Fabry-Perot etalon testing: Zeeman splitting of neon

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Fabry-Perot etalon testing: Zeeman splitting of neon

Post by thesmiths » Fri Feb 12, 2021 10:52 pm

Continuing the exploration of our home-made Fabry-Perot etalon, we use it as an interferometer to observe Zeeman splitting. This phenomenon is the very small effect which splits spectral lines when in a magnetic field. It can, for example, be seen in spectral lines near sunspots, and the magnitude of the splitting gives an indication of the magnetic field strength near the sunspot.

The light source we used was a small neon tube (19cm long, 6mm diameter). This was placed between the poles of two neodymium magnets (25mm diameter, 20mm tall). A magnetic yoke made up of three bars of mild steel (in the shape of a C) channelled the magnetic field to create a strong and fairly uniform field (the bars were each 30mm wide x 25mm high x 100mm long). Two pieces of MDF (one with a hole in it to let out the light) held the bars in place. Assuming the magnets really are N52 (it's hard to tell with Chinese eBay magnets), the field with an 8.0mm spacing should be close to 1 Tesla. Normally a lab experiment demonstrating the Zeeman effect uses a fairly large electromagnet.
neon magnet etalon
neon magnet etalon
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In the photo above, you can see the neon tube on the left, inside the magnet. In the middle is the Fabry-Perot etalon, described in a previous post. Resting on the back of the etalon is an Astronomik 12nm-bandpass hydrogen alpha filter. There are many neon spectral lines in the red so we guessed that one would be close to hydrogen alpha. It turns out the you can tune a H-alpha filter by several nanometres by tilting it. Tilting decreases the bandpass wavelength of interference filters.

On the right is a 150mm camera lens, on the front of which is a linear polarising filter. When Zeeman lines split, different lines have different polarisation. Typically, the unsplit line will be parallel to the magnetic field while the split lines will be polarised perpendicular. The photo below shows the the view of the experiment from the camera side.
zeeman setup
zeeman setup
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The following photos show images of the etalon interference rings, first with the polarising filter parallel to the magnetic field and second with the polarised rotated by 90 degrees. We believe the particular spectral line we are looking at is the 653.3nm line of neon, which has the classical Zeeman triplet splitting (other lines can have more complex splitting patterns). This is 3nm less than the 656.3nm wavelength of hydrogen alpha, consistent with the circa 10 degree tilt of the H-alpha filter.
zeeman filter parallel
zeeman filter parallel
4417s.jpg (485.26 KiB) Viewed 328 times
zeeman filter perpendicular
zeeman filter perpendicular
4416s.jpg (515.21 KiB) Viewed 328 times



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Re: Fabry-Perot etalon testing: Zeeman splitting of neon

Post by thesmiths » Fri Feb 12, 2021 10:57 pm

The effect of rotating the polarising filter is more apparent in the video of the experiment:



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Re: Fabry-Perot etalon testing: Zeeman splitting of neon

Post by p_zetner » Sat Feb 13, 2021 4:13 am

Beautiful experiment!
Will you measure the Zeeman splitting to determine whether your estimate of the field strength is correct?
Cheers.
Peter



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Re: Fabry-Perot etalon testing: Zeeman splitting of neon

Post by marktownley » Sat Feb 13, 2021 8:51 am

Fascinating stuff! Enjoying seeing your experiments!


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Re: Fabry-Perot etalon testing: Zeeman splitting of neon

Post by christian viladrich » Sat Feb 13, 2021 11:50 am

Marvellous. Well done !
Thanks for sharing.


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Re: Fabry-Perot etalon testing: Zeeman splitting of neon

Post by thesmiths » Sat Feb 13, 2021 2:12 pm

To Peter: I tried to build a home-made integrating fluxmeter to measure the field. This involves wrapping some thin copper wire to make a small coil (around 10 turns) then looking at how much charge collects on an integrating op-amp when you move the coil from the peak magnetic field to a zero field region. It's the old school way to measure magnetic field strength. Unfortunately, the answer I got was much smaller than I expected so it's possible the Chinese eBay N52 magnets are not really N52 (maybe more like N42). Or my circuit may have a design flaw somewhere.

The problem with most standard Hall probe chips is they seem to be set up for quite low fields (only 0.1 Tesla or so). The cost of a real gaussmeter seems a bit too high. But you're right, I could work backwards and see what field the Zeeman splitting corresponds to. It might be my fluxmeter calculation is actually correct.



