please tell me how to find the propagation constant of different LP(Linearly polarized) modes inside a multimode step index fiber. Is there any formula to calculate the value of propagation constant?? please suggest me some help
Hi, the best and easiest method would be to use a mode solver (there are some free-ones available on the internet I think like https://www.rp-photonics.com/software/RP_Fiber_Calculator_setup.exe for l,m up to 20) or from a commercial product like Comsol for example. Otherwise you need to develop your own code solving the equations.
optifiber (optiwave product) is best tool for design and analysis of multimode fiber. two design methods are possible (refractive index based or dopant concentration based). Using finite difference method u can calculate supported LP modes(multi modes) and then u can calculate the dispersion, propagation constant, and bending loss of that each LP mode.. trial version is available in optiwave wepsite...
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 normalised 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)
Allan W. Snyder & John D. Love's "Optical Waveguide Theory" has more details in chapter 14. Chapter 12 presents exact solutions which are valid for larger values of refractive index difference where the scalar wave equation and the LP mode description are less accurate.
The Eigenvalue equation must be solved numerically, but this will generally be faster and more accurate than a generic solver using methods such as finite element or finite difference, which are typically required for arbitrary refractive index profiles.
http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html
http://www.amazon.com/Optical-Waveguide-Theory-A-W-Snyder/dp/0412099500
Dear Arijit Datta,
You can determine propagation constant through eigenvalue equation (dispersion relation). Eigenvalue equation is deduced by employing suitable boundary condition at the core-clad interface.
Please follow the following paper:
1.http://link.springer.com/article/10.1007/s00340-014-5858-2#page-1
2.http://www.jpier.org/PIER/pier.php?paper=13041103
3http://www.sciencedirect.com/science/article/pii/S0030402604702016
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