PIERS Proceedings, Beijing, China, March 23–27, 2009
226
Lateral Displacements of an Electromagnetic Beam Transmitted and
Reflected from a Gyrotropic Slab
Hui Huang1, 2 , Yu Fan1 , Bae-Ian Wu2 , and Jin Au Kong2
1
School of Electrical Engineering, Beijing Jiaotong University, Beijing 100044, China
2
Research Laboratory of Electronics, Massachusetts Institute of Technology
Cambridge, MA 02139, USA
Abstract— It is known that for a gyrotropic medium in the Voigt configuration, waves can
be decoupled into TE and TM modes. A detailed study on the lateral displacements of an
electromagnetic beam reflected and transmitted from a gyrotropic slab in Voigt configuration is
presented, for both TM and TE waves. Using the stationary phase approach, analytic expressions
for lateral displacements of the reflected and transmitted waves from a symmetric gyrotropic slab
are obtained, and we also give examples for both cases. It is found that the lateral displacements
for TM and TE waves have different characteristics. Only the TM mode is affected by the
gyrotropy. Due to the external magnetic field, the lateral displacement of a TM wave transmitted
from a gyrotropic slab is not the same as the reflected one, even when the configuration is
symmetric and the media are lossless. We also discuss the phenomena when the incident angle
is near the Brewster angle.
1. INTRODUCTION
The Goos-Hänchen (GH) effect [1, 2] has been studied for many years. GH lateral displacement
refers to the spatial displacement of a reflected wave from the position expected by geometrical reflection. Traditionally, GH shift is always a phenomenon during total reflection, when an
electromagnetic beam is reflected at the interface between media with different reflective indexes.
Moreover, the displacement in this situation is usually positive, i.e., in the same direction of the
component of the incident wave vector along the interface. Recently there are some researches
on negative GH lateral shifts in different systems: strongly reflecting and attenuating media such
as metal at IR frequencies [3, 4], non-absorbing [5], weakly absorbing interfaces [6–10], slabs [11],
metallic gratings [12], transparent dielectric slabs [13], dielectric slabs backed by a metal [14],
photonic crystals [15], and left-handed materials [16–18]. Among them there are some reports on
large lateral shift near Brewster or pseudo-Brewster angle upon reflection from a weakly absorbing
medium [9, 10, 17, 18]. Under those conditions, the reflectance at Brewster angle is not zero, making
the shift of the reflected wave is detectable.
In this paper, we focus on the GH lateral displacements of an electromagnetic beam reflected
and transmitted from a symmetric lossless gyrotropic slab. We consider two cases: incident wave
is TM or TE polarized. We present the analytic expressions for the reflection and transmission
coefficients and mathematical result for the lateral displacement which is obtained by the stationary
phase approach. We find that due to the applied magnetic field in the gyrotropic medium, the lateral
displacement of a TM wave reflected from the slab is not the same as that of the transmitted wave.
And it is shown that for TM waves, within a chosen frequency band, when the incident angle is
near the Brewster angle, a large displacement of the reflected wave with nonzero reflectance can be
obtained. Moreover, for TM waves, we can make the lateral displacement be positive or negative
by adjusting the direction of the incident wave or the applied magnetic field.
2. LATERAL DISPLACEMENTS USING THE STATIONARY PHASE APPROACH
We consider the configuration in Fig. 1, where a plane wave is incident from an isotropic medium
into a infinite gyrotropic slab at an oblique angle θi with respect to the normal of the interface. The
thickness of the gyrotropic slab is d, and an external magnetic field B̄0 is applied in +z direction,
parallel to the interfaces and perpendicular to the direction of the wave propagation, which is the
Voigt configuration. The isotropic media in region 1 and region 3 are the same, with permeability
µ1 and permittivity ε1 . Region 2 is gyrotropic medium with permeability µ2 and permittivity
tensor ε̄¯2 , which takes the following form:
"
#
εxx iεg 0
ε̄¯2 = −iεg εyy 0 ,
(1)
0
0 εzz
Progress In Electromagnetics Research Symposium, Beijing, China, March 23–27, 2009
227
where elements are given by
Ã
εxx = εyy = ε∞
Ã
εzz = ε∞
1−
ωp2
ω2
!
,
(2a)
,
(2b)
#
ωp2 ωc
−
.
ω (ω 2 − ωc2 )
"
εg = ε∞
ωp2
1− 2
ω − ωc2
!
