Unified Theory of Surface-Plasmonic Enhancement
and Extinction of Light Transmission Through Metallic
Nanoslit Arrays
(Supplementary Information)
Jae Woong Yoon1, Jun Hyung Lee2, Seok Ho Song2, and Robert Magnusson1
1Department
of Electrical Engineering, University of Texas at Arlington, Box 19016, Arlington, TX 76019, USA
2Department of Physics, Hanyang University, Seoul 133-791, KOREA
I. Analytic expressions of the SPP resonance wavelength and bandwidth
In this section, we derive analytic expressions of the surface plasmon-polariton (SPP) resonance
wavelength SP and bandwidth SP based on the microscopic pure-SPP model developed by Liu and
Lalanne [R1]. We use elementary scattering coefficients tS, rS, a, and b( ) as defined in Fig. S1(a). Note
that we use symbols different from those in [R1] because some symbols therein are identical to ones in our
main text but have different definitions. In Fig. S1(b), amplitudes of SPPs excited at the n-th slit are
denoted by Un and Vn for SPPs propagating toward the +x and –x directions, respectively. Using the BlochFloquet condition (Un+1, Vn+1) = (Un, Vn) exp(ikx) with kx = (2/)sin [R1], the SPP amplitudes under
surface-normal incidence of transverse-magnetic polarized planewave is reduced to
U n  Vn 
b(0)
,
1   tS  rS  u ( )
(A1)
where u() = exp(inSP/c) with an effective propagation constant of an SPP nSP = [M/(1+M)]1/2, angular
frequency of the incident wave , and speed of light in vacuum c. Equation (A1) describes the SPP
resonance by its pole at the complex frequency SP = SP–itot. Therefore, the SPP resonance condition can
be written by
1   tS  rS  u (SP )  0 .
b( )
(a)
rS
(b)

(A2)
z
x
tS
SPP
a
Vn-1
Un-1 Vn
Un Vn+1 Un+1
slit n-1
slit n
slit n+1
Figure S1| (a) Elementary scattering of an SPP at a single silt. The elementary scattering coefficients
are SPP transmission coefficient tS, SPP reflection coefficient rS, SPP-CM coupling coefficient a, and
SPP-radiation coupling coefficient b( ). (b) Excitation of SPPs on a periodic array of slits by an
external planewave. Un and Vn represent the amplitudes of SPPs propagating along the +x and –x
directions from the n-th slit, respectively.
1
In a slowly decaying regime such that tot <<SP, Eq. (A2) yields
2 c
1  arg  tS  rS  / 2  ,
nSP '  
(A3)
n "
c
ln | tS  rS |1   SP SP ,
nSP ' 
nSP '
(A4)
SP 
 tot 
where nSP′ and nSP″ are real and imaginary parts of nSP, respectively; we assume first-order coupling. On
the right-hand side of Eq. (A4) for the SPP decay rate, the first term represents the radiation decay rate rad
due to scattering at slits while the second term is the non-radiative decay rate nr due to ohmic damping of
free electrons. Finally, analytic expressions for the SPP resonance wavelength SP and bandwidth SP
(full-width at half-maximum) are immediately derived from Eqs. (A3) and (A4) as follows.
SP 
SP 
2 c
SP

SPF
,
1  arg  tS  rS  / 2
 
SP 2
2n " 
 tot   SP ln | tS  rS |1   SP  SP ,
c
nSP ' 
  SPF
(A5)
(A6)
where SPF = nSP′ is the SPP resonance wavelength on a flat, unpatterned metal surface.
II. The SPP resonance wavelength and bandwidth in our case
We estimate SP and SP for our case with a metal dielectric constant M = –5 and a slit width w =
0.05. The SPP transmission tS and reflection rS coefficients are obtained by FEM calculation. Figure
S2(a) shows the FEM calculation result (magnetic field, Hy) of an SPP scattered by a single isolated slit.
The SPP source located at x = 0 launches a unit-amplitude SPP at  = 1.062 to a 0.05-wide slit centered
at x = 5.025. The SPP from the source is transmitted (tS), reflected (rS), and also scattered to the external
radiation and cavity mode. In Fig. S2(a), we observe the reflected SPP for x < 0, coexisting launched and
reflected SPPs for 0  x  5, transmitted SPP for x  5.05, cylindrical pattern of emitted external
radiation, and the cavity mode inside the slit. The fields associated with the cavity mode and external
radiation are negligibly small at the air-metal interface when compared to the SPP amplitude. We therefore
assume that the calculated field at the interface can be expressed solely by the pure SPP amplitude as
 rS exp  ikSP ( x  x0 ) 
( x  0),


