Page 284 - Modern Optical Engineering The Design of Optical Systems

264 Chapter Twelve

functions of wavelength. To do this we refer to the above quantities per
unit interval of wavelength. Thus, if a source emits 5 W of radiant
power in the spectral band between 2 and 2.1 m, it emits 50 W per
micrometer (W/ m) in this region of the spectrum. The standard symbol
for this type of quantity is the symbol given in Fig. 12.5 subscripted
with a , and the name is preceded by “spectral.” For example, the
symbol for spectral radiance is N and its units are watts per steradi-
1
an per square centimeter per micrometer (W ster 1 cm 2 m ).
In many applications it is absolutely necessary to take the spectral
characteristics of sources, detectors, optical systems, filters, and the like
into account. This is accomplished by integrating the particular radia-
tion product function over an appropriate wavelength interval. Since
most spectral characteristics are not ordinary functions, the process of
integration is usually numerical, and thus laborious. As a brief example,
suppose that the irradiance in an image is desired. The spectral radiance
of the object can be described by some function N( ) and the transmis-
sion of the atmosphere, the optical system, and any filters can be com-
bined in a spectral transmission function T( ). Equation 12.11 will give
the irradiance of the image (for any given wavelength); for use over an
extended wavelength interval, we must write
2
2
2
2
H
T( ) N( ) sin d sin
T( ) N ( ) d W/cm 2 (12.12)
1 1
where 1 and 2 , the limits of the integration, may be zero and infinity,
but are usually taken as real wavelengths which encompass the region
of interest. In practice, it is usually necessary to perform the integration
numerically; this process is represented (for this particular example)
by the summation:
2
2
H sin
T( ) N( )
W/cm 2 (12.13)
1
The spectral response of a detector is included in a calculation in the
same manner. For example, the effective power falling on a detector
with an area of A and a relative spectral response R( ), when the
detector is located in the image plane of the system above, would be
(provided that the image completely covered the detector)

2
2
P A sin
R( ) T( ) N( ) d W
1
12.8 Blackbody Radiation
A perfect blackbody is one which totally absorbs all radiation incident
upon it. The radiation characteristics of a heated blackbody are subject