Main Page/PHYS 3220/Interferometer - Revision history

Mgeorge at 19:05, 25 July 2011

2011-07-25T19:05:07Z

Mgeorge: Protected "Main Page/PHYS 3220/Interferometer" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

2011-07-25T19:04:50Z

Protected "Main Page/PHYS 3220/Interferometer" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

Mgeorge at 19:04, 25 July 2011

2011-07-25T19:04:46Z

Mgeorge at 19:03, 25 July 2011

2011-07-25T19:03:05Z

Mgeorge at 19:02, 25 July 2011

2011-07-25T19:02:18Z

Mgeorge at 19:00, 25 July 2011

2011-07-25T19:00:39Z

Mgeorge at 19:00, 25 July 2011

2011-07-25T19:00:07Z

Mgeorge at 18:59, 25 July 2011

2011-07-25T18:59:31Z

Mgeorge at 18:40, 25 July 2011

2011-07-25T18:40:49Z

Mgeorge at 18:28, 25 July 2011

2011-07-25T18:28:36Z

@@ Line 88: / Line 88: @@
 </p>
-<ol style="list-style-type:lower-latin"> >
+<ol style="list-style-type:lower-latin">
 <li><b>Initial adjustment of the Michelson Interferometer</b>
 <p>Verify that the components are in the correct position for the Michelson interferometer. The angular tilt of mirror M<sub>1</sub> is fixed, but it can be displaced using a linkage (lever) to a micrometer. Mirror M<sub>2</sub> can be tilted but not displaced. Place the mercury lamp before the diffusing screen and affix a pointer needle vertically to the screen. </p>

@@ Line 88: / Line 88: @@
 </p>
-<ol >
+<ol style="list-style-type:lower-latin"> >
 <li><b>Initial adjustment of the Michelson Interferometer</b>
 <p>Verify that the components are in the correct position for the Michelson interferometer. The angular tilt of mirror M<sub>1</sub> is fixed, but it can be displaced using a linkage (lever) to a micrometer. Mirror M<sub>2</sub> can be tilted but not displaced. Place the mercury lamp before the diffusing screen and affix a pointer needle vertically to the screen. </p>

@@ Line 88: / Line 88: @@
 </p>
-<ol>
+<ol >
 <li><b>Initial adjustment of the Michelson Interferometer</b>
 <p>Verify that the components are in the correct position for the Michelson interferometer. The angular tilt of mirror M<sub>1</sub> is fixed, but it can be displaced using a linkage (lever) to a micrometer. Mirror M<sub>2</sub> can be tilted but not displaced. Place the mercury lamp before the diffusing screen and affix a pointer needle vertically to the screen. </p>

@@ Line 196: / Line 196: @@
 <table width=400 align=center>
 <tr><td>
-<p align=justify>[[File:Mic-eqn13.png|140px|center]]
+<p align=justify>[[File:Mic-eq13.png|140px|center]]
 </p>
 </td><td> <b>(13)</b></td></tr></table>

@@ Line 140: / Line 140: @@
 <table width=400 align=center>
 <td>
-<p align=justify>[[File:Mic-eqn9c.png|220px|center]]
+<p align=justify>[[File:Mic-eqn9c.png|240px|center]]
 </p>
 </td></table>
@@ Line 147: / Line 147: @@
 <table width=400 align=center>
 <td>
-<p align=justify>[[File:Mic-eqn9d.png|80px|center]]
+<p align=justify>[[File:Mic-eqn9d.png|110px|center]]
 </p>
 </td></table>

← Older revision	Revision as of 19:04, 25 July 2011
(No difference)

