Appendix DVector Representation of the OpticalField: Application to Optical Activity

Appendix D Vector Representation of the Optical Field: Application to Optical Activity We have emphasized the Stokes vector and Jones matrix formulation for polarized light However, polarized light was first represented by another formulation introduced by Fresnel and called the vector representation for polarized light This representation is still much used and for the sake of completeness we discuss it This formulation was introduced by Fresnel to describe the remarkable phenomenon of optical activity in which the ‘‘plane of polarization’’ of a linearly polarized beam was rotated as the optical field propagated through an optically active medium Fresnel’s mathematical description of this phenomenon was a brilliant success After we have discussed the vector representation we shall apply it to describe the propagation of light through an optically active medium For a plane wave propagating in the z direction the components of the optical field in the xy plane are Ex ðz, tÞ ¼ E0x cosðkz À !t þ x Þ À Á Ey ðz, tÞ ¼ E0y cos kz À !t þ y ðD-1aÞ ðD-1bÞ Eliminating the propagator kzÀ!t between (D-1a) and (D-1b) yields E2x ðz, tÞ Ey ðz, tÞ 2Ex ðz, tÞEy ðz, tÞ cos  þ À ¼ sin2  E0x E0y E20y E20x The Stokes vector corresponding to (D-1) is, of course, E20x þ E20y B C B E20x À E20y C C S¼B B C @ 2E0x E0y cos  A ðD-2Þ ðD-3Þ 2E0x E0y sin  In the xy plane we construct the vector E(z, t): Eðz, tÞ ¼ Ex ðz, tÞi þ Ey ðz, tÞj Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ðD-4Þ where i and j are unit vectors in the x and y directions, respectively Substituting (D-1) into (D-4) gives À Á Eðz, tÞ ¼ E0x cosðkz À !t þ x Þi þ E0y cos kz À !t þ y j ðD-5Þ We can also express the optical field in terms of complex quantities by writing Ex ðz, tÞ ¼ E0x cosðkz À !t þ x Þ ¼ RefE0x exp½iðkz À !t þ x ފg À Á À Á Ey ðz, tÞ ¼ E0y cos kz À !t þ y ¼ RefE0y exp½i kz À !t þ y Šg ðD-6aÞ ðD-6bÞ where Re{ .} means the real part is to be taken In complex quantities (D-5) can be written as À Á Eðz, tÞ ¼ E0x expðix Þi þ E0y exp iy j ðD-7Þ In (D-7) we have factored out and then suppressed the exponential propagator [expi(kz-!t)], since it vanishes when the intensity is formed Further, factoring out the term exp(ix) in (D-7), we can write Eðz, tÞ ¼ E0x i þ E0y expðiÞj ðD-8Þ where  ¼ y À x : The exponential propagator [expi(kzÀ!t)] is now restored in (D-8) and the real part taken: Eðz, tÞ ¼ E0x cosðkz À !tÞi þ E0y expðkz À !t þ Þj ðD-9Þ Equation (D-9) is the vector representation for elliptically polarized light There are two special forms of (D-9) The first is for  ¼ 0 or 180 , which leads to linearly polarized light at an angle [see (D-2)] If either E0y or E0x is zero, we have linear horizontally polarized light or linear vertically polarized light respectively For linearly polarized light (D-9) reduces to Eðz, tÞ ¼ ðE0x i Æ E0y jÞ cosðkz À !tÞ ðD-10Þ where Æ corresponds to  ¼ 0 and 180 , respectively The corresponding Stokes vector is seen from (D-3) to be E0x þ E20y B C B E0x À E20y C B C ðD-11Þ S¼B C @ Æ2E0x E0y A The orientation angle tan ¼ of the linearly polarized light is S2 Æ2E0x E0y ¼ S1 E20x À E20y ðD-12Þ From the well-known trigonometric half-angle formulas we readily find that tan ¼ E0y E0x which is exactly what we would expect from inspection of (D-10) Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ðD-13Þ The other special form of (D-9) is for  ¼ À90 or 90 , whereupon the polarization ellipse reduces to the standard form of an ellipse This reduces further to the equation of a circle if E0x ¼ E0y ¼ E0 For  ¼ À90 , (D-9) reduces to Eðz, tÞ ¼ E0 ½cosðkz À !tÞi þ sinðkz À !tÞjŠ ðD-14Þ and for  ¼ 90 Eðz, tÞ ¼ E0 ½cosðkz À !tÞi À sinðkz À !tÞjŠ ðD-15Þ The behavior of (D-14) and (D-15) is readily seen by considering the equations at z ¼ and then allowing !t to take on the values to 2 radians in intervals of /2 One readily