Chapter 4 Wave Equations

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1 Chapter 4 Wave Equations Lecture Notes for Modern Optics based on Pedrotti & Pedrotti & Pedrotti Instructor: Nayer Eradat Spring /11/2009 Wave Equations 1

2 Wave Equation Chapter Goal: developing the mathematical expressions for wave motion. Mostgeneral case Harmonic waves Electromagnetic (EM) waves Light waves Energy delivered by such EM waves Wave: A self sustaining energy carrying disturbance of a medium through which it propagates. Longitudinal wave: the medium is displaced in the direction of motion of the wave. Transverse wave: the medium is displaced in a direction perpendicular to that of the motion of the wave. When a wave propagates, the disturbance advances, not the medium. That is why waves can propagate faster than the medium carrying them (Leonardo da Vinci). 3/11/2009 Wave Equations 2

3 Equation for mathematical form of a pulse Goal: developing a mathematical expression for the most general form of a one-dimensional (1D) traveling wave. Consider: (, ) A stationary coordinate system. '( ', ') A moving coordinate system. Relative speed of motion of O' ( x', y' ) to O ( x, y ) in the + x direction = ( ) ( ) O x y O x y V y' f x' A one dimensionnal pulse of arbitrary time-independent shape fixed to O' x', y' y or y' is the transverse displacement from equilibrium (the disturbance). As O ' moves the pulse maintains its shape. Relationship between the coordinates of an arbitrary ypoint on the pulse x' = x Vt P( x, y) and P( x', y' ) y' = y = f ( x' ) = f ( x Vt) For a pulse moving i n x direction y' = f ( x + Vt ) y' = f ( x Vt) for + x and + for x Equation for mathematical form of a pulse f can be any function for example ( [ ]) y = Asin k x Vt traveling periodic in + x ( ) 2 g y = A x+ Vt traveling in x y k = e ( x Vt ) traveling in + x X 3/11/2009 Wave Equations 3

4 One dimensional wave equation The wave equation is a partial differential equation that any arbitrary wave form will satisfy it. 2 2 y 1 y Verify that any wave of the form y = f ( x Vt) satisfies = the 1D wave equation x V t To show that if a given function represents a traveling wave, it is sufficient to represent it in the ( ) form of y = f x Vt or prove that it satisfies the wave equation. X 3/11/2009 Wave Equations 4

5 Harmonic waves Harmonic waves are smooth patterns that repeat endlesly. They involve the sine and cosine functions: y = Asin ( k[ x Vt] ) or y = Acos( k[ x Vt] ) A and k are constants that specify a wave without altering its harmonic nature. The sin and cos functions form a complete set of functions so a linear combination of them is also a wave. Using this property we can construct complicated waveforms from linear superposition of the simple sin and cos functions. ( ) sin x= cos x π / 2 so sin is equivalent to cosin, shifted by π/2. Amplitude A: the maximum value of the disturbance Wavelength λ: the repetitive spatial unit of the wave ( ) ( ) Asin k x+ λ + Vt = Asin k x+ Vt 2π kλ 2 π or k sin x = sin ( x + = = 2π ) λ Propagation constant: k = 2 π / λ related to spatial period λ k is also called spatial frequency or number of waves in 2 π Period T: the repetitive temporal unit of the wave ( ) ( ) Asin k x+ V t+ T = Asin k x+ Vt kvt = 2 π or VT / λ = 1 Frequency ν : number of oscillations per unit timeν = 1/ T V = νλ Angular frequency: ω = 2 πν related to temporalperiodperiod ν Wave number: κ = 1/ λ number of waves per unit length 3/11/2009 Wave Equations 5

