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Incident-angle-insensitive and polarization
independent polarization rotator
Mingkai Liu, Yanbing Zhang, Xuehua Wang, and Chongjun Jin*
State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-sen
University, Guangzhou 510275, People’s Republic of China
*jinchjun@mail.sysu.edu.cn
Abstract: This paper proposes a method to design an
incident-angle-insensitive polarization-independent polarization rotator. This
polarization rotator is composed of layers of impedance-matched anisotropic
metamaterial (IMAM) with each layer’s optical axes gradually rotating an
angle. Numerical simulation based on the generalized 4 × 4 transfer matrix
method is applied, and the results reveal that the IMAM rotator is not only
polarization-independent but also insensitive to the angle of incidence. A 90°
polarization rotation with tiny ellipticity variation is still available at a wide
range of incident angles from 0 to 40°, which is further confirmed with a
microwave bi-split-ring resonator (bi-SRR) rotator. This may be valuable for
the design of optoelectronic and microwave devices.
©2010 Optical Society of America
OCIS codes: (160.3918) Metamaterials; (230.0230) Optical devices; (230.5440)
Polarization-selective devices.
References and links
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#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11990
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1. Introduction
As an important property of transverse EM waves, polarization has been widely applied in
engineering and scientific researches. Many approaches have been employed to manipulate the
polarization of light. Traditional anisotropic crystals and chiral liquid crystals [1] are commonly
used as wave retarders and polarization rotators. However, the polarization rotator made of
anisotropic crystal is generally polarization-dependent and incident-angle-sensitive, and the
polarization rotator composed of liquid crystals is also incident-angle-sensitive. These
shortcomings limit their applications in microwave and optoelectronic devices. To our
knowledge, how to design and fabricate an incident-angle-insensitive and
polarization-independent polarization rotator still remains a challenge.
Thanks to the advent of metamaterials, this gives a chance to overcome the challenge.
Metamaterials are artificial structures, which usually have periodic arrangements and exhibit
exotic electromagnetic properties [2–8]. These manmade structures provide completely new
mechanisms and novel methods to control light. Early and ongoing researches on metamaterials
have shown that it is possible to obtain strong anisotropy or chirality via deliberate design and
fabrication. This can be used in the design of polarization devices. One of them is polarization
beam splitter achieved by anomalous reflection and transmission [9–12]. Another application is
the polarization rotator based on modes coupling, extraordinary optical transmission (EOT),
chirality of the structure, or phase mutation at resonance frequency [5,13–17]. Up till now,
wide-angle polarizer [18], splitter [10,11] and absorber [19] have been realized with these
artificial structures. In this paper, we proposed a method to design an incident-angle-insensitive
and polarization-independent polarization rotator. The polarization rotator is composed of
layered impendence-matched anisotropic metamaterials (IMAM) with the crystal axes rotated,
as shown in Fig. 1. The cross polarization conversion becomes polarization-independent and
insensitive to the incident direction of the light beam. It can be proved that only two layers of
IMAMs are enough to construct a cross-polarization rotator, which greatly alleviates the
difficulty in fabrication. It might be valuable for the design of optoelectronic and microwave
devices.
2. General formalism of IMAM polarization rotator
To study the transmission and reflection of plane waves in an anisotropic slab, a generalized 4 ×
4 transfer-matrix method is applied [20]. By transforming the permittivity and permeability of
IMAM with a rotation matrix: 1
2 1,ε ε −= ℜ ℜ 1
2 1µ µ −= ℜ ℜ , where
#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11991
Fig. 1. Schematic structure of a multilayered metamaterial rotator. The plane of incidence is in
x-z plane. φ is the angle between the axes of the adjacent layers. ϕ and θ are the
polarization angle and incident angle of the incident wave, respectively.
cos sin 0
sin cos 0 ,
0 0 1
φ φφ φ
ℜ = −
(1)
one can easily obtain the transfer matrix of an anisotropic slab with its crystal axes rotated along
z axis by an angle .φ When many layers are aligned together, the total transfer matrix is given
by
1 11 1( ) ( ) ( ) ( ),total m mm m
P H P h P h P h− −= ⋅⋅ ⋅ (2)
where 1 i
,m
iH h
==∑ which is the total thickness of the stratified slabs. The 2 × 2 Jones
transmission coefficient matrix can be acquired after some tedious algebraic operation that is
not presented here [21]. To simplify the discussion, we restrict the incidence in x-z plane, i.e.
y0,k = and one of the crystal axes is chosen as z axis.
