Wave Propagation 2010 Part 12 doc

12 Electromagnetic Waves For simplicity, let us assume propagation of plasma waves in the direction parallel to the ambient magnetic field, i.e., k ⊥ = 0. Then, we have I 0 [ 0 ] = 1andI n=0 [ 0 ] = 0.Thesealsogives I  ±1 [ 0 ] = 0.5. Thus we obtain K 1,1 = K 2,2 , K 1,3 = 0 3 , K 2,3 = 0 and Eq.(26) becomes ⎧ ⎨ ⎩  ∑ s Π 2 p K 1,1 + ˜ ω 2 −c 2 k 2 ||  2 + ∑ s Π 2 p K 2 1,2 ⎫ ⎬ ⎭  ∑ s Π 2 p K 3,3 + ˜ ω 2  = 0 (27) The first factor is for transverse waves where  k ⊥  E. That is, a wave propagates in the z direction while its electromagnetic fields polarize in the x − y plane. The second factor is for longitudinal waves where  k ||  E. That is, a wave propagates in the z direction and only its electric fields polarize in the z direction. The longitudinal waves are also referred to as compressional waves or sound waves. Especially in the case of  k ||  E, waves are called “electrostatic” waves because these waves arise from electric charge and are expressed by the Poisson equation (3). 3.1 Transverse electromagnetic waves The first factor of Eq.(27) becomes the following equation, ∑ s Π 2 p ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩  V 2 t ⊥ V 2 t || −1  + V 2 t ⊥ V 2 t || ˜ ω −k || V d −Ω c  1 − V 2 t || V 2 t ⊥  √ 2k || V t|| Z 0  ˜ ω −k || V d −Ω c √ 2k || V t||  ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ = c 2 k 2 || − ˜ ω 2 (28) The plasma dispersion function Z 0 [x] (Fried & Conte, 1961) is approximated for |x|1as Z 0 [x] ∼ iσ √ π exp[−x 2 ] − 1 x  1 + 1 2x 2 + 3 4x 4 + ···  (29) with σ = ⎧ ⎨ ⎩ 0, Im [x] > 1/|Re[x]| 1, |Im[x ]| < 1/|Re[x]| 2, Im[x] < −1/|Re[x]| Here the argument of the dispersion function x is a complex value. Let us consider that a phase speed of waves is much faster than velocities of plasma particles. Then, the argument of the plasma dispersion function becomes a larger number. Here, the drift velocity of plasma V d is also neglected. Equation (28) is thus rewritten by using Eq.(29) as 3 K 1,3 = 0ifn = 0, and lim a→0 I n  a 2 2  a = lim a→0 a 2 I n  a 2 2  − I n [ 0 ] a 2 2 −0 = lim a→0 a 2 I  n  a 2 2  = 0ifn = 0. 322 Wave Propagation Electromagnetic Wav e s in Plasma 13 (a) Linear scale. (b) Logarithmic scale. Fig. 1. Linear dispersion relation (frequency ω versus wavenumber k) for electromagnetic waves in plasma. The quantities ω and ck are normalized by Π pe . ∑ s Π 2 p ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩  V 2 t ⊥ V 2 t || −1  − V 2 t ⊥ V 2 t || ˜ ω −Ω c  1 − V 2 t || V 2 t ⊥  ˜ ω −Ω c  1 + k 2 || V 2 t || ( ˜ ω −Ω c ) 2 + ···  ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (30) +iσ √ π ∑ s Π 2 p V 2 t ⊥ V 2 t|| ˜ ω −Ω c  1 − V 2 t || V 2 t ⊥  √ 2k || V t|| exp  − ( ˜ ω −Ω c ) 2 2k 2 || V 2 t||  −c 2 k 2 || + ˜ ω 2 = 0 The solutions to above equation are simplified when we assume that the temperature of plasma approaches to zero, i.e., V t|| →0andV t⊥ →0. Note that this approach is known as the “cold plasma approximation.” Equation (30) is rewritten by the cold plasma approximation as Π 2 pe ˜ ω ˜ ω −Ω ce + Π 2 pi ˜ ω ˜ ω −Ω ci + c 2 k 2 || − ˜ ω 2 = 0 (31) Here an electron-ion pair plasma is assumed (Ω ci > 0andΩ ce < 0). The solutions to the above equation are shown in Figure 1. Note that the imaginary part of the complex frequency (growth/damping rate) becomes zero in the present case. There exist four dispersion curves. The dispersion curves for ω > 0 are