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Electromagnetic waves in vacuum. The discovery of displacement currents entails a peculiar class of solutions of Maxwell equations: travelling waves of electric and magnetic fields in vacuum. In the absence of currents and charges, the equations governing electric and magnetic field are: ∇⋅B = 0 ∇⋅E = 0 ∇×E = − 1 ∂∂B B c ∂t ∇×B = 1 ∂∂E E c ∂t Taking the curl of (3) and using (4): 1 ∂ 2E ∇ × ∇ × E = ∇(∇ ⋅ E) − ∇ E = − 2 2 c ∂t 2 Using (1) we get D’Alambert equation: 1 ∂ 2E ∇ E− 2 2 = 0 c ∂t 2 Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 Electromagnetic waves in vacuum. Each of the two equation pairs lead to the D’Alambert eq. (respectively for Ex and By, and Ey and Bx). For example, the first pair combines into: ∂ 2 Ex 1 ∂ 2 Ex − 2 =0 2 2 c ∂t ∂z ∂ 2 By E x = E x ( z − ct ) B y = B y ( z − ct ) ∂z 2 2 1 ∂ By − 2 =0 c ∂t 2 (E ⊥ B ! ) The linearity of D’Alambert eq. entails that the generic EM plane wave can be seen as the superposition of sinusoidal waves. Let us the assume that each component has the form u = u0 exp[i (kz − ω t + φu )] Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 Electromagnetic waves in vacuum. The pair of equations for (Ex,By) yields kE0 x exp[i (kz − ω t + φ E )] = kB0 y exp[i (kz − ω t + φ B )] = ω c ω c B0 y exp[i (kz − ω t + φ B )] E0 x exp[i (kz − ω t + φ E )] These equations must be verified for any t and z. Therefore E0 x = B0 y φE = φB The same conditions are true for (Ey,Bx). At each (t,z) the magnitude of the electric and magnetic field is the same (in CGS-units) . Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 Polarization of EM waves The two classes of solutions (Ex,By) and (Ey,Bx) are independent: they represent the two polarization modes of EM radiation. As the E and B fields lie on a plane, these modes correspond to linear polarizations. A generic (unpolarized) EM wave is a superposition of the two modes, with different phases and amplitudes. For example, introducing the unit vectors along x and y: E( z , t ) = E0 x cos(kz − ω t + φ x )]xˆ + E0 y cos(kz − ω t + φ y )]yˆ Instead of the linearly polarized modes, one could use two circularly polarized modes: Ex = E y φ y = φx + π 2 Left Circular Polarization (LCP) and Ex = E y φ y = φx − π 2 Right Circular Polarization (RCP) Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 Maxwell Equations in a Plasma The wave propagation is governed by Maxwell equations (CGS-Gauss): ∇ ⋅ E = 4πρ ∇⋅B = 0 1 ∂B ∇×E = − c ∂t ∇×B = 4π 1 ∂E J+ c c ∂t We will solve the combined set of Maxwell equations and electron equation of motion using 1) a perturbative approach (to first order) and 2) normal modes. 1) All quantities are the sum of an unperturbed (background) value and a small perturbation: n = n0 + n1 v= v1 (quiescent plasma) B = B 0 + B1 E= E1 (no external electric field) Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 Maxwell Equations in a Plasma. 2) All perturbations are periodic and depend on space and time as u = u0 exp[i (k ⋅ r − ω t ) The surfaces of constant phase move in space at the phase velocity vφ = ω (suggestion: consider the surfaces with null phase and compute r/t) k The set of differential equations becomes now a set of algebraic equations; the differential operators transform as ∇ → ik ∇⋅ → ik ⋅ ∇× → ik × Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess ∂ → − iω ∂t AA 2007-2008 The equations of motion. Navier-Stokes equation for the electrons (collisionless plasma): 1 ∂v me ne + ( v ⋅ ∇) v = −∇p − ene E + v × B c ∂t The space charge density ρ = -ene and current density J = -enev fullfil the equation of continuity: ∂ρ +∇⋅J = 0 ∂t We assume that the plasma the plasma is magnetized: a constant and homogeneous magnetic field is superimposed to the magnetic field of the wave. The plasma is supposed to be isothermal, so we do not need to consider the energy equation. (Not only: we will see soon that the plasma may be safely supposed to be cold, with T = 0!) Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 The linearized, normal mode equations. The Maxwell equations, charge conservation and electron equation of motion become: ik ⋅ E1 = 4π en1 k × E1 = ω c B1 eω n1 = k ⋅ J (1) k ⋅ B1 = 0 (3) k × B1 = −i (5) (2) 4π ω J − E1 c c (4) 1 − ime n0ω v1 = −en0 E1 + v1 × B 0 c (6) Note that now the first two Maxwell equations (1-2) follow from the last pair (3-4) and charge conservation (5): (2) follows by taking the scalar product of (3) by k; (1) follows by taking the scalar product of (4) by k and using (5). Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 The conductivity tensor. Eq. (6) entails a linear relation between current density and electric field. e 2 n0 e iω J = − E1 + J × B0 me me c or, writing B0=B0n and introducing the plasma and cyclotron frequencies, ω p2 Ω J =i E1 − i c J × n 4πω ω Expanding the vector product, one gets the generalized Ohm’s law and the conductivity tensor σ: J = σ ⋅E Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 The conductivity tensor. 1 2 n e Ω e σ=i i c 2 ω Ω ω m 1 − c ω 0 −i Ω c ω 1 0 0 0 Ω 2 c 1 − ω * The conductivity tensor is therefore anti-hermitean σ ij = −σ ji . Therefore no heat is dissipated, as the real part of σ is skew and Re( Ei ) Im( E j ) = 0 Q = Re(E) ⋅ Re(J ) = 0 Re( J i ) = Re(σ ij ) Re( E j ) − Im(σ ij ) Im( E j ) Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 Plane EM waves in a plasma. The refractive index of a medium is defined as the ratio between c and the phase velocity nr = c ck = vφ ω a × (b × c) = (a ⋅ c)b − (a ⋅ b)c By combining Maxwell eq. k × E1 = ω c k × B1 = −i B1 2 2 r (1 − nr ) Ei + n ki k j k2 E j = −i 4π ω 4π ω J − E1 c c Ji one gets: (7) which gives the current as a function of the electric field. Making use of the relationship J = σ ⋅ E one obtains 2 2 r (1 − nr ) Ei + n ki k j k 2 Ej = i 4π ω σ ij E j kk 4π 2 2 i j δ σ (1 − n ) + n − i r ij r ij E j = 0 2 k ω Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 Plane EM waves in a plasma. The homogeneous set of equations has non-trivial solutions iff kk 4π 2 2 i j det (1 − nr )δ ij + nr 2 − i σ ij = 0 k ω This is the dispersion equation of the plasma. Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 EM waves in a plasma: B=0 If B=0, the conductivity tensor reduces to a scalar: ω p2 σ = −i 4πω Maxwell eq. allow two modes of propagation: (1) k || E1 and (2) k ⊥ E1 2 (1) iω E + 4π J = iω E + 4π σE = i (ω − ω p / ω ) E = 0 ⇒ ω 2 = ω p2 Longitudinal plasma waves occur at the well known plasma frequency. The wavenumber is free! 2 r (2) (1 − n )E = i 4π ω J Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 EM waves in a plasma: B=0 (2) k ⊥ E1 (1 − nr2 )E = i 4π ω J ω p2 Substituting J = σ E = −i E one gets 4πω 2 ω p 2 1 − nr − E = 0 2 ω ω p2 nr = 1 − 2 ω The index of refraction of a plasma is smaller than unity. This implies that the phase velocity of e.m. waves is larger than c: vφ = c ω = >c nr k Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 EM waves in a plasma: B=0 However, information does not travel at the phase velocity: a continuous, monochromatic, plane wave (starting at t=-∞) cannot be used to convey information. Information (e.g. electromagnetic pulses) travels at the group velocity (always smaller or equal to c): vg = dω dk c ω vφ = = > c nr k In an unmagnetized plasma: ω p2 dω nr = = 1 − 2 → ω 2 = c 2 k 2 − ω p2 → v g = = cnr ω ω dk ck Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 Group velocity Let us consider a superposition of two EM waves with slightly different frequency and wavenumber: u1 = cos[(k + ∆k ) z − (ω + ∆ω )t ] u2 = cos[(k − ∆k ) z − (ω − ∆ω )t ] The superposition of the two waves is: u1 + u 2 = cos[(k + ∆k ) z − (ω + ∆ω )t ] + cos[(k − ∆k ) z − (ω − ∆ω )t ] = cos[(kz − ωt ) + (∆kz − ∆ωt )] + cos[(kz − ωt ) − (∆kz − ∆ωt )] = cos(kz − ωt ) cos(∆kz − ∆ωt )] − sin(kz − ωt ) sin(∆kz − ∆ωt )] + cos(kz − ωt ) cos(∆kz − ∆ωt )] + sin(kz − ωt ) sin(∆kz − ∆ωt )] = 2 cos(kz − ωt ) cos(∆kz − ∆ωt )] The low frequency modulation travels at the speed Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess vg = ∆ω ∆k AA 2007-2008 Group velocity For ∆ω , ∆k → 0 v g = dω / dk Let us consider now a “wave packet”, i.e. a more general superposition of EM waves with different frequency and wavenumber: k 0 + ∆k Ψ( z, t ) = ∫ A(k ) exp i[kz − ωt ] dk k 0 − ∆k 1 d2A 2 dω ≅ ∫ A(k0 ) + 2 ξ exp i[(k0 + ξ ) z − ω (k0 )t − ξ t ] dξ 2 dk k dk k0 k 0 − ∆k 0 k 0 + ∆k k 0 + ∆k dω exp i [ z − ξ ξ t ] dξ ∫ dk k0 k 0 − ∆k ≅ A(k 0 ) exp i[k0 z − ω (k0 )t ] dω sin[ z − t] dk k0 ≅ 2 A(k0 ) exp i[k0 z − ω (k0 )t ] dω z − t dk k0 Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess AA 2007-2008 Group velocity This “pulse” travels at the speed v g = The superposition the two waves is:SpazialeAmbienteof Spaziale e Strumentazione dω dk Prof. L.Iess AA 2007-2008