17Radiation Emitted by AcceleratingCharges

17 Radiation Emitted by Accelerating Charges 17.1 STOKES VECTOR FOR A LINEARLY OSCILLATING CHARGE We have shown how Maxwell’s equation gave rise to the equations of the radiation field and the power emitted by an accelerating electron We now discuss the polarization of the radiation emitted by specific electron configurations, e.g., bound charges and charges moving in circular and elliptical paths At the beginning of the nineteenth century the nature of electric charges was not fully understood In 1895 the electron (charge) was discovered by J J Thompson Thus, the long-sought source of the optical field was finally found A year after Thompson’s discovery, P Zeeman performed a remarkable experiment by placing radiating atoms in a constant magnetic field He thereupon discovered that the original single spectral line was split into two, or even three, spectral lines Shortly thereafter, H Lorentz heard of Zeeman’s results Using Maxwell’s theory and his electron theory, Lorentz then treated this problem Lorentz’s calculations predicted that the spectral lines should not only be split but also completely polarized On Lorentz’s suggestions Zeeman then performed further measurements and completely confirmed the predictions in all respects It was only after the work of Zeeman and Lorentz that Maxwell’s theory was accepted and Fresnel’s theory of light replaced Not surprisingly, the importance of this work was immediately recognized, and Zeeman and Lorentz received the Nobel Prize in physics in 1902 We should emphasize that the polarization predictions of the spectral lines played a key part in understanding these experiments This prediction, more than any other factor, was one of the major reasons for the acceptance of Maxwell’s theory into optics In this chapter we build up to the experiment of Zeeman and the theory of Lorentz We this by first applying the Stokes parameters to a number of classical radiation problems These are the radiation emitted by (1) a charge oscillating along an axis, (2) an ensemble of randomly oriented oscillating charges, (3) a charge moving in a circle, (4) a charge moving in an ellipse, and (5) a charge moving in a Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved magnetic field In the following chapter we then consider the problem of a randomly oriented oscillating charge moving in a constant magnetic field—the Lorentz– Zeeman effect We consider a bound charge oscillating along the z axis as shown in Fig 17-1 The motion of the charge is described by d 2z þ !02 z ¼ dt ð17-1Þ The solution of (17-1) is zðtÞ ¼ zð0Þ cosð!0 t þ Þ ð17-2Þ where z(0) is the maximum amplitude and is an arbitrary phase constant Because we shall be using the complex form of the Stokes parameters, we write (17-2) as zðtÞ ¼ zð0Þeið!0 tþÞ ð17-3Þ where it is understood that by taking the real part of (17-3), we recover (17-2); that is, Re½zðtÞ ¼ zð0Þ cosð!0 t þ Þ ð17-4Þ The radiation field equations are given by (16-8) and (16-9) in Section 16.1: E ¼ e ½x€ cos À z€ sin 4"0 c R ð16-8Þ E ¼ e ½ y€ 4"0 c R ð16-9Þ Recall that these equations refer to the observation being made in the xz plane, that is, at ¼ The angle is the polar angle in the observer’s reference frame Performing the differentiation of (17-3) to obtain z€, we have z€ ¼ À!02 zð0Þeið!0 tþÞ Figure 17-1 Motion of a linear oscillating charge Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð17-5Þ Substituting (17-5) into (16-8) yields E ¼ e ½!02 zð0Þ sin eið!0 tþÞ 4"0 c R E ¼ ð17-6aÞ ð17-6bÞ The Stokes parameters are defined in a spherical coordinate system to be S0 ¼ E EÃ þ E EÃ ð16-10aÞ S1 ¼ E EÃ À E EÃ ð16-10bÞ S2 ¼ E EÃ þ E EÃ ð16-10cÞ S3 ¼ iðE EÃ À E EÃ Þ ð16-10dÞ Substituting (17-6a) and (17-6b) into (16-10) yields 2 ezð0Þ !40 sin S0 ¼ 4"0 c R 2 ezð0Þ !40 sin S1 ¼ À 4"0 c R ð17-7aÞ ð17-7bÞ S2 ¼ ð17-7cÞ S3 ¼ ð17-7dÞ We now arrange (17-7) in the form of the Stokes vector: 1 2 B À1 C ezð0Þ C sin !40 B S¼ @ A 4"0 c R ð17-8Þ Equation (17-8) shows that the observed radiation is always linearly vertically polarized light at a frequency !0, the fundamental frequency of oscillation of the bound charge Furthermore, when we observe the radiation parallel to the z axis ( ¼ 0 ), the intensity is zero Observing the radiation perpendicular to the z axis ( ¼ 90 ), we note that the intensity is a