called “L-mode” (left-handed circularly polarized) waves, while the dispersion curves for ω < 0 are called “R-mode” (right-handed circularly polarized) waves. In the present case, a positive frequency corresponds to the direction of ion gyro-motion, which is left-handed (counter-clockwise) circularly polarized against magnetic field lines. The dispersion curves for the high-frequency R-mode and L-mode waves approach to the following frequencies as k || → 0, 323 Electromagnetic Waves in Plasma 14 Electromagnetic Waves ω R =  (Ω ce + Ω ci ) 2 + 4(Π 2 pe + Π 2 pi −Ω ce Ω ci ) −(Ω ce + Ω ci ) 2 (32) ω L =  (Ω ce + Ω ci ) 2 + 4(Π 2 pe + Π 2 pi −Ω ce Ω ci )+(Ω ce + Ω ci ) 2 (33) which are known as the R-mode and L-mode cut-off frequencies, respectively. The two high-frequency waves approach to ω = ck || , i.e., electromagnetic light mode waves as k || →∞. On the other hand, the low-frequency wave approaches to k || V A as k || →0, and approaches to Ω c as k || →∞.NotethatV A ≡cΩ ci /Π pi is called the Alfven velocity. The R-mode and L-mode low-frequency waves are called electromagnetic electron and ion cyclotron wave, respectively, or (electron and ion) whistler mode wave. The temperature of plasma affects the growth/damping rate in the dispersion relation. Assuming ω |γ| (where ˜ ω ≡ ω + iγ), the imaginary part of Eq.(30) gives the growth rate γ as γ ∼− 1 2ω − Ω ce Π 2 pe (ω − Ω ce ) 2 − Ω ci Π 2 pi (ω − Ω ci ) 2 (34) ×σ √ π ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Π 2 pe V 2 te ⊥ V 2 te || ω −Ω ce  1 − V 2 te || V 2 te ⊥  √ 2k || V te|| exp  − ( ω −Ω ce ) 2 2k 2 || V 2 te ||  +Π 2 pi V 2 ti ⊥ V 2 ti || ω −Ω ci  1 − V 2 ti || V 2 te ⊥  √ 2k || V ti|| exp  − ( ω −Ω ci ) 2 2k 2 || V 2 ti ||  ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ Here the higher-order terms are neglected for simplicity. One can find that the growth rate becomes always negative γ ≤ 0 for light mode waves (|ω| > ω R,L ). On the other hand, γ becomes positive for electromagnetic electron cyclotron waves (Ω ce < ω < 0) with ω Ω ce >  1 − V 2 te || V 2 te ⊥  ,2 ω Ω ce < Π 2 pe (ω − Ω ce ) 2 (35) Here ion terms are neglected by assuming |ω|Ω ci and |ω|Π pi . This condition is achieved when V te⊥ > V te|| , which is known as electron temperature anisotropy instability. As a special case, electromagnetic electron cyclotron waves are also excited if the ion contribution (the third line in Eq.(34)) becomes larger than the electron contribution (the second line in Eq.(34)) around |ω|∼Ω ci . The growth rare becomes positive when ω Ω ci >  1 − V 2 ti|| V 2 ti ⊥  , − Π 2 pe Ω ce < Ω ci Π 2 pi (ω − Ω ci ) 2 (36) 324 Wave Propagation Electromagnetic Wav e s in Plasma 15 (a) Electron temperature anisotropy instability with V te⊥ = 4V te|| , T e|| = T i|| = T i⊥ , Π pe = 10|Ω ce |, c = 200V te|| . (b) Ion temperature anisotropy instability with V ti⊥ = 4V ti|| , T i|| = T e|| = T e⊥ , Π pe = 10|Ω ce |, c = 200V te|| . (c) Firehose instability with V ti|| = 16V ti⊥ , T i⊥ = T e|| = T e⊥ , Π pe = 10|Ω ce |, c = 200V te|| . Fig. 2. Linear dispersion relation for electromagnetic instabilities. Here |ω||Ω ce | and |ω|Π pe are assumed. This condition is achieved when V ti||  V ti⊥ , which is known as firehose instability. For electromagnetic ion cyclotron waves (0 < ω < Ω ci ), the growth rare becomes positive when ω Ω ci <  1 − V 2 ti || V 2 ti ⊥  , − Π 2 pe Ω ce > Ω ci Π 2 pi (ω − Ω ci ) 2 (37) Here ω |Ω ce | and ω  Π pe are used. This condition is achieved when V ti⊥ > V ti|| ,whichis known as ion temperature anisotropy instability. Examples of these electromagnetic linear instabilities are shown in Figure 2, which are obtained by numerically solving Eq.(28). 325 Electromagnetic Waves in Plasma 16 Electromagnetic Waves 3.2 Longitudinal electrostatic waves The second factor of Eq.(27) becomes 1 + ∑ s Π 2 p k 2 || V 2 t ||  1 + ˜ ω −k || V d √ 2k || V t|| Z 0  ˜ ω −k || V d √ 2k || V t||  = 0 (38) By using the similar approach, the above equation is rewritten by using Eq.(29) as 1 − ∑ s Π 2 p ( ˜ ω −k || V d ) 2  1 + 3k 2 || V 2 t|| ( ˜ ω −k || V d ) 2 + ···  +iσ √ π ∑ s Π 2 p k 2 || V 2 t || ˜ ω −k || V d √ 2k || V t|| exp  − ( ˜ ω −k || V d ) 2 2k 2 || V 2 t ||  = 0 (39) For a simple case with V d = 0andω ∼ Π pe ,Eq.(39)gives ω 2 = Π 2 pe +  Π 4 pe + 12Π 2 pe k || V 2 te || 2 ∼ Π 2 pe + 3k 2 || V 2 te || (40) which is known as the dispersion relation of Langmuir (electron plasma) waves. The imaginary part in Eq.(39) gives the damping rate as γ ω ∼− σ √ π 2 ω 5 √ 2k 3 || V 3 te || (ω 2 + 6k 2 || V 2 te|| ) exp  − ω 2 2k 2 || V 2 te ||  (41) This means that the damping of the Langmuir waves becomes largest at k || ∼Π pe /V te|| ,which is known as the Landau damping. Note that the second line in Eq.(39) comes from the gradient in the velocity distribution function, i.e., ∂ f 0 /∂v || . Thus electrostatic waves are known to be most unstable where the velocity distribution function has the maximum positive gradient. As an example for the growth of electrostatic waves, let us assume a two-species plasma, and one species drift against the other species at rest. Then, Eq.(39) becomes 0 = 1 − Π 2 p1 ( ˜ ω −k || V d1 ) 2  1 + 3k 2 || V 2 t1 || ( ˜ ω −k || V d1 ) 2  − Π 2 p2 ˜ ω 2  1 + 3k 2 || V 2 t2 || ˜ ω 2  (42) +i σ √ π k 2 ||  Π 2 p1 V 2 t1 || ˜ ω −k || V d1 √ 2k || V t1|| exp  − ( ˜ ω −k || V d1 ) 2 2k 2 || V 2 t1 ||  + Π 2 p2 V 2 t2 || ˜ ω √ 2k || V t2|| exp  − ˜ ω 2 2k 2 || V 2 t2 ||  By neglecting higher-order terms, the growth rate is obtained as γ ∼− ω 3 (ω − k || V d1 ) 3 (ω − k || V d1 ) 3 Π 2 p2 + ω 3 Π 2 p1 (43) × σ √ π 2k 2 ||  Π 2 p1 V 2 t1|| ω −k || V d1 √ 2k || V t1|| exp  − ( ω −k || V d1 ) 2 2k 2 || V 2 t1||  + Π 2 p2 V 2 t2|| ω √ 2k || V t2|| exp  − ω 2 2k 2 || V 2 t2||  326 Wave Propagation Electromagnetic Wav e s in Plasma 17 (a) Electron beam-plasma instability with V te ≡ V t1|| = V t2|| , V d1 = 4T te , Π p2 = 9Π p1 . (b) Electron beam-plasma instability with V te ≡ V t1|| = V t2|| , V d1 = 4T te , Π p1 = 9Π p2 . Fig. 3. Linear dispersion relation for electrostatic instabilities. The frequency is normalized by Π 2 pe ≡ Π 2 p1 + Π 2 p2 . Note that these two cases have the same growth rate, but the maximum growth rate is given at ω ∼ Π p2 . One can find that the growth rate becomes positive when ω −k || V d1 < 0, (ω − k || V d1 ) 3 Π 2 p2 + ω 3 Π 2 p1 < 0, and the second line in Eq.(43) is negative. If V d1  V t1 + V t2 , the growth rate becomes maximum at ω/k || ∼ V d1 − V t1 . It is again noted that the second line in Eq.(43) comes from derivative of the velocity distribution function with respect to velocity with v = ω/k || . Electrostatic waves are excited when the velocity distribution function has positive gradient. This condition is also called the Landau resonance. Since the positive gradient in the velocity distribution function is due to drifting plasma (or beam), the instability is known as the beam-plasma instability. Examples of the beam-plasma instability are shown in Figure 3, which are obtained by numerically solving Eq.