Plane Electromagnetic Wave

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  Electromagnetic Shielding

  Theory and Applications

  • Salvatore Celozzi, Rodolfo Araneo, Paolo Burghignoli, Giampiero Lovat(Authors)
  • 2022(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  73 4 Shielding Effectiveness: Plane Waves Electromagnetic plane waves are the simplest solution of the time-harmonic Maxwell equations in a homogeneous and source-free spatial region. The study of their properties is useful for better understanding the behavior of more complex fields. For instance, the far field radiated by an arbitrary source has, locally and sufficiently far from the source, the characteristics of a plane wave (this usually allows an EM field impinging on a given structure to be approximated as a plane wave). In addition the exact field produced by any source can be expressed in terms of a continuous superposition of elemental plane-wave components (plane-wave spectrum representation). On the other hand, stratified media (in planar, cylindrical, and spherical configurations) are the simplest example of inhomogeneous media and are often considered as models of many shields. 4.1 Electromagnetic Plane Waves: Definitions and Properties A plane wave (with a time-harmonic behavior e j𝜔t , suppressed for the sake of con- ciseness) is a frequency-domain EM field mathematically expressed as E (x, y, z) = E 0 e −j(k x x+k y y+k z z) = E 0 e −jk⋅r , H (x, y, z) = H 0 e −j(k x x+k y y+k z z) = H 0 e −jk⋅r , (4.1) where E 0 and H 0 are constant complex vectors, and k x , k y , and k z are complex scalars that define the wavevector k = k x u x + k y u y + k z u z . The vectors E 0 and H 0 define the polarization of the plane wave, namely, the temporal evolution of the vector direction in the plane on which the vector lies. In particular, according to the locus described by the tip of the vector, polarization can be linear, circular, or, in the most general case, elliptic. For circular and elliptic polarizations, the sense of rotation for increasing time can be clockwise or counterclockwise with respect to Electromagnetic Shielding: Theory and Applications, Second Edition. Salvatore Celozzi, Rodolfo Araneo, Paolo Burghignoli, and Giampiero Lovat.

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  Principles of Optics

  Electromagnetic Theory of Propagation, Interference and Diffraction of Light

  • Max Born, Emil Wolf(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  1.4] B A S I C P R O P E R T I E S OF T H E E L E C T R O M A G N E T I C F I E L D 23 1.4 V E C T O R W A V E S 1.4.1 The general electromagnetic plane wave The simplest electromagnetic field is that of a plane wave; then each Cartesian component of the field vectors and consequently Ε and Η are, according to §1.3.1, functions of the variable u = r . s — vt only : Ε = E(r . s — vt), H=H(r.s-vt), (1) s denoting as before a unit vector in the direction of propagation. Denoting by a dot differentiation with respect to t, and by a prime differentiation with respect to the variable u, we have È = —vE' BE* 2Ε υ I (2) (curl E) x = --f z ? = E'fy -E' y s z = (s A E%.j Substituting these expressions into MAXWELL'S equations § 1.1 (1), (2) withy — 0, and using the material equations § 1.1 (10), (11) we obtain εν s Λ H' + -£ ' = 0,1 c s Λ E' — — H' = 0. (3) If we set the additive constants of integration equal to zero (i.e. neglect a field con-stant in space) and set, as before, vjc = 1 /Vεμ, (3) gives, on integration, (4) Scalar multiplication with s gives Ε . s = H. s = 0. (5) This relation expresses the transversality of the field, i.e. it shows that the electric and magnetic field vectors lie in planes normal to the direction of propagation. * See, for example, F. BORGNIS, Z.f. Phys., 117 (1941), 642; L. J. F. BROER, Appl. Sei. Res., A2 (1951), 329. without appreciable diffusion. In such circumstances, the group velocity, which may be considered as the velocity of the propagation of the group as a whole, will also represent the velocity at which the energy is propagated.* This, however, is not true in general. In particular, in regions of anomalous dispersion (cf. § 2.3.4) the group velocity may exceed the velocity of light or become negative, and in such cases it has no longer any appreciable physical significance.