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Re: Fabry-Perot etalon testing: Zeeman splitting of neon

Post by p_zetner » Sat Feb 13, 2021 3:45 pm

Hi.
I was thinking you would use the observed line splitting to measure the field. I’ve attached a useful formula.
You measure the sigma components with polarizer perpendicular to the field and pi component with polarizer parallel to the field.
Cheers.
Peter
808479B5-8054-41B1-A936-A0558E762447.png
808479B5-8054-41B1-A936-A0558E762447.png (79.21 KiB) Viewed 291 times



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Re: Fabry-Perot etalon testing: Zeeman splitting of neon

Post by p_zetner » Sun Feb 14, 2021 6:49 pm

I couldn't resist an attempt to analyze your interferograms!
Here is the sigma polarization image showing a line along which I plotted intensities (ImageJ) and the plotted intensities.
(Edit Feb15: Ooops. This is the pi interferogram but the line used for plotting was identical and the plotted result is for the sigma case.)
thesmiths_interferogram_plot_direction.jpg
thesmiths_interferogram_plot_direction.jpg (181.39 KiB) Viewed 257 times
sigma plot0.jpg
sigma plot0.jpg (901.65 KiB) Viewed 257 times
I used Microsoft Origin to fit 5 pairs of fringe peaks with Lorentzians and establish the fringe pixel locations.
sigma plot1.jpg
sigma plot1.jpg (1.3 MiB) Viewed 257 times
Plotting the squared fringe radius (px) of the centre of the doublet (average radius) as a function of fringe order gives the expected linear relationship.
thetAVGsq.jpg
thetAVGsq.jpg (935.16 KiB) Viewed 257 times
The doublet splitting in squared distance is constant as well, as expected.
doublet splitting.jpg
doublet splitting.jpg (620.01 KiB) Viewed 257 times
To go beyond this and calculate the Zeeman splitting and the field strength requires a bit more information. Simply knowing the etalon cavity length would be enough, but I'm not sure how precisely this can be determined. Alternatively, knowing the pixel pitch of your camera sensor, the camera focal length (150mm ?) and any scaling you applied to your posted interferograms would allow the Zeeman splitting to be determined.
Last edited by p_zetner on Mon Feb 15, 2021 6:03 pm, edited 2 times in total.



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Re: Fabry-Perot etalon testing: Zeeman splitting of neon

Post by thesmiths » Sun Feb 14, 2021 11:21 pm

To Peter, thank you for you calculation. You're right, measuring the mirror spacing is tricky. Using callipers, I measured the outside spacing of the mirror mounts, subtracted the mirror thickness specified by Thorlabs and measured the amount of recess from the back of the mirror mounts. From this, I estimate the mirror spacing is 3.3 mm. If there is an error, it is probably 0.1mm less (ie 3.2mm).

A magnetic gap calculation gives a theoretical field of 0.99 Tesla for N52 magnetic material and 0.88 Tesla for N42. My integrating fluxmeter measurement gave an implied field of 0.58 Tesla (but I have a low confidence in this measurement).

Do you have an estimate for the finesse? The theoretical maximum for 0.9 reflectivity is 30 and the minimum assuming lambda/10 surface imperfections is 5. The Am. J. Phys. paper by Thomas Moses measures a finesse of 25 using the same type Thorlabs mirrors I used.



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Re: Fabry-Perot etalon testing: Zeeman splitting of neon

Post by christian viladrich » Mon Feb 15, 2021 10:54 am

thesmiths wrote:
Sun Feb 14, 2021 11:21 pm
To Peter, thank you for you calculation. You're right, measuring the mirror spacing is tricky. Using callipers, I measured the outside spacing of the mirror mounts, subtracted the mirror thickness specified by Thorlabs and measured the amount of recess from the back of the mirror mounts. From this, I estimate the mirror spacing is 3.3 mm. If there is an error, it is probably 0.1mm less (ie 3.2mm).

In fact, if you have a monochromatic light source (e.g. Ha) you can calculate the gap thickness, FSR and FWHM of the etalon. This is what I have done here :
viewtopic.php?f=8&t=30693
Main incertainties are due to the optical quality of the camera lens and on the actual focal length of the camera lens (you have to check it star images).


