11 The Jones Matrix Calculus

11
The Jones Matrix Calculus

11.1

INTRODUCTION

We have seen that the Stokes polarization parameters and the Mueller matrix formalism can be used to describe any state of polarization. In particular, if we are dealing
with a single beam of polarized light, then the formalism of the Stokes parameters is
completely capable of describing any polarization state ranging from completely
polarized light to completely unpolarized light. In addition, the formalism of the
Stokes parameters can be used to describe the superposition of several polarized
beams, provided that there is no amplitude or phase relation between them; that
is, the beams are incoherent with respect to each other. This situation arises when
optical beams are emitted from several independent sources and are then superposed.
However, there are experiments where several beams must be added and the
beams are not independent of each other, e.g., beam superposition in interferometers. There we have a single optical source and the single beam is divided by a beam
splitter. Then, at a later stage, the beams are ‘‘reunited,’’ that is, superposed. Clearly,
there is an amplitude and phase relation between the beams. We see that we must
deal with amplitudes and phase and superpose the amplitudes of each of the beams.
After the amplitudes of the beam are superposed, the intensity of the combined
beams is then found by taking the time average of the square of the total amplitude.
If there were no amplitude or phase relations between the beams, then we would
arrive at the same result as we obtained for the Stokes parameters. However, if there
is a relation between the amplitude and the phase of the optical beams, an interference term will arise.
Of course, as pointed out earlier, the description of the polarizing behavior of
the optical field in terms of amplitudes was one of the first great successes of the wave
theory of light. The solution of the wave equation in terms of transverse components
leads to elliptically polarized light and its degenerate linear and circular forms. On
the basis of the amplitude results, many results could be understood (e.g., Young’s
interference experiment, circularly polarized light). However, even using the

amplitude formulation, numerous problems become difficult to treat, such as the
propagation of the field through several polarizing components. To facilitate

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


the treatment of complicated polarization problems at the amplitude level, R. Clark
Jones, in the early 1940s, developed a matrix calculus for treating these problems,
commonly called the Jones matrix calculus. It is most appropriately used when we
must superpose amplitudes. The Jones calculus involves complex quantities contained in 2 Â 1 column matrices (the Jones vector) and 2 Â 2 matrices (the Jones
matrices). At first sight it would seem that the use of the 2 Â 2 matrices would be
simpler than the use of the 4 Â 4 Mueller matrices. Oddly enough, this is not the
case. This is due primarily to the fact that even the matrix multiplication of several
complex 2 Â 2 matrices can be tedious. Furthermore, even after the complete matrix
calculation has been carried out, additional steps are still required. For example, it is
often necessary to separate the real and imaginary parts (e.g., Ex and Ey) and superpose the respective amplitudes. This can involve a considerable amount of effort.
Another problem is that to find the intensity one must take the complex transpose of
the Jones vector and then carry out the matrix multiplication between the complex
transpose of the Jones vector and Jones vector itself. All this is done using complex
quantities, and the possibility of making a computational error is very real. While the
4 Â 4 Mueller matrix formalism appears to be more complicated, all the entries are
real quantities and there are many zero entries, as can be seen by inspecting the
Mueller matrix for the polarizer, the retarder and the rotator. This fact greatly
simplifies the matrix multiplications, and, of course, the Stokes vector is real.
There are, nevertheless, many instances where the amplitudes must be added
(superposed), and so the Jones matrix formalism must be used. There are many
problems where either formalism can be used with success. As a general rule, the
most appropriate choice of matrix method is to use the Jones calculus for amplitude
superposition problems and the Mueller formalism for intensity superposition problems. Experience will usually indicate the best choice to make.
In this chapter we develop the fundamental matrices for the Jones calculus

along with its application to a number of problems.

