16 The Classical Radiation Field 16.1 FIELD COMPONENTS OF THE RADIATION FIELD Equation (15-22a) is valid for any acceleration of the electron However, it is convenient to describe (15-22a) in two different regimes, namely, for nonrelativistic speeds ðv=c ( 1Þ and for relativistic speeds ðv=c ’ 1Þ The field emitted by an accelerating charge observed in a reference frame where the velocity is much less than the speed of light, that is, the nonrelativistic regime, is seen from (15-22a) to reduce to e EðX, tÞ ¼ ½n  ðn  v_ Þ ð16-1Þ 4"0 c2 R where EðX, tÞ is the field vector of the radiated field measured from the origin, e is the charge, c is the speed of light, R is the distance from the charge to the observer, n ¼ R=R is the unit vector directed from the position of the charge to the observation point, and v_ is the acceleration (vector) of the charge The relation between the vectors X and n is shown in Fig 16-1 To apply (16-1), we consider the (radiated) electric field E in spherical coordinates Since the field is transverse, we can write E ¼ E u þ E u ð16-2Þ where u and u are unit vectors in the and directions, respectively Because we are relatively far from the source, we can take n to be directed from the origin and write n ¼ ur , where ur is the radial unit vector directed from the origin The triple vector product in (16-1) can then be expanded and written as ur  ður  v_ Þ ¼ ur ður Á v_ Þ À v_ ð16-3Þ For many problems of interest it is preferable to express the acceleration of the charge v_ in Cartesian coordinates, thus v_ ¼ x€ ux þ y€uy þ z€uz Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð16-4Þ Figure 16-1 Vector relation for a moving charge and the radiation field where the double dot refers to twofold differentiation with respect to time The unit vectors u in spherical and Cartesian coordinates are shown later to be related by ð16-5aÞ ur ¼ sin cos ux þ sin sin uy þ cos uz u ¼ cos cos ux þ cos sin uy À sin uz ð16-5bÞ u ¼ À sin ux þ cos uy ð16-5cÞ ux ¼ sin cos ur þ cos cos u À sin u ð16-6aÞ uy ¼ sin sin ur þ cos sin u þ cos u ð16-6bÞ uz ¼ cos ur À sin u ð16-6cÞ or Using (16-5) and (16-6), we readily find that (16-3) expands to ur ður Á v_ Þ À v_ ¼ Àu ðx€ cos cos þ y€ cos sin À z€ sin Þ þ u ðÀx€ sin þ y€ cos Þ ð16-7Þ We see that ur is not present in (16-7), so the field components are indeed transverse to the direction of the propagation ur An immediate simplification in (16-7) can be made by noting that we shall only be interested in problems that are symmetric in Thus, we can conveniently take ¼ Then, from (16-1), (16-2), and (16-7) the transverse field components of the radiation field are found to be e E ¼ ½x€ cos À z€ sin ð16-8Þ 4"0 c2 R e E ¼ ½y€ ð16-9Þ 4"0 c2 R Equations (16-8) and (16-9) are the desired relations between the transverse radiation field components, E and E , and the accelerating charge described by x€ , y€ , and z€ We note that E , E , and refer to the observer’s coordinate system, and x€ , y€ , and z€ refer to the charge’s coordinate system Because we are interested in field quantities that are actually measured, namely, the Stokes parameters, in spherical coordinates the Stokes parameters are defined by S0 ¼ E E à þ E Eà ð16-10aÞ S1 ¼ E E à À E Eà S2 ¼ E Eà S3 ¼ iðE Eà þ E E à À E E Ã Þ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð16-10bÞ ð16-10cÞ ð16-10dÞ pffiffiffiffiffiffiffi where i ¼ À1 While it is certainly possible to substitute (16-8) and (16-9) directly into (16-10) and find an expression for the Stokes parameters in terms of the acceleration, it is simpler to break the problem into two parts Namely, we first