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Re: Fabry-Perot etalon testing: Zeeman splitting of neon

Post by p_zetner » Mon Feb 15, 2021 5:47 pm

To thesmiths:

I've calculated some results for your Zeeman effect experiment. They are all based on the value of the etalon cavity spacing which I chose to be 3.25mm according to the information you supplied and a source wavelength of 653.3nm. As we've discussed and as Christian has emphasized, this is a somewhat imprecise measurement to base the analysis on but the results seem reasonable. Ideally, you would know precisely the camera lens focal length and the sensor pixel pitch to calculate angles associated with the fringe locations. Knowing these and the wavelength, you would then be able to calculate the etalon spacing accurately. I worked in reverse, using the known etalon spacing to convert pixel values in the interferograms to angle values.

The wavelength splitting of the sigma components worked out to be 0.25 angstroms, giving a resolving power of 26000 for the etalon. The associated B field for this splitting was calculated to be 0.94 Tesla, in line with what you expected.

To calculate the finesse required a measurement of the fringe fwhm when plotted in terms of phase. As suggested by the figure below, such a plot allowed pretty good fitting by Lorentzian lineshapes.
fwhm plot.jpg
fwhm plot.jpg (1.44 MiB) Viewed 211 times
All the fitted lineshapes show nearly equal fwhm, in this case with an average value of 1.05 rad. The finesse is simply 2*pi/fwhm which, in this case, returns a value of about 6. This seems a little low for your setup (implying a mirror reflectance of about 0.5). The fwhm depends on my angle (hence, phase) calibration which, in turn, depends on the supplied value of the etalon length so it's quite likely you are doing better than that. Also, I used the sigma polarized interferogram for this calculation. Due to a little pi "contamination" that I ignored, each doublet is actually a triplet. It would have been a better idea to go back and analyze the pi interferogram (unsplit) for a more robust fitting.

Another point to keep in mind is that the calculations assume collimated light through the etalon. This doesn’t appear to be the case with your setup. It may be worthwhile considering adding a collimator lens between source and etalon for quantitative experiments. In fact, this may be the problem with the finesse calculation. Uncollimated light would have a broadening effect on the fringe thereby reducing the measured finesse.


Let me know if you'd like me to post the details of the calculations.
Cheers.
Peter



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Re: Fabry-Perot etalon testing: Zeeman splitting of neon

Post by thesmiths » Mon Feb 15, 2021 11:34 pm

Dear Peter: That is very thorough and, from my point of view, a very satisfying analysis. You are quite correct, I made little in effort on this first attempt at collimation so the finesse likely suffered for this reason. There is also the lambda/10 issue of the mirrors, but my understanding is mirrors such as these are much better than the spec, especially near the centre.

I am pleased that the magnetic field comes close what was expected. I have reached out to a company in Sweden that makes accurate Hall probes that can be used in high fields https://www.asensor.eu/products/HE244.html. Even uncalibrated, they give a field measurement accurate to within a few percent (they suggest a simple way to make a Hall measurement, which I had never heard of before but which is very clever https://www.asensor.eu/onewebmedia/An%2 ... ements.pdf). If I use a permanent magnet with a hole through the centre (I have one already) and drill a hole through the steel yoke, I can also measure the longitudinal splitting, which exhibits circular polarisation. But then it would be good it have a clearer idea what the field is in this configuration.

As you point out, there is a lot of room to improve the finesse, and therefore the resolution, by improving the optics. In my previous attempt with a sodium lamp, I did use a collimating lens and the fringes were noticeably sharper.

I am also thinking about ordering mercury and helium lamps, which have much fewer lines than neon. The plan would then be to use a transmission diffraction grating and a very wide slit to separate the Zeeman wavelengths (rather than an interference filter).



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Re: Fabry-Perot etalon testing: Zeeman splitting of neon

Post by p_zetner » Tue Feb 16, 2021 12:46 am

Sounds like a terrific program of experiments!
I like the idea of looking at the splitting in the longitudinal direction. In principle, you would then have a known sense of circular polarization available with which you could calibrate the sense of a circular polarizer, something very difficult to do otherwise.
I look forward to seeing your results.
Cheers.
Peter



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Re: Fabry-Perot etalon testing: Zeeman splitting of neon