6 Phase, phase velocity and harmonic waves, initial phase sin y A k( x Vt) = cos sin x Vt sin x t y = 0 for x= 0 and t = 0 y = A 2 A 2 Initial phase 0 cos π π λ λ = cos λ T = y = A for x= 0 and t = 0 sin 2π sin y = A x 2πνt = A kx ωt cos λ cos Phase Phase: argument of the sin or cosine function in a harmonic wave that is a function of space and time. x t φ= k( x Vt) = 2π = kx ωt λ T For any set of x and t that φ stays constant the displacement y = Asin is also constant. Motion of a fixed point on a wavefront is described by motion of a constant phase point. For a point with φ = constant we have x dφ = 0 = kdx Vdt Vphase = General formula t φ Phase velocity of a wave is speed of the motion of a constant-phase point on a disturbance. Value of the pase velocity: for a harmonic wave at any given time φ = kx ωt + ε φ x x ω For constant phase: = k ω = 0 Vphase = Vpha se =± only for harmonic waves t t t k φ 1 y0 Initial phase = φ0 y = Asin k( x Vt) + φ0 ; at t = 0 and x= 0 y = Asinφ0 = y0 φ0 = sin A 3/11/2009 Wave Equations 6 φ ( φ)

7 Exercise For the wavefunction ψ ( xt, ) = 10 sin π(3 10 x t+ 0.5) in SI units find the following gquantities: a) speed, b) wavelength, c) frequency, d) period, e) amplitude, f) phase, g) initial phase, h) phase at t = 10 s, i) phase velocity, j) compare the speed and the phase velocity. k) what is the direction of the motion of the wave. Does phse velocity indicate the direction of motion properly? k) what is ma gnitude of the wave at x = 0 when t = 0, τ /4, τ /2,3 τ /4, τ? l) plot the profile of the wave at t = 0 with initial pahse equal to 0, 0.5 π, π,2 π. 3/11/2009 Wave Equations 7

8 Complex numbers More often wavefunctions are expressed in complex exponentials since complex exponentials can simplify the trigonometric expressions. ( ) ( ) z = z + jy or z = x + iy; i or j = 1 and both x real part and y imaginary part are real numbers. Absolute value or modulus: y Argument: tanφ = x iφ We can also write z = Ae and 2 2 = + ; z x y * iφ z = Ae the complex conjugate of z iφ Euler's formula: e = cosφ + isin φ z = A(cosφ+ isin φ) where x= Acosφ and y = A = z = x + y y φ = tan is the phase x Asinφ is the magnitude or amplitude * Also z = x iy is the complex conjugate of z = x + iy. iφ i ( )( φ) zz = z e z e = z * 2 3/11/2009 Wave Equations 8

9 Eulerformula and harmonics waves as a complex function e iφ = cosφ+ isinφ Using this formula we can express a harmonic wave as a real or imaginary part of a complex function. y = Ae ( -ωt) ikx y = Re y = Acos kx- t ( ω ) y = Im y = Asin kx- t ( ω ) or Any equatrion that involves Re Im will hold odfor y = y and y = y linear terms in y and its derivatives 3/11/2009 Wave Equations 9

10 Plane waves Wavefront: a surface over which the phase of a wave is constant. Plane waves have planar wavefronts that are surfaces perpendicular p to the direction of propagation k. We write equation of a plane that is passing through p0( x0, y0, z0) and is perpendicular to k (direction of propagation). Consider an arbitrary point p on the planar wavefront, we must have: z (r - r 0) ik = 0 kir = kir0 =constnt is equation of such a plane perpendicular to k k ψ ( r) = Asin( kir) is a function defined on a family of planes all perpendicular to k. p(x,y,z) Phase and ψ ( r) are constant on one plane r (wavefront) but they vary sinusoidally p 0 (x 0,y 0,z 0) r from plane to plane. 0 The progressive wave equation is then: ψ (,) r t = Asin( kir ωt) or x ψ ω ω (. ) (,) i ωt t = Ae kr r = Acos( kir t) + iasin( kir t) i ( kr cos θ ωt ) i ( ks ωt ) ψ ( r, t) = Ae = Ae where s is the component of r along the direction of propagation or k. Amplitude of a plane wave stays constant as it propagates. 3/11/2009 Wave Equations 10 y