(2c)
p
Here, ωp = N qe2 /meff ε∞ and ω̄c = qe B̄0 /meff are the plasma and cyclotron frequencies respectively, ε∞ is the background permittivity, N is the electron density, meff is the effective mass, and
qe is the electron charge.
2.1. Case 1: TM Incident Waves
It is known that in gyrotropic medium in the Voigt configuration, waves can be decoupled into TE
and TM modes and with different dispersion relations [19, 20].
We focus on TM waves first. With wave vectors k̄1 = ±x̂k1x + ŷky in the isotropic medium and
TM + ŷk in the gyrotropic medium, the dispersion relations can be expressed as
k̄2TM = ±x̂k2x
y
2
ky2 + k1x
= ω 2 µ1 ε1 ,
¡ TM ¢2
(ky )2 + k2x
= ω 2 µ2 εV .
(3)
(4)
Here εV = (ε2xx − ε2g )/εxx , is the equivalent permittivity of the gyrotropic medium in the Voigt
configuration for TM waves.
y
Region 1
Region 2
µ1 , ε1
Gyrotropic
reflected wave
incident wave
µ1 , ε1
µ2 , ε2
Sr
θi
Region 3
transmitted wave
St
d
0
x
B0
Figure 1: A gyrotropic slab with thickness d in an isotropic medium. Region 1 and Region 3 are the
same isotropic media, with permeability µ1 and permittivity ε1 . Region 2 is a gyrotropic medium in Voigt
configuration with µ2 and ε̄¯2 . An applied magnetic field B̄0 is in +z direction. Sr and St refer to the lateral
displacements of reflected and transmitted waves respectively.
Determined by the Maxwell equations and the boundary conditions, the transmission and re-
PIERS Proceedings, Beijing, China, March 23–27, 2009
228
flection coefficients for TM waves can be written as
1
i ¡
¡ TM ¢
¡ TM ¢ h
¡ TM ¢2
¢,
2
TM
2
cos k2x d − i sin k2x d σ k1x + k2x
+ τ 2 ky2 / 2σk1x k2x
h
i ¡
¡ TM ¢ n
¡
¢
¢o
TM − i σ 2 k 2 − k TM 2 − τ 2 k 2 / 2σk k TM
sin k2x
d τ ky /k2x
1x 2x
y
1x
2x
i ¡
=
¢,
¢
¡ TM ¢
¡ TM ¢ h
¡
2
2 + k TM
2 k 2 / 2σk k TM
cos k2x d − i sin k2x d σ 2 k1x
+
τ
1x
y
2x
2x
T TM =
(5)
RTM
(6)
where the parameters σ and τ are dimensionless, defined as
εg
εV
σ=
, and τ =
.
ε1
εxx
(7)
Here we can see that τ is a direct manifestation of the applied magnetic field B̄0 , arising from the
off-diagonal element εg in the permittivity tensor. Without an external magnetic field, both εg and
τ are zero.
The lateral shift has the well-known expression S = −dφ/dky , which was proposed by Artmann
using the stationary phase method [21]. After some algebra manipulations, and by noting that
the denominators of both coefficients are the same, the lateral displacements of the reflection and
transmission for TM waves can be expressed as
(
£
¤)
£
¤
£ TM ¤ ∂Re T TM
£ TM ¤ ∂Im T TM
1
TM
− Im T
St
= −
Re T
∂ky
∂ky
|T TM |2
n
¡
¢
¡
¢ ¡
¢
TM
sin(2k2x
d)
4 − k TM 4 + k 2 k TM 2 σ 2 + 1 − 2τ 2
−σ 2 k1x
TM
2x
1x
2x
2k2x
dh
h
i
¡
¢ io
¡
¢
2 + k TM 2
2 σ 2 k 2 + k TM 2 + τ 2 k 2
−τ 2 ky2 k1x
+
k
y
2x
1x
1x
2x
= 2σd |tan θi |
h
i2
¡
¢
¡
¢
¡
¢
¡ TM ¢ , (8)
2
2
2
2 k TM
2 k TM d + σ 2 k 2 + k TM
2 k2
4σ 2 k1x
cos
+
τ
sin
k2x d
y
2x
2x
1x
2x
(
£ TM ¤
£ TM ¤ )
£ TM ¤ ∂Im R
£ TM ¤ ∂Re R
1
Re
R
−
Im
R
= ∆Srt + StTM ,
SrTM = −
(9)
2
TM
∂k
∂k
y
y
|R |
where
h
i
h
¡
¢ ¡
¢
¡
¢ i
2 σ 2 + τ 2 − 2 + k TM 2 + τ 2 k 2 + k
2 k 2 − k TM 2
ky |tan θi | k1x
σ
1x
y
2x
1x
2x
∆Srt = −2στ
.
