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

1 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
= Â ð= Â 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
0
1
"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 0
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 0
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 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
! 2 n2
i!2


¼ À 2 Ex þ 2 Ey
2
@z
c
c

ð21-13aÞ

@2 E y
! 2 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
!
! 2 n2
i!2

2
kz À 2 Ex þ 2 Ey ¼ 0
c
c
!
2 2
i!2

!
n
Ex þ k2z À 2 Ey ¼ 0
c
c2

ð21-15aÞ

ð21-15bÞ

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



 2 ! 2 n2
i!2

 kz À


c2
c2
¼0

ð21-16Þ

2
2 2
! n 
 i!

2

kz À 2 
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

2

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

0

E0 ¼ E 0x i þ E 0y j ¼ ðE 00x eix i þ E 00y eiy jÞeÀikz z

0

ð21-20Þ

Now from (21-19a) we see that
E 00x ¼ E 00y

ð21-21aÞ

and

2
Hence, we can write (21-20) as
0x ¼ 0y þ

0

ð21-21bÞ

0

E0 ¼ ðE 00x eix i þ iE 00x eix jÞeÀikz z


0

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 0. 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,
0
1 0
S0 ðzÞ
1
0
B S1 ðzÞ C B 0 cos kz
B
C B
@ S2 ðzÞ A ¼ @ 0 sin kz

S3 ðzÞ
0
0

0
À sin kz
cos kz
0

1
10
0
S0 ð0Þ
B
C
0C
CB S1 ð0Þ C
@
A
S2 ð0Þ A
0
S3 ð0Þ
1

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

2

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 2 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 2 ðzÞ ¼ Æ tan kz

ð21-39aÞ

whence
 

 
1
k0



ðzÞ ¼ Æ kz ¼ Æ
z¼Æ
z
2
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.