88
M. I. Stockman
compensation by gain in metamaterials, which is based on the same principles of
quantum amplification as the spaser. This status exists despite a significant effort
in this direction and numerous theoretical publications, e.g., [267, 305]. There has
been so far a single, not yet confirmed independently, observation of the full loss
compensation in a plasmonic metamaterial with gain [298].
In large periodic metamaterials, plasmonic modes generally are propagating
waves (SPPs) that satisfy Bloch theorem [306] and are characterized by quasiwavevector k. These are propagating waves except for the band edges where
ka = ±π , where a is the lattice vector. At the band edges, the group velocity
v g of these modes is zero, and these modes are localized, i.e., they are SPs. Their
wave function is periodic with period 2a, which may be understood as a result of
the Bragg reflection from the crystallographic planes. Within this 2a period, these
band-edge modes can, indeed, be treated quasistatically because 2a ∪ l s , λ. If any
of the band-edge frequencies is within the range of compensation [where the condition (1.83) [or, (1.85)] is satisfied], the system will spase. In fact, at the band edge,
this metamaterial with gain is similar to a distributed feedback (DFB) laser [307].
It actually is a DFB spaser, which, as all the DFB lasers, generates in a band-edge
mode.
Moreover, not only the SPPs, which are exactly at the band edge, will be localized.
Due to unavoidable disorder caused by fabrication defects in metamaterials, there
will be scattering of the SPPs from these defects. Close to the band edge, the group
velocity becomes small, v g → 0. Because the scattering cross section of any wave is
∝ v −2
g , the corresponding SPPs experience Anderson localization [308]. Also, there
always will be SPs nanolocalized at the defects of the metamaterial, whose local
fields are hot spots—see Fig. 1.10 and, generally, Sect. 1.3.5 and the publications
referenced therein. Each of such hot spots within the bandwidth of conditions (1.83)
or (1.85) will be a generating spaser, which clamps the inversion and precludes the
full loss compensation.
Note that for a 2d metamaterial (metasurface), the amplification of the spontaneous
emission and spasing may occur in SPP modes propagating in plane of the structure,
unlike the signal that propagates normally to it as in Ref. [298].
Acknowledgments This work was supported by Grant No. DEFG02-01ER15213 from the Chemical Sciences, Biosciences and Geosciences Division and by Grant No. DE-FG02-11ER46789 from
the Materials Sciences and Engineering Division of the Office of the Basic Energy Sciences, Office
of Science, U.S. Department of Energy.
References
1. M. Moskovits, Surface-enhanced spectroscopy. Rev. Mod. Phys. 57, 783–826 (1985)
2. M.I. Stockman, V.M. Shalaev, M. Moskovits, R. Botet, T.F. George, Enhanced Raman scattering by fractal clusters: scale invariant theory. Phys. Rev. B 46, 2821–2830 (1992)
3. L. Gunnarsson, S. Petronis, B. Kasemo, H. Xu, J. Bjerneld, M. Kall, Optimizing
nanofabricated substrates for surface enhanced Raman scattering. Nanostruct. Mater. 12,
783–788 (1999)
M. I. Stockman
compensation by gain in metamaterials, which is based on the same principles of
quantum amplification as the spaser. This status exists despite a significant effort
in this direction and numerous theoretical publications, e.g., [267, 305]. There has
been so far a single, not yet confirmed independently, observation of the full loss
compensation in a plasmonic metamaterial with gain [298].
In large periodic metamaterials, plasmonic modes generally are propagating
waves (SPPs) that satisfy Bloch theorem [306] and are characterized by quasiwavevector k. These are propagating waves except for the band edges where
ka = ±π , where a is the lattice vector. At the band edges, the group velocity
v g of these modes is zero, and these modes are localized, i.e., they are SPs. Their
wave function is periodic with period 2a, which may be understood as a result of
the Bragg reflection from the crystallographic planes. Within this 2a period, these
band-edge modes can, indeed, be treated quasistatically because 2a ∪ l s , λ. If any
of the band-edge frequencies is within the range of compensation [where the condition (1.83) [or, (1.85)] is satisfied], the system will spase. In fact, at the band edge,
this metamaterial with gain is similar to a distributed feedback (DFB) laser [307].
It actually is a DFB spaser, which, as all the DFB lasers, generates in a band-edge
mode.
Moreover, not only the SPPs, which are exactly at the band edge, will be localized.
Due to unavoidable disorder caused by fabrication defects in metamaterials, there
will be scattering of the SPPs from these defects. Close to the band edge, the group
velocity becomes small, v g → 0. Because the scattering cross section of any wave is
∝ v −2
g , the corresponding SPPs experience Anderson localization [308]. Also, there
always will be SPs nanolocalized at the defects of the metamaterial, whose local
fields are hot spots—see Fig. 1.10 and, generally, Sect. 1.3.5 and the publications
referenced therein. Each of such hot spots within the bandwidth of conditions (1.83)
or (1.85) will be a generating spaser, which clamps the inversion and precludes the
full loss compensation.
Note that for a 2d metamaterial (metasurface), the amplification of the spontaneous
emission and spasing may occur in SPP modes propagating in plane of the structure,
unlike the signal that propagates normally to it as in Ref. [298].
Acknowledgments This work was supported by Grant No. DEFG02-01ER15213 from the Chemical Sciences, Biosciences and Geosciences Division and by Grant No. DE-FG02-11ER46789 from
the Materials Sciences and Engineering Division of the Office of the Basic Energy Sciences, Office
of Science, U.S. Department of Energy.
References
1. M. Moskovits, Surface-enhanced spectroscopy. Rev. Mod. Phys. 57, 783–826 (1985)
2. M.I. Stockman, V.M. Shalaev, M. Moskovits, R. Botet, T.F. George, Enhanced Raman scattering by fractal clusters: scale invariant theory. Phys. Rev. B 46, 2821–2830 (1992)
3. L. Gunnarsson, S. Petronis, B. Kasemo, H. Xu, J. Bjerneld, M. Kall, Optimizing
nanofabricated substrates for surface enhanced Raman scattering. Nanostruct. Mater. 12,
783–788 (1999)