## My latest adventure in interferometry

This weekend was graduation weekend which meant two days of events in which we, the faculty, basically serve as eye candy. Thus that means listening to lots of speeches. Fortunately, our current Dean is a master at getting through all the graduates’ names quickly. At any rate, this being the first graduation with our newly reduced parking capacity (someone told us we had too much — I’m not kidding), traffic was worse than usual and so I hung out in the lab for a bit after the ceremony ended waiting until I could get out of the parking lot in a timely fashion.

So, let me first say that I have an increasingly god-like reverence for experimentalists. Lesser mortals would go utterly insane from the combination of tedium and unexpected results. As a theorist, I figure that I’m already insane so it doesn’t matter. After an entire semester of getting nothing but parallel lines on my outputs, I ended up getting the “bull’s eye” pattern which is clearly a laser cavity mode (at that point, I was ready to beat my head against the wall).

Curiously (or not?), I got it when the arms were each 8 inches long or 16 inches long but not when the arms were 18 inches long or 20 inches long. In the latter two cases, absolutely nothing I did produced an interference pattern whereas it was pretty easy in the former two cases (the closer it was to a parallelogram, the better the pattern). Now, this summer I’m ordering some fully-gimbaled mirror holders that match the mirror surface up with the center line to make aligning easier. I’m hoping I can quantify some of the nuances of the alignment a bit better with these.

Anyway, that all leads me to two conclusions:

• the interference pattern in an MZI has something to do with cavity modes; and
• textbooks (and even some papers) on optics, particularly on MZIs and Michelsons, are complete crap.

On another note, in reply to my last post on this topic, someone noted that my calculation of the coherence length might be incorrect and should actually be closer to 300 microns. So in doing it again, I got a completely different number. Maybe someone can locate my error. The linewidth of the laser I’ve got (if I’m reading it correctly) is 2 nm. I have no idea if the lineshape is Lorentzian or Gaussian, but I’m just going to guess Lorentzian for now. Thus the coherence time is given by $\tau_{c}=1/\Delta\omega$. Now, I think, upon further reflection, that $\Delta\omega$ is the half width of the lineshape in angular frequency units. Since $\omega = 2\pi c/\lambda$, then $\Delta\omega = 2\pi c \left(\frac{1}{\lambda_{2}}-\frac{1}{\lambda_{1}}\right)$ which, for a half line width of 1 nm gives $\Delta\omega = 5.344\times 10^{14}$ rad/s. The coherence time is then $\tau_{c}=1.87\times 10^{-15}$ s. The coherence length is then $L_{c}=c\tau_{c}=5.61\times 10^{-7}$ m or 561 nm. If my mistake was in the linewidth and it is actually 2 nm, then the coherence length is actually 112 nm.

Now, if it is a Guassian beam instead of a Lorentzian beam, the coherence time is actually $\tau_{c} = (8\pi\ln 2)^{1/2}/\Delta\omega$. This changes the coherence length for a half line width of 1 nm to 234 nm. So it doesn’t seem to matter what I do, I’m consistently getting a coherence length that is in the hundreds of nanometers. As was pointed out, I should be getting a number closer to 300 microns. Where’s my error?

### 6 Responses to “My latest adventure in interferometry”

1. Daniel Oi Says:

Maybe the \pm 1nm spec is not the linewidth but the device to device variability? I’d expect a better linewidth than 1nm, the comments at the ThorLabs site suggests: “Response from Jeremy at Thorlabs: The linewidth for the DJ532-40 should be around 30MHz.”.

2. Daniel Oi Says:

I just saw this at http://www.thorlabs.de/newgrouppage9.cfm?objectgroup_id=5597 “Measuring the DJ532-10 (200mA, 25C) with our SA200 Scanning Fabry Perot Interferometer we calculated a linewidth of ~10.9MHz which translates to a coherence length of ~27.5m.”

• quantummoxie Says:

Ha! Well, that explains it then. You must be right. It must be the variability from device to device. You’d think they’d publish something in the spec sheets about it. Thanks for finding that!

Although, one oddity that I found was that if the optical components (i.e. beam splittlers and mirrors) were too far from the laser source I couldn’t get an interference pattern. I assumed that was because the light was no longer coherent, but it was way less than 27.5 m (though I am using the 40 mW version I think).

3. Hello Ian, I’m glad you keep working on this. Perhaps some about how this will work into testing my proposal is in order to give more on the context. Of course readers can check your previous post, and my blog too: see http://fqxi.org/community, /forum/topic/949. Once you iron out the intra-channel variations it should be possible to move on comparing out put after changing relative phase between photons, and finally adding the extra components of my proposal.

4. First up, I made an experimentalists mistake in the linewidth calculation and ignored the 2*pi term. Thanks for pointing that out.

Your math is a little off there, I think. The radial frequency at 532nm is 3.5e15 rad/s, so a 1nm linewidth must be closer to 6e12 rad/s. I used two equations, including yours, but the easier (I find) is to work out the derivative of omega wrt lambda and then approximate that as a discrete change to get $\Delta$s, which leads to $\Delta\omega \approx \frac{2\pi c}{\lambda^2}\Delta\lambda$. Which will give an answer 2$\pi$ less than my first attempt.

I’m surprised at the difficulty you are having as regards the length of the arms. It shouldn’t be a coherence problem for as long as the arm lengths are the same, and you don’t pass the laser through very turbulent air or some medium, then the phases should still all match up. As was mentioned in a comment on your earlier post, it is possible to make an interferometer with purely white light, but the arms need to be exactly equal in length. But the fact that you can take the interferometer apart and change the arm lengths show that you clearly know what you are doing as regards the set-up.

Interferometers really are fascinating to play with! The thing that always gets me is that virtually every interferometer we use was developed way before the laser, so alignment must have been truly difficult. I was flicking through Newton’s Optics some months back and noticed that in his “Newton’s rings” experiment he could measure the wavelength of yellow light to 1nm scales (see The Second Book Of Opticks: Part I. on http://www.gutenberg.org/files/33504/33504-h/33504-h.htm where he can distinguish between 1/88952th and 1/89063th art of an inch) Which is all the more impressive as he considered it to be a particle …

• quantummoxie Says:

Yep, you’re right. I fell into a trap I routinely admonish my students about — I forgot to convert nm to m. Doh!

Anyway, over the course of the semester I developed a method for aligning the various parts and I will admit I didn’t use them this past weekend. Nevertheless, it was a snap to get the pattern on the 8in and 16in arms. It’s just bizarre since I had gotten so good at getting the patterns that I twice made a portable version so I could roll it into my classes and to a science poster session on campus (the latter required bouncing over some rough sidewalks).

Well, like I said, I plan to order some fully-gimbaled mirrors this summer which should help a bit, though all the stuff we’re using is pretty high-end (mostly ThorLabs stuff).

And, yeah, I am consistently amazed at the fact that this was done with white light. Like I said, I now have a god-like reverence for experimentalists, particularly those who lived in more “primitive” times.