
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b0 ¼ k0 and b00 ¼ 0, that is, the condition at which the attenuation constant d becomes zero. Therefore, to study the surface waves closely confined to the chiral slab the values of b must be kept between k0 and real or imaginary values of k . It is observed that k0 is the lowest value of b in any guided surface wave mode. If we assume the value of b less than k0 , then the value of attenuation constant d in free space becomes imaginary. In that case, we get the so-called radiation mode and the structure behaves like a leakywave antenna [15]. It is worth pointing out that k is the highest value of propagation constant b of any mode in chiral slab. If the value of b is kept higher than the value of k then c1 and c2 become imaginary and all modes associated with the propagation constant would get attenuated inside the chiral slab. Beside, this if b > k , Eq. (23a) becomes negative and imaginary and it cannot be converged to zero. It is noted that when n ¼ 0, the geometry of the problem reduces to PEMC backed achiral slab and the wave number of chiral slab k is reduced to k, i.e., the wave number of ordinary dielectric slab (achiral). In order to analyze the guided surface wave modes in achiral slab the propagation constant must be limited between k0 and real or imaginary values of k.
Throughout
the
discussion,
it
is
assumed
that
l ¼ l0 ð1 þ i0:5Þ; ¼ 20 and the values of chiral and PEMC admittances as interrelated to fractional parameter a given in (53). The plots for dispersion curves and surface waves are obtained by considering real or imaginary values of k and k. In Figs. 2 and 3, the dispersion curves are plotted for phase constant b0 and attenuation constant b00 separately. These plots are constructed for various values of fractional order a. In Fig. 2, when a ¼ 0, chiral and PEMC admittances become zero and the geometry of the problem reduces to PMC backed achiral slab. As an outcome, the dispersion curves represent the standard TE even modes. The plot shown in Fig. 2, for a ¼ 0, is obtained by keeping the values of b0 in the range of k0 and Rfkg (while putting b00 ¼ 0) satisfying Eqs. (23a) and (23b). It is to be noted that for a ¼ 0, the dispersion curves are not shown for the values of b0 between k0 and Ifkg because these values of b0 do not satisfy Eqs. (23a) and (23b). In Fig. 2, the values of b0 are also examined at a ¼ 0:25, in the limit of k0 and Rfkþ g. Whereas for a ¼ 0:25i, the values of b0 are plotted between k0 and Rfk g. The values of b0 corresponding to a ¼ 0:25 and 0:25i represent the intermediate fractional modes solution in the PEMC
Fig. 3. Fractional dispersion diagrams for modes in chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 . In the diagrams b00 is plotted versus frequency, for real and imaginary values of k and k by means of the fundamental values of b0, separately. The dispersion diagrams corresponding to modes of achiral-PMC and chiral-PEMC waveguides where chiral and PEMC admittances are interrelated to fractional order parameter given by (53).
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Fig. 4. 2D and 3D representation of fractional surface waves of Eay ; Hax ; Haz inside achiral-PMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for TE3 mode where a ¼ 0:0 corresponds to zero chiral and PEMC admittances.
Fig. 5. 2D and 3D representation of fractional surface waves of Eax inside chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for hybrid mode HE3 where fractional order parameter is interrelated with chiral and PEMC admittances given by (53).
H. Maab, Q.A. Naqvi / Optics Communications 283 (2010) 2536–2546
2543
Fig. 6. 2D and 3D representation of fractional surface waves of Eay inside chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for hybrid mode HE3 where fractional order parameter is interrelated with chiral and PEMC admittances given by (53).
Fig. 7. 2D and 3D representation of fractional surface waves of Eaz inside chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for hybrid mode HE3 where fractional order parameter is interrelated with chiral and PEMC admittances given by (53).
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Fig. 8. 2D and 3D representation of fractional surface waves of Eay0 ; Hax0 ; Haz0 in free space region of achiral-PMC d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for TE3 mode where a ¼ 0:0 corresponds to zero chiral and PEMC admittances.
waveguide
with
Fig. 9. 2D and 3D representation of fractional surface waves of Eax0 , in free space region of chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for hybrid mode HE3 where fractional order parameter is interrelated with chiral and PEMC admittances given by (53).
H. Maab, Q.A. Naqvi / Optics Communications 283 (2010) 2536–2546
2545
Fig. 10. 2D and 3D representation of fractional surface waves of Eay0 , in free space region of chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for hybrid mode HE3 where fractional order parameter is interrelated with chiral and PEMC admittances given by (53).
Fig. 11. 2D and 3D representation of fractional surface waves of Eaz0 , in free space region of chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for hybrid mode HE3where fractional order parameter is interrelated with chiral and PEMC admittances given by (53).
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backed chiral slab. From the resultant graphs in Fig. 2, it is observed that for real a the values of chiral and PEMC admittances are real and hence Rfkþ g is dominant over Rfk g. Whereas, for imaginary a the values of chiral and PEMC admittances are imaginary and the phenomenon is vice versa. Besides, it is observed from Fig. 2 that for a ¼ 0:25, the number of hybrid modes are greater than that of a ¼ 0:25i. It is noted in Fig. 2, that for a ¼ 0:25; Ifkþ g is more dominant as compared to the negligible values of Ifk g. In the adverse plotting, for a ¼ 0:25i, the values of Ifk g is dominant over Ifkþ g (negligible) where Ifk g is more negative. This phenomena represent some novel property of PEMC backed chiral slab. The phenomenon like this, is helpful in focusing either RCP or LCP plane wave in bi-isotropic media by suppressing one of the CP wave [26]. A similar discussion can be extended for other values of fractional order a. In Fig. 3, the plots are drawn for attenuation constant b00 using the zero critical frequency curves of b0 (first curve in each plot of Fig. 2). The dispersion curves of b00 are investigated at a ¼ 0; 0:25; 0:25i respectively, for real or imaginary values of k and k. The numerical results of fractional surface waves of fractional electromagnetic fields inside and outside of the PEMC backed chiral slab, are shown in Figs. 4–11, in two and three dimensions. These plots are drawn for real and imaginary values of k and k, using TE3 ; HE3 (3rd curve of b0 in each plots of Fig. 2) modes for the following values of a ¼ 0; 0:25; 0:25i respectively. When a ¼ 0, we get Ey ; Hx and Hz field components as shown in Fig. 4. In the same figure, the combined effect of Ey and Hz is standing wave pattern while that of Ey and Hx is propagating wave. The effect of these fields represents standard TE mode (even). In Fig. 5, for a ¼ 0:25; 0:25i with real and imaginary values of k and k, the fractional surface waves of Eax are in phase. In the same figure, Eax wave pattern for a ¼ 0:25i and Rfkþ ; kg is 1800 out of phase with that of Eax for a ¼ 0:25; 0:25i, having real and imaginary values of k and k. It is to be noted that in Fig. 5, for a ¼ 0:25 the fractional surface wave of Eax represents a decaying wave for Ifk ; kg. The situation shown in Fig. 5 reveals that, for a ¼ 0:25i with Rfk ; kg, the wave behavior is more strong, whereas for a ¼ 0:25 and 0:25i with the real and imaginary values of k and k it exhibits a weak behavior. In the light of aforementioned concepts, the other plots in the Figs. 6–11 can be elaborated in a meaningful manner. 