An Examination of Aperture Effects in Cylindrical Nearfield Holography

Abstract

The development and implementation of cylindrical holography is derived from the time independent homogeneous Helmholtz equation for acoustic pressure, p, in cylindrical coordinates

$$({\nabla ^2} + {k^2})p(r,\phi ,z) = 0$$

((1))

, where k = w/c. In solving this equation an e ^-iwt time dependence is assumed and the method of separation of variables is used. Furthermore, it is assumed that the source is confined to a finite cylindrical band of radius, a, and length, l, in an infinite cylindrical baffle. Also, free-space propagation exists external to a so that at any r ≥ a

$$p(r,\phi ,z) = \sum\limits_{n = - \infty }^\infty {{e^{in\phi }}} \int_{ - \infty }^\infty {{A_{n{k_z}}}} {H_n}^{(1)}({k_r}r){e^{i{k_z}z}}d{k_z}]$$

((2))

, where ${k_r} = \sqrt {{k^2} - k_z^2}$ and the Hankel function of the first kind represents outgoing waves. By applying a two-dimensional Fourier transform to both sides of (2) and evaluating at r ⁰, the hologram radius, one obtains

$${P_n}({r_0},{K_z}) = \frac{1}{{2\pi }}{A_{n{k_z}}}H_n^{(1)}({k_r}{r_0})$$

((3))

.