How do I calculate the number of photons emitted?

I am reading through these slides about the transition radiation in the optical range. It says that the spectrum of the intesity of the photons as a function of the frequency is given by:

$$\dfrac{dI}{d\omega}=\dfrac{e^2}{6\pi c} \left( \dfrac{\gamma \omega_p}{\omega} \right)^4$$

where $$\omega_p$$ is the plasma frequency and $$\gamma$$ is the relativistic gamma factor.

How do I calculate from this formula the number of the photons emitted in a given range of frequencies?

• Please explain what this question is about. Commented Apr 25, 2020 at 10:44
• @my2cts I think the question is clear: I have that formula for the intensity of the transition radiation emitted by a particle with a gamma factor $\gamma$ and from that I want to calculate the number of photons emitted instead of the intensity. Commented Apr 25, 2020 at 10:49
• Fine but I did not read the slides. It would help if your question were self contained. Commented Apr 25, 2020 at 12:04

Considering the intensity is in $$W/m^2$$ you will need to consider the area over which you want to calculate the number of photons say $$A$$. You can then multiply the $$dI$$ by the area to get the infinitesimal power $$dP$$
$$dP=dIA=\frac{A\alpha}{\omega^4}d\omega$$ where $$\alpha=\frac{e^2}{6\pi c}\left(\gamma\omega_p\right)^4$$, considering the energy of a photon at a frequency $$\omega$$ is $$E=\hbar\omega$$ the number of photons at $$\omega$$ emmited per unit time for $$dI$$ is then
$$dN=\frac{A\alpha}{\hbar\omega^5}d\omega$$
So to get the number of photons per unit time emitted over a range of frequencies $$\omega_1,\omega_2$$ is
$$N=\int_{\omega_1}^{\omega_2}\frac{A\alpha}{\hbar}\omega^{-5}d\omega=\frac{A\alpha}{4\hbar}\left(\omega_1^{-4}-\omega_2^{-4}\right)$$