13The Interference Laws of Fresneland Arago

13 The Interference Laws of Fresnel and Arago 13.1 INTRODUCTION In this last chapter of the first part, we now turn to the topic that led Stokes to introduce his polarization parameters, namely, the mathematical formulation of unpolarized light and its application to the interference laws of Fresnel and Arago In this section the events that led up to Stokes’ investigation are described We briefly review these events The investigation by Stokes that led to his paper in 1852 began with the experiments performed by Fresnel and Arago in 1817 At the beginning of these experiments both Fresnel and Arago held the view that light vibrations were longitudinal However, one of the results of these experiments, namely, that two rays that are polarized at right angles could in no way give rise to interference, greatly puzzled Fresnel Such a result was impossible to understand on the basis of light vibrations that are longitudinal Young heard of the experiments from Arago and suggested that the results could be completely understood if the light vibrations were transverse Fresnel immediately recognized that this condition would indeed make the experiments intelligible Indeed, as J Strong has correctly pointed out, only after these experiments had been performed was the transverse nature of light as well as the properties of linearly, circularly, and elliptically polarized light fully understood The results of the Fresnel–Arago experiments have been succinctly stated as the interference laws of Fresnel and Arago These laws, of which there are four, can be summarized as follows: Two waves linearly polarized in the same plane can interfere Two waves linearly polarized with perpendicular polarizations cannot interfere Two waves linearly polarized with perpendicular polarizations, if derived from perpendicular components of unpolarized light and subsequently brought into the same plane, cannot interfere Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 4 Two waves linearly polarized with perpendicular polarizations, if derived from the same linearly polarized wave and subsequently brought into the same plane, can interfere The fact that orthogonally polarized rays cannot be made to interfere can be taken as a proof that light vibrations are transverse This leads to a complete understanding of laws and The same confidence in understanding cannot be made with respect to laws and 4, however For these laws involve unpolarized light, a quantity that Fresnel and Arago were unable to understand completely or to characterize mathematically As a consequence, they never attempted a mathematical formulation of these laws and merely presented them as experimental facts Having established the basic properties of unpolarized, as well as partially polarized light, along with their mathematical formulation, Stokes then took up the question of the interference laws of Fresnel and Arago The remarkable fact now emerges that Stokes made no attempt to formulate these laws Rather, he analyzed a related experiment that Stokes states is due to Sir John Herschel This experiment is briefly discussed at the end of this chapter The analysis of the interference laws is easily carried to completion by means of the Mueller matrix formalism The lack of a matrix formalism does not preclude a complete analysis of the experiments, but the use of matrices does make the calculations far simpler to perform We shall first discuss the mathematical statements of unpolarized light With these statements we then analyze the experiments through the use of matrices, and we present the final results in the form of the Stokes vectors The apparatus that was used by Fresnel and Arago is similar to that devised by Young to demonstrate the phenomenon of interference arising from two slits In their experiments, however, polarizers are appropriately placed in front of the light source and behind the slits in order to obtain various interference effects Another polarizer is placed behind the observation screen in two of the experiments in order to bring the fields into the same plane of polarization The optical configuration will be described for each experiment as we go along 13.2 MATHEMATICAL STATEMENTS FOR UNPOLARIZED LIGHT In most optics texts very little attention is paid to the subject of unpolarized light This subject was the source of numerous investigations during the nineteenth century and first half of the twentieth century One of the major reasons for this interest was that until the invention of the laser practically every known optical source emitted only unpolarized light Ironically, when the subject of unpolarized light was finally ‘‘understood’’ in the late 1940s and 1950s, a new optical source, the laser, was invented and it was completely polarized! While there is a natural tendency to