Lecture 3 Geometrical Optics Describe propagation and collection of radiant flux



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Lecture 3



Geometrical Optics

Describe propagation and collection of radiant flux




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/(m2 sr)

Irradiance E =  /A W/m2

Intensity I =  / W/sr


Review of differential areas

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 dA1) (solid angle from dA1 to dA2)





For finite areas




Example 1: infinitesimal area colinear with disk







since cos = D/s

Also,
Where m is angle from dA1 to edge of disk

Example 2: two parallel, colinear disks










where
E
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


Configuration factor


 = Fhem = FA1

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


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.

Two-aperture radiometer



Throughput calculation:

Note:


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


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 input-output 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

si image distance

so 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, Addison-Wesley Publishing, 1987.

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