86
M. I. Stockman
we calculate from Eq. (1.104) a point shown by the magenta solid circle in Fig. 1.31a,
which is significantly above the threshold. Because in such a nanostructure the local
fields are very non-uniform and confined near the metal similar to the spaser, they
likewise cause a feedback. The condition of Eq. (1.95) is likely to be well-satisfied
for Ref. [298]. Thus, the system may spase, which would cause the clamping of
inversion and loss of gain.
In contrast to these theoretical arguments, there is no evidence of spasing indicated
in the experiment [298], which can be explained by various factors. Among them,
the system of Ref. [298] is a gain-plasmonic nanofilm and not a true 3d material.
This system is not isotropic. Also, the size of the unit cell a ≈ 280 nm is significantly
greater than the reduced wavelength λ, which violates the quasistatic conditions and
makes the possibility of homogenization and considering this system as an optical
metamaterial problematic. This circumstance may lead to an appreciable spatial
dispersion. It may also cause a significant radiative loss and prevent spasing for
some modes.
We would also like to point out that the fact that the unit cell of the negativerefracting (or, double-negative) metamaterial of Ref. [298] is relatively large, a ≈
280 nm, is not accidental. As follows from theoretical consideration of Ref. [300],
optical magnetism and, consequently, negative refraction for metals is only possible
if the minimum scale of the conductor feature (the diameter d of the nanowire)
is greater then the skin depth, d l s ≈ 25 nm, which allows one to circumvent
Landau-Lifshitz’s limitation on the existence of optical magnetism [30, 300]. Thus,
a ring-type resonator structure would have a size 2l s (two wires forming a loop)
and still the same diameter for the hole in the center, which comes to the total
of 4l s ≈ 100 nm. Leaving the same distance between the neighboring resonator
wires, we arrive at an estimate of the size of the unit cell a 8l s = 200 nm, which is,
indeed, the case for Ref. [298] and other negative-refraction “metamaterials” in the
optical region. This makes our theory not directly applicable to them. Nevertheless,
if the spasing condition (1.83) [or (1.85), or (1.104)] is satisfied, the system still may
spase on the hot-spot defect modes.
In an experimental study of the lasing spaser [260], a nanofilm of PbS quantum
dots (QDs) was positioned over a two-dimensional metamaterial consisting of an
array of negative split ring resonators. When the QDs were optically pumped, the
system exhibited an increase of the transmitted light intensity on the background of a
strong luminescence of the QDs but apparently did not reach the lasing threshold. The
polarization-dependent loss compensation was only ∼1 %. Similarly, for an array of
split ring resonators over a resonant quantum well, where the inverted electron-hole
population was excited optically [301], the loss compensation did not exceed ∼8 %.
The relatively low loss compensation in these papers may be due either to random
spasing and/or spontaneous or amplified spontaneous emission enhanced by this
plasmonic array, which reduces the population inversion.
A dramatic example of possible random spasing is presented in Ref. [262]. The
system studied was a Kretschmann-geometry SPP setup [302] with an added ∼1 µm
polymer film containing Rodamine 6G dye in the n c = 1.2 × 10 19 cm −3 concentration. When the dye was pumped, there was outcoupling of radiation in a range
M. I. Stockman
we calculate from Eq. (1.104) a point shown by the magenta solid circle in Fig. 1.31a,
which is significantly above the threshold. Because in such a nanostructure the local
fields are very non-uniform and confined near the metal similar to the spaser, they
likewise cause a feedback. The condition of Eq. (1.95) is likely to be well-satisfied
for Ref. [298]. Thus, the system may spase, which would cause the clamping of
inversion and loss of gain.
In contrast to these theoretical arguments, there is no evidence of spasing indicated
in the experiment [298], which can be explained by various factors. Among them,
the system of Ref. [298] is a gain-plasmonic nanofilm and not a true 3d material.
This system is not isotropic. Also, the size of the unit cell a ≈ 280 nm is significantly
greater than the reduced wavelength λ, which violates the quasistatic conditions and
makes the possibility of homogenization and considering this system as an optical
metamaterial problematic. This circumstance may lead to an appreciable spatial
dispersion. It may also cause a significant radiative loss and prevent spasing for
some modes.
We would also like to point out that the fact that the unit cell of the negativerefracting (or, double-negative) metamaterial of Ref. [298] is relatively large, a ≈
280 nm, is not accidental. As follows from theoretical consideration of Ref. [300],
optical magnetism and, consequently, negative refraction for metals is only possible
if the minimum scale of the conductor feature (the diameter d of the nanowire)
is greater then the skin depth, d l s ≈ 25 nm, which allows one to circumvent
Landau-Lifshitz’s limitation on the existence of optical magnetism [30, 300]. Thus,
a ring-type resonator structure would have a size 2l s (two wires forming a loop)
and still the same diameter for the hole in the center, which comes to the total
of 4l s ≈ 100 nm. Leaving the same distance between the neighboring resonator
wires, we arrive at an estimate of the size of the unit cell a 8l s = 200 nm, which is,
indeed, the case for Ref. [298] and other negative-refraction “metamaterials” in the
optical region. This makes our theory not directly applicable to them. Nevertheless,
if the spasing condition (1.83) [or (1.85), or (1.104)] is satisfied, the system still may
spase on the hot-spot defect modes.
In an experimental study of the lasing spaser [260], a nanofilm of PbS quantum
dots (QDs) was positioned over a two-dimensional metamaterial consisting of an
array of negative split ring resonators. When the QDs were optically pumped, the
system exhibited an increase of the transmitted light intensity on the background of a
strong luminescence of the QDs but apparently did not reach the lasing threshold. The
polarization-dependent loss compensation was only ∼1 %. Similarly, for an array of
split ring resonators over a resonant quantum well, where the inverted electron-hole
population was excited optically [301], the loss compensation did not exceed ∼8 %.
The relatively low loss compensation in these papers may be due either to random
spasing and/or spontaneous or amplified spontaneous emission enhanced by this
plasmonic array, which reduces the population inversion.
A dramatic example of possible random spasing is presented in Ref. [262]. The
system studied was a Kretschmann-geometry SPP setup [302] with an added ∼1 µm
polymer film containing Rodamine 6G dye in the n c = 1.2 × 10 19 cm −3 concentration. When the dye was pumped, there was outcoupling of radiation in a range