Light is a superposition of electromagnetic waves, intertwined electric fields E and magnetic fields H. Because these fields are vectors, so are electromagnetic waves. They satisfy vector wave equations similar to the scalar wave equation derived in Section 3.3 for the vibrating string. We usually are most interested in the rate at which radiant energy is transported by electromagnetic waves. The electric and magnetic fields determine this transport rate by way of the Poynting vector

where E and H are the fields at a point. If this point is on a surface (real or mathematical), the rate at which radiant energy is transported across unit area of that surface is

where n is a unit vector normal to the surface. The derivation of Eq. (4.1), boiled down to its essence, is fundamentally no different from, although more complicated than, the derivation in Section 3.3 of the energy flux vector for a vibrating string. We determined the time rate of change of kinetic and potential energy of a finite length of string, then noted that this was

Fundamentals of Atmospheric Radiation: An Introduction with 400 Problems. Craig F. Bohren and Eugene E. Clothiaux Copyright © 2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-40503-8

equal to the difference in energy fluxes at its end points. Similarly, Eq. (4.1) is obtained by determining the time rate of change of electric and magnetic energy within a bounded volume and noting that this is equal to the integral of the Poynting vector over the bounding surface. The energy flux vector for the string, Eq. (3.27), is the product of two functions. Similarly, the energy flux vector for the electromagnetic field, Eq. (4.1), is the (vector) product of two fields.

The scalar quantity Eq. (4.2), with the dimensions of power per unit area, is what we are after, but to get it we need two vector fields (E and H) in order to determine a third (S). This is possible in principle, but, except in very restricted circumstances, is essentially impossible in practice. We don't live in a world of simple electromagnetic waves. Just look around the room in which you are reading these words. You receive light from all directions, differing in amount and in its spectrum for each direction, a complex mosaic that changes when you move. How on earth could anyone ever sum the electric and magnetic fields originating from so many sources: walls, floor, ceiling, furniture, this book itself? Fortunately, we often don't have to determine these vector fields but can go straight to the desired scalar quantity, radiant energy transport. We can circumvent the electromagnetic field because most sources of visible and near-visible radiation are incoherent: there is no fixed phase relation between radiation from the walls and from the floor, from one part of a wall, and from another. Thus we can add the radiant energy from each source and ignore phase differences because they wash out when integrated over space or time. Another reason we can circumvent electromagnetic fields is that the wavelengths of the radiation of interest usually are much smaller than the objects with which the radiation interacts. If our radiation environment consisted of waves with wavelengths of order meters or more we might be in trouble. As we show in Section 3.4.2, the lateral coherence length of a source increases with its wavelength.

Radiometry is based on approximating electromagnetic radiation as a gas of photons. Like gas molecules, photons may be distributed in energy and in direction, and the properties of a photon gas may vary from point to point and from moment to moment. But photons do not interfere (see Sec. 3.4): they don't have phases. Or perhaps it would be more correct to say that phase differences wash out when we take averages over space or time, and so it is as if they had no phases. Ignoring phases results in an enormous simplification. Indeed, it makes radiometry (the measurement of radiation) possible. But all simplification comes with a price. We have to be alert to possible discrepancies between theory and observations. A theory in which phase is tossed out the window cannot possibly account for phenomena in which interference plays an important role.

The fundamental radiometric quantity is radiance, which to understand requires a thorough grasp of solid angle, which we turn to next. Solid angle is almost entirely absent from electromagnetic theory but plays a central role in radiometry.

### 4.1.1 Solid Angle

Any direction in a plane can be specified by a vector. Suppose that a vector r in the plane is rotated about a point O into another direction. The tip of the rotated vector traces out an arc of length s. The angle between these two directions, denoted by is defined as the ratio s/r, and lies between 0 and 2n radians (the circumference of a circle of radius r is 2nr). But another interpretation of angle is that it is the measure (size) of the set of all directions from O to points on the arc. Length is the measure of the set of all points lying on a line between two points. Area is the measure of the set of all points within a closed curve on a surface. And volume is the measure of the set of all points in space within a closed surface. These measures provide a way of comparing the size of one set of points with that of another. Far from being abstract, they are the means by which prices are assigned to rope, parcels of land, and gasoline (the gallon and liter are volumetric measures). Similarly, angle provides a way of comparing sets of directions in the plane. But directions are not confined to two-dimensional space. What is the measure of directions in three-dimensional space?

Consider a spherical surface of radius r on which a closed curve is inscribed (Fig. 4.1); the area of that part of the surface within this curve is A. Every vector from the origin O to a point on A specifies a direction in space. The measure of the set of all these directions is its solid angle

which lies between 0 and 4n steradians (abbreviated as sr). In principle, Q can be determined by evaluating a surface integral:

Q = -i- J J r2 sin ¡9 d;& dlP = Jlsini51 M (4-4)

where and y are spherical polar coordinates (co-latitude and azimuth, respectively) and the limits of integration are determined by A. Let's use Eq. (4.4) to determine the solid angle subtended by the sun at the surface of Earth, by which we mean the solid angle of the set of all directions from a point on Earth to the sun. We need this quantity later. The sun is azimuthally symmetric and its angular width, Ms, is about 0.5° (n/360 rad). The solid angle subtended by the sun is therefore f2n f#s/2 ( M

Qs= J J sin dd dtp = 2tt ^ 1 - cos y ) « 6 x 1(T s sr. (4.5)

Because Ms c 1, we can expand the cosine in Eq. (4.5) and truncate after the first two terms to obtain tts « n&2/Asr. (4.6)

We could have obtained Eq. (4.6) by dividing the area of a small (planar) disc of radius rMs/2 by r2.

Any direction in space can be specified by a unit vector O, and so we sometimes write Eq. (4.4) symbolically as tt = I dO. (4.7)

This does not, however, denote integration over the variable O just as a volume integral written as

Jv does not denote integration over the variable V. Equation (4.7) is simply a more compact way of writing Eq. (4.4).