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


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


• contrast and sharpness of the image (linear s

Similarly from the display subsystem to tvision subsystem

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adiation in


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


• 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


• 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


• 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


• 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


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


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


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)


• 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


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

—

—

—

—


• 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


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)


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


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


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)


• 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


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


(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


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))


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–


• 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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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


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


• 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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• 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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• Flux at receiver

• Irradiance at receiver (W-m-2)

dΦ IdΩ IdA

d2

-------= =

dEdΦdA------- I

d2

-----= =

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Cos Law

• Tilt detector:



dAI d

θ

dΩ dA θcos

d2

-------------------=

dEdΦdA-------

I

d2

----- θcos= =

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receiver plane:


Cos2 Law

• Move tilted detector to off-axis angle θ along


dA

Id

θ

d/cosθ

dΩ dA

d θcos( )⁄[ ]2------------------------------- dA θcos( )2

d2

---------------------------==

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dEdΦdA-------

I

d2

----- θcos( )2= =

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lane:


Cos3 Law

• “Un-tilt” detector so that it’s in the receiver p


dA

Id

θ

d/cosθ

dΩ dA θcos

d θcos( )⁄[ ]2------------------------------- dA θcos( )3

d2

---------------------------==

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dEdΦdA-------

I

d2

----- θcos( )3= =

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surfaces

tion of

rpendicular to

2


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


• 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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Φ12 LA1 θ1Ω12cos LA1 θ1

A2 θ2cos

d2

----------------------cos= =

Φ21 LA2 θ2Ω21cos LA2 θ2

A1 θ1cos

d2

----------------------cos= =

Radiometric Transfer Photometry Radiometrydial/ece425/notes6.pdfPhotometry Radiometry in the context...

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