10 The Mueller Matrices for Dielectric Plates 10.1 INTRODUCTION In Chapter 8, Fresnel’s equations for reflection and transmission of waves at an air– dielectric interface were cast in the form of Mueller matrices In this chapter we use these results to derive the Mueller matrices for dielectric plates The study of dielectric plates is important because all materials of any practical importance are of finite thickness and so at least have upper and lower surfaces Furthermore, dielectric plates always change the polarization state of a beam that is reflected or transmitted One of their most important applications is to create linearly polarized light from unpolarized light in the infrared region While linearly polarized light can be created in the visible and near-infrared regions using calcite polarizers or Polaroid, there are no corresponding materials in the far-infrared region However, materials such as germanium and silicon, as well as others, transmit very well in the infrared region By making thin plates of these materials and then constructing a ‘‘pile of plates,’’ it is possible to create light in the infrared that is highly polarized This arrangement therefore requires that the Mueller matrices for transmission play a more prominent role than the Mueller matrices for reflection In order to use the Mueller matrices to characterize a single plate or multiple plates, we must carry out matrix multiplications The presence of off-diagonal terms of the Mueller matrices create a considerable amount of work We know, on the other hand, that if we use diagonal matrices the calculations are simplified; the product of diagonalized matrices leads to another diagonal matrix 10.2 THE DIAGONAL MUELLER MATRIX AND THE ABCD POLARIZATION MATRIX When we apply the Mueller matrices to problems in which there are several polarizing elements, each of which is described by its own Mueller matrix, we soon discover that the appearance of the off-diagonal elements complicates the matrix Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved multiplications The multiplications would be greatly simplified if we were to use diagonalized forms of the Mueller matrices In particular, the use of diagonalized matrices enables us to determine more easily the Mueller matrix raised to the mth power, Mm, an important problem when we must determine the transmission of a polarized beam through m dielectric plates In this chapter we develop the diagonal Mueller matrices for a polarizer and a retarder To reduce the amount of calculations, it is simpler to write a single matrix that simultaneously describes the behavior of a polarizer or a retarder or a combination of both This simplified matrix is called the ABCD polarization matrix The Mueller matrix for a polarizer is ps þ p2p p2s À p2p 0 C B 2 2 0 C 1B C B ps À pp ps þ pp ð10-1Þ MP ¼ B C 2B 0 2ps pp C A @ and the Mueller B B0 B MC ¼ B B0 @ 0 2ps pp matrix for a phase shifter is 0 C 0 C C C cos  sin  C A 0 À sin  ð10-2Þ cos  where ps and pp are the absorption coefficients of the polarizer along the s (or x) and p (or y) axes, respectively, and  is the phase shift of the retarder The form of (10-1) and (10-2) suggests that the matrices can be represented by a single matrix of the form: A B 0 C B BB A 0C C B ð10-3Þ È¼B C C B0 C D A @ 0 ÀD C which we call the ABCD polarization matrix We see that for a polarizer: A ¼ ðp2s þ p2p Þ ð10-4aÞ B ¼ ðp2s À p2p Þ ð10-4bÞ C ¼ ð2ps pp Þ ð10-4cÞ D¼0 ð10-4dÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved and for the retarder A¼1 ð10-5aÞ B¼0 ð10-5bÞ C ¼ cos  ð10-5cÞ D ¼ sin  ð10-5dÞ If we multiply (10-1) by (10-2), we see that we still obtain a matrix which can be represented by an ABCD matrix; the matrix describes an absorbing retarder The matrix elements ABCD are not all independent; that is, there is a unique relationship between the elements To find this relationship, we see that (10-3) transforms the Stokes parameters of an incident beam Si to the Stokes parameters of an emerging beam S0i so that we have S00 ¼ AS0 þ BS1 ð10-6aÞ S01 ¼ BS0 