1 Nanoplasmonics: From Present into Future
81
Using Eqs. (1.97) and (1.27), (1.34), it is straightforward to show that the effective
permittivity (1.97) simplifies exactly to
¯
ε(ω) = b n [s n ε m (ω) + (1 − s n )ε h (ω)] .
(1.99)
1.5.8 Conditions of Loss Compensation by Gain and Spasing
In the case of the full inversion (maximum gain) and in the exact resonance, the host
medium permittivity acquires the imaginary part describing the stimulated emission
as given by the standard expression
ε h (ω) = ε d − i
4π
3
|d 12 | 2 n c
Γ 12
,
(1.100)
where ε d = Re ε h , d 12 is a dipole matrix element of the gain transition in a chromophore center of the gain medium, Γ 12 is a spectral width of this transition, and n c
is the concentration of these centers (these notations are consistent with those used
above in Sects. 1.5.4.1–1.5.6.3). Note that if the inversion is not maximum, then this
and subsequent equations are still applicable if one sets as the chromophore concentration n c the inversion density: n c = n 2 −n 1 , where n 2 and n 1 are the concentrations
of the chromophore centers of the gain medium in the upper and lower states of the
gain transition, respectively.
The condition for the full electric loss compensation in the metamaterial and
amplification (overcompensation) at the resonant frequency ω = ω n is
Im ¯
ε(ω) ≤ 0
(1.101)
Taking Eq. (1.99) into account, this reduces to
s n Im ε m (ω) −
4π
3
|d 12 | 2 n c (1 − s n )
Γ 12
≤ 0.
(1.102)
Finally, taking into account Eqs. (1.28), (1.47) and that Im ε m (ω) > 0, we obtain
from Eq. (1.102) the condition of the loss (over)compensation as
4π
3
|d 12 | 2 n c [1 − Re s(ω)]
Γ 12 Re s(ω)Im ε m (ω)
≥ 1,
(1.103)
where the strict inequality corresponds to the overcompensation and net amplification. In Eq. (1.100) we have assumed non-polarized gain transitions. If these transitions are all polarized along the excitation electric field, the concentration n c should
be multiplied by a factor of 3.
81
Using Eqs. (1.97) and (1.27), (1.34), it is straightforward to show that the effective
permittivity (1.97) simplifies exactly to
¯
ε(ω) = b n [s n ε m (ω) + (1 − s n )ε h (ω)] .
(1.99)
1.5.8 Conditions of Loss Compensation by Gain and Spasing
In the case of the full inversion (maximum gain) and in the exact resonance, the host
medium permittivity acquires the imaginary part describing the stimulated emission
as given by the standard expression
ε h (ω) = ε d − i
4π
3
|d 12 | 2 n c
Γ 12
,
(1.100)
where ε d = Re ε h , d 12 is a dipole matrix element of the gain transition in a chromophore center of the gain medium, Γ 12 is a spectral width of this transition, and n c
is the concentration of these centers (these notations are consistent with those used
above in Sects. 1.5.4.1–1.5.6.3). Note that if the inversion is not maximum, then this
and subsequent equations are still applicable if one sets as the chromophore concentration n c the inversion density: n c = n 2 −n 1 , where n 2 and n 1 are the concentrations
of the chromophore centers of the gain medium in the upper and lower states of the
gain transition, respectively.
The condition for the full electric loss compensation in the metamaterial and
amplification (overcompensation) at the resonant frequency ω = ω n is
Im ¯
ε(ω) ≤ 0
(1.101)
Taking Eq. (1.99) into account, this reduces to
s n Im ε m (ω) −
4π
3
|d 12 | 2 n c (1 − s n )
Γ 12
≤ 0.
(1.102)
Finally, taking into account Eqs. (1.28), (1.47) and that Im ε m (ω) > 0, we obtain
from Eq. (1.102) the condition of the loss (over)compensation as
4π
3
|d 12 | 2 n c [1 − Re s(ω)]
Γ 12 Re s(ω)Im ε m (ω)
≥ 1,
(1.103)
where the strict inequality corresponds to the overcompensation and net amplification. In Eq. (1.100) we have assumed non-polarized gain transitions. If these transitions are all polarized along the excitation electric field, the concentration n c should
be multiplied by a factor of 3.