Solving Vector Modal Equation

[Aniket1]

I need help solving the vector modal equation for a step index fiber having a constant refractive index in the core and the cladding. (Under the conditions of zero dispersion and absorption.

 

[gtoguy97]

The vector modal equation for a step-index fiber with constant refractive index in the core and cladding is given by:A2∂2E/∂x2 + 2 B ∂E/∂x + (C2 - k2)E = 0where A, B and C are constants that depend on the refractive indices of the core and cladding. k is the wavenumber of light in the fiber.To solve this equation, we need to use a technique called the method of separation of variables. This involves separating the equation into two parts, one involving only the x-dependent variables and the other involving only the y-dependent variables. We then solve each part separately and combine the solutions to obtain the general solution for the equation.For example, if we separate the equation into its x-dependent and y-dependent parts, we get:A2∂2E/∂x2 + 2B∂E/∂x = f(y)and (C2 - k2)E = g(x)where f(y) and g(x) are functions of their respective variables.We can then solve each equation separately. The solution for the first equation is:E = a(x) + b(y) where a(x) and b(y) are arbitrary functions of x and y respectively.For the second equation, the solution is:E = c sin(kx) + d cos(kx)where c and d are constants.Combining the two solutions, we get the general solution for the vector modal equation:E = a(x) + b(y) + c sin(kx) + d cos(kx)This is the general solution to the vector modal equation for a step-index fiber with constant refractive index in the core and cladding under the conditions of zero dispersion and absorption.