PLANE WAVE PROPAGATION AND REFLECTION. David R. Jackson Department of Electrical and Computer Engineering University of Houston Houston, TX

Transcription

1 PLANE WAVE PROPAGATION AND REFLECTION David R. Jackson Department of Electrical and Computer Engineering University of Houston Houston, TX Abstract The basic properties of plane waves propagating in a homogeneous isotropic region are reviewed, for both lossy and lossless media. These basic properties include the nature of the electric and magnetic fields, the properties of the wavenumber vector, and the power flow of the plane wave. Special attention is given to the specific case of a homogenous (uniform) plane wave, i.e., one having real direction angles, since this case is most often met in practice (for example, in the far-field of an antenna). Properties such as wavelength, phase and group velocity, penetration depth, and polarization are discussed for these plane waves. The theory of reflection of plane waves from layered media is then discussed in a unified manner, using a transmission line model called the transverse equivalent network. This method decomposes the incident plane wave into TM and TE parts, and models each separately using a transmission line equivalent circuit. Basic reflection phenomena such as the law of reflection, Snell s law and the Brewster angle effect come directly from this model. More involved reflection problems are also easily treated using this approach. As an example, the reflected and transmitted fields due to a circular-polarized wave incident on the surface of the ocean are calculated.

2 . INTRODUCTION Planes waves are the simplest solution of Maxwell s equations in a homogeneous region of space, such as free-space (vacuum). In spite of their simplicity, plane waves have played an important role throughout the development of electromagnetics, starting from the time of the earliest radio transmissions through the development of modern communications systems. Plane waves are important for several reasons. First, the far-field radiation from any transmitting antenna has the characteristics of a plane wave sufficiently far from the antenna. The incoming wavefield impinging on a receiving antenna can therefore usually be approximated as a plane wave. Second, the exact field radiated by any source in a region of space can be constructed in terms of a continuous spectrum of plane waves via the Fourier transform. Understanding the nature of plane waves is thus important for understanding both the far-field and the exact radiation from sources. The theory of plane wave reflection from layered media is also a well-developed area, and relatively simple expressions suffice for understanding reflection and transmission effects when layers are present. Problems involving reflections from the earth or sea, for example, are easily treated using plane-wave theory. Even when the incident wavefront is actually spherical is shape, as from a transmitting antenna, plane-wave theory may often be approximately used with accurate results. Throughout this article, it will be assumed that the regions of interest are homogeneous (the material properties are constant) and isotropic, which covers most cases of practical interest.

3 . BASIC PROPERTIES OF A PLANE WAVE Definition of a plane wave The most general definition of a plane wave is an electromagnetic field having the form E = E 0 ψ xyz,, b g () H = H 0 ψ xyz,, b g () where E 0 and H 0 are constant vectors, and the wavefunction ψ is defined as where j kxx kyy kzz ψbxyz,, g d + + i = e j = e kr (3) k = x! kx + y! ky + z! kz = β jα (4) r = x! x+ y! y+ z! z, (5) with k x, k y, and k z being complex constants that define a wavenumber vector k. (A time-harmonic dependence of e jωt is assumed and suppressed.) The vector E 0 defines the polarization of the plane wave. The real and imaginary parts of the wavenumber vector k define the phase vector β and an attenuation vector α. The phase vector has units of radians/meter and gives the direction of most rapid phase change, while the attenuation vector has units of nepers/meter and gives the direction of most rapid attenuation. The magnitude of the phase vector gives the phase change per unit length along the direction of the phase vector, while the magnitude of the attenuation vector determines the rate of attenuation along the direction of the attenuation vector. 3

4 Basic properties In a homogeneous lossless space a plane wave must satisfy Maxwell s equations, which in the time-harmonic form are [] H = jωε E (6) E = jωµ H (7) E = 0 (8) H = 0 (9) where ε and µ are the permittivity and permeability of the space. For free space, ε = ε 0 and µ =µ 0, where µ 0 is defined to be 4π 0-7 Henry/meter and ε 0 is determined from the defined velocity of light (or any plane wave) in vacuum [], c = meters/second, since c = / ε 0µ 0. This gives the approximate value ε 0 = Farads/meter. A lossy medium can be modeled using a complex effective permittivity, accounting for conduction loss and/or polarization loss []. The complex effective permittivity is expressed as F σ ε =! ε H G I j ω K J. (0) In this equation!ε is the complex permittivity of the material, accounting for polarization loss (if any), and σ is the conductivity of the medium. Taking the curl of Eq. (7) and then substituting in Eq. (6) gives the vector wave equation b Eg k E = 0, () where k is the (possibly complex) wavenumber defined by k = k' jk'' = ω µε, () 4

