Optics for Engineers Chapter 12

Transcription

1 Optics for Engineers Chapter 12 Charles A. DiMarzio Northeastern University Apr. 214

2 Radiometry Power is Proportional to Area of Aperture Stop Area of Field Stop Brightness of the Source (Radiance) Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 1

3 Radiometry and Photometry Note: Spectral X: X ν, units /Hz or X λ, /µm. Rad. to Phot: 683 lm/w y (λ). y (555nm) 1. φ Radiant Flux, Watt = W Luminous Flux, lumen = lm / A M Radiant Exitance, W/m 2 Luminous Exitance, lm/m 2 = lux = lx / Ω / Ω E Irradiance, W/m 2 Illuminance, lm/m 2 = lux /R 2 / A I Radiant Intensity, W/sr Luminous Intensity, lm/sr = cd L Radiance, W/m 2 /sr Luminance, nit = lm/m 2 /sr 1Ft.Candle = 1lm/ft 2 1Lambert = 1lm/cm 2 /sr/π 1mLambert = 1lm/m 2 /sr/π 1FtLambert = 1lm/ft 2 /sr/π Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 2

4 Radiometric Quantities Quantity Symbol Equation SI Units Radiant Energy Q Joules Radiant Energy Density w w = d3 Q dv 3 Joules/m 3 Radiant Flux or Power P or Φ Φ = dq W Radiant Exitance M M = dφ da W/m 2 Irradiance E E = dφ da W/m 2 Radiant Intensity I I = dφ W/sr dω Radiance L lm = d2 Φ W/m 2 da cos θdω dq Fluence Ψ Fluence Rate F dt J/m 2 da dψ J/m 2 dt Emissivity ɛ ɛ = M M bb Dimensionless d() Spectral () () ν ()/Hz dν d() or () λ ()/µm dλ Luminous Flux or Power P or Φ lm Luminous Exitance M M = dφ da lm/m 2 Illuminance E E = dφ da lm/m 2 Luminous Intensity I I = dφ lm/sr dω Luminance L L = d2 Φ lm/m 2 da cos θdω Spectral Luminous Efficiency V (λ) Dimensionless Color Matching Functions x (λ) ȳ (λ) z (λ) Dimensionless Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 3

5 Irradiance Poynting Vector for Coherent Wave ds dp (sin θ cos ζˆx + sin θ sin ζŷ + cos θẑ) 4πr2 Irradiance (Projected Area, A proj = A cos θ) de = d2 P da = dp ( 4πr 2 ) Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 4

6 Irradiance and Radiant Intensity Solid Angle Ω = A r 2 E I Intensity from Irradiance (Unresolved Source, A) I = Er 2 E = I r 2 di = d2 P dω = d2 P d A r 2 = dp ( 4πr 2 ) Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 5

7 Intensity and Radiance Resolved Source: Many Unresolved Sources Combined I = Radiance di = I A da I L On Axis (x = y = ) L (x, y, θ, ζ) = I (θ, ζ) A I (θ, ζ) L (x, y, θ, ζ) = A = 2 P A Ω Off Axis (Projected Area) I (θ, ζ) = A L (x, y, θ, ζ) dxdy L = I A cos θ = 2 P A Ω cos θ Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 6

8 Radiant Intensity and Radiant Flux Φ I Flux or Power, P or Φ Integrate Intensity Solid Angle Ω = 2π 1 1 ( NA n ) 2 P = Φ = I (θ, φ) sin θdθdζ Constant Intensity P = Φ = IΩ Ω = sin θ dθ dζ Ω = 2π Θ sin θ dθ dζ = 2π (1 cos Θ) = 2π ( 1 ) 1 sin 2 Θ Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 7

9 Radiance and Radiant Exitance M L Radiant Exitance from a Source (Same Units as Irradiance) M (x, y) = [ Radiant Exitance from Radiance 2 P A dω ] sin θdθdζ M (x, y) = L (x, y, θ, ζ) cos θ sin θdθdζ Radiance from Radiant Exitance L (x, y, θ, ζ) = M (x, y) 1 Ω cos θ Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 8

10 Radiance and Radiant Exitance: Special Cases Constant L and Small Solid Angle M (x, y) = LΩ cos θ Ω = 2π 1 1 ( NA n ) 2 or Ω π ( NA n ) 2 Constant L, over Hemisphere M (x, y) = 2π π/2 L cos θ sin θdθdζ = 2πL sin2 π 2 2. M (x, y) = πl (Lambertian Source) Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 9