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  Electromagnetic Field Theory Fundamentals

  • Bhag Singh Guru, Hüseyin R. Hiziroglu(Authors)
  • 2009(Publication Date)
  • Cambridge University Press

    (Publisher)

  The absence of the first-order term signifies that the electromagnetic fields do not decay as they propagate in a lossless medium. We now assume that the components of the field quantities E and H lie in a transverse plane , a plane perpendicular to the direction of propagation of the wave. We refer to such a wave as a plane wave . Let us consider that a plane wave propagates in the z direction. Then, the E and H fields have no components in the longitudinal direction (the direction of wave propagation). That is, E z = 0 and H z = 0. Such a wave is also called a transverse electromagnetic wave (TEM wave) . 354 8 Plane wave propagation In the family of plane waves, the uniform plane wave is one of the simplest to investigate and easiest to understand. The term uniform implies that, at any time, a field has the same magnitude and direction in a plane containing it. Thus, for a uniform plane wave propagating in the z direction, E and H are not functions of x and y . That is, ∂ E ∂ x = 0 ∂ E ∂ y = 0 ∂ H ∂ x = 0 ∂ H ∂ y = 0 For a uniform plane wave propagating in the z direction, the Helmholtz equations can be expressed in scalar form as ∂ 2 E x ∂ z 2 − µ ∂ 2 E x ∂ t 2 = 0 (8.12a) ∂ 2 E y ∂ z 2 − µ ∂ 2 E y ∂ t 2 = 0 (8.12b) ∂ 2 H x ∂ z 2 − µ ∂ 2 H x ∂ t 2 = 0 (8.13a) ∂ 2 H y ∂ z 2 − µ ∂ 2 H y ∂ t 2 = 0 (8.13b) where E x , E y , H x , and H y are the transverse components of the E and H fields. In addition, the field components are functions of z (the direction of propagation) and t (time) only. Each of these four equations is a second-order differential equation with two possible solutions. Because these equations are similar, their solutions must also be similar. In other words, as soon as we know the solution of one of these equations, we immediately know the solutions for the others. There are many possible functions that satisfy these wave equations. However, we are interested only in those functions that result in a trav-eling wave.

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  Waves in Layered Media

  • Leonid Brekhovskikh(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  CHAPTER I PLANE WAVES IN LAYERS T H E theory of wave reflection from interfaces and from layers will be developed in this chapter. Principal attention will be given to plane harmonic waves. The behavior of beams bounded in space and pulses bounded in time are not considered before § 8. I n all cases, the media in which the waves propagate are assumed to be homogeneous and to be bounded by parallel planes. For completeness of presentation, the first sections of the chapter are devoted to relatively simple questions such as plane waves in homo-geneous media, the reflection and refraction of waves a t an interface, etc. However, even in these sections, the reader will find some com-paratively new problems, such as the theory of inhomogeneous plane waves and their refraction and reflection, an analysis of Leontovich's approximate boundary conditions (which are satisfied in cases of so-called locally reacting surfaces), and others. Acoustic and electromagnetic waves will be considered simul-taneously. § 1. P L A N E W A V E S IN HOMOGENEOUS U N B O U N D E D M E D I A 1. Fundamental concepts and definitions The plane wave is the simplest form of wave motion. The most general analytic expression for a plane wave is the function and are the projections on the coordinate axes of the unit vector normal to the wave front, i.e. normal to planes of constant phase. The function (1.1) is a solution of the wave equation where n x , n y , and n z are three numbers which satisfy the condition n 2 x + nl + n* = 1, d 2 F d 2 F d 2 F _ 1 d 2 F (1.2) 1 2 P L A N E W A V E S I N L A Y E R S Φ(ω) = -f + iP(f)e-<*df (1.4) for the spectral density function, j The integrand in Eq. 1.3, corresponding to a definite value of ω / ( ω , χ, y, ζ, t) = Φ(ω) exp (ίωξ) » . x Γ. in x x + n v y + n z z Υ] e . = Φ(ω)θχρ wo!--^- *—-t , (1.5) represents a plane harmonic wave.

