6 Methods of Measuring the Stokes Polarization Parameters

RETARDER POLARIZER METHOD The Mueller matrices for the polarizer diattenuator, retarder phase shifter, androtator can now be used to analyze various methods for measuring the Stokes... W

to Stokes and is probably the best known method; this method was discussed inSection 4.4 There are other methods for measuring the Stokes parameters However,

we have refrained from discussing these methods until we had introduced theMueller matrices for a polarizer, a retarder, and a rotator The Mueller matrixand Stokes vector formalism allows us to treat all of these measurement problems

in a very simple and direct manner While, of course, the problems could havebeen treated using the amplitude formulation, the use of the Mueller matrixformalism greatly simplifies the analysis

In theory, the measurement of the Stokes parameters should be quite simple.However, in practice there are difficulties This is due, primarily, to the fact thatwhile the measurement of S0, S1, and S2is quite straightforward, the measurement

of S3 is more difficult In fact, as we pointed out, before the advent of opticaldetectors it was not even possible to measure the Stokes parameters using Stokes’measurement method (Section 4.4) It is possible, however, to measure the Stokesparameter using the eye as a detector by using a so-called null method; this isdiscussed in Section 6.4 In this chapter we discuss Stokes’ method along withother methods, which includes the circular polarizer method, the null-intensitymethod, the Fourier analysis method, and the method of Kent and Lawson

RETARDER POLARIZER METHOD

The Mueller matrices for the polarizer (diattenuator), retarder (phase shifter), androtator can now be used to analyze various methods for measuring the Stokes

parameters A number of methods are known We first consider the application

of the Mueller matrices to the classical measurement of the Stokes polarizationparameters using a quarter-wave retarder and a polarizer This is the same problemthat was treated in Section 4.4; it is the problem originally considered by Stokes(1852) The result is identical, of course, with that obtained by Stokes However,the advantage of using the Mueller matrices is that a formal method can be used totreat not only this type of problem but other polarization problems as well.The Stokes parameters can be measured as shown in Fig 6-1 An optical beam

is characterized by its four Stokes parameters S0, S1, S2, and S3 The Stokes vector ofthis beam is represented by

1C

@

1C

1C

Figure 6-1 Classical measurement of the Stokes parameters

The Mueller matrix of an ideal linear polarizer with its transmission axis set at

1C

The Stokes vector S00 of the beam emerging from the linear polarizer is found

by multiplication of (6-3) by (6-4) However, we are only interested in theintensity I00, which is the first Stokes parameter S000of the beam incident on the opticaldetector shown in Fig 6-1 Multiplying the first row of (6-4) with (6-3), we thenfind the intensity of the beam emerging from the quarter-wave retarder–polarizercombination to be

Ið
, Þ ¼1

2½S0þS1cos 2
þ S2sin 2
cos  þ S3sin 2
sin  ð6-5ÞEquation (6-5) is Stokes’ famous intensity relation for the Stokes parameters TheStokes parameters are then found from the following conditions on
and :

at
¼ 45
This immediately raises a problem because the retarder absorbs someoptical energy In order to obtain an accurate measurement of the Stokes parametersthe absorption factor must be introduced, ab initio, into the Mueller matrix for theretarder The absorption factor which we write as p must be determined from aseparate measurement and will then appear in (6-5) and (6-6) We can easilyderive the Mueller matrix for an absorbing retarder as follows

The field components Ex and Ey of a beam emerging from an absorbingretarder in terms of the incident field components Exand Eyare

Using (6-7) and (6-8) in the defining equations for the Stokes parameters, we findthe Mueller matrix for an anisotropic absorbing retarder:

M ¼12

1C

Thus, we see that an absorbing retarder behaves simultaneously as a polarizer and

a retarder If we use the angular representation for the polarizer behavior,Section 5.2, equation (5-15b), then we can write (6-9) as

0 0 sin 2 cos  sin 2 sin 

0 0 sin 2 sin  sin 2 cos 

0B

@

1C

where p2xþp2y¼p2 We note that for  ¼ 45
we have an isotropic retarder; that is,the absorption is equal along both axes If p2is also unity, then (6-9) reduces to anideal phase retarder

The intensity of the emerging beam Ið
, Þ is obtained by multiplying (6-1) by(6-10) and then by (6-4), and the result is

Ið
, Þ ¼p

2

2 ½ð1 þ cos 2
cos 2ÞS0þ ðcos 2 þ cos 2
ÞS1

þ ðsin 2 cos  sin 2
ÞS2þ ðsin 2 sin  sin 2
ÞS3 ð6-11Þ

If we were now to make all four intensity measurements with a quarter-waveretarder in the optical train, then (6-11) would reduce for each of the four combina-tions of
and  ¼ 90
to

parameters are measured according to (6-6) The last measurement is done with aquarter-wave retarder in the optical train, (6-12d), so the equations are

we see that the measurement of the first three Stokes parameters is very simple, butthe measurement of the fourth parameter S3 requires a considerable amount

of additional effort

It would therefore be preferable if a method could be devised whereby theabsorption measurement could be eliminated A method for doing this can bedevised, and we now consider this method