6. Conclusions Analytical solution of dispersion relation and electromagnetic field distributions as well as the fractional field solution of hybrid modes in a PEMC backed chiral slab are studied. The numerical results of fractional dispersion behavior and fractional field distribution in PEMC backed chiral slab are examined in detail with reference to real or imaginary values of a; n; M, k and k. For the analysis of dispersion curves and fractional surface waves, fractional parameter is directly proportional to the chiral admittance and PEMC admittance. These relationships yield increasing number of hybrid propagating surface wave modes inside the PEMC backed chiral slab for either real or imaginary values of a. It is observed that for real a and Rfk ; kg, the RCP wave number is greater than the LCP wave number. While for imaginary a and Rfk ; kg the
resultant dispersion curves represent the dominance of LCP wave number over RCP wave number. It is worth pointing out that when a ¼ 0:25 and Ifk ; kg there exists only RCP wave number whereas the value of LCP wave number is negligible. Similarly, when a ¼ 0:25i along with imaginary values of k and k there exists only LCP wave number as compared to the negligible value of RCP wave number. The fractional surface waves are also checked for a ¼ 0; 0:25; 0:25i along with real or imaginary values of n, M, k and k. Those correspond to initial and intermediate fractional surface wave patterns in PMC backed achiral and PEMC backed chiral slabs respectively. When a ¼ 0 and the values of k are real, the fractional surface waves reduce to standard TE mode (even). It is pointed out that when a ¼ 0 and the values of k are imaginary, there exist no mode solution in the specified numerical limit (i.e. Ifkg < b0 < k0 ). It is to be noted that for a ¼ 0:25 and Ifk ; kg, the fractional surface waves of electric fields components represent decaying waves due to negligible LCP wave number. While for a ¼ 0:25i and Ifk ; kg, the fractional surface waves of electric fields components denote propagating waves due to negligible RCP wave number. Acknowledgements We are grateful to the reviewers for their valuable comments. References [1] I.V. Lindell, A.H. Sihvola, S.A. Tretyakov, A.J. Viitanen, Electromagnetic waves in chiral and bi-isotropic media, Artech House, Bosten, 1994. [2] A. Lakhtakia, V.K. Varadan, V.V. Varadan, Time Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics, vol. 335, Springer-Verlag, New York, 1989. [3] S. Bassari, C.H. Papas, N. Engheta, Journal of the Optical Society of America A 5 (1988) 450. Errata: 7 (1990) 2154. [4] A. Lakhtakia, V.K. Varadan, V.V. Varadan, Zeitschrift fuer Natureforschung 45a (1989) 639. [5] P.S. Reese, A. Lakhtakia, Optik 86 (1990) 47. [6] A. Lakhtakia, V.K. Varadan, V.V. Varadan, Optik 87 (1991) 77. [7] C. Eftimiu, L. Wilson Pearson, Radio Science 24 (1989) 351. [8] N. Engheta, P. Pelet, Microwave and Optical Technology Letters 14 (1989) 593. [9] P. Pelet, N. Engheta, IEEE Transactions on Antennas and Propagation 38 (1990) 90. [10] F. Mariotte, Engheta, Journal of Electromagnetic Waves and Applications 7 (1993) 1307. [11] R. Oussaid, B. Haraoubia, Canadian Journal of Physics 82 (2004) 367. [12] V.K. Varadan, A. Lakhtakia, V.V. Varadan, Journal of Wave Material Interaction 3 (1988) 267. see also, ‘‘Erratum,” 3 (1989) 335. [13] F. Mariotte, P. Pelet, N. Engheta, Progress in Electromagnetics Research 9 (1994) 311. [14] H. Cory, Journal of Electromagnetic Waves and Application 9 (1995) 805. [15] J.C. da, S. Lacava, F. Lumini, Progress in Electromagnetics Research 16 (1997) 285. [16] I.V. Lindell, A.H. Sihvola, IEEE Transactions on Antennas and Propagation 53 (2005) 3005. [17] N. Engheta, Microwave and Optical Technology Letters 17 (1998) 86. [18] M.V. Ivakhnychenko, Telecommunications and Radio Engineering 67 (2008) 567. [19] Q.A. Naqvi, G. Murtaza, Optics Communications 178 (2000) 27. [20] A. Hussain, Q.A. Naqvi, Progress in Electromagnetics Research 59 (2006) 199. [21] Q.A. Naqvi, A.A. Rizvi, Progress in Electromagnetics Research 25 (2000) 223. [22] H. Maab, Q.A. Naqvi, Progress In Electromagnetics Research C 1 (2008) 199. [23] H. Maab, Q.A. Naqvi, Progress In Electromagnetics Research B 9 (2008) 69. [24] A. Lakhtakia, Microwave and Optical Technology Letters 28 (2001) 385. [25] V.I. Maksym, Eldar. I. Veliev, Serbia Nis (2007) 26. [26] C. Monzon, D.W. Forester, Physical Review Letters 95 (2005) 123904. [27] S.A. Naqvi, Q.A. Naqvi, A. Hussain, Optics Communications 266 (2006) 404.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Surface wave modes in PEMC backed chiral slab Husnul Maab a,b, Qaisar Abbas Naqvi a,* a b
Department of Electronics, Quaid-i-Azam University, 45320 Islamabad, Pakistan GIK Institute of Engineering Sciences and Technology, 23640 Topi, Pakistan
a r t i c l e
i n f o
Article history: Received 22 August 2009 Received in revised form 16 February 2010 Accepted 16 February 2010
a b s t r a c t The characteristics of surface wave modes in a PEMC backed chiral slab are studied theoretically. First, the analytical solution of electromagnetic fields and dispersion relations are carried out. Then, the fractional field solutions are found using the fractional curl operator. The numerical results are given by assuming that wave numbers k and k are either real or imaginary. These results are also evaluated at real and imaginary values of fractional parameter describing the order of curl operator. The discussion contains fractional dispersion curves at various cut-off frequencies and the fractional surface waves in chiralPEMC and achiral-PMC slabs respectively. For numerical analysis it is assumed that the fractional order of the curl operator is related to chiral admittance, thickness of the slab, and PEMC admittance. For the values of the fractional order equal to 0, 1, and 2 geometry corresponds to PMC backed ordinary dielectric slab, PEMC backed chiral slab, and PEC backed chiral slab respectively. Consequently TE, HE (even), and HE (odd) modes are produced in the respective geometries. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Phenomenons of optical activity and circular dichroism distinguish chiral material from ordinary dielectric material [1,2]. The optical activity and circular dichroism respectively refer to the rotation of the plane of polarization and change in polarization ellipticity of optical waves by a medium. By definition, an object is said to be chiral if it cannot be superimposed on its mirror image by translatory motion or rotation. A chiral medium is characterized by either a left-handedness or a right-handedness in its microstructure. Consequently, in a chiral medium left- and right-circularly polarized (LCP and RCP) waves propagate with different phase velocities [2]. Chiral medium is a special form of bi-isotropic media which has been extensively studied in literature. The reflection and transmission of electromagnetic phenomenon for semiinfinite chiral media and for finite chiral slabs have been analyzed by Bassari et al. [3]. Lakhtakia et al. addressed various issues related to chiral media [2,4–6]. The theory of electromagnetic wave propagation in cylindrical waveguides filled with chiral material, which is also known as chirowaveguides, has been presented and discussed in [7–11]. Lakhtakia [12] analyzed propagation in a parallel-plate waveguide wholly filled with a chiral medium. Mariotte et al. presented an excellent overview of guided waves in chiral medium [13]. Some of canonical problems whose solution eventually useful in chiral devices were reviewed by Cory [14]. Lacava and Lumini proposed an alternative formulation for guided electromagnetic fields in grounded chiral slab [15]. * Corresponding author. E-mail address: [email protected] (Q.A. Naqvi). 0030-4018/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.02.039
The study of chirowaveguides is focused by many researchers due to the existence of hybrid propagation modes [7–11], which is the resultant of coupling effect between electric and magnetic fields in the presence of chirality. Moreover, there exit bifurcated modes with the same cut-off frequencies and unequal propagation characteristics. These phenomena produce several potential applications such as directional couplers, mode converters, electronic switches, and dipole and infinite arrays printed on chiral slab related to chirowaveguides. In literature the chiral slab backed by perfect electric conducting boundary, is analyzed by various researchers [1,2,6]. In the present paper a chiral slab backed by a perfect electromagnetic conducting (PEMC) boundary is analyzed. The problem is further investigated a for fractional solutions using fractional curl operator i.e., curl where a is the order of fractional parameter. The PEMC and chiral admittances are also linked with order of fractional curl operator. In this way changing the order of fractional curl operator effects both PEMC and chiral admittance. It is noted that for a ¼ 0 situation reduces to PMC backed achiral slab while for a ¼ 1 the situation corresponds to PEMC backed chiral slab. The numerical results of fractional surface waves and fractional modes are presented. In electromagnetic theory, PEMC is the generalization of PEC and PMC boundaries [16]. The boundary conditions of PEMC boundary are defined by
~ þ M~ ^ ðH n EÞ ¼ 0; ~ ~ ^ ðD M BÞ ¼ 0; n
ð1Þ ð2Þ
^ denotes the unit normal vector to the boundary and M is where n the PEMC admittance. The value of M can be set as M ¼ tan h [16]
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which leads to M ¼ 0 ðh ¼ 0Þ and M ¼ 1 ðh ¼ p=2Þ for the corresponding PMC and PEC boundaries respectively. Idea of fractional a curl operator ðcurl Þ is introduced by Engheta [17]. The application of fractional curl operator yields new solutions to Maxwell equations and are termed intermediate solutions. Fractional solutions of Maxwell equations may be obtained as
~a Þ ¼ ðikÞa curla ð~ ~ ð~ Ea ; gH E; gHÞ;
ð3Þ
pffiffiffiffiffiffiffiffiffi where k ¼ x l is the propagation constant, g ¼ l= is the ~a Þ are the new solutions to some known intrinsic impedance, ð~ Ea ; gH ~ and the fractional order (FO) a may possess both real fields ð~ E; gHÞ, ~a Þ represents the and complex values. The fractional solution ð~ Ea ; gH intermediate solutions between initial and dual to initial solutions. For a ¼ 0 and a ¼ 1 the corresponding field solutions become pffiffiffiffiffiffi
~ ðE ; gH Þ ¼ ð~ E; gHÞ; 1 1 ~ ~ ~ EÞ: ðE ; gH Þ ¼ ðgH; ~ ~0
~0
ð6Þ
~ ¼ 0; ~ þ k2 H r H þ 2pr H
ð7Þ
2~
pffiffiffiffiffiffi
where k ¼ x l and p ¼ xln with x being the angular frequency of the time harmonic fields [11]. It is identified that in an unbounded chiral medium, a right-circularly polarized and left-circularly polarized plane waves propagate with wave numbers.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 l x2 l2 n2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ xln þ x2 l x2 l2 n2 : kþ ¼ xln þ
ð8Þ ð9Þ
@ In the chiral slab, for no variation along y-axis @y ¼ 0 , the electromagnetic field distribution inside the chiral slab can be described as
Ef ðx; zÞ ¼
( 4 X
) ics z
Hf ðx; zÞ ¼
eibx ;
efs e
s¼1
The fractional operators are very useful because they make simple the description of a transitional and non-ideal situation in the electromagnetic problems. These problems are the extensions of some known and thoroughly studied canonical cases [17]. Naqvi et al. apply the fractional curl operator to: propagation in an infinite chiral medium [19,20,27], the fractional solutions and the corresponding sources [21], fractional surface waveguide, and fractional rectangular cavity resonator [22,23]. Lakhtakia [24] derived a theorem which shows that a dyadic operator commutes with curl operator, that can be used to find new solutions of Faraday and Ampere–Maxwell equations.
( 4 X
ð10Þ
) hfs eics z eibx ;
ð11Þ
s¼1
where f ¼ x; y or z; s ¼ 1; 2; 3 or 4; efs ; hfs are unknown coefficients, cs is the wave number along z-axis, and b ¼ b0 ib00 is the propagation constant along x direction. As the chiral medium supports two bulk plane waves; therefore, the four different wave numbers along z-axis are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kþ b2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ¼ c4 ¼ k2 b2 :
c1 ¼ c3 ¼
ð12Þ ð13Þ
Using Eqs. (10) and (11) in source free Maxwell equations, the relations of unknown coefficients of electromagnetic field may be
2. Formulation Consider a homogeneous, reciprocal, and isotropic chiral slab of uniform thickness d which is placed on a perfect electromagnetic conducting (PEMC) plate of infinite length as shown in Fig. 1. The chiral medium inside the slab is described by complex permittivity , complex permeability l, and complex chiral admittance n. The geometry is divided into two regions. Region 1 ð0 < z < dÞ contains chiral slab whereas region 0 ðz > dÞ is a free space having permittivity 0 and permeability l0 . To assume, the time harmonic dependency eixt , the constitutive relations for isotropic reciprocal chiral medium can be written as [1].