11 Harmonic waves in 3 dimension A general harmonic wave in three-dimension in complex form: ψ i( kr i ωt) ( r, t) = Ae ( r t) The 3-D wave equation that satisfies ψ, ψ ψ ψ 1 ψ + + = x y z V t ψ + + = ψ 2 2 x y z V t x y z ψ = 1 V 2 ψ 3-D wave equation t 2 2 3/11/2009 Wave Equations 11

12 Spherical waves By solving the differential spherical wave equation we can arrive at harmonic spherical wave function: ψ A (,) rt = sin kr ( Vt) r A ik ( r Vt ) A i( kr ωt ) ψ (,) rt = e = e r r that represents cluster of concentric shperes at any instance. On each sphere r is constant so ψ (,) r t is constant. Here A, the source strength is a constant. A is the amplitude and it varies inversly with the distance from the source to r W A conserve e the e nergy. egy The irradiance ada in is the iverse square law of 2 esesqua m r propagation for the spherical waves. Same as plane waves, for spherical waves 2π 2π k =, ω = λ T 3/11/2009 Wave Equations 12 2

13 Cylindrical Waves ψ = A p e ( ρ ω t ) ik ρ is the perpendicular distance from the line of symmetry of the cylindrical wave. If the ethe line eof sylmmetry yis sthe z axis sthen 2 2 ρ = x + y These waves are not exact solutions of the wave function. So they do not exactly represent physical waves. But they are useful to express waves coming out of a slit illuminated by a plane wave. 3/11/2009 Wave Equations 13

14 Hermite Gaussian waves A good representation of the laser beams and an approximate solution to the wave equation. Good for the cases that the irradiance is strongly confined to the direction of propagation. 3/11/2009 Wave Equations 14

15 Electromagnetic waves - The harmonic wave equation can represent any type of disturbance with sinusoidal behavior. - Physical significance of the disturbance is different for different systems (pressure, displacement...) - Maxwell showed that light is composed of electric and magnetic fields oscillating perpendicular to each other and propagating in the direction k, perpendicular to the plane of oscillations. - For light waves the disturbance is the magnitude of varying electric or magnetic fields that are described with the following harmonic wavefunctions: i( kr. ωt) E= E0e = E0sin( kr. ωt), i( kr. ωt) B= B0e = B0sin( k. r ωt) Electric fields are generated by electric charges and time-verying magnetic fields. Magnetic fields are generated by electric currents (charge in motion) and time- varying electric fields. This intedependence of the E and B is a key point in description of the light. Lorentz force: an electric charge q, moving with velocity of v in an area that contains both E and B fileds, feels forces due to existance of both fields. F = q ( E+ v B) 3/11/2009 Wave Equations 15

16 Before Maxwell Faraday's induction law: a time-varying magnetic field will have an B electric field associated with it. E dl = - ds C t 1 Gauss's law-electric: E ds= ρdv, when there are no A ε 0 V sources or sinks of the electric field within the region encompassed by a closed surface, the net flux through the surface equals to zero. Gauss's law-magnetic: Φ = d s = 0, there is no magnetic monopole M B A Ampere's circuital law: dl μ ε E B = + d, a time-varying C J s t E-field or charges in motion (electric current) will generate a B-field A A 16

17 Maxwell equations; integral form Behavior of electric and magnetic fields in a medium with electric permitivity ε, and magnetic permeability μ, in presence of free charges ρ, and current density J, is explained by four integral equations known as Maxwell equations. E dl = - d B s Farady C t A B E dl = + ε d C μ J s Amper t A A A B ds= 0 Gauss magnetic εe ds= ρdv Gauss electric V 17

18 Maxwell equations; differential form E dl = - B d B C s E = t t A B E E dl = d = C + ε μ + ε μ J s B J t t A A A B ds= 0 B= 0 ρ εe ds= ρdv E= ε V We used the following theorems: Stoke's Gauss's s 18 C E dl = E ds A A B d s= B dv V

19 Constitutive relations H = Magnetic field; B= Magnetic induction (effect of the H in the medium B=μH); E= Electric field; D= Electric displacement (effect of the E in the medium D= εe); J = Current density; Constituitive relati ons are: D= DEB (, ); H= HEB (, ); J= JEB (, ) These relations may be nonliner or depend on the past (hysteresis). Linear response: the applied fields are small so they induce electric and magnetic polarizations proportional to the magnitude of the applied field (ferroelectric and ferromagnetic material are exceptions are nonlinear material) D ε α β ε μ α β are the coordinates x,y,z. 1 α = αβeβ; Hα = αβbαβ, β β 1 μ αβ is electric permitivity or dielectric tensor; is inverse magnetic permitivity tensor For material isotropic in space both ε and μ are diagonal matrices and all elements are equal. For isotropic material D= εe and H= μ -1 B At high enough fields every material is nonlinear (nonlinear optics) (1) (2) Dα εαβeβ εαβγeβeγ β β, γ =

20 Electric field in medium 1) P polarization vector: P= ε χe electric dipole moment per unit volume. 0 2) D displacement field: D= ε E+ P electric field within the material 0 D P 3) E internal electric field: E= - and we have D= ε ( E) E ε ε 0 D and E lines begin and end on free charges or polarization cahrges. In absence of free charge field lines close on temselves ( E= D=0) 0). For homogeneous, linear, isotropic dielectrics P and E are in the 0 same direction so D= εe, where ε = ε (1 + χ), and Κ = ε = ε / ε = 1+ χ is the relative dielectric constant or function. 0 e r 0 ( I = V / R) 4) J current density: acording to the Ohm's law electric field intensity determ ines the flow of the cahrge in a conductor J = σ E, true for conductors at constant temperature. 20

21 Constitutive relations: magnetic fields in a medium 1) M magnetic polarization vector: M = Κ m H. 2) H magnetic field intensity: H B M, 1 = μ 0 For homogeneous, linear (nonferromagnetic), isotropic medium B and H are parallel a and proportional, o a, 1 μ H = μ B and μ = μ0( 1 +Κ m) and μr = = (1 +Κm) μ0 For most optical material μ r = 1or Κ m = 0 or no magnetization occures under magnetic field. 21

22 Maxwell equations using the constitutive relations B B E= E= t t ρ E= D= ρ ε E D B = μ J + ε H = J + t t B= 0 B= 0 22

23 The wave equation for the E and M components of the EM waves Maxwell equations in nonconducting medium ( ρ = 0, J = 0) vacuum B ε = ε0, μ = μ0, (1) E=, ( 2) B= μ0ε0 t (3) B= 0, (4) E= 0 Take curl of (1) and use (2) to eliminate B, E= ( B ), t 2 2 E 2 E E= με 0 0, ( E) E= με 2 0 0, using (4) we get 2 t t E 1 E differential wave equation: with E t E= με t = V 2, 2 t V = 1 μ ε 0 0 We can also show: B t 2 2 B =με 0 0 = V 2 t 23 B,

24 The index of refraction Velocity of light based on Maxwell's theoretical treatment in vacuum is 1 1 c = and in medium is V =. εμ εμ 0 0 c εμ Absolute index of refraction defined as: n= = = Keμr= εμ r r. V εμ The E field polarizes the medium rusulting in change of the displacement field. D= ε E+ P= εe. 0 The result is a change in ε and consequently a change in n and V, speed of light in the medium. εω ( ) and n( ω) are functions of the frequency of the EM waves. ε Usually μ μ so n K = ε ; K is the complex dielectric function. 0 = = e r ε 0 e

