Lecture 7: Lambert’s law & reflection

Interaction of Light and Surfaces

Tuesday, 27 January

2.4.3 – 2.6.4 spectra & energy interactions (p.13 – 20), Remote Sensing in Geology, B S Siegal & A R Gillespie, 1980 -- available on class website

Previous lecture: atmospheric effects, scattering

Fresnel’s law

rs = (n-1) 2 + K2

(n+1) 2 + K2

N = refractive indexK = extinction coefficient for the solidrs = fraction of light reflected from the 1st surface

rs

The amount of specular (mirror) reflection is given by Fresnel’s Law

Light is reflected, absorbed , or transmitted (RAT Law)

Transmitted component

Absorption occurs here

Mineral grain

Light passing from one medium to another is refracted according to Snell’s Law

Snell’s law: n1·sin1 =n2·sin2

Beer’s law: (L = Lo e-kz)z = thickness of absorbing materialk = absorption coefficient for the solidLo = incoming directional radianceL = outgoing radiance

Fresnel’s Law describes the reflection rs of light from a surface

rs = ----------------(n -1)2 +K 2

(n+1)2 +K 2

n is the refractive index K is the extinction coefficient

K is not the same as k, the absorption coefficient in Beer’s law (I = Io e-kz) (Beer – Lambert – Bouguer Law)

K and k are related but not identical:

k = ---------4K

K is the imaginary part of the complex index of refraction:m=n-jK

This is the specular ray

Augustin Fresnel Fresnel lens

Complex refractive index

n* = n + i

Consider an electrical wave propagating in the x direction:Ex=E0,x·exp[i·(kx·x·-ωt)]kx = component of the wave vector in the x direction = 2/; = circular frequency

=2v=c/n* = n·λ; v = speed in light in medium; c = speed of light in vacuum; k=2/=·*/c

Substituting,Ex = E0,x·exp[i·(·(n+i·)/c·x·-ω·t)]Ex = E0,x·exp[(i··n·x/c-··x/c-i·ω·t)]Ex = E0,x·exp[-··x/c]·exp[(i·(kx·x·-ω·t))]

If we use a complex index of refraction, the propagation of electromagnetic waves in a material is whatever it would be for a simple real index of refraction times a damping factor (first term) that decreases the amplitude exponentially as a function of x. Notice the resemblance of the damping factor to the Beer-Lambert-Bouguer absorption law. The imaginary part of the complex index of refraction thus describes the attenuation of electromagnetic waves in the material considered.

Surfaces may be

- specular

- back-reflecting

- forward-reflecting

- diffuse or Lambertian

Smooth surfaces (rms<<l) generally are specular or forward-reflecting examples: water, ice

Rough surfaces (rms>>l) generally are diffuse example: sand

Complex surfaces with smooth facets at a variety of orientations are forward- or back-reflecting example: leaves

Reflection envelopes

These styles of reflection from a surface contrast with scattering within the atmosphere

diffusereflection

forward scattering

Types of scattering envelopes

Uniform scattering Forward scattering Back scattering

Forward scattering in snow

ski

Light escapes from snowbecause the absorption coefficient k in e-kz is small

This helps increase the“reflectivity” of snow

You can easily test this: observe the apparentcolor of the snow next to a ski or snowboard with a brightly colored base:What do you see?

snow

How does viewing and illumination geometry affect radiance from Lambertian surfaces?

i

I

I cos i

Illumination

i is the incident angle; I is irradiance in W m-2

The total irradiance intercepted by anextended surface isthe same, but flux density is reduced by 1/cos i --- the total flux per unit area of surface is smaller by cos i

Unit area

Unit area

How does viewing and illumination geometry affect radiance from Lambertian surfaces?

Viewerat zenith Viewer

at viewing angle e

Viewer at zenith seesr -1 I cos i W sr-1 per pixel

angularIFOV

Same IFOV

1 m2 For a viewer off zenith, the same pixel is not filled by the 1 m2 surface element and the measured radiance is

L = r L = r -1-1 I cos i I cos i cos ecos etherefore, point sources look darker as e increases

Unresolved surface element exactly fills the IFOV at nadir, but doesn’t off nadir – part of the pixel “sees” the background instead

How does viewing and illumination geometry affect radiance from Lambertian surfaces?

Viewerat zenith Viewer

at viewing angle e

Viewer at zenith still seesr -1 I cos i W sr-1 per pixel

angularIFOV

Same IFOV

1 m2

For a viewer off zenith, the same pixel now sees a foreshortened surface element with an area of 1/cos e m2 so that the measured radiance is

L = r L = r -1-1 I cos i I cos itherefore, point sources do not change lightness as e increases

Resolved surface element -pixels are filledregardless of e.

How does viewing and illumination geometry affect radiance from Lambertian surfaces?

i

I

I cos i

Reflection

R= I cos ie

i is the incident angle ; I is irradiance in W m-2

e is the emergent angle; R is the radiance in W m -2 sr-1

r

i

I

I cos i

L= I cos ie

i is the incidence angle; I is irradiance in W m-2

e is the emergence angle; L is the radiance in W m -2 sr-1

Specular ray would be at e=i if surface were smooth like glass

r

Lambertian Surfaces

Specular ray

i

L= I cos ir

The total light (hemispherical radiance) reflectedfrom a surface is L = r I cos i W m -2

Lambertiansurface -L is independentof e

Lambertian SurfacesRough at the wavelength of light

Plowed fields

DN=231

222

231

239

231

231239

the brightness of a snow field doesn’t depend on e, the exit angle

ii

Reprise: reflection/refraction of light from surfaces (surface interactions)

Incident ray

Refracted ray

Specular rayReflected light

° amount of reflected light = rr I cos I cos ii° amount is independentindependent of view angle ee° color of specularly reflected light is essentially unchanged° color of the refracted ray is subject to selective absorption° volume scattering permits some of the refracted ray to reach the camera

e

Effect of topography is to change incidence angle

i

Shadow

i’

For topography elements >> l and >> IFOV

This is how shaded relief maps are calculated (“hillshade”)

L= I cos i’r

Imageintensity

Shade vs. Shadow

Shadow – blocking of direct illumination from the sun

Shade – darkening of a surface due to illumination geometry. Does not include shadow.

i

Variable shaded surfaces Shadowed Surface

i’

29

33Confusion of topographic shading and unresolved shadows

Next we’ll consider spectroscopy fundamentals - what happens to light as it is refracted into the surface and absorbed - particle size effects - interaction mechanisms

Light enters a translucent solid - uniform refractive index

Light enters a particulate layer - contrast in refractive index

Light from coarselyparticulate surfaces will have a smaller fraction of specularly reflected light than light from finely particulate surfaces

Surface/volume ratio = lower

Surface/volume ratio = higher

Obsidian Spectra

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

350 850 1350 1850 2350

Rock

16 - 32

32 - 42

42 - 60

60 - 100

100 - 150

150 - 200

Wavelength (nm)

Finest

Coarsest (Rock)

Ref

lect

ance

mesh Rock 16-32 32-42 42-60 60-100 100-150 150-200

Next lecture:

1) reflection/refraction of light from surfaces(surface interactions)

2) volume interactions- resonance- electronic interactions- vibrational interactions

3) spectroscopy- continuum vs. resonance bands- spectral “mining”- continuum analysis

4) spectra of common Earth-surface materials