# Polarization conversion of electromagnetic waves by Faraday chiral media

**ABSTRACT** The reflection of a normally incident plane wave due to the interface between vacuum and a Faraday chiral medium (FCM) supported by a perfect electric conductor was rigorously derived. Numerical results strongly indicated that arbitrary polarization conversions are realizable from the incident plane wave to the reflected plane wave by properly choosing the constitutive parameters of the FCM.

**0**Bookmarks

**·**

**105**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, a low-loss magnetic metamaterial configuration consisting of coupled radiative and dark resonators is proposed based on analog of electromagnetically induced transparency. Full-wave numerical simulations are carried out to validate the metamaterial. Absorptions curves, transmission spectrums, surface current distributions, and effective constitutive parameters for the metamaterial are presented. These results, showing a low-loss transparency window and strong dispersion in the effective permeability with negative values, are in a good agreement with each other.IEEE Transactions on Magnetics 11/2011; · 1.42 Impact Factor

Page 1

Polarization conversion of electromagnetic waves by Faraday chiral media

Fan-Yi Meng (孟繁义?,1,2,a?Kuang Zhang (张狂?,1,a?Qun Wu (吴群?,1,2,a?and

Le-Wei Li (李乐伟?3,b?

1School of Electronics and Information Engineering, Harbin Institute of Technology, Heilongjiang 150001,

China

2State Key Laboratory of Millimeter Waves, Nanjing 210096, China

3Department of Electrical and Computer Engineering, National University of Singapore, Kent Ridge

119260, Singapore

?Received 10 October 2009; accepted 13 January 2010; published online 2 March 2010?

The reflection of a normally incident plane wave due to the interface between vacuum and a Faraday

chiral medium ?FCM? supported by a perfect electric conductor was rigorously derived. Numerical

results strongly indicated that arbitrary polarization conversions are realizable from the incident

plane wave to the reflected plane wave by properly choosing the constitutive parameters of the

FCM. © 2010 American Institute of Physics. ?doi:10.1063/1.3310640?

I. INTRODUCTION

The polarization of the electromagnetic ?EM? fields is

widely concerned in the EM community. Conventionally, op-

tical gratings, dichroic crystals, and the Brewster and bire-

fringence effects are employed to manipulate the polariza-

tion, etc.1–3Here, an alternative approach based on Faraday

chiral mediums ?FCMs? is proposed. FCMs have attracted

considerable attention recently. This increased interest stems

mostly from the desire to understand new response charac-

teristics of complex mediums due to their additional degrees

of freedom and eventually exploit novel phenomena for tech-

nologicalpurposes.4–9

FCM

activity—as exhibited by isotropic chiral mediums10,11—with

Faraday rotation—as exhibited by gyrotropic mediums;12–14

therefore, FCM may be theoretically conceptualized as ho-

mogenized composite mediums15,16arising from blending to-

gether of isotropic chiral mediums with either magnetically

biased ferrite17or a magnetically biased plasma.6

In this paper, we show that a specific FCM reflector can

be employed to manipulate the polarization states of EM

waves. In following sections, reflection formulas of an inci-

dent EM wave from the vacuum to a FCM slab terminated

by perfectly electric conductor ?PEC? wall are rigorously de-

rived, and the deep mathematical analysis for the reflection

formulas is conducted to help propose and understand the

principle of manipulating polarization states of the reflected

wave and realizing conversion between polarization states of

the reflected and incident waves of the FCM reflector. Polar-

ization characteristics of the reflected wave are investigated

and the approach to realize the conversion between polariza-

tion states of the reflected and incident waves is discussed.

Results show that all possible polarization states ?linear, cir-

cular, and elliptic? for the reflected wave and arbitrary polar-

ization conversion are realizable via adjusting constitutive

parameters of the FCM. Particularly, there is a complete con-

combinenaturaloptical

version between two perpendicular linear polarizations of in-

cident and reflected waves under certain conditions. Numeri-

cal calculations are performed to demonstrate some polariza-

tion conversions and results are in good agreement with

theoretical prediction.

