21 The Stokes Parameters and MuellerMatrices for Optical Activityand Faraday Rotation

The Stokes Parameters and Mueller Matrices for Optical Activity and Faraday Rotation In 1811, Arago discovered that the ‘‘plane of polarization’’ of linearly polarized light was rotated

The Stokes Parameters and Mueller

Matrices for Optical Activity

and Faraday Rotation

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 manifesta-tion 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 InChapter 24we 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

1 g of active substance per cubic centimeter (cc) of solution For a solution containing

mg/cc the rotation for a path length l is given by

¼ml

or, in terms of the rotary power ,

 ¼10

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 mag-neto-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 trans-parent 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

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 105for gases under standard conditions and about 102for 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 rota-tion (magneto-optical rotarota-tion), however In the Faraday effect the medium is levorotatory for propagation in the direction of the magnetic field and dextrarota-tory 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

path, as we shall see, not only are k and kþinterchanged but i and i 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

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

=  E ¼ @H

Eliminating H between (21-3a) and (21-3b) leads to

=  ð=  EÞ ¼ @

@t

@D

@t

ð21-4aÞ or

where we have assumed a sinusoidal time dependence for the fields

In an optically active medium the relation between D and E is

where " is a tensor whose form is

" ¼

z "y  x

0

@

1

The parameters "x, "y, and "z correspond to real (on-axis) components of the

of the refractive index For isotropic media the diagonal elements are equal, so

we have

b



and s is a unit vector in the direction of propagation equal to k/k We thus can write (21-5) as

D ¼ n2E þi

Now from (21-3c) we see that

Taking the scalar product of k with D in (21-9), we then see that

Thus, the displacement vector and the electric vector are perpendicular to the pro-pagation 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)

=2E ¼ !2

From the symmetry of this equation we see that we can take the direction of pro-pagation to be along any arbitrary axis We assume that this is the z axis, so (21-12) then reduces to

@2Ex

@z2 ¼ !2n2

c2 Exþi!2

@2Ey

@z2 ¼ !2n2

c2 Eyþi!2

The equation for Ezis trivial and need not be considered further

We now assume that we have plane waves of the form:

Ey ¼E0yeiy ik z z

ð21-14bÞ and substitute (21-14) into (21-13), whereupon we find that

k2!2n2

c2

!

Exþi!2

i!2

c2 Exþ k

2

z!2n2

c2

!

This pair of equations can have a nontrivial solution only if their determinant vanishes:

k2z!2n2

c2

i!2

c2

i!2

c2 k2z!2n2

c2





















so the solution of (21-16) is

where k2¼!2=c2 Because we are interested in the propagation along the positive z axis, we take only the positive root of (21-17), so

Substituting (21-18a) into (21-15a), we find that

while substitution of (21-18b) into (21-15a) yields

For the single primed wave field we can write

E0¼E0xi þ E0yj ¼ ðE00xei0xi þ E00yei0yjÞeikz z 0

ð21-20Þ Now from (21-19a) we see that

and

0x¼0yþ

Hence, we can write (21-20) as

E0¼ ðE00xei0xi þ iE00xei0xjÞeikz z 0

ð21-22aÞ

In a similar manner the double-primed wave field is found to be

E00¼ ðE000xei00xi  iE000xei00xjÞeikz z 00

ð21-22bÞ

To simplify notation let E00x¼E01, 0x¼1, E000x¼E02, and 00x¼2 Then, the fields are

E2¼ ðE02ei2i  iE02ei2jÞeik2 z

ð21-23bÞ where k1¼k0z and k2¼k00z We now add the x and y components of (21-23) and obtain

Ey¼ þiðE01eið1 þk1zÞ

E02eið2 þk2zÞ

The Stokes parameters at any point z in the medium are defined to be

Straightforward substitution of (21-24) into (21-25) leads to

where  ¼ 21 and k ¼ k2k1 We can find the incident Stokes parameters by considering the Stokes parameters at z ¼ 0 We then find the parameters are

