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P. Lalanne and H. Liu
assuming like Fano, that the electromagnetic interaction among the indentations
is only mediated by the SPPs of the flat interfaces between the indentations, the
quasi-CW contribution being neglected. There are two reasons for considering such
a pure-SPP model that only considers SPPs. First the model allows us to define the
SPP scattering coefficients for the individual indentations, and these scattering coefficients are fundamental to understand the multiple scattering process (since the same
scattering coefficients apply to the quasi-CWs as well, as shown in Sect. 10.6). Second, by comparing the predictions of the model with fully-vectorial computational
results, one may directly determine the role of SPP in the electromagnetic property
of subwavelength surfaces [16].
To illustrate our purpose, we consider the text-book case of the extraordinary
optical transmission (EOT). The EOT was first observed in the near infrared with
subwavelength hole arrays perforated in opaque gold and silver films [9], and is an
emblematic example in plasmonics that has sparked a huge amount of research trying
to apply the phenomenon and to unveil the underlying mechanisms, and especially
to unveil the role of SPPs in the transmission. The analysis is performed for a selfsupported membrane (thickness d) in air for the sake of simplicity (the upper and
lower grating interfaces are identical, see Fig. 10.7a).
At a microscopic level, the basic mechanism enabling the EOT is a coherent
diffraction by all the individual holes acting as elementary scatterers. However, it
is more convenient to consider isolated 1D arrays of holes (a periodic hole chain
with periodicity a in the y-direction, see the bottom panels in Fig. 10.7) perforated
in a metal substrate as the elementary scatterers. Provided that the hole separation
distance is subwavelength, the 1D hole chains act as 1D indentations, like in classical
metallic gratings.
The elementary SPP-scattering events used in a pure-SPP model of the EOT
are shown in Fig. 10.7b–d for classical diffraction geometries (the y-component k y
of the in-plane wave vector momentum is zero). Upon interaction with the chain,
the SPP modes are partly excited, transmitted, reflected or scattered into the chain
mode and into a continuum of outgoing plane waves. The interaction defines four
elementary SPP scattering coefficients. Two coefficients, see Fig. 10.7b, namely
the SPP modal reflection and transmission coefficients, ρ SP and τ SP , correspond to
in-plane scattering. The other two, α SP and β SP , correspond to the transformation of
the SPPs into aperture modes or radiation waves, and vice versa. They allow us to
link the local field on the surface to the far field that is transporting light away from
the metal film.
From these elementary SPP scattering coefficients, a coupled-mode model that
provides closed-form expressions for the transmittance and reflectance coefficients
of the fundamental supermode of the 2D hole array, t A and r A , is readily derived
[16]. For instance, the reflection coefficient r A of the fundamental supermode, a very
important physical quantity of the EOT phenomenon [18], can be written
r A = r +
2α 2
SP
u −1 − ( ρ SP + τ SP )
.
(10.2)
P. Lalanne and H. Liu
assuming like Fano, that the electromagnetic interaction among the indentations
is only mediated by the SPPs of the flat interfaces between the indentations, the
quasi-CW contribution being neglected. There are two reasons for considering such
a pure-SPP model that only considers SPPs. First the model allows us to define the
SPP scattering coefficients for the individual indentations, and these scattering coefficients are fundamental to understand the multiple scattering process (since the same
scattering coefficients apply to the quasi-CWs as well, as shown in Sect. 10.6). Second, by comparing the predictions of the model with fully-vectorial computational
results, one may directly determine the role of SPP in the electromagnetic property
of subwavelength surfaces [16].
To illustrate our purpose, we consider the text-book case of the extraordinary
optical transmission (EOT). The EOT was first observed in the near infrared with
subwavelength hole arrays perforated in opaque gold and silver films [9], and is an
emblematic example in plasmonics that has sparked a huge amount of research trying
to apply the phenomenon and to unveil the underlying mechanisms, and especially
to unveil the role of SPPs in the transmission. The analysis is performed for a selfsupported membrane (thickness d) in air for the sake of simplicity (the upper and
lower grating interfaces are identical, see Fig. 10.7a).
At a microscopic level, the basic mechanism enabling the EOT is a coherent
diffraction by all the individual holes acting as elementary scatterers. However, it
is more convenient to consider isolated 1D arrays of holes (a periodic hole chain
with periodicity a in the y-direction, see the bottom panels in Fig. 10.7) perforated
in a metal substrate as the elementary scatterers. Provided that the hole separation
distance is subwavelength, the 1D hole chains act as 1D indentations, like in classical
metallic gratings.
The elementary SPP-scattering events used in a pure-SPP model of the EOT
are shown in Fig. 10.7b–d for classical diffraction geometries (the y-component k y
of the in-plane wave vector momentum is zero). Upon interaction with the chain,
the SPP modes are partly excited, transmitted, reflected or scattered into the chain
mode and into a continuum of outgoing plane waves. The interaction defines four
elementary SPP scattering coefficients. Two coefficients, see Fig. 10.7b, namely
the SPP modal reflection and transmission coefficients, ρ SP and τ SP , correspond to
in-plane scattering. The other two, α SP and β SP , correspond to the transformation of
the SPPs into aperture modes or radiation waves, and vice versa. They allow us to
link the local field on the surface to the far field that is transporting light away from
the metal film.
From these elementary SPP scattering coefficients, a coupled-mode model that
provides closed-form expressions for the transmittance and reflectance coefficients
of the fundamental supermode of the 2D hole array, t A and r A , is readily derived
[16]. For instance, the reflection coefficient r A of the fundamental supermode, a very
important physical quantity of the EOT phenomenon [18], can be written
r A = r +
2α 2
SP
u −1 − ( ρ SP + τ SP )
.
(10.2)