TM-TE Surface Plasmon Polariton Mode
Conversion Induced by a Magneto-optic
Waveguide
M. Khatir
N. Granpayeh
Faculty of Electrical and Computer Engineering
K. N. Toosi University of Technology
Tehran, Iran
Faculty of Electrical and Computer Engineering
K. N. Toosi University of Technology
Tehran, Iran
[email protected]
[email protected]
Abstract— In this paper, we have analyzed the magneto-optic
(MO) effects in three layer surface plasmon polariton (SPP) slab
waveguides in a longitudinal configuration that the applied
magnetic field is parallel to the interfaces and the wave
propagation direction. Two different major configurations
insulator-metal-insulator (IMI) and metal-insulator-metal (MIM)
with dielectric or metal ferromagnetic layers are analyzed.
Because of coupling between SPP modes, there are all
components of the electromagnetic fields, therefore, we do not
have purely TM modes in this configuration. We have shown a
transverse magnetic (TM) to transverse electric (TE) SPP mode
conversion in these anisotropic waveguides. These configurations
could be used to design elements, such as isolators and
modulators in photonic integrated circuits (PICs).
Sepúlveda et al. [3], the MO effects of surface plasmons in
metallic multilayer films have been treated [4]-[9], the MO
Kerr effects in thin films of ferromagnetic metals and
dielectrics have been reported [10]-[12], and the application of
the MO effects of surface plasmons in some devices and
components have been described [13]-[17]. Dispersion
relations of dispersive waveguides with all anisotropic layers in
the transversal and longitudinal configurations have been
extracted [18]-[19]. There is not any deep analysis of metalinsulator-metal (MIM) waveguides in the presence of magnetooptic effects. In this paper, we have analyzed the propagation
properties of the three layer optical waveguides including
surface plasmon polaritons in the presence of magneto-optic
effects for longitudinal configuration that the magnetic field is
parallel to the interfaces and the wave propagation direction,
without any approximation in the calculation of the dispersion
relation and field distribution of the SPP modes. Different
configurations for ferromagnetic dielectric and metallic layers
are considered. We have shown a transverse magnetic (TM) to
transverse electric (TE) SPP mode conversion in these
anisotropic waveguides.
The paper is organized as follows. In section II, we have
derived the general dispersion relation for SPP with
considering MO effects for two side layers. In section III, we
have studied the MO effects for different configuration of the
waveguides, and the paper is concluded in section IV.
Keywords- Magneto-optic (MO) effects, Surface plasmon
polaritons (SPP), Insulator-metal-insulator (IMI) waveguides,
Metal-insulator-metal (MIM) waveguides, Anisotropic plasmonic
slab waveguides.
I.
INTRODUCTION
Reduction of the dimensions of the conventional optical
components to nanooptic scale is restricted by the diffraction
limit of the light. The scientists and researchers have overcome
this limitation by utilizing surface plasmon polaritons (SPPs).
The surface plasmon waves propagate at the interface of a
metal and dielectric and decay exponentially at both sides of
the interface. Therefore, thin films of noble metals and
dielectrics can be used as waveguides for the SPP propagation
[1]. The nanoscale modification of the layers and application of
magneto-optic (MO) effects can lead to achieve desirable
components to control the propagation properties of the SPP,
the same as those of the conventional optical devices, such as
switches, modulators, isolators, and circulators [2][3].
However, there is not considerable work concerning the study
of the MO properties in the propagation of SPPs. For example,
insulator-metal-insulator (IMI) waveguide with different
orientations of the magnetization has been analyzed by
II.
