The mathematical notion of spectral singularity admits a description in terms of purely outgoing solutions of a corresponding linear wave equation. This leads to a nonlinear generalization of this notion for nonlinearities that are confined in space. We examine the nonlinear spectral singularities in arbitrary TE and TM modes of a mirrorless slab laser that involves a weak Kerr nonlinearity. This provides a computational scheme for the determination of the laser output intensity I for these modes. In particular, we offer an essentially mathematical derivation of the linear-dependence of I on the gain coefficient g and obtain an explicit analytic expression for its slope. This shows that if the real part η of the refractive index of the slab does not exceed 3, there is a lower bound on θ below which lasing in both its TE and TM modes requires η to be shifted by a small amount as g surpasses the threshold gain. Our results suggest that lasing in the oblique TM modes of the slab is forbidden if the incidence (emission) angle of the TM mode exceeds the Brewster's angle.

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We investigate linear and nonlinear spectral singularities in the transverse electric and transverse magnetic modes of a slab laser consisting of an active planar slab sandwiched between a pair of graphene or Weyl semimetal thin sheets. The requirement of the presence of linear spectral singularities gives the laser threshold condition while the existence of nonlinear spectral singularities due to an induced weak Kerr nonlinearity allows for computing the laser output intensity in the vicinity of the threshold. The presence of the graphene and Weyl semimetal sheets introduces additional physical parameters that we can use to tune the output intensity of the laser. We provide a comprehensive study of this phenomenon and report peculiarities of lasing in the transverse magnetic (TM) modes of the slab with Weyl semimetal coatings. In particular, we reveal the existence of a critical angle such that no lasing seems possible for TM modes of the slab with the smaller emission angle. Our results suggest that for TM modes with an emission angle slightly exceeding the critical angle, the laser output intensity becomes highly sensitive to the physical parameters of the coating.

We use blow-up solutions of nonlinear Helmholtz equations to introduce a nonlinear resonance effect that is capable of amplifying electromagnetic waves of particular intensity. To achieve this, we propose a scattering setup consisting of a Kerr slab with a negative (defocusing) Kerr constant placed to the left of a linear slab in such a way that a left-incident coherent TE wave with a specific incidence angle and intensity realizes a blow-up solution of the corresponding Helmholtz equation whenever its wavenumber k takes a certain critical value, k⋆. For k = k⋆, the solution blows up at the right-hand boundary of the Kerr slab. For k < k⋆, the setup defines a scattering system with a transmission coefficient that diverges as (k − k⋆) −4 for k → k⋆. By tuning the distance between the slabs we can use this setup to amplify coherent waves with a wavelength in an extremely narrow spectral band. For nearby wavelengths the setup serves as a filter. Our analysis makes use of a nonlinear generalization of the transfer matrix of the scattering theory as well as properties of unidirectionally invisible potentials.

The non-Hermitian models, which are symmetric under parity (P) and time-reversal (T) operators, are the cornerstone for the fabrication of new ultra-sensitive optoelectronic devices. However, providing the gain in such systems usually demands precise control of nonlinear processes, limiting their application. In this paper, to bypass this obstacle, we introduce a class of time-dependent non-Hermitian Hamiltonians (not necessarily Floquet) that can describe a two-level system with temporally modulated on-site potential and couplings. We show that implementing an appropriate non-Unitary gauge transformation converts the original system to an effective one with a balanced gain and loss. This will allow us to derive the evolution of states analytically. Our proposed class of Hamiltonians can be employed in different platforms such as electronic circuits, acoustics, and photonics to design structures with hidden PT-symmetry potentially without imaginary onsite amplification and absorption mechanism to obtain an exceptional point.