4 The Stokes Polarization Parameters 4.1 INTRODUCTION In Chapter we saw that the elimination of the propagator between the transverse components of the optical field led to the polarization ellipse Analysis of the ellipse showed that for special cases it led to forms which can be interpreted as linearly polarized light and circularly polarized light This description of light in terms of the polarization ellipse is very useful because it enables us to describe by means of a single equation various states of polarized light However, this representation is inadequate for several reasons As the beam of light propagates through space, we find that in a plane transverse to the direction of propagation the light vector traces out an ellipse or some special form of an ellipse, such as a circle or a straight line in a time interval of the order 10À15 sec This period of time is clearly too short to allow us to follow the tracing of the ellipse This fact, therefore, immediately prevents us from ever observing the polarization ellipse Another limitation is that the polarization ellipse is only applicable to describing light that is completely polarized It cannot be used to describe either unpolarized light or partially polarized light This is a particularly serious limitation because, in nature, light is very often unpolarized or partially polarized Thus, the polarization ellipse is an idealization of the true behavior of light; it is only correct at any given instant of time These limitations force us to consider an alternative description of polarized light in which only observed or measured quantities enter We are, therefore, in the same situation as when we dealt with the wave equation and its solutions, neither of which can be observed We must again turn to using average values of the optical field which in the present case requires that we represent polarized light in terms of observables In 1852, Sir George Gabriel Stokes (1819–1903) discovered that the polarization behavior could be represented in terms of observables He found that any state of polarized light could be completely described by four measurable quantities now known as the Stokes polarization parameters The first parameter expresses the total intensity of the optical field The remaining three parameters describe the polarization state Stokes was led to his formulation in order to provide a suitable Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved mathematical description of the Fresnel–Arago interference laws (1818) These laws were based on experiments carried out with an unpolarized light source, a quantity which Fresnel and his successors were never able to characterize mathematically Stokes succeeded where others had failed because he abandoned the attempts to describe unpolarized light in terms of amplitude He resorted to an experimental definition, namely, unpolarized light is light whose intensity is unaffected when a polarizer is rotated or by the presence of a retarder of any retardance value Stokes also showed that his parameters could be applied not only to unpolarized light but to partially polarized and completely polarized light as well Unfortunately, Stokes’ paper was forgotten for nearly a century Its importance was finally brought to the attention of the scientific community by the Nobel laureate S Chandrasekhar in 1947, who used the Stokes parameters to formulate the radiative transfer equations for the scattering of partially polarized light The Stokes parameters have been a prominent part of the optical literature on polarized light ever since 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 by taking 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 obtain four parameters, which are exactly the Stokes parameters