Lecture 3 Radiometric and Photometric Properties of Light

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Photon-based Radiometric Quantities

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Number of Photons

• Each photon carries an energy hv or hc/λ • Total energy of optical

Q= Nh Q N = . h

N— the number of photons (unitless) • Define an entirely equivalent set of radiometric quantities • Valuable whenever discrete nature of light is relevant. • Cases: studying photodetection

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Photon Density

• Photon density n • The number of photons per unit volume. N n = n = m-3 V Photon Flux

• Photon flux Pq • The photon quantity analogous to power.

dN -1 P = [P ] = s q dt q q — a photon quantity • Connection between Pq and P (power) is straight forward. 푑푄 푑(푁 ⋅ ℎ휈) 푑푁 푃 = = = ℎ휈 = ℎ휈푃 푑푡 푑푡 푑푡 푞

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Photon Temporal Power Spectrum

• Photon temporal power spectrum Sq • The photon flux per unit temporal frequency.

dPq () -1 S () = [Sq ] = s / Hz q d

Sq(v) — the number of photons per second in the infinitesimal interval (v, v+dv) • Energy-based power spectrum S(v) d P( ) 1 P ( ) SSq ()()= = − d h h 1 For = 2 , S()() = S v qq2 c For = c/ , SS()()=− qq 2

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Photon Intensity

q • Photon intensity I • The photon flux per unit solid angle, Ω.

q dPq () q −1 I = [I ] = s / srad d nˆ dA dk dk d = = x y r 2 k 2

2 q 2 d Pq (k ⊥ ) 2 I = k 2 = k Sq (k ⊥ ) dk ⊥ 2 푑 퐤⊥ = 푑푘푥푑푘푦

Sq (k ⊥ ) — the spatial power spectrum of the photon flux

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Photon Irradiance

• Photon irradiance Iq • The photon flux per unit area. dP s −1 I = q [I ] = q dA q m2 • Energy-based irradiance dP dP I = = hv q = hvI dA dA q Photon Spectral Irradiance

q • Photon spectral irradiance I • The photon flux per unit area, per unit of optical frequency. d 2 P s −1 I q = q [I q ] = dAd m2 Hz

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Photon Radiance

• Photon radiance Lq • The photon flux emitted or absorbed by a surface, per unit solid angle, per unit projected area. d 2 P (, A) L = q [L ] = s −1 / srad m2 q ddAcos q dAcosθ — the projection of the area normal to the direction of interest

Photon Spectral Radiance

q • Photon spectral radiance L • The photon radiance per unit frequency. dL s −1 Lq = q [Lq ] = d m2 srad Hz

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Photon Exitance

• Photon exitance Mq • The amount of photon flux emitted by a source per unit area. −1 dPq s M q = [M q ] = 2 dAS m • Note: Photon exitance describes a source, while the photon irradiance is a property of the field and shares the same units.

Photon Spectral Exitance q • Photon spectral exitance M • The photon exitance per unit frequency. dM s −1 M q = q [M q ] = d m2 Hz

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Photometric Properties of Light

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Photometric properties of light

• Photometry 1 • Properties of the light perceived by the 0.9 daytime dark time human eye (subjective photodetector). 0.8 0.7 • Spectral sensitivity of the eye is taken into 0.6

account. 0.5 • Luminosity function 0.4 • The eye spectral response 0.3 Luminosity function Luminosity 0.2 • Day-time (photopic) adaptation 0.1 • Dark time (scotopic) adaptation 0 • Day time spectral response — Luminous 400 450 500 550 600 650 700 Wavelength [nm] Figure 4.1. Photopic (daytime-adapted, green curve) and scotopic (darkness-adapted blue curve) luminosity functions.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Luminous Energy

• Luminous energy Ql • The total energy weighted according to the luminosity function. • Measured in lumen·second. Lumen is the unit for luminous flux.

[Ql ] = lm s Luminous Flux

• Luminous flux (or luminous power) Pl • The photometric equivalent of power. • Weighted by the luminosity. dQ P = l [P ] = lm l dt l

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Luminous Energy Density

• Luminous energy density Wl • The luminous energy per unit volume dQ lm s W = l [W ] = l dV l m3

Luminous Intensity

• Luminous intensity Il • The luminous flux per unit solid angle dP lm I = l [I ] = = Cd(candela) l d l srad

• Note: Candela (cd) is a primary units comprising the International System of Units (SI).

Prof. Gabriel Popescu Light Emission, Detection, and Statistics 4.4. Luminous Intensity

• Standard of candela 1 • The candela is the luminous intensity in a given 0.9 daytime dark time direction, of a source that emits monochromatic 0.8 radiation of frequency 540 × 1012 Hz and that has 0.7 a radiant intensity in that direction of 1/683 W/rad. 0.6 • For the frequency of 540 × 1012 Hz, λ=c/v=555nm, at 0.5 which the human eye is the most sensitive in bright 0.4 conditions. 0.3 • The luminous intensity with respect to function Luminosity 0.2 wavelength 0.1 0 Ω 400 450 500 550 600 650 700 퐼푙 (휆) = 퐼Ω(휆)푌(휆) 683 푙푚/푊 Wavelength [nm] IΩ — radiometric intensity (in W/srad) Figure 4.1. Photopic (daytime-adapted, green curve) and scotopic (darkness-adapted blue curve) luminosity functions. Y(λ) — the photopic luminosity function evaluated at the particular wavelength

Prof. Gabriel Popescu Light Emission, Detection, and Statistics Illuminance

• Illuminance Il (Equivalent to the photometric irradiance) • The luminous flux per unit area. dP lm I = l [I ] = = lux,lx l dA l m2 Luminance

• Luminance Ll (Equivalent to the photometric radiance) • The luminous power per solid angle, per normal unit area of the source. d 2 P lm cd L = l [L ] = = l ddAcos( ) l srad m2 m2 • Brightness is sometimes used interchangeably with luminance.

Prof. Gabriel Popescu Light Emission, Detection, and Statistics