11 The Jones Matrix Calculus 11.1 INTRODUCTION We have seen that the Stokes polarization parameters and the Mueller matrix formalism can be used to describe any state of polarization In particular, if we are dealing with a single beam of polarized light, then the formalism of the Stokes parameters is completely capable of describing any polarization state ranging from completely polarized light to completely unpolarized light In addition, the formalism of the Stokes parameters can be used to describe the superposition of several polarized beams, provided that there is no amplitude or phase relation between them; that is, the beams are incoherent with respect to each other This situation arises when optical beams are emitted from several independent sources and are then superposed However, there are experiments where several beams must be added and the beams are not independent of each other, e.g., beam superposition in interferometers There we have a single optical source and the single beam is divided by a beam splitter Then, at a later stage, the beams are ‘‘reunited,’’ that is, superposed Clearly, there is an amplitude and phase relation between the beams We see that we must deal with amplitudes and phase and superpose the amplitudes of each of the beams After the amplitudes of the beam are superposed, the intensity of the combined beams is then found by taking the time average of the square of the total amplitude If there were no amplitude or phase relations between the beams, then we would arrive at the same result as we obtained for the Stokes parameters However, if there is a relation between the amplitude and the phase of the optical beams, an interference term will arise Of course, as pointed out earlier, the description of the polarizing behavior of the optical field in terms of amplitudes was one of the first great successes of the wave theory of light The solution of the wave equation in terms of transverse components leads to elliptically polarized light and its degenerate linear and circular forms On the basis of the amplitude results, many results could be understood (e.g., Young’s interference experiment, circularly polarized light) However, even using the amplitude formulation, numerous problems become difficult to treat, such as the propagation of the field through several polarizing components To facilitate Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved the treatment of complicated polarization problems at the amplitude level, R Clark Jones, in the early 1940s, developed a matrix calculus for treating these problems, commonly called the Jones matrix calculus It is most appropriately used when we must superpose amplitudes The Jones calculus involves complex quantities contained in  column matrices (the Jones vector) and  matrices (the Jones matrices) At first sight it would seem that the use of the  matrices would be simpler than the use of the  Mueller matrices Oddly enough, this is not the case This is due primarily to the fact that even the matrix multiplication of several complex  matrices can be tedious Furthermore, even after the complete matrix calculation has been carried out, additional steps are still required For example, it is often necessary to separate the real and imaginary parts (e.g., Ex and Ey) and superpose the respective amplitudes This can involve a considerable amount of effort Another problem is that to find the intensity one must take the complex transpose of the Jones vector and then carry out the matrix multiplication between the complex transpose of the Jones vector and Jones vector itself All this is done using complex quantities, and the possibility of making a computational