10 The Mueller Matrices for DielectricPlates

1C The diagonalized Mueller matrices will play an essential role in the following sectionwhen we determine the Mueller matrices for single and multiple dielectric plates.Before we conclu

in the visible and near-infrared regions using calcite polarizers or Polaroid, there are

no corresponding materials in the far-infrared region However, materials such asgermanium and silicon, as well as others, do transmit very well in the infrared region

By making thin plates of these materials and then constructing a ‘‘pile of plates,’’ it

is possible to create light in the infrared that is highly polarized This arrangementtherefore requires that the Mueller matrices for transmission play a more prominentrole than the Mueller matrices for reflection

In order to use the Mueller matrices to characterize a single plate or multipleplates, we must carry out matrix multiplications The presence of off-diagonalterms of the Mueller matrices create a considerable amount of work We know,

on the other hand, that if we use diagonal matrices the calculations are simplified;the product of diagonalized matrices leads to another diagonal matrix

POLARIZATION MATRIX

When we apply the Mueller matrices to problems in which there are several ing elements, each of which is described by its own Mueller matrix, we soondiscover that the appearance of the off-diagonal elements complicates the matrix

polariz-multiplications The multiplications would be greatly simplified if we were to usediagonalized forms of the Mueller matrices In particular, the use of diagonalizedmatrices enables us to determine more easily the Mueller matrix raised to themth power, Mm, an important problem when we must determine the transmission

of a polarized beam through m dielectric plates

In this chapter we develop the diagonal Mueller matrices for a polarizer and

a retarder To reduce the amount of calculations, it is simpler to write a singlematrix that simultaneously describes the behavior of a polarizer or a retarder or acombination of both This simplified matrix is called the ABCD polarization matrix.The Mueller matrix for a polarizer is

MP¼12

1CC

1CC

where psand ppare the absorption coefficients of the polarizer along the s (or x) and

p (or y) axes, respectively, and  is the phase shift of the retarder

The form of (10-1) and (10-2) suggests that the matrices can be represented by

a single matrix of the form:

1CC

and for the retarder

which in its expanded form is

M ¼

Bcos 2
A2cos22
þ C sin22
ðA  CÞsin 2
cos 2
Dsin 2

Bsin 2
ðA  CÞsin 2
cos 2
Asin22
þ C cos22
Dcos 2

0BB

@

1CCA

@

1C

We now find the diagonalized form of the ABCD matrix This can be done using thewell-known methods in matrix algebra We first express (10-3) as an eigenvalue/eigenvector equation, namely,

ð10-16aÞor

Substituting these eigenvalues into (10-17), we easily find that the eigenvectorcorresponding to each of the respective eigenvalues in (10-19) is

S1¼ 1

ffiffiffi2p

1100

0B

@

1C

1ffiffiffi2p

1

100

0B

@

1C

1ffiffiffi2p

001i

0B

@

1C

1ffiffiffi2p

001

i

0B

@

1CAð10-20ÞThe factor 1= ffiffiffi

2

phas been introduced to normalize each of the eigenvectors

We now construct a new matrix K, called the modal matrix, whose columnsare formed from each of the respective eigenvectors in (10-20):

K ¼ 1ffiffiffi

2p

@

1C

The inverse matrix is easily found to be

K1 ¼ 1

ffiffiffi2p

@

1C

@

1C

@

1C

@

1C

A remarkable relation now emerges From (10-21) and (10-22) one readily seesthat the following identity is true:

Postmultiplying both sides of (10-25) by K , we see that

Equation (10-28) now enables us to find the mth power of the ABCD matrix :

1CC

1C

1 C C C C C C

(10-30)Using (10-30) we readily find that the mth powers of the Mueller matrix of apolarizer and a retarder are, respectively,

Mmp ¼12

@

1CC

1C

The diagonalized Mueller matrices will play an essential role in the following sectionwhen we determine the Mueller matrices for single and multiple dielectric plates.Before we conclude this section we discuss another form of the Muellermatrix for a polarizer We recall that the first two Stokes parameters, S0 and

S1, are the sum and difference of the orthogonal intensities The Stokes parameterscan then be written as

@

1CA

@

1C

12

@

1CA

@

1C

1C

is called the intensity vector The intensity vector is very useful because the 4  4matrix which connects I to I0is diagonalized, thus making the calculations simpler.

To show that this is true, we can formally express (10-35a) and (10-35b) as

where KA and K1A are defined by the 4  4 matrices in (10-35), respectively TheMueller matrix M can be defined in terms of an incident Stokes vector S and anemerging Stokes vector S0:

1CC

1CC

Thus, P is a diagonal polarizing matrix; it is equivalent to the diagonalMueller matrix for a polarizer The diagonal form of the Mueller matrix was firstused by the Nobel laureate S Chandrasekhar in his classic papers in radiative

transfer in the late 1940s It is called Chandrasekhar’s phase matrix in the literature.

