1. PHYS 408 Applied Optics (Lecture 9)
Jan-April 2017 Edition
Jeff Young
AMPEL Rm 113

2. Quick review of key points from last lecture
There are only really two unknown complex amplitudes of the net forward and net backward travelling plane waves inside a thin film.
Together with the overall reflected wave, and the overall transmitted wave amplitudes, there are a total of four unknown complex amplitudes in a single thin-film/plane wave problem.
The continuity of the parallel electric and magnetic fields across the dielectric interfaces gives two equations at each interface, so a total of four independent equations for the four unknowns (linear algebra).
Wavevectors should ultimately be expressed in terms of frequency and refractive indices, not wavelengths (advice that not all physicists or engineers follow!)

3. A reminder Infinite series (homework, very intuitive)
More powerful approach
Remind them of what they did last week at this stage for the single interface problem

4. The solution
Show pdf of handwritten algebra
Note the number of indices: Which terms pertain to interfaces and which to propagation?

5. Moving forward
Cast our result in the form of a matrix equation that yields the reflected and transmitted wave amplitudes (the out-going wave amplitudes) when the incident wave amplitude from the left hand medium is specified.

6. Moving forward
Cast our result in the form of a matrix equation that yields the reflected and transmitted wave amplitudes (the out-going wave amplitudes) when the incident wave amplitude from the left hand medium is specified.
Very general, the box could represent a single interface, or a 100 layer dielectric stack.

7. Moving forward
If you had an incident wave from the right hand side, the corresponding matrix representation of the out-going field amplitudes is;

8. Moving forward out in S, or Scattering Matrix
Combine these to generate a matrix equation that yields the outgoing waves for the general case when you have two in-coming waves, one from each side.
out in
S, or Scattering Matrix

9. The S and M matricies right left
Would this S matrix help you easily solve for the overall reflected and transmitted fields if you had multiple dielectric layers up against each other?
No. Need a “transfer matrix” that can take you from one side to the other, layer by layer, interface by interface.
right
left
No, since can’t concatenate

10. The S and M matricies
This M matrix related to the S matrix via linear algegra:
Just linear algebra
Note the number of indices: Don’t get confused with the overall thin-film expression earlier today. The former can be derived from the latter, as follows.

11. The S and M matricies

12. Let’s see how this worksn1 d1 n2 d2
Find the net, overall M and S matrices assuming d2 is infinite
Hint: start with the (intuitive) individual S matrices for each transition, and convert them to M matrices
What real world problem does this represent? (anti reflection coating on glasses)
Give them 5 minutes to work on this.

13. Step by step n1 d1 n2 d2 ? or or ?

14. Step by step n1 d1 n2 d2
missing

15. Step by step n1 d1 n2 d2
Pause, get them to think about it.
Next Step?

16. Bottom line
Consider Mnet as M, then Snet(A,B,C,D) yields t02,t20,r02,r20
Or don’t be lazy, and just solve for and from

17. n2=1.3, d1=400 nm Ramifications for our application?
can’t use a higher index coating
Really only one thickness/index combination that give you true anti-reflection
Do you still have the flexibility to choose the 0 reflectivity wavelength?

18. Let’s play! … … n1 d1 n2 d2 n1 d1 n2 d2 n1 d1 n3 d3 n1 d1 n2 d2 n1 d1
nlayers-1
nlayers-1

19. Uniform periodic multilayer stack… …

20. Bragg reflection n1=1.3; n2=1.4 d1=580 nm; d2=20 nm 21 periods
Sketch this profile on board, and get them to figure out what condition defines this high reflectivity wavelength.
- Ans: 1/lambda=1/(2n_ave d) (reflections from all periods in phase), since to a good approximation it is a uniform thin film of n=1.3 and thickness 21*600 nm.

21. Physics of peak with max reflectivity