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Научни трудове на Съюза на учените в България-Пловдив. Серия В. Техника и технологии, естествен ии хуманитарни науки, том XVI., Съюз на учените сесия "Международна конференция на младите учени" 13-15 юни 2013. Scientific research of the Union of Scientists in Bulgaria-Plovdiv, series C. Natural Sciences and Humanities, Vol. XVI, ISSN 1311-9192, Union of Scientists, International Conference of Young Scientists, 13 - 15 June 2013, Plovdiv.
Measuring of the index of refraction and the thickness of thin plates by a Michelson interferometer in the student laboratory
Todorka L. Dimitrova(1), Simonas Bulota(2), Diana Adliene(2), George
Ivanov(1)
(1): University of Plovdiv "Paissi Hilendarski", Tzar Assen Str. 24, BG-4000 Plovdiv, Bulgaria (2): Kaunas University of Technology, K. Donelaicio St. 73, LT-44029
Kaunas, Lithuania e-mail(1): [email protected] e-mail(2): [email protected]
Abstract
The Michelson interferometer is an extremely versatile instrument that can be used to make an accurate comparison of wavelengths, to measure the refractive index of gases, liquids and transparent solids, as well to determine small changes in lengths quite precisely. The main purpose of the present research is to develop a Michelson Interferometer setup for measuring the refractive index and/or thickness of thin glass plates in the student laboratory. One of the two parameters of the plate should be defined in a separate measurement. For example, the index of refraction can be evaluated by measuring the Brewster's angle. The plate is mounted on a rotating stage. By rotating the plate toward the laser beam the phase difference between the two interfering beams changes what results in displacement of the fringe pattern. From that the index of refraction (respectively the plate thickness) can be retrieved.
1. Introduction
Interference is fundamental property of waves. It's manifested as in mechanics (water waves, acoustic waves, ultrasonic waves), so in electromagnetic phenomena (radio waves etc.). Interference plays particular role in optics [1]. It allows to measure very precisely different optical and mechanical values due to the short wavelength of the optical waves. This essential characteristic of the elementary particles (like electrons, photons etc.) brings to deep discussions also in quantum mechanics field [2,3].
The Michelson interferometer (MI) is perhaps the best known and most basic in a family of two beams interferometers which includes the Jamin interferometer, the Rayleigh interferometer and the Mach-Zehnder interferometer [4]. It has many advantages such as simple optical scheme, facile aliment, special separation of the two interfering beams etc. what makes it very suitable to be used for teaching purposes [4,5]. Nowadays there are quite practical commercial components for mounting and alignment, allowing to build a small size interferometer in the student laboratory and then adapting it to different kind of measurements.
In this paper is presented a Michelson interferometer build mainly by Thorlabs (http://www.thorlabs.com/) components [5-7] and intended to measure thin plates thickness and index of refraction.
2. Theory a) Optical scheme
The scheme of the MI with sample in one arm is shown in Fig. 1. The incident light is divided by a beam splitter (BS) placed at 45° toward the beam in two beams of equal intensity which, after reflection from the two mirrors (M1, M2) they are recombined by the same beam splitter. The overlapping exiting beams form an interference pattern on a screen. Photo of the interferometer is shown in Fig. 3. The geometry of the MI allows the easy adjustment of equal paths for light passing in the two arms. The optical path length difference can be changed in several ways, e.g., by a small displacement of one of the mirrors, by a small angle change between the mirrors, or by a change of the index of refraction in one path (for example by inserting a thin plate or its rotation toward the beam).
The period of the fringes on the= screen depends K 0 tical scheme ofMI
on the angle between the two mirrors M1 and M2 (see
Fig. 2) when they are projected in one plane on the equivalent optical scheme (not presented here).
a)
b)
c)
d)
Fig.2: Interference pattern: a) zero path length difference; b) three fringes; c, d: many fringes
b) Measurement of the index of refraction of thin plates
Fig. 3 shows the principle of measurement of the index of refraction of thin plates.