5 with the square root chosen so that k lies in the fourth quadrant on the complex plane. Using the definition of the vector Laplacian, E b Eg b Eg, (3) and the fact that the divergence of the electric field is zero for a time-harmonic field in a homogeneous region [], results in the vector Helmholtz equation E+ k E = 0. (4) In rectangular coordinates, the vector Laplacian is expressed as E = x! Ex + y! Ey + z! Ez. (5) Hence, all three rectangular components of the electric field satisfy the scalar Helmholtz equation Ψ+ k Ψ = 0 (6) in a homogeneous region. or Substituting Eq. (3) into Eq. (6) gives the result x y z k + k + k = k k k = k. (7) This is the separation equation that relates the components of the wavenumber vector k defined in Eq. (4). Note that the term on the left side of Eq. (7) is not in general equal to k, since k may be complex. 5

6 Other fundamental relations for a plane wave may be found by substituting Eq. () and () into Maxwell s equations (6)-(9). Noting that jk for a plane wave, Maxwell s equations reduce to k H = ωε E (8) k E =ωµ H (9) k E = 0 (0) k H = 0. () Equations (8) and (9) each imply that E H =0, and together they also imply that k E =0 and k H = 0. That is, all three vectors k, E, and H are mutually orthogonal. Another interesting property that is true for any plane wave, which may be derived directly from Eq. (8) or (9), is that E E = η H H, () where η is the intrinsic impedance of the space (possibly complex), defined from µ η = (3) ε (the principle branch of the square root is chosen so that the real part of η is nonnegative). For vacuum, the intrinsic impedance is often denoted as η 0, and has a value of approximately Ω. Power flow The complex Poynting vector for a plane wave, giving the complex power flow, is (assuming peak notation for phasors) 6

7 * S = E H. (4) Using Eq. (9), the Poynting vector for a plane wave can be written as S = E0 k E0 ωµ ψ * * * e j. (5) Using the triple product rule A bb Cg= ba CgB ba Bg C, this can be rewritten as S = E k E k E ωµ ψ * ωµ ψ * * * * e j. (6) The second term in the above equation is not always zero for an arbitrary plane wave, even though E0 k = 0, since k may be complex in the most general case. If either k or E 0 is proportional to a real vector (i.e., all of the components of the vector have the same phase angle), then it is easily demonstrated that the second term vanishes. In this case the Poynting vector becomes S = E0 k ωµ ψ * *. (7) If the medium is also lossless (µ is real), the time-average power flow (coming from the real part of the complex Poynting vector) is in the direction of the phase vector β. A similar derivation, casting the Poynting vector in terms of the H 0 vector, yields S = H0 k ωε ψ, (8) provided either k or H 0 is proportional to a real vector. If the region is lossless (ε is real), the time-average power flow is then in the direction of the phase vector. Hence, for a lossless region, the time-average power flow is in the direction of the phase vector β if any of the three vectors 7

8 k, E 0, or H 0 is proportional to a real vector. In many practical cases of interest, one of the three vectors will be proportional to a real vector, and hence the conclusion will be valid. However, it is always possible to find exceptions, even for free space. One such example is the plane wave defined by the vectors k = b, j, g, E 0 = b,, jg, and b gb g H 0 = /( ωµ ) j, 4+ j, j, at a frequencyω = c (k 0 = ). This plane wave satisfies Maxwell s equations (8)-(). However for this plane wave the vectors β, α, and b p=rebg S are all in different directions, since the power flow is in the direction of the vector 504,, g One must then be careful to define what is meant by the direction of propagation for such a plane wave. Direction angles One way to characterize a plane wave is through direction angles θφ, b g in spherical coordinates. The direction angles (which are in general complex) are defined from the relations k x = ksin θ cos φ (9) ky = ksinθsinφ (30) kz k. (3) Homogeneous (uniform) plane wave One important class of plane waves is the class of homogeneous or uniform plane waves. A homogeneous plane wave is one for which the direction angles are, by definition, real. A 8

