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Light
In this chapter, we discuss the practical issues of measuring light, usually called
radiometry. The terms that arise in radiometry may at first seem strange and have
terminology and notation that may be hard to keep straight. However, because
radiometry is so fundamental to computer graphics, it is worth studying radiome-
try until it sinks in. This chapter also covers photometry, which takes radiometric
quantities and scales them to estimate how much “useful” light is present. For
example, a green light may seem twice as bright as a blue light of the same power
because the eye is more sensitive to green light. Photometry attempts to quantify
such distinctions.
20.1 Radiometry
Although we can define radiometric units in many systems, we use SI (Interna-
tional System of Units) units. Familiar SI units include the metric units of meter
(m)andgram (g). Light is fundamentally a propagating form of energy, so it is
useful to define the SI unit of energy, which is the joule (J).
20.1.1 Photons
To aid our intuition, we will describe radiometry in terms of collections of large
numbers of photons, and this section establishes what is meant by a photon in this
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518 20. Light
context. For the purposes of this chapter, a photon is a quantum of light that has
a position, direction of propagation, and a wavelength λ. Somewhat strangely,
the SI unit used for wavelength is nanometer (nm). This is mainly for historical
reasons, and 1nm=10
−9
m. Another unit, the angstrom, is sometimes used, and
one nanometer is ten angstroms. A photon also has a speed c that depends only
on the refractive index n of the medium through which it propagates. Sometimes
the frequency f = c/λ is also used for light. This is convenient because unlike
λ and c, f does not change when the photon refracts into a medium with a new
refractive index. Another invariant measure is the amount of energy q carried by
a photon, which is given by the following relationship:
q = hf =
hc
λ
, (20.1)
where h =6.63 × 10
−34
Jsis Plank’s Constant. Although these quantities can
be measured in any unit system, we will use SI units whenever possible.
20.1.2 Spectral Energy
If we have a large collection of photons, their total energy Q can be computed
by summing the energy q
i
of each photon. A reasonable question to ask is “How
is the energy distributed across wavelengths?” An easy way to answer this is to
partition the photons into bins, essentially histogramming them. We then have
an energy associated with an interval. For example, we can count all the energy
between λ = 500 nm and λ = 600 nm and have it turn out to be 10.2 J, and this
might be denoted q[500, 600] = 10.2. If we divided the wavelength interval into
two 50 nm intervals, we might find that q[500, 550] = 5.2 and q[550, 600] = 5.0.
This tells us there was a little more energy in the short wavelength half of the
interval [500, 600]. If we divide into 25 nm bins, we might find q[500, 525] = 2.5,
and so on. The nice thing about the system is that it is straightforward. The bad
thing about it is that the choice of the interval size determines the number.
A more commonly used system is to divide the energy by the size of the
interval. So instead of q[500, 600] = 10.2 we would have
Q
λ
[500, 600] =
10.2
100
=0.12 J(nm)
−1
.
This approach is nice, because the size of the interval has much less impact on
the overall size of the numbers. An immediate idea would be to drive the interval
size Δλ to zero. This could be awkward, because for a sufficiently small Δλ, Q
λ
will either be zero or huge depending on whether there is a single photon or no
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20.1. Radiometry 519
photon in the interval. There are two schools of thought to solve that dilemma.
The first is to assume that Δλ is small, but not so small that the quantum nature of
light comes into play. The second is to assume that the light is a continuum rather
than individual photons, so a true derivative dQ/dλ is appropriate. Both ways of
thinking about it are appropriate and lead to the same computational machinery.
In practice, it seems that most people who measure light prefer small, but finite,
intervals, because that is what they can measure in the lab. Most people who
do theory or computation prefer infinitesimal intervals, because that makes the
machinery of calculus available.
The quantity Q
λ
is called spectral energy, and it is an intensive quantity as op-
posed to an extensive quantity such as energy, length, or mass. Intensive quantities
can be thought of as density functions that tell the density of an extensive quantity
at an infinitesimal point. For example, the energy Q at a specific wavelength is
probably zero, but the spectral energy (energy density) Q
λ
is a meaningful quan-
tity. A probably more familiar example is that the population of a country may
be 25 million, but the population at a point in that country is meaningless. How-
ever, the population density measured in people per square meter is meaningful,
provided it is measured over large enough areas. Much like with photons, popula-
tion density works best if we pretend that we can view population as a continuum
where population density never becomes granular even when the area is small.
