66
M. I. Stockman
always an essentially nonlinear process that involves a noneqilibrium phase transition
and a spontaneous symmetry breaking: establishment of an arbitrary but sustained
phase of the coherent SP oscillations.
A relevant process is spontaneous emission of SPs by a chromophore into a spasing
SP mode. The corresponding rate γ
( p)
2 for a chromophore at a point r p can be found
in a standard way using the quantized field (1.64) as
γ
( p)
2
= 2
A 2
n
γ n
d 12 ∇ϕ n (r p )
2
(Γ 12 + γ n )
2
(ω 12 − ω n )
2
+ (Γ 12 + γ n )
2
.
(1.71)
As in Schawlow-Towns theory of laser-line width [287], this spontaneous emission
of SPs leads to the diffusion of the phase of the spasing state. This defines width γ s
of the spasing line as
γ s =
⎠
p
⎝
1 + n
( p)
21
⎞
γ
( p)
2
2(2N p + 1)
.
(1.72)
This width is small for a case of developed spasing when N p 1. However, for
N p ∼ 1, the predicted width may be too high because the spectral diffusion theory
assumes that γ s γ n . To take into account this limitation in a simplified way,
we will interpolate to find the resulting spectral width Γ s of the spasing line as
Γ s =
γ −2
n + γ −2
s
−1/2 .
We will also examine the spaser as a bistable (logical) amplifier. One of the ways
to set the spaser in such a mode is to add a saturable absorber. This is described by
the same Eqs. (1.67)–(1.70) where the chromophores belonging to the absorber are
not pumped by the external source directly, i.e., for them in Eq. (1.69) one has to set
g = 0.
Numerical examples are given for a silver nanoshell where the core and the external dielectric have the same permittivity of ε d = 2; the permittivity of silver is adopted
from Ref. [32]. The following realistic parameters of the gain medium are used (unless
indicated otherwise): d 12 = 1.5 × 10 −17 esu, Γ 12 = 10 meV, γ 2 = 4 × 10 12 s −1
(this value takes into account the spontaneous decay into SPs), and density of the
gain medium chromophores is n c = 2.4 × 10 20 cm −3 , which is realistic for dye
molecules but may be somewhat high for semiconductor quantum dots that were
proposed as the chromophores [31] and used in experiments [260]. We will assume a
dipole SP mode and chromophores situated in the core of the nanoshell as shown in
Fig. 1.26d. This configuration are of advantage both functionally (because the region
of the high local fields outside the shell is accessible for various applications) and
computationally (the uniformity of the modal fields makes the summation of the
chromophores trivial, thus greatly facilitating numerical procedures).
M. I. Stockman
always an essentially nonlinear process that involves a noneqilibrium phase transition
and a spontaneous symmetry breaking: establishment of an arbitrary but sustained
phase of the coherent SP oscillations.
A relevant process is spontaneous emission of SPs by a chromophore into a spasing
SP mode. The corresponding rate γ
( p)
2 for a chromophore at a point r p can be found
in a standard way using the quantized field (1.64) as
γ
( p)
2
= 2
A 2
n
γ n
d 12 ∇ϕ n (r p )
2
(Γ 12 + γ n )
2
(ω 12 − ω n )
2
+ (Γ 12 + γ n )
2
.
(1.71)
As in Schawlow-Towns theory of laser-line width [287], this spontaneous emission
of SPs leads to the diffusion of the phase of the spasing state. This defines width γ s
of the spasing line as
γ s =
⎠
p
⎝
1 + n
( p)
21
⎞
γ
( p)
2
2(2N p + 1)
.
(1.72)
This width is small for a case of developed spasing when N p 1. However, for
N p ∼ 1, the predicted width may be too high because the spectral diffusion theory
assumes that γ s γ n . To take into account this limitation in a simplified way,
we will interpolate to find the resulting spectral width Γ s of the spasing line as
Γ s =
γ −2
n + γ −2
s
−1/2 .
We will also examine the spaser as a bistable (logical) amplifier. One of the ways
to set the spaser in such a mode is to add a saturable absorber. This is described by
the same Eqs. (1.67)–(1.70) where the chromophores belonging to the absorber are
not pumped by the external source directly, i.e., for them in Eq. (1.69) one has to set
g = 0.
Numerical examples are given for a silver nanoshell where the core and the external dielectric have the same permittivity of ε d = 2; the permittivity of silver is adopted
from Ref. [32]. The following realistic parameters of the gain medium are used (unless
indicated otherwise): d 12 = 1.5 × 10 −17 esu, Γ 12 = 10 meV, γ 2 = 4 × 10 12 s −1
(this value takes into account the spontaneous decay into SPs), and density of the
gain medium chromophores is n c = 2.4 × 10 20 cm −3 , which is realistic for dye
molecules but may be somewhat high for semiconductor quantum dots that were
proposed as the chromophores [31] and used in experiments [260]. We will assume a
dipole SP mode and chromophores situated in the core of the nanoshell as shown in
Fig. 1.26d. This configuration are of advantage both functionally (because the region
of the high local fields outside the shell is accessible for various applications) and
computationally (the uniformity of the modal fields makes the summation of the
chromophores trivial, thus greatly facilitating numerical procedures).