How do i derive the relation between coherence time and bandwidth

[walkingpoet]

I was just curious, sorry if it is already asked,

I was going through a book on photonics and came across the relation

coherence time, t = λ*λ/Δλ

I did not find any proof of this. Can anyone explain me how this relation is found?

 

[DrDu]

The coherence time is the half width of the autocorrelation function. On the other hand, the Fourier transform of the autocorrelation function is the power density. The smaller the coherence time the larger the spectral width of the power density distribution, see: http://en.wikipedia.org/wiki/Wiener–Khinchin_theorem

 

[Andy Resnick]

A clever way (IMO) to derive that is to start with the relation (ΔE)(Δt) ≤ h and E = hω. Then you get the time-bandwidth relation trivially. To convert from frequency to wavelength, simply use c/λ = ω and when you differentiate. you get Δω = -c/λ^2 * Δλ