7 The Measurement of the Characteristics of Polarizing Elements 7.1 INTRODUCTION 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 orthogonal 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 proceeding with any experiment in which polarizing 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 wavelengths, become mixed up Finally, the quality control of manufacturers of polarizing 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 7.2 MEASUREMENT OF ATTENUATION COEFFICIENTS OF 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 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 7-1 Experimental configuration to measure the attenuation coefficients px and py of a polarizer (diattenuator) 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 0 px þ p2y p2x À p2y B C 1B px À p2y p2x þ p2y 0 C B C Mp ¼ B px, y ð7-1Þ C 2@ 0 2px py A 0 2px py It is convenient to rewrite (7-1) as A B 0 BB A 0 C C Mp ¼ B @0 C 0A 0 C ð7-2aÞ where A ¼ ðp2x þ p2y Þ ð7-2bÞ B ¼ ðp2x À p2y Þ ð7-2cÞ C ¼ ð2px py Þ ð7-2dÞ In practice, while we are interested only in determining p2x and p2y , it is useful to measure pxpy as well, because a polarizer satisfies the relation: A2 ¼ B2 þ C2 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð7-3Þ 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 S0 B S1 C C S¼B @ S2 A S3 ð7-4Þ 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 1 B0C C S ¼ I0 B @1A ð7-5Þ where I0 ¼ (1/2)(S0 þ S2) 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), A BBC C S ¼ I0 B @CA ð7-6Þ 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 angle The Mueller matrix of the rotated analyzer is (see Chap 5) 1B B cos 2 MA ¼ B @ sin 2 cos 2 cos2 2 sin 2 cos 2 sin 2 sin 2 cos 2 C C C sin2 2 0A 0 ð7-7Þ The Stokes vector of the beam incident on the optical detector is then seen from multiplying (7-6) by (7-7) to be 1 B cos 2 C I C S0 ¼ ðA þ B cos 2 þ C sin 2 ÞB @ sin 2 A ð7-8Þ and the intensity of the beam is Ið Þ ¼ I0 ðA þ B cos 2 þ C sin 2 Þ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð7-9Þ First method: By rotating the analyzer to ¼ 0 , 45 , and 90 , (7-9) yields the following equations: Ið0 Þ ¼ I0 ðA þ BÞ I0 ðA þ CÞ I Ið90 Þ ¼ ðA À BÞ Ið45 Þ ¼ ð7-10aÞ ð7-10bÞ ð7-10cÞ Solving for A, B, and C, we then find that A¼ Ið0 Þ þ Ið90 Þ I0 ð7-11aÞ B¼ Ið0 Þ À Ið90 Þ I0 ð7-11bÞ C¼ 2Ið45 Þ À Ið0 Þ À Ið90 Þ I0 ð7-11cÞ which are the desired relations From (7-2) we also see that p2x ¼ A þ B ð7-12aÞ p2y ¼ A À B ð7-12bÞ so that we can write (7-10a) and (7-10c) as p2x ¼ 2Ið0 Þ I0 ð7-13aÞ p2y ¼ 2Ið90 Þ I0 ð7-13bÞ Thus, it is only necessary to measure I(0 ) and I(90 ), the intensities in the x and y directions, respectively, to obtain p2x and p2y The intensity I0 of 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 p2x and p2y in (7-13) it is necessary to know I0 However, a relative measurement of p2y =p2x is just as useful We divide (7-12b) by (7-12a) and we obtain p2y Ið90 Þ ¼ Ið0 Þ p2x ð7-14Þ 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 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved In order to obtain A, B, and C and then p2x and p2y 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 A2 ¼ B2 þ C2 ð7-3Þ This suggests that we can write B ¼ A cos ð7-15aÞ C ¼ A sin ð7-15bÞ Substituting (7-15a) and (7-15b) into (7-9), we then have Ið Þ ¼ I0 A ½1 þ cosð2 À Þ ð7-16aÞ and tan ¼ C B ð7-16bÞ where (7-16b) has been obtained by dividing (7-15a) by (7-15b) We see that I( ) leads to a null intensity at null ¼ 90 þ ð7-17Þ where null is