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 ?
Initial conditions for eikonal (16) and transfer (17) equations can be obtained from (13):
The connection to the speed of sound is now established through the eikonal equation [22].
La ecuacion eikonal provee una conveniente descripcion de la variacion de la fase a lo largo del camino del campo de alta frecuencia; sin embargo, tambien se requiere informacion cuantitativa sobre la variacion de la amplitud del campo, esto es, la ecuacion de transporte.
Suppose at some time the Eikonal solution is known at a set of points (denoted Accepted points).
where amplitude [mathematical expression not reproducible] denote any vector functions of position and time and S is the eikonal or phase, any real function of position and time, too.
Menze et al., "Ayache Image guided personalization of reaction-diffusion type tumour growth models using modified anisotropic Eikonal equations," Journal of Medical Imaging, vol.
Menze et al., "Image guided personalization of reaction-diffusion type tumor growth models using modified anisotropic eikonal equations," IEEE Transactions on Medical Imaging, vol.
As regards the scientific review, the eikonal equation is expressed, and some complex-valued solutions are defined corresponding to complex rays and caustics.
Neglecting this term gives the refraction equations (eikonal and transport equations)