Wave propagation Motivation for us

Wave propagation• Waves are important in space research– remote observations from radio waves to gamma rays• the signals propagate through space plasma– wave diagnostics of medium• ionospheric radars, Faraday rotation, etc.• Waves in space plasmas– wide variety of wave modes– natural response of plasma to perturbations– specific frequencies• plasma frequency, gyro frequency and their hybrids– energy transfer• particle acceleration / heating• instabilitiesMotivation for us1

In absence of charges and currentsMaxwell’s equations reduce toEM waves in vacuumhomogeneous wave equationspropagation speed:Consider propagation to ±z – direction and plane wave solutions.For a plane wave there is a plane propagating with the wave wherethe electric field is constant. Such a solution can be written asor in vector formamplitudeangular frequencywave numberphase speedA wave can be considered as a plane wave only far from the source.Sometimes it is necessary to consider spherical waves, i.e., waves,for which there exists a spherical surface where E is constant.An example is the field of a radiating dipole:(this is an approximation up to terms of the order of 1/r 3 ; the exactsolution can be expressed in terms of Bessel functions)Plane waves are most convenientto treat using the complex notationNow the differential operationsreduce to multiplicationsand Maxwell’s equationsbecome algebraic equations2

From vacuum to dielectric mediaIf there are no free charges or currents, but 0 and 0 (but constants)Nowis the index of refractiondispersion equationphase velocity (propagation of the constant phase)group velocity (propagation of energy and information!)If in addition J = E ( constant), the situation is more complicatedone form of so-called telegrapher’s equations;a standard class-room example of using Fouriertransformations to solve partial differential equationsInstead of making the Fourier transformations, let’s start from Maxwell’sequations and make the plane wave assumptionClearly k, E and H are all perpendicularto each other: transversal waveChoose: k || e z , E || e x , H || e ydispersion equationnow !writeThe solution isselect the phase of sothat the wave is damped!impedance of the medium:skin depth3

ExamplesWave polarizationConsider an EM wave propagating in +z directionFocus on the plane z = 0, denote = E y /E x = – H x /H y ;1) If is real, E y and E x are in the same phaseDirection of E is (1,,0) (if = , E is in the y-direction)This is a linearly polarized wave2) If = +i, there is a phase shift (= /2) between E y and E xThis is a right-hand circularly polarized wave (positive helicity)3) If = –i , there is a phase shift ( = –/2) between E y and E xThis is a left-hand circularly polarized wave (negative helicity)4) If is a general complex number, the wave is elliptically polarizedAll polarizations can be obtained from left- and right-hand polarizedwaves by superposition.WARNING: In optics the left- and right-hand polarizations are definedin the opposite way!!4

k is a vectorWave vector:unit vector defining thedirection of the wave normal surface of constant wave phase;it is the direction of wave propagationIn isotropic media k ||If the medium is anistropic, it is possible thati.e., the energy does not need to propagate in the direction of wave normal.Energy and infromationpropagate with the group velocityLetbe the angle between B and k and the frequency a function of k andangle between v g and v pReflection and refractiontransmitted,refractedincidentreflectedThis is vertical polarization (E has a vertical component)The opposite case is horizontal polarizationAll polarizations can be expressed as superposition of these.5

Boundary conditions:Surface currentinduced by the waveAssume K = 0 These conditions must be filled all the time at every point on the interface As the incident and reflected waves are in the same medium, we get theresults familiar from the high-school physicsSnell’s law6

Reflection coefficient for vertical polarizationIn our examplesFresnel’s formulasfor vertical polarizationtransmission coefficient for vertical polarizationFresnel’s formulasfor horizontal polarizationExample: Total reflection from the ionosphereFor radio waves of sufficiently high frequencyThe air below the ionosphere is a good non-conductive dielectricIn the ionosphereFresnel’s formulas for all angles of incidenceThus the horizontal polarization is more efficient in radio communicationsvia the ionosphere than the vertical polarizationFor sufficiently largei.e., the whole wave is reflected;total reflection7

