The present invention relates to optical filters, and more particularly to optical filters using the electric dipole resonance absorption in polarizable particles.
The resonant absorption frequency, f.sub.o, or wavelength, .lambda..sub.o, of polarizable particles which are small in spatial extent with respect to the wavelength of incident radiation depends upon the complex dielectric constant, .epsilon.(.lambda.), where .epsilon.(.lambda.) = .epsilon..sub.1 (.lambda.) + i.epsilon..sub.2 (.lambda.), upon the shape of the particle, and upon the alignment of the particle with respect to the incident radiation field if the particle is not spherical or isotropic. For example, the absorption cross section in vacuum at wavelength .lambda. of a small isotropic spherical particle of a radius a is ##EQU1## WHERE X = KA = 2.pi.A/.lambda.. The scattering cross section is similarly given by Rayleigh's solution ##EQU2## The small particle assumption is equivalent to assuming that both X and X(.epsilon..sup.1/2) &lt;&lt; 1, so that if there is any absorption at all, it will in general dominate any scattering effects as a result of the dependence of C.sub.a and C.sub.s on x: ##EQU3## Hence, for a suspension of small absorbing particles only the absorption contribution to the total extinction need be considered.
If .epsilon..sub.2 (.lambda..sub.o) &lt;&lt; 1 and .epsilon..sub.1 (.lambda..sub.o) .about. -2, then from Equation (1) C.sub.a becomes large. This resonant behavior is related to the resonant surface absorption observed on rough metal surfaces in the ultraviolet (surface plasmon absorption) and on rough dielectric surfaces in the infrared (surface polariton absorption). The condition .epsilon..sub.1 = -2 will occur in a given material at a given wavelength. In this wavelength region .epsilon..sub.1 and .epsilon..sub.2 will be strongly wavelength dependent.
For a spheroidal particle of volume V with depolarization factor L, the absorption cross section is ##EQU4## For an oblate spheroid of large aspect ratio (disc shape) with radiation incident perpendicular to the plane of the disc, ##EQU5## where a and b are respectively the minor and major semi-axes of the ellipse. The new resonant condition is ##EQU6##
If the particles are imbedded in a medium with dielectric constant .epsilon..sub.o, the absorption cross section is recomputed using the relative dielectric constant .epsilon./.epsilon..sub.o and the wavelength in the medium .lambda./.epsilon..sub.o.sup.1/2. For a spherical particle ##EQU7## The resonant condition is now .epsilon..sub.1 (.lambda..sub.o) = -2.epsilon..sub.o, .epsilon..sub.2 (.lambda.) &lt;&lt; .epsilon..sub.o.
If the scattered radiation fields at a particle from all other particles is negligible, then the particles behave independently. In this case, if the suspension consists of N identical spherical particles per unit volume, then the absorption coefficient .alpha. (Lambert's law) = NC.sub.a, or ##EQU8## where m is the mass of suspended material per unit volume of space, and .rho. is the bulk density of the suspension material. Thus, the absorption coefficient is independent of particle size within the assumption x &lt;&lt; 1, X(.epsilon..sup.1/2) &lt;&lt; 1.