Wave Propagation Part 10 potx

The Analysis of Hybrid Reflector Antennas and Diffraction Antenna Arrays on the Basis of Surfaces with a Circular Profile 307 analysis by a method of eigenfunctions and in approximations of GTD is carried out. The spherical diffraction antenna array allows: to control amplitude end phase fields distribution on all aperture of HRA; to provide high efficiency because of active radiating units of feeds do not shade of aperture; to realize a combined amplitude/multibase phase method of direction finding of the objects, polarization selection of signals. The HRA’s provide: increasing of range of radars operation by 8-10 %; reduce the error of measurement of coordinates at 6-8 times; reduction of probability of suppression of radar by active interferences by 20-30 %. On the basis of such antennas use of MMIC technology of fabricate integrated feeds millimeter and centimeter waves is perspective. Embedding the micromodules into integral feeding-source antennas for HRA’s and spherical diffraction antenna arrays for processing of the microwave information can be utilized for long-term evolution multifunctional radars. Future work includes a more detailed investigation the antennas for solving a problem of miniaturization of feeds for these antennas by means of MMIC technologies. 6. References Bucci, O.M., Elia, G.D. & Romito, G. (1996). Synthesis Technique for Scanning and/or Reconfigurable Beam Reflector Antennas With Phase-only Control. IEE Proc Microw. Antennas Propag., Vol. 143, No. 5, October, p.p. 402-412. Chantalat, R., Menudier, C., Thevenot, M., Monediere, T., Arnaud, E. & Dumon, P. (2008). Enhanced EBG Resonator Antenna as Feed of a Reflector Antenna in the Ka Band. IEEE Antennas and Wireless Propag., Vol. 7, p.p. 349-353. Elsherbeni, A. (1989). High Gain Cylingrical Reflector Antennas with Low Sidelobes. AEU, Band 43, Heft 6, p.p. 362-369. Eom, S.Y., Son, S.H., Jung, Y.B., Jeon, S.I., ganin, S.A., Shubov, A.G., Tobolev, A.K. & Shishlov, A.V. (2007). Design and Test of a Mobile Antenna System With Tri-Band Operation for Broadband Satellite Communications And DBS Reception. IEEE Trans. on Antennas and Propag., Vol. 55, No. 11, November, p.p. 3123-3133. Fourikis, N. (1996). Phased Array-Based Systems and Applications, John Willey & Sons., Inc. Gradshteyn, I.S. & Ryzhik, I.M. (2000). Table of Integrals, Series and Products, 930, 8.533, Academic Press, New York. Grase, O. & Goodman, R. (1966). Circumferential waves on solid cylinders. J. Acoust. Soc. America, Vol. 39, No. 1, p.p.173-174. Haupt, R.L. (2008). Calibration of Cylindrical Reflector Antennas With Linear Phased Array Feeds. IEEE Trans. on Antennas and Propag., Vol. 56, No. 2, February, p.p. 593-596. Janpugdee, P., Pathak, P. & Burkholder, R. (2005). A new traveling wave expansion for the UTD analysis of the collective radiation from large finite planar arrays. IEEE AP- S/URSI Int. Symp., Washington, DC, July. Jeffs, B. & Warnick, K. (2008). Bias Corrected PSD Estimation for an Adaptive Array with Moving Interference. IEEE Trans. on Antennas and Propag., Vol. 56, No. 7, July, p.p. 3108-3121. Jung, Y.B. & Park, S.O. (2008). Ka-Band Shaped Reflector Hybrid Antenna Illuminated by Microstrip-Fed Horn Array. IEEE Trans. on Antennas and Propag., Vol. 56, No. 12, December, p.p. 3863-3867. Wave Propagation 308 Jung, Y.B., Shishlov, A. & Park, S.O. (2009). Cassegrain Antenna With Hybrid Beam Steering Scheme for Mobile Satellite Communications. IEEE Trans. on Antennas and Propag., Vol. 57, No. 5, May, p.p. 1367-1372. Keller, J. (1958). A geometrical theory of diffraction. Calculus of variations and its applications. Proc. Symposia Appl. Math., 8, 27-52. Mc Graw-Hill., N.Y. Llombart, N., Neto, A., Gerini, G., Bonnedal, M. & Peter De Maagt. (2008). Leaky Wave Enhanced Feed Arrays for the Improvement of