Chapter 9: Electromagnetic Waves The Wave Equation 9.1 Waves in One Dimension 9.1.1 The Wave Equation How to represent such a “wave” mathematically? What is a “wave”? Hint: The wave at different times, once at t=0, and again at A start: A wave is disturbance of a continuous medium that some later time t --- each point on the wave form simply shifts propagates with a fixed shape at constant velocity. to the right by an amount vt, where v is the velocity. In the presence of absorption, the wave will diminish in initial shapefz ( ,0)= gz ( ) size as it move; If the medium is dispersive different frequencies travel at subsequent formfzt ( , ) = ? different speeds; (capture (mathematically) the Standing waves do not propagate; fzt(,)=− fzvt ( ,0) =− gzvt ( ) essence of wave motion.) Light wave can propagate in vacuum;… The function f(z,t) depends on them only in the very special combination z-vt; When that is true, the function f(z,t) represents a wave of fixed shape traveling in the z direction at speed v. 1 2 The Wave Equation (II) The Wave Equation of a String −−bz() vt2 From Newton’s second law we have fzt1(,)= Ae fzt(,)=− A sin[( bzvt )] ∂2 y Examples: 2 F[sin(θ +∆θ)−sin(θ)]= (µ∆x) A 2 fzt(,)= ∂t 3 bz()1−+ vt2 How about these functions? Small angle approximation: 2 ∂y fzt(,)= Ae−+bz() vt sinθ ≈ tanθ = 4 ∂x f(,) z t= A sin()cos( bz bvt ) ∂2 y ∂2 y 5 = (µ / F) A ∂x2 ∂t2 =++−[sin(bz ( vt )) sin( bz ( vt ))] a standing wave 2 3 4 The Wave Equation 9.1.2 Sinusoidal Waves Derive the wave equation that a disturbance propagates wave speed without changing it shape. (i) Terminology fztgzvt( , )=− ( ); Let u ≡− zvt fzt(,)=−+ A cos[( kz vt )δ ] 22 ∂∂f df udgf ∂ ∂dg 2 d g ==−vvv ⇒ =-()= amplitude wave number phase constant ∂∂t du t du ∂∂ t22 t du du ∂∂f df u dg ∂∂22 f dg d g fzt( , )=−+=−+ A cos[ kz ( vt )δ ] A cos( kzωδ t ) ==⇒ 22= ( )= ∂∂z du z du ∂∂ z z du du 2π k = , λ: wave length dg22211∂∂ f f ∂ 2 f ∂ 2 f λ ==⇒−= 0 qed v duvt222∂∂ z 2 ∂ z 2 vt 22 ∂ ωππ==kv2=2 f λ +−vv or ω : angular frequency fzt( , )=−++ gz ( vthzvt ) ( ) the wave equation is linear. 5 f : frequency 6 Sinusoidal Waves Example 9.1 (ii) Complex notation The advantage of the complex notation. Suppose we want to combine two sinusoidal waves: Euler’s formula eiiθ =+cosθ sinθ ffffffff312=+=Re[ 1 ] + Re[ 2 ] = Re[ 12 + ] = Re[ 3 ] fzt(,)=−+= A cos[( kz vt )δ ] Re[ Aeikz()−+ωδ t ] iikztδω()−− ikzt () ω Simply add the corresponding complex wave functions, and ==Re[Ae ] Re[ Ae ] take the real part. In particular, when they have the same frequency and wave ikz()−ω t fAe ≡ complex wave function number AAe ≡ iδ complex amplitude ikzt()−−−ω ikzt ()ωω ikzt () fAe31=+= Ae 2 Ae 3 iδ3 iiδδ12 fzt(,)= Re[(,)] fzt where AAeAeAe33==+ 1 2 The advantage of the complex notation is that exponentials Try doing this without using the complex notation. are much easier to manipulate than sines and cosines. 