GENERAL DISPERSION RELATION
The schematic view of the three layer magneto-optic
waveguides under study is demonstrated in Fig. 1. The central
layer has thickness of d and dielectric constant of ε 2 ,
surrounded by two semi-infinite layers. This structure acts as a
waveguide for the SPP propagating along the x-direction. It is
assumed that the structure has no spatial variation in the y direction. We assume two side layers have MO effects. The
directions of the magnetization of the magneto-optic materials
are parallel to the SPP propagation direction for longitudinal
___________________________________
978-1-61284-365-0/11/$26.00 ©2011 IEEE
599
configuration. The constructive parameters of MO materials
are given by μ r = 1 and [3],
z
⎛ ε xxi (ω )
0
0 ⎞
⎜
⎟
(1)
ε~ri (ω ) = ⎜ 0
ε yyi (ω ) ε yzi ⎟ ,
⎜
⎟
− ε yzi ε zzi (ω )⎠
⎝ 0
where ε xxi , ε yyi , ε zzi , ε yzi , a , M xi , are respectively the
0
⎛
a11 = ⎜ −
⎜
⎝
⎛
a12 = ⎜ −
⎜
⎝
a 22
a 23
a 32
a 33
a 42
a 43
a14 ⎤
a 24 ⎥⎥
=0
a 34 ⎥
⎥
a 44 ⎦
(2)
ki2± =
a14 = −e k 2+ d
(6)
jε xx 2 k 2+ d
e
k2 +
∓
2
− k02ε yyi − ki2±
ki2±
(β (ε
1
2ε zzi
xxi
+
k02ε xxi
)
(
)
2ε zzi
2
xxi
(16)
(17)
(18)
(
)
)
2
+ ε zzi ) − k02εzzi ε xxi + ε yyi − k02ε yzi
(
(15)
(19)
2
+ ε zzi ) − k02ε zzi ε xxi + ε yyi − k02ε yzi
{(β (ε
( (
(4)
(5)
⎛ ε βp (k − k ) ε βp (k − k ) ⎞ − k d
2+
2+ ⎟
a21 = ⎜ xx21 1+ 1−
e 2+
− xx21 1− 1+
⎟
⎜ k 0 ε yz1 (k1+ − k1− )
(
)
ε
k
k
k
−
0 yz1 1+
1−
⎠
⎝
⎛ ε β p (k + k ) ε βp (k + k ) ⎞ k d
2+
2 + ⎟ 2+
a22 = ⎜ xx21 1+ 1−
e
− xx21 1− 1+
⎟
⎜ k0 ε yz1 (k1+ − k1− )
(
)
ε
k
k
k
−
yz
0
1
1
+
1
−
⎠
⎝
jε xx 2 − k 2+ d
a23 = −
e
k2 +
(β
2
a13 = −e − k 2+ d
a24 =
pi ± =
jβ k1+ p1+ (k1− − k 2 + ) jβk1− p1− (k1+ − k 2 + ) ⎞⎟ − k 2+ d
e
(3)
+ 2
k 02ε yz1 (k1+ − k1− )
k0 ε yz1 (k1+ − k1− ) ⎟⎠
jβk1+ p1+ (k1− + k 2 + ) jβk1− p1− (k1+ + k 2 + ) ⎞⎟ k 2+ d
e
+ 2
k 02ε yz1 (k1+ − k1− )
k0 ε yz1 (k1+ − k1− ) ⎟⎠
(14)
⎛ ε βp (− k − k ) ε βp (− k − k ) ⎞
3−
2+
3+
2+ ⎟
a41 = ⎜ − xx32 3 +
+ xx32 3 −
⎜
(
)
(
) ⎟⎠
k
ε
k
k
k
ε
k
−
k
−
0
yz
3
3
+
3
−
0
yz
3
3
+
3
−
⎝
⎛ ε βp (− k + k ) ε βp (− k + k ) ⎞
3−
2+
3+
2+ ⎟
a42 = ⎜ − xx32 3 +
+ xx32 3 −
⎜
⎟
(
)
(
k
ε
k
k
k
ε
k
−
k
−
0 yz 3 3 +
3−
0 yz 3 3 +
3− ) ⎠
⎝
jε
a43 = − xx 2
k2 +
jε
a44 = xx 2
k2 +
dependence of the TE and TM modes in the Maxwell’s
equations. Hence, there are not purely TM modes in this
configuration The dispersion relation of this configuration is
obtained after applying the boundary conditions at two
interfaces of z = −d and z = 0 which results in the following
relation [19]:
a13
Mx1
a34 = −1
is reversed. The magnetization couples the y and z components
of the electric fields, E y and E z , this coupling induces the
a12
~
ε1 (ω) Layer I
Figure 1.
