Radiometric Transfer Photometry Radiometrydial/ece425/notes6.pdfPhotometry Radiometry in the context...
Transcript of Radiometric Transfer Photometry Radiometrydial/ece425/notes6.pdfPhotometry Radiometry in the context...
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OOLS
DR. ROBERT A. SCHOWENGERDT [email protected]
SECTION II – OPTICAL TIntroduction
Radiometry
Sources of Radiant Energy
Photometry
Radiometric Transfer
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systems
ctronic
sition
humanvision
subsystem
tina*
neuralnetwork
brain
DR. ROBERT A. SCHOWENGERDT [email protected]
Introduction
An imaging system consists of several sub
• * points of signal transduction, optical <—> ele
From the light source to the image acquisubsystem, we’re concerned with:
• how much energy gets through (radiometry)
lightsource
scene
imageacquisitionsubsystem
transmissionsubsystem
displaysubsystem*
optics detector* electronics
coder decoder
optics
re
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ystems)
he human
DR. ROBERT A. SCHOWENGERDT [email protected]
• contrast and sharpness of the image (linear s
Similarly from the display subsystem to tvision subsystem
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adiation in
DR. ROBERT A. SCHOWENGERDT [email protected]
Radiometry
Radiometry involves the propagation of rspace and through optical apertures
Need to use 3-D geometry to describe
3-D spherical coordinate system (r,θ,φ)
φ
θn
P
Q
r
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a conical
cap
DR. ROBERT A. SCHOWENGERDT [email protected]
• 3-D vector with radius r
• angle to the surface normal θ (radians)
• azimuth angle φ (radians)
Radiation propagates from a source intovolume
spherical
r
source
area A
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a subtended by
ponding to a
DR. ROBERT A. SCHOWENGERDT [email protected]
• Define a solid angle Ω (steradians)
• Ω = 1 (unit solid angle) when the spherical arethe cone = radius of the sphere
Example: calculate the solid angles correshemisphere and a sphere
• flat surface sources radiate into a hemisphere
• point sources radiate into a sphere
• set up integration over solid angle
Ω A r2⁄ (steradians)=
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elt”) around sphere
DR. ROBERT A. SCHOWENGERDT [email protected]
• Assume the source P is an isotropic radiator
• Element of solid angle
• circumference of element of solid angle (“bis
• width of “belt” is
rsinθ
r
θ
dθ
P
2πr θsin
rdθ
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lid angle subtended
DR. ROBERT A. SCHOWENGERDT [email protected]
• therefore area of “belt” is and soby “belt” is
• Total solid angle
• For θ = π/2 (hemisphere)
• For θ = π (sphere)
2πr θsin rdθ⋅
dΩ 2πr θsin rdθ⋅
r2
----------------------------------=
2π θdθsin=
Ω θ( ) 2π θsin θd
0
θ
∫=
2π 1 θcos–( )=
Ω 2π steradians=
Ω 4π steradians=
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th sources and
angle
θ
DR. ROBERT A. SCHOWENGERDT [email protected]
Projected Area
• Useful in many radiometric calculations for bodetectors
• Area of surface element dA as viewed from an
dA
dAcosθθθ
n
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abbreviation
J
J m
-3
W
W m
-2
DR. ROBERT A. SCHOWENGERDT [email protected]
Radiant Energy Quantities and Units
• Radiometric (valid for general case)
quantitiy symbol definition common units
radiant energy Q joule
radiant
densityw joule per cubic
meter
radiant flux Φ watt
radiant flux density
M
(exitance)watt per