error is very real While the  Mueller matrix formalism appears to be more complicated, all the entries are real quantities and there are many zero entries, as can be seen by inspecting the Mueller matrix for the polarizer, the retarder and the rotator This fact greatly simplifies the matrix multiplications, and, of course, the Stokes vector is real There are, nevertheless, many instances where the amplitudes must be added (superposed), and so the Jones matrix formalism must be used There are many problems where either formalism can be used with success As a general rule, the most appropriate choice of matrix method is to use the Jones calculus for amplitude superposition problems and the Mueller formalism for intensity superposition problems Experience will usually indicate the best choice to make In this chapter we develop the fundamental matrices for the Jones calculus along with its application to a number of problems 11.2 THE JONES VECTOR The plane-wave components of the optical field in terms of complex quantities can be written as Ex ðz, tÞ ¼ E0x eið!tÀkzþx Þ ð11-1aÞ Ey ðz, tÞ ¼ E0y eið!tÀkzþy Þ ð11-1bÞ The propagator !t À kz is now suppressed, so (11-1) is then written as Ex ¼ E0x eix ð11-2aÞ Ey ¼ E0y eiy ð11-2bÞ Equation (11-2) can be arranged in a  column matrix E: ! ! Ex E0x eix E¼ ¼ Ey E0y eiy Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð11-3Þ called the Jones column matrix or, simply, the Jones vector The column matrix on the right-hand side of (11-3), incidentally, is the Jones vector for elliptically polarized light In the Jones vector (11-3), the maximum amplitudes E0x and E0y are real quantities The presence of the exponent with imaginary arguments causes Ex and Ey to be complex quantities Before we proceed to find the Jones vectors for various states of polarized light, we discuss the normalization of the Jones vector; it is customary to express the Jones vector in normalized form The total intensity I of the optical field is given by I ¼ Ex Exà þ Ey Eyà Equation (11-4) can be obtained by the following matrix multiplication: À Á Ex I ¼ Exà Eyà Ey ð11-4Þ ð11-5Þ The row matrix ðExà EyÃ Þ is the complex transpose of the Jones vector (column matrix E) and is written Ey; thus, À Á Ey ¼ Exà Eyà ð11-6Þ so I ¼ Ey E ð11-7Þ yields (11-4) Carrying out the matrix multiplication of (11-7), using (11-3), yields E20x þ E20y ¼ I ¼ E20 ð11-8Þ It is customary to set E20 ¼ 1, whereupon we say that the Jones vector is normalized The normalized condition for (11-5) can then be written as Ey E ¼ ð11-9Þ We note that the Jones vector can only be used to describe completely polarized light We now find the Jones vector for the following states of completely polarized light Linear horizontally polarized light For this state Ey ¼ 0, so (11-3) becomes E0x eix E¼ ð11-10Þ From the normalization condition (11-9) we see that E20x ¼ Thus, suppressing eix because it is unimodular, the normalized Jones vector for linearly horizontally polarized light is written ð11-11Þ E¼ In a similar manner the Jones vectors for the other well-known polarization states are easily found Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Linear vertically polarized light Ex ¼ 0, so E20y ¼ and E¼ Linear þ45 polarized light Ex ¼ Ey , so 2E20x ¼ and 1 E ¼ pffiffiffi ð11-12Þ Linear À45 polarized light Ex ¼ ÀEy , so 2E20x ¼ and 1 E ¼ pffiffiffi À1 ð11-13Þ ð11-14Þ Right-hand circularly polarized light For this case E0x ¼ E0y and y À x ¼ þ90 Then, 2E20x ¼ and we have 1 E ¼ pffiffiffi ð11-15Þ þi Left-hand circularly polarized light We again have E0x ¼ E0y , but y À x ¼ À90 The normalization condition gives 2E20x ¼ 1, and we have 1 E ¼ pffiffiffi ð11-16Þ Ài Each of