Geometry of the surface plasmon magneto-optic
device under study.
layers. The sign of ε yzi changes if the magnetization direction
⎡ a11
⎢a
det = ⎢ 21
⎢ a 31
⎢
⎣a 41
x
ε 2 (ω ) Layer II
-d
dielectric constants in the x, y, z directions, nondiagonal
element of the dielectric tensor, MO constant, magnetization,
and ε yzi = −aM xi . The i=1, 3 are the number of two side
Mx3
~
ε3 (ω) Layer III
) (
2
− 4ε zzi εxxi β 4 − β 2k02 ε yyi + ε zzi + k04 ε yzi
+ ε yyiε zzi
)
2
(20)
)))}
1
2
where k0, β and k i ± are respectively the vacuum wave
number, propagation constant of travelling waves, decay
constant in transversal direction in three layers of i=1, 2, 3,
and ky is assumed to be zero.
(7)
III.
(8)
MO EFFECTS FOR DIFFERENT CONFIGURATIOS
Now, we present the results for the numerical solution of the
dispersion relation (2) for two different layers' configurations.
All the structures are excited by TM mode. The metallic and
dielectric layers are assumed to be gold, cobalt and YIG,
respectively. In these calculations, for wavelength of 1550 nm
the dielectric constants of gold and the insulator medium are
ε mAu = −115.53 + j11.21 and ε dIns = 4.84 , respectively. We
have also assumed that the diagonal elements of the dielectric
tensor of the YIG and cobalt are ε dYIG = 4.84 and
(9)
(10)
⎛ jβk p (− k − k ) jβk p (− k − k ) ⎞
3+ 3 +
3−
2+
3− 3−
3+
2+ ⎟
a31 = ⎜ −
+
(11)
2
2
⎜
⎟
(
)
(
)
k
ε
k
−
k
k
ε
k
−
k
0 yz 3 3 +
3−
0 yz 3 3 +
3−
⎝
⎠
⎛ jβk p (− k + k ) jβk p (− k + k ) ⎞
3+ 3+
3−
2+
3− 3−
3+
2+ ⎟
a32 = ⎜ −
+
(12)
2
2
⎜
⎟
(
)
(
)
k
ε
k
k
k
ε
k
−
k
−
0
yz
3
3
+
3
−
0
yz
3
3
+
3
−
⎝
⎠
a33 = −1
(13)
ε mCo = −7.96 + j 60.90 , while the nondiagonal elements are
Co
ε YIG
yz = j 0.005 and ε yz = −1.4858 + j 0.9832 , respectively for
a saturated magnetization ( aM x > 0 ) at the same wavelength
[20]-[21].
600
the x-axis, at the wavelength of 1550 nm. It can be observed
that there is the Hz component in these structures, but it
strongly decays because of spreading modes out in metal
layers. As a result, MO effects or mode conversion capability
of structure depends on the configuration of layers. It should
be considered that the propagation length of the SPP waves is
an important parameter to utilize MO effects to design the
nanooptic devices.
A. IMI Configuration
The real and imaginary parts of the propagation constant versus
thickness of the mid layer of an IMI configuration with gold
layer surrounded by insulator media are illustrated in Fig. 2. It
can be observed that the imaginary part of the even modes
decreases when the thickness of the metallic layer is reduced.
This means that the electromagnetic field of the even modes
spreads out in the insulator media and its propagation length
increases. In contrast, the attenuation of odd modes increases
with the reduction of the metallic layer thickness.
(a)
(a)
(b)
(b)
Figure 2.
