- Email: [email protected]
The transmission characteristics of a subwavelength metal slab waveguide
Accepted Manuscript Title: The transmission characteristics of a subwavelength metal slab waveguide Authors: Yan Wang, Yanni Tang, Qingchen Ji, Xiaona Yan, Huifang Zhang PII: DOI: Reference:
S0030-4026(19)30368-7 https://doi.org/10.1016/j.ijleo.2019.03.077 IJLEO 62563
To appear in: Received date: Accepted date:
29 January 2019 15 March 2019
Please cite this article as: Wang Y, Tang Y, Ji Q, Yan X, Zhang H, The transmission characteristics of a subwavelength metal slab waveguide, Optik (2019), https://doi.org/10.1016/j.ijleo.2019.03.077 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The transmission characteristics of a subwavelength metal slab waveguide
Yan Wang*, Yanni Tang, Qingchen Ji, Xiaona Yan and Huifang Zhang
IP T
Department of Physics, Shanghai University, Shanghai 200444, China
SC R
Corresponding author E-mail address: [email protected]
U
Phone number: +86 13701815402
A
N
Abstract
M
The transmission characters of a narrow slit were an interesting topic for a long time.
ED
In this paper, the transmission characters of an air slit carved form a metal film was discussed rigorously. The narrow slit was treated to be a three-layer-waveguide with
PT
two complex refractive index claddings. The transmission character eigen equations
CC E
were derived by solving analytically from Maxwell’s equations and by matching the boundary conditions at the interfaces. The electromagnetic fields in both sides of the claddings and inside of the air slit were obtained. For a given example, numerical
A
simulations were presented. The field distribution and the longitude phase constants at different wavelengths and with different width of the slits were calculated. Five modes were supported, and one surface Plasmon mode was attributed and four guided modes were attributed from their transmission characters. The p-polarized surface
Plasmon mode might play an important role in the extraordinary optical transmission of light through arrayed subwavelength holes.
IP T
Key Words: air slit, metal waveguide, eigen equations, surface Plasmon mode.
1 Introduction
SC R
There was a long history for the study of the transmission characters of a small
hole [1]. In 1998, Ebbesen et. al. discovered in experiments that the transmittance of
U
the metal film in the array of periodic circular holes was much larger than what is
N
estimated from standard aperture theory [2]. This phenomenon is called extraordinary
A
optical transmission (EOT) of subwavelength aperture arrays. It was believed that the
M
enhancement of transmission was due to through surface plasmons formed on each
ED
metal-dielectric interface or by the coupling of incident plane waves with waveguide resonances located in the slit [3, 4]. Many further studies showed that the transmission
PT
coefficient depends on not only the holes’ area, but also on the shapes of the holes
CC E
[5-8]. That is to say, the holes play an important role in the EOT. However, so far, to our limited knowledge, there was no thorough rigorous analysis of the modes inside an air slit carved from a metal film. Therefore, the related mode analysis was carried
A
out, which is the fundamental of varied metal waveguides. It is hoped that the results might give insight into the mechanism and be of importance for device application.
In this paper, the detailed studies of the transmission characters of an air slit
carved from a metal slit were carried out. The system was modeled as an air waveguide with two metal claddings with complex permittivity. The general transition conditions were derived, and two well know circumstance, the dielectric waveguides and the perfect metal waveguides could be covered. The analytic solutions to the
IP T
electromagnetic fields in the air slit and the two metal claddings were provided. The longitude phase constants were calculated at different wavelength and with different
SC R
widths of the slits, which allowed a quantitative description of the transmission character of the waveguides. The influences of the wavelength of incidence,
U
polarization, and the width of the slits on the longitude phase constants were
N
discussed. For a given example, numerical calculation shown that five modes were
A
supported in total and two zero order modes were supported for TM modes. One zero
M
order TM mode was transmitted with a longitude phase constant always faster than
ED
that in the air in the researched range. This mode was attributed to be the Surface Plasmon mode, and named to be TM0p mode. And the guided TM1 mode could be
CC E
conditions.
