4.0 radiometry photometry
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THE UNIVERSITY OF CENTRAL FLORIDA
Geometrical Optics and Image Science (OSE 5203)
James E. Harvey, Instructor
Radiometry and Photometry
FALL Semester 2010
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Radiometry deals with the study and measurement of electromagneticradiation and its transfer from one surface to another. In particular, it is
concerned with how much source radiant power is transmitted through an
optical system and collected at the detector surface.
We will briefly review a few radiometric concepts, and some terminology anddefinitions required to perform radiometric calculations for imaging systems.
The subject of radiometry is somewhat of a neglected stepchild in the field of
physics (most physicists incorrectly refer to radiant power density on asurface as intensity rather than irradiance).
Electrical engineers are even less familiar with radiometric concepts andquantities. It may thus be somewhat of a revelation that diffracted radiance
(not irradiance or intensity) is the fundamental quantity predicted by scalardiffraction theory.*
For a more thorough discussion of the subject, I refer the reader to SPIETutorial Text Volume TT29 entitled Introduction to Radiometryby W. L. Wolfe.
Other excellent references are texts by Boyd, and by Dereniak and Boreman.
Comments about Radiometry
* J. E. Harvey, et. al., Diffracted Radiance: A Fundamental Quantity in Non-paraxial Scalar Diffraction Theory, Appl. Opt. 38 (1 Nov 1999). 4.2
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Outline
Radiometric and Photometric Definitions and Terminology
The Lagrange Invariant
The Solid Angle, Projected Area
The Inverse Square Law
The Fundamental Theorem of Radiometry
Lamberts Cosine Law
The Bidirectional Reflectance Distribution Function (BRDF, BSDF)
Radiometry of Imaging Systems
The Brightness Theorem (Conservation of radiance)
Cosine-fourth Illumination Fall-off
Radiometer and Detector Optics
4.3
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Radiometry(Definitions and Terminology)
area)projectedradian(watts/ste
radian)(watts/ste
a)(watts/are
a)(watts/are
cos
Radiance
IntensityRadiant
ExitanceRadiant
Irradiance
2
ssc
c
s
c
APL
PI
APM
AE
=
=
=
=
(1)
There is an analogous set of quantities based upon the number of photons/secrather than radiant power. This alternate set of units is useful when considering adetector that responds directly to photon events, rather than to thermal energy.
Conversion between the two sets of units is easily done using the following
relationship for the amount of energy per photon, = hc/ (Joule/photon).Note that the conversion factor depends upon the wavelength!
Let us first review the definitions of the main radiometric quantities of interest.
4.4
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Getting Intense about Intensity*
James M. Palmer, Getting Intense about Intensity, Optics & Photonics News, (Feb 1995).*
Jim Palmer has stated that: The term intensityis probably the most misused and
abused word in the technical literature today. It can be found to be used in at least
six (6) contexts: (1) watts per steradian, (2) watts per unit area, (3) watts per unit
area per steradian, (4) just plain watts, and most bizarre (5) cm-1/molecule-cm-2, and
finally (6) cm-2-amagat-1. These last two are used to describe spectral linestrengths.
Intensity is an International System of Units (SI) Base Quantity!
As an SI Base Quantity, it has the same stature as the other six SI BaseQuantities: length (meter), mass (kilogram), time (second), electric current (ampere),
thermodynamic temperature (Kelvin), and amount of substance (mole), and finally,
intensity carries the units of watts per steradian! All other physical quantities
are derived from these seven SI Base Quantities.
Intensity is properly used when describing the radiation emanating from a point
source, or a source small compared to the distance between the source and the
collector. Forextended sources, one must use the radiometric quantity radiance.
And most diffracting apertures should be considered to be extended sources!
4.5
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Photometry
Photometry is a special case of radiometry concerned with measurement ofthe visible part of the electromagnetic spectrum, whereas radiometry considersthe whole spectrum. In the case of photometry, the wavelength sensitivity ofthe human eye is taken into account as a weighting factor.
