Implementation of the Fizeau Aether Drag Experiment for an Undergraduate Physics Laboratory by Bahrudin Trbalic Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2020 c Massachusetts Institute of Technology 2020. All rights reserved. ○
Author...... Department of Physics May 8, 2020 Certified by...... Sean P. Robinson Lecturer of Physics Thesis Supervisor Certified by...... Joseph A. Formaggio Professor of Physics Thesis Supervisor
Accepted by ...... Nergis Mavalvala Associate Department Head, Department of Physics 2 Implementation of the Fizeau Aether Drag Experiment for an Undergraduate Physics Laboratory by Bahrudin Trbalic
Submitted to the Department of Physics on May 8, 2020, in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics
Abstract This work presents the description and implementation of the historically significant Fizeau aether drag experiment in an undergraduate physics laboratory setting. The implementation is optimized to be inexpensive and reproducible in laboratories that aim to educate students in experimental physics. A detailed list of materials, experi- mental setup, and procedures is given. Additionally, a laboratory manual, preparatory materials, and solutions are included.
Thesis Supervisor: Sean P. Robinson Title: Lecturer of Physics
Thesis Supervisor: Joseph A. Formaggio Title: Professor of Physics
3 4 Acknowledgments
I gratefully acknowledge the instrumental help of Prof. Joseph Formaggio and Dr. Sean P. Robinson for the guidance in this thesis work and in my academic life. The Experimental Physics Lab (J-Lab) has been the pinnacle of my MIT experience and I’m thankful for the time spent there. Thanks to Dana and Kaylee for the assistance in making this experiment a reality. I wish to thank Abdurrahman, Harith, Al Barra, Bilal, Amro and all my friends including my roommates at KBL for a wonderful and memorable time at MIT. Thanks to Selver, Nuzdeim, Sladan and Amer for teaching me physics in my early days. I wish to thank my research supervisor and an amazing professor Cardinal Warde. Thanks to Mohammad Ghassemi, who has been an wonderful mentor throughout my MIT time. Lastly, I want to thank everyone in the MIT community for making it such an amazing place to study and live in.
5 6 Contents
1 Introduction 15 1.1 Historical relevance of the Fizeau experiment ...... 15 1.2 Ease of replication of the setup ...... 16
2 History of the Fizeau experiment 17 2.1 Original aether drag experiment ...... 17 2.2 The Experimental Setup and Methods used by Fizeau ...... 18 2.3 Results and their interpretation by Fizeau ...... 20 2.4 Replications of Fizeau’s experiment ...... 21 2.5 The significance of Fizeau’s experiment to the theory of special relativity 22
3 Theoretical background 23 3.1 Special relativity ...... 23 3.1.1 The velocity addition formula ...... 24 3.1.2 Comparison of the nonrelativistic and relativistic velocity ad- dition formulae ...... 25 3.2 Interferometry ...... 26 3.2.1 Fringe patterns ...... 26 3.2.2 Phase difference due to moving water ...... 28 3.3 Hydrodynamics ...... 29 3.3.1 Flow of a viscous fluid ...... 29 3.3.2 Michelson and Morley’s water flow velocity measurement . . 31
7 4 Experimental setup 33 4.1 Physical considerations ...... 33 4.1.1 Cost ...... 34 4.1.2 Space ...... 34 4.1.3 Maintenance ...... 34 4.2 Integral parts of the experiment ...... 34 4.2.1 Optical system ...... 35 4.2.2 Pipe and pump system ...... 37 4.2.3 Peripheral devices ...... 40
5 Experimental procedures and results 41 5.1 Experimental Setup and Procedures ...... 41 5.1.1 Measurements of the geometric properties of the experimental setup ...... 42 5.1.2 Index of refraction of water ...... 43 5.1.3 Water Velocity Meter Calibration ...... 44 5.1.4 Alignment of the laser beams ...... 45 5.1.5 Interference pattern image capturing ...... 46 5.2 Data analysis and results ...... 47 5.2.1 Data acquisition ...... 47 5.2.2 Data analysis ...... 47 5.2.3 Results ...... 49 5.3 Error analysis ...... 51
6 Further improvements 53 6.1 Ultrasonic velocity measurement ...... 53 6.2 Colour Infusion ...... 55
7 Conclusion 57
A Video Analysis Code 59
8 B Student Manual 61
9 10 List of Figures
2-1 The simplified setup of the Fizeau experiment. In this case the wateris flowing counterclockwise, as indicated by the arrows. The light source on the right emits light that goes through a beam splitter 푚′. The two light beams traverse the pipes, reflect at the mirror 푚, and come back to the beam splitter via the pipes...... 19
2-2 The simplified setup of the replica of the Fizeau experiment doneby Michelson and Morley. As with the original experiment, there are two parallel glass pipes through which water can flow. The main difference between the two experimental setups is the positioning of the beam splitter and the light source...... 21
3-1 The interference pattern produced by two plane waves converging onto a screen...... 27
3-2 The pressure difference measurement technique implemented by Michel- son and Morley. The L-shaped probe is facing the water stream. It measures the static and dynamic pressure in the fluid. The other I- shaped probe measures only the static pressure. The difference between the two measurements is the dynamic pressure which depends on the velocity of the water at the 푅 푤 distance from the center of the pipe. − Varying 푤 we can measure the velocity profile...... 31
11 4-1 Shown are a) the Michelson interferometer and b) the modified Michel- son interferometer used for the Fizeau experiment. Note that the colours are used for illustrative purposes to make it easier to follow individual beams...... 35
4-2 The Modified Michelson interferometer with extended distances be- tween the M2-M3 and S-M1 components. Illustrated is the placement of pipes that carry water...... 36
4-3 The pipe and laser system...... 38
4-4 The mounted windows using a union straight connector. They can be easily connected to the ends of the pipes. (Modified from McMaster- Carr [1])...... 39
4-5 The custom made stand for the pipes. It keeps the separation and elevation of the pipes constant...... 39
5-1 Water is driven through two long pipes using a pump. An interfer- ometry setup measures the phase difference caused by the asymmetry of the direction of the water flow and the laser beam propagation. A camera attached to a Raspberry Pi will be used to measure the phase difference...... 42
5-2 The intensities along the fringes were added to produce a single num- ber. In the case of the image here, that was done by summing the columns of the image matrix. The result was an intensity distribu- tion as a function of the horizontal pixel value. This procedure was repeated for every velocity level...... 48
5-3 The position of two interference pattern peaks as a function of phase along the horizontal axis and velocity along the vertical axis. A solid vertical line is drawn as a reference for the maximal shift of the left intensity peak position...... 49
12 5-4 Phase shift data as a function of the velocity of the water flow. The error bars represent the uncertainty in the readings of the central peak of the intensity distribution and the uncertainty in the water velocity flow measurement...... 50 5-5 Same as above. The data is more noisy because the camera was not focused well enough on the fringe pattern. The clustering of data points at near 4 m/s is due to the length of the video of the fringe pattern at that velocity...... 50
6-1 A method for measuring the water velocity between points A and B. We send a sharp ultrasonic pulse at point A and measure the time it takes to detect it at point B...... 54 6-2 A method for measuring the water velocity in the pipe. Left image is the pipe before the colour arrives while the right one is the pipe when the colour has arrived. We infuse red food colour in the pipe inlet and then we compare the delay between the colour arrival between the left and right side. A meter stick is used to measure the distance between the two observation points...... 55 6-3 A method for measuring the water velocity in the pipe. Left image is the pipe before the colour arrives while the right one is the pipe when the colour has arrived. We infuse red food colour in the pipe inlet and then we compare the delay between the colour arrival between the left and right side. A meter stick is used to measure the distance between the two observation points...... 56
13 14 Chapter 1
Introduction
It is widely accepted that experimental physics education is indispensable form school curricula. Experiments are especially important in undergraduate physics education. They enable students to apply their theoretical knowledge, test hypotheses, and ob- serve the natural world with all of its vastness and limitations. Experiments have been the backbone of the development of physics and the only way to challenge hy- potheses. However, there exists a great inequality between theoretical and practical teaching when it comes to allocation of time and resources. Practical education is being neglected despite its pedagogical effectiveness. Most of it is due the impractica- bility of developing, maintaining and supervising hands-on education. Nevertheless, there is a vast number of affordable and implementable experiments that could be used to educate and motivate students. One of them is the adaptation of the Fizeau aether drag experiment described in this thesis work. It is an experiment of seminal value that can easily be implemented in an undergraduate physics lab.
1.1 Historical relevance of the Fizeau experiment
Experimental equipment can have exorbitant costs. Poorly deigned experimental setups take up precious laboratory space and often require specialized equipment that is hard to maintain. But if we look at some of the most influential experiments in history, we can see that their clever makers implemented ingenious designs with
15 rather simple components. The Fizeau aether drag experiment, first performed in 1851 by Fizeau, exemplifies that kind of experiment. Fizeau used his strong theoretical physics knowledge along with his resourcefulness and engineering skills to develop an experiment that was ahead of its time. Quite literally, it was the experiment that proved the theory of special relativity, even before it was conceived by Albert Einstein. One might be baffled by the effects of special relativity and wonder if it’s possible to observe them using simple means and equipment. But centuries ago, physicists devised clever methods to compete with nature’s vastness. Although their eyesight couldn’t reach far, their knowledge helped them see distant galaxies. Although the hydrogen atom is invisibly small, scientists observed it, in some form or another. It seems that for every seemingly insurmountable scale that we are challenged by, Nature leaves a window open for scientists to be able to take a look at it. So it is the case for the speed of light in water. Despite its greatness, we are still able to increase it by letting the water flow. And that change we can observe in an undergraduate lab.
1.2 Ease of replication of the setup
We have demonstrated the replicability of the Fizeau aether drag experiment in an undergraduate laboratory with rather simple and affordable materials. Our aim for implementing the experiment was to demonstrate that high quality experiments are available to many physics students. In addition, this experiment enriched our un- dergraduate physics laboratory repertoire by enabling students to experiment with optics, interference, hydrodynamics, and a novel technique (for our lab) — the exten- sive use of digital cameras and video analysis. In what follows, we will discuss a brief history of the Fizeau experiment, its theoretical background, and experimental setup. Furthermore, we present the methods, data, and results obtained using our setup. Lastly, we discuss several options to improve the setup and adapt it to different labs.
