18 Purcell Effect in PT-Symmetric Waveguides
503
˜
ε satisfies condition ˜
ε(x, y) = ˜
ε
∗
(−x, −y) and the permittivity of the passive system
meets
ε(x, y) = ε
∗
(−x, −y) − 4i
n 0 γ
k 0
.
(18.38)
Assuming that one of the waveguides is transparent (lossless) with ε g = ε
∗
g , one can
easily determine the permittivity of the lossy waveguide as ε l = ε g − 4in 0 γ /k 0 . In
spite of the gain-free system does not have a true PT symmetry, it still possesses
some features inherent in PT-symmetric systems as PT-symmetry breaking observed
in [20]. Passive PT symmetry is a smart technique to ease validation of PT-symmetry
effects and its applicability.
A number of surprising effects arise in guiding systems under gain-loss modulation in a PT-symmetric manner. The modulation shifts positions of exceptional
points resulting in the directional amplification, when the phase transition is made
with a threshold in one direction, being thresholdless in the opposite direction [21].
Similarly, on the boundary between the metallic substrate and PT-symmetrically
modulated dielectric slab, there is a unidirectional excitation of the surface plasmonpolaritons [22]. In a waveguide as an isotropic slab sandwiched between oppositely
biased gyrotropic media, topologically protected guided modes arise. PT symmetry
in such a system introduces exceptional points, where electromagnetic modes are
slow-light and linearly-growing [23]. Slow light phenomenon is known to be associated with degeneracy of the modes (matching of their propagation constants). In
PT-symmetric systems, the degeneration is realised at exceptional points of mode
coalescence [24].
PT-symmetric laser waveguide was fabricated in [25]. Gain and loss are electrically controlled to achieve a lasing threshold in the range of PT-symmetry violation.
By interplay of two guided modes there were distinguished several phases, the lasing
within which being confirmed experimentally.
Two coupled waveguides experience optical forces originating from excited eigenmodes. The forces qualitatively change at exceptional points and may result in pushing and pulling of one waveguide to another. When the forces deflect the waveguides, they may induce the phase transition through changing a gap between them
[26]. An unusual power flow in the PT-symmetric coupled waveguide results in an
off-diagonal stress tensor components causing the shear along the mode propagation
direction [27].
18.3.2 Two-Dimensional Photonic Waveguide Lattices
An array of parallel waveguides can be arranged in nodes of a lattice as demonstrated
in Fig. 18.5b. Such a two-dimensional photonic crystal is a natural generalization of
a pair of coupled waveguides. To engage a PT symmetry in the lattice, the gain and
lossy waveguides should be disposed periodically. As other PT-symmetric systems,
the lattice has exceptional points bordering phases of broken and unbroken PT-
503
˜
ε satisfies condition ˜
ε(x, y) = ˜
ε
∗
(−x, −y) and the permittivity of the passive system
meets
ε(x, y) = ε
∗
(−x, −y) − 4i
n 0 γ
k 0
.
(18.38)
Assuming that one of the waveguides is transparent (lossless) with ε g = ε
∗
g , one can
easily determine the permittivity of the lossy waveguide as ε l = ε g − 4in 0 γ /k 0 . In
spite of the gain-free system does not have a true PT symmetry, it still possesses
some features inherent in PT-symmetric systems as PT-symmetry breaking observed
in [20]. Passive PT symmetry is a smart technique to ease validation of PT-symmetry
effects and its applicability.
A number of surprising effects arise in guiding systems under gain-loss modulation in a PT-symmetric manner. The modulation shifts positions of exceptional
points resulting in the directional amplification, when the phase transition is made
with a threshold in one direction, being thresholdless in the opposite direction [21].
Similarly, on the boundary between the metallic substrate and PT-symmetrically
modulated dielectric slab, there is a unidirectional excitation of the surface plasmonpolaritons [22]. In a waveguide as an isotropic slab sandwiched between oppositely
biased gyrotropic media, topologically protected guided modes arise. PT symmetry
in such a system introduces exceptional points, where electromagnetic modes are
slow-light and linearly-growing [23]. Slow light phenomenon is known to be associated with degeneracy of the modes (matching of their propagation constants). In
PT-symmetric systems, the degeneration is realised at exceptional points of mode
coalescence [24].
PT-symmetric laser waveguide was fabricated in [25]. Gain and loss are electrically controlled to achieve a lasing threshold in the range of PT-symmetry violation.
By interplay of two guided modes there were distinguished several phases, the lasing
within which being confirmed experimentally.
Two coupled waveguides experience optical forces originating from excited eigenmodes. The forces qualitatively change at exceptional points and may result in pushing and pulling of one waveguide to another. When the forces deflect the waveguides, they may induce the phase transition through changing a gap between them
[26]. An unusual power flow in the PT-symmetric coupled waveguide results in an
off-diagonal stress tensor components causing the shear along the mode propagation
direction [27].
18.3.2 Two-Dimensional Photonic Waveguide Lattices
An array of parallel waveguides can be arranged in nodes of a lattice as demonstrated
in Fig. 18.5b. Such a two-dimensional photonic crystal is a natural generalization of
a pair of coupled waveguides. To engage a PT symmetry in the lattice, the gain and
lossy waveguides should be disposed periodically. As other PT-symmetric systems,
the lattice has exceptional points bordering phases of broken and unbroken PT-