11 Radiant Exitance and Flux Φ M Power or Flux from Radiant Exitance P = Φ = M (x, y) dxdy, Radiant Exitance from Power M (x, y) = P A Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 1

12 The Radiance Theorem in Air Solid Angle two ways Ω = A r 2 Ω = A r 2 Power Increment dp = LdAdΩ = LdA da r 2 cos θ dp = LdA dω = LdA da r 2 cos θ Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 11

13 Using the Radiance Theorem: Examples Later Radiance is Conserved in a Lossless System (in Air) Losses Are Multiplicative Fresnel Reflections and Absorption Radiance Theorem Simplifies Calculation of Detected Power Determine Object Radiance Multiply by Scalar, T total, for Loss Find Exit Window (Of a Scene or a Pixel) Find Exit Pupil Compute Power P = L object T total A exit window Ω exit pupil Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 12

14 Etendue and the Radiance Theorem Abbe Invariant: Etendue n x dα = nxdα n 2 AΩ = ( n ) 2 A Ω Power Conservation Ld 2 Ad 2 Ω = L d 2 A d 2 Ω L n 2n2 d 2 Ad 2 Ω = L n 2 ( n ) 2 d 2 A d 2 Ω Radiance Theorem: L n 2 = L (n ) 2 Basic Radiance, L/n 2, Conserved (Thermodynamics Later) Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 13

15 Radiance Theorem Example: Translation Translation Matrix Equation ( ) ( ) [( dx2 1 z12 x1 + dx 1 ) ( x1 )] ( dx1 ) dα 2 = 1 α 1 + dα 1 α 1 = dα 1 dx 2 dα 2 = dx 1 dα 1 Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 14

16 Radiance Theorem Example: Imaging Imaging Matrix Equation M SS = ( m? n n m ) Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 15

17 Radiance Theorem Example: Dielectric Interface Dielectric Interface Matrix Equation ( ) 1 Power Increment n n d 2 Φ = L 1 da cos θ 1 dω 1 = L 2 da cos θ 2 dω 2 L 1 da cos θ 1 sin θ 1 dθ 1 dζ 1 = L 2 da cos θ 2 sin θ 2 dθ 2 dζ 2 Snell s Law and Derivative n 1 sin θ 1 = n 2 sin θ 2 n 1 cos θ 1 dθ 1 = n 2 cos θ 2 dθ 2 Radiance Theorem sin θ 1 sin θ 2 = n 2 n 1 dθ 1 dθ 2 = n 2 cos θ 2 n 1 cos θ 1 L n 2 = L (n ) 2 Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 16

18 Radiance Theorem Example: 1/2 Telecentric Relay Fourier Optics Matrix Equation ( f M F F = 1 f ) det M F F = 1 = n /n n 2 AΩ = ( n ) 2 A Ω Radiance Theorem L n 2 = L (n ) 2 Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 17

19 Idealized Example: 1W Lamp P = 1W, Uniform in Angle Tungsten Filament: Area A = (5mm) 2, Distance, R Radiant Exitance Radiance (Uniform in Angle) Intensity M = 1W/ (5mm) 2 = 4W/mm 2 L = M/ (4π).32W/mm 2 /sr I = 1W/ (4π) 7.9W/sr Reciever: Area A : Received Power and Irradiance P A / ( 4πr 2) = IA /r 2 E = I r 2 8W/sr/ (1m)2 =.8W/m 2 Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 18

20 1W Lamp at the Receiver Using Source Radiance at Receiver E = L A r 2 = LΩ Irradiance Same as Previous Page E.32W/mm 2 /sr (5mm)2 (1m) 2.8W/m2 Useful if r and A are Not Known Only Depends on Ω (Measured at Reciever) Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 19

21 Practical Example: Imaging the Moon Known Moon Radiance L moon = 13W/m 2 /sr Calculation Jones matrices for Transmission (.25), other multiplicative losses (12 Lenses: (.96 2) 12 ). L = 13W/m 2 /sr = 13W/m 2 /sr = 1.2W/m 2 /sr Exit Pupil (NA =.1) and exit window (Pixel: d = 1µm). Irradiance, E = LΩ and Power on a Pixel, P = EA = LAΩ ( ) P = 1.2W/m 2 /sr 2π 1 1 NA 2 ( m ) 2 P = 1.2W/m 2 /sr.315sr 1 1 m 2 = W If desired, multiply by time (1/3sec) to obtain energy. About One Million Photons (and Electrons) Alternative to Solve in Object Space (Need Pixel Size on Moon) Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 2

