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	symbol/s	SI units	Dimension
Object distance	x, s, d, u, ''x₁, s''₁, d₁, u₁	m
Image distance	x', s', d', v, ''x₂, s''₂, d₂, v₂	m
Object height	y, h, ''y₁, h''₁	m
Image height	y', h', H, ''y₂, h''₂, H₂	m
Angle subtended by object	θ, θ_o, ''θ₁	rad	dimensionless
Angle subtended by image	θ', θ_i, θ''₂	rad	dimensionless
Curvature radius of lens/mirror	r, R	m
Focal length	f	m

Quantity	symbol/s	Defining equation	SI units	Dimension
Lens power	P		m⁻¹ = D	⁻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	symbol/s	Defining equation	SI units	Dimension
Poynting vector	S, N		W m⁻²	⁻³
Poynting flux, EM field power flow	Φ_S, Φ_N		W	²⁻³
RMS Electric field of Light	E_rms		N C⁻¹ = V m⁻¹	⁻³⁻¹
Radiation momentum	p, p_EM, p_r		J s m⁻¹	⁻¹
Radiation pressure	P_r, p_r, P_EM		W m⁻²	⁻³

Radiometry

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

Quantity	symbol/s	Defining equation	SI units	Dimension
Radiant energy	Q, E, Q_e, E_e		J	²⁻²
Radiant exposure	H_e		J m⁻²	⁻³
Radiant energy density	ω_e		J m⁻³	⁻³
Radiant flux, radiant power	Φ, Φ_e		W	²⁻³
Radiant intensity	I, I_e		W sr⁻¹	²⁻³
Radiance, intensity	L, L_e		W sr⁻¹ m⁻²	⁻³
Irradiance	E, I, E_e, I_e		W m⁻²	⁻³
Radiant exitance, radiant emittance	M, M_e		W m⁻²	⁻³
Radiosity	J, J_ν, Je, J_eν		W m⁻²	⁻³
Spectral radiant flux, spectral radiant power	Φ_λ, Φ_ν, Φ_eλ, Φ_eν		W m⁻¹ W Hz⁻¹ = J	⁻³⁻³ ⁻²⁻²
Spectral radiant intensity	I_λ, I_ν, I_eλ, I_eν		W sr⁻¹ m⁻¹ W sr⁻¹ Hz⁻¹	⁻³⁻³ ²⁻²
Spectral radiance	L_λ, L_ν, L_eλ, L_eν		W sr⁻¹ m⁻³ W sr⁻¹ m⁻² Hz⁻¹	⁻¹⁻³ ⁻²⁻²
Spectral irradiance	E_λ, E_ν, E_eλ, E_eν		W m⁻³ W m⁻² Hz⁻¹	⁻¹⁻³ ⁻²⁻²

Equations

Luminal electromagnetic waves

Physical situation	Nomenclature	Equations
Energy density in an EM wave	= mean energy density	For a dielectric:
Kinetic and potential momenta		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 source	At a spherical surface:
Doppler effect for light
Cherenkov radiation, cone angle	n = refractive index v = speed of particle θ = cone angle
Electric and magnetic amplitudes	E = electric field H = magnetic field strength	For a dielectric
EM wave components		Electric Magnetic

Geometric optics

Physical situation	Nomenclature	Equations
Critical angle (optics)	n₁ = refractive index of initial medium n₂ = refractive index of final medium θ_c = critical angle
Thin lens equation	f = lens focal length x₁ = object distance x₂ = image distance r₁ = incident curvature radius r₂ = refracted curvature radius	Lens focal length from refraction indices
Image distance in a plane mirror
Spherical mirror	r = curvature radius of mirror	Spherical mirror equation Image 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	Equations
Angle of total polarisation	θ_B = Reflective polarization angle, Brewster's angle
intensity 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 air	n₁ = refractive index of initial medium n₂ = refractive index of final medium	Min: Max:
The grating equation	a = width of aperture, slit width α = incident angle to the normal of the grating plane
Rayleigh's criterion
Bragg's law	d = lattice spacing δ = phase difference between two waves	For 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	d = centre-to-centre separation of slits N = number of slits Phase between N waves emerging from each slit
N-slit diffraction
Circular aperture intensity	a = radius of the circular aperture J₁ is a Bessel function
Amplitude for a general planar aperture	Cartesian and spherical polar coordinates are used, xy plane contains aperture A, amplitude at position r r' = source point in the aperture E_inc, magnitude of incident electric field at aperture	Near-field Far-field
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 aperture
Kirchhoff's diffraction formula

Astrophysics definitions

In astrophysics, L is used for luminosity and F is used for energy flux. They are not new quantities, simply different names.

Quantity	symbol/s	Defining equation	SI units	Dimension
Comoving transverse distance	D_M		pc
Luminosity distance	D_L		pc
Apparent magnitude in band j	m		dimensionless	dimensionless
Absolute magnitude	M		dimensionless	dimensionless
Distance modulus	μ		dimensionless	dimensionless
Colour indices			dimensionless	dimensionless
Bolometric correction	C_bol		dimensionless	dimensionless