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diff --git a/interferometry.tex b/interferometry.tex index 62487ae..e573a91 100644 --- a/interferometry.tex +++ b/interferometry.tex @@ -9,27 +9,27 @@ interferometers, calibrate them using a laser of known wavelength, and then use \section*{Introduction} -Optical interferometers are the instruments that rely on interference of two or more superimposed reflections of the input laser beam. These are one of the most common optical tools, and are used for precision measurements, surface diagnostics, astrophysics, seismology, quantum information, etc. There are many configurations of optical interferometers, and in this lab you will become familiar with two of the more common setups. +Optical interferometers are the instruments that rely on the interference of two or more superimposed reflections of the input laser beam. These are one of the most common optical tools, and are used for precision measurements, surface diagnostics, astrophysics, seismology, quantum information, etc. There are many configurations of optical interferometers, and in this lab you will become familiar with two of the more common setups. The \textbf{Michelson interferometer}, shown in Fig.~\ref{fig1mich.fig}, is based on the interference of two beams: the initial light is split into two arms on a beam splitter, and then these resulting beams are reflected and recombined on the same beamsplitter again. The difference in optical paths in the two arms leads to a changing relative phase of two beams, so when overlapped the two light fields will interfere constructively or destructively. \begin{figure}[h] \centering -\includegraphics[width=0.8\linewidth]{./pdf_figs/fig1} \caption{\label{fig1mich.fig}A Michelson Interferometer setup.} +\includegraphics[width=0.65\linewidth]{./pdf_figs/michelson.png} \caption{\label{fig1mich.fig}A Michelson Interferometer setup.} \end{figure} -Such an interferometer was first 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 fictitious medium for light wave propagation) and providing insight into the true nature of electromagnetic radiation. Michelson interferometers are widely used in many areas of physics and engineering. At the end of this writeup we describe LIGO, the world's largest Michelson interferometer, designed to measure the gravitational waves and thus test general relativity. +Such an interferometer was first 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 fictitious medium for light wave propagation) and providing insight into the true nature of electromagnetic radiation. Michelson interferometers are widely used in many areas of physics and engineering, including the recent discovery of gravitational waves at the LIGO facility (see Sect.~\ref{LIGO} at the end of this lab for more information). Figure~\ref{fig1mich.fig} shows the traditional setting for a Michelson interferometer. 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 (that was initially -transmitted by the beamsplitter) travels to a fixed mirror $M_1$ and back +transmitted by the beamsplitter) travels to a moveable mirror $M_1$ and back again. One-half of this amplitude is then reflected from the partially-silvered surface and directed at 90 degrees toward the observer (you will use a viewing screen). At the same time the second beam -(reflected by the beamsplitter) travels at 90 degrees toward mirror $M_2$ -and back. Since this beam never travels through the glass beamsplitter -plate, its optical path length is shorter than for the first beam. To -compensate for that, it passes twice through a clear glass plate called the compensator plate, that has the same thickness. At the beamsplitter one-half of this light is transmitted to an observer, overlapping with the first beam, and the total amplitude of the light at the screen is a combination of amplitude of the two beams: +(reflected by the beamsplitter) travels at 90 degrees toward an adjustable mirror $M_2$ +and back. Since this beam travels through the glass beamsplitter +plate 3 times, its optical path length is longer than the first beam which only passes through the beamsplitter 1 time. To +compensate for that, the first beam passes twice through a clear glass plate called the compensator plate, that has the same thickness. At the beamsplitter one-half of this light is transmitted to an observer, overlapping with the beam reflected by $M_2$, and the total amplitude of the light at the screen 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} @@ -37,7 +37,7 @@ Here $k\Delta l$ is a phase shift (``optical delay'') between the two wavefronts optical path lengths 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 path length of the light beams in both interferometer arms will be exactly the same. Therefore, the 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: +Mirror $M_1$ 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 path length of the light beams in both interferometer arms will be exactly the same. Therefore, the 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} @@ -47,7 +47,7 @@ Mirror $M_2$ is mounted on a precision traveling platform. Imagine that we adjus It is easy to see that constructive interference happens when the difference between path lengths in the two interferometer arms is equal to the integer number of wavelengths $\Delta l = m\lambda$, and destructive interference corresponds to a 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. -In \textbf{Fabry-Perot configuration} the input light field bounces between two closely spaced partially