@@ Line 116: / Line 116: @@
 <table width=400 align=center>
 <tr><td>
-<p align=justify>[[File:Mic-eqn8.png|140px|center]]
+<p align=justify>[[File:Mic-eqn8.png|240px|center]]
 </p>
 </td><td> <b>(8)</b></td></tr></table>
@@ Line 124: / Line 124: @@
 <table width=400 align=center>
 <tr><td>
-<p align=justify>[[File:Mic-eqn9.png|140px|center]]
+<p align=justify>[[File:Mic-eqn9.png|240px|center]]
 </p>
 </td><td> <b>(9)</b></td></tr></table>
@@ Line 133: / Line 133: @@
 <table width=400 align=center>
 <td>
-<p align=justify>[[File:Mic-eqn9b.png|140px|center]]
+<p align=justify>[[File:Mic-eqn9b.png|180px|center]]
 </p>
 </td></table>
@@ Line 140: / Line 140: @@
 <table width=400 align=center>
 <td>
-<p align=justify>[[File:Mic-eqn9c.png|140px|center]]
+<p align=justify>[[File:Mic-eqn9c.png|180px|center]]
 </p>
 </td></table>

← Older revision		Revision as of 18:40, 25 July 2011
Line 111:		Line 111:

	<p>When a certain point in the visual field is a maximum for both wavelengths (aligned fringes) then the phase differences introduced by the path difference between the mirrors must be integer multiples of 2π. Let δ<sub>1</sub> be the phase difference for wavelength λ<sub>1</sub> between the beams arriving at P from the two mirrors (i.e., δ<sub>1</sub>= φ<sub>2</sub> - φ<sub>i</sub> for λ<sub>1</sub>). Then it follows that δ<sub>1</sub> = (2π/λ)2 d = ''m<sub>1</sub>''(2π), where ''m<sub>1</sub>'' is an integer, and correspondingly for the other wavelength δ<sub>2</sub> = ''m<sub>2</sub>''(2π), where ''m<sub>2</sub>'' is another integer. </p>		<p>When a certain point in the visual field is a maximum for both wavelengths (aligned fringes) then the phase differences introduced by the path difference between the mirrors must be integer multiples of 2π. Let δ<sub>1</sub> be the phase difference for wavelength λ<sub>1</sub> between the beams arriving at P from the two mirrors (i.e., δ<sub>1</sub>= φ<sub>2</sub> - φ<sub>i</sub> for λ<sub>1</sub>). Then it follows that δ<sub>1</sub> = (2π/λ)2 d = ''m<sub>1</sub>''(2π), where ''m<sub>1</sub>'' is an integer, and correspondingly for the other wavelength δ<sub>2</sub> = ''m<sub>2</sub>''(2π), where ''m<sub>2</sub>'' is another integer. </p>
		+
		+	<p>The difference between the phase differences is</p>
		+
		+	<table width=400 align=center>
		+	<tr><td>
		+	<p align=justify>[[File:Mic-eqn8.png\|140px\|center]]
		+	</p>
		+	</td><td> <b>(8)</b></td></tr></table>
		+
		+	<p>where ''m'' is an integer. We denote the mirror separation for which this condition is satisfied by d=d<sub>m</sub>. If we move the mirrors smoothly until the next condition of maximum fringe resolution is obtained, we have two new phase differences δ<sub>1</sub>′ and δ<sub>2</sub>′ for which</p>
		+
		+	<table width=400 align=center>
		+	<tr><td>
		+	<p align=justify>[[File:Mic-eqn9.png\|140px\|center]]
		+	</p>
		+	</td><td> <b>(9)</b></td></tr></table>
		+
		+	<p>The mirror separation at this point is denoted as d<sub>m+1</sub>.</p>
		+
		+	<p>Equation <b>(8)</b> yields </p>
		+	<table width=400 align=center>
		+	<td>
		+	<p align=justify>[[File:Mic-eqn9b.png\|140px\|center]]
		+	</p>
		+	</td></table>
		+
		+	<p>while from equation <b>(9)</b></p>
		+	<table width=400 align=center>
		+	<td>
		+	<p align=justify>[[File:Mic-eqn9c.png\|140px\|center]]
		+	</p>
		+	</td></table>
		+
		+	<p>From the two equations and the geometric mean</p>
		+	<table width=400 align=center>
		+	<td>
		+	<p align=justify>[[File:Mic-eqn9d.png\|140px\|center]]
		+	</p>
		+	</td></table>
		+
		+	<p>one obtains for Δλ=λ<sub>2</sub> - λ<sub>1</sub></p>
		+
		+	<table width=400 align=center>
