# How does the Michelson interferometer measure the self-coherence function of coherent light in an incoherent background?

How does the Michelson interferometer measure the self-coherence function of coherent light in an incoherent background? Detection of coherent light in an incoherent background (Coutinho et al., 1999) is still the best analysis and implementation of this concept despite considerable progress in semiconductor fabrication and photonics since its publication.

The reason I ask this question is because airplane pilots need to see the landing strip, ground landmarks and the horizon while at the same time harmful visible light laser rays are filtered.

• Your last paragraph has absolutely nothing to do with interferometers. – Carl Witthoft Jun 14 '16 at 12:30

As you change the relative arm lengths of a Michelson interferometer, the transmission (or reflection) coefficient of the interferometer ranges from $T=0$ to $T=1$ for coherent light, but, if designed properly, will always have a transmission of $T=0.5$ for incoherent light.

If we define the length of the two arms of the Michelson to be $L_1$ and $L_2+\delta\ell$, where $L_1$ and $L_2$ are macroscopic distances on the order of meters and $\delta\ell$ is a microscopic distance on the order of micrometers, then the interference properties of the Michelson only depend on $\delta\ell$ for highly coherent light like a laser. However, the interference properties of the Michelson for incoherent light depend on the relative macroscopic distances. If $|L_1-L_2|\gg L_c$, where $L_c$ is the coherence length of the incoherent light, then the Michelson will not display any interference regardless of $\delta\ell$.

• Did you mean to write L2 = L1 + δℓ? I would like to extend your analysis where L1 and L2 are macroscopic distances on the order of centimeters so we could miniaturize Coutinho et al's Michelson interferometer to sit directly on the top edge of a Boeing 737 cockpit window. Is is possible to do that? Finally, how do I modify your supern analyis to take into account the angle of incidence? Thank you very much. – Frank Jun 13 '16 at 23:45
• @Frank No, it is important that L1+L2 not be equal for incoherent light to be unaffected by interferometer. The difference in length between them must be greater than the coherence length of the incoherent light (~few mm) in order to work. I do not believe that this will be an effective tool for your application because you will have to hold the interferometer at a point of destructive interference for the coherent light which is different for each wavelength that you might want to reflect. – Chris Mueller Jun 14 '16 at 1:01
• Thank you for your high quality comment.With regard, to your statement " I do not believe that this will be an effective tool for your application ", are you implying that it is physically impossible to build a tunable interferometer (i.e hardware) over the RGB wavelengths? If it is impossible to build such hardware, could we resort to digital signal processing software to cancel the coherent laser pointer beam? – Frank Jun 14 '16 at 1:40
• @Frank You might be able to use DSP techniques to detect the wavelength of the laser and lock the interferometer to reject that particular wavelength. – Chris Mueller Jun 14 '16 at 12:06
• @ChrisMueller but you can't lock a wavelength and all possible entrance angles. – Carl Witthoft Jun 14 '16 at 12:31

The order of the dark fringe of the coherent source to lock to is the first order because the self coherence function's first minimum corresponding to the sinc function's envelope is located at an interval of lambda\4 multiplied by the distance d where lambda equals wavelength and d is equal to the distance between a fixed mirror's image and an moving mirror's image.

The information of the greatest interest is the spectrum in one branch of the interferometer called the interferogram's inverse Fourier transform. As a result, the interferometer's signal as a function of the optical path difference and the spectrum as a function of wavenumber, also known as 1/wavelength,form a Fourier transform pair.

If you wish to explore this topic, please read chapter 6 , Fourier Trahsform Spectroscopy , in the book written by the Finnish professors, J. Kauppinen and J. Partanen , titled Fourier Transforms in Spectroscopy.