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Section Ii – Optical Tools ECE 425 CLASS NOTES – 2000 SECTION II – OPTICAL TOOLS Introduction Radiometry Sources of Radiant Energy Photometry Radiometric Transfer 120 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Introduction An imaging system consists of several subsystems image light transmission display human scene acquisition vision source subsystem subsystem* subsystem subsystem coder decoder optics neural network optics detector* electronics retina* brain • * points of signal transduction, optical <—> electronic From the light source to the image acquisition subsystem, we’re concerned with: • how much energy gets through (radiometry) 121 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 • contrast and sharpness of the image (linear systems) Similarly from the display subsystem to the human vision subsystem 122 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Radiometry Radiometry involves the propagation of radiation in space and through optical apertures Need to use 3-D geometry to describe 3-D spherical coordinate system (r,θ,φ) n θ Q r P φ 123 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 • 3-D vector with radius r • angle to the surface normal θ (radians) • azimuth angle φ (radians) Radiation propagates from a source into a conical volume spherical cap area A r source 124 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 • Define a solid angle Ω (steradians) 2 Ω = Ar⁄ (steradians) • Ω = 1 (unit solid angle) when the spherical area subtended by the cone = radius of the sphere Example: calculate the solid angles corresponding to a hemisphere and a sphere • flat surface sources radiate into a hemisphere • point sources radiate into a sphere • set up integration over solid angle 125 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 θ rsinθ dθ r P • Assume the source P is an isotropic radiator • Element of solid angle • circumference of element of solid angle (“belt”) around sphere is 2πr sinθ • width of “belt” is rdθ 126 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 • therefore area of “belt” is 2πr sinθ ⋅ rdθ and solid angle subtended by “belt” is 2πr sinθ ⋅ rdθ dΩ = ---------------------------------- 2 r = 2πθsin dθ • Total solid angle θ Ωθ()= ∫2πθsin dθ 0 = 2π()1 – cosθ • For θ = π/2 (hemisphere) Ω = 2π steradians • For θ = π (sphere) Ω = 4π steradians 127 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Projected Area • Useful in many radiometric calculations for both sources and detectors • Area of surface element dA as viewed from an angle θ n θ dAcosθ θ dA 128 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Radiant Energy Quantities and Units • Radiometric (valid for general case) quantitiy symbol definition common units abbreviation radiant Q joule J energy radiant ∂Q joule per cubic w w = J m-3 density ∂V meter ∂Q radiant flux Φ Φ = watt W ∂t M ∂Φ M = (exitance) ∂A radiant flux watt per -2 density E square meter W m ∂Φ (irradi- E = ∂A ance) 129 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 quantitiy symbol definition common units abbreviation radiant ∂Φ watt per I I = W sr-1 intensity ∂Ω steradian 2 ∂Φ watt per L = ∂Ω∂Acosθ radiance L steradian and W sr-1 m-2 ∂I = ∂Acosθ square meter 130 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 • Photometric (special to visual sensing) quantity symbol common units abbreviation lumen-second (tal- luminous energy Q lm s bot) lumen-second per luminous density w -3 cubic meter lm s m luminous flux Φ lumen lm M lumen per square (luminous meter exitance) (lux) -2 luminous flux den- lm m (lx) sity lm ft-2 (fc) E lumen per square foot (illuminance) (footcandle) 131 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 quantity symbol common units abbreviation luminous intensity lumen per steradian I lm sr-1 (cd) (candlepower) (candela) candela per square meter (nit) nt luminance L candela per square π foot per steradian fL (footlambert) 132 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 • Radiometric <—> photometric conversion common quantity symbol definition units abbreviation Φ ⁄ Φ -1 luminous KK = v e lumen lm W efficacy per watt luminous VVKK= ⁄ unitless — efficiency maximum • Material properties common quantity symbol definition abbreviation units ε ⁄ emissivity ε = MMblackbody unitless — α αΦ⁄ Φ absorptance = a i unitless — ρΦ⁄ Φ reflectance ρ = r i unitless — τΦ⁄ Φ transmittance τ = t i unitless — 133 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Wavelength Notation • Some quantities are differential with respect to λ, e.g. Lλ and Eλ • Units of “per wavelength interval” • Must be integrated over λ to obtain total quantity within a given wavelength range • Some quantities simply vary with λ, e.g. ρ(λ), V(λ) and τ(λ) • Not integrated alone; used to weight another quantity 134 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Sources of Radiant Energy Blackbody (BB) • Perfect radiator and absorber • Produces maximum M for any source at a given T • Nonattainable, ideal source • spectral radiant exitance M given by Planck’s Equation 2 2πhc M = ------------------------------------------ (wavelength in meters) λBB 5 hc ⁄ ()λkT λ []e –1 (W-m-2-µm-1) C1 = ---------------------------------------⁄ ()λ - (wavelength in micrometers) 5 C2 T λ []e –1 where 135 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 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 8 -2 µ 4 C1 = 3.74151 x 10 W-m - m , and 4 µ C2 = 1.43879 x 10 m-K. • Departure of a given source from a BB is its emittance • Measures the efficiency of a radiator or absorber ε ⁄ ≤≤ε λ =0Mλ MλBB , λ 1 136 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Sun • Solar Irradiance at Top-Of-Atmosphere 2500 5900K BB at earth-sun distance ) -1 2000 MODTRAN m µ - -2 1500 1000 irradiance (W-m 500 0 0.4 0.8 1.2 1.6 2 2.4 wavelength (µm) • Modeled well by a blackbody @ 5900K Solar energy propagates through the atmosphere to Earth’s surface • Atmospheric transmittance creates spectral “windows” through which energy reaches the earth 137 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 • Above 3µm wavelength, the Earth’s self-emitted radiation (BB at 300K) becomes significant • Above 8µm, Earth’s self-emitted radiation dominates and solar radiation is insignifcant wavelength radiation surface property name range source of interest Visible (V) 0.4 – 0.7µm solar reflectance Near InfraRed (NIR) 0.7 – 1.1µm solar reflectance 1.1 – 1.35µm Short Wave InfraRed 1.4 – 1.8µm solar reflectance (SWIR) 2 – 2.5µm Mid Wave 3 – 4µm reflectance, solar, thermal InfraRed (MWIR) 4.5 – 5µm temperature Thermal 8 – 9.5µm thermal temperature InfraRed (TIR) 10 – 14µm temperature (pas- thermal (passive) microwave, radar 1mm – 1m sive) artificial (active) roughness (active) 138 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Wien’s Law • Specifies wavelength at which maximum BB radiation occurs • Differentiate Planck’s equation, set to zero and solve for λ λ = 2898 ⁄ T max where λ is in µm and T is in K λ • As T increases, |max decreases Stefan-Boltzmann’s Law • Specifies total energy radiated by BB over all wavelengths • Integrate Planck’s equation over all wavelengths 139 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 5 4 2π k 4 M = ------------------T tot 2 3 -2 15c h (W-m ) 4 = σT where σ = Stefan-Boltzmann constant = 5.67 x 10-8 (W-m-2-K-4) Examples for Wein’s and Stefan-Boltzmann’s Laws: λ µ -2 source T (K) |max ( m) Mtot (W-m ) earth 300 9.66 (TIR) 4.6 x 102 incandescent 2800 1.04 (NIR) 3.5 x 105 lamp 0.483 (blue- sun 6000 7 green) 7.3 x 10 140 DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 Photometry Radiometry in the context of Human Vision System (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 photometric units • Multiply spectral quantity of interest by photopic visual sensitivity curve • Integrate over λ 141 DR.
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