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A. Karabchevsky et al.
Performing calculation of the modal Purcell factor (18.64) using relations (18.73)–
(18.74) we obtain
F p = F p,diag + F p,off−diag =
6π c
ω 2 μ 0
(|ˆ e p,g |
2
+ |ˆ e p,l |
2
).
(18.76)
Diagonal and off-diagonal terms separately take the form
F p,diag =
3π c
4ω 2 μ 0
4
1 + cos 2θ
(|ˆ e p,g |
2
+ |ˆ e p,l |
2
),
(18.77a)
F p,off−diag = −
3π c
4ω 2 μ 0
2(1 − cos 2θ)
1 + cos 2θ
(|ˆ e p,g |
2
+ |ˆ e p,l |
2
).
(18.77b)
It is curious that although both diagonal and off-diagonal terms (18.77) are singular
at the EP corresponding to θ E P = π/2 and cos 2θ E P = −1, the singularities cancel
each other making the modal Purcell factor finite and independent of θ . The modal
Purcell factor (18.76) depends solely on the mode profiles of the isolated modes in
PT-symmetric regime.
PT-symmetry-broken regime
To obtain the modal Purcell factor in PT-symmetry-broken regime we substitute the
modes in the form (18.33) into expression (18.64).
Calculating the Petermann factors
K 1 = coth
2
θ,
(18.78a)
K 2 = coth
2
θ,
(18.78b)
K 12 = − coth
2
θ,
(18.78c)
K 21 = − coth
2
θ,
(18.78d)
normalised cross-powers
ˆ
P 12 =
1
cosh θ
,
(18.79a)
ˆ
P 21 =
1
cosh θ
,
(18.79b)
and reciprocal modal field projections
ˆ
e p,1 =
1
1
2
(1 + e −2θ )
( ˆ
e p,g + ie
−θ
ˆ
e p,l ),
(18.80a)
ˆ
e p,2 =
1
1
2
(1 + e 2θ )
( ˆ
e p,g + ie
θ
ˆ
e p,l )
(18.80b)