11.2

THE JONES VECTOR

The plane-wave components of the optical field in terms of complex quantities can be
written as
Ex ðz, tÞ ¼ E0x eið!tÀkzþx Þ

ð11-1aÞ

Ey ðz, tÞ ¼ E0y eið!tÀkzþy Þ

ð11-1bÞ

The propagator !t À kz is now suppressed, so (11-1) is then written as
Ex ¼ E0x eix

ð11-2aÞ

Ey ¼ E0y eiy

ð11-2bÞ

Equation (11-2) can be arranged in a 2 Â 1 column matrix E:
!
!
Ex
E0x eix

E¼
¼
Ey
E0y eiy

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

ð11-3Þ


called the Jones column matrix or, simply, the Jones vector. The column matrix on
the right-hand side of (11-3), incidentally, is the Jones vector for elliptically polarized
light.
In the Jones vector (11-3), the maximum amplitudes E0x and E0y are real
quantities. The presence of the exponent with imaginary arguments causes Ex and
Ey to be complex quantities. Before we proceed to find the Jones vectors for various
states of polarized light, we discuss the normalization of the Jones vector; it is
customary to express the Jones vector in normalized form. The total intensity I of
the optical field is given by
I ¼ Ex Exà þ Ey EyÃ
Equation (11-4) can be obtained by the following matrix multiplication:
 
À
Á Ex
I ¼ Exà EyÃ
Ey

ð11-4Þ

ð11-5Þ

The row matrix ðExà EyÃ Þ is the complex transpose of the Jones vector (column matrix
E) and is written Ey; thus,
À
Á
Ey ¼ Exà EyÃ
ð11-6Þ
so
I ¼ Ey E

ð11-7Þ

yields (11-4). Carrying out the matrix multiplication of (11-7), using (11-3), yields
E20x þ E20y ¼ I ¼ E20

ð11-8Þ

It is customary to set E20 ¼ 1, whereupon we say that the Jones vector is normalized.
The normalized condition for (11-5) can then be written as
Ey E ¼ 1

ð11-9Þ

We note that the Jones vector can only be used to describe completely polarized light.
We now find the Jones vector for the following states of completely polarized light.
1.

Linear horizontally polarized light. For this state Ey ¼ 0, so (11-3)
becomes



E0x eix
E¼
ð11-10Þ
0

From the normalization condition (11-9) we see that E20x ¼ 1. Thus, suppressing eix
because it is unimodular, the normalized Jones vector for linearly horizontally polarized light is written
 
1
ð11-11Þ
E¼
0
In a similar manner the Jones vectors for the other well-known polarization states
are easily found.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


Linear vertically polarized light. Ex ¼ 0, so E20y ¼ 1 and
 
0
E¼
1

2.

Linear þ45 polarized light. Ex ¼ Ey , so 2E20x ¼ 1 and
 
1 1

E ¼ pffiffiffi
2 1

ð11-12Þ

3.

Linear À45 polarized light. Ex ¼ ÀEy , so 2E20x ¼ 1 and


1
1
E ¼ pffiffiffi
2 À1

ð11-13Þ

4.

ð11-14Þ

5.

Right-hand circularly polarized light. For this case E0x ¼ E0y and
y À x ¼ þ90 . Then, 2E20x ¼ 1 and we have
 
1
1
E ¼ pffiffiffi
ð11-15Þ

2 þi

6.

Left-hand circularly polarized light. We again have E0x ¼ E0y , but
y À x ¼ À90 . The normalization condition gives 2E20x ¼ 1, and we have
 
1
1
E ¼ pffiffiffi
ð11-16Þ
2 Ài

Each of the Jones vectors (11-11) through (11-16) satisfies the normalization condition (11-9).
An additional property is the orthogonal or orthonormal property. Two vectors A and B are said to be orthogonal if AB ¼ 0 or, in complex notation, Ay B ¼ 0.
If this condition is satisfied, we say that the Jones vectors are orthogonal. For
example, for linearly horizontal and vertical polarized light we find that
 
À
ÁÃ 0
1 0
¼0
ð11-17aÞ
1
so the states are orthogonal or, since we are using normalized vectors, orthonormal.
Similarly, for right and left circularly polarized light:
 
À
ÁÃ 1
1 þi

¼0
ð11-17bÞ
Ài
Thus, the orthonormal condition for two Jones vectors E1 and E2 is
Eyi Ej ¼ 0

ð11-18Þ

We see that the orthonormal condition (11-18) and the normalizing condition
(11-9) can be written as a single equation, namely
Eyi Ej ¼ ij

i; j ¼ 1; 2

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

ð11-19aÞ


where ij is the Kronecker delta and has the property:
ij ¼ 1

i¼j

ð11-19bÞ

ij ¼ 0

i 6¼ j

ð11-19cÞ

In a manner analogous to the superposition of incoherent intensities or
Stokes vectors, we can superpose coherent amplitudes, that is, Jones vectors.
For example, the Jones vector for horizontal polarization is EH and that for vertical
polarization is EV, so