h
i2
¡
¢
2 k 2 + σ 2 k 2 − k TM 2 − τ 2 k 2
4σ 2 τ 2 k1x
y
y
1x
2x
(10)
When there is no applied magnetic field, τ = 0, then the result above is consistent with those in
reference [22]. Furthermore, when the inequality ky2 ≥ ω 2 µ2 εV is satisfied, total reflection occurs
TM becomes imaginary. Let k TM = iαTM , the dispersion relation of
at the boundary x = 0, and k2x
2x
gyrotropic medium becomes ky2 − (αTM )2 = ω 2 µ2 εV , and Eqs. (8) and (10) can be rewritten as
n
¡
¢
¡
¢ ¡
¢
sinh(2αTM d)
4 + αTM 4 + k 2 αTM 2 σ 2 + 1 − 2τ 2
σ 2 k1x
1x
2αTM d
h
h
i
¡
¢ io
¡
¢
2 k 2 k 2 − αTM 2
2 σ 2 k 2 − αTM 2 + τ 2 k 2
+τ
−
k
y
y
1x
1x
1x
0
StTM = 2σd |tan θi |
, (11)
h
i2
2
2
2
2
2
2
2
TM
TM
2
TM
2
2
TM
4σ k1x (α ) cosh (α d) + σ k1x − (α ) + τ ky sinh (α d)
h
i
h
¡
¢ ¡
¢
¡
¢ i
2 σ 2 + τ 2 − 2 − αTM 2 + τ 2 k 2 + k
2 k 2 + αTM 2
ky |tan θi | k1x
σ
1x
y
1x
0
∆Srt
= −2στ
(12)
.
h
i2
2
2
2
2
2
2
2
TM
2
2
4σ τ k1x ky + σ k1x + (α ) − τ ky
It has been known that, for the lossless symmetric configuration, the lateral shifts of both the
reflected and transmitted beams are the same [13, 23–25]. Nevertheless, for TM waves in gyrotropic
0 ) between S (S 0 ) and
case, it is not true. According to Eqs. (10) and (12), a difference ∆Srt (∆Srt
r
r
0
St (St ) arises directly from the external magnetic field B̄0 and the magnitude of the difference is
independence of d, thickness of the slab.
Progress In Electromagnetics Research Symposium, Beijing, China, March 23–27, 2009
229
2.2. Case 2: TE Incident Waves
TE + ŷk in the gyrotropic medium, the
Now we turn to TE modes. With wave vectors k̄2TE = ±x̂k2x
y
dispersion relations for TE waves is
¡ TE ¢2
(ky )2 + k2x
= ω 2 µ2 εzz ,
(13)
where εzz is determined by Eq. (2b), independent of the external magnetic field. Taking the same
process above, we can get the transmission and reflection coefficients for TE waves
1
i ¡
¢,
¡ TE ¢
¡ TE ¢ h¡ TE ¢2
2 / 2qk k TE
k2x + q 2 k1x
cos k2x d − i sin k2x d
1x 2x
¡ TE ¢2 i ¡
¡ TE ¢ h 2 2
¢
TE
−i sin k2x
d q k1x − k2x
/ 2qk1x k2x
i ¡
=
¡ TE ¢
¡ TE ¢ h¡ TE ¢2
¢,
2 / 2qk k TE
cos k2x
d − i sin k2x
d
k2x + q 2 k1x
1x 2x
T TE =
(14)
RTE
(15)
where
q=
µ2
.
