3B Fabry Perot Interferometer Instructions

Experimental introduction to the Fabry-Pérot interferometer using the example of the normal Zeeman effect
Measuring the interference rings of the Fabry-Pérot etalon as a function of the external magnetic field
Determination of the Bohr Magneton

UE5020900
09/24 UD

OVERVIEW

Fig. 1: Experimental setup for the normal Zeeman effect in a longitudinal configuration

GENERAL PRINCIPLES

The Fabry-Pérot interferometer, developed by its name-sakes Charles Fabry and Alfred Pérot, is an optical resonator consisting of two semi-transparent mirrors. A Fabry-Pérot interferometer with a fixed distance between the mir-rors is known as a Fabry-Pérot etalon. As it is designed to fulfill the resonance condition for a specific wavelength, the etalon also acts as an optical filter. An incident light beam is reflected several times in the etalon so that the light beams transmitted with each reflection interfere with each other. This multi-beam interference produces an intensity distribution in transmission with narrow maxima and broad minima. Together with the high interference or-der at correspondingly large resonator dimensions, this results in a high optical quality and correspondingly high resolution. This means that small spectral splittings, such as those present in the normal Zeeman effect at the red Cd line ( = 643.8 nm, = 0.0068 nm at B = 350 mT), can be resolved.

Fig. 2: Beam path in the Fabry-Pérot etalon

A theoretical description of the normal Zeeman effect can be found in the instructions for experiment UE5020850, in which the doublet and triplet splitting is investigated qualitatively.
The focus of this experiment is on spectroscopy with a Fabry-Pérot etalon. The Fabry-Pérot etalon is positioned in front of the camera together with imaging optics, which is used to observe Zeeman splitting.
When the light from the cadmium lamp passes through the Fabry-Pérot etalon, interference rings are created which, like the spectral line, split depending on the external magnetic field and are imaged onto the camera by the optics. Observation parallel or perpendicular to the external magnetic field is made possible by a rotating electromagnet.
The Fabry-Pérot etalon consists of a quartz glass plate with a semi-reflective mirror coating of high reflectivity on both sides (Fig. 2). In this case, the etalon is designed in such a way that the resonance condition for the wavelength = 643,8 nm of the red Cd line is fulfilled. In this sense, the etalon also acts as an optical filter. The thickness d, the refractive index, and the reflection coefficient R of the etalon are as follows:

An incident light beam is reflected several times in the etalon. The light beams transmitted during each reflection interfere with each other. The path difference between two neighbor-ing transmitted light beams, e.g. the light beams emerging at points B and D in Fig. 2, is:

and from this, the condition for the existence of interference maxima:

Overall, an interference pattern of concentric rings is generated. The refraction at the boundary surfaces of the glass plate of the Fabry-Pérot etalon can be neglected as it only shifts the interference pattern in parallel. Therefore, the refraction angle is replaced by the incidence angle , and the interference condition (8) results in

The interference pattern is imaged onto the camera using a convex lens (Fig. 3). The following relationship exists between the anglek at which the kth order interference ring appears, the radius rk of the kth order interference ring and the focal length f of the lens (Fig. 3):

with the small angle approximation tan(x) x. Equation (9) follows for the interference order k and the angle
andAccording to equation (11), because of |cos()| 1, the interference order k is maximum for = 0, i.e. in the center of the interference rings, and corresponds to the parameter k0, which is generally not a whole number. Since the interference rings are counted from the center in the experiment, the interference order k is indexed with a whole number j, which identifies the kth interference order with the jth interference ring counted from the center, in generalization of the parameter k0 already introduced. The first bright interference ring with order k1 appears according to equation (12) at the angle

where k1 is the next whole number that is smaller than k0. As k0 is generally not a whole number, the difference between k0 – and k1 is less than 1. Therefore, a parameter is defined as follows:

For all interference rings with j 2, the order number kj is decreased by 1 in each case, so that for the interference order of the jth interference ring counted from the center the following is generally true

For j = 1, equation (15) just corresponds to the definition from equation (14). Substituting equations (12) (with k = kj) and (15) into equation (10) results in

where, for the sake of simplicity, was set for indexing without restriction of generality. This convention is retained in the following. It follows from equation (16) that the difference between the radius squares of neighboring interference rings is constant

From equations (16) and (17) follows:If the interference rings are split into two very closely spaced components a and b, whose wavelengths differ only slightly from each other, for the first interference ring counted from the center, for example, follows from equation (14):

Since the two components belong to the same interference order, and provided that the interference rings do not overlap by more than one whole order, k1,a = k1,b and thus:

Equation (20) does not explicitly depend on the interference order. If equation (18) is formulated for both components a and b and inserted into equation (20), the following results:

From equation (17) it follows that the difference of the radius squares of the component a or b for neighboring interference orders j and j+1 with j > 0 due to and thus k0,a k0,b are approximately equal:

Accordingly, the following applies for two components a and b of the same interference order j with j > 0:

Substituting equations (22) and (23) into equation (21) results in:Since equation (22) applies to both components a and b of neighboring interference rings and equation (23) applies to all interference rings, mean values

The ratio can be measured as a function of the magnetic flux density B, plotted graphically, and the Bohr magneton B can be determined from the slope a of a linear fit.

EQUIPMENT LIST

SETUP AND SAFETY INSTRUCTIONS

The procedure of this experiment requires that the assembly of the components as well as the experimental setup and adjustment have been carried out according to the instructions for the experiment UE5020850, considering all the safety instructions formulated therein. The maximum current through the coil D with 900 turns is 5 A (7 minutes). It can be doubled for short periods (30 seconds). The coils have an internal reversible thermal fuse that trips at a winding temperature of 85°C. The reset time is 10-20 minutes, depending on the ambient temperature.

