## 12.1 Light Emission

All objects with temperature above absolute zero have moving atoms. In turn, as described by Maxwell’s equations, the motion of atomic particles that hold electrical charges causes objects to emit electromagnetic radiation over a range of wavelengths. As we’ll see shortly, most of the emission is at infrared frequencies for objects at room temperature; objects need to be much warmer to emit meaningful amounts of electromagnetic radiation at visible frequencies.

Many different types of light sources have been invented to convert energy into emitted electromagnetic radiation. Understanding some of the physical processes involved is helpful for accurately modeling light sources for rendering. A number are in wide use today:

- Incandescent (tungsten) lamps have a small tungsten filament. The flow of electricity through the filament heats it, which in turn causes it to emit electromagnetic radiation with a distribution of wavelengths that depends on the filament’s temperature. A frosted glass enclosure is often present to absorb some of the wavelengths generated in order to achieve a desired SPD. With an incandescent light, much of the power in the SPD of the emitted electromagnetic radiation is in the infrared bands, which in turn means that much of the energy consumed by the light is turned into heat rather than light.
- Halogen lamps also have a tungsten filament, but the enclosure around them is filled with halogen gas. Over time, part of the filament in an incandescent light evaporates when it’s heated; the halogen gas causes this evaporated tungsten to return to the filament, which lengthens the life of the light. Because it returns to the filament, the evaporated tungsten doesn’t adhere to the bulb surface (as it does with regular incandescent bulbs), which also prevents the bulb from darkening.
- Gas-discharge lamps pass electrical current through hydrogen, neon, argon, or vaporized metal gas, which causes light to be emitted at specific wavelengths that depend on the particular atom in the gas. (Atoms that emit relatively little of their electromagnetic radiation in the not-useful infrared frequencies are selected for the gas.) Because a broader spectrum of wavelengths are generally more visually desirable than the chosen atoms generate directly, a fluorescent coating on the bulb’s interior is often used to transform the emitted frequencies to a wider range. (The fluorescent coating also helps by converting ultraviolet wavelengths to visible wavelengths.)
- LED lights are based on electroluminescence: they use materials that emit photons due to electrical current passing through them.

For all of these sources, the underlying physical process is electrons colliding with atoms, which pushes their outer electrons to a higher energy level. When such an electron returns to a lower energy level, a photon is emitted. There are many other interesting processes that create light, including chemoluminescence (as seen in light sticks) and bioluminescence—a form of chemoluminescence seen in fireflies. Though interesting in their own right, we won’t consider their mechanisms further here.

*Luminous efficacy* measures how effectively a light source converts
power to visible illumination, accounting for the fact that for human
observers, emission in non-visible wavelengths is of little value.
Interestingly enough, it is the ratio of a photometric quantity (the
emitted luminous flux) to a radiometric quantity (either the total power it
uses or the total power that it emits overall wavelengths, measured in
flux):

where is the spectral response curve introduced in Section 5.4.3.

Luminous efficacy has units of lumens per Watt. If is the power consumed by the light source (rather than the emitted power), then luminous efficacy also incorporates a measure of how effectively the light source converts power to electromagnetic radiation. Luminous efficacy can also be defined as a ratio of luminous exitance (the photometric equivalent of radiant exitance) to irradiance at a point on the surface, or as the ratio of exitant luminance to radiance at a point on a surface in a particular direction.

A typical value of luminous efficacy for an incandescent tungsten lightbulb is around . The highest value it can possibly have is 683, for a perfectly efficient light source that emits all of its light at , the peak of the function. (While such a light would have high efficacy, it wouldn’t necessarily be a pleasant one as far as human observers are concerned.)

### 12.1.1 Blackbody Emitters

A *blackbody* is a perfect emitter: it converts power to
electromagnetic radiation as efficiently as physically possible. While
true blackbodies aren’t physically realizable, some emitters exhibit
near-blackbody behavior. Blackbodies also have a useful closed-form
expression for their emission as a function of temperature and wavelength
that is useful for modeling non-blackbody emitters.

Blackbodies are so-named because they absorb absolutely all incident power, reflecting none of it. Thus, an actual blackbody would appear perfectly black, no matter how much light was illuminating it. Intuitively, the reasons that perfect absorbers are also perfect emitters stem from the fact that absorption is the reverse operation of emission. Thus, if time was reversed, all of the perfectly absorbed power would be perfectly efficiently re-emitted.

*Planck’s law* gives the radiance emitted by a blackbody as a function of
wavelength and temperature measured in Kelvins:

where is the speed of light in the medium ( in a vacuum), is Planck’s constant, , and is the Boltzmann constant, , where Kelvin (K) is the unit of temperature. Blackbody emitters are perfectly diffuse; they emit radiance equally in all directions.

