Page 274 - Modern Optical Engineering The Design of Optical Systems

254 Chapter Twelve

remember that the discussion is equally valid for photometry, if
lumens are read for watts.

12.2 The Inverse Square Law; Intensity

Consider a hypothetical point (or “sufficiently” small) source of radiant
energy, which is radiating uniformly in all directions. If the rate at
which energy is radiated is P watts, then the source has a radiant
intensity J of P/4 watts per steradian,* since the solid angle into
which the energy is radiated is a sphere of 4 steradians. Of course
there are no truly “point” sources and no practical sources which radi-
ate uniformly in all directions, but if a source is quite small relative to
its distance, it can be treated as a point, and its radiation, in the direc-
tions in which it does radiate, can be expressed in watts per steradian.
If we now consider a surface which is S cm from the source, then 1 cm 2
2
of this surface will subtend 1/S steradians from the source (at the
point where the normal from the source to the surface intersects the
surface, if S is large). The irradiance H on this surface is the incident
radiant power per unit area and is obtained by multiplying the inten-
sity of the source in watts per steradian by the solid angle subtended
by the unit area. Thus, the irradiance is given by
1
H J (12.1)
S 2

2
The units of irradiance are watts per square centimeter (W/cm ).
Equation 12.1 is, of course, the “inverse square” law, which is conven-
tionally stated: the illumination (irradiance) on a surface is inversely
proportional to the square of the distance from the (point) source.
Thus, if our uniformly radiating point source emits energy at a rate
1
of 10 W, it will have an intensity J 10/4 0.8 W ster , and the
2
radiation falling on a surface 100 cm away would be 0.8 10 4 W/cm ,
2
or 80 W/cm . If the surface is flat, the irradiance will, of course, be
less than this at points where the radiation is incident at an angle,
since the solid angle subtended by a unit of area in the surface will be
reduced. From Fig. 12.1 it can be seen that the source-to-surface distance

*A steradian is the solid angle subtended (from its center) by 1/4 of the surface area
of a sphere. Thus, a sphere subtends 4 (12.566) steradians from its center; a hemi-
sphere subtends 2 steradians. The size of a solid angle in steradians is found by deter-
mining the area of that portion of the surface of a sphere which is included within the
solid angle and dividing this area by the square of the radius of the sphere. For a small
solid angle, the area of the included flat surface normal to the “central axis” of the angle
can be divided by the square of the distance from the surface to the apex of the angle to
determine its size in steradians. One can visualize a steradian as a cone with an apex
angle of about 65.5°, or 3283 square degrees.