reflecting surfaces, creating a large number of reflections. Interference of these multiple beams produces sharp spikes in the transmission for certain light frequencies. Thanks to the large number of interfering rays, this type of interferometer has extremely high resolution, much better than a Michelson interferometer. For that reason Fabry-Perot interferometers are widely used in telecommunications, lasers and spectroscopy to control and measure the wavelengths of light. In this experiment we will take advantage of the high spectral resolution of the Fabry-Perot interferometer to resolve two very closely-spaces emission lines in Na spectra by observing changes in overlapping interference fringes from the two lines. +In \textbf{Fabry-Perot configuration} the input light field bounces between two closely spaced partially reflecting surfaces, creating a large number of reflections. Interference of these multiple beams produces sharp spikes in the transmission for certain light frequencies. Thanks to the large number of interfering rays, this type of interferometer has extremely high resolution, much better than a Michelson interferometer. For that reason Fabry-Perot interferometers are widely used in telecommunications, lasers and spectroscopy to control and measure the wavelengths of light. In this experiment we will take advantage of the high spectral resolution of the Fabry-Perot interferometer to resolve two very closely-spaced emission lines in Na spectra by observing changes in the overlapping interference fringes from the two lines. \begin{figure}[h] \centering \includegraphics[width=0.8\linewidth]{./pdf_figs/fpfig1} \caption{\label{fpfig1}Sequence of Reflection and Transmission for a ray arriving at the treated inner surfaces $P_1 \& P_2$.} @@ -107,44 +107,54 @@ simply that the mirrors are not parallel, and additional alignment of the interf \begin{figure} \centering -\includegraphics[width=0.8\linewidth]{./pdf_figs/fig2} \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.} +\includegraphics[width=0.8\linewidth]{./pdf_figs/fig2} \caption{\label{fig2mich.fig}Explanation the interference pattern. 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 a known distance, we can determine the wavelength of light in the media using the condition for constructive interference, $\Delta l = 2d\cos -\theta = m\lambda$. +When the path length difference is varied, by using the micrometer to move one of the mirrors a distance $\Delta l$ along the horizontal axis of Figure~\ref{fig2mich.fig}, 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 how many bright fringes pass that spot as we move mirror $M_1$ by a known distance, we can determine the wavelength of light in the media using the condition for constructive interference + +\begin{equation} +\label{eqn:constructive} +2\Delta l \cos\theta = m\lambda. +\end{equation} + +The factor of two comes from from the fact that if I move the mirror $\Delta l$ light has to go an additional distance $\Delta l \cos\theta$ to get to it and then the same distance return from it. For simplicity, we might concentrate on the center of the fringe bull's eye at $\theta = 0$. Equation~\ref{eqn:constructive} for constructive interference then reduces to $2 \Delta l = m\lambda$ (m = integer). If $X_1$ is the initial position of the mirror $M_1$ (as measured on the micrometer) and $X_2$ is the final position after a number of fringes $\Delta m$ has been counted, we have $2\Delta l = 2(X_2-X_1) = \lambda\Delta m$. Then the laser wavelength $\lambda$ is then given as: -For simplicity, we might concentrate on the center of the fringe bull's eye 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} Set up the interferometer as shown in Figure~\ref{fig1mich.fig} using -components from the PASCO interferometry kit. The mirrors $M_{1,2}$ are, correspondingly, a movable and an adjustable mirror from the kit. Align the interferometer with a laser beam. Adjust the beam so that it is impinging on the beamsplitter and on the viewing screen. Try to make the beams to hit 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. Align the beams such that they overlap on the viewing screen, and so that the reflected beam is directed back into the laser. This can be tricky to get right the first time. Be patient, make small changes, think about what you are doing, and get some help from the instructor and TA. +components from the PASCO interferometry kit. The mirrors $M_{1,2}$ are, correspondingly, a movable and an adjustable mirror from the kit. Alignment is not too difficult: + +\begin{enumerate} +\item Begin by installing the movable mirror, adjustable mirror, component holder with viewing screen (magnetically attaches), and the lens holder. Do not install the lens yet. Make sure screws are firmly tightened as we don't want components to move around. +\item Level the table by adjusting its legs and/or adjust the laser height, location and direction to get the beam spot at the center of the movable mirror. It is a good idea to tape your laser down at this point, or make sure it's securely mounted in the optical breadboard (if your station has one). +\item Tweak the laser direction using the two knobs on its back so that the beam reflects back into its aperture. +\item Install the beam splitter, orient it to center the new beam spot on the adjustable mirror. +\item Tweak the adjustable mirror using the two knobs on its back. You want to make the two sets of dots visible on the viewing screen come into alignment. When they are properly aligned you should see