7 The Measurement of theCharacteristics of Polarizing Elements

We now turn our attention to measuring the characteristics of the three major optical polarizing elements, namely, the polarizer diattenuator, retarder, and rotator.. The intensity I0of

The Measurement of the

Characteristics of Polarizing Elements

In the previous chapter we described a number of methods for measuring and characterizing polarized light in terms of the Stokes polarization parameters

We now turn our attention to measuring the characteristics of the three major optical polarizing elements, namely, the polarizer (diattenuator), retarder, and rotator For a polarizer it is necessary to measure the attenuation coefficients of the ortho-gonal axes, for a retarder the relative phase shift, and for a rotator the angle of rotation It is of practical importance to make these measurements Before proceed-ing with any experiment in which polarizproceed-ing elements are to be used, it is good practice to determine if they are performing according to their specifications This characterization is also necessary because over time polarizing components change: e.g., the optical coatings deteriorate, and in the case of Polaroid the material becomes discolored In addition, one finds that, in spite of one’s best laboratory controls, quarter-wave and half-wave retarders, which operate at different wave-lengths, become mixed up Finally, the quality control of manufacturers of polariz-ing components is not perfect, and imperfect components are sold

The characteristics of all three types of polarizing elements can be determined

by using a pair of high-quality calcite polarizers that are placed in high-resolution angular mounts; the polarizing element being tested is placed between these two polarizers A practical angular resolution is 0.1
(60 of arc) or less High-quality calcite polarizers and mounts are expensive, but in a laboratory where polarizing components are used continually their cost is well justified

A POLARIZER (DIATTENUATOR)

A linear polarizer is characterized by its attenuation coefficients px and py along its orthogonal x and y axes We now describe the experimental procedure for

measuring these coefficients The measurement configuration is shown in Fig 7-1.

In the experiment the polarizer to be tested is inserted between the two polarizers

as shown The reason for using two polarizers is that the same configuration can also

be used to test retarders and rotators Thus, we can have a single, permanent, test configuration for measuring all three types of polarizing components

The Mueller matrix of a polarizer (diattenuator) with its axes along the x and y directions is

Mp¼1 2

p2xþp2 p2xp2 0 0

p2xp2 p2xþp2 0 0

0 B B

@

1 C C

It is convenient to rewrite (7-1) as

Mp¼

0 B

@

1 C

where

A ¼1

2ðp

2

B ¼1

2ðp

2

C ¼1

In practice, while we are interested only in determining p2x and p2y, it is useful

to measure pxpyas well, because a polarizer satisfies the relation:

Figure 7-1 Experimental configuration to measure the attenuation coefficients px and py

of a polarizer (diattenuator)

as the reader can easily show from (7-2) Equation (7-3) serves as a useful check

on the measurements The optical source emits a beam characterized by a Stokes vector

S ¼

S0

S1

S2

S3

0 B

@

1 C

In the measurement the first polarizer, which is often called the generating polarizer, is set to þ 45
The Stokes vector of the beam emerging from the generating polarizer is then

S ¼ I0

1 0 1 0

0 B

@

1 C

where I0¼(1/2)(S0þS2) is the intensity of the emerging beam The Stokes vector

of the beam emerging from the test polarizer is found to be, after multiplying (7-2a) and (7-5),

S0¼I0

A B C 0

0 B

@

1 C

The polarizer before the optical detector is often called the analyzing polarizer

or simply the analyzer The analyzer is mounted so that it can be rotated to an

Chap 5)

MA¼1

2

0 B B

1 C

The Stokes vector of the beam incident on the optical detector is then seen from multiplying (7-6) by (7-7) to be

S0¼I0

2ðA þ B

1

0

0 B

@

1 C

and the intensity of the beam is

I0

First method:

