7. Electromagnetic Waves 7A. Plane Waves 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 Complex Notation

Linear Media The reaction of the medium will generally have the same frequency 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

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 c 2  0  0 = 1, so What does phase velocity mean? It’s the speed at which the peaks, valleys, and nodes move

The Electric Field and Magnetic Flux Density Electric field will take the form –E 0 is a constant vector From the first equation, we see that E 0  k The magnetic field can be found from second equation Magnetic field is also transverse Given k, the index of refraction n, and the constant transverse vector E 0, we have completely described the wave

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

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

7B. Polarization and Stokes Parameters 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 of time-averaged Poynting vector) is The Polarization Vectors

Linear, Circular, Elliptical Polarization If E 1 and E 2 are proportional with a real proportionality constant, then we say we have linear polarization E 1 only circular Electric field Magnetic Field E 2 only E 1 = E 2 elliptical If we let E 2 =  iE 1 we get circular polarization Most general case is called elliptical polarization

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 E 0 One of these is the overall phase, corresponding to –These correspond to tiny time shifts The remaining parameters are sometimes described in terms of Stokes Parameters Since there are only three independent parameters, these must be somehow related

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

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: I x y-polarization: I y ; plus circular polarization: I + Predict the intensity if you only allowed minus circular polarization I - Recall the intensity is the magnitude of the Poynting vector: For our three measurements, we have We want to know The Stokes parameter s 0 is given by From which we can easily see Therefore

7C. Refraction and Reflection 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 Boundary Conditions and Waves

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 E 0, 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

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

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

Reflection Amplitudes: Parallel Case Case II: electric field parallel 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  '

Brewster’s Angle and Polarization 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 Perpendicular Parallel

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' z is pure 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

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

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 E 1 and E 2 on either side of the equations Solve for E 2 /E 1 Cross multiply We note that assuming n  n", we can conclude But it must be real, so We therefore have

7D. Wave Packets and Group Velocity 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 wave function by superposing these: If you let t = 0, you see that Or reversing the Fourier transform, we have Wave Packets

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

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 = k 0 Assume  (k) is well approximated by Taylor series: Then we have

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

More About Group Velocity Recall: We therefore have Under most circumstances, this is the speed at which signals can travel –Almost always, v g < 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