16 Chapter 2
History of the Fizeau experiment
In this chapter, we will discuss the aether drag experiment first performed by Hip- polyte Fizeau in 18511 [5]. The discussion will involve the historical motivation of the experiment, experimental setup and methods used by Fizeau, his results, and interpretation of those results. In addition, we will present the improved version of the Fizeau experiment done by Albert Michelson and Edward W. Morley in 1886 [8]. Lastly, we will discuss the relevance of the Fizeau experiment to the development and confirmation of the special theory of relativity proposed by Albert Einstein.
2.1 Original aether drag experiment
In 1851, Fizeau published the work “The Hypotheses Relating to the Luminous Aether, and an Experiment which Appears to Demonstrate that the Motion of Bod- ies Alters the Velocity with which Light Propagates itself in their Interior”, where he investigated the influence moving bodies have on the propagation speed of light [5]. According to the prevalent theory at that time, the properties of aether could be any of the following:
1. Aether, the material that was thought to be present in all of Universe, is at- tached to molecules of materials and moves with them.
1The primary purpose of this experiment was to test the validity of the aether drag theory that was prevalent at that time, hence the name of this experiment.
17 2. Aether is independent and free, not influenced by the motion of other materials.
3. Only a portion of the aether is attached to the molecules of bodies while the rest is independent. In other words, every object has an “aura” of aether that it drags along with its movement.
These propositions were testable under the experiment designed by Fizeau. He devised an experimental setup that was able to measure the small change in the speed of light when it propagates through moving water. He expected to observe one of the following three outcomes of his experiment:
1. The velocity of light propagating through water is increased by the velocity of the water (if the velocity vectors align.)
2. The velocity of light propagating through water is unaltered by the motion of water.
3. The velocity of light propagating through water is partially increased by the velocity of the water.
2.2 The Experimental Setup and Methods used by Fizeau
In order to test the aforementioned propositions, Fizeau needed to observe a change in the velocity of light in water that is eight orders of magnitude less than that velocity itself. According to him, “although the velocity of light is enormous comparatively to such as we are able to impart to bodies, we are at the present time in possession of means of observation of such extreme delicacy, that it seems to me to be possible.”[5] The extremely precise method he refers to is interferometry — a method he inherited from former scientists, specifically M. Arago. Still, the interferometric method was not enough — the relative differences in velocities are so huge that Fizeau had to include one more brilliant method: differ- ential measurement. He needed to make sure that the only cause of the shift in the
18 Figure 2-1: The simplified setup of the Fizeau experiment. In this case the wateris flowing counterclockwise, as indicated by the arrows. The light source on theright emits light that goes through a beam splitter 푚′. The two light beams traverse the pipes, reflect at the mirror 푚, and come back to the beam splitter via the pipes. interference pattern was due to the velocity of the medium that transmits the light waves.
In his experiment, Fizeau used two parallel glass pipes that formed a closed system. The pipes carried water in opposing directions but with the same flow rate (because they were connected at one end). He used a telescope to focus a light beam into a beam splitter (“transparent mirror”, as he called it). After that, the light would propagate through both of the pipes. One light beam propagated with the water flow while the other one against. After they exited the pipe, a telescope with a mirror at the other end exchanged their returning paths so that they continued their journey towards a screen in a different pipe. Note that the beam that travelled with the water flow continued to to do so in the other pipe. The same is true for theother light beam which travelled against the water flow. In a sense, one beam travelled clockwise while the other one counterclockwise. While the two beams traversed the same geometrical distance, they experienced an asymmetric environment caused by the water flow. After the two light beams transversed the system, they recombined to form a fringe pattern. He magnified the fringe pattern and observed its shiftas he increased the flow of water through the pipes. A simplified setup can beseenin Figure 2-1.
The supply of the water was derived from a reservoir. Fizeau measured the velocity of the water by measuring the volume filled up by the water in one second and dividing that by the cross-sectional area of the pipes. To test that his system was immune to physical disturbances in the equipment, he put slabs of glass on the path of the light
19 and observed no change in the fringe pattern. The same happened when he had only one of the pipes filled with water.
2.3 Results and their interpretation by Fizeau
After several observations, Fizeau discovered that only 46% of the water velocity was added to the final speed of light in water. This indicated that the proposition number 3 discussed in Section 2.1 was the most probable. Fizeau investigated the possibility that his water velocity measurement was inaccurate. Adding corrections to the water velocity measurement reduced the fraction of the velocity addition to approximately 40%, which was close to the value calculated using Fresnel’s theory of partial aether drag.
In a different paper “On the Effect of the Motion of a Body upon the Velocity with which it is traversed by Light”, Fizeau discusses the possibility of an increase in aether density due to the motion of a material [6]. The partial aether drag hypothesis postulates that the additional speed of light in water will be higher by (1 1 )푣, − 푛2 where 푛 is the index of refraction of water and 푣 is the velocity of water. For water, this fraction yields a value of approximately 0.43 which is close to the results obtained by Fizeau, especially when the correction due to the imprecise measurement of the water velocity is included.
Although the partial aether drag hypothesis gives an accurate mathematical de- scription, Fizeau displayed skepticism towards it by stating that “...the conception of Fresnel will doubtless still appear both extraordinary and, in some respects, im- probable; and before it can be accepted as the expression of the real state of things, additional proofs will be demanded from the physicist, as well as a thorough examina- tion of the subject from the mathematician” in the conclusion of his paper published in 1860 [6].
20 Figure 2-2: The simplified setup of the replica of the Fizeau experiment doneby Michelson and Morley. As with the original experiment, there are two parallel glass pipes through which water can flow. The main difference between the two experi- mental setups is the positioning of the beam splitter and the light source.
2.4 Replications of Fizeau’s experiment
The most prominent replication of Fizeau’s experiment was done by Albert A. Michel- son and Edward W. Morley in 1886. They had concerns with some aspects of Fizeau’s experiment. Most of the concerns were due to the water velocity issue. Other were due to the shortness of each experimental run [5]. (Fizeau was limited to 3s per observation.)
Michelson and Morley designed a more stable and reliable setup with higher water capacity and velocity. Their experiment could be performed for a longer time and thus have more reliable data. They also used wider pipes so that the velocity profile of the water inside them was easier to measure. The most important improvement they had was the empirical measurement of the water velocity at the center of the pipe. They used a Pitot tube — two thin pipes, one open to the flow, one perpendicular to the flow — to observe the pressure difference created by the water movement. They knew the relationship between the pressure a fluid exerts and its velocity (푣 √1 ). ∝ 푝 They took measurements at different radii of the pipe and obtained a velocity profile. Knowing the average velocity, they calibrated their method and, in that way, got accurate estimates on the velocity of the water at the center of the pipe.
With the improved design and better measurement techniques, Michelson and Morley confirmed Fizeau’s results. But shortly after, they performed their more famous 1887 experiment [9] that disproved the existence of aether, thus opening up the results of the Fizeau experiment to reinterpretation.
21 2.5 The significance of Fizeau’s experiment to the theory of special relativity
Hendrik Lorentz proposed a solution to the quandary established by the lack of a suitable explanation for Fizeau’s experiment. In his proposition the aether is com- pletely stationary. He rederived Fresnel’s dragging coefficient using a novel concept of local time: a modified time variable that depends on time and position simultane- ously. Because of some discrepancies while describing other observations, he adjusted his model with the concept of length contraction, the shrinking of moving bodies along the direction of their movement. With all of this, he was foreshadowing the development of the theory of special relativity. With the theory of special relativity, Albert Einstein successfully laid out the the- oretical basis for explaining the results obtained by Fizeau. In fact, starting from first principles, Einstein confirmed the validity of the mathematical formulation givenby Lorentz. Moreover, the Fizeau experiment was among the first experimental confir- mations of Einstein’s new theory. In 1925 Einstein stated in a public lecture that Fizeau’s 1851 water tube exper- iment was “perhaps the most fundamental to the theory of special relativity.” [4] He even described the experiment as an observable example of relativistic velocity addition.
22 Chapter 3
Theoretical background
The central question in this experiment is whether photons are dragged along by moving water and, if so, what their final velocity is. The fact that the speed of light is a natural barrier and that nothing can go faster than it is only true if we are referring to the speed of light in vacuum, but when it comes to the speed of
light in water (푣푐 = 푐/푛), it is known that certain particles can achieve superluminar velocities through water (see Cherenkov radiation). When light propagates through water, the photons (or electromagnetic waves) are carried along with it so that they acquire additional velocity. A classical approach to calculate the final velocity of the photons would be to use Galilean transformations and perform a simple sum between the water velocity and the speed of light in water. But that can lead us to a problem if the velocity of water is, for example, half of the speed of light in vacuum. Using the classical approach, this would yield a final velocity greater than the speed of light in vacuum. To resolve this problem we have to turn to special relativity and the relativistic velocity addition formula.
3.1 Special relativity
Albert Einstein formulated the theory of special relativity in 1905 [3]. He derived it from two principles: the equivalence principle and the finite speed of light principle. Einstein derived the Lorentz transformation (length contraction and time dilation)
23 from these principles. By adopting the second principle, we don’t need the concept of stationary aether.
3.1.1 The velocity addition formula
Using Lorentz transformations we can derive the relativistic addition formula for light propagating in moving water.