When the polarization of the transmitted wave is perpendicular to a linearly-polarized
incident wave 0
( , ) (cos ,sin ),iM iE
E E E ϕ ϕ= one can obtain
0
sin cos,
sincos
MM ME
t
EM EE
T TE E
T T
ϕ ϕϕϕ
= −
(3)
where ϕ is the polarization angle of the incident wave. MM
T and EE
T are the co-polarization
transmission coefficients of TM and TE waves, respectively; while the cross terms are the
cross-polarization transmission coefficients. By multiplying ( )cos , sinϕ ϕ on both sides of
the equation, we have
2 2
00 [ cos sin ( )sin cos ].
MM EE ME EME T T T Tϕ ϕ ϕ ϕ= + + + (4)
For a polarization-independent rotator, the following condition must be satisfied:
MM EE0,T T= →
EM ME0.T T+ → Generally, it cannot be fulfilled with a single anisotropic
medium but with a layered structure. To give an intuitionistic discussion, we choose an IMAM
slab with the following permittivity and permeability tensors:
#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11992
Fig. 2. Schematic view of the polarization vector projections on x-y plane when the light is
normally incident. The crystal axes of the anisotropic slabs are rotated along z direction by an
angleφ in sequence.
0 0
0 0 ,
0 0
xx
r yy
zz
εε ε
ε
=
(5)
0 0
0 0 ,
0 0
xx
r yy
zz
µµ µ
µ
=
(6)
where
/ / / 1 / .yy xx xx yy sur sur Zε µ ε µ ε µ= = = (7)
Z is the wave impedance, and , ,xx yy xx yy
ε ε µ µ≠ ≠sur
ε and sur
µ are permittivity and
permeability of the surrounding material. When the light strikes normally at the interface
between the lossless surroundings and an IMAM slab with thickness ,h the transmission
coefficients of TE and TM waves can be simplified as
0 0
exp( ' ) exp( '' ),E yy yy
T i k hZ k hZε ε= − (8)
0 0
exp( ' / )exp( '' / ),M yy yy
T i k h Z k h Zµ µ= − (9)
where0
k is the wave vector in vacuum, and 'ε ( 'µ ) and ''ε ( ''µ ) are the real and imaginary
parts of permittivity (permeability). When
2 2
'' '' ''/ ''/ ,yy xx xx yy
Z Zε ε µ µ= = = (10)
TE and TM waves experience the same attenuation, i.e. | | | | 1.E M
T T= ≤
Since the impendence-matched condition is satisfied, reflections at the interface either
between slab and surroundings or between slabs disappear. Then we can give an explicit
expression on the transmission coefficient of the stratified slabs. When the plane wave
( ),iM iE
E E is normally incident onto x-y plane, the projections of the transmitted wave
polarization on the crystal axes (denoted by suffix 1, 2) of the (n + 1)th slab can be written as
1 1
2 1
cos sin cos sin.
sin cos sin cos
n
n iME E E E
M M M Mn iE
E ET T T T
T T T TE E
φ φ δ δφ φ δ δ
< + >
< + >
− − = ×
(11)
#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11993
As shown in Fig. 2, δ is the angle between the x axis and the crystal axis 1 of the first
layer; φ is the angle of the crystal axes between adjacent layers. We once again project
1 1 2 1( , )
n nE E< + > < + > onto the coordinate axes and the Jones matrix can be obtained:
cos( ) sin(n )
sin( ) cos( )
cos sin cos sin.
sin cos sin cos
MM ME
EM EE
n
E E E E
M M M M
T T n
T T n n
T T T T
T T T T
φ δ φ δφ δ φ δ
φ φ δ δφ φ δ δ
+ + = − + +
− − ×
(12)
By carefully tuning the thickness of IMAM slab that makes it as a half-wave retarder, one
can obtain .E M
T T T= − = When n is odd, the Jones matrix can be simplified as follows:
1
cos[( 1) ] sin[( 1) ].
sin[( 1) ] cos[( 1) ]
MM ME n
EM EE
T T n nT
T T n n
φ φφ φ
+ + + = − + +
(13)
It is clear that, if T is not considered, the Jones matrix is a coordinate transformation matrix
by rotating an angle of ( 1)n φ+ anticlockwise. Hence the polarization rotation angle ( 1)n φ+
can be tuned dynamically and is independent of ϕ and .δ When ( 1) / 2,n φ π+ = ±
1
0 1.