the Jones vectors (11-11) through (11-16) satisfies the normalization condition (11-9) An additional property is the orthogonal or orthonormal property Two vectors A and B are said to be orthogonal if AB ¼ or, in complex notation, Ay B ¼ If this condition is satisfied, we say that the Jones vectors are orthogonal For example, for linearly horizontal and vertical polarized light we find that À Áà ¼0 ð11-17aÞ so the states are orthogonal or, since we are using normalized vectors, orthonormal Similarly, for right and left circularly polarized light: À Áà 1 þi ¼0 ð11-17bÞ Ài Thus, the orthonormal condition for two Jones vectors E1 and E2 is Eyi Ej ¼ ð11-18Þ We see that the orthonormal condition (11-18) and the normalizing condition (11-9) can be written as a single equation, namely Eyi Ej ¼ ij i; j ¼ 1; Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð11-19aÞ where ij is the Kronecker delta and has the property: ij ¼ i¼j ð11-19bÞ ij ¼ i 6¼ j ð11-19cÞ In a manner analogous to the superposition of incoherent intensities or Stokes vectors, we can superpose coherent amplitudes, that is, Jones vectors For example, the Jones vector for horizontal polarization is EH and that for vertical polarization is EV, so E0x eix EV ¼ ð11-20Þ EH ¼ E0y eiy Adding EH and EV gives E ¼ EH þ EV ¼ E0x eix E0y eiy ! ð11-21Þ which is the Jones vector for elliptically polarized light Thus, superposing two orthogonal linear polarizations give rise to elliptically polarized light For example, if E0x ¼ E0y and y ¼ x , then, from (11-21), we can write ix ð11-22Þ E ¼ E0x e which is the Jones vector for linear þ45 polarized light Equation (11-22) could also be obtained by superposing (11-11) and (11-12): 1 ð11-23Þ ¼ E ¼ EH þ EV ¼ þ 1 which, aside from the normalizing factor, is identical to (11-13) As another example let us superpose left and right circularly polarized light of equal amplitudes Then, from (11-15) and (11-16) we have 1 1 E ¼ pffiffiffi þ pffiffiffi ¼ pffiffiffi ð11-24Þ Ài i which, aside from the normalizing factor, is the Jones vector for linear horizontally polarized light (11-11) As a final Jones vector example, we show that elliptically polarized light can be obtained by superposing two opposite circularly polarized beams of unequal amplitudes The Jones vectors for two circular polarized beams of unequal amplitudes a and b can be represented by 1 Eþ ¼ a EÀ ¼ b ð11-25Þ þi Ài According to the principle of superposition, the resultant Jones vector for (11-25) is Ex aþb E ¼ Eþ þ EÀ ¼ ¼ ð11-26Þ Ey iða À bÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved In component form (11-26) is written as Ex ¼ a þ b ð11-27aÞ Ey ¼ ða À bÞei=2 ð11-27bÞ We now restore the propagator !t À kz, so (11-27) is then written as Ex ¼ ða þ bÞeið!tÀkzÞ ð11-28aÞ Ey ¼ ða À bÞeið!tÀkzþ=2Þ ð11-28bÞ Taking the real part of (11-28), we have Ex ðz, tÞ ¼ ða þ bÞ cosð!t À kzÞ Ey ðz, tÞ ¼ ða À bÞ cos !t À kz þ ¼ ða À bÞ sinð!t À kzÞ ð11-29aÞ ð11-29bÞ ð11-29cÞ Equations (11-28a) and (11-28b) are now written as Ex ðz, tÞ ¼ cosð!t À kzÞ aþb ð11-30aÞ Ey ðz, tÞ ¼ sinð!t À kzÞ aÀb ð11-30bÞ Squaring and adding (11-30a) and (11-30b) yields E2y ðz, tÞ E2x ðz, tÞ þ ¼1 ða þ bÞ2 ða À bÞ2 ð11-31Þ Equation (11-31) is the equation of an ellipse whose major and minor axes lengths are a þ b and a À b, respectively Thus, the superposition of two oppositely circularly polarized beams of unequal magnitudes gives rise to a (nonrotated) ellipse with its locus vector moving in a counterclockwise direction 11.3 JONES MATRICES FOR THE POLARIZER, RETARDER, AND ROTATOR We now determine the matrix forms for polarizers (diattenuators), retarders (phase shifters), and rotators in the Jones matrix calculus In