Chapter 3 Uniform Plane Waves Dr. Stuart Long

Transcription

1 3-1 Chapter 3 Uniform Plane Waves Dr. Stuart Long

2 3- What is a wave? Mechanism by which a disturbance is propagated from one place to another water, heat, sound, gravity, and EM (radio, light, microwaves, uv,ir) Notice how the media itself is NOT propagated

3 One Dimensional Wave Equation 1 pxt (, ) = x v t 3-3 Given px (,) = f( x) 1 A solution pxt (, ) = [ f( x vt) + f( x+ vt) ] Unique solution depends on physical problem x pxt (, ) = f pxt (, ) = v f t time harmonic case t jω ω + x v px ( ) =

4 3-4 Maxwell s Equations E = H = j i H = i E = j ωε ω μ E H Source Free ρ v = ; J = Time Harmonic case e jωt Time dependent Linear medium B = μ H ; D = ε E

5 3-5 Vector Identity ( E) = ( ie) E jωμ ( H) = ( ie) E jωμ( jωεe) = E E + ω μ ε E = Wave equation for E for E = Exxˆ and E x( z) Ex + ω μ εe x = (1-dim. case) z try soln of form E = xˆ Ee [ k + ω μ ε ] E = k = ω μ ε Dispersion Relation jkz { jωt E } xˆ Ezt (, ) = Re e = Ecos( ωt kz)

6 3-6 E x ω T periodic in time period T E cos( ) x = E ωt ω t x component of the electric field at z= as a function of time -1 pi/ pi π π 3π pi π spatially repeating every wavelength λ angular frequency ω=πf

7 3-7 E cos( ) x = E ωt kz Electric field as a function of z at different times E x 1 ω t = E x π ω t = 1 1 E x ωt ω t = π z. -. λ = π k z. -. z pi/ pi 3pi/ pi π π 3π π -1 pi/ pi 3pi/ pi π π 3π π -1 pi/ pi 3pi/ pi π π 3π π v λ Δz ω = = = Δ t π k ω

8 1 E cos( ) x = E ωt kz Electric field as a function of z at different times 3-8 z 1 π π 3 4 π π 5π 3π λ Δz ω v = = = Δ t π k ω π 4 π

9 3-9 Quick Review - The wave spacially repeats at point z = λ where kλ = π. π - The quantity λ, where λ = is called the wavelenght. k - The number of wavelengths contained in a spatial distribution of π π is given by k = and it is called the wavenumber. λ - The velocity of the peak of the wave (position of constant phase) requires that ωt- kz = constant so the velocity of propagation is z ω given by = v = [m/sec] t k - The velocity in free space is given by ω ω 1 v = = = 3 1 k ω μ ε μ ε 8 [m/sec]

10 3-1 So far we have come across some useful expressions such as: Period T 1 = [sec] Phase Velocity v = f ω k [ m/sec] Angular Frequency ω = π f [rad] Velocity in c 3 1 free space 8 [m/sec] 1 Frequency f = [Hz] Wavenumber k = ω με [1 m] T π Wa velength λ = [m] Not e: f [ GHz] iλ[ cm] 3 k Also, remember that the orientation of the E field of a uniform plane electromagnetic wave is perpendicular to the H field of that wave and that both are perpendicular to the direction from which the wave propagates

11 3-11 Uniform Plane Waves Waves with constant phase fronts (plane waves) and whose amplitude (E ) is uniform Recall E = - jωμh Where the E field of a uniform plane wave is given by jkz E = xˆ Ee The magnetic field is then Ee ˆ H = y η jkz E field is in xˆ direction H field is in yˆ direc tion Wave propagating in + zˆ direction

12 3-1 Or in the time domain { jωt E } ˆ ˆ E(,) zt = Re e x = xe cos( ωt kz) Similarly jωt Ee E ˆ ˆ H(,) zt = Re y = y cos( ωt kz) η η Where the η η is the intrinsic impedance of free space μ = = 1π 377 ε Permittivity ε [Ohms] π 9 F m Permeability μ = 7 4π 1 H m