the angle at which the null is observed Substituting (7-17) into (7-16b) then yields C ¼ tan 2 null B ð7-18Þ 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 Then we see from (7-12) that p2x ¼ þ B ð7-19aÞ p2y ¼ À B ð7-19bÞ 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 ¼ ðp2x À p2y Þ ð7-2cÞ C ¼ ð2px py Þ ð7-2dÞ Substituting (7-2b) and (7-2c) into (7-18) gives tan 2 null ¼ 2px py p2x À p2y Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð7-20Þ The form of (7-20) suggests that we set px ¼ p cos py ¼ p sin ð7-21aÞ tan 2 null ¼ sin ¼ tan cos ð7-21bÞ so and ¼ null ð7-21cÞ This leads immediately to py ¼ tan ¼ tanð null Þ px ð7-22aÞ or, using (7-17) p2y ¼ cot p2x ð7-22bÞ Thus, the shift in the intensity, (7-16a) enables us to determine p2y =p2x directly from We always assume that p2y =p2x A neutral density filter is described by p2x ¼ p2y so the range on p2y =p2x limits to 90 180 ð7-22cÞ For p2y =p2x ¼ 0, an ideal polarizer, ¼ 180 , whereas for p2y =p2x ¼ 1, 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 p2y =p2x ¼  10À6 =0:48 ¼ 4:2  10À6 so from (7-22b) we see that ¼ 179.77 and null ¼ 179.88 , 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 Ið Þ ¼ I0 ðA þ B cos 2 þ C sin 2 Þ ð7-9Þ 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 Z Ið Þ d ð7-23aÞ A¼ I0 Z B¼ Ið Þ cos 2 d ð7-23bÞ I0 Z C¼ Ið Þ sin 2 d ð7-23cÞ I0 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 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 y generating polarizer is 1 C B C I0 B B À1 C S¼ B C 2B C A @ ð7-24Þ The Mueller matrix of the rotated test polarizer (7-2a) is B cos 2 B sin 2 A cos2 2 þ C sin2 2 ðA À CÞ sin 2 cos 2 ðA À CÞ sin 2 cos 2 A sin2 2 þ C cos2 2 0 A B B B cos 2 B M¼B B B sin 2 @ 0 C 0C C C 0C A ð7-25Þ The intensity of the beam emerging from the y analyzing polarizer is IðÞ ¼ I0 ½ðA þ CÞ À 2B cos 2 þ ðA À CÞ cos2 2 ð7-26Þ Equation (7-26) can be solved for its maximum and minimum values by differentiating I() with respect to and setting dI()/d ¼ We then find sin 2½B À ðA À CÞ cos 2 ¼ ð7-27Þ The solutions of (7-27) are sin 2 ¼ ð7-28aÞ and cos 2 ¼ B AÀC Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð7-28bÞ For (7-28a) we have ¼ 0 and 90 The corresponding values of the intensities are then, from (7-26) Ið0 Þ ¼ I0 ½A À B Ið90 Þ ¼ I0 ½A þ B ð7-29aÞ ð7-29bÞ The second solution (7-28b), on substitution into (7-26), leads to I() ¼ 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 ¼ I0 ½A þ C ð7-29cÞ We can again divide (7-29) through by I0 and 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 7.3 MEASUREMENT OF PHASE SHIFT OF A RETARDER 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 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 7-2 Closed polarizer method to measure the phase of a retarder 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 1 0 B cos2 2 þ cos sin2 2 ð1 À cos Þ sin 2 cos 2 À sin sin 2 C C B Mð , ÞB C @ ð1 À cos Þ sin 2 cos 2 sin2 2 þ cos cos2 2 sin cos 2 A sin sin 2 À sin cos 2 cos ð7-30Þ where the phase shift polarizer is B B Æ1 Mx, y ¼ B 2@ 0 is to be determined The Mueller matrix for an ideal linear Æ1 0 0 0 0C C C 0A ð7-31Þ 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 M ¼ My Mð , ÞMx Carrying out the matrix multiplication in (7-32) 1 À1 À1 ð1 À cos Þð1 À cos 4Þ B B M¼ B @ 0 0 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð7-32Þ using (7-30) and (7-31) then yields 0C C ð7-33Þ C 0A Equation (7-33) shows that the polarizing train behaves as a pseudopolarizer The intensity