Spectral pulse shaping (${\tilde{E}}_{\mathrm{out}}(\omega )=M(\omega ){\tilde{E}}_{\mathrm{in}}(\omega )$)

Spectral pulse shaping changes an optical pulse by applying a complex amplitude-and-phase mask $M(\omega )$ to its frequency components. The output in time is the inverse Fourier transform of the masked spectrum. A passive device cannot amplify unavailable spectral components; finite bandwidth and spectral resolution constrain the realizable waveforms and time window.

A 4f pulse shaper uses a diffraction grating to disperse an input pulse, a lens to focus its frequencies at different positions in a shared Fourier plane, a spectral mask or spatial light modulator there to modify them, and a second lens and grating to recombine them. In the symmetric idealization the distances are $f,f,f,f$, with the lenses separated by $2f$. Shaping the spatially resolved spectrum implements spectral pulse shaping rather than a direct time-domain mask.