4.0 radiometry photometry

7/30/2019 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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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.


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

4.0 radiometry photometry

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