21 The Stokes Parameters and MuellerMatrices for Optical Activityand Faraday Rotation

21 The Stokes Parameters and Mueller Matrices for Optical Activity and Faraday Rotation 21.1 INTRODUCTION In 1811, Arago discovered that the ‘‘plane of polarization’’ of linearly polarized light was rotated when a beam of light propagated through quartz in a direction parallel to its optic axis This property of quartz is called optical activity Shortly afterwards, in 1815, Biot discovered, quite by accident, that many liquids and solutions are also optically active Among these are sugars, albumens, and fruit acids, to name a few In particular, the rotation of the plane of polarization as the beam travels through a sugar solution can be used to measure its concentration The measurement of the rotation in sugar solutions is a widely used method and is called saccharimetry Furthermore, polarization measuring instruments used to measure the rotation are called saccharimeters The rotation of the optical field occurs because optical activity is a manifestation of an unsymmetric isotropic medium; that is, the molecules lack not only a center of symmetry but also a plane of symmetry as well Molecules of this type are called enantiomorphic since they cannot be brought into coincidence with their mirror image Because this rotation takes place naturally, the rotation associated with optically active media is called natural rotation In this chapter we shall only discuss the optical activity associated with liquids and solutions and the phenomenon of Faraday rotation in transparent media and plasmas In Chapter 24 we shall discuss optical activity in crystals Biot discovered that the rotation was proportional to the concentration and path length Specifically, for an optically active liquid or for a solution of an optically active substance such as sugar in an inactive solvent, the specific rotation or rotary power g is defined as the rotation produced by a 10-cm column of liquid containing g of active substance per cubic centimeter (cc) of solution For a solution containing Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved m g/cc the rotation for a path length l is given by ¼ ml 10 ð21-1aÞ or, in terms of the rotary power , ¼ 10 ml ð21-1bÞ The product of the specific rotation and the molecular weight of the active substance is known as the molecular rotation In 1845, after many unsuccessful attempts, Faraday discovered that the plane of polarization was also rotated when a beam of light propogates through a medium subjected to a strong magnetic field Still later, Kerr discovered that very strong electric fields rotate the plane of polarization These effects are called either magneto-optical or electro-optical The magneto-optical effect discovered by Faraday took place when lead glass was subjected to a relatively strong magnetic field; this effect has since become known as the Faraday effect It was through this discovery that a connection between electromagnetism and light was first made The Faraday effect occurs when an optical field propagates through a transparent medium along the direction of the magnetic field This phenomenon is strongly reminiscent of the rotation that occurs in an optically active uniaxial crystal when the propagation is along its optical axis; we shall defer the discussion of propagation in crystals