, 45
, and 90
, (7-9) yields the following equations: Ið0
Þ ¼I0

Ið45
Þ ¼I0

Ið90
Þ ¼I0

Solving for A, B, and C, we then find that

A ¼Ið0

Þ þIð90
Þ

B ¼Ið0

Þ Ið90
Þ

C ¼2Ið45

Þ Ið0
Þ Ið90
Þ

which are the desired relations From (7-2) we also see that

so that we can write (7-10a) and (7-10c) as

p2x¼2Ið0
Þ

p2y¼2Ið90
Þ

Thus, it is only necessary to measure I(0
) and I(90
), the intensities in the x and y directions, respectively, to obtain p2xand p2y The intensity I0of the beam emerging from the generating polarizer is measured without the polarizer under test and the analyzer in the optical train

It is not necessary to measure C Nevertheless, experience shows that the additional measurement of I(45
) enables one to use (7-3) as a check on the measurements

In order to determine p2xand p2 in (7-13) it is necessary to know I0 However,

a relative measurement of p2y=p2xis just as useful We divide (7-12b) by (7-12a) and

we obtain

p2y

p2x¼

Ið90
Þ

We see that this type of measurement does not require a knowledge of I0 Thus, measuring I(0
) and I(90
) and forming the ratio yields the relative value of the absorption coefficients of the polarizer

In order to obtain A, B, and C and then px and py in the method described above, an optical detector is required However, the magnitude of p2x and p2y can also be obtained using a null-intensity method To show this we write (7-3) again

This suggests that we can write

Substituting (7-15a) and (7-15b) into (7-9), we then have

I0A

and

tan  ¼C

where (7-16b) has been obtained by dividing (7-15a) by (7-15b)

null¼90
þ

nullis the angle at which the null is observed Substituting (7-17) into (7-16b) then yields

C

Thus by measuring  from the null-intensity condition, we can find B/A and C/A from (7-15a) and (7-15b), respectively For convenience we set A ¼ 1 Then we see from (7-12) that

The ratio C/B in (7-18) can also be used to determine the ratio py/px, which

we can then square to form p2y=p2x From (7-2)

B ¼1

2ðp

2

C ¼1

Substituting (7-2b) and (7-2c) into (7-18) gives

null¼ 2pxpy

p2

The form of (7-20) suggests that we set

so

null ¼sin 2

and

This leads immediately to

py

or, using (7-17)

p2y

p2 x

¼cot2  2

ð7-22bÞ

Thus, the shift in the intensity, (7-16a) enables us to determine p2=p2xdirectly from 

We always assume that p2y=p2x 1 A neutral density filter is described by p2x¼p2y

so the range on p2y=p2x limits  to

For p2y=p2x¼0, an ideal polarizer,  ¼ 180
, whereas for p2y=p2x¼1, a neutral density filter  ¼ 90
as shown by (7-22b) We see that the closer the value of  is to

180
, the better is the polarizer As an example, for commercial Polaroid HN22 at 0.550 mm p2=p2x¼2  106=0:48 ¼ 4:2  106 so from (7-22b) we see that  ¼ 179.77
null¼179.88
, respectively; the nearness of  to 180
shows that it is

an excellent polarizing material

Second method:

The parameters A, B, and C can also be obtained by Fourier-analyzing (7-9), assuming that the analyzing polarizer can be continuously rotated over a half or full cycle Recall that Eq (7-9) is

I0

From the point of view of Fourier analysis A describes a d.c term, and B and C describe second-harmonic terms It is only necessary to integrate over half a cycle, that is, from 0
to , in order to determine A, B, and C We easily find that

A ¼ 2

I0

Z 0

ð7-23aÞ

B ¼ 4

I0

Z  0

ð7-23bÞ

C ¼ 4

I0

Z 0

ð7-23cÞ

Throughout this analysis we have assumed that the axes of the polarizer being measured lie along the x and y directions If this is not the case, then the polarizer under test should be rotated to its x and y axes in order to make the measurement The simplest way to determine rotation angle
is to remove the polarizer under test and rotate the generating polarizer to 0
and the analyzing polarizer to 90