25 Physical meaning of the index of refraction I Consider a general e case where e n is a complex number n= n' + in". Consider the E component of a plane wave, E = E 0 e i( kr ωt) k, the propagation vector in medium is complex. k 0 is propagation p vector in vacuum. V ω ω With P =, the phase velocity, and c= = nv k k ω ωn k = n k 0 = = for one-dimensional case V c nω ( n' + in") ω n" ωx n' x i( x ωt) i( x ωt) iω( t) i( kx ωt) c c c c 0y 0y 0y 0y E = E e = E e = E e = E e e 25 0

26 Physical meaning of the index of refraction II E = E e e e n" ωx c 0 y nx ' i ( t) c e ω n" ωx n' x iω( t ) c c is a real term and decays exponentially as wave propagates. p has a harmonic wave form and propagates without loss This suggests that n", the imaginary part of n n', is associated with absorption. the real part of n is associated with propagation. In absence of n", the case in many dielectrics in the visible part of the EM spectrum, n= n' = c/ V is simply the ratio of the speeds in vacuum and in the medium. n= n' = c/ V is also called absolute index of refraction. 26

27 Frequency dependence of the index of refraction The wavelength dependence of n is stronger at short wavelengths or high frequencies. For most dielectrics the imaginary part of the n is negligible in the visible band 27

28 The electromagnetic spectrum 28

29 Energy and momentum I Physical manifestation of electromagnetic waves is their enegy and momentum. Energy density u, is radiant energy per unit volume In order to calculate the energy content of the EM waves we start from ( E H ) = H ( E ) E ( H ) B D Using E= and H = J+ and J = σ E t t B D B D ( E H) = H ( ) E ( ) EJ. = H ( ) E ( ) σ EE. t t t t B ( μh ) 1 ( μhh. ) 1 2 H = H = = μ H t t 2 t t 2 D ( εe) 1 ( εee. ) 1 2 E = E = = ε E t t 2 t t ( E H) = εe + μh σe t

30 Energy and momentum II ( E H ) = ε E + μ H σ E t 2 2 Integrating both sides over a volume V, and applying the divergence theorem ( E H ) dv = E H ds V E H d s = ε E t 2 S + μh dv σe dv 2 V V power rate leaving the volume from its surface time-rate of change of energy stored in electric and magnetic fields S ohmic power dissipated in the volume as a result of conduction current density σ E in presence of E Thus the E H is a vector representing power flow per unit volume. 30

31 Poynting s Theorem 1 2 S = P = E H = E ( W / m ) B is known as Poynting vector. μ S = power density crossing a surface whose normal is parallel to S. Poynting's theorem: surface integral of P or S over a closed surface S, equals the power leaving the enclosed volume V by S. δ S. d s = ( u ) e + um dv + P σ dv where δt u u e e P S V V * 1 * = ε E = ε EE = DD = Electric energy density 2 2 2ε * 1 * = μh = μh H = B B = Magnetic energy density μ 2 * σ = σe = σ = EE Ohmic power density 31

32 Exercise RV1-16) 16) a) Prove that for a harmonic plane electromagnetic wave with the following E field component traveling through an insulating isotropic medium, E = V B. In vacuum E = c B. E= E0 cos( k r ωt) b) Calculate the magnitude of the poynting vector c) Prove that the energy content of this plane wave, in unit volume, due to electric and magnetic fields is equal. d) Calculate the total energy per unit volume. 32

33 Irradiance Irradiance I, or the amount of light: average energy over unit area per unit time. I is independent of detector area A, and duration of measurement. Time averaged value of the Poynting vector S, or irradiance is: I T t+ 2 1 S = ( ) dt T T E H T t 2 Note that the averaging time T τ, has to be much greater than the period. Irradince is proportional to the square of the amplitude of the E field. For linear, homogeneous, isotropic dielectric c I = εv E In vacuum: I = B = ε 0c E μ T T T 0 33

34 3/11/2009 Wave Equations 34

35 3/11/2009 Wave Equations 35

36 3/11/2009 Wave Equations 36

37 3/11/2009 Wave Equations 37

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