PT
transmitted with a longitude phase constant faster than that in the air at the favorable
A
2 General transmission equations for slab waveguides The eigen equation for a slab waveguide (Shown in Fig.1) can be derived from
the Maxwell equations and by matching the boundary conditions [9].
Fig. 1 Geometry of a slit waveguide carved out of a thin metal film. The width of the waveguide was d, and the permittivity of the substance inside the waveguide was ε0 = εr0 + iεi0 . The
IP T
permittivity of the metals were ε1 = ε1r + iε1i and ε2 = εr2 + iεi2 respectively. θ was the zigzag
SC R
angle between the real part wavevector of the supported modes and the axial of the waveguide.
For a three-layer slab waveguide, the total electric and magnetic fields of TM
U
modes can be derived from Maxwell’s equations and by matching the boundary
N
conditions. The fields in the three regions are written as H2y = H2 eik2x x , x < −d/2
M
H1y = H1 eik1x x , x > 𝑑/2
A
{H0y = Aeik0xx + Be−ik0x x , −d/2 ≤ x ≤ d/2 (1)
where the factor ei(keff z−ωt) was suppressed; k eff = k z .
E0x =
ωε0
< −d/2
(Aeik0xx + Be−ik0x x ), −d/2 ≤ x ≤ d/2 (2) E1x =
keff H1 ik x e 1x , x ωε1
PT
{
keff
keff H2 ik x e 2x , x ωε2
ED
E2x =
> 𝑑/2
−k 2x d H−1 eik2xx , x < − ωε−1 2 −k 0x d d = (Aeik0x x − Be−ik0x x ), − ≤ x ≤ (3) ωε0 2 2 −k1x d E1z = H1 eik1x x , x > ωε1 2
CC E
E2z =
E0z
A
{
Continuity of the tangential fields, Ez and Hy in Eq. (3) at x=±d/2 required that R01 R02 ei2k0x d = 1.
(4)
ε k −ε k
ε k −ε k
Where R01 = ε1k0x +ε0 k1x = r01 eiδ01 and R02 = ε2 k0x+ε0 k2x = r eiδ02 are the Fresnel 1 0x
0 1x
2 0x
0 2x
02
reflection coefficients between the corresponding regions, and k0x = k0xr + ik0xi is the transverse wavevector in the core region, and and k1x = k1xr + ik1xi is the
transverse wavevector in the cladding layer; k2x = k2xr + ik2xi is the one in the subtract layer.
For a symmetric slab waveguide, where ε1 = ε2 , Eq. (4) can be writen as, R01 eik0xd = 1, (5) Eq. (5) has two parts, the phase relationship and the amplitude relationship, which can
IP T
be written as followings, respectively, r01 e−k0xid = 1; or k0xi = ln(r01 )/d (6a)
SC R
δ01 + k0xr d = mπ, (m=0, 1, 2…). (6b)
In (perfect electric conductor) PEC model and in the dielctric waveguide, Eq (6a) is
U
often omitted since in those two typical cases, r01=1 and r02=1 which means k0xi=0,
N
that is to say, the energy of each sorported mode can be fully confined by the
A
waveguide. However, if the finite conductivity and the weakly absorption of a real
M
metal are considered, Eq (6a) can not be omitted since the amplitude of the Fresnel
ED
coefficient are less than one except at 0 degree (See Fig 2).
The above expressions are for arbitrary and . By the principle of duality, the TE
PT
and TM are dual of each other. The expressions for the TE case can be obtained by the replacements
En H n , H n En and n n in Eqs. (1) – (6). But for TE k −k
k −k
CC E
waves, the Fresnel coefficient is R01 = k0x+k1x = r01 eiδ01 and R02 = k0x +k2x = r eiδ02 0x
1x
0x
2x
02
A
for non-magnetic material.
For the waveguide in the setup illustrated in Fig. (1), the coupled eigen equations (6a and 6b) were solved numerically, and the results were shown in Table 1. The air slit waveguide does support 5 modes in total, 3 TM modes and two TE modes (See Table 1 and Fig. 2).