think of lasers as the optical source of choice, the fact is that unpolarized light sources continue to be widely used in optical laboratories This observation is supported by looking into any commercial optics catalog One quickly discovers that manufacturers continue to develop and build many types of optical sources, including black-body sources, deuterium lamps, halogen lamps, mercury lamps, tungsten lamps, etc., all of which are substantially unpolarized Consequently, the subject of unpolarized light is still of major importance not only for understanding the Fresnel– Arago laws but because of the existence and use of these optical sources Hence, we Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved should keep in mind that the subject of unpolarized light is far from being only of academic interest In all of the experiments of Fresnel and Arago an unpolarized source of light is used The mathematical statements that characterize unpolarized light will now be developed, and these expressions will then be used in the analysis of the Fresnel– Arago experiments and the formulation of their laws The Stokes parameters of a beam of light, as first shown by Stokes, can be determined experimentally by allowing a beam of light to propagate through a retarder of retardance and then through a polarizer with its transmission axis at an angle from the x axis The observed intensity I(, ) of the beam is found to be Ið, Þ ¼ ½S0 þ S1 cos 2 þ S2 sin 2 cos À S3 sin 2 sin ð13-1Þ where S0, S1, S2, and S3 are the Stokes parameters of the incident beam In order to use (13-1) to characterize unpolarized light, Stokes invoked the experimental fact that the observed intensity of unpolarized light is unaffected by the presence of the retarder and the orientation of the polarizer In other words, I(, ) must be independent of and This condition can only be satisfied if S1 ¼ S2 ¼ S3 ¼ 0, S0 6¼ ð13-2aÞ so Ið, Þ ¼ S0 =2 ð13-2bÞ The Stokes parameters for a time-varying field with orthogonal components Ex(t) and Ey(t) in a linear basis are defined to be S0 ¼ hEx ðtÞExÃ ðtÞi þ hEy ðtÞEyÃ ðtÞi ð13-3aÞ S1 ¼ hEx ðtÞExÃ ðtÞi À hEy ðtÞEyÃ ðtÞi ð13-3bÞ S2 ¼ hEx ðtÞEyÃ ðtÞi þ hEy ðtÞExÃ ðtÞi ð13-3cÞ S3 ¼ ihEx ðtÞEyÃ ðtÞi À ihEy ðtÞExÃ ðtÞi ð13-3dÞ where hÁ Á Ái means a time average and an asterisk signifies the complex conjugate The Stokes parameters for an unpolarized beam (13-2) can be expressed in terms of the definition of (13-3) so we have hEx ðtÞExÃ ðtÞi þ hEy ðtÞEyÃ ðtÞi ¼ S0 ð13-4aÞ hEx ðtÞExÃ ðtÞi À hEy ðtÞEyÃ ðtÞi ¼ ð13-4bÞ hEx ðtÞEyÃ ðtÞi þ hEy ðtÞExÃ ðtÞi ¼ ð13-4cÞ ihEx ðtÞEyÃ ðtÞi À ihEy ðtÞExÃ ðtÞi ¼ ð13-4dÞ From (13-4a) and (13-4b) we see that hEx ðtÞExÃ ðtÞi ¼ hEy ðtÞEyÃ ðtÞi ¼ S0 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð13-5Þ Thus, we conclude from (13-5) that the time-averaged orthogonal quadratic field components are equal, and so for unpolarized light we tentatively set Ex ðtÞ ¼ Ey ðtÞ ¼ AðtÞ ð13-6Þ This expression indeed satisfies (13-4a) and (13-4b) However, from (13-4c) and (13-4d) we have hEx ðtÞEyÃ ðtÞi ¼ hEy ðtÞExÃ ðtÞi ¼ ð13-7aÞ and this cannot be satisfied by (13-6) Therefore, we must set Ex ðtÞ ¼ Ax ðtÞ ð13-7bÞ Ey ðtÞ ¼ Ay ðtÞ ð13-7cÞ in order to satisfy (13-4a) through (13-4d) We see that unpolarized light can be represented by hAx ðtÞAÃx ðtÞi ¼ hAy ðtÞAÃy ðtÞi ¼ hAðtÞAÃ ðtÞi ð13-8aÞ hAx ðtÞAÃy ðtÞi ¼ hAy ðtÞAÃx ðtÞi ¼ ð13-8bÞ and Equations (13-8) are the classical mathematical statements for unpolarized light The condition (13-8b) is a statement that the orthogonal components of unpolarized light have no permanent phase relation In the language of statistical analysis, (13-8b) states that the orthogonal field components of unpolarized light are uncorrelated We can express (13-8a) and (13-8b) as a single statement by writing hAi ðtÞAÃj ðtÞi ¼ hAðtÞAÃ ðtÞi Á ij i, j ¼ x, y ð13-9aÞ where ij is the Kronecker delta defined by 13.3 ij ¼ if i ¼ j ð13-9bÞ ij ¼ if i 6¼ j ð13-9cÞ YOUNG’S INTERFERENCE EXPERIMENT WITH UNPOLARIZED LIGHT Before we treat the Fresnel–Arago experiments, we consider Young’s interference experiment with an unpolarized light source using the results of the previous section In many treatments of Young’s interference experiments, a discussion of the nature of the light source is avoided In fact, nearly all descriptions of the experiment in many textbooks begin with the fields at each of the slits and then proceed to show that interference occurs because of the differences in path lengths between the slits and the screen It is fortuitous, however, that regardless of the nature of the light source and its state of polarization, interference will always be observed It was fortunate for the science of optics that the phenomenon of interference could be described without having to understand the nature of the optical source Had optical physicists been forced to