1. Chapter 23: Fresnel equations Chapter 23: Fresnel equations

2. Recall basic laws of optics
normal
Law of reflection: qi qr n1 n2
Law of refraction
“Snell’s Law”: qt
Incident, reflected, refracted, and normal in same plane
Easy to derive on the basis of:
Huygens’ principle: every point on a wavefront may be regarded as a secondary source of wavelets
Fermat’s principle: the path a beam of light takes between two points is the one which is traversed in the least time

3. Today, we’ll show how they can be derived when we consider light to be an electromagnetic wave

4. E and B are harmonic Also, at any specified point in time and space,
where c is the velocity of the propagating wave,

5. and the change in the phase upon reflection
We’ll also determine the fraction of the light reflected vs. transmitted
external reflection,  R
1.0 .5 T qi R T
0° ° ° °
Incidence angle, qi
and the change in the phase upon reflection

6. Let’s start with polarization… light is a 3-D vector fieldy
light is a 3-D vector field z x
linear polarization
circular polarization

7. …and consider it relative to a plane interface
Plane of incidence: formed by and k and the normal of the interface plane k
normal

8. Polarization modes (= confusing nomenclature!)
always relative to plane of incidence
TE: transverse electric
s: senkrecht polarized
(E-field sticks in and out of the plane)
TM: transverse magnetic
p: plane polarized
(E-field in the plane) M E E M
perpendicular (  ), horizontal
parallel ( || ), vertical E

9. Plane waves with k along z direction
oscillating electric field y y x x
Any polarization state can be described as linear combination of these two:
“complex amplitude”
contains all polarization info

10. Derivation of laws of reflection and refraction
using diagram from Pedrotti3
boundary point

11. At the boundary point: phases of the three waves must be equal:
true for any boundary point and time, so let’s take or
hence, the frequencies are equal
and if we now consider
which means all three propagation vectors lie in the same plane

12. Reflection focus on first two terms:
incident and reflected beams travel in same medium; same l
hence we arrive at the law of reflection:

13. Refraction now the last two terms:
reflected and transmitted beams travel in different media (same frequencies; different wavelengths!):
which leads to the law of refraction:

14. Boundary conditions from Maxwell’s eqns
for both electric and magnetic fields, components parallel to boundary plane must be continuous as boundary is passed
TE waves
electric fields:
parallel to boundary plane
complex field amplitudes
continuity requires:

15. Boundary conditions from Maxwell’s eqns
for both electric and magnetic fields, components parallel to boundary plane must be continuous as boundary is passed
TE waves
magnetic fields:
continuity requires:
same analysis can be performed for TM waves

16. Summary of boundary conditions
TE waves
TM waves n1 n2
amplitudes are related:

17. Fresnel equations
TE waves
TM waves
Get all in terms of E and apply law of reflection (qi = qr):
For reflection: eliminate Et , separate Ei and Er , and take ratio:
Apply law of refraction and let :

18. Fresnel equations
TE waves
TM waves
For transmission: eliminate Er , separate Ei and Et , take ratio…
And together:

19. Fresnel equations, graphically

20. External and internal reflections

21. External and internal reflections
occur when
external reflection:
internal reflection:

22. External reflections (i.e. air-glass)
n = n2/n1 = 1.5
RTM = 0
(here, reflected light TE polarized;
RTE = 15%)
at normal : 4%
normal
grazing
- at normal and grazing incidence, coefficients have same magnitude
- negative values of r indicate phase change
fraction of power in reflected wave = reflectance =
fraction of power transmitted wave = transmittance =
Note: R+T = 1

23. Glare

24. Internal reflections (i.e. glass-air)
n = n2/n1 = 1.5
total internal reflection
- incident angle where RTM = 0 is:
both and reach values of unity before q=90°
 total internal reflection

25. Internal reflections (i.e. glass-air)

29. Conservation of energy
it’s always true that
and
in terms of irradiance (I, W/m2)
using laws of reflection and refraction, you can deduce
and

30. Summary: Reflectance and Transmittance for an Air-to-Glass Interface
Perpendicular polarization
Incidence angle, qi
1.0 .5
0° ° ° ° R T
Parallel polarization
Incidence angle, qi
1.0 .5
0° ° ° ° R T

31. Summary: Reflectance and Transmittance for a Glass-to-Air Interface
Perpendicular polarization
Incidence angle, qi
1.0 .5
0° ° ° ° R T
Parallel polarization
Incidence angle, qi
1.0 .5
0° ° ° ° R T

32. Back to reflections

33. or the polarizing angle
Brewster’s angle
or the polarizing angle
is the angle qp, at which RTM = 0:

34. Brewster’s angle for internal and external reflections
at qp, TM is perfectly transmitted with no reflection
at Brewster’s angle, “s skips and p plunges”
s-polarized light (TE) skips off the surface; p-polarized light (TM) plunges in

35. Brewster’s angle Punky Brewster Sir David Brewster (1781-1868)
( )

36. Brewster’s other angles: the kaleidoscope

37. Phase changes upon reflection
- recall the negative reflection coefficients
indicates that sometimes electric field
vector reverses direction upon reflection:
-p phase shift
external reflection: all angles for TE and at for TM
internal reflection: more complex…

38. Phase changes upon reflection: internal
in the region , r is complex
reflection coefficients in polar form:
f  phase shift on reflection

39. Phase changes upon reflection: internal
depending on angle of incidence, -p < f < p

40. Exploiting the phase difference
circular polarization
- consists of equal amplitude components of TE and TM linear polarized light, with phases that differ by ±p/2
- can be created by internal reflections in a Fresnel rhomb
each reflection produces a π/4 phase delay

41. Summary of phase shifts on reflection
air
glass
external reflection
TE mode
TM mode
air
glass
internal reflection
TE mode
TM mode

42. A lovely example

43. How do we quantify beauty?

44. Case study for reflection and refraction

45. Exercises
You are encouraged to solve all problems in the textbook (Pedrotti3).
The following may be covered in the werkcollege on 21 September 2011:
Chapter 23:
1, 2, 3, 5, 12, 16, 20