Chapter 6 Propagation of Plane Waves 6 Plane Wave in a Source-Free Homogeneous Medium 62 Plane Wave in a Lossy Medium 63 Interference of Two Plane Waves 64 Reflection and Transmission at a Planar Interface with Normal Incidence 65 Reflection and Transmission at a Planar Interface with Oblique Incidence 66 Critical Angle and Brewster Angle In this Chapter the propagation of electromagnetic wave in space filled with some materials is studied Although the concept of current-induced radiation has been applied with success in the analysis of the Hertzian dipole, it can be mathematically difficult for some more complicated distribution of sources Further, when the sources interact with the fields, the sources themselves may be difficult to evaluate Thus another approach which employs the Helmholtz equations of fields may be helpful This approach allows one to solve the differential equation regionally, usually away from the sources Thereby, some properties of the fields can be obtained from the local solutions, without knowing specific details of the radiating sources Propagation characteristics of a uniform plane wave in a source-free homogeneous media are studied Then reflection and transmission from two different semi-infinite media separated by a planar interface are examined 6 Plane Wave in a Source-Free Homogeneous Medium Vector Helmholtz equation in a general medium Recall Maxwell s equations derived in the previous Chapter In a medium of constitutive parameters µ and ɛ, Faraday s law and Ampere s law become E(r) = jωµh(r) H(r) = J f (r) + jωɛe(r) By applying curl to both sides of Faraday s law, one has E(r) = jωµ H(r) jω µ H(r) = jωµ[j f (r) + jωɛe(r)] + µ µ E(r), where Ampere s law and Faraday s law have been made use of It is noted that when the permeability µ is uniform such that its gradient µ vanishes, the preceding relation becomes simpler Furthermore, E(r) = 2 E(r) + E(r) From the divergence relation of field D and vector identities, one has ρ f (r) = D(r) = [ɛe(r)] = ɛ E(r) + ɛ E(r)
em6 2 Then E(r) = ɛ ɛ E(r) + ρ f(r) ɛ It is noted that when the permittivity ɛ is uniform such that its gradient ɛ vanishes, the preceding relation becomes simpler Thereafter, the Helmholtz equation for the field E in a general medium is as complicated as [ ] ɛ 2 E(r) + ω 2 µɛe(r) + ɛ E(r) + µ [ ] µ E(r) = ρf (r) +jωµj f (r) ɛ This is actually a set of three coupled partial differential equations This Helmholtz equation of electric field is quite complicated and its solutions for general materials and sources may need computer simulation In what follows, simplified forms of the Helmholtz equation suitable for some situations are discussed Vector Helmholtz equation in a simple medium Consider a source-free homogeneous medium where ρ f = 0, J f = 0, and both µ and ɛ are constant Such a simple situation may hold in a small region of a more complicated structure in which wave is generated, propagates, and is then received In this region, the vector Helmholtz equation of field E becomes a simpler form of 2 E(r) + k 2 E(r) = 0, where k = ω µɛ is known as the propagation constant or the wavenumber of that medium It is seen that the three Cartesian components of field E are uncoupled in the equation Ordinary differential equation in field Consider a field component E x (x, y, z) which is uniform over the x-y plane, that is, E x / x = 0 and E x / y = 0 Then the Helmholtz equation for E x reduces to the one-dimensional ordinary differential equation d 2 E x (z) dz 2 + k 2 E x (z) = 0 The solution of the above ODE is readily seen That is, E x (z) = E + 0 e jkz + E 0 e jkz, where E + 0 and E 0 are constants to be determined from the details of the whole structure including the sources generating such a wave Apparently, this solution is not the true solution, but just a linear combination of possible solutions suitable in the region considered However, some general properties of wave can be deduced from a local solution like this In what follows, we discuss the wavelength, phase velocity, propagation direction, and polarization direction deduced from such a solution Wavelength The space-time variation of an electric field E(r, t) with phasor E(z) = ˆxE 0 e jkz is given as E(z, t) = ˆxE 0 Re{e jkz e jωt } = ˆxE 0 cos(ωt kz), where E 0 is supposed to be real At a given position z, the field varies sinusoidally with time t
em6 3 Meanwhile, for a given time, the field varies also sinusoidally over distance z The distance λ between two successive peaks is given as λ = 2π k It is seen that the wavelength changes with the frequency and the constitutive parameters It is noted that the wavelength of a uniform plane wave is determined by the constitutive parameter µ and ε of the medium Ordinarily, µɛ > µ 0 ɛ 0, then the wavelength in a material tends to be shorter than that in free space In a plasma, such as the ionosphere discussed later, the permittivity is given ɛ = ɛ 0 ( ωp/ω 2 2 ) It is seen that the relative permittivity of a plasma is always less than unity and can be even negative, as the frequency ω is below the plasma frequency ω p Phase velocity The peak of the plane wave corresponds to a zero value of the difference ωt kz After a short time interval dt, the position of the peak should displace to make the difference remain zero Obviously, the displacement is dz = ωdt/k Thus the peak of the plane wave travels in the z direction with a fixed speed v p given as v p = dz dt = ω k = µɛ This velocity ẑv p is known as the phase velocity The phase velocity also changes with the constitutive parameters In free space, the phase speed is just the speed of light, since v p = / µ 0 ɛ 0 = c When the permittivity of a dielectric material is negative, the propagation constant k becomes pure imaginary Thereby, the field may be of the form: E(z) = ˆxE 0 e αz, where the attenuation constant α = jk = ω µε The space-time variation of an electric field E(r, t) is then given as E(z, t) = ˆxE 0 Re{e αz e jωt } = ˆxE 0 e αz cos ωt, where E 0 is supposed to be real It is seen that the wave does not oscillate along the propagation z direction Instead, its magnitude decays monotonically along that direction This kind of wave is called an evanescent wave Propagation direction A general form of a uniform plane wave of which the field varies in all directions can be given as E(r) = E 0 e j(kxx+kyy+kzz), where E 0 is a constant vector denoting the direction of the electric field By defining the propagation vector k as k = ˆxk x + ŷk y + ẑk z, the uniform plane wave can be written in a compact form of where r = ˆxx + ŷy + ẑz E(r) = E 0 e jk r,
