Analytical calculation of the excess noise for different aperture shapes

We present an analytical calculation of the transverse excess noise factor which arises when a small aperture of arbitrary shape is placed at an out-c oupling mirror of a laser resonator. We show that for small apertures the e xcess noise can be written in terms of the far-field diffraction pattern of the aperture. We expand the excess noise in the aperture area. The leading term depends on the area but not on the shape of the aperture, whereas the next term is solely determined by the shape. The list of shapes ordered in increasing value of the excess noise based on this analytical calculation agrees with that based on numerical and experimental data for unstable reso nators with large apertures.