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 Circular polarization Linear polarization birefringent
l/4 plate

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

11. Think about that
Linear polarization
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. energy flux (energy per unit time & unit area)
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~ Hz

16. Poynting vector and phasors (II)
Recall: for visible light, ω~ Hz
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 / Refraction Fresnel 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. This angle is known as Brewster’s angle. Under such
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. Why does Brewster happen?
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

51. Phase delay upon reflection

52. Phase delay upon TIR

53. Phase delay upon TIR