II. PLANE WAVE PROPAGATION IN A FCM

In preparation for solving a boundary-value problem in

next step, let us consider plane wave propagation in a FCM

characterizedbythe frequency-domain

relations6,17

constitutive

D?FCM?r ?? = ? ? · E?FCM?r ?? + ??· H?FCM?r ??,

B?FCM?r ?? = − ??? E?FCM?r ?? + ? ? ? H?FCM?r ??,

with constitutive dyadics6,17

?1?

? ? = ?0??tI?− i?gez? I?+ ??z− ?t?ezez?,

??= i??0?0??tI?− i?gez? I?+ ??z− ?t?ezez?,

? ? = ?0??tI?− i?gez? I?+ ??z− ?t?ezez?.

?2?

Here, I?=exex+eyey+ezezis the identity dyadic. Thus, the dis-

tinguished axis of the FCM is chosen to be z-axis.

Let us also confine ourselves to propagation in the

z-direction. Hence, plane waves are with field phasors

E?FCM?z? =?Ex

H?FCM?z? =?Hx

Ey?e−ik0k˜z,

Hy?e−ik0k˜z= ? ?−1??k0k˜/???ez? E?FCM?

+ ??E?FCM?,

?3?

where k˜is the relative wave number. There are four indepen-

dent relative wave number6

k˜1=??t+ ?g·??t+ ?g− ?t− ?g,

a?Authors to whom correspondence should be addressed. Electronic ad-

dresses: blade@hit.edu.cn, zhangkuang@126.com, and qwu@hit.edu.cn.

b?Electronic mail: lwli@nus.edu.sg.

JOURNAL OF APPLIED PHYSICS 107, 054104 ?2010?

0021-8979/2010/107?5?/054104/6/$30.00© 2010 American Institute of Physics

107, 054104-1

Downloaded 14 Oct 2010 to 128.134.54.103. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

Page 2

k˜2= −??t+ ?g·??t+ ?g− ?t− ?g,

k˜3=??t− ?g·??t− ?g+ ?t− ?g,

k˜4= −??t− ?g·??t− ?g+ ?t− ?g.

?4?

Moreover, there is Ex/Ey=i for k˜=k˜1,k˜2and Ex/Ey=−i for

k˜=k˜3,k˜4. Therefore, the complete representation of plane

waves in the FCM is given by

E?FCM?z? =?i

1??E1e−ik0k˜1z+ E2e−ik0k˜2z? +?− i

1??E3e−ik0k˜3z+ E4e−ik0k˜4z? =?

?? ?−1?

i?E1e−ik0k˜1z+ E2e−ik0k˜2z− E3e−ik0k˜3z− E4e−ik0k˜4z?

E1e−ik0k˜1z+ E2e−ik0k˜2z+ E3e−ik0k˜3z+ E4e−ik0k˜4z?,

i?k˜1E1e−ik0k˜1z+ k˜2E2e−ik0k˜2z− k˜3E3e−ik0k˜3z− k˜4E4e−ik0k˜4z??

H?FCM?z? = ? ?−1??k0k˜/???ez? E?FCM? + ??E?FCM? =k0

+ ???

− k˜1E1e−ik0k˜1z− k˜2E2e−ik0k˜2z− k˜3E3e−ik0k˜3z− k˜4E4e−ik0k˜4z

i?E1e−ik0k˜1z+ E2e−ik0k˜2z− E3e−ik0k˜3z− E4e−ik0k˜4z?

E1e−ik0k˜1z+ E2e−ik0k˜2z+ E3e−ik0k˜3z+ E4e−ik0k˜4z?.

?5?

III. FCM REFLECTOR

The studied model system is shown in Fig. 1?a?, which

consists of a FCM layer ?of thickness d? with constitutive

dyadics expressed by Eq. ?2?, put on top of a PEC substrate

?at z=d?. The studied model system is the theoretical model

system for a practical FCM reflector in parallel-plate

waveguides with coaxial line feeding, as shown in Fig. 1?b?.

