Appendix DVector Representation of the OpticalField: Application to Optical Activity

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 th

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 intro-duced by Fresnel and called the vector representation for polarized light This repre-sentation 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

Eyðz, tÞ ¼E0ycos kz  !t þ  y

ðD-1bÞ Eliminating the propagator kz!t between (D-1a) and (D-1b) yields

E2xðz, tÞ

E2

0x

þE2ðz, tÞ

E2 0y

2Exðz, tÞEyðz, tÞcos 

E0xE0y ¼sin

The Stokes vector corresponding to (D-1) is, of course,

S ¼

E20xþE20y

E20xE20y 2E0xE0ycos  2E0xE0ysin 

0 B B

@

1 C C

In the xy plane we construct the vector E(z, t):

where i and j are unit vectors in the x and y directions, respectively Substituting (D-1) into (D-4) gives

E z, tð Þ ¼E0xcos kz  !t þ ð xÞi þ E0ycos kz  !t þ  y

We can also express the optical field in terms of complex quantities by writing

Exðz, tÞ ¼E0xcos kz  !t þ ð xÞ ¼RefE0xexp½i kz  !t þ ð xÞg ðD-6aÞ

Eyðz, tÞ ¼E0ycos kz  !t þ  y

¼RefE0yexp½i kz  !t þ  y

where Re{ .} means the real part is to be taken In complex quantities (D-5) can be written as

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

where  ¼ yx:

The exponential propagator [expi(kz!t)] is now restored in (D-8) and the real part taken:

E z, tð Þ ¼E0xcos kz  !tð Þi þ E0yexp kz  !t þ ð 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 E0yor E0xis zero, we have linear horizontally polarized light or linear vertically polarized light respectively For lin-early polarized light (D-9) reduces to

where  corresponds to  ¼ 0
and 180
, respectively The corresponding Stokes vector is seen from (D-3) to be

S ¼

E20xþE20y

E20xE20y

2E0xE0y 0

0 B B

@

1 C C

The orientation angle of the linearly polarized light is

tan 2 ¼S2

S1 ¼

2E0xE0y

E2 0xE2 0y

ðD-12Þ From the well-known trigonometric half-angle formulas we readily find that tan ¼E0y

which is exactly what we would expect from inspection of (D-10)

The other special form of (D-9) is for  ¼ 90 or 90 , whereupon the polar-ization ellipse reduces to the standard form of an ellipse This reduces further to the equation of a circle if E0x¼E0y¼E0 For  ¼ 90
, (D-9) reduces to

and for  ¼ 90

The behavior of (D-14) and (D-15) is readily seen by considering the equations at

z ¼0 and then allowing !t to take on the values 0 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

Elðz, tÞ þErðz, tÞ ¼2E0cos !t  kzð Þi ¼ Exðz, 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 mathema-tical 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  45
polarized light and (2) right and left circularly polarized light, respectively We decompose an elliptically polar-ized beam into linear 45
states of arbitrary amplitudes A and B (real) and write (D-8) as

E z, tð Þ ¼E0xi þ E0yexp ið Þj ¼ A i þ jð Þ þB i  jð Þ ðD-17aÞ

Taking the vector dot product of the left- and right-hand sides of (D-17) and equat-ing terms yields

Because A and B are real quantities, the left-hand side of (D-18b) can be real only for

 ¼0
or 180
Thus, (D-18) becomes

which leads immediately to

A ¼E0xE0y

B ¼E0xE0y

We see that elliptically polarized light cannot be represented by linear 45
polariza-tion states The only state that can be represented in terms of L  45
light is linear horizontally polarized light This is readily seen by writing

E0xi ¼ E0x

2

i þ E0x 2

i þ E0x 2

j  E0x 2

¼E0x

2 ½i þ j þ

E0x

We see that the right-hand side of (D-21b) consists of linear 45
polarized compo-nents 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,

E0xi ¼ E0x

2

i þ E0x 2

i þ i E0x 2

j  i E0x 2

¼E0x

2 ½i þ ij þ

E0x

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

E z, tð Þ þE0xi þ E0yexp ið Þj ¼ A i þ ijð Þ þB i  ijð Þ ðD-23aÞ

We then find

We see immediately that for  ¼ 90
, (D-24) becomes

so (D-23b) then becomes

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

A ¼E0xE0y

B ¼E0xE0y

The condition  ¼ 90
restricts the polarization ellipse to the standard form of the ellipse [see (D-2)], namely,

