5 The Mueller Matrices for Polarizing Components

All three polarizing elements,polarizer, retarder, and rotator, change the polarization state of an optical beam.In the following sections we derive the Mueller matrices for these polari

of polarization and see that the matrix representation of the Stokes parameters leads

to a very powerful mathematical tool for treating this interaction In Fig 5-1weshow an incident beam interacting with a polarizing element and the emerging beam

In Fig 5-1 the incident beam is characterized by its Stokes parameters Si, where

i ¼0, 1, 2, 3 The incident polarized beam interacts with the polarizing medium, andthe emerging beam is characterized by a new set of Stokes parameters S01, where,again, i ¼ 0, 1, 2, 3 We now assume that S01can be expressed as a linear combination

of the four Stokes parameters of the incident beam by the relations:

1CC

1C

or

where S and S0 are the Stokes vectors and M is the 4  4 matrix known as theMueller matrix It was introduced by Hans Mueller during the early 1940s WhileMueller appears to have based his 4  4 matrix on a paper by F Perrin and a stillearlier paper by P Soleillet, his name is firmly attached to it in the optical literature.Mueller’s important contribution was that he, apparently, was the first to describepolarizing components in terms of his Mueller matrices Remarkably, Muellernever published his work on his matrices Their appearance in the optical literaturewas due to others, such as N.G Park III, a graduate student of Mueller’s whopublished Mueller’s ideas along with his own contributions and others shortlyafter the end of the Second World War.

When an optical beam interacts with matter its polarization state is almostalways changed In fact, this appears to be the rule rather than the exception Thepolarization state can be changed by (1) changing the amplitudes, (2) changing thephase, (3) changing the direction of the orthogonal field components, or (4) trans-ferring energy from polarized states to the unpolarized state An optical element thatchanges the orthogonal amplitudes unequally is called a polarizer or diattenuator.Similarly, an optical device that introduces a phase shift between the orthogonalcomponents is called a retarder; other names used for the same device are wave plate,compensator, or phase shifter If the optical device rotates the orthogonal compo-nents of the beam through an angle
as it propagates through the element, it iscalled a rotator Finally, if energy in polarized states goes to the unpolarized state,the element is a depolarizer These effects are easily understood by writing the trans-verse field components for a plane wave:

Equation (4) can be changed by varying the amplitudes, E0x or E0y, or the phase,

xor yand, finally, the direction of Exðz, tÞ and Eyðz, tÞ The corresponding devicesfor causing these changes are the polarizer, retarder, and rotator The use ofFigure 5-1 Interaction of a polarized beam with a polarizing element

the names polarizer and retarder arose, historically, before the behavior of thesepolarizing elements was fully understood The preferable names would be diattenua-tor for a polarizer and phase shifter for the retarder All three polarizing elements,polarizer, retarder, and rotator, change the polarization state of an optical beam.

In the following sections we derive the Mueller matrices for these polarizingelements We then apply the Mueller matrix formalism to a number of problems

of interest and see its great utility

A polarizer is an optical element that attenuates the orthogonal components of anoptical beam unequally; that is, a polarizer is an anisotropic attenuator; thetwo orthogonal transmission axes are designated px and py Recently, it has alsobeen called a diattenuator, a more accurate and descriptive term A polarizer is some-times described also by the terms generator and analyzer to refer to its use and position

in the optical system If a polarizer is used to create polarized light, we call it agenerator If it is used to analyze polarized light, it is called an analyzer If the ortho-gonal components of the incident beam are attenuated equally, then the polarizerbecomes a neutral density filter We now derive the Mueller matrix for a polarizer

In Fig 5-2 a polarized beam is shown incident on a polarizer along with theemerging beam The components of the incident beam are represented by Exand Ey.After the beam emerges from the polarizer the components are E0xand E0y, and theyare parallel to the original axes The fields are related by