order to this, we assume that the components of a beam emerging from a polarizing element are linearly related to the components of the incident beam This relation is written as E 0x ¼ jxx Ex þ jxy Ey ð11-32aÞ E 0y ¼ jyx Ex þ jyy Ey ð11-32bÞ where E 0x and E 0y are the components of the emerging beam and Ex and Ey are the components of the incident beam The quantities jik , i, k ¼ x, y, are the transforming Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved factors (elements) Equation (11-32) can be written in matrix form as ! ! ! jxx jxy Ex E 0x ¼ jyx jyy Ey E 0y ð11-33aÞ or E0 ¼ JE ð11-33bÞ where J¼ jxx jyx jxy jyy ð11-33cÞ The  matrix J is called the Jones instrument matrix or, simply, the Jones matrix We now determine the Jones matrices for a polarizer, retarder, and rotator A polarizer is characterized by the relations: E 0x ¼ px Ex E 0y ¼ py Ey ð11-34aÞ px, y ð11-34bÞ For complete transmission px, y ¼ 1, and for complete attenuation px, y ¼ In terms of the Jones vector, (11-34) can be written as ! ! ! px Ex E 0x ¼ ð11-35Þ py Ey E 0y so the Jones matrix (11-33c) for a polarizer is px Jp ¼ px, y py ð11-36Þ For an ideal linear horizontal polarizer there is complete transmission along the horizontal x axis and complete attenuation along the vertical y axis This is expressed by px ¼ and py ¼ 0, so (11-36) becomes JPH ¼ ð11-37Þ 0 Similarly, for a linear vertical polarizer, (11-36) becomes 0 JPV ¼ ð11-38Þ In general, it is useful to know the Jones matrix for a linear polarizer rotated through an angle This is readily found by using the familiar rotation transformation, namely, J0 ¼ JðÀÞJJðÞ where JðÞ is the rotation matrix: cos sin JðÞ ¼ À sin cos Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð11-39aÞ ð11-39bÞ and J is given by (11-33c) For a rotated linear polarizer represented by (11-36) and rotated by angle we have from (11-39) that px cos À sin cos sin J0 ¼ ð11-40Þ py sin cos À sin cos Carrying out the matrix multiplication in (11-40) we find that the Jones matrix for a rotated polarizer is ! px cos2 þ py sin2 ðpx À py Þ sin cos ð11-41Þ JP ðÞ ¼ ðpx À py Þ sin cos px sin2 þ py cos2 For an ideal linear horizontal polarizer we can set px ¼ and py ¼ in (11-41), so that the Jones matrix for a rotated linear horizontal polarizer is ! sin cos cos2 JP ðÞ ¼ ð11-42Þ sin cos sin2 The Jones matrix for a linear polarizer rotated through þ45 is then seen from (11-42) to be ! 1 JP ð45 Þ ¼ ð11-43Þ 1 If the linear polarizer is not ideal, then the Jones matrix for a polarizer (11-36) at þ45 is seen from (11-41) to be ! px þ py px À py JP ð45 Þ ¼ ð11-44Þ px À py px þ py We note that for ¼ 0 and 90 , (11-42) gives the Jones matrices for a linear horizontal and vertical polarizer, Eqs (11-37) and (11-38) respectively Equation (11-41) also describes a neutral density (ND) filter The condition for a ND filter is px ¼ py ¼ p, so (11-41) reduces to ! ð11-45Þ JND ðÞ ¼ p Thus, JND() is independent of rotation (), and the amplitudes are equally attenuated by an amount p This is, indeed, the behavior of a ND filter The presence of the unit (diagonal) matrix in (11-45) confirms that a ND filter does not affect the polarization state of the incident beam The next polarizing element of importance is the retarder The retarder increases the phase by þ =2 along the fast (x) axis and retards the phase by À =2, along the slow (y) axis This behavior is described by E 0x ¼ eþi =2 Ex ð11-46aÞ E 0y ¼ eÀi =2 Ey ð11-46bÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved where E 0x and E 0y are the components of the emerging beam and Ex and Ey are the components of the incident beam We can immediately express (11-46) in the Jones formalism as þi =2 Ex