$1)$ A mode$-$locked laser, operating at $547\mathrm{.}5$ nm$,$ generates an average power of $1$ watt, with a pulse repetition rate of $100$ MHz$.$ The gaussian shaped pulse has a duration is $0\mathrm{.}44$ picoseconds.$$ The light is focused into $1$ meter of fiber.$$ Neglecting dispersion, and given that the fiber nonlinearity g $=0\mathrm{.}88$ W$-1$km$-1,$ calculate:

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a$)$ Energy per pulse of the laser

b$)$ The peak power of a single pulse.

c$)$ The amount of spectral broadening of the pulse after propagating in the fiber.

d$)$ The instantaneous frequency impressed across the pulse in units of nm$/$psec$.$

e$)$ The group delay needed for optimal compression.

$$Given a pulse compressor with a groove spacing of $1$ micron, and diffraction angle of $60$ degrees, calculate:

f$)$ The group delay of the compressor in units of ps$/$nm$/$m$.$

g$)$ The distance between gratings for optimal compression using a double pass geometry $($e$.$g$.$ B $=$b$/2).$

h$)$ The compressed pulse duration.