22 Radiometry Summary Five Radiometric Quantities: Radiant Flux Φ or Power P, Radiant Exitance, M, Radiant Intensity, I, Radiance, L, and Irradiance, E, Related by Derivatives with Respect to Projected Area, A cos θ and Solid Angle, Ω. Basic Radiance, L/n 2, Conserved, with the Exception of Multiplicative Factors. Power Calculated from Numerical Aperture and Field Of View in Image (or Object) Space, and the Radiance. Losses Are Multiplicative. Finally Intensity is Not Irradiance. Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 21

23 Spectral Radiometry Definitions Any Radiometric Quantity Resolved Spectrally Put the Word Spectral in Front Use a Subscript for Wavelength or Frequency Modify Units Example: Radiance, L, Spectral Radiance (Watch Units) L ν = dl dν W/m2 /sr/thz or L λ = dl dλ W/m2 /sr/µm Spectral Fraction f λ (λ) = X λ (λ) X for X = Φ, M, I, E, or L Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 22

24 Spectral Radiometric Quantities Quantity Units d/dν Units d/dλ Units Radiant W W/Hz W/µm Flux, Φ Power, P Radiant Exitance, M Radiant Intensity, I Radiance, L W/m 2 W/sr W/m 2 /sr Spectral Radiant Flux, Φ ν Spectral Radiant Exitance, M ν Spectral Radiant Intensity, I ν Spectral Radiance, L ν W/m 2 /Hz W/sr/Hz W/m 2 /sr/hz Spectral Radiant Flux, Φ λ Spectral Radiant Exitance, M λ Spectral Radiant Intensity, I λ Spectral Radiance, L λ W/m 2 /µm W/sr/µm W/m 2 /sr/µm Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 23

25 Spectral Fraction: Sunlight on Earth 5K Black Body (Discussed Later) Defines M λ f λ (λ) = M λ (λ) M M = M λdλ Compute Spectral Irradiance with Known E = 1W/m 2 E λ (λ) = Ef λ (λ), with f λ (λ) = M λ (λ) M = M λ (λ) M λ (λ) dλ In Practice: Use the spectral fraction with any radiometric quantity * Solar Spectral Irradiance is E λ = L λω r 2 e α dr but we need to know a lot to compute it. Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 24

26 Spectral Fraction: Molecular Tag Excitation: Argon 488nm Green Emission Power, Φ Φ λ (λ) = Φf λ (λ), f λ (λ) = S λ (λ) S λ (λ) dλ Dash Dot on Plot Obtained from Experiment (Radiometric Calibration not Needed) Available from Vendor Absorption, Emission, \% λ Fluorescein Spectra Dash: Absorption Spectrum Solid: Emission Spectrum Dash Dot: Spectral Fraction for Emission (nm 1 ).1. Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 25

27 Molecular Tag in Epi Fluorescence Narrow Signal Bad Rejection Narrow Filter Lost Light Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 26

28 Spectral Matching Factor (1) Absorption, Emission, \% λ.1. R, Reflectivity Layers λ, Wavelength, nm Input and Output 1 5 SMF = λ, Wavelength, nm Fluorescein Emission Filter Spectral Matching L λ (λ) R (λ) L λ (λ) R (λ) Reflective Filter (Ch. 8) Spectral Radiance Output of Filter L λ (λ) = L λ (λ) R (λ) Total Radiance Output L = L λ (λ) dλ = L λ (λ) R (λ) dλ Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 27

29 Spectral Matching Factor (2) Total Radiance (Previous Page) L = L λ (λ) dλ = L λ (λ) R (λ) dλ Define Reflectance for This Filter; R = L /L One Number as Opposed to R (λ) Relate to Transmission at Spectral Maximum Characterize SMF for Given Input Spectrum R = R max R (λ) R max f λ (λ) dλ = R max SMF SMF = R R max Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 28

30 Summary of Spectral Radiometry For every radiometric quantity, there exists a spectral radiometric quantity. In the name, no distinction is made between frequency and wavelength derivatives. The notation: subscript ν for frequency or λ for wavelength. The units are the original units divided by frequency or wavelength units. Spectral Fraction, f λ can be applied to any of the radiometric quantities. The spatial derivatives (area and angle) are valid for the spectral quantities, wavelength by wavelength. The behavior of filters is more complicated, and is usually treated with the spectral matching factor. Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 29