Ex e J ¼ ¼ ð11-47Þ Ài =2 Ey E 0y e The Jones matrix for a retarder (phase shifter) is then þi =2 e JR ð Þ ¼ eÀi =2 ð11-48Þ where is the total phase shift between the field components The two most common types of phase shifters (retarders) are the quarter-wave retarder and the half-wave retarder For these devices ¼ 90 and 180 , repectively, and (11-48) becomes i=4 0 e i=4 i=4 JR ð11-49aÞ ¼e ¼e ¼ eÀi=2 Ài eÀi=4 and i=2 i e JR ¼ ¼ i ð11-49bÞ ¼ Ài À1 eÀi=2 The Jones matrix for a rotated retarder is found from (11-48) and (11-39) to be ! ei =2 cos2 þ eÀi =2 sin2 ðei =2 À eÀi =2 Þ sin cos JR ð , Þ ¼ ð11-50Þ ðei =2 À eÀi =2 Þ sin cos ei =2 sin2 þ eÀi =2 cos2 With the half-angle formulas, (11-50) can also be written in the form: i sin sin 2 cos þ i sin cos 2 B C 2 ð11-51Þ JR ð , Þ ¼ @ A i sin sin 2 cos À i sin cos 2 2 For quarter-wave retarder and a half-wave retarder (11-51) reduces, respectively, to 1 i i p ffiffi ffi p ffiffi ffi p ffiffi ffi þ cos 2 sin 2 B C 2 C JR , ¼ B ð11-52Þ @ A i i pffiffiffi sin 2 pffiffiffi À pffiffiffi cos 2 2 and JR cos 2 , ¼ i sin 2 sin 2 À cos 2 ð11-53Þ The factor i in (11-53) is unimodular and can be suppressed It is common, therefore, to write (11-53) simply as cos 2 sin 2 JR , ¼ ð11-54Þ sin 2 À cos 2 Inspecting (11-54) we see that it is very similar to the matrix for rotation, namely, cos sin JðÞ ¼ ð11-39bÞ À sin cos Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved However, (11-54) differs from (11-39b) in two ways First, in (11-54) we have 2 rather than Thus, a rotation of a retarder through rotates the polarization ellipse through 2 Second, a clockwise mechanical rotation in (11-54) leads to a counterclockwise rotation of the polarization ellipse In order to see this behavior clearly, consider that we have incident linear horizontally polarized light Its Jones vector is Ex J¼ ð11-55Þ The components of the beam emerging from a true rotator (11-39b) are then E 0x ¼ ðcos ÞEx ð11-56aÞ E 0y ¼ Àðsin ÞEx ð11-56bÞ The angle of rotation is then tan ¼ E 0y À sin ¼ tanðÀÞ ¼ cos E 0x ð11-57Þ In a similar manner, multiplying (11-55) by (11-54) leads to E 0x ¼ ðcos 2ÞEx ð11-58aÞ E 0y ¼ ðsin 2ÞEx ð11-58bÞ so we now have tan ¼ E 0y sin 2 ¼ tan 2 ¼ E 0x cos 2 ð11-59Þ Comparing (11-59) with (11-57), we see that the direction of rotation for a rotated retarder is opposite to the direction of true rotation Equation (11-59) also shows that the angle of rotation is twice that of a true rotation Because of this similar but analytically incorrect behavior of a rotated half-wave retarder, (11-54) is called a pseudorotator We note that an alternative form of a half-wave retarder, which is the more common form, is given by factoring out i in (11-49b) or simply setting ¼ 0 in (11-54): J ð11-60Þ ¼ À1 The final matrix of interest is the Jones matrix for a rotator The defining equations are E 0x ¼ cos is the angle of rotation Equation (11-61) is written in matrix form as Ex Ex cos sin J ¼ ¼ ð11-62Þ Ey E 0y À sin cos Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved so the Jones matrix for a rotator is cos sin JROT ¼ À sin cos ð11-63Þ It is interesting to see the effect of rotating a true rotator According to (11-39), the rotation of a rotator, (11-63), is given by cos À sin cos sin cos sin JROT ðÞ ¼ ð11-64Þ sin cos À sin cos À sin cos Carrying out the matrix multiplication in (11-64) yields cos sin [...]