Dispersion of the medium(so-called Drude-Lorentz model)In dispersive media (in plasmas we can usually assume )Consider the motion of an electron boundto an atom by a harmonic forceelectronInclude a frictional force to dampthe harmonic oscillationsatomUnder the influence of an external electric field the eq. of motion becomesLet the time dependences be harmonicBy definition the dipole moment associated with this electron isAssume that in a unit volume there are N molecules and Z electrons/moleculeLetelectrons of each molecule have an eigenfrequencyand a damping factor”oscillation strengths”:Þ the density of dipole moments, i.e., electric polarization / unit volumeWrite the electric displacementThe electric permittivity is a complex function and depends on the frequency8

Let the number of free electrons / molecule beTheir eigenfrequencies areAmpère’s law:conductanceelectronsStatic conductivity is a goodapproximation for Cu, ifcompare with FM radioare resonant frequenciesOften almost real, except whenDispersion is calledonly close to resonant frequencies, whereAt resonance energy is transferred between the wave and particlesPolarization current isand the work done by E is9

Averaging over one oscillation periodNote that this model always hasand thus cannot be applied to lasers or plasmainstabilities where the wave gains energy from plasmaenergy transferfrom field to particlesrefractive index:wave number:phase velocity:group velocity:Radio wave propagation in the ionosphere10

Isotropic, lossless ionosphereIsotropic: No effects of B included; OK ifLossless: Effects of collisions neglected; OK ifThese are fulfilled ifConsider the electron motion in the electric field of a (plane) waveassuming that ions form an immobile background.Now we determine and Thus we have formally found Ohm’s law withAssume that except for this conductivity the plasma is vacuum:Now the Ampère-Maxwell lawyields for the plane wavesThus the medium looks like dielectric withIn plasma physics we often writeThe index of refraction isandxEkzBykIf X > 1, the wave does not propagate11

In the ionosphere the maximum electron densities are of the order the maximum plasma frequency in the ionosphere is about 9 MHzEM wave packet (f < 9 MHz) propagates vertically up and returns after time TVirtual reflection height:real reflection height v g becomes zeroIn realityIonosondeLinear density profileReal reflection takes place wherefrom which we get the virtual heightParabolic density profile is also straightforward to calculate (exercise) fromm refers to maximum density12

Oblique propagation: angle between the vertical direction and kThe horizontal distance of wave packet propagationThe vertical motion is obtained by substitutionthe real height at time tEliminating t:This gives the ”ray path”.In isotropic plasma the ray propagates in the direction of nWeakly inhomogeneous ionosphere(the WKB approximation)What happens at the reflection point, i.e., wherethe vertical component of the group velocity is zero?Reflection is expected to take place at each interface of different nConsider a wave with and (for simplicity) vertical propagationÞreflection should be easy to observebut there is no reflection!!Construct a model ionosphere consisting ofof thin layers with upward increasing density, i.e.,decreasing index of refractionFigure is for obliquepropagation but letAt each layerbut this is not the whole story.The phase of the wave ateach layer becomes critical13

Let E 0 be the amplitude for the incident wave in the neutral atmosphereThe electric field of the wave after reflection isFrom each layeris reflectedis refracted the electric field of the refracted wave grows!The wave propagates toward an increasing impedanceFor the wave magnetic fieldAt the limit of continuous profileAt each layer the phase of the wave is shifted byAt the altitude z the accumulated retardation is given byThis is the WKB approximation(Wentzel, Kramers, Brillouin)The electric and magnetic fields are nowNow the Poynting vector is constant:This implies that no net energy is carried by the partially reflected wavesdue to the interference of different phases of waves reflected at differentlayers. This requires that the profile is smooth enough as compared tothe wavength of the wave. Let’s try to quantify this.Amplitude of each partial wave isand the phase differencebetween waves reflected from two consecutive layersConstruct a phase amplitude diagram witharc element representingeach partial waveassociated phase14

