- geometry of propagation
- description of propagation
- relate signal to radiance
- description of entire measurement
Review of radiant quantities
Quantity Symbol Units
Radiant flux W
Radiance L = /(A ) W/(m2 sr)
Irradiance E = /A W/m2
Intensity I = / W/sr
dA = dx dy
dA = r dr d
dA = r sin d d
relates flux to radiance d = dL ·d
also called etendue or geometrical extent
Radiance L is invariant along any ray path
Since flux is also conserved, the throughput is also conserved along any ray path
or infinitesimal areas
d = (projected area of dA1) (solid angle from dA1 to dA2)
For finite areas
Example 1: infinitesimal area colinear with disk
since cos = D/s
Where m is angle from dA1 to edge of disk
Example 2: two parallel, colinear disks
xample 3: infinitesimal area at center of hemisphere
Notice that 2 dA1
For sample with area dA1 and reflectance , reflected flux
= ·E·dA1 = L· => ·E = ·L
= F hem = F A1
For Example 1,
For Example 2,
For Example 3,
F = 1
II. Geometrical Optics
Laws of geometrical optics
1. Law of Transmission – in a region of constant refractive index, light travels in a straight line
2. Law of Reflection – incident angle = reflected angle
3. Law of Refraction – Snell’s Law
n1 sin1 = n2 sin2
Levels of complexity
1. Thin lens
small angles => sin =
optical elements have no thickness
small angles => sin =
We will be limited to the thin lens approximation
Analysis of optical system
1. Lay out the optical system with all the elements, distances, sizes, etc.
2. Draw an axial ray (from source on axis) through the optical system. The first element that the axial ray encounters with increasing angle is the aperture stop. This stop limits the amount of flux collected. Also called the entrance aperture.
3. Draw a chief ray (from the center of the aperture stop on axis) through the optical system. The first element that limits the chief ray encounters with increasing angle is the field stop. This stop limits the extent of the image.
4. The image of the aperture stop in image space is the entrance pupil, its image in object space is the exit pupil.
5. The image of the field stop in image space is the entrance window, its image in object space is the exit window.
6. The ray from the center of the entrance pupil to the edge of the field stop is the field of view.
7. The throughput is calculated from the entrance pupil and window or the exit pupil and window.
Decreasing d1 increases
Relation between source size and field of view (FOV)
= L· = E·A
1. FOV < Source
Detector is field stop
Increasing d0 does not change or
Measuring radiance L
2. FOV > Source
Source is field stop
Increasing d0 decreases and
Measuring irradiance E
Thin lens equation
f is the focal length
so is the object distance
si is the image distance
If si < 0 the image is virtual
If M < 0 the image is inverted
imaging radiometer with aperture stop
Same throughput using either exit or entrance pupils and windows
changing position of lens:
lens is new aperture stop, changing the throughput
III. Measurement Equation
Relates the output signal of a detector to the radiant flux reaching the detector from the source
Ultimately relates the signal to the radiant properties of the source, such as its radiance or irradiance
Basic input-output relation:
S = R ·
where output = S = detector signal
input = = radiant flux
R = response function
spatial – area a and direction
spectral – wavelength
other – time t, polarization , etc.
Propagation from source to detector:
Source – temperature T(a, ) and emissivity (T, , a, )
Propagation – radiance L(T, ) = L(T, , , a, )
Collection – throughput (a, )
d = dL · d
Selection – filter transmittance ()
d = dL · d ·
Detection – detector responsivity R(), amplifier gain G and signal S
dS = d · R · G
combining the expressions from above:
dS = dL(T, , , a, ) · d(a, ) · () · R() · G
the total signal is a multiple integral over all the variables, this is the measurement equation:
this measurement equation relates the thermal properties of the source to the spatial and spectral properties of the detector
note: this measurement equation does not explicitly include other variables on which it might depend, such as time, polarization, etc.
radiance is constant over area and direction
radiance and responsivity vary slowly over the wavelength range of , which is peaked at a wavelength 0.
Often, all the parameters of a radiation thermometer are not known, so it is calibrated with a source of known radiance, yielding
Glossary of Symbols
F configuration factor
r radius in polar and spherical coordinates
x x dimension in Cartesian coordinates
y y dimension in Cartesian coordinates
azimuthal angle in spherical coordinates
polar angle in polar and spherical coordinates
EP entrance pupil
EW entrance window
FOV field of view
f focal length
n index of refraction
si image distance
so object distance
XP exit pupil
XW exit window
C calibration constant
R response function
D. P. DeWitt and G. D. Nutter (eds.), Theory and Practice of Radiation Thermometry, John Wiley and Sons, 1998 (Chpt. 4).
D. C. O’Shea, Elements of Modern Optical Design, John Wiley and Sons, 1985.
C. L. Wyatt, Radiometric System Design, MacMillan Publishing, 1987.
C. L. Wyatt, Radiometric Calibration: Theory and Methods, Academic Press, 1978.
R. W. Boyd, Radiometry and the Detection of Optical Radiation, John Wiley and Sons, 1983.
F. Grum and R. J. Becherer, Optical Radiation Measurements: Vol. 4, Radiometry, Academic Press, 1979.
E. Hecht, Optics, Addison-Wesley Publishing, 1987.
NIST Short Course Material
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