9 homogeneous plane wave enjoys certain special properties that are not true in general for all plane waves. For such a plane wave the wavenumber vector can be written as k = k R! = R! bk' jk'' g, (3) where R! is a real unit vector defined from R! = x! sinθcos φ+ y! sinθsin φ+ z! cosθ. (33) In this case the k vector is proportional to the real vector R!, so the result of Eq. (7) applies. The unit vector R! then gives the direction of time-average power flow, and also points in the direction of the phase and attenuation vectors. That is, all three vectors point in the same direction for a homogeneous plane wave. This direction is, unambiguously, the direction of propagation of the plane wave. The planes of constant phase are also then the planes of constant amplitude, being the planes perpendicular to the! R vector. That is, the plane wave has a uniform amplitude across the plane perpendicular to the direction of propagation. If the plane wave is not homogeneous, corresponding to complex direction angles, then the physical interpretation of the direction angles is not clear. From Eq. (8) or (9) it may be easily proven that the fields of a homogeneous plane wave obey the relation E = η H. (34) (Recall that all plane waves obey Eq. (), which is not in general the same as the above equation.) 9

10 Any homogeneous plane wave can be decomposed into a sum of two plane waves, with electric field vectors polarized perpendicular to each other, and with electric field vectors that are real, to within a multiplicative complex constant. This follows from a simple rotation of coordinates (the direction of propagation is then z', with electric field vectors in the x' and y' directions. the two plane wave then have the fields ( Ex, H y ) and ( y, x) E H, with E / H = E / H = η. This decomposition is used later in the discussion of polarization. x y y x Lossless media Another important special case is when the medium is lossless, so that k ''= 0. If Eq. (4) is substituted into the separation equation (7), the imaginary part of this equation immediately yields the relation β α = 0. (35) Hence, for a lossless region, the phase and attenuation vectors are always perpendicular. In some applications (e.g., a Fourier transform solution of radiation from an aperture in a ground plane at z = 0, or from a planar current source at z = 0 [3]), a plane wave propagating in a lossless region has the characteristic that two of the wavenumbers, e.g., k x and k y, are real (corresponding to x y the transform variables). In this case the third wavenumber k z will be real if k + k < k and will be imaginary if k + k > k. That is, all transverse wavenumbers k, k x y d x y ithat lie within a circle of radius k in the wavenumber plane will be propagating, while all wavenumber outside the circle will be evanescent. In the first case the power flow is in the direction of the (real) k vector dkx, ky, kzi, so that power leaves the aperture from this plane wave. In the second case 0

11 the power flow is in the direction of the transverse wavenumber vector k d x, k,0i, so that no power leaves the aperture for this plane wave component. If the medium is complex, there is no sharp distinction between propagating and evanescent plane waves. In this case all plane waves d x y zi. carry power, in the direction of the vector k, k,imk y Finally, it can be noted that if a homogeneous plane wave is propagating in a lossless region, then the attenuation vector α must be zero, and all wavenumber components k x, k y, and k z are real. Summary of Basic Properties A summary of the basic properties of a plane wave, discussed in the previous sections, is given in Table.

12 Table. Summary of the basic properties of a plane wave. INHOMOGENEOUS HOMOGENEOUS PROPERTY Lossy Lossless Lossy Lossless Direction angles complex complex real real Relation between phase and attenuation vectors β α = kk ' '' β α = 0 β α α = 0 E E = H H Impedance relations E E = η H H E E = η H H E = η H η E E = H H E η = η H Direction of power flow vector Wavenumber vector Orthogonality of vectors not necessarily in β direction β if k, E 0 or H 0 is proportional to a real vector k k = k k k = k k k = k * k k = k E H = 0 E k = 0 H k = 0 E H = 0 E k = 0 H k = 0 β E H = 0 E k = 0 H k = 0 β k k = k * k k = k E H = 0 E k = 0 H k = 0 3. PROPAGATION OF A HOMOGENEOUS PLANE WAVE The case of a homogeneous plane wave is important enough to warrant further attention. Far away from a transmitting antenna, the radiation field always has behaves as a homogeneous plane wave (if the spherical wavefront is approximated as planar). As mentioned, a homogeneous plane wave is one propagating with real direction angles in space, with a direction of propagation described by a real unit vector. By a suitable rotation of coordinate axis, the direction of propagation may be assumed to be along the z axis. Furthermore, the plane way can always be represented as a sum of two separate plane waves, one with the electric field polarized

