
#1
Jun3013, 09:55 PM

P: 1

Hello,
The Fermat principle says that (***) Δt = (1/c) ∫ μ(x,y) √1+y'^{2} dt Say, we are studying a GRIN material where the refraction index is μ = μ(x,y) and want to figure out the shape of the ray trajectory y=y(x). Here is what I know (this is not a homework question) but am unsure if the approach is correct: (1) [itex]\frac{1}{R}[/itex]=[itex]\frac{∂ ln μ}{∂n}[/itex], from Peters' paper . I expect the shape to be a hyperbola say a/x. Can I use this equation (it is derived by seeking stationary time Δt) to calculate μ=μ(x,y)? (R is the radius of curvature, [itex]\frac{∂}{∂n}[/itex], is a derivative along the normal to the curve) (2) Once I have μ=μ(x,y), can I plug it into (***), also plug in y=a/x and derive Δt? I am hoping this can then be minimized to find the optimal parameter a or any similar curve parameter. Does this make sense? (3) or should I write the EulerLagrange equation and try to find out something from it? (4) or should I use a Lagrange multiplier and "constrain" the ray to the hyperbola a/x I am unsure which approach is correct; and also, what of the above known/expected expressions should I plug in into the Lagrangian and then do the analysis. My goal is to find out the extremal curve and its equation. Need help from someone experienced in Fermat/lagrangian formalism... 


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