Consider two events that occur on the common 푥-axes of two mutually aligned inertial coordinate systems A and B. The coordinate system have a uniform relative velocity along their 푥-axes. The two events are described by four coordinates each, three for position and the fourth for time. Let’s call the difference (duration) of each
푥푎, 푦푎, 푧푎, 푡푎 and 푥푏, 푦푏, 푧푏, 푡푏 in coordinate frames A and B respectively. Nonrelativis- tically, the relations between the two coordinate systems are
푥푏 = 푥푎 푣푡푎, −
푦푏 = 푦푎,
푧푏 = 푧푎,
푡푏 = 푡푎, where 푣 is the relative speed, but according to Lorenz transformations, the relations between the two coordinate systems are
푥푏 = 훾(푥푎 훽푐푡푎), −
푦푏 = 푦푎,
푧푏 = 푧푎,
푐푡푏 = 훾(푐푡푎 훽푥푎), −
2 2 where 훽 = 푣/푐 and 훾 = 1/ 1 푣 /푐 . If we divide the the equations for 푥푎 and 푥푏 − √︀ 24 we will obtain the velocity addition equation:
푥 푢 + 푣 푎 = 푟푒푙 B , (3.1) 2 푡푎 1 + 푣B푢푟푒푙/푐 √︀ where 푢rel is the relative velocity of the coordinate systems and 푣B is the velocity of a physical object as measured in the reference frame B. As we see, the upper limit on 푥푎 is 푐. 푡푎 In our experiment the coordinate system A is the lab frame and the coordinate system B is the rest frame of the moving water where the speed of light propagation is 푐/푛 and 푛 is the index of refraction of water. The final velocity of light propagating through moving water, as predicted by Lorentz transformation is:
푐/푛 + 푣 푢 = , (3.2) 1 + 푣/푛푐 √︀ where 푣 is the velocity of water. This formula can be expanded in terms of 푣 (because 푣 푐/푛) so we get a simplified version: ≪ 푐 1 푢 + 푣 1 . (3.3) ≈ 푛 − 푛2 (︂ )︂
3.1.2 Comparison of the nonrelativistic and relativistic veloc- ity addition formulae
We can notice a difference between the relativistic addition formula and the nonrel- ativistic addition formula. That difference depends on the index of refraction ofthe medium. If the index of refraction of the medium is equal to unity then we recover the speed of light. The difference between the two formulations increases with the index of refraction. Comparing the final velocities obtained by using the classical and relativistic theo- ries, we can see that the contribution from the water velocity differs by approximatetly 43% (1 1 0.434). However, that velocity difference is minuscule when we compare − 푛2 ≈ it to the speed of light in water. In order to actually measure it, we need to employ
25 a sensitive measurement technique: interferometry.
3.2 Interferometry
The precision of an interferometry setup, such as the Michelson interferometer, has been widely used to measure small scale changes. The most prominent application is at the Laser Interferometer Gravitational-Wave Observatory (LIGO) where interfer- ometry is used to measure displacement that is 10,000 times smaller than an atomic nucleus [2]. Interferometry hinges on the addition of two coherent light beams. A change in the optical path length of the beams causes their relative phase to shift, which in turn causes their interference pattern to change. The visible light spectrum has wavelengths on the order of 500 nm, which means that if we are carefully observing the fringe pattern changes, we can notice optical path changes up to 100 nm.
3.2.1 Fringe patterns
In order to use interferometry we have to have a source of coherent light. We split that light into two coherent beams, let them interact with the rest of the experimental setup (through propagation or reflections off mirrors) and then recombine them on a screen. From observing the interference pattern shifts on the screen we can infer the relative phase shifts of the two light beams. The phase 휑 that a light wave accumulates while travelling a distance 푑 is
2휋푓 푑 휑 = 0 , 푣
where 푓0 is the frequency of light (given by 푓0 = 푐/휆 for light of wavelength 휆 in vacuum), and 푣 is the propagation velocity of that wave.. We see that the accumulated phase depends on the propagation velocity of light. Particularly in our experiment, that implies a noticeable difference in the phase of light beams that are going with and against the water flow.
26 Figure 3-1: The interference pattern produced by two plane waves converging onto a screen.
The interference pattern produced by two beams of equal intensity and a relative phase shift ∆휑 has an intensity proportional to the square of the cosine function, i.e.
2 퐼Δ휑 = 퐼0 cos (∆휑). But that is only the case if the two beams are parallel. If the two beams are slightly convergent, with an angle of convergence of 2휃, we will observe a striped fringe pattern forming a screen. See Figure 3-1. The wavelength Λ (the distance between two consecutive fringes) of the fringe pattern is given by
휆 Λ = , (3.4) 2 sin(휃)
where 휆 is the wavelength of the laser light and 휃 is the angle of one of the beams with the respect to the bisection of the two converging light beams. Note that the fringe spacing Λ does not depend on the phase difference ∆휑. However, the position of the peaks does depend on ∆휑. For example, if we add a phase of 2휋 to one of the beams, the fringe pattern will shift up or down by exactly Λ, orthogonally to the fringe lines (along the 푦-axis in our case.) That implies that the vertical position changes as
∆휑Λ ∆푦 = . 2휋
The wavelength of the fringe pattern Λ can be measured from the fringe pattern so
27 that we have a linear relation between the 푦-position of the fringe pattern and the phase difference of the two beams.
3.2.2 Phase difference due to moving water
If we denote the water velocity with 푣, then the nonrelativistic and relativistic predic- tions for the beams’ phase difference travelling in a medium of length 푑 and refractive index 푛 in two opposing directions are given by:
푣푑푛2 ∆휑 = 4휋 , (3.5) non-rel 휆푐 푣푑푛2 1 ∆휑rel = 4휋 1 . (3.6) 휆푐 − 푛2 (︂ )︂ We see that the difference between these two is the factor of 1 1 . Let’s look at − 푛2 the orders of magnitude of individual constants and variables:(︀ )︀
푚2 휆red푐 200 , (3.7) ≈ 푠 푚2 4휋푣푑푛2 170휋 , (3.8) ≈ 푠
for 푣 = 2 m/s and 푑 = 6 m and 푛 = 1.33, the approximate parameters of our setup. This means that we expect a phase shift of greater than 휋/2 for the given specifications, which is an observable change in the intensity of the interfering beams. Varying the flow velocity, we will be able to observe the change in the interference intensity. This will yield a linear graph where the slope will be given by the constants in Equation 3.6. The result, along with the independent measurements of the geo- metric path difference 푑, the refractive index of water 푛 and the wavelength of the laser, will yield the speed of light and test the correctness of the relativistic addition formula. The precision of interferometry enables us to accurately measure the phase differ- ence. But we also have to accurately measure the velocity of the water through the pipes. And for that, we need to investigate the properties of the water flow.
28 3.3 Hydrodynamics
Water is a viscous fluid and, as such, it has interesting properties that are relevant to our experiment. We need to be able to measure the velocity of the water at the center of the pipe, since the laser beams will propagate through the center of the pipes. One way to do that is to measure the volume flow of the water and use itto infer the velocity of the water at the center of the pipe. The simplest way to obtain the velocity would be to divide the volume flow by the cross sectional area of the pipe, as done by Fizeau himself [5]. But that would be inaccurate since the velocity distribution is not uniform across the cross-sectional area. The water that is closer to the walls of the pipe is moving slower than the water at the center of the pipe. Nevertheless, we can still use the volume flow to estimate the velocity of the water in the center of the pipe.
3.3.1 Flow of a viscous fluid
Under the assumption that the water flowing through the pipes is laminar, the volume flow 푄 is given by, ∆푃 휋푅4 푄 = , (3.9) 8휂ℓ where ∆푃 is the pressure difference created by the pump, 푅 is the radius of the pipe, 푙 is the length of the pipe and 휂 is the viscosity of water. The same quantity can be defined in terms of the volume of∆ water( 푉 ) that flows in the pipe in a given time (∆푡): ∆푉 푆푣 ∆푡 푄 = = avg = 푆푣 , (3.10) ∆푡 ∆푡 avg where 푆 = 휋푅2 is the cross-sectional area of the pipe. This implies
푄 푣 = . (3.11) avg 휋푅2
The velocity profile of the water, taking the viscosity of water into consideration,
29 is given by ∆푃 푅2 푟 2 푣(푟) = 1 . 4휂ℓ − 푅 [︂ (︂ )︂]︂ The maximum velocity occurs at the center and its given by
∆푃 푅2 푣 = . max 4휂ℓ
By definition, the average velocity is the flow rate divided by the cross-sectional area, 푄 ∆푃 푅2 푣 = = . avg 휋푅2 8휂ℓ We can see that the velocity of the water at the center is twice that of the average velocity. So this analysis yields a simple relation between the velocity of the water at the center and the flow rate: 푄 푣 = 2 . (3.12) max 휋푅2
This results pertains only to laminar flow, which is not necessarily the case for this experiment. Despite the rather large radius of the pipes we use in our experimental setup, and moderate velocities, we cannot expect a perfectly laminar flow. Rather, the coefficient relating the average and maximal velocity of the water in thepipeis bounded by 1 < 휉 < 2, so we have
푣max = 휉푣avr.
Fizeau estimates in his second paper that the coefficient 휉 has one of the following values: 1.1, 1.15, or 1.2 (for his specific setup) [6].
A more rigorous approach to this problem has been done by Michelson and Morley. As discussed in Section 2.4, they measured the pressure difference created by moving water using two thin probes. One was L-shaped and facing the water stream, while the other was I-shaped, standing perpendicular to the pipe.
30 Figure 3-2: The pressure difference measurement technique implemented by Michel- son and Morley. The L-shaped probe is facing the water stream. It measures the static and dynamic pressure in the fluid. The other I-shaped probe measures only the static pressure. The difference between the two measurements is the dynamic pressure which depends on the velocity of the water at the 푅 푤 distance from the center of the pipe. Varying 푤 we can measure the velocity profile.−
3.3.2 Michelson and Morley’s water flow velocity measure- ment
Illustrated in Figure 3-2 is a part of the pipe carrying water flow. Due to the vis- cosity of water, the velocity profile is not uniform. The velocity of the water gets faster towards the center of the pipe. Using Bernoulli’s equation, we can establish a theoretical relation between the dynamic pressure and the velocity of the fluid. The Bernoulli equation states that
휌푣2 푝 + = 푐표푛푠푡, (3.13) 2 from where we can derive 2∆푃 푣 = 퐶 . (3.14) √︃ 휌
where the constant 퐶 can be derived from the following mass conservation rela-
31 tionship: 푅 2 푣(푟)2휋푟푑푟 = 푣avr푅 휋 (3.15) ∫︁0 Michelson and Morley [8] derived from their observations the following empirical velocity profile: 2 푟 0.165 푣(푟) = 푣max 1 , (3.16) − 푅2 (︀ )︀ where the 휉 factor was equal to 1.165. Depending on the geometrical properties of the experimental setup and the water flow velocity, we can expect deviations from this number.
32 Chapter 4
Experimental setup
What follows is a description of the experimental setup needed to perform the Fizeau experiment. As discussed in the introduction, this setup was optimized to be af- fordable and accessible to many physics education laboratories. Inspiration has been taken from the original Fizeau experiment [5] as well as Lahaye, et al [7], and modified in order to bring the cost down and make it easier to replicate.
