# Transverse modes in optical resonators

Im struggling with understanding transverse modes in an optical resonator or laser. Hopefully you can solve this mystery for me. Thank you very much!

As far as I know, there are two types of modes:

(1) Longitudinal modes:

This is the part I understand. The distance of the two resonator mirrors set boundary condition on the possible wavelengths allowed for propagation in the resonator (due to constructive/destructive interference). Where L is the distance of the mirrors and n is any natural number, the allowed wavelengths are $$\lambda=\frac{2L}{n}$$. Basically you have a possible set of nodes in the longitudinal (propagation) direction.

(2) Transverse modes:

Besides of the Gaussian beam, there are other modes, described by Laguerre or Hermite polynomials. Mirros with cylindrical symmetry lead to modes, described by Laguerre polynomials and with rectangular symmetry you describe those modes with Hermite polynomials. Those transverse modes are noted by $$TEM_{nm}$$, depending on the order of the Laguerre or Hermite polynomials describing them. Basically you have a possible set of nodes in the transverse direction (perpendicular to propagation).

Problem or question:

I dont get what boundary conditions lead to the formation of nodes in the transverse direction and thus to formation of transverse modes. As the distance of the mirrors set the longitudinal mode and the shape of them sets the symmetry of the transverse modes (cylindrical/Laguerre or rectangular/Hermite), what parameter is left to set order n,m of the $$TEM_{nm}$$?

• I guess it is because your waves are not plane waves, so there is no uniformity in the transversal direction. Apart from mirrir form, there might be diaphragmes or other wave guiding structures. – Vladimir Kalitvianski Jan 13 at 18:29