Table 1 shows the numerically calculated results of Eq. (6a and 6b). Two even TM0 modes, one TM1 , and two even TE0 modes are surported by the air slit metal slab waveguide. The r01 is the amplitude of the Freshenl coeffiecnt, and Angle is the angle between the real part wavevecter inside the air slit and the axical of the waveguide; K0x=k0xr + ik0xi is the transverse wavevecter inside the airslit, whose real part of
IP T
K0x is k0xr = k0 cos(θ), and the image part is k0xi = ln(r01 )/d, where k0 is the waveve number in the free space, d the width of the air slit, and the θ is the angle
between the wavevector in the air slit and the axial of the air slit, and 00≤ θ ≤ 900
SC R
should be saticfied (See Fig. 1), and r01 is the amplidude of the possible Fresnel
coefficient, and 0≤r01≤1. k1x=k1xr + ik1xi is the transverse wavevector in the metal
U
cladding; and Kz is the longitude wavevector.
N
TM0 and TE0 modes are the fundamental modes passing through the waveguide
A
directly, which are never cut-off. This is in coincident with the character of a
M
symmetry waveguide. There is the other zero order TM mode, TM0p mode, which travels faster than the two fundamental modes, and with a very tiny angle with the
ED
axial line. In order to distinguish it from the fundamental TM0 mode, it is named to be TM0p mode, because it is a kind of Surface Plasmon mode. Its longitude attenuation
PT
coefficient is less than that of the guided TM1 mode, which means it can travel longer distance than the TM1 mode. The real part of its traverse wavevector is much less
CC E
than that of the TM1 mode, but the image part of its traverse wavevector is much greater than that of the TM1 mode, which means it is well confined near the surface between the air and the metal cladding, however the TM1 mode leaks much energy
A
because it penetrates into the cladding deeply.
And the electromagnetic fields distributions were shown in Fig. 2.
2
1.5
X (nm)
250
1
0.5
0
0
-0.5
-250
-1
(a) 250
500
Z (nm)
750
1,000
IP T
0
-1.5
1 0.8
250
0.6
SC R
X (nm)
0.4 0.2
0
0 -0.2 -0.4
250
-0.6 -0.8
(b) 500
Z (nm)
750
1,000
A
3
2
M
X (nm)
250
1
0
0
(c) 250
500
Z (nm)
750
-2
-3
1,000
PT
0
ED
-1
-250
2
CC E
1.5
X (nm)
250
1
0.5
0
0
-0.5
A
-250
-1
-1.5
(d) 0
U
250
N
0
-1
250
500
Z (nm)
750
1,000
250
1
X (nm)
1.5
0.5
0
0
-0.5
-250
-1
(e) 250
500
750
Z (nm)
1,000
IP T
0
-1.5
Fig. 2 The field distributions of the 5 sorpported modes. (a)-(c) show the Hy compenents of the
SC R
TM0 mode, the TM1 mode, the TM0p mode, respectively. And (d)-(e) show the Ey compenents of the TE0 mode and TE1 mode, respectively.
U
In order to understand the transmission of the modes deeply, the longitudenal
N
phase constant of the three non-fundmental modes were calculated at different
620
640
1.100
M TM0p
1.080
(a) 1.02
500nm r 500nm i 600nm i
PT
1.085
600nm r 580
600
580
600
1.00 0.98
0.0095
0.0090
400nm r 400nm i
1.095 1.090
680
ED
1.105
620
640
660
680
660
680
0.0085
0.0080
0.0075
wavelength (nm) 620
640
CC E A
660
600nm r
0.007 0.006
500nm r
0.96
0.005
0.94 0.92 0.90
400nm i 400nm r
0.004
TM1
0.003
0.88 0.86 0.84 0.82
(b)
500nm i
0.002
600nm i 580
600
image part of the longitudinal phase constant image part of the longitudinal phase constant
600
1.110
real part of the longitudinal phase constant r
real part of the longitudinal phase constant r
580 1.115
A
wavelength with different width of the slit (See Fig. 3).
