Eikonal, (from the Greek eikon — image) in geometrical optics, a function that determines the optical path length of a ray of light between two arbitrary points, one of which, A, belongs to the space of objects and the other, A`, to the space of images (see Optical image) Depending on the choice of parameters, a distinction is drawn between: the point eikonal, or Hamilton's eikonal (Hamilton's characteristic function of the coordinates x, y, z; x`, y`, z` of the points A and A`); the angular Bruns eikonal (a function of the angular coefficients μ, ν; μ`,ν` of the ray); the more complicated Schwarzschild eikonal and a number of others. Applying the eikonal in the calculation of optical systems makes it possible, by differentiating it with respect to definite parameters, to find expressions for some of the main aberrations of optical systems. The functions called the eikonal are widely used in electron and ion optics, within the general analogy that exists between it and classical optics, and also in describing the scattering of particles and waves in quantum mechanics and quantum field theory (the eikonal approximation), where analogies with optics also arise.