1 August 1998 Ž . Optics Communications 153 1998 207–210 The beam propagation factor for higher order Gaussian beams S. Saghafi, C.J.R. Sheppard 1 Physical Optics Department, School of Physics, UniÕersity of Sydney, Sydney NSW 2006, Australia Received 2 September 1997; revised 16 February 1998; accepted 6 May 1998 Abstract The beam propagation factor M 2 for coherent, higher-order, single mode beams can be calculated in a number of different ways, for example directly from information concerning the waist, or from far field information. The factor M 2 is calculated for higher order ordinary and elegant Hermite– and Laguerre–Gaussian modes. q 1998 Elsevier Science B.V. All rights reserved. 1. Introduction Any beam can be expressed as a combination of Her- Ž . mite–Gaussian modes for rectangular coordinates or La- Ž . guerre–Gaussian modes for cylindrical coordinates , wx which are eigenmodes of the paraxial wave equation 1 . In the ordinary Hermite–Gaussian function, the Gaussian part has a complex argument, but that of the Hermite part is wx purely real. Siegman 2 introduced new Hermite–Gaus- sian solutions that satisfy the paraxial wave equation but have a more symmetrical form. The argument of the Hermite part is then complex. These new solutions are not wx orthogonal in the usual sense 2 . There are analogous elegant Laguerre–Gaussian solutions, which satisfy the w x paraxial wave equation in cylindrical coordinates 3,4 . An important property of optical beams is the beam propaga- tion factor M 2 , defined as the normalized product of the second moments of the intensities in the waist and far field wx 2 5 . The factor M can describe beam propagation as well as quality, in the sense of how localized the beam is. The factor M 2 for the different types of Hermite– and La- guerre–Gaussian beams is investigated. In some cases it is convenient to calculate M 2 directly from information con- cerning the waist, or from far field information. Appropri- ate expressions for M 2 are thus considered. 1 Also with Australia Key Centre for Microscopy and Micro- analysis, University of Sydney, Sydney, Australia. 2. The beam propagation factor M 2 For beams which have a circularly symmetric intensity variation, if the field in the waist is C , then the effective w x spot size w of the beam is defined by 6,7 e ` ` 2 2 2 3 w s2 C r d r C r d r , 1 Ž. H H e 0 0 where r is the radial coordinate. It should be noted that some papers omit the factor 2 in the definition. We can express M 2 as the normalized product of the spot sizes in w x the waist and far field. It can also be characterized 5,8 by the normalized product of the waist radius w and the far e field divergence u e w u e e 2 M s , 2 Ž. w u G G where w , u represent the analogous properties for a G G fundamental Gaussian beam with the same wavelength as the beam under study. For a purely real amplitude, in the wx context of propagation from optical fibres, Petermann 6 also defined a different spot size w , related to waveguide d dispersion, 2 d C ` ` 2 2 w s2 C r d r r d r . 3 Ž. H H d ž / d r 0 0 Again the factor 2 is sometimes omitted. This so-called Ž . Petermann spot size sometimes called Petermann 2 was 0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. Ž . PII S0030-4018 98 00256-9