18 Purcell Effect in PT-Symmetric Waveguides
513
18.4.2 Modal Purcell Factor Within the Coupled Mode
Theory
PT-symmetric regime
To find the modal Purcell factor for the coupled waveguide system in PT-symmetric
regime we substitute the modes in the form (18.29) into expression (18.64).
One more assumption is introduced for the sake of simplicity:
g|g
∗
= =l|l
∗
= 1.
(18.72)
It implies that the Hermitian norms of the isolated modes are equal to the nonHermitian norms or, in other words, the Petermann factors for the modes equal unity.
Then the quantities K kl and ˆ
P kl can be written in the closed form as
K 1 =
||1|1
∗
|
2
||1|1| 2 =
2
1 + cos 2θ
,
(18.73a)
K 2 =
||2|2
∗
|
2
||2|2| 2 =
2
1 + cos 2θ
,
(18.73b)
K 12 =
1|1
∗
1|1
2|2
∗
∗
2|2 ∗ =
2(1 + e
−i2θ
)
2
(1 + cos 2θ) 2 ,
(18.73c)
K 21 =
2|2
∗
2|2
1|1
∗
∗
1|1 ∗ =
2(1 + e
+i2θ
)
2
(1 + cos 2θ) 2 ,
(18.73d)
ˆ
P 12 =
1|2
∗
√
1|1 ∗ 2|2 ∗
=
1
2
(1 − e
i2θ
),
(18.74a)
ˆ
P 21 =
2|1
∗
√
1|1 ∗ 2|2 ∗
=
1
2
(1 − e
−i2θ
).
(18.74b)
Normalised field projections ˆ
e p,k in the basis of isolated modes read
ˆ
e p,1 =
1
1
2
1|1 ∗
( ˆ
e p,g + e
iθ
ˆ
e p,l ) = ˆ
e p,g + e
iθ
ˆ
e p,l ,
(18.75a)
ˆ
e p,2 =
1
1
2
2|2 ∗
( ˆ
e p,g − e
−iθ
ˆ
e p,l ) = ˆ
e p,g − e
−iθ
ˆ
e p,l .
(18.75b)
In above expressions ˆ
e p,g and ˆ
e p,l denote projections of the fields of backwardpropagating isolated modes onto dipole orientation. If the emitter dipole moment is
perpendicular to ˆ
z, projections of backward-propagating modal fields are equal to
the projections of forward-propagating ones.