maximum This behavior is shown in Fig 17-2 In order to plot the intensity behavior as a function of , we set IðÞ ¼ sin ð17-9aÞ In terms of x() and z() we then have xðÞ ¼ IðÞ sin ¼ sin sin zðÞ ¼ IðÞ cos ¼ sin cos ð17-9bÞ ð17-9cÞ The term ez(0) in (17-8) is recognized as a dipole moment A characteristic of dipole radiation is the presence of the sin2 term shown in (17-8) Hence, (17-8) Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 17-2 Plot of the intensity behavior of a dipole radiation field describes the Stokes vector of a dipole radiation field This type of field is very important because it appears in many types of radiation problems in physics and engineering Finally, we note that a linearly oscillating charge gives rise to linearly polarized light Thus, the state of polarization is a manifestation of the fundamental motion of the electron This observation will be confirmed for other types of radiating systems 17.2 STOKES VECTOR FOR AN ENSEMBLE OF RANDOMLY ORIENTED OSCILLATING CHARGES In the previous section, we considered the radiation field emitted by a charge or electron oscillating with an angular frequency !0 about an origin Toward the end of the nineteenth century a model was proposed for the atom in which an oscillating electron was bound to a positively charged atom The electron was believed to be negative (from work with ‘‘free’’ electrons in gases and chemical experiments) The assumption was made that the electron was attracted to the positively charged atom, and the force on the electron was described by Hooke’s law, namely, -kr This model was used by H Lorentz to solve a number of longstanding problems, e.g., the relation between the refractive index and the wavelength, the so-called dispersion relation The motion of the electron was described by the simple force equation: m€r ¼ Àkr ð17-10aÞ or r€ þ !02 r ¼ ð17-10bÞ where m is the mass of the electron, k is the restoring force constant, and the angular frequency is !02 ¼ k=m We saw in Part I that the nature of unpolarized light was not Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved well understood throughout most of the nineteenth century We shall now show that this simple model for the motion of the electron within the atom leads to the correct Stokes vector for unpolarized light The treatment of this problem can be considered to be among the first successful applications of Maxwell’s equations in optics This simple atomic model provides a physical basis for the source term in Maxwell’s equations The model leads to the appearance of unpolarized light, a quantity that was a complete mystery up to the time of the electron Thus, an ensemble of oscillating charges bound to a positive nucleus and randomly oriented gives rise to unpolarized light We now determine the Stokes vector of an ensemble of randomly oriented, bound, charged oscillators moving through the origin This problem is treated by first considering the field emitted by a single charge oriented at the polar angle and the azimuthal angle in the reference frame of the charge An ensemble average is then taken by integrating the radiated field over the solid angle sin d d The diagram describing the motion of a single charge is given in Fig 17-3 The equations of motion of the charged particle can be written immediately from Fig 17-3 and are xðtÞ ¼ A sin sin ei!0 t ð17-11aÞ yðtÞ ¼ A sin sin ei!0 t ð17-11bÞ zðtÞ ¼ A cos ei!0 t ð17-11cÞ where !0 is the angular frequency of natural oscillation Differentiating (17-11) twice with respect to time gives x€ ðtÞ ¼ À!02 A sin cos ei!0 t ð17-12aÞ y€ ðtÞ ¼ À!02 A sin sin ei!0 t ð17-12bÞ z€ðtÞ ¼ À!02 A cos ei!0 t ð17-12cÞ Figure 17-3 Instantaneous motion of an ensemble of oscillating charges Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Substituting (17-12) into the radiation field equations, we find that E ¼ À eA!02 ei!0 t ðsin cos cos À cos sin Þ 4"0 c R ð17-13aÞ E ¼ À eA!02 ei!0 t ðsin sin Þ 4"0 c R ð17-13bÞ where is the observer’s viewing angle measured from the z axis Recall that the Stokes parameters are defined by S0 ¼ E EÃ þ E EÃ ð16-10aÞ S1 ¼ E EÃ À E EÃ ð16-10bÞ S2 ¼ E EÃ þ E EÃ ð16-10cÞ S3 ¼ iðE EÃ À E EÃ Þ ð16-10dÞ Substituting (17-13) in (16-10), we then find that the Stokes parameters are S0 ¼ C ½sin sin þ sin cos cos À sin cos cos cos sin þ cos sin ð17-14aÞ S1 ¼ C ½sin sin À sin cos cos þ sin cos cos cos sin À cos sin ð17-14bÞ S2 ¼ C ½2ðsin sin cos cos À cos sin sin sin Þ ð17-14cÞ S3 ¼ ð17-14dÞ where 2 eA !40 C¼ 4"0 c R ð17-14eÞ The fact that S3 is zero in (17-14d) shows that the emitted radiation is always linearly polarized, as we would expect from a model in which the electron only undergoes linear motion In order to describe an ensemble of randomly oriented charges we integrate (17-14) over the solid angle sin d d: Z 2 Z hÁ Á Ái ¼ ðÁ Á ÁÞ sin d d ð17-15Þ 0 where hÁ Á Ái is the ensemble average and ðÁ Á ÁÞ represents (17-14a), etc Carrying out the integration of (17-14) by using (17-15) and forming the Stokes vector, we find that 1 2 B C 8 eA 4B C S¼ !0 @ A ð17-16Þ 4"0 c R Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved which is the Stokes vector for unpolarized light This is exactly what is observed from natural light sources Note that the polarization state is always independent of the observer’s viewing angle ; the observed light always appears to be unpolarized Thus, this simple model explains the appearance of unpolarized light from optical sources Unpolarized light can only arise from an ensemble of randomly oriented accelerating charges, which can be the case for bound electrons Electrons moving at a constant velocity, even if the motion is random, cannot give rise to unpolarized light This simple atomic model received further support when it was used by Lorentz to explain the Lorentz–Zeeman effect, namely, the radiation field emitted by a bound electron moving in a constant magnetic field We emphasize that the motion of a free accelerating electron gives rise to a different result, as we shall see 17.2.1 Note on Use of Hooke’s Law for a Simple Atomic System At first glance the use of Hooke’s law to describe the motion of a negative electron bound to a positive charge (nucleus) within an atom may appear to be quite arbitrary The use of Hooke’s law is based, however, on the following simple atomic model The force of attraction between two opposite but equal charges e separated by a distance r is given by F¼ ðþeÞðÀeÞ ur 4"0 r ð17-17Þ where ur is a unit radius vector The positive charge is located at the origin of a spherical coordinate system We now assume that the positive charge is distributed over a sphere of volume V and radius r, so the charge density is ¼ þe þe ¼ V 4r3 =3 ð17-18Þ or þe ¼ 4r3 ð17-19Þ Substituting (17-19) into (17-17) gives F ¼ Àkr ð17-20Þ where r ¼ rur, and k ¼ e/3"0 Equation (17-20) is Hooke’s law Thus, on the basis of this very simple atomic model the motion of the electron is expected to undergo simple harmonic motion 17.3 STOKES VECTOR FOR A CHARGE ROTATING IN A CIRCLE We now continue with our application of the Stokes parameters to describe radiation problems In this section we turn our attention to the determination of the field Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 17-4 Motion of a charge moving counterclockwise in a circle of radius a in the xy plane with an angular frequency !0 radiated by a charge moving in a circle This is shown in Fig 17-4 The coordinates of the charge are xðtÞ ¼ a cos !0 t ð17-21aÞ yðtÞ ¼ a sin !0 t ð17-21bÞ zðtÞ ¼ ð17-21cÞ That (17-21) describe counterclockwise motion is easily checked by first setting ¼ !0t Then, as t increases, increases Choosing ¼ 0, =2, and 3=2, the reader will easily see that plotting the position of the charge describes a counterclockwise motion as it moves in a circle of radius a To use the complex form of the Stokes parameters, the coordinates (17-21) must also be expressed in complex form We have (Euler’s relation) ei!0 t ¼ cos !0 t þ i sin !0 t ð17-22Þ The real part of (17-22) is cos !0t We can also express sin !0t in terms of the real part of (17-22), Re{ }, by multiplying (17-22) by Ài Then, we see that È É Re ei!0 t ¼ cos !0 t ð17-23aÞ È i!0 t É Re Àie ð17-23bÞ ¼ sin !0 t Thus, in complex notation (17-21a) and (17-21b) become xðtÞ ¼ aei!0 t ð17-24aÞ yðtÞ ¼ Àiaei!0 t ð17-24bÞ and the acceleration is then x€ ðtÞ ¼ Àa!02 ei!0 t ð17-25aÞ y€ ðtÞ ¼ þia!02 ei!0 t ð17-25bÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Substituting (17-25a) and (17-25b) into the radiation field equations (16-8) and (16-9) we find that Â Ã e E ¼ Àa!02 cos ei!0 t ð17-26aÞ 4"0 c R Â Ã e ia!02 ei!0 t E ¼ ð17-26bÞ 4"0 c R Again, we express (17-26a) and (17-26b) in terms of the Stokes