18 Purcell Effect in PT-Symmetric Waveguides
509
|ψ 1 (z 1 ) =
i
A i,z 1 |i, z 1 ,
|ψ 1 (z n ) =
i
A −i,z n | − i, z n .
(18.46)
Here A i,z 1 and A −i,z n are the amplitudes of the modes propagating forward to port z 1
and backward to port z n , respectively, |i, z 1 , | − i, z n are respectively eigenmodes
of ports z 1 and z n propagating from the cavity.
In our notations the Lorentz reciprocity theorem
δV
(E 1 × H 2 − E 2 × H 1 ) · ˆ
zdxdy =
V
(E 2 · J 1 − E 1 · J 2 ) dV.
(18.47)
should be rewritten as
ψ 1 (z 1 )|ψ 2 (z 1 ) − −ψ 2 (z 1 )|ψ 1 (z 1 ) − −ψ 1 (z n )|ψ 2 (z n ) + +ψ 2 (z n )|ψ 1 (z n )
=
V
(E 2 · J 1 − E 1 · J 2 ) dV, (18.48)
where δV is the surface enclosing the cavity volume V between two planes z = z 1
and z = z n . In (18.48), J 1 and |ψ 1 are defined above, while the source J 2 and the
fields |ψ 2 produced by it can be chosen as we need. Let the source current J 2 , being
outside the volume V (J 2 = 0), excite a single mode | − k, z 1 . In general, this mode
is scattered by the cavity V and creates the set of transmitted and reflected modes as
discussed in [59]:
|ψ 2 (z 1 ) =B −k,z 1 | − k, z 1 +
i
B i,z 1 |i, z 1 ,
(18.49)
|ψ 2 (z n ) =
i
B −i,z n | − i, z n .
(18.50)
Using the orthogonality condition (18.17) and the symmetry relations (18.20) we
obtain the inner products of the fields
ψ 1 (z 1 )|ψ 2 (z 1 ) =
i
A i,z 1 B −k,z 1 i, z 1 | − k, z 1 +
i, j
A i,z 1 B j,z 1 i, z 1 | j, z 1
= −2 A k,z 1 B −k,z 1 N k + 2
i
A i,z 1 B i,z 1 N i,z 1 . (18.51a)