2. Correlation analysis

The Wiener theorem, as cited in [49], forms the basis of correlation spectroscopy:

This relates the desired spectrum A_f(ω) of a time-varying signal to the inverse Fourier transform F⁻¹ and the autocorrelation function Φ:

Applying relation (3.62) to Eq. (4.36) and evaluating the Fourier transform yields an expression for the autocorrelation function containing the free parameter Γ that is to be determined:

In principle, this result could be used directly during measurement. The difficulty is that the photocurrent is not continuous. Statistical methods must therefore be applied.

3. Consideration of photon statistics

Let P(n|I) be the probability of detecting n photoelectric pulses for a given signal intensity I. Two assumptions can then be made. First, n is proportional to I:

Second, n(t) and n(t + τ) are assumed to be uncorrelated. Consequently, the probability of recording n₁ events at time t and n₂ events at time t + τ is the product of the individual probabilities. Taking these assumptions into account, and assuming that P(n(t)|I(t)) follows a Poisson distribution, the autocorrelation function becomes:

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