Problems

7.1. Show that an unpolarized beam can be considered to be the incoherent superposition of two orthogonally elliptically polarized (100%) beams of equal irradiance.

HINT: This problem is a simple application of the Stokes parameters.

7.2. Show that the electric and magnetic fields of a plane harmonic wave are perpendicular to the direction of propagation only if the wave is homogeneous, and that the electric and magnetic fields are perpendicular to each other only if the medium is nonabsorbing.

7.3. Show by simple arguments that transmission of a beam of unpolarized light by an ideal linear retarder cannot change the state of polarization of the beam. No fancy mathematics is necessary, just simple arguments based on the nature of unpolarized light and the function of a retarder.

7.4. Show that the reflectivities for specular reflection, R| and R±, are both equal to 1 for an angle of incidence of 90° regardless of the refractive index of the medium.

7.5. Show that as m becomes indefinitely large, the degree of polarization of specularly reflected light (given unpolarized incident light) approaches zero for all angles of incidence.

7.6. Many years ago a scheme for reducing glare from the headlights of oncoming automobiles was seriously considered. Linear polarizing filters were to be placed on the headlights and windshields of all automobiles. How should these filters be oriented so that drivers do not see the light from oncoming cars but do see light reflected by the headlights of their own cars? You may assume that the state of polarization of light from headlights is not changed upon reflection. Can you think of at least one reason why this scheme was never adopted?

7.7. Estimate the extent to which the maximum degree of polarization of skylight is reduced because of the finite width (about half a degree) of the sun. You may assume that air molecules are spherically symmetric.

7.8. By how much does an ideal linear polarizing filter reduce the irradiance of an incident unpolarized beam? By how much does the filter reduce the irradiance of an incident beam partially polarized with degree P perpendicular to the transmission axis?

HINT: The first question is easier than the second. You can guess the answer to the second by considering the two limiting cases. To check your guess note that the Mueller matrix for an ideal linear polarizing filter has the same form as the Mueller matrix for reflection in which one of the reflectivities is 1, the other 0.

7.9. It should be evident on physical grounds that incident 100% polarized light yields specularly reflected light that also is 100% polarized although with possibly different degrees of linear and circular polarization. Prove this. That is, show that if /¡2 = Q + U2 + Vj2 it follows that Ir2 = Q + Ur2 + Vr2. In constructing this proof you will also discover that the Mueller matrix elements for specular reflection are not independent (indeed, this result is an essential part of the proof).

7.10. The transmission axis of a linearly polarizing filter in, for example, polarizing sunglasses, is not perceptible to the human eye. How would you quickly determine the direction of this axis?

7.11. We discussed only the degree of polarization of specularly reflected light, for incident unpolarized light, as a function of angle of incidence for reflection by an infinitely thick medium. Such media are thin on the ground. How do you expect the degree of polarization to vary for light incident on a medium of finite thickness? Take it to be a nonabsorbing plate of uniform thickness sufficiently large relative to the wavelength that the consequences of interference need not be taken into account. For sake of visualization, consider the plate to be a microscope slide. Equations in Section 5.1 will help you derive an expression for the degree of polarization of the plate as a function of the reflectivities of the infinite medium. But you should first try to determine by physical reasoning if the degree of polarization increases, decreases, or remains the same (relative to the infinite medium) and how this depends on angle of incidence.

7.12. Find the degree of polarization of light specularly reflected by an incoherent pile of N = 2,4,... identical plates like those in the previous problem. The plates are sufficiently far apart (relative to the wavelength) that coherence need not be taken into account. Find the degree of polarization of the reflected light and the reflectivity for unpolarized light as a function of the angle of incidence in the limit of indefinitely large N. You can do this by analysis or by physical reasoning. Sketch the reflectivity and degree of polarization as a function of angle in this limit.

7.13. Determine the degree of polarization of light transmitted by the pile of plates in the previous problem as a function of the angle of incidence in the limit of an indefinitely large number of plates. Sketch the degree of polarization of the transmitted light and reflectivity for unpolarized light as function of angle of incidence in this limit. First try to make these sketches by only physical reasoning. Compare these sketches with those for reflection. What do you conclude about using a pile of plates as a polarizing filter?

7.14. Show that the surfaces of constant amplitude for an inhomogeneous plane wave that results from illumination of a planar, optically smooth interface between a negligibly absorbing medium and an absorbing medium are planes parallel to the interface. Show that when the imaginary part of the refractive index of the absorbing medium is small compared with the real part, the spatial rate of attenuation of the transmitted field is approximately exp(-2nnis/A), where ni is the imaginary part of the complex refractive index of the absorbing medium and s is the distance into this medium along the direction of refraction.

