We demonstrate the presence of parity-time (PT) symmetry for the non-Hermitian two-state Hamiltonian of a dissipative microwave billiard in the vicinity of an exceptional point (EP). The shape of the billiard depends on two parameters. The Hamiltonian is determined from the measured resonance spectrum on a fine grid in the parameter plane. After applying a purely imaginary diagonal shift to the Hamiltonian, its eigenvalues are either real or complex conjugate on a curve, which passes through the EP. An appropriate basis choice reveals its PT symmetry. Spontaneous symmetry breaking occurs at the EP.

We present measurements of transmission and reflection spectra of a microwave photonic crystal composed of 874 metallic cylinders arranged in a triangular lattice. The spectra show clear evidence of a Dirac point, a characteristic of a spectrum of relativistic massless fermions. In fact, Dirac points are a peculiar property of the electronic band structure of graphene, whose properties consequently can be described by the relativistic Dirac equation. In the vicinity of the Dirac point, the measured reflection spectra resemble those obtained by conductance measurements in scanning tunneling microscopy of graphene flakes. PACS numbers: 42.70.Qs, 73.22.Pr, 42.25.Fx Graphene is a monolayer of carbon atoms arranged in a honeycomb lattice [1,2]. Due to its peculiar electronic properties this carbon allotrope recently attracted a lot of attention in condensed matter physics. The conduction and the valence band of the electronic energy in graphene form conically shaped valleys that touch each other at the corners of the Brillouin zone [3]. As a consequence, close to these touch points the energy of the electron depends linearly on its quasi-momentum vector. This linear dispersion relation implies an energy independent velocity, and the related wave equation is the Dirac equation [4]. Thus, although the Fermi velocity is typically 300 times smaller than that of light, the energy spectrum of the electrons in graphene is similar to that of massless relativistic fermions [4][5][6][7]. However, graphene with its Dirac spectrum is not an exception. Photonic crystals, an optical analogue of ordinary crystals, possess similar particular properties [8,9]. The unusual transmission properties near a Dirac point predicted in [9] were observed experimentally both in sonic and in microwave photonic crystals [10,11].The photonic crystal considered in the present work is two-dimensional and composed of rows of metallic cylinders which are arranged to form the triangular lattice schematically shown in the left part of Fig. 1. Electromagnetic waves propagating in such a periodic structure exhibit a dispersion relation with a band structure similar to the electronic band structure in a solid. That for the triangular lattice of metallic cylinders with radius R = 0.25a, where a is the lattice constant, is shown on the right part of Fig. 1. It was obtained by solving numerically the Helmholtz equation with the finite difference method [12]. In this so-called band diagram the frequency f is plotted as function of the quasi momentum components (k x , k y ) for the first two propagating modes. Around the corners of the first Brillouin zone the bands have the shape of cones and the band structure resembles that of the electronic energy in graphene [2,5]. In fact FIG. 1: (Color online) Left: Triangular lattice of metallic cylinders. The arrows indicate the two different directions ΓM and ΓK in the triangular lattice; the radius R of the metallic cylinders equals R = 0.25a, where a is the lattice constant. Right: The plot of the numerically dete...

The invention of lasers 60 years ago is one of the greatest breakthroughs in modern optics. Throughout the years, lasers have enabled major scientific and technological advancements, and have been exploited in numerous applications due to their advantages such as high brightness and high coherence. However, the high spatial coherence of laser illumination is not always desirable, as it can cause adverse artifacts such as speckle noise in imaging applications. To reduce the spatial coherence of a laser, novel cavity geometries and alternative feedback mechanisms have been developed. By tailoring the spatial and spectral properties of cavity resonances, the number of lasing modes, the emission profiles and the coherence properties can be controlled. This technical review presents an overview of such unconventional, complex lasers, with a focus on their spatial coherence properties. Laser coherence control not only provides an efficient means for eliminating coherent artifacts, but also enables new applications.

Random numbers are widely used for information security, cryptography, stochastic modeling, and quantum simulations. Key technical challenges for physical random number generation are speed and scalability. We demonstrate a method for ultrafast generation of hundreds of random bit streams in parallel with a single laser diode. Spatiotemporal interference of many lasing modes in a specially designed cavity is introduced as a scheme for greatly accelerated random bit generation. Spontaneous emission, caused by quantum fluctuations, produces stochastic noise that makes the bit streams unpredictable. We achieve a total bit rate of 250 terabits per second with off-line postprocessing, which is more than two orders of magnitude higher than the current postprocessing record. Our approach is robust, compact, and energy-efficient, with potential applications in secure communication and high-performance computation.

Spatiotemporal instabilities are widespread phenomena resulting from complexity and nonlinearity. In broad-area edge-emitting semiconductor lasers, the nonlinear interactions of multiple spatial modes with the active medium can result in filamentation and spatiotemporal chaos. These instabilities degrade the laser performance and are extremely challenging to control. We demonstrate a powerful approach to suppress spatiotemporal instabilities using wave-chaotic or disordered cavities. The interference of many propagating waves with random phases in such cavities disrupts the formation of self-organized structures such as filaments, resulting in stable lasing dynamics. Our method provides a general and robust scheme to prevent the formation and growth of nonlinear instabilities for a large variety of high-power lasers.

This article presents experimental results on properties of waves propagating in an unbounded and a bounded photonic crystal consisting of metallic cylinders which are arranged in a triangular lattice. First, we present transmission measurements of plane waves traversing a photonic crystal. The experiments are performed in the vicinity of a Dirac point, i.e., an isolated conical singularity of the photonic band structure. There, the transmission shows a pseudodiffusive 1/L dependence, with L being the thickness of the crystal, a phenomenon also observed in graphene. Second, eigenmode intensity distributions measured in a microwave analog of a relativistic Dirac billiard, a rectangular microwave billiard that contains a photonic crystal, are discussed. Close to the Dirac point states have been detected which are localized at the straight edge of the photonic crystal corresponding to a zigzag edge in graphene.

Open dielectric resonators of different shapes are widely used for the manufacture of microlasers. A precise determination of their resonance frequencies and widths is crucial for their design. Most microlasers have a flat cylindrical geometry, and a two-dimensional approximation, the so-called method of the effective index of refraction, is commonly employed for numerical calculations. Our aim has been an experimental test of the precision and applicability of a model based on this approximation. We performed very thorough and accurate measurements of the resonance frequencies and widths of two passive circular dielectric microwave resonators and found significant deviations from the model predictions. From this we conclude that the model generally fails in the quantitative description of three-dimensional dielectric resonators.

Resonance spectra of two-dimensional dielectric microwave resonators of circular and square shapes have been measured. The deduced length spectra of periodic orbits were analyzed and a trace formula for dielectric resonators recently proposed by Bogomolny [Phys. Rev. E 78, 056202 (2008)] was tested. The observed deviations between the experimental length spectra and the predictions of the trace formula are attributed to a large number of missing resonances in the measured spectra. We show that by taking into account the systematics of observed and missing resonances the experimental length spectra are fully understood. In particular, a connection between the most long-lived resonances and certain periodic orbits is established experimentally.