A CIRCULAR POLARIZER

The problem of absorption by a retarder can be completely overcome by using

a single polarizing element, namely, a circular polarizer; this is described below.The beam is allowed to enter one side of the circular polarizer, whereby the firstthree parameters can be measured The circular polarizer is then flipped 180
, andthe final Stokes parameter is measured A circular polarizer is made by cementing

a quarter-wave retarder to a linear polarizer with its axis at 45
to the fast axis ofthe retarder This ensures that the retarder and polarizer axes are always fixed withrespect to each other Furthermore, because the same optical path is used in allfour measurements, the problem of absorption vanishes; the four intensities arereduced by the same amount

The construction of a circular polarizer is illustrated in Fig 6-2

The Mueller matrix for the polarizer–retarder combination is

@

1CA

@

1C

and thus

M ¼12

1C

Equation (6-14b) is the Mueller matrix of a circular polarizer The reason for calling(6-14b) a circular polarizer is that regardless of the polarization state of the incidentbeam the emerging beam is always circularly polarized This is easily shown byassuming that the Stokes vector of an incident beam is

@

1CC

1

0BB

1C

which is the Stokes vector for left circularly polarized light (LCP) Thus, regardless

of the polarization state of the incident beam, the output beam is always leftcircularly polarized Hence, the name circular polarizer Equation (6-14b) defines

a circular polarizer

Next, consider that the quarter-wave retarder–polarizer combination

is ‘‘flipped’’; that is, the linear polarizer now follows the quarter–wave retarder.The Mueller matrix for this combination is obtained with the Mueller matricesFigure 6-2 Construction of a circular polarizer using a linear polarizer and a quarter-waveretarder

in (6-14a) interchanged; we note that the axis of the linear polarizer when it isflipped causes a sign change in the Mueller matrix (seeFig 6-2).Then

@

1CA

@

1C

@

1C

10

0B

@

1C

which is the Stokes vector for linear 45
polarized light Regardless of thepolarization state of the incident beam, the final beam is always linear þ45
polarized It is of interest to note that in the case of the ‘‘circular’’ side of thepolarizer configuration, (6-15), the intensity varies only with the linear component,

S2, in the incident beam On the other hand, for the ‘‘linear’’ side of the polarizer,(6-17), the intensity varies only with S3, the circular component in the incident beam.The circular polarizer is now placed in a rotatable mount We saw earlierthat the Mueller matrix for a rotated polarizing component, M, is given by therelation:

@

1C

and Mð2
Þ is the Mueller matrix of the rotated polarizing element The Muellermatrix for the circular polarizer with its axis rotated through an angle
is thenfound by substituting (6-14b) into (5-51) The result is

@

1C

where the subscript C refers to the fact that (6-18) describes the circular side of thepolarizer combination We see immediately that the Stokes vector emerging fromthe beam of the rotated circular polarizer is, using (6-18) and (6-1),

SC¼1

2ðS0S1sin 2
þ S2cos 2
Þ

100

1

0BBB

1CC

Thus, as the circular polarizer is rotated, the intensity varies but the polarizationstate remains unchanged, i.e., circular We note again that the total intensity depends

on S0and on the linear components, S1and S2, in the incident beam

The Mueller matrix when the circular polarizer is flipped to its linear side is,from (6-16b) and (5-51),

1CC

where the subscript L refers to the fact that (6-20) describes the linear side of thepolarizer combination The Stokes vector of the beam emerging from the rotatedlinear side of the polarizer, multiplying, (6-20) and (6-1), is

SL¼1

2ðS0S3Þ

1sin 2

cos 2

0

0BBB

1CC

Under a rotation of the circular polarizer on the linear side, (6-21) shows that thepolarization is always linear The total intensity is constant and depends on S0andthe circular component S3in the incident beam

The intensities detected on the circular and linear sides are, respectively, from(6-19) and (6-21),

In order to obtain the Stokes parameters, we first use the circular side of thepolarizing element and rotate it to
¼ 0
, 45
, and 90
, and then flip it to the linearside The measured intensities are then

to each other

6.4 THE NULL-INTENSITY METHOD

In previous sections the Stokes parameters were expressed in terms of measuredintensities These measurement methods, however, are suitable only for use withquantitative detectors We pointed out earlier that before the advent of solid-statedetectors and photomultipliers the only available detector was the human eye It canonly measure the presence of light or no light (a null intensity) It is possible, as

we shall now show, to measure the Stokes parameters from the condition of anull-intensity state This can be done by using a variable retarder (phase shifter)followed by a linear polarizer in a rotatable mount Devices are manufacturedwhich can change the phase between the orthogonal components of an opticalbeam They are called Babinet–Soleil compensators, and they are usually placed in

a rotatable mount Following the compensator is a linear polarizer, which is alsoplaced in a rotatable mount This arrangement can be used to obtain a null intensity

In order to carry out the analysis, the reader is referred toFig 6-3

The Stokes vector of the incident beam to be measured is

1C

in Section 4.3 is used:

