I have a complex vector of the transverse amplitude and phase distribution of a laser beam, derived from experimental data. When modelling these field distributions, ordinarily the eigenmodes of the optical setup are found and used in a linear superposition to determine which field shapes are supported. For example, the field distribution of a cylindrically symmetric laser cavity can be expressed as a linear superposition of Laguerre-Gauss modes. Similarly, Bessel modes are used to describe the transverse field profile in (some) optical fibres.
I am trying to decompose my experimentally derived data set into a superposition of such modes, which I have done by taking the inner product of the data set with each of the modes within a defined range (the first 100 modes, for example). However, when changing basis set I find it hard to determine which is the most appropriate to use.
My question: Is there a (computationally efficient) method for finding the most appropriate basis without invoking arguments relating to the experimental geometry?
I realise that this question could arguably be categorised as a physics, linear algebra, or scientific programming question. If this is the wrong place to ask, I will move it elsewhere.
Thanks.
After some research it doesn't seem as though there is a straight forward way to pre-determine whether a given basis set is the appropriate one to use. Instead, it is necessary to take the inner product of the data set with all defined modes in each basis set to retrieve the (in this case) complex projection amplitudes. The sparsity of the matrix of these complex projection amplitudes will then provide some indication of which basis set is the most appropriate. For example, the Hoyer measure may be used, and becomes more positive for increasing sparsity. The more sparse the matrix of projection amplitudes is, the better the corresponding basis set describes the experimental data.
See here for definition of Hoyer sparsity (and other measures).