~ D ¼ ~ E in~ B; ~ ~ B ¼ lH þ in~ E:
2 E þ 2pr ~ E ¼ 0; r2~ Eþk ~
ð4Þ ð5Þ
Upon substituting the constitutive relations in source free Maxwell equations, the wave equations for electromagnetic fields inside the chiral slab take the form [15].
written
in
terms
of
ezs
as
exs ¼ cbs ezs ; eys ¼ ð1Þsþ1 ikbs ezs ;
ezs ; hzs ¼ ð1Þsþ1 gi ezs ; k1 ¼ k3 ¼ kþ ; c pffiffiffiffiffiffiffiffiffiffi k2 ¼ k4 ¼ k . Quantity gc ¼ l=c ; is the intrinsic impedance of
hxs ¼
ð1Þsþ1 bicgs ezs ; hys c
¼
bkgs c
chiral medium and c ¼ þ ln2 , the permittivity of chiral slab. Introducing n ¼ 0; l ¼ l0 and ¼ 0 in (6) and (7) reduced to wave equations of free space 2 E0 þ k0~ E0 ¼ 0; r2~ 2~
r H0 þ
2~ k0 H 0
ð14Þ
¼ 0;
ð15Þ
pffiffiffiffiffiffiffiffiffiffi
where k0 ¼ x l0 0 . The electromagnetic field expressions in free space (region 0) may be expressed as
Ef0 ðx; zÞ ¼ ef0 eðdzibxÞ ; Hf0 ðx; zÞ ¼ hf0 e
ðdzibxÞ
ð16Þ ;
ð17Þ
where ef0 ; hf0 are unknown coefficients and d is a positive valued decay rate guided mode quantity in transversal z-direction and is given by
d¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 k0 :
ð18Þ
Using the same procedure as discussed above for the unknown coefficient in chiral slab, the relations of the unknown coefficients of electromagnetic field outside (in region 0) the chiral slab in terms of electric and magnetic z-coefficients are ex0 ¼ idb ez0 ; ey0 ¼ xbl0 hz0 ; hx0 ¼ idb hz0 ; hy0 ¼ xb0 ez0 , where ez0 and hz0 are the un-
Fig. 1. Geometry of PEMC backed chiral slab.
known coefficients in free space region outside the chiral slab, which will be determined using boundary conditions at z ¼ d. Application of the appropriate tangential boundary conditions at z ¼ 0 (a planar interface between PEMC and chiral medium) [16] and z ¼ d (a planar interface between chiral medium and free space) yield the following unknown coefficients inside and outside the PEMC backed chiral slab.
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h
2
2
ez1 ¼ jaj2 fka c1 c2 k þ idk c2 gc2 þ jaj2 fika c22 kþ dk c1 gs2 i Þ2 k c2 B1 eic1 d ez4 ; þ ða ð19Þ h 2 2 ez2 ¼ jaj2 fka c1 c2 kþ idk c1 gc1 þ jaj2 fika c21 k þ dkþ c2 gs1 i ðaÞ2 kþ c1 B2 eic2 d ez4 ; ð20Þ ez0 ¼
2edd
Y 1 e z4 ;
ð21Þ
2edd Y 2 ez4 ; ixl0 D
ð22Þ
ix0 gc D
hz0 ¼
¼ g1 ðMgc iÞ; ka ¼ ez3 ¼ ez1 ; D ¼ ez2 =ez4 ; a ¼ g1 ðMgc þ iÞ; a
where
c
c
2 idk ; Y 1 ¼ ðkþ 22 2 2 2
2 k 21 Þjaj2 s1 s2 þ 2 c1 c2 dk jaj2 ½ 1
x0 gc ; B1 ¼ ka c1 þ idkþ ; B2 ¼ ka c2 þ c c c1 c2 k2 ½ðaÞ2 ðaÞ2 ; Y 2 ¼ ka c1 c2 k2 ½ðaÞ ðaÞ jaj c 2 2 2 2 2 k c1 s2 þ c2 kþ s1 c2 þ ka jaj ½kþ c2 þ k c1 s1 s2 ; c1 ¼ cosðc1 dÞ; c2 ¼ cosðc2 dÞ; s1 ¼ sinðc1 dÞ; s2 ¼ sinðc2 dÞ. Note that in the above relations the bar over quantities represent the complex conjugate. The dispersion relation for the surface wave modes [for detail see, [15]], guided in chiral slab is obtained as follow
h gc c2 k ðk20 c21 d2 k2þ Þs1 c2 dxk2 c1 c2 ð0 g2c þ l0 Þc1 c2 2aa þ 0:5dx 2
2 ðk 21
2
2 2 2 kþ Þð 0
c c
2 2 c 1 kþ ðk0 2
g2c þ l0 Þs1 s2 þ g c
c
2 d2 k Þc1 s2
2
þ ða þ a Þdxk c1 c2 ð0 gc l0 Þ ¼ 0: 0
i
ð23aÞ
00
When b ¼ k0 and b ¼ 0, the attenuation propagation constant outside the chiral slab becomes equal to zero (i.e., d ¼ 0) and (23a) takes the following form
k c1c sinðc1c dÞ cosðc2c dÞ þ kþ c2c sinðc2c dÞ cosðc1c dÞ ¼ 0 where c1c
ð23bÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 ¼ kþ k0 ; c2c ¼ k k0 . Eq. (23b) is similar to the one
reported in [15] and the references therein. Eqs. (23a) and (23b) can be numerically evaluated to find the propagation constant b and cut-off frequencies of the guided surface wave modes inside the chiral slab. 3. Expressions for guided electromagnetic fields The field expressions for region 1 are obtained by inserting (19) and (20) in the expressions containing the unknown coefficients defined for the inside of the chiral slab. Introducing the resultant expressions in Eqs. (10) and (11) , the following set of expressions is obtained