The factors px and pyare the amplitude attenuation coefficients along orthogonaltransmission axes For no attenuation or perfect transmission along an orthogonalaxis pxðpyÞ ¼1, whereas for complete attenuation pxðpyÞ ¼0 If one of the axes has

an absorption coefficient which is zero so that there is no transmission along thisaxis, the polarizer is said to have only a single transmission axis

Figure 5-2 The Mueller matrix of a polarizer with attenuation coefficients pxand py

The Stokes polarization parameters of the incident and emerging beams are,respectively,

@

1CCA

¼12

@

1CCA

@

1CC

The 4  4 matrix in (5-8) is written by itself as

M ¼12

@

1CC

Equation (5-9) is the Mueller matrix for a polarizer with amplitude attenuationcoefficients px and py In general, the existence of the m33 term shows that thepolarization of the emerging beam of light will be elliptically polarized

For a neutral density filter px¼py¼p and (5-9) becomes

1C

which is a unit diagonal matrix Equation (5-10) shows that the polarization state isnot changed by a neutral density filter, but the intensity of the incident beam isreduced by a factor of p2 This is the expected behavior of a neutral density filter,

since it only affects the magnitude the intensity and not the polarization state.According to (5-10), the emerging intensity I0 is then

where I is the intensity if the incident beam

Equation (5-9) is the Mueller matrix for a polarizer which is described byunequal attenuations along the pxand pyaxes An ideal linear polarizer is one whichhas transmission along only one axis and no transmission along the orthogonal axis.This behavior can be described by first setting, say, py¼0 Then (5-9) reduces to

M ¼p

2 x

@

1C

Equation (5-12) is the Mueller matrix for an ideal linear polarizer which polarizesonly along the x axis It is most often called a linear horizontal polarizer,arbitrarily assigning the horizontal to the x direction It would be a perfect linearpolarizer if the transmission factor pxwas unity ð px¼1Þ Thus, the Mueller matrixfor an ideal perfect linear polarizer with its transmission axis in the x direction is

@

1C

If the original beam is completely unpolarized, the maximum intensity of theemerging beam which can be obtained with a perfect ideal polarizer is only 50%

of the original intensity It is the price we pay for obtaining perfectly polarized light

If the original beam is perfectly horizontally polarized, there is no change inintensity This element is called a linear polarizer because it affects a linearlypolarized beam in a unique manner as we shall soon see

In general, all linear polarizers are described by (5-9) There is only one knownnatural material that comes close to approaching the perfect ideal polarizer described

by (5-13), and this is calcite A synthetic material known as Polaroid is also used as

a polarizer Its performance is not as good as calcite, but its cost is very low incomparison with that of natural calcite polarizers, e.g., a Glan–Thompson prism.Nevertheless, there are a few types of Polaroid which perform extremely well as

‘‘ideal’’ polarizers We shall discuss the topic of calcite and Polaroid polarizers in

@

1C

which is the Mueller matrix for a linear vertical polarizer

Finally, it is convenient to rewrite the Mueller matrix, of a general linearpolarizer, (5-9), in terms of trigonometric functions This can be done by setting

1C

where 0  90
For an ideal perfect linear polarizer p ¼ 1 For a linear horizontalpolarizer  ¼ 0, and for a linear vertical polarizer  ¼ 90
The usefulness of thetrigonometric form of the Mueller matrix, (5-16), will appear later

The reason for calling (5-13) and (5-14) linear polarizers is due to the followingresult Suppose we have an incident beam of arbitrary intensity and polarization sothat its Stokes vector is

1C

1C

C¼12

1CC

1C

1C

C¼1

2ðS0S1Þ

1

100

0BB

1C

Inspecting (5-19), we see that the Stokes vector of the emerging beam is alwayslinearly horizontally (þ) or vertically () polarized Thus an ideal linear polarizeralways creates linearly polarized light regardless of the polarization state of theincident beam; however, note that because the factor 2pxpy in (5-9) is never zero,

in practice there is no known perfect linear polarizer and all polarizers createelliptically polarized light While the ellipticity may be small and, in fact, negligible,there is always some present