E2xðz, tÞ

E2

0x

þE2ðz, tÞ

E2 0y

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

A þ B

y2

A  B

and the RCP and LCP circles as

Figure D-1 Superposition of oppositely circularly polarized light of unequal amplitudes to form elliptically polarized light

where we have set Ex¼x and Ey¼y Straightforward algebra shows the points of intersection (xR, yR) for the RCP circle are

xR¼ A þ B

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A  B A

r

ðD-30aÞ

yR¼ A  B

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A þ B A

r

ðD-30bÞ and the points of intersection (xL, yL) for the LCP circle are

xL¼ A þ B

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2B  A B

r

ðD-31aÞ

yL¼ A  B

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2B þ A B

r

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

as expected

As a numerical example of these results consider that we have an ellipse where

A ¼3 and B ¼ 1 From (D-30) and (D-31) we then find that

xR¼2 ffiffiffi

5 p

yR¼  ffiffiffi

5

p

ðD-33bÞ and the points of intersection (xL, yL) for the LCP circle are

yL¼  ffiffiffi

5

p

ðD-34bÞ Thus, as we can see fromFig 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 do 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

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 levorotatoryor 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 repre-sented 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 refrac-tion, 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 phenom-enon 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 opti-cally active medium with field components Ex and Ey After the beam has propa-gated through the medium the field components are E0xand E0y

Fresnel suggested that in an optically active medium a right circularly polarized beam propagates with a wavenumber kRand a left circularly polarized beam pro-pagates 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

For the Ex(z) component we can write this in terms of circular components as

ERxð Þ ¼z Ex

ELxð Þ ¼z Ex

Adding (D-35a) and (D-35b) we see that, at z ¼ 0,

which shows that (D-35) represents the x component of the incident field Similarly, for the Ey(z) component we can write

ERyð Þ ¼z Ey

ELyð Þ ¼z Ey

Adding (D-37a) and (D-37b) we see that, at z ¼ 0,

so (D-37) corresponds to the y component of the incident field The total field E0ðzÞin the optically active medium is

E0ð Þ ¼z E0xi þ E0yjþ ¼ ERxþELxþERyþELy ðD-39Þ Substituting (D-35) and (D-37) into (D-39) we have

E0ð Þ ¼z i Ex

2 ½cos kRz þcos kLz þ

Ey

2 ½sin kRz þsin kLz

þj Ex

2 ½sin kRz sin kLz þ

Ey

2 ½cos kRz þsin kLz

ðD-40Þ Hence, we see that

E0xð Þ ¼z Ex

2 ½cos kRz þcos kLz þ

Ey

E0yð Þ ¼ z Ex

2 ½sin kRz sin kLz þ

Ey

2 ½cos kRz þcos kLz ðD-41bÞ Equations (D-41a) and (D-41b) can be simplified by rewriting the terms:

Let

a ¼ðkRþkLÞz

b ¼ðkRkLÞz

and (D-42) then becomes

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

cos kRz þcos kLz ¼2 cos ðkRþkLÞz

2

cos ðkRkLÞz

2

ðD-46aÞ

sin kRz sin kLz ¼2 cos ðkRþkLÞz

2

sin ðkRkLÞz

2

ðD-46bÞ

The term cos(kRþkL)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

E0xð Þ ¼z Ex

2 cos

kRkL

Ey

2 sin

kRkL

E0yð Þ ¼ z Ex

2 sin

kRkL

Ey

2 cos

kRkL

We see that (D-47) are the equations for rotation of Exand Ey We can write (D-47)

in terms of the Stokes vector and the Mueller matrix as

S00

S01

S02

S03

0

B

B

@

1 C C

0 sin 2 cos 2 0

0 B B

@

1 C C A

S0

S1

S2

S3

0 B B

@

1 C C

where

 ¼ðkRkLÞz

The angle of rotation  can be expressed in terms of the refractive indices nRand nL

kR¼k0nR¼2nR

ðD-49aÞ

kL¼k0nL¼2nL

ðD-49bÞ

and k0¼ R nL the medium is d-rotatory, and if nR nL the medium

is l-rotatory Substituting (D-49) into (D-48), we then have

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 |nRnL| ¼ 7.1105 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 right-handed 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 biochem-istry 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 under-stood 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 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