List of optics equations

This article summarizes equations used in optics, including geometric optics, physical optics, radiometry, diffraction, and interferometry.

Definitions

Geometric optics (luminal rays)

General fundamental quantities

Quantity (common name/s) (Common) symbol/s SI units Dimension Object distance x, s, d, u, x1, s1, d1, u1 m [L] Image distance x', s', d', v, x2, s2, d2, v2 m [L] Object height y, h, y1, h1 m [L] Image height y', h', H, y2, h2, H2 m [L] Angle subtended by object θ, θo, θ1 rad dimensionless Angle subtended by image θ', θi, θ2 rad dimensionless Curvature radius of lens/mirror r, R m [L] Focal length f m [L]

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Lens power P m−1 = D (dioptre) [L]−1 Lateral magnification m dimensionless dimensionless Angular magnification m dimensionless dimensionless

Physical optics (EM luminal waves)

There are different forms of the Poynting vector, the most common are in terms of the E and B or E and H fields.

Quantity (common name/s) (Common) symbol/s Defining equation SI units DimensionPoynting vector S, N W m−2 [M][T]−3Poynting flux, EM field power flow ΦS, ΦN W [M][L]2[T]−3RMS Electric field of Light Erms N C−1 = V m−1 [M][L][T]−3[I]−1Radiation momentum p, pEM, pr J s m−1 [M][L][T]−1Radiation pressure Pr, pr, PEM W m−2 [M][T]−3

Radiometry

For spectral quantities two definitions are in use to refer to the same quantity, in terms of frequency or wavelength.

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Radiant energy Q, E, Qe, Ee J [M][L]2[T]−2 Radiant exposure He J m−2 [M][T]−3 Radiant energy density ωe J m−3 [M][L]−3 Radiant flux, radiant power Φ, Φe W [M][L]2[T]−3 Radiant intensity I, Ie W sr−1 [M][L]2[T]−3 Radiance, intensity L, Le W sr−1 m−2 [M][T]−3 Irradiance E, I, Ee, Ie W m−2[M][T]−3 Radiant exitance, radiant emittance M, Me W m−2 [M][T]−3 Radiosity J, Jν, Je, Jeν W m−2 [M][T]−3 Spectral radiant flux, spectral radiant power Φλ, Φν, Φeλ, Φeν W m−1 (Φλ) W Hz−1 = J (Φν) [M][L]−3[T]−3 (Φλ) [M][L]−2[T]−2 (Φν) Spectral radiant intensity Iλ, Iν, Ieλ, Ieν W sr−1 m−1 (Iλ) W sr−1 Hz−1 (Iν) [M][L]−3[T]−3 (Iλ) [M][L]2[T]−2 (Iν) Spectral radiance Lλ, Lν, Leλ, Leν

W sr−1 m−3 (Lλ) W sr−1 m−2 Hz−1 (Lν) [M][L]−1[T]−3 (Lλ) [M][L]−2[T]−2 (Lν) Spectral irradiance Eλ, Eν, Eeλ, Eeν W m−3 (Eλ) W m−2 Hz−1 (Eν) [M][L]−1[T]−3 (Eλ) [M][L]−2[T]−2 (Eν)

Equations

Luminal electromagnetic waves

Physical situation Nomenclature EquationsEnergy density in an EM wave = mean energy density For a dielectric:Kinetic and potential momenta (non-standard terms in use) Potential momentum:

Kinetic momentum:

Canonical momentum:Irradiance, light intensity = time averaged poynting vector I = irradiance I0 = intensity of source P0 = power of point source Ω = solid angle r = radial position from sourceAt a spherical surface: Doppler effect for light (relativistic) Cherenkov radiation, cone angle n = refractive index v = speed of particle θ = cone angleElectric and magnetic amplitudesE = electric field H = magnetic field strengthFor a dielectricEM wave componentsElectric

Magnetic

Geometric optics

Physical situation Nomenclature EquationsCritical angle (optics)n1 = refractive index of initial medium n2 = refractive index of final medium θc = critical angleThin lens equation f = lens focal length x1 = object distance x2 = image distance r1 = incident curvature radius r2 = refracted curvature radiusLens focal length from refraction indices distance in a plane mirrorSpherical mirrorr = curvature radius of mirror Spherical mirror equation

distance in a spherical mirror

Subscripts 1 and 2 refer to initial and final optical media respectively.

These ratios are sometimes also used, following simply from other definitions of refractive index, wave phase velocity, and the luminal speed equation:

where: ε = permittivity of medium, μ = permeability of medium, λ = wavelength of light in medium, v = speed of light in media.

Polarization

Physical situation Nomenclature EquationsAngle of total polarisationθB = Reflective polarization angle, Brewster's angleintensity from polarized light, Malus's law I0 = Initial intensity, I = Transmitted intensity, θ = Polarization angle between polarizer transmission axes and electric field vector

Diffraction and interference

Property or effect Nomenclature Equation Thin film in airn1 = refractive index of initial medium (before film interference) n2 = refractive index of final medium (after film interference) Min: Max:The grating equation a = width of aperture, slit width α = incident angle to the normal of the grating planeRayleigh's criterionBragg's law (solid state diffraction) d = lattice spacing δ = phase difference between two wavesFor constructive interference: For destructive interference:

where Young's double slit experiment △y = difference between 2 fringes D = Slit-screen distance a = Slit separation Single slit diffraction intensity I0 = source intensity Wave phase through apertures N-slit diffraction (N ≥ 2) d = centre-to-centre separation of slits N = number of slits Phase between N waves emerging from each slit N-slit diffraction (all N) Circular aperture intensity a = radius of the circular aperture J1 is a Bessel functionAmplitude for a general planar apertureCartesian and spherical polar coordinates are used, xy plane contains aperture

A, amplitude at position r r' = source point in the aperture Einc, magnitude of incident electric field at aperture Near-field (Fresnel)

Far-field (Fraunhofer)Huygens–Fresnel–Kirchhoff principle r0 = position from source to aperture, incident on it r = position from aperture diffracted from it to a point α0 = incident angle with respect to the normal, from source to aperture α = diffracted angle, from aperture to a point S = imaginary surface bounded by aperture = unit normal vector to the apertureKirchhoff's diffraction formula

Astrophysics definitions

In astrophysics, L is used for luminosity (energy per unit time, equivalent to power) and F is used for energy flux (energy per unit time per unit area, equivalent to intensity in terms of area, not solid angle). They are not new quantities, simply different names.

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Comoving transverse distance DM pc (parsecs) [L] Luminosity distance DL pc (parsecs) [L] Apparent magnitude in band j (UV, visible and IR parts of EM spectrum) (Bolometric) m dimensionless dimensionless Absolute magnitude (Bolometric) M dimensionless dimensionless Distance modulus μ dimensionless dimensionless Colour indices (No standard symbols) dimensionless dimensionless Bolometric correction Cbol (No standard symbol) dimensionless dimensionless

List of optics equations

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