1 Nanoplasmonics: From Present into Future
63
This universal unsaturable behavior can be very simply understood qualitatively—
cf. Ref. [285]. The excitation rate ˙
N e of the upper spasing level is linearly proportional
to pumping intensity I p , ˙
N e = σ e I p , where σ e is the total excitation cross section into
the conduction band of the semiconductor gain medium. In the developed spasing
regime, plasmon population N n of the spasing eigenmode becomes large, asymptotically N n → ∞. Correspondingly, the stimulated decay rate, which is ∝ N n , becomes
large and dominates over any spontaneous decay rate. Thus, all the excitation events
to the conduction band end up with the emission of a SP into the spasing mode
whose SP population becomes N n = ˙
N e /γ n , where γ n is the SP decay rate—see
above Eq. (1.48). Finally, radiation rate ˙
N r for a spaser becomes
˙
N r = σ e γ
(r )
γ n ,
(1.62)
where γ (r ) is the SP radiative decay rate, which for a plasmonic metal sphere is given
by Eq. (1.16) and in, general case, by Eq. (1.56). Of course. in reality the straight-line,
unsaturable L–L curves will end when the pumping intensities become so high that
the nonlinearity in the spaser metal develops (including, but not limited to, thermal
nonlinearity), or optical breakdown occurs, or heat production will physically damage
the spaser.
As theory shows (see below Sect. 1.5.6.1 and Fig. 1.30a), under steady pumping,
the generating spaser reaches its stationary regime within ∼100 fs. Correspondingly,
we expect that any fluctuation in the emission radiated by the generating spaser
relaxes back to the mean level within the same time. A measure of the fluctuations
of the spaser-radiation intensity I (t) with time t is the second-order autocorrelation
function
g
(2)
(τ ) =
I (t + τ )I (t)
I (t)
2
,
(1.63)
where τ is the delay time, and · · · denotes quantum-mechanical (theory) or temporal (experiment) averaging.
Experimentally, g (2) (τ ) has been measured for a single spaser in Ref. [283]. The
result is reproduced in Fig. 1.27d. The upper curve is recorded below the spasing
threshold; at the zero delay, it shows a peak, which is characteristic of incoherent
radiation. If such radiation is produced by many independent emitters, it has Gaussian
statistics, and the peak value should be g (2) (0) = 2—this effect was introduced
by Hanbury Brown and Twiss and used by them for stellar interferometry [286].
For the upper curve of Fig. 1.27d, g (2) (0) is significantly less. This may be due to
various reasons, in particular, insufficient temporal resolution of the photodetection
or partial coherence between the individual emitters of the gain medium induced by
their interaction via plasmonic fields.
In sharp contrast, above the spasing threshold, the autocorrelation function in
Fig. 1.27d is a constant at all delays. As we have already pointed out this is due to
the fact that after an emission of a photon, the number of plasmons in the spaser
is restored within ∼100 fs, while the temporal resolution of the photodetection in
63
This universal unsaturable behavior can be very simply understood qualitatively—
cf. Ref. [285]. The excitation rate ˙
N e of the upper spasing level is linearly proportional
to pumping intensity I p , ˙
N e = σ e I p , where σ e is the total excitation cross section into
the conduction band of the semiconductor gain medium. In the developed spasing
regime, plasmon population N n of the spasing eigenmode becomes large, asymptotically N n → ∞. Correspondingly, the stimulated decay rate, which is ∝ N n , becomes
large and dominates over any spontaneous decay rate. Thus, all the excitation events
to the conduction band end up with the emission of a SP into the spasing mode
whose SP population becomes N n = ˙
N e /γ n , where γ n is the SP decay rate—see
above Eq. (1.48). Finally, radiation rate ˙
N r for a spaser becomes
˙
N r = σ e γ
(r )
γ n ,
(1.62)
where γ (r ) is the SP radiative decay rate, which for a plasmonic metal sphere is given
by Eq. (1.16) and in, general case, by Eq. (1.56). Of course. in reality the straight-line,
unsaturable L–L curves will end when the pumping intensities become so high that
the nonlinearity in the spaser metal develops (including, but not limited to, thermal
nonlinearity), or optical breakdown occurs, or heat production will physically damage
the spaser.
As theory shows (see below Sect. 1.5.6.1 and Fig. 1.30a), under steady pumping,
the generating spaser reaches its stationary regime within ∼100 fs. Correspondingly,
we expect that any fluctuation in the emission radiated by the generating spaser
relaxes back to the mean level within the same time. A measure of the fluctuations
of the spaser-radiation intensity I (t) with time t is the second-order autocorrelation
function
g
(2)
(τ ) =
I (t + τ )I (t)
I (t)
2
,
(1.63)
where τ is the delay time, and · · · denotes quantum-mechanical (theory) or temporal (experiment) averaging.
Experimentally, g (2) (τ ) has been measured for a single spaser in Ref. [283]. The
result is reproduced in Fig. 1.27d. The upper curve is recorded below the spasing
threshold; at the zero delay, it shows a peak, which is characteristic of incoherent
radiation. If such radiation is produced by many independent emitters, it has Gaussian
statistics, and the peak value should be g (2) (0) = 2—this effect was introduced
by Hanbury Brown and Twiss and used by them for stellar interferometry [286].
For the upper curve of Fig. 1.27d, g (2) (0) is significantly less. This may be due to
various reasons, in particular, insufficient temporal resolution of the photodetection
or partial coherence between the individual emitters of the gain medium induced by
their interaction via plasmonic fields.
In sharp contrast, above the spasing threshold, the autocorrelation function in
Fig. 1.27d is a constant at all delays. As we have already pointed out this is due to
the fact that after an emission of a photon, the number of plasmons in the spaser
is restored within ∼100 fs, while the temporal resolution of the photodetection in