until Chapter 24 The magnitude of the rotation angle  for the Faraday effect is given by  ¼ VHl ð21-2Þ where H is the magnetic intensity, l is the path length in the medium, and V is a constant called Verdet’s constant, a ‘‘constant’’ that depends weakly on frequency and temperature In (21-2) H can be replaced by B, the magnetic field strength If B is in gauss, l in centimeters, and  in minutes of arc (0 ), then Verdet’s constant measured with yellow sodium light is typically about 10À5 for gases under standard conditions and about 10À2 for transparent liquids and solids Verdet’s constant becomes much larger for ferromagnetic solids or colloidal suspensions of ferromagnetic particles The theory of the Faraday effect can be easily worked out for a gas by using the Lorentz theory of the bound electron This analysis is described very nicely in the text by Stone However, our interest here is to derive the Mueller matrices that explicitly describe the rotation of the polarization ellipse for optically active liquids and the Faraday effect Therefore, we derive the Mueller matrices using Maxwell’s equations along with the necessary additions from Lorentz’s theory In addition to the Faraday effect observed in the manner described above, namely, rotation of the polarization ellipse in a transparent medium, we can easily extend the analysis to Faraday rotation in a plasma (a mixture of charged particles) There is an important difference between natural rotation and Faraday rotation (magneto-optical rotation), however In the Faraday effect the medium is levorotatory for propagation in the direction of the magnetic field and dextrarotatory for propagation in the opposite direction If at the end of the path l the light ray is reflected back along the same path, then the natural rotation is canceled while the magnetic rotation is doubled The magnetic rotation effect is because, for the return Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved path, as we shall see, not only are kÀ and kþ interchanged but i and Ài are also interchanged The result is that the vector direction of a positive rotation is opposite to the direction of the magnetic field Because of this, Faraday was able to multiply his very minute rotation effect by repeated back-and-forth reflections In this way he was then able to observe his effect in spite of the relatively weak magnetic field that was used 21.2 OPTICAL ACTIVITY In optically active media there are no free charges or currents Furthermore, the permeability of the medium is, for all practical purposes, unity, so B ¼ H Maxwell’s equations then become =ÂE¼À =ÂH¼ @H @t @D @t ð21-3aÞ ð21-3bÞ =ÁD¼0 ð21-3cÞ =ÁB¼0 ð21-3dÞ Eliminating H between (21-3a) and (21-3b) leads to   @ @D =  ð=  EÞ ¼ À @t @t ð21-4aÞ or =ð= Á EÞ À =2 E ¼ !2 D ð21-4bÞ where we have assumed a sinusoidal time dependence for the fields In an optically active medium the relation between D and E is D ¼ "E where " is a tensor whose form is "x Àiz iy "y Àix A " ¼ @ iz Àiy ix "z ð21-5Þ ð21-6Þ The parameters "x, "y, and "z correspond to real (on-axis) components of the refractive index and x, y, and z correspond to imaginary (off-axis) components of the refractive index For isotropic media the diagonal elements are equal, so we have " x ¼ " y ¼ " z ¼ n2 where n is the refractive index The vector quantity  can be expressed as   b ¼ s  Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð21-7Þ ð21-8Þ where b is a constant (actually a pseudoscalar) of the medium,  is the wavelength, and s is a unit vector in the direction of propagation equal to k/k We thus can write (21-5) as i ð21-9Þ D ¼ n2 E þ ðk  EÞ k where ¼ b/ Now from (21-3c) we see that = Á D ¼ ik Á D ¼ ð21-10Þ Taking the scalar product of k with D in (21-9), we then see that k Á D ¼ n2 k Á E ¼ ð21-11Þ Thus, the displacement vector and the electric vector are perpendicular to the propagation vector k This fact is quite important since the formation of the Stokes parameters requires that the direction of energy flow (along k) and the direction of the fields be perpendicular With these results (21-4) now becomes (replacing k/k by s) !2 ðn E þ is  EÞ ð21-12Þ c2 From the symmetry of this equation we see that we can take the direction of propagation to be along any arbitrary axis We assume that this is the z axis, so (21-12) then reduces to =2 E ¼ À @2 E x ! n2 i!2 ¼ À Ex þ Ey @z c c ð21-13aÞ @2 E y ! n2 i!2 ¼ À E þ Ex y @z2 c2 c2 ð21-13bÞ The equation for Ez is trivial and need not be considered further We now assume that we have plane waves of the form: Ex ¼ E0x eix Àikz z ð21-14aÞ Ey ¼ E0y eiy Àikz z ð21-14bÞ and substitute (21-14) into (21-13), whereupon we find that ! ! n2 i!2 kz À Ex þ Ey ¼ c c ! 2 i!2 ! n Ex þ k2z À Ey ¼ c c2 ð21-15aÞ ð21-15bÞ This pair of equations can have a nontrivial solution only if their determinant vanishes:    ! n2 i!2   kz À   c2 c2 ¼0  ð21-16Þ  2 2 ! n   i!  kz À  c c2 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved so the solution of (21-16) is k2z ¼ k20 ðn2 Æ iÞ ð21-17Þ where k20 ¼ !2 =c2 Because we are interested in the propagation along the positive z axis, we take only the positive root of (21-17), so k0z ¼ k0 ðn2 À iÞ1=2 k00z ð21-18aÞ 1=2 ¼ k0 ðn þ iÞ ð21-18bÞ Substituting (21-18a) into (21-15a), we find that E 0y ¼ þE 0x ð21-19aÞ while substitution of (21-18b) into (21-15a) yields E 00y ¼ ÀE 00x ð21-19bÞ For the single primed wave field we can write 0 E0 ¼ E 0x i þ E 0y j ¼ ðE 00x eix i þ E 00y eiy jÞeÀikz z ð21-20Þ Now from (21-19a) we see that E 00x ¼ E 00y ð21-21aÞ and  Hence, we can write (21-20) as 0x ¼ 0y þ ð21-21bÞ E0 ¼ ðE 00x eix i þ iE 00x eix jÞeÀikz z ð21-22aÞ In a similar manner the double-primed wave field is found to be 00 00 E00 ¼ ðE 000x eix i À iE 000x eix jÞeÀikz z E 00x To simplify notation let fields are 00 ¼ E01 , ð21-22bÞ 0x ¼ 1 , E 000x ¼ E02 , and 00x ¼ 2 Then, the E1 ¼ ðE01 ei1 i þ iE01 ei1 jÞeik1 z ð21-23aÞ E2 ¼ ðE02 ei2 i À iE02 ei2 jÞeik2 z ð21-23bÞ where k1 ¼ k0z and k2 ¼ k00z We now add the x and y components of (21-23) and obtain Ex ¼ E01 eið1 þk1 zÞ þ E02 eið2 þk2 zÞ Ey ¼ þiðE01 e ið1 þk1 zÞ À E02 e ið2 þk2 zÞ ð21-24aÞ Þ ð21-24bÞ The Stokes parameters at any point z in the medium are defined to be S0 ðzÞ ¼ Ex ðzÞExà ðzÞ þ Ey ðzÞEyà ðzÞ S1 ðzÞ ¼ Ex ðzÞExà ðzÞ S2 ðzÞ ¼ Ex ðzÞEyà ðzÞ S3 ðzÞ ¼ iðEx ðzÞEyà ðzÞ ð21-25aÞ À Ey ðzÞEyà ðzÞ ð21-25bÞ þ Ey ðzÞExà ðzÞ ð21-25cÞ À Ey ðzÞExà ðzÞÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð21-25dÞ Straightforward substitution of (21-24) into (21-25) leads to S0 ðzÞ ¼ 2ðE201 þ E202 Þ ð21-26aÞ S1 ðzÞ ¼ 4E01 E02 cosð þ kzÞ ð21-26bÞ S2 ðzÞ ¼ 4E01 E02 sinð þ kzÞ ð21-26cÞ S3 ðzÞ ¼ 2ðE201 À E202 Þ ð21-26dÞ where  ¼ 2 À 1 and k ¼ k2 À k1 We can find the incident