H y ( x, z  0)  ei  exp ikSP ( x  x0 )   rS exp  ikSP ( x  x0 )  (0  x  5),

tS exp ikSP ( x  x0 ) 
( x  5.05 ),
2
(A7)
1
(a)
Hy / H0
rS
air
0
external radiation
tS
SPP source
metal (M = -5)
-1
1.0
single slit (w = 0.05)
CM
(b)
Hy /H0
0.5
0.0
-0.5
FEM calculation
model fit
-1.0
-5
-4
-3
-2
-1
0
1
2
x
3
4
5
6
7
8
9
10
Figure S2| (a) FEM calculation result of SPP scattered by a single slit. An SPP is launched from the
SPP source at x = 0 and scattered by a single slit centered at x = x0 = 5.025 with width w = 0.05. (b)
Magnetic field profile at the metal surface. Red circles (●) are obtained by FEM calculation in (a) while
the black solid curve (─) is due to the unmixed pure SPP model in Eq. (A8) with tS = 0.925896–
0.14069i and rS = –0.073074–0.150357i. The phase factor  = 1.08.
where the in-plane wave vector of the SPP kSP = nSP/c, the slit center position x0 = 5.025 and the  is
arbitrary phase constant. Fitting the pure SPP amplitude in Eq. (A7) to the field at the interface due to the
FEM calculation yields tS = 0.925896–0.14069i and rS = –0.073074–0.150357i as confirmed in Fig. S2(b).
Applying these values to Eqs. (A5) and (A6), we finally obtain the SPP resonance wavelength and
bandwidth
SP = 1.0624  and SP = 0.03346 .
(A8)
These values quantitatively agree with SP = 1.062 and SP = 0.03211 from the surface excitation
spectrum in Fig. 2c of the main text.
III. Formal consistency of our theory with the microscopic theory of EOT
The microscopic theory of EOT developed by Liu and Lalanne yields the single-interface transmission
coefficient [R1]
 ( )   D 
2ab
u ( )   tS  rS 
1
(A9)
 SP
for surface-normal incidence. On the right-hand side of Eq. (A9), the first and second terms represent the
non-resonant and resonant contributions, respectively. Near the SPP resonance condition  = SP+,
3
where  << SP, SP can be rewritten by a Lorentzian function. For the reinterpretation of SP in terms of
Lorentzian resonance parameters, we apply the Taylor series expansion of u–1() as
u 1 (SP   )  u 1 (SP    i tot )
 u 1 (SP ) 
du 1 ( )
d   
   i tot   . . .
(A10)
SP
n 


  tS  rS  1  i SP    i tot    . . .
c


In this expansion, Eqs. (A3) and (A4) for SP were used. For a slowly decaying resonance that satisfies 
<< SP, SP has a significant amplitude only within a narrow frequency range around SP; therefore, we can
take the first order of the expansion in Eq. (A10) as a reasonable approximation. Finally, we obtain
 ( )   D 
c
2ab
,
nSP   tS  rS  1  i
(A11)
where reduced frequency  = /tot. Formal consistency of Eq. (A11) with
 ( )   D 
2 inex
1  i
ei ,
(A12)
i.e., Eq. (2) in the main text, is obtained by directly comparing these two expressions and including a
relation in/ex = |a/b|2 according to the definitions of in, ex, a, and b. We finally obtain
in 
c
| a |2
c
| b |2
and ex 
,
| nSP |  tot | tS  rS |
| nSP |  tot | tS  rS |
  arg(a)  arg(b)  arg(tS  rS )  arg(nSP ) .
(A13)
(A14)
For these expressions to have practical meaning in a quantitative manner, normalization of modes should
be identical in two different approaches. In defining the elementary scattering coefficients tS, rS, a, and b,
the external planewave and slit-guided mode should be normalized so that they carry unit surface-normal
power within a period. The SPP mode should be normalized so that it has unit energy when its energy
density is integrated over a period.
IV. The surface-plasmonic Fano resonance theory vs. previous interpretations
The excitation of an SPP induces various features such as antiresonant extinction, a null-field at the
aperture opening, an abrupt change in the cavity resonance condition, and resonance peak narrowing. As
elaborated in detail throughout the main text, these features are all rooted in a single resonance interaction
caused by the surface-plasmonic Fano resonance at the interface. In Table S1, we summarize the main
issues under debate regarding the role of SPPs, various interpretations describing such issues, and our
unified explanations based on the surface-plasmonic Fano resonance theory.
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Table 1. SPP issues in previous debates and the unified interpretations due to our theory.
Issues
Previous interpretations
Description
Enhanced absorption at the SPP
resonance condition on a flat metal
surface.
Transmission
peak extinction
Null-field at the
aperture
opening
Peak
narrowing
effect
Destructive interference between the
surface and cavity resonances.
Refs.
R2, R3.
The unified interpretation
based on the surface-plasmonic Fano resonance
Closed-cavity (high-Q) regime
metal
R4.
Surface-plasmonic bandgap effect.
R5, R6.
Zero of the interface transmittance
associated with the SPP excitation.
R7-R9.
Excitation of the nonresonant SPP.
R8, R9.
Destructive interference between
multiply scattered SPPs by the slit
array.
R10,
R11.
Transition of the operative
resonance mode between the SPP
and cavity modes.
R12,
R13.
Guided-mode-like resonance at the
surface-plasmonic band edge.
R5.
Effect of the Rayleigh anomaly.
R14.
narrow cavity
metal resonance
Associated resonance processes
The Fano-type interference between the SPP-resonant and
direct coupling at the interface is
- constructive in the internal reflection,
- totally destructive in the transmission.
Consequences
- The slits act as closed cavities.
- Very narrow, high-Q cavity resonance.
- The transmission peaks are highly sensitive to the
dissipative losses.
- A null-field at the slit opening is observed under the
interface excitation by an external planewave as a
result of destructive interference in the transmission to
the cavity mode.
Open-cavity (low-Q) regime
metal
Abrupt shift of
the cavity (slit)
resonance
condition
Transition of the operative
resonance mode between the SPP
and cavity modes.
Phase change in the internal
reflection of the cavity mode.
Peak
broadening
effect
Strong coupling to the cavity mode
at the surface-plasmonic band edge.
R6, R7,
R13,
R15.
R15,
R16.
metal
broad cavity
resonance
Associated resonance processes
The Fano-type interference between the SPP-resonant and
direct coupling at the interface is
- destructive in the internal reflection,
- constructive in the transmission.
Consequences
- The slits act as open cavities.
- Relatively broad, low-Q cavity resonance.
- The transmission peaks are insensitive to the
dissipative losses.
- The SPP is maximally excited at this condition.
- Strong SPP excitation leads to resonant phase change
in the internal reflection of the cavity mode, causing an
abrupt shift of the cavity resonance condition.
R5.
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