Main Page/PHYS 4210/Zeeman Effect - Revision history

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Mgeorge at 18:20, 14 April 2015

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Mgeorge at 15:27, 18 November 2013

2013-11-18T15:27:01Z

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Mgeorge at 15:08, 18 November 2013

2013-11-18T15:08:11Z

@@ Line 115: / Line 115: @@
 <ul>
 <li><b>Data for the Lummer-Gehrcke plate</b> [from Leybold’s manual]:  ''d'' = 4.04 mm	η = 1.4567 </li>
 <li>Grotrian diagrams for H, Cd, Hg, and He taken from <ref>Radzig A.A., Smirnov B.M.,[https://www.library.yorku.ca/find/Record/243579 ''Reference Data on Atoms Molecules and Ions''], Springer 1985.</ref> are available below.
 <ul>

@@ Line 115: / Line 115: @@
 <ul>
 <li><b>Data for the Lummer-Gehrcke plate</b> [from Leybold’s manual]:  ''d'' = 4.04 mm	η = 1.4567 </li>
-<li>Grotrian diagrams for Cd, Hg, and He taken from <ref>Radzig A.A., Smirnov B.M.,[https://www.library.yorku.ca/find/Record/243579 ''Reference Data on Atoms Molecules and Ions''], Springer 1985.</ref> are available below.
+<li>Grotrian diagrams for H, Cd, Hg, and He taken from <ref>Radzig A.A., Smirnov B.M.,[https://www.library.yorku.ca/find/Record/243579 ''Reference Data on Atoms Molecules and Ions''], Springer 1985.</ref> are available below.
 <ul>
 <li>[[Media:GrotrianH.pdf| Hydrogen]]</li>

@@ Line 115: / Line 115: @@
 <ul>
 <li><b>Data for the Lummer-Gehrcke plate</b> [from Leybold’s manual]:  ''d'' = 4.04 mm	η = 1.4567 </li>
-<li>Grotrian diagrams for Cd, Hg, and He  taken from <ref>Radzig A.A., Smirnov B.M.,[https://www.library.yorku.ca/find/Record/243579 ''Reference Data on Atoms Molecules and Ions''], Springer 1985.</ref>. A copy of these are in the binder in the laboratory.
+<li>Grotrian diagrams for Cd, Hg, and He taken from <ref>Radzig A.A., Smirnov B.M.,[https://www.library.yorku.ca/find/Record/243579 ''Reference Data on Atoms Molecules and Ions''], Springer 1985.</ref> are available below.
 <ul>
 <li>[[Media:GrotrianH.pdf| Hydrogen]]</li>

@@ Line 85: / Line 85: @@
 <table width=400 align=center>
 <tr><td>
-<p align=justify>[[File:Zee-eqn2v2.PNG|220px|center]]
+<p align=justify>[[File:Zee-eqn2v2.PNG|280px|center]]
 </p>
 </td><td> <b>(2)</b></td></tr></table>

@@ Line 85: / Line 85: @@
 <table width=400 align=center>
 <tr><td>
-<p align=justify>[[File:Zee-eqn2.png|220px|center]]
+<p align=justify>[[File:Zee-eqn2v2.PNG|220px|center]]
 </p>
 </td><td> <b>(2)</b></td></tr></table>

