Analysis of planar waveguides by perturbation method

V.M. Fitio, V.V. Romakh, Ya.V. Bobitski

Lviv Polytechnic National University

One of the important tasks today in the field of communications and information technologies is the ability to provide high-speed data transmission using fiber-optic communication lines. The presence of existing homo- and heterostructures in semiconductor lasers cannot implement this task due to the fact that at high-frequency modulation these structures generate a large number of modes, i.e., wavelengths. As an alternative, distributed feedback lasers (DFB lasers) are used which active medium is similar to a waveguide that includes a periodic structure, namely a grating. A waveguide indicates that active layer has greater refractive index than layers surround it. In a grating which is formed at layer interface of structure, Bragg reflection occurs providing single-mode generation in spite of spatial failure due to a standing wave in active medium.
As to the theory of distributed feedback lasers, it is well developed and studied when first order Bragg conditions are carried out. Instead, there is another situation for higher order Bragg conditions (second, third). In particular, the theory for analysis of waveguide lasers to satisfy second order conditions is described, but it is only applied to active waveguides with surface relief gratings; in addition, representation of the fields in active waveguide is not quite complete. Also, the fields are not sufficiently presented for waveguides with bulk gratings resulting radiation is expected only from waveguide top, but at small angles to the axes x (i.e., almost perpendicular to a waveguide) and z (beam generated is propagating in a substrate) it is not provided by the theory.
The theory of propagation of plane only reflective wave in periodic medium with gain and without it is worked out, and generation conditions at implementation of second and higher order Bragg conditions are obtained. But there is no possible to use this theory directly for waveguide DFB lasers because it is not clear how fields and waveguide mode constants of planar waveguides are calculated.
Therefore, the aim of this work is a usage of method to find propagation constants of waveguide modes at perturbation presence which is determined by change of waveguide permittivity during pumping. This method is based on fact that field of unperturbed waveguide is represented as linear combination of unperturbed waveguide fields with coefficients which depends on coordinate along which waveguide modes propagate. As a result, systems of second order differential equations are obtained. In this paper, both passive (permittivity ε(x) is a real function) and active (ε(x) becomes imaginary values) waveguides without grating in a structure are considered. Using both the perturbation method and approach based on Fourier transform, symmetric and asymmetric waveguides are analyzed, and propagation constants of guided modes are found. Good coincidence of results obtained by these methods is observed. Difference is only in last digits of propagation constants; it appears due to rounding. Theory used is approximate because only guided modes are taken into account, but it can be accurate when there is no interaction between modes of unperturbed waveguide.