13 in the x direction, and the other polarized in the y direction. Considering the wave polarized in the x direction, the fields are E x = E e jk z 0 (36) H y = E e 0 η jk z. (37) The wavenumber k and the intrinsic impedance η of the space are given by Eqs. () and (3), respectively. Wavelength, phase velocity, and group velocity The wavelength λ is defined at the distance required for the plane wave to change phase by π radians, and is given by π λ = k '. (38) The phase velocity of the plane wave is defined as the velocity at which a point of constant phase travels (such as the crest of the wave, where the electric field is the maximum). The phase velocity is given by ω v p = = ' k ω Reeω µεj. (39) If the permittivity ε is not a function of frequency (true for vacuum, and often approximately true for nonconducting media, over a certain frequency range), then the phase velocity is constant with frequency (the plane wave propagates without dispersion). For vacuum the phase velocity is 3

14 c = meters/second. The group velocity (velocity of energy flow, and often a good approximation to the velocity of signal propagation [4], is given by dω vg = = ' dk d Re ω µε dω e j. (40) For a nondispersive lossless media (constant real-valued permittivity) the group velocity is equal to the phase velocity, and both are equal to v p = vg = µε. (4) Depth of penetration (skin depth) For a lossy medium, the plane wave decays as it propagates, since k '' > 0. The depth of penetration d p is defined as the distance required to reduce the field level by a factor of e. 78 8, so that the field is % of the starting value (the power is reduced by a factor of e, or is at a level of % of the starting value). The depth of penetration is given as d p = = '' k Im ω µε e j. (4) The permittivity is complex for a lossy media, and is represented by Eq. (0). Special cases are of particular interest. For a low loss medium, ε '' ' << ε, a simple binomial approximation of the square root in Eq. (4) gives d p = k '' ' 05. k tanδ (43) where 4

15 k ' ' ω µε (44) and the loss tangent is defined as '' ε tanδ = '. (45) ε If the loss is due purely to medium conductivity (no polarization loss, so that! number), the penetration depth formula (43) for the low-loss case becomes ε ' = ε is a real d p σ ' ε. (46) µ ' For a highly lossy medium, ε << ε, in which case we have '' k ' '' '' k ω µε. (47) For a good conductor the conductivity is very large, so that ε '' σ / ω from Eq. (0). In this case the penetration depth is more commonly referred to as the skin depth δ, since the value may be very small. (There is no relation to the δ symbol appearing in the loss tangent symbol.) For a good conductor, δ = d p. (48) ωµσ For pure copper (σ = S/m, µ µ 0 ) at.45 GHz, δ = 335µ. m. Whether a particular material falls into the low loss or highly lossy categories may depend on frequency. For example, consider typical seawater, which has roughly!ε r = 78 5

16 (ignoring polarization losses) and σ = 40. S/m. Table shows the loss tangent versus frequency. It is seen that the loss tangent is on the order of unity at microwave frequencies. For much lower frequencies the seawater is highly lossy, and for much higher frequencies it is a low loss media. Note that the depth of penetration continues to decrease as the frequency is raised, even though the medium becomes more of a low loss material at higher frequencies, according to the definition used (a low loss tangent). At low frequencies the penetration depth is increasing inversely as the square root of frequency, according to Eq. (48). At high frequencies the penetration depth approaches a constant, according to Eq. (46). Table. Loss tangent and penetration depth for typical seawater, versus frequency. frequency (GHz) loss tangent '' ' ( ε / ε ) penetration depth d p (meters) Summary of Homogeneous Plane Wave Properties A summary of the properties relating to wavelength, velocity, and depth of penetration for a homogeneous plane wave is given in Table 3. 6

17 Table 3. Propagation Properties of a homogeneous plane wave. General Highly Conducting Low Loss Lossless wavelength π π k ' π ωµσ ω µε ' π ω µε phase velocity ω k ' ω µσ µε ' µε group velocity dω ' dk ω µσ µε ' µε depth of penetration / '' k ωµσ σ ' ε µ Polarization The above discussion has assumed a homogeneous plane wave propagating in the z direction, polarized with the electric field in the x direction. The most general polarization of a wave propagating in the z direction is one having both x and y components of the field, d x0 y0i. (49) j( kz) E = x! E + y! E e The field components are represented in polar form as φ E = E e, (50) x0 x0 j x jφ y E = E e. (5) y0 y0 The phase difference between the two components is defined as 7

