The more times you see an explanation of a physical phenomenon or a statement about physical reality, especially in the form of an invariable mantra, especially in a textbook unaccompanied by any qualifications, the more certain you can be that it is wrong. Stated more succinctly, repetition increases the probability of incorrectness. This is a law of almost universal validity. One example is the assertion that the electric and magnetic fields of light waves are always perpendicular to each other and to the direction of propagation. Because this assertion has been made so often, without qualification, you can be certain it is wrong. And indeed it is. Ask any electrical engineer who knows something about near fields. In Section 4.1 we noted that the Poynting vector

Fundamentals of Atmospheric Radiation: An Introduction with 400 Problems. Craig F. Bohren and Eugene E. Clothiaux Copyright © 2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-40503-8

specifies the magnitude and direction of energy transport by any electromagnetic field at any point. Both the electric field E and the magnetic field H are necessarily perpendicular to S, although they are not necessarily perpendicular to each other or to the direction of propagation (if by which is meant the normal to a surface of constant phase). For example, the fields within an illuminated sphere (indeed, any particle) are not perpendicular to each other, and the concept of a surface of constant phase is meaningless. The fields scattered by a sphere also are not perpendicular to each other except approximately at sufficiently large (compared with the wavelength) distances; and the concept of a surface of constant phase has its limitations. All we can be certain of is that the electric and magnetic fields lie in a plane to which the Poynting vector is perpendicular. It has become the custom to specify the polarization properties of electromagnetic waves by the electric field, although the magnetic field would serve just as well, and you occasionally come across works (especially by British authors) in which polarization is based on the magnetic field.

7.1.1 Vibration Ellipse and Ellipsometric Parameters

The only assumption we make at this point is that the electric field is time-harmonic:

where E is the complex representation of the electric field (see Sec. 2.5). To specify the polarization state of E, however, we need the real field. Because E lies in a plane perpendicular to S, only two components are needed. We denote two orthogonal unit vectors as and e|| chosen such that x e|| is in the direction of the Poynting vector. It will become apparent when we discuss applications why the coordinate axes are denoted as perpendicular (±) and parallel (||). The field components are the real parts of

E± = a± exp{—i(^ + ut)}, E\\ = a\\ exp{-i(tf || + ut)}, (7.3)

where the amplitudes a and phases tf are real functions that may depend on position but not time. Without loss of generality we may take the amplitudes as positive because the field components can be negative by virtue of the phases (i.e., cos n = -1).

At a fixed point in space the tip of the electric vector (the point with coordinates given by the real parts of E± and E|) endlessly traces out a closed, bounded curve. When the two phases are equal, this curve is a straight line with slope equal to the ratio of amplitudes. When the phases differ by n/2, the curve is an ellipse with principal axes aligned along the coordinate axes, where the lengths of the two semi-axes are and a| .A circle results when these two amplitudes are equal.

In general, Eq. (7.3) describes an arbitrarily oriented ellipse of arbitrary ellipticity (not to be confused with eccentricity), defined as the ratio of the minor to major axis lengths (Fig. 7.1). The azimuth of this vibration ellipse is the angle between the major axis and a reference axis (e.g., one of the coordinate axes). One more ellipsometric parameter of the vibration ellipse is its handedness, the rotational sense in which it is traced out in time. There is no universal convention for what is meant by right- and left-handed rotation. Moreover, investigators in a particular field often assume that everyone knows what their convention is so feel no need to state it. We adopt the convention of calling a field right-handed if the vibration ellipse is traced

out clockwise as imagined to be seen looking into the Poynting vector. With this convention the helix traced out in space by the tip of the electric field vector is what all the world calls right-handed.

