4
M. I. Stockman
E
mirror
H
Ideal mirror
Ideal mirror
λ /2 ~250 nm
Skin depth ~25 nm
Size ~10 nm
La ttic e ion s
Neutral gold
D is p la c e d e le c t r o n s
E
-
-
-
+
+
+
(a)
(b)
(c)
Fig. 1.1 a Localization of optical fields by ideal mirrors and b by a gold nanoparticle. c Schematic
of charge separation
One may think that it is impossible to achieve a localization of the optical energy
to smaller volume than λ 3 /8 by any means, because the ideal mirrors provide the
best confinement of electromagnetic waves. There are two implied assumptions:
(i) The optical energy is electromagnetic energy, and (ii) The best confinement is
provided by ideal mirrors. Both these assumptions must be abandoned to achieve
nanolocalization of optical energy.
Consider a nanoplasmonic system whose size is less than or comparable to the
skin depth
l s = λ
Re
−ε 2
m
ε m + ε d
1/2 −1
,
(1.2)
where λ = λ/(2π) = ω/c is the reduced vacuum wavelength. For single-valence
plasmonic metals (silver, gold, copper, alkaline metals) l s ≈ 25 nm in the entire
optical region.
For such a plasmonic nanosystem with R l s , the optical electric field penetrates
the entire system and drives oscillations of the metal electrons. The total energy of
the system in this case is a sum of the potential energy of the electrons in the electric field and their mechanical kinetic energy. While the magnetic field is present,
non-relativistic electrons’ interaction with it is weak proportional to a small parameter v F /c ∼ α ∼ 10 −2 , where v F is the electron speed at the Fermi surface, c
is speed of light, and α = e 2 /c is the fine structure constant. Thus in this limit,
which is conventionally called quasistatic, the effects of the magnetic component of
the total energy is relatively small. Hence, this total energy is mostly electromechanical (and not electromagnetic) energy. [At this point, it may be useful to refer to
Eq. (1.107), which expresses the Brillouin formula for total energy E of a system in
such a quasistatic case.] This is why the wavelength, which determines the length
M. I. Stockman
E
mirror
H
Ideal mirror
Ideal mirror
λ /2 ~250 nm
Skin depth ~25 nm
Size ~10 nm
La ttic e ion s
Neutral gold
D is p la c e d e le c t r o n s
E
-
-
-
+
+
+
(a)
(b)
(c)
Fig. 1.1 a Localization of optical fields by ideal mirrors and b by a gold nanoparticle. c Schematic
of charge separation
One may think that it is impossible to achieve a localization of the optical energy
to smaller volume than λ 3 /8 by any means, because the ideal mirrors provide the
best confinement of electromagnetic waves. There are two implied assumptions:
(i) The optical energy is electromagnetic energy, and (ii) The best confinement is
provided by ideal mirrors. Both these assumptions must be abandoned to achieve
nanolocalization of optical energy.
Consider a nanoplasmonic system whose size is less than or comparable to the
skin depth
l s = λ
Re
−ε 2
m
ε m + ε d
1/2 −1
,
(1.2)
where λ = λ/(2π) = ω/c is the reduced vacuum wavelength. For single-valence
plasmonic metals (silver, gold, copper, alkaline metals) l s ≈ 25 nm in the entire
optical region.
For such a plasmonic nanosystem with R l s , the optical electric field penetrates
the entire system and drives oscillations of the metal electrons. The total energy of
the system in this case is a sum of the potential energy of the electrons in the electric field and their mechanical kinetic energy. While the magnetic field is present,
non-relativistic electrons’ interaction with it is weak proportional to a small parameter v F /c ∼ α ∼ 10 −2 , where v F is the electron speed at the Fermi surface, c
is speed of light, and α = e 2 /c is the fine structure constant. Thus in this limit,
which is conventionally called quasistatic, the effects of the magnetic component of
the total energy is relatively small. Hence, this total energy is mostly electromechanical (and not electromagnetic) energy. [At this point, it may be useful to refer to
Eq. (1.107), which expresses the Brillouin formula for total energy E of a system in
such a quasistatic case.] This is why the wavelength, which determines the length