How to get circular fringes?

[Joe Farina]

With regards to generating fringes with a Mach-Zehnder interferometer, Saxby mentioned:

"If you line the beams up very precisely, you can obtain circular fringes which move in and out with any change in optical path lengths, making the detection of drift easy." (Practical Holography, 2nd ed., p. 163)

In all my years of generating fringes (either Michelson or Mach-Zehnder) I have only seen straight, parallel fringes, sometimes with a slight curve. I was wondering if I'm missing something with regards to generating circular fringes.

[lobaz]

You can get circular interference fringes by interfering two spherical waves, e.g. from two spatial filters. The center of the circles will be at the line connecting centers of the waves, i.e. the pinholes. So to make nice circles on the screen, you should add them together using beam splitter to make proper on-axis setup.
As an example I created the Octave script that simulates two spherical waves (wavelength 532 nm); the center of the first is located at 1000 mm from the screen, the second at 1001 mm. The extent of the image is 200x200 mm so the fringes should be perfectly visible.

Interference fringes

circfringes.jpg (11.29 KiB) Viewed 4684 times

And the script:

# all distances are in meters
lambda = 532e-9;
x=(-100:100)*1e-3;
y=(-100:100)*1e-3;
[xx,yy] = meshgrid(x, y);
z0 = 1;
z1 = 1.001;
beam1 = exp(j*2*pi/lambda*sqrt(xx.*xx + yy.*yy + z0*z0));
beam2 = exp(j*2*pi/lambda*sqrt(xx.*xx + yy.*yy + z1*z1));
wave = beam1 + beam2;
interference = wave .* conj(wave);
imagesc(x, y, interference);
colormap(gray)

[Joe Farina]

Thank you, Lobaz, now maybe I understand.

In the past, I've always used an un-diverged beam to combine on the beamsplitter/beam-combiner (i.e., the two raw beams striking a point on the beam-combiner), prior to expanding optics to show the fringes.

So if I understand correctly, if a beam-expanding lens is inserted (i.e., as a substitute for the spatial filter you mentioned above) into each beam, immediately before the beam-combiner, then (perhaps) the resulting fringes will be circular. Need to give that a try, thanks again.

[lobaz]

You are right, Joe. The beams must have different radii of curvature. It is even enough if one beam is collimated (i.e. it is a spherical wave with infinite radius) and the second one is diverging or converging.
If you have a flat beam splitter or a plane-parallel plate, you can see circular fringes with one beam only: take a diverging beam (after a spatial filter or diverging lens) and put the plate into the beam. Try to make it perpendicular to the beam. The beam is transmitted, of course, but a bit is reflected back at the second surface, and back again at the first surface. So in fact behind the plate, there are the directly transmitted spherical wave plus a weaker one that has a bit longer optical path (longer by a distance there-and-back inside the plate). The situation is actually the same as if there were two diverging beams with origins on axis, a little bit apart from each other. (In fact, there will be other reflections as the light bounces multiple times between the plate surfaces, but they get weaker and weaker. It is a sort of a Fabry-Perot interferometer.)
I wanted to make photograph of actual experiment, but I don't have a flat enough plate. The flat beam splitter I used is quite uneven and the fringes were far from being "circular". But there was definitely some center of fringes.
Try it!

Petr

[Joe Farina]

Thank you, Petr

[dannybee]

http://www.ebay.com/itm/Skink-Pinhole-P ... 460c20ed5b they use this as a lens setup in a pinhole camera too very interesting I think they call them "zone plates"

[Joe Farina]

So far, no luck in getting circular fringes.

My (faulty) method was:

1. Have two beams strike point on beamsplitter as usual, expand w/negative lens. Straight fringes, so far so good.
2. Insert positive lens in each beam, just before beamsplitter. Beams focus, spread out somewhat, then strike same point on beamsplitter. Just got some spurious straight fringes (not subject to movement) but no circular fringes.

A photo is attached, beamsplitter in center, positive lenses to top and left, negative beam expanding lens to bottom.

[Joe Farina]

I found some interesting pages in the old book by Winston E. Kock, "Lasers and Holography" regarding circular "zone plate" fringes. I tried to post the relevant pages from my copy, but it seems very difficult to post papers here on the forum, and the .pdf format is not allowed anymore. But the full copy is available online as a 16MB file:

https://ia700809.us.archive.org/30/item ... graphy.pdf

The relevant parts are on pages 9 through 15, and plates 4 and 5 (which is past page 32). The author favors a pinhole (or aperture, I guess) for the spherical wave, interfering with a plane wave (method of producing plane wave unknown, perhaps collimated), to generate the circular fringe pattern or zone plate.

[Joe Farina]

Ok, got it. Plane wave + spherical wave.

[Joe Farina]

A couple photos. It was important to add linear polarizers in each beam (downstream) to get good fringes. This may have been part of my mistake before. If the polarization is wrong, there is nothing, of course. The locking is nice, and seems better than with straight fringes.