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Ploblem 1.4 (37.5 points) (Graded-index waveguide) The figure below shows a and and B, starting planar...

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Ploblem 1.4 (37.5 points) (Graded-index waveguide) The figure below shows a and and B, starting planar...

Ploblem 1.4 (37.5 points) (Graded-index waveguide) The figure below shows a and and B, starting planar dielectric waveguide in which the refractive index changes from n to n to nj so along y at y -62, 382, 582,.... Consider the paths of two rays,A from point 0. The launching angle for the ray A is such that 0, is the critical angle for TIR at the 1-2 interface. Ray B launched at a smaller angle penetrates in medium point B and then travels toward point B' at the boundary between media 2 and 3. Ray is launched such that at interface. The indexes of refraction and the layer thicknesses are chosen so that both ra meet at point O point B, its angle ? is the critical angle for TIR at the 2.3 -362 Medrum 3 Medien 2 - 62 52 Ray B Miedium (a) Show that the time taken by the ray A to travel from O to O'is: o (b) Show that the time taken by the ray B to travel from O to O' is: (c) Let the step variations of n at y-2, 382,.. obey: where ? (0.5) and ? are two parameters used to describe the spatial dependence of the refraction index. Show that where ? 2a (d) Show that the condition for the rays to arrive at the same time at point O' is: 20-c), 1 (e) If the thickness of the different index of refraction layers is very small, i.e. 8-0 show that ? 2.067 ensures that the travel time of different rays between points O and O' is the same. Sketch the index of refraction dependence on y and discuss, what would be the advantage of an optical fiber that has this type of graded refractive index.

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