82
M. I. Stockman
Equation (1.103) is a fundamental condition, which is precise [assuming that the
requirement (1.95) is satisfied, which is very realistic for metamaterials] and general.
Moreover, it is fully analytical and, actually, very simple. Remarkably, it depends
only on the material characteristics and does not contain any geometric properties
of the metamaterial system or the local fields. (Note that the system’s geometry
does affect the eigenmode frequencies and thus enters the problem implicitly.) In
particular, the hot spots, which are prominent in the local fields of nanostructures
[78, 158], are completely averaged out due to the integrations in Eqs. (1.90) and
(1.97).
The condition (1.103) is completely non-relativistic (quasistatic)—it does not
contain speed of light c, which is characteristic of also of the spaser. It is useful to
express this condition also in terms of the total stimulated emission cross section
σ e (ω) (where ω is the central resonance frequency) of a chromophore of the gain
medium as
cσ e (ω)
√
ε d n c [1 − Re s(ω)]
ωRe s(ω)Im ε m (ω)
≥ 1.
(1.104)
We see that Eq. (1.103) exactly coincides with a spasing condition expressed by
Eq. (1.83). This brings us to an important conclusion: the full compensation (overcompensation) of the optical losses in a metamaterial [which is resonant and dense
enough to satisfy condition (1.95)] and the spasing occur under precisely the same
conditions.
We have considered above in Sect. 1.5.4.2 the conditions of spasing, which are
equivalent to (1.104). These are given by one of equivalent conditions of Eqs. (1.83),
(1.85), (1.103). It is also illustrated in Fig. 1.28. We stress that exactly the same
conditions are for the full loss compensation (overcompensation) of a dense resonant
plasmonic metamaterial with gain.
We would like also to point out that the criterion given by the equivalent conditions
of Eqs. (1.83), (1.85), (1.103), or (1.104) is derived for localized SPs, which are
describable in the quasistatic approximation, and is not directly applicable to the
propagating plasmonic modes (SPPs). However, we expect that very localized SPPs,
whose wave vector k l s , can be described by these conditions because they are,
basically, quasistatic. For instance, the SPPs on a thin metal wire of a radius R l s
are described by a dispersion relation [12]
k ≈
1
R
−
ε m
2ε d
ln
−
4ε m
ε d
− γ
−1/2
,
(1.105)
where γ ≈ 0.57721 is the Euler constant. This relation is obviously quasistatic
because it does not contain speed of light c.
M. I. Stockman
Equation (1.103) is a fundamental condition, which is precise [assuming that the
requirement (1.95) is satisfied, which is very realistic for metamaterials] and general.
Moreover, it is fully analytical and, actually, very simple. Remarkably, it depends
only on the material characteristics and does not contain any geometric properties
of the metamaterial system or the local fields. (Note that the system’s geometry
does affect the eigenmode frequencies and thus enters the problem implicitly.) In
particular, the hot spots, which are prominent in the local fields of nanostructures
[78, 158], are completely averaged out due to the integrations in Eqs. (1.90) and
(1.97).
The condition (1.103) is completely non-relativistic (quasistatic)—it does not
contain speed of light c, which is characteristic of also of the spaser. It is useful to
express this condition also in terms of the total stimulated emission cross section
σ e (ω) (where ω is the central resonance frequency) of a chromophore of the gain
medium as
cσ e (ω)
√
ε d n c [1 − Re s(ω)]
ωRe s(ω)Im ε m (ω)
≥ 1.
(1.104)
We see that Eq. (1.103) exactly coincides with a spasing condition expressed by
Eq. (1.83). This brings us to an important conclusion: the full compensation (overcompensation) of the optical losses in a metamaterial [which is resonant and dense
enough to satisfy condition (1.95)] and the spasing occur under precisely the same
conditions.
We have considered above in Sect. 1.5.4.2 the conditions of spasing, which are
equivalent to (1.104). These are given by one of equivalent conditions of Eqs. (1.83),
(1.85), (1.103). It is also illustrated in Fig. 1.28. We stress that exactly the same
conditions are for the full loss compensation (overcompensation) of a dense resonant
plasmonic metamaterial with gain.
We would like also to point out that the criterion given by the equivalent conditions
of Eqs. (1.83), (1.85), (1.103), or (1.104) is derived for localized SPs, which are
describable in the quasistatic approximation, and is not directly applicable to the
propagating plasmonic modes (SPPs). However, we expect that very localized SPPs,
whose wave vector k l s , can be described by these conditions because they are,
basically, quasistatic. For instance, the SPPs on a thin metal wire of a radius R l s
are described by a dispersion relation [12]
k ≈
1
R
−
ε m
2ε d
ln
−
4ε m
ε d
− γ
−1/2
,
(1.105)
where γ ≈ 0.57721 is the Euler constant. This relation is obviously quasistatic
because it does not contain speed of light c.