G. P. Karman

CREATION AND ANNIHILATION OF PHASE SINGULARITIES IN A FOCAL FIELD

It is well known that uniform illumination of a lens leads to a focal field with a pattern of dark Airy rings in the focal plane, whereas this is not the case for Gaussian illumination. We show theoretically and experimentally that in the transition between the two cases the Airy rings, being phase singularities, reorganize themselves by means of a creation-annihilation process leading to new dark rings outside the focal plane.

Fractal structure of eigenmodes of unstable-cavity lasers

We show that the eigenmodes of unstable-cavity lasers have fractal structure, in contrast with the well-known stable-cavity eigenmodes. As with all fractals, the dynamic range over which self-similarity holds is limited; in this case the range is set by diffraction, i.e., by the Fresnel number of the resonator. We determine the fractal dimension of the mode profiles and show that it is related to the aperture shape.

Fractal modes in unstable resonators

One of the simplest optical systems, consisting of two mirrors facing each other to form a resonator, turns out to have a surprising property. Stable resonators, in which the paths of the rays are confined between the two mirrors, have a well known mode structure (hermite–gaussian), but the nature of the modes that can occur in unstable reson-ant cavities (from which the rays ultimately escape) are harder to calculate, particularly for real three-dimensional situations. Here we show that these peculiar eigenmodes of unstable resonators are fractals, a finding that may lead to a better understanding of phenomena such as chaotic scattering and pattern formation. Our discovery may have practical application to lasers based on unstable resonators.

Airy pattern reorganization and subwavelength structure in a focus

An early result of optical focusing theory is the Lommel field, resulting from a uniformly illuminated lens; the dark rings in the focal plane, the Airy rings, have been recognized as phase singularities. On the other hand, it is well known that Gaussian illumination leads to a Gaussian beam in the focal region without phase singularities. We report a theoretical and experimental study of the transition between the two cases. Theoretically, we studied this transition both within and outside the paraxial limit by means of diffraction theory. We show that in the gradual transition from uniform toward Gaussian illumination, the Airy rings reorganize themselves by means of a creation/annihilation process of the singularities. The most pronounced effect is the occurrence of extra dark rings (phase singularities) in front of and behind the focal plane. We demonstrate theoretically that one can bring these rings arbitrarily close together, thus leading to structures on a scale arbitrarily smaller than 1 wavelength, although at low intensities. Experimentally, we have studied the consequences of the reorganization process in the paraxial limit at optical wavelengths. To this end, we developed a technique to measure the three-dimensional intensity (3D) distribution of a focal field. We applied this technique in the study of truncated Gaussian beams; the experimentally obtained 3D intensity distributions confirm the existence and the reorganization of extra dark rings outside the focal plane.

Measurement of the three-dimensional intensity distribution in the neighborhood of a paraxial focus.

We describe a technique for measuring the three-dimensional(3D) intensity distribution of a paraxial focus, based on scanning a CCD image sensor along the optical axis and on subsequently analyzing the data. We demonstrate the possibility of measuring high-resolution 3D intensity maps of the focal field, down to intensities of more than 5 orders of magnitude below that in the focal point, and show the excellent agreement with scalar diffraction theory. Further applications of the technique are indicated.

How phase and amplitude aberrations destabilize the phase singularities in the focal field of a lens

The output field of a uniformly illuminated lens contains points of zero intensity on the optical axis and zero-intensity Airy rings in the focal plane, as formed by diffraction. These intensity zeros have been recognized as phase singularities or wave dislocations. Recently it was shown that, under the influence of a perturbation, the axial singularities may transform into rings or disappear and that the Airy rings may split into triplets. Starting from optical diffraction theory, we identify the physical perturbations that can induce such topological transformations. The basic perturbations are phase and amplitude aberrations of the wave front that is incident on the lens; we show that their different natures have consequences for the possible dislocation reactions.

Observation of a stronger focus due to spherical aberration

Abstract Recently it was predicted that addition of spherical aberration to a focusing system with a small Fresnel number can increase the value of the maximum intensity on the optical axis. In this paper we report experiments which confirm this prediction. To this end we have measured the three-dimensional intensity distribution in the focal region of a lens with spherical aberration. We also show that this effect cannot be used to increase the intensity at a fixed distance from the lens.

Excess-noise dependence on intracavity aperture shape.

In lasers with nonorthogonal eigenmodes the excess-noise factor K can be large, especially in unstable-cavity lasers with hard-edged intracavity apertures. To the best of our knowledge, we report the first detailed study of the dependence of K on aperture shape. Calculations and measurements of K for unstable-cavity lasers with variable-size apertures of triangular, square, pentagonal, hexagonal, octagonal, and rhomboid symmetries are summarized. It is shown that both the magnitude of K and its resonant behavior strongly depend on aperture shape and that many aspects of this dependence can be explained in terms of one-dimensional resonance lengths.

Observed factorization of excess quantum noise that is due to both polarization and spatial mode nonorthogonality

When the eigenmodes of a laser cavity are nonorthogonal, the quantum-limited linewidth of the laser is larger by an excess-noise factor than the standard Schawlow-Townes expression. Mode nonorthogonality can exist in the spatial domain, as in unstable-cavity resonators, but also in the polarization domain when the two polarization eigenmodes are nonorthogonal. We show experimentally that these two contributions are independent of each other, i.e., that the excess-noise factor factorizes as a product of the spatial and polarization excess-noise factors.

Unfolding of an unstable singularity point into a ring

It is well known that phase singularities, in general lines in space, are topologically stable features of a wave field. An exception is a pointlike singularity, which is unstable and deforms into a ring or disappears when a small perturbation is applied. Recently, Nye showed how such an event can be understood as an unfolding of a higher-order dislocation [J. Opt. Soc. Am. A (to be published)]. We present an optical implementation of this model and show experimentally that the focal region of a lens contains points of zero intensity on the optical axis that deform into rings when a small amount of spherical aberration is applied to the system.