o 2iez4 n c1 c2 ½W 1 C 1 W 11 C 2 þ ½k c1 W 2 S1 þ W 12 kþ c2 S2 eibx ; bD ð24Þ o 2iez4 n 2 k ½W 2 C 1 W 22 C 2 ½kþ c2 W 1 S1 þ W 21 k c1 S2 eibx ; Ey ¼ bD ð25Þ o 2ez4 n ½k W 2 C 1 þ kþ W 12 C 2 ½c2 W 1 S1 c1 W 11 S2 eibx ; ð26Þ Ez ¼ D n o 2e gc Hx ¼ z4 c1 c2 ½W 1 C 1 þ W 11 C 2 þ ½k c1 W 2 S1 W 12 kþ c2 S2 eibx ; bD ð27Þ o 2ez4 n 2 1 1 ibx gc Hy ¼ k ½W 2 C 1 þ W 2 C 2 ½kþ c2 W 1 S1 W 1 k c1 S2 e ; bD ð28Þ Ex ¼
gc Hz ¼
2iez4 f½k W 2 C 1 kþ W 12 C 2 ½c2 W 1 S1 þ c1 W 11 S2 geibx ; D ð29Þ
where 2
Þ2 c1 þ jaj2 c2 ka ½ða Þ2 k c1 s1 þ jaj2 kþ c2 c2 ; W 1 ¼ dk ½ða Þ2 c1 jaj2 c2 þ d½ða Þ2 kþ c2 s1 jaj2 k c1 c2 ; W 2 ¼ ka c1 c2 ½ða 2
W 11 ¼ dk ½jaj2 c1 þ ðaÞ2 c2 ka ½jaj2 k c1 s1 þ ðaÞ2 kþ c2 c2 ; W 12
¼ ka c1 c2 ½jaj2 c1 ðaÞ2 c2 þ d½jaj2 kþ c2 s1 ðaÞ2 k c1 c1 ; 2
Þ2 c2 ka ½jaj2 k c1 s1 þ ða Þ2 kþ c2 c2 ; W 21 ¼ dk ½jaj2 c1 þ ða Þ2 c2 þ d½jaj2 kþ c2 s1 ða Þ2 k c1 c2 ; W 22 ¼ ka c1 c2 ½jaj2 c1 ða C 1 ¼ cosðc1 zÞ; C 2 ¼ cosðc2 zÞ; S1 ¼ sinðc1 zÞ; S2 ¼ sinðc2 zÞ: The electromagnetic field expressions in free space are obtained by substituting (21) and (22) into the unknown coefficient relations defined for free space, which are then, plugged-in into (16) and (17), yielding
2idez4 Y 1 edd1 ibx ; Db 2iez4 Y 2 edd1 ibx ; Ey0 ¼ Db 2ez4 Ez0 ¼ Y 1 edd1 ibx ; D 2de g0 Hx0 ¼ z4 Y 2 edd1 ibx ; Dbk0 2k0 ez4 g0 Hy0 ¼ Y 1 edd1 ibx ; Db 2ie g0 Hz0 ¼ z4 Y 2 edd1 ibx ; Dk0 Ex0 ¼
ð30Þ ð31Þ ð32Þ ð33Þ ð34Þ ð35Þ
where d1 ¼ d z. It is interesting to note that the expressions (24)– (35) and the dispersion relation given in Eq. (23a) along with cut-off frequency expression (23b) reduce to standard TE even modes, when n ¼ 0 and M ¼ 0 which is a planar waveguide composed of PMC backed achiral slab. However, when n ¼ 0 and M ¼ 1 the geometry of the problem reduces to PEC backed achiral slab. For these values, the field expressions (24)–(35) including Eqs. (23a) and (23b) convert to standard TE odd modes that compromise with the results shown in reference [15].
4. Fractional dual expressions for guided electromagnetic fields a
In this section fractional curl operator, curl , is used to fraction~a Þ alize the principle of duality that is, new set of solutions ð~ Ea ; gH to the Maxwell equations are obtained. These new solutions are known as intermediate or fractional solutions between the original solution and dual to original solution. According to [17], the fractional electromagnetic fields may be obtained as
~a Þ ¼ ðikÞa curla ð~ ~ ð~ Ea ; gH E; gHÞ;
ð36Þ
where g and k are the intrinsic impedance and wave number of unbounded dielectric medium respectively. In PEMC backed chiral ~ take the folslab, fractional solutions with original solution ð~ E; gc HÞ lowing form
~a Þ ¼ ðik Þa curla ð~ ~ ð~ Ea ; gc H E; gc HÞ;
ð37Þ
where gc is the intrinsic impedance of chiral material and k represent the right and left circular wave numbers in chiral slab. Eq. (37) can easily be reduce to Eq. (36) by setting n ¼ 0 (i.e., non-chiral material). Using above relations, the fractional electromagnetic fields both inside and outside the PEMC backed chiral slab are obtained as: In region 1 ð0 < z < dÞ
H. Maab, Q.A. Naqvi / Optics Communications 283 (2010) 2536–2546
4iez4 a ~ ^ Eax3 sðaÞ^zeibx ; Eax ¼ ½E CðaÞ^x Eax2 SðaÞy bD x1 4iez4 a ~ ^ þ Eay3 SðaÞ^zeibx ; ½E SðaÞ^x þ Eay2 CðaÞy Eay ¼ bD y1 4ez4 a ~ ^ þ Eaz3 CðaÞ^zeibx ; Eaz ¼ ½E sðaÞ^x Eaz2 SðaÞy D z1
4ez4 a ^ Hax3 sðaÞ^zeibx ; ½H CðaÞ^x Hax2 SðaÞy bD x1 4ez4 a ^ þ Hay3 SðaÞ^zeibx ; ¼ ½H SðaÞ^x þ Hay2 CðaÞy bD y1 4iez4 sðaÞ ^x Haz2 SðaÞy ^ þ Haz3 CðaÞ^z eibx ; ¼ Haz1 b D
2539
ð38Þ
~a ¼ gc H x
ð41Þ
ð39Þ
~a gc H y
ð42Þ
ð40Þ
~a gc H z
ð43Þ
Fig. 2. Fractional dispersion diagrams for modes in chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 . In the diagrams b0 is plotted versus frequency, for real and imaginary values of k; k separately. The dispersion diagrams corresponding to modes of achiral-PMC and chiral-PEMC waveguides where chiral and PEMC admittances are interrelated to fractional order parameter given by (53).
