New Theoretical Method For Nonuniform Gratings Investigation

Abstract

Uniform Fiber Bragg Gratings are wavelength selective reflectors obtained by a periodic modulation of core refractive index along the fiber. They have many applications as narrow-band elements in optical fiber systems. Recently the NG are used for many purposes [1]. For their investigations the numerical solutions of coupled mode are used traditionally. This appoach requires a much machine time, and is not convenient for NG design. Therefore the analytical formulaes for characteristics of NG will be of benefit in NG design, syntezing, and optimization. The first step in solving this problem has been undertake in [2] where the WKB- approximation for solving coupled mode has been used. The drawback of this approximation is that it is not uniform in all frequency interval and is not valid near so called turning points. Then, an unknown phase shift occurs in formulaes which can not be determined within the WKB- approximation, and one needs to evaluate this unknown parameter from qualitative estimations. To avoide these drawbacks of WKB- approximation we elaborated another more suitable method which is valid uniformely in all frequency interval. This method, called R-approximation is the generalization of asymptotic solution of second order differential having one turning point [3] for the case when there are two or more turning points, in the result the analitical formulaes for characteristics of arbitrary NG obtained. R-approximtion is more exact and common than WKB-approximation. The last come to R-approximation by removal from turning points. As an uniform asymtotic formulaes R-approximation has a relative error ~ O(1/H) in all frequency interval, where H=πμN describes the gratings strength, μ-is the depth of effective permittivity modulation, N-is the number of grating periods (for uniform grating the maximum reflectivity is |R|2=th2(H/4)). On the basis of obtained common formulaes a special case of linearly chirped gratings (LCG) is investigated in detail and a designing software “LCG” is created. This software provides all characteristics of LCG versus strength, detuning, and chirping rate, and is a powerfull and convinient tool for designers.