Suppose we have a PDE, for example the Helmholtz paraxial equation: $$ \nabla_\perp^2A+2ik\frac{\partial A}{\partial z}=0 $$

Solutions depend on the coordinate system we are using, i.e. we obtain Hermite-Gaussian modes if we use Cartesian coordinates or we obtain Laguerre-Gaussian modes if we study the equation in cylindrical coordinates. I know there is a relation between Hermite-Gaussian modes and Laguerre-Gaussian modes.

In general my question is: if we study an equation in different coordinate system, can we always find a relation between the solutions? Can I be sure there is a relation between the solutions?

  • $\begingroup$ Sure, as long as you have a solution expressed in one coordinate system you can always transform it to another coordinate system by transforming the coordinates. $\endgroup$ – Maxim Umansky Mar 10 '14 at 13:32
  • $\begingroup$ Transforming coordinates won't be enough. Existence of these distinct types of modes are result of degeneracy of the Laplacian spectrum. To transform from one type of modes to another one has to compute linear combination of partner modes (i.e. those belonging to the same eigenvalue) of the original type. In infinite space this would mean integrating a continuum of partner eigenmodes. $\endgroup$ – Ruslan Mar 10 '14 at 15:41
  • $\begingroup$ I know we have a relation that allows us to compute a Laguerre-Gaussian mode as a linear combination of Hermite-Gaussian modes and vice-versa. I don't understand if a relation between the solutions is a trivial consequence of the fact of being solutions of the same equation or not. $\endgroup$ – Danny Mar 10 '14 at 19:37

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