Eikonal


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Eikonal

 

(also characteristic function, characteristic), in geometrical optics, a function that specifies the optical path length of a ray of light between two arbitrary points, one (point A) in object space and the other (point A’) in image space (seeIMAGE, OPTICAL).

Depending on the choice of parameters, several different types of eikonals are distinguished. The point characteristic, which is also referred to as Hamilton’s characteristic or the characteristic function of Hamilton, is the eikonal of an optical system with respect to the coordinates x, y, and z of A and x’, y’, and z’ of A’. The angle characteristic, which was discovered by H. E. Bruns, is the eikonal of an optical system with respect to a ray’s direction cosines μ, v and μ′, v′. An example of a more complex eikonal is the characteristic function that was discovered by K. Schwarzschild.

In the designing of an optical system, the use of an eikonal makes it possible to obtain expressions for the system’s transverse aberrations by differentiating the eikonal with respect to certain parameters. Functions called eikonals are widely used in charged-particle optics in the framework of the general analogy that exists between charged-particle and classical optics. Such functions are also used to describe particle and wave scattering processes in cases where analogies with optics arise. Examples of the techniques employed in such cases include the Hamiltonian method and the eikonal, or slowly varying amplitude, approximation in quantum mechanics and quantum field theory.

REFERENCES

Born, M., and E. Wolf. Osnovy optiki. Moscow, 1973. (Translated from English.)
Kel’man, V. M., and S. Ia. Iavor. Elektronnaia optika, 3rd ed. Leningrad, 1968.
Goldberger, M., and K. Watson. Teoriia stolknovenii. Moscow, 1967. (Translated from English.)
References in periodicals archive ?
Suppose at some time the Eikonal solution is known at a set of points (denoted Accepted points).
For the function S(x, y) we have the eikonalequationInitial conditions for the eikonal S for the horizontalcase are defined on the line L : x (a ), y (a ) : 0 0 construct the rays, that is, the equation (18) with characteristics (rays)element of the ray.
As explained by Sethian (1996), solving the Eikonal equation (Eq.
1), the Eikonal can be rewritten in terms of ray path s according to:
Such an extension is of crucial importance to GPS signal propagation modeling, especially for numerical methods of the eikonal solving.
First of all, synthetic traveltimes are calculated using the ray tracing theory, which traces the fastest ray between a source-receiver pair, using either ray tracing or a finite differences solution of the eikonal equation.
He covers some variational problems in Hilbert spaces, iterative methods in Hilbert spaces, operator-splitting and alternating directions methods, augmented Lagrangians and alternating direct methods of multipliers, the least-squares solution of linear and nonlinear problems, obstacle problems and Bingham flow with applications to control, nonlinear eigenvalue problems, Eikonal equations, and fully nonlinear elliptic equations.
Depending on the specific ansatzes for the eikonal s and the amplitude series, there are three different asymptotic methods applied to fusion problems: Ray tracing, emanating from geometric optics, quasi-optical ray tracing and paraxial beam tracing, products of the complex eikonal theory.
The topics include hydrogen photo-ionization with strong lasers, the role of trajectories in quantum chemistry and chemical physics, beyond the eikonal approximation in classic optics and quantum physics, and a subquantum accelerating universe.
A key mathematical equation in ray theory is the Eikonal equation which is reformulated in 2-D as:
A complementary wave equation is also given there in the form of a completely geometric eikonal equation:
Offering six chapters and eight appendices, this book develops new heuristic approaches to diffraction theory, focusing on applications of the generalized eikonal.