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In region 0 ðz > dÞ
Eax ¼ CðaÞEx þ SðaÞðgc Hx Þ; ðgc Hax Þ ¼ SðaÞEx þ CðaÞðgc Hx Þ;
o 4ez4 n 1þa a a 1þa ~ ^ þ ika0 bsðaÞ^z edd1 ibx ; ð44Þ Eax0 ¼ Y 1 di k0 CðaÞ^x i k0 SðaÞy bD ( ) a 1þa 4e di i b z4 1þa a a ~ ^ ^ ^ Ey0 ¼ Y 2 1a SðaÞx þ i k0 CðaÞy þ 1a SðaÞz edd1 ibx ; bD k0 k0
Eay ¼ CðaÞEy þ SðaÞðgc Hy Þ; ðgc Hay Þ ¼ CðaÞðgc Hx Þ SðaÞEy ;
(
ð45Þ
)
1þa
4ez4 d a ak ~ ^ þ ia ka0 CðaÞ^z edd1 ibx ; ð46Þ Eaz0 ¼ Y 1 ik0 sðaÞ^x i 0 SðaÞy b D b n o ~a ¼ 4ez4 Y 2 dia ka CðaÞ^x i1þa k1þa SðaÞy ^ bsðaÞ^z edd1 ibx ; g0 H x0 0 0 bDk0 ð47Þ ( ) 1þa a 4k e di i b a a ~ a ¼ 0 z4 Y 1 ^ 1a SðaÞ^z edd1 ibx ; g0 H SðaÞ^x i k0 CðaÞy y0 1a bD k0 k0 ð48Þ ( ) 1þa 1þa ~a ¼ 4ez4 Y 1 i d ka sðaÞ^x i k0 SðaÞy ^ þ i1þa ka0 CðaÞ^z edd1 ibx ; g0 H z0 b 0 k0 D b ð49Þ where n o a a a a a Eax1 ¼ i c1 c2 ðW 1 kþ C 1 W 11 k C 2 Þ þ ðkþ k c1 W 2 S1 þ W 22 k kþ c2 S2 Þ ; n o aþ1 aþ1 aþ1 aþ1 aþ1 Eax2 ¼ i ðkþ c2 W 1 S1 k c1 W 11 S2 Þ ðkþ k W 2 C 1 þ k kþ W 12 C 2 Þ ; n o a a a a Eax3 ¼ b ðkþ c2 W 1 S1 k c1 W 11 S2 Þ ðkþ k W 2 C 1 þ k kþ W 12 C 2 Þ ; ( !) Eay1 ¼ i
a
aþ1
a
a
c1 c2 ðkþ W 1 C 1 þ k W 21 C 2 Þ þ k a
2
a
c1
2
1a
kþ
1þa
W 2 S1 þ
c2
1a
k
W 22 S2
;
1þa
Eay2 ¼ i fk ðkþ W 2 C 1 k W 22 C 2 Þ ðkþ c2 W 1 S1 þ k c1 W 21 S2 Þg; ! ( ) W2 W 22 2 a a a 2 ðk Eay3 ¼ i b k C C c W S þ k c W S Þ ; 1 2 1 1 2 2 1 þ 1 1a 1a kþ k i a a a a Eaz1 ¼ fðkþ k c1 W 2 C 1 þ k kþ c2 W 12 C 2 Þ c1 c2 ðkþ W 2 S1 k W 12 S2 Þg; b a
i 1þa 1þa 1þa 1þa Eaz2 ¼ fðkþ k W 2 C 1 þ k kþ W 12 C 2 Þ ðkþ c2 W 1 S1 k c1 W 11 S2 Þg; b a
a
a
a
a
Eaz3 ¼ i fðkþ k W 2 C 1 þ k kþ W 12 C 2 Þ ðkþ c2 W 1 S1 k c1 W 11 S2 Þg; a
a
a
a
a
Hax1 ¼ fi c1 c2 ðW 1 kþ C 1 þ W 11 k C 2 Þ þ ðkþ k c1 W 2 S1 W 12 k kþ c2 S2 Þg; Hax2 ¼ i
1þa
1þa
1þa
1þa
1þa
fðkþ c2 W 1 S1 þ k c1 W 11 S2 Þ ðkþ k W 2 C 1 k kþ W 12 C 2 Þg; a
a
a
a
Hax3 ¼ bfðkþ c2 W 1 S1 þ k c1 W 11 S2 Þ ðkþ k W 2 C 1 k kþ W 12 C 2 Þg; ( Hay1 ¼ i
c1 c2 ðkaþ W 1 C 1 ka W 11 C 2 Þ þ k2
1þa
a
2
a
a
c1
W 2 S1 þ 1a
kþ
1þa
c2
!)
W 12 S2 1a
k
;
1þa
Hay2 ¼ i fk ðkþ W 2 C 1 þ k W 12 C 2 Þ ðkþ c2 W 1 S1 k c1 W 11 S2 Þg; ( ) ! W2 W 12 2 a a a 1 C þ C c W S k c W S Þ ; ðk Hay3 ¼ i b k 1 1 þ 2 1 1 2 1a 1 1a 2 kþ k a
a
a
a
Haz1 ¼ fðkþ k c1 W 2 S1 k kþ c2 W 12 S2 Þ þ c1 c2 ðkþ W 1 C 1 þ k W 11 C 2 Þg;
Eaz ¼ CðaÞEz þ SðaÞðgc Hz Þ; ðgc Haz Þ ¼ CðaÞðgc Hz Þ SðaÞEz :
ð50Þ
Similarly in region 0 ðz > dÞ
Eax0 ¼ CðaÞEx0 þ SðaÞðg0 Hx0 Þ; ðg0 Hax0 Þ ¼ SðaÞEx0 þ CðaÞðg0 Hx0 Þ; Eay0 ¼ CðaÞEy0 þ SðaÞðg0 Hy0 Þ; ðg0 Hay0 Þ ¼ CðaÞðg0 Hx0 Þ SðaÞEy0 ; Eaz0 ¼ CðaÞEz0 þ SðaÞðg0 Hz0 Þ; ðg0 Haz0 Þ ¼ CðaÞðg0 Hz0 Þ SðaÞEz0 :
ð51Þ
The discussion can be extended to show that how the characteristic parameters of chiral media and PEMC boundary are correlated with fractional order parameter a. Let assume an infinite chiral slab of thickness L. For normal incidence, the transmitted field in chiral b 31 , is defined by medium in term of transmission matrix T pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 31 b b 31 ¼ T 31 Rðjr k2 LÞ ¼ T 31 RðwÞ; jr ¼ j 0 l =l; ~ Et ¼ T E i where T 0 pffiffiffiffiffiffiffiffiffiffi k2 ¼ x l2 2 ; and RðwÞ denotes a matrix of rotation by an angle of w [1,18]. Using the fractional curl operator given in (36) for achiral medium, the fractional solution for field transmitted through an b a~ infinite achiral slab becomes ~ Ea;t ¼ T Ea;i where T a ¼ T 31 Rðap=2Þ [18]. Comparing the non-fractional transmitted field for chiral slab and fractional transmitted fields for achiral slab, we get the following relation.
pa=2 ¼ jr k2 L:
ð52Þ
It follows that the fractional field solution in chiral medium can be constructed as the field traverses through a non-chiral medium. From the above relation, it is suggested that for a wave transmission in chiral material, the fractional order a may be expressed in terms of chirality and the thickness of the chiral slab. In case of reflected field from an interface between PEC and bi-isotropic medium, the fractional order a, is associated with Tellegen parameter but independent of chirality parameter [18,25]. In view of the above statement, we assume that the fractional order (FO) a can be adjusted through chiral admittance and slab thickness for the guided electromagnetic fields in PEMC backed chiral slab and is given by
ap 2
¼ nd:
ð53Þ
The relationship between fractional order a and PEMC admittance is considered as M ¼ tanðap=4Þ [see for instance, [18]]. According to this relation, when a ¼ 0 the PEMC boundary reduces to PMC boundary whereas when a ¼ 1, and a ¼ 2, one can get the corresponding PEMC and PEC boundaries. Using the interrelationship among chiral admittance, PEMC admittance, and fractional order, we deduced the following observations. When a ¼ 0, the corresponding chiral and PEMC admittances become zero and the hybrid modes (HE) in PEMC backed chiral slab transform to standard TE even modes in PMC backed achiral slab. When 0 < a 6 1, the corresponding chiral and PEMC admittances reduce to some nonzero values. Therefore, the modes in the slab correspond to fractional intermediate solutions and the geometry of the problem behaves as PEMC backed chiral slab.