Stokes parameters by considering the Stokes parameters at z ¼ We then find the parameters are S0 ð0Þ ¼ 2ðE201 þ E202 Þ ð21-27aÞ S1 ð0Þ ¼ 4E01 E02 cos  ð21-27bÞ S2 ð0Þ ¼ 4E01 E02 sin  ð21-27cÞ 2ðE201 ð21-27dÞ S3 ð0Þ ¼ À E202 Þ We now expand (21-26), using the familiar trignometric identities and find that S0 ðzÞ ¼ 2ðE201 þ E202 Þ ð21-28aÞ S1 ðzÞ ¼ ð4E01 E02 cos Þ cos kz À ð4E01 E02 sin Þ sin kz ð21-28bÞ S2 ðzÞ ¼ ð4E01 E02 sin Þ cos kz þ ð4E01 E02 cos Þ sin kz ð21-28cÞ S3 ðzÞ ¼ 2ðE201 À E202 Þ ð21-28dÞ which can now be written in terms of the incident Stokes parameters, as given by (21-27), as S0 ðzÞ ¼ S0 ð0Þ ð21-29aÞ S1 ðzÞ ¼ S1 ð0Þ cos kz À S2 ð0Þ sin kz ð21-29bÞ S2 ðzÞ ¼ S1 ð0Þ sin kz þ S2 ð0Þ cos kz ð21-29cÞ S3 ðzÞ ¼ S3 ð0Þ ð21-29dÞ or, in matrix form, S0 ðzÞ B S1 ðzÞ C B cos kz B C B @ S2 ðzÞ A ¼ @ sin kz S3 ðzÞ 0 À sin kz cos kz 10 S0 ð0Þ B C 0C CB S1 ð0Þ C @ A S2 ð0Þ A S3 ð0Þ ð21-30Þ Thus, the optically active medium is characterized by a Mueller matrix whose form, corresponds to a rotator The expression for k in (21-30) can be rewritten with the aid of (21-18) as k ¼ k2 À k1 ¼ k00z À k0z ¼ k0 ðn2 À Þ1=2 À k0 ðn2 þ Þ1=2 ð21-31Þ Since ( n (21-31) can be approximated as k’ k0 n Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð21-32Þ The degree of polarization at any point in the medium is defined to be PðzÞ ¼ ðS21 ðzÞ þ S22 ðzÞ þ S23 ðzÞÞ1=2 S0 ðzÞ ð21-33Þ On substituting (21-29) into (21-33) we find that PðzÞ ¼ ðS21 ð0Þ þ S22 ð0Þ þ S23 ð0ÞÞ1=2 ¼ Pð0Þ S0 ð0Þ ð21-34Þ that is, the degree of polarization does not change as the optical beam propogates through the medium The ellipticity of the optical beam is given by sin 2ðzÞ ¼ ðS21 ðzÞ S3 ðzÞ þ S22 ðzÞ þ S23 ðzÞÞ1=2 ð21-35Þ Substituting (21-29) into (21-35) then shows that the ellipticity is sin 2ðzÞ ¼ S3 ð0Þ ¼ sin 2ð0Þ ðS21 ð0Þ þ S22 ð0Þ þ S23 ð0ÞÞ1=2 ð21-36Þ so the ellipticity is unaffected by the medium Finally, the orientation angle of the polarization ellipse is given by tan ðzÞ ¼ ¼ S2 ðzÞ S1 ðzÞ ð21-37aÞ S1 ð0Þ sin kz þ S2 ð0Þ cos kz S1 ð0Þ cos kz À S2 ð0Þ sin kz ð21-37bÞ When the incident beam is linearly vertically or horizontally polarized, the respective Stokes vectors are ð1, À 1, 0, 0Þ and ð1, 1, 0, 0Þ ð21-38Þ so S1(0) ¼ Æ1, S2(0) ¼ 0, and (21-37b) reduces to tan ðzÞ ¼ Æ tan kz ð21-39aÞ whence     k0  ðzÞ ¼ Æ kz ¼ Æ z¼Æ z 2n n ð21-39bÞ Thus, the orientation angle (z) is proportional to the distance traveled by the beam through the optically active medium and inversely proportional to wavelength, in agreement with the experimental observation We can now simply equate (21-39b) with (21-1a) and relate the measured quantities of the medium to each other As a result we see that Maxwell’s equations completely account for the behavior of the optical activity Before, we conclude this section one question should still be answered In section 21.1 we pointed out that for natural rotation the polarization of the beam is unaffected by the optically active medium when it is reflected back through the Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 21-1 Reflection of a polarized beam propagating through an optically active medium medium To study this problem, we consider Fig 21-1 The Mueller matrix of the optically active medium is, from (21-30) 1 0 B cos kz À sin kz C C MðkzÞ ¼ B ð21-40Þ @ sin kz cos kz A 0 Now for a reflected beam we must replace