Add arc elements to each other by turningthe next one with respect to the previousone by the phase angleMove to the continuous limitrdIf no reflection would take place, this wouldgive a circle. In case of reflection the radiusincreases resulting in a spiralThe increase of the radius can be neglected ifThus the WKB approximation breaks if• k 0 is small (long wave wrt. the gradient scale length• n 0 (close to the actual reflection point)• at local gradients with otherwise smooth profileAbove the reflection region n 2 < 0 and thus n is imaginaryAt the reflection pointthe wave is dampedthe WKB approximation breaksThe field can, can however, be calculated analytically if the profileclose to the reflection point is linear.Now Maxwell’s equations can be written asassuming linear profilechange of variable 15

is known as Airy’s differential equationwhose solutions areAiry integralsBiAiAs the wave must vanish abovethe reflection point, C 2 = 0The amplitude and period increase–whenThereafter the field approachesrapidly 0Ai can be given in the integral formTo determine the constant C 1 the Airy function solution must join the WKBsolution at large negative . Finding the asymptotic behaviour of Ai fornegative argument is technically difficult (roots of negative numbers).The result isMatching this with the WKB solution givesupward propagatingWKB solutionphase shift /4 + /4 = /2coming from the non-WKB regiondownward propagatingWKB solution16

The reflection coefficient isi comes from the phase shiftIn the reflection regiongradient scale lengthLet L 100 km ffor a perfect mirror:and increases by a factor of 14If the wave is strong enough, it can couple to the oscillation modes of plasmaand lead to heating of the medium.For oblique propagation substituteInclude collisionsModel the effect of collisions as friction to electron motionHarmonic time-dependence Denotethe WKB solution isnow complicated thanin the loss-free case17

Polar ionosphereInclude background magnetic fieldThe unmagnetized theory islimited to frequenciesDenote”magnetoionic theory”xEkzBypolarization:refractiveindex:Appleton-Hartree equations+ in & – in n 2 : ordinary mode– in & + in n 2 : extraordinary modeMore of this with a more modernformalism in the following.Cold plasma wavesEM wave propagation through and interaction with plasmas belong to centralissues of plasma physics.- diagnostics of the medium(recognize p density, recognize c magnetic field, and other much morecomplicated methods)- understanding of the effects of the medium to the ”original” signalMeaning of cold:Strategy: look forrequires physicsunderstandingand find its zeros dispersion equationsolutions often requirenumerical methods18

Cold plasma dispersion equationStarting from Maxwell’s equations and Ohm’s law (where is a tensor)we can derive the wave equationThe dielectric tensorIf there are no background fieldsthis reduces towhereis a dimensionless quantity:In the absence of B 0 we already know two solutions of the wave equation:Assume then that the plasma is in a homogeneousbackground B 0 and consider small perturbations:Now we also include ions, thus the current is:Note:As plasma is assumed cold, the average velocity is the same asthe velocity of individual particlesAssuming harmonic time dependence of V:the equation of motion for each species isThis equation is easiest to study in the coordinate system with base vectorswhereThat is, the plane perpendicular to B 0 is considered as a complex plane(recall the discussion of wave polarization)Letindex the components in this frame.Introduce the shorthand notation(here is positive and is the sign of the charge of the species)Now the components of the currentareand the dielectric tensorin this frame is19

The elements of the tensor can be written asThe denominator of R is zero when the wavefrequency equals to electron gyro frequency:Resonance with electronsThe denominator of L is zero when the wavefrequency equals to positive ion gyro frequency:Resonance with ionsPlasma oscillationR and L refer to right-hand and left-hand polarized waves whencan be transformed back to the basewhereConsider the index of refraction as a vectorThus the wave equation can be written asChoose coordinates asxznB 0yThe non-trivial solutions of this wave equation are given by(this is the dispersion equation!)whereThis is a quite convenient formulation, as it readily allows for consideration ofpropagation to different directions with respect to background magnetic field.Solving forwe get20

Principal modes:The principal mode solutions forCut-offareNote that = / 2refers to the anglebetween k and B 0 !The wave cannot propagate through the cut-off (reflection)ResonanceFor EM-wave k B 1Wave energy is absorbed by the plasma (acceleration, heating)Examples of resonances for parallel propagation ( = 0):R resonance with right-hand polarized wave (electrons)L resonance with left-hand polarized wave (positive ions)Parallel propagationWave modes in cold plasmaThe right-hand polarized mode:(consider here and belowone positive ion species)resonance with electron gyro motionCut-off:At low frequency limitAt high frequency limitThe left-hand polarized mode:small n limitlarge n limit(Alfvén wave)(EM wave)Cut-off:small n limitlarge n limitresonance with ion gyro motionSmall and large density limits mean pe > ceIn non-magnetized plasma the cut-off was at p , now it is split to two branches.21

