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Instructions for 3B models including: Fabry Perot Interferometer, Interferometer
Oct 5, 2024 — The Fabry-Pérot interferometer, developed by its name- sakes Charles Fabry and Alfred Pérot, is an optical reso- nator consisting of two semi-transparent ...
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DocumentDocumentAtomic and Nuclear Physics
Atomic Shell
Fabry-Pérot Interferometer, Determination of the Bohr Magneton
SPECTROSCOPY WITH A FABRY-PÉROT ETALON
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
Fig. 1: Experimental setup for the normal Zeeman effect in longitudinal configuration
GENERAL PRINCIPLES
The Fabry-Pérot interferometer, developed by its namesakes Charles Fabry and Alfred Pérot, is an optical resonator consisting of two semi-transparent mirrors. A FabryPérot interferometer with a fixed distance between the mirrors 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 order 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.
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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 FabryPérot 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 n and the reflection coefficient R of the etalon are as follows:
d 4mm (1) n 1.4567
R 0.85
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 s between two neighboring transmitted light beams, e.g. the light beams emerging at points B and D in Fig. 2, is:
(2) s n BC CP .
and from this, the condition for the existence of interference maxima:
(8) k 2 d n2 sin2 k 2 d n cos k .
k: Whole number, interference order k: Incidence angle of the kth interference order k: Refraction angle of the kth interference order
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
(9)
k
2
d
n
cos
k
2
d
n
1
k 2 2
,
with the expansion cos(x) (1 x2 / 2) of the cosine function.
The interference pattern is imaged onto the camera using a convex lens (Fig. 3). The following relationship exists between the angle k 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):
(10) rk f tan k f k ,
with the small angle approximation tan(x) x. From equation (9) follows for the interference order k and the angle k
(11)
k
k0
cos
k
k0
1
k2 2
with
k0
2d n
and
(12) k
2k0 k
k0
.
From
(3) CP BC cos 2 , (4) d BC cos ,
Snellius' law of refraction (nair 1)
(5) sin n sin
and the addition theorems
cos 1 sin2
(6)
cos 2 1 2 sin2
the path difference results in
(7) s 2 d n2 sin2 2 d n cos
Fig. 3: Imaging the interference rings of the Fabry-Pérot etalon onto the digital camera
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According to equation (11), because of |cos(k)| 1, the interference order k is maximum for k = 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
(13) k1
2 k0
k0
k1
,
where k1 is the next whole number that is smaller than k0. As k0 is generally not a whole number, the difference k0 k1 is less than 1. Therefore, a parameter is defined as follows:
(14) k0 k1 with 0 1
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:
(15) kj k0 j 1
For j = 1, equation (15) just corresponds to the definition of from equation (14). Substituting equations (12) (with k = kj) and (15) into equation (10) results in
(16) rj
2f 2 k0
j 1 ,
where, for the sake of simplicity, rkj rj 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:
(17)
r2 j 1
r2 j
2f2 k0
const. .
From equations (16) and (17) follows:
(18)
r2 j 1
r
2
j 1
r
2
j
j
.
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):
(19)
a b
k0,a k0,b
k1,a k1,b
2d n
a 2d n
b
k1,a k1,b
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:
(20)
a
b
k0,a
k0,b
2d
n
1 a
1 b
.
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:
(21)
1
a
1 b
1 2d n
r2 j+1,a
r r 2
2
j+1,a j,a
r2 j+1,b
r r 2
2
j+1,b j,b
.
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 a b and thus k0,a k0,b are approximately equal:
(22)
j+1,j a
r2 j+1,a
r2 j,a
r2 j+1,b
r2 j,b
j+1,j b
.
Accordingly, the following applies for two components a and b of the same interference order j with j > 0:
(23)
j a,b
r2 j,a
r2 j,b
r2 j+1,a
r2 j+1,b
j+1 a ,b
.
Substituting equations (22) and (23) into equation (21) results in:
(24)
1
a
1 b
1 2d n
j+1 a,b
j+1,j a
for all j 0
Since equation (22) applies to both components a and b of neighboring interference rings and equation (23) applies to all interference rings, mean values
(25)
j a,b
and
(26)
j+1,j a
can be calculated and inserted into equation (24):
(27)
1 a
1 b
1 2d n
.
With
(28)
Ea,b
h
c
1 a
1 b
B
B
follows from equation (27):
(29)
2
d h
n c
B
B
a
B
with
a
2
d h
n c
B
.
