Optimal concentration of electromagnetic radiation

For complete spherical concentration of light the maximum theoretically possible total energy density for a given power input can be, in principle, achieved by appropriate choice of polarization and angular amplitude variation. Illumination of a focusing system with a plane-polarized wave creates at the focus equal electric and magnetic energy densities . By appropriate choice of radial variation this energy density can be maximized . For hemispherical concentration the electric energy density can be seven sixteenths of the maximum possible for a given power input, and the total energy density can be seven eighths of the maximum possible . Focusing by optical systems satisfying the sine condition amongst others is also considered . For a system satisfying the sine condition, the total energy density can be is of the maximum possible . 1 . Introduction Bassett [1] has established an upper bound to the energy density which is attainable by passive concentration at any point 0 for a given input power . He assumes that the point is many wavelengths from any object and that any radiation which is subsequently reflected or scattered back to the point can be neglected. This is equivalent to assuming a high Fresnel number in a focusing system, that is that the (electromagnetic generalization of the) Debye theory is appropriate . Richards and Wolf [2] have considered the electric and magnetic energy density in the focal region of a lens satisfying the sine condition in the Debye approximation. It is interesting to consider how close a real focusing system can come to the maximum attainable energy density given by Bassett . Later workers have generalized the method of Richards and Wolf to other illumination distributions, including Gaussian weighting, annular pupils, parabolic mirrors and so on [3-6] . Bassett's method is to expand the field in terms of a multipole expansion . These multipole fields are Green functions, that is they satisfy Maxwell's equations for any point other than the point O . The field at 0 vanishes for any other than the electric and magnetic dipole components. Bassett's upper bound is the energy density produced from a combination of these dipoles, each oriented in three orthogonal directions . Thus this upper bound can be achieved by illumination with a combination of these dipole fields . Sheppard [6] has shown that the polarization, as given by Richards and Wolf [2], produced when a plane-polarized wave is refracted to produce a spherically convergent wave is the same as that for the sum of an electric and a magnetic dipole whose axes are two orthogonal directions in the 0950-0340/94 $10 . 00 © 1994 Taylor & Francis Ltd . 1496 C. J. R. Sheppard and K. G. Larkin focal plane . Thus with appropriate weighting the plane-polarized wave can be coupled completely into electric and magnetic dipole fields . 2 . Energy density at the focus for mixed-dipole fields Consider a focusing system in which the monochromatic illumination over the Gaussian reference sphere has a polarization given by Richards and Wolf [2] for focusing of a plane-polarized wave, that is it is the sum of two orthogonal electric and magnetic dipole fields and has an electric field amplitude E;(B) which is axially symmetric . Then, if the radius of the reference sphere is f, and a is the angular semiaperture of the optical system (figure 1) the power flow into the system is /E\ 1/2 S = of 21 1 f IE ;(B)I2 sin B dB . \μ o The electric field amplitude at the focus 0 is given by Richards and Wolf [2] as f2,7E2 a Eo = 2 J E ;(B)(1 + cos 0) sin B dB . (2) 0 The electric energy density at focus is We introduce the factor Fe (the normalized electric energy density) given by We k 2 Fe=-= S 16rtc We = 4EOE z0