Surface Plasmon Coupling Efficiency from Nanoslit Apertures to
Metal-Insulator-Metal Waveguides
Haifeng Hu 1,2, Xie Zeng2, Lina Wang1, Yun Xu1*, Guofeng Song1, Qiaoqiang Gan2†,
1. Institute of Semiconductors, Chinese Academy of Science, Beijing, China, 100083
2. Department of Electrical Engineering, The State University of New York at Buffalo, Buffalo, NY
14260
Supporting Material
Coupled mode method (CMM) formalism for the analysis of the
slit-MIM network
In this section, we derive the CMM formalism [S1] to solve the scattering problem
shown in Fig.1(a). To simplify the analytical model, we assume that the metal is a perfect
electric conductor (PEC). The electromagnetic (EM) fields are expanded on the eigenmodes
of each region as follows: In the vertical metal-insulator-metal (MIM) slot region, the EM
fields can be expressed as a linear combination of various modes supported by the MIM
waveguide [see Eq.(S1)]:
Ex  x,0   Ex,TM 0  x    R Ex ,   x 
(S1a),
H y  x,0   H y ,TM 0  x    R H y ,   x 
(S1b).


To simplify the calculation, only the TM0 mode in the vertical slot is considered (i.e. the
single mode assumption). Therefore the EM field can be expressed as follows:
Ex  x, 0   Ex ,TM 0  x   RTM 0 Ex ,TM 0  x 
(S2a),
H y  x, 0   H y ,TM 0  x   RTM 0 H y ,TM 0  x 
(S2b).
In the region of the horizontal MIM waveguide, the eigenmodes can be treated as plane waves
towards all directions. The total fields in the horizontal MIM structure can be described by Eq.
(S3).
Ex  x, z    dkEx ,k  x   ak exp  ik z z   bk exp  ik z z  
1
(S3a)
H y ( x, z )   dkH y ,k  x   ak exp  ik z z   bk exp  ik z z  
(S3b)
Here k and kz=(εdω2/c2-k2)1/2 are components of the wave vector along x-direction and
z-direction, respectively. By matching the boundary conditions on the two interfaces of the
horizontal MIM waveguide, the scattering coefficients, RTM0, ak and bk can be obtained using
Eq. (S4), where G2 is defined as the coupling term at the opening of the vertical slit, which
can be calculated by Eq.(S5).
RTM 0  
G2  i
G2  i
(S4a)
ak 
w  exp  ik z d 
sin c  kw / 2 
2 sin  k z d  GT  i 
(S4b)
bk 
exp  ik z d 
w
sin c  kw / 2 
2 sin  k z d  GT  i 
(S4c)
G2  
wk



0
cot

k2  k 2 d

sin c 2  kw / 2 
k2  k 2
dk
(S5)
By substituting ak and bk into Eq.(S3), the scattering fields in the horizontal MIM waveguide
can be obtained. As will be discussed in Section 3, these scattering fields will be employed to
determine the coupling efficiency of SPPs.
Modal analysis on MIM waveguides
It is known that MIM waveguides support two kinds of bound modes in the core layer, i.e.
plasmonic modes with the exponential nature and photonic modes with the trigonometric
behavior [S2-S4], which can be obtained by solving Maxwell’s equations. At the wavelength
of 1μm, the dielectric constants of gold and air are εm=-46.5+3.5i and εd=1. The modal
propagation constant, β, can be calculated by solving the eigenvalue equation of the MIM
waveguide using Eq.(S6).
tan   d w  
2T
, T   d  m /  m d
1 T 2

where w is the width of the MIM waveguide,  d  k2 d   2

1/2
(S6)

and  m   2  k2 m

1/2
.
kω=ω/c is the wave vector in the vacuum. Moreover, the field of the plasmonic mode can be
described in Eq. (S7)[S4].
H y , sp