The above behavior of linear polarizers allows us to develop a test to determine

if a polarizing element is actually a linear polarizer The test to determine if we have alinear polarizer is shown in Fig 5-3 In the test we assume that we have a linearpolarizer and set its axis in the horizontal (H ) direction We then take anotherpolarizer and set its axis in the vertical (V ) direction as shown in the figure TheStokes vector of the incident beam is S, and the Stokes vector of the beam emergingfrom the first polarizer (horizontal) is

We now carry out the multiplication in (5-22) and write, using (5-13) and (5-14),

1CC

1C

1C

Thus, we obtain a null Mueller matrix and, hence, a null output intensity regardless

of the polarization state of the incident beam The appearance of a null Muellermatrix (or intensity) occurs only when the linear polarizers are in the crossed polar-izerconfiguration Furthermore, the null Mueller matrix always arises whenever thepolarizers are crossed, regardless of the angle of the transmission axis of the firstpolarizer

Figure 5-3 Testing for a linear polarizer

5.3 THE MUELLER MATRIX OF A RETARDER

A retarder is a polarizing element which changes the phase of the optical beam.Strictly speaking, its correct name is phase shifter However, historical usagehas led to the alternative names retarder, wave plate, and compensator Retardersintroduce a phase shift of  between the orthogonal components of the incidentfield This can be thought of as being accomplished by causing a phase shift ofþ=2 along the x axis and a phase shift of =2 along the y axis These axes ofthe retarder are referred to as the fast and slow axes, respectively In Fig 5-4 we showthe incident and emerging beam and the retarder The components of the emergingbeam are related to the incident beam by

@

1C

@

1CA

@

1C

The Mueller matrix for a retarder with a phase shift  is, from (5-26),

@

1C

@

1C

The quarter-wave retarder has the property that it transforms a linearly polarizedbeam with its axis at þ 45
or  45
to the fast axis of the retarder into a right or leftcircularly polarized beam, respectively To show this property, consider the Stokesvector for a linearly polarized  45
beam:

S ¼ I0

10

10

0B

@

1C

Multiplying (5-29) by (5-28) yields

S0¼I0

1001

0B

@

1C

which is the Stokes vector for left (right) circularly polarized light The formation of linearly polarized light to circularly polarized light is an importantapplication of quarter-wave retarders However, circularly polarized light isobtained only if the incident linearly polarized light is oriented at  45

trans-On the other hand, if the incident light is right (left) circularly polarizedlight, then multiplying (5-30) by (5-28) yields

S0¼I0

1010

0B

@

1C

which is the Stokes vector for linear  45 or þ 45 polarized light The quarter-waveretarder can be used to transform linearly polarized light to circularly polarized light

or circularly polarized light to linearly polarized light

The other important type of wave retarder is the half-wave retarder ð ¼ 180
Þ.For this condition (5-27) reduces to

@

1C

A half-wave retarder is characterized by a diagonal matrix The terms m22¼m33¼

1 reverse the ellipticity and orientation of the polarization state of the incidentbeam To show this formally, we have initially

@

1C

@

1C

@

1C

where

tan 2 0¼S02

S0 1

ð5-34aÞ

sin 20¼S03

S0 0

ð5-34bÞSubstituting (5-33) into (5-34) yields

Half-wave retarders also possess the property that they can rotate the polarizationellipse This important property shall be discussed in Section 5.5.