z by –z and k by –k We thus obtain (21-40) From a physical point of view we must obtain the same Mueller matrix regardless of the direction of propagation of the beam Otherwise, we would have a preferential direction! The Mueller matrix for a perfect reflector is 1 0 B0 0 C C MR ¼ B ð21-41Þ @ 0 À1 A 0 À1 Thus, from Fig 21-1 the Mueller matrix for propagation through the medium, reflection, and propagation back through the medium, is M ¼ MðkzÞMR MðkzÞ 10 0 0 B cos kz À sin kz CB B CB ¼B CB @ sin kz cos kz A@ 0 À1 0 Carrying out the matrix 0 B0 M¼B @ 0 À1 0 0 0 ð21-42aÞ B C C CB cos kz À sin kz C C CB A@ sin kz cos kz A À1 10 0 0 multiplication in (21-42b), we obtain 0 C C A À1 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð21-42bÞ ð21-43Þ Thus, (21-43) shows that the forward and backward propagation, as well as polarization of the beam, are completely unaffected by the presence of the optically active medium 21.3 FARADAY ROTATION IN A TRANSPARENT MEDIUM Natural rotation of the plane of polarization was first observed in quartz by Arago in 1811 With the development of electromagnetism, physicists began to investigate the effects of the magnetic field on materials and, in particular, the possible relationship between electromagnetism and light In 1845, Michael Faraday discovered that when a linearly polarized wave is propagating in a dielectric medium parallel to a static magnetic field the plane of polarization rotates This phenomenon is known as the Faraday effect The behavior is similar to that taking place in optically active media However, there is an important difference If, at the end of a path l the radiation is reflected backwards, then the rotation in optically active media is opposite to the original direction and cancels out; this was shown at the end of the previous section For the magnetic case, however, the angle of rotation is doubled This behavior along with some other important observations, will be shown at the end of this section In the present problem we take the direction of the magnetic field to be along the z axis In addition, the plane waves are propagating along the z axis, and the directions of the electric (optical) vibrations are along the x and y axes In such a medium (transparent, isotropic, and nonconducting) the displacement current vector is D ¼ "0 E þ P ð21-44Þ where P is the polarization vector (this vector refers to the electric polarizibility of the material) and is related to the position vector r of the electron by P ¼ ÀNer ð21-45Þ Maxwell’s equation (21-3) then become =  E ¼ Ài!H ð21-46aÞ =  H ¼ i!ð"0 E þ PÞ ð21-46bÞ Eliminating H between (21-46a) and (21-46b), we find that =2 E þ !2 "0 E ¼ À!2 P ð21-47Þ or, in component form, =2 Ex þ !2 "0 Ex ¼ À!2 Px ð21-48aÞ =2 Ey þ !2 "0 Ey ¼ À!2 Py ð21-48bÞ The position of the electron can readily be found from the Lorentz force equation to be  e  eH! 2 E Æ ¼ ! À !0 Ç ð21-49aÞ m Æ m Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved where Æ ¼ x Æ iy ð21-49bÞ EÆ ¼ Ex Æ iEy ð21-49cÞ The polarization vector is then expressed as PÆ ¼ NeÆ ¼ Px Æ iPy ð21-50Þ Solving for Px and Py, we find that Px ¼ AEx þ iBEy ð21-51aÞ Py ¼ AEy þ iBEx ð21-51bÞ where "   #À1 Ne2 eH! 2 2 A¼ ð! À !0 Þ ð! À !0 Þ À m m "   #À1 Ne3 H! eH! 2 2 ð! À !0 Þ À B¼ m m ð21-52aÞ ð21-52bÞ With Px and Py now known, (21-48a) and (21-48b) become À Á À Á @2 E