1 0
MM ME n
EM EE
T TT
T T
+ ± =
∓ (14)
This is the condition for the cross polarization conversion. It is clear that only two (n = 1)
IMAM slabs are enough to construct a polarization-independent rotator.
Equation (14) indicates that the eigen polarization states of the rotator are left and right
circular polarizations. Although it is the same for planar chiral structures with four-fold
symmetry, it is very difficult to achieve linear polarization rotation in planar metallic chiral
structures due to dichroism; while in dielectric chiral structures, one can achieve linear
polarization rotation but this property strongly depends on incident direction, as has been
pointed out [22].
3. Incident-angle-insensitivity and polarization independency
It is more interesting to find that the IMAM rotator is not only polarization-independent, but
also incident-angle-insensitive. This is distinct from conventional anisotropic or chiral rotators.
Without loss of generality, we choose air as surroundings, and a lossless IMAM bilayered
rotator with parameters as 2,xx yy
ε µ= = 1,yy xx
ε µ= = 1.zz zz
ε µ= = The thickness h of each
layer is chosen as0
/ 1.5h λ = so that exp[ ( )]E M x
T T i kϑ= − = is satisfied at normal incidence
and each layer of slab acts as a half-wave retarder. To construct a polarization-independent
rotator, two half-wave plates are aligned together in the above mentioned way, i.e. 2 90 .φ = °
To study the effects of incident direction on the performance of the polarization rotator, incident
angle θ (denoted by0
/x
k k ), angleδ and incident polarization angle ϕ should be considered.
Figures 3(a) to 3(d) show the polarization changes of the transmitted light. The ellipticity ∆
and the polarization rotation anglet
ψ of the transmitted light are defined as
/ ,a b
E E∆ = (15)
,t t
ψ ϕ ϕ= − (16)
where the electric field amplitudesa
E and b
E are the minor and major polarization
components of the transmitted light, respectively; t
ϕ is the transmitted polarization angle.
#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11994
Fig. 3. The projections of (a) the ellipticity ∆ and (b) the polarization rotation angle t
ψ with
respect to 0
/x
k k and δ when ϕ = 45°. The projection views of (c) ∆ and (d) t
ψ with
respect to δ and ϕ at 40° incident angle, in which ϕ ranges from −90° to 90°. The insets
demonstrate the three-dimensional relations. The lossless IMAM bilayered polarization rotator is
characterized with 2,xx yy
ε µ= = 1,yy xx
ε µ= = 1,zz zz
ε µ= = and a total thickness
02 / 3.h λ =
Since we have restricted the incident beam in x-z plane, the variation of δ is actually
equivalent to the change of incidence in polar direction. Now we look about how the
performance of the polarization rotator depends on δ when ϕ = 45°. Figure 3(a) is the
projection of the three-dimensional (3D) curvature of ellipticity ∆ as a function of δ and
0/
xk k onto the
0~ /
xk k∆ plane; Fig. 3(b) is the projection of the 3D curvature of
polarization rotation angle t
ψ as a function of δ and 0
/x
k k onto the 0
~ /t x
k kψ plane. It
is clear that, at normal incidence (0
/ 0x
k k = ), 0∆ → and 90 ,t
ψ → − � the bilayered structure
acts as a perfect cross-polarization rotator. As the incident angle increases, the variation ranges
of ∆ and t
ψ become larger but still acceptable within a broad range of incident angles. It can be
seen in Fig. 3(a) and Fig. 3(b) that when δ ranges from 0° to 360°, ( 0.077, 0.077)∆∈ −
(indicating a value of 0.006 as an intensity contrast of two polarization components) and
tψ ∈ ( 90.69 , 89.05 )− ° − ° even at a 40° incident angle (
0/ 0.64
xk k ≈ ). As the incident angle
decreases to 30°, variations can drop to ( 0.024, 0.024)∆∈ − and t
ψ ∈ ( 90.4 , 89.6 ).− ° − °
#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11995
Figures 3(c) and 3(d) depict the effect of incident polarization angle ϕ on the polarization
rotation at the oblique incidence of 40°. The insets plot the variations of ∆ and t
ψ with
respect to δ and ϕ at a 40° incident angle. The projections of these curvatures onto ~ δ∆
and ~t
ψ δ planes are shown in these figures. When ϕ changes from −90° to 90°, the
maximum variation range of ∆ is ( 0.077, 0.077),− while the maximum variation of t
ψ is
( 90.69 , 89.05 ).− ° − ° After studying the effects of incident direction under various0
/ ,x
k k ϕ
and ,δ we confirm that the IMAM rotator is more insensitive to incident direction when the
incident angle is smaller.