The final way to change the polarization state of an optical field is to allow a beam topropagate through a polarizing element that rotates the orthogonal field components

Ex(z, t) and Ey(z, t) through an angle
In order to derive the Mueller matrix forrotation, we consider Fig 5-5 The angle
describes the rotation of Exto E0xand of

Eyto E0y Similarly, the angle  is the angle between E and Ex In the figure the point

Pis described in the E0x, E0y coordinate system by

Expanding the trigonometric functions in (5-37) gives

Collecting terms in (5-39) using (5-38) then gives

Figure 5-5 Rotation of the optical field components by a rotator

Equations (5-40a) and (5-40b) are the amplitude equations for rotation In order tofind the Mueller matrix we form the Stokes parameters for (5-40) as before and findthe Mueller matrix for rotation:

@

1C

We note that a physical rotation of
leads to the appearance of 2
in (5-41) ratherthan
because we are working in the intensity domain; in the amplitude domain wewould expect just

Rotators are primarily used to change the orientation angle of the polarizationellipse To see this behavior, suppose the orientation angle of an incident beam is Recall that

0

In the derivation of the Mueller matrices for a polarizer, retarder, androtator, we have assumed that the axes of these devices are aligned along the Exand Ey(or x, y axes), respectively In practice, we find that the polarization elementsare often rotated Consequently, it is also necessary for us to know the form of theMueller matrices for the rotated polarizing elements We now consider this problem

5.5 MUELLER MATRICES FOR ROTATED POLARIZING

where we have used (5-47) Finally, we must take the components of the emergingbeam along the original x and y axes as seen in Fig 5-6 This can be described by acounterclockwise rotation of S00 through 
and back to the original x, y axes, so

S0 0 0¼MRð2
ÞS0 0

where MR(2
) is, again, the Mueller matrix for rotation and S000is the Stokes vector

of the emerging beam Equation (5-49) can be written as

Equation (5-51) is the Mueller matrix of a rotated polarizing component We recallthat the Mueller matrix for rotation MR(2
) is given by

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1C

The rotated Mueller matrix expressed by (5-51) appears often in the treatment

of polarized light Of particular interest are the Mueller matrices for a rotatedpolarizer and a rotated retarder The Mueller matrix for a rotated ‘‘rotator’’

is also interesting, but in a different way We recall that a rotator rotates thepolarization ellipse by an amount
If the rotator is now rotated through an

R(2
); that is, the rotator

is unaffected by a mechanical rotation Thus, the polarization ellipse cannot

be rotated by rotating a rotator! The rotation comes about only by the intrinsicbehavior of the rotator It is possible, however, to rotate the polarization ellipsemechanically by rotating a half-wave plate, as we shall soon demonstrate

The Mueller matrix for a rotated polarizer is most conveniently found byexpressing the Mueller matrix of a polarizer in angular form, namely,

@

1C

Carrying out the matrix multiplication according to (5-51) and using (5-52), theMueller matrix for a rotated polarizer is

M ¼12

cos 2 cos 2
cos22
þ sin 2 sin22
ð1  sin 2Þ sin 2
cos 2
0cos 2 sin 2
ð1  sin 2Þ sin 2
cos 2
sin22
þ sin 2 cos22
0

0BB

1CC

ð5-53Þ

In (5-53) we have set p2to unity We note that  ¼ 0
, 45
, and 90
correspond to alinear horizontal polarizer, a neutral density filter, and a linear vertical polarizer,respectively

The most common form of (5-53) is the Mueller matrix for an ideal linearhorizontal polarizer ( ¼ 0
) For this value (5-53) reduces to

1C

In (5-54) we have written MP(2
) to indicate that this is the Mueller matrix for

a rotated ideal linear polarizer The form of (5-54) can be checked immediately by

Equations (5- 40a) and (5- 40b) are the amplitude equations for rotation In order tofind the Mueller matrix we form the Stokes parameters for (5- 40) as before and findthe Mueller matrix for rotation:

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5. 5 MUELLER MATRICES FOR ROTATED POLARIZING< /p>

where we have used (5- 47) Finally, we must take the components of the emergingbeam along the original x and y axes as seen in Fig 5- 6 This... polarizer,respectively

The most common form of (5- 53) is the Mueller matrix for an ideal linearhorizontal polarizer ( ¼ 0
) For this value (5- 53) reduces to

1C

In (5- 54) we have