(38). 327 Electromagnetic Waves in Plasma 18 Electromagnetic Waves 3.3 Cyclotron resonance Since the Newton-Lorentz equation (6) and the Vlasov equation (7) cannot treat the relativism (such that c  V d ), plasma particles cannot interact with electromagnetic light mode waves. On the other hand, drifting plasma can interact with electromagnetic cyclotron waves when a velocity of particles is faster than the Alfven speed V A but is slow enough such that electrostatic instabilities do not take place. For isotropic but drifting plasma, Eq.(27) is rewritten as 0 = ˜ ω 2 −c 2 k 2 || −Π 2 pe ˜ ω −k || V de ˜ ω −k || V de −Ω ce −Π 2 pi ˜ ω −k || V di ˜ ω −k || V di −Ω ci (44) +iσ √ π  Π 2 pe ˜ ω −k || V de √ 2k || V te|| exp  − ( ˜ ω −k || V de −Ω ce ) 2 2k 2 || V 2 te ||  +Π 2 pi ˜ ω −k || V di √ 2k || V ti|| exp  − ( ˜ ω −k || V di −Ω ci ) 2 2k 2 || V 2 ti ||  Here higher-order terms are neglected. The imaginary part of the above equation gives the growth rate as γ ∼− 1 2ω − Ω ce Π 2 pe (ω − k || V de −Ω ce ) 2 − Ω ci Π 2 pi (ω − k || V di −Ω ci ) 2 (45) ×σ √ π  Π 2 pe ω −k || V de √ 2k || V te|| exp  − ( ω −k || V de −Ω ce ) 2 2k 2 || V 2 te ||  +Π 2 pi ω −k || V di √ 2k || V ti|| exp  − ( ω −k || V di −Ω ci ) 2 2k 2 || V 2 ti ||  In the case of V de = 0andV di = 0, ion cyclotron waves have positive growth rate at ω ∼ Ω ci when ω k || −V de < 0, − Π 2 pe Ω ce > Ω ci Π 2 pi (ω − Ω ci ) 2 (46) Here ω |Ω ce | and ω  Π pe are assumed. The maximum growth rate is obtained at ω ∼ k || (V de −V A ). In the case of V de = 0andV de = 0, electron cyclotron waves have positive growth rate at ω ∼−Ω ci when ω k || −V di > 0, − Π 2 pe Ω ce < Ω ci Π 2 pi (ω − k || V di −Ω ci ) 2 (47) 328 Wave Propagation Electromagnetic Wav e s in Plasma 19 (a) Ion beam-cyclotron instability with V di = 2V A = 0.005c, T e|| = T e⊥ = T ti|| = T i|| , Π pe = 10|Ω ce |. (b) Electron beam-cyclotron instability with V de = 2V A = 0.005c, T e|| = T e⊥ = T ti|| = T i|| , Π pe = 10|Ω ce |. Fig. 4. Linear dispersion relation for beam-cyclotron instabilities. Here |ω||Ω ce | and |ω|Π pe are assumed. The maximum growth rate is obtained at ω ∼ k || (V di + V A ). These conditions are called the cyclotron resonance. Note that electron cyclotron waves are excited by drifting ions while ion cyclotron waves are excited by drifting electrons. These instabilities are known as the beam-cyclotron instability. Examples of the beam-cyclotron instability are shown in Figure 4, which are obtained by numerically solving Eq.(28). 4. Summary In this chapter, electromagnetic waves in plasma are discussed. The basic equations for electromagnetic waves and charged particle motions are given, and linear dispersion relations of waves in plasma are derived. Then excitation of electromagnetic waves in plasma is discussed by using simplified linear dispersion relations. It is shown that electromagnetic cyclotron waves are excited when the plasma temperature in the direction perpendicular to an ambient magnetic field is not equal to the parallel temperature. Electrostatic waves are excited when a velocity distribution function in the direction parallel to an ambient magnetic field has positive gradient. Note that the former condition is called the temperature