@@ Line 42: / Line 42: @@
 <h1>Introduction</h1>
-<p>The Zeeman effect is a powerful demonstration of the splittings of magnetic sublevels in an angular momentum multiplet. Many aspects of the emission of light by excited atoms, particularly when exposed to strong magnetic fields ('''B''') were understood by Lorentz in a classical model <ref name="Jenkins">Jenkins F.A., White H.E., ''Fundamentals of Optics'', McGraw-Hill</ref><ref name="Mullin">Brehm J.J., Mullin W.J., [https://www.library.yorku.ca/find/Record/1126595 ''Introduction to the Structure of Matter''], Wiley </ref> before the advent of quantum mechanics. It is possible to understand the changes to classical electron orbits due to the Lorentz force in a 3D harmonic oscillator model. When one complements this with the idea that electromagnetic waves are transverse (the associated electric and magnetic fields of the EM wave oscillate in a plane perpendicular to the propagation direction of the wave), one can understand why circularly polarized light emerges as the atoms are observed in a direction longitudinal with the external '''B''' field, and why they appear as plane-polarized as viewed from the transverse direction. The understanding in the classical framework helps to build an intuition about the problem.</p>
+<p>The Zeeman effect is a powerful demonstration of the splittings of magnetic sublevels in an angular momentum multiplet. Many aspects of the emission of light by excited atoms, particularly when exposed to strong magnetic fields ('''B''') were understood by Lorentz in a classical model <ref name="Jenkins">Jenkins F.A., White H.E., [https://www.library.yorku.ca/find/Record/48797 ''Fundamentals of Optics''], McGraw-Hill</ref><ref name="Mullin">Brehm J.J., Mullin W.J., [https://www.library.yorku.ca/find/Record/1126595 ''Introduction to the Structure of Matter''], Wiley </ref> before the advent of quantum mechanics. It is possible to understand the changes to classical electron orbits due to the Lorentz force in a 3D harmonic oscillator model. When one complements this with the idea that electromagnetic waves are transverse (the associated electric and magnetic fields of the EM wave oscillate in a plane perpendicular to the propagation direction of the wave), one can understand why circularly polarized light emerges as the atoms are observed in a direction longitudinal with the external '''B''' field, and why they appear as plane-polarized as viewed from the transverse direction. The understanding in the classical framework helps to build an intuition about the problem.</p>
 <p>In the modern quantum mechanical description <ref name="Mullin"/><ref>Merzbacher E., [https://www.library.yorku.ca/find/Record/47405 ''Quantum Mechanics''], Wiley</ref><ref name="Bethe">Bethe H.A., Salpeter E.E., [https://www.library.yorku.ca/find/Record/2227072 ''Quantum Mechanics of One- and Two-Electron Systems''], Springer</ref> one has to take into account that the presence of the '''B''' field singles out an axis. The additional interaction term between the magnetic moment of the electronic state (proportional to the ''z'' component of the total angular momentum) and '''B''' serves to split the magnetic sublevels of states with non-zero angular momentum. The additional interaction forces the use of this axis as a quantization axis. Without an external field one usually picks a ''z'' axis, but should arrive at results that are independent of this choice. To obtain the observed result that the light emanating from spontaneous transitions without an external field is unpolarized, one has to average over random orientations of the quantization axis. The observation that a definite orientation is singled out as quantization axis in the Zeeman effect is sometimes referred to as ‘space quantization’.</p>
 <p>The problem can be illustrated by using pure orbital angular momentum states, i.e.,ignoring spin, and considering an np - ms transition. This transition is an allowed electric dipole transition, since a single unit of orbital angular momentum is changed, and this difference of one unit is carried away in the form of the spin for the spontaneously radiated photon. The important quantity to watch is the change in the projection of the orbital angular momentum, which can be +1, 0, -1 depending on the choice of the magnetic sublevel.</p>

@@ Line 111: / Line 111: @@
 <p>Once you are able to see nice clear lines as shown in Figure 3, use the ''Region of Interest'' tool to focus in on a few lines in center. You can now use the ''Zoom'' to expand the image. One the left margin of the video image, there is a scale showing the pixel number. You can use this as a fixed reference point- as you increase the applied magnetic field, the line will split into sublevels, check the dial gauge, and then use the adjusting screw to place the shifted line back to the pixel number of the original line, then check the new reading of the dial gauge and record the measurement.</p>
 <h1>References</h1>
 <references/>
 <ol>
 <li>Jenkins F.A., White H.E., ''Fundamentals of Optics'', McGraw-Hill</li>
@@ Line 123: / Line 138: @@
 <li>Radzig A.A., Smirnov B.M., ''Reference Data on Atoms Molecules and Ions'', Springer 1985.</li>
 </ol>
+-->