S ¼ I0

10B

@

1C

@

1CA

10B

@

1C

BB

1C

Figure 6-3 Null intensity measurement of the Stokes parameters

Two important observations on (6-26) can be made The first is that (6-26) can

be transformed to linearly polarized light if S03 can be made to be equal to zero.This can be done by setting    to 0
If we then analyze S0with a linear polarizer,

we see that a null intensity can be obtained by rotating the polarizer; at the nullalways used to obtain a null intensity The null-intensity method works because

in (6-25) is simply transformed to    in (6-26) after the beam propagates throughthe compensator (retarder) For the moment we shall retain the form of (6-26)and not set    to 0
The function of the Babinet–Soleil compensator in thiscase is to transform elliptically polarized light to linearly polarized light

Next, the beam represented by (6-26) is incident on a linear polarizer withits transmission axis at an angle
The Stokes vector S00 of the beam emergingfrom the rotated polarizer is now

1CC

10

BB

1C

C ð6-27Þ

where we have used the Mueller matrix of a rotated linear polarizer, Equation (5-54)Section 5.5 We are interested only in the intensity of the beam emerging from therotated polarizer; that is, S000¼Ið
, Þ Carrying out the matrix multiplication withthe first row in the Mueller matrix and the Stokes vector in (6-27) yields

values for the orientation angle and the ellipticity of the incident beam.

, namely,

ð4-40aÞð4-40bÞSubstituting (6-31) into (4-40), we see that and can be expressed in the terms ofthe measured values of
and :

Remarkably, (6-32) is identical to (4-40) in form It is only necessary to take themeasured values of
and  and insert them into (6-32) to obtain and EquationsSection 5.6, and we have

in the measurement the intensity will not necessarily be zero, only a minimum, as wesee from (6-29b),

Ið
, Þ ¼I0

Next, the linear polarizer is rotated through an angle
until a null intensity

is observed; the setting at which this angle occurs is then measured In theory thiscompletes the measurement In practice, however, one finds that a small adjustment

in phase of the compensator and rotation angle of the linear polarizer are almostalways necessary to obtain a null intensity Substituting the observed angularsettings on the compensator and the polarizer into (6-32) and (6-33), we then findthe Stokes vector (4-38) of the incident beam We note that (4-38) is a normalizedrepresentation of the Stokes vector if I0is set to unity

6.5 FOURIER ANALYSIS USING A ROTATING

QUARTER-WAVE RETARDERAnother method for measuring the Stokes parameters is to allow a beam topropagate through a rotating quarter-wave retarder followed by a linear horizontalpolarizer; the retarder rotates at an angular frequency of ! This arrangement isshown inFig 6-4

The Stokes vector of the incident beam to be measured is

@

1C

The Mueller matrix of the rotated quarter-wave retarder (Section 5.5) is

M ¼

0 cos22
sin 2
cos 2
sin 2

0 sin 2
cos 2
sin22
cos 2

0B

@

1C

and for a rotating retarder we consider
¼ !t Multiplying (6-1) by (5-72) yields

S0¼

S0

S1cos22
þ S2sin 2
cos 2
 S3sin 2

S1sin 2
cos 2
þ S2sin22
þ S3cos 2

S1sin 2
 S2cos 2

0B

@

1C

@

1C

Figure 6-4 Measurement of the Stokes parameters using a rotating quarter-wave retarderand a linear polarizer

The Stokes vector of the beam emerging from the rotating quarter-wave retarder–horizontal polarizer combination is then found from (6-34) and (5-13) to be

0B

@

1C

2
þ S2sin 2
cos 2
 S3sin 2
Þ ð6-36Þ

Equation (6-36) can be rewritten by using the trigonometric half-angle formulas:

Ið
Þ ¼1

2 S0þ

S12

Solving (6-38) for the Stokes parameters gives

B ¼4

N

XN n¼1

C ¼ 4

N

XN n¼1

D ¼ 4

N

XN n¼1

As an example of (6-41), consider the rotation of a quarter-wave retarderthat makes a complete rotation in 16 steps, so N ¼ 16 Then the step size is

j¼2=N ¼ 2=16 ¼ =8 Equation (6-41) is then written as

A ¼1

8

X16 n¼1

I n8

I n8

sin n4

I n8

cos n2

I n8

sin n2

ð6-42dÞ

Thus, the data array consists of 16 measured intensities I1 through I16 Wehave written each intensity value as Iðn=8Þ to indicate that the intensity is measured

at intervals of =8; we observe that when n ¼ 16 we have Ið2Þ as expected

At each step the intensity is stored to form (6-42a), multiplied by sinðn=4Þ toform B, cosðn=2Þ to form C, and sinðn=2Þ to form D The sums are then performedaccording to (6-42), and we obtain A, B, C, and D The Stokes parameters are thenfound from (6-40) using these values

j¼2=N ¼ 2= 16 ¼ =8 Equation (6- 41)... form C, and sinðn=2Þ to form D The sums are then performedaccording to (6- 42), and we obtain A, B, C, and D The Stokes parameters are thenfound from (6- 40) using these values