attack the problem of the polarization of sources before proceeding, the difficulties might have been insurmountable and, possibly, greatly Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved impeded further progress Fortunately, this did not occur Nevertheless, the problem of characterizing the polarization of light remained a problem well into the twentieth century as a reading of the papers in the references at the end of this chapter show Many beginning students of physical optics sometimes believe that Young’s experiment must be performed with light that is specially prepared; i.e., initially the light source is unpolarized and then is transformed to linear polarized light before it arrives at the slits The fact is, however, that interference phenomena can be observed with unpolarized light This can be easily shown with the mathematical statements derived in the previous section In Young’s experiment an unpolarized light source is symmetrically placed between the slits A and B as shown in Fig 13-1 The Stokes vector of the unpolarized light can again be decomposed in the following manner: 1 1 1 B0C B C B C Ã B1C Ã B À1 C C S ¼ hAAÃ iB ð13-10Þ @ A ¼ hAx Ax i@ A þ hAy Ay i@ A 0 The Stokes vector at slit A is 1 1 B1C B C Ã B À1 C C SA ¼ hAx AÃx iB @ A þ hAy Ay i@ A A A ð13-11aÞ and at slit B is 1 1 B1C B C C þ hAy AÃy iB À1 C SB ¼ hAx AÃx iB @ @ A A 0 2 B B Figure 13-1 Young’s interference experiment with unpolarized light Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð13-11bÞ where the subscripts A and B remind us that these are the Stokes vectors of the field at the respective slits The fields which satisfy the Stokes vector SA are Ax ðtÞ ffiffiffi ExA ðtÞ ¼ p Ay ðtÞ EyA ðtÞ ¼ pffiffiffi ð13-12aÞ Ay ðtÞ EyB ðtÞ ¼ pffiffiffi ð13-12bÞ and SB Ax ðtÞ ffiffiffi ExB ðtÞ ¼ p The field components at point C on the screen arising from the field propagating from slit A is Ax ðtÞ ffiffiffi expðiA Þ ExA ðtÞ ¼ p ð13-13aÞ Ay ðtÞ EyA ðtÞ ¼ pffiffiffi expðiA Þ ð13-13bÞ and, similarly, that due to slit B Ax ðtÞ ffiffiffi expðiB Þ ExB ðtÞ ¼ p ð13-14aÞ Ay ðtÞ EyB ðtÞ ¼ pffiffiffi expðiB Þ ð13-14bÞ The total field in the x and y directions is Ax ðtÞ ffiffiffi ½expðiA Þ þ expðiB Þ Ex ðtÞ ¼ ExA ðtÞ þ ExB ðtÞ ¼ p ð13-15aÞ Ay ðtÞ Ey ðtÞ ¼ EyA ðtÞ þ EyB ðtÞ ¼ pffiffiffi ½expðiA Þ þ expðiB Þ ð13-15bÞ Ax ðtÞ ffiffiffi ð1 þ ei Þ Ex ðtÞ ¼ p ð13-16aÞ Ay ðtÞ Ey ðtÞ ¼ pffiffiffi ð1 þ ei Þ ð13-16bÞ and or where ¼ B À A and the constant factor expðiA Þ has been dropped Equation (13-16) describes the field components at a point C on the observing screen It is interesting to note that it is not necessary at this point to know the relation between the slit separation and the distance between the slits and the observing screen Later, this relation will have to be known to obtain a quantitative description of the interference phenomenon We shall see shortly that interference is predicted with the information presented above Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved The Stokes vector for (13-16) is now formed in accordance with (13-3) and applying the conditions for unpolarized light (13-8) or (13-9) We then find that the Stokes vector for the field at C is 1 B 0C C ð13-17Þ S ¼ hAAÃ ið1 þ cos ÞB @0A Thus, we see from (13-17) that light observed on the screen is still unpolarized Furthermore, the intensity is I ¼ hAAÃ ið1 þ cos Þ ð13-18Þ Equation (13-18) is the familiar statement for describing interference According to (13-18), the interference pattern on the screen will consist of bright and dark (null intensity) lines In order to use (13-18) for a quantitative measurement, the specific relation between the slit separation and the distance from the slits to the screen must be known This is described by ¼ B À A ¼ kÁl, where k ¼ 2/ and Ál is the path difference between the fields propagating from A and B to C The phase shift can be expressed in terms of the parameters shown in Fig 13-1 a l22 ¼ d þ y þ ð13-19aÞ a l12 ¼ d þ y À ð13-19bÞ Subtracting (13-19b) from (13-19a) yields l22 À l12 ¼ 2ay ð13-20Þ We can assume that a is small, d ) a, and c is not far from the origin so that l2 þ l1 ffi 2d ð13-21Þ so (13-20) becomes Ál ¼ l2 À l1 ¼ ay d ð13-22Þ The phase shift is then ¼ B À A ¼ kÁl ¼ 2ay d ð13-23Þ where k ¼ 2/ is the wavenumber and is the wavelength of the optical field The maximum intensities are, of course, observed when cos ¼ 1, ¼ 2m so that d y¼ m m ¼ 0, 1, 2, ::: ð13-24aÞ a Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved and the minimum (null) intensities are observed when cos ¼ À1, ¼ ð2m þ 1Þ so that d ð13-24bÞ y¼ m m ¼ , , , ::: a 2 One can easily show that, regardless of the state of polarization of the incident beam, interference will be observed Historically, this was first done by Young and then by Fresnel and Arago, using unpolarized light We now consider the mathematical