In photometry the unit of luminous power is the lumen. The lumen (watt) isthe only fundamental unit in photometry (radiometry), in the sense that all other
units are are defined in terms of lumens (watts), areas, and solid angles:
Luminous (Radiant) Intensity
Luminous (Radiant) Exitance
Illuminance (Irradiance)
Luminance or Brightness (Radiance)
The official name for this sensitivityfunction is spectral luminous efficiency,and is shown here for normal daylightconditions. The inverse of this curvegives the number of watts of radiant
power at any given wavelength that isrequired to produce a constant sensationof brightness.
4.6
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The Lagrange Invariant
We have previously discussed marginal and chief rays. Recall that a ray originating atan axial object point and grazing the edge of the entrance pupil is called the marginal ray.A ray originating at the edge of the object field and passing through the center of theentrance pupil is called a chief ray.
Thereafter, every time the marginal ray crosses the optical axis, we have an image
plane; and every time the chief ray crosses the optical axis, we have a pupil plane.
MarginalRayChief
Ray
PupilPlane
ImagePlane
Optical
Axis
y
u
y u
The marginal ray height (y) and angle (u) in any arbitrary plane will be distinguishedfrom the chief ray height (y)and angle (u) by placing a bar over the chief ray parameters.
The Lagrange Invariant
is an invariant quantity throughout the entire system, not just at conjugate planes. In the
special case of conjugate planes, the Lagrange invariant reduces to the HelmholtzInvariant.
)()( nuyunyH =
4.7
(2)
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The Solid Angle(Steradians)
4.8
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Projected Area
If the area, A, is tilted at an angle (measured from the surface normal to the
line of sight), its solid angular subtense is reduced by a factor of cos fromwhat it would be if it were not tilted.
For the special case of a small plane area A (such as a detector element)inclined at an angle from the line-of-sight from a point P, the solid angle
subtended by the detector is given by
4.9
PThe quantity A cos
is referred to as aprojected area.
The solid angle, , subtended by an arbitrary surface area, A (not on thesurface of a sphere) as seen by a point P, is defined as the ratio of the projected
area, A, upon a sphere divided by the square of the radius of that sphere.
2r=P
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The Inverse Square Law
The Inverse Square Law, which is usually stated as the illumination (irradiance) on asurface is inversely proportional to the square of the distance from a point source,follows almost intuitively from the definition of radiant intensity from a point source andthe law of conservation of energy. It can be readily verified with any small source and aphotographic exposure meter.
Clearly the irradiance on a spherecentered upon a luminous pointsource is given by:
Likewise, it is true in general that,
Light doth decrease in duplicateproportion to its distance of
propagation from the luminousbody. Robert Hooke (1635-1703).
2
1r
E
24 rP
E =
i.e., radiant power density (irradiance) is inversely proportional to the square of thedistance from the source to the surface of interest. 4.10
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The Fundamental Theorem of Radiometry
22211
coscos2rdAdALPd =
2112212 coscos ddALdAdLd ==
Assuming small angles, we can group the r2 with either the source or the receiverprojected area and obtain two equivalent expressions
Dropping the differentials, the radiant power transmitted to the collector from the source is
given by the product of the source radiance and either of the two projected area, solidangleAp products.
EitherAp product can be used. This is often a convenient flexibility in making calculations.
211
221
cos
cos
AL
L
=
=
1 2
2dA1dA
1A 2A1
2 12
r(3)
4.11
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Johann Heinrich Lambert was born the son of a tailor in
Mulhasen, Sundgau, Switzerland (now Mulhouse, Alsace,
France) in 1728. At the age of 12 he was taken out of school in a
spurious effort to teach him a tailors skills. Instead his
younger brother became a tailor and Johann used his spare
time to acquire knowledge in an autodidactic manner, studying
literature, the Latin and French languages, calculus andelementary sciences. He also became interested in astronomy
and began observing the night sky. In 1743 be was employed
as a bookkeeper for an ironworks company. He observed
the comet of 1744 and tried to calculate its orbit. In 1745 he
Johann Heinrich Lambert (1728-1777)
became Secretary to Professor Iselin, the editor of a newspaper at Basel, who three years later
recommended him as a private tutor to the family of Count A. von Salis of Coire. Coming thus intovirtual possession of a good library, Lambert had peculiar opportunities for improving himself in his
literary and scientific studies. He eventually became a member of the Berlin Academy of Sciences by
Fredrich II, the King of Prussia. He died of consumption in 1777 in Berlin at the age of 49. He was
honored during his life by a number of academic memberships, and is now remembered in the names of
several physical laws and units.