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  Engineering Electromagnetics

  Pergamon Unified Engineering Series

  • David T. Thomas, Thomas F. Irvine, James P. Hartnett, William F. Hughes(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  10 Electromagnetic Plane Waves ON THE NATURE OF WAVES Each of you has observed countless examples of wave motion in nature. In some cases the waves could be seen: a guitar string when plucked, the ripples on a pond when a stone is dropped, waves in the ocean. Other waves cannot be seen but nonetheless we know they are there: sound waves, light waves (the wave cannot be seen because of its small size), television and radio waves. Light, radio and television waves are all electromagnetic waves which we shall describe in this chapter. In discussing waves of any kind, we use three terms to describe the wave: phase, amplitude, and velocity. As these terms are widely misunderstood, we shall now discuss each of them. Consider, for example, ocean waves. If the waves are evenly distributed (rolling in at a uniform rate) and each wave is similar to the preceding one, the view to Surfer Sam waiting to catch the big wave might look like this: Fig. 10-1. The adventures of Surfer Sam. While waiting around to catch a wave, Sam notices that the wave crests are about six feet above the troughs. This measurement is, of course, subject to usual comments concerning accuracy, repeatability, etc. But if we accept Sam's observation, we conclude the wave amplitude is three feet. The amplitude 325 326 Electromagnetic Plane Waves of these waves is the height of the crest above the average ocean level (the depth of the trough below the average ocean level). This form of measurement is necessary since wave motion is often superimposed on other phenomena or conditions, so that absolute measurements (as for example the depth of the ocean here) would be useless in describing the wave. So in any wave, the ampli-tude is the maximum deviation from some average, in units or dimensions of the measurement (feet here). As luck would have it, a big wave happened along that very instant and Surfer Sam caught it.

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  Analysis and Modeling of Radio Wave Propagation

  • Christopher John Coleman(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  The corre- sponding magnetic field is then given by H = ˆ p × E η (2.26) where η = √ μ/ is the impedance of the propagation medium. The Poynting vector for the above field will take the form ˆ p η E · E (2.27) from which it can be seen that the plane wave will transport energy in the propagation direction (Figure 2.1). x y z E H Speed c EM pulse Figure 2.1 A plane wave consisting of an electromagnetic pulse. 2.2 Plane Electromagnetic Waves 19 An important specialization of the electromagnetic field is the case where it oscillates in time on a single frequency (a time harmonic field), i.e. H(r, t) =  {H(r) exp( jωt)} (2.28) and E (r, t) =  {E(r) exp( jωt)} (2.29) with nonitalic capital letters representing the fields with the exp( jωt) term factored out. (In theory, through Fourier transform techniques, a general electromagnetic field can be analyzed in terms of time harmonic fields.) The time harmonic fields H(r) and E(r) will satisfy the equations ∇ × E = −jωμH (2.30) and ∇ × H = jω E + J (2.31) where J =   Je jωt  . (It should be noted that the divergence Maxwell equations are automatically satisfied providing the time harmonic continuity equation jωρ +∇ · J = 0 is satisfied.) For a time harmonic field, we measure the energy flow in terms of the average flow over a wave period, giving the time harmonic Poynting vector P = 1 2   E × H ∗  (2.32) A time harmonic plane wave has the form E =   E 0 exp  jω  t − ˆ p · r c  (2.33) and H =   H 0 exp  jω  t − ˆ p · r c  (2.34) where ˆ p is the unit vector in the direction of propagation. Fields E 0 and H 0 will be related through H 0 = ˆ p × E 0 η (2.35) and E 0 will satisfy ˆ p · E 0 = 0. From these relations, it is obvious that H 0 , E 0 and ˆ p form a mutually orthogonal triad. For a plane wave, the time harmonic Poynting vector simplifies to P = ˆ p 2η E · E ∗ (2.36) Without loss of generality, we now consider propagation in the z direction ( ˆ p = ˆ z), then E = E 0 exp(−jβ z) (2.37)

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  Theory of Reflectance and Emittance Spectroscopy

