18 Purcell Effect in PT-Symmetric Waveguides
505
direct connection of the scattering matrix ˆ
S with the PT-symmetric Hamiltonian ˆ
H of
the one-dimensional multilayer system as ˆ
S = exp(i ˆ
H (t − t 0 )/). Correct positions
of the exceptional points then read as s 1,2 = t ±
√
r L r R , where the scattering matrix
defined by the right-hand equation in (18.39) is employed. Exceptional points of
the scattering matrix with permuted channels given by the another scattering matrix
approximate the lasing onset.
PT-symmetric multilayer systems are widely studied with the aim of enhancement
of physical phenomena near exceptional points. A PT-symmetric bilayer exhibits
a giant Goos-Hänhen shift at specific angles of incidence [36]. The enhancement
is explained by excitation of surface modes at the interface between the gain and
lossy slabs facilitating the quasi-BIC (quasi-bound state in the continuum) states. An
ordinarily weak spin-orbit interaction of light can be also significantly intensified
in vicinity of exceptional points of the PT-symmetric bilayer. Interaction of light
spin and orbital momenta is coined as the spin Hall or Imbert-Fedorov effect and
results in a lateral shift of a light beam. Near exceptional points of the PT-symmetric
bilayer the reflection coefficient experiences negligible values and abrupt phase shift
enhancing the lateral beam displacement, though it takes zero value at the exceptional
point [37]. Graphene sheets in PT-symmetric multilayer systems can be used for
modulation of an exceptional point position via tuning their surface conductivity
[38]. Light also makes a bilayer to move. Optical forces exerting on PT-symmetric
multilayer structures can be both pushing and pulling depending on the direction of
light and realization of the broken or unbroken PT-symmetric phase [39].
A PT-symmetric multilayer structure can be used as a laser. More precisely, the
laser must be simultaneously a coherent perfect absorber [40, 41]. In the broken
PT-symmetry phase, an illumination from one side is normally amplified, while a
coherent illumination from both sides of the multilayer system is absorbed owing
to interference. Laser-absorber modes arise, when a pole and a zero of a scattering matrix approach each other on the real axis in the plane of complex frequency.
Finite-difference-time-domain (FDTD) calculations basically confirm predictions of
the transfer-matrix and scattering-matrix approaches and show an enormous increase
in the output intensity, when the laser threshold is achieved [35]. If the pole is not on
the real axis, the lasing onset is still feasible at a greater threshold. In realistic PTsymmetric systems, realization of the gain occurs in quantum systems and requires
accounting for the saturation effect. This means that the PT-symmetric system has
to be nonlinear. In [42] it was considered a two-level resonant medium described
by the Maxwell-Bloch equations. Due to the saturation, the condition for PT symmetry is approximately valid and the system can be named as a non-Hermitian one
[43]. Saturation may result in novel effects, e.g., locking of the light propagation by
the PT-symmetry breaking [42]. Steady-state solutions for PT-symmetric multilayer
structures with nonlinear refractive indices of gain and loss media are investigated
in [44] using a modified transfer-matrix method. The bistable behavior of the transmitted and reflected intensities was studied together with unidirectional invisibility
and coherent perfect absorption versus the input and saturation intensities.
505
direct connection of the scattering matrix ˆ
S with the PT-symmetric Hamiltonian ˆ
H of
the one-dimensional multilayer system as ˆ
S = exp(i ˆ
H (t − t 0 )/). Correct positions
of the exceptional points then read as s 1,2 = t ±
√
r L r R , where the scattering matrix
defined by the right-hand equation in (18.39) is employed. Exceptional points of
the scattering matrix with permuted channels given by the another scattering matrix
approximate the lasing onset.
PT-symmetric multilayer systems are widely studied with the aim of enhancement
of physical phenomena near exceptional points. A PT-symmetric bilayer exhibits
a giant Goos-Hänhen shift at specific angles of incidence [36]. The enhancement
is explained by excitation of surface modes at the interface between the gain and
lossy slabs facilitating the quasi-BIC (quasi-bound state in the continuum) states. An
ordinarily weak spin-orbit interaction of light can be also significantly intensified
in vicinity of exceptional points of the PT-symmetric bilayer. Interaction of light
spin and orbital momenta is coined as the spin Hall or Imbert-Fedorov effect and
results in a lateral shift of a light beam. Near exceptional points of the PT-symmetric
bilayer the reflection coefficient experiences negligible values and abrupt phase shift
enhancing the lateral beam displacement, though it takes zero value at the exceptional
point [37]. Graphene sheets in PT-symmetric multilayer systems can be used for
modulation of an exceptional point position via tuning their surface conductivity
[38]. Light also makes a bilayer to move. Optical forces exerting on PT-symmetric
multilayer structures can be both pushing and pulling depending on the direction of
light and realization of the broken or unbroken PT-symmetric phase [39].
A PT-symmetric multilayer structure can be used as a laser. More precisely, the
laser must be simultaneously a coherent perfect absorber [40, 41]. In the broken
PT-symmetry phase, an illumination from one side is normally amplified, while a
coherent illumination from both sides of the multilayer system is absorbed owing
to interference. Laser-absorber modes arise, when a pole and a zero of a scattering matrix approach each other on the real axis in the plane of complex frequency.
Finite-difference-time-domain (FDTD) calculations basically confirm predictions of
the transfer-matrix and scattering-matrix approaches and show an enormous increase
in the output intensity, when the laser threshold is achieved [35]. If the pole is not on
the real axis, the lasing onset is still feasible at a greater threshold. In realistic PTsymmetric systems, realization of the gain occurs in quantum systems and requires
accounting for the saturation effect. This means that the PT-symmetric system has
to be nonlinear. In [42] it was considered a two-level resonant medium described
by the Maxwell-Bloch equations. Due to the saturation, the condition for PT symmetry is approximately valid and the system can be named as a non-Hermitian one
[43]. Saturation may result in novel effects, e.g., locking of the light propagation by
the PT-symmetry breaking [42]. Steady-state solutions for PT-symmetric multilayer
structures with nonlinear refractive indices of gain and loss media are investigated
in [44] using a modified transfer-matrix method. The bistable behavior of the transmitted and reflected intensities was studied together with unidirectional invisibility
and coherent perfect absorption versus the input and saturation intensities.