parameters and form the Stokes vector The result is 1 þ cos 2 B À cos C ea C S¼ ð17-27Þ !40 B @ A 4"0 c R cos Equation (17-27) is the Stokes vector for elliptically polarized light Thus, we see that the radiation is elliptically polarized and is characterized by a frequency !0, the frequency of rotation of the electron Furthermore, we see that we have the factor ea in (17-27), the familiar expression for the dipole moment We observe that (17-27) shows that the orientation angle of the polarization ellipse is always zero Similarly, the ellipticity angle is À1 S3 ¼ sin ð17-28Þ S0 so from (17-27) we have cos ¼ sinÀ1 þ cos ð17-29Þ The ellipticity angle is a function of the observation angle We see that for ¼ 0 , that is, we view the rotating electron along the z axis, (17-29) becomes ¼ 45 and we observe right circularly polarized light The Stokes vector (17-27) reduces to 1 2 B C ea 4B C ð17-30Þ !0 @ A S¼2 4"0 c R If we now view the rotating electron perpendicular to the z axis, that is, ¼ 90 , we find that ¼ 0 and we observe linearly horizontally polarized light The corresponding Stokes vector is 1 2 B ea 1C C !40 B S¼ ð17-31Þ @ 0A 4"0 c R These results agree with our earlier observation that the polarization of the emitted radiation is a manifestation of the motion of the charge Thus, if we look along the z axis we would see an electron moving counterclockwise in a circle, so Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved we observe right circularly polarized light If we look perpendicular to the z axis, the electron appears to behave as a linear oscillator and we observe linearly horizontally polarized light, in agreement with our earlier conclusion The linear polarization is to be expected, because if we view the motion of the charge as it moves in a circle at ¼ 90 it appears to move from left to right and then from right to left, identical to the behavior of a linear oscillator described in Section 17.1 Finally, for ¼ 180 we see that (17-29) becomes ¼ À45 , so we observe left circularly polarized light Also observe that (17-27) satisfies the equality: S02 ¼ S12 þ S22 þ S32 ð17-32Þ The equals sign shows that the emitted radiation is always completely polarized Furthermore, the degree of polarization is independent of the observation angle 17.4 STOKES VECTOR FOR A CHARGE MOVING IN AN ELLIPSE It is of interest to consider the case where an electron moves in an elliptical orbit The equations of motion are xðtÞ ¼ a cos !0 t ð17-33aÞ yðtÞ ¼ b sin !0 t ð17-33bÞ where a and b are the semimajor and semiminor axes lengths, respectively In complex notation (17-33) becomes xðtÞ ¼ aei!0 t i!0 t yðtÞ ¼ Àibe ð17-34aÞ ð17-34bÞ The acceleration is then x€ ðtÞ ¼ Àa!02 ei!0 t ð17-35aÞ y€ ðtÞ ¼ ib!02 ei!0 t ð17-35bÞ Again using the radiation field equations (16-8) and (16-9), the radiated fields are found to be ! e!02 ð17-36aÞ E ¼ ei!0 t ½Àa cos 4"0 c R ! e!02 ð17-36bÞ ei!0 t ½ib E ¼ 4"0 c R We now form the Stokes vector for (17-36) and find that b þ a cos 2 B 2 C e B b À a cos C S¼ ! B C @ A 4"0 c R 2ab cos Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð17-37Þ Equation (17-37) is the Stokes vector for elliptically polarized light We see immediately that if a ¼ b then (17-37) reduces to the Stokes vector for an electron moving in a circle The orientation angle of the polarization ellipse is seen from (17-37) to be 0 The ellipticity angle is À1 2ab cos ¼ sin ð17-38Þ b þ a cos The radiation is always elliptically polarized with one exception; the exception will be discussed in a moment We see that for ¼ 0 , (17-37) reduces to b þ a2 2 B 2C e 4B b À a C S¼ ! ð17-39Þ B C @ A 4"0 c R 2ab which is the Stokes vector for elliptically polarized light The other case of interest is to observe the radiation perpendicular to the z axis, that is, ¼ 90 For this angle (17-37) reduces to 1 2 B C e 2B C S¼ ð17-40Þ !0 b @ A 4"0 c R which is the Stokes vector for linear horizontally polarized light Again, this is perfectly understandable, because at this angle the moving charge appears to be oscillating in a straight line as it moves in its elliptical path The Stokes vectors derived here will reappear when we discuss the Lorentz– Zeeman effect REFERENCES Books Jackson, J D., Classical Electrodynamics, Wiley, New York, 1962 Lorentz, H A., Theory of Electrons, Dover reprint, New York, 1952 Sommerfeld, A., Lectures on Theoretical Physics, Vols I–V, Academic Press, New York, 1952 Jeans, J H., Mathematical Theory of Electricity and Magnetism, 5th ed., Cambridge Unversity Press, Cambridge, UK, 1948 Wood, R W., Physical Optics, 3rd ed., Optical Society of America, Washington, DC, 1988 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved [...]