x þ !2 "0 Ex þ !2 A Ex þ i !2 B Ey ¼ @z ð21-53aÞ À Á À Á @2 E y þ !2 "0 Ey þ !2 A Ey þ i !2 B Ex ¼ @z ð21-53bÞ Since we are assuming that there is propagation only along the z axis, we can rewrite (21-53) as À Á À Á Àk þ !2 "0 þ !2 A Ex þ i !2 B Ey ¼ ð21-54aÞ À Á À Á Àk þ !2 "0 þ !2 A Ey þ i !2 B Ex ¼ ð21-54bÞ If we now compare (21-53) with (21-13), we see that the forms of the equations are identical Hence, we can proceed directly with the writing of the Mueller rotation matrix and the remaining relations In addition, we find the wavenumber for the propagating waves to be " #1=2 ! Ne2 =m 0;00 1À k ¼ ð21-55Þ c ð! À !20 Þ Æ eH!=m where the single and double primes correspond to the (þ) and (À) solutions in (21-55), respectively The orientation angle for linearly polarized radiation is then determined from (21-37) to be ¼ ðk00 À k0 Þz Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð21-56Þ Since the second term under the square root in (21-55) is small compared with unity, we easily find that k00 À k0 ’ 2Ne3 !2 H m2 !2 À !20 ð21-57Þ so the orientation angle of the radiation is ’ Ne3 !2 Hz ¼ VHz m2 ð!2 À !20 Þ ð21-58Þ where Verdet’s constant V is V¼ Ne3 !2 m2 ð!2 À !20 Þ ð21-59Þ We thus see that the Mueller matrix for 0 B cos VHz À sin VHz MðzÞ ¼ B @ sin VHz cos VHz 0 the Faraday effect is 0C C 0A ð21-60Þ Thus, the rotation (21-58), is proportional to the path length, in agreement with the experimental observation Before concluding, let us again consider the problem where the beam propagates through the magneto-optical medium and is reflected back toward the optical source For convenience, we replace VHz with  and we write (21-60) as 1 0 B cos  À sin  C C ð21-61Þ MðzÞ ¼ B @ sin  cos  A 0 Now for a reflected beam we must replace z by –z However, VH is unaffected Unlike natural rotation, in the Faraday effect we have superposed an asymmetry in the problem with the unidirectional magnetic field Thus,  transforms to –, and the Mueller matrix M(z) for the beam propagating back to the source becomes 1 0 B cos  sin  C C ð21-62Þ MðÀzÞ ¼ B @ À sin  cos  A 0 The Mueller matrix for a 0 B0 MR ¼ B @ 0 À1 0 0 reflector (mirror) is 0 C C A À1 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð21-63Þ Thus, the Mueller matrix for the propagation through the medium, reflection, and propagation back through the medium, is M ¼ MðÀzÞMR MðzÞ 0 B cos  sin  B ¼B @ À sin  cos  0 10 B C CB CB A@ 0 0 À1 0 0 10 B C CB CB A@ 0 À1 0 cos  sin  ð21-64aÞ 0 À sin  C C C cos  A ð21-64bÞ Carrying out the matrix multiplication in (21-64b), we find that B0 B M¼B @0 0 0 cos 2 À sin 2 À sin 2 À cos 2 0 C C C A 0 À1 ð21-65Þ Since  is arbitrary, we can replace  by – in (21-65) without changing its meaning, and we then have B0 B M¼B @0 0 cos 2 sin 2 sin 2 À cos 2 0 C C C A ð21-66Þ À1 Equation (21-66) is recognized as the Mueller matrix for a pseudorotator That is, the rotation as well as the ellipticity are opposite to the true behavior of a rotator Thus, unlike natural rotation the angle of rotation is doubled upon reflection, so that for n reflections, 2 in (21-66) is replaced by 2n and a relatively large rotation angle can then be measured 21.4 FARADAY ROTATION IN A PLASMA While we have used Maxwell’s equations to describe the propagation and polarization of light in optical media, the fact is that Maxwell’s equations are universally applicable In this section we briefly wish to show that the phenomenon of Faraday rotation appears when waves propagate in plasmas Plasmas are gaseous matter consisting of charged particles They appear not only in the laboratory but throughout the universe In a plasma the fields are again described by Maxwell’s equations, which we write here as =  E ¼ Ài!H ð21-67aÞ =  H ¼ i!" Á E ð21-67bÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved where " is the plasma dielectric tensor For a plasma having a static magnetic field along the z axis we have H ¼ Hk, and the tensor " is then found to be (see the book by Bekefi) "xx "xy " ¼ @ À"xy "xx A ð21-68Þ 0 "zz where !2p !2 À !2g ð21-69aÞ Ài!g !2p !ð!2 À !2g Þ ð21-69bÞ "xx ¼ "yy ¼ À "xy ¼ À"yx ¼ "zz ¼ À !2p !2 ð21-69cÞ and !2p ¼ e2 ¼ plasma frequency "0 m ð21-69dÞ !g ¼ eH ¼ electron gyrofrequency m ð21-69eÞ Eliminating H between (21-67a) and (21-67b) gives =ð= Á EÞ À =2 E ¼ !2 "E ð21-70Þ We now consider the wave to be propagating along the z axis, i.e., in the direction of the static magnetic field For this case it is not difficult to show that = Á E ¼ Equation (21-70) then reduces to @2 E x ¼ À!2 ½"xx Ex þ "xy Ey Š @z2 ð21-71aÞ @2 E y ¼ À!2 ½"xx Ey À "xy Ex Š @z2 ð21-71bÞ These equations are identical to (21-54), and again we obtain the same results as in that section In the present problem we now find that the wavenumbers for individual waves are ! k0 ¼ ð"xx þ i"xy Þ1=2 ð21-72aÞ c ! k00 ¼ ð"xx þ i"xy Þ1=2 ð21-72bÞ c so k0 À k00 ’ !g !2p cð!2 À !2g Þ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð21-73Þ and the angle of rotation is   !g !2p z k À k00 ¼ z¼ 2cð!2 À !2g Þ ð21-74Þ Thus, we see that for a plasma we obtain the rotation Mueller matrix: 1 0 B cos sin 0C C M¼B ð21-75Þ @ À sin cos 0A 0 The subject of optical activity and magneto-optical phenomena is vast Many of the details of particular aspects as well as general treatments of the subject can be found in the references REFERENCES Paper Condon, E U., Rev Mod Phys., 9, 432 (1937) Books Born, M., Optik, Springer Verlag, Berlin, 1933 Sommerfeld, A., Lectures on Theoritical Physics, Vols I-V, Academic Press, New York, 1952 Stone, J M., Radiation and Optics, McGraw-Hill, New York, 1963 Papas, C H., Theory of Electromagnetic Wave Propagation, McGraw-Hill, New York, 1965 Bekefi, G., Radiation Processes in Plasmas, Wiley, New York, 1966 Wood, R W., Physical Optics, 3rd ed., Optical Society of America, Washington, DC, 1988 Jenkins, F S., and White, H E., Fundamentals of Optics, McGraw-Hill, New York, 1957 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved [...]... ¼ 0 21- 54bÞ If we now compare (21- 53) with (21- 13), we see that the forms of the equations are identical Hence, we can proceed directly with the writing of the Mueller rotation matrix and the remaining relations In addition, we find the wavenumber for the propagating waves to be " #1=2 ! Ne2 =m 0;00 1À 2 k ¼ 21- 55Þ c ð! À !20 Þ Æ eH!=m where the single and double primes correspond to the (þ) and (À)... refers to the electric polarizibility of the material) and is related to the position vector r of the electron by P ¼ ÀNer 21- 45Þ Maxwell’s equation (21- 3) then become =  E ¼ Ài!H 21- 46aÞ =  H ¼ i!