4500HWG01 - Group Velocity A pulse of light of freespace...

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Unformatted text preview: Group Velocity A pulse of light of freespace wavelength A is propagating in a bulk dielectric of refractive index n. The duration of the pulse is T- The dispersion in this dielectric mate rial is given by dn/dA. In freespace, this same pulse would travel at a velocity of c. The phase velocity is defined as up = w / k, while the grOup velocity is defined as 129 = dw/dk. In the class notes, vg in the bulk dielectric is obtained by differentiating k with respect to w, thus obtaining dk/dw. The reciprocal of dk/dw is 119. Perform the follow— ing alternative operations. Evaluate dw/dA. Evaluate dk/dA. Divide dw/dA/dk/d)‘. In each case, express your answers as functions of c, n, A, dn/dA, and constants only. Put your final answers in the spaces provided. How does dw/dA/dk/dA compare to the 129 found in the class notes? E _ dA _ dk dA dw/dA dk/dA Group Velocity The group velocity is defined by v _E g _ dk Differentiating _ E - 212 _ -2719 w — A gives d/\ —. A2 Differentiating _nw , dk___1_d(nw)__1_ E d_n _ k“ T glves EX _ c dA _ C("dA +wdA) _ dk 1 27rc 27rc dn 27r dn n ax = 2[" ‘W + (T a] = 7K5) — X] and so dw dk 27m 21r dn n c dn n C vs = a; ii = -v/fl(a) - x] = ‘X [(5) - x] = —_(n—Ag; ...
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