  • Bruce Hapke(Author)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  (2.27) Two independent orthogonal solutions to the wave equation are possible in which the electric vectors are perpendicular to each other. If the positive x direction is chosen to be parallel to E e , then the component of B m corresponding to this solution points in the positive y direction. Figure 2.2 illustrates this solution. If the positive y direction is chosen to be parallel to the electric vector, then the accompanying magnetic vector points in the negative x direction. In free space, neither E e nor B m has a component parallel to z. x E e B m Plane of constant phase y v z Figure 2.2 Relation between the fields and the propagation velocity vector in a Plane Electromagnetic Wave. 10 Electromagnetic wave propagation 2.2.2 Huygens’s principle Three hundred years ago when it was realized that light had a wave nature, the question arose as to what medium the waves were propagating through. It was pos- tulated that all space was filled with an invisible fluid called the “aether.” However, in the modern view of an electromagnetic wave no aether is required. According to Maxwell’s equations, a changing electric field generates a magnetic field, and a changing magnetic field generates an electric field. Thus, a propagating electro- magnetic wave may be regarded as generating itself; that is, the changing fields on the wave front continually regenerate the wave. This concept leads to Huygens’s principle, in which each point on a wave front may be considered to be a source of spherical wavelets that travel radially outward and combine coherently with wavelets from all the other points to produce a new wave front. If the wave front is plane and infinite in lateral extent, this process simply produces another plane wave front. However, if part of the wave front is obstructed, Huygens’s principle predicts that fields still exist behind the obstructing object.

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  Problems and Solutions on Electromagnetism

  • Yung-Kuo Lim(Author)
  • 1993(Publication Date)
  • WSPC

    (Publisher)

  Eo for the two components are respectively Electromagnetic Waver 465 4022 A Plane Electromagnetic Wave of angular frequency w is incident nor- mally on a slab of non-absorbing material. The surface lies in the t y plane. The material is anisotropic with E,, = n2.0 Eyy = +o , etY = cY2 = E ~ , = 0 , n, # ny . (a) If the incident plane wave is linearly polarized with its electric field at 45' to the t and y axes, what will be the state of polarization of the reflected wave for an infinitely thick slab? (b) For a slab of thickness d , derive an equation for the relative ampli- tude and phase of the transmitted electric field vectors for polarization in the t and directions. (UC, Berkeley) Solution: Consider a Plane Electromagnetic Wave incident from an anisotropic medium 1 into another anisotropic medium 2, and choose coordinate axes so that the incidence takes place in the zz plane, the interface being the t o y plane, as shown in Fig. 4.8. The incident, reflected, and transmitted waves are represented as follows: incident wave: ei(K.r-w') , reflected wave: ei(Kr-w'f) , transmitted wave: ei(K'r-wt) . The boundary condition on the interface that the tangential components of E and H are continuous requires that I<, = I<: = I<; , I<, = I<; = I<: w = w' = W I I . From these follow the laws of reflection and refraction: I< (e) sin e = I<'( el) sin el , q e ) sin e = zP(el1) sin e/' . 466 Problems d Solutions on Elecltomognetirm Fig. 4.8 (a) As the medium 1 given is air or vacuum, we have K = K' = = C ' For normal incidence e = el = elt = 0, so that K = Kn , K' = -Kn, K = Kn. (n = e,) From Maxwell's equation V x H = D, we have (Problem 4004) K x H = -wD . (1) As K is parallel to e,, D and H are in the t y plane. Take the Bxes along the principal axes of the dielectric, then and El' = E:e, + E!ey, Dy = E; = 0 . If the incident wave is linearly polarized with its electric field at 45' to the z and y axes, we have E = E,e, + E y e , , with E,+ Ei = E 2 , E, = Ey = 5.

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  Theory and Computation of Electromagnetic Fields

