# 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, x$_1$, s$_1$, d$_1$, u$_1$ | m | [L] |
| Image distance | x', s', d', v, x$_2$, s$_2$, d$_2$, v$_2$ | m | [L] |
| Object height | y, h, y$_1$, h$_1$ | m | [L] |
| Image height | y', h', H, y$_2$, h$_2$, H$_2$ | m | [L] |
| Angle subtended by object | θ, θ$_o$, θ$_1$ | rad | dimensionless |
| Angle subtended by image | θ', θ$_i$, θ$_2$ | 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 | $P=1/f\,\!$ | m$^{−1}$ = D (dioptre) | [L]$^{−1}$ |
| Lateral magnification | m | $m=-x_{2}/x_{1}=y_{2}/y_{1}\,\!$ | dimensionless | dimensionless |
| Angular magnification | m | $m=\theta _{2}/\theta _{1}\,\!$ | 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 | Dimension |
| Poynting vector | S, N | $\mathbf {N} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} =\mathbf {E} \times \mathbf {H} \,\!$ | W m$^{−2}$ | [M][T]$^{−3}$ |
| Poynting flux, EM field power flow | Φ$_S$, Φ$_N$ | $\Phi _{N}=\int _{S}\mathbf {N} \cdot \mathrm {d} \mathbf {S} \,\!$ | W | [M][L]$^2$[T]$^{−3}$ |
| RMS Electric field of Light | E$_{rms}$ | $E_{\mathrm {rms} }={\sqrt {\langle E^{2}\rangle }}=E/{\sqrt {2}}\,\!$ | N C$^{−1}$ = V m$^{−1}$ | [M][L][T]$^{−3}$[I]$^{−1}$ |
| Radiation momentum | p, p$_{EM}$, p$_r$ | $p_{EM}=U/c\,\!$ | J s m$^{−1}$ | [M][L][T]$^{−1}$ |
| Radiation pressure | P$_r$, p$_r$, P$_{EM}$ | $P_{EM}=I/c=p_{EM}/At\,\!$ | 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, Q$_e$, E$_e$ | | J | [M][L]$^2$[T]$^{−2}$ |
| Radiant exposure | H$_e$ | $H_{e}=\mathrm {d} Q/\left(\mathbf {\hat {e}} _{\angle }\cdot \mathrm {d} \mathbf {A} \right)\,\!$ | J m$^{−2}$ | [M][T]$^{−3}$ |
| Radiant energy density | ω$_e$ | $\omega _{e}=\mathrm {d} Q/\mathrm {d} V\,\!$ | J m$^{−3}$ | [M][L]$^{−3}$ |
| Radiant flux, radiant power | Φ, Φ$_e$ | $Q=\int \Phi \mathrm {d} t$ | W | [M][L]$^2$[T]$^{−3}$ |
| Radiant intensity | I, I$_e$ | $\Phi =I\mathrm {d} \Omega \,\!$ | W sr$^{−1}$ | [M][L]$^2$[T]$^{−3}$ |
| Radiance, intensity | L, L$_e$ | $\Phi =\iint L\left(\mathbf {\hat {e}} _{\angle }\cdot \mathrm {d} \mathbf {A} \right)\mathrm {d} \Omega$ | W sr$^{−1}$ m$^{−2}$ | [M][T]$^{−3}$ |
| Irradiance | E, I, E$_e$, I$_e$ | $\Phi =\int E\left(\mathbf {\hat {e}} _{\angle }\cdot \mathrm {d} \mathbf {A} \right)$ | W m$^{−2}$ | [M][T]$^{−3}$ |
| Radiant exitance, radiant emittance | M, M$_e$ | $\Phi =\int M\left(\mathbf {\hat {e}} _{\angle }\cdot \mathrm {d} \mathbf {A} \right)$ | W m$^{−2}$ | [M][T]$^{−3}$ |
| Radiosity | J, J$_ν$, Je, J$_{eν}$ | $J=E+M\,\!$ | W m$^{−2}$ | [M][T]$^{−3}$ |
