# Geometry of Light

#### Point sources of light

From far enough away every source of light is a point source

• geometry of the problem has spherical symmetry

Consider a point source of light,

• emitting steadily, which means with constant power,,
• surrounded by concentric spheres.

Energy

• is conserved
• is neither created nor destroyed by the medium through which the light is transmitted

In any unit of time the same amount of light passes through every sphere

• The total power passing through each spherical surface is equal
and it's equal to the power emitted by the point source
• By spherical symmetry, the power is constant across each surface.

#### Power density of light

The area of the surface of a sphere is (4 pi ) r^2.

• For a sphere at distance r, the power of the light passing through a unit area is
• (the power of the source) / (4 pi r^2)
• Newton found this falling off of gravity using just this argument
• Another way of saying this is the a point source emits constant power per solid angle.
• The area of a unit solid angle at a distance r grows as r^2
• The power density of the point source must be infinite.

#### Power passing from one area to another

Consider an emitting area, dA1,as an array of point radiators

• Radiation into a solid angle dw is (power density) * dA1 * dw

The receiving area, dA2, is r^2 dw

• Radiation passing from one area to another is (power density) * dA1 * dA2 / (r^2)

## Units in which Light is Measured

Considering wavelength, power emiited by a point source,, P_l, is watts per unit wavelength

• Integrate with the luminous efficiency function, V(l), F = \int V(l) (P_l) dl, to get luminous flux.
• Unit is lumen.
• To measure it surround the source by a detector

From a distance the amount of energy captured depends on the size of the detector

• Call the amount captured the luminous intensity, I = F / dw
• Angular size is what's important
• To measure it use a detector with a finite aperture, and divide by the aperture
• Unit is the candela = lumen per steradian

Normally we are measuring an area and want to know the emission per unit area of the surface

• Two factors to consider
1. area of the surface, dA
2. inclination of the surface to the line of sight, cos(t)
• This is luminance.
• L = I / ( dA * cos(t))
• The unit is candela / (m^2)

We often want to measure the amount of light falling on a surface, which we call illuminance, because the surface is being illuminated.

• It is E = F / dA
• Its units are lux = lumen / (m^2)
• This is a concept used extensively in illuminating engineering.

Similarly, we have a name for the amount of light emitted by the surface, luminous emittance.

• Its units are also lux.

#### Important ideas

2. Solid angle, steradian, and its relationship to area
• Use solid angle to remove r^2 factors
3. Dualism between incoming and outgoing light.
• Time-reversal invariance of the dynamical laws governing light.

# Colour Measurement

Two types of measurements

1. Visual measurement
• guaranteed to measure things that people see, but
• no guarantee that two people are looking for the same quality because language is imprecise
• e.g. heterochromatic brightness
• Introduce the luminous efficiency function, which was used above to convert between tadiant and luminous quantities..
• That is, define equal luminance as a way of building a bridge between physical and visual measurement.
• flicker photometry
• easy to create the measurement
• measures to within 1%
• extremely good person to person reproducibility
• except for uncertainty in blue, which is probably related to aging
• minimum motion tests
• spatial fusion tests
• linearity
• one dimensional subspace of colour space
2. Instrumental measurement.
There are two things that we can measure
1. psychophysical response to light
• reproduce and improve on visual measurement
• using filters and detectors to do optical integration
2. physical properties of light
• use results of psychophysical experiments
• such as colour matching functions

to calculate psychophysical response from physical measurements

## Measuring physical properties

Almost always energy in the past,

• but now is increasingly photon counting
• photon counting must be calibrated
• by energy measurement, of course

Ultimate calibration is to heat

• Shine a light onto something
• How much does it heat up?
You need to know
1. the mass of the material
2. the specific heat of the material
3. how much heat is lost
• Use this to calibrate a detector
• most sensitive is a photomultiplier
• most common is a solid state detector (CCD = charge-coupled detector)
• Need to convert energy calibration to power calibration

You now have a detector and a calibration.

• When the meter on the detector reads A (for amps)
• The voltage across which the current is flowing is
• high for photomultipliers
• low for CCDs
• and the wavelength is \lambda
• Then the power of the light source is W (for watts)

#### Measuring a spectral power distribution

In principle, it is straightforward

1. Split the light into a spectrum using
• a prism
• a diffraction grating
2. Spectrum can be spread out in
• space, which requires moving detector
• time, which can use a stationary detector
3. Get a stream of measurements
• correct for effects of wavelength non-linearity,
• because you are really measuring \Phi(\lambda) \Delta\lambda

Current technology uses an array of detectors, but

Two aspects of calibration are hard

• wavelength: spectral lines used for wavelength calibration
• detector response
• all detectors must be the same size
• far from true in your digital camera
• detectors much be low noise
• far from true in your digital camera
• dark current
• gain

If an instrument is inexpensive they most likely skimped on calibration.

## What can be measured?

Power of light emitted from a source in all directions

• integrated with luminous intensity function
• called luminous intensity
• unit is candela

Power of light enitted from a source in a particular direction

• called luminous flux
• unit is lumen = candela per steradian
• need to talk about solid geometry

Power of light falling on a surface

• called illuminance
• unit is lumen per square metre

Power of light falling on a surface from a particular direction

• called luminance
• unit is lumen per square metre per steradian

Power of light leaving a surface

• called luminous exitance
• unit is lumen per square metre

Power of light leaving a surface in a particular direction

• called luminance
• Why?