4.1 Physical considerations
There are several factors that we need to take into consideration when we design a new experiment for a physics education laboratory. First and foremost, we need to consider the educational value of the new experiment. Is the experiment bringing new challenges, physical concepts, or data acquisition and processing techniques to the lab repertoire? If so, is the experiment suitable and safe for undergraduate students? Such questions must be considered and weighed against the material cost of the new experimental equipment. More specifically, we need to consider the monetary cost, space, and maintenance requirements of the experiment.
33 4.1.1 Cost
We are aware that expensive equipment is the main obstacle in enriching physics labs with new experiments, so we sought materials that are reasonably affordable. For some components of the experimental equipment we suggest alternatives that don’t have to come from a high precision manufacturing company but still serve their purpose. The total estimated cost will be given below, but it will vary depending on factors such as location, material suppliers, etc.
4.1.2 Space
The needed lab space for this experiment is approximately a 3 m 0.4 m table or × shelf accessible by students. It is even possible to attach the long pipes under a table so that the space occupied by this experiment is negligible. Due to the properties of the differential measurements used in this experiment, we don’t need an optical table. A reasonably rigid lab bench will suffice.
4.1.3 Maintenance
Once built, this experiment requires little to no maintenance. The only expendable supplies are water and optional food colour additives. Electricity consumption of the setup is on par with a few light bulbs. However, instructors and lab guardians will need to make sure that the water is removed from the system after each use to prevent corrosion and sedimentation. In addition, they need to monitor the system for leaks which could damage equipment in the proximity of this experimental setup.
4.2 Integral parts of the experiment
The Fizeau experimental equipment can be divided into its optical system, the pipe and pump system, and the peripheral devices.
34 Figure 4-1: Shown are a) the Michelson interferometer and b) the modified Michelson interferometer used for the Fizeau experiment. Note that the colours are used for illustrative purposes to make it easier to follow individual beams.
4.2.1 Optical system
The optical setup for the Fizeau experiment can be described as a modified Michel- son interferometer1. Usually, an interferometer measures geometrical changes of the length of one of its arms, but, in our experiment, we measure optical changes in the environment, while keeping the geometrical distances constant. Figure 4-1 compares the Michelson interferometer and the modified Michelson interferometer used in the Fizeau experiment. In the case of the Michelson interfer- ometer, coherent light from the laser gets divided into two beams at the beam splitter B. The two light beams traverse the distances to mirrors M1 and M2, return to the beam splitter and combine to produce a interference pattern on the screen S. The intensity on the screen will depend on the difference of paths the two beams have taken. This configuration is sensitive to movements of both of the mirrors. For the modified Michelson interferometer we have added a third mirror M3and rotated the mirrors M1 and M2 by 45∘. In this configuration, a beam of light that goes through the beam splitter towards mirror M1 gets redirected towards mirrors M3 and M2 and finally reaches the screen S. We refer to this path as the counterclockwise path. In a similar fashion, the beam that gets redirected by the beam splitter towards the mirror M2 traverses a clockwise path and ends up at the screen S to form an
1It could also be described as a modified Mach-Zehnder interferomter.
35 Figure 4-2: The Modified Michelson interferometer with extended distances between the M2-M3 and S-M1 components. Illustrated is the placement of pipes that carry water. interference pattern. It is important to note that the geometrical paths are identical for the two beams regardless of the index of refraction of the environment. This fact makes this experiment immune to external vibrations. In order to create a phase difference between the clockwise and the counterclock- wise beams, we need to create an asymmetry between those two pathways. Inserting a water flow with either clockwise or counter clockwise direction will do the trick. In that case, one of the beams will propagate with the water flow and the other one against it. As discussed in Chapter 3, this will increase the velocity of the beam going with the water flow and decrease the velocity of the beam going against the water flow, creating a difference in phase. The precise dependence between thephase difference and the water flow velocity is given in Equation 3.6. Illustrated in Figure 4-2 are most of the components needed for the optical setup. We need a laser with a coherence length of at least 6 m. Since the phase difference depends inversely with the wavelength of the light produced by the laser, we recom- mend a purple 405 nm source, but any powerful enough laser should suffice. The laser mount should have two degrees of rotational freedom for the purposes of alignment of the optical system. If the laser is fixed in place, the alignment procedure becomes burdensome. Next on the list of needed optical components are a 50/50 beam splitter and three
36 mirrors, all mounted on kinematic mounts with two rotational degrees of freedom. It is important for the mirrors to be flat so that their reflected light does not diverge. In addition, a convex lens with a small focal length is needed to magnify the recombined beams onto a translucent screen. The relative positioning and orientation of the components is given in Figure 4-2. The setup does not need to be mounted on an optical table. A rigid lab bench or shelf should suffice for the purposes of this experiment.
4.2.2 Pipe and pump system
To ensure the asymmetry in propagation of the light beams, we need a controllable, steady water flow that coincides with the laser beams. In order to optimize thespace usage, we use two long pipes. Specification of the pipes are: thick-wall (schedule 80), clear, unthreaded PVC pipe for water, 8 feet long, 3/4 pipe size. We use thick-wall piping to ensure they can endure the pressure, although that might not be required. To decide on the length and the diameter of the pipes, we had to take into consid- eration the following: head loss (pressure drop) due to the pipe’s length and diameter, velocity of the water, intensity loss of the laser light, and the ease of aligning the opti- cal elements. Through a back-of-the-envelope calculation we concluded that a length of 2.42 m and an inner radius of 9.4 mm would serve our purposes well. The cru- cial calculation needed to be done is the water pump’s flow output divided by the cross-sectional area of the pipe, to estimate the average velocity. We used a submersible pump (Jebao DCT-15000) with a power of 155 W, maximal flow rate of 3962 gph 4.2 L , and maximal head loss of 5 m. It had a controller ≈ s with 10 levels of power output. To control the flow rate even further, we installed a continuous valve at the outlet of the pump. A crucial part of this experiment is the water flow velocity measurement. For that we have used a flow meter that was capable to measure the maximal flow outputof the pump. Unfortunately, it turned out that the flow meter needed to be calibrated to work properly. Alternative ways of measuring the water flow velocities are discussed in Chapters 3 and 6.
37 Flow meter Mirror
22L/min
v Beam splitter Pump
Laser
22L/min Camera Flow meter L
Pump power control Screen
Figure 4-3: The pipe and laser system.
To connect the two pipes to the pump, we used flexible garden hoses and appropri- ate fittings. Depending on the pump and the diameter of the pipes, reducers maybe required to connect them because they might have different diameters. A connection to the pipe is established by a “Y-tee” connector. The straight end of it is sealed by a transparent window: Either a circular piece of Plexiglas or a more expensive anti- reflective coated fused silica disk can be used. Plexiglas reflects a significant amount of the laser light back to the screen, which can cause problems when acquiring the data. The AR coated fused silica windows transmit most of the light, but they are quite expensive. A 45∘ connector is added to the “Y-tee” connector in order to make the flow out of the pipe as smooth as possible.
A good way to mount the windows onto the pipe endings is to use a union straight connector. They usually come with an isolating o-ring at their end and they can be attached to the end of the pipe. See Figure 4-4.
To hold the pipes fixed to the lab table, we have made custom stands. As seenin Figure 4-5, it is a rectangular piece of acrylic or wood, with holes for the two pipes. At least five of them are needed to keep the separation and elevation of thepipes constant.
38 Figure 4-4: The mounted windows using a union straight connector. They can be easily connected to the ends of the pipes. (Modified from McMaster-Carr [1]).
Figure 4-5: The custom made stand for the pipes. It keeps the separation and elevation of the pipes constant.
39 4.2.3 Peripheral devices
The easiest way to capture the fringe pattern is using a video camera. It can be any digital camera, such as a regular web camera, Raspberry Pi camera, or any kind of smartphone. The only requirement is that it is well attached to the table for the duration of the data acquisition and that the imaging screen is well in focus. A piece of cark cloth can help improve the visibility of the fringe pattern. In addition, a convex lens can be used to magnify the fringe pattern or to aid the camera focus while it is very close to the screen. Finally, a useful tool for measuring the index of refraction of water is a time-of- flight device called a laser measure. Its use is described in Chapter5.
40 Chapter 5
Experimental procedures and results
This chapter will present the recommended procedures for setting up the experimental apparatus and data acquisition. The first part will describe all the steps needed to obtain some of the variables which are present in the phase-velocity relationship described in Equation 3.6. More specifically, we will describe how to measure the geometrical and optical length of the device and the index of refraction of water. Additionally, the calibration process of the velocity reading and the proper alignment approach will be described. At the end of the chapter, we will present an example of obtained data together with the analysis and final results.
5.1 Experimental Setup and Procedures
Looking back at Equation 3.6, we can see the phase difference depends on the velocity of the water, the length of the pipe system, and the index of refraction of water. As a first step we will make measurements of the pipe system using a) a meter stickto measure the geometric length and b) a time-of-flight sensor to measure the optical length. From these two measurements we can calculate the index of refraction of water (but we recommend using the common reference value of 푛 = 1.33 for the remainder of the experiment.) We will treat velocity as an independent variable which will be controlled by adjusting the speed of the pump. A change in the water velocity will change the
41 Flow meter Mirror
22L/min
v Beam splitter Pump
Laser
22L/min Camera Flow meter L
Pump power control Screen
Figure 5-1: Water is driven through two long pipes using a pump. An interferometry setup measures the phase difference caused by the asymmetry of the direction ofthe water flow and the laser beam propagation. A camera attached to a Raspberry Pi will be used to measure the phase difference.
interference pattern on our screen from which we can extract the phase difference of the two interfering beams. The experimental setup is shown in Figure 5-1. The central part of the experi- ment involves propagating two laser beams in opposite directions — one upstream, one downstream. The two beams will traverse the same distance but have different velocities with respect to the lab frame and thus accumulate different phases.
5.1.1 Measurements of the geometric properties of the exper- imental setup
The first measurement of this experiment is the radius of the clear, long pipe.We need to make sure to measure the inner radius of the pipe while being as accurate as possible, because the velocity calculation will depend on this value. With this measurement we have the radius 푅 of the clear pipe. In addition, we need to determine the the geometric and optical length of the pipes. One way to do that is using a meter stick. Note that we need to measure only the length of the pipe that will have moving water and laser light in it. In other words, we will ignore the parts of the setup that have light propagating in air
42 or that have no laser light going through them. Also, notice that the length 푑 in Equation 3.6 refers to the total geometric length (i.e. twice the length of one pipe.) With this measurement we have the geometric length 푑 of the system.
