4 The Stokes Polarization Parameters

Stokes also showed that his parameters could be applied notonly to unpolarized light but to partially polarized and completely polarized light as well.. Chandrasekhar in 1947, who used t

us to follow the tracing of the ellipse This fact, therefore, immediately prevents usfrom ever observing the polarization ellipse Another limitation is that the polariza-tion ellipse is only applicable to describing light that is completely polarized.

It cannot be used to describe either unpolarized light or partially polarizedlight This is a particularly serious limitation because, in nature, light is very oftenunpolarized or partially polarized Thus, the polarization ellipse is an idealization ofthe true behavior of light; it is only correct at any given instant of time Theselimitations force us to consider an alternative description of polarized light inwhich only observed or measured quantities enter We are, therefore, in the samesituation as when we dealt with the wave equation and its solutions, neither ofwhich can be observed We must again turn to using average values of the opticalfield which in the present case requires that we represent polarized light in terms

mathematical description of the Fresnel–Arago interference laws (1818) Theselaws were based on experiments carried out with an unpolarized light source,

a quantity which Fresnel and his successors were never able to characterize matically Stokes succeeded where others had failed because he abandoned theattempts to describe unpolarized light in terms of amplitude He resorted to

mathe-an experimental definition, namely, unpolarized light is light whose intensity isunaffected when a polarizer is rotated or by the presence of a retarder of anyretardance value Stokes also showed that his parameters could be applied notonly to unpolarized light but to partially polarized and completely polarized light

as well Unfortunately, Stokes’ paper was forgotten for nearly a century Its tance was finally brought to the attention of the scientific community by the Nobellaureate S Chandrasekhar in 1947, who used the Stokes parameters to formulate theradiative transfer equations for the scattering of partially polarized light The Stokesparameters have been a prominent part of the optical literature on polarized lightever since

impor-We saw earlier that the amplitude of the optical field cannot be observed.However, the quantity that can be observed is the intensity, which is derived bytaking a time average of the square of the amplitude This suggests that if we take

a time average of the unobserved polarization ellipse we will be led to the observables

of the polarization ellipse When this is done, as we shall show shortly, we obtainfour parameters, which are exactly the Stokes parameters Thus, the Stokes par-ameters are a logical consequence of the wave theory Furthermore, the Stokesparameters give a complete description of any polarization state of light Mostimportant, the Stokes parameters are exactly those quantities that are measured.Aside from this important formulation, however, when the Stokes parameters areused to describe physical phenomena, e.g., the Zeeman effect, one is led to a veryinteresting representation Originally, the Stokes parameters were used only todescribe the measured intensity and polarization state of the optical field But byforming the Stokes parameters in terms of a column matrix, the so-called Stokesvector, we are led to a formulation in which we obtain not only measurables but alsoobservables, which can be seen in a spectroscope As a result, we shall see that theformalism of the Stokes parameters is far more versatile than originally envisionedand possesses a greater usefulness than is commonly known

We consider a pair of plane waves that are orthogonal to each other at a point inspace, conveniently taken to be z ¼ 0, and not necessarily monochromatic, to berepresented by the equations:

where E0x(t) and E0y(t) are the instantaneous amplitudes, ! is the instantaneousangular frequency, and x(t) and y(t) are the instantaneous phase factors At alltimes the amplitudes and phase factors fluctuate slowly compared to the rapidvibrations of the cosinusoids The explicit removal of the term !t between (4-1a)

and (4-1b) yields the familiar polarization ellipse, which is valid, in general, only at

a given instant of time:

Z T

0

Multiplying (4-4a) by 4E20xE20y, we see that

4E20yhE2xðtÞi þ4E20xhE2yðtÞi 8E0xE0yhExðtÞEyðtÞicos 

Substituting (4-6a), (4-6b), and (4-6c) into (4-5) yields

2E20xE20yþ2E20xE20y ð2E0xE0ycos Þ2¼ ð2E0xE0ysin Þ2 ð4-7ÞSince we wish to express the final result in terms of intensity this suggests that

we add and subtract the quantity E0x4 þE0y4 to the left-hand side of (4-7); doing this

leads to perfect squares Upon doing this and grouping terms, we are led to thefollowing equation:

