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diff --git a/unused_chapters/michelson.tex b/unused_chapters/michelson.tex new file mode 100644 index 0000000..101fa64 --- /dev/null +++ b/unused_chapters/michelson.tex @@ -0,0 +1,311 @@ +%\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 |