The real (a) and imaginary (b) parts of the
propagation constant vs. mid layer thickness for the even and odd SPP
modes in IMI configuration with gold layer surrounded by insulator
media with εd = 4.84, at the wavelength of 1550 nm.
(c)
Figure 3.
Field distribution of (a) Hy, (b) real part of Hy,
and (c) Hz for the odd SPP mode in IMI configuration with the
middle gold layer of 50 nm surrounded by YIGs with magnetizations
parallel to the positive direction of the x-axis, at the wavelength of
1550 nm.
In Fig. 3, the field distributions of Hy, real part of Hy and Hz
are demonstrated for the odd SPP mode in IMI configuration
with gold layer of 50 nm surrounded by YIGs with
magnetizations parallel to the positive direction of the x-axis,
at the wavelength of 1550 nm. It can be observed that the TE
mode component, Hz, is induced with two maximum points in
the side layers. This means that mode conversion from TM to
TE has happened because of the coupling between y and z
components of magnetic field in longitudinal bias
configuration. In Fig. 4, the results of the same configuration
are illustrated for the even SPP modes which have propagation
length longer than that of the odd SPP modes.
IV.
CONCLUSION
In this paper, we have analyzed the three layer surface Plasmon
polariton waveguides. The effective refractive index and the
dispersion relation of the structure are derived in the cases that
two side layers show magneto-optic behavior by longitudinal
magnetization. The MO effects have been studied for the SPP
modes guided by IMI and MIM waveguides. We have shown
TE mode component (Hz) is induced by magneto-optic
waveguide in longitudinal bias configuration which TM mode
is converted to TE mode. The propagation length of the even
modes in IMI waveguides is greater than odd modes. We have
studied the guided modes in MIM waveguides with
ferromagnetic layers which are propagated in short length.
However, We think, there is a optimum plan to design a MO
device based on SPPs with considering different parameters
B. MIM Configuration
In Fig. 5, we have shown the field distributions of Hy and Hz
for the even SPP mode in MIM configuration with insulator
media with εd=4.84 and 100 nm thickness surrounded by Co
layers with magnetizations parallel to the positive direction of
601
such as confinement factor, propagation length and spatial 3.5
cmextension.
REFERENCES
[1]
[2]
[3]
[4]
[5]
(a)
[6]
[7]
[8]
(b)
[9]
[10]
[11]
[12]
(c)
Figure 4.
Field distribution of (a) Hy, (b) real part of Hy,
and (c) Hz for the even SPP mode in IMI configuration with the
middle gold layer of 50 nm surrounded by YIGs with magnetizations
parallel to the positive direction of the x-axis, at the wavelength of
1550
[13]
[14]
[15]
[16]
(a)
[17]
[18]
[19]
(b)
[20]
Figure 5.
Field distribution of (a) Hy and (b) Hz for the
even SPP mode in MIM configuration with insulator media with
εd=4.84 and 100 nm thickness surrounded by Co layers with
magnetizations parallel to the positive direction of the x-axis, at the
wavelength of 1550 nm.
[21]
602
W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon
subwavelength optics,” Nature., Vol. 424, pp. 824-830, 2003.
R. Gordon, “Surface Plasmon nanophotonics: a tutorial,” IEEE
Nanotechnol Mag., Vol. 2, pp. 12-18, 2008.
B. Sepúlveda, L. M. Lechuga, and G. rmelles, “Magnetooptic effects in
surface-plasmon-polaritons slab waveguides,” IEEE J. Lightwave
Technol., Vol. 24, pp. 945-955, 2006.
V. I. Safarov, V. A. Kosobukin, C. Hermann, G. Lampel, and J. Peretti.
“Magneto-optical effects enhanced by surface plasmons in metallic
multilayer films,” Appl. Phys. Lett., Vol. 73, pp. 3584-3587, 1994.
S. Uba, L. Uba, A. N. Yaresko, A. Ya. Perlov, V. N. Antonov, and R.
Gontarz, “Optical and magneto-optical properties of Co/Pt multilayers,”
Phys. Rev. B, Vol. 53, pp. 6526-6535, 1996.
