2 Physical System, Initial Problem, and Multipole Expansions
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6
L.G. Grechko et al.
in the regions a (inside ambient, outside spheres), i (inside i th sphere), and s (inside
substrate) together with the standard boundary conditions
ψσuu−b = ψ v , εu
∂ψ u
∂ψ v
= εv
,
∂nu σu−v
∂nv
(1.2)
where εu is the permittivity of the matter filling the u th region (u = a, i, s) , ψ u
is the resulting field potential in the u th region, subscript σu− v under the equal
sign denotes that the expression is valid for observation points lying on the common
boundary surface σu− v of regions u and v.
Using the superposition principle to represent the resulting potential in regions
a and s, together with the image method and multipole expansion techniques for
solving electrostatic problems, we seek a solution of problems (1.1) and (1.2) in the
following form [10–13]:
a
+
ψ a = ψext
a
ψiath sphere + ψsubstrate
i
= − E0 × r +
Ailm Flm (ρi ) +
ilm
Ailm Flm (ρ i )
(1.3)
ilm
in the region a,
Bilm Glm ρi
ψi =
(1.4)
lm
in the region i, and
s
s
+ ψindused
= −E0 × r + ψ0s +
ψ s = ψext
Cilm Flm ρ i
(1.5)
ilm
a = −E × r is the potential of the given external field
in the region s, where ψext
o
s
E0 , in the ambient, ψext = −E 0 × r + ψ0s is that in the substrate, E 0 is a constant
vector representing the given external field in the substrate (note that E 0 = E0
due to “refraction” of the force lines on the ambient–substrate boundary), ψ0s is a
s related with a choice of the origin point
constant contribution to the potential ψext
a
location, ψith sphere = lm Ailm Flm (ρi ) is the contribution to ψ a due to the induced
a
=
charge distribution of the i th sphere, ψsubstrate
ilm Ailm Flm ρ i is that due
s
to the induced charge distribution of the substrate, ψinduced
=
ilm Cikm Flm ρ i
is the contribution to ψ s due to all the induced charges (of both the substrate and
all the spheres), Flm (r) ≡ r−l−1 Ylm rˆ and Glm (r) ≡ rl Ylm rˆ are spherical
harmonics regular, respectively, at infinity and at zero, Ylm rˆ is a spherical function
normalized in the usual way [13 ,14], rˆ ≡ r/r is a unit vector in the r direction,
ρi ≡ r− ri is the radius vector of an observation point with respect to the center of
the i th sphere (see Fig.1.1), ρ i ≡ r− r i is that with respect to the image of the i th
sphere, ri is the radius vector of the i th sphere’s center.
1
Surface Plasmons in Assemblies of Small Particles
7
Fig. 1.1 Spheres, images,
and vector description of the
considered system
It should be mentioned that all the individual terms in Eqs. (1.3), (1.4),
and (1.5) automatically satisfy Laplace’s equation (1.1), so the unknown values
Ailm , Ailm , Bilm , Cilm , E 0 , and ψ0s can be obtained after applying only the boundary conditions (1.2) to expansions (1.3), (1.4), and (1.5). Also, it is assumed that
∞
l
lm ≡
l=0
m=−l throughout this chapter.
The earlier obtained [10] equations form a full set to determine unknown coefficients Ailm and Bilm (recall that the values Ailm are expressed in terms of Ailm and the
explicit form of the expression was found earlier [10]). After some transformations
the equations noted can be reduced to the form
δjlilm
Ailm = Ujl1 m1 ,
+ Kjlilm
1 m1
1 m1
(1.6)
ilm
Bilm = f (Ailm ),
(1.7)
where
j m
≡ ajl1 Tlm1
Kjlilm
1 m1
ail =
FLM ri − rj
l (εi − εa )
R2 l+1 , Uilm =
lεi + (l + l) εa i
εa − εs
(1.8)
FLM r i − r j ,
εa + εs
εi − εa
4π/3ail E0m δl1 = 3/ (4π )
Vi E0m δl1 ,
εi + 2εa
(1.9)
+ (−1)l+m
2l + 1
(L + M)! (L − M)!
,
(2l1 + 1) (2L + 1) (l + m)! (l − m)! (l1 + m1 )! (l1 − m1 )!
(1.10)
and Vi = (4π/3)R3i is the volume of the i th sphere, L = l + l1 , M = m − m1
(l1 = 0,1,2,..., m1 = −l1 ,...,l1 ).
