G. W. Forbes

Wigner functions for Helmholtz wave fields

We investigate a general form of the Wigner function for wave fields that satisfy the Helmholtz equation in two-dimensional free space. The momentum moment of this Wigner function is shown to correspond to the flux of the wave field. For a forward-propagating wave field, the negative regions of the Wigner function are seen to be associated with small regions of backward flux in the field. We also study different projections of the Wigner function, each corresponding to a distribution in a reduced phase space that fully characterizes the wave field. One of these projections is the standard Wigner function of the field at a screen. Another projection introduced by us has the added property of being conserved along rays and is better suited to the description of nonparaxial wave fields.

VALIDITY OF THE FRESNEL APPROXIMATION IN THE DIFFRACTION OF COLLIMATED BEAMS

Most studies of the validity of the Fresnel approximation have relied principally on numerical results, because cancellation due to the oscillatory integrands suggests that the resulting field errors are difficult to analyze. A simple analysis is shown here, however, to give an excellent prediction of the associated errors in modeling the diffraction of a collimated beam. Further, the error estimates are presented as a universal contour map where only the contour labels depend on the aperture-size-to-wavelength ratio. The inaccuracy of the Kirchhoff boundary conditions effectively sets error bounds that are essential in deriving this universal map.

Differential ray tracing in inhomogeneous media

Differential ray tracing determines an optical systems first-order properties by finding the first-order changes in the configuration of an exiting ray in terms of changes in that rays initial configuration. When one or more of the elements of a system is inhomogeneous, the only established procedure for carrying out a first-order analysis of a general ray uses relatively inefficient finite differences. To trace a ray through an inhomogeneous medium, one must, in general, numerically integrate an ordinary differential equation, and Runge–Kutta schemes are well suited to this application. We present an extension of standard Runge–Kutta schemes that gives exact derivatives of the numerically approximated rays.

Using rays better. I. Theory for smoothly varying media.

We present a method for computing ray-based approximations to optical fields that not only offers unprecedented accuracy but is also accompanied by accessible error estimates. The basic elements of propagation through smooth media, refraction and reflection at interfaces, and diffraction by obstacles give the foundations for the new framework, and the first of these is treated here. The key in each case is that the wave field and any relevant derivatives are expressed consistently as a superposition of delocalized ray contributions. In this way, the mysteries surrounding the sometimes perplexing tenaciousness of ray-based estimates are clearly resolved. Further, an essential degree of freedom in this approach offers an attractive resolution of part of the apparent conflict of particle/wave duality.

Reducing canonical diffraction problems to singularity-free one-dimensional integrals

The oscillatory integrands of the Kirchhoff and the Rayleigh–Sommerfeld diffraction solutions mean that these two-dimensional integrals typically lead to challenging computations. By adoption of the Kirchhoff boundary conditions, the domain of the integrals is reduced to cover only the aperture. For perfect spherical (both diverging and focused) and plane incident fields, closed forms are derived for vector potentials that allow each of these solutions to be further simplified to just a one-dimensional, singularity-free integral around the aperture rim. The results offer easy numerical access to exact—although, given the approximate boundary conditions, not rigorous—solutions to important diffraction problems. They are derived by generalization of a standard theorem to extend previous results to the case of focused fields and the Rayleigh–Sommerfeld solutions.

New approach to semiclassical analysis in mechanics

A new method is proposed for constructing approximate solutions to the Schrodinger equation. In place of the wave function, its Gaussian-windowed Fourier transform is used as the fundamental entity. This allows an intuitively attractive connection to be made with a family of classical trajectories and, at all times, the wave function is inferred from the present state of these trajectories. The fact that the connection between the wave function and the classical trajectories is consistently constructed in phase space allows this method to be free of the limitations of other methods.

Using rays better. II. Ray families to match prescribed wave fields

A key step in any ray-based method for propagating waves is the choice of a family of rays to be associated with the initial wave field. We develop some basic prescriptions for constructing initial ray families to match two particular types of waves. Various Gaussian and Bessel beams are separately given special treatment because of their general interest. These ideas are directly useful for a newly developed method for ray-based wave modeling. The new method expresses the wave as a superposition of ray contributions that is independent of the width of the field element associated with each ray. This insensitivity is investigated here even when the elemental width varies from ray to ray. The results increase the applicability of the new wave-modeling scheme.

Beyond the Fresnel approximation for focused waves

By extension of the transitional operator method developed by Wunsche, the Rayleigh–Sommerfeld and Kirchhoff solutions to the diffraction of a converging spherical (or cylindrical) wave are expressed in terms of a series of derivatives of the field estimate that follows from the Fresnel approximation. This result allows a systematic assessment of the error associated with the paraxial wave model for focused fields and offers simple corrections to this model. In particular, for simple diffracting masks, the Fresnel approximation leads to estimates of the field that have a relative error near focus that is of the order of one on the square of the f-number. The number of significant digits in the field estimate is shown to be doubled by retaining just the first of the series of corrections derived here.

Algebraic corrections for paraxial wave fields

Asymptotic analysis of the angular spectrum solution for diffraction is used to establish the validity of a standard, formal series for nonparaxial wave propagation. The lowest term corresponds to the field in the Fresnel approximation, and this derivation clarifies some of the remarkable aspects of Fresnel validity for both small and large propagation distances. This asymptotic approach is extended to derive simple, generic algebraic corrections to the field estimates found by using the paraxial model, i.e., the Fresnel approximation. Contour maps of the field errors associated with the diffraction of collimated beams—both uniform and Gaussian—in two and three dimensions demonstrate the effectiveness of these corrections.

Generalization of Hamilton’s formalism for geometrical optics

A generalization of Hamilton’s formalism for geometrical optics is given to provide more convenient descriptions of the optical properties of certain classes of systems. This generalization is made by replacement of the usual points and planes that are effectively used as references from which to measure optical path length in the definition of characteristic functions by more general surfaces. In this way an unlimited number of options are made available. Some particular cases that are well suited to the study of asymmetric systems are investigated.