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diff --git a/manual/chapters/michelson.tex b/manual/chapters/michelson.tex deleted file mode 100644 index 101fa64..0000000 --- a/manual/chapters/michelson.tex +++ /dev/null @@ -1,311 +0,0 @@ -%\chapter*{Michelson Interferometer} -%\addcontentsline{toc}{chapter}{Michelson Interferometer} -\documentclass{article} -\usepackage{tabularx,amsmath,boxedminipage,epsfig} - \oddsidemargin 0.0in - \evensidemargin 0.0in - \textwidth 6.5in - \headheight 0.0in - \topmargin 0.0in - \textheight=9.0in - -\begin{document} -\title{Michelson Interferometer} -\date {} -\maketitle - -\noindent - \textbf{Experiment objectives}: Assemble and align a Michelson -interferometer, and use it to measure wavelength of unknown laser, and the -refractive index of air. - -\section*{History} - -Michelson interferometer is an extremely important apparatus. It was used by Michelson and Morley in 1887 to -determine that electromagnetic waves propagate in vacuum, giving the first strong evidence against the theory of -a \textit{luminiferous aether} (a fictious medium for light wave propagation) and providing the insight into -the true nature of electromagnetic radiation. Nowedays, Michelson interferometer remains a widely used tool in -many areas of physics and engineering. In this laboratory you will use the interferometer to accurately measure -the wavelength of laser light and the index of refraction of air. -\begin{figure}[h] -\centerline{\epsfig{width=0.8\linewidth,file=fig1.eps}} \caption{\label{fig1mich.fig}A Michelson Interferometer -setup.} -\end{figure} - -\section*{Theory} - - The interferometer works by combining two light waves - traversing two path lengths. A diagram of this type of - interferometer is shown in Figure!\ref{fig1mich.fig} - A beamsplitter (a glass - plate which is partially silver-coated on the front surface - and angled at 45 degrees) splits the laser beam into two parts of equal - amplitude. One beam (reflected by the - beamsplitter) travels at 90 degrees toward mirror $M_2$ and back - again, passing twice through a clear glass plate called the - compensator plate. At the beamsplitter one-half of - this light is transmitted to an observer (you will use a - viewing screen). At the same time the other beam (that was initially transmitted by the beamsplitter) - travels to - a fixed mirror $M_1$ and back again. One-half of this amplitude - is reflected from the partially-silvered surface and directed - at 90 degrees toward the observer. Thus, the total amplitude of the light the observer - records is a combination of amplitude of the two beams: -\begin{equation} -\mathbf{E}_{total} = \mathbf{E}_1 + \mathbf{E}_2 = \frac12 \mathbf{E}_0 + \frac12 \mathbf{E}_0\cos(k\Delta l) -\end{equation} - -Here $k\Delta l$ is a phase shift (``optical delay'') between the two wavefronts caused by the difference in -pathlengths for the two beams $\Delta l$ , $k=2\pi n/\lambda$ is the wave number, $\lambda$ is the wavelength -of light in vacuum, and $n$ is the refractive index of the optical medium (in our case - air). - -Mirror $M_2$ is mounted on a precision traveling platform. Imagine that we adjust its position (by turning the -micrometer screw) such that the distance traversed on both arms is exactly identical. Because the thickness of -the compensator plate and the beamsplitter are the same, both wavefronts pass through the same amount of glass -and air, so the pathlength of the light beams in both interferometer arms will be exactly the same. Therefore, -two fields will arrive in phase to the observer, and their amplitudes will add up constructively, producing a -bright spot on the viewing screen. If now you turn the micrometer to offset the length of one arm by a half of -light wavelength, $\Delta l = \lambda/2$, they will acquire a relative phase of $\pi$, and total destructive -interference will occur: -\begin{displaymath} -\mathbf{E}_1 +\mathbf{E}_2=0\;\;\mathrm{or} \;\;\mathbf{E}_1 = -\mathbf{E}_2. -%\end{displaymath} -% or -%\begin{displaymath}\mathbf{E}_1(t) = -\mathbf{E}_2(t). -\end{displaymath} -It is easy to see that constructive interference happens when the difference between pathlengths in two -interferometer arms is equal to the integer number of wavelengths $\Delta l = m\lambda$, and destructive -interference corresponds to the half-integer number of wavelengths $\Delta