µ1
(16)
Using the stationary phase method, the lateral displacements of the reflection and transmission
for TE waves are the same, and can be expressed as
h`
i
` TE ´4
` TE ´2 ` 2
´i
´
4
2
2
TE 2
2
−q 2 k1x
− k2x
+ k1x
k2x
q + 1 + k1x
k2x
+ q 2 k1x
h
i2
2
TE 2
TE
TE 2
2
TE
4q 2 k1x
(k2x
) cos2 (k2x
d) + (k2x
) + q 2 k1x
sin2 (k2x
d)
TE
sin(2k2x
d)
StTE
=
SrTE
= 2qd |tan θi |
TE d
2k2x
h
(17)
TE
When the inequality ky2 ≥ ω 2 µ2 εzz fulfills, total reflection occurs at the boundary, and k2x
TE
TE
2
becomes imaginary. Let k2x = iα , the dispersion relation of gyrotropic medium becomes ky −
(αTE )2 = ω 2 µ2 εzz , and the lateral displacements of the reflection and transmission in Eq. (17) can
be rewritten as
h
`
´4
` TE ´2 ` 2
´i
`
´2 i
4
2
2
2
q 2 k1x
+ αTE + k1x
α
q + 1 − k1x
q 2 k1x
− αTE
ˆ
˜2
2
2
4q 2 k1x
(αTE )2 cosh2 (αTE d) + q 2 k1x
− (αTE )2 sinh2 (αTE d)
sin(2αTE d)
0
0
StTE = SrTE = 2qd |tan θi |
2αTE d
h
(18)
3. RESULT AND DISCUSSION
Here, we consider an indium antimony (InSb) slab in vacuum. The material parameters used in
the computation are: µ1 = µ2 = µ0 , ε1 = ε0 , ε∞ = 15ε0 , N = 1022 m−3 , and meff = 0.015m0 =
0.13664 × 10−31 kg. Hence, ωp = 1.19 × 1013 rad/s and ωc /ωp = 0.98B0 . For the lossless symmetric
gyrotropic slab, we can plot the lateral displacements in the (ω, θi ) plane with a color scale proportional to the magnitude, shown in Fig. 2. Fig. 2(a) is for the TE incident waves, and Fig. 2(b),
Fig. 2(c) and Fig. 2(d) are for incident TM waves. The frequencies are normalized to the plasma
frequency ωp , and the displacements are normalized to λ0 = 2πc/ω, the incident wavelength in
vacuum.
For TE waves, from Fig. 2(a), we can see that, corresponding to the change of εzz with frequency,
there are two regions (A and B) separated by a white solid line in the (ω, θi ) plane. In region A,
total reflection occurs on the boundary at x = 0; while in region B there is no total reflection.
Since TE modes is not affected by the gyrotropy, for the lossless symmetric configuration, the
lateral shifts of both the reflected and transmitted beams are the same, independent of the external
magnetic field.
For TM waves, without any applied magnetic field, there still are two regions (A and B) in
Fig. 2(b) corresponding to the change of equivalent permittivity εV . But as shown in Figs. 2(c)
and 2(d), the existence of the applied magnetic field splits each region up into two, marked with
subscript 1 and 2.
Figure 2(b) shows magnitude of displacement of both transmitted and reflected TM waves
0 )
without any external magnetic field. Under this situation, since εg = 0 and τ = 0, ∆Srt (∆Srt
0
0
is zero [see Eqs. (10) and (12)]. Hence St = Sr (St = Sr ), i.e., the lateral displacements of waves
transmitted and reflected from the lossless symmetric gyrotropic slab are always the same in the
PIERS Proceedings, Beijing, China, March 23–27, 2009
230
absence of external magnetic field. However, it is not true when an external magnetic field is
applied. From Eqs. (10) and (12), the presence of an applied magnetic field B̄0 results in a nonzero
0 ), a difference between S (S 0 ) and S (S 0 ), which means that
εg and τ , and gives rise to ∆Srt (∆Srt
t
r
t
r
if there is an applied magnetic field B̄0 , the lateral shifts of TM waves transmitted and reflected
from the gyrotropic slab are not the same even for the lossless symmetric configuration. We show
them in Fig. 2(c) and Fig. 2(d) respectively.