In practice, the PEC boundaries and perfect magnetic con-

ductor boundaries of parallel-plate waveguides can be real-

ized by metallic sheets and artificial magnetic conductor sur-

faces, respectively. For the practical FCM reflector shown in

Fig. 1?b?, the output wave is the vector sum of the direct and

reflected waves. We consider the reflection and refraction

properties of the structure shown in Fig. 1?a?, when a mono-

chromatic EM wave with a wave vector of k?in=k0ezand

polarization in xy-plane strikes on the surface. The incident

wave phasors are given by

E?in?z? =?Eix

Eiy?e−ik0z,

H?in?z? =

k0

?0?ez? E?in?z? =

k0

?0??− Eiy

Eix?e−ik0z,

?6?

where Eixand Eiyare the x and y components of the incident

electric field, respectively. The reflected wave phasors are

given by

E?r?z? =?Erx

Ery?eik0z,

H?r?z? =

k0

?0??− ez? ? E?r?z? =

k0

?0??Ery

− Erx?eik0z,

?7?

where Erxand Eryare the x and y components of the reflected

electric field. Moreover, the reflected field can be related

with the incident field using the reflection factors as follows:

?Erx

Ery?=?Rxx Rxy

Ryx Ryy??Eix

Eiy?.

?8?

Here, Rxxand Rxyare x-direction reflection factors arising

from Eixand Eiy, respectively; Ryxand Ryyare y-direction

reflection factors arising from Eixand Eiy, respectively.

in

k

in

H

in

E

r

E

r

H

r k

ix

E

iy

E

x

y

z

d

FCM

PEC

rx

E

ry

E

μ ξ

ε (

)

(a)

PEC

PEC

PEC

Coaxial Line

Output

Wave

Input Wave

Reflected Wave

Direct Wave

y

z

x

(b)

FCM

PEC

PMC

FIG. 1. ?Color online? ?a? Geometry of the studied model system in this

paper. ?b? Geometry of a practical FCM reflector in parallel-plate

waveguides with coaxial line feeding.

054104-2Meng et al. J. Appl. Phys. 107, 054104 ?2010?

Downloaded 14 Oct 2010 to 128.134.54.103. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

Page 3

In addition, according to Eq. ?5?, in the FCM region, the

refracted wave phasors are given by

E?FCM?z? =?

j=1

4

E?FCM,0jeik0k˜j?z−d?,

H?FCM?z? =?

j=1

4

H?FCM,0jeik0k˜j?z−d?.

?9?

Because the FCM is terminated by the PEC wall, there is

E?FCM?d? · ?ex+ ey? = 0.

Substituting Eq. ?9? into Eq. ?10? leads to

?10?

E?FCM,01= − E?FCM,02,

E?FCM,03= − E?FCM,04,

T1= − T2,

T3= − T4.

?11?

Moreover, following boundary conditions at the interface be-

tween the FCM and vacuum should be satisfied:

E?FCM?0? = E?in?0? + E?r?0?,

H?FCM?0? = H?in?0? + H?r?0?.

Inserting Eqs. ?6?, ?7?, and ?9? into Eq. ?12?, the reflection

factors can be calculated as follows:

?12?

Rxx= Ryy= −B?D??g

2− ?t

2? + EC + 2G??t− ?g??

GBL − ?DL + FC???t+ ?g? + EBC,

Rxy= − Ryx

= i?− FC + 2D??g− ?t????g+ ?t? − G??g− ?t?B

GBL − ?DL + FC???t+ ?g? + EBC

.

?13?

Here, D, E, F, and G are introduced to simplify the expres-

sions of reflection factors, and they are given by

D = ?A2 − 1??A1 − 1?,

E = ?A2 + 1??A1 + 1?,

F = ?A2 + 1??A1 − 1?,

G = ?A2 − 1??A1 + 1?,

A1 = ei4?Bd/?0,

A2 = ei4?Cd/?0,

B =???g+ ?t???g+ ?t?,

C =???g− ?t???g− ?t?,

L = 2?t− 2?g+ ?g− ?t.