l = (m + 1/2) \lambda$ (here $m$ is -an integer number). Since the wavelength of light is small ($600-700$~nm for a red laser), Michelson -interferometers are able to measure distance variation with very good precision. - - - - -%Figure 1. The Michelson Interferometer - -To simplify the alignment of a Michelson interferometer, it is convenient to work with diverging optical beams. -In this case an interference pattern will look like a set of concentric bright and dark circles, since the -components of the diverging beam travel at slightly different angles, and therefore acquire different phase, as -illustrated in Figure \ref{fig2mich.fig}. Suppose that the actual pathlength difference between two arms is $d$. -Then the path length difference for two off-axis rays arriving at the observer is $\Delta l = a + b$ where $a = -d/\cos \theta$ and $b = a\cos 2\theta$: -\begin{equation} -\Delta l=\frac{d}{\cos \theta}+\frac{d}{\cos \theta}\cos 2\theta -\end{equation} -Recalling that $\cos 2\theta = 2(\cos \theta)^2-1$, we obtain $ \Delta l=2d\cos\theta$. The two rays interfere -constructively for any angle $\theta_c$ for which $ \Delta l=2d\cos\theta = m\lambda$ ($m$=integer); at the same -time, two beams traveling at the angle $\theta_d$ interfere destructively when $ \Delta l=2d\cos\theta = -(m+1/2)\lambda$ ($m$=integer). Because of the symmetry about the normal direction to the mirrors, this will mean -that interference ( bright and dark fringes) appears in a circular shape. If fringes are not circular, it means -simply that the mirrors are not parallel, and additional alignment of the interferometer is required. - -\begin{figure} -\centerline{\epsfig{width=0.8\linewidth,file=fig2.eps}} \caption{\label{fig2mich.fig}Explanation of circular -fringes. Notice that to simplify the figure we have ``unfold'' the interferometer by neglecting the reflections -on the beamsplitter.} -\end{figure} - -When the path length difference $\Delta l$ is varied by moving one of the mirrors using the micrometer, the -fringes appear to "move". As the micrometer is turned, the condition for constructive and destructive -interference is alternately satisfied at any given angle. If we fix our eyes on one particular spot and count, -for example, how many bright fringes pass that spot as we move mirror $M_2$ by known distance, we can determine -the wavelength of light in the media using the condition for constructive interference, $\Delta l = 2d\cos -\theta = m\lambda$. - -For simplicity, we might concentrate on the center of the fringe bullseye at $\theta = 0$. The equation above -for constructive interference then reduces to $2\Delta l = m\lambda$ (m = integer). If $X_1$ is the initial -position of the mirror $M_2$ (as measured on the micrometer) and $X_2$ is the final position after a number of -fringes $\delta m$ has been counted, we have $2(X_2-X_1) = \lambda\delta m$. Then the laser wavelength, -$\lambda$, is then given as: -\begin{equation}\label{old3} -\lambda = 2(X_2-X_1)/\delta m. -\end{equation} - -\section*{Procedure} - -\subsection*{Laser Safety} -While this is a weak laser caution should still be used. \textbf{Never look directly at the laser beam!} Align -the laser so that it is not at eye level. - -\subsection*{Set Up} -\textbf{Equipment needed}: Pasco precision interferometry kit, a laser, -adjustable-hight platform. - -Set up the interferometer as shown in Figure~\ref{fig1mich.fig} using the components of Pasco precision -interferometry kit. A mirrors $M_{1,2}$ are correspondingly a movable and an adjustable mirror from the kit. -Make initial alignment of the interferometer with a non-diverging laser beam. Adjust the beams so that it is -impinging on the beamsplitter and on the viewing screen. Make sure the beam is hitting near the center of all -the optics, including both mirrors, the compensator plate and beam splitter. The interferometer has leveling -legs which can be adjusted. - -Then insert a convex lens after the laser to spread out the beam (ideally the laser beam should be pass through -the center of the lens to preserve alignment). After the beams traverse through the system, the image of the -interfering rays will be a circular pattern projected onto a screen. The two beam reflected off the mirrors -should be aligned as parallel as possible to give you a circular pattern. - -\subsection*{ Measurement of laser