em6 4 From the Helmholtz equation for each Cartesian component of the field, it is seen that in a source-free homogeneous medium, the components of the propagation vector should satisfy the relation k 2 x + k 2 y + k 2 z = k 2 That is, k = k Thus these three components are not entirely arbitrary The values of the components k x, k y, and k z determine the direction of wave propagation The wave with k x = 0 and k y = 0 propagates in the z direction, as described previously The propagation direction of the wave with k x 0, k z 0, and k y = 0 lies in the x-z plane It makes an angle of θ = tan (k x /k z ) with the z axis Polarization direction The direction of polarization of a plane wave is also not entirely arbitrary As discussed previously, in a source-free homogeneous region, field E satisfies the relation E(r) = 0 That is, field E is divergence-free By a direct expansion in Cartesian coordinates, this relation implies that ( jk x E 0x jk y E 0y jk z E 0z )e j(k xx+k y y+k z z) = j(k E 0 )e j(k xx+k y y+k z z) = 0 Since this relation holds for an arbitrary position r in a source-free homogeneous region, one has k E 0 = 0 This relation states that for a uniform plane wave propagating in a source-free homogeneous medium, the polarization direction is always orthogonal to the propagation direction There are two independent polarizations for each k For example, if k lies in the x-z plane, the vector E 0 is in the direction of either ŷ or (ˆxk z ẑk x ) For a uniform plane wave the associated field H is related to field E in a simple manner From Faraday s law, one has H(r) = jωµ E(r) = jωµ ( jk) E(r) = η ˆk E(r), where η = µ/ɛ is known as the intrinsic impedance of the medium It is seen that the electric field of a uniform plane wave is perpendicular to the magnetic field and both of the fields in turn are perpendicular to the propagation direction Thus the uniform plane wave is a transverse electromagnetic (TEM) wave, which means that both the electric and magnetic fields are transverse to the direction of propagation For example, the magnetic field H associated with field E(r) = ˆxE 0 + e jkz is given as H(r) = ŷ η E+ 0 e jkz 62 Plane Wave in a Lossy Medium In a lossy medium, the induced current contains the conduction current, in addition to the polarization current and the magnetization current That is, the induced current density J is given as J = jω(ɛ ɛ 0 )E + σe + ( ) B, µ 0 µ
em6 5 where σ is called the conductivity For lossy media, a complex permittivity ɛ c can be defined, such that the polarization current incorporates the conduction current That is, Thus the complex permittivity ɛ c is given as ( J p = jω(ɛ c ɛ 0 )E = jω ɛ j σ ) ω ɛ 0 E, ɛ c = ɛ j σ ω = ɛ jɛ, where both ɛ and ɛ are positive real and the imaginary part ɛ = σ/ω It is seen that the effect of conductivity tends to be more significantly at low frequencies The ratio ɛ /ɛ, known as the loss tangent, can be expressed as where δ c is known as the loss angle tan δ c = ɛ ɛ = σ ωɛ, Effect of conductivity on power absorption and field attenuation Note that owing to the conductivity (or to the imaginary part ɛ ), the conduction current density will be in phase with the electric field which induces the current itself Due to this particular phase relation, the time-average absorption power density P a will be nonzero, since P a = 2 Re{E J } = 2 ωɛ E 2 Meanwhile, the polarization current due to the real part ɛ has a phase difference of 90 with respect to the field and hence the corresponding time-average absorption power density is zero **Another effect of the imaginary part ɛ is that the electric field tends to attenuate in a lossy material Recall that due to a current sheet of surface current density J s (x, y) = ˆxJ s0, the electric field is given as E(r) = ˆx 2 η 0J s0 e jk 0 z Note that the phase difference between the current density and the electric field generated by the current is 80 Meanwhile, the phase difference between the conduction current density and the electric field which induces the current is zero Thus the regenerated electric field is out of phase by 80 with the impressed electric field and tends to cancel in part the impressed electric field Phase constant and attenuation constant To investigate the propagation properties quantitatively, recall the Helmholtz equation For a homogeneous, homogeneous, lossy medium, the source-free Helmholtz equation is still given by 2 E(r) + k 2 E(r) = 0, where the propagation constant k (= ω µɛ) is a complex number (here, the subscript c in ɛ c is omitted) Explicitly, complex propagation constant k can be given as k = ω ( ) e µɛ = ω µɛ ( j tan δ c ) /2 jδ c /2 = ω µɛ cos δ c
em6 6 Let the complex propagation constant be given as k = β jα It can be shown that β = ω ( ) µɛ + tan 2 δ c + 2 α = ω µɛ 2 ( ) + tan 2 δ c It is seen that β > α The constants β and α are real and called phase constant and attenuation constant, respectively The attenuation constant α determines how fast the wave is decaying For a uniform plane wave polarized in the x direction and propagating in the z direction, the electric field is given as E(z) = ˆxE 0 e αz e jβz The space-time variation of the field with a real E 0 is given as E x (z, t) = E 0 e αz cos(ωt βz) It is seen that the wave behaves as a damped oscillation That is, at a given instant, the wave varies sinusoidally along z with its amplitude decaying with z Time-average power flux density in a lossy medium For a plane wave with E(z) = ˆxE 0 e αz e jβz, the magnetic field is given as H(z) = ŷe 0 e αz e jβz /η, where the intrinsic impedance η (= µ/ɛ = η e jθ η ) of a lossy medium is complex Thus, for a uniform plane wave propagating in a lossy medium, it is seen that the electric field is no longer in phase with the magnetic field Instead, the electric field leads the magnetic field by a phase of θ η The time-average power flux density of a uniform plane wave in a lossy medium is then given by S av (z) = 2 Re{E(z) H (z)} = ẑ E 0 2 { } 2 e 2αz Re = ẑ E 0 2 η 2 η e 2αz cos θ η It is seen that the time-average power flux density decays with propagation distance and depends on the phase angle θ η of the intrinsic impedance In what follows we consider two extreme cases, low-loss dielectric and good conductor, where some approximations on the propagation constant and intrinsic impedance can be made Low-loss dielectric A perfect dielectric material has a zero ɛ It is usually desired that the imaginary part ɛ of the dielectric foil sandwiched between the electrodes in a capacitor is close to zero Otherwise, the leakage current will degrade the performance of the capacitor For a low-loss material, ɛ ɛ (σ/ωɛ or tan δ c ) Thus β ω ( ) ( ) µɛ + 8 tan2 δ c = ω µɛ ɛ + 2 8 α ω µɛ 2 tan δ c = ω µɛ ɛ 2ɛ = 2 σ µ ɛ An alternative derivation of the constants can be given as [ k = ω µ(ɛ jɛ ) = ω µɛ [ jɛ /ɛ ] 2 = ω µɛ j 2 ɛ /ɛ + 8 (ɛ /ɛ ) 2 + ], ɛ
