Acoustic Field in a Luneberg Lens

Abstract

A Luneberg lens, which is a perfectly focusing omnidirectional spherical lens, has been subjected to a wave analysis. Plane radiation incident on a fluid lens was assumed. The separability of the scalar Helmholtz equation for the lens region permits the representation of the internal acoustic field as an expansion in Legendre polynomials and radial functions. The radial equation obtained from the separation was solved in integral form from which the necessary recurrence relations were obtained. Beam patterns at and relatively near the focal point were obtained for moderate wave number-aperture products (k0∼a80). Calculations for high frequencies are hampered by the lack of sufficiently general schemes for function generation. The beam patterns so obtained are quite similar to those obtained for comparable spherical fluid lenses of uniform refractive index. Lens gains and beamwidths do not differ significantly in the two cases. In general, the side lobes for the Luneberg lens are relatively high and decrease relatively slowly in magnitude as the polar angle increases.