There are two sets of orthonormal modes that are widely used in optics: Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) ones. Several first-order optical systems are special with respect to HG and LG modes. Indeed, HG and LG modes are self-reciprocal under the propagation through fractional Fourier transform systems (in the case of LG modes, the fractional Fourier transformer should be isotropic). Other first-order optical systems - mode converters - transform HG modes into LG ones and vice versa, or into other modes. In general, after propagation through a first-order optical system, the orthonormal set of Hermite-Gaussian modes maps into an other orthonormal set.

In this contribution a compact expression for these so-called ABCD-HG modes, obtained at the output of a lossless optical ABCD system with the HG (or LG) modes at its input, is derived. In a particular case, the ABCD-HG modes reduce to two-dimensional HG functions. The relationship between the ABCD-HG modes and two-variable HG functions is established. Some properties of the ABCD-HG modes are underlined.

The introduction of a novel class of orthonormal modes is useful for the description of the evolution of an arbitrary complex field during its propagation through first-order systems. It also simplifies the design of mode converters and information processing systems. The application of orthonormal mode mapping for phase recovery from intensity distributions by non-interferometric techniques is discussed.

To: Papers by Martin J. Bastiaans