em6 7 where the binomial expansion has been made use of Note that the phase constant β is close to ω µɛ and the attenuation constant α is proportional to the conductivity σ If the conductivity σ is independent of ω, the attenuation constant is then independent of the frequency The associated wavelength is given by λ = 2π β 2π ω ( ) ɛ 2, µɛ 8 ɛ which is slightly shorter than the one in the lossless medium with an identical ɛ The intrinsic impedance η of a low-loss dielectric is complex ) µ µ η = ( ɛ + j ɛ µ = ɛ 2ɛ ɛ ( + j tan δ 2 c) It is seen that the real part η r µ/ɛ And the attenuation constant can be given as α = ση r /2, which is proportional to the conductivity σ The phase velocity v p in the lossy medium is which is slightly lower v p = ω β µɛ 8 ( ) ɛ 2, ɛ Penetration depth and microwave oven The inverse of the attenuation constant α in a lossy medium is defined as the penetration depth of the electromagnetic wave in that material That is, the penetration depth δ is defined as δ = α The penetration depth determines the distance with which the field intensity of a plane wave decays to the extent of e (0367879) For the wave decaying to 0 db, the distance is 23 δ And for the wave propagating a distance of 3 δ, it decays to about 005 For a low-loss material, the penetration depth δ 2/η r σ In terms of the wavelength, the penetration depth can be expressed as δ λ/π tan δ c When the loss tangent is much less than unity in a low-loss material, the penetration depth can be much longer than the wavelength The decay in field and hence in power flux density corresponds to the power absorption in a lossy material within a few penetration depths Thus electromagnetic wave can be utilized in the induction heating of a lossy material If the penetration depth of the electromagnetic wave in a material is close to the linear size of that material, the induction heating can achieve the capabilities of fast- and uniform-heating, as compared with traditional ovens that rely on heat conduction Meanwhile, a too large penetration depth leads to a poor efficiency, since the power absorption is difficult A microwave oven cooks food by irradiating the food with microwave generated by a magnetron which converts electric power at 60 Hz into microwave power at 245 GHz (free-space wavelength λ 0 = 22 cm) Suppose that a beef steak has a relative permittivity ɛ /ɛ 0 = 40 and a conductivity σ = 2 /m at that frequency The corresponding loss tangent is 0367 The penetration depth of such a steak is δ λ/π tan δ c = 67 cm (exact value of δ = 70 cm), which is suitable in the microwave, in consideration of fastness, uniformity, and efficiency
em6 8 In wireless communication, the penetration depth of the propagation medium should not be much shorter than the path length Note that the penetration depth is proportional to the wavelength, if the loss tangent is not a strong function of frequency Thus, for propagation through the atmosphere containing cloud, rain, or fog, the penetrating ability of microwave is better than visible light Thus, in satellite and long-distance terrestrial communications and in radar, microwave is more suitable than visible light Good conductor For a good conductor ɛ ɛ (σ/ωɛ or tan δ c ) Thus ɛ β = α = ω µɛ 2 tan δ c = ω µɛ = 2ɛ 2 ωµσ 2 ω µɛ = 2 An alternative derivation of the constants can be given as ( ) 2 2 k = ω µ(ɛ jɛ ) ω µ( jɛ ) = ω µɛ 2 j 2 = πfµσ Note that in a good conductor the attenuation constant α is identical to the phase constant β Further, it is noted that these constants increases with the frequency as ω, if the conductivity σ is a very slow function of ω The intrinsic impedance η of a good conductor is η = µ ɛ ωµ jσ = ( + j)α σ, which has a phase angle of 45 The phase velocity v p in a good conductor is then v p = ω β 2ω µσ, which grows with increasing frequency, if the σ is not a strong function of frequency The wavelength in a good conductor is given as λ = 2π β = 2π 2 ωµσ For copper with σ = 58 0 7 /m and µ r =, the phase velocity is as slow as v p = 720 m/s at 3 MHz And the corresponding wavelength is as short as λ = 024 mm, while λ 0 = 00 m Thus conductor is a quite dense medium for electromagnetic wave Skin depth of a metal A high-frequency wave will attenuate rapidly when it propagates through a conductor of high conductivity and permeability The skin depth δ (= /α) in a metal can be given as δ = 2 ωµσ Obviously, the skin depth is proportional to the wavelength simply as δ = λ/2π and decreases with frequency For a wave propagating in a metal with a distance of one wavelength, its amplitude decays to as small as about 0002
em6 9 In a good conductor, the induced current is given as J = σe This current tends to be confined to the surface of the conductor with a few skin depths and can be presented by a surface current density J s Consider an electric field E(z) = ˆxE 0 e jkz within the conductor (z > 0) It can be shown that the surface current density J s can be given in terms of the current density at the conductor s surface J(0) as J s = J(z)dz = e jπ/4 J(0)δ 2 0 Thus the current can be viewed as being distributed uniformly over a distance of δ with a current density of J(0)e jπ/4 / 2, the current density at the conductor s surface time the factor e jπ/4 / 2 Then, in terms of the surface current density J s, it can be shown that the power absorbed in the metal plate of unit area is given as P a = 2 0 E(z) J (z)dz = 2δσ J s 2 It is seen that for a given amount of J s, the absorption power decreases with the square root of the conductivity This formula will be used in the calculation of the metallic power loss in