78
M. I. Stockman
(iii) Yet another proposed direction is using highly doped semiconductors where
the Ohmic losses can be significantly lower due to much lower free carrier
concentrations [294]. However, a problem with this approach may lie in the fact
that the usefulness of plasmonic modes depends not on the loss per se but on
the quality factor Q, which for doped semiconductors may not be higher than
for the plasmonic metals.
(iv) One of the alternative approaches to low-loss plasmonic metamaterials is based
on our idea of the spaser: it is using a gain to compensate the dielectric (Ohmic)
losses [295, 296]. In this case the gain medium is included into the metamaterials. It surrounds the metal plasmonic component in the same manner as in the
spasers. The idea is that the gain will provide quantum amplification compensating the loss in the metamaterials quite analogously to the spasers.
We will consider theory of the loss compensation in the plasmonic metamaterials
using gain [140, 141]. Below we show that the full compensation or overcompensation of the optical loss in a dense resonant gain metamaterial leads to an instability
that is resolved by its spasing (i.e., by becoming a generating spaser). We further
show analytically that the conditions of the complete loss compensation by gain and
the threshold condition of spasing—see Eqs. (1.83) and (1.85)—are identical. Thus
the full compensation (overcompensation) of the loss by gain in such a metamaterial will cause spasing. This spasing limits (clamps) the gain—see Sect. 1.5.5—and,
consequently, inhibits the complete loss compensation (overcompensation) at any
frequency.
1.5.7.2 Permittivity of Nanoplasmonic Metamaterial
We will consider, for certainty, an isotropic and uniform metamaterial that, by definition, in a range of frequencies ω can be described by the effective permittivity
¯
ε(ω) and permeability ¯
μ(ω). We will concentrate below on the loss compensation
for the optical electric responses; similar consideration with identical conclusions
for the optical magnetic responses is straightforward. Our theory is applicable for
the true three-dimensional (3d) metamaterials whose size is much greater than the
wavelength λ (ideally, an infinite metamaterial).
Consider a small piece of such a metamaterial with sizes much greater that the
unit cell but much smaller than λ. Such a piece is a metamaterial itself. Let us subject
this metamaterial to a uniform electric field E(ω) = −∇φ(r, ω) oscillating with
frequency ω. Note that E(ω) is the amplitude of the macroscopic electric field inside
the metamaterial. We will denote the local field at a point r inside this metamaterial
as e(r, ω) = −∇ϕ(r, ω). We assume standard boundary conditions
ϕ(r, ω) = φ(r, ω),
(1.86)
for r belonging to the surface S of the volume under consideration.
M. I. Stockman
(iii) Yet another proposed direction is using highly doped semiconductors where
the Ohmic losses can be significantly lower due to much lower free carrier
concentrations [294]. However, a problem with this approach may lie in the fact
that the usefulness of plasmonic modes depends not on the loss per se but on
the quality factor Q, which for doped semiconductors may not be higher than
for the plasmonic metals.
(iv) One of the alternative approaches to low-loss plasmonic metamaterials is based
on our idea of the spaser: it is using a gain to compensate the dielectric (Ohmic)
losses [295, 296]. In this case the gain medium is included into the metamaterials. It surrounds the metal plasmonic component in the same manner as in the
spasers. The idea is that the gain will provide quantum amplification compensating the loss in the metamaterials quite analogously to the spasers.
We will consider theory of the loss compensation in the plasmonic metamaterials
using gain [140, 141]. Below we show that the full compensation or overcompensation of the optical loss in a dense resonant gain metamaterial leads to an instability
that is resolved by its spasing (i.e., by becoming a generating spaser). We further
show analytically that the conditions of the complete loss compensation by gain and
the threshold condition of spasing—see Eqs. (1.83) and (1.85)—are identical. Thus
the full compensation (overcompensation) of the loss by gain in such a metamaterial will cause spasing. This spasing limits (clamps) the gain—see Sect. 1.5.5—and,
consequently, inhibits the complete loss compensation (overcompensation) at any
frequency.
1.5.7.2 Permittivity of Nanoplasmonic Metamaterial
We will consider, for certainty, an isotropic and uniform metamaterial that, by definition, in a range of frequencies ω can be described by the effective permittivity
¯
ε(ω) and permeability ¯
μ(ω). We will concentrate below on the loss compensation
for the optical electric responses; similar consideration with identical conclusions
for the optical magnetic responses is straightforward. Our theory is applicable for
the true three-dimensional (3d) metamaterials whose size is much greater than the
wavelength λ (ideally, an infinite metamaterial).
Consider a small piece of such a metamaterial with sizes much greater that the
unit cell but much smaller than λ. Such a piece is a metamaterial itself. Let us subject
this metamaterial to a uniform electric field E(ω) = −∇φ(r, ω) oscillating with
frequency ω. Note that E(ω) is the amplitude of the macroscopic electric field inside
the metamaterial. We will denote the local field at a point r inside this metamaterial
as e(r, ω) = −∇ϕ(r, ω). We assume standard boundary conditions
ϕ(r, ω) = φ(r, ω),
(1.86)
for r belonging to the surface S of the volume under consideration.