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For an optical source with power spectrum as 𝐼(𝜔), the autocorrelation function is related to power by a Fourier relation (Wiener–Khinchin theorem).

F{𝑔(𝜏)} = 𝐼(𝜔)

However, the above relation does not include any factor (if exist) which could change the phase of light randomly but only consider frequency(wavelength) broadening. The random phase should also change the value of autocorrelation since randomization is destructive to correlation.

So, Does random phase variation not exist? If exist, is it included in Wiener–Khinchin theorem? If Yes, then how? If exist, if not included in Wiener–Khinchin theorem, then how to incorporate this random phase variation in autocorrelation value?

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  • $\begingroup$ Your question is unclear; please, define what you mean by $g(\tau)$. That the autocorrelation $g(\tau)$ is the Fourier transform of the spectrum $G(\omega)$ is just a definition, the interesting part is that if $X(t)$ is the stochastic process and $g_X(\tau)=\mathbf E[X(t)X(t+\tau)]$ then $G_X(\omega)=\lim_{A\to\infty}\frac{1}{2A}\mathbf E[|\int_{-A}^A dt X(t)e^{-j\omega t}|^2]$, which actually explains the meaning of the definition. $\endgroup$
    – hyportnex
    Commented Aug 7 at 12:44
  • $\begingroup$ Ok. With the new definition you provided, could you understand the question? $\endgroup$
    – MohitKumar Singh
    Commented Aug 10 at 1:52
  • $\begingroup$ The magnitude square removes the phase information from the spectrum; any sensible definition of a power spectrum estimate should remove phase because power of a sinusoid is phase independent. $\endgroup$
    – hyportnex
    Commented Aug 10 at 2:06

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Power spectral density $S_\nu(f)$ is directly related to the phase noise by $S_\nu(f) = f^2 S_\varphi(f)$. This relation comes from the fact that $\nu(t) = \frac{1}{2\pi}\frac{\text{d}\varphi}{\text{d}t}$ and $\mathcal{F}(\frac{\text{d}f}{{\text{d}t}})=2\pi i \omega \text{F}(\omega)$. So indeed the information about random phase fluctuations is already encoded in the shape of your PSD.

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  • $\begingroup$ Could you please share some easy references where I can learn this. Thank you. $\endgroup$
    – MohitKumar Singh
    Commented Aug 8 at 11:11
  • $\begingroup$ @MohitKumarSingh do you mean the math standing behind the relation I mentioned? $\endgroup$
    – Noct
    Commented Aug 8 at 17:26
  • $\begingroup$ Yes. The relationship mentioned in the first line between Power spectral density and phase noise. Where does it come from? And I couldn't get what 'nu' and 'phi' symbols mean here. I assume f and omega are frequencies. Could you share any book article where I can learn from basics. In the books of optics, the phase term is never mentioned when discussing relation between power and autocorrelation. $\endgroup$
    – MohitKumar Singh
    Commented Aug 10 at 1:49
  • $\begingroup$ Can you elaborate on the context in which you need that information? Then I would be able to provide better sources. BTW, Nu is a natural frequency and a phi a phase, see section 3.1 here: tf.nist.gov/general/pdf/2220.pdf . The third equation is a property of the Fourier transform. $\endgroup$
    – Noct
    Commented Aug 10 at 11:59

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