waveguide induction heating While the conductivity in a metal may cause dissipation as a waste, it can be utilized in the induction heating Note that the skin depth (δ = 2/ωµσ) decreases with increasing permeability This has an effect to lower the operating frequency in induction heating Suppose an iron plate has a relative permeability of µ r = 2000, and a conductivity of σ = 0 7 /m, then the skin depth δ = 046 mm at 60 Hz Thus most of the power will be absorbed in an iron plate thicker than about 5 mm On the other hand, for aluminum with µ r = and σ = 354 0 7 /m, the skin depth is as thick as δ = cm at 60 Hz Thus the heating efficiency is low, since most of the power will propagate through a thin aluminum plate as a waste, if the plate thickness is less than a few mm Obviously, the induction heating of a thin nonmagnetic metal plate requires a higher frequency source which is usually more costly For the aluminum in a microwave wave at f = 245 GHz, the skin depth is as thin as δ = 7 µm A too thin skin depth tends to concentrate the power sharply Thus the heating is extremely nonuniform and spark may even occur Thus avoid to place an aluminum foil or other metallic objects in a microwave oven On the other hand, at a low frequency of 60 Hz, the penetration depth for beef steak, still supposed to have a relative permittivity ɛ /ɛ 0 = 40 and a conductivity σ = 2 /m at this frequency, is as large as 459 m Obviously, it is too large for an efficient heating skin depth effect in propagation For an iron plate with σ = 0 7 /m and µ r = 2000, the skin depth δ = 356 and 0356 µm for a typical AM and FM wave, respectively, which are much shorter than the thickness of the iron plate of a car Thus an outdoor antenna is needed for radio reception in a car, especially for the FM radio The difference in the skin depths also explains why an FM radio wave is less viable to a metal plate than an AM wave Wireless communication through seawater may be difficult The conductivity of seawater is as high as σ = 4 /m and the penetration depth in seawater is as short as δ = 02 m at 5 MHz or δ = 25 m at 0 khz (λ = 30 km) of the VLF band Thus, owing to the high attenuation of seawater, the communication of submarine must use the VLF band or lower However, the radiation efficiency would be very low at such low frequencies, since the antenna size is limited and is inevitably much smaller than the operating wavelength
em6 0 63 Interference of Two Plane Waves In this section the interference between two plane waves of different polarizations, propagation directions, frequencies, propagation constants, or of phases are discussed Circular polarization due to phase shift The polarization of a uniform plane wave describes the direction of the electric field at a given point and instant For example, the wave with field E(z) = ˆxE 0 e jkz or E(z, t) = ˆxE 0 cos(ωt kz), where E 0 is a real constant, is linearly polarized in the x direction at any instant of time Consider the polarization of the wave composed of two plane waves of different polarizations If the phases of the two components are identical, such as in the field E(z) = E 0 (ˆx + ŷ)e jkz, the wave is also linearly polarized But the direction of the polarization has been rotated by an angle of 45 in the x-y plane Then consider the case where the two components are 90 out of phase, such as in the field E(z) = E 0 (ˆx jŷ)e jkz The corresponding space-time variation is given as E(z, t) = E 0 [ˆx cos(ωt kz) + ŷ sin(ωt kz)] At z = 0 and t = 0, the polarization points in the x direction Meanwhile, at t = π/2ω = /4f, the polarization points in the y direction at the same location During the time interval between t = 0 and /4f, the polarization changes gradually from x to y counterclockwise as viewed in front of the wave The wave is called as right-hand or positive circularly polarized The handiness is defined according to that the fingers follow the rotation of direction and the thumb points to the propagation direction of the wave The wave corresponding to E(z) = E 0 (ˆx + jŷ)e jkz is then left-hand or negative circularly polarized A more general situation is the elliptically polarized wave, which can be given as E(z) = (ˆxE 0x + ŷe 0y )e jkz, where E 0x and E 0y is arbitrary complex numbers The ratio E 0x /E 0y determines the polarization property Apparently, a circularly or elliptically polarized wave is composed of two linearly polarized waves Conversely, a linearly polarized wave can be resolved into two circularly polarized waves of opposite senses of rotation For example, E(z) = ˆxE 0 e jkz = 2 E 0(ˆx jŷ)e jkz + 2 E 0(ˆx + jŷ)e jkz Faraday rotation In some crystal placed in a magnetic field, the left-hand circularly polarized wave travels with a propagation constant different from that of the right-hand wave Thus, after a linearly-polarized wave passed through the crystal, the direction of the field tends to change This phenomenon is known as the Faraday rotation Consider a crystal slab of thickness d Suppose the wave enters the crystal via the surface at z = 0 and exits it at the other surface z = d And at z = 0, the field is x-polarized and given as E(0) = ˆxE 0 Then the field upon exiting the crystal at z = d is given as E(d) = 2 E 0(ˆx jŷ)e jk Rd + 2 E 0(ˆx + jŷ)e jk Ld = 2 E 0e jk Rd [(ˆx jŷ) + (ˆx + jŷ)e j kd ]
em6 where k R and k L are respectively the propagation constants of the right- and left-hand circularly polarized waves in the crystal and k = k R k L If the crystal thickness d = π/ k, then E(d) = 2 E 0e jk Rd [(ˆx jŷ) (ˆx + jŷ)] = ŷje 0 e jk Rd That is, the wave becomes y-polarized after exiting from the crystal Wave packet with different frequencies and propagation constants Any information-carrying wave actually consists of a frequency band of waves forming a wave packet Consider a wave packet consists of only two components: one of (ω, k ) and the other of (ω 2, k 2 ) Suppose these two components are of identical magnitude Its field is then given as E x (z, t) = E 0 cos(ω t k z) + E 0 cos(ω 2 t k 2 z) = 2E 0 cos( ωt kz) cos(ωt kz), where ω = 2 (ω + ω 2 ), k = 2 (k + k 2 ), ω = 2 (ω ω 2 ), and k = 2 (k k 2 ) (Note that the field can not be written in the phasor form) Suppose that ω ω, and k k, then the wave can be viewed as the plane wave cos(ωt kz) amplitude-modulated by the slowly-varying signal cos( ωt kz) In communication, a