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.
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EQUIPMENT LIST
1 Cadmium lamp with accessories @230 V or 1 Cadmium lamp with accessories @115 V 1 Fabry-Pérot etalon 644 nm 1 DC power supply, linear regulated,
1 30V, 0 10A @230V or 1 DC power supply, 0 20 V, 0 5 A @115 V 1 U Core D 2 Coil D, 900 turns 1 Electromagnet accessory for Zeeman effect 1 Microscope camera BRESSER MikroCam SP
3.1 1 Lens 12 mm for Bresser microscope camera 1 Stainless steel rod with ¼ inch thread, 100 mm 1 Red filter mounted on holder 2 Convex lens on stem f =+100 mm 1 Quarter-wavelength filter on stem 1 Polarising attachment 1 Polarisation filter on stem 1 Optical precision bench D, 1000 mm 1 Support for optical bench D, set 1 Optical base D 3 Optical rider D, 90/36 2 Optical rider D, 60/36 1 Safety experiment leads, 75 cm, blue, red, (2
pcs) 1 Safety experiment leads, 75 cm, black, (2 pcs)
1021366
1021747 1020903 1025380
1003311 1000979 1012859 1021365 1024060
1024059 1025431 1025376 1003023 1021353 1021364 1008668 1002628 1012399 1009733 1012401 1002639 1017718
1002849
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 coils 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 which 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 electromagnet 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 convex 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 differently 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 teslameter 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, measure 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.
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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 screenshot:
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 tool bar 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.
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 C4C1, C5C2, C6C3, C7C4, C8C5, C9C6).
Calculate the area differences of neighboring components of the same interference orders (Eq. (23), Tab. 4; circles C2C1, C3C2, C5C4, C6C5, C8C7, C9C8).
Calculate the mean values from all area differences in Tab. 3 and 4 (Eq. (25), (26)) and enter it in the tables.
Calculate the ratio / of the mean values for all set currents 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. 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
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
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Determine the Bohr magneton from the slope a = 0.53 / T The value corresponds to the literature value 9.3 10 24 J/T ex-
of the fitted straight line using equation (29):
cept for approx. 3%.
B
1 2
hc d n
a
(30)
1 2
6.6 1034 Js 3.0 108 4 mm 1.4567
m/s
0.53
/
T
.
9.0 1024 J T
Tab. 2: Areas A enclosed by the interference rings determined with the help of the camera software
I / A
Area A / Pixel
C1
C2
C3
C4
C5
C6
C7
3.0 167734 200055 229205 367830 398701 430412 559306
3.5 161486 200196 234474 365742 400854 434853 554225
4.0 157753 199493 238088 358148 398737 439637 552909
4.5 151447 200768 241074 354744 399174 442546 548700
5.0 146500 201657 248223 352695 398436 448720 546544
C8 592777 592457 592559 591057 591877
C9 620040 622683 624921 629975 633671
Tab. 3: Area differences of the corresponding components of neighboring interference orders
I / A
3.0 3.5 4.0 4.5 5.0
C4,C1
200096 204256 200395 203297 206195
C5,C2
198646 200658 199244 198406 196779
Area difference / Pixel
C6,C3
C7,C4
201207
191476
200379 201549
188483 194761
201472
193956
200497
193849
C8,C5
194076 191603 193822 191883 193441
C9,C6
189628 187830 185284 187429 184951
Mittelwert
195855 195535 195843 196074 195952
Tab. 4: Area differences of neighboring components of the same interference orders
I / A
3.0 3.5 4.0 4.5 5.0
C2,C1
32321 38710 41740 49321 55157
C3,C2
29150 34278 38595 40306 46566
Area difference / Pixel
C5,C4
C6,C5
30871 35112
31711 33999
40589
40900
44430 45741
43372 50284
C8,C7
33471 38232 39650 42357 45333
C9,C8
27263 30226 32362 38918 41794
Mean value
30798 35093 38973 43117 47479
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Tab. 5: Ratio / of the area differences for different currents I or magnetic flux densities B, respectively
I / A
B / T
/
3.0
0.288
0.157
3.5
0.334
0.179
4.0
0.377
0.199
4.5
0.422
0.220
5.0
0.458
0.242
3B SCIENTIFIC® PHYSICS EXPERIMENT
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