the Edge of Coverage Gain in Multibeam Reflector Antennas. IEEE Trans. on Antennas and Propag., Vol. 56, No. 5, May, p.p. 1280-1291. Love, A. (1962). Spherical Reflecting Antennas with Corrected Line Sources. IRE Trans. on Antennas and Propagation, Vol. AP-10, September, No. 5-6, p.p.529-537. Miller, M. & Talanov, V. (1956). Electromagnetic Surface Waves Guided by a Boundary with Small Curvature. Zh. Tekh. Fiz, Vol. 26, No. 12, p.p. 2755-2765. Ponomarev, O. (2008). Diffraction of Electromagnetic Waves by Concave Circumferential Surfaces: Application for Hybrid Reflector Antennas. Bull. of the Russian Academy of Sciences: Physics, Vol. 72, No. 12, p.p. 1666-1670. Rayleigh, J.W.S. (1945). The Theory of Sound, 2-nd ed., Vol. 2, Sec. 287, Dover Publication, ISBN 0-486-60293-1, New York. Schell, A. (1963). The Diffraction Theory of Large-Aperture Sp-herical Reflector Antennas. IRE Trans. on Antennas and Propagation, July, p.p. 428-432. Shevchenko, V. (1971). Radiation losses in bent waveguides for surface waves. Institute of Radioengineering and Electronics, Academy of Sciences of the USSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 14, No. 5, p.p. 768-777, May. Spencer, R., Sletten, C. & Walsh, J. (1949). Correction of Spherical Aberration by a Phased Line Source. Proc. N.E.C., Vol. 5, p.p. 320-333. Tap, K. & Pathak, P.H. (2006). A Fast Hybrid Asymptotic and Numerical Physical Optics Analysis of Very Large Scanning Cylindrical Reflectors With Stacked Linear Array Feeds. IEEE Trans. on Antennas and Propag., Vol. 54, No. 4, April, p.p. 1142-1151. Tingye, L. (1959). A Study of Spherical Reflectors as Wide-Angle Scanning Antennas. IRE Trans. on Antennas and Propagation, July, p.p. 223-226. Part 4 Wave Propagation in Plasmas 1. Introduction No less than 99.9% of the matter in the visible Universe is in the plasma state. The plasma is a gas in which a certain portion of the particles are ionized, and is considered to be the “fourth” state of the matter. The Universe is filled with plasma particles ejected from the upper atmosphere of stars. The stream of plasma is called the stellar wind, which also carries the intrinsic magnetic field of the stars. Our solar system is filled with solar-wind-plasma particles. Neutral gases in the upper atmosphere of the Earth are also ionized by a photoelectric effect due to absorption of energy from sunlight. The number density of plasma far above the Earth’s ionosphere is very low ( ∼100cm −3 or much less). A typical mean-free path of solar-wind plasma is about 1AU 1 (Astronomical Unit: the distance from the Sun to the Earth). Thus plasma in Geospace can be regarded as collisionless. Motion of plasma is affected by electromagnetic fields. The change in the motion of plasma results in an electric current, and the surrounding electromagnetic fields are then modified by the current. The plasma behaves as a dielectric media. Thus the linear dispersion relation of electromagnetic waves in plasma is strongly modified from that in vacuum, which is simply ˜ ω = kc where ˜ ω, k,andc represent angular frequency, wavenumber, and the speed of light, respectively. This chapter gives an introduction to electromagnetic waves in collisionless plasma 2 , because it is important to study electromagnetic waves in plasma for understanding of electromagnetic environment around the Earth. Section 2 gives basic equations for electromagnetic waves in collisionless plasma. Then, the linear dispersion relation of plasma waves is derived. It should be noted that there are many good textbooks for linear dispersion relation of plasma waves. However, detailed derivation of the linear dispersion relation is presented only in a few textbooks (e.g., Stix, 1992; Swanson, 2003; 2008). Thus Section 2 aims to revisit the derivation of the linear dispersion relation. Section 3 discusses excitation of plasma waves, by providing examples on the excitation of plasma waves based on the linear dispersion analysis. Section 4 gives summary of this chapter. It is noted that the linear dispersion relation can be applied for small-amplitude plasma waves only. Large-amplitude plasma waves sometimes result in nonlinear processes. Nonlinear processes are so complex that it is difficult to provide their analytical expressions, and computer simulations play important roles in studies of nonlinear processes, which should be left as a future study. 