The explicit form of the function f in Eq. (1.7) is sort of cumbersome. In this
chapter, however, we are not interested in coefficients Bilm representing the resulting
l1 m1
Tlm
= (−1)l+m1 4π
8
L.G. Grechko et al.
field inside the spheres, so the function f is not needed for further consideration and
hence not given here. The expression for Uilm is presented in two equivalent forms
(due to ail = R3i (εi − εa )/(εi + 2εa )). Both the forms are useful.
It is remarkable that the transition from a 2D array of identical spheres located
on a substrate [9] to the system considered here (i.e., a spatial system of different
spheres (above a substrate)) leads, formally, only to the appearance of the ail values
(known as the multipolar polarizabilities of a single sphere in the expressions for
and Uilm (Eqs. (1.8) and (1.10)).
Kjlilm
1 m1
Thus, we have obtained an infinite set of coupled linear algebraic equations
(in indices l,m, and, possibly, i) for calculating the multipole coefficients Ailm of
the induced field for each sphere in the ambient. The remaining coefficients (Ailm ,
Cilm , and Bilm ) are expressed in terms of Ailm . Consequently, the initial problem is
reduced to the determination of Ailm from Eq. (1.6). Having determined the values
of Ailm , one can then determine all the remaining multipolar coefficients and, by
using the initial expansions (1.3), (1.4), and (1.5), one can in principle calculate the
electrostatic potential at any point and all other values of interest.
1.3 Two Spheres Above a Substrate: Spheres’
Polarizability Tensor
Let us consider the case of two spheres located above a substrate in such a way
that the line connecting the centers of the spheres is perpendicular to the substrate
surface (see Fig.1.2a). By varying the parameters of this system, one can obtain
Fig. 1.2 A system of two
different spheres above a
substrate, and its particular
cases
1
Surface Plasmons in Assemblies of Small Particles
9
the particular cases shown in Fig. 1.2b–d. The more general case when the line of
spheres’ centers is inclined turns out to be far more analytically complicated and is
not examined in this paper.
In the work [10] we obtained the expression for the mm1 th component of the i th
sphere’s polarizability tensor
a
m
αim
1
= ai1
1 + ( − 1)m ηm d¯i31 +
2 a11 a21 +
1 − ηm
d6
m
¯im m
δm1 ,
(1.11)
where
¯i ≡
1, m = ±1
,
2, m = 0
1, if i = 2
,η ≡
2, if i = 1 m
m
2
≡Hm
a11 a21
− Hm
im
a
Hm ≡ ηm εεss −ε
+εa ,
1
3
≡ Hm ai1
)3
−
1
(h1 + h2 )
3
−
1
(2hi )3
,
1
(h1 + h2 )6
(2h1 ) (2h2
a11
2a11 a21
a21
,
+
+ ( − 1)m ηm 3
3
3
(2h1 )
(2h2 )
d (h1 + h2 )3
and the geometrical parameters h1 , h2 , and d are defined in Fig. 1.2. The values
m
lm and m are additions to the numerator and denominator of αlm1 , respectively,
describing the substrate’s influence on the sphere’s polarizability and vanishing
when there is no substrate (formally, when εs = εa ).
Starting from Eq. (1.11), expressions for αˆ i can be easily obtained for the
particular cases shown in Fig. 1.2b–d; namely, in the following cases:
(1) Two spheres without a substrate (Fig. 1.2b): by setting εs = εa (or, alternatively,
moving the substrate away from the spheres to infinity) we find
m
= αi1
αim
1
1 + ( − 1)m ηm a¯i1 /d3 m
δm1 .
2 a a /d 6
1 − ηm
11 21
(1.12)
(2) A single sphere above a substrate (Fig. 1.2c): by setting ε¯i = εa (or d = ∞ )
we find
m
=
αim
1
ai1
m
ai1 δm1 .
1 − Hm (2h
3
)
(1.13)
i
(3) A single sphere in a homogeneous medium with no substrate (Fig. 1.2d): by
setting εs = εi = εa (or d = hi = ∞ ) we find the classical result
m
m
= ai1 δm
=
αim
1
1
3V εi − εa m
δ .
4π εi + 2εa m1
(1.14)
10
L.G. Grechko et al.
Note that tensors (1.12), (1.13), and (1.14) are diagonal in the Cartesian frame
with z axis along the system’s axis of symmetry and have two different diago1 or
nal components. These components are either transversal to z axis αi⊥ ≡ αi1
0
longitudinal to it αi ≡ αi0 .