determine the acceleration and the field components and then form the Stokes parameters according to (16-10) 16.2 RELATION BETWEEN THE UNIT VECTOR IN SPHERICAL COORDINATES AND CARTESIAN COORDINATES We derive the relation between the vector in a spherical coordinate system and a Cartesian coordinate system The rectangular coordinates x, y, z are expressed in terms of spherical coordinates r, , by the equations: x ¼ xðr, , Þ y ¼ yðr, , Þ z ¼ zðr, , Þ ð16-11Þ Conversely, these equations can be expressed so that r, , can be written in terms of x, y, z Then, any point with coordinates (x, y, z) has corresponding coordinates (r, , ) We assume that the correspondence is unique If a particle moves from a point P in such a way that and are held constant and only r varies, a curve in space is generated We speak of this curve as the r curve Similarly, two other coordinate curves, the curves and the curves, are determined at each point as shown in Fig 16-2 If only one coordinate is held constant, we determine successively three surfaces passing through a point in space, these surfaces intersecting in the coordinate curves It is generally convenient to choose the new coordinates in such a way that the coordinate curves are mutually perpendicular to each other at each point in space Such coordinates are called orthogonal curvilinear coordinates Let r represent the position vector of a point P in space Then r ¼ xi þ yj þ zk ð16-12Þ From Fig 16-2 we see that a vector vr tangent to the r curve at P is given by @r @r dsr v¼ ¼ Á ð16-13Þ @r @sr dr Figure 16-2 Determination of the r, , and curves in space Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved where sr is the arc length along the r curve Since @r=@sr is a unit vector (this ratio is the vector chord length Ár, to the arc length Ásr such that in the limit as Ásr ! the ratio is 1), we can write (16-13) as v r ¼ hr ur ð16-14Þ where ur is the unit vector tangent to the r curves in the direction of increasing arc length From (16-14) we see then that hr ¼ dsr =dr is the length of vr Considering now the other coordinates, we write v r ¼ hr ur v ¼ h u v ¼ h u ð16-15Þ so (16-14) can be simply written as v k ¼ hk uk k ¼ r, , ð16-16Þ where uk ðk ¼ r, , Þ is the unit vector tangent to the uk curve Furthermore, we see from (16-13) that dsr @r ð16-17aÞ hr ¼ ¼ dr @r ds @r h ¼ ¼ d @ ds @r h ¼ ¼ @ d ð16-17bÞ ð16-17cÞ Equation (16-17) can be written in differential form as dsr ¼ hr dr ds ¼ h d ds ¼ h d ð16-18Þ We thus see that hr , h , h are scale factors, giving the ratios of differential distances to the differentials of the coordinate parameters The calculations of vk from (16-15) leads to the determination of the scale factors from hk ¼ vk and the unit vector from uk ¼ vk =hk We now apply these results to determining the unit vectors for a spherical coordinate system In Fig 16-3 we show a spherical coordinate system with unit vectors ur , u , and u The angles and are called the polar and azimuthal angles, Figure 16-3 Unit vectors for a spherical coordinate system Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved respectively We see from the figure that x, y, and z can be expressed in terms of r, and by x ¼ r sin cos y ¼ r sin sin z ¼ r cos ð16-19Þ Substituting (16-19) into (16-12) the position vector r becomes r ¼ ðr sin cos Þi þ ðr sin sin Þj þ ðr cos Þk ð16-20Þ From (16-13) we find that vr ¼ @r ¼ sin cos i þ sin sin j þ cosk @r ð16-21aÞ v ¼ @r ¼ r cos cos i þ r cos sin j À r sin k @ ð16-21bÞ v ¼ @r ¼ Àr sin sin i þ r sin cos j @ ð16-21cÞ The scale factors are, from (16-17), @r hr ¼ ¼ @r @r h ¼ ¼ r @ @r h ¼ ¼ r sin @ Finally, from (16-21) and (16-22) the unit vectors are v ur ¼ r ¼ sin cos i þ sin sin j þ cos k hr ð16-22aÞ ð16-22bÞ ð16-22cÞ ð16-23aÞ u ¼ v ¼ cos cos i þ cos sin j À sin k h ð16-23bÞ u ¼ v ¼ À sin i þ cos j h ð16-23cÞ which corresponds to the result given by (16-6) (it is customary to express ux , uy , uz as i, j, k) We can easily check the direction of the unit vectors shown in Fig 16-3 by considering (16-23) at, say, ¼ 0 and ¼ 90 For this condition (16-23) reduces to ur ¼ k ð16-24aÞ u ¼ j ð16-24bÞ u ¼ Ài ð16-24cÞ which is exactly what we would expect according to Fig 16-3 An excellent discussion of the fundamentals of vector analysis can be found in the text of Hilderbrand given in the references at the end of this chapter The material presented here was adapted from his Chapter Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 16.3 RELATION BETWEEN THE POYNTING VECTOR AND THE STOKES PARAMETERS Before we proceed to use the Stokes parameters to describe the field radiated by accelerating charges, it is useful to see how the Stokes parameters are related to the Poynting vector and Larmor’s radiation formula in classical electrodynamics In Chapter 13, in the discussion of Young’s interference experiment the fact was pointed out that two ideas were borrowed from mechanics The first was the wave equation Its solution alone, however, was found to be insufficient to arrive at a mathematical description of the observed interference fringes In order to describe these fringes, another concept was borrowed from mechanics, namely, energy Describing the optical field in terms of energy or, as it is called in optics, intensity, did lead to results in complete agreement with the observed fringes with respect to their intensity and spacing However, the wave equation and the intensity formulation were accepted as hypotheses In particular, it was not at all clear why the quadratic averaging of the amplitudes of the optical field led to the correct results In short, neither aspect of the optical field had a theoretical basis With the introduction of Maxwell’s equations, which were a mathematical formulation of the fundamental laws of the electromagnetic field, it was possible to show that these two hypotheses were a direct consequence of his theory The first success was provided by Maxwell himself, who showed that the wave equation of optics arose directly from his field equations In addition, he was surprised that his wave equation showed that the waves were propagating with the speed of light The other hypothesis, namely, the intensity formed by taking time averages of the quadratic field components was also shown around 1885 by Poynting to be a direct consequence of Maxwell’s equations We now show this by returning to Maxwell’s equations in free space [see Eqs.(15-1)], =  E ¼ À =ÂH¼" @H @t ð16-25aÞ @E @t ð16-25bÞ =ÁE¼0 ð16-25cÞ =ÁB¼0 ð16-25dÞ and where we have also used the constitutive equations, (15-6) First, we take the scalar product of (16-25a) and H so that we have H Á =  E ¼ ÀH Á @H @t ð16-26aÞ Next, we take the scalar product of (16-25b) and E so that we have E Á =  H ¼ "E Á @E @t ð16-26bÞ We now subtract (16-26b) from (16-26a): H Á =  E À E Á =  H ¼ ÀH Á Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved @H @E À "E Á @t @t ð16-27Þ The left-hand side of (16-27) is recognized as the identity: = Á ðE  HÞ ¼ H Á ð=  EÞ À E Á ð=  HÞ ð16-28Þ The terms on the right-hand side of (16-27) can be written as HÁ @H @ ¼ ðH Á HÞ @t @t ð16-29aÞ EÁ @E @ ¼ ðE Á EÞ @t @t ð16-29bÞ and Then, using (16-28) and (16-29), (16-27) can be written as ! @ ðH Á HÞ þ "ðE Á EÞ = Á ðE  HÞ þ ¼0 @t