5.1.2 Index of refraction of water
A more sophisticated way to measure the optical path of our system and the refractive index of water is to use a time-of-flight device. The so-called laser measure uses the delay between emitting and receiving a pulsed light signal. It multiplies that number by 푐 and divides by 2 to get the distance measurement. First, we use the device to take some measurements around the lab to test its accuracy. Next, we put it in front of one of the windows into the clear pipe, emptied of water. We place a screen on the other end of the pipe, as close to the window as possible and write down the measurement and check if it makes sense. We also repeat the same for the other pipe. We repeat the same set of measurements with water inside the pipe. Throughout this part we made sure to be consistent with the placement of the Laser Measure and the screen.
Startup procedure for the water flow
We fill up 3/4 of a 5 gallon bucket with water and then submerge the pump and the returning end of the hose into the bucket. The bucket should be on the floor, below the optical apparatus. Check that the system is closed: all four windows of the clear pipes pipes are present, all hoses are properly connected to each other, and the cap on the midsection close to the laser is on. As a matter of precaution, we have an extra bucket ready for possible leaks of the system. After we are confident that the system is ready, we flip the switch that turns on the pump. After the flow has stabilized and there are no bubbles floating around, it istimeto repeat the measurements with the Laser Measure. The second set of measurements will yield a larger distance since light needs more time to get back to the receiver due to the refraction index of water. The difference between the two readings ∆푦 is
43 related to the index of refraction,
∆푦 = (푛 1)푑 − where 푑 is the geometrical length the laser light takes through water. From these measurements, you can calculate the index of refraction of the water 푛.
5.1.3 Water Velocity Meter Calibration
The water velocity through the pipe system can be adjusted by partially closing the valve close to the pump and by changing the power output of the pump. There are several levels of power the pump can produce and they are indicated on the dashboard connected to it. We expect the water velocity to monotonically increase with the pumping power. However, it is a difficult task to measure the velocity of the water. Attached next to the pump is a flow meter that activates as soon as the water starts flowing. The readings on the flow meter are inaccurate but consistent. That means that we can still use it as long as we calibrate it properly. We take a 5 L or 10 L container and measure the time needed to fill up a certain volume of the container. We do this for all of the following settings:
lowest pumping power and the valve 1/4, 1/2, and 3/4 open ∙ valve fully open, every level of the pumping power. ∙ We write down the measured flow rates and the flow rates displayed on the flowmeter (after making sure to leave time for the water flow to stabilize between measurements). We convert the measured flow rates into maximum velocities using Equation 3.11. From these measurements we can convert the flow rate 푄 displayed on the flow meter
into the average velocity of the water 푣avg.
The Calibration Data
In order to calibrate the reading shown on our flow meter, we needed to measure the flow by timing the time needed to fill up a 5 L container. Taking severaldata
44 points and fitting them to a linear plot, we obtained the link between the flow meter’s reading and the average flow velocity. For our pipe of radius 9.4 mm (cross-sectional
2 area 279.36 mm ) the link between the average flow velocity 푣avg and the flow meter reading 푄fm is given by:
m/s 푣avg = (0.111 0.002) 푄fm (5.1) ± L [︁ ]︁ 5.1.4 Alignment of the laser beams
The most challenging and time consuming part of this experiment is the alignment of the laser beams. The laser light goes through a beam splitter, then the two beams traverse two long pipes, bounce off two mirrors, go back through the two pipes and reunite at the beam splitter to produce a fringe pattern at the screen. In order to get a good interference pattern we need to do the following:
Follow the procedure to start the pump. Do the alignment while the water is ∙ flowing on the lowest power level of the pump. Give the system at least five minutes to settle down and for the small bubbles to disappear.
Turn on the laser (휆 = 405 nm, 5 mW power). ∙
Align the laser so that the beam which passes through the beam splitter is ∙ clearly visible upon exiting the pipe. Use a white piece of paper to aid you in aligning the laser. Use adjustments on the laser stand to rotate the laser vertically and horizontally. Ensure that the laser beam enters and exits the pipe through the center.
Using the beam splitter and the mirror (4 degrees of freedom) direct the second ∙ beam into the other pipe. As before, make sure the beam enters and exits the pipe through the center. Use a meter stick to confirm that the two light beams are parallel by measuring their distance before they enter and after they exit the pipes. Also make sure that the beam height relative to the table is constant.
45 Now that we have two parallel beams coming out of the pipes, we need to take ∙ them back to the beam splitter but using the opposite pipes. The two laser beams will need to swap the pathways from where they came from. Look at the Figure 5-1 above. Use the two mirrors attached to the rail to direct one of the beams to enter the opposite pipe. We need to set the mirrors to a 45∘ angle and then fine tune them. We can first focus on one of the light beams. Block the other one at the beam splitter and try to get the first one to come back through the second pipe. If you are able to get one of the beams to have a round trip and show up on the screen, then the other one should be already aligned. Adjusting the beam splitter slightly brings the two returning spots together and improves the fringe pattern.
The following are some troubleshooting tips:
If you cannot get the beam through the pipe, it might be that the water absorbs ∙ the laser light and you need to change the water. Usually the first load of water is used to clean the system from dust and impurities. When you add the new water, wait for the system to settle and the tiny white bubbles to disappear.
If you have the urge to look into the pipe to see where the laser light is stuck ∙ — don’t! Use your phone camera.
Have plenty of white screens around to block, trace and display the light beams. ∙
5.1.5 Interference pattern image capturing
After the beams come back through the beam splitter, they will recombine and create an interference pattern on a screen. Since the beam recombination occurs at a right angle to the table, we need to redirect the produced beam with a mirror onto a screen mounted on the optical board. The recombined beam passes through a mounted lens in order to be magnified. Then it is projected onto a translucent sheet of paper.At least three maxima and two minima should be visible on the screen. An example of the interference pattern can be seen in Figure 5-2.
46 Once satisfied with the interference pattern, we take the Raspberry Pi camera and start capturing the fringe pattern. In the command line of the Raspberry Pi, we enter raspivid -o or raspistill -o to display the camera input. Once the interference pattern is visible and the picture is focused, we secure the whole setup to the optical table and cover the screen and camera area with a dark cloth to isolate them from outside light and improve image contrast. The interference pattern quality should improve significantly and we are ready to start the main measurement.
5.2 Data analysis and results
After the successful calibration of the flow meter, proper alignment of the optical components, and the preparing the camera to take images and videos, we are ready to start the main experiment. If we look back at Equation 3.6, we see that it predicts a linear relationship between the velocity of the water and the phase difference.
5.2.1 Data acquisition
The data acquisition proceeded while taking two videos simultaneously: one was cap- turing the fringe pattern while the other was capturing the flow meter reading. They were synchronized by their audio output. The power output of the pump was in- crementally increased while giving enough time for the system to reach equilibrium between every step. After all the levels of the power output of the pump were ex- hausted, we swapped the inlet and outlet that were connected to the pump in order to be able to circulate the water in reverse. This doubled the range of our velocities. We assumed that the calibration from above remained the same.
5.2.2 Data analysis
We extracted all the frames (individual images) from the fringe capturing video and annotated them with the average velocity derived using the calibration formula in Equation 5.1 and the readings from the flow meter. After that, the images were
47 Figure 5-2: The intensities along the fringes were added to produce a single number. In the case of the image here, that was done by summing the columns of the image matrix. The result was an intensity distribution as a function of the horizontal pixel value. This procedure was repeated for every velocity level.
divided into velocity groups depending on their velocity annotation. Images with the same velocity annotation were added (on top of each other) to produce a statistically more significant image. This was done for every velocity level. Note that onlya window of three vertical fringes near the center was taken into consideration. See Figure 5-2. The intensities along the fringe were added to produce a single number. In the case of Figure 5-2, that was done by summing the columns of the image matrix. The result was an intensity distribution as a function of the horizontal pixel value. This procedure was repeated for every velocity level. The result can be seen on a 2D plot of the intensity distribution as a function of phase on the horizontal axis and the velocity on the vertical axis. The phase is obtained by normalizing the horizontal pixel value by the average distance between the peaks of the two fringes (i.e. the wavelength Λ). As can be seen in Figure 5-3, the peak of the intensity distribution shifts as we change the velocity. A more rigorous approach was to take every intensity distribution and fit a Gaus- sian function on all the peaks and extract their positions (together with their uncer-
48 counterclockwise water flow 1.75
2.00 ] s
/ 2.25 m 2.50
2.75
3.00 Velocity [ 3.25
3.50 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Phase Figure 5-3: The position of two interference pattern peaks as a function of phase along the horizontal axis and velocity along the vertical axis. A solid vertical line is drawn as a reference for the maximal shift of the left intensity peak position.
tainty). The results can be seen on the two plots Figure 5-4 for counterclockwise flow and Figure 5-5 for clockwise flow. They present the phase shift as a function ofthe velocity of the water flow. A linear fit was performed on the data to obtain thea linear relationship between the velocity and the phase shift.
5.2.3 Results
From the linear interpolations we can see that the average slope is
0.05 0.015 s/m. (5.2) ±
Theoretically predicted slopes (using Equation 3.6, classically and relativistically respectively):
∆휑nonrel = (0.131 0.003) 2휋푣, (5.3) ± ×
∆휑rel = (0.057 0.001) 2휋푣. (5.4) ± ×
The error estimates in the theoretically predicted numerical values are due to the
49 Counterclockwise water flow
0.375
0.400
0.425
e 0.450 c n e r e f
f 0.475 i D
e s Slope: (0.063 ± 0.01) s/m a 0.500 h P
0.525
0.550
0.575 1.5 2.0 2.5 3.0 3.5 Velocity [m/s]
Figure 5-4: Phase shift data as a function of the velocity of the water flow. The error bars represent the uncertainty in the readings of the central peak of the intensity distribution and the uncertainty in the water velocity flow measurement.
Clockwise water flow
1.05
1.00
0.95
e c
n 0.90 e r e f f i D
0.85 e s a h P 0.80
0.75 (Slope: 0.033 ± 0.02) s/m
0.70
1.0 1.5 2.0 2.5 3.0 3.5 4.0 Velocity [m/s]
Figure 5-5: Same as above. The data is more noisy because the camera was not focused well enough on the fringe pattern. The clustering of data points at near 4 m/s is due to the length of the video of the fringe pattern at that velocity.