The electric field described by Eq. (7.3) is 100% or completely polarized in that it has a definite and fixed vibration ellipse. The general state of complete polarization is elliptical, special cases being linear and circular. But some textbooks, and, even more so, books on popular science, convey the notion that by polarization is meant linear polarization, no other kind being conceivable. To make matters worse, linearly polarized light is sometimes called plane polarized, especially in older works. This is a poor choice of terminology on several grounds. If a plane electromagnetic wave [Eq. (7.4)] is linearly polarized we would have the awkward designation plane-polarized plane wave (and polarized parallel or perpendicular to yet another plane). The first plane is defined by the electric field vector and the direction of propagation (equivalently, the plane surface traced out by the field as it propagates), the second plane is a surface of constant phase. To be consistent we would have to describe elliptically polarized light as elliptical-helicoidally polarized light because its electric field traces out an elliptical helicoid in space. Yet this is unnecessary because the polarization state of a plane wave (indeed, any wave) is specified by the ellipsometric parameters, which have nothing essential to do with surfaces.

Our experience has been that people who were taught at an impressionable age that light is plane polarized then find it difficult to understand elliptically polarized light and even more difficult to understand partially polarized light. Indeed, they sometimes confuse unpolarized light with circularly polarized light. And yet partially polarized light is readily understood beginning with a firm grasp of completely polarized light. The essential property of such light is complete correlation between two orthogonal components of the electric field. They may fluctuate in time, but if they do so synchronously (i.e., the ratio of amplitudes is constant as is the phase difference), the vibration ellipse has a definite and fixed form. Partially polarized light results when there is partial correlation between the two orthogonal components; unpolarized light results when there is no correlation.

We are aware of the existence of polarization only because two beams, identical in all respects except in one or more ellipsometric parameters (ellipticity, azimuth, handedness) can interact with matter in observably different ways. Were it not for this, the polarization state of time-harmonic fields would be a kind of non-functional adornment, like the fins on 1959 Cadillacs. Only two things can happen to a field when it interacts with matter: its amplitude or phase (or both) are changed. If two orthogonal components of the field are changed differently, the polarization state is changed. By "changed" here is meant that the polarization state of an incident (or exciting) wave is different from the polarization state of waves it gives rise to.

7.1.2 Orthogonally Polarized Waves do not Interfere

The general plane harmonic (complex) electric wave has the form

where E0 is constant in space and time and the wave vector k may be complex; the magnetic field H is given by a similar expression. These fields must satisfy where C is a frequency-dependent parameter (possibly complex if the propagation medium is absorbing) characteristic of the medium in which the wave propagates; its value is of no consequence here. Keep in mind that the real and imaginary parts of k need not be parallel to each other; that is, the surfaces of constant phase and the surfaces of constant amplitude need not coincide (see Prob. 7.52). If they do, the wave is said to be homogeneous; if not, it is inhomogeneous. Inhomogeneous waves are not the product of an unbridled imagination. They can be produced readily by illuminating an absorbing medium at oblique incidence. Only if a wave is homogeneous, that is, its wave vector has the form k = k e, where k may be complex but e is a real unit vector, are the (real) electric and magnetic fields perpendicular to e, the direction of propagation. And only if k is real are the (real) electric and magnetic fields perpendicular to each other. From now on we assume, unless stated otherwise, that the waves of interest are homogeneous (k = k e) and the medium in which they propagate is nonabsorbing (k is real). Absolutely plane waves and nonabsorbing media do not exist. We can get away with assuming they do because measurements of polarization are almost always made in negligibly absorbing media (e.g., air) and at sufficiently large distances (compared with the wavelength) from finite sources (e.g., bounded scatterers) that the fields from them are approximately planar over the detector. But if we were to inquire about polarization of waves in absorbing media or close to sources, much of the following analysis would not be strictly applicable.

Given the assumptions in the previous paragraph, the Poynting vector is k • E = 0, H = Ck x E, k • H = 0,

which follows from Eqs. (7.1) and (7.5) and the identity

Equation (7.6) is a generalization to vector waves of a result in Section 3.3.2, namely, energy propagation by a scalar plane harmonic wave (on a string) is proportional to the square of a wave function. To avoid cluttered notation we do not use different symbols for fields and their complex representations, trusting that context indicates which is meant.