anisotropy instability. The latter condition is achieved when a high-speed charged-particle beam propagates along the ambient magnetic field, and is called the beam-plasma instability. Charged-particle beams can also interact with electromagnetic cyclotron waves, which is called the beam-cyclotron instability. These linear instabilities take place by free energy sources existing in velocity space. It is noted that plasma is highly nonlinear media, and the linear dispersion relation can be applied for small-amplitude plasma waves only. Large-amplitude plasma waves sometimes result in nonlinear processes, which are so complex that it is difficult to provide their analytical expressions. Therefore computer simulations play essential roles in studies of nonlinear processes. One can refer to textbooks on kinetic plasma simulations (e.g., Birdsall & Langdon, 2004; Hockney & Eastwood, 1988; Omura & Matsumoto, 1994; Buneman, 1994) for further reading. 329 Electromagnetic Waves in Plasma 20 Electromagnetic Waves 5. References Birdsall, C. K., & Langdon, A. B. (2004), Plasma Physics via Computer Simulation, 479pp., Institute of Physics, ISBN 9780750310253, Bristol Buneman, O. (1994), TRISTAN, In: Computer Space Plasma Physics: Simulation Techniques and Software, Matsumoto, H. & Omura,Y. (Ed.s) pp.67–84, Terra Scientific Publishing Company, ISBN 9784887041110, Tokyo Fried, B. D. & Conte, S. D. (1961), The Plasma Dispersion Function, 419pp., Academic Press, New York Hockney, R. W. & Eastwood, J. W. (1988) Computer Simulation Using Particles, 540pp., Institute of Physics, ISBN 9780852743928, Bristol Omura, Y. & Matsumoto, H. (1994), KEMPO1, In: Computer Space Plasma Physics: Simulation Techniques and Software, Matsumoto, H. & Omura,Y. (Ed.s) pp.21–65, Terra Scientific Publishing Company, ISBN 9784887041110, Tokyo Stix, T. H. (1992), Waves in Plasmas, 584pp., Springer-Verlag, ISBN 9780883188590, New York Swanson, D. G. (2003), Plasma Waves, Second Edition, 400pp., Institute of Physics, ISBN 9780750309271, Bristol Swanson, D. G. (2008), Plasma Kinetic Theory, 344pp., Crc Press, ISBN 9781420075809, New York 330 Wave Propagation [...]... obtained several methods to derive propagation of waves in two-dimensional plasma structures, and analytical solution of surface waves with the effects of significant ne gradient Combining these results, we verified wave propagation as localized surface modes Clearly, the properties of wave propagation are different from those of surface waves on metals as well as those in waves propagating solid photonic... characteristic wave propagation in and around plasmas 2.1 Complex permittivity in a plasma To describe wave transmission and absorption as well as phase shift and reflection of the propagating waves, we here introduce a new drawing of dispersion relation in the 3D space of three coordinates consisting of wave frequency ω / 2π , real wavenumber kr , and imaginary wavenumber k i A propagating wave which... the complex frequency for a real wave number, like in the cases of the plane -wave expansion method described in Section 3.1.1 If we consider spatial wave damping of the static propagation in a finite region, a complex wave number is derived for a real value of frequency This method enables us to take such a flexible approach The relation of complex wavenumber and complex wave frequency was well investigated... kxa/2π = 0.50 and ky = 0 