Section 3: Electromagnetic Waves 1 EM waves in vacuum In regions of space where there are no charges and currents, Maxwell equations read ∇ ⋅E = 0 (3.1) ∇ ⋅B = 0 (3.2) ∂B ∇×E =− (3.3) ∂t ∂E ∇×B = µ ε (3.4) 0 0 ∂t They are a set of coupled, first order, partial differential equations for E and B. They can be decoupled by applying curl to eqs. (3.3) and (3.4): ∂B ∂ ∂ 2 E ∇×∇×()()EEE =∇∇⋅ −∇2 =−∇× =−() ∇× B =− µ ε (3.5) ∂t ∂ t0 0 ∂ t 2 ∂E ∂ ∂ 2 B ∇×∇×()()BBB =∇∇⋅ −∇2 =∇×µε = µε() ∇× E =− µε (3.6) 00∂t 00 ∂ t 00 ∂ t 2 Since ∇ ⋅E = 0 and ∇ ⋅B = 0 we have ∂2E ∇2E = µ ε (3.7) 0 0 ∂t 2 ∂2B ∇2B = µ ε (3.8) 0 0 ∂t 2 We now have separate equations for E and B, but they are of second order; that's the price you pay for decoupling them. In vacuum, then, each Cartesian component of E and B satisfies the three-dimensional wave equation ∂2 f ∇2 f = µ ε (3.9) 0 0 ∂t 2 The solution of this equation is a wave. So Maxwell’s equations imply that empty space supports the propagation of electromagnetic waves, traveling at a speed 1 c= =3.00 ⋅ 108 m / s (3.10) µ ε 0 0 which happens to be precisely the velocity of light, c. The implication is astounding: light is an electromagnetic wave. Of course, this conclusion does not surprise anyone today, but imagine what a ε µ revelation it was in Maxwell's time! Remember how 0 and 0 came into the theory in the first place: they were constants in Coulomb's law and the Biot-Savart law, respectively. You measure them in experiments involving charged pith balls, batteries, and wires—experiments having nothing whatever to do with light. And yet, according to Maxwell's theory you can calculate c from these two numbers. Notice the crucial role played by Maxwell's contribution to Ampere's law; without it, the wave equation would not emerge, and there would be no electromagnetic theory of light. 1 Monochromatic plane waves Since different frequencies in the visible range correspond to different colors, such waves are called monochromatic . Suppose that the waves are traveling in the z direction and have no x or y dependence ; these are called plane waves, because the fields are uniform over every plane perpendicular to the direction of propagation. We are interested, then, in fields of the form = ikz(−ω t ) E(z , t ) E 0 e (3.11) = i( kz−ω t ) B(z , t ) B 0 e , (3.12) where E0 and B0 are the (complex) amplitudes (the physical fields, of course, are the real parts of E and B). Substituting eqs.(3.11) and (3.12) to eqs. (3.7) and (3.8) respectively we find that ω c = . (3.13) k Here, k is the wave number, which is related to the wavelength of the wave by the equation 2π λ = , (3.14) k and ω is the angular frequency of EM wave. Fig. 1 This is the paradigm for a monochromatic plane wave. The wave is polarized in the x direction (by convention, we use the direction of E to specify the polarization of an electromagnetic wave). Now, the wave equations for E and B were derived from Maxwell's equations. However, whereas every solution to Maxwell's equations (in empty space) must obey the wave equation, the converse is not true; Maxwell's equations impose extra constraints on E0 and B0. In particular, since ∇ ⋅E = 0 and ∇ ⋅B = 0 , it follows that = = E0z B 0 z 0 (3.15) That is, electromagnetic waves are transverse: the electric and magnetic fields are perpendicular to the ∂B direction of propagation. Moreover, Faraday's law, ∇×E =− , implies a relation between the electric ∂t and magnetic amplitudes: xˆ y ˆ zˆ ∇×=∇×() ikzt()−ω −×∇ ikzt () − ω =−×∇ ikzt () − ω =− ikzt ()− ω = EE0e E 0 e E 0 e EEe 00x y 0 (3.16) 0 0 ik −+i() kz−ω t =+ω ω i () kz− ω t ()()xyˆˆikE00yx ikEexy ˆˆ iB 00 xy iBe which results in 2 − =ω = ω kE0y B 0 x, kE 0 x B 0 y , (3.17) or, more compactly: k 1 B=() zEˆ × =() zEˆ × (3.18) 0ω 0c 0 Evidently, E and B are in phase and mutually perpendicular; their (real) amplitudes are related by k 1 B= E = E (3.19) 0ω 0c 0 ∂E The fourth of Maxwell's equations, ∇×B = µ ε , does not yield an independent condition; it simply 0 0 ∂t reproduces Eq. (3.16). There is nothing special about the z direction, of course—we can easily generalize to monochromatic plane waves traveling in an arbitrary direction. The