square meterE
(irradi-ance)
wV∂
∂Q=
Φt∂
∂Q=
MA∂
∂Φ=
EA∂
∂Φ=
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W sr-1
W sr-1 m-2
abbreviation
DR. ROBERT A. SCHOWENGERDT [email protected]
radiant
intensityI
watt per
steradian
radiance Lwatt per
steradian and square meter
quantitiy symbol definition common units
IΩ∂
∂Φ=
LΩ A θcos∂
2
∂∂ Φ=
A θcos∂∂I=
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abbreviation
lm s
lm s m-3
lm
lm m-2 (lx)
lm ft-2 (fc)
DR. ROBERT A. SCHOWENGERDT [email protected]
• Photometric (special to visual sensing)
quantity symbol common units
luminous energy Q lumen-second (tal-bot)
luminous density w lumen-second per cubic meter
luminous flux Φ lumen
luminous flux den-sity
M
(luminous
exitance)
lumen per square meter
(lux)
lumen per square foot
(footcandle)
E
(illuminance)
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lm sr-1 (cd)
nt
fL
abbreviation
DR. ROBERT A. SCHOWENGERDT [email protected]
luminous intensity
(candlepower)I
lumen per steradian
(candela)
luminance L
candela per square meter (nit)
candela per square foot per π steradian
(footlambert)
quantity symbol common units
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breviation
lm W-1
—
bbreviation
—
—
—
—
DR. ROBERT A. SCHOWENGERDT [email protected]
• Radiometric <—> photometric conversion
• Material properties
quantity symbol definition common units ab
luminous
efficacy
K lumen per watt
luminous
efficiencyV unitless
quantity symbol definition commonunits a
emissivity ε unitless
absorptance α unitless
reflectance ρ unitless
transmittance τ unitless
K Φv Φe⁄=
V K Kmaximum⁄=
ε M Mblackbody⁄=
α Φa Φi⁄=
ρ Φr Φi⁄=
τ Φt Φi⁄=
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to λ, e.g. Lλ and Eλ
antity within a
(λ) and τ(λ)
r quantity
DR. ROBERT A. SCHOWENGERDT [email protected]
Wavelength Notation
• Some quantities are differential with respect
• Units of “per wavelength interval”
• Must be integrated over λ to obtain total qugiven wavelength range
• Some quantities simply vary with λ, e.g. ρ(λ), V
• Not integrated alone; used to weight anothe
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y
n T
Equation
m-2-µm-1)
DR. ROBERT A. SCHOWENGERDT [email protected]
Sources of Radiant Energ
Blackbody (BB)
• Perfect radiator and absorber
• Produces maximum M for any source at a give
• Nonattainable, ideal source
• spectral radiant exitance M given by Planck’s
(W-
where
MλBB2πhc
2
λ5e
hc λkT( )⁄1–[ ]
------------------------------------------ (wavelength in meters)=
C1
λ5e
C2 λT( )⁄1–[ ]
---------------------------------------- (wavelength in micrometers)=
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ittance
orber
DR. ROBERT A. SCHOWENGERDT [email protected]
T is the blackbody’s temperature in Kelvin (K),
h = 6.6256 x 10-34 (W-s2) Planck’s Constant k = 1.38054 x 10-23 (W-s-K-1) Boltzmann’s Constant c = 2.997925 x 108 (m-s-1) velocity of light λ = wavelength of radiation
C1 = 3.74151 x 108 W-m-2-µm4, and C2 = 1.43879 x 104 µm-K.
• Departure of a given source from a BB is its em
• Measures the efficiency of a radiator or abs , ελ Mλ MλBB⁄= 0 ελ 1≤ ≤
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osphere to
windows”
4
DR. ROBERT A. SCHOWENGERDT [email protected]
Sun
• Solar Irradiance at Top-Of-Atmosphere
• Modeled well by a blackbody @ 5900K
Solar energy propagates through the atmEarth’s surface
• Atmospheric transmittance creates spectral “through which energy reaches the earth
0
500
1000
1500
2000
2500
0.4 0.8 1.2 1.6 2 2.
