coherenceinterferometrylengthvisible-light
I have measured the spectrum of the LED in my interferometry set-up, and now I want to calculate the coherence length from it. A commonly found formula is $l_{coh} = \frac{c}{\Delta f}$, sometimes with an additional factor for the shape of the spectrum. However, I have only ever found factors for Gaussian and Lorentzian lineshapes, and mine is neither.
I am looking for either a way to determine this factor, or derive the coherence length from the spectrum in a different way.
Thanks in advance.
The coherence length is just the coherence time multiplied by the propagation speed.
To understand the coherence time, say you have a wave described, in complex notation, by $$ E(t) = A(t) e^{i \omega t} $$ where $A(t)$ is a slowly varying complex amplitude. You make this wave interfere with a delayed version of itself and collect the intensity $$ |E(t) + E(t-\tau)|^2 = |E(t)|^2 + |E(t-\tau)|^2 + 2\Re\big(E(t)E^*(t-\tau)\big). $$ where $\Re$ means real part and $^*$ means complex conjugate. The interference term is $$ 2\Re\big(E(t)E^*(t-\tau)\big) = 2\Re\big(A(t)A^*(t-\tau)e^{i \omega \tau}\big) $$ If $A(t)$ is constant, or roughly constant within a time interval $\tau$, then this becomes $$ 2|A(t)|^2 \cos(\omega \tau) $$ which is the interference pattern. On the other hand, if $A(t)$ fluctuates sufficiently fast, and $\tau$ is larger than its correlation time, then $A(t)A^*(t-\tau)$ averages to zero and there is no interference. Thus, the coherence time can be simply seen as the correlation time of the complex amplitude $A(t)$.
Now, I'm not sure there is a very quantitative definition of the correlation time. You could define it as the delay where the autocorrelation function drops below some arbitrary threshold. This is equivalent to setting a threshold on the visibility of the interference pattern. The relationship with the shape of the spectral line should also be apparent: the squared modulus of the Fourier transform of $A(t)$ is the shape of the line (the spectrum of the wave shifted by $-\omega$). It is also the Fourier transform of the autocorrelation function of $A(t)$. Thus, when the line is wide, the autocorrelation function is narrow,and the coherence time is short.
The beam of white light is refracted by the prism, but the different wavelengths in the light are refracted by different angles. That means when the light falls on the screen the different wavelengths are spread out over a region of the screen. This is what is meant by dispersion.
The dispersion is linear if the different wavelengths of light are spread out evenly across the screen. So the wavelength at the point I've shown as $d$ would be given by:
$$ \lambda_d = \lambda_{red} - \frac{d \left(\lambda_{red} - \lambda_{blue}\right)}{L} $$
In general the dispersion won't be linear, but for glass across the range of red to blue light it's not a bad approximation. The reason this matters is when you try to calculate the coherence length from the size of the slit. The coherence length is given by:
$$ L = \frac{2\space ln(2)}{\pi n} \frac{\lambda^2}{\Delta \lambda} $$
Because the slit has a finite width the image is casts on the screen has a finite width and therefore covers a range of frequencies. Going back to my diagram above, suppose the image of the slit on the screen has a width of $w$, then if the dispersion can be assumed linear the range of frequencies is simply:
$$ \Delta \lambda = \frac{w}{L} \left(\lambda_{red} - \lambda_{blue} \right) $$
Put this value for $\Delta \lambda$ back into the equation for the coherence length to get the effect of the slit width on coherence length.
Best Answer
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:
$$\sqrt{\frac{\int_0^\infty x^2\,\Gamma(x)^2\,dx}{\int_0^\infty \Gamma(x)^2\,dx}}$$