Equivalence of the Huygens–Fresnel and Debye approach for the calculation of high aperture point-spread functions in the presence of refractive index mismatch

As discussed in recent work (Sheppard, C. J. R. & Török, P., J. Microsc., 185, 366–384; Török et al., J. Microsc., 188, 158– 172), two approaches have been used extensively for vectorial computations of high aperture confocal pointspread functions when focusing through a dielectric interface. Whereas the equation by Hell, Reiner, Cremer & Stelzer (J. Microsc., 169, 391–405) is based on the Huygens–Fresnel principle, the more recent approach by Török, Varga & Booker (J. Opt. Soc. Am. A, 12, 325–332; J. Opt. Soc. Am. A, 12, 2136–2144) is based on the Debye approximation. While the earlier theory considers a large but finite focal length the second theory is derived for an infinitely high Fresnel number. In a high aperture microscope, a high Fresnel number is equivalent to assuming that the focal length be infinitely large with respect to the wavelength. So far, the two theories are regarded as different, with the one by Török et al. being rigorous. In this paper, we demonstrate that, if the same conditions are applied, the equation by Török et al. can be analytically derived from that by Hell et al. Producing the same results, the benefit brought about by the equation by Török et al. is improved flexibility and computational speed for cases with azimuthal symmetry.