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.

Full text

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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
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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?
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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
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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?
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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?
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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?.
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