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diff --git a/manual_source/chapters/interferometry.tex b/manual_source/chapters/interferometry.tex index 112e493..2170fd9 100644 --- a/manual_source/chapters/interferometry.tex +++ b/manual_source/chapters/interferometry.tex @@ -10,7 +10,7 @@ % \textheight=9.0in %\begin{document} -\chapter{Optical Interferometery} +\chapter{Optical Interferometry} \setcounter{figure}{1} \setcounter{table}{1} \setcounter{equation}{1} @@ -31,24 +31,24 @@ The \textbf{Michelson interferometer}, shown in Fig.~\ref{fig1mich.fig}, is base \centering \includegraphics[width=0.8\linewidth]{./pdf_figs/fig1} \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 fictious 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 graviational 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. 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. 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 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 passing 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: \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 -optical pathlengths for the two beams $\Delta l$ , $k=2\pi n/\lambda$ is the wave number, $\lambda$ is the +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 pathlength 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 $M_2$ 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: \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 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. +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. @@ -96,7 +96,7 @@ laser beam can be dangerous for your eyes. \textbf{Equipment needed}: Pasco precision interferometry kit, a laser, Na lamp, adjustable-height platform (or a few magazines or books). -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 = +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 path length 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 @@ -140,7 +140,7 @@ Once the interferometer is aligned, insert a convex lens ($f=\unit[18]{mm}$ work \subsection*{ Calibration of the interferometer} -Record the initial reading on the micrometer. Focus on a 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: pickout 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. +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 $<X_2-X_1>$. @@ -158,9 +158,13 @@ Each lab partner should make at least two independent measurements, starting fro \subsection*{Measurement of the Na lamp wavelength} -A calibrated Michelson interferometer can be used as a \textbf{wavemeter} to determine the wavelength of different light sources. In this experiment you will use it to measure the wavelength of strong yellow sodium fluorescent light, produced by the discharge lamp.\footnote{Actually, sodium might be a better calibration source than a HeNe laser, since it has well known lines, whereas a HeNe can lase at different wavelenths. Perhaps an even better calibration source might be a line from the Hydrogen Balmer series, which can be calculated from the Standard Model.} +A calibrated Michelson interferometer can be used as a \textbf{wavemeter} +to determine the wavelength of different light sources. In this experiment +you will use it to measure the wavelength of strong yellow sodium +fluorescent light, produced by the discharge lamp.\footnote{Actually, +sodium might be a better calibration source than a HeNe laser, since it has well known lines, whereas a HeNe can lase at different wavelengths. Perhaps an even better calibration source might be a line from the Hydrogen Balmer series, which can be calculated from the Standard Model.} -Without changing the alignment of the interferometer (i.e. without touching any mirrors), remove the focusing lens and carefully place the interferometer assembly on top of an adjustable-hight platform such that it is at the same level as the output of the lamp. Since the light power in this case is much weaker than for a laser, you won't be able to use the viewing screen. You will have to observe the interference looking directly to the output beam - unlike laser radiation, the spontaneous emission of a discharge is not dangerous\footnote{In the ``old days'' beams in high energy physics were aligned using a similar technique. An experimenter would close his eyes and then put his head in a columated 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 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. +Without changing the alignment of the interferometer (i.e. without touching any mirrors), remove the focusing lens and carefully place the interferometer assembly on top of an adjustable-hight platform such that it is at the same level as the output of the lamp. Since the light power in this case is much weaker than for a laser, you won't be able to use the viewing screen. You will have to observe the interference looking directly to the output beam - unlike laser radiation, the spontaneous emission of a discharge is not dangerous\footnote{In the ``old days'' beams in high 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 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}. @@ -184,7 +188,7 @@ Align the interferometer one more time such that the distance between two mirror \begin{enumerate} \item Turn off the laser, remove the viewing screen and the lens, and place the interferometer on the adjustable-height platform, or alternatively place the Na lamp on it's side and plan to adjust it's height with books or magazines. With the diffuser sheet in front of the lamp, check that you see the interference fringes when you look directly to the lamp through the interferometer. If necessary, adjust the knobs on the adjustable mirror to get the best fringe pattern. -\item Because the Na emission consists of two light at two close wavelengthes, the interference picture consists of two sets of rings, one corresponding to fringes of $\lambda_1$, the other to those for $\lambda_2$. Move the mirror back and forth (by rotating the micrometer) to identify two sets of ring. Notice that they move at slightly different rate (due to the wavelength difference). +\item Because the Na emission consists of two light at two close wavelengths, the interference picture consists of two sets of rings, one corresponding to fringes of $\lambda_1$, the other to those for $\lambda_2$. Move the mirror back and forth (by rotating the micrometer) to identify two sets of ring. Notice that they move at slightly different rate (due to the wavelength difference). \item Seek the START condition illustrated in Fig.(\ref{fpfig4.fig}), such that all bright fringes are evenly spaced. Note that alternate fringes may be of somewhat different intensities. Practice going through the fringe conditions as shown in Fig.(\ref{fpfig4.fig}) by turning the micrometer and viewing the relative movement of fringes. Do not be surprised if you have to move the micrometer quite a bit to return to the original condition again. |