a
i 1þa 1þa 1þa 1þa Haz2 ¼ fðkþ k W 2 C 1 k kþ W 12 C 2 Þ ðkþ c2 W 1 S1 þ k c1 W 11 S2 Þg; b a
a
a
a
a
Haz3 ¼ i fðkþ k W 2 C 1 k kþ W 12 C 2 Þ ðkþ c2 W 1 S1 þ k c1 W 11 S2 Þg; ap a p ;SðaÞ ¼ sin ;sðaÞ ¼ sinðapÞ: CðaÞ ¼ cos 2 2
It may be noted that the fractional curl operator describes the polarization reversal of electromagnetic field [18]. Therefore, we can also represent the fractional field expressions, inside and outside the PEMC backed chiral slab, in terms of original electric and magnetic field solutions. In region 1 ð0 < z < dÞ
5. Numerical results The propagation constants c1 and c2 in chiral slab while the attenuation constant d in free space along the transversal direction (z-axis) depend on b ¼ b0 jb00 . The real and imaginary parts of propagation constant b represent phase constant and attenuation constant along the direction of propagation (x-axis). The values of these constants can be obtained numerically using the dispersion and critical expressions in (23a) and (23b) over a given frequency range of interest. It is to be noted that the critical condition for PEMC backed chiral slab geometry is obtained when
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b0 ¼ k0 and b00 ¼ 0, that is, the condition at which the attenuation constant d becomes zero. Therefore, to study the surface waves closely confined to the chiral slab the values of b must be kept between k0 and real or imaginary values of k . It is observed that k0 is the lowest value of b in any guided surface wave mode. If we assume the value of b less than k0 , then the value of attenuation constant d in free space becomes imaginary. In that case, we get the so-called radiation mode and the structure behaves like a leakywave antenna [15]. It is worth pointing out that k is the highest value of propagation constant b of any mode in chiral slab. If the value of b is kept higher than the value of k then c1 and c2 become imaginary and all modes associated with the propagation constant would get attenuated inside the chiral slab. Beside, this if b > k , Eq. (23a) becomes negative and imaginary and it cannot be converged to zero. It is noted that when n ¼ 0, the geometry of the problem reduces to PEMC backed achiral slab and the wave number of chiral slab k is reduced to k, i.e., the wave number of ordinary dielectric slab (achiral). In order to analyze the guided surface wave modes in achiral slab the propagation constant must be limited between k0 and real or imaginary values of k.
Throughout
the
discussion,
it
is
assumed
that
l ¼ l0 ð1 þ i0:5Þ; ¼ 20 and the values of chiral and PEMC admittances as interrelated to fractional parameter a given in (53). The plots for dispersion curves and surface waves are obtained by considering real or imaginary values of k and k. In Figs. 2 and 3, the dispersion curves are plotted for phase constant b0 and attenuation constant b00 separately. These plots are constructed for various values of fractional order a. In Fig. 2, when a ¼ 0, chiral and PEMC admittances become zero and the geometry of the problem reduces to PMC backed achiral slab. As an outcome, the dispersion curves represent the standard TE even modes. The plot shown in Fig. 2, for a ¼ 0, is obtained by keeping the values of b0 in the range of k0 and Rfkg (while putting b00 ¼ 0) satisfying Eqs. (23a) and (23b). It is to be noted that for a ¼ 0, the dispersion curves are not shown for the values of b0 between k0 and Ifkg because these values of b0 do not satisfy Eqs. (23a) and (23b). In Fig. 2, the values of b0 are also examined at a ¼ 0:25, in the limit of k0 and Rfkþ g. Whereas for a ¼ 0:25i, the values of b0 are plotted between k0 and Rfk g. The values of b0 corresponding to a ¼ 0:25 and 0:25i represent the intermediate fractional modes solution in the PEMC
Fig. 3. Fractional dispersion diagrams for modes in chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 . In the diagrams b00 is plotted versus frequency, for real and imaginary values of k and k by means of the fundamental values of b0, separately. The dispersion diagrams corresponding to modes of achiral-PMC and chiral-PEMC waveguides where chiral and PEMC admittances are interrelated to fractional order parameter given by (53).
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Fig. 4. 2D and 3D representation of fractional surface waves of Eay ; Hax ; Haz inside achiral-PMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for TE3 mode where a ¼ 0:0 corresponds to zero chiral and PEMC admittances.
Fig. 5. 2D and 3D representation of fractional surface waves of Eax inside chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for hybrid mode HE3 where fractional order parameter is interrelated with chiral and PEMC admittances given by (53).
H. Maab, Q.A. Naqvi / Optics Communications 283 (2010) 2536–2546
2543
Fig. 6. 2D and 3D representation of fractional surface waves of Eay inside chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for hybrid mode HE3 where fractional order parameter is interrelated with chiral and PEMC admittances given by (53).
Fig. 7. 2D and 3D representation of fractional surface waves of Eaz inside chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for hybrid mode HE3 where fractional order parameter is interrelated with chiral and PEMC admittances given by (53).
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Fig. 8. 2D and 3D representation of fractional surface waves of Eay0 ; Hax0 ; Haz0 in free space region of achiral-PMC d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for TE3 mode where a ¼ 0:0 corresponds to zero chiral and PEMC admittances.
waveguide
with
Fig. 9. 2D and 3D representation of fractional surface waves of Eax0 , in free space region of chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for hybrid mode HE3 where fractional order parameter is interrelated with chiral and PEMC admittances given by (53).
H. Maab, Q.A. Naqvi / Optics Communications 283 (2010) 2536–2546
2545
Fig. 10. 2D and 3D representation of fractional surface waves of Eay0 , in free space region of chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for hybrid mode HE3 where fractional order parameter is interrelated with chiral and PEMC admittances given by (53).
Fig. 11. 2D and 3D representation of fractional surface waves of Eaz0 , in free space region of chiral-PEMC waveguide with d ¼ 32 mm; l ¼ l0 ð1 þ i0:5Þ; ¼ 20 ; f ¼ 5 GHz. The plots are drawn for hybrid mode HE3where fractional order parameter is interrelated with chiral and PEMC admittances given by (53).