Parallel propagating wave modesin -spaceNote the differences to thenon-magnetized case:Cut-offsRLresonances high-density limitThe EM-branch issplit to left- and righthandedcomponentsthat have differentphase velocities!kCut-offsRresonances Llow-density limitFaraday rotationAn important consequence of the different phase velocities of theL- and R-modes is the so-called Faraday rotation. Consider a linearly polarizedwave and express its electric field as a sum of L- and R-polarized components:For a given , k R k L andFor a linearly polarized waveThe rotation anglei.e., the polarization plane rotatesas a function of zdepends on n and BIn astrophysics the plasma and gyro frequencies are small as comparedto the frequency of EM waves we use in our observation. Thus thedispersion equation can be approximated as22

the differential rotation of the plane of polarization isIntegrate this from the source to the observer:In astrophysics the rotation measure RM is defined byNumericallycm –3nTHzmBeware of unitsin the literature!!Note that the rotation is determined modulo observations at several frequencies neededDispersion measureAssume a distant broad-band source (e.g., a pulsar) at frequencies higherthan plasma and gyro frequencies. From the dispersion equation:The propagation time of the signal to the observer isD is the dispersion measureDetermining the time delay between two frequenciesgives average density along the propagation pathExample:1. Determine the distance from the redshift2. Estimate the density from D n e3. Using n e and Faraday rotation, estimate B23

Whistler modeThe R-mode propagates also in the frequency rangewhere the dispresion equation can be approximated asFrom the group velocitywe can calculate the propagationtime (as a function of frequency)Thus the lower frequencies arriveto an observer after a longer timethan the higher frequences.Easily observable with a simpleantenna and audio amplifier(kHz frequency range)The whistlers were first observed on telegraph lines during World War I.The phenomenon was not explained until 1953, when L. R. O. Storeydescribed the phenomenon to be due to lightning strokes on the conjugatehemisphere and propagating along the magnetic field to the observer.Lightning dischargespace is not empty!Number of lightning strokes / km 2 / yearThere are 16 million lightning storms per year!Thus there are whistlers in space all the time.Two-hop whistler24

Perpendicular propagationModes propagating perpendicular to the background magnetic fieldare called ordinary (O) and extraordinary (X) modes. Unfortunatelythe convention in plasma physics is opposite to optics!O-mode:This corresponds to the EM mode in isotropic plasma with cut-off at(to remember: B is not included in dispersion eq. of the O-mode)X-mode: (in the following we make use of )The X-mode has two hybrid resonancesThe upper hybrid resonance:The lower hybrid resonance:The lower hybrid resonance is particulary important because there the wavecan be in a resonance of both electrons and ions, which provides efficientmeans for energy transfer between particle populationsLow-density limit:High-density limit:X-mode has also two cut-offs:Low-density limitHigh-density limit:When 0the X-modeapproaches the magnetosonicmode of MHDXOXXIn MHD finite T assumed(not in cold theory)Effect of both finite T and displacement currentcorrection due to the displacementcurrent; v A can be a considerablefraction of c25

Propagation atarbitrary anglesClemmow-Mullaly-Allis (CMA) diagramTopologies of wave-normal surfacesB is upwardThe R, L, O, X division isunambiguous for principalmodes only ( = 0, / 2).Wave normal surfaces:- calculate the phase velocityfor different :- e.g., MHD waves (next Ch.)Presentation of wave modes in(here electron-related modes only)spacea) Whistlerb) L-modec) Upper hybrid moded) Plasma frequency(Langmuir wave)e) R-modef) Bernstein modes(not discussed here)This picture has beencomputed in Vlasovtheory and thus ismore exact than thecold and MHDapproximationsExcercise: Identify O- and X-modes, where do the thermaleffects on Langmuir wave start to play role, etc. ?26