7.15. An article in Science News (July 3, 2003) about beetles navigating by moonlight begins with the assertion that "an international team of researchers has turned up evidence that the insect aligns its path by detecting the polarization of moonlight." The article goes on to say that the researchers "found that beetles active during the day depend on sunlight polarization patterns." These statements, taken literally (which is the only way we can take them) are incorrect. Why? How would you rewrite them to make them correct?

7.16. On what kind of imaginary but physically allowable planet with what kind of atmosphere illuminated by what kind of sun would skylight be 100% polarized at 90° from the sun?

7.17. Show that in the limit m ^ 1 Eq. (7.108) for the degree of polarization of specularly reflected light (given incident unpolarized light) approaches Eq. (7.115), the degree of polarization of light scattered by a spherically symmetric electric dipole if by the scattering angle in Eq. (7.108) is meant that between the reflected and transmitted waves.

HINTS: You will need L'Hospital's rule, basic theorems about limits of products and quotients, and trigonometric identities for sums of angles and half angles.

7.18. The angular dependence of scattering (in a given plane) of unpolarized light by a spherically symmetric dipolar scatterer is given by Eq. (7.116). Because this scattering is az-imuthally symmetric, you should be able to determine the corresponding (normalized) phase function (the probability of scattering, per unit solid angle, in any direction).

7.19. Show that any optical element that transforms both amplitudes and phases of perpendicular electric field components differently can transform unpolarized light only into partially linearly polarized light. This is a more complicated version of Problem 7.3.

HINT: A proof follows from the definition of the Stokes parameters for light of arbitrary polarization state together with simple trigonometric identities.

7.20. We state without proof in Section 2.2.1 that "for many materials over many wavelength intervals, reflectivity changes hardly at all even with huge increases in absorption coefficient. And if there is a change, it is likely to result in a decrease in absorptivity..." You now have all the ingredients to prove this statement, namely the Fresnel coefficients and the discussion of the complex refractive index (see Sec. 3.5.2).

HINT: Consider specular reflection at normal incidence.

7.21. A plane harmonic wave incident on the interface between two negligibly absorbing media and originating in the (first) medium of higher refractive index is totally internally reflected at an angle of incidence, called the critical angle $c, such that the angle of transmission is n/2 (see Sec. 4.2.1). Show that the reflectivity at the critical angle for incident light polarized parallel or perpendicular to the plane of incidence is 1. Take the second medium to be air with refractive index 1.

HINT: Snel's law [Eq. (7.90)] and the cosine form of the Fresnel coefficients [Eqs. (7.91) and (7.92)] should be helpful.

7.22. This problem is an extension of the previous one. There is no law prohibiting waves from being incident at angles greater than the critical angle. What are the reflectivities for incident waves perpendicular and parallel to the plane of incidence at these angles?

HINT: The key to this problem is to recognize that $t in sin and cos in Eq. (7.88) does not necessarily correspond to a real angle. That is, kt sin dt and kt cos dt are simply the components of what is called the transmitted wavevector, the only requirement being that sin2 dt + cos2 dt = 1, and Snel's law is a mathematical relation that is always satisfied but not always easy to interpret geometrically. The cosine form of the Fresnel coefficients again should be helpful.

7.23. This problem is an extension of the previous one. Show that for angles of incidence greater than the critical angle, the reflected light is elliptically polarized for incident light linearly polarized obliquely to the plane of incidence.

7.24. We stated in Section 7.1 that the concept of a surface of constant phase is, in general, meaningless. Show this.

HINT: By a surface of constant phase is meant a single such surface. Write down the expression for an arbitrary (complex) vector field and the proof should be obvious.

7.25. Blackbody radiation is unpolarized and isotropic but can be considered the incoherent superposition of two sources of equal magnitude but orthogonally linearly polarized (see Prob. 7.1). From this result, derive an expression for the emissivity of an opaque slab, optically smooth and homogeneous (in air, say), in terms of the reflectivities for incident radiation polarized parallel (R) and perpendicular (R±) to the plane of incidence.

7.26. Derive an expression for the degree of polarization as a function of direction for radiation emitted by the slab in the previous problem. It may help to write the Stokes parameters of the emitted radiation. What is the largest degree of polarization (and in what direction) of 10 pm radiation emitted by a layer of water in air?

