502
A. Karabchevsky et al.
ˆ
P ˆ
T |1 = |l − ie
−θ
|g = ie
−θ
|2,
(18.34)
ˆ
P ˆ
T |2 = |g − ie
θ
|g = −ie
θ
|1.
(18.35)
Typical mode profiles for the coupled waveguide system (see Fig. 18.5a) in PTsymmetric and PT-broken regimes are shown in Fig. 18.3 and in Fig. 18.4.
When the parameter α approaches the value α c = 2κ corresponding to the EP,
eigenmodes |1, 2 merge to become |g + i|l. Interestingly, that the modes become
self-orthogonal as 1|1 = =2|2 = 0 at the EP. Self-orthogonality is responsible for
singularity of Petermann factor due to zero in denominator in (18.21).
18.3 PT-Symmetric Photonic Devices
Photonics is an excellent platform for experimental verification of the fundamental
concept of the parity-time symmetry discussed earlier. Novel photonic devices can be
fabricated using several basic types of PT-symmetric structures, such as waveguides,
multilayer systems and photonic crystals. A number of remarkable applications of
the PT symmetry have been proposed and well studied including unidirectional invisibility, lasing, sensing and coherent perfect absorption. In this section, we overview
the recent PT-symmetric photonic devices with the application perspective.
18.3.1 Coupled Waveguide Systems
Coupled waveguides were the first candidates for observation of the parity-time symmetry. In [19] it was fabricated a gradient-index system with balanced loss and gain,
the gain being guaranteed by the photorefractive nonlinearity of Fe-doped LiNbO 3 .
The detailed description of the physics of coupled waveguides has been provided
in the previous section. Here we discuss a gain-free technique for observation of
the PT symmetry demonstrated in practice in [20]. The idea behind the passive PT
symmetry is to carry out transformation of the fields in a purely lossy system with
the aim of reducing the governing equation to that describing PT-symmetric systems.
In fact, using the gauge transformation |ψ = exp(−γ z)| ˜
ψ, we can rewrite (18.11)
of the passive system as
i
∂
∂z
| ˜
ψ = ˆ ˜
H | ˜
ψ
(18.36)
Now we claim that (18.36) describes a PT-symmetric system with the effective Hamiltonian ˆ ˜
H which corresponds to the system with the effective permittivity
˜
ε(x, y) = ε(x, y) + 2i
n 0 γ
k 0
.
(18.37)
A. Karabchevsky et al.
ˆ
P ˆ
T |1 = |l − ie
−θ
|g = ie
−θ
|2,
(18.34)
ˆ
P ˆ
T |2 = |g − ie
θ
|g = −ie
θ
|1.
(18.35)
Typical mode profiles for the coupled waveguide system (see Fig. 18.5a) in PTsymmetric and PT-broken regimes are shown in Fig. 18.3 and in Fig. 18.4.
When the parameter α approaches the value α c = 2κ corresponding to the EP,
eigenmodes |1, 2 merge to become |g + i|l. Interestingly, that the modes become
self-orthogonal as 1|1 = =2|2 = 0 at the EP. Self-orthogonality is responsible for
singularity of Petermann factor due to zero in denominator in (18.21).
18.3 PT-Symmetric Photonic Devices
Photonics is an excellent platform for experimental verification of the fundamental
concept of the parity-time symmetry discussed earlier. Novel photonic devices can be
fabricated using several basic types of PT-symmetric structures, such as waveguides,
multilayer systems and photonic crystals. A number of remarkable applications of
the PT symmetry have been proposed and well studied including unidirectional invisibility, lasing, sensing and coherent perfect absorption. In this section, we overview
the recent PT-symmetric photonic devices with the application perspective.
18.3.1 Coupled Waveguide Systems
Coupled waveguides were the first candidates for observation of the parity-time symmetry. In [19] it was fabricated a gradient-index system with balanced loss and gain,
the gain being guaranteed by the photorefractive nonlinearity of Fe-doped LiNbO 3 .
The detailed description of the physics of coupled waveguides has been provided
in the previous section. Here we discuss a gain-free technique for observation of
the PT symmetry demonstrated in practice in [20]. The idea behind the passive PT
symmetry is to carry out transformation of the fields in a purely lossy system with
the aim of reducing the governing equation to that describing PT-symmetric systems.
In fact, using the gauge transformation |ψ = exp(−γ z)| ˜
ψ, we can rewrite (18.11)
of the passive system as
i
∂
∂z
| ˜
ψ = ˆ ˜
H | ˜
ψ
(18.36)
Now we claim that (18.36) describes a PT-symmetric system with the effective Hamiltonian ˆ ˜
H which corresponds to the system with the effective permittivity
˜
ε(x, y) = ε(x, y) + 2i
n 0 γ
k 0
.
(18.37)