Lambertian Sources
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RADIOMETRY OF LAMBERTIAN SOURCESRelate M and L Valuable exercise in spherical coordinate integrationL dA2
rdA1
Area element of sphere, radius rd A 2 = rdr sin d = r sin dd2
Start with radiance to get flux within area dA2157DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
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d L ( , ) = -----------------------------d A 1 cos d d = L ( , ) cos d A 1 d d A2 = L ( , ) cos d A 1 --------2 r = L ( , ) cos d A 1 sin dd2
2
Total flux in hemisphere above source2 2
= d A1
d 0 0
L ( , ) cos sin d
Now, in general L = L(,), but for a Lambertian source
158DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
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L = constant Therefore( sin ) = d A 1 2L ----------------2 = Ld A 12 2 0
Radiant exitance definition M = --------d A1 = L
159DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
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Include spectral variationM = L or L = M
Can think of as having units of sr, which cancels the sr-1 units of L Note: M 2L, as one might guess since hemisphere is 2 srWhy?
160DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
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TRANSMITTANCEDefined as ratio of transmitted output flux to input fluxfilter t() in out = in()
t ( ) =
out
in
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REFLECTANCEThree types:L L E E specular (mirror) E directional diffuse (Lambertian) L
Most natural surfaces are approximately Lambertian
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for < 40, snow and sand for < 60 At larger , natural surfaces tend to become directional
Reflectance of Lambertian surfaceL dA2
E
rdA1
Reflectance defined similarly to transmittance ( ) = out in
From earlier derivation163DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
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out
= L d A 1
and = E d A1in
Therefore, ( ) = L E , 0 ( ) 1
Given reflectance of surface and incident irrradiance, can get radianceL = ( )E
164DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
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BAND-AVERAGED IRRADIANCECascade spectral quantitites from source-to-surface-tosurface . . .E = (geometric factors) L t 1 ( ) t 2 ( ) 1 ( )
until arriving at a spectrally-integrating element, i.e. a detector,E total =
E S ( ) d
where S() is the detectors spectral sensitivity
Example of a specific detector S() is human vision system sensitivity V()
Etotal is the effective irradiance, because it is what is
165DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
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detected and measured Band-averaged spectral irradianceE total E b = --------------------
S ( ) d0
166DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
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RADIOMETRY OF OPTICAL SYSTEMStelescope collects light from point source light bucketaperture (diameter d) z0 point source (radiant intensity I) zi detector lens (or mirror)
assume: source and detector on optical axis object-to-sensor distance much greater than aperture167DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
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diameter
z0 d
no transmission losses irradiance at lens aperture flux collected by aperture c = E A aperture d = E -------4 I d = ---- -------2 4 z02 2
E = I z0
2
(inverse square law)
flux at detector (no losses)
i = c
proportional to radiant intensity of source proportional to square of aperture diameter168DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
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inversely proportional to square of object-to-sensor distanceWhat is the only way to increase the amount of light collected by a telescope?
Camera assume: Lambertian source and detector plane normal to optical axis magnificationm = hi ho = zi zo
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aperture (diameter d) z0 Lambertian source (radiance Lo, height ho, area Ao) zi image (height hi, area Ai) lens (or mirror)
flux collected by aperturec = Lo Ao A = L o A o ---2 zo L o A o d 2 = ------------ -------2 4 zo
NOTE: similarity to light bucket equation flux at detector (no losses)i = c
170DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
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irradiance at detector
E i = c Ai c = ------------2 m Ao d = L o --------------2 2 m 4z o2
by definition of magnification (see Geometrical Optics)d E i = L o -------2 4z i2
define effective f-number N of sensor, then, E i = L o --------2 4N
N = zi d
Camera Equation
NOTE: proportional to radiance of source171DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
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inversely proportional to square of sensor f-number does not depend on zo, the source-to-sensor distance f-number N typically preset on camera to 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22What is rationale for the above preset values of N?
172DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
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Camera imaging reflecting object Irradiance on object E Reflectance of object
source
aperture (diameter d) irradiance E image (height hi, area Ai) lens (or mirror)
Lambertian reflector (reflectance o, height ho, area Ao)
From earlier derivation:
Eo E i = --------2 4N173
DR. ROBERT A. SCHOWENGERDT
520 621-2706 (voice), 520 621-8076 (fax)
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Camera imaging extended reflecting object
source
aperture (diameter d) irradiance E
image (height hi, area Ai) Lambertian reflector (reflectance o, height ho, area Ao) lens (or mirror)
Cos4 Law applies:Eo 4 E i ( ) = --------- ( cos ) 2 4N
With object irradiance and reflectance fixed, what is the only way to increase the174DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
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amount of light collected by a camera?
175DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)