formulation of the Fresnel–Arago interference laws 13.4 THE FIRST EXPERIMENT: FIRST AND SECOND INTERFERENCE LAWS We consider a source of unpolarized light symmetrically placed between slits A and B as shown in Fig 13-2 A linear polarizer P with its transmission axis parallel to the x axis is placed in front of the light source A pair of similar polarizers PA and PB are also placed behind slits A and B, respectively The transmission axes of these polarizers PA and PB are at angles and with respect to the x axis, respectively We wish to determine the intensity and polarization of the light on the screen Æ The Stokes vector for unpolarized light of intensity AA* can be represented by 1 B 0C C S ¼ hAðtÞAÃ ðtÞiB ð13-25Þ @0A Figure 13.2 The first experiment The transmission axis of the P is parallel to the x axis The transmission axes of PA and PB are at angles and from the x axis Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Equation (13-25) can be decomposed into two orthogonally linearly polarized beams We then write 1 B0C B C S ¼ hAAÃ iB C @0A 0 1 1 B1C B À1 C B C B C ¼ hAx AÃx iB C þ hAy AÃy iB C @ A @ 2 0 A ð13-26Þ where we have used (13-8a) The Mueller matrix for P is 1 0 1B1 0C C M ¼ B 2@0 0 0A 0 0 ð13-27Þ The output beam from P is obtained from the multiplication of (13-27) and (13-26): 1 B 1C C SP ¼ hAx AÃx iB ð13-28Þ @0A Thus, the polarizer P transmits the horizontal and rejects the vertical component of the unpolarized light, (13-26) The light is now linearly horizontally polarized The matrix of a polarizer, MP, with its transmission axis at an angle from the x axis, is determined from MP ð2Þ ¼ MðÀ2ÞMP Mð2Þ ð13-29Þ where MP(2) is the matrix of the rotated polarizer and M(2) is the rotation matrix: 1 0 B cos 2 sin 2 C B C Mð2Þ ¼ B ð13-30Þ C @ À sin 2 cos 2 A 0 The Mueller matrix for PA is then found by setting ¼ in (13-30) and then substituting (13-27) into (13-29) The result is 1 cos 2 sin 2 cos 2 cos 2 sin 2 C 1B B cos 2 C MP A ¼ B ð13-31Þ C @ sin 2 cos 2 sin 2 0A sin 2 0 0 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved A similar result holds for MPB with replaced by The Stokes vector SA that emerges from PA is obtained by the multiplication of (13-28) by (13-31): 1 B cos 2 C B C SA ¼ hAx AÃx i cos B ð13-32aÞ C @ sin 2 A In a similar manner the Stokes vector SB is found to be 1 B cos C B C SB ¼ hAx AÃx i cos B C @ sin A ð13-32bÞ Inspection of (13-32a) and (13-32b) shows that both beams are linearly polarized at slits A and B In order to describe interference phenomena at the screen Æ, we must now determine the fields at slits A and B in the following manner From the definition of the Stokes vector given by (13-3) and the Stokes vector that we have just found at slit A, Eq (13-32a), we can write hEx ðtÞExÃ ðtÞiA þ hEy ðtÞEyÃ ðtÞiA ¼ hAx ðtÞAÃx ðtÞi cos hEx ðtÞExÃ ðtÞiA À hEy ðtÞEyÃ ðtÞiA ¼ hAx ðtÞAÃx ðtÞi cos cos 2 hEx ðtÞEyÃ ðtÞiA þ hEy ðtÞExÃ ðtÞiA ¼ hAx ðtÞAÃx ðtÞi cos sin 2 ð13-33bÞ ihEx ðtÞEyÃ ðtÞiA À ihEy ðtÞExÃ ðtÞiA ¼ ð13-33dÞ ð13-33aÞ ð13-33cÞ where the subscript A on the angle brackets reminds us that we are at slit A We now solve these equations and find that hEx ðtÞExÃ ðtÞiA ¼ hAx ðtÞAÃx ðtÞi cos4 hEy ðtÞEyÃ ðtÞiA ¼ hAx ðtÞAÃx ðtÞi cos sin hEx ðtÞEyÃ ðtÞiA ¼ hEy ðtÞExÃ ðtÞiA ¼ hAx ðtÞAÃx ðtÞi cos sin 2 ð13-34aÞ ð13-34bÞ ð13-34cÞ We see that the following fields will then satisfy (13-34): Ax ðtÞ ffiffiffi cos ExA ðtÞ ¼ p ð13-35aÞ Ax ðtÞ ffiffiffi cos sin EyA ðtÞ ¼ p ð13-35bÞ where Ax(t) is the time-varying amplitude The quantity Ax(t) is assumed to vary slowly in time In view of the fact that the Stokes vector at slit B is identical in form Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved with that at slit A, the field at slit B, following (13-35), will be Ax ðtÞ ffiffiffi cos ExB ðtÞ ¼ p ð13-36aÞ Ax ðtÞ ffiffiffi cos sin EyB ðtÞ ¼ p ð13-36bÞ The propagation of the beams along the paths AC and BC as shown in Fig 13-2 increases the phase of the fields by an amount A ¼ kl1 and B ¼ kl2 , respectively, where k ¼ 2/ and is the wavelength Thus, at point C on the screen Æ, the s and p field components will be, by the principle of superposition, Ex ðtÞ ¼ ExA ðtÞ expðiA Þ þ ExB ðtÞ expðiB Þ ð13-37aÞ Ey ðtÞ ¼ EyA ðtÞ expðiA Þ þ EyB ðtÞ expðiB Þ ð13-37bÞ Ex ðtÞ ¼ expðiA Þ½ExA ðtÞ þ ei ExB ðtÞ ð13-38aÞ or i Ey ðtÞ ¼ expðiA Þ½EyA ðtÞ þ e EyB ðtÞ ð13-38bÞ where ¼ B À A ¼ kðl2 À l1 Þ: The factor expðiA Þ will disappear when the Stokes parameters are formed, and so it can be dropped We now substitute (13-35) and (13-36) into (13-38), and we find that Ax ðtÞ ffiffiffi ðcos þ ei cos Þ Ex ðtÞ ¼ p ð13-39aÞ Ax ðtÞ ffiffiffi ðcos sin þ ei cos sin Þ Ey ðtÞ ¼ p ð13-39bÞ The Stokes parameters for Ex ðtÞ and Ey ðtÞ are now formed in the same manner as in (13-3) The Stokes vector observed on the screen will then be cos þ cos þ cosð À Þ cos cos cos B cos cos 2 þ cos cos þ cosð þ Þ cos cos cos C B C S ¼ hAAÃ iB C @ cos sin 2 þ cos sin þ ðcos sin þ cos sin 2Þ cos A ðcos sin À cos sin 2Þ sin ð13-40Þ We now examine the Stokes vector, (13-40), for some special cases Case I The transmission axes of the polarizers PA and PB are parallel For this condition ¼ and (13-40) reduces to 1 B cos 2 C C S ¼ hAAÃ ið1 þ cos Þ cos B ð13-41Þ @ sin 2 A The factor þ cos tells us that we will always have perfect interference Furthermore, the beam intensity is proportional to ð1 þ cos Þ cos and the light Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved is always linearly polarized Thus, (13-41) is the mathematical statement of the first interference law of Fresnel and and Arago There are two further subcases of interest Case I(a) The axes of the polarizers PA and PB are parallel to the axis of the polarizer P Then ¼ 0, and (13-41) reduces to 1 B1C B C S ¼ hAAÃ ið1 þ cos ÞB C @0A ð13-42aÞ The beam is linearly horizontally polarized, and the intensity is at a maximum Case I(b) The axes of the polarizers PA and PB are perpendicular to the axis of the polarizer P Then ¼ =2, and (13-41) reduces to 1 B1C B C S ¼ hAAÃ ið1 þ cos Þð0ÞB C @0A ð13-42bÞ Thus the observed intensity of the beam will be zero at all points on the observation screen Case II The transmission axes of PA and PB are perpendicular to each other For this condition ¼ þ =2 and (13-40) reduces to 1 B cos 2 þ sin 2 cos C B C S ¼ hAAÃ iB C @ sin 2 cos 2ð1 þ cos Þ A À sin 2 sin ð13-43Þ We now see that the interference term þ cos is missing in S0, the intensity Equation (13-43) is the mathematical statement of the second law of Fresnel and Arago, i.e., we not have interference in this case In general, the light is elliptically polarized as the presence of S3 in (13-43) shows Again there are some interesting subcases of (13-43) Case II(a) The axis of PA is parallel to the axis of P For this condition ¼ 0, and (13-43) reduces to 1 B1C B C S ¼ hAAÃ iB C @0A ð13-44Þ There is no interference, and the intensity and polarization of the observed beam and the polarized light from the source are identical Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Case II(b) The axis of PA is perpendicular to the axis of P In this case the axis of PB is parallel to P Then (13-43) reduces again to (13-44) Case II(c) The transmission axes of PA and PB are at þ=4 and À=4 from the transmission axis of P For this last case (13-43) reduces to 1 B cos C C S ¼ hAAÃ iB @ A À sin ð13-45Þ Again, there will be no interference, but the light is elliptically polarized The Stokes vector degenerates into circularly or linearly polarized light for ¼ ðm Æ 1=2Þ and Æ m, respectively, where m ¼ 0, Æ 1, Æ 2, :::: 13.5 THE SECOND EXPERIMENT: THIRD INTERFERENCE LAW In order to determine the mathematical statement that corresponds to the third law, we consider the following experiment represented by Fig 13-3 The polarizer P is placed, with its transmission axis at an angle from the x axis, behind the screen Æ to enable the fields that orginate at A and B to be brought into the same plane of polarization Here, is again an unpolarized light source and the transmission axes of PA and PB are placed parallel and perpendicular, respectively, to the x axis The Figure 13.3 The transmission axes of PA and PB are along the x and y axes, respectively The transmission axis of P is at an angle from the x axis Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved matrices for PA and PB are then [we set ¼ and then =2 in (13-31)] 1 0 1B1 0C C MPA ð ¼ 0Þ ¼ B 2@0 0 0A 0 0 1 À1 0 B À1 0 C C M PB ¼ ¼ B @ 0 0A 2 0 0 ð13-46Þ ð13-47Þ The Stokes vector at slit A is then found by multiplication of the Stokes vector for unpolarized light, (13-10), by (13-46) The result is 1 B 1C C SA ¼ hAx AÃx iB ð13-48Þ @ 0A Similarly, the Stokes vector at slit B is obtained by multiplication of the Stokes vector for unpolarized light by (13-47): 1 B À1 C C ð13-49Þ SB ¼ hAy AÃy iB @ A Thus, the beams are linearly and orthogonally polarized; they are derived from the perpendicular components of the unpolarized light The fields which satisfy (13-48) and (13-49) are, respectively, Ax ðtÞ ffiffiffi :ExA ðtÞ ¼ p ExB ðtÞ ¼ EyA ðtÞ ¼ Ay ðtÞ EyB ðtÞ ¼ pffiffiffi ð13-50aÞ ð13-50bÞ These fields now propagate to the screen Æ, where they are intercepted by the polarizer P At the polarizer the fields are Ax ðtÞ ffiffiffi expðiA Þ Ex ðtÞ ¼ p ð13-51aÞ Ay ðtÞ Ey ðtÞ ¼ pffiffiffi expðiB Þ ð13-51bÞ Ax ðtÞ ffiffiffi Ex ðtÞ ¼ p ð13-52aÞ Ay ðtÞ Ey ðtÞ ¼ pffiffiffi expðiÞ ð13-52bÞ or Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved where again ¼ B À A and we have dropped the factor exp(iA) The transmission axis of the polarizer P is at an angle with respect to the x axis Since we are now dealing with fields, we can conveniently use the Jones calculus to find the field on the screen after the beam has passed