The common peculiarity among tensors (1.12), (1.13), and (1.14) is that they
all differ from the spherical one because of the axial symmetry of the corresponding physical system (Fig. 1.2a–c), while the tensor, Eq. (1.14), for a single sphere
(Fig. 1.2d) is simply proportional to the unit one due to the point symmetry of a
sphere. This last statement means that the induced dipole moments of the spheres in
the systems depicted in Fig. 1.2a–c are not parallel to E0 , in contrast to the case of
a single sphere. Thus, the dipole moment of a single sphere is changed in both its
value and direction in the presence of a substrate or another sphere. These changes
are caused, of course, by the interaction between the induced dipole moment of
the sphere and those of neighboring objects (i.e., substrate and/or other spheres)
and depend on the values as well as the relative orientation of the moments. This
simple physical picture of the presence of anisotropy for initially isotropic spheres
helps us to understand some peculiarities of the sphere’s polarizability behavior
considered below.
1.4 Investigations Surface Plasmons for Specific Systems
1.4.1 Substrate Influence on the Optical Properties
of a Small Sphere
We turn now to developing analytical results and subsequent insight into the influence of a substrate on the optical properties of a sphere. To accomplish this, we shall
consider a single sphere above a substrate. Its polarizability we rewrite here in the
following form (from hereon, the index i is omitted):
m
= R3
αm
1
(ε − εa )(εs + εa )
(ε
3
+ 2εa )(εs + εa ) − ηm 2Rh (ε
− εa )(εs − εa )
m
.
δm
1
(1.15)
In the optical region, all the permittivities in Eq. (1.15) become, in general,
frequency dependent and complex, as does the polarizability itself. In order to analytically investigate the behavior of α,
ˆ we shall assume ε and εs to be Lorentzian
dielectric functions [3]
ε(ω)
= ε∞ +
ωp2
ω02 − ω2 − iγ ω
, εs (ω) = ε∞s +
2
ωps
2 − ω2 − iγ ω
ω0s
s
,
(1.16)
while εa is assumed to be constant and equal to unity (i.e., having vacuum or rare
gases as the ambient). The index s in Eqs. (1.16) denotes the values characterizing
the substrate material.
1
Surface Plasmons in Assemblies of Small Particles
11
Despite only a few materials being described quite well by the Lorentzian model,
it often gives universal results [3, 7]. Therefore, we shall use this model here to
be satisfied not so much with the quantitative fitness but in clarifying the physical
picture. To accomplish this, we will first obtain the resonant frequencies for a sphere
located near a substrate.
Defining the resonant frequencies as those at which the polarizability of the
sphere becomes infinite (and, correspondingly, the denominator in Eq. (1.15) equals
zero [5]), we find from Eq. (1.15), when accounting for Eqs. (1.16), the following
algebraic equation for the resonant frequencies (in the case ε∞ = ε∞s = εa = 1):
ω4 + a3 ω3 + a2 ω2 + a1 ω + a0 = 0
(1.17)
with
2
2
a3 = I(γ + γs ), a2 = −(ω˜ 02 + ω˜ 0s
+ γ γs ), a1 = −i(ω˜ 02 γs + ω˜ 0s
γ ),
2
a0 = ω˜ 02 ω˜ 0s
− xm
2
ωp2 ωps
3
2
, xm ≡ ηm
R
2h
3
,
(1.18)
where
ω˜ 02 ≡ ω02 +
ωp2
3
2
2
and ω˜ 0s
≡ ω0s
+
2
ωps
2
2 , and x ≡ η (R/2 h)3 .
are “shifted” squared frequencies of ω02 and ω0s
m
m
Equation (1.17) is of fourth order with complex coefficients and has, in general,
four complex roots. Consequently, the resonant frequencies are, generally speaking, complex values. However, resonance occurs at real frequencies close to the
real part of the corresponding complex frequencies. The exact complex solutions to
Eq. (1.17) are analytically too complicate and not of interest to us here. Instead, one
can determine the real roots by neglecting damping ( γ = γs = 0). In this case,
Eq. (1.17) is reduced to a biquadratic form with the solutions
± 2
ωm
2
ω˜ 2 + ω˜ 0s
±
= 0
2
2
ω˜ 02 − ω˜ 0s
2
2
+ xm
2
ωp2 ωps
3 2
.
(1.19)
This expression simple enough contains useful information on the sphere resonances that allows one to trace how the substrate influences the absorption peak
position of a sphere brought near it.