		+	<tr><td>
		+	<p align=justify>[[File:Mic-eqn10.png\|140px\|center]]
		+	</p>
		+	</td><td> <b>(10)</b></td></tr></table>
		+
		+	<p>This procedure is also applicable between conditions of minimum fringe contrast; it is up to the experimenter to decide which yields more accurate results.</p>
		+	<p>By using the results and the technique of the calibration experiment the mean wavelength λ can be found (you should make 5 measurements of λ); also Δd = d<sub>m+1</sub> - d<sub>m</sub> can be measured for a number of successive positions at which the pattern is least distinct (measure Δd for 10 successive positions and average the results). From this, Δλ, λ<sub>1</sub> and λ<sub>2</sub> can be evaluated.</p>
		+	<p>Note that you should use the M.I. in the same region of the micrometer screw as used while calibrating the instrument. Also, be careful about "backlash" in micrometer readings.</p>
		+	</li>
		+
		+	<li><p><b>Observation of inhomogeneities of the refractive index</b></p>
		+	<p>The fringe pattern is sensitive to changes of the refractive index of the medium in the light path. You can put a piece of glass into one arm of the interferometer and observe the variation of fringes from monochromatic light due to inhomgeneities in the glass. The variation in the fringe pattern from the change in the medium is commonly used in interferometry, e.g., to determine the refractive index of a medium, or (if the latter is known) to study the physical properties of a gas (e.g., density). You can see this effect by using a blow dryer to induce changes in one arm of the M.I.</p>
		+	</li>
		+	<li><p><b>White light fringes - Measurement of the refractive index</b></p>
		+	<p>If a source of white light is used, no fringes will be seen except for a path difference so small that it does not exceed a few wavelengths. In observing these fringes, the mirrors are tilted slightly as for localized fringes, and the position of M<sub>1</sub> is found where it intersects M<sub>2</sub>' (cf. ref. 2, pp248-50). From eq. 5 the condition ''d'' = 0 implies ''m'' = 0 for all wavelengths resulting in a central fringe that is either dark (no colour) or bright (all colours superimposed). In our experimental set-up the central fringe is bright bordered by dark bands (for the reasons discussed in the theory section).</p>
		+	<p>The ''d'' = 0 position is usually hard to find using white light only. It is best to use a Hg source to locate the position where the localized fringes become straight in the field of view. The lamp fixture has two light sources: a mercury discharge tube and a white light tungsten lamp, that are controlled by separate switches. Use the Hg source to locate straight parallel fringes and then turn on the white light. The mercury fringes will now appear against a white background. Turn the path length control slowly until a distinctive central white band flanked by dark fringes appears in the middle of the field. Now switch off the mercury lamp. The fringes will remain visible. </p>
		+	<p>Discuss the spectral variation that you observe in the field of view, i.e., why the first few higher-order fringes are coloured followed by a broad white illumination. What is the wavelength range for visible light?</p>
		+	<p>Using white light fringes you will measure the refractive index of a piece of plastic. When you insert the plastic into the path between the fixed mirror and the dividing mirror you will see that the fringes disappear. Note where (on the micrometer) they appeared before inserting the plastic and then carefully and slowly search a few turns on either side of this position until you find white-light fringes. Do not try to re-adjust the mirror tilt controls.</p>