@@ Line 43: / Line 43: @@
 <h1>Introduction</h1>
 <p>The Zeeman effect is a powerful demonstration of the splittings of magnetic sublevels in an angular momentum multiplet. Many aspects of the emission of light by excited atoms, particularly when exposed to strong magnetic fields ('''B''') were understood by Lorentz in a classical model <ref name="Jenkins">Jenkins F.A., White H.E., ''Fundamentals of Optics'', McGraw-Hill</ref><ref name="Mullin">Brehm J.J., Mullin W.J., [https://www.library.yorku.ca/find/Record/1126595 ''Introduction to the Structure of Matter''], Wiley </ref> before the advent of quantum mechanics. It is possible to understand the changes to classical electron orbits due to the Lorentz force in a 3D harmonic oscillator model. When one complements this with the idea that electromagnetic waves are transverse (the associated electric and magnetic fields of the EM wave oscillate in a plane perpendicular to the propagation direction of the wave), one can understand why circularly polarized light emerges as the atoms are observed in a direction longitudinal with the external '''B''' field, and why they appear as plane-polarized as viewed from the transverse direction. The understanding in the classical framework helps to build an intuition about the problem.</p>
-<p>In the modern quantum mechanical description <ref name="Mullin"/><ref>Merzbacher E., [https://www.library.yorku.ca/find/Record/47405 ''Quantum Mechanics''], Wiley</ref><ref>Bethe H.A., Salpeter E.E., [https://www.library.yorku.ca/find/Record/2227072 ''Quantum Mechanics of One- and Two-Electron Systems''], Springer</ref> one has to take into account that the presence of the '''B''' field singles out an axis. The additional interaction term between the magnetic moment of the electronic state (proportional to the ''z'' component of the total angular momentum) and '''B''' serves to split the magnetic sublevels of states with non-zero angular momentum. The additional interaction forces the use of this axis as a quantization axis. Without an external field one usually picks a ''z'' axis, but should arrive at results that are independent of this choice. To obtain the observed result that the light emanating from spontaneous transitions without an external field is unpolarized, one has to average over random orientations of the quantization axis. The observation that a definite orientation is singled out as quantization axis in the Zeeman effect is sometimes referred to as ‘space quantization’.</p>
+<p>In the modern quantum mechanical description <ref name="Mullin"/><ref>Merzbacher E., [https://www.library.yorku.ca/find/Record/47405 ''Quantum Mechanics''], Wiley</ref><ref name="Bethe">Bethe H.A., Salpeter E.E., [https://www.library.yorku.ca/find/Record/2227072 ''Quantum Mechanics of One- and Two-Electron Systems''], Springer</ref> one has to take into account that the presence of the '''B''' field singles out an axis. The additional interaction term between the magnetic moment of the electronic state (proportional to the ''z'' component of the total angular momentum) and '''B''' serves to split the magnetic sublevels of states with non-zero angular momentum. The additional interaction forces the use of this axis as a quantization axis. Without an external field one usually picks a ''z'' axis, but should arrive at results that are independent of this choice. To obtain the observed result that the light emanating from spontaneous transitions without an external field is unpolarized, one has to average over random orientations of the quantization axis. The observation that a definite orientation is singled out as quantization axis in the Zeeman effect is sometimes referred to as ‘space quantization’.</p>
 <p>The problem can be illustrated by using pure orbital angular momentum states, i.e.,ignoring spin, and considering an np - ms transition. This transition is an allowed electric dipole transition, since a single unit of orbital angular momentum is changed, and this difference of one unit is carried away in the form of the spin for the spontaneously radiated photon. The important quantity to watch is the change in the projection of the orbital angular momentum, which can be +1, 0, -1 depending on the choice of the magnetic sublevel.</p>
 <p>A transition 2p<sub>0</sub> - 1s is associated with the emission of linearly polarized light with the oscillating electric field vector aligned with the z axis, with the wave propagation vector being orthogonal to this axis. This can be understood from the fact that the only non-vanishing matrix element for the dipole operator is <1s|''z''|2p<sub>0</sub>>. Similar calculations show that the transitions originating in the ''m'' = 1 and ''m'' = -1 sublevels result in circularly polarized light being emitted, which can propagate in the z direction only. One of the fascinating aspects of the Zeeman experiment is the following. For field-free atoms no axis is singled out, and thus, one has to include all possible orientations of the ''z'' axis, which results in the prediction that the light emitted from free atoms is unpolarized. However, once a homogeneous magnetic field is applied, an axis is singled out in space, which becomes the natural quantization axis. By probing the polarization of the spontaneously emitted light of atoms in the presence of a magnetic field one can verify that indeed the turn-on of the field causes a repopulation of the magnetic sublevels in a way that corresponds to the classical predictions of the Lorentz model. Thus, it is necessary to observe the light emitted longitudinally and transverse to the magnetic field.</p>
@@ Line 90: / Line 90: @@
 <p>One common method of determining the electron charge-to-mass ratio is through eq (2). Our interest is, however, to determine the energy splitting as a function of the magnetic field strength ''B''.</p>
 <p>In our version of the Zeeman experiment the intense red Cd line at 643.8 nm is used. For optical transitions Cd acts as an effective two-electron system, i.e., it has a He-like configuration, as has Hg. Compare the Grotrian diagram shown in Fig. 1 to the one for mercury provided in the appendix (cf.. the Franck-Hertz experiment). In order to understand the selection rules for allowed electric dipole transitions follow the arguments given in <ref name="Preston"/> in the context of the HeNe laser experiment. The two active electrons have combined orbital angular momentum ''L'' = l<sub>1</sub>+l<sub>2</sub>, spin S = s<sub>1</sub>+s<sub>2</sub>, and total angular momentum J = j<sub>1</sub>+j<sub>2</sub>. Allowed transitions require a change of one unit in ''L'' and ''J'' to make up for the spin of the photon, considering that spin flip is unlikely. Using <sup>(2''S''+1)</sup>''L<sub>J</sub>'' notation (with ''L'' = 0 denoted as S, ''L'' = 1 as P, ''L'' = 2 as D, etc) we have for the relevant line a <sup>1</sup>D<sub>2</sub> - <sup>1</sup>P<sub>1</sub> transition. This means that the spins are paired up (spin singlet) and that in nonrelativistic notation a 5s5d to 5s5p transition takes place. In Hg the equivalent line at the n = 6 level is the yellow line at 579 nm. The shift of the same line towards yellow is the result of having an additional electron shell in the core. What wavelength is associated with this transition in He?</p>
-<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>''  =  -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. <ref name="Melissinos"/> Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. <ref name="Mellisinos"/><ref name="Melissinos"/>[3,6]). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
+<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>''  =  -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. <ref name="Melissinos"/> Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. <ref name="Melissinos"/><ref name="Bethe"/>). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
 <p>To verify these predictions about the polarization states of the light when the B field is turned on you need to recall some optical properties of polarizers and of quarter-wave plates (cf.. Ref.1). By placing these in the correct order you can verify the following:</p>
 <p>In the transverse direction, use a linear polarizer to identify the polarization states of the three components with '''B''' turned on; what happens for ''B'' = 0?</p>