Moreover, we studied the effect of material loss. As is expected, the loss has a small impact
on the performance of the IMAM rotator that satisfies Eq. (10). We set
'' '' '' '' 0.2yy xx xx yy
ε ε µ µ= = = = while other parameters maintain the same. The results show
that for a lossy rotator, the corresponding curvatures of Figs. 3(a), 3(b) and 3(c) are similar to
the lossless ones; while for the one correspond to Fig. 3(d), the variation range of t
ψ is
enlarged to (−91.66°,-88.37°). Even so, a less than 2° deviation in polarization rotation at 40°
incidence is still quite acceptable.
4. Comparison and discussion
To make a comparison, we choose three lossless anisotropic slabs (surrounded by air) with
parameters as 2 ,xx
bε = 1,yy zz
ε ε= = 2 / ,yy
bµ = 1xx zz
µ µ= = , and the thickness h of each
layer is chosen as 0
/ 1.5h λ = , where 1, 2, 0.5b = respectively. Only the case b = 1 is
impedance-matched among the three cases. Since 1/2
y 0( )
xx yk hε µ = 6π for TM waves and
1/2
yy xx 0( ) k hε µ = 3π for TE waves, exp[ ( )]
E M xT T i kϑ= − = is satisfied for all three
half-wave plates at normal incidence; when two identical plates are stacked together in the
aforementioned way, i.e. 2 90 ,φ = ° they all act as polarization-independent rotators at normal
incidence. However, the impedance-matched conditions are not satisfied at oblique incidence,
and this will bring in negative influences on the performance of the rotators. Nevertheless, as is
shown above, the structure composed of IMAM slabs (b = 1) is more insensitive to incident
angle.
Figures 4(a) to 4(d) demonstrate the cross-polarization transmission coefficients and
polarization rotations with respect to 0
/x
k k when 45 ,δ = ° ϕ = 45°. It is clear that for the
impendence-matched polarization rotator, the incident angle region with | | | | 1EM ME
T T≈ ≈
( | |EE
T and | |MM
T tend to zero, which are not shown here) and EM ME
T T π∠ −∠ = ± is much
wider than the impendence-mismatched ones. Thus, the polarization rotation angle | | 90t
ψ → °
and the polarization ellipticity | | 0∆ → can be realized within a much wider incident angle as
well. For the IMAM polarization rotator, when the incident angle varies from 0° to 40°
(0
/ 0.64x
k k ≈ ), ( 0.077, 0.077)∆∈ − and t
ψ ∈ ( 90.69 , 89.05 )− ° − ° . While for the
impendence-mismatched rotators, the polarization rotation and ellipticity become very instable
as the incident angle increases. To take the nonmagnetic rotator with b = 2 as an example, the
polarization rotation becomes −105° and the ellipticity is around-0.6 at a 40° incidence.