of the optical beam on the detector is then ð1 À cos Þð1 À cos 4Þ Ið, Þ ¼ I0 ð7-34Þ where I0 is 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 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) ID ¼ I0 ð7-35Þ so (7-34) can be written as Ið, Þ ¼ ID ð1 À cos Þð1 À cos 4Þ ð7-36Þ 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 ¼ cos À1 Ið45 , Þ 1À 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 analyzer can be rotated through an angle The Stokes vector of the beam emerging from the generating polarizer is (again let us assume that unpolarized light enters the generating polarizer) 1 I0 B 1C C S¼ B @0A Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð7-38Þ Multiplication of (7-38) by (7-30) yields 1 B C 2 I B cos 2 þ cos sin 2 C C S0 ¼ B C 2B @ ð1 À cos Þ sin 2 cos 2 A ð7-39Þ sin sin 2 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 1 B C I B1C C ð7-40Þ S0 ¼ B C 2B @0A 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 1 B C cos C I0 B B C ð7-41Þ S ¼ B 2@ C A sin 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 1 cos 2 sin 2 B C cos 2 cos2 2 sin 2 cos 2 C 1B B C ð7-42Þ Mð Þ ¼ B @ sin 2 sin 2 cos 2 sin2 2 0C A 0 0 The Stokes vector of the beam emerging from the analyzer is then 1 B C B cos 2 C I C S ¼ ð1 þ cos cos 2 ÞB B sin 2 C @ A ð7-43Þ so the intensity is Ið , Þ ¼ I0 ð1 þ cos cos 2 Þ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð7-44Þ In order to find , (7-44) is evaluated at ¼ 0 and 90 , and I0 ð1 þ cos Þ I Ið90 , Þ ¼ ð1 À cos Þ Ið0 , Þ ¼ ð7-45aÞ ð7-45bÞ Equation (7-45a) is divided by (7-45b) and solved for cos : cos ¼ Ið0 , Þ À Ið90 , Þ Ið0 , Þ þ Ið90 , Þ ð7-46Þ We note that in this method the source intensity need not be known We can also determine the direction of the fast axis of the retarder in a ‘‘dynamic’’ fashion The intensity of the beam emerging from the analyzer when it is in the y position is (see (7-39) and (7-42)) Iy ¼ I0 ½1 À ðcos2 2 þ cos sin2 2Þ ð7-47aÞ where is the angle of the fast axis measured from the horizontal x axis We now see that when the analyzer is in the x position: Ix ¼ I0 ½1 þ ðcos2 2 þ cos sin2 2Þ ð7-47bÞ Adding (7-47a) and (7-47b) yields Ix þ Iy ¼ I0 ð7-48aÞ Next, subtracting (7-47a) from (7-47b) yields Ix À Iy ¼ I0 ðcos2 2 þ cos sin2 2Þ ð7-48bÞ We see that when ¼ the sum and difference intensities (7-48) are equal Thus, one can measure Ix and Iy continuously as the retarder is rotated and the analyzer is flipped between the horizontal and vertical directions until (7-48a) equals (7-48b) When this occurs, the amount of rotation that has taken place determines the magnitude of the rotation angle of the fast axis from the x axis Third method: Finally, if a compensator is available, the phase shift can be measured as follows Figure 7-3 shows the measurement method The compensator is placed between the retarder under test and the analyzer The transmission axes of the generating and analyzing polarizers are set at þ45 and þ135 , that is, in the crossed position The Stokes vector of the beam incident on the test retarder is 1 I0 B 0C C S¼ B ð7-49Þ @1A Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 7-3 Measurement of the phase shift of a wave plate using a Babinet–Soleil compen- sator The Mueller matrix of the test retarder is B0 M¼B @0 0 0 0 cos À sin 0 C C sin A cos ð7-50Þ Multiplying (7-49) by (7-50) yields 1 I B C C S ¼ 0B @ cos A À sin ð7-51Þ The Mueller matrix of the Babinet–Soleil compensator is B0 M¼B @0 0 0 0 cos Á À sin Á 0 C C sin Á A cos Á ð7-52Þ Multiplying (7-51) by (7-52) yields the Stokes vector of the beam incident on the linear À45 polarizer: 1 C I B C S ¼ 0B @ cosðÁ þ Þ A À sinðÁ þ Þ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð7-53Þ Finally, the Mueller matrix at À45 (þ135 ) is À1 1B 0 M¼ B @ À1 0 for the ideal linear polarizer with its transmission axis 0C C 0A ð7-54Þ Multiplying (7-53) by the first row of (7-54) gives the intensity on the detector, namely, IðÁ þ Þ ¼ I0 ½1 À cosðÁ þ Þ ð7-55Þ We see that a null intensity is found at Á ¼ 360 À ð7-56Þ from which we then find There are still other methods to determine the phase of the retarder, and the techniques developed here can provide a useful starting point However, the methods described here should suffice for most problems 7.4 MEASUREMENT OF ROTATION ANGLE OF A ROTATOR The final type of polarizing element that we wish to characterize is a rotator The Mueller matrix of a rotator is 1 0 B cos 2 sin 2 C C M¼B ð7-57Þ @ À sin 2 cos 2 A 0 First method: The angle can be determined by inserting the rotator between a pair of polarizers in which the generating polarizer is fixed in the y position and the analyzing polarizer can be rotated This configuration is shown in Fig 7-4 The Stokes vector of the beam incident on the rotator is 1 I B À1 C C ð7-58Þ S ¼ 0B 2@ A The Stokes vector of the beam incident on the analyzer is then found by multiplying (7-58) by (7-57) 1 C I B B À cos 2 C S0 ¼ B ð7-59Þ C @ sin 2 A Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 7-4 Measurement of the rotation angle of a rotator The Mueller matrix of the analyzer is cos 2 sin 2 B cos 2 sin 2 cos 2 B cos 2 M¼ B @ sin 2 sin 2 cos 2 sin2 2 0 0C C C 0A ð7-60Þ The intensity of the beam emerging from the analyzer is then seen from the product of (7-60) and (7-59) to be Ið Þ ¼ I0 ½1 À cosð2 þ 2Þ ð7-61Þ The analyzer is rotated and, according to (7-61), a null intensity will be observed at ¼ 180 À ð7-62aÞ or, simply, ¼ 180 À ð7-62bÞ Second method: Another method for determining the angle is to rotate the generating polarizer sequentially to 0 , 45 , 90 , and 135 The rotator and the analyzing polarizer are fixed with their axes in the horizontal direction The intensities of the beam emerging from the analyzing polarizer for these four angles are then I0 ð1 þ cos 2Þ I Ið45 Þ ¼ ð1 þ sin 2Þ I Ið90 Þ ¼ ð1 À cos 2Þ I Ið135 Þ ¼ ð1 À sin 2Þ Ið0 Þ ¼ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð7-63aÞ ð7-63bÞ ð7-63cÞ ð7-63dÞ Subtracting (7-63c) from (7-63a) and (7-63d) from (7-63b) yields I0 cos 2 ¼ Ið0 Þ À Ið90 Þ I0 sin ¼ Ið45 Þ À Ið135 Þ ð7-64aÞ ð7-64bÞ Dividing (7-64b) by (7-64a) then yields the angle of rotation : ¼ tanÀ1 ½ðIð45 Þ À Ið135 ÞÞ=ðIð0 Þ À Ið90 ÞÞ ð7-65Þ In the null-intensity method an optical detector is not required, whereas in this second method a photodetector is needed However, one soon discovers that even a null measurement can be improved by several orders of magnitude below the sensitivity of the eye by using an optical detector–amplifier combination Finally, as with the measurement of retarders, other configurations can be considered However, the two methods described here should, again, suffice for most problems REFERENCE Book Clark, D and Grainger, J F., Polarized Light and Optical Measurement, Pergamon Press, Oxford, 1971 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved [...]