through the polarizer P The Jones matrix of the rotated polarizer, JðÞ, is ! ! ! cos sin cos À sin JðÞ ¼ À sin cos 0 sin cos cos sin cos @ A ¼ ð13-53Þ sin cos sin The field which is now at the screen can be obtained if we write (13-52) as a column matrix Multiplication of the vector composed of Eqs (13-52) by (13-53) then gives the field at the screen as Ay ðtÞ Ax ðtÞ ffiffiffi cos þ pffiffiffi ei cos sin Ex ðtÞ ¼ p 2 ð13-54aÞ Ay ðtÞ Ax ðtÞ ffiffiffi cos sin þ pffiffiffi ei sin Ey ðtÞ ¼ p 2 ð13-54bÞ We now form the Stokes vector and apply the conditions for unpolarized light given by (13-3) and (13-8) and find that 1 B C B cos 2 C B C ð13-55Þ S ¼ hAAÃ iB C B sin 2 C @ A Thus, we see that under no circumstances can there be interference Equation (13-55) is the mathematical statement of the third interference law of Fresnel and Arago In general, the light is linearly polarized In particular, for ¼ the light is linearly horizontally polarized, and for ¼ =2 it is linearly vertically polarized, as expected 13.6 THE THIRD EXPERIMENT: FOURTH INTERFERENCE LAW In this final experiment the arrangement of the polarizers is identical to the previous experiment except that a linear polarizer P , with its transmission axis at þ=4 from the x axis, is placed in front of the unpolarized light source (see Fig 13-4) In this case we take the axes of the unpolarized light source to be at an angle of þ=4 from the horizontal x axis The Stokes vector of the unpolarized light for this new direction is related to the old direction by the transformation: S0 ¼ Mð2Þ Á S Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð13-56Þ Figure 13-4 The third experiment The transmission axis of P is at þ=4 from the x axis The directions of the axes of PA, PB, and PC are identical to the second experiment where Mð2Þ is the rotation matrix We now multiply out (13-56) (and set ¼ /4), and we find, after decomposing the unpolarized light in the familiar manner, that 0 1 1 B B C C 1 Ã B0C C S0 ¼ hAx AÃx iB ð13-57Þ @ À1 A þ hAy Ay i@ A 0 Another way to arrive at (13-57) is to use the fact that unpolarized light is equivalent to two independent beams of light of equal intensities and polarized in orthogonal directions Then we could simply take the statement for unpolarized light, (13-10) directly and resolve it into (13-57) without the introduction of (13-56) Either way we obtain (13-57) The Mueller matrix for the polarizer P [with a set to /4 in (13-31)] is 1 1B0 0 0C C ð13-58Þ My ¼ B 2@1 0A 0 0 We now multiply (13-57) and (13-58), and the beam that emerges from P is 1 B C Ã B0C Sy0 ¼ hAy Ay i@ A ð13-59Þ Thus, the light is linearly polarized (þ/4 preference) and derived from a single orthogonal component of the unpolarized light The beam (13-59) now passes Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved through PA and PB, and the Stokes vectors at the slits are 1 B C Ã B1C SA ¼ hAy Ay i@ A 0 1 B À1 C C SB ¼ hAy AÃy iB @ A ð13-60aÞ ð13-60bÞ Thus, both beams are orthogonally linearly polarized but are derived from the same component of the unpolarized light The fields at slits A and B which satisfy (13-60a) and (13-60b) are then Ay ðtÞ ExA ðtÞ ¼ pffiffiffi ExB ðtÞ ¼ EyA ðtÞ ¼ ð13-61aÞ Ay ðtÞ EyB ðtÞ ¼ pffiffiffi ð13-61bÞ The fields at the polarizer PC will then be Ay ðtÞ Ex ðtÞ ¼ pffiffiffi ð13-62aÞ Ay ðtÞ Ey ðtÞ ¼ pffiffiffi ei ð13-62bÞ After the field passes through the polarizer PC the components become Ay ðtÞ Ay ðtÞ Ex ðtÞ ¼ pffiffiffi cos þ pffiffiffi ei cos sin 2 ð13-63aÞ Ay ðtÞ Ay ðtÞ Ey ðtÞ ¼ pffiffiffi cos sin þ pffiffiffi ei sin 2 ð13-63bÞ The Stokes vector observed on the screen is then, from (13-63), 1 B cos 2 C C S ¼ hAAÃ ið1 þ sin 2 cos ÞB @ sin 2 A ð13-64Þ An inspection of this Stokes vector shows that interference can be observed Equation (13-64) is the mathematical statement of the fourth and last of the interference laws of Fresnel and Arago There are again some interesting subcases Case III(a) The axis of PC is parallel to the axis of PA and orthogonal to the axis of PB Then ¼ and (13-64) reduces to 1 B 1C C S ¼ hAAÃ iB ð13-65aÞ @0A Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved The light is linearly horizontally polarized, and there is no interference (the beam from PB is not contributing to the field) Case III(b) The axis of PC is þ/4 from the axis of PA In this case ¼ /4, and so (13-64) reduces to 1 B C B0C ð13-65bÞ S ¼ hAAÃ ið1 þ cos ÞB C @1A The light shows maximum interference and is linearly polarized (þ/4 preference) Case III(c) The axis of PC is perpendicular to the axis of PA so ¼ /2 Then (13-64) becomes 1 B À1 C B C ð13-65cÞ S ¼ hAAÃ iB C @ A The light is linearly vertically polarized, an again there is no interference (now the beam from PA is not contributing) At this point we can summarize the Fresnel–Arago laws However, we defer this in order to consider one more interesting