4.4 Light Emission

The atoms of an object with temperature above absolute zero are moving. 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 will see shortly, at room temperature most of the emission is at infrared frequencies; 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. An object that emits light is called a lamp or an illuminant, though we avoid the latter terminology since we generally use “illuminant” to refer to a spectral distribution of emission (Section 4.4.2). A lamp is housed in a luminaire, which consists of all the objects that hold and protect the light as well as any objects like reflectors or diffusers that shape the distribution of light.

Understanding some of the physical processes involved in emission is helpful for accurately modeling light sources for rendering. A number of corresponding types of lamps 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 diffuse the emission over a larger area than just the filament and to absorb some of the wavelengths generated in order to achieve a desired distribution of emission by wavelength. With an incandescent light, much of the emitted power 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 is 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 does not 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 is generally more visually desirable than wavelengths that the chosen atoms generate directly, a fluorescent coating on the bulb’s interior is often used to transform the emitted wavelengths to a broader range. (The fluorescent coating also improves efficiency 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 will not 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 over all wavelengths, measured in flux):

StartFraction integral normal upper Phi Subscript normal e Baseline left-parenthesis lamda right-parenthesis upper V left-parenthesis lamda right-parenthesis normal d lamda Subscript Baseline Over integral normal upper Phi Subscript normal i Baseline left-parenthesis lamda right-parenthesis normal d lamda Subscript Baseline EndFraction comma

where upper V left-parenthesis lamda right-parenthesis is the spectral response curve that was introduced in Section 4.1.4.

Luminous efficacy has units of lumens per watt. If normal upper Phi Subscript normal i 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 a 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 15 normal l normal m slash normal upper W . The highest value it can possibly have is 683, for a perfectly efficient light source that emits all of its light at lamda equals 555 normal n normal m , the peak of the upper V left-parenthesis lamda right-parenthesis function. (While such a light would have high efficacy, it would not necessarily be a pleasant one as far as human observers are concerned.)

4.4.1 Blackbody Emitters

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

Blackbodies are so-named because they absorb absolutely all incident power, reflecting none of 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 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 lamda and temperature upper T measured in kelvins:

upper L Subscript normal e Baseline left-parenthesis lamda comma upper T right-parenthesis equals StartFraction 2 h c squared Over lamda Superscript 5 Baseline left-parenthesis normal e Superscript h c slash lamda k Super Subscript b Superscript upper T Baseline minus 1 right-parenthesis EndFraction comma

where c is the speed of light in the medium ( 299,792,458 normal m slash normal s in a vacuum), h is Planck’s constant, 6.62606957 times 10 Superscript negative 34 Baseline normal upper J normal s , and k Subscript b is the Boltzmann constant, 1.3806488 times 10 Superscript negative 23 Baseline normal upper J slash normal upper K , where kelvin (K) is the unit of temperature. Blackbody emitters are perfectly diffuse; they emit radiance equally in all directions.

Figure 4.12 plots the emitted radiance distributions of a blackbody for a number of temperatures.

Figure 4.12: Plots of emitted radiance as a function of wavelength for blackbody emitters at a few temperatures, as given by Equation (4.17). Note that as temperature increases, more of the emitted light is in the visible frequencies (roughly 380 nm–780 nm) and that the spectral distribution shifts from reddish colors to bluish colors. The total amount of emitted energy grows quickly as temperature increases, as described by the Stefan–Boltzmann law in Equation (4.19).

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

<<Spectrum Function Declarations>>= 
Float Blackbody(Float lambda, Float T) { if (T <= 0) return 0; const Float c = 299792458.f; const Float h = 6.62606957e-34f; const Float kb = 1.3806488e-23f; <<Return emitted radiance for blackbody at wavelength lambda>> 
Float l = lambda * 1e-9f; Float Le = (2 * h * c * c) / (Pow<5>(l) * (FastExp((h * c) / (l * kb * T)) - 1)); return Le;

The wavelength passed to Blackbody() is in nm, but the constants for Equation (4.17) are in terms of meters. Therefore, it is necessary to first convert the wavelength to meters by scaling it by 10 Superscript negative 9 .