In particular, for the Mueller matrix of a polarizer we see that (10-44) becomes

1CC

@

1CC

where
is the observation angle in spherical coordinates and is measured from the

zaxis (
¼ 0
) Transforming (10-46) to Chandrasekhar’s phase matrix, we find

@

1CC

DIELECTRIC PLATES

In the previous sections, Fresnel’s equations for reflection and transmission at

an air–dielectric interface were cast into the form of Mueller matrices In this section

we use these results to derive the Mueller matrices for dielectric plates We firsttreat the problem of determining the Mueller matrix for a single dielectric plate.The formalism is then easily extended to multiple reflections within a single dielectricplate and then to a pile of m parallel transparent dielectric plates

For the problem of transmission of a polarized beam through a single dielectricplate, the simplest treatment can be made by assuming a single transmissionthrough the upper surface followed by another transmission through the lowersurface There are, of course, multiple reflections within the dielectric plates, and,strictly speaking, these should be taken into account While this treatment ofmultiple internal reflections is straightforward, it turns out to be quite involved In

the treatment presented here, we choose to ignore these effects The completely correcttreatment is given in the papers quoted in the references at the end of this chapter Thedifference between the exact results and the approximate results is quite small, andvery good results are still obtained by ignoring the multiple internal reflections.Consequently, only the resulting expressions for multiple internal reflections arequoted We shall also see that the use of the diagonalized Mueller matrices developed

in the previous section greatly simplifies the treatment of all of these problems

In Fig 10-1 a single dielectric (glass) plate is shown The incident beam isdescribed by the Stokes vector S Inspection of the figure shows that the Stokesvector S0of the beam emerging from the lower side of the dielectric plate is related

to S by the matrix relation:

where MTis the Mueller matrix for transmission and is given by (8-13) in Section 8.3

We easily see, using (8-13), that M2T is then

M2T¼12

sin 2
isin 2
rðsin
þcos
Þ2

@

1CC

where
iis the angle of incidence,
ris the angle of refraction, and
¼
i
r.Equation (10-49) is the Mueller matrix (transmission) for a single dielectricplate We can immediately extend this result to the transmission through m paralleldielectric plates by raising M2T to the mth power, this is, M2mT The easiest way to

do this is to transform (10-49) to the diagonal form and raise the diagonal matrix tothe mth power as described earlier After this is done we transform back to theFigure 10-1 Beam propagation through a single dielectric plate

Mueller matrix form Upon doing this we then find that the Mueller matrix fortransmission through m parallel dielectric plates is

M2mT ¼1

2

sin 2
isin 2
rðsin
þcos
Þ2

1CCCC

ð10-50Þ

Equation (10-50) includes the result for a single dielectric plate by setting m ¼ 1 Wenow consider that the incident beam is unpolarized Then, the Stokes vector of abeam emerging from m parallel plates is, from (10-50),

S0¼1

2

sin 2
isin 2
rðsin
þcos
Þ2

cos4m
þ1cos4m
100

0BBB

1CC

InFig 10-2a plot of (10-52) is shown for the degree of polarization as a function

of the incident angle
i The plot shows that at least six or eight parallel platesare required in order for the degree of polarization to approach unity At normalincidence the degree of polarization is always zero, regardless of the number ofplates

The use of parallel plates to create linearly polarized light appears veryoften outside the visible region of the spectrum In the visible and near-infraredregion (<2 m) Polaroid and calcite are available to create linearly polarizedlight Above 2 m, parallel plates made from other materials are an importantpractical way of creating linearly polarized light Fortunately, natural materialssuch as germanium are available and can be used; germanium transmits morethan 95% of the incident light up to 20 m

According to (10-51) the intensity of the beam emerging from m parallel plates,

IT, is

IT¼1

2

sin 2
isin 2
rðsin
þcos
Þ2

Figure 10-3shows a plot of (10-53) for m dielectric plates

Figure 10-2 Plot of (10-52), the degree of polarization P versus incident angle and thenumber or parallel plates The refractive index n is 1.5.

Figure 10-3 The intensity of a beam emerging from m parallel plates as a function of theangle of incidence The refractive index is 1.5

At the Brewster angle the Mueller matrix for transmission through m dielectricplates is readily shown from the results given inChapter 8and Section 10.2 to be

@

1CCCAð10-54Þ

For a single dielectric plate m ¼ 1, (10-54) reduces to

M2T, B ¼1

2

sin42
iBþ1 sin42
iB1 0 0sin42
iB1 sin42
iBþ1 0 0

0BBB

@

1CCCA

0BBB

1CC

A plot of (10-57) is shown inFig 10-4for m dielectric plates

The intensity of the transmitted beam is given by S0in (10-56) and is

IT¼1

2ð1 þ sin

Equation (10-58) has been plotted inFig 10-5

From Figs 10-4 and 10-5 the following conclusions can be drawn In Fig 10-4,there is a significant increase in the degree of polarization up to m ¼ 6 Figure 10-5,

on the other hand, shows that the intensity decreases and then begins to ‘‘level off’’for m ¼ 6 Thus, these two figures show that after five or six parallel plates there isvery little to be gained in using more plates to increase the degree of polarization andstill maintain a ‘‘constant’’ intensity In addition, the cost for making such largeassemblies of dielectric plates, the materials, and mechanical alignment becomesconsiderable

Dielectric plates can also rotate the orientation of the polarization ellipse

At first this behavior may be surprising, but this is readily shown Consider the

Figure 10-4 Plot of the degree of polarization P versus number of dielectric plates at theBrewster angle for refractive indices of 1.5, 2.0, and 2.5.