Third method:

Finally, another method to determine A, B, and C is to place the test polarizer in

a rotatable mount between polarizers in which the axes of both are in the y direction The test polarizer is then rotated until a minimum intensity is observed from which A, B, and C can be found The Stokes vector emerging from the

ygenerating polarizer is

S ¼I0

2

1

1 0 0

0 B B B

1 C C

The Mueller matrix of the rotated test polarizer (7-2a) is

M ¼

Bcos 2
Acos22
þ C sin22
ðA  CÞsin 2
cos 2
0

Bsin 2
ðA  CÞsin 2
cos 2
Asin22
þ C cos22
0

0 B B B

1 C C

The intensity of the beam emerging from the y analyzing polarizer is

Ið
Þ ¼I0

4½ðA þ CÞ 2B cos 2
þ ðA  CÞ cos

2

Equation (7-26) can be solved for its maximum and minimum values by differentiat-ing I(
) with respect to
and settdifferentiat-ing dI(
)/d
¼ 0 We then find

The solutions of (7-27) are

and

cos 2
¼ B

For (7-28a) we have
¼ 0 and 90 The corresponding values of the intensities are then, from (7-26)

Ið0
Þ ¼I0

Ið90
Þ ¼I0

The second solution (7-28b), on substitution into (7-26), leads to I(
) ¼ 0 Thus, the minimum intensity is given by (7-29a) and the maximum intensity by (7-29b) Because both the generating and analyzing polarizers are in the y direction, this is exactly what one would expect We also note in passing that at
¼ 45
, (7-26) reduces to

Ið45
Þ ¼I0

We can again divide (7-29) through by I0and then solve (7-29) for A, B, and C

We see that several methods can be used to determine the absorption coefficients of the orthogonal axes of a polarizer In the first method we generate

a linear þ45
polarized beam and then rotate the analyzer to obtain A, B, and C

of the polarizer being tested This method requires a quantitative optical detector However, if an optical detector is not available, it is still possible to determine A, B, and C by using the null-intensity method; rotating the analyzer until a null is observed leads to A, B, and C On the other hand, if the analyzer can be mounted

in a rotatable mount, which can be stepped (electronically), then a Fourier analysis

of the signal can be made and we can again find A, B, and C Finally, if the transmission axes of the generating and analyzing polarizers are parallel to one another, conveniently chosen to be in the y direction, and the test polarizer is rotated, then we can also determine A, B, and C by rotating the test polarizer

to 0
, 45
, and 90

There are numerous occasions when it is important to know the phase shift of

a retarder The most common types of retarders are quarter-wave and half-wave retarders These two types are most often used to create circularly polarized light and to rotate or reverse the polarization ellipse, respectively

Two methods can be used for measuring the phase shift using two linear polarizers following the experimental configuration given in the previous section First method:

In the first method a retarder is placed between the two linear polarizers mounted in the ‘‘crossed’’ position Let us set the transmission axes of the first and second polarizers to be in the x and y directions, respectively By rotating the retarder, the direction (angle) of the fast axis is rotated and, as we shall soon see, the phase can be found The second method is very similar to the first except that the fast axis

of the retarder is rotated to 45
In this position the phase can also be found We now consider both methods

For the first method we refer to Fig 7-2 It is understood that the correct wavelength must be used; that is, if the retarder is specified for, say 6328 A˚, then the optical source should emit this wavelength In the visible domain calcite polarizers are, as usual, best However, high-quality Polaroid is also satisfactory, but its optical bandpass is much more restricted In Fig 7-2 the transmission axes of the polarizers (or diattenuators) are in the x (horizontal) and y (vertical) directions, respectively The Mueller matrix for the retarder rotated through an angle
is

Mð,
Þ

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

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

0 B B

1 C C

ð7-30Þ where the phase shift  is to be determined The Mueller matrix for an ideal linear polarizer is