0
E0x eix
EV ¼
ð11-20Þ
EH ¼
E0y eiy
0
Adding EH and EV gives
E ¼ EH þ EV ¼

E0x eix
E0y eiy

!
ð11-21Þ

which is the Jones vector for elliptically polarized light. Thus, superposing two
orthogonal linear polarizations give rise to elliptically polarized light. For example,
if E0x ¼ E0y and y ¼ x , then, from (11-21), we can write
 

ix 1
ð11-22Þ
E ¼ E0x e
1
which is the Jones vector for linear þ45 polarized light. Equation (11-22) could also
be obtained by superposing (11-11) and (11-12):
     
1
0
1
ð11-23Þ
¼
E ¼ EH þ EV ¼
þ
1
1
0
which, aside from the normalizing factor, is identical to (11-13).
As another example let us superpose left and right circularly polarized light of
equal amplitudes. Then, from (11-15) and (11-16) we have
 
 
 
1
1 1
2 1
1
E ¼ pffiffiffi
þ pffiffiffi
¼ pffiffiffi

ð11-24Þ
2 Ài
2 i
2 0
which, aside from the normalizing factor, is the Jones vector for linear horizontally
polarized light (11-11).
As a final Jones vector example, we show that elliptically polarized light can be
obtained by superposing two opposite circularly polarized beams of unequal amplitudes. The Jones vectors for two circular polarized beams of unequal amplitudes
a and b can be represented by
 
 
1
1
Eþ ¼ a
EÀ ¼ b
ð11-25Þ
þi
Ài
According to the principle of superposition, the resultant Jones vector for (11-25) is

  
Ex
aþb
E ¼ Eþ þ EÀ ¼
¼
ð11-26Þ
Ey
iða À bÞ

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

In component form (11-26) is written as
Ex ¼ a þ b

ð11-27aÞ

Ey ¼ ða À bÞei=2

ð11-27bÞ

We now restore the propagator !t À kz, so (11-27) is then written as
Ex ¼ ða þ bÞeið!tÀkzÞ

ð11-28aÞ

Ey ¼ ða À bÞeið!tÀkzþ=2Þ

ð11-28bÞ

Taking the real part of (11-28), we have
Ex ðz, tÞ ¼ ða þ bÞ cosð!t À kzÞ


Ey ðz, tÞ ¼ ða À bÞ cos !t À kz þ
2
¼ ða À bÞ sinð!t À kzÞ

ð11-29aÞ
ð11-29bÞ

ð11-29cÞ

Equations (11-28a) and (11-28b) are now written as
Ex ðz, tÞ
¼ cosð!t À kzÞ
aþb

ð11-30aÞ

Ey ðz, tÞ
¼ sinð!t À kzÞ
aÀb

ð11-30bÞ

Squaring and adding (11-30a) and (11-30b) yields
E2y ðz, tÞ
E2x ðz, tÞ
þ
¼1
ða þ bÞ2 ða À bÞ2

ð11-31Þ

Equation (11-31) is the equation of an ellipse whose major and minor axes lengths
are a þ b and a À b, respectively. Thus, the superposition of two oppositely circularly polarized beams of unequal magnitudes gives rise to a (nonrotated) ellipse with
its locus vector moving in a counterclockwise direction.

11.3

JONES MATRICES FOR THE POLARIZER,
RETARDER, AND ROTATOR

We now determine the matrix forms for polarizers (diattenuators), retarders (phase
shifters), and rotators in the Jones matrix calculus. In order to do this, we assume
that the components of a beam emerging from a polarizing element are linearly
related to the components of the incident beam. This relation is written as
E 0x ¼ jxx Ex þ jxy Ey

ð11-32aÞ

E 0y ¼ jyx Ex þ jyy Ey

ð11-32bÞ

where E 0x and E 0y are the components of the emerging beam and Ex and Ey are the
components of the incident beam. The quantities jik , i, k ¼ x, y, are the transforming

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


factors (elements). Equation (11-32) can be written in matrix form as
!
!
!
jxx jxy
Ex
E 0x
¼
jyx jyy