interference fringes appear in the dot. +\item It is possible to see interference without the compensator (ask yourself why). But, it can potentially make the pattern better and more distinct. Try inserting it and aligning to improve the pattern. +\item Install the lens to blow up the pattern. +\item Further tweak the adjustable mirror to bring the bullseye on screen. You will find it's very sensitive and a little tweaking goes a long way. +\end{enumerate} -Once the interferometer is aligned, insert a convex lens ($f=\unit[18]{mm}$ -works well) after the laser to spread out the beam (ideally the laser beam -should be pass through the center of the lens to preserve alignment). -Adjust the adjustable mirror slightly until you see the interference -fringes in the screen. Continue to make small adjustments until you see a clear bull's eye circular pattern. \emph{A word of caution: sometimes dust on a mirror or imperfections on optical surfaces may produce similar intensity patterns. True interference disappears if you block one arm of the interferometer. Try it!} + Be patient, make small changes, think about what you are doing, and get some help from the instructor and TA. It helps to start the alignment procedure with the micrometer and the movable mirror near the center of their ranges. Also, the alignment knobs on the laser and adjustable mirror should be at the middle of their ranges, so that the plane of each is perpendicular to the table. + +\emph{A word of caution: sometimes dust on a mirror or imperfections on optical surfaces may produce similar intensity patterns. True interference disappears if you block one arm of the interferometer. Try it!} \textbf{Note}: before starting the measurements, make sure you understand how to read the micrometer properly! \begin{figure}[h] \centering %\includegraphics[width=0.7\columnwidth]{./pdf_figs/fig3} -\subcaptionbox{Reading = 211~$\mu$m} { +%\subcaptionbox{Reading = 211~$\mu$m} { \includegraphics[height=1.3in]{./pdf_figs/micrometer1} -} -\subcaptionbox{Reading = 345~$\mu$m} { +%} +%\subcaptionbox{Reading = 345~$\mu$m} { \includegraphics[height=1.3in]{./pdf_figs/micrometer2} -} -\subcaptionbox{Reading = 166~$\mu$m} { +%} +%\subcaptionbox{Reading = 166~$\mu$m} { \includegraphics[height=1.3in]{./pdf_figs/micrometer3} -} -\caption{\label{fig3mich.fig}Micrometer readings. The coarse division equals to 100~$\mu$m, and smallest division on the rotary dial is 1~$\mu$m (same as 1 micron). The final measurements is the sum of two. } +%} +\caption{\label{fig3mich.fig}Micrometer readings. The coarse division equals to 100~$\mu$m, and smallest division on the rotary dial is 1~$\mu$m (same as 1 micron). The final measurements is the sum of two. So, from left to right, the figures above show 211~$\mu$m, 345~$\mu$m, and 166~$\mu$m.} \end{figure} \section*{Wavelength measurements using Michelson interferometer} @@ -153,7 +163,7 @@ fringes in the screen. Continue to make small adjustments until you see a clear Record the initial reading on the micrometer. Focus on the central fringe and begin turning the micrometer. You will see that the fringes move. For example, the central spot will change from bright to dark to bright again, that is counted as one fringe. A good method: pick out a reference line on the screen and then softly count fringes as they pass the point. Count a total of about $\Delta m = 50$ fringes and record the new reading on the micrometer. -Each lab partner should make at least two independent measurements, starting from different initial positions of the micrometer. For each trial, approximately 50 fringes should be accurately counted and tabulated with the initial $X_1$ and final $X_2$ micrometer settings. Do this at least five times (e.g., $5\times 50$ fringes). Consider moving the mirror both forward and backward. Make sure that the difference $X_2-X_1$ is consistent between all the measurements. Calculate the average value of the micrometer readings $<X_2-X_1>$. +Each lab partner should make at least two independent measurements, starting from different initial positions of the micrometer. For each trial, approximately 50 fringes should be accurately counted and tabulated with the initial $X_1$ and final $X_2$ micrometer settings. Do this at least five times (e.g., $5\times 50$ fringes). Consider moving the mirror both forward and backward. Make sure that the difference $X_2-X_1$ is consistent between all the measurements. Calculate the average value and standard deviation of the micrometer readings $<X_2-X_1>$. What value do you obtain for the laser wavelength, and what are your uncertainties on that measurement? Does this match what you expect for a red laser? %When your measurements are done, ask the instructor how to measure the wavelength of the laser using a commercial wavemeter. Using this measurement and Equation~\ref{old3} calculate the true distance traveled by the mirror $\Delta l$, and calibrate the micrometer (i.e. figure out precisely what displacement corresponds to one division of the micrometer screw dial). @@ -187,22 +197,23 @@ energy physics were aligned using a similar technique. An experimenter would close his eyes and then put his head in a collimated particle beam. Cerenkov radiation caused by particles traversing the experimenter's eyeball is visible as a blue glow or flashes. This is dangerous but various -people claim to have done it... when a radiation safety officer isn't +people claim to have seen it done... when a radiation safety officer wasn't around.