PHYS 408 Applied Optics (Lecture 3) JAN-APRIL 2016 EDITION JEFF YOUNG AMPEL RM 113

Quiz #1 1)The electric field that enters the Maxwell equations in the form that involves a material susceptibility , or relative dielectric constant , represents the true electric field at every point inside the material (T or F) 2)The Poynting vector of the electromagnetic field is a unit vector that indicates the direction of wave propagation (T or F) 3)A spherical wave can be approximated as a plane wave at distances far from the source of the spherical wave (T or F) 4)Professor Young’s research involves quantum optics (T or F)

Quick review of key points from last lecture Many versions of “Maxwell’s equations” One of the challenges is to understand them well enough to distinguish which are the relevant ones to use for your particular problem In this course, will deal almost exclusively with non-magnetic dielectric materials characterized by a susceptibility that may be either homogenous,  a constant  or inhomogeneous in space,  r  Today we will continue with understanding the wave equation, and how Maxwell’s equations in uniform dielectric media constrain solutions of the wave equation for electromagnetic waves, which describe the properties of macroscopically averaged field quantities in those media

Vacuum Equations First step towards actually solving them What do you now get if take the curl of the first two equations? Write down these equations without the vector notation. Hint: use cartesian coordinates.

The Homogeneous Wave Equation Write a sentence that relates what you discovered on the last slide, that includes the phrase “the homogeneous wave equation”? Each Cartesian component of the electric and magnetic fields in vacuum is a solution to “the homogeneous wave equation”.  

Harmonic Solutions of the Homogeneous Wave Equation  Derive a defining equation for U(r) assuming this complex harmonic form of 

Con’t Find a solution that doesn’t vary at all in the y or z directions. where A is a complex number

Con’t  Using the following 2 equations, show that the real and imaginary parts of  are each independently valid solutions of the homogeneous wave equation (note, A is a complex number). x Real fields must be represented by real functions, and by convention the real field is related to the complex field via its real part. x x

Con’t k x 2 +k y 2 +k z 2 =  2 /c 2 More generally show under what conditions the following is a solution.

Con’t What type of wave is this, when substituted back into  ? A plane wave travelling in the k direction.

Wavefronts What is the phase of our general complex wave solution using this form of U(r) ?  Arg[A exp(-j k. r) exp(j  t)] = Arg[exp(j (  t-k. r+Arg(A))]=  t-k. r+Arg(A) What geometric object is defined by setting  t-k. r+Arg(A) = a constant value? What is special about values of  t-k. r+Arg(A) that differ by a multiple of 2  ?

Con’t The wavefront, or phasefront of a plane wave is defined by  t-k. r+Arg(A) = 2  n where n is an integer Sketch phase fronts in two dimensions (x-y plane) assuming k z =0, k x 2 =k y 2 = 0.5 (  /c) 2 Hint: do everything in units of = 2  k How would you calculate the velocity of these wavefronts? How would you calculate the direction in which these wavefronts are moving? What is the effect of changing Arg(A)?

Pause The wave amplitude, A, or more specifically |A|, since we included Arg(A) in the phase. We will see later why |A| 2 is referred to as the optical intensity of the wave. This is a good seque for returning to the Maxwell Eqns. Have dealt with wavefronts, their direction of propagation, and the phase velocity of these plane waves: what other important parameter of the wave has been ignored so far?

First generalize to waves in uniform material Derive the relevant wave equations for E and H?

What is the only difference? The speed of propagation of the field components is now reduced by n=  r 1/2 When have a sequence of materials with different values of n, {n i }, what changes and what stays the same as the wave propagates from some laser source through the different materials? n1n1 n2n2 White space is vacuum with n 0 = 1

To summarize

Further constraints on E and H from the Maxwell Equations While each component of E and H must satisfy the wave equation, one cannot superimpose arbitrary combinations of such solutions and claim to have a solution of the Maxwell equations.

Full plane wave solutions Using this form for E and H, what do the Maxwell Equations impose as a condition on the relationship between E o and H o ? k. E 0 = k. H 0 = 0

Physical implications/interpretation? E and H each separately orthogonal to k and to each other, hence

Furthermore

Superposition An general harmonic solution of the Maxwell equations in a uniform dielectric medium free of external sources can be expressed as a linear superposition of these transverse plane wave solutions in that medium