Ey
E 0y

ð11-33aÞ

or
E0 ¼ JE

ð11-33bÞ

where

J¼

jxx
jyx

jxy
jyy


ð11-33cÞ

The 2 Â 2 matrix J is called the Jones instrument matrix or, simply, the Jones matrix.
We now determine the Jones matrices for a polarizer, retarder, and rotator.
A polarizer is characterized by the relations:
E 0x ¼ px Ex
E 0y ¼ py Ey

ð11-34aÞ

0

px, y

1

ð11-34bÞ

For complete transmission px, y ¼ 1, and for complete attenuation px, y ¼ 0. In terms
of the Jones vector, (11-34) can be written as
!
!
!
px 0
Ex
E 0x
¼
ð11-35Þ
0 py
Ey
E 0y
so the Jones matrix (11-33c) for a polarizer is


px 0
Jp ¼
0 px, y 1
0 py

ð11-36Þ

For an ideal linear horizontal polarizer there is complete transmission along
the horizontal x axis and complete attenuation along the vertical y axis. This is
expressed by px ¼ 1 and py ¼ 0, so (11-36) becomes


1 0
JPH ¼
ð11-37Þ
0 0
Similarly, for a linear vertical polarizer, (11-36) becomes


0 0
JPV ¼
0 1

ð11-38Þ

In general, it is useful to know the Jones matrix for a linear polarizer rotated
through an angle . This is readily found by using the familiar rotation transformation, namely,
J0 ¼ JðÀÞJJðÞ
where JðÞ is the rotation matrix:


cos  sin 
JðÞ ¼
À sin  cos 

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

ð11-39aÞ

ð11-39bÞ


and J is given by (11-33c). For a rotated linear polarizer represented by (11-36) and
rotated by angle  we have from (11-39) that




px 0
cos  À sin 
cos  sin 
J0 ¼
ð11-40Þ
0 py
sin  cos 
À sin  cos 
Carrying out the matrix multiplication in (11-40) we find that the Jones matrix for a
rotated polarizer is
!
px cos2  þ py sin2  ðpx À py Þ sin  cos 
ð11-41Þ
JP ðÞ ¼
ðpx À py Þ sin  cos  px sin2  þ py cos2 
For an ideal linear horizontal polarizer we can set px ¼ 1 and py ¼ 0 in (11-41), so
that the Jones matrix for a rotated linear horizontal polarizer is
!

sin  cos 
cos2 
JP ðÞ ¼
ð11-42Þ
sin  cos 
sin2 
The Jones matrix for a linear polarizer rotated through þ45 is then seen from
(11-42) to be
!
1 1 1

JP ð45 Þ ¼
ð11-43Þ
2 1 1
If the linear polarizer is not ideal, then the Jones matrix for a polarizer (11-36) at
þ45 is seen from (11-41) to be
!
1 px þ py px À py

JP ð45 Þ ¼
ð11-44Þ
2 px À py px þ py
We note that for  ¼ 0 and 90 , (11-42) gives the Jones matrices for a linear
horizontal and vertical polarizer, Eqs. (11-37) and (11-38) respectively.
Equation (11-41) also describes a neutral density (ND) filter. The condition for
a ND filter is px ¼ py ¼ p, so (11-41) reduces to
!
1 0
ð11-45Þ
JND ðÞ ¼ p

0 1
Thus, JND() is independent of rotation (), and the amplitudes are equally attenuated by an amount p. This is, indeed, the behavior of a ND filter. The presence of the
unit (diagonal) matrix in (11-45) confirms that a ND filter does not affect the
polarization state of the incident beam.
The next polarizing element of importance is the retarder. The retarder
increases the phase by þ=2 along the fast (x) axis and retards the phase by À=2,
along the slow (y) axis. This behavior is described by
E 0x ¼ eþi=2 Ex

ð11-46aÞ

E 0y ¼ eÀi=2 Ey

ð11-46bÞ

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


where E 0x and E 0y are the components of the emerging beam and Ex and Ey are the
components of the incident beam. We can immediately express (11-46) in the Jones
formalism as
 0   þi=2
 