Two plane harmonic waves are said to be orthogonally polarized if they are opposite in handedness and the azimuths of their vibration ellipses are perpendicular. Orthogonally polarized waves do not interfere in that the Poynting vector of their sum is the sum of their Poynting vectors. To prove this, consider two such waves:

Ei = a\ cos wt e|| + b\sin wt e^, E2 = b2 sin wt e|| + a2 cos wt e^, (7.8)

where aj and bj are positive but otherwise arbitrary. The Poynting vector corresponding to the sum of these two waves is

S = Ck[(ai + a2)cos2 wt + (bf + b2)sin2 wt + 2(a1b2 + a2b1)sin wt cos wt]. (7.9)

If the two fields are orthogonal in the restricted sense that E1 •E2 = 0 then a1b2+a2b1 = 0 and the waves do not interfere at any instant. Regardless of their state of orthogonal polarization the time-averaged Poynting vectors are additive because (sin wt cos wt) = 0:

We are usually interested in Poynting vectors averaged over times large compared with the period (inverse frequency) of waves. Although the fields in Eq. (7.1) must be real, we can determine time-averaged Poynting vectors directly from the complex representations of fields:

This equation is valid for any time-harmonic electromagnetic field. A more restricted version, applicable only to plane homogeneous waves in nonabsorbing media, is

This equation is at the heart of what follows.

7.1.3 Stokes Parameters and the Ellipsometric Parameters

Although we can imagine watching the tip of an electric vector rotating at, say, 1015 Hz, to think that we could actually do so is pure fantasy. All that we can measure, usually, is time-averaged irradiances. Such measurements of the magnitude of the Poynting vector must therefore be the route to ellipsometric parameters, and given that Eq. (7.3) is the equation of an ellipse, they must depend only on the amplitudes a« and a±_and the phases fin and

From Eqs. (7.3) and (7.12) the time-averaged irradiance of a beam, denoted here by I, is the sum of squares of amplitudes

Missing from this equation is a constant factor, which we ignore here because absolute measurements are not needed to determine ellipsometric parameters. To obtain the separate amplitudes we need the help of an ideal linear polarizer (or linear polarizing filter). Such a filter completely transmits light linearly polarized in a particular direction but does not transmit light linearly polarized in the orthogonal direction. As its name implies, an ideal linear polarizer does not exist but we can come close, at least over a restricted range of wavelengths. An example is the sheet polarizers used in polarizing sunglasses or in polarizing filters for cameras (the function of which is explained in Sec. 7.4). Absorption by such a sheet polarizer is asymmetric in that nd C 1, where k is the absorption coefficient and d the thickness of the sheet, for light linearly polarized along the transmission axis, whereas nd > 1 for light linearly polarized perpendicular to this axis. At visible and near-visible wavelengths this difference in absorption coefficients is a consequence of anisotropy of the sheet material on a molecular scale. We cannot see the transmission axis, although we might be able to see that of a polarizing filter for microwave radiation. A medium with different absorption coefficients for different orthogonal linear states of polarization is said to be linearly dichroic.

Now we imagine inserting an ideal linear polarizing filter in the beam and measuring transmitted irradiances, first for the transmission axis along e||, then along e±_, and then subtracting these two irradiances:

We now have done enough to obtain the amplitudes:

What about the phases? From Eqs. (7.3) and (7.12) it would seem that to obtain phases we must transmit a bit of both orthogonal components of an electric field. For example, if we align a linear polarizing filter with its transmission axis at 45° to e j, the transmitted amplitude is

Rotate the filter by 90° and the transmitted amplitude is

The difference in the irradiances corresponding to Eqs. (7.16) and (7.17) is

Measurement of I, Q, and U is sufficient to obtain cos S, but because cos S = cos(-S) is not sufficient to determine the handedness of the wave. Given cos S we cannot say if S is positive or negative, which determines handedness. To find this quantity requires the help of ideal circular polarizers (or circular polarizing filters), devices that completely transmit circularly polarized light of one handedness but do not transmit circularly polarized light of the opposite handedness. Such circular polarizers are much more difficult to find than linear polarizers. Media with different absorption coefficients for different states of circularly polarized light, said to be circularly dichroic, exist. For example, our bodies and all organic matter are chock full of helical molecules (e.g., the double helix of DNA), and helices are not superposable on their mirror images: the reflection of a right-handed helix is a left-handed helix. Because of this mirror asymmetry we expect absorption by such molecules to be different for different states of circular polarization. And indeed this is so, but the difference is usually greatest at ultraviolet frequencies and, moreover, media with greatly different absorption coefficients are difficult to find. Nevertheless, we can imagine thought experiments with ideal circular polarizers.