620
640
wavelength (nm)
660
680
0.001
0.85
600
650
0.007
600nm r
0.006
500nm r 0.005
0.80
0.75
400nm i 400nm r
0.004
TE1
0.003 0.70
0.65
0.002
600
wavelength (nm)
0.001
650
IP T
(c)
500nm i 600nm i
image part of the longitudinal phase constant
real part of the longitudinal phase constant r
0.90
SC R
Fig. 4 shows the longitudinal phase constants of the TM0p, TM1 and TE1 modes.
Fig. 4 shows the transmission character of the TM0p, TM1 and TE1 modes. Although TE1 mode and the TM1 mode have different polarization, they share almost
N
U
the same transmission characters. The real part of the longitudinal phase constants
A
decrease with the increase of the wavelength and the decrease of the slit width.
M
However, the transmission character of the TM0p mode is on the opposite side, that is to say, its real part of the longitudinal phase constant increases with the increase of the
ED
wavelength and the decrease of the slit width. And furthermore, the image part of the
PT
longitudinal phase constants of the TM0p mode increase with the increase of its real part. However, the image part of the longitudinal phase constants of the TM1 mode
CC E
and TE1 mode increase with the decrease of their real part. Therefore, it can be concluded that the TM1 mode and the TE1 mode are two guided modes of the
A
waveguide. However, the TM0p mode is the Plasmon mode of the waveguide. Comparing experiments showed that the transmission features of the array are
consistent with p-polarized resonant modes of the structure [4], the TM0p mode might play an constructive role in the EOT. The transverse phase constants (k1x) of the supported modes in the metal cladding
are all complex numbers (see Table 1), which means all the modes are leaky. That is to say, the energy can penetrate into the metal cladding. And there might be good chance for the leaked energy to be coupled to the surface plasmons formed on the
IP T
incident and output metal-air or metal-subtract interface.
In conclusion, the general eigen equation of slab waveguide is derived, and the
SC R
field distribution were calculated. For a given example, the calculation shows that 5
modes were supported, and one of them was attributed to be a Surface Plasmon mode,
U
which might play an important role in EOT since it travels with a very tiny angle with
A
N
the axial line, which might mean that it is easily to be excited.
1.
ED
M
References
M. Born and E. Wolf E, Principles of Optics 7th Ed (Cambridge University Press,
PT
Cambridge, England, 1999)
CC E
2. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. L. Wolff, Extraordinary optical transmission through sub-wavelength hole arrays, Nature
391 (1998)
667-669
A
3. J. A. Porto, F. J. Garcı´a-Vidal, and J. B. Pendry, Transmission resonances on metallic gratings with very narrow slits, Phys. Rev. Lett. 83 (1999) 2845-2848
4. W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, Surface plasmon polaritons and their role in the enhanced transmission of light through
periodic arrays of subwavelength holes in a metal film, Phys. Rev. Lett. 92 (2004) 107401 5. A. Degiron, H. J. Lezec, W. L. Barnes, and T. W. Ebbesen, Effects of hole depth on enhanced light transmission through subwavelength hole arrays, Appl. Phys.
IP T
Lett. 81 (2002) 4327-4328 6. K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers,
SC R
Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes, Phys. Rev. Lett. 92 (2004) 183901
U
7. R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L.
N
Kavanagh, Strong polarization in the optical transmission through elliptical
A
nanohole arrays, Phys. Rev. Lett. 92 (2004) 037401
M
8. Nicolas Bonod, Stenfan Enoch, Lifeng Li, Evgeny Popov and Michel Nevière,
ED
Resonant optical transmission through thin metallic films with and without holes, Opt. Exp. 11 (2003) 482-490
A
CC E
(2000)
PT
9. Electromagnetic Wave Theory, JIN AU KONG, EMW Publishing, Cambridge,
Table 1 The parameters of the modes surpored by a gold metal slab waveguide, the width of the slit is 500nm, the wavelength 632.8nm, and the refractive index of gold is nau =0.18+i2.99.
