Calculating the coherence length from a spectrum I have measured the spectrum of the LED in my interferometry set-up, and now I want to calculate the coherence length from it. A commonly found formula is $l_{coh} = \frac{c}{\Delta f}$, sometimes with an additional factor for the shape of the spectrum. However, I have only ever found factors for Gaussian and Lorentzian lineshapes, and mine is neither. 
I am looking for either a way to determine this factor, or derive the coherence length from the spectrum in a different way.
Thanks in advance.
 A: Presumably you have measured your spectrum as a function of wavelength, so you have $\mathscr{F}(\lambda)$, which is an power per unit wavelength. You must now convert this power per unit frequency spectrum.
So we seek $\mathscr{G}(f)$ where $\mathscr{G}(f)\,|df| = \mathscr{F}(\lambda)\,|d\lambda|$; given $c = f\,\lambda$ we have:
$$d\lambda = -\frac{df}{f}\,\lambda = - \frac{c}{f^2} df$$
so that 
$$\mathscr{G}(f) = \frac{c}{f^2} \mathscr{F}\left(\frac{c}{f}\right)$$
So now we have derived our power spectral density $\mathscr{G}(f)$ from your experimental $\mathscr{F}$ spectrum as a function of wavelength. This is then converted to an autocorrelation function of time by the Wiener-Khinchin theorem:
$$\tilde{\Gamma}(t) = \int_{-\infty}^\infty e^{2\pi\,i\,f\,t} \mathscr{G}(f) \,df=2\,\mathrm{Re}\left(\int_{f_{min}}^{f_{max}} e^{2\pi\,i\,f\,t} \frac{c}{f^2} \mathscr{F}\left(\frac{c}{f}\right) \,df\right)$$
where $[f_{min},\,f_{max}]$ is your experimental measurement interval.
So now you convert your autocorrelation as a function of time to autocorrelation as a function of shift displacement $x = c\,t$, so your final autocorrelation function will be:
$$\Gamma(x) = \tilde{\Gamma}\left(\frac{x}{c}\right) = 2\,\mathrm{Re}\left(\int_{f_{min}}^{f_{max}} \exp\left(2\pi\,i\,f\,\frac{x}{c}\right) \frac{c}{f^2} \mathscr{F}\left(\frac{c}{f}\right) \,df\right)$$
So now you have to decide how you will define your coherence length: common definitions include (1) the shift displacement $x$ at which $\Gamma(x)$ is $1/e$ times $\Gamma(0)$ and (2) the rms spread:
$$\sqrt{\frac{\int_0^\infty x^2\,\Gamma(x)^2\,dx}{\int_0^\infty \Gamma(x)^2\,dx}}$$
A: A detector generally provides values that are proportional to the power spectral density (PSD) of the radiation hitting it.  Energy per time interval (collection time), per spectral interval (pass band, slit width, pixel width, ...).
The Fourier transform of the PSD is the time-domain autocorrelation function of the radiation. The autocorrelation function gives you a direct window on the correlation time.  The FT will be peaked at zero (perfect correlation when the signal is on top of itself) and decline from there as the correlation reduces (imperfect correlation when the signal is offset from itself).  Once you have the autocorrelation function you will have to make some judgement about how to get the correlation time from it.  The time at which the correlation falls to 1/2 its peak might be a good choice.   Multiply that by $c$ and you have your correlation length.
Note that you will get slightly different answers if you choose some other criterion for the correlation time, for example if you choose $1/e$ of peak value.  Also, the spectral sensitivity of your detector will change things a bit.  Moral:  at the end of the day you might do no better than $c/\Delta f$.  It depends on what you want to do with the information once you have it.
Update 
I see that @Wet...  has posted a more detailed response.  He or she correctly adds the necessity of converting your wavelength data to frequency data.