We now expand (21-26), using the familiar trignometric identities and find that

S1ðzÞ ¼ ð4E01E02cos Þ cos kz  ð4E01E02sin Þ sin kz ð21-28bÞ

S2ðzÞ ¼ ð4E01E02sin Þ cos kz þ ð4E01E02cos Þ sin kz ð21-28cÞ

which can now be written in terms of the incident Stokes parameters, as given by (21-27), as

or, in matrix form,

S0ðzÞ

S1ðzÞ

S2ðzÞ

S3ðzÞ

0 B

@

1 C

0 cos kz sin kz 0

0 B

@

1 C A

S0ð0Þ

S1ð0Þ

S2ð0Þ

S3ð0Þ

0 B

@

1 C

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 k2k1¼k00zk0z¼k0ðn2Þ1=2k0ðn2þÞ1=2 ð21-31Þ

2

(21-31) can be approximated as

k ’k0

The degree of polarization at any point in the medium is defined to be PðzÞ ¼ðS

2

1ðzÞ þ S22ðzÞ þ S23ðzÞÞ1=2

On substituting (21-29) into (21-33) we find that

PðzÞ ¼ðS

2ð0Þ þ S2ð0Þ þ S2ð0ÞÞ1=2

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

ðS2

1ðzÞ þ S2

2ðzÞ þ S2

Substituting (21-29) into (21-35) then shows that the ellipticity is

ðS2

1ð0Þ þ S2

2ð0Þ þ S2

so the ellipticity is unaffected by the medium

Finally, the orientation angle of the polarization ellipse is given by

tan 2 ðzÞ ¼S2ðzÞ

¼S1ð0Þ sin kz þ S2ð0Þ cos kz

When the incident beam is linearly vertically or horizontally polarized, the respective Stokes vectors are

so S1(0) ¼ 1, S2(0) ¼ 0, and (21-37b) reduces to

whence

ðzÞ ¼ 1

2kz ¼ 

k0 2n



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

medium To study this problem, we consider Fig 21-1 The Mueller matrix of the optically active medium is, from (21-30)

MðkzÞ ¼

0 cos kz sin kz 0

0 B

@

1 C

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

0 B

@

1 C

Thus, from Fig 21-1 the Mueller matrix for propagation through the medium, reflection, and propagation back through the medium, is

¼

0 cos kz  sin kz 0

0 B B

1 C C

0 B B

1 C C

0 cos kz  sin kz 0

0 B B

1 C C ð21-42bÞ Carrying out the matrix multiplication in (21-42b), we obtain

M ¼

0 B

@

1 C

Figure 21-1 Reflection of a polarized beam propagating through an optically active medium

Thus, (21-43) shows that the forward and backward propagation, as well as polar-ization of the beam, are completely unaffected by the presence of the optically active 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

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

Maxwell’s equation (21-3) then become

Eliminating H between (21-46a) and (21-46b), we find that

or, in component form,

The position of the electron can readily be found from the Lorentz force equation to be

¼ e

mE

!2!20eH!

m

ð21-49aÞ

The polarization vector is then expressed as

Solving for Px and Py, we find that

where

2

2

!20Þ ð!2!20Þ2 eH!

m

ð21-52aÞ

B ¼Ne

3

H!

2

!20Þ2 eH!

m

ð21-52bÞ With Pxand Pynow known, (21-48a) and (21-48b) become

@2Ex

@z2 þ!2"0Exþ !2A

Exþi ! 2B

@2Ey

@z2 þ!2"0Eyþ !2A

Eyþi ! 2B

Since we are assuming that there is propagation only along the z axis, we can rewrite (21-53) as

k2þ!2"0þ!2A

Exþi ! 2B

k2þ!2"0þ!2A

Eyþi ! 2B

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

k0;00¼!

Ne2=m ð!2!2Þ eH!=m

ð21-55Þ

where the single and double primes correspond to the (þ) and () solutions in (21-55), respectively The orientation angle for linearly polarized radiation is then determined from (21-37) to be

¼1

2ðk

00

Since the second term under the square root in (21-55) is small compared with unity,

we easily find that

k00k0’2Ne3!2

m2

H

so the orientation angle of the radiation is

3

!2Hz

where Verdet’s constant V is

3!2

m2ð!2!2

We thus see that the Mueller matrix for the Faraday effect is

MðzÞ ¼

0 B

@

1 C

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 propa-gates 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

MðzÞ ¼

0 cos
sin
0

0 B

@

1 C

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

MðzÞ ¼

0 sin
cos
0

0 B

@

1 C

The Mueller matrix for a reflector (mirror) is

MR¼

0 B

@

1 C