50 uncertainty in the length of the experimental setup (about 3%). The experimental results are within
1휎 of the relativistic prediction and, ∙ 5휎 of the nonrelativistic prediction. ∙ Thus we conclude that the relativistic velocity addition formula is the more appro- priate one.
5.3 Error analysis
As with the original Fizeau experiment, the source of the most systematic uncertainty is the water flow velocity measurement. In our case, we had to calibrate ourflow meter which added additional systematic error on top of the inability to measure the velocity of the water in the center of the pipe. Possible improvements of the water flow velocity will be discussed in Chapter 6. We have estimated that the water velocity measurement has a 20% uncertainty associated with it. In addition, the length of the pipes has a 2% uncertainty due to our inability to tell where the flow starts when we look at the corners connecting the pipes. At those sharp edges of the system the water has an acceleration period that is on the order of the diameter of the pipe. An additional concern is the change in the index of refraction due to dispersion. The wavelength of the light in the water is Doppler shifted, either increased or de- creased, depending on if it propagates with or against the water stream. The change in the index of refraction 푛 with wavelength 휆 is given by
휕푛 푛 = 푛 + ∆휆 , (5.5) 0 휕휆
where Lahaye et al. estimate that the linear factor in Equation 3.6 is increased by an additional 3.8% [7]. The interference pattern measurement technique allows for further increase in accuracy. With longer exposure times the images would have sharper fringes which
51 would enable better tracking of the shift in phase due to the velocity change. In addition, with changing the velocity, the pressure inside pipes increases which could cause the whole system to shift or produce asymmetrical movement. Although the measurements are immune to vibrations and physical changes, it can happen that the wavelength of the interference pattern changes as a function of velocity. That should be taken into consideration when calculating the phase shift.
52 Chapter 6
Further improvements
The prevailing source of error in the original Fizeau experiment, as well as in our adaptation of it, is the water flow velocity measurement. Michelson and Morley improved the measurement technique by measuring the dynamic pressure across the pipe (as discussed in Section 2.4). However, their advance is insufficient since it still produced large systematic uncertainties. Discussed below are few attempts to adopt a better measurement technique for the water flow velocity through the center of the pipe.
6.1 Ultrasonic velocity measurement
The proposed method for measuring the water velocity between points A and B (in Figure 6-1) is as follows: We send a sharp ultrasonic pulse at point A and measure the time it takes to detect it at point B. If the water was not flowing the pulse would arrive in time 푡 given by 2.5 m 푡 = m = 1.655 ms, (6.1) 1510 s where the speed of the pulse sound through water is approximately 1510 m/s. How- ever, the pulse propagating through the solid PVC pipe comes a bit later, since the speed of sound through PVC material is approximately 1060 m/s, so we need to track the arrival of the pulse only, disregarding the rest of the signal. On the other hand,
53 Figure 6-1: A method for measuring the water velocity between points A and B. We send a sharp ultrasonic pulse at point A and measure the time it takes to detect it at point B.
if the water flows through the pipe with velocity 푣, the sound wave will be carried by the water, obtaining an effective velocity of (1510 + 푣) m/s. That would be a noticeable difference in the arrival time,
2.5 m 푣 푡 = m = 1.655 1 ms (6.2) 푣 + 1510 s − 1510 m/s (︁ )︁ If we want to obtain a resolution of 0.2 m/s in the water velocity measurement, we would have to have a resolution of 0.22 휇s in the time measurement. Given that 1 5 MHz, we would a need a device capable of producing and measuring such 0.22 휇푠 ≈ frequencies. In addition, we need an effective transducer and receiver to generate and receive the acoustic signal. We have tried using piezo-electric devices which have proven effective in receiving signals but ineffective for generating signals. We havetried knocking on one of the ends of the pipe to generate a “sharp" acoustic sound, but the exact timing of the signal production is a challenge. We also used the high symmetry of the setup: having two receivers in the pipe caps on the pump side and producing the sound at the middle of the other side. In that case, one acoustic signal propagates against the water stream while the other one propagates with the stream. We used an oscilloscope to trigger on the arrival of the first signal and observe the second.
54 Figure 6-2: A method for measuring the water velocity in the pipe. Left image is the pipe before the colour arrives while the right one is the pipe when the colour has arrived. We infuse red food colour in the pipe inlet and then we compare the delay between the colour arrival between the left and right side. A meter stick is used to measure the distance between the two observation points.
Despite our best efforts, this method was inconclusive.
6.2 Colour Infusion
Another approach to measure the water flow velocity we have tried is infusing food colour into the inlet of the pump. Since we had clear, transparent pipes, we could observe the colour propagate thought the system. We took videos of the colour propagation and analyzed them by looking at two different and distant points. The camera outputs a series of images, which are essentially matrices of RGB (red, green and blue) values. An example of the analysis is done using a video recording of the red colour propagation is shown in Figure 6-2. We have recorded the values of the red pixel values at the left and right end of the pipe in the video. The distance between these two points is 21 0.4 cm. That measurement was done by comparing the points ± with the meter stick present in the images. The video recording was done with a smartphone with a framerate of 240 fps. The values of the red pixels at the left and right edge of the image are given in Figure 6-3. We see that the blue line is the first one to increase, shortly followed by the red line. Both of them have been normalized
55 Figure 6-3: A method for measuring the water velocity in the pipe. Left image is the pipe before the colour arrives while the right one is the pipe when the colour has arrived. We infuse red food colour in the pipe inlet and then we compare the delay between the colour arrival between the left and right side. A meter stick is used to measure the distance between the two observation points.
on a unitary scale. By shifting one of the lines on the horizontal axis and evaluating the overlap of the two functions, we have found that the graphs are delayed by 35 2 frames which ± corresponds to a delay in time of 0.154 0.008 s. Dividing the distance between the ± two points of measurement by this time delay, we obtain a velocity of 1.35 0.1 m/s. ± This is a reasonable estimate since the pump was in its lowest power setting and the velocities measured by other means coincide with this one. This is a promising technique that could be used to improve the accuracy of this experiment.
56 Chapter 7
Conclusion
The adaptation of the Fizeau aether drag experiment described this thesis work is an example of an experiment that had seminal value in the history of physics and is easily replicable in the undergraduate physics laboratory. It adds to the quality of experiences of physics students in their practical education since it enables them to interact with optics, set up complex interference schemes, and explore hydrody- namics. This experiment encourages the use of digital cameras and video processing techniques, often neglected but powerful methods of data acquisition. In addition, it is one of the increasingly rare table-top experiments, where students can see all the moving parts and nothing is concealed in a black box. The data presented in this work was taken at the first run of the experiment. Despite having satisfactory accuracy, we believe that it could be improved with more careful measurements and better measurement techniques.
57 58 Appendix A
Video Analysis Code
import numpy as np import pylab as plt import random, time import cv2 count = 0 path ="" #path to the video cap = cv2.VideoCapture(path) length = int(cap.get(cv2.CAP_PROP_FRAME_COUNT)) print(’Number of frames:’,length) ret, frame = cap.read() imS = cv2.resize(frame, (960, 540)) #resize the video #select the fringes r = cv2.selectROI(imS) # r = (x0,y0,xwidth,ywidth) HorizontalSegments = [] howmuch = 20 #average over the number of fringes while cap.isOpened(): ret, frame = cap.read() imS = cv2.resize(frame, (960, 540)) imS = imS[int(r[1]):int(r[1]+r[3]), int(r[0]):int(r[0]+r[2])] #cv2.imshow(’frame’,imS) if count%howmuch==0:
59 vektor = np.zeros(len(np.sum(imS[:,:,0],axis=0))) vektor = vektor + np.sum(imS[:,:,0],axis=0) if cv2.waitKey(1) & 0xFF == ord(’q’) or count==length: break count+=1 if count%howmuch==0: print(count) HorizontalSegments.append(vektor) cv2.imshow(’frame’,imS) cap.release() cv2.destroyAllWindows() plt.imshow(HorizontalSegments)
60 Appendix B
Student Manual
61 Relativistic Velocity Addition Using the Fizeau Experiment
MIT Department of Physics (Dated: May 8, 2020) The purpose of this experiment is to demonstrate the validity of the relativistic addition formula using moving water and light propagation through water. Laser light that propagates against the water stream has a smaller speed than the light that propagates with the water stream. This discrepancy in speed intensity will result in a interference pattern when the two beams combine. Using that interference pattern we can infer how the light was “carried” by water.
PREPARATORY QUESTIONS medium, so that the measured speed of the light would be a simple sum of its speed through the medium and Please visit the Fizeau Experiment on the 8.13x web- the speed of the medium. Fizeau confirmed the fact that site at mitx.mit.edu to review the background material light ”rides along” with the flowing water, but his find- for this experiment. Work out the solutions in your lab- ings indicated that the final speed was lower than the oratory notebook; submit your answers on the web site. simple sum of the water velocity through the pipes and the light velocity in water. It was only with the advent of Albert Einstein’s theory of special relativity that this WHAT YOU WILL MEASURE discrepancy was resolved. Einstein stressed the impor- tance of this experiment for special relativity because his theory correctly predicts the outcome of the experiment This experiment touches upon several sub-fields of performed by Fizeau [4]. physics: Relativistic addition of velocities, interferomet- ric phenomena of light, time of flight measurements, and hydrodynamics. To observe the relativistic effect on the velocity addition between the water and laser light, we I.1. Problem and Relevant Theory must employ interferometric techniques which are known for their sensitivity and precision. You will measure the The central question in this experiment is whether pho- phase difference of two interfering light beams that is tons are dragged along by moving water and, if so, what caused by the water flow. The speed of the water will be their final velocity is. The fact that the speed of light measured by flowmeters which you will calibrate by mea- is a natural barrier and that nothing can go faster than suring the time needed to fill up a bucket of water. Then it is only true if we are referring to the speed of light in the volume flow has to be associated with the speed of the vacuum, but when it comes to the speed of light in water water through the center of the pipe. A brief overview of (vc = c/n), it is known that certain particles can achieve measurements needed to execute this experiment: superluminar velocities through water (see Cherenkov ra- diation). When light propagates through water, the pho- At least four flow rate measurements to calibrate • tons (or electromagnetic waves) are carried along with the readings of the flow meter for different power it so that they acquire additional velocity. A classical settings of the pump approach to calculate the final velocity of the photons would be to use Galilean transformations and perform a Optical length of the pipe system (with and without simple sum between the water velocity and the speed of • the water in the pipes) light in water. But that can lead us to a problem if the Interference pattern measurements using a Rasp- velocity of water is, for example, half of the speed of light • berry Pi camera and dedicated software. in vacuum. Using the classical approach, this would yield a final velocity greater than the speed of light in vacuum. To resolve this problem we have to turn to special rela- I. INTRODUCTION tivity and the relativistic velocity addition formula.