sees that (D-14) describes a vector E(z, t) which rotates clockwise at an angular frequency of ! Consequently, (D-14) is said to describe left circularly polarized light Similarly, in (D-15), E(z, t) rotates counterclockwise as the wave propagates toward the viewer and, therefore, we have right circularly polarized light Equations (D-14) and (D-15) lead to a very interesting observation If we label E(z, t) in (D-14) and (D-15) as El(z, t) and Er(z, t), respectively, and add the two equations we see that El ðz, tÞ þ Er ðz, tÞ ¼ 2E0 cosð!t À kzÞi ¼ Ex ðz, tÞi ðD-16Þ Thus, a linearly polarized wave can be synthesized from two oppositely polarized circular waves of equal amplitude This property played a key role in enabling Fresnel to describe the propagation of a beam in an optically active medium The vector representation introduced by Fresnel revealed for the first time the mathematical existence of circularly polarized light; before Fresnel no one suspected the possible existence of circularly polarized light Before we conclude this section another important property of the vector formulation must be discussed Elliptically polarized light can be decomposed into two orthogonal polarized states (coherent decomposition) We consider the form of the polarization ellipse which can be represented in terms of (1) linearly Æ 45 polarized light and (2) right and left circularly polarized light, respectively We decompose an elliptically polarized beam into linear Æ45 states of arbitrary amplitudes A and B (real) and write (D-8) as Eðz, tÞ ¼ E0x i þ E0y expðiÞj ¼ Aði þ jÞ þ Bði À jÞ ¼ ðA þ BÞi þ ðA À BÞj ðD-17aÞ ðD-17bÞ Taking the vector dot product of the left- and right-hand sides of (D-17) and equating terms yields E0x ¼ A þ B ðD-18aÞ E0y ei ¼ A À B ðD-18bÞ Because A and B are real quantities, the left-hand side of (D-18b) can be real only for  ¼ 0 or 180 Thus, (D-18) becomes E0x ¼ A þ B ðD-19aÞ ÆE0y ¼ A À B ðD-19bÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved which leads immediately to A¼ E0x Æ E0y ðD-20aÞ B¼ E0x Ç E0y ðD-20bÞ We see that elliptically polarized light cannot be represented by linear Æ45 polarization states The only state that can be represented in terms of L Æ 45 light is linear horizontally polarized light This is readily seen by writing         E0x E0x E0x E0x E0x i ¼ iþ iþ jÀ j ðD-21aÞ 2 2 ¼ E0x E ½i þ jŠ þ 0x ½i À jŠ 2 ðD-21bÞ We see that the right-hand side of (D-21b) consists of linear Æ45 polarized components of equal amplitudes It is also possible to express linearly polarized light, E0xi, in terms of right and left circularly polarized light of equal amplitudes We can write, using complex quantities,         E0x E0x E E E0x i ¼ iþ i þ i 0x j À i 0x j ðD-22aÞ 2 2 ¼ E0x E ½i þ ijŠ þ 0x ½i À ijŠ 2 ðD-22bÞ We see that (D-22b) describes two oppositely circularly polarized beams of equal amplitudes We now represent elliptically polarized light in terms of right and left circularly polarized light of amplitudes (real) A and B We express (D-8) as Eðz, tÞ þ E0x i þ E0y expðiÞj ¼ Aði þ ijÞ þ Bði À ijÞ ¼ ðA þ BÞi þ iðA À BÞj ðD-23aÞ ðD-23bÞ We then find E0x ¼ A þ B ðD-24aÞ E0y ei ¼ iðA À BÞ ðD-24bÞ We see immediately that for  ¼ Æ90 , (D-24) becomes E0x ¼ A þ B ðD-25aÞ ÆE0y ¼ ðA À BÞ ðD-25bÞ so (D-23b) then becomes Eðz, tÞ ¼ E0x i Æ iE0y j Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ðD-25cÞ Equation (D-25c) is the vector representation of the standard form of the polarization ellipse For convenience we only consider the þ value of (D-25b) so the amplitudes (that is, the radii) of the circles are A¼ E0x Æ E0y ðD-26aÞ B¼ E0x À E0y ðD-26bÞ The condition  ¼ Æ90 restricts the polarization ellipse to the standard form of the ellipse [see (D-2)], namely, E2x ðz, tÞ Ey ðz, tÞ þ ¼1 E20y E20x ðD-27Þ Thus, only the nonrotated form of the polarization ellipse can be represented by right and left circularly polarized light of unequal amplitudes, A and B (D-26) In Fig D-1 we show elliptically polarized light as the superposition of the right (R) and left (L) circularly polarized light We can determine the points where the circles (RCP) and (LCP) intersect the polarization ellipse We write (D-27) as x2 y2 þ ¼1 ðA þ BÞ2 ðA À BÞ2 ðD-28Þ and the RCP and LCP circles as x2 þ y2 ¼ A2 ðD-29aÞ x2 þ y2 ¼ B2 ðA-29bÞ Figure D-1 Superposition of oppositely circularly polarized light of unequal amplitudes to form elliptically polarized light Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved where we have set Ex ¼ x and Ey ¼ y Straightforward algebra shows the points of intersection (xR, yR) for the RCP circle are rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ B 2A À B ðD-30aÞ xR ¼ Æ A rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A À B 2A þ B yR ¼ Æ ðD-30bÞ A and the points of intersection (xL, yL) for the LCP circle are rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ B 2B À A xL ¼ Æ B rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A À B 2B þ A yL ¼ Æ B ðD-31aÞ ðD-31bÞ Equations (D-30) and (D-31) can be confirmed by squaring and adding (D-30a) and (D-30b) and, similarly, (D-31a) and (D-31b) We then find that x2R þ y2R ¼ A2 ðD-32aÞ x2L þ y2L ¼ B2 ðD-32bÞ as expected As a numerical example of these results consider that we have an ellipse where A ¼ and B ¼ From (D-30) and (D-31) we then find that pffiffiffi Æ2 ðD-33aÞ xR ¼ pffiffiffi yR ¼ Æ ðD-33bÞ and the points of intersection (xL, yL) for the LCP circle are xL ¼ Æ2i pffiffiffi yL ¼ Æ ðD-34aÞ ðD-34bÞ Thus, as we can see from Fig D-1, the RCP circle intersects the polarization ellipse, whereas the existence of the imaginary number in (D-34a) shows that there is no intersection for the LCP circle We now use these results to analyze the problem of the propagation of an optical beam through an optically active medium Before we this, however, we provide some historical and physical background to the phenomenon of optical activity Optical activity was discovered in 1811 by Arago, when he observed that the plane of vibration of a beam of linearly polarized light underwent a continuous rotation as it propagated along the optic axis of quartz Shortly thereafter Biot (1774–1862) discovered this same effect in vaporous and liquid forms of various substances, such as the distilled oils of turpentine and lemon and solutions of sugar Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved and camphor Any material that causes the E field of an incident linear plane wave to appear to rotate is said to be optically active Moreover, Biot discovered that the rotation could be left- or right-handed If the plane of vibration appears to revolve counterclockwise, the substance is said to be dextrorotatory or d-rotatory (Latin dextro, right) On the other hand, if E rotates clockwise it is said to be levorotatory or l-rotatory (Latin levo, left) The English astronomer and physicist Sir John Herschel (1792–1871), son of Sir William Herschel, the discoverer of the planet Uranus, recognized that the d-rotatory and l-rotatory behavior in quartz actually corresponded to two different crystallographic structures Although the molecules are identical (SiO2), crystal quartz can be either right-or left-handed, depending on the arrangement of these molecules In fact, careful inspection shows that there are two forms of the crystals, and they are the same in all respects except that one is the mirror image of the other; they are said to be enantiomorphs of each other All transparent enantiomorphic structures are optically active In 1825, Fresnel, without addressing himself to the actual mechanism of optical activity, proposed a remarkable solution Since an incident linear wave can be represented as a superposition of R- and L-states, he suggested that these two forms of circularly polarized light propagate at different speeds in an optically active medium An active material shows circular