1. Optical Properties of Materials
Index of refraction
… reflection
… refraction (Snell’s law)
… index of refraction
Absorption

2. Maxwell’s Equations Materials equations 𝐸 … electric field
𝐻 … magnetic field
𝐷 … electric displacement field
𝐵 … magnetic induction
𝑗 … current density
𝜌 … electric charge density
𝜎 … electrical conductivity
𝜖 … permittivity
𝜇 … permeability
Materials equations

3. Maxwell’s Equations
… no free charge
… wave equation

4. The Wave Equation

5. Refraction and Absorption
k … wave vector
 … angular frequency
c … velocity of light
n … index of refraction
 … electrical conductivity
Complex permittivity: permittivity and losses
Complex index of refraction: refraction and absorption

6. Amplitude and Intensity of the Propagating Wave

7. Relationship between Dielectric and Optical Constants
* dielectric constant = permittivity

8. Insulator … non-conducting material … no absorption, no losses
… the index of refraction is a real quantity

9. Penetration Depth
… dependent on frequency (wavelength) and absorption

10. Penetration Depth and Absorption (Examples)
𝑘  𝜅
* absorption = damping

11. Reflection and Transmission1
𝜃 t 2
Same amplitude and phase of wave at the point “0”
Reflection:
Transmission:
(Snell’s law)

12. Electric and Magnetic Field
The vectors of the electric and magnetic fields are perpendicular to the propagation direction of the wave. 𝐼
𝜃 i
𝜃 r 𝑅 𝑬 𝒔 𝑇 𝑯
The original wave:

13. Electric and Magnetic Field
The transmitted wave:
The reflected wave:

14. Fresnel Equations
… are obtained from the boundary conditions: Tangential components of 𝐸 and 𝐻 have to be continuous at the interface (surface).

15. Fresnel Coefficients
Snell

16. Index of Refraction (Experimental Examples)

17. Materials with different refractive indices are very important for complex optical systems

18. Transmission and Reflection
Vacuum  Glass: n=1.5
Brewster angle – complete polarization of reflected electromagnetic wave (polarization of light)
Vacuum  Glass (n=1,5)

19. Transmission and Reflection
Vacuum  Germanium: n=5,3
Vacuum  Germanium (n=5,3)

20. Optical Reflection Total internal reflection Glass (n=1,5)  Vacuum

21. Total Internal Reflection
Glass (n = 1,5): c = 41,8°
Water (n = 2): c = 30°

22. Transmission and Reflection with Complex Index of Refraction

23. Transmission and Reflection with an Incident Angle of 0°
Interface material - vacuum:

24. 𝛼… absorption coefficient 𝑛… index of refraction … wavelength
Table 11.2
Refractive index 𝑛 and absorption index 𝜅 of some materials with 𝜆=589 nm
𝜅… absorption index
𝛼… absorption coefficient
𝑛… index of refraction
… wavelength

25. Transmission and Reflection with Complex Index of Refraction
Vacuum  Copper (n= i)
Copper
n = 0.14
k = 3.35
R = 95.6 %

26. Transmission and Reflection with Complex Index of Refraction
Vacuum  Sodium (n= i)
Sodium
n = 0.048
k = 1.86
R = 95.8 %

27. Transmission and Reflection with Complex Index of Refraction
Vacuum  Gallium (n= i)
Gallium
n = 3.69
k = 5.43
R = 71.3 %

28. Transmission and Reflection with Complex Index of Refraction
Vacuum  Cobalt (n= i)
Cobalt
n = 2.0
k = 4.0
R = 68.0 %

30. Reflection with Complex Index of Refraction
Influence of absorption (weakening, damping) on the reflection

31. Reflection with Complex Index of Refraction
Total external reflection vanishes

32. Reflectivity as function of Refractive Index and Absorption
Reflectivity increases with increasing index of refraction and an increasing absorption index
Fig. 11.2
Reflectivity as function of absorption and refractive index

33. Refractive Index as function of Wavelength
Material
(Sphalerite)
Color of Materials
(Rutile)
Fig. 11.5
Refractive index as function of absorption index and absorption coefficient as function of wavelength for Si (a), KCl (b) and Cu (c).

34. Reflection and Transmission of a Thin Film
Fresnel coefficients at the interfaces:
Phase shift:

35. Reflection and Transmission of a Thin Film
Vacuum  Glass (n = 1.5, t = 6 μm)  Vacuum, λ = 600 nm
Constant wavelength (monochromatic radiation)
Thickness of the film is ten times of the wavelength
Reflection
Intensity (%)
Angle of incidence (degree)

36. Reflection and Transmission of a Thin Film
Vacuum  Glass (n = 1.5, t = 1.2 μm)  Vacuum, λ = 600 nm
Constant wavelength (monochromatic radiation)
Thickness of the film is two times of the wavelength
Reflection
Intensity (%)
Angle of incidence (degree)

37. Reflection and Transmission of a Thin Film
Vacuum  Glass (n = 1.5, t = 24 μm)  Vacuum, λ = 600 nm
Constant wavelength (monochromatic radiation)
Thickness of the film is 40 times of the wavelength
Reflection
Intensity (%)
Angle of incidence (degree)

38. Reflection and Transmission of a Thin Film
Vacuum  Glass (n = 1.5, t = 1.2 μm)  Vacuum, λ = nm
Different wavelengths (polychromatic radiation)
Thickness of film is 1.2 m
Different “Colors” are reflected and transmitted differently.
Intensity (%)
Angle of incidence (degree)