3.1. Influence of the Sign of the Incident Angle on the GH Lateral Shift for TM Case
It is important to note that the shifts in Fig. 2(a), Figs. 2(b) and 2(c) are all symmetric with respect
to the incident angle θi while asymmetric in Fig. 2(d). This is due to the nonreciprocal nature of
the gyrotropic medium. For TM cases, if we change the sign of incident angle θi from positive to
negative while keeping the magnitude, i.e., the wave is incident from the upper left instead of the
lower left, the magnitude of the field in gyrotropic medium changes and causes the modification of
reflection and transmission coefficients. In fact, we can obtain the expressions for negative θi case
by the replacement τ → −τ in all equations which include τ . For TM cases, from Eqs. (8) and (11),
we can see that lateral shift for transmission St (St0 ) remains the same because it is the function
(a)
(b)
(c)
(d)
Figure 2: Lateral displacements of TE or TM waves transmitted and reflected from a lossless symmetric
gyrotropic slab, the Brewster angle is also shown for TM cases (in white dashed line). The incident and
reflected waves are both on the left side of the slab. We consider the angle as positive when the incident
wave comes from the lower left. The frequencies are normalized to ωp , and the displacements are normalized
to λ0 = 2πc/ω, the incident wavelength. The thickness of the slab d = 0.5λp = πc/ωp . (a) The lateral
displacement of both transmitted and reflected TE waves, independent of external magnetic field. St = Sr ;
(b) For TM cases, St = Sr with no applied magnetic field; (c) St for TM waves with an applied magnetic
field B̄0 = +ẑ0.4 T; (d) Sr for TM waves with the same applied magnetic field as (c).
Progress In Electromagnetics Research Symposium, Beijing, China, March 23–27, 2009
231
0 ) will change sign. That is why shifts for
of τ 2 ; while Eqs. (10) and (12) show that ∆Srt (∆Srt
transmission in Fig. 2(b) are symmetric however those for reflection in Fig. 2(c) are not. Without
an applied magnetic field, τ = 0, so the sign of the incident angle does not affect the lateral shift
at all, resulting in lateral shifts for both transmission and reflection in Fig. 2(b) are symmetric too.
3.2. Phenomenon when the Incident Angle is near the Brewster Angle for TM Case
It is also of interest to note that in regions B1 and B2 of Fig. 2(d), there is a positively or negatively
large displacement of reflected TM but no such phenomenon in Fig. 2(c) for the displacement of
transmitted waves. This is due to the Brewster angle. We also show the Brewster angle in the white
dashed line. (In region B1 , since the equivalent permittivity εV changes greatly with frequency,
the Brewster angle is sensitive to the frequency. Furthermore, the frequency band of region B1 is
narrow, so we focus on the region B2 instead.) For TM waves, there is an abrupt phase change near
the Brewster angle [9, 10, 17] which causes a large lateral shift for the reflected beam. Nevertheless,
there is no abrupt change for the phase of the transmission coefficient at the Brewster angle.
Therefore, we expect a large lateral shift of the reflected beam near the Brewster angle but no such
phenomenon for the transmitted beam. Without an applied magnetic field, the lateral shift of the
reflected wave is of no interest near the Brewster angle because the reflection coefficient vanishes.
However due to the presence of an applied magnetic field B̄0 , the existence of τ makes it reasonable
to expect a large finite slope of the phase change with a nonzero reflection. Under this situation,
∆Srt plays an important role in the total lateral shift for reflection, hence the sign of incident
angle θi will affect the sign of lateral shift Sr greatly. Fig. 3(a) shows the dependence of the lateral
shifts of the reflected waves Sr and those of the transmitted waves St on the incident angle θi , and
Fig. 3(b) shows the corresponding reflectance |R|2 . In Fig. 4, we show the displacement of reflection
Sr and the reflectance |R|2 versus frequency when the incident angle is equal to the Brewster angle.
From Fig. 3 and Fig. 4, we can see that for TM waves, because of the change of the equivalent
permittivity εV with frequency, the Brewster angle θB changes, and so do the lateral displacements
of the reflected wave Sr and the reflectance |R|2 . It is shown that near the Brewster angle, the
lateral displacement can be positively or negatively large depending on the sign of the incident
angle. And when the condition k2x d = mπ is satisfied, magnitude of the reflection coefficient
vanishes. For instance, when the working frequency is near 1.18ωp , 1.237ωp , 1.287ωp , 1.383ωp , and
1.528ωp , |R| = 0, as shown in Fig. 4. Furthermore, from Fig. 4, we can see that within the chosen
frequency band (for example, 1.22ωp to 1.50ωp ) we can obtain a larger reflectance (for instance,
ω = 1.32ωp , |R|2 = 54.45 × 10−3 ) than those in weakly absorbing media [9], making it easier to
detect. However, when the frequency is very high, the effect of the magnetic field vanishes, causing
the zero reflection at the Brewster angle.