In 1925 Einstein stated in a public lecture that Fizeau’s 1851 water tube experiment was “perhaps the most fun- I.2. Special relativity damental to the theory of special relativity” [3]. The Fizeau experiment was carried out to measure the rel- Albert Einstein formulated the theory of special rela- ative speeds of light in moving water. Fizeau used an tivity in 1905 [2]. He derived it from two principles: the interferometer to measure the slight changes of the speed equivalence principle and the finite speed of light princi- of light in water when the water is flowing. ple. Einstein derived the Lorentz transformation (length According to the aether drag theory, light traveling contraction and time dilation) from these principles. By through a moving medium would be dragged along by the adopting the second principle, we don’t need the concept
62 2 of stationary aether. I.2.2. Comparison of the nonrelativistic and relativistic velocity addition formulae
I.2.1. The velocity addition formula We can notice a difference between the relativistic ad- dition formula and the nonrelativistic addition formula. Using Lorentz transformations we can derive the rela- That difference depends on the index of refraction of the tivistic addition formula for light propagating in moving medium. If the index of refraction of the medium is equal water. to unity then we recover the speed of light. The difference Consider two events that occur on the common x-axes between the two formulations increases with the index of of two mutually aligned inertial coordinate systems A and refraction. B. The coordinate system have a uniform relative veloc- Comparing the final velocities obtained by using the ity along their x-axes. The two events are described by classical and relativistic theories, we can see that the four coordinates each, three for position and the fourth contribution from the water velocity differs by approx- for time. Let’s call the difference (duration) of each imatetly 43% (1 1 0.434). However, that velocity − n2 ≈ xa, ya, za, ta and xb, yb, zb, tb in coordinate frames A and difference is minuscule when we compare it to the speed B respectively. Nonrelativistically, the relations between of light in water. In order to actually measure it, we the two coordinate systems are need to employ a sensitive measurement technique: in- terferometry. x = x vt , b a − a yb = ya, I.3. Interferometry zb = za,
tb = ta, The precision of an interferometry setup, such as the where v is the relative speed, but according to Lorenz Michelson interferometer, has been widely used to mea- transformations, the relations between the two coordi- sure small scale changes. The most prominent applica- nate systems are tion is at the Laser Interferometer Gravitational-Wave Observatory (LIGO) where interferometry is used to x = γ(x βct ), b a − a measure displacement that is 10,000 times smaller than yb = ya, an atomic nucleus [1]. Interferometry hinges on the addition of two coherent zb = za, light beams. A change in the optical path length of the ct = γ(ct βx ), b a − a beams causes their relative phase to shift, which in turn causes their interference pattern to change. The visible where β = v/c and γ = 1/ 1 v2/c2. If we divide the light spectrum has wavelengths on the order of 500 nm, the equations for x and x we− will obtain the velocity a b which means that if we are carefully observing the fringe addition equation: p pattern changes, we can notice optical path changes up x u + v to 100 nm. a = rel B , (1) 2 ta 1 + vBurel/c where urel is the relativep velocity of the coordinate sys- I.3.1. Fringe patterns tems and vB is the velocity of a physical object as mea- sured in the reference frame B. As we see, the upper limit In order to use interferometry we have to have a source on xa is c. ta of coherent light. We split that light into two coherent In our experiment the coordinate system A is the lab beams, let them interact with the rest of the experimental frame and the coordinate system B is the rest frame of setup (through propagation or reflections off mirrors) and the moving water where the speed of light propagation is then recombine them on a screen. From observing the c/n and n is the index of refraction of water. The final interference pattern shifts on the screen we can infer the velocity of light propagating through moving water, as relative phase shifts of the two light beams. The phase φ predicted by Lorentz transformation is: that a light wave accumulates while travelling a distance d is c/n + v u = , (2) 2πf d 1 + v/nc φ = 0 , v where v is the velocity ofp water. This formula can be expanded in terms of v (because v c/n) so we get a where f0 is the frequency of light (given by f0 = c/λ for simplified version: light of wavelength λ in vacuum), and v is the propa- gation velocity of that wave.. We see that the accumu- c 1 lated phase depends on the propagation velocity of light. u + v 1 . (3) ≈ n − n2 Particularly in our experiment, that implies a noticeable
63 3
We see that the difference between these two is the factor 1 of 1 n2 . Let’s look at the orders of magnitude of individual− constants and variables: m2 λ c 200 , (7) red ≈ s m2 4πvdn2 170π , (8) ≈ s for v = 2 m/s and d = 6 m and n = 1.33, the approximate FIG. 1. The interference pattern produced by two plane waves parameters of our setup. This means that we expect a converging onto a screen. phase shift of greater than π/2 for the given specifica- tions, which is an observable change in the intensity of the interfering beams. difference in the phase of light beams that are going with Varying the flow velocity, we will be able to observe the and against the water flow. change in the interference intensity. This will yield a lin- The interference pattern produced by two beams of ear graph where the slope will be given by the constants equal intensity and a relative phase shift ∆φ has an in- in equation6. The result, along with the independent tensity proportional to the square of the cosine function, measurements of the geometric path difference d, the re- 2 i.e. I∆φ = I0 cos (∆φ). But that is only the case if the fractive index of water n and the wavelength of the laser, two beams are parallel. If the two beams are slightly will yield the speed of light and test the correctness of convergent, with an angle of convergence of 2θ, we will the relativistic addition formula. observe a striped fringe pattern forming a screen. See The precision of interferometry enables us to accu- Figure I.3.1. The wavelength Λ (the distance between rately measure the phase difference. But we also have two consecutive fringes) of the fringe pattern is given by to accurately measure the velocity of the water through the pipes. And for that, we need to investigate the prop- λ erties of the water flow. Λ = , (4) 2 sin(θ) where λ is the wavelength of the laser light and θ is the I.4. Hydrodynamics angle of one of the beams with the respect to the bisection of the two converging light beams. Note that the fringe Water is a viscous fluid and, as such, it has interesting spacing Λ does not depend on the phase difference ∆φ. properties that are relevant to our experiment. We need However, the position of the peaks does depend on ∆φ. to be able to measure the velocity of the water at the For example, if we add a phase of 2π to one of the beams, center of the pipe, since the laser beams will propagate the fringe pattern will shift up or down by exactly Λ, through the center of the pipes. One way to do that is to orthogonally to the fringe lines (along the y-axis in our measure the volume flow of the water and use it to infer case.) That implies that the vertical position changes as the velocity of the water at the center of the pipe. The simplest way to obtain the velocity would be to divide ∆φΛ ∆y = the volume flow by the cross sectional area of the pipe, as 2π done by Fizeau himself [4]. But that would be inaccurate . The wavelength of the fringe pattern Λ can be measured since the velocity distribution is not uniform across the from the fringe pattern so that we have a linear relation cross-sectional area. The water that is closer to the walls between the y-position of the fringe pattern and the phase of the pipe is moving slower than the water at the center difference of the two beams. of the pipe. Nevertheless, we can still use the volume flow to estimate the velocity of the water in the center of the pipe. I.3.2. Phase difference due to moving water
If we denote the water velocity with v, then the nonrel- I.5. Flow of a viscous fluid ativistic and relativistic predictions for the beams’ phase difference travelling in a medium of length d and refrac- Under the assumption that the water flowing through tive index n in two opposing directions are given by: the pipes is laminar, the volume flow Q is given by, 4 vdn2 ∆P πR ∆φ = 4π , (5) Q = , (9) non-rel λc 8η` vdn2 1 ∆φ = 4π 1 . (6) where ∆P is the pressure difference created by the pump, rel λc − n2 R is the radius of the pipe, l is the length of the pipe and
64 4
η is the viscosity of water. The same quantity can be time-of-flight sensor to measure the optical length. From defined in terms of the volume of water (∆V ) that flows these two measurements we can calculate the index of in the pipe in a given time (∆t): refraction of water (but we recommend using the value of n = 1.33 for the remainder of the experiment). We ∆V Svavg∆t will treat velocity as an independent variable which will Q = = = Svavg, (10) ∆t ∆t be controlled by adjusting the power of the pump. A change in the water velocity will change the interference where S = πR2 is the cross-sectional area of the pipe. pattern on our screen, from where we can extract the This implies phase difference of the two interfering beams. Q The experimental setup is shown in Figure2. The cen- v = . (11) avg πR2 tral part of the experiment involves propagating two laser beams in opposite directions: one upstream, one down- The velocity profile of the water, taking the viscosity stream. The two beams will traverse the same distance of water into consideration, is given by but will effectively experience different refractive indices. ∆PR2 r 2 v(r) = 1 . 4η` − R II.1. Measurements of the geometric properties of the experimental setup The maximum velocity occurs at the center and its given by The easiest measurement in this experiment is the ∆PR2 radius of the clear, long pipe. Make sure to measure v = . max 4η` the inner radius of the pipe and to be as accurate as possible, as the velocity calculation will depend on this By definition, the average velocity is the flow rate di- value. With this measurement you will have the radius vided by the cross-sectional area, R of the clear pipe. Q ∆PR2 vavg = = You will need to determine the geometric and optical πR2 8η` length of the pipes. One way to do so is using a meter stick. Please note that you need to measure only the . We can see that the velocity of the water at the center is length of the pipe that will have moving water and laser twice that of the average velocity. So this analysis yields light in it. In other words, ignore the parts of the setup a simple relation between the velocity of the water at the that have light propagating in air, or that have no laser center and the flow rate: light going through them. Also notice that the length Q d in Equation (6) refers to the total geometric length v = 2 . (12) max πR2 (i.e. twice the length of one pipe). With this measure- ment you will have the geometric length d of your system. This results pertains only to laminar flow, which is not necessarily the case for this experiment. Despite the rather large radius of the pipes we use in our experi- mental setup, and moderate velocities, we cannot expect II.2. Index of Refraction of Water a perfectly laminar flow. Rather, the coefficient relating the average and maximal velocity of the water in the pipe is bounded by 1 < ξ < 2, so we have A more sophisticated way to measure the optical path of your system and the refractive index of water is to use
vmax = ξvavr. a time-of-flight device. The so-called Laser Measure uses the delay between emitting and receiving pulsed light sig- Fizeau estimates in his second paper that the coefficient nals. It multiplies that number by c and divides by 2 to ξ has one of the following values: 1.1, 1.15, or 1.2 (for his get the distance measurement. Use the device to take specific setup) [5]. some measurements around the lab. Test its accuracy. What is the error of its measurement? Next, put it in front of one of the windows into the clear pipe. Have your II. EXPERIMENTAL SETUP AND lab partner hold a screen on the other end of the pipe, as PROCEDURES close to the window as possible. Make sure to adjust the Laser Measure so that its light does not touch the walls Look back at Equation (6). As you can see, the phase of the pipe. Write down the measurement and check if it difference depends on the velocity of the water, the length makes sense. Repeat the same for the other pipe. You of the pipe system, and the index of refraction of water. will do the same set of measurements with water inside We will make measurements of the pipe system using a) the pipe. Make sure to be consistent with the placement a meter stick to measure the geometric length and b) a of the Laser Measure and the screen.
