84
M. I. Stockman
We are interested in a resonant case when the metamaterial possesses a resonance
at some eigenfrequency ω n ≈ ω. For this to be true, the system’s behavior must be
plasmonic, i.e., Re ¯
ε(ω) < 0. Then the dominating contribution to ¯
ε comes from a
resonant SP eigenmode n with a frequency ω n ≈ ω. In such a case, the dielectric
function [78] ¯
ε(ω) has a simple pole at ω = ω n . As a result, ∂ (ωRe ¯
ε) /∂ω ≈
ω∂Re ¯
ε/∂ω and, consequently, Γ = γ n , where γ n is the SP decay rate given by
Eqs. (1.3) or (1.48), and the metal dielectric function ε m is replaced by the effective
permittivity ¯
ε of the metamaterial. Thus, Eq. (1.109) is fully consistent with the
spectral theory of SPs—see Sect. 1.3.4.
If the losses are not very large so that energy of the system is meaningful, the
Kramers-Kronig causality requires [30] that ∂(ωRe ¯
ε)/∂ω > 0. Thus, Im ¯
ε < 0 in
Eq. (1.109) would lead to a negative decrement,
Γ < 0,
(1.110)
implying that the initial small fluctuation starts exponentially grow in time in its field
and energy, which is an instability. Such an instability is indeed not impossible: it
will result in spasing that will eventually stabilize |E| and E at finite stationary (CW)
levels of the spaser generation.
Note that the spasing limits (clamps) the gain and population inversion making the
net gain to be precisely zero [139] in the stationary (continuous wave or CW) regime
see Sect. 1.5.6 and Fig. 1.29b. Above the threshold of the spasing, the population
inversion of the gain medium is clamped at a rather low level n 21 ∼ 1 %. The
corresponding net amplification in the CW spasing regime is exactly zero, which is
a condition for the CW regime. This makes the complete loss compensation and its
overcompensation impossible in a dense resonant metamaterial where the feedback
is created by the internal inhomogeneities (including its periodic structure) and the
facets of the system.
Because the loss (over) compensation condition (1.103), which is also the spasing
condition, is geometry-independent, it is useful to illustrate it for commonly used
plasmonic metals, gold and silver whose permittivity we adopt from Ref. [32]. For
the gain medium chromophores, we will use a reasonable set of parameters: Γ 12 =
5 × 10 13 s −1 and d 12 = 4.3 × 10 −18 esu. The results of computations are shown
in Fig. 1.31. (Note that this figure expresses a condition of spasing equivalent to
that of Fig. 1.28). For silver as a metal and n c = 6 × 10 18 cm −3 , the corresponding
lower (black) curve in panel (a) does not reach the value of 1, implying that no
full loss compensation is achieved. In contrast, for a higher but still very realistic
concentration of n c = 2.9 × 10 19 cm −3 , the upper curve in Fig. 1.31a does cross
the threshold line in the near-infrared region. Above the threshold area, there will be
the instability and the onset of the spasing. As Fig. 1.31b demonstrates, for gold the
spasing occurs at higher, but still realistic, chromophore concentrations.
M. I. Stockman
We are interested in a resonant case when the metamaterial possesses a resonance
at some eigenfrequency ω n ≈ ω. For this to be true, the system’s behavior must be
plasmonic, i.e., Re ¯
ε(ω) < 0. Then the dominating contribution to ¯
ε comes from a
resonant SP eigenmode n with a frequency ω n ≈ ω. In such a case, the dielectric
function [78] ¯
ε(ω) has a simple pole at ω = ω n . As a result, ∂ (ωRe ¯
ε) /∂ω ≈
ω∂Re ¯
ε/∂ω and, consequently, Γ = γ n , where γ n is the SP decay rate given by
Eqs. (1.3) or (1.48), and the metal dielectric function ε m is replaced by the effective
permittivity ¯
ε of the metamaterial. Thus, Eq. (1.109) is fully consistent with the
spectral theory of SPs—see Sect. 1.3.4.
If the losses are not very large so that energy of the system is meaningful, the
Kramers-Kronig causality requires [30] that ∂(ωRe ¯
ε)/∂ω > 0. Thus, Im ¯
ε < 0 in
Eq. (1.109) would lead to a negative decrement,
Γ < 0,
(1.110)
implying that the initial small fluctuation starts exponentially grow in time in its field
and energy, which is an instability. Such an instability is indeed not impossible: it
will result in spasing that will eventually stabilize |E| and E at finite stationary (CW)
levels of the spaser generation.
Note that the spasing limits (clamps) the gain and population inversion making the
net gain to be precisely zero [139] in the stationary (continuous wave or CW) regime
see Sect. 1.5.6 and Fig. 1.29b. Above the threshold of the spasing, the population
inversion of the gain medium is clamped at a rather low level n 21 ∼ 1 %. The
corresponding net amplification in the CW spasing regime is exactly zero, which is
a condition for the CW regime. This makes the complete loss compensation and its
overcompensation impossible in a dense resonant metamaterial where the feedback
is created by the internal inhomogeneities (including its periodic structure) and the
facets of the system.
Because the loss (over) compensation condition (1.103), which is also the spasing
condition, is geometry-independent, it is useful to illustrate it for commonly used
plasmonic metals, gold and silver whose permittivity we adopt from Ref. [32]. For
the gain medium chromophores, we will use a reasonable set of parameters: Γ 12 =
5 × 10 13 s −1 and d 12 = 4.3 × 10 −18 esu. The results of computations are shown
in Fig. 1.31. (Note that this figure expresses a condition of spasing equivalent to
that of Fig. 1.28). For silver as a metal and n c = 6 × 10 18 cm −3 , the corresponding
lower (black) curve in panel (a) does not reach the value of 1, implying that no
full loss compensation is achieved. In contrast, for a higher but still very realistic
concentration of n c = 2.9 × 10 19 cm −3 , the upper curve in Fig. 1.31a does cross
the threshold line in the near-infrared region. Above the threshold area, there will be
the instability and the onset of the spasing. As Fig. 1.31b demonstrates, for gold the
spasing occurs at higher, but still realistic, chromophore concentrations.