13 The Electromagnetic Spectrum 3-13 Source Freq.[Hz] Freq. (common units) Wavelength [m] Wavelength (common units) U.S A-C Power 6 6 Hz 5x1 6 5 Km ELF Subm. Comm. 5 5 Hz 6x1 5 6 Km AM radio Hz 3 3 m CB radio.7x1 7 7 MHz m Cordless phone 4.9x MHz m TV ch. 6x1 7 6 MHz 5 5 m FM radio MHz 3 3 m TV ch x MHz m UHF Aircraft Comm. 5x1 8 5 MHz.6 6 cm TV ch x1 8 6 MHz cm Cellular phone 8.7x MHz cm μ-wave oven.45x GHz.1 1 cm "C" band 6x1 9 6 GHz.5 5 cm Police radar 1.5x GHz cm mm wave GHz.3 3 mm He-Ne Laser 4.7x x Å Light x1-7 3 Å X-ray x1-7 3 Å

14 The Electromagnetic Spectrum

15 The Electromagnetic Spectrum 3-15

16 Polarization 3-16 The polarization of a wave is described by the locus of the tip of the E vector as time progresses at a fixed point in space. If locus is a straight line the wave is said to be Linearly Polarized If locus is a circle the wave is said to be Circularly Polarized If locus is an ellipse the wave is said to be Elliptically Polarized

17 Polarization 3-17 If locus is a straight line the wave is said to be Linearly Polarized

18 Polarization 3-18 If locus is a circle the wave is said to be Circularly Polarized

19 Polarization 3-19 If locus is an ellipse the wave is said to be Elliptically Polarized

20 Polarization 3- Consider a plane wave propagating in the positive z direction. E = E cos( ωt kz) The associated electric field can be expressed in the form of E = xˆe + yˆe x where the two components are, in general terms, y E = acos( ωt kz + φ ) x E = bcos( ωt kz + φ ) y a b The polarization of this plane wave is determined by the quantity Where E E y x = A φ A Ey b and = = φ= φb φa E a x

21 Polarization Classification If E field is traveling in the positive yˆ, xˆ or zˆ direction A φ can be found respectively by Ex φx Ez φ E z y φy or or E φ E φ E φ z z y y x x 3-1 A = ; φ = or ± π Linear Polarization (LP) A Linear Polarization (LP) A= 1 ; φ = π Left-Hand Circular Polarization (LHCP) A = 1 ; φ = π Right-Hand Circular Polarization (RHCP) < φ < π Left-Hand Elliptical Polarization (LHEP) π < φ < Right-Hand Elliptical Polarization (RHEP)

22 Polarization Classification If E field is traveling in the negative yˆ, xˆ or zˆ direction A φ can be found respectively by Ez φ Ey φ x z Ex φy or or E φ E φ E φ x z z y y x 3- A = ; φ = or ± π Linear Polarization (LP) A Linear Polarization (LP) A= 1 ; φ = π Left-Hand Circular Polarization (LHCP) A = 1 ; φ = π Right-Hand Circular Polarization (RHCP) < φ < π Left-Hand Elliptical Polarization (LHEP) π < φ < Right-Hand Elliptical Polarization (RHEP)

23 Polarization 3-3 Consider a plane wave propagating in the positive z direction. E = E cos( ωt kz) The associated electric field can be expressed in the form of E = xˆe + yˆe x where the two components are, in general terms, y E = acos( ωt kz + φ ) x E = bcos( ωt kz + φ ) y a b The complex representation is given can be expressed by -jkz ( - φ ) -jkz ( - φ ) a b E = xˆae + yˆbe

24 Polarization 3-4 Look at z = and φ = φ ; φ = E = acosωt x E = bcos( ωt + φ ) y Ex EE x y Ey cosφ sin a + = ab b b b a φ Recall that the general quadratic equation is given by Ax + Bxy + cy + Dx + Ey + F = where 1 cosφ 1 A= ; B = ; C = ; D = ; E = ; F = sin ab a b φ

25 Polarization 3-5 Ax + Bxy + cy + Dx + Ey + F = where 1 cosφ 1 A= ; B = ; C = ; D = ; E = ; F = sin ab a b φ If B 4AC < this becomes equ of an ellipse φ ( φ 1) cos 4 = cos ab a b a b rt o ated by an angle θ cot θ = A- C B 1 1 ab cot θ = a b cosφ

26 Example 3-6 Let φ = ( or φ = π) Ex EE x y Ey cosφ sin a + = ab b Ex EE x y Ey E E x y a + = = ab b a b φ E E x y b = Ey = a b a E x Linear polarization (line of slope ba)

27 Example 3-7 Let a = b ; φ = π Ex EE x y Ey cosφ sin a + = ab b E E x y + = 1 a a φ Circular polarization (circle of radius "") a