		+	<p>You can measure the distance the mirror moved to re-establish the fringe system. (Several readings will give an average value x). If n1 is the index of the plastic and n<sub>o</sub> ~ 1 is the index for air (n<sub>0</sub> = 1 for vacuum), then the optical path length in the plastic (thickness t) is n<sub>1</sub> t, while the path length in the same distance t in air is no t. The additional path length introduced in the arm of the interferometer is thus</p>
		+
		+	<table width=400 align=center>
		+	<tr><td>
		+	<p align=justify>[[File:Mic-eqn11.png\|140px\|center]]
		+	</p>
		+	</td><td> <b>(11)</b></td></tr></table>
		+
		+	<p>This path length is equal to the amount by which the mirror must be reset (after inserting the plastic) to regain the white-light fringes:</p>
		+
		+	<table width=400 align=center>
		+	<tr><td>
		+	<p align=justify>[[File:Mic-eqn12.png\|140px\|center]]
		+	</p>
		+	</td><td> <b>(12)</b></td></tr></table>
		+
		+	<p>From this condition, find n<sub>1</sub> after measuring t using a micrometer.</p>
		+	</li>
		+
		+	<li><p><b>The index of refraction of air at atmospheric pressure</b></p>
		+	<p>It is possible to see the change in the optical path that occurs as air is removed from a cell positioned in one arm of the Michelson interferometer. This is used for a quantitative measurement of the index of refraction of air.</p>
		+	<p>Suppose that ''m'' fringes pass the reference point in the visual field as one changes the pressure in the cell by ΔP. A passage of one fringe corresponds to a change in the optical path length by λ. Let the index of refraction of air at pressure P be μ(P). At zero pressure, i.e., for a vacuum, μ(0)=1. Thus, the change in the optical path due to the presence of air at pressure P in a cell of length t in one arm, as compared to vacuum, corresponds to a total passage of N fringes:</p>
		+	<table width=400 align=center>
		+	<tr><td>
		+	<p align=justify>[[File:Mic-eqn13.png\|140px\|center]]
		+	</p>
		+	</td><td> <b>(13)</b></td></tr></table>
		+	<p>Place the cell in one arm of the interferometer and attach the hand pump. Align the interferometer to see monochromatic fringes from the mercury lamp with a green filter. Then verify that the fringes move when air is exhausted from the cell. Use the pump gently. This pump cannot provide a very good vacuum. You will need to extrapolate your results to a perfect vacuum condition. Note that the gauge starts at zero corresponding to atmospheric pressure, and as you evacuate the cell the pressure reading increases. (You may have to tap the pressure gauge gently to ensure that the needle is not sticking). Begin evacuating the cell, and record the pressure changes associated with the passage of ''m'' = 10 fringes. Are the changes in pressure (for a fixed number of fringes) approximately equal? Calculate the index of refraction at room pressure. Discuss the sources of error in your measurement.</p>
		+	</li>
		+	</ol>
		+
		+	<h1>References</h1>
		+
		+	<ol>
		+	<li>Halliday, Resnick, Walker, ''Fundamentals of Physics'', chapter 40.</li>
		+	<li>Jenkins, White, ''Fundamentals of Optics'', McGraw-Hill 1957</li>
		+	<li>Serway, Moses, Moyer, ''Modern Physics'', Saunders 1989, page 229</li>
		+	<li>Hecht, E. , ''Optics'', Addison-Wesley 1987.</li>
		+	<li>Tenquist D.W., Whittle R.M., Yarwood J., ''University Optics'', Iliffe Books, London 1969.</li>
		+	<li>Lipson S.G., Lipson H., Tannhauser D.S., ''Optical Physics'', 3rd ed., Cambridge Univ. Press,1995.</li>
		+	</ol>