Fig. 3: Principle of measuring of the index of refraction of thin plates
Let us indicate by L1 and L2 the distances between the beam splitter and the relative mirrors M1 and M2 (not indicated but easily visible at Fig. 1) When the plate is parallel to the incident beam the path length between the interfering beams at the interferometer exit is:
A1 = 2L + 2nAB - 2L2 = 2L1 + 2nd - 2L2
(1)
When the plate is rotated by a small angle a toward the beam, the path length difference becomes:
A2 = 2L1 + 2nAD + 2DE - 2L2 = 2L1 +
The fringes move proportionally to: 282
2nd cos P
(
+ 2
d-
nd cos P
cos(a - P) I- 2L2
(2)
A2 -A! = NfnngesÁ,
(3)
where A is the light wave length and Nfringes is the number of fringes passing through a fix point on the screen. By substitution of the equations (1) and (2) in (3), considering the Snell's law n0 sin a = n sin p (n0 = 1 for air) and doing some approximations for small incident angles, one gets to the formula for the refraction index obtained from Monk in 1937:
n =
(2d - Nfnnge¿ll - cosa) 2d(1 - cosa) - NfrmgeA
(4)
3. Experimental setup and results a) Components and setup
Photo of the interferometer with a sample in one arm is shown in Fig. (4). The positions of the components are corresponding to those from the optical scheme shown in Fig. 1. All mechanical and optical components used to build the interferometer are produced by the company Thorlabs. The light source is green laser pointer (model RLDD532-1-3, 532 nm, 3 VDC, 5 mW) from Roithner Lasertechnik (Austria). The interferometer is mounted on an aluminum board (300x300mm2). One inch diameter optics is used. The additional mirror M0 is used to facilitate the alignment. The (negative focal length) lens L serves to project the interference picture on the screen. The principle of mounting and aligning an interferometer using commercial Thorlabs components is explained in [7].
The sample is mounted on a rotating stage allowing precise rotation by 10' and rotating by handed knob. Two thin quartz plates (BK7) with catalogue index of refraction and thickness are use to demonstrate the accuracy of the proposed experimental method. The catalogue index of refraction of the used samples is n*=1,515 and the thickness is relatively di=0,14 mm for Sample 1 and d2=0,91mm for Sample 2. In the case of unknown thickness is can be measured by an ordinary micrometer.
Fig. 4: Photo of MI with sample in one arm with sample in one arm
b) Experimental results
For any sample with thickness d student puts the experimental results in Table 1 where a'and a" are the clockwise and the anticlockwise rotation angles at a given number of fringes Nfringes passing through a fix point.
On the base of the obtained results for any sample the student calculates the average refractive index n = n ± An, where An is the standard deviation, as well as the relative error
avg.
s=--.100% and the absolute error Anabs toward the catalogue index of refraction (if
n * known).
Table 1 Experimental results
№ Nfringes a a" a'+a" a =- 2 n =(2d - Nfringes^X1 - cos a) i 2d(1 - cosa) - Nfringes^ An = navg. ni An2
1 5
2 10
k Nk
For example, for Sample 1 the obtained results are: n = 1,512 ± 0,025, e = 0,2% and
hnabs = -0,003. The results for Sample 2 are similar.
When measuring the thickness of thin plates by MI, it can be obtained by formula (4). In this case, the index of refraction should be known or defined in a separate measurement, for example, by defining the Brewster's angle.
Acknowledgements
The authors acknowledge financial support from the Scopes program (grant no. IZ73Z0-127942-1) of the Swiss National Science Foundation.
The authors kindly thank Prof. A. Weis from the University of Fribourg, Switzerland, for his competent advices.
4. Summary
A student laboratory experiment for bachelors is presented in this work. The experiment is designed to measure the index of refraction and the thickness of thin plates. It can be used also for research and practical measurement of the index of refraction of transparent solid materials (glasses, liquids, birefringent materials, etc.). The measurement may be automatized by an oscilloscope recording of the intensity variation using a photodiode and a motor-controlled displacement of the rotating stage via a motor placed on a separate support to avoid eventual vibration of the interferometer.
References
1. P. Hariiharan, "Optical interferometry", Second edition, Academic Press, Elsevier, 2003.
2. T. L. Dimitrova and A. Weis - The wave-particle duality of light: a demonstration experiment -Am. J. Phys., (2008), 76 (2), 137-142..
3. T. L. Dimitrova and A. Weis - Lecture demonstrations of interference and quantum erasing with single photons - Phys. Scr., (2009), T134, 014003 (4pp).
4. L. Augulis, Taikomoji optika ir fotonika, 55-59, KTU (2007).
5. T. L. Dimitrova, A. Weis - Measuring the wave length of a laser and the index of refraction of air by a Michelson interferometer in the student laboratory - Scientific Research of the Union of Scientists Plovdiv, Series B. Natural Sciences and Humanities, (2009), XII, 67-70.
6. A. Weis, T. L. Dimitrova, "Interférométrie", Expérience AP, Instructions, Université de Fribourg, Suisse, (2008).
7. T. L. Dimitrova, "Building of a Max-Zehnder interferometer for teaching purposes", XXXVII National Conference on the Physic Education, p. 237-240, Ruse, Bulgaria, 2-4 April, (2009).
8. Monk, George S. Light-Principles and experiments. McGraw Hill Book Company, NY & London (1937).
