1. Atilla Ozgur Cakmak, PhD
Nanophotonics
Atilla Ozgur Cakmak, PhD

2. Lecture 9: Propagation over wells and barriers
Unit 2
Lecture 9: Propagation over wells and barriers

3. Outline Transfer Matrix Transmission Coefficient E > V0
Wave function

4. A couple of words…
Since we have learned the transfer matrix formalism, we are ready to proceed with the wells and barriers. The propagation over the wells and barriers will be covered in this lecture. Both analytical formulations and numerical analysis will be presented.
Suggested reading: Peter Markos, Costas M. Soukoulis Wave Propagation From Electrons to Photonic Crystals and Left-Handed Materials, 2nd Chapter.

5. Transfer Matrix

6. Transfer Matrix

7. Transfer Matrix

8. Transfer Matrix

9. Transfer Matrix (problem)
Find the transfer matrix if we have free space with no potential?
a) M12=M21=0, and M11=exp(2ika), M22=exp(-2ika)
b) M12=M21=0, and M11=exp(-2ika), M22=exp(2ika)
c) M11=M22=0, and M12=exp(2ika), M21=exp(-2ika)
d) M11=M22=0, and M12=exp(-2ika), M21=exp(2ika)

10. Transfer Matrix (solution)
Find the transfer matrix if we have free space with no potential?

11. Transmission Coefficient E > V0

12. Transmission Coefficient E > V0
V0 > E > 0
β=10
E > V0 > 0
E > 0
V0 < 0

13. Transmission Coefficient E > V0
β=0.25
β=1
β=10
β=100

14. Transmission Coefficient E > V0(pr)
Find the β range that shows 5 ripples (modes) within the E/V0 range of -5 to 5. Use MATLAB.
a) 2.25 < β < 2.5
b) 2.15 < β < 3
c) 0.5 < β < 1.5
d) 3.35 < β < 4

15. Transmission Coefficient E > V0(sol)
Find the β range that shows 5 ripples (modes) within the E/V0 range of -5 to 5. Use MATLAB.

16. Transmission Coefficient E > V0(sol)
Find the β range that shows 5 ripples (modes) within the E/V0 range of -5 to 5. Use MATLAB.
β=2.25
β=2.5

17. Wave Function

18. Wave Function

19. Wave Function
n=1
0 < V0 < E
β=10

20. Wave Function
n=7
V0 < 0 < E
β=10

21. Wave Function (problem)
Find the max. of the |ψ(x)| inside a barrier with T=1, β=5 and incident wavefunction is 1.
a) 1.132
b) 2.345
c) 0.587
d) 1.392

22. Wave Function (solution)
Find the max. of the |ψ(x)| inside a barrier with T=1, β=5 and incident wavefunction is 1 for the 6th mode.

23. Wave Function (solution)
Find the max. of the |ψ(x)| inside a barrier with T=1, β=5 and incident wavefunction is 1 for the 6th mode.

24. Wave Function (solution)
Find the max. of the |ψ(x)| inside a barrier with T=1, β=5 and incident wavefunction is 1 for the 6th mode.

25. Wave Function (problem)
Find T if E/V0=1.32 and β=10 using the wave functions and looking at the transmitted wave at the right hand side. (Hint: t=C/A) Compare that value with the one that you attained from the previous section (transmission coefficient). a) b) c) d)

26. Wave Function (problem)
Find T if E/V0=1.32 and β=10 using the wave functions and looking at the transmitted wave at the right hand side. (Hint: t=C/A) Compare that value with the one that you attained from the previous section (transmission coefficient).

27. Wave Function (problem)
Find T if E/V0=1.32 and β=10 using the wave functions and looking at the transmitted wave at the right hand side. (Hint: t=C/A) Compare that value with the one that you attained from the previous section (transmission coefficient).

28. Wave Function (solution)
Find T if E/V0=1.32 and β=10 using the wave functions and looking at the transmitted wave at the right hand side. (Hint: t=C/A) Compare that value with the one that you attained from the previous section (transmission coefficient).

29. Wave Function (solution)
Find T if E/V0=1.32 and β=10 using the wave functions and looking at the transmitted wave at the right hand side. (Hint: t=C/A) Compare that value with the one that you attained from the previous section (transmission coefficient).