Biomedical subjects

G W Forbes

Publications and source records attributed to G W Forbes.

6 recordsLinked to original sources

PubMed2001

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.

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PubMed2001

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.

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PubMed2001

Using rays better. III. Error estimates and illustrative applications in smooth media.

A new method for computing ray-based approximations to optical wave fields is demonstrated through simple examples involving wave propagation in free space and in a gradient-index waveguide. The analytic solutions that exist for these cases make it easy to compare the new estimates with exact results. A particularly simple RMS error estimate is developed here, and corrections to the basic field estimate are also discussed and tested. A key step for any ray-based method is the choice of a family of rays to be associated with the initial wave field. We show that, for maximal accuracy, not only must the initial field be considered in choosing the rays, but so too must the medium that is to carry the wave.

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PubMed2001

Using rays better. IV. Theory for refraction and reflection.

A new ray-based method is extended to include the modeling of optical interfaces. The essential idea is that the wave field and its derivatives are always expressed as a superposition of ray contributions of flexible width. Interfaces can be analyzed in this way by introducing a family of surfaces that smoothly connects them. Even though the ray-to-wave link may appear to be obscured at caustics, the standard Fresnel coefficients (for plane waves at flat interfaces between homogeneous media) are shown to be universally applicable on a ray-by-ray basis. Thus, in the interaction at the interface, the surface's curvature and any gradients in the refractive indices influence only the higher asymptotic corrections. Further, this method finally gives access to such corrections.

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PubMed2000

Phase-space distributions for high-frequency fields.

The Wigner distribution function and various windowed Fourier transforms are examples of phase-space distributions that are used, among other things, to formalize the link between ray and wave optics. It is well known that, in the limit of high frequencies, these distributions become localized for simple wave fields and therefore that the localization can be used to define the associated ray families. This localized form is characterized here for both the Wigner distribution function and a Gaussian windowed Fourier transform. Aside from the greater understanding of the distributions themselves, these results promise a clearer intuition of phase-space-based methods for optical modeling. In particular, regardless of the context, the geometric construction that is presented for estimating the Wigner distribution function gives a valuable appreciation of its highly structured and sometimes surprising form.

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PubMed2000

Uncertainty products for nonparaxial wave fields.

Although maximal localization is a basic notion in the consideration of phase-space representations of fields, it has not yet been pursued for general wave fields. We develop measures of spatial and directional spreads for nonparaxial waves in free space. These measures are invariant under translation and rotation and are shown to reduce to the conventional ones when applied to paraxial fields. The associated uncertainty relation sets limits to joint localization in coordinate and frequency space. This relation provides a basis for the definition of a joint localization measure that is analogous to the beam propagation factor (i.e., M2) of paraxial optics. The results are first developed for two-dimensional fields and then generalized to three dimensions.

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