Figure 10-5 Plot of the transmitted intensity of a beam propagating through m parallelplates at the Brewster angle
i

B The refractive indices are 1.5, 2.0, and 2.5, respectively

situation when the incident beam is linear þ45 polarized light The normalized Stokesvector of the beam emerging from m dielectric plates is then, from (10-54),

S0¼1

2

sin4m2
iBþ1sin4m2
iB1

2 sin2m2
iB0

0BBB

1CC

!

ð10-60Þ

We note that for m ¼ 0 (no dielectric plates) the absolute magnitude of theangle of rotation is ¼ 45
, as expected Figure 10-6 illustrates the change in theangle of rotation as the number of parallel plates increases For five parallel platesthe orientation angle rotates from þ45
to þ24.2

Equation (10-57) can also be expressed in terms of the refractive index, n Werecall that (10-57) is

At the Brewster angle we have

as inspection ofFig 10-7shows, the curves are identical to those inFig 10-4except

in the former figure the abscissa begins with m ¼ 1

In the beginning of this section we pointed out that the Mueller matrixformalism can also be extended to the problem of including multiple reflectionswithin a single dielectric plate as well as the multiple plates G G Stokes (1862)was the first to consider this problem and showed that the inclusion ofmultiple reflections within the plates led to the following equation for the degree

of polarization for m parallel plates at the Brewster angles:

It is of interest to compare (10-62) and (10-63) In Fig 10-9 we haveplotted these two equations for n ¼ 1.5 We see immediately that the degree ofpolarization is very different Starting with 0 parallel plates, that is, the unpolarizedlight source by itself, we see the degree of polarization is zero, as expected Asthe number of parallel plates increases, the degree of polarization increases forboth (10-62) and (10-63) However, the curves diverge and the magnitudes differ

by approximately a factor of two so that for 10 parallel plates the degree ofpolarization is 0.93 for (10-62) and 0.43 for (10-63) In addition, for (10-63), the

Figure 10-8 Plot of the degree of polarization as a function of the number of parallel platesfor the case where multiple reflections are included

lower curve is almost linear with a very shallow slope, and shows that there is verylittle to be gained by increasing the number of parallel plates in order to increase thedegree of polarization.

A final topic that we discuss is the use of a simpler notation for theMueller matrices for reflection and transmission by representing the matrix elements

in terms of the Fresnel reflection and transmission coefficients These coefficients aredefined to be

s¼ Rs

Es

2

¼ sin
sin
þ

ð10-64bÞand

s¼ncos
rcos
i

Ts

Es

2

¼tan
itan
r

2 sin
rcos
isin
þ

¼sin 2
isin 2
r

p¼ncos
rcos
i

Tp

Ep

2

¼tan
itan
r

2 sin
rcos
isin
þcos


¼ sin 2
isin 2
rsin2
þcos2


ð10-65bÞFigure 10-9 Degree of polarization for m parallel plates for n ¼ 1.5 The upper curvecorresponds to (10-62), and the lower corresponds to (10-63)

One can readily show that the following relations hold for Fresnel coefficients:

@

1CC

@

1CC

The reflection coefficients sand p, (10-64a) and (10-64b), are plotted as a function

of the incident angle for a range of refractive indices inFigs 10-10and10-11.Similarplots are shown inFigs 10-12 and10-13for sand p, (10-65a) and (10-65b)

In a similar manner the reflection and transmission coefficients at the Brewsterangle, (10-67) and (10-68), are plotted as a function of the refractive index n inFigs 10-14and10-15

The great value of the Fresnel coefficients is that their use leads to simplerforms for the Mueller matrices for reflection and transmission For example, instead

Figure 10-10 Plot of the Fresnel reflection coefficient s as a function of incidenceangle
i, (10-64a).

Figure 10-11 Plot of the Fresnel reflection coefficient p as a function of incidenceangle
i, (10-64b)

Figure 10-12 Plot of the Fresnel reflection coefficient sas a function of incidence angle

i, (10-65a)

Figure 10-13 Plot of the Fresnel reflection coefficient p as a function of incidence angle

i, (10-65b)

of the complicated matrix entries given above, we can write, say, the diagonalizedform of the Mueller matrices as

@

1C

Figure 10-14 Plot of the reflection coefficients at the Brewster angle, (10-67)

Figure 10-15 Plot of the transmission coefficients at the Brewster angle, (10-68)