<<Return emitted radiance for blackbody at wavelength lambda>>= 
Float l = lambda * 1e-9f; Float Le = (2 * h * c * c) / (Pow<5>(l) * (FastExp((h * c) / (l * kb * T)) - 1)); return Le;

The emission of non-blackbodies is described by Kirchhoff’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

upper L prime Subscript normal e Baseline left-parenthesis upper T comma omega comma lamda right-parenthesis equals upper L Subscript normal e Baseline left-parenthesis upper T comma lamda right-parenthesis left-parenthesis 1 minus rho Subscript normal h normal d Baseline left-parenthesis omega right-parenthesis right-parenthesis comma

where upper L Subscript normal e Baseline left-parenthesis upper T comma lamda right-parenthesis is the emitted radiance given by Planck’s law, Equation (4.17), and rho Subscript normal h normal d Baseline left-parenthesis omega right-parenthesis is the hemispherical-directional reflectance from Equation (4.12).

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

upper M left-parenthesis normal p Subscript Baseline right-parenthesis equals sigma upper T Superscript 4 Baseline comma

where sigma is the Stefan–Boltzmann constant, 5.67032 times 10 Superscript negative 8 Baseline normal upper W normal m Superscript negative 2 Baseline normal upper K Superscript negative 4 . Note that the total emission over all frequencies grows very rapidly—at the rate upper T Superscript 4 . Thus, doubling the temperature of a blackbody emitter increases the total energy emitted by a factor of 16.

The blackbody emission distribution provides a useful metric for describing the emission characteristics of non-blackbody emitters through the notion of color temperature. If the shape of the emitted spectral distribution 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 Wien’s displacement law, which gives the wavelength where emission of a blackbody is maximum given its temperature:

lamda Subscript normal m normal a normal x Baseline equals StartFraction b Over upper T EndFraction comma

where b is Wien’s displacement constant, 2.8977721 times 10 Superscript negative 3 Baseline normal m normal upper K .

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.”

4.4.2 Standard Illuminants

Another useful way of categorizing light emission distributions is a number of “standard illuminants” that have been defined by Commission Internationale de l’Éclairage (CIE).

The Standard Illuminant A was introduced in 1931 and was intended to represent average incandescent light. It corresponds to a blackbody radiator of about 2856 normal upper K . (It was originally defined as a blackbody at 2850 normal upper K , but the accuracy 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 4.13 shows a plot of the spectral distribution of the A illuminant.

Figure 4.13: Plot of the CIE Standard Illuminant A’s Spectral Power Distribution as a Function of Wavelength in nm. This illuminant represents incandescent illumination and is close to a blackbody at 2856 normal upper K .

(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 having a constant spectral distribution 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 spectra, 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 6504 normal upper K color temperature (not 6500 normal upper K —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 4.14.) The CIE recommends that this illuminant be used for daylight unless there is a specific reason not to.

Figure 4.14: Plot of the CIE Standard D65 Illuminant Spectral Distribution as a Function of Wavelength in nm. This illuminant represents noontime daylight at European latitudes and is commonly used to define the whitepoint of color spaces (Section 4.6.3).

Finally, the F series of illuminants describes fluorescents; it is based on measurements of a number of actual fluorescent lights. Figure 4.15 shows the spectral distributions of two of them.

Figure 4.15: Plots of the F4 and F9 Standard Illuminants as a Function of Wavelength in nm. These represent two fluorescent lights. Note that the distributions are quite different. Spikes in the two distributions correspond to the wavelengths directly emitted by atoms in the gas, while the other wavelengths are generated by the bulb’s fluorescent coating. The F9 illuminant is a “broadband” emitter that uses multiple phosphors to achieve a more uniform spectral distribution.