  • Jian-Ming Jin(Author)
  • 2015(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  Before we proceed further, let us examine the physical meaning of the solution in Equations (4.2.11) and (4.2.12). Since 𝜸 is a complex number, it can be written as 𝜸 = 𝜶 + j 𝜷 , where both 𝜶 and 𝜷 are real vectors. With this, the instantaneous electric field corresponding to Equation (4.2.11) can be written as E ( 𝐫 , t ) = Re [ 𝐄 0 e j 𝜔 t ±( 𝜶 +j 𝜷 ) ⋅ 𝐫 ] = 𝐄 0 e ± 𝜶 ⋅ 𝐫 cos ( 𝜔 t ± 𝜷 ⋅ 𝐫 ) (4.2.13) where for simplicity 𝐄 0 is assumed to be a real vector here. Since cos ( 𝜔 t ± 𝜷 ⋅ 𝐫 ) represents a wave propagating along the ∓ ̂ 𝛽 direction whose phase is constant in the plane defined by 𝜷 ⋅ 𝐫 = constant, e ± 𝜸 ⋅ 𝐫 represents a plane wave. The amplitude of the wave is also uniform in the plane defined by 𝜶 ⋅ 𝐫 = constant. In the case that 𝜶 and 𝜷 have the same direction, the equiphase and equiamplitude planes are parallel to each other, and the wave is called a 146 TRANSMISSION LINES AND PLANE WAVES uniform plane wave . If 𝜶 and 𝜷 have different directions, the wave is called a nonuniform plane wave [4]. Since Equations (4.2.11) and (4.2.12) are valid for any 𝛾 x and 𝛾 y , the general solution to Equation (4.2.2) is the linear superposition of all possible solutions, which can be expressed as 𝐄 ( 𝐫 ) = ∫ ∞ −∞ ∫ ∞ −∞ 𝐄 0 ( 𝛾 x , 𝛾 y ) e ± 𝜸 ⋅ 𝐫 d 𝛾 x d 𝛾 y (4.2.14) 𝐇 ( 𝐫 ) = ∫ ∞ −∞ ∫ ∞ −∞ 𝐇 0 ( 𝛾 x , 𝛾 y ) e ± 𝜸 ⋅ 𝐫 d 𝛾 x d 𝛾 y . (4.2.15) Note that 𝛾 z is constrained by 𝛾 2 x + 𝛾 2 y + 𝛾 2 z = 𝛾 2 . Equations (4.2.14) and (4.2.15) indicate that any field at a source-free point can be expanded as a linear superposition of an infinite number of plane waves. 4.2.2 Characteristics of a Plane Wave Let us consider the plane wave represented by Equations (4.2.11) and (4.2.12) further and study its fundamental characteristics.

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  Microwave Engineering

  • David M. Pozar(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  This result shows that the magnetic field vector ¯ H lies in a plane normal to ¯ k , the direction of propagation, and that ¯ H is perpendicular to ¯ E . See Figure 1.8 for an illustration of these vector relations. The quantity η 0 = √ µ 0 / 0 = 377  in (1.76) is the intrinsic impedance of free-space. The time domain expression for the electric field can be found as ¯ E (x , y , z , t ) = Re  ¯ E (x , y , z )e j ωt  = Re  ¯ E 0 e − j ¯ k · ¯ r e j ωt  = ¯ E 0 cos( ¯ k · ¯ r − ωt ), (1.77) 1.5 General Plane Wave Solutions 23 E n z y x H ˆ FIGURE 1.8 Orientation of the ¯ E , ¯ H , and ¯ k = k 0 ˆ n vectors for a general plane wave. assuming that the amplitude constants A, B , and C contained in ¯ E 0 are real. If these constants are not real, their phases should be included inside the cosine term of (1.77). It is easy to show that the wavelength and phase velocity for this solution are the same as obtained in Section 1.4. EXAMPLE 1.3 CURRENT SHEETS AS SOURCES OF PLANE WAVES An infinite sheet of surface current can be considered as a source for plane waves. If an electric surface current density ¯ J s = J 0 ˆ x exists on the z = 0 plane in free- space, find the resulting fields by assuming plane waves on either side of the current sheet and enforcing boundary conditions. Solution Since the source does not vary with x or y , the fields will not vary with x or y but will propagate away from the source in the ±z direction. The boundary conditions to be satisfied at z = 0 are ˆ n × ( ¯ E 2 − ¯ E 1 ) = ˆ z × ( ¯ E 2 − ¯ E 1 ) = 0, ˆ n × ( ¯ H 2 − ¯ H 1 ) = ˆ z × ( ¯ H 2 − ¯ H 1 ) = J 0 ˆ x , where ¯ E 1 , ¯ H 1 are the fields for z < 0, and ¯ E 2 , ¯ H 2 are the fields for z > 0. To satisfy the second condition, ¯ H must have a ˆ y component. Then for ¯ E to be or- thogonal to ¯ H and ˆ z , ¯ E must have an ˆ x component.

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