P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of
finite width: Bound modes of symmetric structures,” Phys. Rev. B, Vol.
61, pp. 10484- 10503, 2000.
N. Bond, R. Reinisch, E. Popov, and M. Neviere, “Optimization of
surface-plasmon-enhanced magneto-optical effects,” J. Opt. Soc. Am.
B., Vol. 21, pp. 791-797, 2004.
Y. M. Strelniker, D. Stroud, and A. O. Voznesenskaya, “Magnetooptical features and extraordinary light transmission through perforated
metal films filled with liquid crystals,” J. Appl. Phys., Vol. 99, pp. 70217023, 2006.
E. F. Vila, X. M. B. Sueiro, J. B. González-Díaz, A. García-Martín, J.
M. García-Martín, A. C. Navarro, G. A. Reig, D. M. Rodríguez, and E.
M. Sandova, “Surface plasmon resonance effects in the magneto-optical
activity of Ag–Co–Ag trilayers,” IEEE Trans. Magnetics, Vol. 44, pp.
3303-3306, 2008.
N. Richard, “Magneto-optical Kerr effect in bilayer structures,” Eur.
Phys. J. B., Vol. 16, pp. 411–422, 2000.
Z. Q. Qiu and S. D. Bader, “Surface magneto-optic Kerr effect,” Rev.
Sci. Instrum., Vol. 71, pp. 1243-1255, 2000.
S. Tomita, T. Kato, S. Tsunashima, S. Iwata, M. Fujii, and S. Hayashi.
“Magneto-optical kerr effects of yttrium-iron garnet thin films
incorporating gold nanoparticles,” Appl. Phys. Lett., Vol. 96, pp.
167402(1-4), 2006.
K. J. Chau, S. E. Irvine, and A. Y. Elezzabi, “A gigahertz surface
magneto-plasmon optical modulator ,” IEEE J. Quantum Electron., Vol.
40, pp. 571-579, 2004.
R. L. Espinola, T. Izuhara, M. Ch. Tsai, R. M. Osgood, and H. Dötsch,
“Direct Magneto-optical nonreciprocal phase shift in garnet_silicon-oninsulator waveguides,” Opt. Lett., Vol. 29, pp. 941–943, 2004.
M. Vanwolleghem, P. Gogol, P. Beauvillain, W. Van Parys, and R.l.
Baets, “Design and optimization of a monolithically integratable InPbased optical waveguide isolator,” J. Opt. Soc. Am. B., Vol. 24,
pp. 1-12, 2007
F. M. Kong , K. Li, H. Huang, B.-I. Wu, and J. A. Kong, “Analysis of
the surface magnetoplasmon modes in the semiconductor slit waveguide
at terahertz frequencies,” Prog. in Electromag. Research, PIER, Vol. 82,
pp. 257-270, 2008.
J. Montoya, J. Hensley, K. Parameswaran, M. Allen, and R. Ram
“Surface plasmon isolator based on nonreciprocal coupling,” J. Appl.
Phys., Vol. 106, pp. 1063 (1-9), 2009
M. Khatir and N. Granpayeh,, “An exact analysis method of SPP
propagation in the anisotropic magneto-optic slab waveguides, in
transversal configuration,” unpublished.
M. Khatir and N. Granpayeh,, “An exact analysis method of SPP
propagation in the anisotropic magneto-optic slab waveguides, in
longitudinal configuration,” unpublished.
J. H. Weaver, C. Krafka, D. W. Lynch, and E. E. Kock, “Optical
Properties of Metals,”. Appl. Opt. Vol. 20, pp. 1124_1-1125, 1981.
T. Shintaku and T. Uno, “Preparation of Ce-substituted yttrium iron
garnet films for magneto-optic waveguide devices,” Jpn. J. Appl. Phys.,
Vol. 35, pp. 4689–4691, 1996.

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