Using Hocker and Burns numerical method to analyze the dispersion properties of a planar waveguide with an exponentially graded-index core layer for s-polarized light

Although not realistic, the theory of step-index waveguide has been discussed in most published works. Electric and magnetic fields can be written as well-known functions such as exponential, sine and cosine functions. However, the index distribution of the most practical waveguides is better described as graded. Despite certain similarities, step-index and graded-index waveguide structures also have considerable differences. The characteristic equation of s-polarized wave propagating in a planar waveguide with an exponentially graded-index thin core layer is examined in this work using the Hocker and Burns numerical approach. This method utilizes the effective index method of analyzing waveguides with 2D confinement. It was efficiently applied to problems of channel waveguides formed by diffusion. The technique depends on finding the phase shift of the curved optical path in the graded index zone as stacked infinite thin layers. Three factors contribute to the total transverse phase shift: (1) film-cladding interface phase delay, (2) film-substrate interface phase delay, and (3) phase delay caused by the zigzag optical path of the guiding film. The findings revealed the following intriguing observations. The dispersion curves of the graded-index waveguide structure are in the normal shape in which the generalized guide index (GeGI) increases with the rise of the normalized frequency. At high values of the normalized frequency, the GeGI displays less dependence on it. The dispersion curves show cut-off thicknesses which increase for higher asymmetry measure values. A comparison between graded-index and step-index waveguide structures is carried out.