exp  m  x  w / 2  

  A exp  d  x  w / 2    B exp   d  x  w / 2  

C exp   m  x  w / 2  

2
x  w / 2
w / 2  x  w / 2
x  w/ 2
(S7a)
Ex , sp


exp  m  x  w / 2  

k


m

 A
B

exp  d  x  w / 2   
exp   d  x  w / 2  
k  d
 k  d

C
exp   m  x  w / 2  

k  m

x

w
2
w
w
x
2
2
w
x
2
(S7b)
It should be noticed that the MIM waveguide is along the z-axis in Eq.(S7). The coefficients,
A, B and C, can be obtained by matching the boundary conditions at the two interfaces of the
MIM waveguide, i.e. at x=-w/2 and x=w/2. Based on Eq. (S6) and (S7), the propagation
constant and the field distribution of the mode supported in the MIM waveguide can be
calculated. For example, when w=300nm, only one mode can be calculated using Eq. (S6) in
the MIM waveguide, whose propagation constant is β=6.7809+0.0194i. By substituting the
value of β into Eq. (S7), the field of the plasmonic mode can be modeled. As shown in Fig.
S1(a), one can see that the modal field (Hy) can penetrate into the metallic cladding layers.
The penetration depth, δ, is defined as the depth where the field density has diminished to 1/e
of the value at the surface. According to Eq. (S7), the penetration depth equal to 1/γm [see the
spacing between the black and gray dashed lines in Fig.S1(a)]. When we employ the
analytical formalism [i.e. Eq. (S3-S5)] to model the scattering process at the T-junction, the
geometrical width of the vertical slit, w, is replaced by the effective width, weff=w+2δ to
modify the PEC approximation. In this case, as shown by Fig. 1(b) in the main text, the
modeled dependence of the coupling efficiency on the slit width, w, has a better fit to the
finite difference time domain (FDTD) modeling results compared with the model employed in
the previous pure SPP picture [S7] in visible and near infrared spectral regions.
o demonstrate the modal structure of the MIM waveguide further, the geometric
dispersion curves of TM0 and TM1 modes are shown in Fig. S1(b). The plasmonic modes and
photonic modes are separated by the light line (see the dashed line): i.e. plasmonic modes are
1/2
1/2
in the region of   k  d , and photonic modes are in the region of   k  d . In addition,
when w<450nm [see the red arrow in Fig.S1(b)], only the plasmonic mode can be supported
by this MIM waveguide.
Fig. S1 (a) The Hy component of the TM0 mode when w=300nm (blue line). The dashed black
lines represent the metallo-dielectric interfaces. The spacing between the black and gray lines is
3
the penetration depth. (b) The geometric dispersion of propagating constants in the MIM
waveguide. The blue and red lines represent the TM0 and TM1 modes, respectively. The dashed
line is the light line.
Orthogonality condition for absorbing waveguides
According to the reciprocity theorem of Maxwell’s equations, every two modes in the
absorbing waveguide should satisfy the unconjugate general form of orthogonality condition,
as described in Eq.(S8) [S5].



Ez,  H y,    H y,  Ez,   dz  0 , for    
(S8)
Here we use  and   to represent each two modes in the waveguide. The ‘+’ and ‘-’ are used
to distinguish the forward-propagating and backward-propagating modes along the x-axis
[S6]. The total EM field in the MIM waveguide can be expressed as:
Ez  x, z     Ez, sp  x, z     Ez, sp  x, z  
 c E   x, z 
  sp
H y  x, z     H y, sp  x, z     H y,sp  x, z  
z,
 c H   x, z 
  sp
y,
(S9a)
(S9b)
The first two terms on the right hand side of Eq.(S9a) and (S9b) correspond to the filed
components of the plasmonic mode supported in the MIM waveguide [i.e. TM0 mode in Fig.
S1(b)] which can be obtained by Eq.(S7), and the last term describes the contribution of
higher order modes. However, it should be noticed that Eq.(S7) is used to describe the vertical
MIM waveguide, while Ez,sp and Hy,sp in Eq.(S9) represent the field components of plasmonic
mode in the horizontal waveguide. Based on the expansion of the total filed, Ez and Hy,
described in Eq. (S9), one can get