#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11996
Fig. 4. (a) The amplitudes of transmission coefficients EM
| |T and | |ME
T , (b) the phase
difference EM ME
T T∠ −∠ , (c) the polarization rotation angle t
ψ and (d) the ellipticity ∆
with respect to the incident transversal vector 0
/x
k k for three different anisotropic bilayered
structures characterized as 2xx
bε = , 1yy zz
ε ε= = , 2 /yy
bµ = , 1xx zz
µ µ= = , and
the total thickness 0
2 / 3h λ = , when 45δ = ° , 45ϕ = ° . (e) Longitudinal wave vector
difference MZ EZ
( )k k− between TM and TE components and (f) p + 1/p as a function of
0/
xk k for different b values.
The basic mechanism of the incident-angle-insensitivity of the IMAM polarization rotator
can be understood by a simple argument. A lossless incident-angle-insensitive polarization
rotator actually requires that ( ) ( ) exp[ ( )]E x M x x
T k T k i kϑ≈ − ≈ can be satisfied for each single
slab even at oblique incidence. When 0,x
k ≠ the transmission coefficient of TM wave through
a homogeneous slab surrounded by air (the transmission coefficient of TE wave can be obtained
by duality) can be written as
1
,2 2cos( ) ( 1/ )sin( )M MZ MZ
M MT k d i p p k d
− = × − + (17)
where
2 2 1/ 2 2 2 1/2
yy 0 zz 0
[( / ) / ( )] / ( ) .M xx x xx x
p k k k kµ ε ε ε= − − (18)
The key issue is that for the impendence-mismatched slab, yy
( / ) 1xx
µ ε ≠ and
thus 1 2.M M
p p −+ > Calculations show that (not shown here), the reflection increases quickly
and become fluctuating when 0;x
k > meanwhile, ( )E x
T k and ( )M x
T k change periodically with
respect to kx, indicating the existence of high-order Fabry-Perot interference. While for the
IMAM slab, yy
( / ) 1,xx
µ ε = 1M
p ≈ (i.e. 1 / 2M M
p p+ ≈ ) can be maintained at a much larger
,x
k as depicted in Fig. 4(f), and thus ( )E x
T k and ( )M x
T k vary smoothly as a function of
#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11997
.x
k Moreover, it can be seen from the dispersion curves in Fig. 4(e) that 0MZ EZ
k k k− ≈ is
maintained better in the IMAM polarization rotator as 0
/x
k k increases, which indicates a more
stable phase difference. Hence, ( ) ( )E x M x
T k T k≈ − ≈ exp[ ( )]x
i kϑ can be obtained, and Eq. (14)
can be fulfilled even at a large-angle incidence. This is the reason that a bilayered polarization
rotator composed of such impedance-matched half-wave retarders will thus exhibit
incident-angle-insensitivity.
It should be noted that we mainly consider about the effects of transversal parameters
(xx
ε ,yy
ε ,xx
µ andyy
µ ) in the above discussion. Our calculations disclosed that the ratio between
zzε and
zzµ will also affect the property of angle-insensitivity. Numerical simulations show that
the optimized ratio generally locates around one when the surroundings is air, i.e.
zz zzε µ≈ .Since
2 2 1/ 2
yy 0( / )
EZ xx xx x zzk k kε µ µ µ= − and
2 2 1/2
yy 0 zz( / ) ,
MZ xx xx xk k kµ ε ε ε= − as the
longitudinal parameters zz
ε and zz
µ increase, the dispersion curve z
~x
k k of the metamaterial
will become flatter. Thus, the IMAM polarization rotator will become more insensitive to
incident angle and even exhibits a self-collimating property when /xx zz
ε ε and /xx zz
µ µ are
sufficiently small [5]. Then, the ratio deviation betweenzz
ε and zz
µ can be larger within the
same tolerable variation range of ∆ and t
ψ . To take the lossless IMAM polarization rotator
discussed in Fig. 3 as an example, within the same tolerable variation range of ∆ and t
ψ (i.e.
|∆| < 0.077 and t
ψ ∈ ( 90.69 , 89.05 )− ° − ° at 40° incidence), when 5,zz zz
ε µ≈ = the maximum
ratio deviation can be up to 20% (i.e. ,6.0 5.0zz zz
ε µ= = ).