}. However, your eyes will get tired quickly! Placing a diffuser plate in front of the lamp will make the observations easier. Since the interferometer is already aligned, you should see the interference picture. Make small adjustments to the adjustable mirror to make sure you see the center of the bull's eye. Repeat the same measurements as in the previous part by moving the mirror and counting the number of fringes. Each lab partner should make at least two independent measurements, recording initial and final position of the micrometer, and you should do at least five trials. Calculate the wavelength of the Na light for each trial. Then calculate the average value and its experimental uncertainty. Compare with the expected value of \unit[589]{nm}. In reality, the Na discharge lamp produces a doublet - two spectral lines that are very close to each other: \unit[589]{nm} and \unit[589.59]{nm}. Do you think your Michelson interferometer can resolve this small difference? Hint: the answer is no - we will use a Fabry-Perot interferometer for that task. +\clearpage + \section*{Alignment of the Fabry-Perot interferometer} \begin{figure} \centering -\includegraphics[width=0.8\linewidth]{./pdf_figs/fpfig3} \caption{\label{fpfig3.fig}The Fabry-Perot Interferometer. For initial alignment the laser and the convex lens are used instead of the Na lamp.} +\includegraphics[width=0.8\linewidth]{./pdf_figs/fpfig3} \caption{\label{fpfig3.fig}The Fabry-Perot Interferometer. \textbf{For initial alignment the laser and the convex lens are used instead of the Na lamp.}} \end{figure} Disassemble the Michelson Interferometer, and assemble the Fabry-Perot interferometer as shown in -Figure~\ref{fpfig3.fig}. First, place the viewing screen behind the two -partially-reflecting mirrors ($P1$ and $P2$), and adjust the mirrors such +Figure~\ref{fpfig3.fig}, initially using the laser for alignment instead of the sodium lamp. First, place the viewing screen behind the two partially-reflecting mirrors ($P1$ and $P2$), and adjust the mirrors such that the multiple reflections on the screen overlap. Then place a convex lens after the laser to spread out the beam, and make small adjustments until you see the concentric circles. Is there any difference between the @@ -269,7 +280,7 @@ valid the approximation that $\lambda_1\lambda_2\approx \lambda^2$ \boxed{\Delta\lambda \approx \frac{\lambda^2}{2(d_2-d_1)}} \end{equation} -Use this equation and your experimental measurements to calculate average value of Na doublet splitting and its standard deviation. Compare your result with the established value of $\Delta \lambda_{Na}=0.598$~nm. +Use this equation and your experimental measurements to calculate average value of Na doublet splitting and its standard deviation, as well as any experimental uncertainties. Compare your result with the established value of $\Delta \lambda_{Na}=0.598$~nm. \begin{figure}[h] \centering @@ -277,25 +288,25 @@ Use this equation and your experimental measurements to calculate average value \end{figure} \newpage -\section*{Detection of Gravitational Waves} - -\textbf{A Michelson interferometer can help to test the theory of relativity!} +\section*{Observation of Gravitational Waves (LIGO)} +\label{LIGO} +\textbf{A Michelson interferometer can help to test the theory of general relativity!} % 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). -% +charges, and often travel to Earth. Until recently the only indirect evidence for these waves +was the observation of the rotation of a binary pulsar (for which the 1993 Nobel Prize was awarded). + + \begin{figure}[h] \centering \includegraphics{./pdf_figs/LIGO} \caption{\label{LIGO.fig}For more details see http://www.ligo.caltech.edu/} \end{figure} -Laser Interferometry Gravitational-wave Observatory (LIGO) sets the ambitious -goal of the direct detection of a gravitational wave. The measuring tool in this -project is a giant Michelson interferometer. Two mirrors hang $2.5$~mi apart, + +The Laser Interferometry Gravitational-wave Observatory (LIGO) Michelson interferometer +built with the goal of directly detecting gravitational wave. 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 @@ -306,18 +317,14 @@ 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. -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} -%\centering -%\includegraphics{LISA.eps} \caption{\label{LISA.fig}For more details see http://lisa.nasa.gov/} -%\end{figure} +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 the photodetector 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, shown in Fig.~\ref{LIGO.fig}. -%\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)! -% -%} +\begin{figure}[h] +\centering +\includegraphics[width=0.6\linewidth]{./pdf_figs/LIGO_data.png} \caption{\label{LIGO_data.fig} Data from LIGO's two sites from the first detection of gravitational waves caused by the collision of two black holes (from \href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.061102}{Physical Review Letters \textbf{116}, 061102).}} +\end{figure} + +In 2015, LIGO observed the first direct detection of gravitational waves from the collision of two black holes. Figure~\ref{LIGO_data.fig} shows data from the black hole merger event where the `strain' reflects the observed change in distance in the interferometer arm due to the passing gravitation wave, which oscillates as a function of time. For a more information you can browse the \href{http://www.ligo.caltech.edu/}{LIGO webpage} and view a \href{https://www.ligo.caltech.edu/video/ligo20170601v2}{simulation of a black hole merger}. \end{document} |