signal is usually put on a carrier which is at a higher frequency and can be easier to transmit and propagate beat due to frequency shift At a given position, say z = 0, the field of the wave packet behaves as E x (0, t) = 2E 0 cos( ωt) cos(ωt) It appears that the amplitude of the time-harmonic wave cos ωt is not a constant but varies temporally with time slowly with cos( ωt) In acoustics, the phenomenon is known as beat The beat frequency f determines the number of cycles with which the amplitude of a tone, a sound at a particular frequency f, changes per second If f is low enough, the change in the amplitude can be perceived by human s ears And this may sound as inharmonic Thus it is not a common practice in music to make an accord with two close keys group speed At a given instant, say t = 0, the field of the wave packet behaves as E x (z, 0) = 2E 0 cos( kz) cos(kz) It appears that the amplitude of the space-harmonic wave cos kz is not a constant but varies spatially with z slowly as cos( kz) By connecting the tips of the field along the z direction at a given time, two artificial curves symmetric about the z axis can be obtained They form an envelope within which the magnitude of the electric field at that given time is confined The function cos( ωt kz) is associated with space-time variation of the envelope It is seen that the envelope itself also moves with a speed v g given as v g = ω k = k/ ω = dk/dω
em6 2 This velocity of the envelope is known as the group velocity The group speed v g of the wave packet in general is different from the phase speed v p of each individual component wave In free space, the propagation constant grows linearly with frequency as k = ω µ 0 ɛ 0 Then the group speed is equal to the phase speed, that is, v g = (dk/dω) = / µ 0 ɛ 0 = v p And the medium is called dispersionless Thus in a dispersionless medium, the signal propagates at the same speed with the carrier In a dispersive medium, the constitutive parameters µ or ɛ depends on frequency and hence the dispersion occurs In a medium if the phase constant grows superlinearly with frequency, k/ ω > µɛ, then v g < v p This is known as normal dispersion, which occurs for visible light in glass whose permittivity increases with increasing frequency Conversely, if the phase constant grows sublinearly with frequency, k/ ω < µɛ, then v g > v p This is known as anomalous dispersion For example, consider a good conductor in which β = ωµσ/2 If the variation of σ over the band of frequency of a wave packet under consideration is ignored, the group velocity is given as v g = ( ) dβ 2ω = 2 dω µσ = 2v p Thus a good conductor is an anomalous dispersive medium, since the group speed is higher than the phase speed In a dispersive medium, the various frequency components comprising a signal-carrying pulse travel at different phase speeds Then the pulse tends to broaden in its shape while traveling through the medium The pulse broadening will limit the bit rate in digital communication in a dispersive medium It is not easy to figure out how can the interference among a couple of independent waves of slightly different phase speeds make the motion of the wave packet quite different A computer animation is helpful for visualization Standing wave due to opposite propagation directions One extreme case of the wave combination is the one with ω = ω 2 = ω and k = k 2 = k Since ω = 0, the group speed of the wave packet is zero Thus the envelope of the wave packet does not move, while each component wave moves fast at the phase velocity This kind of wave packet is called standing wave The standing wave occurs in the reflection from a metal plate and in the mechanical vibration of a string fixed at both ends For example, consider the field which is composed of two waves of equal amplitude but travelling antiparallel That is, E x (z) = E 0 (e jkz + e jkz ) = 2E 0 cos kz The space-time variation of the wave is then given as E x (z, t) = 2E 0 cos kz cos ωt, where E 0 is supposed to be real It is seen that the peaks are stationary and do not propagate along z However, its magnitude varies sinusoidally with time If the two waves are of unequal amplitude, the situation is more complicated and will be discussed in a later section 64 Reflection and Transmission at a Planar Interface with Normal Incidence Consider the propagation of a plane wave in a space composed of two semiinfinite media separated by a planar interface at z = 0 The constitutive parameters of the two semiinfinite media
em6 3 are µ = µ, ɛ = ɛ z < 0 µ = µ 2, ɛ = ɛ 2 z > 0 Suppose that a wave E i (r) = ˆxE 0 e jk z is incident from z 0 It will be seen that due to such a planar interface, there will result in a reflected and a transmitted wave in the regions of z < 0 and z > 0, respectively In general the reflection and transmission is a scattering problem, where medium 2 serves as the scatterer and the scattered field which forms both the reflected and the transmitted fields is generated from the current induced in the scatterer However, even for such a simple case, the solution of the scattering problem is rather difficult However, we can draw some useful features of the wave without actually solving the scattering problem From the concept of the scattering, multiple sheets of current are induced by the incident field Since the incident field is a uniform plane wave, the induced currents are also uniform in the x-y plane and at the identical frequency These induced currents will generate uniform plane waves, which in turn induce other uniform sheets of current, and so on Thus, without any calculation, we may deduce that the resultant electric field remains to be an x-polarized uniform plane wave Thus the field has variation only in the z direction and its partial derivatives with respect to x and y vanish Recall that in a source-free homogeneous region, where ρ f = 0, J f = 0, ɛ = 0, and µ = 0, the Helmholtz equation for the electric field polarized in the z direction and uniform in the x-y plane is d 2 dz 2 E x(z) + ω 2 µɛe x (z) = 0 However, for the present problem, there exists a permittivity gradient at the interface z = 0 Thus, over the entire domain, the governing equation is quite complicated A simpler approach to handle the problem is to solve the equation separately for the two regions z < 0 and z > 0 By so doing, one obtains possible