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(a) Show that Fourier transform of a Gaussian function is also a Gaussian function. (b) Consider a 3 x 3 spatial mask that averages the four closet neighbors of a point (x,y) but excludes the point itself from the average. Calculate equivalent filter H(u,v) in frequency domain. (c) Calculate the equivalent filter H(u,v) that implements in the frequency domain the spatial operations performed by Laplacian mask

1. Can we construct a thermocouple from non-metallic materials? Why? 2. Why do PMT and MCP need to operate in vacuum? 3. In Raman spectroscopy, if the resolution of the spectral measurement is 0.1 cm , what would be the bandwidth of the exciting laser which has the center wavelength of 488.0 nm. Please give the bandwidth in the unit of nm. 4. Can we get Lamb dip spectroscopy using ordinary (not laser) light sources? Why? 5. What are the requirements/arrangement for the laser beams if we want to do three-photon Doppler's free spectroscopy? 6. Prove that convolution of two Gaussian functions is a Gaussian function. If the duration of the input Gaussian functions are T1 and T2, what would be the duration of the convoluted function? 7. (Optional) Why does the photodiode's response fall into nanosecond/picosecond regime and not femtosecond regime?Problem 4 (Kittel, page 458, p.2) Approach to Gaussian distribution. Show that the Poisson function P(M) =^ exp(- )/ M! closely approaches a Gaussian function in form, for large . That is, show when / is close to that P(N) = A exp[-B(N-)'], where A, B are quantities to be determined by you.The following function is called a Gaussian Function. Gaussian functions are used in statistics, image processing, sig processing, and in mathematics to solve heat and diffusion equations. -(x-b)2 f(x) = ae 202 Find the first and second derivative of the given Gaussian function when a = 4, b = -1, and c = 1 f' () = f" (x) =<><><>