Fermat's Principle Of Least Time Research Articles

A Ray-Tracing Algorithm Based on the Computation of (Exact) Ray Paths With Bidirectional Ray-Tracing

A novel ray-tracing algorithm to cope with the phase errors due to incorrect ray path computations in ray-launching approaches is presented. The algorithm utilizes bidirectional ray-tracing to collect information about wavefronts incident on an interaction surface and yields considerable improvements in accuracy compared to conventional unidirectional ray-tracing. The points, where exact ray paths intersect with the surface, are obtained according to the Fermat principle of least time. If the interaction surface is aligned with diffraction edges, the corresponding critical points of the second kind can also be retrieved and complicated diffraction treatments by shooting diffracted rays on Keller cones can be avoided. Thus, a substantial reduction in the number of rays can be achieved. Furthermore, a typical problem encountered in traditional ray-tracing due to the reception sphere mechanism, i.e., incorrect ray contributions, can mostly be evaded. Numerical results demonstrate the capabilities of the new algorithm and its advantages against traditional techniques.

Bernoulli's Light Ray Solution of the Brachistochrone Problem Through Hamilton's Eyes

This article contains a review of the brachistochrone problem as initiated by Johann Bernoulli in 1696–1697. As is generally known, the cycloid forms the solutions to this problem. We follow Bernoulli's optical solution based on the Fermat principle of least time and later rephrase this in terms of Hamilton's 1828 paper. Deliberately an anachronistic style is maintained throughout. Hamilton's solution recovers the cycloid in a way that is reminiscent of how Newton's mathematical principles imply Kepler's laws.

Derivation of the Evans Wave Equation from the Lagrangian and Action: Origin of the Planck Constant in General Relativity

The Evans wave equation is derived from the appropriate Lagrangian and action, identifying the origin of the Planck constant ħ in general relativity. The classical Fermat principle of least time, and the classical Hamilton principle of least action, are expressed in terms of a tetrad multiplied by a phase factor exp(iS/ħ), where S is the action in general relativity. Wave (or quantum) mechanics emerges from these classical principles of general relativity for all matter and radiation fields, giving a unified theory of quantum mechanics based on differential geometry and general relativity. The phase factor exp(iS/ħ) is an eigenfunction of the Evans wave equation and is the origin in general relativity and geometry of topological phase effects in physics, including the Aharonov-Bohm class of effects, the Berry phase, the Sagnac effect, related interferometric effects, and all physical optical effects through the Evans spin field B(3) and the Stokes theorem in differential geometry. The Planck constant ħ is thus identified as the least amount possible of action or angular momentum or spin in the universe. This is also the origin of the fundamental Evans spin field B(3), which is always observed in any physical optical effect. It originates in torsion, spin and the second (or spin) Casimir invariant of the Einstein group. Mass originates in the first Casimir invariant of the Einstein group. These two invariants define any particle.

I. Geometrical optics of variable-frequency light rays:Theoretical basis

A variable-frequency light ray (VF ray) is a new concept in optics; it is a ray that is monochromatic at every point of the path (no frequency spread) but its frequency changes along the path due to interactions occurring along that path. For example, the frequency of the light ray can be affected by gravitational time dilation, reflection from a moving mirror, or coherent Raman scattering. Since the Fermat principle of least time (PLT) implicitly assumes that the frequency of the light ray is an irrelevant constant, the PLT-based geometrical optics is insufficient to handle propagation of VF rays. In this paper, we derive a new extremum principle (NEP) that is sensitive to changes of frequency along the path of the light ray and constitutes a basis for geometrical optics of VF rays. The NEP is derived directly from the principle of least action applied to the quantized electromagnetic field associated with the light ray propagating in a vacuum or in a transparent material medium in the presence of the gravitational field. The relationship between the NEP, the classical version of the Fermat principle, and the general relativistic generalization of the Fermat principle (the extremum principle for null geodesics) is discussed in detail. Applications of the NEP are suggested in the nonrelativistic, special relativistic, and general relativistic regimes. PACS Nos.: 42.15-i, 12.20-m, 04.20-q

An investigation of pulse-timing techniques for broadband ultrasonic velocity determination in cancellous bone: a simulation study

Berlage wavelets are used to simulate ultrasonic pulses in an unbounded, homogeneous, isotropic and absorptive medium. Intrinsic absorption of the medium is properly described by Kolsky's attenuation, which considers velocity dispersion to meet the causality condition. Several current time-domain velocity measurement techniques have been investigated using numerically simulated pulses for three normalized BUA values: 20, 40 and 60 dB , which mimic experimentally determined values for cancellous bone. The velocities, calculated using first motion transit times, are used as references supported by the Fermat principle of least time. The simulated results for fixed sample thickness indicate that pulse-broadening increases with the transit time of the reference point and the intrinsic absorption of the medium. Comparison shows that the first zero-crossing method yields 3-6% errors in velocity results, better than the cross-correlation method. However, the zero-crossing method gives inconsistent velocity measurement for a medium of 40 dB absorption and three different thicknesses: 0.2, 0.4 and 0.6 cm. A novel technique for velocity measurement is presented using the peak of the envelope of a signal as a reference point to measure transit time difference. The envelope of a signal represents the instantaneous amplitude of the associated analytic signal. The velocities derived using this method differ from the true velocities by only 1.2-2.4%, more accurate than those obtained by the first zero-crossing method. The envelope peak has the additional merits of easy detection and robustness. Most importantly, the envelope technique may be used to yield accurate velocity measurement in cases where an accurate determination of the first motion transit time is sometimes prohibited due to the presence of noise.