Lecture 3
Geometrical Optics
Topics:
Throughput
 geometry of propagation
Geometrical Optics
 description of propagation
Measurement Equation
 relate signal to radiance
 description of entire measurement
Review of radiant quantities
Quantity Symbol Units
Radiant flux W
Radiance L = /(A ) W/(m^{2} sr)
Irradiance E = /A W/m^{2}
Intensity I = / W/sr
Rectangular coordinates
dA = dx dy
Polar coordinates
dA = r dr d
Spherical coordinates
dA = r sin d d
I. Throughput 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 F or infinitesimal areas d = (projected area of dA_{1}) (solid angle from dA_{1} to dA_{2})
For finite areas
Example 1: infinitesimal area colinear with disk
since cos_{} = D/s
Also,
Where _{m} is angle from dA_{1} to edge of disk
Example 2: two parallel, colinear disks
where
E xample 3: infinitesimal area at center of hemisphere
Notice that 2 dA_{1}
For sample with area dA_{1} and reflectance , reflected flux
= ·E·dA_{1} = L· => ·E = ·L
Configuration factor
= F _{hem} = F A_{1}
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
n_{1} sin_{1} = n_{2} sin_{2}
Levels of complexity
1. Thin lens
small angles => sin =
optical elements have no thickness
2. Paraxial
small angles => sin =
3. Exact
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.
Twoaperture radiometer
Throughput calculation:
Note:
Decreasing d_{1} increases
Relation between source size and field of view (FOV)
= L· = E·A
1. FOV < Source
Detector is field stop
Increasing d_{0} does not change or
Measuring radiance L
2. FOV > Source
Source is field stop
Increasing d_{0} decreases and
Measuring irradiance E
Thin lens equation
f is the focal length
s_{o} is the object distance
s_{i} is the image distance
If s_{i} < 0 the image is virtual
Magnification
If M < 0 the image is inverted
Imaging radiometer
Throughput calculation:
imaging radiometer with aperture stop
Throughput calculation:
Same throughput using either exit or entrance pupils and windows
changing position of lens:
lens is new aperture stop, changing the throughput
vignetting
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 inputoutput relation:
S = R ·
where output = S = detector signal
input = = radiant flux
R = response function
Simple radiometer:
Variables:
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.
simplifying assumptions:
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
Symbol Definition
Review
E irradiance
I intensity
L radiance
radiant flux
Throughput
A area
D distance
d distance
F configuration factor
r radius in polar and spherical coordinates
s distance
x x dimension in Cartesian coordinates
y y dimension in Cartesian coordinates
azimuthal angle in spherical coordinates
throughput
solid angle
reflectance
polar angle in polar and spherical coordinates
Geometrical Optics
A aperture
D detector
EP entrance pupil
EW entrance window
FOV field of view
f focal length
L lens
M magnification
n index of refraction
s_{i} image distance
s_{o} object distance
XP exit pupil
XW exit window
Measurement Equation
a area
C calibration constant
G gain
R response function
R responsivity
S signal
T temperature
t time
emissivity
wavelength
direction
polarization
transmittance
References
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, AddisonWesley Publishing, 1987.
NIST Short Course Material
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