Polychromatic propagation of EM wave

Question

Say I have a continuous light source emitting a flat light spectrum $\hat G(\nu)$ centered at $\nu_o$ with bandwidth $\Delta \nu$ (i.e. $\hat G(\nu)=\mathrm{rect}((\nu-\nu_o)/\Delta \nu)$) and want to see its time evolution. My understanding is I just need to take the Fourier transform. This can be done easily enough and evaluates to $$G(t)=\frac{\Delta \nu}{\sqrt{2\pi}}\mathrm{sinc}{(\frac{\Delta \nu t }{2})}\exp{(i\nu_o t)}.$$

I don't understand why the result is a decaying wave. When you have a polychromatic light source (like a light bulb) I don't see it decaying in time. I know this type of solution makes sense when talking about mutual coherence (which loses its coherence with time delays for a polychromatic source). But here we are just talking about the time propagation of a polychromatic field. Have I made a mistake with my thinking?

Answer 1

The solution you have given is completely correct: if you have a finite pulse, which starts off at zero, has an on-ramp and an off-ramp and then decays to zero, the spectrum will be polychromatic, and you can easily disperse the light source into a rainbow using a prims. Pulsed laser sources are extremely common (cf. e.g. this thread), and when in operation they typically give off rainbow hues at various points.

When you want to describe a light bulb, you don't just want to describe a polychromatic source – you want an incoherent source (which must by definition be polychromatic). The error in your thinking is the assumption that a measurement of the power spectral density, which gives you $$\left|\hat G(\nu)\right|=\mathrm{rect}((\nu-\nu_o)/\Delta \nu),$$ can be extended into an assignment of $\hat G(\nu)=\mathrm{rect}((\nu-\nu_o)/\Delta \nu)$ with a flat spectral phase – or even any given definite assignment for the spectral phase. For dealing with incoherent light, you need to use a statistical approach for dealing with the signal, both on the time domain and on the frequency domain.