To discuss circularly polarized light it is convenient to introduce a set of complex basis vectors:

which are orthonormal in that eR • eR = 1, eL • eR = 1, eR • eR = 0. (7.20)

eR corresponds to a right-circularly polarized wave of unit amplitude, eL to a left-circularly polarized wave. An arbitrary electric field thus can be written

where the circularly polarized (complex) amplitudes are related to the linearly polarized amplitudes by

Now imagine that an ideal right-circular polarizer is inserted in the beam and the transmitted irradiance ERER is measured, then the irradiance ElEL transmitted by an ideal left-circular polarizer is measured, and the second irradiance subtracted from the first:

V = ErER - ELEL = i(E^E*± - E±E*\) = 2^^^} = 2a^a± sinS. (7.23)

Knowing both sin S and cos S we can determine the sign of S and hence the handedness of the wave.

The four quantities {I, Q, U, V}, which are no more than sums and differences of irradi-ances, are called the Stokes parameters, first set down by Sir George Gabriel Stokes in 1852. Even more than 150 years later his paper "On the composition and resolution of streams of

polarized light from different sources" is still worth reading. You are likely to encounter different symbols for the Stokes parameters (he used A, B, C, and D), and linear combinations of Stokes parameters are also valid Stokes parameters. They sometimes are written compactly as a column matrix

/ |
I |
\ | |

Q | |||

U | |||

V |
V |
) |

EnE* + El El E\\E*\ — ElEl EnEl + ElE* y i(E\\E*± — EE)

EnE* + El El E\\E*\ — ElEl EnEl + ElE* y i(E\\E*± — EE)

called the Stokes vector, although it does not have the same rights and privileges as proper vectors. For example, the Stokes parameters are not independent in that

Stokes's paper appeared a dozen years before the publication of Maxwell's famous electromagnetic theory of light and 32 years before the publication of Poynting's work. Much was known about the properties of light waves even before they were fully grounded in an adequate theory.

Now we have to show that the Stokes parameters determine the ellipsometric parameters. In what follows fields are real. The real parts of the fields in Eq. (7.3) can be expanded using the identity for the cosine of the sum of angles and written in matrix form as

an cos fin —an sin aj_ cos —a± sin cos wt sin wt

In a coordinate system for which the field components are

the field traces out a right-handed ellipse with minor axis b, major axis a (if a> b), which follows from solving Eq. (7.27) for cos wt and sin wt, then squaring and adding them. The transformation from the original coordinate system to the primed coordinate system is (Fig. 7.2)

E« \ = ( cos y sin y \ / E« E± J y — sin y cos y / \ E'±

a cos y b sin y \ i cos wt —a sin y b cos y J y sin wt

Equality of Eqs. (7.26) and (7.28) requires that a« cos = a cos y, (7.29)

Square and add the left sides of these equations and set the result equal to the sum of the squares of the right side:

where 0 < n < n/4 and tan n = b/a. Now multiply Eq. (7.29) by Eq. (7.30), Eq. (7.31) by Eq. (7.32), add the result, and use the identities for the sine and cosine of the sum of angles:

U = 2a«a± cos S = (b2 — a2) sin 2y = —A2 cos 2nsin 2y. (7.35)

Now square Eqs. (7.32) and (7.30), take their difference, square Eqs. (7.29) and (7.31), take their difference, and finally take the difference of the result:

which yields

Q = a2 — a\ = A2 cos2n cos2y. (7.37) Finally, to obtain V, multiply Eqs. (7.29) and (7.32), Eqs. (7.30) and (7.31), and add to obtain ana± sin S = ab, (7.38)

which from Eq. (7.34) yields

If we go through the same steps for a left-circularly polarized wave, I, Q, and U are unchanged whereas V becomes

