1 Nanoplasmonics: From Present into Future
25
geometric eigenmodes. However, their physical frequencies, of course, depend on
the material composition of the system.
For weak relaxation, γ n ∪ ω n , one finds that this real surface plasmon frequency
satisfies an equation
Re[s(ω n )] = s n ,
(1.47)
and that the surface plasmon spectral width is expressed as
γ n =
Im[s(ω n )]
s
n
, s
n ≡
∂Re[s(ω)]
∂ω
ω=ω n
.
(1.48)
In terms of the dielectric permittivity as functions of frequency
s
(ω) =
ε d
|ε d − ε(ω)| 2 Re
∂ε m (ω)
∂ω
, γ(ω) =
Imε m (ω)
Re
∂ε m (ω)
∂ω
.
(1.49)
This expression has been given in Sect. 1.2.1 as Eq. (1.3). Importantly, the spectral
width γ is a universal function of frequency ω and does not explicitly depend on
the eigenmode wave function ϕ n (r) or system’s geometry. However, the system’s
geometry does, of course, define the plasmon eigenfrequencies ω n . This property has
been successfully used in Ref. [176] where a method of designing nanoplasmonic
systems with desired spectra has been developed. Note also that the classical SPs have
been quantized in Ref. [31] in connection with the prediction of spaser, a nanoscale
counterpart of laser (see Sect. 1.5).
As follows from Eq. (1.28), external frequency ω is within the range of the physical surface plasmon frequencies and, therefore, can be close to a surface plasmon
resonance [pole of Green’s function (1.36) as given by Eq. (1.45)] under the following
conditions
0 ≤ Re s(ω) ≤ 1, Im s(ω) ∪ Re s(ω).
(1.50)
These conditions are equivalent to
ε d > 0, 0 ≤ Re ε m (ω) < 0, Im ε m (ω) ∪ |Re ε m (ω)| .
(1.51)
These conditions, in fact, constitute a definition of a plasmonic system, i.e., a system
where a position of surface plasmon resonance can be physically approached: the
dielectric permittivity of the metal component should be negative and almost real,
while the permittivity of the second constituent (dielectric) should be positive, as
assumed.
It is useful to write down an expression for Green’s function (1.36) that is asymptotically valid near its poles, which can be obtained from Eqs. (1.47) and (1.48) as
G
r
(r, r
; ω) =
1
s (ω)
n
ϕ n (r) ϕ n (r ) ∗
ω − ω n + iγ n
,
(1.52)
25
geometric eigenmodes. However, their physical frequencies, of course, depend on
the material composition of the system.
For weak relaxation, γ n ∪ ω n , one finds that this real surface plasmon frequency
satisfies an equation
Re[s(ω n )] = s n ,
(1.47)
and that the surface plasmon spectral width is expressed as
γ n =
Im[s(ω n )]
s
n
, s
n ≡
∂Re[s(ω)]
∂ω
ω=ω n
.
(1.48)
In terms of the dielectric permittivity as functions of frequency
s
(ω) =
ε d
|ε d − ε(ω)| 2 Re
∂ε m (ω)
∂ω
, γ(ω) =
Imε m (ω)
Re
∂ε m (ω)
∂ω
.
(1.49)
This expression has been given in Sect. 1.2.1 as Eq. (1.3). Importantly, the spectral
width γ is a universal function of frequency ω and does not explicitly depend on
the eigenmode wave function ϕ n (r) or system’s geometry. However, the system’s
geometry does, of course, define the plasmon eigenfrequencies ω n . This property has
been successfully used in Ref. [176] where a method of designing nanoplasmonic
systems with desired spectra has been developed. Note also that the classical SPs have
been quantized in Ref. [31] in connection with the prediction of spaser, a nanoscale
counterpart of laser (see Sect. 1.5).
As follows from Eq. (1.28), external frequency ω is within the range of the physical surface plasmon frequencies and, therefore, can be close to a surface plasmon
resonance [pole of Green’s function (1.36) as given by Eq. (1.45)] under the following
conditions
0 ≤ Re s(ω) ≤ 1, Im s(ω) ∪ Re s(ω).
(1.50)
These conditions are equivalent to
ε d > 0, 0 ≤ Re ε m (ω) < 0, Im ε m (ω) ∪ |Re ε m (ω)| .
(1.51)
These conditions, in fact, constitute a definition of a plasmonic system, i.e., a system
where a position of surface plasmon resonance can be physically approached: the
dielectric permittivity of the metal component should be negative and almost real,
while the permittivity of the second constituent (dielectric) should be positive, as
assumed.
It is useful to write down an expression for Green’s function (1.36) that is asymptotically valid near its poles, which can be obtained from Eqs. (1.47) and (1.48) as
G
r
(r, r
; ω) =
1
s (ω)
n
ϕ n (r) ϕ n (r ) ∗
ω − ω n + iγ n
,
(1.52)