ð"0 E þ PÞ 21- 46bÞ Eliminating H between (21- 46a) and (21- 46b), we find that =2 E þ !2 "0 E ¼ À!2 P 21- 47Þ or, in component form, =2 Ex þ !2 "0 Ex ¼ À!2 Px 21- 48aÞ =2 Ey þ !2 "0 Ey ¼ À!2 Py 21- 48bÞ The position... now simply equate (21- 39b) with (21- 1a) and relate the measured quantities of the medium to each other As a result we see that Maxwell’s equations completely account for the behavior of the optical activity Before, we conclude this section one question should still be answered In section 21. 1 we pointed out that for natural rotation the polarization of the beam is unaffected by the optically active medium... 21- 42bÞ 21- 43Þ Thus, (21- 43) shows that the forward and backward propagation, as well as polarization of the beam, are completely unaffected by the presence of the optically active medium 21. 3 FARADAY ROTATION IN A TRANSPARENT MEDIUM Natural rotation of the plane of polarization was first observed in quartz by Arago in 1811 With the development of electromagnetism, physicists began to investigate the. .. However, there is an important difference If, at the end of a path l the radiation is reflected backwards, then the rotation in optically active media is opposite to the original direction and cancels out; this was shown at the end of the previous section For the magnetic case, however, the angle of rotation is doubled This behavior along with some other important observations, will be shown at the end... end of this section In the present problem we take the direction of the magnetic field to be along the z axis In addition, the plane waves are propagating along the z axis, and the directions of the electric (optical) vibrations are along the x and y axes In such a medium (transparent, isotropic, and nonconducting) the displacement current vector is D ¼ "0 E þ P 21- 44Þ where P is the polarization vector... in the medium is defined to be PðzÞ ¼ ðS21 ðzÞ þ S22 ðzÞ þ S23 ðzÞÞ1=2 S0 ðzÞ 21- 33Þ On substituting (21- 29) into (21- 33) we find that PðzÞ ¼ ðS21 ð0Þ þ S22 ð0Þ þ S23 ð0ÞÞ1=2 ¼ Pð0Þ S0 ð0Þ 21- 34Þ that is, the degree of polarization does not change as the optical beam propogates through the medium The ellipticity of the optical beam is given by sin 2ðzÞ ¼ ðS21 ðzÞ S3 ðzÞ þ S22 ðzÞ þ S23 ðzÞÞ1=2 21- 35Þ... parameters by considering the Stokes parameters at z ¼ 0 We then find the parameters are S0 ð0Þ ¼ 2ðE201 þ E202 Þ 21- 27aÞ S1 ð0Þ ¼ 4E01 E02 cos  21- 27bÞ S2 ð0Þ ¼ 4E01 E02 sin  21- 27cÞ 2ðE201 21- 27dÞ S3 ð0Þ ¼ À E202 Þ We now expand (21- 26), using the familiar trignometric identities and find that S0 ðzÞ ¼ 2ðE201 þ E202 Þ 21- 28aÞ S1 ðzÞ ¼ ð4E01 E02 cos Þ cos kz À ð4E01 E02 sin Þ sin kz 21- 28bÞ S2 ðzÞ ¼... polarized, the respective Stokes vectors are ð1, À 1, 0, 0Þ and ð1, 1, 0, 0Þ 21- 38Þ so S1(0) ¼ Æ1, S2(0) ¼ 0, and (21- 37b) reduces to tan 2 ðzÞ ¼ Æ tan kz 21- 39aÞ whence     1 k0  ðzÞ ¼ Æ kz ¼ Æ z¼Æ z 2 2n n 21- 39bÞ Thus, the orientation angle (z) is proportional to the distance traveled by the beam through the optically active medium and inversely proportional to wavelength, in agreement with the. .. !20 Þ 21- 58Þ where Verdet’s constant V is V¼ Ne3 !2 m2 ð!2 À !20 Þ 21- 59Þ We thus see that the Mueller matrix for 0 1 0 0 B 0 cos VHz À sin VHz MðzÞ ¼ B @ 0 sin VHz cos VHz 0 0 0 the Faraday effect is 1 0 0C C 0A 1 21- 60Þ Thus, the rotation (21- 58), is proportional to the path length, in agreement with the experimental observation Before concluding, let us again consider the problem where the beam