Post by p_zetner » Wed Feb 17, 2021 4:03 pm

I’d like to make some concluding comments about my rough analysis of the interferograms posted in this thread and suggest an analysis procedure one might want to consider for future measurements of this type. Using the etalon spacing, as I did in this analysis, is a poor method but was used because of the limited information available to me. Knowledge of the camera lens focal length along with sensor (more accurately, image) pixel pitch would also have allowed an analysis of the interferograms and with better precision. In my opinion, the best method is to make measurements with a reference source (notably sodium) and use these results to measure the Zeeman splitting of the target spectral line. I think Christian alluded to this in his post above and I’ll make some of these ideas explicit here. The proposed scheme bypasses the need for measurements of etalon spacing, camera focal length or image pixel pitch. To start with, here’s a generic diagram of the etalon setup (without polarizer or interference filter).
SCpost_fig1.jpg
SCpost_fig1.jpg (110.66 KiB) Viewed 155 times
The etalon spacing is D and the quantity, x , in the figure, is the distance, in image pixels, measured in the interferogram with respect to the geometric centre of the fringe pattern. This is related to the light propagation angle, theta, inside the etalon by a scale factor, eta.
SCpost_eqn1.jpg
SCpost_eqn1.jpg (4.13 KiB) Viewed 155 times
The scale factor could be determined by measuring the camera focal length and image pixel pitch but we can circumvent this direct determination and achieve a result of higher precision by first conducting a measurement using the sodium reference source. If we measure the squared radii of the two components of a particular Na fine structure doublet fringe we get eta directly. Say xA and xB are the fringe radii (in pixels) of the Na fringe doublet components, A and B. Define the difference in squared radii as:
SCpost_eqn2.jpg
SCpost_eqn2.jpg (6.7 KiB) Viewed 155 times
One can show that the wavelength splitting of the Na doublet is given by:
SCpost_eqn3.jpg
SCpost_eqn3.jpg (9.11 KiB) Viewed 155 times
The quantities on the left side of the equation are known, with the numerator equal to 0.5974 nm and the denominator equal to 589.294 nm (centre wavelength of the doublet). On the right hand side of the equation, the difference of squared radii is measured so that this equation can be used to determine the scale factor, eta.
SCpost_eqn4.jpg
SCpost_eqn4.jpg (10.72 KiB) Viewed 155 times
Once the scale factor is established by the sodium measurement, the measurement on the Zeeman split line is conducted, without changing the imaging optics. Squared radii of the two components of the Zeeman split fringe are measured and yield the wavelength splitting by the same equation used in the Na measurement:
SCpost_eqn5.jpg
SCpost_eqn5.jpg (6.93 KiB) Viewed 155 times
The above procedure assumes that the camera optics remain the same when changing from the Na wavelength to the wavelength of interest. For most wavelengths and camera optics that are well colour corrected, this is probably an excellent assumption. The procedure is insensitive to changes in the source optics assuming the collimation is properly adjusted when changing sources.


I also wanted to comment on the calculation of etalon finesse. This requires a determination of the fwhm of an interference fringe measured in terms of “phase”, alpha. I’m using the term “phase” as a nickname for “the phase change undergone by the optical field through one back-and-forth bounce inside the etalon”. Each bright fringe in the interferogram is produced when the phase becomes an even multiple of pi.

Here’s an outline of a procedure I’d suggest for accomplishing the measurement. Step one is to plot the fringe radius squared (in image pixels) versus fringe order (counted with respect to the unknown central fringe order, m0). Here is an example of such a plot:
SCpost_fig2.jpg
SCpost_fig2.jpg (31.08 KiB) Viewed 96 times
Knowing the slope of the line in the plot above, the phase can be calculated (to within a constant that doesn't depend on the propagation angle, theta, and will be ignored). The connection between phase and radial distance squared (in pixels squared) in the image plane is:
SCpost_eqn6.jpg
SCpost_eqn6.jpg (12.39 KiB) Viewed 155 times
You can set the constant equal to zero (or your favourite number) and recognize the term in brackets as the inverse of the line slope measured in the previous plot. Plotting the interferogram intensities in terms of the calculated phase should yield Lorentzian shaped fringes of equal fwhm. Once the fwhm (in phase) is measured, either by fitting (as I've done in the post above) or by hand, the cavity finesse is simply given by:
SCpost_eqn7.jpg
SCpost_eqn7.jpg (7.99 KiB) Viewed 133 times
Hope this is useful.
Cheers.
Peter
Last edited by p_zetner on Thu Feb 18, 2021 7:51 pm, edited 2 times in total.



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Re: Fabry-Perot etalon testing: Zeeman splitting of neon

Post by thesmiths » Thu Feb 18, 2021 1:34 am

To Peter: calibration with the sodium doublet is an excellent proposal. I have ordered helium and mercury tubes and will do measurements with them in due course.



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