1. PHYS 408 Applied Optics (Lecture 9) JAN-APRIL 2016 EDITION JEFF YOUNG AMPEL RM 113

2. Quiz #4 1)There are an infinite number of unknown wave amplitudes to solve for when a plane wave bounces back and forth inside a thin film (T/F). 2)The phase accumulated by a plane wave that bounces back and forth between the two interfaces inside a film depends on the Fresnel reflection coefficients at the two interfaces (T/F). 3)There is money to be made by understanding the equations that govern the reflectivity of plane waves of different frequencies that encounter planar stacks of dielectric materials (T/F). 4)There is only one conceptual way to formulate the solution to the thin-film reflection problem (T/F).

3. 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!)

4. A reminder Infinite series (homework, very intuitive) More powerful approach

5. The solution Note the number of indicies: Which terms pertain to interfaces and which to propagation?

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.

7. 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. 0 0 0 Very general, the box could represent a single interface, or a 100 layer dielectric stack.

8. Moving forward If you had an incident wave from the right hand side, how might you express the corresponding matrix representation of the out-going field amplitudes? 0 0 0

9. Moving forward Can you 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? S, or Scattering Matrix outin

10. The S and M matricies 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? What would the ideal “transfer matrix” be that would allow you to get the overall transmission simply by multiplying matrices for each boundary? right left

11. The S and M matricies Can you obtain this desired matrix, call it the M matrix, from the S matrix? Note the number of indicies: Don’t get confused with the overall thin-film expression earlier today. The former can be derived from the latter, as follows.

12. The S and M matricies What is then the algorithm for obtaining the overall transmission and reflection amplitude for plane waves incident on an arbitrary number of dielectric films arranged in a planar stack?

13. Let’s see how this works n1n1 d1d1 n2n2 d2d2 Find the net, overall M and S matricies assuming d 2 is infinite Hint: start with the (intuitive) individual S matricies for each transition, and convert them to M matricies

14. Step by step n1n1 d1d1 n2n2 d2d2 ? or ?

15. Step by step n1n1 d1d1 n2n2 d2d2 missing

16. Step by step n1n1 d1d1 n2n2 d2d2 Next Step?

17. Bottom line Consider M net as M, then S net (A,B,C,D) yields t 02,t 20,r 02,r 20 Or don’t be lazy, and just solve for and from

18. n 2 =1.3, d 1 =400 nm

19. Let’s play! n1n1 d1d1 n2n2 d2d2 n1n1 d1d1 n1n1 d1d1 n2n2 d2d2 n1n1 d1d1 n2n2 d2d2 n3n3 d3d3 n1n1 d1d1 n2n2 d2d2 … … nlayers-1

20. Bragg reflection n 1 =1.3; n 2 =1.4 d 1 =400 nm; d 2 =200 nm 21 periods

21. Bragg reflection n 1 =1.3; n 2 =1.4 d 1 =580 nm; d 2 =20 nm 21 periods