The photometrical unit "Lambert" for luminosity density (1 la = 0.3183099 candela / cm2).
The Lambert-Beer law for extinction of light in solutions.
Lambert's law of illumination density, Lamberts Cosine Law
The Bouguer-Lambert law of exponential decrement (e.g. of radiation in opaque media).
Lambertian Surface: Ideally white surface which reflects all light in diffuse manner.
Perhaps best known for being the first to prove that is an irrational number.
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Lamberts Cosine Law (1760)
oo LLL ==
cos
cos
In optics, Lambert's cosine law says that the radiant intensityobserved froma "Lambertian" surface is directly proportional to the cosine of the angle
between the observer's line of sight and the surface normal. The law is also known
as the cosine emission law orLambert's emission law. It is named after Johann
Heinrich Lambert, from his Photometria, published in 1760.
An important consequence of Lambert's cosine law is that when a Lambertiansurface is viewed from any angle, it has the same apparent radiance. This means,for example, that to the human eye it has the same apparent brightness (or
luminance). It has the same radiance because, although the emitted power from a
given area element is reduced by the cosine of the emission angle, the size of the
observed area is decreased by a corresponding amount. Therefore, its radiance(power per unit solid angle per unit projected source area) is the same.
Io
Iocos
Lo=Io/dA
oo L
dA
I
dA
IL ===
cos
cos
cos
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Lamberts Cosine Law and Lambertian Sources
cos)( o=
Many extended sources of radiation (most thermal sources, for example)
obey, approximately at least, Lamberts Cosine Law, which can be stated as:the radiant intensity emitted from a Lambertian source decreases with thecosine of the angle from the normal to the surface
Hence, although the emitted radiation per stradian (intensity) falls off with
cosine in accordance with Lamberts Law, the projected area falls off at
exactly the same rate. The result is that the radiance of a Lambertian surface is
constant with respect to . This is readily observable by noting that the
brightness of a Lambertian source (or reflecting surface) is the same regardless
of the angle from which it is viewed.
It should be noted here that there are many diffusely reflecting (no specularreflection) surfaces. However, a Lambertian surface is an ideal or perfectlydiffuse surface that strictly obeys Lamberts Cosine Law. There are also, of
course, partially diffusesurfaces whose reflected radiation consists of both aspecular component and a diffuse (or scattered) component. The bi-directional
reflectance distribution function (BRDF) is a very general function used by theradiometric community for describing these situations. 4.12
(4)
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Radiation into a Hemisphere
For a perfectly diffuse (Lambertian) source radiating into a hemisphere, thefollowing relationships exist between the radiant exitance (M), radiantintensity (I), radiance (L), and the source area (A):
4.13
Note that the relationship between radiance and radiant exitance is L = M/ ,
a consequence of Lamberts cosine law, and not L = M/2 (as we mightreason from the fact that there are 2 steradians in a hemisphere).
I/A/LA LAII/A,L / ======
Geometry of a Lambertian sourceradiating into a hemisphere.