The main peculiarity of Eq. (1.19) is that it predicts four positive nonzero resonances for a sphere. Indeed, at each fixed value of m (m = 0,1) we have two resonant
+ and ω− corresponding to different signs of the square root in Eq.
frequencies ωm
m
(1.19). Thus, there are two resonances for transversal (m = ±1) and two for longitudinal (m = 0) components of α.
ˆ Therefore, in the general case of the external field
12
L.G. Grechko et al.
ω2
Fig. 1.3 Location of the
resonant frequencies
according to Eq. (1.19). The
location is shown
schematically for one of the
components of αˆ . For another
component, the frequency
splitting is qualitatively the
same, but with another value
of the ( ωm )2 being equal to
the square root from Eq.
(1.29). Note that the
magnitude of the
2 )/2
half-difference (ω˜ 02 − ω˜ 0s
(short arrows) is always less
than ( ωm )2 (long arrows)
2
(ω||+ )
2
(ω⊥+ )
2
ω∼0
2
2
(Δω)||
(Δω)⊥
2
2
ω∼0 + ω∼0s
2
2
2
ω∼0s
(Δω)⊥
2
(Δω)||
2
(ω⊥– )
2
(ω||– )
arbitrarily oriented with respect to the substrate’s normal direction, the polarizability
of the sphere drastically increases at four frequencies.
+ and ω− are simultaneously dependent
Second, the resonant frequencies ωm
m
on both sets of parameters (ω0 , ωp ) and (ω0s , ωps ), which characterize the sphere
itself and the substrate material. Consequently, there is a “coupling” of corresponding material oscillators. Only in the limiting case of a single isolated
sphere (no substrate) or an isolated substrate (no sphere) we obtain, as it must
be, the single-object resonance occurring at ω = ω˜ 0 ≡
ω02 + ωp2 /3
1/2
or
1/2
2 /2
ω = ω˜ 0s ≡ ω02 + ωps
, respectively. For a sphere and substrate of fine
metal (ω0 = ω0s = 0) it gives
the well-known √
Frohlich frequencies of surface
√
plasmons [11] ωF = ωp / 3 and ωF = ωps / 2. In the general case, however, the Fröhlich resonances become entangled with those of ω0 and ω0s for bulk
materials.
Third, the resonant frequency locations obey the following regularities (Fig. 1.3).
(1) The resonances for each component of the sphere’s polarizability ( α ⊥ or α )
are located symmetrically with respect to the square root of the arithmetic mean
2 )/2 ) of the shifted squared frequencies ω
2 .
˜ 02 and ω˜ 0s
( (ω˜ 02 + ω˜ 0s
2 )/2 for the
(2) The up- or down-shifts in frequency from the mean (ω˜ 02 + ω˜ 0s
+ and ω− are defined by the value of the square root in
resonances ωm
m
Eq. (1.19). They are the same for both resonances (at fixed value of m), while
being different for transversal and longitudinal components. The shifts for α ⊥
(m = ±1, ηm = 1) are less than those for α (m = 0, ηm = 2).
1
Surface Plasmons in Assemblies of Small Particles
13
(3) Because the radicand in Eq. (1.19) is always greater than the half-difference
2 /2, the “upper” resonances ω+ (m = 0, ± 1) are always
magnitude ω˜ 02 − ω˜ 0s
m
located above the greater of the two frequencies ω˜ 0 and ω˜ 0S , while the “lower”
− are located below the smaller of ω
˜ 0 and ω˜ 0S .
ones ωm
(4) These shifts, being dependent on the value h/R (see the expression for xm ),
decrease with an increase in the height h and are the same for spheres of the
same material but of different radii lying on a substrate (when h = R), i.e., they
are scaling invariant.
Note, finally, that the locations of these sphere resonances with respect to
the resonant frequency ω˜ 0 of an isolated sphere define the red and blue shifts of
the single-sphere eigenmode. As is clear (see Fig. 1.3), these shifts are not equal.
− ) is greater than the blue shift (from ω
˜ 0 to
The red shift (i.e., the shift from ω˜ 0 to ωm
2
2
+
ωm ) in the case when ω˜ 0 > ω˜ 0s , and less in the opposite case.
It should be stressed that no matter how small the blue shift may be, it always
exists in principle. This result is quite surprising and differs from the commonly
accepted viewpoint that the substrate can cause only the red shift of an isolated
sphere resonant frequency, and the blue-shifted resonances appear (if any) due to
either higher multipoles [8] or nonlocality of the permittivity. From our results, such
statements should be considered as erroneous. Moreover, the appearance of the four
resonances due to splitting and shifting of the single-sphere resonance in the presence of a substrate is quite analogous to the same dipole approximation for a system
of two unequal spheres, as well as to the production of the combination frequencies
in a system of two coupled oscillators in the classical mechanics.
