UNIFIED AND GENERALIZED FRESNEL NUMBERS

The concept of Fresnel number is discussed and expressions are derived for misaligned optical systems. For the case of perfectly aligned optical elements, the usual number, N, based on a Fresnel zone concept is found to be given by N = (a2/lambda) (D1/B1 + A2/B2), where B1, D1 and B2, A2 are the transfer matrix elements of the optical systems before and after a circular aperture of radius a respectively. A modified definition of the Fresnel number is proposed for Gaussian beam propagation. This parameter N(G)', is related to the complex beam parameter and may be represented by the angle theta = tan-1 N(G)', found in the familiar Collins chart and its dual representation. A general relation for the transformation of this Fresnel number is found. The expressions for Gaussian beam transformation are thus simplified, since the waist-waist transform is given by N(G1)' = N(G2)' = 0. Finally, these two kinds of Fresnel numbers are written as tensors when applied to cases involving elliptical apertures, astigmatic beams and nonsymmetrical systems.