Figure 12.1 plots the emitted radiance distributions of a blackbody for a number of temperatures.

The `Blackbody()` function computes emitted radiance at the given
temperature `T` in Kelvin for the `n` wavelengths in
`lambda`.

`lambda[i]`>>

The `Blackbody()` function takes wavelengths in nm, but the
constants for Equation (12.1) are in terms of meters.
Therefore, we must first convert the wavelength to meters by scaling it by
.

`lambda[i]`>>=

The *Stefan–Boltzmann law* gives the radiant exitance (recall that
this is the outgoing irradiance) at a point for a blackbody
emitter:

where is the Stefan–Boltzmann constant, . Note that the total emission over all frequencies grows very rapidly—at the rate . Thus, doubling the temperature of a blackbody emitter increases the total energy emitted by a factor of 16.

Because the power emitted by a blackbody grows so quickly with temperature,
it can also be useful to compute the normalized SPD for a blackbody where
the maximum value of the SPD at any wavelength is 1. This is easily done
with *Wien’s displacement law*, which gives the wavelength where emission of
a blackbody is maximum given its temperature:

where is Wien’s displacement constant, .

`Le`values based on maximum blackbody radiance>>

Wien’s displacement law gives the wavelength in meters where emitted
radiance is at its maximum; we must convert this value to nm before calling
`Blackbody()` to find the corresponding radiance value.

`Le`values based on maximum blackbody radiance>>=

The emission behavior of non-blackbodies is described by *Kirchoff’s
law*, which says that the emitted radiance distribution at any frequency is
equal to the emission of a perfect blackbody at that frequency times the
fraction of incident radiance at that frequency that is absorbed by the
object. (This relationship follows from the object being assumed to be in
thermal equilibrium.) The fraction of radiance absorbed is equal to 1
minus the amount reflected, and so the emitted radiance is

where is the emitted radiance given by Planck’s law, Equation (12.1), and is the hemispherical-directional reflectance from Equation (8.1).

The blackbody emission distribution provides as useful metric for
describing the emission characteristics of non-blackbody emitters through
the notion of *color temperature*. If the shape of the SPD of an
emitter is similar to the blackbody distribution at some temperature, then
we can say that the emitter has the corresponding color temperature. One
approach to find color temperature is to take the
wavelength where the light’s emission is highest
and find the corresponding temperature using
Equation (12.3).

Incandescent tungsten lamps are generally around 2700 K color temperature, and tungsten halogen lamps are around 3000 K. Fluorescent lights may range all the way from 2700 K to 6500 K. Generally speaking, color temperatures over 5000 K are described as “cool,” while 2700–3000 K is described as “warm.”

### 12.1.2 Standard Illuminants

Another useful way of categorizing light emission distributions are a number of “standard illuminants” that have been defined by Commission Internationale de l’Éclairage (CIE), which also specified the XYZ matching curves that we saw in Section 5.2.1.

The Standard Illuminant A was introduced in 1931 and was intended to represent average incandescent light. It corresponds to a blackbody radiator of about . (It was originally defined as a blackbody at , but the precision of the constants used in Planck’s law subsequently improved. Therefore, the specification was updated to be in terms of the 1931 constants, so that the illuminant was unchanged.) Figure 12.2 shows a plot of the SPD of the A illuminant.

(The B and C illuminants were intended to model daylight at two times of day and were generated with an A illuminant in combination with specific filters. They are no longer used. The E illuminant is defined as a constant-valued SPD and is used only for comparisons to other illuminants.)

The D illuminant describes various phases of daylight. It was defined based on characteristic vector analysis of a variety of daylight SPDs, which made it possible to express daylight in terms of a linear combination of three terms (one fixed and two weighted), with one weight essentially corresponding to yellow-blue color change due to cloudiness and the other corresponding to pink-green due to water in the atmosphere (from haze, etc.). D65 is roughly color temperature (not —again due to changes in the values used for the constants in Planck’s law) and is intended to correspond to mid-day sunlight in Europe. (See Figure 12.3.) The CIE recommends that this illuminant be used for daylight unless there’s a specific reason not to.

Finally, the F series of illuminants describes fluorescents; it is based on measurements of a number of actual fluorescent lights. Figure 12.4 shows the SPDs of two of them.

The files named `cie.stdillum.*` in the `scenes/spds` directory
in the `pbrt` example scenes distribution have the SPDs of the standard illuminants,
measured at 5-nm increments from 300 nm to 830 nm.