?14?

It can be seen that above reflection formulas are very com-

plicated; so it is necessary to discuss their basic mathemati-

cal characteristics first to make the analysis of polarization

characteristics easy and clear.

Firstly, if we define

r ˆ =Rxx

Rxy

,

?? = arg?r ˆ? = arg?Rxx

Rxy?,

?? = ?t− ?g,

?15?

it can be found through mathematical analysis that ?? is

closely related to ??. Particularly, in the nondissipative case,

??=0 leads to

arg?G

F?= 0 or ?,arg?D

F?= ??

2,

arg?E

F?

= ??

2.

?16?

Therefore,

???0? = ???????=0

=?arg?

=?

= 0 or ?.

B?D??g

2− ?t

2? + EC?

i?FC??g+ ?t? + GB??g− ?t???????=0

arg?

B?D

F??g

2− ?t

2? +E

FC?

i?C??g+ ?t? +G

FB??g− ?t????

???=0

?17?

Equation ?17? corresponds to conditions where r ˆ=Rxx/Rxyis

a real number. In order to demonstrate the relation between

???? and ????, ???? is calculated with different ???? when

?t=0.5, ?z=0.5, ?t=1.17, ?g=0.4, ?t=2, and ?g=0.1. Calcu-

lated results are shown as a function of the normalized sub-

strate thickness d/?0in Fig. 2?a?, where ?0is the wavelength

of the incident wave in free space. In Fig. 2?a?, the solid line

represents ???0?, the dot-dashed line represents ???? with

????=0.3, and the dashed line represents ???? with ????

=0.7. It can be seen that, when ??=0, ???? either is 0 or ?.

Moreover, ???? is gradually far from ???0? as the absolute

value of ???? increases. The ???0? for a dissipative FCM with

??=0, ?t=1.17−0.117i, ?g=0.4−0.04i, ?t=2−0.2i, and ?g

=0.1−0.01i is also calculated and shown as a function of

d/?0in Fig. 2?b?. According to Ref. 18, the ratio of the

imaginary part to the real part of constitutive parameters of

chiral materials is usually less than 0.1. Therefore, the ratio

is set to 0.1 in all the calculations for dissipative FCM in this

paper. The ???0? in Fig. 2?a? is reploted with dot-dashed line

in Fig. 2?b?. From Fig. 2?b?, it can be seen that although the

gap of ???0? between the dissipative FCM and the nondissi-

pative FCM widens as d/?0increases because of the effect

of dissipation, it is very small when d/?0?0.4.

Secondly, it can be derived from Eq. ?14? that when

054104-3Meng et al. J. Appl. Phys. 107, 054104 ?2010?

Downloaded 14 Oct 2010 to 128.134.54.103. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

Page 4

A2 = − A1 = 1,

?18?

there is D=E=G=0, which leads to

Rxx= 0,

Rxy= i,

r ˆ = 0.

?19?

Similarly, when

A2 = A1 = 1,

?20?

there is D=F=G=0, which leads to

Rxx= − 1,

Rxy= 0,

?r ˆ? → ?.

?21?

From Eq. ?14?, it is found that conditions ?18? and ?20? can

be satisfied by same groups of FCM’s constitutive dyadics ? ?,

? ?, and ??with different normalized FCM substrate thickness

d/?0.

Thirdly, it can also be derived from Eq. ?14? that when

A2 = − A1 = − 1,

?22?

there is D=E=F=0, which leads to

r ˆ = i

2??

?t− ?g

.

?23?

Similarly, when

A2 = A1 = − 1,

?24?

there is D=F=G=0, which leads to

r ˆ = iB??g− ?t?

2??

.

?25?

Obviously, in the two cases, r ˆ is an imaginary number when

the FCM is nondissipative.

IV. POLARIZATION STATES OF THE REFLECTED

WAVE

Define the ratio of the x component to y component of

the incident electric field as

e ˆi=Eix

Eiy

.