28 Example 3-8 Let b = a ; φ = π Ex EE x y Ey cosφ sin a + = ab b Ex Ey 1 a + = a φ Elliptical polarization (equ of an ellipse with and minor radius = a) major radius = a

29 Polarization Example 3-9 Find the polarization of the following field: x y - jkz (a) E = ( jxˆ+ yˆ) e E = (1 kz + 9 E = (1 kz) ) E y (1 kz) 1 A φ = = = kz E (1 kz + 9 ) 1 x ( kz + 9 ) A φ = 1 9 RHCP

30 Polarization Example 3-3 Find the polarization of the following field: x z z - jky (b) E = (( + jx ) ˆ + (3 j) zˆ ) e E = ( 5 ky ) E = ( 1 ky ) E x ( 5 ky ) 5 A φ = = = ky ( ky ) E ( 1 ky ) 1 1 A φ = 45 LHEP

31 Polarization Example 3-31 Find the polarization of the following field: (c) E = ((1 + j) yˆ + (1 j) zˆ ) e E = ( kx + 45) y E = ( kx 45) z y - jkx E z ( kx 45 ) A φ = = = kx 45 ( kx 45 ) E ( kx + 45 ) A φ = 1 9 RHCP

32 3-3 Plane Waves in Dissipative Media For isotropic conductors Ohm's Law states that Jc =σe where Jc conduction current ; σ conductivity J source current Consequently Ampere's Law becomes H= jωd+ Jc + J σ H= jω ε j + ω E J m Where ε = ε σ j ω Complex Permittivity

33 3-33 Plane Waves in Dissipative Media ( ω με) ( J=) In a source free conducting medium Ampere's Law states H = jωεe As derived earlier, the wave equation is given by + E = As we have seen, ε is complex for a conducting medium. Note: The wave number and the intrinsic impedance are now complex numbers. k = ω με ; η = μ ε

34 3-34 Plane Waves in Dissipative Media The wave number and the intrinsic impedance can also be written as k = k R - jk I jφ η = η e The electromagnetic fields of a uniform plane wave in a dissipative medium are given by E = xe ˆ e jkz H = yˆ Ee η jkz

35 3-35 Plane Waves in Dissipative Media The electromagnetic fields can also be written as E = H = xe ˆ e Ee yˆ k I z jk R z e k I z e η jk R z jφ e Or in the time domain k z ( ω ) I E ( z, t) = E e cos t k z x R H y ( z,) t k z ( ω φ) I Ee cos t krz = η

36 3-36 Plane Waves in Dissipative Media From the electromagnetic fields we can observe that 1) The wave travels in the + zˆ direction with a velocity ω v = k R where k is called the wavenumber. R ) The amplitude is attenuated exponentially at the rate k I nepers per meter, where constant. is the attenuation 3) The magnetic field H is out of phase by φ. y k I

37 3-37 Attenuation One neper attenuation if Amplitude = Amplitude end start ln 1 The attenuation in nepers after lenght d is given by Attenuation[nepers] -kz I Ee = ln -ki ( z+ d) = Ee kd I The relationship between nepers and db is given by 1[neper] = [db]

38 3-38 Example The electric field is decreased by a factor of.77. Find the attenuation in nepers and db E = [ ] = E I F ln ln [nepers] db.3467[nepers] i8.686 = 3.1 [db] nepers or E = ( ) = E I F log log [db]

39 3-39 Note on db Scale If dealing with electric field use E log E If dealing with power use P 1log P F I F I This is because P ~ E when E =.77E F I then P =.77 P P =.5P F I F I [ ] [ ] log.77 = 1log.5 = 3.1 [db]

40 3-4 General Medium 1 The penetration depth ( d p ) such that E ( ) = ( z ) is given by z= d p E = e k d I p = 1 Where for a conducting media k ω με 1 j σ = ω ωε = σ ωε με 1 j = kr - jki Keep in mind that If a>>1 then a 1 + ja 1 ( + j ) or If a< < 1 then 1+ ja 1+ j a

41 3-41 Slightly Conducting Media σ ( Good Dielectric) << 1 ωε k ω με 1 j σ ωε k = k k ; k = R - j I I σ μ ε σ ε μ dp = ; kr = ω με

42 3-4 Highly Conducting Media k ω μσ ( 1 j) ( Good Conductor) σ ωε >> 1 k = ; = kr - jki ki ωμσ d p ωμσ = δ ; R = k δ Also called the skin depth ωμσ