ðE20xþE20yÞ2 ðE20xE20yÞ2 ð2E0xE0ycos Þ2 ¼ ð2E0xE0ysin Þ2 ð4-8Þ

We now write the quantities inside the parentheses as

S2 describes the amount of linear þ45
or 45
polarization, and the parameter

S3describes the amount of right or left circular polarization contained within thebeam; this correspondence will be shown shortly We note that the four Stokesparameters are expressed in terms of intensities, and we again emphasize that theStokes parameters are real quantities

If we now have partially polarized light, then we see that the relations given by(4-9) continue to be valid for very short time intervals, since the amplitudes andphases fluctuate slowly Using Schwarz’s inequality, one can show that for any state

of polarized light the Stokes parameters always satisfy the relation:

ð3-33bÞ

Inspecting (4-9) we see that if we divide (4-9c) by (4-9b), can be expressed in terms

of the Stokes parameters:

tan 2 ¼S2

Similarly, from (3-40) and (3-41) inChapter 3the ellipticity angle was given bysin 2 ¼2E0xE0ysin 

To obtain the Stokes parameters of an optical beam, one must always take atime average of the polarization ellipse However, the time-averaging process can beformally bypassed by representing the (real) optical amplitudes, (4-1a) and (4-1b), interms of complex amplitudes:

As examples of the representation of polarized light in terms of the Stokesparameters, we consider (1) linear horizontal and linear vertical polarized light,(2) linear þ45
and linear 45
polarized light, and (3) right and left circularlypolarized light.

4.2.1 Linear Horizontally Polarized Light (LHP)

For this case E0y¼0 Then, from (4-9) we have

4.2.2 Linear Vertically Polarized Light (LVP)

For this case E0x¼0 From (4-9) we have

4.2.3 Linear Q45 Polarized Light (L Q 45)

The conditions to obtain L þ 45 polarized light are E0x¼E0y¼E0and  ¼ 0
Usingthese conditions and the definition of the Stokes parameters (4-9), we find that

4.2.4 Linear 45 Polarized Light (L 45)

The conditions on the amplitude are the same as for L þ 45 light, but the phasedifference is  ¼ 180
Then, from (4-9) we see that the Stokes parameters are

4.2.5 Right Circularly Polarized Light (RCP)

The conditions to obtain RCP light are E0x¼E0y¼E0and  ¼ 90
From (4-9) theStokes parameters are then

4.2.6 Left Circularly Polarized Light (LCP)

For LCP light the amplitudes are again equal, but the phase shift between theorthogonal, transverse components is  ¼ 90
The Stokes parameters from (4-9)are then

The four Stokes parameters can be arranged in a column matrix and written as

1CC

The column matrix (4-24) is called the Stokes vector Mathematically, it is not avector, but through custom it is called a vector Equation (4-24) should correctly be

called the Stokes column matrix The Stokes vector for elliptically polarized light isthen written from (4-9) as

S ¼

E20xþE20y

E20xE20y2E0xE0ycos 2E0xE0ysin 

0BB

1C

Equation (4-25) is also called the Stokes vector for a plane wave

The Stokes vectors for linearly and circularly polarized light are readily foundfrom (4-25) We now derive these Stokes vectors

4.3.1 Linear Horizontally Polarized Light (LHP)

For this case E0y¼0, and we find from (4-25) that

S ¼ I0

1100

0B

@

1C

where I0¼E20x is the total intensity

4.3.2 Linear Vertically Polarized Light (LVP)

For this case E0x¼0, and we find that (4-25) reduces to

S ¼ I0

1

100

0B

@

1C

where, again, I0is the total intensity

4.3.3 Linear Q45 Polarized Light (L Q 45)

In this case E0x¼E0y¼E0and  ¼ 0, so (4-25) becomes

S ¼ I0

1010

0B

@

1C

where I0¼2E20

4.3.4 Linear 45 Polarized Light (L 45)

Again, E0x¼E0y¼E0, but now  ¼ 180
Then (4-25) becomes

S ¼ I0

10

10

0B

@

1C

and I0¼2E20

4.3.5 Right Circularly Polarized Light (RCP)

In this case E0x¼E0y ¼E0and  ¼ 90
Then (4-25) becomes

S ¼ I0

1001

0B

@

1C

and I0 ¼2E20

4.3.6 Left Circularly Polarized Light (LCP)

Again, we have E0x¼E0y, but now the phase shift  between the orthogonalamplitudes is  ¼  90
Equation (4-25) then reduces to

