SoSe 19 Laboratory of Nano-Optics
Ultrafast and Nonlinear Optics
Exercise 4
1 - Pulse compression
Consider fiber and grating pulse compression of pulses from a Nd:YAG laser. The input pulses are Gaussian with a 75-ps FWHM pulse duration, a 100-MHz repetition rate, a 1.06-microns center wavelength, and 1 W of average power (coupled into the fiber). The single-mode fiber is 250 m long and has normal dispersion with = 23 ps2km-1 and nonlinear coefficient = 4 W-1km-1. The grating pair compressor is setup in a double-pass configuration with gratings ruled at 1200 lines/
mm. Assume that pulses propagate in the fiber without loss and without dispersion.
• Compute and plot the power spectrum subsequent to fiber propagation. Compare your result (spectral broadening, number of peaks) to that expected theoretically based on the
instantaneous frequency picture.
• Estimate the chirp (derivative of the instantaneous frequency with time) at the center of the pulse, and use this information to predict the grating spacing required for optimum pulse
compression. Assume that the gratings are used at the Littrow condition, , and consider only the quadratic spectral phase of the grating pair (neglect higher-order spectral phase).
• Compute and plot the intensity profile obtained as a result of this quadratic compressor.
Compare the pulse duration to that expected based on the bandwidth of the spectrally
broadened pulse. Furthermore, select an appropriate gate function and then calculate and plot a spectrogram of the compressed pulse. Repeat for the case of an ideal compressor. Attempt to relate features present in your spectrograms to features observed in the compressed intensity profiles.
2 - Nonlinear Schrödinger equation and solitons
Write a code to solve the nonlinear Schrödinger equation based on the split-step Fourier method.
Confirm your code by stimulating pulse propagation for known test cases (e.g., dispersion only, self-phase modulation only, N = 1 solitons). Then apply your code for the following:
• Compute the temporal and spectral evolution of N = 4 and N = 5 solitons within one soliton period.
• Simulate the propagation of input pulses of the form for various values of in the range . Plot the temporal intensity profiles after a propagation length sufficient that soliton and dispersive wave components can be distinguished visually (possibly using a logarithmic intensity scale). Compute the energies of soliton and dispersive wave components and compare to the prediction Usoliton = Uinput
• Simulate the evolution of a N = 1 soliton with loss. The loss coefficient should be small enough that the solution propagates adiabatically. Plot the variation in pulse duration with pulse energy (energy will decrease with propagation length) and compare to the prediction area =
and .
Exercises selected from chapter 6 of Ultrafast Optics by A.M. Wiener (J. Wiley & Sons, 2009).