Thus, the Stokes parameters are a logical consequence of the wave theory Furthermore, the Stokes parameters give a complete description of any polarization state of light Most important, the Stokes parameters are exactly those quantities that are measured Aside from this important formulation, however, when the Stokes parameters are used to describe physical phenomena, e.g., the Zeeman effect, one is led to a very interesting representation Originally, the Stokes parameters were used only to describe the measured intensity and polarization state of the optical field But by forming the Stokes parameters in terms of a column matrix, the so-called Stokes vector, we are led to a formulation in which we obtain not only measurables but also observables, which can be seen in a spectroscope As a result, we shall see that the formalism of the Stokes parameters is far more versatile than originally envisioned and possesses a greater usefulness than is commonly known 4.2 DERIVATION OF THE STOKES POLARIZATION PARAMETERS We consider a pair of plane waves that are orthogonal to each other at a point in space, conveniently taken to be z ¼ 0, and not necessarily monochromatic, to be represented by the equations: Ex ðtÞ ¼ E0x ðtÞ cos½!t þ x ðtފ ð4-1aÞ Ey ðtÞ ¼ E0y ðtÞ cos½!t þ y ðtފ ð4-1bÞ where E0x(t) and E0y(t) are the instantaneous amplitudes, ! is the instantaneous angular frequency, and x(t) and y(t) are the instantaneous phase factors At all times the amplitudes and phase factors fluctuate slowly compared to the rapid vibrations of the cosinusoids The explicit removal of the term !t between (4-1a) Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved and (4-1b) yields the familiar polarization ellipse, which is valid, in general, only at a given instant of time: E2y ðtÞ 2Ex ðtÞEy ðtÞ E2x ðtÞ cos ðtÞ ¼ sin2 ðtÞ þ À E0x ðtÞ E0y ðtÞ E0x ðtÞE0y ðtÞ ð4-2Þ where ðtÞ ¼ y ðtÞ À x ðtÞ For monochromatic radiation, the amplitudes and phases are constant for all time, so (4-2) reduces to E2x ðtÞ Ey ðtÞ 2Ex ðtÞEy ðtÞ þ À cos  ¼ sin2  E0x E0y E0y E20x ð4-3Þ While E0x, E0y, and  are constants, Ex and Ey continue to be implicitly dependent on time, as we see from (4-1a) and (4-1b) Hence, we have written Ex(t) and Ey(t) in (4-3) In order to represent (4-3) in terms of the observables of the optical field, we must take an average over the time of observation Because this is a long period of time relative to the time for a single oscillation, this can be taken to be infinite However, in view of the periodicity of Ex(t) and Ey(t), we need average (4-3) only over a single period of oscillation The time average is represented by the symbol hÁ Á Ái, and so we write (4-3) as       Ey ðtÞ Ex ðtÞEy ðtÞ Ex ðtÞ þ À cos  ¼ sin2  ð4-4aÞ E0x E0y E20x E0y where hEi ðtÞEj ðtÞi ¼ lim T!1 T Z T Ei ðtÞEj ðtÞ dt i, j ¼ x, y ð4-4bÞ Multiplying (4-4a) by 4E20x E20y , we see that 4E20y hE2x ðtÞi þ 4E20x hE2y ðtÞi À 8E0x E0y hEx ðtÞEy ðtÞi cos  ¼ ð2E0x E0y sin Þ2 ð4-5Þ From (4-1a) and (4-1b), we then find that the average values of (4-5) using (4-4b) are hE2x ðtÞi ¼ E20x ð4-6aÞ hE2y ðtÞi ¼ E20y ð4-6bÞ hEx ðtÞEy ðtÞi ¼ E0x E0y cos  ð4-6cÞ Substituting (4-6a), (4-6b), and (4-6c) into (4-5) yields 2E20x E20y þ 2E20x E20y À ð2E0x E0y cos Þ2 ¼ ð2E0x E0y sin Þ2 ð4-7Þ Since we wish to express the final result in terms of intensity this suggests that 4 þ E0y to the left-hand side of (4-7); doing this we add and subtract the quantity E0x Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved leads to perfect squares Upon doing this and grouping terms, we are led to the following equation: ðE20x þ E20y Þ2 À ðE20x À E20y Þ2 À ð2E0x E0y cos Þ2 ¼ ð2E0x E0y sin Þ2 ð4-8Þ We now write the quantities inside the parentheses as S0 ¼ E20x þ E20y ð4-9aÞ S1 ¼ E20x À