S ¼ I0

100

1

0B

@

1C

0BB

@

1CC

Equation (4-35) suggests Fig 4-1 From Fig 4-1 we see that

S ¼ I0

1

0

0BB

1C

where I0¼E20is the total intensity Equation (4-36) can also be used to represent theStokes vector for elliptically polarized light, (4-25) Substituting (4-36) into (4-25)gives

S ¼ I0

10BB

1C

1C

Figure 4-1 Resolution of the optical field components

The orientation angle and the ellipticity angle of the polarization ellipse aregiven by (4-33a) and (4-33b) Substituting S1, S2, and S3into (4-39) into (4-33a) and(4-33b) gives

ð4-40aÞð4-40bÞwhich are identical to the relations we found earlier

, (4-39) reduces to

S ¼

10cos sin 

0B

@

1C

Thus, the polarization ellipse is expressed only in terms of the phase shift  betweenthe orthogonal amplitudes The orientation angle is seen to be always 45
Theellipticity angle, (4-40b) however, is

so ¼ /2 The Stokes vector (4-41) expresses that the polarization ellipse is rotated

45
from the horizontal axis and that the polarization state of the light can varyfrom linearly polarized ( ¼ 0, 180
) to circularly polarized ( ¼ 90
, 270
).Another unique polarization state occurs when  ¼ 90
or 270
For thiscondition (4-39) reduces to

S ¼

10



0B

@

1C

We see that we now have a Stokes vector and a polarization ellipse, which dependsHowever, (4-40b) and (4-43) show that the ellipticity angle is now given by

ð4-44Þ

45
and 45
we obtain right and left circularly polarized light Similarly, for

0
and 90
we obtain linear horizontally and vertically polarized light

The Stokes vector can also be expressed in terms of S0, , and To show this

we write (4-33a) and (4-33b) as

In Section 4.2 we found that

Substituting (4-45a) and (4-45b) into (4-10), we find that

0B

@

1C

The Stokes parameters (4-46) are almost identical in form to the well-knownequations relating Cartesian coordinates to spherical coordinates We recall thatthe spherical coordinates r,
, and  are related to the Cartesian coordinates x, y,and z by

on a sphere; the radius of the sphere is taken to be unity The representation

of the polarization state on a sphere was first introduced by Henri Poincare´ in

1892 and is, appropriately, called the Poincare´ sphere However, at that time,Poincare´ introduced the sphere in an entirely different way, namely, by representingthe polarization equations in a complex plane and then projecting the plane on to asphere, a so-called stereographic projection In this way he was led to (4-46) Hedoes not appear to have known that (4-46) were directly related to the Stokesparameters Because the Poincare´ sphere is of historical interest and is still used todescribe the polarization state of light, we shall discuss it in detail later It isespecially useful for describing the change in polarized light when it interacts withpolarizing elements

The discussion in this chapter shows that the Stokes parameters and the Stokesvector can be used to describe an optical beam which is completely polarized

We have, at first sight, only provided an alternative description of completely ized light All of the equations derived here are based on the polarization ellipsegiven in Chapter 3,that is, the amplitude formulation However, we have pointedout that the Stokes parameters can also be used to describe unpolarized andpartially polarized light, quantities which cannot be described within an amplitude

polar-formulation of the optical field In order to extend the Stokes parameters tounpolarized and partially polarized light, we must now consider the classicalmeasurement of the Stokes polarization parameters.

