The potentials themselves are solutions of the scalar Helmholtz equation, and the particular solution is found by observing the boundary conditions imposed by physical considerations on **E **and **h**. The potentials are supposed to form modes, so a solution where the variables are separated is appropriate

and a similar expression applies for the other Debye potential, ϕ . The Helmholtz equation (8) is expanded in cylindrical co-ordinates, and then (15) is substituted.For regions where ε is constant, the radial functions follow the Ordinary or Modified Bessel equation, and so the solutions are linear combinations of Bessel functions of integer order v. In any given layer,

where . In layers where the propagation constant squared is larger than k^{2}ε , the Bessel functions *J* and *Y* are replaced by the Modified Bessel functions *I* and *K*, respectively.,

where . Equations (15), (16), and (17) are substituted in (6) and (7) to find the electromagnetic fields. It is the tangential components θ and z that are needed explicitly, since it is the tangential components of the electric and magnetic fields that should match at layer boundaries. These field components are related to the coefficients *A*, *B*, *C*, and *D* by a 4×4 matrix.

where *n *= β ⁄ *k *is the modal index, and the common factor exp [ *j*( νθ – β*z *) ] is suppressed. In layers where the propagation constant squared is larger than *k*^{2}ε , the Bessel functions *J *and *Y *are replaced by the Modified Bessel functions *I *and *K*, respectively, and the *u *is replaced by *w*.

A similar equation to (18) applies in adjacent layers, with different constants *A*, *B*, *C*, *D*. Given the constants in one layer, the constants in the adjacent layer can be found by solving the linear system created by the field matching condition. The **f **found by the two matrices should be the same field vector at the boundary between layers. The difference between this formulation and that in Ref. [2] and [3] is that this formulation uses real numbers only.