The required size of the detector and its distance from the edge may be estimated as follows: An aperture of a diameter 4 w passes over 99.9% of a circular Gaussian beam of spot size w. As far as diffraction effects due to the edge are concerned, the power of the cylindrical waves in the shadow region diminishes as (kz)−1 [M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 11], which amounts to ∼10−5 for λ = 633 nm at r = 1 cm off the edge: with a detector placed more than 1 cm behind the edge we can safely disregard diffraction effects.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 7.

Ref. 3, Chap. 26. The approximation 26.2.16 therein is over 2 orders of magnitude more accurate than that used by Khosrofian and Garetz (see Ref. 1); however the following approximation, which is similar to the latter in structure is also accurate to ±1 × 10−5 for −∞⩽x⩽+∞:Q(x)≃{1+exp[f(x)]}−1,f(x)=a1x+a2x3+a3x5+a4x7,a1=1.595700,a2=0.072953,a3=−0.000324,a4=−0.0000350.

Error function paper is available as graph paper no. 46800 from Keuffel & Esser, covering relative magnitudes from 0.01 to 99.99; a better-suited variant of it, covering 0.15–0.85, was devised by D. Preonas, Dayton Research Institute, in 1980. This latter graph paper makes use of the central portion of the Gaussian only, thus avoiding errors due to deviations of the beam from Gaussian shape in the tails.

W. C. Hamilton, Statistics in Physical Science (Ronald Press, New York, 1964), Chaps. 4 and 5.

The program is available on request.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976), Sec.9.9.

Ref. 8, p. 169, Eq. (9q).

This situation prevails in eigenmodes of a ring laser.