þ AS1 ð10-6bÞ S02 ¼ CS2 þ DS3 ð10-6cÞ S03 ¼ ÀDS2 þ CS3 ð10-6dÞ We know that for completely polarized light the Stokes parameters of the incident beam are related by S20 ¼ S21 þ S22 þ S23 ð10-7Þ and, similarly, 02 02 02 S02 ¼ S1 þ S2 þ S3 ð10-8Þ Substituting (10-6) into (10-8) leads to ðA2 À B2 ÞðS20 À S21 Þ ¼ ðC2 þ D2 ÞðS22 þ S23 Þ ð10-9Þ But, from (10-7), S20 À S21 ¼ S22 þ S23 ð10-10Þ Substituting (10-10) into the right side of (10-9) gives ðA2 À B2 À C2 À D2 ÞðS20 À S21 Þ ¼ ð10-11Þ A2 ¼ B þ C2 þ D2 ð10-12Þ and We see that the elements of (10-4) and (10-5) satisfy (10-12) This is a very useful relation because it serves as a check when measuring the Mueller matrix elements The rotation of a polarizing device described by the ABCD matrix is given by the matrix equation: M ¼ MðÀ2ÞÈMð2Þ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð10-13Þ which in its expanded form is A B cos 2 B 2 B B cos 2 A cos 2 þ C sin2 2 M¼B B B sin 2 ðA À CÞ sin 2 cos 2 @ D sin 2 B sin 2 ðA À CÞ sin 2 cos 2 A sin2 2 þ C cos2 2 ÀD cos 2 C ÀD sin 2 C C D cos 2 C A C ð10-14Þ In carrying out the expansion of (10-13), we used B0 Mð2Þ ¼ B @0 0 cos 2 À sin 2 0 sin 2 cos 2 0C C 0A ð10-15Þ We now find the diagonalized form of the ABCD matrix This can be done using the well-known methods in matrix algebra We first express (10-3) as an eigenvalue/ eigenvector equation, namely, ÈS ¼ S ð10-16aÞ ðÈ À ÞS ¼ ð10-16bÞ or where  and S are the eigenvalues and the eigenvectors corresponding to È In order to find the eigenvalues and the eigenvectors, the determinant of (10-3) must be taken; that is,   A À  B 0    B AÀ 0   ð10-17Þ ¼0  0 C À  D     0 ÀD C À  The determinant is easily expanded and leads to an equation called the secular equation: ½ðA À Þ2 À B2 Š½ðC À Þ2 þ D2 Š ¼ ð10-18Þ The solution of (10-18) yields the eigenvalues: 1 ¼ A þ B ð10-19aÞ 2 ¼ A À B ð10-19bÞ 3 ¼ C þ iD ð10-19cÞ 4 ¼ C À iD ð10-19dÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Substituting these eigenvalues into (10-17), we easily find that the eigenvector corresponding to each of the respective eigenvalues in (10-19) is 0 1 1 0 B À1 C B0C B C C B 1 1 C C C C S ¼ pffiffiffi B S ¼ pffiffiffi B S ¼ pffiffiffi B S ¼ pffiffiffi B 2@0A 2@ A 2@1A 2@ A 0 i Ài ð10-20Þ pffiffiffi The factor 1= has been introduced to normalize each of the eigenvectors We now construct a new matrix K, called the modal matrix, whose columns are formed from each of the respective eigenvectors in (10-20): 1 0 B À1 0 C C K ¼ pffiffiffi B ð10-21aÞ 2@0 1 A 0 i Ài The inverse matrix is easily found to be 1 0 B À1 0 C C KÀ1 ¼ pffiffiffi B @ 0 Ài A 0 i ð10-21bÞ We see that KKÀ1 ¼ I, where I is the unit matrix We now construct a diagonal matrix from each of the eigenvalues in (10-19) and write AþB 0 B AÀB 0 C C MD ¼ B ð10-22Þ @ 0 C þ iD A 0 C À iD From (10-4) the diagonal Mueller matrix for a polarizer MD,P is then ps 0 B p2p 0 C C MD, P ¼ B @ 0 ps pp A 0 ps pp and from (10-5) the diagonal matrix for a retarder is 1 0 B0 0 C C MD, C ¼ B @ 0 ei A 0 eÀi ð10-23Þ ð10-24Þ A remarkable relation now emerges From (10-21) and (10-22) one readily sees that the following identity is true: ÈK ¼ KMD Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð10-25Þ Postmultiplying both sides of (10-25) by KÀ1, we see that È ¼ KMD KÀ1 ð10-26aÞ MD ¼ KÀ1 ÈK ð10-26bÞ or where we have used KKÀ1 ¼ I We now square both sides of (10-26a) and find that È2 ¼ KM2D KÀ1 ð10-27Þ which shows that Èm is obtained from À1 Èm ¼ KMm DK ð10-28Þ Thus, by finding the eigenvalues and the eigenvectors of È and then constructing the diagonal matrix and the modal matrix (and its inverse), the mth power of the ABCD matrix È can be found from (10-28) Equation (10-26b) also allows us to determine the diagonalized ABCD matrix È Equation (10-28) now enables us to find the mth power of the ABCD matrix È: A B 0 m BB A 0 C B C Èm ¼ B C @ 0 C ÀD A 0 D ðA þ BÞm B B ¼ KB @ 0 C ðA À BÞm 0 0 0 ðC þ iDÞm 0 ðC À iDÞm C C À1 CK A ð10-29Þ Carrying out the matrix multiplication using (10-21) then yields 0h i h i 0 C B C Bh i h i C B m m C B ðA þ BÞm À ðA À BÞm