ð16-30Þ Inspection of (16-30) shows that it is identical in form to the continuity equation for current and charge: =Ájþ @ ¼0 @t ð16-31Þ In (16-31) j is a current, that is, a flow of charge Thus, we write the corresponding term for current in (16-30) as S ¼ ðE  HÞ ð16-32Þ The vector S is known as Poynting’s vector and represents, as we shall show, the flow of energy The second term in (16-30) is interpreted as the time derivative of the sum of the electrostatic and magnetic energy densities The assumption is now made that this sum represents the total electromagnetic energy even for time–varying fields, so the energy density w is w¼ H2 þ "E2 ð16-33aÞ where H Á H ¼ H2 ð16-33bÞ E Á E ¼ E2 ð16-33cÞ Thus, (16-30) can be written as =ÁSþ @w ¼0 @t ð16-34Þ The meaning of S is now clear It is the flow of energy, analogous to the flow of charge j (the current) Furthermore, if we write (16-34) as =ÁS¼À @w @t Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð16-35Þ then the physical meaning of (16-35) (and (16-34)) is that the decrease in the time rate of change of electromagnetic energy within a volume is equal to the flow of energy out of the volume Thus, (16-34) is a conservation statement for energy We now consider the Poynting vector further: S ¼ ðE  HÞ ð16-32Þ In free space the solution of Maxwell’s equations yields plane-wave solutions: Eðr, tÞ ¼ E0 eiðkÁrÀ!tÞ ð16-36aÞ Hðr, tÞ ¼ H0 eiðkÁrÀ!tÞ ð16-36bÞ We can use (16-25a) to relate E to H: =  E ¼ À @H @t ð16-25aÞ Thus, for the left-hand side of (16-25a) we have, using (16-36a), =  E ¼ =  ½E0 eiðkÁrÀ!tÞ ¼ ik  E ð16-37aÞ where we have used the vector identity =  ð aÞ ¼ r  a þ r  a ð16-38Þ Similarly, for the right-hand side we have À @H ¼ i!H @t ð16-39Þ Thus (16-25a) becomes nÂE¼ H c"0 ð16-40aÞ where n¼ k k ð16-40bÞ since k ¼ !=c The vector n is the direction of propagation of S Equation (16-40a) shows that n, E, and H are perpendicular to one another Thus, if n is in the direction of propagation, then E and H are perpendicular to n, that is, in the transverse plane We now substitute (16-40a) into (16-32) and we have S ¼ c"0 ½E  ðn  EÞ ð16-41Þ From the vector identity: a  ðb  cÞ ¼ ða Á cÞb À ða Á bÞc ð16-42Þ we see that (16-41) reduces to S ¼ c"0 ðE Á EÞn Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð16-43Þ In Cartesian coordinates the quadratic term in (16-43) is written out as E Á E ¼ Ex Ex þ Ey Ey ð16-44Þ Thus, Maxwell’s theory leads to quadratic terms, which we associate with the flow of energy For more than 20 years after Maxwell’s enunciation of his theory in 1865, physicists constantly sought to arrive at other well-known results from his theory, e.g., Snell’s law of refraction, or Fresnel’s equations for reflection and transmission at an interface Not only were these fundamental formulas found but their derivations led to new insights into the nature of the optical field Nevertheless, this did not give rise to the acceptance of this theory An experiment would have to be undertaken which only Maxwell’s theory could explain Only then would his theory be accepted If we express E and H in complex terms, then the time-averaged flux of energy is given by the real part of the complex Poynting vector, so hSi ¼ ðE  HÃ Þ ð16-45Þ From (16-40) we have n  Eà ¼ Hà ð16-46Þ and substituting (16-46) into (16-45) leads immediately to hSi ¼ c"0 ðE Á Eà Þn ð16-47Þ Thus, Maxwell’s theory justifies the use of writing the intensity I as I ¼ Ex Exà þ Ey Eyà ð16-48Þ for the time-averaged intensity of the optical field In spherical coordinates the field is written as E ¼ E u þ E u ð16-49Þ so the Poynting vector (16-47) becomes