local solutions with undetermined coefficients as a e jkz + Γ e jk z z < 0 E x (z) = τ e jk2z + b e jk2z, z > 0 where k i = ω µ i ɛ i, i = or 2 Obviously, the local solution in each medium satisfies the homogeneous Helmholtz equation for that medium It is easily seen that if both µ 2 ɛ 2 and µ 2 ɛ 2 are real and µ 2 ɛ 2 is greater than µ ɛ, then the wavelength of the transmitted field becomes shorter By using the radiation condition it is inferred that a = and b = 0, where the constant E 0 is made to be unity for simplicity Then there remain two undetermined coefficients Γ and τ These two coefficients should be chosen in such way that the linear combination of the local solutions will satisfy the associated boundary conditions at the interface The coefficients Γ and τ are dimensionless quantities which correspond to the magnitudes of the reflected and the transmitted waves and thus are known as reflection coefficient and transmission coefficient, respectively By imposing boundary conditions at the interface, that is, enforcing the continuity for the field components E x and H y [= ( /jωµ)de x /dz] tangential to the interface, one has + Γ = τ η ( Γ) = η 2 τ,
em6 4 respectively, where η i = ωµ i /k i It is easy to see that the solutions for the coefficients Γ and τ are Γ = η 2 η η 2 + η τ = 2η 2 η 2 + η Coefficients Γ and τ can be complex when η or η 2 is complex It can be shown that Γ, that is, the reflected field can not be stronger than the incident field The case with Γ = 0 becomes reflectionless This is possible when η 2 = η or when the following condition is met µ = µ 2 ɛ ɛ 2 For this reflecionless case, τ = That is, the wave is totally transmitted If medium 2 is a metal, η 2 = ( + j)/σδ = ( + j) ωµ/2σ 0, then Γ = and τ = 0 Thus reflection from a metallic surface with η 2 = 0 is perfect Total reflection with Γ = is also possible when η 2 is purely imaginary and η is real This total reflection occurs in the case where region is air and region 2 is a plasma, of which the permittivity can be negative The total field in medium is composed of two wave traveling antiparallel Thus they form a standing wave Quantitatively, the field in region is given as E (z) = ˆx(e jk z + Γe jk z ) = ˆxe jk z ( + Γ e j2k z e jθ Γ ) The space-time dependence of the electric field in a lossless media is then given as E (z, t) = ˆx[(cos(ωt k z) + Γ cos(ωt + k z + θ Γ )] At a fixed instant of time, it is seen that the field varies periodically That is, E (z, t) = E (z λ /2, t) = E (z λ, t) and so on standing-wave ratio For reflection from a perfect conductor, Γ Then the field in a lossless medium is given as E (z) = ˆx(e jk z e jk z ) = ˆxj2 sin k z and E (z, t) = ˆx2 sin k z sin ωt It is seen that the field is a standing wave, that is, the field varies sinusoidally with time and its amplitude varies sinusoidally along the position Note that the amplitude varies periodically along the position with a period of π/k There are repeated points at which the amplitudes are zero The locations of these node points are fixed The air-metal interface is a node for the electric field For a general value of Γ, the wave behavior is more complicated At any fixed position, the field also varies sinusoidally with time For a real Γ, the field at the planar interface z = 0 is E (0, t) = ˆx( + Γ) cos ωt And at the position z = z quarter-wavelength away from the interface (z = π/2k ), the field becomes E (z, t) = ˆx( Γ) sin ωt
em6 5 And at the position z = z 2 half-wavelength away from the interface (z 2 = π/k ), the field becomes E (z 2, t) = ˆx( + Γ) cos ωt As well as at z = 0, the ampltude is + Γ For a positive Γ, the maximum and the minimum amplitudes are + Γ and Γ, respectively Note that the minimum amplitude is not zero, unless Γ = At an arbitrary z, the amplitude of the electric field is given by E (z) = e jk z + Γe jk z = { [( + Γ) cos k z] 2 + [( Γ) sin k z] 2} /2 = { + Γ 2 + 2Γ cos 2k z } /2 Again, it is seen that the maximum amplitude is + Γ and the minimum is Γ The maximum amplitude occurs repeatedly at z = mλ /2, while the minimum at z = (m + 05)λ /2, where m = 0,, 2 For a general value of Γ, the amplitude is given as E (z) = + Γ e j2k z e jθ Γ The situation of spatial variation in field amplitude may occur in a mobile phone placed near a building or a container The maximum and the minimum amplitudes are + Γ and Γ, respectively The maximum amplitude occurs repeatedly at e j2kz e jθ Γ = or z = (m + θ Γ /2π)λ /2, while while the minimum at e j2kz e jθ Γ = or z = (m + 05 + θ Γ /2π)λ /2 Again, the maximum or minimum amplitude repeats every half wavelength Note that the locations of a maximum or minimum amplitude depends on the phase angle θ Γ and location of the interface may not be a maximum or minimum amplitude The ratio of the maximum amplitude to the minimum one is known as the standing-wave ratio (SWR) given by S = E (z) max = + Γ E (z) min Γ Conversely, the magnitude of the reflection coefficient Γ can be determined from the SWR as Γ = S S + The value of S varies from to, as the value of Γ varies from 0 (reflectionless) to (total reflection) During the interference of a forward wave with a backward wave, the resultant wave can still be deemed as a forward traveling wave but its amplitude changes both with space and time A computer animation is helpful in visualizing the wave motion magnetic field The magnetic field can be found from Faraday s law That is, H(z) = ŷ jωµ de x (z) dz ŷ (e jkz Γe jkz ) z < 0 η = ŷ τe jk2z z > 0 η 2
em6 6 Note that in region, at the position where E(z) is a maximum + Γ (Γ 0), H(z) is a minimum Γ ; and vice versa For reflection from a perfect conductor, Γ = Thus The space-time behavior of this standing wave is H (z) = ŷ η (e jk z + e jk z ) = ŷ 2 η cos k z H (z, t) = ŷ 2 η cos k z cos ωt The amplitude of magnetic field is maximum at the interface (z = 0), while that of the electric field is a minimum there power relation in lossless media Remark that the complex Poynting vector is defined as S(r) = E(r) 2 H (r) For lossless media, the complex Poynting vector becomes S(z) = ẑ 2 E x(z)hy(z) = ẑ 2η { Γ 2 + j2γ sin 2k z} z < 0 2η 2 τ 2, z > 0 where the pure imaginary term is due to the interference between the incident and the reflected waves Remark that the time-average power flux density is given by the real part of the complex Poynting vector That is, S av (z) = Re{S(z)} = ẑ 2η { Γ 2 } z < 0 2η 2 τ 2 z > 0 Note that the time-average power flux density in lossless media is independent of position z, which states that the power is conserved when the wave is traveling through the media Further, it can be shown that η { Γ 2 } = η 2 τ 2 This relation states that the complex Poynting vector is continuous at the interface and that the time-average incident power flux density is equal to the sum of the time-average reflected and transmitted power flux densities at arbitrary positions **power relation in lossy media Further, it can be shown that for general media {e 2αz Γ 2 e 2αz + Γe j2βz Γ e j2βz } z < 0 2η S(z) = ẑ τ 2 e 2α2z z > 0 2η 2 If µ 2 ɛ 2 is complex (such as in a lossy medium), the transmitted field propagates as a damped oscillation