1
0.9
6
ω=1.34ωp
ω=1.40ωp
0.8
4
0.7
2
2
reflectance (|R| )
lateral shifts of reflected wave or transmitted wave
8
0
Sr / λ0 (ω=1.34ωp )
-2
-4
-6
-8
-80
-60
-40
0.6
0.5
0.4
St / λ0 (ω=1.34ωp )
0.3
Sr / λ0 (ω=1.40ωp )
0.2
St / λ0 (ω=1.40ωp )
0.1
-20
0
20
incident angle (in degree)
(a)
40
60
80
0
-80
-60
-40
-20
0
20
incident angle (in degree)
40
60
80
(b)
Figure 3: For TM waves, (a) dependence of lateral displacements for transmission and for reflection Sr on the
incident angle θi . St and Sr are normalized to λ0 = 2πc/ω; (b) dependence of magnitude of the reflectance
|R|2 on the incident angle θi . The thickness d = 0.5λp = πc/ωp and the applied magnetic field B̄0 = +ẑ0.4 T.
The red and blue lines are for cases of ω = 1.34ωp and ω = 1.40ωp respectively.
PIERS Proceedings, Beijing, China, March 23–27, 2009
232
lateral shift of reflected wave, reflectance
100
|R| 2 x 1000
80
Sr / λ0 ( θ i = + θ B )
60
Sr / λ0 ( θ i = − θ B )
40
20
0
-20
-40
-60
1.2
1.3
1.4
1.5
1.6
1.7
1.8
frequency (normalized to plasma frequency)
1.9
2
Figure 4: For TM waves, when the incident angle equals to the Brewster angle θB , the lateral displacements
of reflection Sr (normalized to λ0 ), and the reflectance |R|2 (multiplied by 1000) versus frequency. The
thickness d = 0.5λp = πc/ωp and the applied magnetic field B̄0 = +ẑ0.4 T.
Furthermore, we want to mention that, due to nonreciprocal characteristics of the gyrotropic
medium, for reflection near the Brewster angle within the chosen frequency band, there are several
methods to realize the negative displacement. As shown in Fig. 5, when the applied magnetic field
is in +z direction, if both the incident and reflected waves are at the left side of the slab and the
incident wave comes from the lower left (case 1), the lateral shift reflected by the slab is positive;
while it becomes negative when the incident wave comes from the upper left (case 2). If the incident
and reflected waves move to the right side of the slab and incident wave comes from the lower right
(case 3), the lateral shift is negative, the same as case 2. Moreover, we can change the signs of all
the lateral shifts by adjusting B̄0 from +z direction to −z direction.
y
reflected wave 1
µ2 , ε2
incident wave 2
Sr2<0
θi <0
θ i >0
reflected wave 3
Sr1>0
d
x
0
Sr3<0
reflected wave 2
incident wave 1
B0
incident wave 3
Figure 5: Comparison of several incident TM wave cases. Case 1: incident wave and reflected wave are both
on the left side of the slab, and the incident wave comes from the lower left. Sr1 > 0. Case 2: incident
wave and reflected wave are also on the left side, but the wave is incident from the upper left. Due to the
nonreciprocal nature of the gyrotropic medium, the shift in case 1 is different from that in case 2. Sr2 < 0.
Case 3: incident wave also comes from below, but on the right side of the slab. Sr3 < 0. Case 2 and case 3
are equivalent, and lateral shifts are the same.
Progress In Electromagnetics Research Symposium, Beijing, China, March 23–27, 2009
233
4. CONCLUSION
This paper investigates the lateral displacements of waves transmitted and reflected from a symmetric gyrotropic slab, for both TE and TM modes. For TM waves, due to the external magnetic
field and the nonreciprocal nature of the gyrotropic medium, the displacements for transmission
beam are different from those for reflection, even for the lossless symmetric configuration. Furthermore, within the chosen frequency band, when incident angle is near the Brewster angle, the shift
of reflection can be positively or negatively large while the reflectance can be larger than weakly
absorbing media, making it easier to detect. Meanwhile, several cases for negative lateral shift are
shown. The results here may have potential applications in magnetic modulations. For instance,
switching the direction of the external magnetic field from +z to −z can result in a significant variation of the difference between the lateral displacements of the transmitted wave and the reflected
wave.
ACKNOWLEDGMENT
This work is sponsored by the Office of Naval Research under Contract N00014-06-1-0001, the Department of the Air Force under Air Force Contract F19628-00-C-0002, the Chinese National Foundation under Contract 60531020, and the Grant 863 Program of China under Contracts 2002AA529140 and 2004AA529310.
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