5900K BB at earth-sun distanceMODTRAN
irra
dian
ce (W
-m-2
-µm
-1)
wavelength (µm)
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d radiation (BB
inates and solar
ce property interest
ectance
ectance
ectance
ectance,
perature
perature
rature (pas-sive)
ess (active)
DR. ROBERT A. SCHOWENGERDT [email protected]
• Above 3µm wavelength, the Earth’s self-emitteat 300K) becomes significant
• Above 8µm, Earth’s self-emitted radiation domradiation is insignifcant
name wavelength range
radiationsource
surfaof
Visible (V) 0.4 – 0.7µm solar refl
Near InfraRed (NIR) 0.7 – 1.1µm solar refl
Short Wave InfraRed (SWIR)
1.1 – 1.35µm
1.4 – 1.8µm
2 – 2.5µm
solar refl
Mid Wave
InfraRed (MWIR)
3 – 4µm
4.5 – 5µmsolar, thermal
refl
tem
Thermal
InfraRed (TIR)
8 – 9.5µm
10 – 14µmthermal tem
microwave, radar 1mm – 1mthermal (passive)
artificial (active)
tempe
roughn
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diation occurs
d solve for λ
wavelengths
ths
DR. ROBERT A. SCHOWENGERDT [email protected]
Wien’s Law
• Specifies wavelength at which maximum BB ra
• Differentiate Planck’s equation, set to zero an
where λ is in µm and T is in K
• As T increases, λ|max decreases
Stefan-Boltzmann’s Law
• Specifies total energy radiated by BB over all
• Integrate Planck’s equation over all waveleng
λmax
2898 T⁄=
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n’s Laws:
ot (W-m -2)
4.6 x 102
3.5 x 105
7.3 x 107
DR. ROBERT A. SCHOWENGERDT [email protected]
(W-m-2)
where σ = Stefan-Boltzmann constant =
5.67 x 10-8 (W-m-2-K-4)
Examples for Wein’s and Stefan-Boltzman
source T (K) λ|max ( µm) Mt
earth 300 9.66 (TIR)
incandescent
lamp2800 1.04 (NIR)
sun 6000 0.483 (blue-green)
Mtot2π5
k4
15c2h
3------------------T
4=
σT4
=
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n System
metric units
opic visual
DR. ROBERT A. SCHOWENGERDT [email protected]
Photometry
Radiometry in the context of Human Visio(HVS)
Luminous flux Φ in lumens (lm)
• Corresponds to radiometric flux in Watts (W)
• Incorporates the HVS sensitivity to radiation
Conversion of radiometric units to photo
• Multiply spectral quantity of interest by photsensitivity curve
• Integrate over λ
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d by the sun
unitless
k of V(λ))
DR. ROBERT A. SCHOWENGERDT [email protected]
Ex: Find the effective visual lm-m-2 emitte
• Photopic visual sensitivity: V(λ), 0 ≤ V(λ) ≤ 1,
• Scaling factor is 683 lm-W-1 at λ = 555nm (pea
0
2 107
4 107
6 107
8 107
1 108
1.2 108
0
0.2
0.4
0.6
0.8
1
400 450 500 550 600 650 700
solar Mrelative visual M photopic visual sensitivity
sola
r M
(w
-m-2
- µm
-1)
photopic visual sensitivity
wavelength (nm)
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diometric
c quantity)
version
2–
DR. ROBERT A. SCHOWENGERDT [email protected]
• Spectral radiant exitance: MλΒΒ (T = 6000K, raquantity)
• Total effective luminous exitance (photometri
(W-m-2)
• Using data in following table,
Why isn’t this 1011?
Table for radiometric —> photometric con
λ (nm) Mλ (W•m-2•µm-1) V(λ) V(λ)•Mλ(W•m-2•µm-1)
400 9.13e+07 4.00e-04 3.65e+04410 9.34e+07 1.20e-03 1.12e+05420 9.52e+07 4.00e-03 3.81e+05430 9.67e+07 1.16e-02 1.12e+06440 9.79e+07 2.30e-02 2.25e+06450 9.88e+07 3.80e-02 3.75e+06460 9.95e+07 6.00e-02 5.97e+06470 9.99e+07 9.10e-02 9.09e+06480 1.00e+08 1.39e-01 1.39e+07
MeffV
683V λ( )MλBB λd∫=
MeffV
6.88 109lm m–×=
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DR. ROBERT A. SCHOWENGERDT [email protected]