related problem 13.7 THE HERSCHEL–STOKES EXPERIMENT In Section 13.1 we pointed out that Stokes did not formulate the Fresnel–Arago interference laws, but treated a related experiment suggested by Sir John Herschel This experiment is represented in Fig 13-5 In this experiment an unpolarized source of light, , is again used The transmission axis of polarizer PB is fixed in the direction of the x axis, while the polarizer PA is rotated through an angle a The Stokes vector on the screen Æ is to be determined The Stokes vector at the slit B is, following the methods developed earlier, 1 B C B1C ð13-66aÞ SB ¼ hAx AÃx iB C @0A while the Stokes vector at slit A is 0 1 1 B B C C B cos 2 C B cos 2 C C þ hAy AÃy i sin B C SA ¼ hAx AÃx i cos B B sin 2 C B sin 2 C @ @ A A 0 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð13-66bÞ Figure 13-5 The Herschel–Stokes experiment The transmission axis of PB is fixed along the x axis, and the transmission axis of PA is rotated through an angle from the x axis The fields at B and A that satisfy (13-66a) and (13-66b) are Ax ðtÞ ffiffiffi ExB ðtÞ ¼ p EyB ðtÞ ¼ ð13-67aÞ Ax ðtÞ ffiffiffi cos E0xA ðtÞ ¼ p Ax ðtÞ ffiffiffi cos sin E 0yA ðtÞ ¼ p ð13-67bÞ Ay ðtÞ E 00xA ðtÞ ¼ pffiffiffi cos sin Ay ðtÞ E 00yA ðtÞ ¼ pffiffiffi sin ð13-67cÞ The primed and double primed fields correspond to the first and second Stokes vector in (13-66b) and arise because the Stokes vectors in (13-66b) are independent The fields at the screen Æ are then Ay ðtÞ Ax ðtÞ i ffiffiffi ðe þ cos Þ þ pffiffiffi sin cos Ex ðtÞ ¼ p 2 ð13-68aÞ Ay ðtÞ Ax ðtÞ ffiffiffi cos sin þ pffiffiffi sin Ey ðtÞ ¼ p 2 ð13-68bÞ We now form the Stokes vector in the usual way and apply the condition for unpolarized light and find that 1 þ cos cos B C B cos ð1 þ cos Þ C S ¼ hAAÃ iB ð13-69Þ C @ sin cos ð1 þ cos Þ A À sin cos sin Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Stokes actually obtained only S0 and took to be equal to zero or We examine (13-69) at some special values of Case IV(a) The transmission axis of the polarizer PA is parallel to the transmission axis of PB, so ¼ Then (13-69) reduces to 1 B1C B C S ¼ hAAÃ ið1 þ cos ÞB C @0A ð13-70Þ We have perfect interference, and the light is linearly horizontally polarized Case IV(b) The transmission axis of PA is perpendicular to the transmission axis of PB, so a ¼ /2 Then (13-69) reduces to 1 B C Ã B0C S ¼ hAA i@ A ð13-71Þ 0 There is no interference, and the light is unpolarized This problem shows why it was selected by Stokes Within the confines of a single problem he was able to show that one could obtain complete interference along with completely polarized light, (13-70), and, conversely, no interference and completely unpolarized light, (13-71) It was this ‘‘peculiar’’ behavior of polarized light which was a source of great confusion Stokes, however, by his investigation was able to show that with his parameters all these questions could be answered, and, equally important, this could be done within the structure of the wave theory of light 13.8 SUMMARY OF THE FRESNEL–ARAGO INTERFERENCE LAWS In view of the rather lengthy analysis required to obtain the mathematical statements for the Fresnel–Arago interference laws, it is worthwhile to summarize these results We restate each of the laws and the corresponding Stokes vector 13.8.1 The First Interference Law Two waves, linearly polarized in the same plane, can interfere 1 B cos 2 C C S ¼ hAAÃ ið1 þ cos Þ cos B @ sin 2 A ð13-41Þ The angle a refers to the condition when the transmission axes of the two polarizers behind the slits are parallel We see that the light is always linearly polarized and there will always be interference Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 13.8.2 The Second Interference Law Two waves, linearly polarized with perpendicular polarizations, cannot interfere 1 B cos 2 þ sin 2 cos C B C S ¼ hAAÃ iB C @ sin 2 cos 2ð1 þ cos Þ A ð13-43Þ À sin 2 sin The interference term þ cos is missing in S0, the intensity Equation (13-43) shows that the light is always elliptically polarized, but there is never any interference 13.8.3 The Third Interference Law Two waves, linearly polarized with perpendicular polarizations, if derived from perpendicular components of unpolarized light and subsequently brought into the same plane, cannot interfere 1 B cos 2 C B C S ¼ hAAÃ iB C @ sin 2 A ð13-55Þ Equation (13-55) shows that interference is never seen under these conditions 13.8.4 The Fourth Interference Law Two waves, linearly polarized with perpendicular polarizations, if derived from the same linearly polarized wave and subsequently brought into the same plane, can interfere 1 B cos 2 C B C S ¼ hAAÃ ið1 þ sin 2 cos ÞB C @ sin 2 A ð13-64Þ Only if ¼ 0 or 90 does the interference term in (13-64) vanish; otherwise interference will always be observed; the Stokes vector is always