ðA þ BÞ þ ðA À BÞ 0 C B C B Èm ¼ B C i h i h C 2B m m m m C B þ ðC À iDÞ þ iðC À iDÞ À iðC þ iDÞ 0 ðC þ iDÞ C B C B @ h i h i A m m m m 0 iðC þ iDÞ À iðC À iDÞ ðC þ iDÞ þ ðC À iDÞ ðA þ BÞm þ ðA À BÞm ðA þ BÞm À ðA À BÞm (10-30) Using (10-30) we readily find that the mth powers of the Mueller matrix of a polarizer and a retarder are, respectively, 2m 2m p2m 0 ps þ p2m p s À pp B 2m C 2m 1B ps À p2m p2m 0 C p s þ pp B C Mm ð10-31Þ ¼ p C m 2B 0 2pm A @ s pp m 0 2pm s pp Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved and B0 m MC ¼ B @0 0 0 0 cos m À sin m 0 C C sin m A cos m ð10-32Þ The diagonalized Mueller matrices will play an essential role in the following section when we determine the Mueller matrices for single and multiple dielectric plates Before we conclude this section we discuss another form of the Mueller matrix for a polarizer We recall that the first two Stokes parameters, S0 and S1, are the sum and difference of the orthogonal intensities The Stokes parameters can then be written as S ¼ Ix þ Iy ð10-33aÞ S ¼ Ix À Iy ð10-33bÞ S2 ¼ S2 ð10-33cÞ S3 ¼ S3 ð10-33dÞ where Ix ¼ Ex Exà Iy ¼ Ey Eyà ð10-33eÞ We further define Ix ¼ I0 ð10-34aÞ Iy ¼ I1 ð10-34bÞ S ¼ I2 ð10-34cÞ S ¼ I3 ð10-34dÞ Then, we can relate 1 S0 B S1 C B B C¼B @ S2 A @ 0 S3 or I to S, 0 1 I0 B I1 C B B C¼ B @ I2 A @ 0 I3 S to I by À1 0 À1 0 0 0 10 I0 B I1 C 0C CB C A@ I A I3 10 S0 B S1 C 0C CB C A@ S2 A S3 The column matrix: I0 BI C B 1C I¼B C @ I2 A I3 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð10-35aÞ ð10-35bÞ ð10-36Þ is called the intensity vector The intensity vector is very useful because the  matrix which connects I to I is diagonalized, thus making the calculations simpler To show that this is true, we can formally express (10-35a) and (10-35b) as S ¼ KA I ð10-37aÞ I ¼ KÀ1 A S ð10-37bÞ where KA and KÀ1 A are defined by the  matrices in (10-35), respectively The Mueller matrix M can be defined in terms of an incident Stokes vector S and an emerging Stokes vector S0 : S0 ¼ M S ð10-38Þ Similarly, we can define the intensity vector relationship: I0 ¼ P I ð10-39Þ where P is a  matrix We now show that P is diagonal We have from (10-37a) S0 ¼ KA I0 ð10-40Þ Substituting (10-40) into (10-38) along with (10-37a) gives I0 ¼ ðKÀ1 A M KA ÞI ð10-41Þ or, from (10-39) P ¼ KÀ1 A M KA ð10-42Þ We now show that for a polarizer P is a diagonal matrix The Mueller matrix for a polarizer in terms of the ABCD matrix elements can be written as A B 0 C B BB A 0 C C B ð10-43Þ M¼B C B0 C 0C A @ 0 C Substituting (10-43) into (10-42) and using KA and KÀ1 A from (10-35), we readily find that AþB 0 B C B AÀB 0C B C ð10-44Þ P¼B C B C C @ A 0 C Thus, P is a diagonal polarizing matrix; it is equivalent to the diagonal Mueller matrix for a polarizer The diagonal form of the Mueller matrix was first used by the Nobel laureate S Chandrasekhar in his classic papers in radiative Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved transfer in the late 1940s It is called Chandrasekhar’s phase matrix in the literature In particular, for the Mueller matrix of a polarizer we see that (10-44) becomes 0 p2s C B B p2 0 C p C B ð10-45Þ P¼B C C B 0 ps pp A @ 0 ps pp which is identical to the diagonalized Mueller matrix given by (10-23) In Part II we shall show that the Mueller matrix for scattering by an electron is proportional to 1 þ cos2  À sin2  0 B C 2 0 C B À sin  þ cos  C Mp ¼ B ð10-46Þ C 2B 0 cos  A @ 0 cos  where  is the observation angle in spherical coordinates and is measured from the z axis ( ¼ 0 ) Transforming (10-46) to Chandrasekhar’s phase matrix, we find 0 cos2  B C B 0 C B C P¼B ð10-47Þ C cos  A @ 0 0 cos  which is the well-known representation for Chandrasekhar’s phase matrix for the scattering of polarized light by an electron Not surprisingly, there are other interesting and useful transformations which can be developed However, this development would take us too far from our original goal, which is to determine the Mueller matrices for single and multiple dielectric plates We now apply the results in this section to the solution