... Reserved 7- 32Þ using (7- 30) and (7- 31) then yields 1 0 0C C 7- 33Þ C 0A 0 Equation (7- 33) shows that the polarizing train behaves as a pseudopolarizer The intensity of the optical beam on the detector is then ð1 À cos Þð1 À cos 4Þ 4 Ið, Þ ¼ I0 7- 34Þ where I0 is 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. .. fashion The intensity of the beam emerging from the analyzer when it is in the y position is (see (7- 39) and (7- 42)) Iy ¼ I0 ½1 À ðcos2 2 þ cos sin2 2Þ 4 7- 47aÞ where is the angle of the fast axis measured from the horizontal x axis We now see that when the analyzer is in the x position: Ix ¼ I0 ½1 þ ðcos2 2 þ cos sin2 2Þ 4 7- 47bÞ Adding (7- 47a) and (7- 47b) yields Ix þ Iy ¼ I0 2 7- 48aÞ... Rights Reserved Figure 7- 4 Measurement of the rotation angle of a rotator The Mueller matrix of the analyzer is 0 1 cos 2 sin 2 2 B cos 2 sin 2 cos 2 1 B cos 2 M¼ B 2 @ sin 2 sin 2 cos 2 sin2 2 0 0 0 1 0 0C C C 0A 7- 60Þ 0 The intensity of the beam emerging from the analyzer is then seen from the product of (7- 60) and (7- 59) to be Ið Þ ¼ I0 ½1 À cosð2 þ 2Þ 4 7- 61Þ The analyzer is rotated... polarizers in which the generating polarizer is fixed in the y position and the analyzing polarizer can be rotated This configuration is shown in Fig 7- 4 The Stokes vector of the beam incident on the rotator is 0 1 1 I B À1 C C 7- 58Þ S ¼ 0B 2@ 0 A 0 The Stokes vector of the beam incident on the analyzer is then found by multiplying (7- 58) by (7- 57) 0 1 1 C I B B À cos 2 C S0 ¼ 0 B 7- 59Þ C 2 @ sin 2... magnitude of the rotation angle of the fast axis from the x axis Third method: Finally, if a compensator is available, the phase shift can be measured as follows Figure 7- 3 shows the measurement method The compensator is placed between the retarder under test and the analyzer The transmission axes of the generating and analyzing polarizers are set at þ45 and þ135 , that is, in the crossed position The. .. @ À1 0 1 0 0 0 for the ideal linear polarizer with its transmission axis 1 0 0C C 0A 0 7- 54Þ Multiplying (7- 53) by the first row of (7- 54) gives the intensity on the detector, namely, IðÁ þ Þ ¼ I0 ½1 À cosðÁ þ Þ 4 7- 55Þ We see that a null intensity is found at Á ¼ 360 À 7- 56Þ from which we then find There are still other methods to determine the phase of the retarder, and the techniques developed... 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) ID ¼ I0 2 7- 35Þ so (7- 34) can be written as Ið, Þ ¼ ID ð1 À cos Þð1 À cos 4Þ 2 7- 36Þ The retarder is now reinserted into the polarizing train The maximum intensity,... (7- 47a) from (7- 47b) yields Ix À Iy ¼ I0 ðcos2 2 þ cos sin2 2Þ 2 7- 48bÞ We see that when ¼ 0 the sum and difference intensities (7- 48) are equal Thus, one can measure Ix and Iy continuously as the retarder is rotated and the analyzer is flipped between the horizontal and vertical directions until (7- 48a) equals (7- 48b) When this occurs, the amount of rotation that has taken place determines the. .. 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 p2y =p2x ¼ 2  10À6 =0:48 ¼ 4:2  10À6 so from (7- 22b) we see that ¼ 179 .77 and null ¼ 179 .88 , respectively; the nearness of to 180 shows that it is an excellent polarizing material Second method: The parameters A, B, and C can... Stokes vector of the beam incident on the test retarder is 0 1 1 I0 B 0C C S¼ B 7- 49Þ 2 @1A 0 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 7- 3 Measurement of the phase shift of a wave plate using a Babinet–Soleil compen- sator The Mueller matrix of the test retarder is 0 1 B0 M¼B @0 0 0 1 0 0 0 0 cos À sin 1 0 0 C C sin A cos 7- 50Þ Multiplying (7- 49) by (7- 50) yields