← Older revision		Revision as of 18:28, 25 July 2011
Line 84:		Line 84:

	<h1>Experimental Procedure</h1>		<h1>Experimental Procedure</h1>
		+	<p><b>NOTE:</b>The interferometer is a precise optical system that can be easily damaged.
		+	<b>DO NOT</b> put your fingers on any reflecting surface.
		+	</p>
		+
		+	<ol>
		+	<li><b>Initial adjustment of the Michelson Interferometer</b>
		+	<p>Verify that the components are in the correct position for the Michelson interferometer. The angular tilt of mirror M<sub>1</sub> is fixed, but it can be displaced using a linkage (lever) to a micrometer. Mirror M<sub>2</sub> can be tilted but not displaced. Place the mercury lamp before the diffusing screen and affix a pointer needle vertically to the screen. </p>
		+
		+	<p><b><u>CAUTION</u>: The mercury lamp emits UV radiation. You have to wear safety glasses to protect your eyes. Never operate the mercury lamp without a glass plate attached.</b></p>
		+
		+	<p>Turn the mercury lamp on and align the eye with the viewing axis about ten inches (25 cm) from the instrument. Two pairs of reflected images of the pointer will be seen, one coming from reflection at the front surface of A, and the other from reflection at its back surface. When the tilting screws on M<sub>2</sub> are now turned until one pair of images falls exactly on the other, the interference fringes should appear. Acquaint yourself with the instrument, and try to obtain a circular, as well as a parallel pattern. Try to produce a circular pattern with a small number of fringes? </p>
		+
		+	<p>As mirror M<sub>1</sub> is moved a distance d, the pattern of bright (dark) fringes shifts according to eq 5 (eq. 6). Thus, if one changes d and counts how many bright (dark) fringes pass a reference point (e.g. the image of the pointer), the wavelength λ of the monochromatic light can be determined. On the other hand, one can use this method with light of a known wavelength in order to calibrate the mechanism for the mirror movement, i.e., to determine the distance d that corresponds to a given reading on the micrometer. In the following steps you will first perform a calibration of the linkage, and then determine λ for a source of unknown wavelength.</p>
		+
		+	<p><b>Calibration of the mirror movement</b></p>
		+
		+	<p>The mercury lamp produces a spectrum of lines of discrete wavelengths (consult a spectral chart to verify, cf.. Table I in Exp. 9) .The green filter used in front of the Hg lamp isolates the 5461Ǻ (=546.1 nm) line. Measure the distance on the micrometer screw that produces a shift of 50 fringes. Repeat the measurement for ten sets of 50 fringes. Carefully deduce the mechanical ratio of the micrometer-mirror linkage with its error.</p>
		+	</li>
		+
		+	<li><b>Determination of the wavelength of the Na doublet lines</b>
		+	<p>It is possible to use the Michelson interferometer to measure the wavelength difference between two lines closely spaced in wavelength and of similar intensities, without finding the wavelength of either line to high precision. In this part we determine the separation and average wavelength of the sodium doublet, known for its characteristic yellow lines.</p>
		+
		+	<p>The two lines with wavelengths λ<sub>1</sub> and λ<sub>2</sub> are closely spaced for the following reason. At the level of nonrelativistic quantum mechanics a single line is predicted for the transition corresponding to a de-excitation of the Na(3p) to the Na(3s) ground-state level. Note that in contrast to the hydrogen spectrum (a pure -Ze<sup>2</sup>/r potential) the Na energy spectrum does have a dependence on the principal quantum number n as well as on the magnitude of angular momentum ''l'' , i.e., 3s and 3p are not degenerate. In a more refined treatment the orbital angular momentum (''l'' = 1) state Na(3p) can couple with the electron spin (''s'' = 1/2) in two ways: a state with total angular momentum projection ''j<sub>z</sub>'' = 3/2, and another state with projection ''j<sub>z</sub>'' = 1/2 along the z-axis. The energies for these states are slightly different. The separation of the two wavelengths (that you determine in this experiment) is a measure of the effect of the spin-orbit coupling: a tiny energy offset that arises as a result of the interaction between the magnetic moments associated with the orbital and spin angular momenta of the electron (cf.. ref. 3).</p>
		+
		+	<p>The presence of a superposition of two monochromatic waves in the M.I. complicates the analysis somewhat. One obtains a superposition of two slightly different fringe patterns due to the difference between the wavelengths λ<sub>1</sub> and λ<sub>2</sub>. For some choices of d the fringes for the two wavelengths are aligned, while for other values the maximum of intensity for one wavelength lies between the maxima of the other wavelength, leading to uniform illumination (as opposed to a fringe pattern). The difference in the mirror displacements for which a uniform field is observed can be used to measure the wavelength difference as shown below.</p>
		+
		+	<p>When a certain point in the visual field is a maximum for both wavelengths (aligned fringes) then the phase differences introduced by the path difference between the mirrors must be integer multiples of 2π. Let δ<sub>1</sub> be the phase difference for wavelength λ<sub>1</sub> between the beams arriving at P from the two mirrors (i.e., δ<sub>1</sub>= φ<sub>2</sub> - φ<sub>i</sub> for λ<sub>1</sub>). Then it follows that δ<sub>1</sub> = (2π/λ)2 d = ''m<sub>1</sub>''(2π), where ''m<sub>1</sub>'' is an integer, and correspondingly for the other wavelength δ<sub>2</sub> = ''m<sub>2</sub>''(2π), where ''m<sub>2</sub>'' is another integer. </p>