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1 (a) What is the transmission of a stack of three thin plane parallel plates
of glass (n = 1.5) at normal incidence?
(b) What percentage of the incident light is transmitted directly (i.e., with-
out any intervening reflections)?
: (a) 80 percent, (b) (0.96)6 = 78 percent
2 If a 1-cm thickness of a material transmits 85 percent and 2-cm thickness
transmits 80 percent, (a) what percentage will a 3-cm thickness transmit? (b)
What is the absorption coefficient of the material? (Neglect all multiple reflec-
: (a) 75.3 percent; (b) 0.06062 cm-1
3 Determine the coefficients for the dispersion equations given in Section 7.1
for one of the optical glasses listed in Fig. 7.4. Evaluate the accuracy of the
equations by comparing the index values given by the equations with those
listed in the table (for wavelengths not used in determining the constants).
4 Using the spectral transmission curves of Fig. 7.10, plot the spectral trans-
mission which would result from a combination of filters (c) and (f).
5 Plot, in the manner of Fig. 7.14, the curve of reflection against angle of inci-
dence for a single surface of glass (n = 1.52) coated with a quarter wavelength
thickness of magnesium fluoride (n = 1.38).
Optical Materials and Interface Coatings
In concept, both radiometry and photometry are quite straightfor-
ward; however, both have been cursed with a jungle of often bewilder-
ing terminology. Radiometry deals with radiant energy (i.e.,
electromagnetic radiation) of any wavelength. Photometry is restrict-
ed to radiation in the visible region of the spectrum. The basic unit of
power (i.e., rate of transfer of energy) in radiometry is the watt; in pho-
tometry, the corresponding unit is the lumen, which is simply radiant
power as modified by the relative spectral sensitivity of the eye (Fig.
5.10) per Eq. 8.18. Note that watts and lumens have the same dimen-
sions, namely energy per time.
All radiometry must take into account the variation of characteris-
tics with wavelength. Examples are the spectral variation of emission,
the variation of transmission of the atmosphere and optics with wave-
length, and the differences in detector and film response with
wavelength. A convenient way to deal with this is to multiply, wave-
length by wavelength, all such factors together so as to arrive at one
unified spectral weighting function. Thus, all radiometry is spectrally
weighted and it should be apparent that photometry is simply one par-
ticular spectral weighting. See Sec. 8.9.
The principles of radiometry and photometry are readily understood
when one thinks in terms of the basic units involved, rather than the
special terminology which is conventionally used. The next five sec-
tions will discuss radiation in terms of watts; the reader should
remember that the discussion is equally valid for photometry, if
lumens are read for watts.
8.2 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/4π watts per steradian,* since the solid angle into
which the energy is radiated is a sphere of 4π 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
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
The units of irradiance are watts per square centimeter (W/cm
Equation 8.1 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
of 10 W, it will have an intensity J = 10/4π = 0.8 W ster
, and the
radiation falling on a surface 100 cm away would be 0.8 × 10
or 80 μW/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. 8.1 it can be seen that the source-to-surface dis-
tance is increased to S/cos θ and that the effective area (normal to the
*A steradian is the solid angle subtended (from its center) by 1/4π of the surface area
of a sphere. Thus, a sphere subtends 4π (12.566) steradians from its center; a hemi-
sphere subtends 2π 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.
direction of the radiation) is reduced by a cos θ factor. Thus, the solid
angle subtended, and the irradiance, are reduced by a cos
8.3 Radiance and Lambert’s Law
An extended source, that is, one whose dimensions are significant,
must be treated differently than a point source. A small area of the
source will radiate a certain amount of power per unit of solid angle.
Thus, the radiation characteristics of an extended source are
expressed in terms of power per unit solid angle per unit area. This is
called radiance; the usual units for radiance are watts per steradian
per square centimeter (W ster
) and the symbol is N. Note that
the area is measured normal to the direction of radiation, not in the
Most extended sources of radiation follow, at least approximately,
what is known as Lambert’s law of intensity,
is the intensity of a small incremental area of the source in a
direction at an angle θ from the normal to the surface, and J
intensity of the incremental area in the direction of the normal. For
example, a heated metal disk with a total area of 1 cm
and a radiance
of 1 W ster
will radiate 1 W/ster in a direction normal to its sur-
face. In a direction 45° to the normal, it will radiate only 0.707 W/ster
(cos 45° = 0.707).
Notice that although radiance is given in terms of watts per stera-
dian per square centimeter, this should not be taken to mean that the
radiation is uniform over a full steradian or over a full square cen-
timeter. Consider a source consisting of a 0.1-cm square incandescent
Radiometry and Photometry
Geometry of a point
source irradiating a plane,
showing that irradiance (or illu-
mination) varies with cos
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