Before tackling radiance we need one more result. Consider a monodirectional, monochromatic, uniform beam of light. To obtain a measure of the amount of radiant energy transported by the beam we imagine a surface A to be placed in the beam with the normal to the surface parallel to it (Fig. 4.2). We can in principle determine how many photons in unit time No cross A ; No multiplied by the photon energy is the amount of radiant power (energy per unit time) crossing A. Divide this quantity by A to obtain the radiant power crossing unit area. This quantity is not solely a property of the beam (the radiation field): it also depends on the orientation of A. Tilt A so that its normal makes an angle M with the direction of the beam (chosen so that cos M is positive). Now we measure N photons crossing A per unit time. This is related to the previous number by

It follows from this equation that

No N

Figure 4.2: The rate at which radiant energy is transported across an area A in a monodirectional radiation field depends on the orientation of A.

and hence the quantity

where N is the rate at which photons cross A for any is a property solely of the radiation field. Geometrically, A cos is the area of the surface projected onto a plane perpendicular to the direction of the beam.

We use the term radiation field to describe the beam. For our purposes a field is any physical quantity that varies in space and time, usually continuously except possibly across surfaces. Field quantities often satisfy differential equations.

The (scalar) radiation field (not to be confused with the underlying vector electromagnetic field) is specified by the radiance L, a non-negative distribution function much like the distribution functions discussed in Section 1.2. As we show in Section 6.1.2, L satisfies an integro-differential equation, another reason for saying that L specifies a radiation field. Like all distribution functions, L is defined by its integral properties, and in general depends on position, direction, frequency, and time, so we sometimes write it as L(x, Q,u,t) to explicitly indicate these dependencies. At any point in space consider a planar surface of area A, a set of directions with solid angle Q, a set of frequencies between and w2, and a time interval between t1 and t2. The total amount of radiant energy confined to this set of frequencies and directions, and crossing this surface in the specified time interval is given by pt2 p /■

Jt 1 Ju1 JaJQ

where © is the angle between the normal to the surface and the direction n. The cosine factor is introduced so that L is a property solely of the radiation field, not of the orientation of A [see Eq. (4.11)]. The dimensions of L are power per unit area, per unit solid angle, per unit frequency. The radiance defined by Eq. (4.12) is sometimes called the spectral or monochromatic radiance, and its dependence on frequency or, equivalently, wavelength sometimes indicated by a subscript: Lu, Lv, Lx . The total or integrated radiance is the integral of L over a range of frequencies. Unless specified otherwise, by radiance we mean spectral radiance.

4 Radiometry and Photometry: What you Get and What you See £2

Figure 4.3: Radiant energy E(A, Q, Aw, At), confined to a solid angle Q around the direction O, is incident on a detector with area A. The unit vector n is normal to A.

Before writing Eq. (4.12) in a different way, we review the mean-value theorem of integral calculus, which we invoked in Section 1.2. According to this theorem (for a one-dimensional integral), between any two values x\ and x2, there is some intermediate value, call it x, such that where f is any continuous and bounded function. The mean-value theorem doesn't tell us how to find x, only that it exists. The geometrical interpretation of this theorem is straightforward. The integral in Eq. (4.13) is the area of the region bounded by the continuous curve y = f (x), the x-axis between xi and x2, and lines perpendicular to the x-axis of length yi = f (xi) and V2 = ). According to the mean-value theorem there is some value y on this curve such that the area is y(x2 - x\).

The mean-value theorem also holds for multiple integrals such as Eq. (4.12), which therefore is equal to

where L and cos © indicate some value of L and cos © over the domain of integration, Aw = w2 — w1, and At = t2 —11. Equation (4.14) provides a means by which we can (in principle) measure the radiance at a point and in a particular direction. Place a detector with area A at the point where L is to be measured (Fig. 4.3). The detector is collimated in that it receives radiation only over a set of directions with solid angle Q, around some direction O, and is equipped with a filter that passes only radiation in some frequency interval Aw. Measure the total radiant energy E(A, Q, Aw, At) received by this detector over some time interval At. Divide this energy by Acos ©QAwAt to obtain an estimate for L. (A can be oriented relative to O so that cos © has the limiting value 1 as Q shrinks.) Form the quotient

for ever-decreasing values of A, l, Aw, and At until it no longer changes, then stop. At this point a fractional change in either A, l, Aw, or At leads to the same fractional change in E(A, i, Aw, At). The radiation field is now uniform over the geometric quantities A and l and the intervals At and Aw. The quotient so obtained is L at space point x, in the direction O, for frequency w, at time t.

We cannot let A become indefinitely small because at some point (A < A2) the concept of a continuous radiance becomes invalid. But this is hardly cause for concern because we run into this kind of limitation all the time. For example, the density p at a point in a fluid often is defined as

where V is a volume containing the point and M the mass of fluid within this volume. But the limit of V here is not literally 0. When V shrinks to molecular dimensions (or smaller), the quotient in Eq. (4.16) undergoes wild fluctuations depending on whether V contains molecules or not. So the limit in Eq. (4.16) is interpreted to mean that we shrink V until the quotient no longer changes, then stop and call that quotient the density at a point. But a better definition of density, in our view, is that it is a distribution function. Its integral over any arbitrary (within limits) volume is the mass enclosed by that volume.

The only way to truly grasp radiance, or indeed any physical concept, is to become familiar with its properties, to observe how it behaves in as many contexts as possible. Defining radiance is only a first small step toward understanding it. One essential property of radiance is that it is additive: if several incoherent sources contribute to the radiance at a particular point and in a particular direction, the total radiance is the sum of the radiances from each source as if it were acting alone. Another property of radiance is its invariance, which we turn to next.

If absorption and scattering by the medium in which radiation propagates is negligible, radiance is invariant along a particular direction. By this is meant the following. Go to any point in the radiation field and determine the radiance there in the direction O. Now proceed along this direction. At any point on this path the radiance in the direction O is the same. The proof is as follows.

At any point insert a planar surface with area A, sufficiently small that L in the direction O is the same over every point of A, oriented so that its normal is parallel to O. The quantity LA is therefore the amount of radiation crossing A per unit solid angle around the direction O. A surface A" at a distance r from A receives an amount of radiant energy

A"

where Q' is the solid angle subtended by A" at A (both of which are C r2 ). This is shown schematically in Fig. 4.4. If A'' is sufficiently small, the power intercepted by it (per unit area) is uniform and equal to