... at a constant velocity, even if the motion is random, cannot give rise to unpolarized light This simple atomic model received further support when it was used by Lorentz to explain the Lorentz–Zeeman effect, namely, the radiation field emitted by a bound electron moving in a constant magnetic field We emphasize that the motion of a free accelerating electron gives rise to a different result, as we shall... of the real part of (17-22), Re{ }, by multiplying (17-22) by Ài Then, we see that È É Re ei!0 t ¼ cos !0 t ð17-23aÞ È i!0 t É Re Àie ð17-23bÞ ¼ sin !0 t Thus, in complex notation (17-21a) and (17-21b) become xðtÞ ¼ aei!0 t ð17-24aÞ yðtÞ ¼ Àiaei!0 t ð17-24bÞ and the acceleration is then x€ ðtÞ ¼ Àa!02 ei!0 t ð17-25aÞ y€ ðtÞ ¼ þia!02 ei!0 t ð17-25bÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved... oscillating charges Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Substituting (17-12) into the radiation field equations, we find that E ¼ À eA!02 ei!0 t ðsin cos cos À cos sin Þ 4"0 c 2 R ð17-13aÞ E ¼ À eA!02 ei!0 t ðsin sin Þ 4"0 c 2 R ð17-13bÞ where is the observer’s viewing angle measured from the z axis Recall that the Stokes parameters are defined by S0 ¼ E EÃ þ E EÃ ð16-10aÞ... atom may appear to be quite arbitrary The use of Hooke’s law is based, however, on the following simple atomic model The force of attraction between two opposite but equal charges e separated by a distance r is given by F¼ ðþeÞðÀeÞ ur 4"0 r 2 ð17-17Þ where ur is a unit radius vector The positive charge is located at the origin of a spherical coordinate system We now assume that the positive charge is... describe radiation problems In this section we turn our attention to the determination of the field Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 17-4 Motion of a charge moving counterclockwise in a circle of radius a in the xy plane with an angular frequency !0 radiated by a charge moving in a circle This is shown in Fig 17-4 The coordinates of the charge are xðtÞ ¼ a cos !0 t ð17-21aÞ... d d ð17-15Þ 0 0 where hÁ Á Ái is the ensemble average and ðÁ Á ÁÞ represents (17-14a), etc Carrying out the integration of (17-14) by using (17-15) and forming the Stokes vector, we find that 0 1 1 2 B C 8 eA 4B 0 C S¼ !0 @ A ð17-16Þ 0 3 4"0 c 2 R 0 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved which is the Stokes vector for unpolarized light This is exactly what is observed from natural... !40 B S¼ ð17-31Þ 2 @ 0A 4"0 c R 0 These results agree with our earlier observation that the polarization of the emitted radiation is a manifestation of the motion of the charge Thus, if we look along the z axis we would see an electron moving counterclockwise in a circle, so Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved we observe right circularly polarized light If we look perpendicular... cos 2 sin 2 ð17-14bÞ S2 ¼ C ½2ðsin 2 sin cos cos À cos sin sin sin Þ ð17-14cÞ S3 ¼ 0 ð17-14dÞ where 2 eA !40 C¼ 4"0 c 2 R ð17-14eÞ The fact that S3 is zero in (17-14d) shows that the emitted radiation is always linearly polarized, as we would expect from a model in which the electron only undergoes linear motion In order to describe an ensemble of randomly oriented charges we integrate... C ea C S¼ ð17-27Þ !40 B @ A 2 0 4"0 c R 2 cos Equation (17-27) is the Stokes vector for elliptically polarized light Thus, we see that the radiation is elliptically polarized and is characterized by a frequency !0, the frequency of rotation of the electron Furthermore, we see that we have the factor ea in (17-27), the familiar expression for the dipole moment We observe that (17-27) shows that... circle This is shown in Fig 17-4 The coordinates of the charge are xðtÞ ¼ a cos !0 t ð17-21aÞ yðtÞ ¼ a sin !0 t ð17-21bÞ zðtÞ ¼ 0 ð17-21cÞ That (17-21) describe counterclockwise motion is easily checked by first setting ¼ !0t Then, as t increases, increases Choosing ¼ 0, =2, and 3=2, the reader will easily see that plotting the position of the charge describes a counterclockwise motion as it