wavelength} - -Note the reading on the micrometer. Focus on a particular fringe (the center is a good place). Begin turning the -micrometer so that the fringes move (for example, from bright to dark to bright again is the movement of 1 -fringe). Count a total of about 100 fringes and record the new reading on the micrometer. Calculate the -wavelength from Eq. \ref{old3} above, remembering that you may need to convert the distance traveled on the -micrometer to the actual distance traveled by the mirror. - - - Each lab group must make at least four (4) measurements of $\lambda$. Each - partner must do at least one. For each trial, a minimum of 100 - fringes should be accurately counted, and related to an - initial $X_1$ and final $X_2$ micrometer setting. A final mean - value of $\lambda$ and its uncertainty should be - generated. Compare your value with the accepted value (given - by the instructor). - -\textbf{\emph{Experimental tips}}: -\begin{enumerate} -\item Avoid touching the face of the front-surface mirrors, the beamsplitter, and any other optical elements! -\item Engage the micrometer with both hands as you turn, maintaining -positive torque. -\item The person turning the micrometer should also do the counting of -fringes. It can be easier to count them in bunches of 5 or 10 (\textit{i.e.} -100 fringes = 10 bunches of 10 fringes). -\item Before the initial position $X_1$ is read make sure that the micrometer has engaged the -drive screw (There can be a problem with "backlash"). -\item Before starting the measurements make sure you understand how to read a -micrometer! See Fig.\ref{fig3mich.fig}. -\item Move the travel plate to a slightly different location for the -four readings. This can done by loosening the large nut atop the traveling -plate,and then locking again. -\item Avoid hitting the table which can cause a sudden jump in the -number of fringes. - -\end{enumerate} - -\begin{figure}[h] -\centerline{\epsfig{width=0.7\columnwidth,file=fig3.eps}} \caption{\label{fig3mich.fig}Micrometer readings. The -course scale is in mm, and smallest division on the rotary dial is 1~$\mu$m (same as 1 micron). The final -measurements is the sum of two. } -\end{figure} - - -\subsection*{Measurement of the index of refraction of air} - - If you recall from the speed of light experiment, the value -for air's index of refraction $n_{air}$ is very close to unity: -$n_{air}$=1.000293. Amazingly, a Michelson interferometer is precise enough to -be able to make an accurate measurement of this quantity! - -Let's remind ourselves that a Michelson interferometer is sensitive to a phase -difference acquired by the beams travelling in two arms -\begin{equation}\label{phase} -k\Delta l=2\pi n\Delta l/\lambda. -\end{equation} -In previous calculations we assumed that the index of refraction of air $n$ is exactly one, like in vacuum. -However, it is actually slightly varies with air pressure, as shown in Fig.~\ref{fig4mich.fig}. Any changes in -air pressure affect the phase $k\Delta l$. -% -\begin{figure} -\centerline{\epsfig{file=macfig1add.eps}} \caption{\label{fig4mich.fig}Index of refraction as a function of air -gas pressure} -\end{figure} - -To do the measurement, place a cylindrical gas cell which can be evacuated in -the path of light heading to mirror $M_1$ and correct alignment of the -Michelson interferometer, if necessary. Make sure that the gas cell is -initially at the atmospheric pressure. - -Now pump out the cell by using a hand pump at your station and count the number of fringe transitions $\delta -m$ that occur. When you are done, record $\delta m$ and the final reading of the vacuum gauge $p_{fin}$. -\textbf{Note}: most vacuum gauges display the difference between measured and atmospheric pressure . If -absolute pressure is needed, it should be found by subtracting the gauge reading from the atmospheric pressure -($p_0=76$~cm Hg). For example, if the gauge reads $23$~cm Hg, the absolute pressure is $53$~cm Hg. -Alternatively, you can pump out the air first, and then admit air is slowly to the cell while counting the -number of fringes that move past a selected fixed point. - -The shifting fringes indicate a change in relative optical phase difference for the two arms caused by the the -difference in refractive indices of the gas cell at low and atmospheric pressures $\Delta n$. According to -Eq.