31 Photometry and Colorimetry Spectral Luminous Efficiency, ȳ (λ) Source Spectral Radiance, L λ (λ, x, y) Eye Response Y (x, y) = ȳ (λ) L λ (λ, x, y) dλ Four LEDs: Equal Radiance Blue, 4 Appears Weak Green, 55 Appears Strong Red, 63 Moderately Weak IR, 98 Invisible ybar Wavelength, nm. Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 3

32 Lumens Power or Radiant Flux (Watts) P = P λ (λ) Eye Response Y = ȳ (λ) P λ (λ) dλ Luminous Flux (Lumens, Subscript V 683 lumens/watt P (V ) = max (ȳ) for Clarity) ȳ (λ) P λ (λ) dλ Luminous Efficiency P (V ) P = 683 lumens/watt ȳ (λ) P λ (λ) max (ȳ) P dλ Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 31

33 Some Typical Radiance and Luminance Values Object W/m 2 /sr nits = lm/m 2 /sr Footlamberts lm/w Minimum Visible Green Dark Clouds.2 Vis Lunar disk 13 Vis Clear Sky 27 Vis Bright Clouds 13 Vis All 82 Solar disk Vis All Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 32

34 Tristimulus Values: Three is Enough X, Y, Z X = Y = Z = Example: 3 Lasers Krypton λ red = 647.1nm X R = x (λ) L λ (λ) dλ ȳ (λ) L λ (λ) dλ z (λ) L λ (λ) dλ x (λ) δ (λ 647.1nm) dλ = x (647.1nm) xbar,ybar,zbar x z Wavelength, nm. y x xbar ybar zbar X R =.337 Y R =.134 Z R =. Krypton λ green = 53.9nm X G =.171 Y G =.831 Z G =.35 xbar,ybar,zbar 1 Argon λ blue = 457.9nm.5 X B =.289 Y B =.31 Z B = Wavelength, nm. Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 33

35 Chromaticity Coordinates (1) Three Laser Powers, R, G, and B, Watts Tristimulus Values X Y Z = X R X G X B Y R Y G Y B Z R Z G Z B R G B Chromaticity Coordinates (Normalized X, Y x = X X + Y + Z y = Y X + Y + Z Monochromatic Light x = x (λ) x (λ) + ȳ (λ) + z (λ) y = ȳ (λ) x (λ) + ȳ (λ) + z (λ) Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 34

36 Chromaticity Coordinates (2) Monochromatic Light (Boundary) x = y = x (λ) x (λ) + ȳ (λ) + z (λ) ȳ (λ) x (λ) + ȳ (λ) + z (λ) 3 Lasers (Triangles) and Their Gamut (Black - -) y K 3K 35K λ=457.9 nm λ=53.9 nm λ=647.1 nm RGB Phosphors (White) Thermal Sources (Later) x Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 35

37 Generating Colored Light Given P (V ), x, and y Required Tristimulus Values Y = X = P (V ) y 683lm/W xp (V ) y 683lm/W Z = (1 x y) P (V ) y 683lm/W Powers of Three Lasers R G = B X R X G X B Y R Y G Y B Z R Z G Z B 1 X Y Z Projector: 1lux White x = y = 1/3 4lm on (2m) 2 = 4m 2 X = Y = Z = 4lm/(1/3)/683lm/W R = 1.2W, G =.5W, B =.33W Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 36

38 Another Example: Simulating Blue Laser Light Monochromatic Light from Blue Argon Laser Line λ = 488nm X =.65 Y =.2112 Z =.563 Solution with Inverse Matrix R =.2429 G =.2811 B =.3261 Old Bear Hunters Saying: Sometimes You Eat the Bear,... and Sometimes the Bear Eats You. Best Compromise (Right Hue, Insufficient Saturation) R = G =.2811 B =.3261 Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 37

39 Summary of Color Three color matching functions x, ȳ, and z Tristimulus Values: 3 Integrals X, Y, and Z Describe Visual Response Y Related to Brightness. 683lm/W, Links Radiometric to Photometric Units Chromaticity Coordinates, x and y, Describe Color. Gamut of Human Vision inside the Horseshoe Curve. Any 3 sources R, G, B, Determine Tristimulus Values. Sources with Different Spectra May Appear Identical. Light Reflected or Scattered May Appear Different Under Different Sources Of Illumination. R, G, and B, to X, Y, Z Conversion Can Be Inverted over Limited Gamut. Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 38