Ez  x, z  H y, sp  H y  x, z  Ez, sp dz

    ( Ez, sp H y, sp  H y, sp Ez, sp )dz 

 c 
  sp


Ez, H y, sp  H y, Ez, sp dz
(S10)
Due to the orthogonality condition described in Eq.(S8), the last term on the right hand side of
Eq.(S10) is zero. Consequently,   can be expressed in a compact expression as shown in
Eq.(S11).

 x   


 Ez  x, z  H y , sp  H y  x, z  Ez ,sp  dz
 E



z , sp
H

y , sp
H

y , sp
E

z , sp
 dz
(S11)
According to Eq. (S3) listed in Section 1, the total fields, Ez(x,z) and Hy(x,z), in the horizontal
MIM waveguide can be obtained. Consequently, One can use Eq.(S11) (i.e. Eq. 1 in the main
text) to estimate the intensity of the surface plasmon polaritons (SPP) in the MIM waveguide
base on the analytical method.
4
The coupling efficiency of the T-junction with the finite-length slit
In the T-junction with a finite length input slit shown in Fig. S2(a), the coupling
mechanism can be explained by the combination of four basic scattering processes as
illustrated in Fig.S2(b)-(d). In the vertical slot, the TM0 mode is reflected back and forth
between the two ends of the slot. Similar with the transmission of asymmetric FP etalon [S7],
the total coupling efficiency can be expressed by Eq.(2) in the main text, where the
parameters ηin, R1, R2 and ηinf are related to the processes described in Fig. S2(b)-(d)
respectively. Here we consider these four basic processes separately: (1) The transmission
through the interface between the slot and the semi-infinite free space is shown in Fig.S2(b).
The coupling efficiency from the normal incident plane wave impinging on the opening of the
input slit to the TM0 mode is ηin=|T1|2. (2) The reflection on the interface between the slit and
the semi-infinite free space is shown in Fig.S2(c), with a reflection coefficient of R1. (3) The
reflection on the interface between the input slit and the horizontal MIM waveguide, R2, has
been derived in Section 1 [i.e. RTM0 in Eq.(S4a)]. (4) The coupling efficiency of the T-junction
with the infinitely long vertical slot, ηinf=|α+|2+|α-|2, has been calculated in Fig.1(b) analytically.
Considering all these scattering coefficients, an analytical formalism can be developed, as
described by Eq. (2) in the main text.
Fig. S2 (a) The multi-scattering process in the T-junction with the finite length input slit. (b) The
transmission process at the interface between the free-space and the input slit. (c) The reflection
process at the interface between the free-space and the slit. (d) The reflection (the solid green arrow)
and the transmission process (the dashed green arrows) at the interface between the slit and the
horizontal MIM waveguide.
Reference
[S1] F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen and L. Kuipers, Light passing through
subwavelength apertures, Rev. Mod. Phys. 82, 729 (2010)
[S2] B.Prade, J. Y. Vinet, and A. Mysyrowicz, Guided optical waves in planar heterostructures
with negative dielectric constant, Phys. Rev. B 44,13556 (1991)
[S3] P. Ginzburg, D. Arbel, and M. Orenstein, Gap plasmonpolariton structure for very efficient
microscale-to-nanoscale interfacing, Opt. Lett. 31, 3288 (2006)
[S4] S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, New York, 2007)
[S5] A.W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York,
1983)
[S6] P. Lalanne, J.P. Hugonin, H.T. Liu, B. Wang, A microscopic view of the electromagnetic
properties of sub-λ metallic surfaces, Surf. Sci. Rep. 64, 453(2009)
[S7] A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford
University Press, New York 2007).
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