?26?

Therefore, when e ˆiis a real number, the incident wave is of

linear polarizations, and the value of e ˆiis determined by the

polarization angle. When e ˆi=?i, the incident wave is of

circular polarizations and +i denotes the right-circular polar-

ization ?the incident wave propagates in +z-direction? while

−i denotes the left-circular polarization. When e ˆiis other

complex number, the incident wave is of elliptical polariza-

tions.

Assume

Erx

Ery

= pEix

Eiy

= pe ˆi,

?27?

where p is a complex number and used to describe the de-

gree of the conversion between polarization states of incident

and reflected waves. After substituting Eq. ?8? into Eq. ?27?,

the relation between p and Ruv?u,v=x,y? is obtained as

e ˆir ˆ + 1

r ˆ − e ˆi

p =1

e ˆi

,

r ˆ =Rxx

Rxy

.

?28?

Below we will discuss the conversion between polarization

states of incident and reflected waves through analyzing the

possible value range of p.

A. Conversion of linear polarizations to linear

polarizations with different polarization directions

In this case, both e ˆiand p are real numbers, which re-

quires that r ˆ must be a real number. Because the value range

of p is determined by that of r ˆ, it is necessary to analyze the

possible value range of r ˆ. According to Eqs. ?20? and ?21?,

when A2=−A1=1 or A2=A1=1, ?r ˆ? has the value range of

?0,+??. According to Eq. ?17?, when ??=0, r ˆ is a real num-

ber. Because A1 and A2 are independent on ?gand ?t, con-

ditions A2=A1=1 and ??=0 can be simultaneously satis-

fied, which means that r ˆ can be a real number and have the

value range of ?−?,+??. Consequently, according to Eq.

?28?, p can take any value in the range of ?−?,+?? through

choosing proper parameters of the FCM reflector. This

means that the FCM reflector can realize an arbitrary conver-

sion between the linear polarization directions of incident

and reflected waves, including a complete conversion be-

tween two perpendicular linear polarizations. Figure 3 shows

the calculated magnitude of p as a function of d/?0when

e ˆi=1, d=30 mm, ?t=0.5−0.05i, ?g=0.5−0.05i, ?t=1.17

−0.117i, ?g=0.4−0.04i, ?t=2−0.2i, and ?g=0.1−0.01i. The

constitutive parameters will satisfy ??=0 and A2=A1=1 if

their imaginary parts equal zero. From Fig. 3, it can be seen

that the maximum ?p? can reach is about 25 rather than in-

0.00.40.6 0.81.0

0.0

1.0

2.0

3.0

d/λ0

|Δφ|

|Δξ|=0.7

|Δξ|=0.3

|Δξ|=0

|Δξ|=0.7

|Δξ|=0.3

|Δξ|=0

0

0.3

|Δξ|=0.7

0.2

(a)

0.00.4 0.6 0.81.0

d/λ0

0.2

0.0

1.0

2.0

3.0

|Δφ0|

（b）

FIG. 2. ?Color online? ?a? Calculated results of ???? as a function of d/?0

when ????=0.7 ?dash line?, 0.3 ?dot-dashed line?, and 0 ?solid line?. ?b?

Comparison of ???0? between the dissipative FCM and the nondissipative

FCM.

054104-4Meng et al. J. Appl. Phys. 107, 054104 ?2010?

Downloaded 14 Oct 2010 to 128.134.54.103. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

Page 5

finity because of the effect of the dissipation. However, in

practice, the values range of ?0, 25? of ?p? can be used to

realize almost all the linear polarization conversions. Figure

4 shows the phase of p as a function of d/?0. It can be seen

that the dissipation results in the phase of p deviating from 0

or ?, which will cause the axial ratio of the reflected wave to

drop from infinity ?linear polarizations? to a finite value ?el-

liptical polarizations?. The deviation gradually increased

with the increment of d/?0. However, it also can be seen that

when d/?0?0.4 the phase of p is very close to 0 or ?, so the

effect of the dissipation on the axial ratio of the reflected

wave is very small.

