1 Nanoplasmonics: From Present into Future
69
Next, we give approximate expressions for the spectral parameter (1.4), which are
very accurate for the realistic case of Q 1,
Im s(ω) =
s 2
n
ε d
Im ε m (ω) =
1
Q
s n (1 − s n ) ,
(1.82)
where definition (1.6) is used. Taking into account Eqs. (1.47), (1.48) and (1.81),
(1.82), we obtain from Eq. (1.80) a necessary condition of spasing at a frequency
ω as
4π
3
|d 12 | 2 n c [1 − Re s(ω)]
Γ 12 Re s(ω)Im ε m (ω)
≥ 1,
(1.83)
For the sake of comparison, consider a continuous gain medium comprised of the
same chromophores as the gain shell of the spaser. Its gain g (whose dimensionality
is cm −1 ) is given by a standard expression
g =
4π
3
ω
c
√
ε d |d 12 | 2 n c
Γ 12
.
(1.84)
Substituting it into Eq. (1.83), we obtain the spasing criterion in terms of the gain as
g ≥ g th , g th =
ω
c
√ ε d
Re s(ω)
1 − Re s(ω)
Im ε m (ω),
(1.85)
where g th has a meaning of the threshold gain needed for spasing. Importantly, this
gain depends only on the dielectric properties of the system and spasing frequency
but not on the geometry of the system or the distribution of the local fields of the
spasing mode (hot spots, etc.) explicitly. However note that the system’s geometry
(along with the permittivities) does define the spasing frequencies.
In Figs. 1.28a, b, correspondingly, we illustrate the analytical expression (1.85)
for gold and silver embedded in a dielectric with ε d = 2 (simulating a light glass)
and ε d = 10 (simulating a semiconductor), correspondingly. These are computed
from Eq. (1.85) assuming that the metal core is embedded into the gain medium with
the real part of the dielectric function equal to ε d . As we see from Fig. 1.28, the
spasing is possible for silver in the near-ir communication range and the adjacent red
portion of the visible spectrum for a gain g < 3000 cm −1 (regions below the red line
in Fig. 1.28), which is realistically achievable with direct band-gap semiconductors
(DBDSs).
1.5.5 Spaser in CW Mode
The “spasing curve” (a counterpart of the light–light curve, or L–L curve, for lasers),
which is the dependence of the coherent SP population N n on the excitation rate
69
Next, we give approximate expressions for the spectral parameter (1.4), which are
very accurate for the realistic case of Q 1,
Im s(ω) =
s 2
n
ε d
Im ε m (ω) =
1
Q
s n (1 − s n ) ,
(1.82)
where definition (1.6) is used. Taking into account Eqs. (1.47), (1.48) and (1.81),
(1.82), we obtain from Eq. (1.80) a necessary condition of spasing at a frequency
ω as
4π
3
|d 12 | 2 n c [1 − Re s(ω)]
Γ 12 Re s(ω)Im ε m (ω)
≥ 1,
(1.83)
For the sake of comparison, consider a continuous gain medium comprised of the
same chromophores as the gain shell of the spaser. Its gain g (whose dimensionality
is cm −1 ) is given by a standard expression
g =
4π
3
ω
c
√
ε d |d 12 | 2 n c
Γ 12
.
(1.84)
Substituting it into Eq. (1.83), we obtain the spasing criterion in terms of the gain as
g ≥ g th , g th =
ω
c
√ ε d
Re s(ω)
1 − Re s(ω)
Im ε m (ω),
(1.85)
where g th has a meaning of the threshold gain needed for spasing. Importantly, this
gain depends only on the dielectric properties of the system and spasing frequency
but not on the geometry of the system or the distribution of the local fields of the
spasing mode (hot spots, etc.) explicitly. However note that the system’s geometry
(along with the permittivities) does define the spasing frequencies.
In Figs. 1.28a, b, correspondingly, we illustrate the analytical expression (1.85)
for gold and silver embedded in a dielectric with ε d = 2 (simulating a light glass)
and ε d = 10 (simulating a semiconductor), correspondingly. These are computed
from Eq. (1.85) assuming that the metal core is embedded into the gain medium with
the real part of the dielectric function equal to ε d . As we see from Fig. 1.28, the
spasing is possible for silver in the near-ir communication range and the adjacent red
portion of the visible spectrum for a gain g < 3000 cm −1 (regions below the red line
in Fig. 1.28), which is realistically achievable with direct band-gap semiconductors
(DBDSs).
1.5.5 Spaser in CW Mode
The “spasing curve” (a counterpart of the light–light curve, or L–L curve, for lasers),
which is the dependence of the coherent SP population N n on the excitation rate