40 The Radiometer or Photometer Aperture Stop Field Stop Measured Power P = LAΩ = LA Ω Adjustable Stops (Match FOV) Sighting Scope? Calibration Required Spectrometer on Output? Spectral Filter? Photometric Filter? Computer Radiance,P/(ΩA) Luminance (All Units) Spectral Quantities Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 39

41 Integrating Sphere Power from Intensity Integrate over Solid Angle Goniometry Information Rich Time Intensive Integrating Sphere Easy Single Measurement Applications Wide Angle Sources Diffuse Materials Variations Two Spheres Spectroscopic Detector More Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 4

42 Blackbody Radiation Outline Background Equations, Approximations Examples Illumination Thermal Imaging Polar Bears, Greenhouses m λ, Spect. Radiant Exitance, W/m 2 /µ m λ, Wavelength, µ m Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 41

43 Blackbody Background Cavity Modes 2πν x l c 2πν x l c = N x π = N y π Waves in a Cavity 2πν x l c = N z π Mode Frequencies ν = ν 2 x + ν 2 y + ν 2 z ν = c 2l 2 E x E y E z 2 = 1 2 E c 2 t 2 ν = ν N 2 x + N 2 y + N 2 z Boundary Conditions E (x, y, z, t) = sin 2πν xx c sin 2πν yy c sin 2πν zz c sin 2πνt Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 42

44 Counting Modes Q, Energy Density, J/Hz Temperature = 5K λ, Wavelength, µ m Mode Density (Sphere) dn = 4πν2 dν/8 c 3 /(8l 3 ) N ν = dn dν = 4πν2 l 3 c 3 2 for Polarization N ν = 8πν2 l 3 c 3 Energy Distribution Rayleigh Jeans: Equal Energy, k B T (Dash Dot) J ν = k B T 8πν2 l 3 c 3 Ultraviolet Catastrophe Wein hν exp (hν/kt ) per Mode J ν = hν exp (hν/kt ) 8πν2 l 3 Dashed c 3 Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 43

45 Planck Got it Right Q, Energy Density, J/Hz Temperature = 5K λ, Wavelength, µ m Planck Mode Energy Quantized J N = Nhν N Random Average Energy Density N= J N P (J N ) J = N= P (J N ) Boltzmann Distribution J N P (J N ) = e k B T Average Energy J = N= hν Nhν k Nhνe B T N= e Nhν k B T N= Nhν k Ne B T N= e Nhν k B T With Some Work... J = hν e hν/k B T (1 e hν/k B T ) 2 1 (1 e hν/k B T ) hν e hν/k BT 1 = = Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 44

46 Planck s Result (1) Previous Page hν e hν/k BT 1 Mode Density N ν J = 8πν2 l 3 hν c 3 e hν/kbt 1 Solid Curve (Previous Page) Energy Per Volume d w dν = N ν J l 3 = 8πν2 c 3 hν e hν/k BT 1 1. Radiance Independent of Direction 2. Emssion Balanced by Absorption M(T ) emitted = ɛ a Ē(T ) incident 3. Balance at Every λ 4. Light Moves at c w λ = 1 c E from L E λ = sphere LdΩ = 4πl c L λ cos θ sin θdθdζ = πl λ Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 45

47 Planck s Result (2) E from w: E λ = 4cw λ Wall Balance & ɛ a = 1: M(T ) bb = E(T ) incident Planck s Law M ν(bb) (λ, T ) = hνc 2πν2 c 3 1 e hν/k BT 1 = 2πhν3 c 2 1 e hν/k BT 1 Emissivity Detailed Balance M ν (λ, T ) = ɛm ν(bb) (λ, T ) ɛ (λ) = ɛ a (λ) Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 46

48 The Planck Equation Fractional Linewidth dν/ν = dλ/λ M λ = dm dλ = dν dm dλ dν = ν dm λ dν = c dm λ 2 dν Planck Law vs. Wavelength M λ (λ, T ) = 2πhc2 λ 5 1 e hν/k BT 1 Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 47

49 Useful Blackbody Equations Wien Displacement Law λ peak T = 2898µm K Stefan Boltzmann Law m λ, Spect. Radiant Exitance, W/m 2 /µ m M (T ) = 2π5 k 4 T h 3 c 2 = σ e T λ, Wavelength, µ m Stefan Boltzmann Constant σ e = W/cm 2 /K 4 = W/m 2 /K 4 Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 48