1 1AU∼150,000,000km 2 This work was supported by Grant-in-Aid for Scientific Research on Innovative Areas No.21200050 from MEXT of Japan. The author is grateful to Y. Kidani for his careful reading of the manuscript. Takayuki Umeda Solar-Terrestrial Environment Laboratory, Nagoya University Japan Electromagnetic Waves in Plasma 15 2 Electromagnetic Wa ves 2. Linear dispersion relation 2.1 B asic equations The starting point is Maxwell’s equations (1-4) ∇×  E = − ∂  B ∂t ,(1) ∇×  B = μ 0  J + 1 c 2 ∂  E ∂t ,(2) ∇·  E = ρ  0 ,(3) ∇·  B = 0, (4) where  E,  B,  J,andρ represent electric field, magnetic field, current density, and charge density, respectively. Here a useful relation  0 μ 0 = 1/c 2 is used where  0 and μ 0 are dielectric constant and magnetic permeability in vacuum, respectively. The motion of charged particles is described by the Newton-Lorentz equations (5,6) d x dt = v,(5) d v dt = q m   E +v ×  B  ,(6) where x and v represent the position and velocity of a charged particle with q and m being its charge and mass. The motion of charged particles is also expressed in terms of microscopic distribution functions ∂ f ∂t +v · ∂ f ∂x + q m   E +v ×  B  · ∂ f ∂v = 0, (7) where f [x,v, t] represents distribution function of plasma particles in a position-velocity phase space. Equation (7) is called the Vlasov equation or the collisionless Boltzmann equation (collision terms of the Boltzmann equation in right hand side is neglected). The zeroth momentum and the first momentum of the distribution function give the charge density and the current density ρ = q  f d 3 v,(8)  J = q  vfd 3 v.(9) 2.2 D erivation of linear dispersion equation Let us “linearize” the Vlasov equation. That is, we divide physical quantities into an equilibrium part and a small perturbation part (for the distribution function f = n( f 0 + f 1 ) with f 0 and f 1 being the equilibrium and the small perturbation parts normalized to unity, respectively). Then the Vlasov equation (7) becomes ∂ f 1 ∂t +v · ∂ f 1 ∂x + q m  v ×  B 0  · ∂ f 1 ∂v = − q m   E 1 +v ×  B 1  · ∂ f 0 ∂v . (10) 312 Wave Propagation Electromagnetic Waves in Plasma 3 Here, the electric field has only the perturbed component (  E 0 = 0) and the multiplication of small perturbation parts is neglected ( f 1  E 1 → 0andf 1  B 1 → 0). Let us evaluate the term  v ×  B 0  · ∂ f 0 ∂v by taking the spatial coordinate relative to the ambient magnetic field and writing the velocity in terms of its Cartesian coordinate v =[v ⊥ cosφ,v ⊥ sinφ,v || ]. Here, v || and v ⊥ represent velocity components parallel and perpendicular to the ambient magnetic field, and φ = Ω c t + φ 0 represents the phase angle of the gyro-motion where Ω c ≡ q m B is the cyclotron angular frequency (with sign included). Then, we obtain  v ×  B 0  · ∂ f 0 ∂v = ⎡ ⎣ v y B 0 −v x B 0 0 ⎤ ⎦ · ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ∂v ⊥ ∂v x ∂ f 0 ∂v ⊥ ∂v ⊥ ∂v y ∂ f 0 ∂v ⊥ ∂ f 0 ∂v || ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎣ v y B 0 −v x B 0 0 ⎤ ⎦ · ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ v x v ⊥ ∂ f 0 ∂v ⊥ v y v ⊥ ∂ f 0 ∂v ⊥ ∂ f 0 ∂v || ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = 0. This means that the distribution function must not be changed during the gyration of plasma particles around the ambient magnetic field at an equilibrium state. By using the total derivative, Eq.