Understanding Electromagnetic Wave Behavior: Maxwell's Equations in Complex Notation

1. 7. Electromagnetic Waves 7A. Plane Waves Complex Notation • Consider Maxwell’s Equations with no sources • We are going to search for waves of the form • To make things as general as possible, we write • To save ourselves work, we will simply keep track of • Just remember to take the real part at the end • Space derivatives of expressions like this become ik • Time derivatives of expressions like this become –i • Maxwell equations are now

2. Linear Media • The reaction of the medium will generally have the samefrequency as the fields only if the material is linear • We therefore assume the medium is linear • In general,  and  will depend on frequency • It is possible for there to be a phase shift between D and E or B and H • Similar to a phase shift for a damped, driven harmonic oscillator • This can be show up as complex  and  • We will (for now) assume they are both real • Most common situation is  = 0 and  > 0 • With these assumptions, our equations become

3. Finding the Wave Velocity • Multiply second equation by  • Substitute third equation • Use kB = 0 • We therefore have • We define the phase velocity v and the index of refraction n as • We therefore have • Recall that c200 = 1, so • What does phase velocity mean? • It’s the speed at which the peaks, valleys, and nodes move

4. The Electric Field and Magnetic Flux Density • Electric field will take the form • E0 is a constant vector • From the first equation, we see that E0  k • The magnetic field can befound from second equation • Magnetic field is also transverse • Given k, the index of refraction n, and the constant transverse vector E0, we have completely described the wave

5. Time-Averaged Energy Density • First rewrite B • Energy density is • The last terms oscillate at frequency 2 - too fast to measure • If we do a time average, these terms go away, so

6. Time-Averaged Poynting Vector • The Poynting vector is • If we time average, we get • We note that: • Energy moves in direction of k at phase velocity v

7. 7B. Polarization and Stokes Parameters The Polarization Vectors • The electric field is transverse • We define two polarization vectors • They are chosen to be orthogonal to k and to each other: • If we use real polarization vector 1, typically define the other to be • For example, if k is in z-direction,then we could pick • An arbitrary wave is then described by two complex numbers • That means four real parameters • The magnetic field is then given by • The intensity (magnitude oftime-averaged Poynting vector) is

8. Linear, Circular, Elliptical Polarization Electric field Magnetic Field • If E1 and E2 are proportional with a real proportionalityconstant, then we say we have linear polarization E1 = E2 circular E1 only elliptical E2 only • If we let E2 =  iE1 we get circular polarization • Most general case is called elliptical polarization

9. Polarization and Stokes Parameters • Instead of using real polarization vectors,we could use complex ones • These are also called positive and negative helicity polarizations • Then we would write • Any way you look at it, there are four real numbers describing E0 • One of these is the overall phase, corresponding to • These correspond to tiny time shifts • The remaining parameters are sometimesdescribed in terms of Stokes Parameters • Since there are only three independentparameters, these must be somehow related

10. Measuring Polarization and Stokes Parameters • There are a variety of ways of measuring polarization, but one of the easiest is to put it through a polarizer • Blocks all the light of one polarization, lets much of the other polarization through • Easiest to only allow through one linear polarization, but you can also make them to only allow through one circular polarization

11. Sample Problem 7.1 A pure wave moving in the z-direction is put through a variety of polarizers, and its intensity measured. The types of polarizers and the resulting intensities measured are x-polarization: Ixy-polarization: Iy; plus circular polarization: I+ Predict the intensity if you only allowed minus circular polarization I- • Recall the intensity is themagnitude of the Poynting vector: • For our threemeasurements,we have • We want to know • The Stokes parameter s0 is given by • From which we can easily see • Therefore

12. 7C. Refraction and Reflection Boundary Conditions and Waves • What happens if our linear medium is not uniform? • We will consider only the case of a planar barrier at z = 0 • To simplify, we will assume  = ' = 0 • We therefore have • In each region, we will have waves • We have to match boundary condition at z = 0 • These must match at all t, x, and y • Since  = ' = 0, last two conditions simplify to

