74
M. I. Stockman
Starting with the stationary regime, we assume that the pumping at a rate (per
one chromophore) of g = 5 × 10 12 s −1 starts at a moment of time t = 0 and stays
constant after that. Immediately at t = 0, a certain number of SPs are injected into
the spaser. We are interested in its temporal dynamics from this moment on.
The dynamical behavior of the spaser under this pumping regime is illustrated in
Figs. 1.30a, b. As we see, the spaser, which starts from an arbitrary initial population
N n , rather rapidly, within a few hundred femtoseconds approaches the same stationary (“logical”) level. At this level, an SP population of N n = 67 is established, while
the inversion is clamped at a low level of n 21 = 0.02. On the way to this stationary state, the spaser experiences relaxation oscillations in both the SP numbers and
inversion, which have a trend to oscillate out of phase (compare panels a and b). This
temporal dynamics of the spaser is quite complicated and highly nonlinear (unharmonic). It is controlled not by a single relaxation time but by a set of the relaxation
rates. Clearly, among these are the energy transfer rate from the gain medium to the
SPs and the relaxation rates of the SPs and the chromophores.
In this mode, the main effect of the initial injection of the SPs (described theoretically as different initial values of N n ) is in the interval of time it is required for
the spaser to reach the final (CW) state. For very small N n , which in practice can
be supplied by the noise of the spontaneous SP emission into the mode, this time is
approximately 250 fs (cf.: the corresponding SP relaxation time is less then 50 fs). In
contrast, for the initial values of N n = 1–5, this time shortens to 150 fs.
Now consider the second regime: pulse pumping. The gain-medium population
of the spaser is inverted at t = 0 to saturation with a short (much shorter than 100 fs)
pump pulse. Simultaneously, at t = 0, some number of plasmons are injected (say,
by an external nanoplasmonic circuitry). In response, the spaser should produce an
amplified pulse of the SP excitation. Such a function of the spaser is illustrated in
Figs. 1.30c, d.
As we see from panel (c), independently from the initial number of SPs, the spaser
always generates a series of SP pulses, of which only the first pulse is large (at or
above the logical level of N n ∼ 100). (An exception is a case of little practical
importance when the initial N n = 120 exceeds this logical level, when two large
pulses are produced.) The underlying mechanism of such a response is the rapid
depletion of the inversion seen in panel (d), where energy is dissipated in the metal
of the spaser. The characteristic duration of the SP pulse ∼100 fs is defined by this
depletion, controlled by the energy transfer and SP relaxation rates. This time is
much shorter than the spontaneous decay time of the gain medium. This acceleration
is due to the stimulated emission of the SPs into the spasing mode (which can be
called a “stimulated Purcell effect”). There is also a pronounced trend: the lower is
initial SP population N n , the later the spaser produces the amplified pulse. In a sense,
this spaser functions as a pulse-amplitude to time-delay converter.
M. I. Stockman
Starting with the stationary regime, we assume that the pumping at a rate (per
one chromophore) of g = 5 × 10 12 s −1 starts at a moment of time t = 0 and stays
constant after that. Immediately at t = 0, a certain number of SPs are injected into
the spaser. We are interested in its temporal dynamics from this moment on.
The dynamical behavior of the spaser under this pumping regime is illustrated in
Figs. 1.30a, b. As we see, the spaser, which starts from an arbitrary initial population
N n , rather rapidly, within a few hundred femtoseconds approaches the same stationary (“logical”) level. At this level, an SP population of N n = 67 is established, while
the inversion is clamped at a low level of n 21 = 0.02. On the way to this stationary state, the spaser experiences relaxation oscillations in both the SP numbers and
inversion, which have a trend to oscillate out of phase (compare panels a and b). This
temporal dynamics of the spaser is quite complicated and highly nonlinear (unharmonic). It is controlled not by a single relaxation time but by a set of the relaxation
rates. Clearly, among these are the energy transfer rate from the gain medium to the
SPs and the relaxation rates of the SPs and the chromophores.
In this mode, the main effect of the initial injection of the SPs (described theoretically as different initial values of N n ) is in the interval of time it is required for
the spaser to reach the final (CW) state. For very small N n , which in practice can
be supplied by the noise of the spontaneous SP emission into the mode, this time is
approximately 250 fs (cf.: the corresponding SP relaxation time is less then 50 fs). In
contrast, for the initial values of N n = 1–5, this time shortens to 150 fs.
Now consider the second regime: pulse pumping. The gain-medium population
of the spaser is inverted at t = 0 to saturation with a short (much shorter than 100 fs)
pump pulse. Simultaneously, at t = 0, some number of plasmons are injected (say,
by an external nanoplasmonic circuitry). In response, the spaser should produce an
amplified pulse of the SP excitation. Such a function of the spaser is illustrated in
Figs. 1.30c, d.
As we see from panel (c), independently from the initial number of SPs, the spaser
always generates a series of SP pulses, of which only the first pulse is large (at or
above the logical level of N n ∼ 100). (An exception is a case of little practical
importance when the initial N n = 120 exceeds this logical level, when two large
pulses are produced.) The underlying mechanism of such a response is the rapid
depletion of the inversion seen in panel (d), where energy is dissipated in the metal
of the spaser. The characteristic duration of the SP pulse ∼100 fs is defined by this
depletion, controlled by the energy transfer and SP relaxation rates. This time is
much shorter than the spontaneous decay time of the gain medium. This acceleration
is due to the stimulated emission of the SPs into the spasing mode (which can be
called a “stimulated Purcell effect”). There is also a pronounced trend: the lower is
initial SP population N n , the later the spaser produces the amplified pulse. In a sense,
this spaser functions as a pulse-amplitude to time-delay converter.