1 Introduction to Radiometry Mathieu Hébert, Thierry Lépine
Program 2 Radiometry and Color science IOGS CIMET MINASP 3DMT Introduction to radiometry Advanced radiometry (2 nd semester) x x x x x o o Color Science x x Color measurement and assessment x
Radiometry 3 Radiometry: science of emission, propagation and detection of radiations
Radiometry / Photometry 4 Radiometry: science of emission, propagation and detection of radiations Photometry: optical radiometry, i.e. emission, propagation and detection of light OPTICS
Radiometry / Photometry 5 Radiometry: science of emission, propagation and detection of radiations Photometry: optical radiometry, i.e. emission, propagation and detection of light based mainly upon: the particle-like (photon) behavior of light the laws of geometrical optics (linear propagation along rays) but wave-like behavior of light (diffraction and interference) must be taken into account in image forming systems, not necessarily in more general lighting systems
Applications 6 Space Navigation Earth observation Satellite communication
Applications 7 Space Industry Laser soldering Fiber optic communications Laser cleaning Electronic circuit it controls Chemical analysis Lighting (domestic, urban) Thermal application in building Bar code readers
Applications 8 Space Industry Media Camera and video Video projector cd reader
Main concepts to remember 9 Geometrical concepts Areas on surfaces Solid angles directions in space Geometrical extent areas and directions Four basic quantities Flux Intensity Radiance Irradiance Unit systems Energetic, visual, photon Spectral
Geometrical concepts 10 Areas Area on a planar surface Area on a sphere Apparent or projected area
Geometrical concepts 11 Directions Spherical coordinates Polar angle θ :from0toπ π Azimuthal angle φ : from 0 to 2π Hemisphere : Polar angle θ : from 0toπ/2 Azimuthal angle φ : from 0 to 2π
Geometrical concepts 12 Solid angle Area on a sphere / (radius of the sphere) 2, in steradian (sr) Denotes a direction (when very small) or a set of directions Whole sphere : 4π sr Hemisphere : 2π sr Infinitesimal solid angle in some direction: d 2 ω = sinθ d θ d φ Annular solid angle dω = 2πsinαdα
Geometrical concepts 13 Geometrical extent Transfer volume between two surfaces (Apparent area S A ) (apparent area S B ) / distance² (in sr.m 2 ) S A θ A θ B cosθ cosθ L ( )( ) S B S A A B B G = 2 L S Extends the notion of solid angle G= ω S cosθ = S cosθ ω ( ) ( ) A B B A A B
Geometrical concepts 14 Geometrical extent Change of extent due to refraction at some interface 2 dgi 2 dgt = dsdωicosθ1 = dsdωtcosθ2 n sinθ = n sinθ 1 1 2 2 n cosθ d θ = n cosθ d θ 1 1 1 2 2 2 ndω cosθ = ndω cosθ 2 2 1 i 1 2 t 2 n dg = n dg 2 2 2 2 1 i 2 t
Geometrical concepts 15 Geometrical extent No change of extent through optical systems (from air to air) ( ) 1 1 2 2 3 3 4 4 2 2 2 2 2 2 2 2 2 2 dg = ndg = ndg = ndg = ndg = dg A B
The four radiometric quantities 16 Flux Used to define amounts of radiation Homogeneous to: Energy per unit time = Energetic power, in Watt (W) Number of photons per unit time (s -1 ) Visual stimulus Flux concept in itself is regardless of direction and position in space Energetic unit W Visual unit lm (lumen) Photon unit s -1
The four radiometric quantities 17 Visual Flux Spectral sensitivity of the human visual system V(λ) F λ ( ) visual F ( λ ) F energetic = KV m λ ( ) CIE (1998) Colorimetry; CIE Technical Report, 3rd Edition.