B. Conversion from linear, circular, and elliptical

polarizations to circular polarizations

From Eqs. ?27? and ?28?, it is derived that r ˆ=i leads to

Erx/Ery=pe ˆi=−i, which means that the reflected wave is al-

ways of the left-circular polarization under the condition of

r ˆ=i no matter what the polarization states of the incident

wave are. Similarly, it is also derived that r ˆ=−i leads to

Erx/Ery=pe ˆi=i, which means that the reflected wave is al-

ways of the right-circular polarization under the condition of

r ˆ=−i. On the other hand, from Eqs. ?22?–?25? it is known

that when A2=−A1=−1 or A2=A1=−1 r ˆ is an imaginary

number. Moreover, the imaginary part of r ˆ can be adjusted

by ??=?t−?g, which is independent on A1 and A2. There-

fore, the constitutive parameters, which cause the conversion

from arbitrary polarization to circular polarizations, can be

easily obtained through solving the equation group of A2=

−A1=−1 ?or A2=A1=−1? and Im?r ˆ?=?1 ?the plus is for the

left-circular polarization while the minus is for the right-

circular polarization?. Figure 5 shows the calculated r ˆ as a

function of ???? when ?t=Re??t?−0.1 Re??t?i, ?g=0.5−0.05i,

?t=1.54−0.154i,

?g=0.36−0.36i,

−0.01i, and d?=0.5. The constitutive parameters will satisfy

A2=−A1=−1 if their imaginary parts equal zero. In Fig. 5,

the solid line and the dashed line represent the imaginary

part and the real part of r ˆ, respectively. It can be seen that

?t=2−0.2i,

?g=0.1

Im?r ˆ? linearly varies as ???? increases, and Re?r ˆ? is very

small although the dissipation is considered. Therefore, r ˆ

=?i can be approximately obtained in this case.

C. Change from linear polarizations to elliptical

polarizations

According to Eqs. ?27? and ?28?, it is derived that

=?

Therefore, according to Fig. 5, when e ˆi=0 or e ˆi→? the FCM

reflector can realize the conversion from linear to elliptical

polarizations. Moreover, the constitutive parameters which

lead to the polarization conversion can be easily calculated

through solving the equation group of A2=−A1=−1 and

Im?r ˆ?=−Erx/Eryfor e ˆi→? ?or Im?r ˆ?=−Ery/Erxfor e ˆi=0? or

the equation group of A2=A1=−1 and Im?r ˆ?=−Erx/Eryfor

e ˆi→? ?or Im?r ˆ?=−Ery/Erxfor e ˆi=0?.

Erx

Ery

1

r ˆ,

e ˆ = 0

− r ˆ, e ˆ → ??.

?29?

D. Some special types of polarization states change

From Eqs. ?27? and ?28?, it is found that when r ˆ takes

some values there are some special types of polarization

states change.

?i?

When r ˆ=0 ?Rxx=0? there is Erx/Ery=−Eiy/Eix, which

means that, in this case, there always is a complete

conversion between two perpendicular linear polariza-

tions no matter what the polarization angles of the

incident wave are.

When ?r?→? ?Rxy=0? there is Ery/Erx=Eiy/Eix. In

this case, the FCM reflector conducts as a magnetic

conductor, which is widely applied in miniaturization

design of microwave devices, such as resonators and

antennas.

?ii?

V. CONCLUSION

The four independent wave numbers associated with

plane wave propagation in a FCM can give rise to a host of

complex EM behavior. In this paper, we proposed to use

FCM to manipulate EM wave polarizations after deeply ana-

lyzing EM characteristics of a FCM reflector, and showed

that all possible polarization states ?linear, circular, and ellip-

tic? for the reflected wave and arbitrary polarization conver-

sion are realizable via adjusting constitutive parameters of

0

5

10

15

20

0.00.40.6 0.81.0

d/λ0

0.2

Magnitude of p

FIG. 3. The magnitude of p as a function of d/?0.