5. Construct a microwave IMAM rotator with bi-SRR structure
To demonstrate an incident-angle-insensitive and polarization-independent polarization rotator,
we need to construct an IMAM half-wave retarder first, and then stack two retarders together
with their optical axes rotated 45° to form such a polarization rotator. Various metamaterial
structures can be employed to fabricate an IMAM retarder; however, metamaterials with both
electric and magnetic resonances are preferred, since the effective permeability of nonmagnetic
structures generally lies around one, which indicates that high refractive index and
impedance-matching condition can hardly be achieved simultaneously. Moreover, spatial
dispersion and anisotropy are inevitable because metamaterials are artificial mesostructures
[23]. In order to diminish such effects, electrically small non-bianisotropic microwave
bi-split-ring resonator (bi-SRR) is selected to construct an IMAM half-wave retarder [24].
Figures 5(a) and 5(b) are the schematic layouts of the IMAM retarder. The metallic bi-SRR
patterns are fabricated on one side of the FR-4 (lossless) substrate characterized as 04.9ε ε= ,
0µ µ= ,the substrate thickness t = 0.5 mm. The other dimensions of a unit cell are as follows:
lattice constant az = ay = 5 mm; the length of metal slices (perfect electric conductor) in Z and Y
directions z = y = 4 mm, the separation distance of the metal slices p = 0.12 mm, the gap g = 0.2
mm, the separation distance between adjacent unit cells s = 1.0 mm, and the width w and
thickness of metal are 0.2 mm and 0.08mm respectively. The design principle of the IMAM
half-wave retarder is that only TM component can excite magnetic resonances while TE wave
propagates “quietly” through the structure. Thus nearly full transmission and low effective
refractive index can be obtained for TE wave, while high effective index is available for TM
component at the impedance-matched frequency near the resonance. By tuning the dimensions
or the substrate material, one can change the impedance-matched frequency and the phase
difference between the two orthogonal components of the transmitted waves.
#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11998
Fig. 5. (a) and (b) are schematic layouts of the bi-SRR half-wave retarder. az = ay = 5 mm, z = y
= 4 mm, w = 0.2 mm, p = 0.12 mm, g = 0.2 mm, the substrate thickness t = 0.5 mm, the separation
distance between adjacent layers s = 1.0 mm, and the thickness of metal (perfect electric
conductor) is 0.08mm. (c) The structure of the polarization rotator, where yellow arrow indicates
the incident waves, pink arrows indicate the direction of electric field.
The simulation is performed using the software package CST Microwave Studio, in which
periodic boundary conditions are applied. Figure 6(a) depicts the amplitudes of co-polarization
terms of transmission coefficient 1T ( )x
k and reflection coefficient 1R ( )x
k for the first retarder,
where the cross-polarization terms are negligible. There are two dips in the TM wave reflection
coefficient curve corresponding to two possible impedance-matched frequencies at 3.12 GHz
and 3.25 GHz. Here the working frequency is chosen to be 3.25GHz. Because the ratio of unit
cell size to wavelength is less than 1/9, the effective medium description is valid, and the
retrieval effective parameters at normal incidence are: 1.156,TE
n = 3.457,TM
n = 0.854TE
Z =
and 1.001TM
Z = . Though the structure is not perfectly impedance-matched for TE wave at the
working frequency, the transmission is still very high (>99%) and it will have less influence on
the transmitted polarization state.
Note that in Fig. 6(b), the phase difference between two polarization states at 3.25GHz is
180°, which indicates that the structure acts as a half-wave retarder. Figure 6(c) reveals the
amplitudes of transmission coefficients for TE and TM waves with respect to 0
/x
k k at
3.25GHz, and Fig. 6(d) shows the transmitted phase difference between the two orthogonal
components, in which a 174° phase difference is still available at a 40° incidence
(0
/ 0.64x
k k ≈ ). It is clear that high transmission and a near 180° phase difference can be
maintained within a large incident angle.
Then, we construct a polarization rotator by aligning two such half-wave retarders in the
aforementioned way, i.e. 2 90φ = ° , as depicted in Fig. 5(c). In the simulation, the polar angle
of incident plane is rotated by 45° to perform an equivalent rotation of the IMAM retarder.
