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The correlation functions of the transmission coefficients for scalar wave propagation through disordered media are calculated by use of both diagrammatic techniques and numerical simulations. The calculation is valid in the diff'usive regime: multiple elastic scattering with negligible absorption or inelastic scattering and a scattering length much longer than the wavelength. In addition to the familiar large local intensity fluctuations we find a novel memory eA'ect and long-range correlations in the transmission coefficients which decay to a positive background value. Implications for light-scattering experiments are discussed.PACS numbers: 42.20. -y Coherent waves propagating through a disordered medium' will emerge from that medium with a phase that varies in an effectively random manner along the wave front. If one assumes that the outgoing wave amplitude at a given point (or in a given mode) is a sum of a large number of uncorrelated amplitudes and that the total amplitudes at points separated by more than a few wavelengths are uncorrelated, then it is easy to show that the wave intensity will fluctuate by an amount of the order of its average. 2 3 The most familiar example of this is the speckle pattern created by a laser beam reflected off' a rough surface.It has recently been understood, however, that waves propagating through an inhomogeneous medium by multiple elastic scattering will create a fluctuating intensity pattern which is not nearly as "random" as intuition suggests. First, such patterns will show on average a higher intensity for backscattering as a result of constructive interference of time-reversed pairs of scattering sequences, '4 an eff'ect that has been observed recently in several experiments. 56 Second, although there are large intensity fluctuations observed, 67 the fluctuations at different points (or in different modes) must be statistically correlated. The theoretical and experimental evidence for this comes from the study of conductance fluctuations in small metal circuits.It is now understood that these fluctuations occur because of the high sensitivity of the complex interference pattern of the (coherent) electron wave function to changes in impurity scattering potentials. It is found that the fluctuations in the total transmission of electrons through the circuit summed over all incident and outgoing channels (which is proportional to the conductance) is always of order unity. s ' It is easy to show that such a result cannot be obtained if the fluctuations in each mode are uncorrelated. " It should be stressed that the statistical behavior encountered here is only quantum mechanical in that it derives from the wavelike behavior of electrons, and thus analogous statistical correlations should be present in any system in which waves propagate by coherent multiple elastic scattering. In this Letter we explicitly calculate the statistical correlation function in different transmitted modes and uncover both the correlations responsible for the "universal conductance fluctuations" ...

We demonstrate the existence of resistance fluctuations in experimentally realizable ballistic conductors due to scattering from geometric features. The magnetic-field and energy correlation functions are calculated both semiclassically and exactly numerically, and are found to have a scale determined by the underlying chaotic classical scattering. These systems provide a test of the "random" quantum behavior of classically chaotic systems.

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We analyze the optical properties of one-dimensional (1D) PT -symmetric structures of arbitrary complexity. These structures violate normal unitarity (photon flux conservation) but are shown to satisfy generalized unitarity relations, which relate the elements of the scattering matrix and lead to a conservation relation in terms of the transmittance and (left and right) reflectances. One implication of this relation is that there exist anisotropic transmission resonances in PT -symmetric systems, frequencies at which there is unit transmission and zero reflection, but only for waves incident from a single side. The spatial profile of these transmission resonances is symmetric, and they can occur even at PT -symmetry breaking points. The general conservation relations can be utilized as an experimental signature of the presence of PT -symmetry and of PT -symmetry breaking transitions. The uniqueness of PT -symmetry breaking transitions of the scattering matrix is briefly discussed by comparing to the corresponding non-Hermitian Hamiltonians.

High-power and highly directional semiconductor cylinder-lasers based on a new optical resonator with deformed cross-section are reported. In the favorable directions of the far-field, a power increase of up to three orders of magnitude over the conventional circularly symmetric lasers was obtained. A "bowtie"-shaped resonance is responsible for the improved performance of the lasers in the higher range of deformations, in contrast to "whispering-gallery"-type modes of circular and weakly deformed lasers. This new resonator design, although demonstrated here in mid-infrared quantum-cascade lasers, should be applicable to any laser based on semiconductors or other high-refractive index materials.Lasers consist of two basic components. First, the active material in which light of a certain wavelength range is generated from an external energy source, such as electric current; second, the laser resonator, which contains the active material, provides feedback for the stimulated emission of light. The resonator largely influences the special features of the emitted light: power, beam directionality, and spectral properties, as well as the laser's physical features such as size and shape. Semiconductor lasers are the most widely used and versatile class of lasers. Their most common resonators are FabryPerot cavities, in which two cleaved semiconductor crystal planes act as parallel mirrors, reflecting the light back and forth through the active material.There have been many attempts to improve resonator properties. In particular, an increase of the reflectivity of the resonator mirrors is highly desirable. This allows low thresholds for the onset of laser action and a smaller volume of active material with concomitant moderate energy requirements and the ability to pack the lasers in a small space. * To whom correspondence should be addressed; email:fc@lucent One excellent example is the development of microdisk semiconductor lasers (1). These lasers exploit total internal reflection of light to achieve a mirror reflectivity near unity. Micro-disk, -cylinder or -droplet lasers form a class of lasers based on circularly symmetric resonators, which lase on "whispering-gallery modes" of the electromagnetic field (2,3,4). In such a mode light circulates around the curved inner boundary of the resonator, reflecting from the walls of the resonator with an angle of incidence always greater than the critical angle for total internal reflection, thus remaining trapped inside the resonator. There are only minute losses of light caused by evanescent leakage (tunneling) and scattering from surface roughness. This principle allowed the fabrication of the world's smallest lasers (2). Besides possible future applications in optical computing and networking, micro-lasers are of strong interest for research problems of cavity quantum electrodynamics, such as resonatorenhanced spontaneous emission and threshold-less lasers (5). Small resonators may also serve as model systems for the study of wave phenomena in mesoscopic systems, parti...

Unlike conventional lasers, diffusive random lasers (DRLs) have no resonator to trap light and no high-Q resonances to support lasing. Because of this lack of sharp resonances, the DRL has presented a challenge to conventional laser theory. We present a theory able to treat the DRL rigorously and provide results on the lasing spectra, internal fields, and output intensities of DRLs. Typically DRLs are highly multimode lasers, emitting light at a number of wavelengths. We show that the modal interactions through the gain medium in such lasers are extremely strong and lead to a uniformly spaced frequency spectrum, in agreement with recent experimental observations.