of this problem 10.3 MUELLER MATRICES FOR SINGLE AND MULTIPLE DIELECTRIC PLATES In the previous sections, Fresnel’s equations for reflection and transmission at an air–dielectric interface were cast into the form of Mueller matrices In this section we use these results to derive the Mueller matrices for dielectric plates We first treat the problem of determining the Mueller matrix for a single dielectric plate The formalism is then easily extended to multiple reflections within a single dielectric plate and then to a pile of m parallel transparent dielectric plates For the problem of transmission of a polarized beam through a single dielectric plate, the simplest treatment can be made by assuming a single transmission through the upper surface followed by another transmission through the lower surface There are, of course, multiple reflections within the dielectric plates, and, strictly speaking, these should be taken into account While this treatment of multiple internal reflections is straightforward, it turns out to be quite involved In Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 10-1 Beam propagation through a single dielectric plate the treatment presented here, we choose to ignore these effects The completely correct treatment is given in the papers quoted in the references at the end of this chapter The difference between the exact results and the approximate results is quite small, and very good results are still obtained by ignoring the multiple internal reflections Consequently, only the resulting expressions for multiple internal reflections are quoted We shall also see that the use of the diagonalized Mueller matrices developed in the previous section greatly simplifies the treatment of all of these problems In Fig 10-1 a single dielectric (glass) plate is shown The incident beam is described by the Stokes vector S Inspection of the figure shows that the Stokes vector S0 of the beam emerging from the lower side of the dielectric plate is related to S by the matrix relation: S0 ¼ M2T S ð10-48Þ where MT is the Mueller matrix for transmission and is given by (8-13) in Section 8.3 We easily see, using (8-13), that M2T is then " #2 sin 2i sin 2r MT ¼ ðsin þ cos À Þ2 cos4 À þ cos4 À À 0 B C B cos4 À À cos4 À þ C 0 B C ð10-49Þ B C 0 cos À @ A 0 cos2 À where i is the angle of incidence, r is the angle of refraction, and Æ ¼ i Æ r Equation (10-49) is the Mueller matrix (transmission) for a single dielectric plate We can immediately extend this result to the transmission through m parallel dielectric plates by raising M2T to the mth power, this is, M2m T The easiest way to this is to transform (10-49) to the diagonal form and raise the diagonal matrix to the mth power as described earlier After this is done we transform back to the Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Mueller matrix form Upon doing this we then find that the Mueller matrix for transmission through m parallel dielectric plates is " #2m sin 2i sin 2r 2m MT ¼ ðsin þ cos À Þ2 cos4m À þ cos4m À À B B B cos4m À À cos4m À þ B B B 0 B @ 0 0 cos2m À 0 cos2m À C C C C C C C A ð10-50Þ Equation (10-50) includes the result for a single dielectric plate by setting m ¼ We now consider that the incident beam is unpolarized Then, the Stokes vector of a beam emerging from m parallel plates is, from (10-50), cos4m À þ C " #2m B B cos4m  À C sin 2i sin 2r À C B S ¼ ð10-51Þ C B C B ðsin þ cos À Þ2 A @ The degree of polarization P of the emerging beam is then   1 À cos4m    À P¼  1 þ cos4m À  ð10-52Þ In Fig 10-2 a plot of (10-52) is shown for the degree of polarization as a function of the incident angle i The plot shows that at least six or eight parallel plates are required in order for the degree of polarization to approach unity At normal incidence the degree of polarization is always zero, regardless of the number of plates The use of parallel plates to create linearly polarized light appears very often outside the visible region of the spectrum In the visible and near-infrared region ([...]