Optics for Engineers Chapter 12 Charles A. DiMarzio Northeastern University Apr. 2014 Radiometry • Power is Proportional to { Area of Aperture Stop { Area of Field Stop { \Brightness" of the Source (Radiance) Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{1 Radiometry and Photometry φ Radiant Radiant Flux, M Note: \Spectral X:" Xν, Exitance, Watt = W units =Hz or Xλ, /µm. W=m2 Rad. to Phot: Luminous Flux, Luminous 683 lm=W × y (λ). Exitance, lumen = 2 y (555nm) ≈ 1. lm lm=m = @=@A ! lux = lx # @=@Ω # # @=@Ω # ·=R2 @=@A ! E Irradiance, W=m2 Illuminance, lm=m2 = lux I Radiant L Radiance, Intensity, W m2 sr W sr = = = Luminance, Luminous nit = Intensity, 2 lm=sr = cd lm=m =sr 1Lambert = 1lm=cm2=sr/π 1Ft:Candle = 1lm=ft2 1mLambert = 1lm=m2=sr/π 1FtLambert = 1lm=ft2=sr/π Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{2 Radiometric Quantities Quantity Symbol Equation SI Units Radiant Energy Q Joules 3 Radiant Energy Density = d Q Joules m3 w w dV 3 = Radiant Flux or Power or Φ Φ = dQ W P dt Radiant Exitance = dΦ W m2 M M dA = Irradiance = dΦ W m2 E E dA = Radiant Intensity = dΦ W sr I I dΩ = Radiance = d2Φ W m2 L lm dA cos θdΩ = Fluence Ψ dQ J m2 dA = Fluence Rate dΨ J m2 F dt = Emissivity = M Dimensionless Mbb Spectral () () d() () Hz ν dν = or () d() () m λ dλ /µ Luminous Flux or Power P or Φ lm Luminous Exitance = dΦ lm m2 M M dA = Illuminance = dΦ lm m2 E E dA = Luminous Intensity = dΦ lm sr I I dΩ = Luminance = d2Φ lm m2 L L dA cos θdΩ = Spectral Luminous Efficiency V (λ) Dimensionless Color Matching Functions x¯ (λ) y¯(λ) z¯(λ) Dimensionless Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{3 Irradiance • Poynting Vector for Coherent Wave dP dS ≈ (sin θ cos ζx^ + sin θ sin ζy^ + cos θz^) 4πr2 • Irradiance (Projected Area, Aproj = A cos θ) 2 d P ( dP ) dE = 0 = dA 4πr2 Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{4 Irradiance and Radiant Intensity • Solid Angle A0 ·· Ω = r2 EI · • Intensity from Irradiance (Unresolved Source, A) I I = Er2 E = r2 2 2 d P d P ( dP ) dI = = 0 = dΩ dA 4πr2 r2 Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{5 Intensity and Radiance • Resolved Source: Many Unresolved Sources ·· Combined Z Z · IL @I I = dI = dA @A • Radiance • On Axis (x0 = y0 = 0) @I (θ; ζ) @I (θ; ζ) @2P L (x; y; θ; ζ) = L (x; y; θ; ζ) = = @A @A @A@Ω • Off Axis (Projected Area) 2 Z @I @ P L = = I (θ; ζ) = L (x; y; θ; ζ) dxdy @A cos θ @A@Ω cos θ A Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{6 Radiant Intensity and Radiant Flux • Solid Angle · 0 s 1 Φ ( ) · I · NA 2 Ω = 2π @1 − 1 − A • Flux or Power, P or Φ n • Integrate Intensity ZZ P = Φ = I (θ; φ) sin θdθdζ • Constant Intensity P = Φ = IΩ ZZ 0 0 Ω = sin θ dθ dζ Z Z ( q ) 2π Θ 0 0 Ω = sin θ dθ dζ = 2π (1 − cos Θ) = 2π 1 − 1 − sin2 Θ 0 0 Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{7 Radiance and Radiant Exitance · M ·· L • Radiant Exitance from a Source (Same Units as Irradiance) ZZ " # @2P M (x; y) = sin θdθdζ @A@dΩ • Radiant Exitance from Radiance ZZ M (x; y) = L (x; y; θ; ζ) cos θ sin θdθdζ • Radiance from Radiant Exitance @M (x; y) 1 L (x; y; θ; ζ) = @Ω cos θ Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{8 Radiance and Radiant Exitance: Special Cases • Constant L and Small Solid Angle M (x; y) = LΩ cos θ 0 s 1 ( ) ( ) NA 2 NA 2 Ω = 2π @1 − 1 − A or Ω ≈ π n n • Constant L, over Hemisphere Z Z 2π π=2 sin2 π M (x; y) = L cos θ sin θdθdζ = 2πL 2 : 0 0 2 M (x; y) = πL (Lambertian Source) Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{9 Radiant Exitance and Flux Φ M ··· • Power or Flux from Radiant Exitance ZZ P = Φ = M (x; y) dxdy; • Radiant Exitance from Power @P M (x; y) = @A Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{10 The Radiance Theorem in Air • Solid Angle two ways A0 A Ω = Ω0 = r2 r2 • Power Increment 0 dA 0 0 0dA dP = LdAdΩ = LdA cos θ dP = LdA dΩ = LdA cos θ r2 r2 Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{11 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, Ttotal, for Loss { Find Exit Window (Of a Scene or a Pixel) { Find Exit Pupil { Compute Power P = LobjectTtotalAexit windowΩexit pupil Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{12 Etendue and the Radiance Theorem • Abbe Invariant: 0 0 0 n x dα = nxdα • Etendue ( ) 0 2 0 0 n2AΩ = n A Ω • Power Conservation ZZZZ ZZZZ 0 0 0 Ld2Ad2Ω = L d2A d2Ω ZZZZ ZZZZ 0 ( ) L L 0 2 0 0 n2d2Ad2Ω = n d2A d2Ω n2 n2 L L0 Radiance Theorem: = n2 (n0)2 • Basic Radiance, L=n2, Conserved (Thermodynamics Later) Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{13 Radiance Theorem Example: Translation • Translation Matrix Equation ! ! " ! !# ! dx 1 z x + dx x dx 2 = 12 1 1 − 1 = 1 dα2 0 1 α1 + dα1 α1 dα1 dx2dα2 = dx1dα1 Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{14 Radiance Theorem Example: Imaging • Imaging Matrix Equation ! m 0 M 0 = SS ? n n0m Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{15 Radiance Theorem Example: Dielectric Interface • Dielectric Interface Matrix Equation ! 1 0 0 n n0 • Power Increment 2 d Φ = L1dA cos θ1dΩ1 = L2dA cos θ2dΩ2 L1dA cos θ1 sin θ1dθ1dζ1 = L2dA cos θ2 sin θ2dθ2dζ2 • Snell's Law and Derivative n1 sin θ1 = n2 sin θ2 n1 cos θ1dθ1 = n2 cos θ2dθ2 sin θ n dθ n cos θ 1 = 2 1 = 2 2 sin θ2 n1 dθ2 n1 cos θ1 • Radiance Theorem L L0 = n2 (n0)2 Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{16 Radiance Theorem Example: 1/2 Telecentric Relay • Fourier{Optics Matrix Equation ! 0 M f FF 0 = 1 f 0 ( )2 M 0 2 0 0 0 det FF 0 = 1 = n =n n AΩ = n A Ω • Radiance Theorem L L0 = n2 (n0)2 Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{17 Idealized Example: 100W Lamp • P = 100W, Uniform in Angle • Tungsten Filament: Area A = (5mm)2, Distance, R • Radiant Exitance M = 100W= (5mm)2 = 4W=mm2 • Radiance (Uniform in Angle) L = M= (4π) ≈ 0:32W=mm2=sr • Intensity I = 100W= (4π) ≈ 7:9W=sr • Reciever: Area A0: Received Power and Irradiance ( ) 0 0 I PA = 4πr2 = IA =r2 E = ≈ 8W=sr= (10m)2 = 0:08W=m2 r2 Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{18 100W Lamp at the Receiver • Using Source Radiance at Receiver A 0 E = L = LΩ r2 • Irradiance Same as Previous Page (5mm)2 E ≈ 0:32W=mm2=sr × ≈ 0:08W=m2 (10m)2 • Useful if r and A are Not Known { Only Depends on Ω0 (Measured at Reciever) Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{19 Practical Example: Imaging the Moon 2 • Known Moon Radiance Lmoon = 13W=m =sr • Calculation { Jones matrices for Transmission (0:25), other multiplica- ( )12 tive losses (12 Lenses: 0:962 ). L = 13W=m2=sr×0:25×0:9624 = 13W=m2=sr×0:25×0:367 = 1:2W=m2=sr { Exit Pupil (NA = 0:1) and exit window (Pixel: d = 10µm). { Irradiance, E = LΩ and Power on a Pixel, P = EA = LAΩ ( q ) ( ) − 2 P = 1:2W=m2=sr × 2π 1 − 1 − NA2 × 10 × 10 6m − − P = 1:2W=m2=sr × 0:315sr × 10 10m2 = 3:8 × 10 12W { If desired, multiply by time (1=30sec) to obtain energy. ∗ About One Million Photons (and Electrons) { Alternative to Solve in Object Space (Need Pixel Size on Moon) Apr. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{20 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 Pro- jected Area, A cos θ and Solid Angle, Ω. • Basic Radiance, L=n2, Conserved, with the Exception of Mul- tiplicative 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. 2014 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides12r1{21 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) dL 2 dL 2 Lν = W=m =sr=THz or L = W=m =sr/µm dν λ dλ • Spectral Fraction X (λ) f (λ) = λ for X = Φ; M; I; E; or L λ X Apr.