68
M. I. Stockman
This expression is fully consistent with Ref. [31]. The following order of magnitude estimate of this spasing condition has a transparent physical meaning and is of
heuristic value,
d 2
12 Q N c
Γ 12 V n
1,
(1.79)
where Q = ω/γ n is the quality factor of SPs, V n is the volume of the spasing SP
mode, and N c is the of number of the gain medium chromophores within this volume.
Deriving this estimate, we have neglected the detuning, i.e., set ω 21 − ω n = 0. We
also used the definitions of A n of Eq. (1.64) and ˜
Ω
( p)
12 given by Eq. (1.68), and the
estimate |∇ϕ n (r)|
2
∼ 1/V following from the normalization of the SP eigenmodes
|∇ϕ n (r)|
2 d 3 r = 1 of Ref. [78]. The result of Eq. (1.79) is, indeed, in agreement
with Ref. [31] where it was obtained in different notations.
It follows from Eq. (1.79) that for the existence of spasing it is beneficial to have a
high quality factor Q, a high density of the chromophores, and a large transition dipole
(oscillator strength) of the chromophore transition. The small modal volume V n (at
a given number of the chromophores N c ) is beneficial for this spasing condition:
physically, it implies strong feedback in the spaser. Note that for the given density of
the chromophores n c = N c /V n , this spasing condition does not explicitly depend on
the spaser size, which opens up a possibility of spasers of a very small size limited
from the bottom by only the nonlocality radius l nl ∼ 1 nm. Another important
property of Eq. (1.79) is that it implies the quantum-mechanical nature of spasing
and spaser amplification: this condition essentially contains the Planck constant
and, thus, does not have a classical counterpart. Note that in contrast to lasers, the
spaser theory and Eqs. (1.78), (1.79) in particular do not contain speed of light, i.e.,
they are quasistatic.
Now we will examine the spasing condition and reduce it to a requirement for the
gain medium. First, we substitute all the definitions and assume the perfect resonance
between the generating SP mode and the gain medium, i.e., ω n = ω 21 . As a result,
we obtain from Eq. (1.78),
4π
3
s n |d 12 | 2
γ n Γ 12 ε d s
n
V
[1 − Θ(r)] |E n (r)|
2 d
3 r ≥ 1,
(1.80)
where the integral is extended over the volume V of the system, and the Θ-function
takes into account a simplifying realistic assumption that the gain medium occupies
the entire space free from the core’s metal. We also assume that the orientations of
the transition dipoles d
( p)
12 are random and average over them, which results in the
factor of 3 in the denominator in Eq. (1.80). From Eqs. (1.27) and (1.34), it follows
that
V
[1 − Θ(r)] |E n (r)|
2 d
3 r = 1 − s n .
(1.81)
M. I. Stockman
This expression is fully consistent with Ref. [31]. The following order of magnitude estimate of this spasing condition has a transparent physical meaning and is of
heuristic value,
d 2
12 Q N c
Γ 12 V n
1,
(1.79)
where Q = ω/γ n is the quality factor of SPs, V n is the volume of the spasing SP
mode, and N c is the of number of the gain medium chromophores within this volume.
Deriving this estimate, we have neglected the detuning, i.e., set ω 21 − ω n = 0. We
also used the definitions of A n of Eq. (1.64) and ˜
Ω
( p)
12 given by Eq. (1.68), and the
estimate |∇ϕ n (r)|
2
∼ 1/V following from the normalization of the SP eigenmodes
|∇ϕ n (r)|
2 d 3 r = 1 of Ref. [78]. The result of Eq. (1.79) is, indeed, in agreement
with Ref. [31] where it was obtained in different notations.
It follows from Eq. (1.79) that for the existence of spasing it is beneficial to have a
high quality factor Q, a high density of the chromophores, and a large transition dipole
(oscillator strength) of the chromophore transition. The small modal volume V n (at
a given number of the chromophores N c ) is beneficial for this spasing condition:
physically, it implies strong feedback in the spaser. Note that for the given density of
the chromophores n c = N c /V n , this spasing condition does not explicitly depend on
the spaser size, which opens up a possibility of spasers of a very small size limited
from the bottom by only the nonlocality radius l nl ∼ 1 nm. Another important
property of Eq. (1.79) is that it implies the quantum-mechanical nature of spasing
and spaser amplification: this condition essentially contains the Planck constant
and, thus, does not have a classical counterpart. Note that in contrast to lasers, the
spaser theory and Eqs. (1.78), (1.79) in particular do not contain speed of light, i.e.,
they are quasistatic.
Now we will examine the spasing condition and reduce it to a requirement for the
gain medium. First, we substitute all the definitions and assume the perfect resonance
between the generating SP mode and the gain medium, i.e., ω n = ω 21 . As a result,
we obtain from Eq. (1.78),
4π
3
s n |d 12 | 2
γ n Γ 12 ε d s
n
V
[1 − Θ(r)] |E n (r)|
2 d
3 r ≥ 1,
(1.80)
where the integral is extended over the volume V of the system, and the Θ-function
takes into account a simplifying realistic assumption that the gain medium occupies
the entire space free from the core’s metal. We also assume that the orientations of
the transition dipoles d
( p)
12 are random and average over them, which results in the
factor of 3 in the denominator in Eq. (1.80). From Eqs. (1.27) and (1.34), it follows
that
V
[1 − Θ(r)] |E n (r)|
2 d
3 r = 1 − s n .
(1.81)