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I am studying ultrafast spectroscopy (pump-probe) and I know that ultrashort laser pulses are used for the pump and probe. These pulses often contain multiple frequencies (i.e. polychromatic pulse) that are generated by pulse broadening. Broadening in the frequency domain also means broadening in the time domain, so usually pulse shaping techniques are used to compress the pulse to the Fourier limit (i.e. uncertainty principle).

If I understand correctly, the time domain and frequency domains can be interconverted with Fourier transform. Now, usually pulse shaping involves applying a "mask" on the input light pulse. In my lectures, the professor showed us several examples of amplitude masks and phase masks. For example-

amplitude mask

In this case the amplitude of specific wavelengths are being reduced in an input pulse containing a range of wavelengths. And that changes the shape of the pulse in the time domain.

An example of a phase mask-

phase mask

Here the phases are being changed for different wavelengths (i.e. frequency domain) and the shape of the pulse changes in the time domain.

My problem is that I have no idea how to get to the shape of the output pulse by looking at the shape of the mask (which is what we are supposed to learn from the lecture). I did not find any books on this topic, but I did find some research papers, which did not help me.

I want to know what is the intuitive way to predict the output pulse shape from the shape of the phase or amplitude mask. Any help is appreciated. Also recommend any book or paper you know that is at the undergraduate reading level on this topic.

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  • $\begingroup$ Hint: how can you go from a signal in the time domain to one in the frequency domain? How do you go back? $\endgroup$
    – Jon Custer
    May 25, 2022 at 21:35
  • $\begingroup$ @JonCuster Fourier transform and inverse Fourier transform? But how does that help? I know that already, I just need to figure out how to intuitively predict the shape after Fourier transform. I can't do that in my head. $\endgroup$
    – S R Maiti
    May 26, 2022 at 9:10
  • $\begingroup$ Well, do a half dozen or so different shapes and see what you think. That is how you build intuition. It doesn’t magically just happen (sadly). $\endgroup$
    – Jon Custer
    May 26, 2022 at 12:11
  • $\begingroup$ There is no intuitive way. No one can make fourier transforms in their heads. With experience you memorize a few rules/transforms and you can start making educated guesses about the output, mostly about widths, but there is no way other than using a computer and making the fourier transform of the filter. $\endgroup$ May 29, 2022 at 13:57

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First of all, I think the OP has some misconceptions, so I will start by addressing them "These pulses often contain multiple frequencies (i.e. polychromatic pulse) that are generated by pulse broadening." It is correct to think the time domain and frequency domain are Fourier conjugates. I have an issue with this sentence because ultrashort pulses do contain many many frequencies, not just "often". I think the OP needs some basic understanding of pulses and associated spectra as well the concept of transform limited pulses.

"Broadening in the frequency domain also means broadening in the time domain, so usually pulse shaping techniques are used to compress the pulse to the Fourier limit (i.e. uncertainty principle)."

Broadening in frequency domain means narrowing in the time domain for transform limited pulses. This goes back to my first objection: to be able to obtain a ultrashort pulse in time, one, by definition needs a broad spectrum.

Now to answer the OPs question: guessing the temporal outcome of the amplitude masks applied on the frequency domain is simple. Whatever the shape of the mask, the output temporal shape will be the convolution of the input temporal pulse and the Fourier transform of the mask shape. This clearly shown in the example the OP has provided, Fourier transform of a rect function is a sinc function, and so the output is the convolution of the sinc and the input pulse. Generally people assume the input pulse of Gaussian shape for simplicity.

When it come to phase masks, the situation is a bit more complicated, one needs to take the Fourier transform to come up with a definite answer. Although, there are simple cases that are well known and the OP could benefit from deriving these themselves once and then just knowing/remembering what they would get for a few standard cases (just as the other answers for this question stated). One of these cases would be the example shown by the OP: a linear phase in spectrum (similar to group velocity dispersion accrued upon propagation in a medium) would affect a Gaussian beam as broadening while preserving the Gaussian shape (albeit with a broader Gaussian width of course). This is related to the fact that the Fourier transform of a Gaussian is still a Gaussian, also meaning a transform limited (0-phase difference across the spectrum) pulse of Gaussian spectral shape has a Gaussian temporal shape as well. It could be in the OPs interest to look up what a quadratic or 3rd order phase would do to the temporal pulse shape or looking at the what a linear (or quadratic or 3rd order, etc.) phase in spectrum would do to a originally rectangular (or triangular etc.) pulse.

I hope this helps.

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