linearly polarized This concludes our discussion of the fundamental properties of polarized light At this point the reader can certainly see that a great deal of knowledge can be obtained about the properties and behavior of polarized light without having to resort to the equations of the electromagnetic field However, this is as far as we can go Ultimately, we must deal with the source of the polarized radiation fields In order to this, we must now turn to the theory of the electromagnetic field, i.e., Maxwell’s equations and the source of polarized light We shall see that the Stokes parameters and Mueller formalism play a major and very interesting role Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved REFERENCES Papers 10 Stokes, G G., Trans Camb Phil Soc., 9, 399 (1852) Reprinted in Mathematical and Physical Papers, Cambridge University Press, London, 1901, Vol 3, p 233 Soleillet, P., Ann Phys., 12 (10) 23 (1929) Langsdorf, A and DuBridge, L., J Opt Soc Am., 24, (1934) Birge, R T., J Opt Soc Am., 25, 179 (1935) Perrin, F J Chem Phys., 10, 415 (1942) Hurwitz, H J Opt Soc Am., 35, 525 (1945) Parke, N G., III, Statistical Optics II: Mueller Phenomenological Algebra, RLE TR-119, Research Laboratory of Elect at M.I.T (1949) Wolf, E., Nuovo Cimento, 12, 884 (1954) Hannau, R., Am J Phys., 31, 303 (1962) Collett, E., Am J Phys., 39, 1483 (1971) Books Fresnel, A J., L’Oeuvres Comple`tes, Publicés par Henri de Senarmont, Emile Verdet et Leonor Fresnel, Paris, 1866, Vol I Whittaker, E., A History of the Theories of Aether and Electricity, Philosophical Society of New York, 1951, Vol I Wood, R W., Physical Optics, 3rd ed., Optical Society of America, Washington, DC, 1988 Born, M., and Wolf, E., Principles of Optics, 3rd ed., Pergamon Press, New York, 1965 Ditchburn, R W., Light, 2nd ed., Blackie, London, 1963 Strong, J., Concepts of Classical Optics, Freeman, San Francisco, 1959 Jenkins, F S., and White, H E., Fundamental of Optics, McGraw-Hill, New York, 1957 Shurcliff, W A., Polarized light, Harvard University Press, Cambridge, MA, 1962 Hecht, E and Zajac, A., Optics, Addison-Wesley, Reading, MA, 1974 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved [...]... 0 0 0 0 ð13-27Þ The output beam from P is obtained from the multiplication of (13-27) and (13-26): 0 1 1 B 1 1C C SP ¼ hAx AÃx iB ð13-28Þ @0A 2 0 Thus, the polarizer P transmits the horizontal and rejects the vertical component of the unpolarized light, (13-26) The light is now linearly horizontally polarized The matrix of a polarizer, MP, with its transmission axis at an angle from the x axis,... obtained by the multiplication of (13-28) by (13-31): 0 1 1 B cos 2 C 1 B C SA ¼ hAx AÃx i cos 2 B ð13-32aÞ C @ sin 2 A 2 0 In a similar manner the Stokes vector SB is found to be 0 1 1 B cos 2 C 1 B C SB ¼ hAx AÃx i cos 2 B C @ sin 2 A 2 0 ð13-32bÞ Inspection of (13-32a) and (13-32b) shows that both beams are linearly polarized at slits A and B In order to describe interference phenomena at the... in time In view of the fact that the Stokes vector at slit B is identical in form Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved with that at slit A, the field at slit B, following (13-35), will be Ax ðtÞ ffiffiffi cos 2 ExB ðtÞ ¼ p 2 ð13-36aÞ Ax ðtÞ ffiffiffi cos sin EyB ðtÞ ¼ p 2 ð13-36bÞ The propagation of the beams along the paths AC and BC as shown in Fig 13-2 increases the phase of the fields by... both beams are linearly polarized at slits A and B In order to describe interference phenomena at the screen Æ, we must now determine the fields at slits A and B in the following manner From the definition of the Stokes vector given by (13-3) and the Stokes vector that we have just found at slit A, Eq (13-32a), we can write 1 hEx ðtÞExÃ ðtÞiA þ hEy ðtÞEyÃ ðtÞiA ¼ hAx ðtÞAÃx ðtÞi cos 2 2 1 hEx ðtÞExÃ ðtÞiA... linearly horizontally polarized The matrix of a polarizer, MP, with its transmission axis at an angle from the x axis, is determined from MP ð2Þ ¼ MðÀ2ÞMP Mð2Þ ð13-29Þ where MP(2) is the matrix of the rotated polarizer and M(2) is the rotation matrix: 0 1 1 0 0 0 B 0 cos 2 sin 2 0 C B C Mð2Þ ¼ B ð13-30Þ C @ 0 À sin 2 cos 2 0 A 0 0 0 1 The Mueller matrix for PA is then found by setting ... the phase of the fields by an amount A ¼ kl1 and B ¼ kl2 , respectively, where k ¼ 2/ and is the wavelength Thus, at point C on the screen Æ, the s and p field components will be, by the principle of superposition, Ex ðtÞ ¼ ExA ðtÞ expðiA Þ þ ExB ðtÞ expðiB Þ ð13-37aÞ Ey ðtÞ ¼ EyA ðtÞ expðiA Þ þ EyB ðtÞ expðiB Þ ð13-37bÞ Ex ðtÞ ¼ expðiA Þ½ExA ðtÞ þ ei ExB ðtÞ ð13-38aÞ or i Ey ðtÞ ¼ expðiA... 2 þ cos 2 sin 2 þ ðcos 2 sin 2 þ cos 2 sin 2Þ cos A 2 ðcos 2 sin 2 À cos 2 sin 2Þ sin ð13-40Þ We now examine the Stokes vector, (13-40), for some special cases Case I The transmission axes of the polarizers PA and PB are parallel For this condition ¼