18 Purcell Effect in PT-Symmetric Waveguides
497
components of SVE fields e t and h t , hence, satisfy first-order equations with respect
to the z derivative. The above can be summarised in a Schrödinger-like equation
i
∂
∂z
|ψ = ˆ
H |ψ,
(18.11)
for an optical state-vector |ψ defined as
|ψ =
e t
h t
.
(18.12)
ˆ
H is an optical Hamiltonian governing the z-evolution of SVE fields. Generally, ˆ
H
is represented by a 4 × 4 matrix joining operators describing evolution of e t and h t .
Explicit form of these operators found in [7]. For waveguide structures with very
small index contrast in both transverse directions equation (18.11) can be reduced
to a scalar equation. Within the scalar approximation the Hamiltonian ˆ
H takes the
form
ˆ
H =
1
2k 0 n 0
∂
2
∂ x 2 +
∂
2
∂ y 2 + V (x, y)
.
(18.13)
Quantity
V (x, y) = k
2
0
ε(x, y) − n
2
0
(18.14)
can be associated with a potential of the Schödinger equation. From the condition
of the PT symmetry in quantum mechanics V (x, y) = V
∗
(−x, −y) we arrive at the
similar condition in optics ε(x, y) = ε
∗
(−x, −y). Therefore, in optical systems the
PT symmetry can be established by judiciously incorporating gain and loss. Thus,
the refractive index profile now plays the role of the complex potential.
18.2.3 Inner Product for PT-Symmetric Optical Systems
We define the inner product as a cross product of the bra-electric and ket-magnetic
fields integrated over the cross-section z = const:
φ 1 |φ 2 ≡
(E 1 × H 2 ) · ˆ
zdxdy
(18.15)
Such a definition is justified by the non-Hermitian nature of PT-symmetric systems.
In the above and following relations we can drop t subscripts because z component of
the vector products depends only on transverse components and
E t,1 × H t,2
· ˆ
z =
(E 1 × H 2 ) · ˆ
z.
It is well known that the modes of Hermitian systems are orthogonal in the sense
497
components of SVE fields e t and h t , hence, satisfy first-order equations with respect
to the z derivative. The above can be summarised in a Schrödinger-like equation
i
∂
∂z
|ψ = ˆ
H |ψ,
(18.11)
for an optical state-vector |ψ defined as
|ψ =
e t
h t
.
(18.12)
ˆ
H is an optical Hamiltonian governing the z-evolution of SVE fields. Generally, ˆ
H
is represented by a 4 × 4 matrix joining operators describing evolution of e t and h t .
Explicit form of these operators found in [7]. For waveguide structures with very
small index contrast in both transverse directions equation (18.11) can be reduced
to a scalar equation. Within the scalar approximation the Hamiltonian ˆ
H takes the
form
ˆ
H =
1
2k 0 n 0
∂
2
∂ x 2 +
∂
2
∂ y 2 + V (x, y)
.
(18.13)
Quantity
V (x, y) = k
2
0
ε(x, y) − n
2
0
(18.14)
can be associated with a potential of the Schödinger equation. From the condition
of the PT symmetry in quantum mechanics V (x, y) = V
∗
(−x, −y) we arrive at the
similar condition in optics ε(x, y) = ε
∗
(−x, −y). Therefore, in optical systems the
PT symmetry can be established by judiciously incorporating gain and loss. Thus,
the refractive index profile now plays the role of the complex potential.
18.2.3 Inner Product for PT-Symmetric Optical Systems
We define the inner product as a cross product of the bra-electric and ket-magnetic
fields integrated over the cross-section z = const:
φ 1 |φ 2 ≡
(E 1 × H 2 ) · ˆ
zdxdy
(18.15)
Such a definition is justified by the non-Hermitian nature of PT-symmetric systems.
In the above and following relations we can drop t subscripts because z component of
the vector products depends only on transverse components and
E t,1 × H t,2
· ˆ
z =
(E 1 × H 2 ) · ˆ
z.
It is well known that the modes of Hermitian systems are orthogonal in the sense