HINTS: We chose our words carefully here: "largest degree" rather than maximum degree. The easiest way to do this problem is to use the Fresnel coefficients Eq. (7.91) and Eq. (7.92) to obtain an explicit expression for the degree of polarization as a function of the angles and $t. Use the form of these equations containing trigonometric functions of the sum and differences of angles. Even though water is absorbing at 10 pm, the imaginary part of its refractive index is sufficiently small compared with its real part (about 1.2) that you can ignore the imaginary part. With a bit of algebra and a simple trigonometric identity, you can obtain a fairly simple expression for the degree of polarization.

7.27. Our proof of the inequality Eq. (7.57) was based on the (unproven) assumption that Ip < I, whereas a rigorous proof must start from Eqs. (7.51)-(7.54). You will need the

Cauchy-Schwarz inequality in the form (/ fg dt) < J f2 dt J g2 dt and the identity cos2 S + sin2 S =1. We confess that we needed the help of two mathematicians, George Greaves and V. I. Burenkov, with a proof, which need not be long.

7.28. Convince yourself that Eq. (7.8) does indeed correspond to two orthogonally polarized waves. All that is needed is a crude sketch.

7.29. We once observed through a polarizing (camera) filter daylight reflected by a polished floor near the Brewster angle. As we rotated the filter, the brightness of the reflection greatly diminished, as expected. But the reflection could not be made to completely disappear. We always observed a dark reflection of a strikingly pure blue. At first we thought this had something to do with illumination by the blue sky. But this hypothesis was quickly discarded when we noticed that the source of illumination was light from an overcast sky. Explain.

HINT: Although it is not necessary to write down the Mueller matrix for an non-ideal linear polarizing filter, doing so, or at least thinking about doing so, is likely to help.

7.30. The Umov effect or Umov's law (rule) is a reciprocal relationship between reflectivity and degree of polarization of light reflected by rough surfaces or granular media (e.g., soils, snow, powders): the higher the reflectivity the lower the degree of polarization and vice versa. Although this rule appears to be fairly well known to (planetary) astronomers, a simple, short explanation of it is hard to find. And yet it can be explained adequately in a sentence or two. Provide such an explanation, concise, correct, and easy to understand.

7.31. Obtaining reflectivities from the Fresnel coefficients (ratios of fields, not irradiances) for reflection is straightforward because the incident and reflected waves make the same angle with the normal to the interface and both are in the same medium. Indeed, we presented these reflectivities without fanfare. Transmissivities require a bit more care. Derive expressions for the two transmissivities (ratio of transmitted irradiance to incident irradiance) as a function of angle of incidence for an infinite negligibly absorbing medium in air.

HINTS: You need the Fresnel coefficients for transmission as well as the relation between the Poynting vector and irradiance. We previously ignored a constant of proportionality between the magnitude of the Poynting vector (irradiance) and the square of electric fields. But for this problem we cannot ignore this constant because it is different for different media. For a plane, homogeneous wave in a negligibly absorbing medium |S| = n|E • E*|/2Zo, where n is the (real) refractive index of the medium and Zo is a universal constant called the impedance of free space.

7.32. To check the correctness of the result obtained in the previous problem, show that R + T =1, where R is the reflectivity and T is the transmissivity (either polarization state). A further check is the reciprocal relation T($¡,n) = T($t, 1/n), which follows from the reversibility of rays.

7.33. In Section 7.2.2 we assert that to obtain elliptically polarized light by specular reflection requires obliquely polarized light incident on a metal (or a material with appreciable absorption at the wavelength of interest) at non-normal incidence. This statement is strictly true only for an infinite medium because if it is nonabsorbing R34 = 0. Show by simple arguments that for oblique incidence R34 is not necessarily zero for a finite nonabsorbing single layer or many layers (the form of the Mueller matrix is the same as that for an infinite medium). This problem is a variation on the theme that high reflectivity can be obtained either by a metal (appreciably absorbing) or by a multi-layer interference filter (negligibly absorbing).

7.34. Derive the Mueller matrix for an ideal linear polarizing filter with its transmission axis at an arbitrary angle to the reference coordinate system in which the Stokes parameters are defined.

7.35. Derive the Mueller matrix for an ideal linear retarder with arbitrary retardance and with its fast axis at an arbitrary angle to the reference coordinate system in which the Stokes parameters are defined. This is considerably more difficult than the previous problem. You have to resolve the field components in the reference system along the fast and slow axes of the retarder, introduce different phase shifts, then transform back to the reference system. Trigonometric identities for the cosine and sine of the sum of angles can be used to simplify results.