We will follow the convention of graphics where spectral energy is almost al-
ways used, and energy is rarely used. This results in a proliferation of λ subscripts
if “proper” notation is used. Instead, we will drop the subscript and use Q to de-
note spectral energy. This can result in some confusion when people outside of
graphics read graphics papers, so be aware of this standards issue. Your intuition
about spectral power might be aided by imagining a measurement device with an
energy sensor that measures light energy q. If you place a colored filter in front of
the sensor that allows only light in the interval [λ − Δλ/2,λ+Δλ/2], then the
spectral power at λ is Q =Δq/Δλ.
20.1.3 Power
It is useful to estimate a rate of energy production for light sources. This rate is
called power, and it is measured in watts, W , which is another name for joules
per second. This is easiest to understand in a steady state, but because power is
an intensive quantity (a density over time), it is well defined even when energy
production is varying over time. The units of power may be more familiar, e.g., a
100-watt light bulb. Such bulbs draw approximately 100 J of energy each second.
The power of the light produced will actually be less than 100 W because of
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520 20. Light
heat loss, etc., but we can still use this example to help understand more about
photons. For example, we can get a feel for how many photons are produced in a
second by a 100 W light. Suppose the average photon produced has the energy of
a λ = 500 nm photon. The frequency of such a photon is
f =
c
λ
=
3 × 10
8
ms
−1
500 × 10
−9
m
=6× 10
14
s
−1
.
The energy of that photon is hf ≈ 4 × 10
−19
J. That means a staggering 10
20
photons are produced each second, even if the bulb is not very efficient. This
explains why simulating a camera with a fast shutter speed and directly simulated
photons is an inefficient choice for producing images.
As with energy, we are really interested in spectral power measured in
W(nm)
−1
. Again, although the formal standard symbol for spectral power is
Φ
λ
, we will use Φ with no subscript for convenience and consistency with most
of the graphics literature. One thing to note is that the spectral power for a light
source is usually a smaller number than the power. For example, if a light emits
apowerof100 W evenly distributed over wavelengths 400 nm to 800 nm, then
the spectral power will be 100 W/400 nm = 0.25 W(nm)
−1
. This is something to
keep in mind if you set the spectral power of light sources by hand for debugging
purposes.
The measurement device for spectral energy in the last section could be mod-
ified by taking a reading with a shutter that is open for a time interval Δt centered
at time t. The spectral power would then be ΔQ/(ΔtΔλ).
20.1.4 Irradiance
The quantity irradiance arises naturally if you ask the question “How much light
hits this point?” Of course the answer is “none,” and again we must use a density
function. If the point is on a surface, it is natural to use area to define our density
function. We modify the device from the last section to have a finite ΔA area
sensor that is smaller than the light field being measured. The spectral irradiance
H is just the power per unit area ΔΦ/ΔA. Fully expanded this is
H =
Δq
ΔA ΔtΔλ
. (20.2)
Thus, the full units of irradiance are Jm
−2
s
−1
(nm)
−1
. Note that the SI units for
radiance include inverse-meter-squared for area and inverse-nanometer for wave-
length. This seeming inconsistency (using both nanometer and meter) arises be-
cause of the natural units for area and visible light wavelengths.
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20.1. Radiometry 521
When the light is leaving a surface, e.g., when it is reflected, the same quantity
as irradiance is called radiant exitance, E. It is useful to have different words
for incident and exitant light, because the same point has potentially different
irradiance and radiant exitance.
20.1.5 Radiance
Although irradiance tells us how much light is arriving at a point, it tells us little
about the direction that light comes from. To measure something analogous to
what we see with our eyes, we need to be able to associate “how much light” with
a specific direction. We can imagine a simple device to measure such a quantity
(Figure 20.1). We use a small irradiance meter and add a conical “baffler” which
limits light hitting the counter to a range of angles with solid angle Δσ.The
response of the detector is as follows:
Figure 20.1. By adding
a blinder that shows only
a small solid angle Δσ to
the irradiance detector, we
measure radiance.
response =
ΔH
Δσ
=
Δq
ΔA Δσ Δt Δλ
.
This is the spectral radiance of light travelling in space. Again, we will drop the
“spectral” in our discussion and assume that it is implicit.
Δ
Δ
Figure 20.2. The signal a radiance detector receives does not depend on the distance to
the surface being measured. This figure assumes the detectors are pointing at areas on the
surface that are emitting light in the same way.
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