Six-parameter Gaussian modes in free space

A derivation is given of free-space Gaussian modes described by six parameters. For a beam traveling in the $z$ direction, three parameters describe the beam properties in the planes of constant $y$ and three parameters describe the beam properties in the planes of constant $x$. It is further shown that the beam waist for the properties in the planes of constant $y$ need not coincide with that for the properties in the planes of constant $x$. For special values of the parameters the six-parameter modes reduce to the familiar three-parameter modes.