Computational Photonics Seminar 06 Fiber mode solver (Bessel’s differential equation) Learn how to combine analytical solutions with numerical routines to find semi-analytical solutions Use your lecture knowledge to specify the general cylindrical problem solution to a step-index fiber Program a mode solver for guided modes of this fiber and find their mode-field diameters develop own numerical algorithm

Mathematical solution for step index fiber modes core refractive index , cladding , radius Helmholtz equation in radial coordinates ansatz: separation of variables: azimuthal equation simply solved by harmonic sine/cosine dependence => two sets of modes exist: even with A=0, B=1; odd with A=1, B=0 radial dependence solved by Bessel functions of 1st and 2nd kind (Hankel functions) boundary condition: and continuous at 2a

Mathematical solution for step index fiber modes These two conditions lead to the characteristic equation for the propagation constant of the different modes. For every given , several solutions exist which are indicated by the index , i.e. modes . characteristic equation: for U and V, the following relation holds: here, is the numerical aperture of the fiber 2a

Task: Implementation of step-index fiber mode solver physical problem step index fiber n1=1.460 n2=1.455 core diameter 15 µm wavelength l = 570 nm result output find the propagation constants of all guided modes for implement your own root finding algorithm to solve characteristic equation plot the field distribution of all calculated modes calculate the mode-field diameters of all calculated modes: modes normalized as:

Hints imaginary part of guided modes obey for , no guided modes exist any more  to find all modes, start at m=0, find all modes for this m and then increase m until no mode is found any more to solve the characteristic equation, the observation that solutions must lie on a circle with can help structure your root-search solving f(x)=0 is equivalent to finding the points where analytical knowledge about zeros of Bessel functions is never wrong MATLAB implements the functions besselj() and besselk() as well as bessely() and besseli(). first modes with m=0:

Function header function [beta, m_res, p_res] = FindStepIndexFiberModes(a, lambda, n1, n2, precision) % [beta, m_res, p_res] = FindStepIndexFiberModes(a,lambda,n1,n2,precision) % calculates propagation constants of all modes of a step-index fiber with given parameters % % INPUT: % <a> ... fiber core radius [m] % <lambda> ... wavelength in free space [m] % <n1> ... refractive index of core % <n2> ... refractive index of cladding, n2 < n1 % <precision>.. precision of calculation % OUTPUT: % <beta> ... found mode propagation constants [1/m] % <m_res> ... vector with found azimuthal mode numbers % <p_res> ... vector with found radial mode numbers

Homework 5 (due 27 June 2016) Solve the task I. Prepare a short report about your solution with the required figured and quantities. Submit your m-files of your program together with your report electronically to teaching-nanooptics@uni-jena.de by 27 June 2016. Please put everything together in one single email which contains your name (FAMILY NAME, Given Name) and matriculation number. (Late submissions will not be accepted!) 27 June, the solutions of the tasks will be available online at the lectures homepage www.iap.uni-jena.de/teaching >>> Computational Photonics. You are expected to solve the task yourself and a declaration of independent work must be signed by every student at the end of the semester.