13. Setting Up the Waves • We will consider a wave coming in from the +z direction in the xz-plane, reflecting in the xz-plane, and refracting in the xz-plane • Call the wave number for the incoming, refracted,and reflected wave k, k', and k", respectively • Call their constant vector E0, E'0, and E"0 respectively • Then we have • To make them match on the boundary, we need • These must be valid at all x and all t • The only way to make this work is to have • Then we have

14. Snell’s Law and Law of Reflection • Recall we also have • Combining these,we see that • And therefore • Define the angles as , ', and " • Then we have • We also have • It is then easy to see that • We also have

15. Reflection Amplitudes: Perpendicular Case • We still have to find the magnitudes of the reflected and refracted waves • Case I: electric fieldperpendicular to the xz-plane: • One boundary condition: • Another boundary condition: • Rewrite using our expressions for k'z

16. Reflection Amplitudes: Parallel Case • Case II: electric fieldparallel to the xz-plane: • One boundary condition: • Another boundary condition: • First equation times n, plus second times cos: • So we have • Solve for E" • Rewrite using our expressions for cos'

17. Brewster’s Angle and Polarization Perpendicular Parallel • Are there any cases where nothing is reflected? • For perpendicular, only if index of refraction matches • For parallel: • Consider light reflected at Brewster’s Angle, defined by • At this angle, the reflected light is completely polarized • Evan at other angles, reflected light is partially polarized

18. Total Internal Reflection • Suppose we are going from high index to low index • Snell’s Law • If n sin > n', this would yield sin ' > 1 • What do we make of this? • We previously found • This implies k'zispure imaginary • Substituting this in, we find • Wave falls off exponentially in the disallowed region • The evanescent wave • The reflection amplitude in each case is • These numbers are both complex numbers of magnitude one

19. Sample Problem 7.2 (1) Light of frequency  is normally incident from a region of index n to a region of index n"..In order to avoid reflection, a coating of index n' of thickness d is placed between them. Show that this works for appropriate choice of n' and d. • Start by writing down electric field in each region • Let’s pick polarization in the x-direction • Fields going both directions in the middle region • We also need magnetic fields from • Have to match E||, D and B at the boundaries • Eliminate E"and E

20. Sample Problem 7.2 (2) Light of frequency  is normally incident from a region of index n to a region of index n"..In order to avoid reflection, a coating of index n' of thickness d is placed between them. Show that this works for appropriate choice of n' and d. • Gather E1 and E2 on either side of the equations • Solvefor E2/E1 • Cross multiply • We note that assuming n  n", we can conclude • But it must be real, so • We therefore have

21. 7D. Wave Packets and Group Velocity Wave Packets • No wave is truly monochromatic • If it were, then the plane wave would go for all time and all space • To simplify our understanding, let’s work in one dimension • We’ll combine a number of waves of the form • Assume (k) is a known function • We then make a wavefunction by superposing these: • If you let t = 0, you see that • Or reversing the Fourier transform, we have

22. Uncertainty Relation for Arbitrary Waves • At any given time, we can define the average position or average wave number • We can similarly define the uncertainty in the position or the wave number • There is an uncertainty relation between them • Same relationship as in quantum mechanics • Any wave that is finite in extent has some spread in wave number

23. Dispersion and Group Velocity • Each mode has a phase velocity given by • Speed of the peaks and valleys of the modes • If this is bigger than c, can we transmit information faster than light? • Assume we have a nearly monochromatic wave, so f is only non-zero for a small region of k near k = k0 • Assume (k) is wellapproximated by Taylor series: • Then we have

24. Dispersion and Group Velocity (2) • Now substitute • Fundamental theorem of Fourier transforms: • And therefore we have • Define the group velocity as • Then we have

25. More About Group Velocity • Recall: • We therefore have • Under most circumstances, this is the speed at which signals can travel • Almost always, vg< c • In circumstances where n'() is large and negative, this may be violated • Under such circumstances, Taylor series approximation may be invalid • In situations where n'() is large, usually you get lots of absorption as well • This leads to additional complications