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.

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.

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)

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

- Call this
*radiant flux.* - Integrate with the luminous efficiency function, V(l), F = \int V(l)
(P_l) dl, to get
*luminous flux*.- Unit is
*lumen*.

- Unit is
- 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
- area of the surface, dA
- 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.

- Luminous versus radiant
- Solid angle, steradian, and its relationship to area
- Use solid angle to remove r^2 factors

- Dualism between incoming and outgoing light.
- Time-reversal invariance of the dynamical laws governing light.

Two types of measurements

- 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

- guaranteed to measure things that people see, but
- Instrumental measurement.

There are two things that we can measure- psychophysical response to light
- reproduce and improve on visual measurement
- using filters and detectors to do optical integration

- physical properties of light
- use results of psychophysical experiments

- such as colour matching functions

to calculate psychophysical response from physical measurements

- use results of psychophysical experiments

- psychophysical response to light

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- the mass of the material
- the specific heat of the material
- 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

- The voltage across which the current is flowing is
- and the wavelength is \lambda
- Then the power of the light source is W (for watts)

In principle, it is straightforward

- Split the light into a spectrum using
- a prism
- a diffraction grating

- Spectrum can be spread out in
- space, which requires moving detector
- time, which can use a stationary detector

- 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

- readout is still sequential

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

- all detectors must be the same size

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

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?

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