@@ Line 76: / Line 76: @@
 <p>The LG plate shown as part of the apparatus in Fig. 2 consists of a precisely polished quartz glass plate of given thickness d with a prism attached at one end so that light entering from the slit has an angle of incidence on the plate that is near the critical angle. This results in some refractive transmission and mostly reflection at the glass/air surface. The reflected light inside the glass plate undergoes multiple ‘bounces’ of this type (interior reflection and partial refractive transmission). Two different interference patterns emerge when looking at a grazing angle at the top or bottom of the LG plate. The pattern formed at the top shows sharp bright lines on a dark background. In contrast to a Michelson interferometer a multiple-beam interferometer such as the FP and LG can produce an uneven interference pattern <ref name="Jenkins"/><ref name="Melissinos"/>.</p>
-<p>The separation between the fringes that appear without the magnetic field depends on the angle of observation. This spacing ''ΔA'' has to be determined for the particular fringe chosen for observation. As a magnetic field is applied each bright fringe splits either into two or into three depending on the orientation with respect to the magnetic field. To obtain a quantitative measure of the Zeeman effect, one needs to determine the '''B''' dependent splitting ''ΔS'' relative to ''ΔA''. Making use of the ratio eliminates the need to know the optical magnification, observation angle and distance from the plate. The frequency splitting depends also on the LG plate thickness d (as in the FP case), and additionally on the index of refraction η of the quartz glass. In the FP case this would be equal to 1, but there are versions of the experiment where the evacuation of a sealed FP interferometer is used to produce a scanning effect in the fringe pattern <ref>Preston D.W. Dietz E.R., [https://www.library.yorku.ca/find/Record/1038893 ''The Art of Experimental Physics''],Wiley</ref>[4]. The frequency splitting can be written as</p>
+<p>The separation between the fringes that appear without the magnetic field depends on the angle of observation. This spacing ''ΔA'' has to be determined for the particular fringe chosen for observation. As a magnetic field is applied each bright fringe splits either into two or into three depending on the orientation with respect to the magnetic field. To obtain a quantitative measure of the Zeeman effect, one needs to determine the '''B''' dependent splitting ''ΔS'' relative to ''ΔA''. Making use of the ratio eliminates the need to know the optical magnification, observation angle and distance from the plate. The frequency splitting depends also on the LG plate thickness d (as in the FP case), and additionally on the index of refraction η of the quartz glass. In the FP case this would be equal to 1, but there are versions of the experiment where the evacuation of a sealed FP interferometer is used to produce a scanning effect in the fringe pattern <ref name="Preston">Preston D.W. Dietz E.R., [https://www.library.yorku.ca/find/Record/1038893 ''The Art of Experimental Physics''],Wiley</ref>. The frequency splitting can be written as</p>
 <table width=400 align=center>
 <tr><td>
@@ Line 89: / Line 89: @@
 </td><td> <b>(2)</b></td></tr></table>
 <p>One common method of determining the electron charge-to-mass ratio is through eq (2). Our interest is, however, to determine the energy splitting as a function of the magnetic field strength ''B''.</p>
-<p>In our version of the Zeeman experiment the intense red Cd line at 643.8 nm is used. For optical transitions Cd acts as an effective two-electron system, i.e., it has a He-like configuration, as has Hg. Compare the Grotrian diagram shown in Fig. 1 to the one for mercury provided in the appendix (cf.. the Franck-Hertz experiment). In order to understand the selection rules for allowed electric dipole transitions follow the arguments given in ref. 4 in the context of the HeNe laser experiment. The two active electrons have combined orbital angular momentum ''L'' = l<sub>1</sub>+l<sub>2</sub>, spin S = s<sub>1</sub>+s<sub>2</sub>, and total angular momentum J = j<sub>1</sub>+j<sub>2</sub>. Allowed transitions require a change of one unit in ''L'' and ''J'' to make up for the spin of the photon, considering that spin flip is unlikely. Using <sup>(2''S''+1)</sup>''L<sub>J</sub>'' notation (with ''L'' = 0 denoted as S, ''L'' = 1 as P, ''L'' = 2 as D, etc) we have for the relevant line a <sup>1</sup>D<sub>2</sub> - <sup>1</sup>P<sub>1</sub> transition. This means that the spins are paired up (spin singlet) and that in nonrelativistic notation a 5s5d to 5s5p transition takes place. In Hg the equivalent line at the n = 6 level is the yellow line at 579 nm. The shift of the same line towards yellow is the result of having an additional electron shell in the core. What wavelength is associated with this transition in He?</p>
+<p>In our version of the Zeeman experiment the intense red Cd line at 643.8 nm is used. For optical transitions Cd acts as an effective two-electron system, i.e., it has a He-like configuration, as has Hg. Compare the Grotrian diagram shown in Fig. 1 to the one for mercury provided in the appendix (cf.. the Franck-Hertz experiment). In order to understand the selection rules for allowed electric dipole transitions follow the arguments given in <ref name="Preston"/> in the context of the HeNe laser experiment. The two active electrons have combined orbital angular momentum ''L'' = l<sub>1</sub>+l<sub>2</sub>, spin S = s<sub>1</sub>+s<sub>2</sub>, and total angular momentum J = j<sub>1</sub>+j<sub>2</sub>. Allowed transitions require a change of one unit in ''L'' and ''J'' to make up for the spin of the photon, considering that spin flip is unlikely. Using <sup>(2''S''+1)</sup>''L<sub>J</sub>'' notation (with ''L'' = 0 denoted as S, ''L'' = 1 as P, ''L'' = 2 as D, etc) we have for the relevant line a <sup>1</sup>D<sub>2</sub> - <sup>1</sup>P<sub>1</sub> transition. This means that the spins are paired up (spin singlet) and that in nonrelativistic notation a 5s5d to 5s5p transition takes place. In Hg the equivalent line at the n = 6 level is the yellow line at 579 nm. The shift of the same line towards yellow is the result of having an additional electron shell in the core. What wavelength is associated with this transition in He?</p>
-<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>''  =  -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. 3 Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. [3,6]). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
+<p>For the advanced student: note that the initial level splits into 5 sublevels ''M<sub>J</sub>''  =  -2,-1,0,1,2 , while the final state has ''M<sub>J</sub>'' = -1,0,1. They split equidistantly and one can group the nine possible transitions according to the allowed ''ΔM'' = -1,0,1 (cf.. Ref. <ref name="Mellisinos"/> Fig. 7.3). To understand the polarization of the emitted light in the presence of the external B field note that for an electromagnetic wave its electric, magnetic fields and the wave propagation vector '''k''' form a right-handed coordinate system. Understand the validity of the dipole approximation (the wavelength λ is much longer than atomic dimensions) and how the electric field of the EM wave can be replaced by a constant vector times a temporary oscillatory factor (ref. <ref name="Mellisinos"/>[3,6]). Convince yourself why no linearly polarized light can be observed in the longitudinal direction as the magnetic field is turned on. Correspondingly understand why circularly polarized light as observed in the longitudinal direction must appear as linearly polarized when observed in a direction transverse to the ''B'' field. Why can all three components associated with the ''ΔM'' = -1,0,1 selection rule be observed in the transverse direction?</p>
 <p>To verify these predictions about the polarization states of the light when the B field is turned on you need to recall some optical properties of polarizers and of quarter-wave plates (cf.. Ref.1). By placing these in the correct order you can verify the following:</p>
 <p>In the transverse direction, use a linear polarizer to identify the polarization states of the three components with '''B''' turned on; what happens for ''B'' = 0?</p>