Instead of having separate sets of equations for left- and right-circularly polarized light we can combine them by allowing n to lie in the range (-n/4, n/4), where negative angles correspond to left-circularly polarized light, positive angles to right-circularly polarized light. To recapitulate:

I = A2, Q = A2 cos 2ncos 2y, U = -A2 cos 2nsin 2y, V = A2 sin 2n, (7.41)

where 0 < y < n and —n/4 < n < n/4. Because tan 2y does not uniquely determine y we need additional information: if U < 0, 0 < y < n/2, whereas if U > 0, n/2 < y < n. From Eq. (7.41) it follows that I and V do not depend on the coordinate system (i.e., on y), Q and U do, but the sum of their squares does not.

The surfaces of constant phase and amplitude for the plane wave Eq. (7.4) are infinite in extent. Thus the electric field of this wave occupies all space, which, of course, is physically unrealistic. To apply the previous analysis to real beams finite in lateral extent their properties have to be more or less laterally uniform. The Stokes parameters [Eq. (7.24)] were obtained by way of thought experiments that are easy to state but not all of them readily done in practice. Nevertheless, once we know the form of these parameters we can devise feasible ways of measuring them with readily available linear retarders and polarizing filters (see Prob. 7.36 at the end of this chapter).

An electric wave described by Eq. (7.2) or Eq. (7.3) is necessarily completely polarized in that its vibration ellipse is traced out with monotonous regularity from the beginning until the end of time (actually, this time interval need not span eternity, just much longer than the period of the wave). Radiation from a microwave or radio antenna might closely fit this description because an antenna is a coherent object, its parts fixed relative to each other (on the scale of the wavelength), driven by electric currents that are more or less time-harmonic. It would take some ingenuity to make a microwave or radio antenna that did not radiate completely polarized waves. Radiation at much shorter wavelengths, however, often originates from vast arrays of tiny antennas (molecules) emitting more or less independently of each other, and hence we would not expect the same degree of regularity of the radiation from such sources. The extreme example of irregularity is unpolarized light whereas the extreme example of regularity is completely polarized light, both idealizations never strictly realized in nature. But what is unpolarized light?

Perhaps the simplest way to define such light is operationally, subject to previous caveats about ideal linear and circular polarizers. What kind of experimental tests can we devise to determine if a beam is unpolarized? Suppose that we transmit it through an ideal linear polarizer and discover that regardless of the orientation of its transmission axis, the transmitted irradiance is the same. This implies that there is no preferred direction of the electric field, for if there were the irradiance would vary. According to our operational definition of the Stokes parameters, Q = U = 0 for this beam. But wait! A circularly-polarized beam would yield the same result. So we now have to determine if the beam exhibits a preferential handedness. First transmit the beam through an ideal left-circular polarizer, then through a right-circular polarizer. If the two transmitted irradiances are equal, V = 0, and the electric field of the beam exhibits no preference for left-handed over right-handed rotation. Thus our operational definition of unpolarized light is that for which Q = U = V = 0. The Stokes parameters of partially polarized light also do not satisfy Eq. (7.25) but Q, U, V are not all zero.

We can put more theoretical flesh onto these bare bones by extending Eq. (7.3) to quasi-monochromatic radiation with (real) electric field components

E\\ (t) = a\\ (t)cos{$||(t)+wt], El (t) = a±(t)cos{tf±(t)+ wt], (7.43)

where the amplitudes and phases now vary with time but much more slowly than cos wt. With this restriction the electric field Eq. (7.43) and its associated magnetic field approximately satisfy Eq. (7.5). The instantaneous Poynting vector corresponding to Eq. (7.43) is

the magnitude of which (within a constant factor) is

—2(a| cos-1| sin+ al cos -1 sin -i) sin wt cos wt, where the amplitudes and phases may depend on time (not explicit to keep the notation uncluttered). To determine the time average of Eq. (7.45) requires evaluating integrals of the form