M
M
M
M
M = L
R2
(5)
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The bidirectional reflectance distribution function (BRDF) is a fundamental quantitythat completely describes the scattering properties of a surface. It is defined as thereflected radiance (radiant power per unit solid angle per unit projected area) in agiven direction divided by the incident irradiance(radiant power per unit area)
Bidirectional Reflectance Distribution Function(Scattered Radiance)
The BRDF is a function of the angles
of reflectance, ( r, r), and the angles of
incidence, ( i, i), as well as thewavelength of the incident radiation and
the state of polarization of both the
incident and reflected waves. The
angles used in the above definition are
illustrated here for a narrow beam at afixed angle of incidence, we can drop
the differentials and approximate the
resulting quantity as
),(),;,(),;,(
iii
iirrriirr
dEdLfBRDF
==
AP
AddP
E
LBRDF
o
rr
iii
iirrrii /
)cos(/),(
),(
),;,( ,
==
(6)
(7)4.14
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Radiometry of Imaging Systems
Assuming axial symmetry (i.e., a circular object and a circular aperture stop),we can sketch a marginal ray and a chief ray in both object space and imagespace
From the fundamental theorem of radiometry, the power collected by theentrance pupil is given by
2
222
o
eoe
t
yyLP
=
4.15
Object Space Image Space
(8)
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The Brightness Theorem
.22 const =
= nL
nL
Note that the Lagrange invariant can be readily evaluated in the object plane,the entrance pupil plane, the exit pupil plane, and the image plane
uynuynunyuyn oeeo ==== 4.16
(9)
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Image Irradiance
The quantity referred to as the throughput or etendue of the system is alsoproportional to an Area-solid angle product
Now, from the brightness theorem we know that the radiance of the exit pupil
is the same as the radiance of the source (if n = n), and from the aboveexpression for the square of the Lagrange invariant in the image plane, we can
write the irradiance in the image plane as the radiant power collected by the
entrance pupil divided by the area of the image
eooeoeeoeo
o
eo AnAnAnAntyy ===== 2
2
2
2
2
2
2
2
22
22
ObjectPlane
ImagePlane
EntrancePupil
ExitPupil
eLE
nA
LAPE
o
oe
=
==
2
2
Therefore, the irradiance in the final image plane is given by the radiance ofthe source times the solid angle subtended by the exit pupil at the image. 4.17
(10)
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Equivalence of Solid Angle, F#, and Lagrange Invariant
Recall that the front effective F# is a measure of the marginal ray angle inobject space
Hence, for paraxial angles the solid angle subtended by the entrance pupil at theobject is related to the front effective F#
enefffront
efffrontD
dF
Fu
#
# ,
2
1tan ==
,2#
2
ear4 effrFu
e ==
,2#
2
4 efffrontFu
e ==
and the solid angle subtended by the exit pupil at the image is related to the reareffective F#
Hence, we have
2#
2
2#
2
2
2
2
2
2
2
2
2
4
4
effrearefffront F
An
F
AnAnAnAnAn ooeooeoeeo
======
ObjectPlane
ImagePlane
EntrancePupil
ExitPupil 4.18
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Cosine-Fourth Illumination Fall-Off
When imaging an extended object plane of uniform radiance, the off-axis image points willexhibit a cosine-to-the-fourth illumination fall-off. This is readily shown by applying thefundamental theorem of radiometry, first to the on-axis image point A, then to the off-axisimage point H.
A
O
H
ExitPupil
ImagePlane
2
The illumination at an image point is proportional to the solid angle subtended by the exit
pupil at the image point. We get a factor of cos because the distance OHis greater thanthe distance OA by a factor of 1/cos . The exit pupil is viewed obliquely from the point Hand the projected area is reduced by the factor cos Thus the illumination at point H isreduced by a factor of cos . This is, however, true for illumination on a plane normal tothe line OH (indicated by the dashed line in the figure). We want the illumination on theplane AH, which is reduced by another factor of cos because the illumination on thedashed plane is spread out over a larger area on the AH plane. This results in acosine-fourth fall-off which can be quite a severe reduction of some wide-angle cameras. 4.19
3
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A radiometer is a device or instrument for measuring the radiation from a source. Insimple form, it may consist of an objective lens (or mirror) which collects the radiationfrom the source and images it upon the detector. In many applications of radiometers,the following characteristics are desired:
#, hence
2 22
sin
ss DsDf f
D
NA N u s
= = =
= =
The numerical aperture is thus given by
In order to collect a large quantity of
radiant power from the source, thediameter, D, should be as large aspossible.
In order to increase the signal-to-noiseratio, the size of the detector, s, should beas small as possible.
In order to cover a practical field-of-view,the field angle, , should be of reasonablesize (and often as large as possible).
The semi field-of-view is given by Since the F# cannot exceed 0.5 and sin ucannot
exceed 1.0, the objective diameter, half field angle,and detector size are related by
0.1
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