birefringence; i.e., it possesses two indices of refraction, one for the R-state (nR) and one for the L-state (nL) In propagating through an optically active medium, the two circular waves get out of phase and the resultant linear wave appears to rotate We can see this behavior by considering this phenomenon analytically for an incident beam that is elliptically polarized; linearly polarized light is then a degenerate case In Fig D-2 we show an incident elliptically polarized beam entering an optically active medium with field components Ex and Ey After the beam has propagated through the medium the field components are E 0x and E 0y Fresnel suggested that in an optically active medium a right circularly polarized beam propagates with a wavenumber kR and a left circularly polarized beam propagates with a different wavenumber kL In order to treat this problem analytically we consider the decomposition of Ex(z, t) and Ey(z, t) separately Furthermore, we suppress the factor !t in the equations because the time variation plays no role in the final equations Figure D-2 Field components of an incident elliptically polarized beam propagating through an optically active medium Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved For the Ex(z) component we can write this in terms of circular components as Ex ½cosðkR zÞi À sinðkR zÞjŠ E ELx ðzÞ ¼ x ½cosðkL zÞi þ sinðkL zÞjŠ ERx ðzÞ ¼ ðD-35aÞ ðD-35bÞ Adding (D-35a) and (D-35b) we see that, at z ¼ 0, ERx ð0Þ þ ELx ð0Þ ¼ Ex i ðD-36Þ which shows that (D-35) represents the x component of the incident field Similarly, for the Ey(z) component we can write Ey ½sinðkR zÞi þ cosðkR zÞjŠ Ey ½À sinðkL zÞi þ cosðkL zÞjŠ ELy ðzÞ ¼ ERy ðzÞ ¼ ðD-37aÞ ðD-37bÞ Adding (D-37a) and (D-37b) we see that, at z ¼ 0, ERy ð0Þ þ ELy ð0Þ ¼ Eyx j ðD-38Þ so (D-37) corresponds to the y component of the incident field The total field E0 ðzÞ in the optically active medium is E0 ðzÞ ¼ E0x i þ E0y jþ ¼ ERx þ ELx þ ERy þ ELy ðD-39Þ Substituting (D-35) and (D-37) into (D-39) we have ! Ey Ex ½cos kR z þ cos kL zŠ þ ½sin kR z þ sin kL zŠ 2 ! Ey ÀEx ½sin kR z À sin kL zŠ þ ½cos kR z þ sin kL zŠ þj 2 E0 ðzÞ ¼ i ðD-40Þ Hence, we see that Ey Ex ½cos kR z þ cos kL zŠ þ ½sin kR z þ sin kL zŠ 2 Ey E E y ðzÞ ¼ À x ½sin kR z À sin kL zŠ þ ½cos kR z þ cos kL zŠ 2 E x ð zÞ ¼ ðD-41aÞ ðD-41bÞ Equations (D-41a) and (D-41b) can be simplified by rewriting the terms: cos kR z þ cos kL z ðD-42aÞ sin kR z À sin kL z ðD-42bÞ Let a¼ ðkR þ kL Þz ðD-43aÞ b¼ ðkR À kL Þz ðD-43bÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved so kR z ¼ a þ b ðD-44aÞ kL z ¼ a À b ðD-44bÞ and (D-42) then becomes cos kR z þ cos kL z ¼ cosða þ bÞ þ cosða À bÞ ðD-45aÞ sin kR z À sin kL z ¼ sinða þ bÞ À sinða À bÞ ðD-45bÞ Using the familiar sum and difference formulas for the cosine and sine terms of the right-hand sides of (D-45a) and (D-45b) along with (D-43), we find that ! ! ðk þ kL Þz ðk À kL Þz cos kR z þ cos kL z ¼ cos R cos R ðD-46aÞ 2 ! ! ðkR þ kL Þz ðkR À kL Þz sin kR z À sin kL z ¼ cos sin ðD-46bÞ 2 The term cos(kR þ kL)z/2 in (D-46a) and (D-46b) plays no role in the final equations and can be dropped Substituting the remaining cosine and sine term in (D-46) into (D-41), we finally obtain Ex ðk À kL Þz Ey ðk À kL Þz þ sin R cos R 2 2 E ðk À kL Þz Ey ðk À kL Þz þ cos R E0y ðzÞ ¼ À x sin R 2 2 E0x ðzÞ ¼ ðD-47aÞ ðD-47bÞ We see that (D-47) are the equations for rotation of Ex and Ey We can write (D-47) in terms of the Stokes vector and the Mueller matrix as 10 01 S0 0 S0 CB C B 0C B B S1 C B cos sin CB S1 C CB C B C¼B ðD-48aÞ B S0 C B À sin cos CB S C A@ A @ 2A @ 0 0 S3 S03 where ¼ ðkR À kL Þz ðD-48bÞ The angle of rotation can be expressed in terms of the refractive indices nR and nL of the medium and the wavelength  of the incident beam by writing k R ¼ k nR ¼ 2nR  ðD-49aÞ kL ¼ k0 nL ¼ 2nL  ðD-49bÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure D-3 Fresnel’s construction of a composite prism