0.0

0.5

1.0

1.5

2.0

2.5

0.00.40.60.8 1.0

d/λ0

0.2

Phase of p

FIG. 4. The phase of p as a function of d/?0.

>

∆ξ

-3

-2

-1

0

1

2

3

0

-1

-2

1

2

r

Im[r]

Re[r]

>

>

FIG. 5. ?Color online? Imaginary and real parts of r ˆ as a function of ??.

054104-5Meng et al. J. Appl. Phys. 107, 054104 ?2010?

Downloaded 14 Oct 2010 to 128.134.54.103. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

Page 6

the FCM. The ideas were demonstrated by some numerical

examples, and the calculation results are in a good agreement

with analytical calculation results.

ACKNOWLEDGMENTS

This work is supported by the National Natural Science

Foundation of China ?Grant Nos. 60801015 and 60971064?,

Aviation Science Funds ?Grant Nos. 20080177013 and

20090177002?, the Open Project Program of State Key

Laboratory of Millimeter Wave ?Grant Nos. K201006 and

K201007?, Ph.D. programs foundation of Ministry of Educa-

tion of China ?Grant No. 200802131075?, Development Pro-

gram for Outstanding Young Teachers in Harbin Institute of

Technology ?Grant No. HITQNJS.2008.07?, and Natural Sci-

entific Research Innovation Foundation in Harbin Institute of

Technology ?Grant No. HIT.NSRIF2009119?.

1J. Hao, Y. Yuan, L. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou,

Phys. Rev. Lett. 99, 063908 ?2007?.

2M. Born and E. Wolf, Principles of Optics ?Cambridge University Press,

Cambridge, England, 1999?.

3E. Hecht, Optics ?Addison-Wesley, New York, 2002?.

4T. G. Mackay and A. Lakhtakia, J. Appl. Phys. 100, 063533 ?2006?.

5A. J. Duncan, T. G. Mackay, and A. Lakhtakia, Opt. Commun. 271, 470

?2007?.

6T. G. Mackay and A. Lakhtakia, Phys. Rev. E 69, 026602 ?2004?.

7C.-W. Qiu, L.-W. Li, H.-Y. Yao, and S. Zouhdi, Phys. Rev. B 74, 115110

?2006?.

8W. S. Weiglhofer and S. O. Hansen, IEEE Trans. Antennas Propag. 47,

807 ?1999?.

9N. Engheta, D. L. Jaggard, and M. W. Kowarz, IEEE Trans. Antennas

Propag. 40, 367 ?1992?.

10T. G. Mackay and A. Lakhtakia, Microwave Opt. Technol. Lett. 49, 1245

?2007?.

11A. Lakhtakia, Beltrami Fields in Chiral Media ?World Scientific, Sin-

gapore, 1994?.

12B. Lax and K. J. Button, Microwave Ferrites and Ferrimagnetics

?McGraw-Hill, New York, 1962?.

13H. C. Chen, Theory of Electromagnetic Waves ?McGraw-Hill, New York,

1983?.

14R. E. Collin, Foundations for Microwave Engineering ?McGraw-Hill,

New York, 1966?.

15T. G. Mackay, Electromagnetics 25, 461 ?2005?.

16T. G. Mackay, A. Lakhtakia, and W. S. Weiglhofer, Phys. Rev. E 62, 6052

?2000?.

17W. S. Weiglhofer, A. Lakhtakia, and B. Michel, Microwave Opt. Technol.

Lett. 18, 342 ?1998?.

18Á. Gómez, A. Lakhtakia, J. Margineda, G. J. Molina-Cuberos, M. J.

Núñez, J. A. S. Ipiña, A. Vegas, and M. A. Solano, IEEE Trans. Micro-

wave Theory Tech. 56, 2815 ?2008?.

054104-6 Meng et al.J. Appl. Phys. 107, 054104 ?2010?

Downloaded 14 Oct 2010 to 128.134.54.103. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions