A Appendix: Radiometry and Photometry This appendix deals with the photometric and radiometric variables that are useful in setting and understanding problems about color and energy exchange. All eletromagnetic radiation transfers energy, called radiant energy. Radiometry is the science of the measurement of the physical variables as- sociated with the propagation and exchange of radiant energy. Photometry is the branch of radiometry that deals with these variables from the viewpoint of visual responses; that is, it studies how radiant energy is perceived by an observer. Thus, radiometry in general deals with physical processes, while photometry deals with psychophysical ones. A.1 Radiometry Although an understanding of the physics of electromagnetic waves is impor- tant in the study of light–matter interaction, we need not be concerned with the nature of electromagnetism in the study of radiometry and photometry. It is enough to know that radiant energy flows through space; the fundamental variable involved is the flux through a surface, that is, the rate at which radi- ant energy is transferred through a surface. This notion is therefore analogous to the notion of an electric current or to the flow of matter in fluid dynamics. In contrast with the situation in fluid dynamics, however, there exist point sources of radiant energy, and indeed they are of great importance in radio- metry. To define the radiometric variables associated with point sources, we introduce the notion of solid angles. Solid Angles In the plane, angles can be regarded as a measure of apparent length from an observer’s viewpoint. To compute the angle subtended by an object when 440 A Appendix: Radiometry and Photometry Fig. A.1. Angles and radial projection. observed from a point O, we project the object radially onto the unit circle centered at O and measure the arc determined by this projection; this is how big the object looks to someone stationed at O (Figure A.1). We can also use a circle of radius other than 1, but in this case we must divide the length of the projected arc by the radius of the circle. Thus, angles are dimensionless quantities. Radians and other units of angle measurement are a notational device to avoid confusion when working with such measure- ments: a radian is simply the number 1; a degree is the number π/180; and so on. This way of looking at angles can be extended to higher dimensions, leading to the definition of a solid angle, that is, a measurement of the apparent area as seen from a point. Consider a subset A of space and a viewpoint O.The solid angle ω determined by A (with respect to O) is the area of the radial projection of A onto the unit sphere centered at O, which we call the visual sphere (Figure A.2). As in the plane case, we can take as the visual sphere Fig. A.2. Measuring a solid angle. A.1 Radiometry 441 a sphere of arbitrary radius r = 1, but then we must divide the area of the projection by the square of the radius. Thus, the solid angle is given by Area(A) ω = . (A.1) r2 The radial projection of A onto the visual sphere determines a cone with vertex O and base A, that is, the solid formed by all the rays (half-lines) in space starting at O and going through points of A (Figure A.3). The solid angle defined by (A.1) can also be regarded as a measurement of this cone. Two subsets of the plane that subtend the same angle (from a fixed point of view O) are called perceptually congruent (with respect to O); this is illustrated in Figure A.4, left. Similarly, subsets of space that determine the same cone (or, which is the same, the same projection on the visual sphere) are called perceptually congruent; see Figure A.4, right. Since (A.1) is a ratio of two areas, solid angles are dimensionless. However, just as in the case of plane angles, it is comforting to be able to use a unit when Fig. A.3. Cone with vertex O and base A. Fig. A.4. Perceptual congruence. 442 A Appendix: Radiometry and Photometry Fig. A.5. Element of solid angle. discussing measurements of solid angles. The standard unit, representing the number 1, is the steradian, abbreviated sr. The whole visual sphere has solid angle 4π sr, since the area of a sphere of radius r is 4πr2. When integrating over the visual sphere (or part thereof), we will consider an infinitesimal element of solid angle dω. Pictorially, we represent dω by a vector pointing radially away from O; see Figure A.5. A.1.1 Radiometric Magnitudes Suppose a light bulb is turned on, left on for a while, then turned off. There are several measurements that one might be interested in: the total energy emitted by the bulb during this period; the energy emitted per second; the energy that reaches a certain target; the brightness as seen from that target; and so on. Radiant Flux The total energy emitted is denoted by Qe, and it is measured in units of energy: joules in the MKS system. The radiant flux Φe is the rate at which energy is being emitted: dQ Φ = e . e dt In the MKS system it is measured in joules per second, also known as watts. Irradiance The energy emitted by the light source can also be considered to be going through a closed surface surrounding the source, so it makes sense to consider the flux density—that is, flux per unit area—at points of such a closed surface. Flux density, also called irradiance and denoted Ee, is vector-valued: given a point P and an element of surface containing P ,havingareadA, the flux through that surface element is Ee cos θdA, where Ee is the magnitude of the irradiance vector at P and θ is the angle between the irradiance vector and the normal to the area element (see Figure A.6, where the flux through the A.1 Radiometry 443 Fig. A.6. The amount of radiant energy crossing a surface per unit time depends on which way the surface faces. surface is negative on the left, zero in the middle, and positive on the right). Loosely, we can write dΦ E = e . e dA Irradiance is measured (in the MKS system) in watts per square meter. Radiant Intensity The irradiance depends, of course, on how far the point of measurement is from the source (and usually also on the direction as seen from the source). It is often useful to work instead with a magnitude that is associated with the source itself. For a point light source, this is easy: we just consider flux per solid angle instead of flux per area. The radiant intensity Ie of a point source is defined as dΦ I = e , e dω where dω is the element of solid angle as seen from the source and is measured in watts per steradian. (Note that this does not make sense unless the light source has negligible extension, since the notion of solid angle depends essen- tially on the choice of an origin.) The radiant intensity is a function of the direction as seen from the source. When we integrate it over all directions, we recover the total flux, Φe = Iedω. For a source that sheds light uniformly in all directions, Ie is constant and Φe =4πIe. Clearly, the flux density an observer perceives at a distance d from the 2 source is Ee = Ie/d . Radiance This concept can be adapted to the case of nonpoint light sources, as follows. Consider a surface S that delimits the source in question—the surface of a light bulb, say, or a sphere around it. (The light need not be generated on S; it is the flux through S that concerns us.) If we take an element of the surface, of area dA, we can regard it as a point source and look at its radiant intensity 444 A Appendix: Radiometry and Photometry Fig. A.7. Radiance of a light source. dIe in a certain direction. The ratio dIe/dA is the density of radiant intensity at the given point of the source surface, in the given direction. Actually, we must take into account that the area of the surface element as seen from the chosen direction is not dA, but dA cos θ, where θ is as in Figure A.7. This leads us to the following definition: the radiance (not to be confused with the irradiance defined earlier) is dI d2Φ L = e = e . e dA cos θ dω dA cos θ We stress that this is a function of the chosen point P on the source surface S and of the chosen direction as seen from P . (In mathematical terms, it is a function on the unit tangent bundle to S.) Radiance is measured in watts per square meter per steradian. Integrating the radiance over the points of the source gives the flux density. More precisely, Le(P, u)cosθdA Ee = uP 2 , S r where P ranges over the surface S, u is the unit vector in the direction from P to the observer, θ is the angle between u and the normal to S at P ,and r is the distance from P to the observer. This equation is a generalization of 2 the earlier formula Ee = Ie/d for point sources. A.1.2 Spectral Distribution So far we have ignored the fact that light is composed of many wavelengths.