consisting of R-quartz and L-quartz to demonstrate optical activity and the existence of circularly polarized light and k0 ¼ 2/ If nR nL the medium is d-rotatory, and if nR ! nL the medium is l-rotatory Substituting (D-49) into (D-48), we then have ¼ ðnR À nL Þz  ðD-50Þ The quantity /d is called the specific rotatory power For quartz it is found to be 21.7 /mm for sodium light, from which it follows that |nR À nL| ¼ 7.1Â10À5 Thus, the small difference in the refractive indices shows that at an optical interface the two oppositely circularly polarized beams will be very difficult to separate Fresnel was able to show the existence of the circular components and separate them by an ingenious construction of a composite prism consisting of R- and L-quartz, as shown in Fig D-3 He reasoned that since the two component traveled with different velocities they should be refracted by different amounts at an oblique interface In the prism the separation is increased at each interface This occurs because the righthanded circular component is faster in the R-quartz and slower in the L-quartz The reverse is true for the left-handed component The former component is bent down and the latter up, the angular separation increasing at each oblique interface If the two images of a linearly polarized source are observed through the compound prism and then examined with a linear polarizer the respective intensities are unaltered when the polarizer is rotated Thus, the beams must be circularly polarized The subject of optical activity is extremely important In the field of biochemistry a remarkable behavior is observed When organic molecules are synthesized in the laboratory, an equal number of d- and l-isomers are produced, with the result that the mixture is optically inactive One might expect in nature that equal amounts of d- and l-stereoisomers would exist This is by no means the case Natural sugar (sucrose, C12H22O6) always appears in the d-rotatory form, regardless of where it is grown or whether it is extracted from sugar cane or sugar beets Moreover the sugar dextrose of d-glucose (C6H12O11) is the most important carbohydrate in human metabolism Evidently, living cells can distinguish in a manner not yet fully understood between l- and d-molecules One of the earliest applications of optical activity was in the sugar industry, where the angle of rotation was used as a measure of the quality of the sugar (saccharimetry) In recent years optical activity has become very important in other branches of chemistry For example, the artificial sweetener aspartame and the pain reducer ibuprofen are optically active In the pharmaceutical industry it has been estimated that approximately 500 out of the nearly 1300 commonly used drugs are optically active The difference between the l- and d-forms can, it is believed, lead Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved to very undesirable consequences For example, it is believed that the optically active sedative drug thalidomide when given in the l-form acts as a sedative, but the d-form is the cause of birth defects Interest in optical activity has increased greatly in recent years Several sources are listed in the references Of special interest is the stimulating article by Applequist, which describes the early investigations of optical activity by Biot, Fresnel, and Pasteur, as well as recent investigations, and provides a long list of related references REFERENCES Hecht, E and Zajac, A., Optics, Addison-Wesley, Reading, MA, 1979 Wood, R W., Physical Optics, 3rd ed., Optical Society of America, Washington, DC, 1988 Callewaert, D M and Genuyea, J., Basic Chemistry, Worth, New York, 1980 Applequist, J., American Scientist, 75, 59 (1987) Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved [...]