Today’s summary

1. Today’s summary • Polarization • Energy / Poynting’s vector • Reflection and refraction at a dielectric interface: • wave approach to derive Snell’s law • reflection and transmission coefficients • total internal reflection (TIR) revisited

2. Polarization

3. Propagation and polarization In isotropic media(e.g. free space, amorphous glass, etc.) planar wavefront electric field vector E More generally, (reminder :Anisotropic in media, e.g. crystals, one could E not have parallel to D) wave-vector k

4. Linear polarization (frozen time)

5. Linear polarization (fixed space)

6. Circular polarization (frozen time)

7. Circular polarization:linear components

8. Circular polarization (fixed space)

9. /4 plate Linearpolarization Circularpolarization birefringent l/4 plate

10. λ/2 plate Linearpolarization Linear (90o-rotated) polarization birefringent λ/2 plate

11. Think about that Linearpolarization birefringent λ/4 plate mirror

12. Relationship between E and B Vectors k, E, B form a right-handed triad. Note: free space or isotropic media only

13. Energy

14. The Poynting vector S has units of W/m2 so it represents energy flux (energy per unit time & unit area)

15. Poynting vector and phasors (I) For example, sinusoidal field propagating along z Recall: for visible light, ω~1014-1015Hz

16. Poynting vector and phasors (II) Recall: for visible light, ω~1014-1015Hz So any instrument will record the average incident energy flux where T is the period (T=λ/c) is called the irradiance, aka intensity of the optical field (units: W/m2)

17. Poynting vector and phasors (III) 2 For example: sinusoidal electric field, Then, at constant z:

18. Poynting vector and phasors (IV) Recall phasor representation: complex amplitude or " phasor": Can we use phasors to compute intensity?

19. Poynting vector and phasors (V) Consider the superposition of two fields of the same frequency: Now consider the two corresponding phasors:

20. Poynting vector and phasors (V) Consider the superposition of two fields of the same frequency: Now consider the two corresponding phasors: and the quantity

21. Poynting vector and irradiance

22. Reflection / RefractionFresnel coefficients

23. Reflection & transmission@ dielectric interface

24. Reflection & transmission@ dielectric interface

25. Reflection & transmission@ dielectric interface I. Polarization normal to plane of incidence

26. Reflection & transmission@ dielectric interface I. Polarization normal to plane of incidence

27. Reflection & transmission@ dielectric interface I. Polarization normal to plane of incidence Continuity of tangential electric field at the interface: Since the exponents must be equal for all x, we obtain

28. Reflection & transmission@ dielectric interface I. Polarization normal to plane of incidence Continuity of tangential electric field at the interface: law of reflection Snell’s law of refraction so wave description is equivalent to Fermat’s principle!! ☺

29. Reflection & transmission@ dielectric interface I. Polarization normal to plane of incidence Incident electric field: Reflected electric field: Transmitted electric field: Need to calculate the reflected and transmitted amplitudes E0r, E0t i.e. need two equations

30. Reflection & transmission@ dielectric interface I. Polarization normal to plane of incidence Continuity of tangential electric field at the interface gives us one equation: which after satisfying Snell’s law becomes

31. Reflection & transmission@ dielectric interface I. Polarization normal to plane of incidence The second equation comes from continuity of tangential magnetic field at the interface:

32. Reflection & transmission@ dielectric interface I. Polarization normal to plane of incidence So continuity of tangential magnetic field Bx at the interface y=0 becomes:

33. Reflection & transmission@ dielectric interface I. Polarization normal to plane of incidence

34. Reflection & transmission@ dielectric interface II. Polarization parallel to plane of incidence Following a similar procedure ...

35. Reflection & transmission@ dielectric interface

36. Reflection & transmission of energy@ dielectric interface Recall Poynting vector definition: different on the two sides of the interface

37. Energy conservation

38. Reflection & transmission of energy@ dielectric interface

39. Normal incidence Note: independent of polarization

40. Brewster angle Recall Snell’s Law This angle is known as Brewster’s angle. Under such circumstances, for an incoming unpolarized wave, only the component polarized normal to the incident plane will be reflected.

41. elemental Why does Brewster happen? dipole radiator excited by the incident field

42. Why does Brewster happen?

43. Why does Brewster happen?

44. Why does Brewster happen?

45. Turning the tables Is there a relationship between r, t and r’, t’ ?

46. Relation between r, r’ and t, t’ Proof: algebraic from the Fresnel coefficients or using the property of preservation of the preservation of the field properties upon time reversal

47. Proof using time reversal

48. Total Internal Reflection Happens when Substitute into Snell’s law no energy transmitted

49. Total Internal Reflection Propagating component no energy transmitted

50. Total Internal Reflection Pure exponential decay º evanescent wave It can be shown that: no energy transmitted