252
M. B. Raschke et al.
In order to describe the radiative emission of a oscillating charge it must be
recognized that the radiation field in turn influences the motion of the charge itself,
termed radiation reaction. Assuming the radiation reaction force F r as the only
damping term, the equation of motion can be written as:
m
d 2 r
dt 2 + ω
2
0 m r = F r = −mΓ 0
dr
dt
=
q 2
6πε 0 c 3
d 3 r
dt 3 ,
(7.17)
with the Abraham-Lorentz equation to describe the reaction force coefficient:
Γ 0 =
1
4πε 0
2q 2 ω 2
0
3mc 3 .
(7.18)
This gives rise to radiative lifetimes τ = 1/Γ 0 20 ns for optical frequencies. An
additional term, conventionally introduced to describe the damping of a Lorentzian
oscillator of the form Γ dr/dt, contains both radiative and non-radiative contributions.
Similarly to Eq.7.17, one can start with the induced optical polarization of the
form:
P(ω) = χ(ω)
E inc + i
2k 3
0
3
P(ω)
(7.19)
with particle susceptibility χ(ω). The second term corresponds to the radiation reaction field with:
F r = e E rad =
2
3
e 2
c 3 ¨
v = i
2
3
ω 3
c 3 ex = i
2
3
k
3 P,
(7.20)
using x = e −iωt and ¨
v = iω 3 x for the harmonic oscillator. Hence, both approaches
are equivalent, with the difference that the damping for the resonant denominator
for χ(ω) already contains the a priori indistinguishable radiative and non-radiative
terms.
Interestingly, the quantum description for the spontaneous emission of a two level
system provides a qualitative intuition for the high radiative emission rate as derived
from Mie theory in the femtosecond regime. The transition rate follows from Fermi’s
golden rule as
Γ sp =
πω 0
3ε 0
||a| ˆ
μ|b|
2
ρ μ (r 0 , ω 0 ),
(7.21)
with transition dipole moment operator ˆ
μ and ρ μ the partial local density of states
(LDOS) at the location r 0 of the system, given by ρ v (ω) = ω 2
0 /π 2 c 3 in vacuum.
With μ ba
2 = ||a| ˆ
μ|b| 2 = q 2 r 2
21 the spontaneous emission rate becomes:
Γ sp =
ω 3
0
3πε 0 c 3 μ ba
2
.
(7.22)
M. B. Raschke et al.
In order to describe the radiative emission of a oscillating charge it must be
recognized that the radiation field in turn influences the motion of the charge itself,
termed radiation reaction. Assuming the radiation reaction force F r as the only
damping term, the equation of motion can be written as:
m
d 2 r
dt 2 + ω
2
0 m r = F r = −mΓ 0
dr
dt
=
q 2
6πε 0 c 3
d 3 r
dt 3 ,
(7.17)
with the Abraham-Lorentz equation to describe the reaction force coefficient:
Γ 0 =
1
4πε 0
2q 2 ω 2
0
3mc 3 .
(7.18)
This gives rise to radiative lifetimes τ = 1/Γ 0 20 ns for optical frequencies. An
additional term, conventionally introduced to describe the damping of a Lorentzian
oscillator of the form Γ dr/dt, contains both radiative and non-radiative contributions.
Similarly to Eq.7.17, one can start with the induced optical polarization of the
form:
P(ω) = χ(ω)
E inc + i
2k 3
0
3
P(ω)
(7.19)
with particle susceptibility χ(ω). The second term corresponds to the radiation reaction field with:
F r = e E rad =
2
3
e 2
c 3 ¨
v = i
2
3
ω 3
c 3 ex = i
2
3
k
3 P,
(7.20)
using x = e −iωt and ¨
v = iω 3 x for the harmonic oscillator. Hence, both approaches
are equivalent, with the difference that the damping for the resonant denominator
for χ(ω) already contains the a priori indistinguishable radiative and non-radiative
terms.
Interestingly, the quantum description for the spontaneous emission of a two level
system provides a qualitative intuition for the high radiative emission rate as derived
from Mie theory in the femtosecond regime. The transition rate follows from Fermi’s
golden rule as
Γ sp =
πω 0
3ε 0
||a| ˆ
μ|b|
2
ρ μ (r 0 , ω 0 ),
(7.21)
with transition dipole moment operator ˆ
μ and ρ μ the partial local density of states
(LDOS) at the location r 0 of the system, given by ρ v (ω) = ω 2
0 /π 2 c 3 in vacuum.
With μ ba
2 = ||a| ˆ
μ|b| 2 = q 2 r 2
21 the spontaneous emission rate becomes:
Γ sp =
ω 3
0
3πε 0 c 3 μ ba
2
.
(7.22)