We can now describe the physical pictureof the single-resonance splitting as follows. The interaction of the sphere and substrate with the external field leads at first
√
to exciting and√coupling of the corresponding bulk and surface modes ω0 , ωp / 3
and ω0s , ωps / 2, respectively, resulting in the natural modes of the sphere and
1/2
1/2
2 /2
substrate, ω˜ 0 ≡ ω02 + ωp2 /3
and ω˜ 0s ≡ ω02 + ωps
. Then, the natural
modes are coupled via mutual electromagnetic interaction (dipole–dipole, in our
±.
case). The latter is what causes the splitting of ω˜ 0 into the set of resonances ωm
Increasing the distance between the sphere and the substrate leads to a weakening
of their mutual interaction. In the limit h → ∞, we have noninteracting modes with
eigenfrequencies ω˜ 0 and ω˜ 0s with no splitting.
This description of the splitting and shifting of the sphere resonance, being
dependent on the combination of the values ω0 , ωp , ω0s , ωps , and h/R, can lead
to various pictures of absorption band localization with respect to the fundamental
frequencies ω0 and ω0s of the bulk materials, as well as to the plasma frequencies
ωp and ωps . Particularly, for a metallic sphere (ω0 = 0) near a dielectric subωp ), if only ω0s
ωp , one can derive the approximate expressions
strate (ωps
(presented in [11], but with misprints)
3
+
ωm
≈ 1 + xm
4
ωps
ωp
2
ωps
ωp −
≈ ω02 + (1 − xm )
√ , ωm
2
3
2
.1/2
(1.20)
14
L.G. Grechko et al.
One can see from Eq. (1.20), for example, that the blue shift for a metallic sphere
is a small value of the order of (ωps /ωp )2 in the presence of a dielectric substrate,
but is substantial in the case of a metallic substrate.
A problem is analyzed when a small particle and substrate are metals
(ω0 = ω0s = 0) and in general case we assume that ε∞ = ε∞s = εa . For
± 2 , an expression can be obtained:
frequencies ωm
± 2
) =
(ωm
(ω¯ f2
1
2(1 − xm ααs )
+ ω¯ fs2 )2
ω¯ f2
±
− ω¯ fs2
2
+ 4ω¯ f2 ω¯ fs2 xm
(1 − α)(1 − αs )
(1 − xm α)(1 − αs xm )
(1.21)
1
2
,
where
ω¯ f2 =
ωfs2 =
2
ωps
ε∞s + εa
ωf2 (1 − αs ηm )
1 − ηm ααs
, ωf2 =
ωp2
ε∞ + 2εa
,ω¯ fs2 =
,α=
ωfs2 (1 − αs ηm )
1 − ηm ααs
,
(1.22)
ε∞ − εa
ε∞s − εa
, αs =
.
ε∞ + 2εa
ε∞s + εa
(1.23)
Here ωf and ωfs are independent frequencies of surface plasmons of small particles and the substrate; if d→∞, ω¯ f2 and ω¯ fs2 are the same frequencies taking into
consideration dipole interaction of the substrate and small particle for finite d values
(Fig. 1.2b). In case when ε∞ = ε∞s = εa = 1 (Eq. (1.20)) at ω0 = ω0s = 0
we have
1
± 2
2(ωm
) = ωf2 + ωfs2 ± (ωf2 − ωfs2 ) + 4ωf2 ωfs2 Am . 2
(1.24)
In case when material of particle and the substrate is the same (ωp = ωps ), it
follows
±
ωm
2
⎧
ωp2 ⎨
=
5 ± 1 + 3ηm
12 ⎩
R
h
3
1
2
⎫
⎬
⎭
.
(1.25)
To determine optically active modes, it is necessary to evaluate relative oscillator
− 2
strengths fm+ and fm− [10]. It follows from the analysis that only two modes ωm
appear to be optically active in absorption spectra. Oscillator strengths of other two
+ )2 tend to zero.
modes (ωm
1
Surface Plasmons in Assemblies of Small Particles
15
1.4.2 Two Metallic Spherical Particles in External Electric Field
Let us analyze the case of two metallic spherical parts with distance d between them
(Fig. 1.2) in external (variable in time) electric field with wave length λ0 which is
greatly more than the particles’ dimension and d. In this case it is necessaryto assign
εa = εs , polarizability tensor of i th particle can be formed as
1 + (−1)m ηm a¯i1
,
2 a11 a21
1 − ηm
d6
εi − εa 3
ai =
R , i = 1,2,
εi + 2εa i
m
αim
= ai1
1
(1.26)
(1.27)
where Ri is i th particle radius.