Modes r01 (a. u.) Angle
K0x (/k0)
K1x (/k0)
Kz (/k0)
0
0.1707+3.152i
1
(degree) 1
0
TM0p
0.1109 1.1575
0.2006-4.3983i 0.1662+3.1830i
1.0936+0.0082i
TM1
0.7697 14.822
2.5401-0.5235i 0.1670+3.142i
0.9683+0.0139i
TE0
1
0
1
TE1
0.9829 34.1196 5.5695-0.0345i 0.1729+3.102i
0.1707+3.152i
A
CC E
PT
ED
M
A
N
U
SC R
0
IP T
TM0
0.8279+0.0024i
S0030-4026(19)30368-7 https://doi.org/10.1016/j.ijleo.2019.03.077 IJLEO 62563
To appear in: Received date: Accepted date:
29 January 2019 15 March 2019
Please cite this article as: Wang Y, Tang Y, Ji Q, Yan X, Zhang H, The transmission characteristics of a subwavelength metal slab waveguide, Optik (2019), https://doi.org/10.1016/j.ijleo.2019.03.077 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The transmission characteristics of a subwavelength metal slab waveguide
Yan Wang*, Yanni Tang, Qingchen Ji, Xiaona Yan and Huifang Zhang
IP T
Department of Physics, Shanghai University, Shanghai 200444, China
SC R
Corresponding author E-mail address: [email protected]
U
Phone number: +86 13701815402
A
N
Abstract
M
The transmission characters of a narrow slit were an interesting topic for a long time.
ED
In this paper, the transmission characters of an air slit carved form a metal film was discussed rigorously. The narrow slit was treated to be a three-layer-waveguide with
PT
two complex refractive index claddings. The transmission character eigen equations
CC E
were derived by solving analytically from Maxwell’s equations and by matching the boundary conditions at the interfaces. The electromagnetic fields in both sides of the claddings and inside of the air slit were obtained. For a given example, numerical
A
simulations were presented. The field distribution and the longitude phase constants at different wavelengths and with different width of the slits were calculated. Five modes were supported, and one surface Plasmon mode was attributed and four guided modes were attributed from their transmission characters. The p-polarized surface
Plasmon mode might play an important role in the extraordinary optical transmission of light through arrayed subwavelength holes.
IP T
Key Words: air slit, metal waveguide, eigen equations, surface Plasmon mode.
1 Introduction
SC R
There was a long history for the study of the transmission characters of a small
hole [1]. In 1998, Ebbesen et. al. discovered in experiments that the transmittance of
U
the metal film in the array of periodic circular holes was much larger than what is
N
estimated from standard aperture theory [2]. This phenomenon is called extraordinary
A
optical transmission (EOT) of subwavelength aperture arrays. It was believed that the
M
enhancement of transmission was due to through surface plasmons formed on each
ED
metal-dielectric interface or by the coupling of incident plane waves with waveguide resonances located in the slit [3, 4]. Many further studies showed that the transmission
PT
coefficient depends on not only the holes’ area, but also on the shapes of the holes
CC E
[5-8]. That is to say, the holes play an important role in the EOT. However, so far, to our limited knowledge, there was no thorough rigorous analysis of the modes inside an air slit carved from a metal film. Therefore, the related mode analysis was carried
A
out, which is the fundamental of varied metal waveguides. It is hoped that the results might give insight into the mechanism and be of importance for device application.
In this paper, the detailed studies of the transmission characters of an air slit
carved from a metal slit were carried out. The system was modeled as an air waveguide with two metal claddings with complex permittivity. The general transition conditions were derived, and two well know circumstance, the dielectric waveguides and the perfect metal waveguides could be covered. The analytic solutions to the
IP T
electromagnetic fields in the air slit and the two metal claddings were provided. The longitude phase constants were calculated at different wavelength and with different
SC R
widths of the slits, which allowed a quantitative description of the transmission character of the waveguides. The influences of the wavelength of incidence,
U
polarization, and the width of the slits on the longitude phase constants were
N
discussed. For a given example, numerical calculation shown that five modes were
A
supported in total and two zero order modes were supported for TM modes. One zero
M
order TM mode was transmitted with a longitude phase constant always faster than
ED
that in the air in the researched range. This mode was attributed to be the Surface Plasmon mode, and named to be TM0p mode. And the guided TM1 mode could be
CC E
conditions.
PT
transmitted with a longitude phase constant faster than that in the air at the favorable
A
2 General transmission equations for slab waveguides The eigen equation for a slab waveguide (Shown in Fig.1) can be derived from
the Maxwell equations and by matching the boundary conditions [9].
