Presumably you have measured your spectrum as a function of wavelength, so you have $\mathscr{F}(\lambda)$, which is an power per unit wavelength. You must now convert this power per unit frequency spectrum.
So we seek $\mathscr{G}(f)$ where $\mathscr{G}(f)\,|df| = \mathscr{F}(\lambda)\,|d\lambda|$; given $c = f\,\lambda$ we have:
$$d\lambda = -\frac{df}{f}\,\lambda = - \frac{c}{f^2} df$$
so that
$$\mathscr{G}(f) = \frac{c}{f^2} \mathscr{F}\left(\frac{c}{f}\right)$$
So now we have derived our power spectral density $\mathscr{G}(f)$ from your experimental $\mathscr{F}$ spectrum as a function of wavelength. This is then converted to an autocorrelation function of time by the Wiener-Khinchin theorem:
$$\tilde{\Gamma}(t) = \int_{-\infty}^\infty e^{2\pi\,i\,f\,t} \mathscr{G}(f) \,df=2\,\mathrm{Re}\left(\int_{f_{min}}^{f_{max}} e^{2\pi\,i\,f\,t} \frac{c}{f^2} \mathscr{F}\left(\frac{c}{f}\right) \,df\right)$$
where $[f_{min},\,f_{max}]$ is your experimental measurement interval.
So now you convert your autocorrelation as a function of time to autocorrelation as a function of shift displacement $x = c\,t$, so your final autocorrelation function will be:
$$\Gamma(x) = \tilde{\Gamma}\left(\frac{x}{c}\right) = 2\,\mathrm{Re}\left(\int_{f_{min}}^{f_{max}} \exp\left(2\pi\,i\,f\,\frac{x}{c}\right) \frac{c}{f^2} \mathscr{F}\left(\frac{c}{f}\right) \,df\right)$$
So now you have to decide how you will define your coherence length: common definitions include (1) the shift displacement $x$ at which $\Gamma(x)$ is $1/e$ times $\Gamma(0)$ and (2) the rms spread:
$$\sqrt{\frac{\int_0^\infty x^2\,\Gamma(x)^2\,dx}{\int_0^\infty \Gamma(x)^2\,dx}}$$
To start with the double slit experiment gives interference even when the beam is composed by one photon at a time. The spot on the screen a photon/particle the statistical accumulation the interference seen as expected classically too.
The joint comes because the photon as a quantum mechanical entity has a wavefunction that is the solutions of Maxwell's equation, treated as operators operating on the wave function. The E=h*nu identity the photon carries is the same as the frequency of the classical solution of Maxwell's equation and that , together with the phase attached to the wavefunction allow the continuity of classical down to quantum single photon level For a formal treatment how classical emerges from an ensemble of photons have a look at this blog entry.
There exists a fundamental difference between classical and quantum equations and their solutions, but also a continuity at the interface.
Edit after question edit
How is the theory of partial coherent light related to quantum-mechanics?
This needs somebody familiar with the formalism of both, but I believe the connection should follow the method in the link, how classical electromagnetic beams emerge from an ensemble of photons.
]>... the amplitude of a wave function ... But is this really a fundamental difference, or just a difference in the common practices of the respective theories?
the square of the wavefunction is the connection with predictions and experiments in quantum mechanics, it is the probabilistic nature that makes the difference with the classical framework, as far as I know.
How much of the strange phenomena of quantum-mechanics can be explained by the theory of partial coherent light alone, without any reference to particles or measurement processes?
Phenomenon is " an observable" , observing something implies a measurement process, measurement implies interaction, picking up a point that will contribute to the quantum mechanical probability distribution (or building up the distribution itself by continuous observations) so there is an inherent contradiction in this part of the question.
Best Answer
Superconductivity is about the appearance of an energy gap $\Delta$ in the excitation spectrum of the electron quasi-particles, which become paired in the form of a Cooper pair below the critical temperature $T_{c}$.
An other important energy scale in metals (superconducting or not) is the Fermi energy $E_{F}$, which represents the baseline energy of all conduction electrons. So one can generate many interesting criteria by manipulating the energy gap and the Fermi energy. For instance $\Delta/E_{F}$ represents the strength of the superconducting binding of electrons in the form of Cooper pairs.
From these energies : the energy gap $\Delta$ and the Fermi energy $E_{F}=mv_{F}^{2}/2$ with $v_{F}$ the Fermi velocity and $m$ the conduction band mass, one can generate a natural characteristic length $\ell_{b.}\sim\frac{\hbar v_{F}}{\Delta}$. This characteristic length naively represents the size of the Cooper pair, and it is called the coherence length.
This was for clean system. In diffusive systems, there is in addition the diffusion constant $D$ which represents an area explored by unit of time. So the natural length scale constructed from $\Delta$ is $\ell_{d.}\sim\sqrt{\frac{\hbar D}{\Delta}}$. Since one usually takes $D\sim v_{F}\cdot l$, with $l$ the mean-free path, this is still related to the Fermi velocity, though different exponent.
Coherence length depends on temperature (because $\Delta$ depends on temperature, as e.g. $\Delta\sim\sqrt{T-T_{c}}$ in the Ginzburg-Landau regime), and it's related to the phase rigidity of the superconducting condensate : one needs to tilt the superconducting phase over the size of the superconducting coherence length to get some current. Also, coherence length is related to the penetration length a superconductor exhibits in contact with a normal metal. Since $\Delta\rightarrow0$ as $T\rightarrow T_{c}$, the coherence length diverges at the critical temperature, a hallmark of a critical phenomena (i.e. a second order phase transition here).
Superconducting coherence length appears naturally in many circonstances, as e.g. the penetration of the superconducting properties over non-superconducting materials. For instance, the Josephson current $j$ behaves as $j\sim \ell_{b.}/x$ in ballistic systems and as $j\sim e^{-x/\ell_{d.}}$ in diffusive systems at distance $x$ from the superconducting interface.