50 Some Examples: Thermal Equilibrium Earth Temperature Heating 1kw/m 2 Half of Surface Cooling Radiation Result M (T ) = σ e T 4 T 36K Body Temperature Cooling (31K) M (T ) = σ e T 4 = 524W/m 2 Heating (Room=295K) M (T ) = σ e T 4 = 429W/m 2 Excess Cooling M A 1W 2kCal/day (Many Approximations) Because d ( T 4) = 4T 3 dt, small changes in temperature have big effects on heating or cooling. Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 49

51 Temperature and Color Radiant Exitance M (R) = M (R)λ dλ Luminous Exitance M, Radiance, W/m 2 /sr M (V ) = M (V )λ dλ M (V ) = 683lm/W Solar Disk Min. Vis. Lunar Disk ȳm (R)λ dλ Solar Disk 1 1 Lunar Disk Min. Vis T, Temperature, K M, Luminance, L/m 2 /sr Tristimulus Values X Y Z = x ȳ z Integrands for 2 3, 35K xs,ys,zs xs,ys,zs xs,ys,zs M (R)λ dλ λ, Wavelength, nm λ, Wavelength, nm λ, Wavelength, nm Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 5

52 Chromaticity Coordinates of Blackbody Chromaticity.9.8 x = y = X X + Y + Z Y X + Y + Z x y y K 3K 35K λ=457.9 nm λ=53.9 nm λ=647.1 nm RGB x,y, Color Coordinates T, Temperature, K x Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 51

53 Solar Spectrum Exo Atmospheric 6K, 148W/m 2 Sea Level 5K, 1W/m 2 E λ (λ) = 156W/m 2 E λ (λ) = 125W/m 2 f λ (λ,6k) f λ (λ,5k) 25 2 e λ, W/m 2 /µ m e λ, W/m 2 /µ m λ, Wavelength, µ m λ, Wavelength, µ m Constants are higher than total irradiance to account for absorption in certain regions of the spectrum. Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 52

54 Outdoor Radiance Sunlit Cloud E incident = 1W/m 2 5K White Cloud M cloud = E incident Lambertian L cloud = M cloud /π m λ, Spect. Radiant Exitance, W/m 2 /µ m Blue Sky: M λ = M sky f λ (5K) λ Sun Cloud Blue Sky Night λ, Wavelength, µ m Radiant Exitance, M of Object Surface Illuminated with E M λ (λ) = R (λ) E λ (λ) Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 53

55 Thermal Illumination Thermal Sources (e.g. Tungsten) M (R) = M (R)λ dλ M (V ) = 683lm/W ȳm (R)λ dλ Luminous Efficiency M (V ) /M (R) = 683lm/W ȳm (R)λdλ M (R)λdλ Highest Efficiency λ = 555nm Bad Color Efficiency, lm/w T, Temperature, K Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 54

56 Illumination Thanks to Dr. Joseph F. Hetherington Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 55

57 IR Thermal Imaging Infrared Imaging λ2 ρ (λ) ɛ (λ) M λ (λ, T ) dλ λ 1 ρ is Responsivity ɛ is Emissivity Maximize Sensitivity M λ (λ, T ) T Work in Atmospheric Pass Bands Shorter Wavelength for Higher Temperatures Hard to Calibrate (Emissivity, etc.) Reflectance and Emission M λ /Delta T m λ, Spect. Radiant Exitance, W/m 2 /µ m M λ /Delta T T = 3 K T = 5 K λ, Wavelength, µ m 1 1 Sun Cloud Blue Sky Night 1 2 λ, µ m Wavelength, Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 56

58 Polar Bears Thermal Equilibrium Heating by Sun High Temperature High Radiance Small Solid Angle Cooling to Surroundings Body Temperature Low Radiance Ω = 2π Extra Heat from Metabolism Short Pass Filter Pass Visible E incident = 5W/m 2, 2W/m 2 6W/m 2, top to bottom W/m 2 /µ m M λ M λ λ, Wavelength, µ m λ, Wavelength, µ m λ, Wavelength, µ m 15 2 Heating ( ), Cooling (- -), Net ( ) Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 57

59 Wavelength Filtering 15 1 Perfect Bear Bare Bear 8nm LongPass 2.5µm LongPass Best Bear Net Cooling for Two Best µm ShortPass Dynamic Filter M net, W/m 2 5 M lost, W/m E sun, W/m 2 Bare Bear: No Filter 8nm Short Pass 2.5nm Short Pass Like Glass Best Bear is a Dynamic Filter E sun, W/m 2 Dynamic Filter Follows Zero Crossing of Net M λ Perfect: No Cooling (Impossible: Nothing Out Means Nothing In) Apr. 214 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1 58