2 2 2 2 2 2 2 2 -iω t -τ /4 (ω - ω ) -τ /4 (ω - ω ) nω 2 -iω t -i(β z - ω t) (β z + t) /4 (i/2 β z - τ /4) nω 2 -t /τ o τ o τ o -iβ(ω) z o dβ(ω) 1 d β(ω) 2 o ∞ iΔωt τ π o o 1 2 o dβ(ω) d β(ω) StyleBox[Cell[TextData[{Pulses in a Dispersive Medium, <br />, <br />The dispersion induced by a fiber on an optical pulse is a the result of the frequency dependent variation in the index of refraction. This results in the frequency dependent group velocity. If we consider a Gaussian pulse described by the equation: <br /><br />, Cell[BoxData[A(0, t) = e e ], Background -> RGBColor[0, 1, 0]], <br /><br />The Gaussian pulse has a frequency spectrum given by the following expression:<br /><br />, Cell[BoxData[A(0, ω) = ------- e ], Background -> RGBColor[0, 1, 0]], <br /><br />The spectrum of the pulse at any point along the fiber can be described by the following expression:<br /><br />, Cell[BoxData[A(z, ω) = ------- e e ], Background -> RGBColor[0, 1, 0]], <br /><br />If dispersion effects higher than second order are neglected, the propogation constant can be expanded in a Taylor series to yield the following expression for β(ω)<br /><br />, Cell[BoxData[β(ω) = -------- + --------------- (ω - ω ) + - ----------------- (ω - ω ) ], Background -> RGBColor[0, 1, 0]], <br /><br />The temporal behavior of the pulse as it propogates down the fiber can be determined by defining a parameter <br /><br />, Cell[BoxData[Δω = ω - ω ], Background -> RGBColor[0, 1, 0]], <br /><br />and taking the inverse Fourier transform of the pulse. This yields the following integral for A(z,t):<br /><br />, Cell[BoxData[A(z, t) = e ∫ A(z, Δω) e d Δω], Background -> RGBColor[0, 1, 0]], <br /><br />This integral is performed by completing the square (Robinett, Appendix B.33, B.34). The resulting function for the temporal behavior of the pulse is:<br /><br />, Cell[BoxData[A(z, t) = ------- Sqrt[---------------------] e e ], Background -> RGBColor[0, 1, 0]], <br /><br />Where the following definitions have been used:<br /><br />, Cell[BoxData[β = --------, β = ---------------, β = -----------------], Background -> RGBColor[0, 1, 0]], <br /><br />The results are used in the next cell to illustrate the behavior for pulses in a dispersive medium. The various parameters can be adjusted in the model given below to observe how parameters such as pulse width effect the overall broadening of the pulse.<br />}]], Background -> RGBColor[0., 1., 0.]] Sqrt[2] Sqrt[2] c dω o 2 2 o o -∞ Sqrt[2] 2 o c 1 dω 2 2 d ω i τ dω - β z - ------- 2 2 4 Cell[]

λ = 1.55 * 10^(-6) ; "Center Wavelength" ; c = 3 * 10^8 ; "Speed of light in vacuum" ; <br /> Dp = 170 ; "Dispersion parameter of medium (s/m^2). This is abnormally high. Typically the value is around 17 ps/nm-km." ; n = 1.5 ; "Mean index of refraction of medium" ; ω _ o = c/λ ; β _ 1 = n/c ; "1/Group Velocity)" ; β _ 2 = Dp * -λ^2/(2 * π * c) ; A[z_, t_, τ_] := Re[τ/2^(1/2) π/(i/2 β _ 2 z + τ^2/4)^(1/2) e^(i * ((-n * ω _ o)/c z - ω _ o t)) e^-(-β _ 1 z + t)^2/(4 (i/2 β _ 2 z + τ^2/4))] ; "Plot of Pulse Width =5.4 ns at t=0 and t=0.5 μs" ; Show[Plot[A[z, 0, 5.4 * 10^(-9)], {z, -10, 250}, PlotRange -> {-2, 2}, DisplayFunction -> Identity, PlotStyle -> RGBColor[0, 1, 0]], Plot[A[z, 0.5 * 10^-6, 5.4 * 10^(-9)], {z, -10, 250}, PlotRange -> {-2, 2}, DisplayFunction -> Identity, PlotStyle -> RGBColor[1, 0, 0]], Frame -> True, DisplayFunction -> $DisplayFunction, FrameLabel -> {"Distance (m)", Amplitude }] "Plot of Pulse Width =1.4 ns at t=0 and t=0.5 μs" ; Show[Plot[A[z, 0, 1.4 * 10^(-9)], {z, -10, 250}, PlotRange -> {-2, 2}, DisplayFunction -> Identity, PlotStyle -> RGBColor[0, 1, 0]], Plot[A[z, 0.5 * 10^-6, 1.4 * 10^(-9)], {z, -10, 250}, PlotRange -> {-2, 2}, DisplayFunction -> Identity, PlotStyle -> RGBColor[1, 0, 0]], Frame -> True, DisplayFunction -> $DisplayFunction, FrameLabel -> {"Distance (m)", Amplitude}]

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