1. D=electric displacement B=magnetic induction E=electric field
Chapter 2: Electromagnetic Theory, Refractive Index, and Definitions of Radiance, Irradiance.
Gauss’ law
Gauss’ law for B
Faraday’s law induction
Ampere’s law
D=electric displacement
B=magnetic induction
E=electric field
H=magnetic field
 = free charge density
Qenclosed = free charge enclosed by Gaussian surface S
dS=closed boundary on S
Gauss’s law to get the E field of a charge in vacuum?

2. Boundary Conditions at Interfaces
Used along with boundary conditions to calculate the single scattering properties of aerosols and hydrometeors (cloud droplets, rain drops, ice crystals, snow flakes, etc), from first principles if possible. {Mie theory for homogeneous spheres, coupled dipole theory for general particles, T-Matrix method, etc}
Are not used to calculate the radiation field arriving at the surface from the complex atmosphere. Multiple scattering theory is used.
Which case is Mie Theory?
Which refer to normal and tangential components of the fields?

3. Constitutive Relationships: Material Properties and .
Homogeneous Media
J=E =electric conductivity
(like Ohm’s Law, V=IR)
B=H =magnetic permeability
D= 0(1+ ) E
0 =permittivity of free space
 =electric susceptibilty (to polarization)
f, f=frequency of time harmonic wave (next slides).
= 0(1+ ) + i= complex permittivity

4. Seek Plane Wave Solutions to Maxwell’s Equations
E0 and H0 are complex constants.
What is f for wall current, radio stations?

5. Dispersion Relationship: Relationship between  and k.
Comes from putting the assumed solutions into Maxwell’s equations.
At 550 nm, what is nr for water? For glass?
What is nr for ice at 2.85 um?
What is ni for ice at 2.85 um?

6. In which medium is the speed of light less?
Trace velocity matching principle: Snell’s law (continuity of the wavefront at a boundary) “slow is more normal”
Why do we sometimes see lightning but not hear thunder?
Here assume
n1=n1r, n1i=0,
n2=n2r, n2i=0.
In which medium is the speed of light less?
n1sin(1)= n2sin(2)
MIRAGES z
For a gas, (nr-1) ≈ 
=gas density.
d/dz > 0 for this type or mirage.
What does this say about the likelihood of convection?

7. Snell’s Law: Kinematics

8. Poynting Vector: Direction and magnitude of electromagnetic irradiance (power / area or energy/second / area).
Consider a time harmonic wave traveling in the x direction.
Why does the navy typically use acoustic methods under water instead of radar to find submarines from other countries and other things?

9. Some Basics, Electromagnetic Skin Depth

10. Particle Diameter << Wave Skin Depth

11. Particle Diameter >> Electromagnetic Skin Depth

12. Particle Radius Equal to the Skin Depth
(Rigor needed in the electromagnetic theory to get the right answer).

13. Aerosol Optical Properties: Absorbing particles.
F0 (W/m2)
Pext (W) = F0 ext
Pabs (W) = F0 abs
Psca (W) = F0 sca
Optical power removed by ext=abs+sca.
For small optical depths,
and D < 0.1 µm:
I(L)/I(0) = e(-L L),
L(1/m) ≈ S.O.C (m2/g) x r (g/m3),
L = path length,
r = aerosol concentration by mass. 
particle
mass
Absorption dominates for D < 0.1 µm (Rayleigh scattering).
Aside: For non-absorbing aerosols, Extinction=Scattering. Note the strong dependence of the scattering coefficient on diameter!
1/r
Rayleigh

14. Simple Collapsed Sphere Absorption Analysis

15. Example of Dry Chamise Particle SEM Image`

16. Another Example of Dry Chamise Particle SEM Image`

17. Example of Chamise Particle SEM Image After H20 Vapor Applied at 85%`

18. Another Example of Chamise Particle SEM Image After H20 Vapor Applied at 85%`

19. Complex Refractive Index of Water in the IR
500 1/cm = 20 microns
5000 1/cm = 2 microns
Minima in nr are associated with minima in scattering by water droplets.
Peaks in ni are associated with strong absorption phenomena in water, intermolecular vibration, rotation, etc.