... problem can be expanded further by allowing the beam emerging from the polarizer (11- 68) to be incident on a linear vertical polarizer The Jones matrix is found by setting ¼ 90 in (11- 42): 0 0 JP ð90 Þ ¼ 11- 78Þ 0 1 The Jones vector of the beam emerging from the second linear polarizer, found by multiplying (11- 68) by (11- 78) is 0 0 E p ¼ Ex cos sin 11- 79Þ 1 and the intensity is immediately... problems is to determine the Jones vector for a beam emerging from a rotated linear polarizer and its intensity The Jones vector of the incident beam is ! Ex E¼ 11- 66Þ Ey The Jones matrix of a rotated (ideal) linear polarizer was shown in (11- 42) to be ! sin cos cos2 JP ðÞ ¼ 11- 42Þ sin cos sin2 While it is straightforward to determine the Jones vector and the intensity of the emerging beam,... using the relations: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ sin 2 sin pffiffiffi cos ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À sin 2 sin pffiffiffi sin ¼ 2 11- 87Þ 11- 88Þ 11- 89aÞ 11- 89bÞ Squaring (11- 89a) and (11- 89b) and subtracting, we find that cos 2 ¼ sin 2 sin 11- 90Þ We now write (11- 87) and (11- 90) as the pair cos 2 ¼ sin 2 sin 11- 91aÞ cot ¼ À tan cos 2 11- 91bÞ Equations (11- 91a) and (11- 91b)... immediately from (11- 72b) and ¼ 0 , so (11- 72a) is b sin ¼ a cos 11- 73Þ The polarization ellipse corresponding to (11- 69) is x2 y2 2xy cos ¼ sin2 þ À ab a2 b2 11- 74Þ For ¼ 0 , (11- 74) reduces to b sin x y¼ x¼ a cos 11- 75Þ Thus, the Jones vector (11- 68) describes a beam that is linearly polarized with a slope equal to m ¼ tan ¼ tan 11- 76Þ The intensity of the emerging beam... aeix 0 E ¼ 11- 69Þ beiy where a and b are real Equating (11- 68) and (11- 69), we have E 0x ¼ Ex cos2 ¼ aeix 11- 70aÞ E 0y ¼ Ex cos sin ¼ beiy 11- 70bÞ Dividing (11- 70b) by (11- 70a) then gives E 0y sin b i ¼ ¼ e a E 0x cos 11- 71Þ where ¼ y À x Finally, taking the real and imaginary parts of (11- 71) leads to sin b ¼ cos cos a b 0 ¼ sin a 11- 72aÞ b 6¼ a 11- 72bÞ We conclude... 11- 61aÞ E 0y ¼ À sin Ex þ cos Ey 11- 61bÞ where is the angle of rotation Equation (11- 61) is written in matrix form as 0 Ex Ex cos sin 0 J ¼ ¼ 11- 62Þ Ey E 0y À sin cos Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved so the Jones matrix for a rotator is cos sin JROT ¼ À sin cos 11- 63Þ It is interesting to see the effect of rotating a true rotator According to (11- 39), the. .. elliptically polarized beam We can also treat the problem in the amplitude domain and apply the Kent–Lawson method to determine the phase and orientation of the beam The incident beam can be written in the form: cos E¼ 11- 81Þ sin ei The beam (11- 81) is incident on a retarder of arbitrary phase oriented at an angle The phase and orientation of the retarder are now adjusted until circularly... not affect the total intensity Thus, it is easy to see that taking the complex transpose of (11- 83) for each Jones vector and multiplying by its normal Jones vector gives a unit intensity as required We now equate the components in each of the column matrices in (11- 83) and divide these equations to find tan ei ¼ sin þ iei cos cos À iei sin 11- 84Þ Rationalizing the denominator in (11- 84),... intrinsic to the rotator, which is the rotation angle We conclude that the only way to create a rotation of the polarization ellipse mechanically is to use a half-wave retarder placed in a mechanical rotating mount 11. 4 APPLICATIONS OF THE JONES VECTOR AND JONES MATRICES We now turn our attention to applying the results of Sections 11. 2 and 11. 3 to several problems of interest One of the first problems... ð1 À cos 4Þ 8 11- 80Þ where I ¼ Exà Ex Thus, as the second polarizer is rotated, a null intensity is observed at ¼ 0 , 90 , 180 , and 270 Equation (11- 80) is, of course, the same as obtained using the Mueller–Stokes calculus We now apply the Jones formalism to several other problems of interest We recall from Section 6.6 that we used the method of Kent and Lawson to determine the Stokes parameters