hSi ¼ c"0 ðE E à þ E Eà Þn ð16-50Þ The quantity within parentheses is the total intensity of the radiation field, i.e., the Stokes parameter S0 Thus, the Poynting vector is directly proportional to the first Stokes parameter Another quantity of interest is the power radiated per unit solid angle, written as dP c"0 ¼ ðE Á Eà ÞR2 d ð16-51Þ We saw that the field radiated by accelerating charges is given by E¼ e ½n  ðn  v_ Þ 4"0 c2 R Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð16-1Þ Expanding (16-1) by the vector triple product: E¼ e ½nðn Á v_ Þ À v_ 4"0 c2 R ð16-52Þ We denote n Á v_ ¼ jnjjv_ j cos  ð16-53Þ where  is the angle between n and v_ and jÁ Á Áj denotes that the absolute magnitude is to be taken Using (16-52) and (16-53), we then find (16-51) becomes dP ¼ e2 jv_ j sin2  d ð16-54Þ We saw that the field radiated by accelerating charges is given by E ¼ e ðx€ cos cos þ y€ cos sin À z€ sin Þ 4"0 c2 R ð16-55aÞ E ¼ e ðÀx€ sin þ y€ cos Þ 4"0 c2 R ð16-55bÞ The total radiated power over the sphere is given by integrating (16-51) over the solid angle: Z Z c"0 2 P¼ ðE E à þ E Eà ÞR2 sin d d ð16-56Þ 0 We easily find that Z 2 Z ðE E à ÞR2 sin d d ¼ 0 4e2 ðjx€ j2 þjy€ j2 Þ 162 "20 c4 ð16-57aÞ 4e2 ðjx€ j2 þ jy€ j2 þ 4jz€j2 Þ 3ð162 "20 c4 Þ ð16-57bÞ and Z 2 Z ðE Eà ÞR2 sin d d ¼ where j ÁÁ j2 ð ÁÁ Þð ÁÁ Þà Thus, adding (16-57a) and (16-57b) yields Z 2 Z e2 ðE E à þ E Eà ÞR2 sin d d ¼ ðj€rj2 Þ 4"0 c4 0 ð16-58aÞ where r€ ¼ x€ ux þ y€ uy þ z€uz ð16-58bÞ Substituting (16-58a) into (16-56) yields the power radiated by an accelerating charge: P¼ e2 j€rj2 4"0 c3 ð16-59Þ Equation (16-59) was first derived by J J Larmor in 1900 and, consequently, is known as Larmor’s radiation formula Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved The material presented in this chapter shows how Maxwell’s equations led to the Poynting vector and then to the relation for the power radiated by the acceleration of an electron, that is, Larmor’s radiation formula We now apply these results to obtain the polarization of the radiation emitted by accelerating electrons Finally, very detailed discussions of Maxwell’s equations and the radiation by accelerating electrons are given in the texts by Jackson and Stratton REFERENCES Books Jackson, J D., Classical Electrodynamics, Wilcy, New York, 1962 Sommerfeld, A., Lectures on Theoretical Physics, Vols I–V, Academic Press, New York, 1952 Hildebrand, F B., Advanced Calculus for Engineers, Prentice-Hall, Englewood Cliffs, NJ, 1949 Stratton, J A., Electromagnetic Theory, McGraw-Hill, New York, 1941 Schott, G A., Electromagnetic Radiation, Cambridge University Press, Cambridge, UK, 1912 Jeans, J H., Mathematical Theory of Electricity and Magnetism, 5th ed., Cambridge University Press, 1948 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved [...]... acceleration of an electron, that is, Larmor’s radiation formula We now apply these results to obtain the polarization of the radiation emitted by accelerating electrons Finally, very detailed discussions of Maxwell’s equations and the radiation by accelerating electrons are given in the texts by Jackson and Stratton REFERENCES Books 1 Jackson, J D., Classical Electrodynamics, Wilcy, New York, 1962... 1952 3 Hildebrand, F B., Advanced Calculus for Engineers, Prentice-Hall, Englewood Cliffs, NJ, 1949 4 Stratton, J A., Electromagnetic Theory, McGraw-Hill, New York, 1941 5 Schott, G A., Electromagnetic Radiation, Cambridge University Press, Cambridge, UK, 1912 6 Jeans, J H., Mathematical Theory of Electricity and Magnetism, 5th ed., Cambridge University Press, 1948 Copyright © 2003 by Marcel Dekker,