How do i derive the relation between coherence time and bandwidth

AI Thread Summary
The discussion centers on the relationship between coherence time and wavelength in photonics, specifically the formula t = λ²/Δλ. Participants seek clarification on the derivation of this relation, noting that coherence time represents the half-width of the autocorrelation function. The Fourier transform of the autocorrelation function relates to power density, indicating that shorter coherence times correspond to broader spectral widths. A suggested approach to derive the relation involves using the uncertainty principle and converting frequency to wavelength through differentiation. Understanding these concepts is crucial for grasping the principles of coherence in photonics.
walkingpoet
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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?
 
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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
 
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
 
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