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Home , Intensity (physics), Radiance, Radiometry

Definitions and Sources of

Emmett Ientilucci, Ph.D. Digital Imaging and Laboratory Chester F. Carlson Center for Imaging Science 13 March 2007 Radiometry Lab

• Wednesdays, 6-9pm, Room 3125 • Lab website – www.cis.rit.edu/class/simg401

• This Wednesday: – Topic: of a Radiometer – Handouts are on website

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometry Lecture Overview

• What is Radiometry? • What is ? • Radiometric / Photometric Definitions • Sources – Blackbody radiation –Gas – Fluorescent – Photodiode – – Carbon Arc – Beam

R.I.T Digital Imaging and Remote Sensing Laboratory What is Radiometry?

• Measurement or characterization of EM radiation and its interaction with matter

R.I.T Digital Imaging and Remote Sensing Laboratory What is Photometry?

• Measurement or characterization of EM radiation which is detectable by the human eye

R.I.T Digital Imaging and Remote Sensing Laboratory Why develop these concepts?

• Given an optical system, for example – Camera, telescope, etc – Any optical radiation source, a surface, detector, etc

• Can calculate how much radiation gets to the detector array or film in the image plane

• Can calculate the value of the Signal-to-Noise (SNR) or

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometry Definitions: Summary

• Units can be divided into two conceptual – Those having to do with energy or • Energy, Q ( or [J] ) • Power or , Φ ( or [W] )

– Those that are geometric in nature • , E [W/m2] • Exitance, M [W/m2] • , I [W/sr] • , L [W/m2 sr]

R.I.T Digital Imaging and Remote Sensing Laboratory Photometry Definitions: Summary

• Units can be divided into two conceptual areas – Those having to do with energy or power • Energy, Q ( or [lm s] or Talbots) • Power or flux, Φ, (lumen or [lm] = [cd sr] )

– Those that are geometric in nature • [lm/m2 = or lx] • Emittance [lm/m2 = lux or lx] • Intensity [lm/sr = or cd] • [lm/m2 sr = cd/m2 = nit] – Sometimes called Luminosity

R.I.T Digital Imaging and Remote Sensing Laboratory Systeme International d’Unites (SI)

– SI developed in 1960 – 7 SI Base Units • [kg] • Second [s] • Meter [m] • Ampere [A] •Kelvin [K] • Mole [mol] • Candela [cd] – All others are SI derived units • Previous slide radiometric definitions are all SI derived units

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

• Energy, q – We think of energy as being transferred in terms of energy packets or quanta – The energy carrier is a “photon” – Each photon carries energy,

hc q = hν = [] λ

– Shorter carry more energy than longer wavelength photons

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

• , Q –The total energy (Q) in a beam is a function of: • or wavelength of the photons, ν or λ • Number of photons, n of a particular ν or λ

Q = ∑ qi = ∑ nihν i [joules] i

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

• or Power, Φ – Quantity of energy propagating onto, through, or emerging from, a specified surface of a given in a given period of time

2

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

• Radiant Flux or Power, Φ – Quantity of energy propagating onto, through, or emerging from, a specified surface of a given area in a given period of time

9 dQ 1 J Φ = = ∑ Qi [ s = watt or W ] dt ∆t i = 4

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

• Irradiance, E (flux density) – Rate at which radiant flux, Φ is delivered onto a surface (e.g., a detector surface)

dΦ E=E(x, y) = [wm−2] dA

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

• Projected Area

Esin

θ Eo Eo Hyp θ cos θ = Adj/Hyp = E / E cos o Adj

Ecos = Eo cos θ Ecos

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

• , M – Rate at which radiant flux, Φ is delivered away from a surface (e.g., a diffuser, reflected surface)

dΦ M=M(x,y)= [w m−2] dA

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

• , I – Rate at which radiant flux, Φ is incident on, passing through, or emerging from a point in space in a given direction

dΦ I = I(θ,φ) = [wsr−1] dΩ

“”

