  In the framework of linear optics, light fields do not interact with each
other in a medium. Yet, when their field amplitude becomes comparable to the
electron binding energies of matter, the nonlinear motion of these electrons
emits new dipole radiation whose amplitude, frequency and phase differ from the
incoming fields. Such high fields are typically achieved with ultra-short,
femtosecond (1fs = 10-15 sec.) laser pulses containing very broad frequency
spectra. Here, the matter not only couples incoming and outgoing fields but
also causes different spectral components to interact and mix through a
convolution process. In this contribution, we describe how frequency domain
nonlinear optics overcomes the shortcomings arising from this convolution in
conventional time domain nonlinear optics1. We generate light fields with
previously inaccessible properties because the uncontrolled coupling of
amplitudes and phases is turned off. For example, arbitrary phase functions are
transferred linearly to the second harmonic frequency while maintaining the
exact shape of the input power spectrum squared.
  This nonlinear control over output amplitudes and phases opens up new avenues
for applications based on manipulation of coherent light fields. One could
investigate c.f. the effect of tailored nonlinear perturbations on the
evolution of discrete eigenmodes in Anderson localization2. Our approach might
also open a new chapter for controlling electronic and vibrational couplings in
2D-spectroscopy3 by the geometrical optical arrangement.