Thus, we can get the transmission and reflection Jones matrices of the two IMAM
retarders 1T ( )x
k , 1R ( )x
k and 2T ( )x
k , 2R ( )x
k directly from the simulation rather than from the
effective parameters. The cross-polarization terms of the transmission coefficient for the second
retarder 2T ( )x
k satisfy ( ) ( )EM x ME x
T k T k≈ ≈ exp[ ( )]x
i kϑ at 3.25GHz, as expected, and are not
shown here. The total transmission Jones matrix of the rotator can then be written
as 2 1T ( ) T ( )T ( ) ( )( ),R x x x xk k k O T k= + where 2 1 2 1( ) T ( )R ( )R ( )T ( ) ,
x x x xO T k k k k= +⋯ which is
the small quantity due to multiple reflections between the slabs. The first term of the expression
of O(T) represents the first order approximation. The ellipticity and polarization rotation angle
with respect to incident wave polarization under different incident angles can then be worked
out. For clarity but without loss of generality, we only give out the results for incident polar
angle o0δ = in Figs. 6(e) and 6(f). The results under zeroth order and first order
approximations are compared, it is clear that the variation ranges of ellipticity and polarization
rotation can be well described under zeroth order approximation. The variations of ellipticity
#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11999
Fig. 6. (a) The amplitudes of the transmission and reflection coefficients, (b) the phase of the
transmission coefficient of TE and TM waves at normal incidence for polarization rotator with
lossless substrate. (c) The transmission coefficients and (d) the phase difference between the two
orthogonal polarized waves with respect to 0
/x
k k at 3.25 GHz.(e) and (g) Polarization
rotation angle, (f) and (h) ellipticity variations with respect to incident wave polarization under
different angles of incidence, where (e) and (f) are for lossless substrate with 4.9,r
ε = (g) and
(h) are for lossy substrate with 4.9 0.01.r
ε = +
#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 12000
and polarization rotation are still quite acceptable even at the angle of incidence 40θ = ° ,
showing that the structure can act as an incident-angle-insensitive and polarization-independent
polarization rotator.
When a lossy substrate with a permittivity 4.9 ''r r
ε ε= + is considered in the polarization
rotator, it is found that the imaginary term may lead to a stronger absorption for TM wave near
resonant frequencies. However, we can still find the two dips in reflection curve of the
half-wave retarder. Although they are not as sharp as that in the lossless case, but they are small
enough to suppress the influence of multiple reflections between the two retarders in a
polarization rotator. For a commercial available low loss substrate with '' 0.01,r
ε ≤ our
simulations show that the phase difference has a tiny change (less than 2°), and there is a small
decrease in the transmission amplitude of TM wave due to absorption. The influences of the
substrate loss are illustrated in Figs. 6(g) and 6(h) when '' 0.01r
ε = and the polarization rotator
still works at 3.25GHz. Compared with the lossless rotator, the variations of ellipticity and
polarization rotation angle are slightly increased. But we have to emphasize that the depicted
results for the lossy rotator above are just for comparison, without any structural adjustment
according to the loss, so the performance can be further improved after a proper structural
optimization.
6. Conclusion
In conclusion, we proposed a design method to realize broad–angle and polarization-
independent polarization rotator based on impedance-matched anisotropic metamaterial
(IMAM). The mechanism and the influencing factors of the polarization rotator are discussed
analytically, in conjunction with the generalized 4 × 4 transfer matrix method. Also, we
illustrate how to construct a microwave IMAM rotator by using a bi-SRR structure. Compared
with traditional polarization rotators, the IMAM polarization rotator is more insensitive to
incident direction. This may offer the possibility to unyoke the limitation of the narrow working
angle and enhance the compactness and stability of microwave and optoelectronic systems.
Acknowledgement
The authors acknowledge the financial support from the National Natural Science Foundation
of China (NSFC) under the contracts 10774195, U0834001 and 10974263. The work is also
partially supported by Program for New Century Excellent Talents in University and the
Chinese National Key Basic Research Special Fund (2010CB923200).
#126263 - $15.00 USD Received 30 Mar 2010; revised 8 May 2010; accepted 10 May 2010; published 21 May 2010(C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 12001
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