7.36. As noted at the end of Section 7.1.3, although the Stokes parameters were defined by way of a set of hypothetical measurements not all of which are feasible, once these parameters are defined we can devise ways to measure them. Suppose that you have an irradiance detector, a linear polarizing filter, and a linear retarder with variable retardance. How many measurements and which kind would you have to make in order to determine the Stokes parameters of an arbitrary beam? Do you need both a polarizing filter and a retarder? Try to devise the simplest (easiest to describe) set of measurements. The results of the previous two problems can help you specify in detail what kind of measurements to make.

7.37. Show that if an incoherent suspension of spheres, regardless of their distribution in size and composition but sufficiently thin (optically) that multiple scattering is negligible, is illuminated by light linearly polarized perpendicular (parallel) to the scattering plane, the scattered light in this plane is 100% polarized. You can show this by simple physical arguments, but a mathematical proof (by way of the scattering matrix) leads you into the second part of this problem. By measuring the polarization properties of light scattered by a suspension of particles, how can it be determined if they are nonspherical even if randomly oriented? What scattering matrix elements or combination of elements should be measured and how (we can think of at least two possibilities)?

HINT: Results in Section 7.4 are necessary, and although the solution to Problem 7.34 is not, it might be helpful. For what scattering angles is the quantity measured likely to be most sensitive to departures of the particles from sphericity? You might review Section 3.4.8 before addressing this last question.

7.38. Based on the discussion of fluorescence in the references at the end of Chapter 1 and the discussion in Section 7.4 of the essential role that asymmetry plays in yielding polarized light, what do you expect the state of polarization (at any angle) of fluorescent light excited in gases and liquids to be? What about solids (with other than cubic symmetry)? No mathematical analysis is necessary. This problem tests your physical understanding of polarization and emission by gases and solids.

7.39. Does it seem contradictory that the light scattered at, say, 90°, by randomly oriented nonspherical molecules (for incident unpolarized light) can be partially polarized?

HINT: Consider a spheroidal molecule that differs only slightly from a spherical molecule.

7.40. Show that for negligible absorption, the reflection coefficient Eq. (7.91) or (7.92) reverses sign when the rays are reversed. That is, if we denote f12 as the reflection coefficient for light incident at angle from medium 1 onto medium 2, and f21 as the reflection coefficient for light incident from medium 2 onto medium 1, where the angles of incidence and refraction are reversed, show that f21 = —f12. Then show that 1 + f12f21 = i12i21, where i12 is the transmission coefficient [Eq. (7.98) or (7.99)], corresponding to f12. What is the physical interpretation of this equation?

7.41. We did not derive Eq. (5.24), but you now should be able to do so in a way similar to summing the infinite series in Section 1.4. Although this equation was for normal incidence, no more effort is necessary to derive it for arbitrary incidence.

HINTS: Equation (5.7) suggests the form of the solution. For this problem you have to add fields, taking due account of phase shifts, then take the product of the resultant with its complex conjugate to obtain the reflectivity. Problem 7.40 also is needed for this problem. And you will need the difference in phase between a transmitted wave at z = 0 and at z = h, which you can obtain from Eq. (7.88).

7.42. With the results of Problem 7.41 you now should be able to explain why different thin film interference colors sometimes are seen in different directions.

7.43. Suppose that the thin film of Problem 7.41 is illuminated by a laser beam of diameter d at oblique incidence. What criterion must be satisfied for the reflectivity obtained in that problem to be applicable to reflection of the laser beam? The purpose of this problem is to underscore the (possible) differences between reflection of an infinite plane wave and a finite laser beam.

7.44. A single particle is a coherent object, and hence if illuminated by 100% polarized light, the scattered light is also 100% polarized (although not necessarily with all the same ellipso-metric parameters). This is not difficult to understand. But we also argue in Section 3.4.2 that a piece of paper is a coherent object, and yet if it is illuminated by 100% polarized light the reflected light is at best weakly polarized. Why the difference? Devise a simple experiment to show that light reflected by a piece of white paper illuminated by 100% polarized light is essentially unpolarized. Try the same experiment with an optically smooth object such as glassware.

7.45. This problem is related to the previous one. If 100% polarized light illuminates a piece of white paper, the reflected light is weakly polarized but not with a degree of polarization of exactly 0%. If 100% polarized light illuminates a piece of glass, the reflected light is not exactly 100% polarized. Explain. Why do we specify "white" paper? What happens when black paper is illuminated by 100% polarized light (see Prob. 7.30)? Do a simple experiment to find out.