... expected As the number of parallel plates increases, the degree of polarization increases for both (10- 62) and (10- 63) However, the curves diverge and the magnitudes differ by approximately a factor of two so that for 10 parallel plates the degree of polarization is 0.93 for (10- 62) and 0.43 for (10- 63) In addition, for (10- 63), the Figure 10- 8 Plot of the degree of polarization as a function of the number... C A 0 1=2 ðs p Þ 10- 71bÞ For treating problems at angles other than the Brewster angle it is much simpler to use either (10- 71a) or (10- 71b) rather than the earlier forms of the Mueller matrices because the matrix elements s, p,  s, and  p are far easier to work with In this chapter we have applied the Mueller matrix formalism to the problem of determining the change in the polarization of light... A 10- 70bÞ The reflection coefficients s and p, (10- 64a) and (10- 64b), are plotted as a function of the incident angle for a range of refractive indices in Figs 10- 10 and 10- 11 Similar plots are shown in Figs 10- 12 and 10- 13 for  s and  p, (10- 65a) and (10- 65b) In a similar manner the reflection and transmission coefficients at the Brewster angle, (10- 67) and (10- 68), are plotted as a function of the. .. refractive index n in Figs 10- 14 and 10- 15 The great value of the Fresnel coefficients is that their use leads to simpler forms for the Mueller matrices for reflection and transmission For example, instead Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 10- 10 Plot of the Fresnel reflection coefficient s as a function of incidence angle i, (10- 64a) Figure 10- 11 Plot of the Fresnel reflection... of the transmitted beam is given by S0 in (10- 56) and is 1 IT ¼ ð1 þ sin4m 2iB Þ 2 10- 58Þ Equation (10- 58) has been plotted in Fig 10- 5 From Figs 10- 4 and 10- 5 the following conclusions can be drawn In Fig 10- 4, there is a significant increase in the degree of polarization up to m ¼ 6 Figure 10- 5, on the other hand, shows that the intensity decreases and then begins to ‘‘level off’’ for m ¼ 6 Thus, these... Reserved 10- 65aÞ 10- 65bÞ One can readily show that the following relations hold for Fresnel coefficients: s þ s ¼ 1 10- 66aÞ p þ p ¼ 1 10- 66bÞ and At the Brewster angle, written as iB , Fresnel’s reflection and transmission coefficients (10- 65) and (10- 66) reduce to s, B ¼ cos2 2iB 10- 67aÞ p, B ¼ 0 10- 67bÞ s, B ¼ sin2 2iB 10- 68aÞ p, B ¼ 1 10- 68bÞ We see immediately that s, B þ s, B ¼ 1 10- 69aÞ... that the inclusion of multiple reflections within the plates led to the following equation for the degree of polarization for m parallel plates at the Brewster angles:     m   10- 63Þ P¼ 2 2 2 m þ ½2n =ðn À 1ފ The derivation of (10- 63) along with similar expressions for completely and partially polarized light has been given by Collett (1972), using the Jones matrix formalism (Chapter 11) and the. .. Reserved as inspection of Fig 10- 7 shows, the curves are identical to those in Fig 10- 4 except in the former figure the abscissa begins with m ¼ 1 In the beginning of this section we pointed out that the Mueller matrix formalism can also be extended to the problem of including multiple reflections within a single dielectric plate as well as the multiple plates G G Stokes (1862) was the first to consider this... C C C A 10- 55Þ If the incident beam is unpolarized, the Stokes vector for the transmitted beam after passing through m parallel dielectric plates will be 1 0 4m sin 2iB þ 1 C B 4m C 1B B sin 2iB À 1 C 0 S ¼ B 10- 56Þ C C 2B 0 A @ 0 The degree of polarization is then   1 À sin4m 2   iB  P¼  1 þ sin4m 2i  B 10- 57Þ A plot of (10- 57) is shown in Fig 10- 4 for m dielectric plates The intensity... 11) and the Mueller matrix formalism In Fig 10- 8, (10- 63) has been plotted as a function of m and n, the refractive index It is of interest to compare (10- 62) and (10- 63) In Fig 10- 9 we have plotted these two equations for n ¼ 1.5 We see immediately that the degree of polarization is very different Starting with 0 parallel plates, that is, the unpolarized light source by itself, we see the degree of