Parameters used are similar to Fig 5 (Sakai & Tachibana, 2007) 344 Wave Propagation 3.1.4 Examples of wave propagation in periodic plasma structures Now, we demonstrate some specific examples of electromagnetic wave propagation (Sakai & Tachibana, 2007), especially with coupling of surface waves on the interface of 2dimensional structures The method to derive field profiles as... temperature, we expect a very wide range waveguide composed of plasma chains; an example was demonstrated in Fig 4 350 Wave Propagation Figure 12 shows conceptual dispersion relations of surface wave modes on surfaces of isolated plasmas, as a summary of the discussion here In a case of the gradual profile of ne shown in Fig 12( a), the propagating modes are on ω − z plane Wave fields are distributed around... electromagnetic waves in and around plasmas 2 Fundamentals of new aspects for wave propagation In this Section, we demonstrate features and importance of complex permittivity which is usual in a low-temperature partially-ionized plasma Section 2.1 includes general 333 Propagation of Electromagnetic Waves in and around Plasmas description which is also applicable to other lossy materials (Sakai et al., 2010( 1)),... 2007), where the region with no detection of flat bands ranges from 20 GHz to ωpe / 2π The structures of wave propagation are too fine to be detected in the modified plane -wave method, and therefore, an increase of assumed plane waves might be required to detect them in this method In summary, wave propagation on the flat bands of a 2-D columnar plasma array is mainly attributed to the dispersion of... 2007) 346 Wave Propagation So far, we have investigated wave propagation in an array of plasma columns The next target is antiparallel structure, which is an infinite-size plasma with periodic holes Above ωpe / 2π , periodic dielectric constant in space will contribute to form a similar band diagram When the frequency is low enough, since there is no continuous vacuum space in this structure, wave propagation. .. discharge plasma 3.2 Surface wave propagation in a plasma with spatially gradient electron density In Section 3.1.4, several features of the localized surface waves in plasma periodic structures have been demonstrated Some features are in common with the cases of light propagation on metal particles, but others are not; in this section, we clarify the different points from the surface waves or the surface... surface wave modes in a gas-discharge plasma Up to now, we have concentrated on surface waves on an infinite flat interface Usually the excitation of surface waves on such a flat surface requires some particular methods such as ATR configuration or periodic structure like fluctuating surface If we generate an isolated plasma from the others whose size is less than the wavelength, localized surface waves . aspects of plasma wave propagation Since wave propagation in a magnetized plasma is well described elsewhere (Stix, 1962; Ginzburg, 1964; Swanson 1989), we here focus on the propagation in and. so this treatment is beneficial Propagation of Electromagnetic Waves in and around Plasmas 337 when we consider wave propagation on the edge as a surface wave. Ignoring the friction term. of  k ||  E, waves are called “electrostatic” waves because these waves arise from electric charge and are expressed by the Poisson equation (3). 3.1 Transverse electromagnetic waves The first