(\ref{phase}), this difference is -\begin{equation} \label{delta_n} -\Delta n=\delta m \frac{\lambda}{2d_{cell}} -\end{equation} -where $d_{cell}=3$~cm is the length of the gas cell. - -Since the change in the refractive index $\Delta n$ is linearly depends on the -air pressure $\Delta p=p_0-p_{fin}$, it is now easy to find out the -proportionality coefficient $\Delta n/\Delta p$ and calculate the value of the -refractive index at the atmospheric pressure $n_{air}$. - -Each partner should make one measurement of the fringe shift quantity $\delta m$. Use Eq.(\ref{delta_n}) to find -mean values of the relative change of the refractive index $\Delta n$, proportionality coefficient $\Delta -n/\Delta p$ and $n_{air}$ with corresponding uncertainties. Compare your measurements to the following -accepted experimental values: \\ -Index of Refraction of Air(STP) = 1.000293 \\ - - -\subsection*{\emph{Detection of Gravitational Waves}} - -\textbf{\emph{A Michelson interferometer can help to test the theory of -relativity!}} \emph{ -% -Gravity waves, predicted by the theory of relativity, are ripples in the fabric -of space and time produced by violent events in the distant universe, such as -the collision of two black holes. Gravitational waves are emitted by -accelerating masses much as electromagnetic waves are produced by accelerating -charges, and often travel to Earth. The only indirect evidence for these waves -has been in the observation of the rotation of a binary pulsar (for which the -1993 Nobel Prize was awarded).} -% -\begin{figure}[h] -\centerline{\epsfig{file=LIGO.eps}} \caption{\label{LIGO.fig}For more details see http://www.ligo.caltech.edu/} -\end{figure} -\emph{ -% -Laser Interferometry Gravitational-wave Observatory (LIGO) sets the ambitious -goal to direct detection of gravitational wave. The measuring tool in this -project is a giant Michelson interferometer. Two mirrors hang $2.5$~mi apart, -forming one "arm" of the interferometer, and two more mirrors make a second arm -perpendicular to the first. Laser light enters the arms through a beam splitter -located at the corner of the L, dividing the light between the arms. The light -is allowed to bounce between the mirrors repeatedly before it returns to the -beam splitter. If the two arms have identical lengths, then interference -between the light beams returning to the beam splitter will direct all of the -light back toward the laser. But if there is any difference between the lengths -of the two arms, some light will travel to where it can be recorded by a -photodetector.} - -\emph{ -%The space-time ripples cause the distance measured by a light beam to change as -the gravitational wave passes by. These changes are minute: just $10^{-16}$ -centimeters, or one-hundred-millionth the diameter of a hydrogen atom over the -$2.5$ mile length of the arm. Yet, they are enough to change the amount of -light falling on the photodetector, which produces a signal defining how the -light falling on changes over time. LlGO requires at least two widely separated -detectors, operated in unison, to rule out false signals and confirm that a -gravitational wave has passed through the earth. Three interferometers were -built for LlGO -- two near Richland, Washington, and the other near Baton -Rouge, Louisiana.} -% -\begin{figure} - \centerline{\epsfig{file=LISA.eps}} \caption{\label{LISA.fig}For -more details see http://lisa.nasa.gov/} -\end{figure} - -\emph{ -% -LIGO is the family of the largest existing Michelson interferometers, but just -wait for a few years until LISA (Laser Interferometer Space Antenna) - the -first space gravitational wave detector - is launched. LISA is essentially a -space-based Michelson interferometer: three spacecrafts will be arranged in an -approximately equilateral triangle. Light from the central spacecraft will be -sent out to the other two spacecraft. Each spacecraft will contain freely -floating test masses that will act as mirrors and reflect the light back to the -source spacecraft where it will hit a detector causing an interference pattern -of alternating bright and dark lines. The spacecrafts will be positioned -approximately 5 million kilometers from each other; yet it will be possible to -detect any change in the distance between two test masses down to 10 picometers -(about 1/10th the size of an atom)! -% -} - -\end{document} -\newpage |