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7. Physics of Light

Dr. Hamish Carr

COMP 5812M: Foundations of Modelling & Rendering
Photons
Light is a particle – a photon
But also a wave with a frequency
Energy of a photon is related to frequency:

(Planck’s constant)
is the frequency
is speed of light in a vacuum

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Units for Light
Energy () depends on wavelength
Power () is energy/unit time
Irradiance () is power/unit area
Also known as radiosity
Radiance () is irradiance/unit angle
Conserved along a ray
Defers inverse-square law to surface

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Why This Matters
Thin films generate rainbows of colour
Many other phenomena also affected

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Atomic Orbitals

Electrons orbit atoms
Fixed radii called orbitals
Related to electron energy
Electrons capture photons
Jump to higher orbitals
Or they release photons
Drop to lower orbitals

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Photon Frequencies
Single atom – very strict orbitals
Photons have specific frequencies
Blocks of material have roaming electrons
So they have bands of frequencies
But photons still restricted to these bands

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Electron Effects
Electrons may be tightly or loosely bound
Tight: photon released at same frequency
Loose: photon released at lower frequency
Spare energy absorbed as heat
Reflective material  conductors
Transparent material  insulators

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Fluorescence
Energy absorbed, transformed to heat
Electron releases lower-energy photon
Lower energy => lower frequency
Lower frequency => different colour
UV light can drop to visible light
Fluorescence

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Time Delays
Photon re-radiation is not immediate
There is a time lag
Small lag + re-emission of identical photon
Means the effective speed of light is slower
Different materials => different speeds
Known as virtual transition
Large lag + energy loss
Phosphorescence

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Carbon
Carbon has loosely bound electrons
Can absorb photons of many frequencies
Store energy in vibrations (heat)
Dark materials warm up
Reverse this to generate photons
Heat up soot so atoms vibrate
They start kicking out photons

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Black-body Radiation
Heat up a black body
Radiates lots of photons
Radiation depends on temperature
Stefan-Boltzmann law

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Wave Nature of Light
Photons also behave like waves
wave length – distance between peaks
wave velocity – speed of peaks

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Planar Waves
Generalisation of sine waves to space
Light wave has an associated electric field
So, a wave along x-direction is:

All others are just rotations of this

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Diffraction
Waves passing through a slit diffract
depending on width of slit

At one distance D
At all distances

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Polarisation
Change the origin in x or t
will change
But their difference will not
This is the basis of polarisation

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Circular Polarisation
y,z are out of phase
Represents a helical light wave
Projection sideways gives sine, cosine
Projection forward gives a circle

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Linear Polarisation
y, z are in phase
Light wave is planar in nature
Projects to a straight line
And can have any axis

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Elliptical Polarisation
Any other case
a combination of circular and linear

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Polarising Material
Some material is polarising
Transparent to one polarisation
Opaque to opposite polarisation
Commonly used for driving glasses
Also used in LCD screens
And in monitor glass
More in a few minutes …

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Refractive Index
Recall that light speed depends on material
The refractive index is the ratio with c
Vacuum:
Air:
Water:
Diamond:
Actually depends on wavelength

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering

Wave Retardation
As waves slow down, they bunch up
Frequency is constant, wavelength changes
Slowing one side changes the direction

Uniform
Slowed on One Side

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Snell’s Law
Waves retard based on angle
So this rotates the wave
Based on indices of refraction

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Reflection & Refraction
Not all light can refract
So energy must go somewhere
Reflection
Refraction
Absorption
Governed by Fresnel equations

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Polarised Reflection
Light reflected gets polarised
p-polarisation:
parallel to surface
perpendicular to page
s-polarisation:
perpendicular to light
in plane of page

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Total Internal Reflection
If angle is too steep, light cannot transmit
All light reflects internally
Basis of lenses, &c.

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Refractive Effects

Rainbows
Sundogs

CD-ROMs

Caustics

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Conductors
Insulators tend to refract light
Conductors tend to absorb then release
Rate of release affects reflected light
Can be expressed as complex number
Real part = index of refraction
Imaginary part = coefficient of extinction

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Simpler Approach
Treat refraction and extinction separately
Fresnel reflectance then becomes

But this doesn’t cover everything
Phenomena such as birefringence

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Without Polarisation
Graphics usually assumes no polarisation
Makes computations much easier
Simplifies Fresnel equations to:

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Moving to Calculus
All of this is per photon
But there are many photons
At least 1023 in practice
So we assume a continuous function
called a density function
Provided that the integral comes out right

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Solid Angles
A radian subtends a unit arc

A steradian subtends a unit area

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Change of Variables
We often have rectangular patches
We want to integrate in steradians
This involves a change of variables

becomes
becomes

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Integrating Radiance
Energy is an integral of radiance
Over wavelengths from to
Over all incoming directions
Over a given pixel/patch
And over a time interval

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
But . . .
The amount of light depends on the angle
Dependence is as with diffuse Phong
On the dot product with the angle

becomes
becomes

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
A Lambertian Emitter
As P gets further from sphere S
Angle and
Substitute and get

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering
Irradiance
Similarly, we can simplify the irradiance
By substituting assumptions

COMP 5812M: Foundations of Modelling & Rendering

COMP 5812M: Foundations of Modelling & Rendering

Brief Article

The Author

November 6, 2017

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November 6, 2017

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The Author

November 6, 2017

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Brief Article

The Author

November 6, 2017

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sin


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