Fig. 1 Geometry of a slit waveguide carved out of a thin metal film. The width of the waveguide was d, and the permittivity of the substance inside the waveguide was ε0 = εr0 + iεi0 . The
IP T
permittivity of the metals were ε1 = ε1r + iε1i and ε2 = εr2 + iεi2 respectively. θ was the zigzag
SC R
angle between the real part wavevector of the supported modes and the axial of the waveguide.
For a three-layer slab waveguide, the total electric and magnetic fields of TM
U
modes can be derived from Maxwell’s equations and by matching the boundary
N
conditions. The fields in the three regions are written as H2y = H2 eik2x x , x < −d/2
M
H1y = H1 eik1x x , x > 𝑑/2
A
{H0y = Aeik0xx + Be−ik0x x , −d/2 ≤ x ≤ d/2 (1)
where the factor ei(keff z−ωt) was suppressed; k eff = k z .
E0x =
ωε0
< −d/2
(Aeik0xx + Be−ik0x x ), −d/2 ≤ x ≤ d/2 (2) E1x =
keff H1 ik x e 1x , x ωε1
PT
{
keff
keff H2 ik x e 2x , x ωε2
ED
E2x =
> 𝑑/2
−k 2x d H−1 eik2xx , x < − ωε−1 2 −k 0x d d = (Aeik0x x − Be−ik0x x ), − ≤ x ≤ (3) ωε0 2 2 −k1x d E1z = H1 eik1x x , x > ωε1 2
CC E
E2z =
E0z
A
{
Continuity of the tangential fields, Ez and Hy in Eq. (3) at x=±d/2 required that R01 R02 ei2k0x d = 1.
(4)
ε k −ε k
ε k −ε k
Where R01 = ε1k0x +ε0 k1x = r01 eiδ01 and R02 = ε2 k0x+ε0 k2x = r eiδ02 are the Fresnel 1 0x
0 1x
2 0x
0 2x
02
reflection coefficients between the corresponding regions, and k0x = k0xr + ik0xi is the transverse wavevector in the core region, and and k1x = k1xr + ik1xi is the
transverse wavevector in the cladding layer; k2x = k2xr + ik2xi is the one in the subtract layer.
For a symmetric slab waveguide, where ε1 = ε2 , Eq. (4) can be writen as, R01 eik0xd = 1, (5) Eq. (5) has two parts, the phase relationship and the amplitude relationship, which can
IP T
be written as followings, respectively, r01 e−k0xid = 1; or k0xi = ln(r01 )/d (6a)
SC R
δ01 + k0xr d = mπ, (m=0, 1, 2…). (6b)
In (perfect electric conductor) PEC model and in the dielctric waveguide, Eq (6a) is
U
often omitted since in those two typical cases, r01=1 and r02=1 which means k0xi=0,
N
that is to say, the energy of each sorported mode can be fully confined by the
A
waveguide. However, if the finite conductivity and the weakly absorption of a real
M
metal are considered, Eq (6a) can not be omitted since the amplitude of the Fresnel
ED
coefficient are less than one except at 0 degree (See Fig 2).
The above expressions are for arbitrary and . By the principle of duality, the TE
PT
and TM are dual of each other. The expressions for the TE case can be obtained by the replacements
En H n , H n En and n n in Eqs. (1) – (6). But for TE k −k
k −k
CC E
waves, the Fresnel coefficient is R01 = k0x+k1x = r01 eiδ01 and R02 = k0x +k2x = r eiδ02 0x
1x
0x
2x
02
A
for non-magnetic material.
For the waveguide in the setup illustrated in Fig. (1), the coupled eigen equations (6a and 6b) were solved numerically, and the results were shown in Table 1. The air slit waveguide does support 5 modes in total, 3 TM modes and two TE modes (See Table 1 and Fig. 2).