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

• Plain angle or linear angle, θ – Length of arc, s divided by the radius, r – Plain angle is dimensionless s – SI assigns unit of measure: s s • radian C = 2πr θ For any s r=1 s θ = []m = rad r m

2π radians in a full circle

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

• Plain angle or linear angle, θ –A straight line or even a curved line can subtend the same angle as an arc on the circle

– DEF: Plane angle is the projection of a line on a unit circle, and the line need not be straight

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

• , Ω – 3D equivalent of a plane angle – Projection of a area (or a closed curve in space) onto a unit sphere – “square radians” or steradian SA = 4πr2 dA For any A dΩ

dA 2 dΩ= []m = sr r=1 r2 m2

4π in a sphere

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

• Solid angle example – A that radiates equally well in all directions (isotropic), and whose output Intensity is 1 W sr-1, has a total output power of 4π . dΦ I = []w sr-1 dΩ dΦ= I dΩ dΦ=()1 w ()4 sr sr π Φ= 4π ≈13watts

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

• Radiance, L – Combine the concepts of irradiance and intensity – Function of both position and direction – Flux, Φ incident on, passing through, or emerging in a specified direction from a specified point in a specified surface

d2Φ d2Φ L = = dΩdA dΩ(dA0 cosθ)

Where dA = dAo cos θ is the projected area Derive Expression In Class

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

• Radiance, L Element of Flux d2 Φ

Element of d2Φ ⎡ w ⎤ Solid Angle dΩ L = ⎢ 2 ⎥ dΩdA ⎣sr m ⎦

Element of Area dA

R.I.T Digital Imaging and Remote Sensing Laboratory Radiometric Definitions

Element of • Radiance, L Flux d2 Φ

θ

Element of Area in the Element of Surface dA0 Solid Angle dΩ

Element of 2 d Φ Projected Area dA = dA0 cos θ L = φ dΩ(dA0 cosθ)

The “area” in the units (m2) is now with respect to the projected area dA.

R.I.T Digital Imaging and Remote Sensing Laboratory Constancy of Radiance

- A basic principle in - Assume a beam of energy with constant radiance across the profile - Assume lossless media

dΦ1 = dΦ2 = dΦ

beam

Earth Sensor

R.I.T Digital Imaging and Remote Sensing Laboratory Constancy of Radiance

- How is the radiance at surface 1 (L1) related to the radiance at surface 2 (L2)?

2 d Φ1 12L1 = dA1cosθ1dΩ12

- Radiance from the Earth, p1

Earth Sensor

R.I.T Digital Imaging and Remote Sensing Laboratory Constancy of Radiance

- How is the radiance at surface 1 (L1) related to the radiance at surface 2 (L2)?

12

- Radiance at the sensor, p2

2 d Φ2 L2 = dA2 cosθ2dΩ21

Earth Sensor

R.I.T Digital Imaging and Remote Sensing Laboratory Constancy of Radiance

2 2 d Φ1 d Φ2 L1 = L2 = dA1cosθ1dΩ12 dA2 cosθ2dΩ21

- Let “r” be an arbitrary distance between p and p θ 1 2 dA cosθ τ = dA cos dΩ = dA cosθ 2 2 1 1 θ1 12 1 1 r 2 dA cosθ τ = dA cos dΩ = dA cosθ 1 1 2 2 2 21 2 2 r 2

- We see that,

τ1 =τ 2 =τ R.I.T Digital Imaging and Remote Sensing Laboratory Constancy of Radiance

- Re-write radiance using this information

2 2 τ =τ =τ d Φ1 d Φ2 1 2 L1 = L2 = τ1 τ2 dΦ1 = dΦ 2 = dΦ

d 2Φ L = L = 1 2 τ - This tells us that radiance along a ray is constant over distance in a lossless media

R.I.T Digital Imaging and Remote Sensing Laboratory