Ex
Ex
0
e
0
J ¼
¼

ð11-47Þ
Ài=2
Ey
E 0y
0
e
The Jones matrix for a retarder (phase shifter) is then
 þi=2

e
0
JR ðÞ ¼
0
eÀi=2

ð11-48Þ

where  is the total phase shift between the field components. The two most common
types of phase shifters (retarders) are the quarter-wave retarder and the half-wave
retarder. For these devices  ¼ 90 and 180 , repectively, and (11-48) becomes





   i=4

0
0
0

e
i=4 1
i=4 1
JR
ð11-49aÞ
¼e
¼e
¼
0 eÀi=2
0 Ài
4
0
eÀi=4
and
 



   i=2

i 0
1 0
e
0
JR
¼
¼
i
ð11-49bÞ
¼

0 Ài
0 À1
2
0
eÀi=2
The Jones matrix for a rotated retarder is found from (11-48) and (11-39) to be
!
ei=2 cos2  þ eÀi=2 sin2  ðei=2 À eÀi=2 Þ sin  cos 
JR ð, Þ ¼
ð11-50Þ
ðei=2 À eÀi=2 Þ sin  cos  ei=2 sin2  þ eÀi=2 cos2 
With the half-angle formulas, (11-50) can also be written in the form:
0
1



i sin sin 2
cos þ i sin cos 2
B
C
2
2
2
ð11-51Þ
JR ð, Þ ¼ @
A




i sin sin 2
cos À i sin cos 2
2
2
2
For quarter-wave retarder and a half-wave retarder (11-51) reduces, respectively, to
0
1
1
i
i


p
ffiffi
ffi
p
ffiffi
ffi
p
ffiffi
ffi
þ
cos
2
sin
2
B 2
C


2
2
C
JR ,  ¼ B
ð11-52Þ
@
A
i
1
i
4
pffiffiffi sin 2
pffiffiffi À pffiffiffi cos 2
2
2
2
and
JR





cos 2
, ¼ i
sin 2
2

sin 2
À cos 2


ð11-53Þ

The factor i in (11-53) is unimodular and can be suppressed. It is common, therefore,
to write (11-53) simply as


 

cos 2
sin 2
JR ,  ¼
ð11-54Þ
sin 2 À cos 2
2
Inspecting (11-54) we see that it is very similar to the matrix for rotation, namely,


cos  sin 
JðÞ ¼
ð11-39bÞ
À sin  cos 

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


However, (11-54) differs from (11-39b) in two ways. First, in (11-54) we have 2
rather than . Thus, a rotation of a retarder through  rotates the polarization ellipse
through 2. Second, a clockwise mechanical rotation  in (11-54) leads to a counterclockwise rotation of the polarization ellipse. In order to see this behavior clearly,

consider that we have incident linear horizontally polarized light. Its Jones vector is
 
Ex
J¼
ð11-55Þ
0
The components of the beam emerging from a true rotator (11-39b) are then
E 0x ¼ ðcos ÞEx

ð11-56aÞ

E 0y ¼ Àðsin ÞEx

ð11-56bÞ

The angle of rotation  is then
tan  ¼

E 0y À sin 
¼ tanðÀÞ
¼
cos 
E 0x

ð11-57Þ

In a similar manner, multiplying (11-55) by (11-54) leads to
E 0x ¼ ðcos 2ÞEx

ð11-58aÞ

E 0y ¼ ðsin 2ÞEx

ð11-58bÞ

so we now have
tan  ¼

E 0y sin 2
¼ tan 2
¼
E 0x cos 2

ð11-59Þ

Comparing (11-59) with (11-57), we see that the direction of rotation for a rotated
retarder is opposite to the direction of true rotation. Equation (11-59) also shows
that the angle of rotation is twice that of a true rotation. Because of this similar but
analytically incorrect behavior of a rotated half-wave retarder, (11-54) is called a
pseudorotator. We note that an alternative form of a half-wave retarder, which is the
more common form, is given by factoring out i in (11-49b) or simply setting  ¼ 0 in
(11-54):

  

1 0
J
ð11-60Þ
¼
0 À1

2
The final matrix of interest is the Jones matrix for a rotator. The defining
equations are
E 0x ¼ cos
Ex þ sin
Ey

ð11-61aÞ

E 0y ¼ À sin
Ex þ cos
Ey

ð11-61bÞ

where
is the angle of rotation. Equation (11-61) is written in matrix form as
 0  
 
Ex
Ex
cos
sin

0
J ¼
¼
ð11-62Þ
Ey
E 0y

À sin
cos


Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


so the Jones matrix for a rotator is


cos
sin

JROT ¼
À sin
cos


ð11-63Þ

It is interesting to see the effect of rotating a true rotator. According to (11-39),
the rotation of a rotator, (11-63), is given by




cos  À sin 
cos
sin