... understood between l- and d-molecules One of the earliest applications of optical activity was in the sugar industry, where the angle of rotation was used as a measure of the quality of the sugar (saccharimetry) In recent years optical activity has become very important in other branches of chemistry For example, the artificial sweetener aspartame and the pain reducer ibuprofen are optically active In the. .. then examined with a linear polarizer the respective intensities are unaltered when the polarizer is rotated Thus, the beams must be circularly polarized The subject of optical activity is extremely important In the field of biochemistry a remarkable behavior is observed When organic molecules are synthesized in the laboratory, an equal number of d- and l-isomers are produced, with the result that the. .. ðD-48bÞ The angle of rotation can be expressed in terms of the refractive indices nR and nL of the medium and the wavelength  of the incident beam by writing k R ¼ k 0 nR ¼ 2nR  ðD-49aÞ kL ¼ k0 nL ¼ 2nL  ðD-49bÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure D-3 Fresnel’s construction of a composite prism consisting of R-quartz and L-quartz to demonstrate optical activity and the. .. the existence of circularly polarized light and k0 ¼ 2/ If nR nL the medium is d-rotatory, and if nR ! nL the medium is l-rotatory Substituting (D-49) into (D-48), we then have ¼ ðnR À nL Þz  ðD-50Þ The quantity /d is called the specific rotatory power For quartz it is found to be 21.7 /mm for sodium light, from which it follows that |nR À nL| ¼ 7.1Â10À5 Thus, the small difference in the refractive... In the prism the separation is increased at each interface This occurs because the righthanded circular component is faster in the R-quartz and slower in the L-quartz The reverse is true for the left-handed component The former component is bent down and the latter up, the angular separation increasing at each oblique interface If the two images of a linearly polarized source are observed through the. .. indices shows that at an optical interface the two oppositely circularly polarized beams will be very difficult to separate Fresnel was able to show the existence of the circular components and separate them by an ingenious construction of a composite prism consisting of R- and L-quartz, as shown in Fig D-3 He reasoned that since the two component traveled with different velocities they should be refracted... approximately 500 out of the nearly 1300 commonly used drugs are optically active The difference between the l- and d-forms can, it is believed, lead Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved to very undesirable consequences For example, it is believed that the optically active sedative drug thalidomide when given in the l-form acts as a sedative, but the d-form is the cause of birth defects... result that the mixture is optically inactive One might expect in nature that equal amounts of d- and l-stereoisomers would exist This is by no means the case Natural sugar (sucrose, C12H22O6) always appears in the d-rotatory form, regardless of where it is grown or whether it is extracted from sugar cane or sugar beets Moreover the sugar dextrose of d-glucose (C6H12O11) is the most important carbohydrate... d-form is the cause of birth defects Interest in optical activity has increased greatly in recent years Several sources are listed in the references Of special interest is the stimulating article by Applequist, which describes the early investigations of optical activity by Biot, Fresnel, and Pasteur, as well as recent investigations, and provides a long list of related references REFERENCES 1 Hecht, E... recent investigations, and provides a long list of related references REFERENCES 1 Hecht, E and Zajac, A., Optics, Addison-Wesley, Reading, MA, 1979 2 Wood, R W., Physical Optics, 3rd ed., Optical Society of America, Washington, DC, 1988 3 Callewaert, D M and Genuyea, J., Basic Chemistry, Worth, New York, 1980 4 Applequist, J., American Scientist, 75, 59 (1987) Copyright © 2003 by Marcel Dekker, Inc All