(10) can be rewritten as d f 1 dt = − q m   E 1 +v ×  B 1  · ∂ f 0 ∂v , and the solution to which can be obtained as f 1 [x,v , t]=− q m  t −∞   E 1 [x  ,t  ]+v  ×  B 1 [x  ,t  ]  · ∂ f 0 [v  ] ∂v  dt  , (11) where [x  ,v  ] is an unperturbed trajectory of a particle which passes through the point [x,v ] when t  = t. Let us Fourier analyze electromagnetic fields, E 1 (x,t ) ≡ E 1 exp[i  k ·x −i ˜ ωt], B 1 (x,t ) ≡ B 1 exp[i  k ·x −i ˜ ωt]. where ˜ ω ≡ ω + iγ is complex frequency and  k is wavenumber vector. Then Maxwell’s equations yield  k ×  E 1 = ˜ ω  B 1 , (12)  k ×  B 1 = −iμ 0  J 1 − ˜ ω c 2  E 1 . (13) Inserting Eq.(12) into Eq.(13), we obtain  k ×(  k ×  E 1 )=(  k ·  E 1 )  k −|  k| 2  E 1 = −i ˜ ωμ 0  J 1 − ˜ ω 2 c 2  E 1 , 0 =   k  k −|  k| 2 ←→ I  c 2 ˜ ω 2  E 1 +  E 1 + i c 2 ˜ ω μ 0  J 1 , (14) 313 Electromagnetic Waves in Plasma 4 Electromagnetic Wa ves where ←→ I represents a unit tensor anda  b denotes a tensor such that a  b = ⎡ ⎣ a x b x a x b y a x b z a y b x a y b y a y b z a z b x a z b y a z b z ⎤ ⎦ = ⎡ ⎣ a x a y a z ⎤ ⎦ ⎡ ⎣ b x b y b z ⎤ ⎦ T . By using Eqs.(9), (11) and (12), the last term in the right hand side of Eq.(14) yields i c 2 ˜ ω μ 0  J 1 = −i Π 2 p ˜ ω  t −∞   E 1 +v  ×  k ×  E 1 ˜ ω  · ∂ f 0 ∂v  exp[i  k ·x  −i ˜ ωt  ]dt  vd 3 v, (15) where Π p ≡  q 2 n m 0 represents the plasma angular frequency. It follows that   E 1 +v  ×  k ×  E 1 ˜ ω  · ∂ f 0 ∂v  = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ E x1  1 − v  y k y + v  z k z ˜ ω  + E y1 v  y k x ˜ ω + E z1 v  z k x ˜ ω E x1 v  x k y ˜ ω + E y1  1 − v  x k x + v  z k z ˜ ω  + E z1 v  z k y ˜ ω E x1 v  x k z ˜ ω + E y1 v  y k z ˜ ω + E z1  1 − v  x k x + v  y k y ˜ ω  ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ · ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ v  x v ⊥ ∂ f 0 ∂v ⊥ v  y v ⊥ ∂ f 0 ∂v ⊥ ∂ f 0 ∂v || ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = E x1 h x + E y1 h y + E z1 h z where h x = v  x v ⊥  1 − v  z k z ˜ ω  ∂ f 0 ∂v ⊥ + v  x k z ˜ ω ∂ f 0 ∂v || h y = v  y v ⊥  1 − v  z k z ˜ ω  ∂ f 0 ∂v ⊥ + v  y k z ˜ ω ∂ f 0 ∂v || h z = v  z (v  x k x + v  y k y ) ˜ ωv ⊥ ∂ f 0 ∂v ⊥ +  1 − v  x k x + v  y k y ˜ ω  ∂ f 0 ∂v || ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ . (16) Now, let us consider transforming from Lagrangian coordinate along the unperturbed trajectory [x  ,v  ,t  ] to Eulerian coordinate [x,v, t] in a stationary frame. We define the velocity as v  x = v ⊥ cos[Ω c (t −t  )+φ 0 ] v  y = v ⊥ sin[Ω c (t −t  )+φ 0 ] v  z = v || ⎫ ⎬ ⎭ , and integrate the velocity in the polar coordinate over time to obtain the position x  = x − v ⊥ Ω c { sin[Ω c (t −t  )+φ 0 ] −sinφ 0 } y  = y + v ⊥ Ω c { cos[Ω c (t −t  )+φ 0 ] −cosφ 0 } z  = z −v || (t −t  ) ⎫ ⎬ ⎭ . 314 Wave Propagation Electromagnetic Waves in Plasma 5 Further taking the wavenumber vector k x = k ⊥ cosθ, k y = k ⊥ sinθ, k z = k || , we obtain exp [i  k ·x  −i ˜ ωt  ]=exp[i  k ·x −i ˜ ωt]exp[i( ˜ ω −v || k || )(t −t  )] × exp  −i v ⊥ k ⊥ Ω c  sin [Ω c (t −t  )+φ 0 −θ] −sin[φ 0 −θ]   = exp[i  k ·x −i ˜ ωt] ∞ ∑ l,n=−∞ J l  v ⊥ k ⊥ Ω c  J n  v ⊥ k ⊥ Ω c  ×exp[i(l −n)(φ 0 −θ)]exp[i( ˜ ω −v || k || −nΩ c )(t −t  )] (17) where J n [x] is the Bessel function of the first kind of order n with exp [ia sin ψ]= ∞ ∑ n=−∞ J n [a]exp[inψ]. Eq.(16) also becomes h x = cos[Ω c (t −t  )+φ 0 ]   1 − v || k || ˜ ω  ∂ f 0 ∂v ⊥ + v ⊥ k || ˜ ω ∂ f 0 ∂v ||  h y = sin[Ω c (t −t  )+φ 0 ]   1 − v || k || ˜ ω  ∂ f 0 ∂v ⊥ + v ⊥ k || ˜ ω ∂ f 0 ∂v ||  h z = v || k ⊥ ˜ ω cos [Ω c (t −t  )+φ 0 −θ] ∂ f 0 ∂v ⊥ +  1 − v ⊥ k ⊥ ˜ ω cos [Ω c (t −t  )+φ 0 −θ]  ∂ f 0 ∂v || ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ . For the time integral in Eq.