notation is facilitated by the introduction of the wave vector, k , pointing in the direction of propagation, whose magnitude is the wave number k. The scalar product k⋅ r is the appropriate generalization of kz , so = i(k⋅ r − ω t ) E( r ,t ) E0 e e , (3.20) 1⋅ − ω 1 Br( ,t ) = E() ne × e i(k r t ) =×() nE . (3.21) c0 c k Where vector n = is the unit vector in the direction of propagation of the EM wave and e is the k polarization vector. Because E is transverse, n⋅ e = 0 . (3.22) Linear and circular polarizations The plane wave (3.20) and (3.21) is a wave with its electric field vector always in the direction e. Such a = wave is said to be linearly polarized with polarization vector e1 e . Evidently there exists another wave ≠ which is linearly polarized with polarization vector e2 e 1 and is linearly independent of the first. Thus the two waves are E( r ,t ) = E e e i(k⋅ r − ω t ) 1 1 1 (3.23) = i(k⋅ r − ω t ) E2( r ,t ) E 2 e 2 e 1 with B=() n × E . 1,2c 1,2 They can be combined to give the most general homogeneous plane wave propagating in the direction k= k n : = + =+( ) i(k⋅ r − ω t ) Er(,)t Er1 (,) t Er 2 (,) tEEe 1122 e e . (3.24) The amplitudes E1 and E2 are complex numbers, to allow the possibility of a phase difference between waves of different linear polarization. If E1 and E2 have the same phase wave (3.24) represents a linearly polarized wave, with its polarization θ = =2 + 2 vector making an angle tanE2 / E 1 with respect to e1 and a magnitude E E2 E 1 as shown in Fig.2a. 3 Fig. 2 (a) Electric field of a linearly polarized wave. (b) Electric field of a circularly polarized wave. If E1 and E2 have different phases, the wave (3.24) is elliptically polarized. To understand what this means let us consider the simplest case, circular polarization. Then E1 and E2 have the same magnitude, but differ in phase by 90°. The wave (3.24) becomes: =( ± ) i(k⋅ r − ω t ) Er( ,tE ) 0 ee 1 ie 2 (3.25) with E0 the common real amplitude. We imagine axes chosen so that the wave is propagating in the positive z direction, while e1 and e2 are in the x and y directions, respectively. Then the components of the actual electric field, obtained by taking the real part of (3.25), are = − ω Ex (,)r t E0 cos( kz t ) = − ω (3.26) Ey (,)r t∓ E0 sin( kz t ) At a fixed point in space, the fields (3.26) are such that the electric vector is constant in magnitude, but ω ( + ) sweeps around in a circle at a frequency , as shown in Fig.2b. For the upper sign e1i e 2 , the rotation is counterclockwise when the observer is facing into the oncoming wave. This wave is called left circularly polarized in optics. In the terminology of modern physics, however, such a wave is said to have positive helicity. The latter description seems more appropriate because such a wave has a positive ( − ) projection of angular momentum on the z axis. For the lower sign e1i e 2 , the rotation of E is clock- wise when looking into the wave; the wave is right circularly polarized (optics): it has negative helicity. Figure 3. Electric field and magnetic induction for an elliptically polarized wave. The two circularly polarized waves (3.26) form an equally acceptable set of basic fields for description of a general state of polarization. If we introduce the complex orthogonal unit vectors: 1 e± =() e ± i e , (3.27) 2 1 2 then a general representation, equivalent to (3.24), is 4 i(k⋅ r − ω t ) Er( ,t ) =( E++ e + E −− e ) e (3.28) where E+ and E− are complex amplitudes. If E+ and E− have different magnitudes, but the same phase, (3.28) represents an elliptically polarized wave with principal axes of the ellipse in the directions of e1 + − = and e2 . The ratio of semimajor to semiminor axis is (1r ) /(1 r ) , where r E−/ E + . If the amplitudes iα have a phase difference between them, E−/ E + = re , then the ellipse traced out by the E vector has its axes rotated by an angle α/2. Fig.3 shows the general case of elliptical polarization and the ellipse traced out by E at a given point in space. For r = ± 1 we get back a linearly polarized wave. Energy and momentum of electromagnetic waves Energy per unit volume stored in electromagnetic fields is 1 1 u=ε E2 + B 2 .