18 φ = φ y φx. (5) Without any real loss of generality, φ x may be chosen as zero, so that φ = φ y. In the time domain, the field components are then E x = E 0 cos ω t x b g (53) E y = E y 0 cosbωt + φg. (54) Using trigonometric identities, Eq. (54) may be expanded into a sum of sinbωtg and cosbωtg functions, and then Eq. (53) may be used to put both cosbωtg and sinbωtg in terms of E x (using sin = cos ). After simplification, the result is AE + BE E + CE = D (55) x x y y where A = E E y0 x0 (56) B = E E y0 x0 cosφ (57) C = (58) D = E y 0 sin φ. (59) After simplification, the discriminant of this quadratic curve is E y0 = B 4AC = 4 sin φ E, (60) x0 8

19 which is always negative. This curve thus always represents an ellipse. The general form of the ellipse is shown in Fig.. The tilt angle of the ellipse is τ, and the axial ratio AR is defined as the ratio of the major axis of the ellipse to the minor axis ( AR ). In the time domain the electric field vector rotates with the tip of the vector lying on the ellipse. Right-handed elliptical polarization, or RHEP, corresponds to counterclockwise rotation (the thumb of the right hand aligns with the direction of propagation, the fingers of the right hand align with the direction of rotation in time). (This is the IEEE definition, opposite to the usual optics convention.) LHEP corresponds to rotation in the opposite direction. y B A τ x A B AR = A B A B Figure. Geometry of the polarization ellipse. 9

20 A convenient way to represent the polarization state is with the Poincaré sphere [5]. Using spherical trigonometric relations, the following results may be derived for the tilt angle of the ellipse and the axial ratio. tan τ = tan γ cosφ (6) sinξ = sinγ sinφ, (6) where the parameter ξ is related to the axial ratio as ( AR) ξ = cot, 45 ξ for LHEP for RHEP. (63) The phase angle φ is defined in Eq. (5). The parameter γ characterizes the ratio of the fields along the x and y axes, and is defined from E y0 o γ = tan, 0 γ 90. (64) E x0 0 0 There is no ambiguity in Eq. (6) for ξ, since 90 ξ However, Eq. (6) gives an ambiguity for τ, since adding multiples of 80 o does not change the tangent. To resolve this ambiguity, Table 4 may be used to determine the appropriate quadrant (,, 3 or 4) that the angle τ is in, based on the Poincaré sphere [5]. Table 4. Quadrant that the angle τ is in. cosφ > 0 cosφ < 0 cosγ > 0 4 cosγ < 0 3 0

21 The following special cases are important. ) Linear polarization: φ = 0 or 80 0, or either E x0 or E y0 is zero. In this case AR = (the ellipse degenerates to a straight line). ) Circular polarization: E = E and φ =±90 0 (to within any multiple of 80 o ). y0 x0 In this case the ellipse becomes a circle (either RHCP or LHCP), and AR =. It can also be noted that a wave of arbitrary polarization can be represented as a sum of RHCP and a LHCP waves, by noting that the arbitrarily polarized wave in Eq. (49) can be written as L NM d O L O QP + x0 y0i x0 y0 NM d i (65) QP E = r! + l! E je E je where the unit-amplitude RHCP and LHCP waves are b g e jkz (66) r! = x! jy! b g e jkz. (67)! l = x! + jy! 5. PLANE WAVE REFLECTION AND TRANSMISSION A general plane wave reflection/transmission problem consists of an incident plane wave impinging on a multilayer structure, as shown in Fig.. An important special case is the tworegion problem shown in Fig. 3. The incident plane wave may be either homogeneous or inhomogeneous, and any of the regions may have an arbitrary amount of loss. The vector

22 representing the direction of propagation for the incident wave in Figs. and 3 is a real vector (as shown) if the incident wave is homogeneous, but the analysis is valid for the general case. inc ref θ i inc ref n Z 0 (n) θ t trans N trans Figure. A plane wave reflecting from a multilayer stack of different materials. Each layer has a uniform (constant) set of material parameters. On the right side the equivalent transmission-line model (transverse equivalent network) is shown. inc ref θ i inc Z 0 () ref θ t trans trans Z 0 () Figure 3. Reflection and transmission from a single interface between two different media. The transmission-line model (transverse equivalent network) for the two-region reflection problem is shown on the right.