Mx, y¼1

2

0 B B

1 C

where the plus sign corresponds to a horizontal polarizer and the minus sign to a vertical polarizer The Mueller matrix for Fig 7-2 is then

Carrying out the matrix multiplication in (7-32) using (7-30) and (7-31) then yields

M ¼ð1  cos Þð1  cos 4
Þ

8

0 B B

1 C

Figure 7-2 Closed polarizer method to measure the phase of a retarder

Equation (7-33) shows that the polarizing train behaves as a pseudopolarizer The intensity of the optical beam on the detector is then

Ið
, Þ ¼ I0ð1  cos Þð1  cos 4
Þ

where I0is the intensity of the optical source

Equation (7-34) immediately allows us to determine the direction of the fast axis of the retarder When the retarder is inserted between the crossed polarizers, the intensity on the detector should be zero, according to (7-34), at
¼ 0
If it

is not zero, the retarder should be rotated until a null intensity is observed After this angle has been found, the retarder is rotated 45
according to (7-34) to obtain the maximum intensity In order to determine , it is necessary to know I0 The easiest way to do this is to rotate the x polarizer (the first polarizer) to the y position and remove the retarder; both linear polarizers are then in the y direction The intensity

ID on the detector is then (let us assume that unpolarized light enters the first polarizer)

ID¼I0

so (7-34) can be written as

Ið
, Þ ¼ IDð1  cos Þð1  cos 4
Þ

The retarder is now reinserted into the polarizing train The maximum intensity, Ið
, Þ, takes place when the retarder is rotated to
¼ 45
At this angle (7-36) is solved for , and we have

 ¼cos1 1 Ið45

, Þ

ID

ð7-37Þ

The disadvantage of using the crossed-polarizer method is that it requires that

we know the intensity of the beam, I0, entering the polarizing train This problem can be overcome by another method, namely, rotating the analyzing polarizer and fixing the retarder at 45
We now consider this second method

Second method:

The experimental configuration is identical to the first method except that the from the generating polarizer is (again let us assume that unpolarized light enters the generating polarizer)

S ¼I0 2

1 1 0 0

0 B

@

1 C

Multiplication of (7-38) by (7-30) yields

S0¼I0

2

1 cos22
þ cos  sin22

ð1  cos Þ sin 2
cos 2

sin  sin 2

0 B B

@

1 C C

We assume that the fast axis of the retarder is at
¼ 0
If it is not, the retarder should be adjusted to
¼ 0
by using the crossed-polarizer method described in the first method; we note that at
¼ 0
, (7-39) reduces to

S0¼I0

2

1 1 0 0

0 B B

@

1 C C

so that the analyzing polarizer should give a null intensity when it is in the y direction Assuming that the retarder’s fast axis is now properly adjusted, we rotate the retarder counterclockwise to
¼ 45
Then (7-39) reduces to

S0¼I0

2

1 cos  0 sin 

0 B B

@

1 C C

This is a Stokes vector for elliptically polarized light The conditions  ¼ 90
and 180
correspond to right circularly polarized and linear vertically polarized light, respectively We note that the linear vertically polarized state arises because for  ¼ 180
the retarder behaves as a pseudorotator The Mueller matrix of the analyzing polarizer is

MðÞ ¼1

2

0 B B

@

1 C C

The Stokes vector of the beam emerging from the analyzer is then

S ¼I0

4ð

1

0

0 B B

@

1 C C

so the intensity is

, Þ ¼I0

Equation (7- 34) immediately allows us to determine the direction of the fast axis of the retarder When the retarder is inserted between the crossed...

ID

? ?7- 37? ?

The disadvantage of using the crossed-polarizer method is that it requires that

we know the intensity of the beam, I0, entering the polarizing train... rotate the x polarizer (the first polarizer) to the y position and remove the retarder; both linear polarizers are then in the y direction The intensity

ID on the detector is then