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H. Maab, Q.A. Naqvi / Optics Communications 283 (2010) 2536–2546
backed chiral slab. From the resultant graphs in Fig. 2, it is observed that for real a the values of chiral and PEMC admittances are real and hence Rfkþ g is dominant over Rfk g. Whereas, for imaginary a the values of chiral and PEMC admittances are imaginary and the phenomenon is vice versa. Besides, it is observed from Fig. 2 that for a ¼ 0:25, the number of hybrid modes are greater than that of a ¼ 0:25i. It is noted in Fig. 2, that for a ¼ 0:25; Ifkþ g is more dominant as compared to the negligible values of Ifk g. In the adverse plotting, for a ¼ 0:25i, the values of Ifk g is dominant over Ifkþ g (negligible) where Ifk g is more negative. This phenomena represent some novel property of PEMC backed chiral slab. The phenomenon like this, is helpful in focusing either RCP or LCP plane wave in bi-isotropic media by suppressing one of the CP wave [26]. A similar discussion can be extended for other values of fractional order a. In Fig. 3, the plots are drawn for attenuation constant b00 using the zero critical frequency curves of b0 (first curve in each plot of Fig. 2). The dispersion curves of b00 are investigated at a ¼ 0; 0:25; 0:25i respectively, for real or imaginary values of k and k. The numerical results of fractional surface waves of fractional electromagnetic fields inside and outside of the PEMC backed chiral slab, are shown in Figs. 4–11, in two and three dimensions. These plots are drawn for real and imaginary values of k and k, using TE3 ; HE3 (3rd curve of b0 in each plots of Fig. 2) modes for the following values of a ¼ 0; 0:25; 0:25i respectively. When a ¼ 0, we get Ey ; Hx and Hz field components as shown in Fig. 4. In the same figure, the combined effect of Ey and Hz is standing wave pattern while that of Ey and Hx is propagating wave. The effect of these fields represents standard TE mode (even). In Fig. 5, for a ¼ 0:25; 0:25i with real and imaginary values of k and k, the fractional surface waves of Eax are in phase. In the same figure, Eax wave pattern for a ¼ 0:25i and Rfkþ ; kg is 1800 out of phase with that of Eax for a ¼ 0:25; 0:25i, having real and imaginary values of k and k. It is to be noted that in Fig. 5, for a ¼ 0:25 the fractional surface wave of Eax represents a decaying wave for Ifk ; kg. The situation shown in Fig. 5 reveals that, for a ¼ 0:25i with Rfk ; kg, the wave behavior is more strong, whereas for a ¼ 0:25 and 0:25i with the real and imaginary values of k and k it exhibits a weak behavior. In the light of aforementioned concepts, the other plots in the Figs. 6–11 can be elaborated in a meaningful manner. 6. Conclusions Analytical solution of dispersion relation and electromagnetic field distributions as well as the fractional field solution of hybrid modes in a PEMC backed chiral slab are studied. The numerical results of fractional dispersion behavior and fractional field distribution in PEMC backed chiral slab are examined in detail with reference to real or imaginary values of a; n; M, k and k. For the analysis of dispersion curves and fractional surface waves, fractional parameter is directly proportional to the chiral admittance and PEMC admittance. These relationships yield increasing number of hybrid propagating surface wave modes inside the PEMC backed chiral slab for either real or imaginary values of a. It is observed that for real a and Rfk ; kg, the RCP wave number is greater than the LCP wave number. While for imaginary a and Rfk ; kg the
resultant dispersion curves represent the dominance of LCP wave number over RCP wave number. It is worth pointing out that when a ¼ 0:25 and Ifk ; kg there exists only RCP wave number whereas the value of LCP wave number is negligible. Similarly, when a ¼ 0:25i along with imaginary values of k and k there exists only LCP wave number as compared to the negligible value of RCP wave number. The fractional surface waves are also checked for a ¼ 0; 0:25; 0:25i along with real or imaginary values of n, M, k and k. Those correspond to initial and intermediate fractional surface wave patterns in PMC backed achiral and PEMC backed chiral slabs respectively. When a ¼ 0 and the values of k are real, the fractional surface waves reduce to standard TE mode (even). It is pointed out that when a ¼ 0 and the values of k are imaginary, there exist no mode solution in the specified numerical limit (i.e. Ifkg < b0 < k0 ). It is to be noted that for a ¼ 0:25 and Ifk ; kg, the fractional surface waves of electric fields components represent decaying waves due to negligible LCP wave number. While for a ¼ 0:25i and Ifk ; kg, the fractional surface waves of electric fields components denote propagating waves due to negligible RCP wave number. Acknowledgements We are grateful to the reviewers for their valuable comments. References [1] I.V. Lindell, A.H. Sihvola, S.A. Tretyakov, A.J. Viitanen, Electromagnetic waves in chiral and bi-isotropic media, Artech House, Bosten, 1994. [2] A. Lakhtakia, V.K. Varadan, V.V. Varadan, Time Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics, vol. 335, Springer-Verlag, New York, 1989. [3] S. Bassari, C.H. Papas, N. Engheta, Journal of the Optical Society of America A 5 (1988) 450. Errata: 7 (1990) 2154. [4] A. Lakhtakia, V.K. Varadan, V.V. Varadan, Zeitschrift fuer Natureforschung 45a (1989) 639. [5] P.S. Reese, A. Lakhtakia, Optik 86 (1990) 47. [6] A. Lakhtakia, V.K. Varadan, V.V. Varadan, Optik 87 (1991) 77. [7] C. Eftimiu, L. Wilson Pearson, Radio Science 24 (1989) 351. [8] N. Engheta, P. Pelet, Microwave and Optical Technology Letters 14 (1989) 593. [9] P. Pelet, N. Engheta, IEEE Transactions on Antennas and Propagation 38 (1990) 90. [10] F. Mariotte, Engheta, Journal of Electromagnetic Waves and Applications 7 (1993) 1307. [11] R. Oussaid, B. Haraoubia, Canadian Journal of Physics 82 (2004) 367. [12] V.K. Varadan, A. Lakhtakia, V.V. Varadan, Journal of Wave Material Interaction 3 (1988) 267. see also, ‘‘Erratum,” 3 (1989) 335. [13] F. Mariotte, P. Pelet, N. Engheta, Progress in Electromagnetics Research 9 (1994) 311. [14] H. Cory, Journal of Electromagnetic Waves and Application 9 (1995) 805. [15] J.C. da, S. Lacava, F. Lumini, Progress in Electromagnetics Research 16 (1997) 285. [16] I.V. Lindell, A.H. Sihvola, IEEE Transactions on Antennas and Propagation 53 (2005) 3005. [17] N. Engheta, Microwave and Optical Technology Letters 17 (1998) 86. [18] M.V. Ivakhnychenko, Telecommunications and Radio Engineering 67 (2008) 567. [19] Q.A. Naqvi, G. Murtaza, Optics Communications 178 (2000) 27. [20] A. Hussain, Q.A. Naqvi, Progress in Electromagnetics Research 59 (2006) 199. [21] Q.A. Naqvi, A.A. Rizvi, Progress in Electromagnetics Research 25 (2000) 223. [22] H. Maab, Q.A. Naqvi, Progress In Electromagnetics Research C 1 (2008) 199. [23] H. Maab, Q.A. Naqvi, Progress In Electromagnetics Research B 9 (2008) 69. [24] A. Lakhtakia, Microwave and Optical Technology Letters 28 (2001) 385. [25] V.I. Maksym, Eldar. I. Veliev, Serbia Nis (2007) 26. [26] C. Monzon, D.W. Forester, Physical Review Letters 95 (2005) 123904. [27] S.A. Naqvi, Q.A. Naqvi, A. Hussain, Optics Communications 266 (2006) 404.