PARAMETERS

The Stokes polarization parameters are immediately useful because, as we shall nowsee, they are directly accessible to measurement This is due to the fact that they are

an intensity formulation of the polarization state of an optical beam In this section

we shall describe the measurement of the Stokes polarization parameters This isdone by allowing an optical beam to pass through two optical elements known as aretarder and a polarizer Specifically, the incident field is described in terms of itscomponents, and the field emerging from the polarizing elements is then used todetermine the intensity of the emerging beam Later, we shall carry out this sameproblem by using a more formal but powerful approach known as the Muellermatrix formalism In the following chapter we shall also see how this measurementmethod enables us to determine the Stokes parameters for unpolarized and partiallypolarized light

We begin by referring to Fig 4-3, which shows an monochromatic opticalbeam incident on a polarizing element called a retarder This polarizing element isthen followed by another polarizing element called a polarizer The components ofthe incident beam are

Figure 4-2 The Poincare´ representation of polarized light on a sphere

In Section 4.2 we saw that the Stokes parameters for a plane wave written in complexnotation could be obtained from

In order to measure the Stokes parameters, the incident field propagatesthrough a phase-shifting element which has the property that the phase of the

x component (Ex) is advanced by =2 and the phase of the y component Ey isretarded by =2, written as =2 The components E0x and E0y emerging from thephase-shifting element component are then

an angle
only the components of E0x and E0y in this direction can be transmittedperfectly; there is complete attenuation at any other angle A polarizing elementwhich behaves in this manner is called a polarizer This behavior is described in

Fig 4-4.The component of E0xalong the transmission axis is E0xcos
Similarly, thecomponent of E0y is E0ysin
The field transmitted along the transmission axis is thesum of these components so the total field E emerging from the polarizer is

Figure 4-3 Measurement of the Stokes polarization parameters

Substituting (4-51) into (4-52), the field emerging from the polarizer is

The intensity of the beam is defined by

Taking the complex conjugate of (4-53) and forming the product in accordance with(4-54), the intensity of the emerging beam is

Ið
, Þ ¼ ExExcos2
þ EyEysin2

þExEyeisin
cos
þ ExEyeisin
cos
ð4-55ÞEquation (4-55) can be rewritten by using the well-known trigonometric half-angleformulas:

first introduced in the optical literature Replacing the terms in (4-57) by thedefinitions of the Stokes parameters given in (4-17), we arrive at

Ið
, Þ ¼1

2½S0þS1cos 2
þ S2cos  sin 2
þ S3sin  sin 2
 ð4-58ÞEquation (4-58) is Stokes’ famous intensity formula for measuring the four Stokesparameters Thus, we see that the Stokes parameters are directly accessible tomeasurement; that is, they are observable quantities

The first three Stokes parameters are measured by removing the retarderð ¼0
Þ and rotating the transmission axis of the polarizer to the angles
¼ 0
,þ45
, and þ90
, respectively The final parameter, S3, is measured by reinserting aso-called quarter-wave retarder ð ¼ 90
Þ into the optical path and settingthe transmission axis of the polarizer to
¼ 45
The intensities are then foundfrom (4-58) to be

Solving (4-59) for the Stokes parameters, we have

Equation (4-60) is really quite remarkable In order to measure theStokes parameters it is necessary to measure the intensity at four angles We mustremember, however, that in 1852 there were no devices to measure the intensityquantitatively The intensities can be measured quantitatively only with an opticaldetector But when Stokes introduced the Stokes parameters, such detectors did notexist The only optical detector was the human eye (retina), a detector capable

of measuring only the null or greater-than-null state of light, and so the abovemethod for measuring the Stokes parameters could not be used! Stokes didnot introduce the Stokes parameters to describe the optical field in terms of observ-ables as is sometimes stated The reason for his derivation of (4-58) was not tomeasure the Stokes polarization parameters but to provide the solution to an entirelydifferent problem, namely, a mathematical statement for unpolarized light We shallsoon see that (4-58) is perfect for doing this It is possible to measure all four Stokesparameters using the human eye, however, by using a null-intensity technique Thismethod is described in Section 6.4

Unfortunately, after Stokes solved this problem and published his great paper

on the Stokes parameters and the nature of polarized light, he never returned to

4. 2 .4 Linear 45  Polarized Light (L 45 )

The conditions on the amplitude are the same... 90
The Stokes parameters from (4- 9)are then

The four Stokes parameters can be arranged in a column matrix and written as

1CC

The column matrix (4- 24) is called the Stokes. .. now have a Stokes vector and a polarization ellipse, which dependsHowever, (4- 40b) and (4- 43) show that the ellipticity angle is now given by

? ?4- 44? ?

45
and 45