em6 7 For a perfect conducting plate, ɛ 2 j and η 2 0, thus Γ = and τ = 0 The associated transmitted power is zero For the case where µ 2 ɛ 2 < 0 (as in a plasma), the transmitted field becomes evanescent and the time-average transmitted power is also zero This is because that η 2 is pure imaginary and hence τ 2 Re{/2η2} = 0, although τ itself is not equal to zero Thus the incident power can be totally reflected from a perfect conducting plate or from a plasma For a lossless medium, it can be shown that { } ( Γ 2 ) = Re τ 2 η This relation states that the time-average incident power flux density is equal to the sum of the time-average reflected and transmitted power flux densities at the interface 65 Reflection and Transmission at a Planar Interface with Oblique Incidence Consider the incident electric field E i of a uniform plane wave which is spatially harmonic in the x direction, in addition to the z direction That is, η 2 E i (x, z) = E 0 e jk zz e jk xx, where k 2 z + k 2 x = k 2 and E 0 is a vector representing the polarization of the incident field Such a wave propagates with an angle θ i from the z axis, the normal direction to the interface, where θ i = tan (k x /k z ) = sin (k x /k ), This angle θ i is known as the angle of incidence For the previous case of normal incidence, k x = 0 and the angle θ i = 0 Since the polarization of a plane wave is orthogonal to its propagation direction, there are generally two polarizations E 0 = ŷ or (ẑk x ˆxk z ) For the former case E 0 = ŷ, the polarization is perpendicular to the plane of incidence (the x-z plane, parallel both to the propagation vector and the normal of the interface plane) and is called perpendicular polarization This case is also known as TE (transverse electric), since the electric field is transverse to ẑ, the normal of the interface For the case with E 0 = ẑk x ˆxk z, the polarization is parallel to the plane of incidence and is called parallel polarization This case is also known as TM (transverse magnetic), since the magnetic field is transverse to z Oblique incidence with perpendicular polarization Consider the incident electric field E i of a y-polarized uniform plane wave spatially harmonic both in the z and x directions That is, E i (x, z) = ŷe jk zz e jkxx The field tends to induce y-directed current in the media The induced current as well as the field reradiated by this current should preserve the spatial harmonic variation e jkxx Thus the resultant field will also be y-directed and preserve this harmonic variation Note that the electric field has a spatial harmonic variation in the x direction and no variation in the y direction Thus, in a source-free and homogeneous region, the PDE for this field reduces to the ODE d 2 dz 2 E y(x, z) k 2 xe y (x, z) + ω 2 µɛe y (x, z) = 0
em6 8 Solving the equation regionally for the two separate regions z < 0 and z > 0, one obtains possible local solutions e jkzz e jkxx + Γ e jkzz e jk xx z < 0 E y (x, z) = τ e jkz2z e jkxx, z > 0 where k 2 z2 + k 2 x = k 2 2 and the amplitude of the incident electric field is made to be unity Note that all the local solutions should have the same spatial harmonic e jk xx (called the phase-matching condition); otherwise, the matching of boundary conditions at the interface is a problem By imposing boundary conditions at the interface z = 0, that is, enforcing the continuity of both tangential components E y and H x [= (/jωµ) E y / z] there, one has + Γ = τ k z µ ( Γ ) = k z2 µ 2 τ It is easy to solve for the coefficients Γ and τ The results read Γ = k z/µ k z2 /µ 2 k z /µ + k z2 /µ 2 τ = 2k z /µ k z /µ + k z2 /µ 2 The above equations can be obtained by replacing the k i by k zi (index i =, 2) in the corresponding equations for the case of normal incidence In lossless media with k x < k, k 2, then both k z and k z2 are real The corresponding angles of reflection and refraction for the reflected and the transmitted waves can be given as θ r = tan (k x /k z ) = sin (k x /k ) θ t = tan (k x /k z2 ) = sin (k x /k 2 ) It is seen that θ r = θ i, which is known as Snell s law of reflection Furthermore, angles θ i and θ t are related as k x = k sin θ i = k 2 sin θ t For dielectric media, this relation is just Snell s law of refraction n sin θ i = n 2 sin θ t, where the refractive index n i = ɛ i /ɛ 0 If n 2 < n, then θ t > θ i Physically, Snell s laws come from the phase-matching condition For the reflection from a perfect conducting plate, Γ = and the total field is E (x, z) = ŷ(e jk zz e jk xx e jk zz e jk xx ) = jŷ2 sin k z ze jk xx The space-time dependence of such a field is then E (x, z, t) = ŷ2 sin k z z sin(ωt k x x) The field is a standing wave in the z direction and a traveling wave in the x direction In other words, this field is a plane wave travelling in the x direction, but the amplitude varies along z Thus it is a nonuniform plane wave A computer animation is helpful in visualization
