How do I find the propagation constant for different LP modes for step index multimode fiber?

Hi,

In the scalar wave approximation, LP modes of a cylindrical core step-index waveguide are described by Bessel functions in the core and modified Bessel functions in the infinite, uniform refractive index cladding.  The propagation constants are found by solving a simple transcendental equation involving these Bessel functions.
For LPm,n modes, the core field at radius r is proportional to Jm(U r/a) where U is the transverse wavevector normalized to the core radius, a.  Jm(x) is a Bessel function of the first kind of order m.
The cladding field varies as Km(W r/a), where W is the cladding transverse evanescent propagation constant, normalised to the core radius.  Km(x) is a modified Bessel function of the second kind of order m.
The Eigenvalue equation is derived by matching the core and cladding fields and gradients at the core-cladding boundary:  U Jm+1(U) / Jm(U) = W Km+1(W) / Km(W); U2 + W2 = V2
The normalised frequency:  V = 2 pi / wavelength sqrt(n2core - n2clad)

Thanks!