1. PHYS 408 Applied Optics (Lecture 2)
Jan-April 2016 Edition
Jeff Young
AMPEL Rm 113

2. The Maxwell Equations In Vacuum (from your text)
What are E, H?
What generates them?
Electric and Magnetic fields
Why aren’t there any charges or currents in these equations?

3. More Generally What are the differences?
B = mu_o H
Source terms
Sketch some moving charges and then a box without any included

4. What are r and j inside a uniform medium?
Obviously neutral, but if all electrons moving together in one direction, j not zero
Why does this make sense?

5. Maxwell equations inside a uniform medium with no external sources or free currents
Substitute these into

6. Compare with text Are they the same? Yes, assuming chi constant
We deal only with electric polarization density (non-magnetic materials)
Are they the same?

7. More generally What is the difference?
The main difference, and a very important one, is that if the material is not uniform (even if it has a boundary), \Del \cdot \vec P will not in general be zero, so it has to be del dot D, not del dot E that is zero, and same if generalize to magnetic materials with del dot B not del dot H
What is the difference?

8. And more generally still!
E, B, D, H, J are all MACROSCOPICALLY AVERAGED FIELDS, not the true microscopic e and b fields!!!
This is a crucial point, often glossed over in texts, that becomes important when dealing with textured dielectrics, and self-consistently solving ME and Lorentz force equations (eg deriving chi)
Pause: How do all of these field quantities differ from the actual fields?

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

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

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

12. Con’t
Find a solution that doesn’t vary at all in the y or z directions.
where A is a complex number
More generally show under what conditions the following is a solution.
kx2+ky2+kz2=w2/c2

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

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