E20y ð4-9bÞ S2 ¼ 2E0x E0y cos  ð4-9cÞ S3 ¼ 2E0x E0y sin  ð4-9dÞ and then express (4-8) as S20 ¼ S21 þ S22 þ S23 ð4-10Þ The four equations given by (4-9) are the Stokes polarization parameters for a plane wave They were introduced into optics by Sir George Gabriel Stokes in 1852 We see that the Stokes parameters are real quantities, and they are simply the observables of the polarization ellipse and, hence, the optical field The first Stokes parameter S0 is the total intensity of the light The parameter S1 describes the amount of linear horizontal or vertical polarization, the parameter S2 describes the amount of linear þ45 or À45 polarization, and the parameter S3 describes the amount of right or left circular polarization contained within the beam; this correspondence will be shown shortly We note that the four Stokes parameters are expressed in terms of intensities, and we again emphasize that the Stokes 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 and phases fluctuate slowly Using Schwarz’s inequality, one can show that for any state of polarized light the Stokes parameters always satisfy the relation: S20 ! S21 þ S22 þ S23 ð4-11Þ The equality sign applies when we have completely polarized light, and the inequality sign when we have partially polarized light or unpolarized light In Chapter 3, we saw that orientation angle of the polarization ellipse was given by tan ¼ 2E0x E0y cos  E20x À E20y Inspecting (4-9) we see that if we divide (4-9c) by (4-9b), of the Stokes parameters: tan ¼ S2 S1 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð3-33bÞ can be expressed in terms ð4-12Þ Similarly, from (3-40) and (3-41) in Chapter the ellipticity angle  was given by 2E0x E0y sin  sin 2 ¼ ð4-13Þ E20x þ E20y Again, inspecting (4-9) and dividing (4-9d) by (4-9a), we can see that  can be expressed in terms of the Stokes parameters: S sin 2 ¼ ð4-14Þ S0 The Stokes parameters enable us to describe the degree of polarization P for any state of polarization By definition, P¼ Ipol ðS21 þ S22 þ S23 Þ1=2 ¼ Itot S0 P ð4-15Þ where Ipol is the intensity of the sum of the polarization components and Itot is the total intensity of the beam The value of P ¼ corresponds to completely polarized light, P ¼ corresponds to unpolarized light, and < P < corresponds to partially polarized light To obtain the Stokes parameters of an optical beam, one must always take a time average of the polarization ellipse However, the time-averaging process can be formally bypassed by representing the (real) optical amplitudes, (4-1a) and (4-1b), in terms of complex amplitudes: Ex ðtÞ ¼ E0x exp½ið!t þ x ފ ¼ Ex expði!tÞ ð4-16aÞ Ey ðtÞ ¼ E0y exp½ið!t þ y ފ ¼ Ey expði!tÞ ð4-16bÞ where Ex ¼ E0x expðix Þ and Ey ¼ E0y expðiy Þ ð4-16cÞ ð4-16dÞ are complex amplitudes The Stokes parameters for a plane wave are now obtained from the formulas: S0 ¼ Ex Exà þ Ey Eyà ð4-17aÞ Ey Eyà ð4-17bÞ S2 ¼ Ex Eyà þ Ey Exà ð4-17cÞ iðEx Eyà ð4-17dÞ S1 ¼ S3 ¼ Ex Exà À À Ey ExÃ Þ We shall use (4-17), the complex representation, henceforth, as the defining equations for the Stokes parameters Substituting (4-16c) and (4-16d) into (4-17) gives S0 ¼ E20x þ E20y ð4-9aÞ S1 ¼ E20x À E20y ð4-9bÞ S2 ¼ 2E0x E0y cos  ð4-9cÞ S3 ¼ 2E0x E0y sin  ð4-9dÞ which are the Stokes parameters obtained formally from the polarization ellipse Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved As examples of the representation of polarized light in terms of the Stokes parameters, we consider (1) linear horizontal and linear vertical polarized light, (2) linear þ45 and linear À45 polarized light, and (3) right and left circularly polarized light 4.2.1 Linear Horizontally Polarized Light (LHP) For this case E0y ¼ Then, from (4-9) we have S0 ¼ E20x ð4-18aÞ E20x ð4-18bÞ S1 ¼ 4.2.2 S2 ¼ ð4-18cÞ S3 ¼ ð4-18dÞ Linear Vertically Polarized Light (LVP) For this case E0x ¼ From (4-9) we have 4.2.3 S0 ¼ E20y ð4-19aÞ S1 ¼ ÀE20y ð4-19bÞ S2 ¼ ð4-19cÞ S3 ¼ ð4-19dÞ Linear Q45 Polarized Light (L Q 45) The conditions to obtain L þ 45 polarized light are E0x ¼ E0y ¼ E0 and  ¼ 0 Using these conditions and the definition of the Stokes parameters (4-9), we find that S0 ¼ 2E20 ð4-20aÞ S1 ¼ ð4-20bÞ S2 ¼ 2E20 S3 ¼ 4.2.4 ð4-20cÞ ð4-20dÞ Linear À45 Polarized Light (L À 45) The conditions on the amplitude are the same as for L þ 45 light, but the phase difference is  ¼ 180 Then, from (4-9) we see that the Stokes parameters are S0 ¼ 2E20 ð4-21aÞ S1 ¼ ð4-21bÞ S2 ¼ À2E20 ð4-21cÞ S3 ¼ ð4-21dÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 4.2.5 Right Circularly Polarized Light (RCP) The conditions to obtain RCP light are E0x ¼ E0y ¼ E0 and  ¼ 90 From (4-9) the Stokes parameters are then 4.2.6 S0 ¼ 2E20 ð4-22aÞ S1 ¼ ð4-22bÞ S2 ¼ ð4-22cÞ S3 ¼ 2E20 ð4-22dÞ Left Circularly Polarized Light (LCP) For LCP light the amplitudes are again equal, but the phase shift between the orthogonal, transverse components is  ¼ À90 The Stokes parameters from (4-9) are then S0 ¼ 2E20 ð4-23aÞ S1 ¼ ð4-23bÞ S2 ¼ ð4-23cÞ S3 ¼ À2E20 ð4-23dÞ Finally, the Stokes parameters for elliptically polarized light are, of course, given by (4–9) Inspection of the four Stokes parameters suggests that they can be arranged in the form of a column matrix This column matrix is called the Stokes vector This step, while simple, provides a formal method for treating numerous complicated problems involving polarized light We now discuss the Stokes vector 4.3 THE STOKES VECTOR The four Stokes parameters can be arranged in a column matrix and written as S0 B C B S1 C B C S¼B C B S2 C @ A ð4-24Þ S3 The column matrix (4-24) is called the Stokes vector Mathematically, it is not a vector, but through custom it is called a vector Equation (4-24) should correctly be Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved called the Stokes column matrix The Stokes vector for elliptically polarized light is then written from (4-9) as E20x þ E20y B C 2 B E0x À E0y C ð4-25Þ S¼B C @ 2E0x E0y cos  A 2E0x E0y sin  Equation (4-25) is also called the Stokes vector for a plane wave The Stokes vectors for linearly and circularly polarized light are readily found from (4-25) We now derive these Stokes vectors 4.3.1 Linear Horizontally Polarized Light (LHP) For this case E0y ¼ 0, and we find from (4-25) that 1 B1C C S ¼ I0 B @0A ð4-26Þ where I0 ¼ E20x is the total intensity 4.3.2 Linear Vertically Polarized Light (LVP) For this case E0x ¼ 0, and we find that (4-25) reduces to 1 B À1 C C S ¼ I0 B @ 0A ð4-27Þ where, again, I0 is the total intensity 4.3.3 Linear Q45 Polarized Light (L Q 45) In this case E0x ¼ E0y ¼ E0 and  ¼ 0, so (4-25) becomes 1 B0C C S ¼ I0 B @1A ð4-28Þ where I0 ¼ 2E20 4.3.4 Linear À45 Polarized Light (L À 45) Again, E0x ¼ E0y ¼ E0, but now  ¼ 180 Then (4-25) becomes 1 B 0C C S ¼ I0 B @ À1 A and I0 ¼ 2E20 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð4-29Þ 4.3.5 Right Circularly Polarized Light (RCP) In this case E0x ¼ E0y ¼ E0 and  ¼ 90 Then (4-25) becomes 1 B0C C S ¼ I0 B @0A ð4-30Þ and I0 ¼ 2E20 4.3.6 Left Circularly Polarized Light (LCP) Again, we have E0x ¼ E0y, but now the phase shift  between the orthogonal amplitudes is  ¼ À 90 Equation (4-25) then reduces to 1 B 0C C S ¼ I0 B ð4-31Þ @ 0A À1 and I0 ¼ 2E20 We also see from (4-25) that if  ¼ 0 or 180 , then (4-25) reduces to E20x þ E20y B C B E0x À E20y C C S¼B B C @ Æ2E0x E0y A ð4-32Þ We recall that the ellipticity angle  and the orientation angle ellipse are given, respectively, by sin 2 ¼ S3 S0 À S2 S1 tan ¼   partial polarization ð4-125bÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved One can readily determine the coherency matrices for various states of polarized light, using (4-121) We easily find for unpolarized light that   S ð4-126aÞ J¼ for linearly horizontally polarized light   J ¼ S0 0 and for right circularly polarized light   S Ài J¼ i The degree of polarization is readily found to be sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi det J P¼ 1À ðTr J Þ2 ð4-126bÞ ð4-126cÞ ð4-127Þ where Tr J is the trace of the matrix J and is defined as the sum of the diagonal elements; that is Tr J ¼ Jxx þ Jyy ð4-128Þ The coherency matrix elements can also be introduced by considering the measurement of the polarization state of an optical beam We recall that the intensity of a beam emerging from a retarder/polarizer combination is Ið, Þ ¼ Ex Exà cos2  þ Ey Eyà sin2  þ Exà Ey eÀi sin  cos  þ Ex Eyà ei sin  cos  ð4-55Þ The Stokes parameters were then found by expressing the sinusoidal terms in terms of the half-angle trigonometric formulas If we had a quasi-monochromatic wave, then we could time-average the quadratic field terms and express (4-55) as Ið, Þ ¼ hEx Exà i cos2  þ hEy Eyà i sin2  þ hExà Ey ieÀi sin  cos  þ hEx Eyà iei sin  cos  ð4-129aÞ or Ið, Þ ¼ Jxx cos2  þ Jyy sin2  þ Jxy eÀi sin  cos  þ Jyx ei sin  cos  ð4-129bÞ where the Jij are defined to be Jij ¼ hEi Ejà i which are the coherency matrix elements Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð4-129cÞ Finally, there is a remarkable relation between the Stokes parameters and the coherency matrix We first note from (4-121) that Jxx ¼ S0 þ S1 ð4-130aÞ Jyy ¼ S0 À S1 ð4-130bÞ Jxy ¼ S2 À iS3 ð4-130cÞ Jyx ¼ S2 þ iS3 ð4-130dÞ so we can express (4-130) in matrix form as !   Jxx Jxy S0 þ S1 S2 À iS3 J¼ ¼ S2 þ iS3 S0 À S1 Jyx Jyy ð4-131Þ One can easily decompose (4-131) into  matrices such that J¼ 1X S i¼0 i i ð4-132aÞ where  0 ¼  1 ¼  2 ¼  3 ¼  0 1 0 i Ài 0 À1 ð4-132bÞ  ð4-132cÞ  ð4-132dÞ  ð4-132eÞ The remarkable fact about this decomposition is that 1 , 2 , the 3 are the three Pauli spin matrices of quantum mechanics with the addition of the identity matrix,  This connection between the coherency matrix, the Stokes parameters, and the Pauli spin matrices appears to have been first pointed out by U Fano in 1954 What is even more surprising about the appearance of the Pauli spin matrices is that they were introduced into quantum mechanics by Pauli in order to describe the behavior of the spin of the electron, a particle Indeed, in quantum mechanics the wave function that describes a pure state of polarization can be expanded in a complete set of orthonormal eigenfunctions; it has the same form for electromagnetic radiation and particles of spin 1/2 (the electron) The coherency matrix is treated in an elegant manner by Born and Wolf and the reader is referred to their text for further information on this subject Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved REFERENCES Papers Stokes, G G., Trans Camb Phil Soc., 9, 399 (1852) Reprinted in Mathematical and Physical Papers, Vol 3, Cambridge University Press, London, 1901, p 233 Wolf, E., Nuovo Cimento, 12, 884 (1954) Walker, M J., Am J Phys., 22, 170 (1954) McMaster, W H., Rev Mod Phys., 33, (1961) Collett, E., Am J Phys., 36, 713 (1968) Fano, U., Phys Rev., 93, 121 (1954) Wolf, E., Nuovo Cimento, 13, 1165 (1959) Books Born, M and Wolf, E., Principles of Optics, 3rd ed., Pergamon Press, New York, 1965 Shurcliff, W., Polarized Light, Harvard University Press, Cambridge, MA, 1962 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved [...]... (4- 113) Not surprisingly there are other interesting relations between the Stokes parameters and the parameters of the polarization ellipse These relations are fundamental to the development of the Poincare´ sphere, so we shall discuss them in Chapter 12 4. 