23 The key to obtaining a simple solution to such reflection and transmission problems is the use of a transmission line model of the layered structured, which is based on TE-TM decomposition of the plane waves, discussed next. TE-TM Decomposition According to a basic electromagnetic theorem [], the fields in a source-free homogeneous region can be represented as the sum of two types of fields, a field that is transverse magnetic to z (TM z ) and a field that is transverse electric to z (TE z ). The TM z field is defined as one that has H z = 0, while the TE z field by definition has E z = 0. A general plane wave field may therefore be written as the sum of a TM z plane wave and a TE z plane wave. In free space, the direction z is rather arbitrary. When a layered media is present, the preferred direction z is perpendicular to the layers, because this allows for the TM z and TE z plane waves in each region to be modeled as waves on a transmission line. Standard transmission-line theory may then be conveniently used to solve plane-wave reflection and transmission problems in a relatively simple manner, without having to solve the electromagnetic boundary-value problem of matching fields at the interfaces. The TM z plane wave is commonly referred to as one that is polarized with the electric field in the place of incidence, while the TE z place wave is polarized with the electric field perpendicular to the plane of incidence. (The plane of incidence is the y-z plane.) For a homogeneous plane wave with an incident wave vector in the y-z plane, the polarizations are shown in Fig. 4. 3

24 H x TM z E TE z E H y z Figure 4. Polarizations of incident TM z and TE z plane waves. The field components may be found by using Maxwell s equations to write the transverse (x and y) field components in terms of the longitudinal components E z and H z. The nonzero longitudinal component is assumed to be proportional to the wavefunction ( jkx x ky y kz z ) ψ = exp ( + ± ). (68) The plus sign in this equation is chosen for plane waves propagating or decaying in the positive z direction, while the negative sign is for propagation or decay in the minus z direction. This representation is convenient for plane-wave reflection problems, since the characteristic impedance of a transmission line that models a particular region will then have a positive characteristic impedance for both upward and downward propagating plane waves, in agreement with the usual transmission line convention. For the TM z plane wave, the normalized field components may then be written as 4

25 b g b g b g b g c h b g E = k k Ψ x, y, z H = ωε k Ψ x, y, z x x z x y E = k k Ψ x, y, z H = ωε k Ψ x, y, z y y z y x E = k k Ψ x, y, z H = 0. z z z (69) For a TE z plane wave the corresponding results are b g b g b g b g c h b g E = ωµ kψ xyz,, H = kkψ xyz,, x y x x z E = ωµ kψ xyz,, H = kkψ xyz,, y x y y z E = 0 H = k k Ψ x, y, z. z z z (70) Note that a plane wave propagating in the z direction (k z = k) has both longitudinal field components that are zero. Such a plane wave is TEM (transverse electric and magnetic) to the z direction) It may be seen from the above equations that the transverse fields obey the relations where TM TM E = Z 0 z! H t TE TE E = Z 0 z! H t c TM t h (7) c TE t h (7) TM kz Z0 = ωε (73) TE Z0 = ωµ. (74) k z 5

26 Transverse Equivalent Network From the above relations (7) and (7), the transverse (perpendicular to z) field components behave as voltages and currents on a transmission line model called the transverse equivalent network. In particular, the correspondence is given through the relations (i = TM or TE) E H i t i t = e! xyv, t i z ψ b g bg (75) = h! xyi, t i z ψ b g bg (76) where the unit vectors are chosen so that eˆ hˆ =± z ˆ (77) for a plane wave propagating in the ±z ˆ direction. The transverse wavefunction jkx x ky y ψ t bxy, g d i = e + (78) is a common transverse phase term that must be the same for all regions (if the transverse wavenumbers k x or k y were different between two regions, a matching of transverse fields at the boundary would not be possible). The fact that the transverse wavenumbers are the same in all regions leads to the law of reflection, which states that the direction angle θ for a reflected plane wave must equal that for the incident plane wave. It also leads to Snell s law, which states that the direction angles θ inside each of the regions are related to each other, through ni sinθ i = nsin θ, i =,, N, (79) where n i is the index of refraction (possibly complex) of region i, defined as n = k / k = ε µ. It is often convenient to express the characteristic impedance for region i i i 0 ri ri from Eqs. (73) and (74) in terms of the medium intrinsic impedance η i as 6

27 Z0 TM = ηi cos θi (80) Z TE 0 = ηi sec θi. (8) Using Snell s law, the impedances then become Z TM 0 F n = i H G I η n K J i sin θ (8) Z TE 0 = η i F n H G I ni K J sin θ. (83) The above expressions remain valid for lossy media. The square roots are chosen so that the real part of the characteristic impedances are positive. The functions V(z) and I(z) behave as voltage and current on a transmission line, with characteristic impedance Z TM 0 or Z TE 0, depending on the case. Hence any plane-wave reflection and transmission problem reduces to a transmission line problem, giving the exact solution that satisfies all boundary conditions. One consequence of this is that TM z and TE z plane waves do not couple at a boundary. If the incident plane wave is TM z, for example, the waves in all regions will remain TM z plane waves. Hence, the motivation for the TM z -TE z decomposition. The transverse equivalent network for the multilayer and two-region problems are shown on the right sides of Figs. and 3, respectively. The network model is the same for either TM z or TE z polarization, except that the characteristic impedances are different. If an incident plane wave is a combination of both TM z and TE z waves, the two parts are solved separately and then summed to get the total reflected or transmitted field. 7

28 Special Case: Two-Region Problem For the simple two-region problem of Fig. 3, the reflected and transmitted voltages (modeling the transverse electric fields) are represented as bg bg () ref inc + V z V e jk z = Γ 0 z bg bg ( ) trans inc V z TV e jk z = 0 z (84) (85) where the reflection and transmission coefficients are given by the standard transmission line equations Γ= Z Z ( ) ( ) 0 Z0 ( ) ( ) (86) 0 + Z0 T = + Γ. (87) Note that a plus sign is used in the exponent of Eq. (84) to account for upward propagation of the reflected wave. A. Critical angle If regions and are lossless, and region is more dense than region ( n incident angle θ = ( ) θ c will exist for which k z θ c F = H G n n sin I KJ = 0. From Snell s law, the angle is > n ), an. (88) When θ > ( ) θ c, the wavenumber k z will be purely imaginary, of the form k ( ) ( ) z z = jα. In this case there is no power flow into the second region, since the phase vector in region has no z component. In this case 00% of the incident power is reflected back from the interface. There 8

29 are, however, still fields present in the second region, decaying exponentially with distance z. In the second region the power flow is in the horizontal direction only. B. Brewster angle For lossless layers, it is possible to have 00% of the incident power transmitted into the second region, with no reflection. This corresponds to a matched transmission line circuit, with Z () ( ) 0 0 = Z. (89). For nonmagnetic layers, it may be easily shown that this matching equation can only be satisfied in the TM z case (there is always a nonzero reflection coefficient in the TE z case, unless the trivial case of identical medium is considered). The angle θ b at which no reflection occurs is called the Brewster angle. For the TM z case, a simple algebraic manipulation of Eq. (89) yields the result tanθ b = ε ε. (90) Orthogonality When treating reflection problems, there is more than one plane wave in at least one of the regions, a wave traveling in the positive z direction (focusing on the z variation) and a wave traveling in the negative z direction. Often, it is desired to calculate power flow in such a region. For a single plane wave, the complex power density in the z direction (the z component of the complex Poynting vector) is equal to the complex power flowing on the corresponding transmission line of the transverse equivalent network. When both an incident and a reflected wave are present, or both TM z and TE z waves are present the following orthogonality results are 9

30 useful. These theorems related to power flow in the z direction may be proven by direct calculation using the field components in rectangular coordinates.. An orthogonality exists between a homogeneous TM z plane wave and a homogeneous TE z plane wave propagating in the same direction, in the sense that the complex power density in the z direction, S z, is the sum of the two complex power densities S z and S z. This is true for a lossy or lossless media.. An orthogonality exists between an incident wave and a reflected wave in a lossless media (both are either TE z or TM z ), provided the wavenumber component k z of the two waves is real. The two waves are orthogonal in the sense that the time-average power density in the z direction, Re S z, is the sum of the two time-average power densities Re S z and Re S z. To illustrate property, consider an incident plane wave traveling in a glass region, impinging on an air gap that separates the glass region from another identical glass region, as shown in Fig. 5. inc ref θ glass h air glass trans 30