20. Complex Refractive Index of Ice in the IR
500 1/cm = 20 microns
5000 1/cm = 2 microns
Minima in nr are associated with minima in scattering by ice crystals.
Peaks in ni are associated with strong absorption phenomena in ice, intermolecular vibration, rotation, etc.
Arnott, W. P., Y. Y. Dong, and J. Hallett, 1995: Extinction efficiency in the IR (2 µm to 18 µm) of laboratory ice clouds: Observations of scattering minima in the Christiansen bands of ice. Applied Optics 34 ,

21. Consequences of Refractive Index for Water and Ice: Carl Schmitt Senior Thesis, 1995, using FTIR
Experimental setup:
Cloud box filled held water vapor, water droplets, or ice crystals.
Measure light transmission for 635 nm and the range 1.27 um to 4.2 um.
2 meter per side

22. Optical Depth of Water Vapor: Note the sharp, discrete lines

23. Extinction Efficiency for Water Droplet Cloud
OD IR / 0.5 OD Visible

24. Ice Cloud Microphysics: Relevant to fresh contrails, ice fogs, freshly nucleated cirrus clouds, and probably mesospheric clouds
Hexagonal column crystals

25. Comparison of extinction on a per water vapor molecule basis
Liquid water light gray
Ice solid black
Notes:
Water and ice refractive indices are different; allows for remote sensing each using solar radiation, near IR.
Minima in extinction are slightly left of the minima in the real refractive index due to absorption.
Results depend on the hydrometeor size distribution.
water droplets
Ice crystals

26. Radiant Intensity or Radiance: Watts / (m2 Sr)

27. Flux (also Irradiance) and Radiant Intensity (Radiance)

28. Spherical Coordinate System: z axis is the vertical component in the atmosphere.
SOLID ANGLE
What angle is latitude?

29. Spherical Coordinate System: z axis is the vertical component in the atmosphere: Another view.

30. Flux (irradiance) as a distribution function and broadband quantity
Flux (irradiance) as a distribution function and broadband quantity. Purpose: Describe radiation in particular direction such as net downward, net upward, etc.

31. Radiant Intensity Definition (also known as Radiance) Purpose: Describe radiation from all and any direction. It is also a distribution function with respect to wavelength (or frequency, or wavenumber, depending on the orientation).

32. Flux and Radiant Intensity Relationships
Prove this relation…

33. Irradiance - Radiance Relations
Special case: I isotropic, same in all directions, like black body radiation from a surface.

34. THE BIG PICTURE: Radiation Heating of the Atmosphere
From Oort and Peixoto

35. ATMOSPHERE HEATING BY RADIATION: The heating rate is the divergence of the net irradiance (or net flux if you prefer).
From Oort and Peixoto

36. ATMOSPHERE HEATING BY RADIATION: The heating rate is the divergence of the net irradiance (or net flux if you prefer).
From Oort and Peixoto

37. FTIR Radiance: Atmospheric IR Window
13 microns
8 microns

38. DEFINITION OF THE BRIGHTNESS TEMPERATURE TB
Measured Radiance at wavenumber v =
Theoretical Radiance of a Black Body at temperature TB

39. FTIR Brightness Temperatures

40. Solar Radiance at the Top of the Atmosphere

41. Solar Flux S0
Earth
SUN

42. Regional and Seasonal Insolation at the TOA
Normal Flux:
What is the range in Reno?
In Mexico City?
In Barrow Alaska?
Where is the peak? Why?

43. Regional and Seasonal Insolation at the TOA
Normal Flux:
What is the range in Reno?
In Mexico City?
In Barrow Alaska?
Where is the peak? Why?

44. Insolation at the Two Solstices and the Annual Average
What is the average insolation over all latitudes?

45. Sun Cross Section, Sunspots, and Nuclear Fusion
A sunspot is a region on the Sun's surface (photosphere) that is marked by a lower temperature than its surroundings and has intense magnetic activity, which inhibits convection, forming areas of reduced surface temperature. They can be visible from Earth without the aid of a telescope. Although they are at temperatures of roughly K, the contrast with the surrounding material at about 5800 K leaves them clearly visible as dark spots, as the intensity of a heated black body (closely approximated by the photosphere) is a function of T (temperature) to the fourth power. If a sunspot was isolated from the surrounding photosphere it would be brighter than an electric arc. Source: Wikipedia.
4 1H + 2 e --> 4He + 2 neutrinos + 6 photons

46. Sun’s Atmosphere: Region above the photosphere.
Chromosphere,
Corona.

47. Solar Corona

48. Number of Sun Spots Observed as a function of Year …

49. Geometry of Earth and Sun

50. Sun and Satellite Perspective: How do the properties of the surface affect what we see?

51. Radiance and Irradiance: How do we define radiation?
Types of reflection: Can also think of the reflected light as emitted light from different types of surfaces.

52. Geometry for the BRDF (bidirectional reflection distribution function)
S is solar irradiance coming in.
I is the reflected radiance.
BRDF