7 8 9.1.3 Boundary Conditions: Sinusoidal Waves (III) Reflection and Transmission (iii) Linear combinations of sinusoidal waves ikz()1 −ω t Incident wave: fztII(,)= Ae ∞ ikzt()−−1 ω ikz()−ω t Reflected wave: fztAe (,)= fzt( , )== Ake ( ) dk , where ωω( k ) RR ∫−∞ ikz()2 −ω t Transmitted wave: fztAeTT(,)= Ak( ) can be obtained in terms of the initial conditions ∗ All parts of the system are oscillating at the same frequency ω. fz( ,0) and fz( ,0) from the theory of Fourier transforms. The wave velocities are different in two vkλ regimes, which means the wave lengths 121== vkλ Any wave can be written as a linear combination of and wave numbers are also different. 212 sinusoidal waves. The waves in the two regions: ikzt()11−−−ωω ikzt ( ) So from now on we shall confine our attention to sinusoidal AeIR+ Ae for z < 0 fzt(,)= waves. Ae ikz()2 −ω t for z > 0 9 T 10 Boundary Conditions Determine the Complex Boundary Conditions Amplitudes Mathematically, f(z,t) is continuous at z=0. ftft(0−+ , )=⇒+= (0 , ) AAA ftft(0−+ , )= (0 , ) IRT df df The derivative of f(z,t) must also be continuous at z=0. =⇒−= kA( A ) kA dz dz 12IR T 00−+ df df Why? kk−− vv = A==()()12AA 21 dz00−+ dz RII AA+= A kk12++ vv 21 IRT ⇒ kA()−= A kA 22kv 12IR T A==()()12AA TII kk12++ vv 21 When v2>v1, all three waves have the same phase angle. The complex wave function obeys the same rules: Why? When v2<v1 the reflected wave is out of phase by 180°. df df ftft(0−+ , )= (0 , ); = Consider two extreme cases, open end and fixed end. dz dz 11 12 00−+ The Open End and Fixed End 9.1.4 Polarization Superposition of the actual pulse and an imaginary pulse. Transverse waves: the displacement of the wave is perpendicular to the direction of propagation, e.g. EM waves. Longitudinal waves: the displacement of the wave is along the direction of propagation, e.g. sound waves. Transverse waves occur in two independent states of polarization: ikz()−ω t ikz()−ω t ˆ fxv (,)zt= Ae ˆ fyh (,)zt= Ae General form: fnnxy(zt , )==+ Aeikz()−ω t ˆˆ , where cosθ ˆˆ sinθ 13 14 9.2 Electromagnetic Waves in Vacuum Right and Left Hand Circular Polarizations 9.2.1 The Wave Equation for E and B In regions of space where there is no charge or current, Maxwell’s equations read ∂B (i) ∇⋅EE =0 (iii) ∇× =− ∂t ∂E (ii) ∇⋅BB =0 (iV) ∇× =µε 00∂t ∂∇×()BE ∂2 ∇×( ∇×EEE ) =− ⇒ ∇∇⋅( ) −∇2 =−µε ∂∂tt00 2 2 ∂∇×()EB2 ∂ ∇×( ∇×BBB ) =µε00 ⇒ ∇( ∇⋅ ) −∇ =− µε 00 2 ∂∂tt2 2 ∂ E ∇=E µε00 2 ∇⋅E =0 ∂t since 2 ∇⋅B =0 ∂ B ∇=2B µε 15 00∂t 2 16 The Wave Equation for E and B Hertz’s Experiment In vacuum, each Cartesian component of E and B satisfies When Maxwell’s work was published in 1867 it did not the three-dimensional wave equation receive immediate acceptance. It is Hertz who conclusively 2 2 ∂ E demonstrated the existence of electromagnetic wave. ∇=E µε00 2 2 ∂t 2 1 ∂ f ⇒∇ f = ∂2B vt22∂ ∇=2B µε 00∂t 2 Maxwell’s equations imply that empty space supports the propagation of electromagnetic waves, traveling at a speed 1 v ==×3108 m/s the speed of light µε00 17 18 9.2.2 Monochromatic Plane Waves The Electromagnetic Spectrum Since different frequencies in the visible range correspond to Electromagnetic waves span an immerse range of different