65 5
Flow meter Mirror
22L/min
v Beam splitter Pump
Laser
22L/min Camera Flow meter L
Pump power control Screen
FIG. 2. Water is driven through two long pipes using a pump. An interferometry setup measures the phase difference caused by the asymmetry of the direction of the water flow and the laser beam propagation. A camera attached to a Raspberry Pi will be used to measure the phase difference.
Startup procedure for the water flow: Fill up 3/4 We expect the water velocity to monotonically increase of a 5-gallon bucket with water. Put the pump and the with the pumping power. However, it is a difficult task returning hose end into the bucket. The bucket should to measure the velocity of the water. be on the floor. Check that the system is closed (all four windows of the clear pipes pipes are present, all hoses Attached next to the pump is a flow meter that acti- are properly connected to each other, and the cap on vates as soon as the water starts flowing. The readings the midsection close to the laser is on). As a matter of on the flow meter are inaccurate but consistent. That precaution, have an extra bucket ready for possible leaks means that we can still use it as long as we calibrate it of the system. Have paper towels ready, too. If you are properly. confident that the system is ready, flip the switch that turns on the pump. Be ready to turn off the pump and Take a 5 L or 10 L container and measure the time shout down the valve if you see a leakage in the system! needed to fill up a certain volume of the container. Do If you see lights on the main controller of the pump but this for the following settings: no water is flowing, make sure to open the valve that is lowest pumping power and the valve 1/4, 1/2, and attached close to the pump. • 3/4 open, After the flow has stabilized and there are no bubbles floating around, it is time to repeat the measurements valve fully open, every level of the pumping power. with the Laser Measure. The second set of measurements • Write down the measured flow rates and the flow rates will yield a larger distance since light needs more time to displayed on the flow meter. Make sure to leave time get back to the receiver due to the refraction index of for the water flow to stabilize between measurements. water. The difference between the two readings is Convert the measured flow rates into maximum velocities ∆y = (n 1)d using Eqyation (11). From these measurements you can − convert the flow rate Q displayed on the flow meter into where d is the geometrical length the laser light takes the velocity of the water v at the center of the pipe. through water. From these measurements you can calcu- late the index of refraction n of the water. II.4. Alignment of the Laser Beams
II.3. Water Velocity Meter Calibration The most challenging and time consuming part of this experiment is the alignment of the laser beams. The laser The water velocity through the pipe system can be light goes through a beam splitter, then the two beams adjusted by partially closing the valve close to the pump traverse two long pipes, bounce off two mirrors, go back and by changing the power output of the pump. There through the two pipes and reunite at the beam splitter are several levels of power the pump can produce and to produce a fringe pattern at the screen. In order to get they are indicated on the dashboard connected to it. a good interference pattern you need to do the following:
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Follow the procedure to start the pump. Do the • alignment while the water is flowing on the low- est power level of the pump. Give the system at least five minutes to settle down and for the small bubbles to disappear. Turn on the laser (λ = 405 nm). Use the key to • switch on the laser and press the button through the circular opening of the laser stand. Align the laser so that the beam which passes • through the beam splitter is clearly visible upon exiting the pipe. Use a white piece of paper to aid you in aligning the laser. The two knobs on the laser stand can rotate the laser vertically and hori- zontally. Make sure the laser beam enters and exits the pipe through the center. Using the beam splitter and the mirror (4 degrees • of freedom) direct the second beam into the other FIG. 3. The magnified interference pattern while the water pipe. As before, make sure the beam enters and was flowing. Dark and bright spots are visible. The shape of exits the pipe through the center. Use a meter stick your interference pattern might differ. to confirm that the two light beams are parallel by measuring their distance before they enter and after they exit the pipes. Also make sure that the II.5. Interference Pattern Capturing beam height relative to the table is constant. Only proceed if you managed to have parallel beams! After the beams come back through the beam splitter, they will recombine and create a interference pattern on Now that you have two parallel beams coming out a screen. Since the beam recombination occurs at a right • of the pipes, we need to take them back to the angle to the table. we need to redirect with a mirror beam splitter but using different pipes. The two onto a screen mounted on the optical board where the laser beams will need to swap the pathways they laser is fixed too. You can move around the optical came from. Look at the figure 1 above. Use the components to optimize the space usage. Use a mounted two mirrors attached to the rail to direct one of the lens to magnify the combined beams and project them beams to enter the opposite pipe. You will need to onto a screen. Tweak all optical components until set the mirrors to a 45 angle and then fine tune ◦ you are satisfied with the quality of your interference them. You can first focus on one of the light beams: pattern. At least three maxima and two minima should block the other one at the beam splitter and try be visible on the screen. to get the first one come back through the second pipe. If you are able to get one of the beams have a round trip and show up on the screen, the other Once satisfied with the interference pattern, take the one should be straightforward. Try adjusting the Raspberry Pi camera close to the screen from the same beam splitter slightly to bring the two returning side the laser light is coming from. In the command line spots together. of the raspberry pi enter raspivid -o or raspistill -o to display the camera input. Once the interference Troubleshooting tips: pattern is visible and the picture is focused, you can secure the whole setup to the optical table. Cover the If you cannot get the beam through the pipe, it • screen and camera area with a dark cloth (isolate them might be that the water absorbs the laser light and from outside light). The interference pattern quality you need to change the water. Usually the first load should improve significantly and you are ready to start of water is used to clean the system from dust and the main measurement of the experiment. impurities. When you add the new water, wait for the system to settle and the tiny white bubbles to The data acquisition proceeded while taking two videos disappear. simultaneously: one was capturing the fringe pattern If you have the urge to look into the pipe to see while the other was capturing the flow meter reading. • where the laser light is stuck — don’t! Use your They were synchronized by their audio output. The phone camera. power output of the pump was incrementally increased while giving enough time for the system to reach equi- Have plenty of white screens around to block, trace, librium between every step. After all the levels of the • and display the light beams. power output of the pump were exhausted, we swapped
67 7
doubled the range of our velocities. We assumed that the calibration from above remained the same.
II.6. Video analysis
We extract all the frames (individual images) from the fringe capturing video and annotated them with the av- erage velocity derived using the calibration formula in Equation 11 and the readings from the flow meter. The code for doing that is in appendix A. After that, the im- ages should be divided into velocity groups depending on their velocity annotation. Images with the same velocity annotation were added (on top of each other) to produce FIG. 4. The intensities along the fringes were added to pro- a statistically more significant image. This should be duce a single number. In the case of the image here, that done for every velocity level. Note that only a window was done by summing the columns of the image matrix. The of three vertical fringes near the center was taken into result was an intensity distribution as a function of the hor- consideration. See Figure4. izontal pixel value. This procedure was repeated for every The intensities along the fringe were added to produce velocity level. a single number. In the case of Figure4, that was done by summing the columns of the image matrix. The re- sult was an intensity distribution as a function of the the inlet and outlet that were connected to the pump in horizontal pixel value. This procedure was repeated for order to be able to circulate the water in reverse. This every velocity level.
[1] LIGO Caltech. Facts. Laser Interfer- ometer Gravitational Wave Observatory. https://www.ligo.caltech.edu/page/facts . (5/8/2020). [2] Albert Einstein. ”Zur Elektrodynamik bewegter K¨orper” [on the electrodynamics of moving bodies]. Annalen der Physik (in German). 17 (10): 891–921., 1905. [3] Albert Einstein. ”Lectures at the University of Buenos Aires. Published in Spanish in the 29 and 31 March, 2, 4, 7, 16, 18, and 21 April” [collected papers, 14, appendix f, 915-940; trans. 941-965.]., 1925. [4] Hippolyte Fizeau. ”The Hypotheses Relating to the Lu- minous Aether, and an Experiment which Appears to Demonstrate that the Motion of Bodies Alters the Ve- locity with which Light Propagates itself in their Inte- rior”. Philosophical Magazine, Series 4, vol. 2, pp. 568- 573, 1851. [5] Hippolyte Fizeau. ”On the Effect of the Motion of a Body upon the Velocity with which it is traversed by Light.”. Philosophical Magazine, Series 4, vol. 19, pp. 245-260,, 1860.
68 Bibliography
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[3] Albert Einstein. "Zur Elektrodynamik bewegter Körper" [on the electrodynamics of moving bodies]. Annalen der Physik (in German). 17 (10): 891–921., 1905.
[4] Albert Einstein. "Lectures at the University of Buenos Aires. Published in Spanish in the 29 and 31 March, 2, 4, 7, 16, 18, and 21 April" [collected papers, 14, appendix f, 915-940; trans. 941-965.]., 1925.
[5] Hippolyte Fizeau. "The Hypotheses Relating to the Luminous Aether, and an Experiment which Appears to Demonstrate that the Motion of Bodies Alters the Velocity with which Light Propagates itself in their Interior". Philosophical Magazine, Series 4, vol. 2, pp. 568-573, 1851.
[6] Hippolyte Fizeau. "On the Effect of the Motion of a Body upon the Velocity with which it is traversed by Light.". Philosophical Magazine, Series 4, vol. 19, pp. 245-260,, 1860.
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[8] Edward W. Michelson, Albert A. Morley. "Influence of Motion of the Medium on the Velocity of Light". 1886.
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