Zeeman Image Position to Frequency Calibration
The Fabry-Perot in use has an approximately quadratic variation of transmitted frequency with
respect to angle from the normal for small angles. The angles are small enough that the angle can be
approximated to be proportional to distance form the center of the pattern. The Fabry-Perot spacing
of d = 2 mm gives a round trip mirror separation of 4 mm for a free spectral range of c/2d = 75 GHz.
A single frequency is transmitted through the device in a series of concentric ring with the ring to
ring spacing representing 75 GHz. The function to fit is: f(x) = fo +  (D – x)2. D is the distance
along the x axis from the edge of the image to the center of the pattern The position is not known as
there may be no transmission at the center, and, if there is, it phase is uncertain. The use of an
undetermined parameter D removes the requirement to know the center position, and the
undetermined parameter  allows a fit to remove the requirement to carefully measure distances.
It may be easier to use the equivalent form: f(x) = a + b x + c x2.
If a scientific camera is used, set the camera to use a linear A/D conversion of intensity ( = 1,
black level = 0).
Calibration data set. Find position x for each frequency peak. Note that x may be expressed in
distance units or pixels. The innermost ring is declared to be the 75 GHz ring and each ring outward
has a frequency that is incremented by + 75 GHz. The image below identifies x = 477 pixels as the
position of the 150 GHz peak.1 A complete data set would be of the form:
X (pixels)





f(x) (Hz)
75 GHz
150 GHz
225 GHz
300 GHz
375 GHz
Beware: the x data in this table is not valid for your experiment. Take and analyze new data.
1
The image as viewed and analyzed using the freeware program ImageJ. See the Color Adjustment appendix.
Zeeman Image Position to Frequency Calibration
Fig. 1: A zero field ring pattern with a plot profile generated in ImageJ.
The deep black background of the image suggests that the black level has not been set to zero.
*** Be sure that the arrow keys are used to move the data rectangle against the left (or right)
edge of the image. Do it consistently.
The procedure is to align the optics to provide sharp rings with as near to uniform intensity around
each individual ring as possible. In order to preserve this optical arrangement for all images, it might
be best to set the polarizer at 450 for the zero field calibration image and then to take images with
field at 0, 45 and 90 degrees. The image below illustrates an analysis using ImageJ. A rectangle is
drawn around the rectangle in the image of data to be analyzed that runs from one edge inward far
enough to enclose the center. Starting at one edge provides a solid reference position when one
compares images taken at different field strengths and polarizer position.
For a calibration image that crosses
the center, assign frequencies
symmetrically in multiples of the free
spectral range, 75 GHZ. Do not use 0
for the center. It may differ from zero
by tenths of a radian. Omit the point.
Data taken using a research grade etalon.
Theory: In a separate document, the path length difference  between the direct ray and the
reflected once ray for an air spaced etalon with separation d is 2 d cos. For visible light, 2d/ > 5000.
For the kth ring, cos is less than (5000 – k)/5000 so the small angle approximation is clearly valid for ring
numbers < 25. Assuming a precisely constructive interference at  - 0, m  = 2 d.
fo = m c/2d = m ffsr.
As the frequency is increased (wavelength is decreased), the mth interference ring occurs for an angle
.
c

m  m fo 
f  2d cos 
2
2
f o  f  2dmc
cos  f o [1  ½  ]; f  ½ f o 
fo 2
r
2F 2
The focusing lens forms ring of radius r =  F where F is the focal length of the lens. We are using r
as the distance from the center of the pattern. As it is very difficult to locate the exact center of the
pattern and that the condition for perfect constructive interference may not be realized at that point,
we add an adjustable parameter to the expression so r = x – D, or we use the alternate form of the fit
function: f(x) = a + b x + c x2.

f  ½
Zeeman Image Position to Frequency Calibration
Calibration: Fit the data to f(x) = a + b x + c x2 using a quadratic fit procedure. Try to find one that
includes uncertainty estimates. The calibration must be repeated if any optic other than the polarizer
is adjusted.
The frequency shift due to the field for a peak that moves from xB=0 to xB=B is:
f(xB=B) – f(xB=0).

Apply the fit to the remaining images in your data set to label the various rings as shifted by nn.n
GHz relative to the zero field frequency position of the zero-field ring from which the new rings
separate. Those separations are the input data for the Zeeman analysis.
Safety Consideration:
WARNING: The mercury source has a fused silica glass envelope which transmits strong corneadamaging ultraviolet radiation at 185.0 and 253.6 nm. Do not look at the lamp directly.
Preliminary Survey: The students should look directly into the output face of the Fabry-Perot to
observe the ring pattern as the magnetic field is varied. A mirror should be available if needed to
facilitate the viewing. The various optical components between the output face and the lamp do
block the harmful UV radiation.
UV-grade Fused Silica (SiO2)
Chemical Formula – SiO2
Refractive Index (@ 588nm) – 1.46
(Transmission) –
175nm to 2.2μm & 2.9μm to 3.6μm
Reviewing the free spectral range concept:
Zeeman Image Position to Frequency Calibration
With light of a single frequency, the Fabry-Perot etalon gives pattern for which the ‘center spot’
represents an nth order constructive interference. (We are assuming that the very center just happens
to be a maximum bright. The first ring outward is an (n + 1)st order constructive interference; the
next ring outward is an (n + 2)nd order constructive interference and so on for light all of exactly the
same frequency. Next, we look for the nth order interference of a higher frequency. For our etalon, it
occurs at the position of our first ring outward if we increase the frequency by 75 GHz. (The mirrors
are spaced by 2 mm and the light travels back and for at c. ffsr =c/(2 * 2 mm) = 75 GHz.) In fact, all of
the rings effectively move out to the position of the next ring outward if the frequency is increased
by 75 GHz. The pattern only spreads the light out by frequency uniquely for shifts of 75/2 GHz. At
that point the outward shifted ring from a ring overlaps the inward shifted ring from the zero field
ring initially next outside of it. The pattern is free of spectral ambiguities for a net range of 75 GHz,
the free spectral range of the etalon which is set by its air gap separation of 2 mm.) We expect
symmetric spreading from the original zero field frequencies.