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Almost every textbook and a journal article that deals with thermal noise in the context of optical fields (such as signal processing, micro-ring resonators, laser sources, etc.) assumes that the autocorrelation function of the thermal noise in the system (whether it be studied from a quantum or classical point of view) is given by a delta-function of the kind $\langle x(t)x(t')\rangle = \delta(t-t')$. This implies that the spectral density of the noise in frequency space is constant and thus contains all possible frequencies in an equal amount.

However, we know that in physical systems, such as an elastic material or a dielectric, there is a frequency cutoff on the higher end which is given by either the spacing between atoms (in the case of space-correlations) or by the Raman shift (in the case of optical fields in time-domain), so the spectrum of the noise does not actually contain infinite frequencies in it.

How then, do we justify the use of delta-correlated noise in modelling these physical systems? What makes the model appropriate in theoretical terms? Is it that the frequency cutoff occurs at such high frequencies that we can almost think of it as "infinite"?

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Is it that the frequency cutoff occurs at such high frequencies that we can almost think of it as "infinite"?

Yes.

In physics, "infinite" is generally always shorthand for "finite, but so big (as compared to the other scales in the problem under consideration) that the details of the size stop mattering and it might as well be infinite".

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