All other designations are the same.
Condition for obtaining frequencies of surface plasmons (zero value of denominator in Eq. (1.27)) in this case looks like
2
ηm
R31 R32
d6
ε1∞ − εa
ε1∞ + 2εa
ε2∞ − εa
ε2∞ + 2εa
2
ω2 − ωe1
2
ω2 − ωe2
ω2 − ωf21
ω2 − ωf22
= 1.
(1.28)
Permittivities of metal spheres were taken in Drude formulation [3]
ε1 (ω) = ε1∞ −
2
ωp1
ω (ω + iγ1 )
ε2 (ω) = ε2∞ −
;
2
ωp2
ω (ω + iγ2 )
,
(1.29)
and in case of realization expression (1.28) γ1 , γ2 should come to zero.
Additionally, in Eq. (1.28) next designations are introduced:
ωfi2 =
2
ωpi
εi∞ + 2εa
2
ωei
=
;
2
ωpi
εi∞ − εa
, i = 1,2.
(1.30)
Taking into consideration these remarks, frequencies of surface plasmons are
1
2
± 2
ωm
= ω¯ f21 + ω¯ f22 ±
ω¯ f21 − ω¯ f22
2
+ 4ω¯ f21 × ω¯ f22
2
2
(1 − α1 ) (1 − α2 ) xm
,
2
1 − α1 A2m 1 − α2 xm
(1.31)
where
2
2
xm
= ηm
ω¯ f21 = ωf21
R31 R32
εi∞ − εa
; α12 = α1 α2 ; α1 =
6
εi∞ + 2εa
d
2
1 − α2 xm
;
2
1 − α12 xm
ω¯ f22 = ωf22
2
1 − α1 xm
; i = 1,2.
2
1 − α12 xm
16
L.G. Grechko et al.
Expression (1.31) represents the basic formula for calculating frequencies of
surface plasmons in a system of two different metallic spherical particles which
disposed in external electric field at distance d. Corresponding frequencies of the
“oscillators” force can be obtained using the method displayed in [10].
Let us analyze a particular case when particles consist of the same material
ε1 (ω) = ε2 (ω) but have different dimensions R1 = R2 . Then from Eq. (1.31)
frequencies of surface plasmons in a system of two particles are
± 2
ωm
ε(ω) = ε∞ −
=
ωp2
ω2
,
(1 ± xm )ωf2
1 ± α0 xm
, ωf =
ωp2
ε∞ + 2ε0
= R2 /R1 (R2 ≤ R1 ), α0 =
ηm =
, ε1 (ω) = ε2 (ω) ≡ ε(ω),
ε∞ − ε0
, xm = ηm
ε∞ + 2ε0
2, when m = (0; ||)
.
1, when m = ( ± 1; ⊥)
3/2
(R1 /d)3 ,
(1.32)
The sign means that field E0 is directed along a straight line connecting the
particles’ centers, and longitudinal field is directed at right angle to the line:
( ω)2 = (ω0+ )2 − (ω0− )2 = 3ωp2
(R1 R2 )1/2
d
3
.
(1.33)
At ε∞ = ε0 , α0 = 0 expression for four frequencies of surfaces plasmons looks
like
±
ωm
2
= ωf2 1 ± ηm
3/2
(R1 /d)3 ,
and expression for the tensor of a polarizability of the first particle (i = 1) can be
presented in the form
m
αm
= R3i ωf2
1
fm+
fm−
m
δm
+
,
+ 2
− 2
1
(ωm
) − ω2
(ωm
) − ω2
± are f + = (1/2)(1 + ( −
where oscillator strengths, corresponded frequencies ωm
m
m
3/2
−
m
3/2
+
−
), fm = (1/2)(1 − ( − 1)
) with fm + fm = 1.
1)
For longitudinal modes (m = 0), more optically active is the mode ω0+ , and for
transversal (m = ±1) it is ω1− . At = 1 and identical radii of particles there will
be two modes in the spectrum of surface modes, thus a change of frequency will be
equal to
( ω)2 ≡ (ω0+ )2 − (ω1− )2 = 3ωp2
(R1 R2 )1/2
d
3
.