colors, such waves are called monochromatic. frequencies, from very long wavelength to extremely high This definition can be applied to the whole spectrum. A wave energy with frequency 1023 Hz. There is no theoretical limit of single frequency is called a monochromatic wave. to the high end. 19 20 Mainly Heating Effect in Micro/mm-Wave Spectrum Windows for Research and Application Opportunities 21 22 Spectrum to Be Exploited Monochromatic Plane Waves --- Significance of the Electron Cyclotron Maser Consider a monochromatic wave of frequency ω and the one photon multiple-photon multiple photon wave is traveling in the z direction and has no x or y per excitation, per electron, per electron, dependence, called plane waves. Plane waves: the fields are uniform over every plane large interaction large interaction interaction space space space ~ wavelength perpendicular to the direction of propagation. ↓ ↓↓ Are these waves common? Yes, very common. EE(,)zt= eikz()−ω t 0 where EB and are the complex amplitudes. ikz()−ω t 00 BB(,)zt= 0 e THz gap 23 24 Transverse Electromagnetic Waves Transverse Electromagnetic Waves (II) ∂E Q: What is the relation between E and B? Ampere’s law with Maxwell’s correction: ∇×B = µε 00∂t ∂E ∇⋅E =0z = (Eike )ikz()−ω t = 0 ⇒( E ) = 0 ∂∂()BE∂()B () ∂z 00zz xˆ : 00zx−=0 y µε ⇒ kB( ) = µεω ( E ) ∂∂yz00 ∂ t 0yx 00 0 ∂B ∇⋅B =0z = (Bike )ikz()−ω t = 0 ⇒( B ) = 0 ∂()E 00zz ∂∂()BB00xz () 0 y ∂z yˆ : −=µε00 ⇒ kB( 0 )x =− µεω 00 ( E 0 ) y That is, electromagnetic waves are transverse: the electric ∂∂zx ∂ t ∂()B ∂∂()BE () and magnetic fields are perpendicular to the direction of zˆ : 0 y −=00xzµε ⇒= 0 0 ∂B 00 propagation. Moreover, Faraday’s law ∇×E =− ∂∂xy ∂ t ∂t ω 1 ∂∂()EB∂()E () In free space, the speed of light is c == xˆ : 00zx−=−⇒0 y kE( ) =−ω ( B ) k µ ε ∂∂yz ∂ t 00yx 00 k 1 ∂()B More compactly, BzEzEEB=×=×⇒⊥()()ˆˆ ∂∂()EE00xz () 0 y 00 0 yˆ : −=−⇒ kE(00 )x =ω ( B ) y ω c ∂∂zx ∂ t 1 amplitude relation: BE= ∂()E0 y ∂∂()EB () 00 zˆ : −=−⇒=00xz 0 0 25 c 26 ∂∂xy ∂ t Example 9.2 Plane Waves Traveling in an Arbitrary Direction Prove: If E points in the x direction, There is nothing special about the z then B points in the y direction. direction---we can generalize to monochromatic plane waves traveling in ikz()−ω t Sol: Ex(,)zt= Ee0 ˆ an arbitrary direction. k 11ikz()−−ωω t ikz () t BzE000=×=()ˆˆˆˆEe () zx ×= Ee 0 y The propagation (or wave) vector, k: pointing in the ω cc direction of propagation. Take the real part: Generalization of kz: using the scalar product k·r. it()kr⋅−ω Ex(,)zt=−+ E0 cos( kzωδ t )ˆ Er(,)tEe= 0 nˆ ← the polarization vector 1 11 B(,)zt=−+ E cos( kzω t δ )yˆ Br(,)tEe=×=×it()kr⋅−ω ( kˆˆ nˆ ) k E c 0 cc0 Q: Why not use sin function? Q: Can you write down the real electric and magnetic fields? 27 28 9.2.3 Energy and Momentum in Energy Transport and the Poynting Vector Electromagnetic Waves The energy per unit volume stored in the electromagnetic Consider two planes, each of area A, a distance dx apart, field is and normal to the direction of propagation of the wave.