43 k I or k R (1/m) σ ωε Behavior of k I and k R as a Function of Loss Tangent Exact kr Exact ki Good Conductor appx. kr=ki Good Dielectric appx. kr 3 Good Dielectric appx. ki 1 Seawater μ = μ ; ε = 81ε ; σ = 4 mho m

44 Conductors 3-44

45 Good Conductor Ordinary metal with very high values of σ approximate perfect conductors J = OHM'S LAW σe σ [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Superconductive lead σ =.7 1 mho/m 7 Silver Copper Gold Aluminum Brass Solder Stainless steel σ = 6. 1 mho/m 7 σ = mho/m 7 σ = mho/m 7 σ = mho/m 7 σ = mho/m 7 σ =.7 1 mho/m 7 σ =.1 1 mho/m 4 Graphite σ = 7 1 mho/m 3 Silicon Sea water σ = 1. 1 mho/m σ = 4 mho/m 4 Distilled water σ = 1 mho/m 5 Sandy soil σ = 1 mho/m 6 Granite σ = 1 mho/m 9 Bakelite σ = 1 mho/m 13 Diamond σ = 1 mho/m 16 Polystyrene σ = 1 mho/m 17 Quartz σ = 1 mho/m A perfect conductor is an idealized material in which no electric field can exits 3-45

46 Lossy Dielectrics Can dissipate energy in oscillations of bound charge in a dielectric. D= ε E ε = ε jε Can define an effective conductivity σe = ωε Same effect as σ but from a different source ε σ e Table gives [tan δ ] = = ε ωε ε = εr tanδ ε Ice 4..1 Dry soil.8.7 Distilled water 8.4 Nylon 4.1 Teflon.3 Glass 4 7. Dry wood Styrofoam Steak 4.3 Phase Lag caused by bound charge not keeping up with E Field 3-46

47 3-47 Skin Effect The skin effect is the tendency of an alternating electric current to distribute itself within a conductor so that the current density near the surface of the conductor is greater than that at its core. That is, the electric current tends to flow at the "skin" of the conductor. For EM wves a kz I jkz R Ee e Since σ E = xˆ J = E kz ˆ I jkz R J = xσ Ee e Current is exponentially damped into material

48 3-48 Plane Waves in a Plasma Plasma is a collection of (+) and (-) charged particles for which <ρ v >= For low density plasma (few collisions) ω p μ = μ ; ε = ε 1 ω p Plasma freq. ω "Cold Plasma" Note: ε is a function of ω Dispersive medium For ω > ω p 1 ω p k = ω μ ε 1 ω v = k ω

49 3-49 Plane Waves in a Plasma For ω < ω the wavenumber becomes imaginary p 1 ω p 1 k = jα = jω μ ε ω Then and jk z α z ˆ E(z) = xe ˆ e = xe e α α z H(z) = yˆ Ee jωμ Since Eand Hare both imaginary Evanescent Waves Attenuation occurs but no real power is dissipated s = 1 R e E H = *

50 3-5 Phase vs. Group Velocity The phase velocity is the speed of the individual wave crests, whereas the group velocity is the speed of the wave packet as a whole (the envelope). In this case, the phase velocity is greater than the group velocity.

51 3-51 Phase vs. Group Velocity Consider a plane wave propagating in the + x ˆ ( ω ) E(x, t) = E cos t kx direction with two frequencies ω = ω Δ ω and ω = ω +Δω 1 and with wavenumbers k = k Δ k and k = k +Δk 1 For For ( Δ Δ ) ω E cos ( ω ω) t ( k k) x 1 ( +Δ +Δ ) ω E cos ( ω ω) t ( k k) x Sum to get total field { cos (( ω ω) ( Δ ) ) + cos (( ω +Δω) ( +Δ ) )} E(x, t) = E Δ t k k x t k k x total

52 Phase vs. Group Velocity 3-5 Using trig identities ( ω ) ( ω Δ ) E(x, t) = E cos t k x cos Δ t kx total The cosine factors give a slow variation superimposed over a more rapid one Constant phase on rapid (1 cos) term st δ x ω ωt kx= constant = = v δt k p Phase Velocity Constant argument on slower variation nd δx Δω δω Δωt Δ kx= constant = = = v δt Δk δ k g Group Velocity

53 Phase vs. Group Velocity