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Photon has a specific wavelength $\lambda$. Imagine we created a mode-locked pulse, with $80\: \text{MHz}$ repetition rate, i.e. pulse are separated by $13\: \text{ns}$. The pulse duration is $4\: \text{ps}$, I understand that pulse has a very broad frequency range. One can imagine, a pulse is composed of many monochromatic wave with different wavelengths adding up together in phase (in dispersion-less medium). So, if the peak power is $100\: \text{W}$ and I wish to calculate the number of photons in a pulse, How am I supposed to take the weighting of each wavelength? Or Should One simply calculate using the center wavelength? I do think other components play a role in different energy.

The whole idea of this question is that I have to do single photon correlation experiment by combining a single photon (from the weak signal) with a pulse (from the strong pump), However, if one detect the pulse, How could one which wavelength upconverts the single photon? I imagined pulse is composed of many photons adding together.

Update: My friend proposed that If the pump pulse combined with the photon from a weak signal, you have the center wavelength of the pulse combined with the center wavelength of the photon, to get a new frequency, and you could filter out other wavelength components, to do a single photon detection.

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Lasing is a quantum mechanical effect, and the frequency has a very narrow distribution in frequency from the width of the energy level lines in transitions. See this link for line widths.

So the way I would treat to find the energy of a time interval on a laser beam is to integrate the classical electric field squared folded with the frequency distribution, i.e. get the energy for that time interval. Find the average photon frequency, using the same distribution, and divide the energy in the pulse by the average photon E=h*nu energy. That should give the number of photons with an error given by the width of the Lorentzian distribution.

A pulse would be composed by an enormous number of photons, ( a photon belongs to the quantum mechanics framework), in superposition of their wavefunctions making up the classical field. If you know QED how this happens is discussed here.

Single photon measurements are shown here.

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An easy approach is to take the total energy of the pulse and divide it by the center optical pulsation times $\hbar$ : $$N_{photons} \approx \frac{\text{Total Energy of One Pulse}}{\hbar\omega_{center}} = \frac{\int_0^{+\infty}dt P_{opt}(t)}{\hbar\omega_{center}}$$

This approximation holds when the spectral width of the pulse $\Delta\omega$ is small compared to the center pulsation $\omega_{center}$.

When you start to work with ultra-short pulses (the duration of the pulse decrease and its spectral width increases), you might need to account for the spectral pulsation distribution of your photons, which you could measure via an optical spectrum analyzer for instance.

Cheers

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I worked as firmware engineer for the femto-second laser Maitai. This is the automated version of the tsunami, a well known laser in the industry.

The frequency or wavelength is adjusted by moving a slit in the path of a prism and the bandwidth is adjusted by modifying the opening of the slit. The maximum efficiency is at 800 nm.

The distribution of frequency is gaussian, describing a symetrical distribution above and below 800 nm and a shape similar to any fair dice as seen in statistical math. This mean that you can compute the number of photons like if they were all at the same frequency.

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    $\begingroup$ I'm not sure I agree that this is a non-answer. Part of the question is "how do I account for the spread in the wavelengths, can I just use the central value?" and this is an answer which defines some circumstances under which using the central value of the wavelength/frequency distribution is okay. $\endgroup$ – rob Jul 17 '18 at 15:42

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