Isaac Freund

Second harmonic generation in collagen

Ordered collagen in native, intact rat tail tendon has been studied by means of optical second harmonic generation (SHG). The two independent hyperpolarizability tensor elements of the collagen molecule, β3 and β1, have been found to be of opposite sign and roughly equal in absolute magnitude. Evidence is presented for a systematic variation in the ratio β3/β1 with age which may reflect the effects of progressive cross linking. Our experimental findings are in reasonably good agreement with a simple theory constructed on the basis of current models of the tendon structure, and thereby confirm this structure in the native, intact tissue. The many advantages of SHG for the structural study of intact partially ordered biological tissues are outlined.

Optical dislocation networks in highly random media

Abstract Networks of optical dislocations in gaussian speckle patterns are studied both theoretically and experimentally. A simple product wavefunction is suggested to describe an arbitrary dislocation network. Phase maps predicted by this wavefunction are tested against measured interferograms for a variety of different dislocations and good agreement is found between theory and experiment. Simple arguments based upon the sampling theorem are presented which suggest that in the gaussian limit the dislocation network contains all the information needed to reconstruct the total wavefield, and alternatively, that the wavefield may also be reconstructed from a small fragment of the phase map using the method of zero crossings. It is also argued that there must exist some degree of correlation between neighboring dislocations.

Second-harmonic microscopy of biological tissue.

We present the first reported optical second-harmonic microscope images of a biological sample—rattail tendon—and discuss, also for the first time, the need to distinguish between coherent and incoherent second-harmonic imaging. Our data show that the currently unexplained macroscopic polar order of this classic representative of connective tissue is due both to a coherent network containing a large number of fine, polar, filamentlike structures that permeate the entire tendon volume and to a small number of intensely polar surface patches.

Looking through walls and around corners

It is shown theoretically that under appropriate conditions a visually opaque, multiply scattering optical barrier can be made to serve as a thin lens which produces a near perfect, real, paraxial image of objects lying behind the barrier. Preliminary experimental results are described which verify the validity of the underlying assumptions. The barrier can also be made to serve as various other types of optical instruments, such as mirrors, polarizers, optical Fourier analyzers, theodolites, etc. Thus it is now clear that multiply scattering media should no longer be considered barriers to optical propagation, but are more properly to be regarded as potential high-precision optical instruments.

Polarization singularity indices in Gaussian laser beams

Abstract Two types of point singularities in the polarization of a paraxial Gaussian laser beam are discussed in detail. V-points, which are vector point singularities where the direction of the electric vector of a linearly polarized field becomes undefined, and C-points, which are elliptic point singularities where the ellipse orientations of elliptically polarized fields become undefined. Conventionally, V-points are characterized by the conserved integer valued Poincare–Hopf index η , with generic value η =±1, while C-points are characterized by the conserved half-integer singularity index I C , with generic value I C =±1/2. Simple algorithms are given for generating V-points with arbitrary positive or negative integer indices, including zero, at arbitrary locations, and C-points with arbitrary positive or negative half-integer or integer indices, including zero, at arbitrary locations. Algorithms are also given for generating continuous lines of these singularities in the plane, V-lines and C-lines. V-points and C-points may be transformed one into another. A topological index based on directly measurable Stokes parameters is used to discuss this transformation. The evolution under propagation of V-points and C-points initially embedded in the beam waist is studied, as is the evolution of V-dipoles and C-dipoles.

OPTICAL VORTICES IN GAUSSIAN RANDOM WAVE FIELDS : STATISTICAL PROBABILITY DENSITIES

Simple, closed-form analytical expressions are given for the statistical probability densities of the six parameters that define an optical vortex (phase singularity) in a Gaussian random wave field. Good agreement is found between calculation and a computer simulation that generates these vortices.

Elliptic critical points in paraxial optical fields

Abstract Generic critical points of elliptically polarized, paraxial optical fields include (i) C -points, which are isolated points of circular polarization, (ii) stationary points of the azimuthal angle that measures the orientation of the major axes of the ellipses, and (iii) stationary points of the form factor that measures the ratio of the minor to major axes of the ellipses. These newly defined elliptic stationary points are introduced, and using a mapping of ellipse fields onto a complex Stokes field, topological constraints are formulated that tie them to C -points and to a -lines, which are lines of constant azimuthal orientation, as well as to L -lines, which are lines of linear polarization. Experiments on random ellipse fields are described that verify the most important of these constraints, the sign rule. A mapping of C -points onto phase vortices is made that permits construction of a large variety of C -points with predetermined topological charges, Poincare–Hopf indices, and three-dimensional trajectories. This mapping also shows that C -points in random ellipse fields should exhibit essentially complete screening of their topological charges. This result is verified by large-scale computer simulations.

Optical polarization singularities and elliptic stationary points

Polarization singularities and elliptic stationary points (collectively, elliptic critical points) were measured experimentally via the complex Stokes field S1 + iS2, where S1 and S2 are Stokes parameters. This new, easily implemented method yielded detailed, high-resolution experimental data for all elliptic critical points. These data confirm with high precision the elliptic-field topological sign rule, loop rules, and Stokes singularity relations introduced recently.

Stokes singularity relations.

Polarization singularities in paraxial vector optical fields are analyzed in terms of the phase singularities of complex Stokes scalar fields. Six independent relationships are obtained that connect the topological charges of these singularities on special closed contours with the charges of singularities that are enclosed by these contours. These relationships, which have been confirmed by experimental data and computer simulations, imply topological polarization correlations of an infinite range.

Critical point explosions in two-dimensional wave fields

Abstract We discuss degenerate critical points in two-dimensional fields described by complex wave functions. Typical examples include the familiar optical field in a plane normal to the direction of propagation, electrons in two-dimensional heterojunctions, the Abrikosov lattice in superconductors, thin films of liquid helium at low temperatures, etc. Using simple topological arguments, we show that under the influence of small perturbations a degenerate critical point can decay explosively into a large number of irreducible (nondegenerate) components, and that this explosion may trigger a chain reaction that fragments other aspects of the wave field. We also discuss controlled explosions designed to produce particular decay products. We show that vortices (phase singularities) may be divided into two classes, generic and nongeneric, and that each class has its own special properties and modes of decay. The fallout from a critical point explosion can be extensive. The mound of debris resulting from decay of a single 5th-order optical vortex can easily contain over 150 new critical points. In superconductors vortex orders can approach 103, and the number of new critical points generated during explosion of vortices of this order can exceed 5×106.