The four radiometric quantities 18 Visual Flux Spectral sensitivity of the human visual system depends upon luminance of the scene R. Sève, Physique de la couleur, Masson 1996
The four radiometric quantities 19 Photon Flux / Energetic flux Each has got some energy, related to the frequency ν of the associated wave (or the corresponding wavelength λ 0 in vacuum): hc E = hν = h = 6.62 10 34 Js (Planck constant) λ0 c = 3 10 8 m/s Radiant power of a flux of photons: P= Fp hν P = radiant power (energy per second) F p = Flux of photons (number of photons per second) hν = energy of one photon
The four radiometric quantities 20 Flux of some sources Tungsten lamp Green LED Nd-Yag laser Power supply (W) 100 2 20 Energetic flux F v (W) 90 1 1 Photon flux (s -1 ) 10 21 2.5 10 18 5 10 18 Visual flux F e (lm) 1500 100 0 Radiating efficiency 90% 50% 5% L i ffi it Luminous efficacity F v /F e (lm/w) 17 100 0
The four radiometric quantities 21 Intensity Density of flux per unit solid angle (W. sr) May depend upon direction I(θ,φ) I = df dω Energetic units W.sr -1 Visual units cd (candela) Photon units s -1.sr -1
The four radiometric quantities 22 Intensity I(θ,φ) ( = Intensity diagram for point sources (e.g. LED) θ,φ Isotropic source I( θ,φ)= cste Directional source F = I θ,φ sinθ d θ d φ π Flux from intensity: ( ) 4 sr
The four radiometric quantities 23 Radiance (luminance) Density of flux per unit geometrical extent L = 2 df 2 dg May depend upon position on a surface (x,y) and direction (θ, φ) L( x, y,θ,φ) θ ds dω 2 dg = dscosθdω Apparent area Energetic units Visual units Photon units W.sr -1.m -2 cd.m -2 s -1.m -2.sr -1 W 1 2
The four radiometric quantities 24 Radiance (luminance) Radiance characterizes light rays Adapted to surface sources Homogeneously illuminated area Radiance is independent of position (x,y) L( x, y)= cste Perfectly diffuse light (Lambert law) Radiance is independent of direction (θ,φ) Lambertian source L L L L L L( θ,φ)= cste A surface able to perfectly diffuse light is a Lambertian surface
The four radiometric quantities 25 Irradiance (illuminance) Density of flux (received by a surface) per unit area It may depend upon position (x,y) on the surface E = df ds Energetic unit Visual unit Photon unit Wm W.m -2 lux s -1.m -2 (1 lux = 1 lumen / m 2 )
The four radiometric quantities 26 Irradiance (illuminance) Examples Surfaces illuminated by: Total starlight, overcast sky 10 44 Quarter moon 0.01 Full moon on a clear night 027 0.27 Full moon overhead at tropical latitudes 1 Very dark overcast day 100 Illuminance (lux) Office lighting 320 500 Sunrise or sunset on a clear day 400 Overcast day 1000² Full daylight (not direct sun) 10,000 25,000 Direct sunlight 32,000 130,000
Other radiometric quantities 27 Exitance Density of flux (emitted by a surface) per unit area It may depend upon position (x,y) on the surface M = df ds Energetic unit Visual unit Photon unit Wm W.m -2 lm.mm -2 s -1.m -2
Other radiometric quantities 28 Energy (Integrated flux) Amount of flux collected during a finite time (accumulated energy) Φ = F Δ t Energetic unit Visual unit Photon unit J lm.s Nb photons
Other radiometric quantities 29 Fluence (exposure) Accumulated energy per unit area In visual unit : illuminance multiplied by exposition time H = EΔt Energetic unit Visual unit Photon unit J.m -2 lx.s Nb photons / m 2
Overview of the radiometric quantities 30 Energetic Visual Photon Radiant power, Flux Flux Photon flux W lumen (lm) s -1 Intensity Intensity Intensity W sr -1 candela (cd) s -1 sr -1 Radiance Luminance W sr -1 m -2 cd m -2 Radiance s -1 m -2 sr -1 Exitance W m -2 Irradiance W m -2 Energy J Fluence J m -2 Exitance lm m -2 Illuminance lux (lx) Integrated flux lm s Exposure lx s Exitance s -1 m -2 Irradiance s -1 m -2 Nb of photons Nb of photons m -2
Spectral quantities 31 Spectrum spectral density of radiometric quantity y( (flux, intensity, radiance, irradiance) per unit wavelength (or wavenumber or frequency) Spectral density of Name Notation Unit Power Spectral power F λ = df / dλ W µm -1 Intensity Spectral intensity I λ = di / dλ W sr -1 µm -1 Radiance Spectral radiance L λ = dl / dλ W m -2 sr -1 µm -1 Irradiance Spectral Irradiance E λ = de / dλ W m -2 µm -1 Example : Total flux over the spectral domain [λ 1, λ 2 ]: F = 2 F dλ λ λ 1 λ
Relationship between quantities 32 Flux and intensity Infinitesimal solid angle dω (θ,φ) df = I( θ,φ) dω Finite (small) solid angle Δω ΔF = I average Δω Large solid angle Ω F = I( θ,φ) sinθdθdφ Ω NB: isotropic source I = cst F = IΩ
Relationship between quantities 33 Flux and radiance pencil of light (quasi collimated beam) ds θ L dω 2 df Ld G Lds d = = cosθ ω beam of finite extent 2 F L x, y, z,θ,φ d G = S, s 1 2 ( ) NB: Lambertian source L = cst F = LG
Relationship between quantities 34 Radiance and intensity ( ) = L xy,,θ,φθ φ dacosψψ dω Contribution of all da on the source: ( ) = ( y ) df θ,φ, dω L x, y,θ,φ,φ dacosψψ dω source Corresponding intensity: i I ( θ,φ ) = df = source ( θ,φ, dω) dω ( ) L x, y,θ,φ dacosψ
Relationship between quantities 35 Radiance and exitance ( ) = ( ) 2 df L da d θ,φ θ,φ cosθ ω Flux emitted over the hemisphere: ( ) = ( ) df θ,φ L θ,φ θφ dacosθ dω hemisphere Corresponding exitance: ( θ,φ) df M= = L( θ,φ) cosθdω da hemisphere
Relationship between quantities 36 Irradiance and intensity : Bourguer s law Irradiance of a flat surface from a source of intensity I Point source θ da dacosθ Flux received by da: df = I 2 x E df Icosθ = = 2 da x Resulting irradiance: (valid if x is long!)
Lambertian radiations 37 Radiance Irradiance ΔΩ Exitance: M= L cosθsinθdθφ d ΔΩ Special case of hemispherical solid angle: M = πl Radiance Intensity I ( θ) = LAcosθ