1. PHYS 408 Applied Optics (Lecture 3)
Jan-April 2017 Edition
Jeff Young
AMPEL Rm 113

2. 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, c = a constant, or inhomogeneous in space, c =c(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

3. Vacuum Equations First step towards actually solving them
Aim to eliminate H
Using cartesian coordinates…

4. The Homogeneous Wave Equationz z
Each Cartesian component of the electric and magnetic fields in vacuum is a solution to “the homogeneous wave equation”.

5. Harmonic Solutions of the Homogeneous Wave Equation
A defining equation for U(r) assuming this complex harmonic form of z, is z

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

7. Con’t
The real and imaginary parts of z are each independently valid solutions of the homogeneous wave equation (note, A is a complex number). z 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.

8. Con’t More generally the following is a solution, provided that
kx2+ky2+kz2=w2/c2

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

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

11. Con’t
The wavefront, or phasefront of a plane wave is defined by wt-k.r+Arg(A) = 2p n where n is an integer
Sketch phase fronts in two dimensions (x-y plane) assuming kz=0, kx2=ky2= 0.5 (w/c)2
Hint: do everything in units of l=2p/k
What is the effect of changing Arg(A)?
How would you calculate the velocity of these wavefronts?
How would you calculate the direction in which these wavefronts are moving?

12. Pause
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?
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 segue for returning to the Maxwell Eqns.

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

14. What is the only difference?
The speed of propagation of the field components is now reduced by n=er1/2
When have a sequence of materials with different values of n, {ni}, what changes and what stays the same as the wave propagates from some laser source through the different materials? n1 n2
White space is vacuum with n0 = 1

15. To summarize

16. 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.

18. Full plane wave solutions
Using this form for E and H, what do the Maxwell Equations impose as a condition on the relationship between Eo and Ho?
k.E0= k.H0= 0

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

20. Furthermore

21. 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.