16 This idea has guided my research: for matter, just as much as for radiation, in particular light, we must introduce at one and the same time the corpuscle concept and wave concept Louis de Broglie
17 Waves move A wave is… a disturbance in a medium f(x)t0t1 > t0t2 > t1t3 > t2vxA wave is…a disturbance in a medium-propagating in space with velocity v-transporting energy-leaving the medium undisturbed
18 1D traveling wave f(x) v x 0 1 2 3 t1 > t0t2 > t1t3 > t2vxTo displace f(x) to the right: x x-a, where a >0.Let a = vt, where v is velocity and t is time-displacement increases with time, and-pulse maintains its shape.So f(x -vt) represents a rightward, or forward, propagating wave.f(x+vt) represents a leftward, or backward, propagating wave.
19 works for anything that moves! The wave equationLet y = f (x'), where x' = x ± vt So andNow, use the chain rule:So Þ and ÞCombine to get the 1D differential wave equation:works for anything that moves!
20 Harmonic waves periodic (smooth patterns that repeat endlessly) generated by undamped oscillators undergoing harmonic motioncharacterized by sine or cosine functions-for example,-complete set (linear combination of sine or cosine functionscan represent any periodic waveform)
21 Snapshots of harmonic waves at a fixed time:at a fixed point:TAApropagation constant: k = 2p/lwave number: k = 1/lfrequency: n = 1/Tangular frequency: w = 2pnn ≠ vv = velocity (m/s)n = frequency (1/s)v = nlNote:
22 The phase of a harmonic wave The phase, j, is everything inside the sine or cosine (the argument).y = A sin[k(x ± vt)]j = k(x ± vt)For constant phase, dj = 0 = k(dx ± vdt)which confirms that v is the wave velocity.
23 Absolute phase y = A sin[k(x ± vt) + j0] A = amplitude j0 = initial phase angle (or absolute phase) at x = 0 and t =0y = A sin[k(x ± vt) + j0]p
24 How fast is the wave traveling? The phase velocity is the wavelength/period: v = l/TSince n = 1/T:In terms of k, k = 2p/l, andthe angular frequency, w = 2p/T, this is:v = lnv = w/k
25 Phase velocity vs. Group velocity Here, phase velocity = group velocity (the medium is non-dispersive).In a dispersive medium, the phase velocity ≠ group velocity.
29 Complex numbers make it less complex! P = (x,y)x: real and y: imaginaryP = x + i y= A cos(j) + i A sin(j)where
30 “one of the most remarkable formulas in mathematics” Euler’s formula“one of the most remarkable formulas in mathematics”Links the trigonometric functions and the complex exponential functioneij = cosj + i sinjso the point, P = A cos(j) + i A sin(j), can also be written:P = A exp(ij) = A eiφwhereA = Amplitudej = Phase
31 Harmonic waves as complex functions Using Euler’s formula, we can describe the wave asy = Aei(kx-wt)so that y = Re(y) = A cos(kx – wt)y = Im(y) = A sin(kx – wt)~~~Why?Math is easier with exponential functions than with trigonometry.
32 Plane wavesA plane wave’s contours of maximum field, called wavefronts, are planes. They extend over all space.The wavefronts of a wave sweep along at the speed of light.equally spacedseparated by one wavelength apartperpendicular to direction or propagationUsually we just draw lines; it’s easier.
33 represents direction of propagation Wave vector krepresents direction of propagationY = A sin(kx – wt)Consider a snapshot in time, say t = 0:Y = A sin(kx)Y = A sin(kr cosq)If we turn the propagation constant (k = 2p/l) into a vector, kr cosq = k · rY = A sin(k · r – wt)wave disturbance defined by rpropagation along the x axis
34 General case Y = Aei(k·r - wt) k · r = xkx = yky + zkz = kr cosq ks arbitrary directionIn complex form:Y = Aei(k·r - wt)Substituting the Laplacian operator:
35 Spherical waves harmonic waves emanating from a point source wavefronts travel at equal rates in all directions
39 Derivation of the wave equation from Maxwell’s equations
40 E and B are perpendicular Perpendicular to the direction of propagation:I. Gauss’ law (in vacuum, no charges):soII. No monopoles:Always!soAlways!
41 E and B are perpendicular Perpendicular to each other:III. Faraday’s law:soAlways!
42 E and B are harmonic Also, at any specified point in time and space, where c is the velocity of the propagating wave,
43 The 1D wave equation for light waves where E is the light electric fieldWe’ll use cosine- and sine-wave solutions:where:E(x,t) = B cos[k(x ± vt)] + C sin[k(x ± vt)]kx ± (kv)tE(x,t) = B cos(kx ± wt) + C sin(kx ± wt)The speed of light in vacuum, usually called “c”, is 3 x 1010 cm/s.
44 EM wave propagation in homogeneous media The speed of an EM wave in free space is given by:0 = permittivity of free space, 0 = magnetic permeability of free spaceTo describe EM wave propagation in other media, two properties of the medium are important, its electric permittivity ε and magnetic permeability μ. These are also complex parameters. = 0(1+ ) + i = complex permittivity= electric conductivity= electric susceptibility (to polarization under the influence of an external field)Note that ε and μ also depend on frequency (ω)
45 Simple case— plane wave with E field along x, moving along z: y
46 General case— along any direction which has the solution:whereand
47 Arbitrary direction= wave vectorxPlane of constant , constant Ezy
48 Electromagnetic waves transmit energy Energy density (J/m3) in an electrostatic field:Energy density (J/m3) in an magnetostatic field:Energy density (J/m3) in an electromagnetic wave is equally divided:
49 Rate of energy transport Rate of energy transport: Power (W)AcDtPower per unit area (W/m2):PoyntingPointing vector
50 Poynting vector oscillates rapidly Take the time average:“Irradiance” (W/m2)
51 Light is a vector fieldA 2D vector field assigns a 2D vector (i.e., an arrow of unit length having a unique direction) to each point in the 2D map.
52 Light is a 3D vector field A light wave has both electric and magnetic 3D vector fields:A 3D vector field assigns a 3D vector (i.e., an arrow having both direction and length) to each point in 3D space.
53 Polarization corresponds to direction of the electric field determines of force exerted by EM wave on charged particles (Lorentz force)linear,circular, eliptical
54 Evolution of electric field vector linear polarizationyx
55 Evolution of electric field vector circular polarizationyx
56 ExercisesYou are encouraged to solve all problems in the textbook (Pedrotti3).The following may be covered in the werkcollege on 14 September 2011:Chapter 4:3, 5, 7, 13, 14,17, 18, 24