Fourier Transformation and the Coherence of Light

The Fourier transformation is one of the most frequently used techniques in mathematical physics, which allows for instance to transform any time dependent function f(t) into its frequency spectrum

(1) F(ω) = _-∞∫^{^{^∞}}dt f(t)^.e^-iωt .

It is often overlooked however that even for f(t) being a real function, F(ω) will in general be a complex function

(2) F(ω) = |F(ω)|^.e^iφ(ω) ,

where φ(ω) is a certain phase shift.
It is only the inclusion of the phase factor which in general allows the inverse Fourier transform

(3) f(t) = 1/2π ^._-∞∫^{^{^∞}}dω F(ω)^.e^iωt .

Inserting instead only the power spectral density |F(ω)| into Eq.(3) will in general not lead to the correct function f(t). So a knowledge of merely the intensity spectrum of a light source does not allow to make any conclusions regarding the temporal (phase) coherence of the radiation as it is often done by means of the relationship

(4) Δω^.Δt = 1/2 .

where Δω and Δt are the equivalent widths of the frequency and time spectrum respectively.
For instance, if one has a Gaussian frequency spectrum centered at frequency ω₀ i.e.

(5) |F(ω)| ~ e^{^{-((ω-ω₀)/ Δω)²}} .

then a Fourier transformation of this spectrum would suggest that the corresponding time series also has a Gaussian envelope i.e.

(6) f(t) ~ e^{^iω₀t}^.e^{^-(t/Δt)²} .

but this would obviously require that the phase shift φ(ω) in Eq.(2) is correspondingly fixed at all times for the Fourier components at all frequencies. This condition does however in general not hold for time series occurring in nature. Natural light for instance is composed of the random and uncorrelated emissions of a large number of atoms, and thus for the resultant total electromagnetic wave field no fixed phase relationship between any two frequencies exists. In this sense, the thermal Doppler shift of the individual emissions of atoms for instance produces (for negligible natural linewidths) exactly a Gaussian spectrum as given by Eq.(5), and here it is obvious that the time dependence of the radiation field is not at all given by Eq.(6). This is not only if the emissions are so infrequent that they don't overlap at all (in which case it should be obvious that the temporal and spatial coherence of the radiation field is given by that of the individual emissions), but also if the random emissions overlap to constitute the resultant radiation field. Consider for instance the emission of wavetrains ('photons') with a fundamental frequency ω₀ and a damping constant γ (assumed here independent of frequency) subject to Doppler broadening with a width Δω: if the individual wave trains are offset in time by a (random) amount t₀(ω), then their superposition yields the time signal (leaving out the normalization constants here)

(7) f(t) ~ _-∞∫^{^{^∞}}dω e^{^{-((ω-ω₀)/ Δω)²}}^.e^{^{-γ^.(t-t₀(ω))}}^.e^{^{iω(t-t₀(ω))}} ,

where the complex notation for the wave train has been used.
Now, since t₀(ω) is a (random) function of ω, it is obvious that t-t₀(ω) can not simply be replaced by a variable t' such that Eq.(7) could be interpreted as the Fourier transform of exp[-(ω-ω₀)/ Δω)²] in terms of t'. On the other hand, we know that for those wave trains that overlap significantly at all, |t₀(ω)| has on average a value of the order of 1/γ. So if we consider only times t<<1/γ, we can approximately replace iω(t-t₀(ω)) by -iωt₀(ω), and thus we have

(8) <f(t)> ~ e^{^-γt}^._-∞∫^{^{^∞}}dω e^{^{-((ω-ω₀)/ Δω)²}}e^{^γt₀(ω)}^.e^{^-iωt₀(ω)} (t<<1/γ) .

where the brackets should indicate the statistical ensemble average over the time (phase) offsets.
The integral over the individual emissions is now not only independent of t, but, since t₀(ω) is a random function, independent of the Doppler broadening Δω altogether. This is because the 'phase factor' exp[-iωt₀(ω)] is now not only not sinusoidal anymore (which it would only be for t₀ independent of ω), but merely provides a random factor for each emitted wave train with a value between -1 and 1 (for the real component). Since the Gaussian Doppler broadening distribution is normalized (if one includes the here neglected normalization factor), this means that if the distribution is made up of a large number N of overlapping wavetrains, the integral will, according to the laws of statistics, be of the order of √N (the factor exp[γt₀(ω)] will on average be of the order of 1), i.e.

(9) <f(t)> ~ e^{^-γt}^.√N .

This shows that for an incoherent (random) superposition of wave trains, the temporal coherence of the combined wave field does not at all depend on the overall width of the spectrum (the 'spectral coherence') Δω (here given by the Doppler broadening) but only on the frequency width γ of the individual wavetrain ('photon') (which of course may not only be given by the natural line width but also the collisional line width).

One should also note that the the bandwidth γ of the individual wave train is in fact the narrowest bandwidth to which the spectrum can be reduced: if one multiplies for instance a delta function δ(ω'-ω) to the integrand in Eq.(7) (representing thus an ideal monochromator at frequency ω'), the resultant signal is

(10) f(t) ~ e^{^{-((ω'-ω₀)/ Δω)²}}^.e^{^{-γ^.(t-t₀(ω'))}}^.e^{^{iω(t-t₀(ω'))}}

and the bandwidth of this signal would thus still be determined by the decay constant γ (which in general may of course depend on frequency). This is also intuitively obvious, because in order to increase the coherence even further, one would need an interaction time between the monochromator and the original signal that is longer than the length of the latter, which is a logical impossibility. Reversely, the observed width of an absorption line in a given spectrum indicates thus that the temporal coherence at the corresponding frequency is at least as high as given by this width.
On the other hand, the coherency can of course be arbitrarily decreased (i.e. the bandwidth increased) by 'chopping' the signal into smaller segments (e.g. by means of an electro-optical shutter).