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

As discussed in recent work (Sheppard, C. J. R. & Torok, P., I. Microsc.. 1 85, 366-384; Torok et al., J. Microsc., 188, 158-172), two approaches have been used extensively for vectorial computations of high aperture confocal point-spread functions when focusing through a dielectric interface. Wherea s the equation by Hell, Reiner, Cremer & Stelzer (I. Microsc., 169, 391-405 ) is based on the Huygens-Fresnel principle, the more recent approach by To rok. Varga Br Booker (J. Opt. Sec. Aln. A, 12, 325-332; J. Opt. SOC. Am. A, 12, 2136-2144) is based on the Debye approximation. While the earlier theo ry considers a large but finite focal length the second theory is derived f or an infinitely high Fresnel number, In a high aperture microscope, a high Fresnel number is equivalent to assuming that the focal length be infinite ly large with respect to the wavelength. So far, the two theories are regar ded as different, with the one by Torok et al, being rigorous, In this pape r, we demonstrate that, if the same conditions are applied, the equation by Torok et ttl, can be analytically derived from that by Hell et nl. Produci ng the same results, the benefit brought about by the equation by Torok el nl. is improved flexibility and computational speed for cases with azimutha l symmetry.