490 1.00e+08 2.08e-01 2.08e+07500 9.98e+07 3.23e-01 3.22e+07510 9.93e+07 5.03e-01 4.99e+07520 9.88e+07 7.10e-01 7.01e+07530 9.80e+07 8.62e-01 8.45e+07540 9.72e+07 9.54e-01 9.27e+07550 9.62e+07 9.95e-01 9.57e+07560 9.52e+07 9.95e-01 9.47e+07570 9.40e+07 9.52e-01 8.95e+07580 9.28e+07 8.70e-01 8.07e+07590 9.14e+07 7.57e-01 6.92e+07600 9.01e+07 6.31e-01 5.69e+07610 8.87e+07 5.03e-01 4.46e+07620 8.72e+07 3.81e-01 3.32e+07630 8.57e+07 2.65e-01 2.27e+07640 8.42e+07 1.75e-01 1.47e+07650 8.27e+07 1.07e-01 8.85e+06660 8.11e+07 6.10e-02 4.95e+06670 7.95e+07 3.20e-02 2.54e+06680 7.80e+07 1.70e-02 1.33e+06690 7.64e+07 8.20e-03 6.26e+05
700 7.48e+07 4.10e-03 3.07e+05
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propagation
DR. ROBERT A. SCHOWENGERDT [email protected]
Radiometric Transfer
Radiance
• Fundamental quantity for extended sources
• Flux per source area per solid angle radiated
• Measured normal to the source
dΩdA
LΩ A∂
2
∂∂ φ=
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ce normal to cted solid angle)
propagation
DR. ROBERT A. SCHOWENGERDT [email protected]
• Measured at an angle to the source
• equivalent to a tilted source
• use projected area
• NOTE: dΩ must be calculated for receiver surfapropagation direction (sometimes called proje
Inverse Square Law
• Point source emits radiation in all directions
dΩ
dAL
Ω A θcos∂
2
∂∂ φ=
θ
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DR. ROBERT A. SCHOWENGERDT [email protected]
• Radiant intensity I (W-sr–1)
• Receiver normal to propagation:
• Solid angle subtended by receiver at source
dAI d
dΩ dA
d2
-------=
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DR. ROBERT A. SCHOWENGERDT [email protected]
• Flux at receiver
• Irradiance at receiver (W-m-2)
dΦ IdΩ IdA
d2
-------= =
dEdΦdA------- I
d2
-----= =
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DR. ROBERT A. SCHOWENGERDT [email protected]
Cos Law
• Tilt detector:
• Solid angle subtended by receiver at source
• Irradiance at receiver (W-m-2)
dAI d
θ
dΩ dA θcos
d2
-------------------=
dEdΦdA-------
I
d2
----- θcos= =
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receiver plane:
DR. ROBERT A. SCHOWENGERDT [email protected]
Cos2 Law
• Move tilted detector to off-axis angle θ along
• Solid angle subtended by receiver at source
dA
Id
θ
d/cosθ
dΩ dA
d θcos( )⁄[ ]2------------------------------- dA θcos( )2
d2
---------------------------==
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DR. ROBERT A. SCHOWENGERDT [email protected]
• Irradiance at receiver (W-m-2)
dEdΦdA-------
I
d2
----- θcos( )2= =
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lane:
DR. ROBERT A. SCHOWENGERDT [email protected]
Cos3 Law
• “Un-tilt” detector so that it’s in the receiver p
• Solid angle subtended by receiver at source
dA
Id
θ
d/cosθ
dΩ dA θcos
d θcos( )⁄[ ]2------------------------------- dA θcos( )3
d2
---------------------------==
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DR. ROBERT A. SCHOWENGERDT [email protected]
• Irradiance at receiver (W-m-2)
dEdΦdA-------
I
d2
----- θcos( )3= =
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surfaces
tion of
rpendicular to
2
DR. ROBERT A. SCHOWENGERDT [email protected]
Example: optical throughput between two
• Total flux through system is invariant to direcpropagation
• Case I: surfaces parallel to each other, and penormal direction
A1 A
Ω21 Ω12
d
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lel to each other,
A2
DR. ROBERT A. SCHOWENGERDT [email protected]
• Case II: source and receiver surfaces not paralor perpendicular to normal direction
LΦ
AΩ--------=
Φ12 LA1Ω12
LA1A2
d2
-----------------= =
Φ21 LA2Ω21
LA2A1
d2
-----------------= =
A1
Ω21 Ω12
dθ1 θ2
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DR. ROBERT A. SCHOWENGERDT [email protected]
Φ12 LA1 θ1Ω12cos LA1 θ1
A2 θ2cos
d2
----------------------cos= =
Φ21 LA2 θ2Ω21cos LA2 θ2
A1 θ1cos
d2
----------------------cos= =