- / f(t) cos2 wt dt, - / g(t) sin2 wt dt, - / h(t) sin wt cos wt dt. (7.46)

We need consider only the first of these integrals because it sets the pattern for the other two. Divide the range of integration into N equal intervals At:

(/ cos2 wt) = - f(t) cos2 wt dt = —— Y / /(t) cos2 wt dt. (7.47)

From the mean-value theorem of integral calculus

where U < U < U + At. By the definition of quasi-monochromatic light we can choose At » 1/uj such that f(t) is approximately constant [call the value /(¿¿)] over this time interval, and hence the integral of the cosine squared is approximately At/2 and the time average is approximately

Similarly,

(ffsin2 cut) » gjtj), (h sin cut cos cut) « 0. (7.50)

From Eqs. (7.45), (7.49), and (7.50) it therefore follows that the time-averaged irradiance for quasi-monochromatic light is

As previously, all common factors are omitted. Because the Stokes parameters are sums and differences of irradiances, the other three parameters for such light are given by similar expressions:

An example of a quasi-monochromatic beam of light is collimated sunlight passed through an ordinary (as opposed to a polarizing) filter, a device that transmits light only over a band of frequencies. The spectral width of this transmitted light may be quite narrow but the amplitudes and phases of its orthogonal field components fluctuate over times large compared with the period and small compared with the response time of the detector.

According to Eq. (7.42) the ellipsometric parameters depend only on ratios of Stokes parameters, which in turn implies that they depend only on the ratio of the amplitudes a| and and the difference in phases S = — Suppose that these amplitudes and phases fluctuate in time but do so synchronously, that is, they are correlated, the ratio of amplitudes and the difference in phases constant in time. It then follows from Eqs. (7.51)-(7.54) that 12 = Q2 + U2 + V2, and hence the light is completely polarized despite the fluctuations. Correlation is essential to understanding polarized light. Complete correlation corresponds to completely polarized light, no correlation to unpolarized light, and partial correlation to partially polarized light.

We may visualize Eq. (7.43) as follows. Over a time interval of several periods the electric field vector traces out a more or less definite vibration ellipse, but with the passage of time the vibration ellipse changes. If all vibration ellipses are traced out over the response time of the detector the light is unpolarized.

7.1.5 Degree of Polarization

Any beam with Stokes parameters I, Q, U, V may be considered the incoherent superposition of two beams, one unpolarized, one completely polarized:

where

Ip = Q2 + U2 + V2. Because Ip < I it follows (but see Prob. 7.27) that

equality holding for completely polarized light. We define the degree of (elliptical) polarization of this beam as the ratio of the irradiance of the polarized component to the total irradiance:

often multiplied by 100 and expressed as a percentage (< 100%).

We can go further and imagine the beam to be a superposition of three beams, one unpolarized, one linearly polarized, and one circularly polarized:

Iu 0

where

/ip = VQ2 + u2, /cp = \v\. This naturally leads to definitions of the degree of linear polarization hp = VQ2 + U2 I I

and the degree of circular polarization

Ic cp I

a signed quantity: positive values correspond to right-circular polarization, negative to left-circular polarization.

To determine the degree of linear polarization of a beam, insert an ideal linear polarizer in it and measure the irradiance of the transmitted light. Suppose that the transmission axis of the polarizer makes an angle £ (between 0 and n) with the en axis. The transmitted amplitude along this axis is

where the subscript i denotes components of the incident field. The components of this transmitted field are therefore

From these equations and the definition of the Stokes parameters it follows that the transmitted irradiance is

The maximum and minimum of It occur for tan2£=--^. (7.67)

The two solutions to Eq. (7.67) are separated by n/2. Without loss of generality we may take Ui and Qi to be positive, in which instance the maximum occurs when the cosine is negative and the sine positive, whereas the minimum occurs when the cosine is positive and the sine negative:

where £ is the solution to Eq. (7.67) for which the cosine is negative. Subtract these two equations, add them, and take their ratio:

/max ~ /"min _ COS 2£ + Uj sin 2£ T 4- T ■ T

Because of Eq. (7.67) we can write

where

This then yields

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