The integral representation for the electromagnetic field at the focus of a Gaussian beam was derived as an extension of the work of Richards and Wolf (1959). Using this derivation we have calculated the effect of underfilling the back aperture of an objective on the resolution of an MPE microscope. A previous form (Yoshida and Asakura, 1974) was found to be incorrect and not consistent with experimental measurements due to omission of the sine condition of the objective lens in the formulation. The left figure shows the fractional power at the sample as a function of the aperture to 1/e beam waist ratio for various modes of excitation. The figure on the right shows the effect on the axial resolution as this ratio changes at high (1.2) and low (0.3) NA. The two plots below compare the formulation with experimental measurements of the axial resolution (circles) for two different aperture to beam waist ratios. Solid (integrated over x and y) and dotted ( x = y= 0) lines were calculated numerically, as was the solid line fitting the data points which is a running numerical integration of the xy integral. The axial resolution was measured by stepping into a solid fluorescent plane (hence it is fit to the running integral).

B. Richards and E. Wolf., Proc. Royal Soc. 253:358-379, 1959.

A. Yoshida and T. Asakura. Optik 41:281, 1974.