7 STOKES PARAMETERS AND WOLF’S COHERENCY MATRIX We have demonstrated that the state of polarization is specified completely by the four Stokes parameters. .. Rights Reserved Figure 4- 2 The Poincare´ representation of polarized light on a sphere formulation of the optical field In order to extend the Stokes parameters to unpolarized and partially polarized light, we must now consider the classical measurement of the Stokes polarization parameters 4. 4 CLASSICAL MEASUREMENT OF THE STOKES POLARIZATION PARAMETERS The Stokes polarization parameters are immediately... Expressing (4- 39) in this manner shows that there are two unique polarization states For  ¼ 45  , (4- 39) reduces to 0 1 1 B 0 C C S¼B 4- 41Þ @ cos  A sin  Thus, the polarization ellipse is expressed only in terms of the phase shift  between the orthogonal amplitudes The orientation angle is seen to be always 45  The ellipticity angle, (4- 40b) however, is sin 2 ¼ sin  4- 42Þ so  ¼ /2 The Stokes. .. 4- 112Þ We can now express (4- 112) in terms of the Stokes parameters from (4- 97b), (4- 97c) and (4- 97d) and we find that (4- 112) becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S0 À S21 þ S22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R¼ 4- 113Þ S0 þ S21 þ S22 Thus, we have found the relation between the length of the major and minor axes of the polarization ellipse and the Stokes parameters This can be expressed directly by using (4- 110) and (4- 97)... coordinates x, y, and z by x ¼ r sin  cos  4- 48aÞ y ¼ r sin  sin  4- 48bÞ z ¼ r cos  4- 48cÞ Comparing (4- 48) with (4- 46), we see that the equations are identical if the angles are related by  ¼ 90 À 2 4- 49aÞ ¼2 4- 49bÞ In Fig 4- 2 we have drawn a sphere whose center is also at the center of the Cartesian coordinate system We see that expressing the polarization state of an optical beam in terms... 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 S3 ¼ S0 sin 2 4- 45aÞ S2 ¼ S1 tan 2 4- 45bÞ In Section 4. 2 we found that S20 ¼ S21 þ S22 þ S23 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 4- 10Þ Substituting (4- 45a) and (4- 45b) into (4- 10), we find that S1 ¼ S0 cos 2 cos 2 4- 46aÞ S2 ¼ S0 cos 2 sin 2 4- 46bÞ S3 ¼... whereas the measurement was made in terms of intensities, the Stokes parameters In other words, there were two distinct ways to describe the polarization of the optical field We have seen, however, that the Stokes parameters are actually a consequence of the wave theory and arise naturally from the polarization ellipse It is only necessary to transform the nonobservable polarization ellipse to the observed... ð ¼ 90 Þ into the optical path and setting the transmission axis of the polarizer to  ¼ 45  The intensities are then found from (4- 58) to be 1 Ið0 , 0 Þ ¼ ½S0 þ S1 Š 2 1 I 45  , 0 Þ ¼ ½S0 þ S2 Š 2 1 Ið90 , 0 Þ ¼ ½S0 À S1 Š 2 1 I 45  , 90 Þ ¼ ½S0 þ S3 Š 2 4- 59aÞ 4- 59bÞ 4- 59cÞ 4- 59dÞ Solving (4- 59) for the Stokes parameters, we have S0 ¼ Ið0 , 0 Þ þ Ið90 , 0 Þ    4- 60aÞ  S1 ¼ Ið0... Substituting (4- 51) into (4- 52), the field emerging from the polarizer is E ¼ Ex ei=2 cos  þ Ey eÀi=2 sin  4- 53Þ The intensity of the beam is defined by I ¼ E Á Eà 4- 54 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 Ex Exà cos2  þ Ey Eyà sin2  þ Exà Ey eÀi sin  cos  þ Ex Eyà ei sin  cos  4- 55Þ Equation (4- 55)... rather than the amplitude formulation enables us to deal directly with the quantities measured in an optical experiment Thus, we carry out the analysis in the amplitude domain and then transform the amplitude results to the Stokes parameters, using the defining equations When this is done, we can easily relate the experimental results to the theoretical results Furthermore, when we obtain the Stokes parameters,