Regular papers

Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture

¹ Philips Research Europe, HTC-36, NL 5656 AE Eindhoven, The Netherlands
² Faculty of Electrical Engineering, Eindhoven University of Technology, NL-5600 MB Eindhoven, The Netherlands
³ Optics Research Group, Faculty of Applied Sciences, Technical University Delft, Lorentzweg 1, NL-2628 CJ Delft, The Netherlands
⁴ Philips Research Europe, HTC-04, NL-5656 AE Eindhoven, The Netherlands

^* s.vanhaver@tudelft.nl

Received: 12 July 2008

Abstract

We present a derivation of the analytic result for on-axis field values of the Rayleigh diffraction integral, a result that was originally presented in a paper by Osterberg and Smith (1961). The method on which our derivation is based is then applied to other diffraction integrals used in acoustics and optics, e.g., the far-field Rayleigh integral, the Debye integral and the separate near-field part of the Rayleigh integral. Having available our on-axis analytic or semi-analytic solutions for these various cases, we compare the various integrals for wave numbers k pertaining to low-frequency acoustic applications all the way up to high-frequency optical applications. Our analytic results are compared to numerical results presented in the literature.