em6 9 magnetic field and power The magnetic field can be found from Faraday s law That is, H(r) = [ E(r) = ẑ E y(x, z) ˆx E ] y(x, z) jωµ jωµ x z The transmitted magnetic field is then given as k x H 2z (x, z) = τ e j(k z2z+k x x) ωµ 2 H 2x (x, z) = τ k z2 ωµ 2 e j(k z2z+k x x) Thus the time-average transmitted power flux density flowing in the z direction is given by ẑ S av = Re{E 2 yhx}, which becomes Re{k 2 z2/ωµ 2 } τ 2 at the interface Note that the z-directed transmitted power can vanish in time average, if τ = 0 or if k z2 /µ 2 is pure imaginary The case of τ = 0 corresponds to that medium 2 is a perfect conductor For a lossless medium 2, a pure imaginary k z2 is possible so long as the variation in the x direction is strong enough such that k > k x > k 2 This property of total reflection can be used for field confinement in optical fiber discussed later For a lossless medium and k x < k, it can be shown from the boundary conditions that k z µ ( Γ 2 ) = Re { } kz2 µ 2 τ 2 This relation states that the time-average incident power flux density is equal to the sum of the time-average reflected and transmitted power flux densities at the interface Oblique incidence with parallel polarization Next, we consider the incident electric field with parallel polarization given by E i (x, z) = (ẑk x ˆxk z )e j(k zz+k x x) The associated magnetic field can be given from Faraday s law That is, H i (x, z) = ŷ [ Ex (x, z) E ] z(x, z) = ŷ k2 e j(k zz+k x x) jωµ z x ωµ It is seen that H y is the only component of the magnetic field Thus the field is TM (transverse magnetic) to z From the mechanism of scattering, it can be expected that the dependence of the resultant field in the x direction is also the spatial harmonic e jkxx Thus it is more convenient to deal with the TM case by solving the differential equation for the H field It can be shown that in a source-free homogeneous region, the magnetic field H satisfies the Helmholtz equation 2 H(r) + ω 2 µɛh(r) = 0, which is identical to the one for electric field Further, as the magnetic field H y has a spatial harmonic variation in the x direction and no variation in the y direction, the PDE for this field reduces to the ODE d 2 dz 2 H y(x, z) k 2 xh y (x, z) + ω 2 µɛh y (x, z) = 0
em6 20 By solving the equation regionally for the two separate homogeneous regions z < 0 and z > 0, one obtains possible local solutions (e jkzz e jkxx Γ e jkzz e jkxx ) z < 0 η H y (x, z) = τ e jkz2z e jkxx, z > 0 η 2 where the amplitude of the incident magnetic field is supposed to be /η such that the electric field is unity in amplitude Note that due to the presence of η and η 2, both the reflection coefficient Γ and the transmission coefficient τ are referred to electric field, rather than magnetic field Again, k 2 z2 + k 2 x = k 2 2 The corresponding angles of reflection and refraction are identical to those given in the TE case In other words, Snell s laws are independent of the polarization By imposing boundary conditions at the interface, that is, enforcing the continuity of both tangential components H y and E x [= ( /jωɛ) H y / z], one has ( Γ ) = η η 2 τ k z ɛ ( + Γ ) = k z2 ɛ 2 η η 2 τ The solutions for the coefficients Γ and η η 2 τ can follow those for the TE case, by replacing k zi /µ i with k zi /ɛ i in the formulas for the TE Γ and τ The results read Γ = k z/ɛ k z2 /ɛ 2 k z /ɛ + k z2 /ɛ 2 or η η 2 τ = 2k z /ɛ k z /ɛ + k z2 /ɛ 2 Γ = k z/ɛ k z2 /ɛ 2 k z /ɛ + k z2 /ɛ 2 τ = η 2 η 2k z /ɛ k z /ɛ + k z2 /ɛ 2 Note that for a perfect conducting plate (ɛ 2 = ɛ 0 jσ/ω), one has k z2 /ɛ 2 0, η 2 0 Thus τ 0 but (η /η 2 )τ 2 Therefore the amplitude of the transmitted magnetic field is a finite value equal to 2/η, while the transmitted electric field vanishes electric field and power The electric field can be found from Ampere s law That is, E(r) = jωɛ H(r) = [ ẑ H y(x, z) ˆx H ] y(x, z) jωɛ x z The transmitted electric field is then given as k x E 2z (x, z) = τ e j(k z2x+k xx) k 2 E 2x (x, z) = τ k z2 k 2 e j(k z2z+k x x)
em6 2 Thus the z component of the time-average power flux density of the transmitted wave is given by ẑ S av = 2 Re{E xh y}, which becomes 2 τ 2 Re{k z2 /k 2 η 2} at the interface The transmitted power flux density can be equal to zero in time average, if τ = 0 or if k z2 /k 2 η 2 is pure imaginary For a lossless medium 2, a pure imaginary k z2 /k 2 η 2 is possible so long as the variation in the x direction is strong enough such that k > k x > k 2 This condition of total reflection is identical to that in the TE case In other words, the condition for the total reflection from a lossless medium is independent of the polarization For a lossless medium and k x < k, it can be shown that { } k z ( Γ 2 kz2 ) = Re τ k η k 2 η2 2 This relation states that the time-average incident power flux density is equal to the sum of the time-average reflected and transmitted power flux densities at the interface reflectance of perpendicular and parallel polarization For the trivial case with θ i = 0 (k x = 0 and k z = k ), then Γ = Γ = Γ Meanwhile, for the grazing case with θ i = 90 (k x = k and k z = 0), then Γ = Γ = For dielectric media, it can be shown that the reflectance of perpendicular polarization Γ 2 is always greater than the one of parallel polarization Γ 2 over all the incident angle, except for the extreme cases of θ i = 0, 90 (n < n 2 ) Thus, if the incident field is unpolarized, the electric field of a wave reflected from an interface is predominantly parallel to the interface plane Reflection is an important part in making our vision of everyday s life The reflection from common objects with generally more complex structure tends to be substantially unpolarized On the other hand, the sun glare due to the reflection from the earth is dominantly perpendicular polarization Thus Polaroid sunglasses made with polarizers can filter out the reflection from the ground significantly, but generally has less effect on the reflection from other objects 66 Critical Angle and Brewster Angle In this section we consider two special angles of incidence: one is the critical angle corresponding to total reflection and the other is the Brewster angle corresponding to total transmission The media is supposed to be lossless Critical angle of total internal reflection If the media have constitutive parameters with k > k 2 and if the variation of the field in the x direction is strong enough such that k > k x > k 2, then k z is real and k z2 = jα becomes pure imaginary, where the attenuation constant α = kx 2 k2 2 The spatial variation of the transmitted field over the x-z plane is then E 2 (x, z) = E 0 e αz e jkxx, which is evanescent in the z direction while traveling in the x direction Note that the reflectance Γ 2 becomes unity, since k z2 /µ 2 in Γ and k z2 /ɛ 2 in Γ are pure imaginary That is, the incident power is totally reflected Thus total internal reflection is possible for k x > k 2 or θ i > θ c, where θ i = sin (k x /k ) and the critical angle θ c is then given by ( ) k2 θ c = sin k