Errors incurred in a plane-wave-type expansion of a Gaussian beam

The multipole expansion obtained by Morita et al. (1968) of the Gaussian laser beam used to levitate an aerosol particle in order that its complete phase matrix may be measured is compared with that of Tsai and Pogorzelski (1975) in order to demonstrate the effect of the incorrect expansion used by Morita. Errors incurred by the use of an equation in which one side satisfies the scalar wave equation while the other side does not and can be reduced to a plane wave amplitude are calculated as functions of the inverse of the wave number times the beam waist, the wave number times the radial spherical coordinate and the angular spherical coordinate. Errors on the order of a few percent, considered undetectable are obtained in the squared-field amplitudes due to the expansion, however, they are found to become significant (several tens of percent) when the angle is zero. It is concluded that the expansion of Morita should only be used in the regions where the spherical angle is less than 0.01 and its product with the wave number and the radial spherical coordinate is less than unity.