31 Figure 5. An incident wave traveling in a semi-infinite region of lossless glass impinges on an air gap separating the glass region from an identical region below. Incident, reflected and transmitted plane waves are shown. If the incident plane wave is beyond the critical angle, the plane waves in the air region will be evanescent, with an imaginary vertical wavenumber k z. Each of the two plane waves in the air region (upward and downward) that constitute a standing-wave field do not, individually, have a time-average power flow in the z direction, since k z is imaginary. However, there is an overall power flow in the z direction inside the air region, since there is a transmitted field in the lower region. The total power flow in the z direction is thus not the sum of the two individual power flows. The two waves in the air region are not orthogonal, and property does not apply since the wavenumber k z is not real. VI. EXAMPLE A RHCP plane wave at a frequency of.0 GHz is incident on the surface of the ocean at an angle of θ = 30 o. Determine the percentage of power that is reflected from the ocean, and characterize the polarization of the reflected wave (axial ratio, tilt angle, and handedness). Also, determine the field of the inhomogeneous transmitted plane wave. The parameters of the ocean water are assumed to be!ε = 78 and σ = 4 S/m. Solution The geometry of the incident and reflected waves is shown in Fig. 6. The incident plane wave is represented as E inc + = E x+ u j e jky y k z z 0!!b g d i (9) where 3

32 ky = k0sinθ = k0 (9) 3 kz = k0cosθ = k0 (93) u! = y!cos θ z! sinθ. (94) The reflected wave is represented as E ref = E Ax+ Bv e jky y k z0 z 0!! d i (95) where v! = y!cos θ z! sinθ. (96) u v inc ref θ ocean y z Figure 6. Geometry that defines the coordinate system for the example of a plane wave reflecting from the surface of the ocean. The x component of the incident and reflected waves corresponds to TE z waves, while the u and v components correspond to TM z waves. (The u and v directions substitute for the y direction in the previous discussion on polarization.) The transverse equivalent network is shown in Fig. 3. From Eqs. (8) and (83), the impedances are Z TM =. Ω, Z TE =. Ω, 3

33 TM TE Z = j( 3. 69) Ω, Z = j( ) Ω. The reflection coefficients are then Γ TM = j( ) and Γ TE = j( ). The coefficient A in Eq. (94) is determined directly from TE A= Γ = j( ). (97) To determine the coefficient B, recall that the voltages in the transverse equivalent network model only the transverse (horizontal) component of the electric field (the y component in the TM case). Hence, ref TM 0 0 BE cosθ = E j cosθ inc ref inc Γ b g. (98) Since θ = θ = θ, B is determined directly as B TM TM = Γ bg j = Γ = j( ). (99) The percent power reflected is calculated from P ref % = which yields 00 F H G L NM L NM A Z Z TE TE B cosθ + TM Z OI QP O J QP K j cos + θ (00) TM J Z P ref % = (0) Hence % of the incident power is transmitted into the ocean. The phase angle between the v and x components of the reflected field is φ = B A= o (0) Hence, the reflected wave is a left-handed elliptically polarized wave. 33

34 The parameter γ is determined from the ratio of the magnitudes of the v and x components of the reflected field as F HG I KJ = B o γ = tan (03) A From the formulas in the Polarization section, the tilt angle τ of the polarization ellipse from the x axis (towards the positive v axis) is then o τ = , (04) and the polarization parameter ξ is ξ = o. (05) The axial ratio of the reflected wave is then AR =+ cot ξ = (06) The reflected wave is nearly CP, since the ocean is highly reflecting (if the ocean were replaced by a perfect conductor, the reflected wave would be LHCP). The transmitted field in the ocean is represented as where kz E trans = E xc+ yd+ ze e jky y k z z 0!!! d i. (07) is the vertical wavenumber in the ocean, found from Eq. (3) as z θt 0 θ 0 n ( ) k = k cos = k n sin = k j From the transverse equivalent network, TE TE 34. (08) C = T = + Γ = j( ) (09)

35 and TM TM b g e jb g D = T jcosθ = + Γ jcos θ = j( ) (0) The z component of the transmitted field can be determined from the electric Gauss law (0), which gives E = kz Hence, dky Di. () E = j( ). () From the wavenumbers k y and k z the direction angles of the transmitted wave may be determined from Eqs. (9) (3). The results are φ = 90 o (3) θ t = j( ) radians. (4) The transmitted plane wave is inhomogeneous, since the angle θ t is complex. REFERENCES [] R. F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw Hill, 96. [] W. H. Hayt, Engineering Electromagnetics, 5 th Ed., McGraw-Hill, 989, Appendix B. 35

36 [3] P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields, IEEE Press, 996, pp [4] R. E. Collin, Field Theory of Guided Waves, nd Ed., IEEE Press, 99, pp [5] J. D. Kraus, Antennas, McGraw-Hill, 988, pp