(15), we use the following relationship,  t −∞ ∞ ∑ n=−∞ J n [λ] ⎡ ⎣ cos [Ω c (t −t  )+φ 0 ] sin[Ω c (t −t  )+φ 0 ] 1 ⎤ ⎦ exp [−inφ 0 ]exp[i( ˜ ω −v || k || −nΩ c )(t −t  )]dt  = ∞ ∑ n=−∞ J n [λ] 2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ i exp [−i(n −1)φ 0 ] ˜ ω −v || k || −(n −1)Ω c + i exp[−i(n + 1)φ 0 ] ˜ ω −v || k || −(n + 1)Ω c exp[−i(n − 1)φ 0 ] ˜ ω −v || k || −(n −1)Ω c − exp[−i(n + 1)φ 0 ] ˜ ω −v || k || −(n + 1)Ω c 2i exp[−inφ 0 ] ˜ ω −v || k || −nΩ c ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ∞ ∑ n=−∞ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ nΩ c v ⊥ k ⊥ J n [λ] i exp[−inφ 0 ] ˜ ω −v || k || −nΩ c −J  n [λ] exp[−inφ 0 ] ˜ ω −v || k || −nΩ c J n [λ] i exp[−inφ 0 ] ˜ ω −v || k || −nΩ c ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (18) Here, 315 Electromagnetic Waves in Plasma [...]... ω 2 − c2 k2 ⊥ (26) 3 Excitation of electromagnetic waves Eq.(26) tells us what kind of plasma waves grows and damps in arbitrary Maxwellian plasma This section gives examples on the excitation of plasma waves based on the linear dispersion analysis 12 322 Electromagnetic Waves Wave Propagation For simplicity, let us assume propagation of plasma waves in the direction parallel to the ambient magnetic... 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... 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... 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., 2 010( 1)),... solving Eq.(38) 18 328 Electromagnetic Waves Wave Propagation 3.3 Cyclotron resonance Since the Newton-Lorentz equation (6) and the Vlasov equation (7) cannot treat the relativism (such that c Vd ), 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... plasma -wave interactions, especially arising from discontinuities in both space and time Progress of techniques to control shape and parameters of plasmas enables us to make discontinuities in a clear and stable state 1.2 Emerging 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. .. discussion, this form is valid for sinusoidal waves Secondly, we take a look on the contributions of the pressure term in equation (2.2.1) The pressure term will be significant on the edge of a plasma, 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 in equation... i.e., electromagnetic light mode waves as k || → ∞ On the other hand, the low-frequency wave approaches to k || VA as k || → 0, and approaches to Ωc as k || → ∞ Note that VA ≡ cΩci /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... al., 2008; Sakai et al., 2 010( 1); Sakai et al., 2 010( 2)), includes novel physical aspects which have not been described in usual textbooks of plasma physics Also, complex dielectric constant or permittivity whose imaginary part is significantly large is observed and easily controlled in a plasma as a macroscopic value (Naito et al., 2008; Sakai et al., 2 010( 1)) This imaginary part strongly depends on... electromagnetic waves In Section 3.1, various methods to describe effects of periodic spatial discontinuities are demonstrated, including both analytical and numerical ones, and specific examples of band diagrams of 2D structures are shown In Section 3.2, another aspect of the spatial discontinuity associated with surface wave propagation is described, in which propagation of surface waves on the interface . f 0 ∂v . (10) 312 Wave Propagation Electromagnetic Waves in Plasma 3 Here, the electric field has only the perturbed component (  E 0 = 0) and the multiplication of small perturbation parts is. 0. 322 Wave Propagation Electromagnetic Waves in Plasma 13 (a) Linear scale. (b) Logarithmic scale. Fig. 1. Linear dispersion relation (frequency ω versus wavenumber k) for electromagnetic waves. + v ⊥ Ω c { cos[Ω c (t −t  )+φ 0 ] −cosφ 0 } z  = z −v || (t −t  ) ⎫ ⎬ ⎭ . 314 Wave Propagation Electromagnetic Waves in Plasma 5 Further taking the wavenumber vector k x = k ⊥ cosθ, k y = k ⊥ sinθ, k z = k || ,