Massachusetts Institute of Technology - EECS 6.977Aps1

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.977 Ultrafast Optics Spring 2005 Problem Set 1 Issued: Feb. 3, 2005. Due: Feb. 15, 2005. Problem... 1.1: Time-Bandwidth Product The time-bandwidth product links the full width at half maximum (FWHM) in the time domain to the corresponding width in the frequency domain. The values are pulse-shape specific, and follow from the Fourier transform relation or the uncertainty principle, as the case may be. The following are amplitude functions of a pulse in the time domain, in complex notation: f(t) = f0 · �1 − τt22� eiω0t for |t| ≤ τ (1) f(t) = 0 for |t| ≥ τ (2) (a) Sketch the intensity function | | f(t) 2 and calculate the full width at half maximum (FWHM) Δt of the intensity function. (b) Calculate the Fourier transform f˜(ω). Sketch the power spectrum | | f˜(ω) and identify the full width at half maximum Δν = Δω/2π. (Hint: Introduce the variable x = (ω − ω0)τ and calculate Δx numerically.) (c) Calculate the time-bandwidth product Δν · Δt for this pulse shape. Problem 1.2: Gires-Tournois Interferometer Gires-Tournois Interferometer (GTI) is essentially a Fabry-Perot resonator with a 100% reflector. As with an ideal high-reflectivity mirror, the whole reflectivity of the device stays 100%. In contrast, the phase delay is, as with a Fabry-Perot, frequency-dependent. Thus the GTI can be used in a laser resonator for dispersion compensation. Using r1 = −√R1, r2 = −√R2 = −1 and assuming that medium 2 has a refractive index 1, the following expression for the amplitude reflectivity can be found: [Show More]

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