7.46. Unlike scattering by a small particle (see Sec. 3.5), reflection because of an interface (i.e., the Fresnel coefficients in Sec. 7.2) does not depend explicitly on wavelength (although it does depend implicitly on wavelength by way of the refractive index). Give a simple explanation why.

HINT: This is not a problem in electromagnetic theory but rather requires invoking a fundamental characteristic of all equations in which the variables are dimensional.

7.47. We note at the end of the chapter that neglecting polarization results in errors in radiance calculations for clear air illuminated by sunlight. Based strictly on physical reasoning you should be able to estimate the (normal) optical thickness for which the error is a maximum.

7.48. In the references for Chapters 2 and 3 we cite (quite favorably) Tony Rothman's Everything's Relative. But that doesn't mean that we'll let him get away with the footnote on page 22 of this book: "every time you look through a pair of Polaroid sunglasses you are using Malus' discovery. Polaroids work because reflected light viewed through them loses half its intensity." What's wrong with this (other than the use of intensity for luminance)?

7.49. If you did Problem 1.40 then you already have done almost everything you need to find the specific entropy of blackbody radiation. Do so.

7.50. In what simple, intuitive sense can specular reflection be said to be reversible whereas diffuse reflection is irreversible.

HINT: Consider rays.

7.51. Based strictly on your intuition about entropy determine what happens to the entropy of radiation upon specular reflection and refraction, diffuse reflection, scattering of a beam by a particle, and the incoherent superposition of partially polarized beams.

7.52. Section 7.2 begins with a discussion of reflection and refraction because of illumination of a smooth interface between two different media, the first of which can be taken to be air. Equation (7.86) is the wave vector of the transmitted wave in an arbitrary illuminated medium. Show that if this medium is absorbing, the surfaces of constant amplitude are parallel to the interface and the surfaces of constant phase are not, except for normal incidence.

7.53. Show that for a plane harmonic wave the time-averaged Poynting vector is not, in general, parallel to the real part of the complex wave vector. Under what conditions is the Poynting vector parallel to the real part of the wave vector? You may take the constant C in Eq. (7.5) to be real.

7.54. Consider reflection and refraction because of a smooth interface between two dissimilar infinite media. A wave is incident at an arbitrary direction from medium 1, giving rise to a refracted wave in medium 2 and a reflected wave in medium 1. Denote by 112 the amplitude of the wave transmitted from 1 to 2 and by r 12 the amplitude of the reflected wave (for unit incident amplitude). The polarization state is either parallel or perpendicular to the plane of incidence. Now consider the reverse: a wave is incident from medium 2 along the direction of the refracted wave. Denote by i21 and f21 the corresponding ratios of amplitudes. Show that tutu = 1 - r212, r 12 = -r21.

These relations seem to have been derived first by Stokes. You can obtain them by physical arguments about the reversibility of the waves or analytically from the Fresnel equations. A simple diagram is essential. You may find statements in textbooks that these relations are valid only for nonabsorbing media. Show that this is not true.

7.55. With the results of the previous problem derive Eq. (5.24), the normal-incidence reflectivity of a nonabsorbing slab of uniform thickness, by adding all the multiply reflected waves taking account of all phase shifts. The equation you obtain for the reflected amplitude will be valid even for an absorbing medium, illuminated at arbitrary angle of incidence, and for either polarization state (parallel or perpendicular).

7.56. Find the transmitted amplitude for the slab considered in the previous problem. As a check on your result, show that the corresponding transmittivity plus the reflectivity is equal to 1.

7.57. We implicitly assumed in the previous problems that the medium on both sides of the slab is the same. Find the reflectivity of a slab by the series summation method for medium 1 on one side, medium 3 on the other side (all media negligibly absorbing).

7.58. Consider a negligibly absorbing slab between two different infinite media, also negligibly absorbing. Denote the media by subscripts 1, 2, and 3, where 2 denotes the slab. Show that the reflectivity for radiation incident from medium 1 is the same as that for radiation from medium 3. Keep in mind that this reciprocity principle requires the angle of incidence for radiation from medium 3 to be the same as the angle of transmission for the slab when illuminated from medium 1.

7.59. Use the results of Problem 7.57 to design an anti-reflection coating. That is, for normal-incidence radiation of wavelength A, determine the refractive index n2 and thickness h of a thin layer to be deposited on a substrate with refractive index n3 such that the reflectivity of the system is zero.

7.60. We state that infinite, monodirectional plane electromagnetic waves do not exist. Prove this.

HINT: About the only physical law that we can always depend upon is conservation of energy.

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