Table 1 shows the numerically calculated results of Eq. (6a and 6b). Two even TM0 modes, one TM1 , and two even TE0 modes are surported by the air slit metal slab waveguide. The r01 is the amplitude of the Freshenl coeffiecnt, and Angle is the angle between the real part wavevecter inside the air slit and the axical of the waveguide; K0x=k0xr + ik0xi is the transverse wavevecter inside the airslit, whose real part of
IP T
K0x is k0xr = k0 cos(θ), and the image part is k0xi = ln(r01 )/d, where k0 is the waveve number in the free space, d the width of the air slit, and the θ is the angle
between the wavevector in the air slit and the axial of the air slit, and 00≤ θ ≤ 900
SC R
should be saticfied (See Fig. 1), and r01 is the amplidude of the possible Fresnel
coefficient, and 0≤r01≤1. k1x=k1xr + ik1xi is the transverse wavevector in the metal
U
cladding; and Kz is the longitude wavevector.
N
TM0 and TE0 modes are the fundamental modes passing through the waveguide
A
directly, which are never cut-off. This is in coincident with the character of a
M
symmetry waveguide. There is the other zero order TM mode, TM0p mode, which travels faster than the two fundamental modes, and with a very tiny angle with the
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axial line. In order to distinguish it from the fundamental TM0 mode, it is named to be TM0p mode, because it is a kind of Surface Plasmon mode. Its longitude attenuation
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coefficient is less than that of the guided TM1 mode, which means it can travel longer distance than the TM1 mode. The real part of its traverse wavevector is much less
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than that of the TM1 mode, but the image part of its traverse wavevector is much greater than that of the TM1 mode, which means it is well confined near the surface between the air and the metal cladding, however the TM1 mode leaks much energy
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because it penetrates into the cladding deeply.
And the electromagnetic fields distributions were shown in Fig. 2.
2
1.5
X (nm)
250
1
0.5
0
0
-0.5
-250
-1
(a) 250
500
Z (nm)
750
1,000
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0
-1.5
1 0.8
250
0.6
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X (nm)
0.4 0.2
0
0 -0.2 -0.4
250
-0.6 -0.8
(b) 500
Z (nm)
750
1,000
A
3
2
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X (nm)
250
1
0
0
(c) 250
500
Z (nm)
750
-2
-3
1,000
PT
0
ED
-1
-250
2
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1.5
X (nm)
250
1
0.5
0
0
-0.5
A
-250
-1
-1.5
(d) 0
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250
N
0
-1
250
500
Z (nm)
750
1,000
250
1
X (nm)
1.5
0.5
0
0
-0.5
-250
-1
(e) 250
500
750
Z (nm)
1,000
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0
-1.5
Fig. 2 The field distributions of the 5 sorpported modes. (a)-(c) show the Hy compenents of the
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TM0 mode, the TM1 mode, the TM0p mode, respectively. And (d)-(e) show the Ey compenents of the TE0 mode and TE1 mode, respectively.
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In order to understand the transmission of the modes deeply, the longitudenal
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phase constant of the three non-fundmental modes were calculated at different
620
640
1.100
M TM0p
1.080
(a) 1.02
500nm r 500nm i 600nm i
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1.085
600nm r 580
600
580
600
1.00 0.98
0.0095
0.0090
400nm r 400nm i
1.095 1.090
680
ED
1.105
620
640
660
680
660
680
0.0085
0.0080
0.0075
wavelength (nm) 620
640
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660
600nm r
0.007 0.006
500nm r
0.96
0.005
0.94 0.92 0.90
400nm i 400nm r
0.004
TM1
0.003
0.88 0.86 0.84 0.82
(b)
500nm i
0.002
600nm i 580
600
image part of the longitudinal phase constant image part of the longitudinal phase constant
600
1.110
real part of the longitudinal phase constant r
real part of the longitudinal phase constant r
580 1.115
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wavelength with different width of the slit (See Fig. 3).
620
640
wavelength (nm)
660
680
0.001
0.85
600
650
0.007
600nm r
0.006
500nm r 0.005
0.80
0.75
400nm i 400nm r
0.004
TE1
0.003 0.70
0.65
0.002
600
wavelength (nm)
0.001
650
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(c)
500nm i 600nm i
image part of the longitudinal phase constant
real part of the longitudinal phase constant r
0.90
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Fig. 4 shows the longitudinal phase constants of the TM0p, TM1 and TE1 modes.