Carry out the measurement quickly enough to prevent the thermal fuse from tripping due to high currents flowing for too long.
Do not operate the coils without a transformer core.

EXPERIMENT PROCEDURE
Measurement

Establish the transversal configuration by rotating the electromagnetic as described in the instruction manual for the experiment UE5020850.

Fig. 4: Calibration curve of the electromagnet

Focus the 12 mm lens so that the three interference rings of the innermost order, for which they appear clearly separated from each other, are in focus. Do not move the con-vex lenses (imaging and condenser lens) and do not refocus the 12 mm lens, otherwise, the evaluation will give incorrect results.
Note:
Due to the temperature sensitivity of Fabry-Pérot etalons, the center of the interference rings may look different depending on the ambient temperature and may therefore differ from the screenshots in this manual.
Switch on the DC power supply unit, increase the current through the coils first to 3 A, then in 0.5 A steps to 5 A. At each step, take a screenshot (“snapshot”) with the camera software and save it as a “JPEG”.
Note:
When increasing the current, make sure that the interference rings do not overlap by more than one whole order.

Calibration of the electromagnet
The values for the magnetic flux densities B, which correspond to the set currents I, can be taken from the calibration curve in Fig. 4 or Tab. 1. Alternatively, the calibration curve can be measured as follows:

Remove the Cd lamp on the housing from the base plate.
Place a Tesla meter in the air gap between the two pole pieces (approx. 10 mm) so that the magnetic field sensor is centered.
Switch on the DC power supply unit and increase the current I through the coils in 0.5 A steps. At each step, meas-ure the values for the magnetic flux density B, note them and plot them against the set currents.
Reduce the current to zero and switch off the DC power supply unit.
Insert the Cd lamp back into the base plate.

Tab 1: Calibration of the electromagnet. Set currents I and measured magnetic flux densities B

I / A	B / mT	I / A	B / mT
0.0	0	5.0	458
0.5	46	5.5	489
1.0	101	6.0	518
1.5	148	6.5	540
2.0	194	7.0	556
2.5	239	7.5	574
3.0	288	8.0	589
3.5	334	8.5	602
4.0	377	9.0	614
4.5	422	9.5	625

MEASUREMENT EXAMPLE AND EVALUATION

The following steps are to be carried out for each saved screen-shot:

Open a screenshot in the camera software (click on “File” in the menu bar and select “Open image”).
Click on “Options” in the menu bar, then on “Measurement”, select “Length Unit” in the window that opens, tick “Pixel” under “Current” and confirm the setting by clicking on “OK”.
Click on the “Circle” button in the toolbar and select “3 Points”. Place a circle on the innermost interference ring. This is referred to as “C1” in the following.

The “Measurement” window opens automatically.

Fig. 5: Triplet splitting of the red cadmium line (I = 5.0 A ≙ B = 458 mT). Interference rings marked with circles to determine the enclosed areas

If necessary, adjust the appearance under “Appearance” (e.g. line width/color, show/hide label type).
Under “Geometry”, note the numerical value for the area in pixels (Tab. 2). Mark further interference rings in the same way (C2-C9, Fig. 5) and note the areas (Tab. 2). Click on the “Track” button (hand symbol) to complete the process.
Click on “Layer” in the menu bar, select “Merge to image” and click on “OK”.
Click on “File” in the menu bar, select “Save as” and save the image as a JPEG with a meaningful name.

Note:
The unit of the area is irrelevant for further evaluation, as only relative values and ratios are calculated, not absolute values. The absolute values of the areas (Tab. 2) can deviate significantly depending on the position of the optics.

Calculate the area differences of the corresponding components of neighboring interference orders (Eq. (22), Tab. 3; circles C4↔C1, C5↔C2, C6↔C3, C7↔C4, C8↔C5, C9↔C6).
Calculate the area differences of neighboring components of the same interference orders (Eq. (23), Tab. 4; circles C2↔C1, C3↔C2, C5↔C4, C6↔C5, C8↔C7, C9↔C8).
Calculate the mean values from all area differences in Tab. 3 and 4 (Eq. (25), (26)) and enter them in the tables.
Calculate the ratio of the mean values for all set cur-rents or magnetic flux densities, respectively (Tab. 5). Take the corresponding values for the magnetic flux density from the calibration curve of the electromagnet (Fig. 4, Tab. 1).
Plot the ratio as a function of the magnetic flux density B and fit a straight line through the origin (Fig. 6).Fig. 6: Ratio of the area differences as a function of the magnetic flux density B. The slope of the fitted straight line through the origin is a = 0.53 / T

Determine the Bohr magneton from the slope a = 0.53 / T of the fitted straight line using equation (29):

The value corresponds to the literature value 9.3 10 24 J/T except for approx. 3%.

Tab. 2: Areas A enclosed by the interference rings determined with the help of the camera software

I / A	Area A / Pixel
I / A	C1	C2	C3	C4	C5	C6	C7	C8	C9
3.0	167734	200055	229205	367830	398701	430412	559306	592777	620040
3.5	161486	200196	234474	365742	400854	434853	554225	592457	622683
4.0	157753	199493	238088	358148	398737	439637	552909	592559	624921
4.5	151447	200768	241074	354744	399174	442546	548700	591057	629975
5.0	146500	201657	248223	352695	398436	448720	546544	591877	633671