Fig. 4 shows the transmission character of the TM0p, TM1 and TE1 modes. Although TE1 mode and the TM1 mode have different polarization, they share almost
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the same transmission characters. The real part of the longitudinal phase constants
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decrease with the increase of the wavelength and the decrease of the slit width.
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However, the transmission character of the TM0p mode is on the opposite side, that is to say, its real part of the longitudinal phase constant increases with the increase of the
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wavelength and the decrease of the slit width. And furthermore, the image part of the
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longitudinal phase constants of the TM0p mode increase with the increase of its real part. However, the image part of the longitudinal phase constants of the TM1 mode
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and TE1 mode increase with the decrease of their real part. Therefore, it can be concluded that the TM1 mode and the TE1 mode are two guided modes of the
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waveguide. However, the TM0p mode is the Plasmon mode of the waveguide. Comparing experiments showed that the transmission features of the array are
consistent with p-polarized resonant modes of the structure [4], the TM0p mode might play an constructive role in the EOT. The transverse phase constants (k1x) of the supported modes in the metal cladding
are all complex numbers (see Table 1), which means all the modes are leaky. That is to say, the energy can penetrate into the metal cladding. And there might be good chance for the leaked energy to be coupled to the surface plasmons formed on the
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incident and output metal-air or metal-subtract interface.
In conclusion, the general eigen equation of slab waveguide is derived, and the
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field distribution were calculated. For a given example, the calculation shows that 5
modes were supported, and one of them was attributed to be a Surface Plasmon mode,
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which might play an important role in EOT since it travels with a very tiny angle with
A
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the axial line, which might mean that it is easily to be excited.
1.
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References
M. Born and E. Wolf E, Principles of Optics 7th Ed (Cambridge University Press,
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Cambridge, England, 1999)
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2. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. L. Wolff, Extraordinary optical transmission through sub-wavelength hole arrays, Nature
391 (1998)
667-669
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3. J. A. Porto, F. J. Garcı´a-Vidal, and J. B. Pendry, Transmission resonances on metallic gratings with very narrow slits, Phys. Rev. Lett. 83 (1999) 2845-2848
4. W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, Surface plasmon polaritons and their role in the enhanced transmission of light through
periodic arrays of subwavelength holes in a metal film, Phys. Rev. Lett. 92 (2004) 107401 5. A. Degiron, H. J. Lezec, W. L. Barnes, and T. W. Ebbesen, Effects of hole depth on enhanced light transmission through subwavelength hole arrays, Appl. Phys.
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Lett. 81 (2002) 4327-4328 6. K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers,
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Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes, Phys. Rev. Lett. 92 (2004) 183901
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7. R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L.
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Kavanagh, Strong polarization in the optical transmission through elliptical
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nanohole arrays, Phys. Rev. Lett. 92 (2004) 037401
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8. Nicolas Bonod, Stenfan Enoch, Lifeng Li, Evgeny Popov and Michel Nevière,
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Resonant optical transmission through thin metallic films with and without holes, Opt. Exp. 11 (2003) 482-490
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(2000)
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9. Electromagnetic Wave Theory, JIN AU KONG, EMW Publishing, Cambridge,
Table 1 The parameters of the modes surpored by a gold metal slab waveguide, the width of the slit is 500nm, the wavelength 632.8nm, and the refractive index of gold is nau =0.18+i2.99.
Modes r01 (a. u.) Angle
K0x (/k0)
K1x (/k0)
Kz (/k0)
0
0.1707+3.152i
1
(degree) 1
0
TM0p
0.1109 1.1575
0.2006-4.3983i 0.1662+3.1830i
1.0936+0.0082i
TM1
0.7697 14.822
2.5401-0.5235i 0.1670+3.142i
0.9683+0.0139i
TE0
1
0
1
TE1
0.9829 34.1196 5.5695-0.0345i 0.1729+3.102i
0.1707+3.152i
A
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PT
ED
M
A
N
U
SC R
0
IP T
TM0
0.8279+0.0024i