| Spectral radiant flux, spectral radiant power | Φ$_λ$, Φ$_ν$, Φ$_{eλ}$, Φ$_{eν}$ | $Q=\iint \Phi _{\lambda }{\mathrm {d} \lambda \mathrm {d} t}$ $Q=\iint \Phi _{\nu }\mathrm {d} \nu \mathrm {d} t$ | W m$^{−1}$ (Φ$_λ$) W Hz$^{−1}$ = J (Φ$_ν$) | [M][L]$^{−3}$[T]$^{−3}$ (Φ$_λ$) [M][L]$^{−2}$[T]$^{−2}$ (Φ$_ν$) |
| Spectral radiant intensity | I$_λ$, I$_ν$, I$_{eλ}$, I$_{eν}$ | $\Phi =\iint I_{\lambda }\mathrm {d} \lambda \mathrm {d} \Omega$ $\Phi =\iint I_{\nu }\mathrm {d} \nu \mathrm {d} \Omega$ | W sr$^{−1}$ m$^{−1}$ (I$_λ$) W sr$^{−1}$ Hz$^{−1}$ (I$_ν$) | [M][L]$^{−3}$[T]$^{−3}$ (I$_λ$) [M][L]$^2$[T]$^{−2}$ (I$_ν$) |
| Spectral radiance | L$_λ$, L$_ν$, L$_{eλ}$, L$_{eν}$ | $\Phi =\iiint L_{\lambda }\mathrm {d} \lambda \left(\mathbf {\hat {e}} _{\angle }\cdot \mathrm {d} \mathbf {A} \right)\mathrm {d} \Omega$ $\Phi =\iiint L_{\nu }\mathrm {d} \nu \left(\mathbf {\hat {e}} _{\angle }\cdot \mathrm {d} \mathbf {A} \right)\mathrm {d} \Omega \,\!$ | W sr$^{−1}$ m$^{−3}$ (L$_λ$) W sr$^{−1}$ m$^{−2}$ Hz$^{−1}$ (L$_ν$) | [M][L]$^{−1}$[T]$^{−3}$ (L$_λ$) [M][L]$^{−2}$[T]$^{−2}$ (L$_ν$) |
| Spectral irradiance | E$_λ$, E$_ν$, E$_{eλ}$, E$_{eν}$ | $\Phi =\iint E_{\lambda }\mathrm {d} \lambda \left(\mathbf {\hat {e}} _{\angle }\cdot \mathrm {d} \mathbf {A} \right)$ $\Phi =\iint E_{\nu }\mathrm {d} \nu \left(\mathbf {\hat {e}} _{\angle }\cdot \mathrm {d} \mathbf {A} \right)$ | W m$^{−3}$ (E$_λ$) W m$^{−2}$ Hz$^{−1}$ (E$_ν$) | [M][L]$^{−1}$[T]$^{−3}$ (E$_λ$) [M][L]$^{−2}$[T]$^{−2}$ (E$_ν$) |

## Equations
### Luminal electromagnetic waves

| Physical situation | Nomenclature | Equations |
| Energy density in an EM wave | $\langle u\rangle \,\!$ = mean energy density | For a dielectric: $\langle u\rangle ={\frac {1}{2}}\left(\varepsilon \mathbf {E} ^{2}+{\mathbf {B} ^{2} \over \mu }\right)\,\!$ |
| Kinetic and potential momenta (non-standard terms in use) | | Potential momentum: $\mathbf {p} _{\mathrm {p} }=q\mathbf {A} \,\!$ Kinetic momentum: $\mathbf {p} _{\mathrm {k} }=m\mathbf {v} \,\!$ Canonical momentum: $\mathbf {p} =m\mathbf {v} +q\mathbf {A} \,\!$ |
| Irradiance, light intensity | $\langle \mathbf {S} \rangle \,\!$ = time averaged poynting vector I = irradiance I$_0$ = intensity of source P$_0$ = power of point source Ω = solid angle r = radial position from source | $I=\langle \mathbf {S} \rangle =E_{\mathrm {rms} }^{2}/c\mu _{0}\,\!$ At a spherical surface: $I={\frac {P_{0}}{\Omega \left|r\right|^{2}}}\,\!$ |
| Doppler effect for light (relativistic) | | $\lambda =\lambda _{0}{\sqrt {\frac {c-v}{c+v}}}\,\!$ $v=|\Delta \lambda |c/\lambda _{0}\,\!$ |
| Cherenkov radiation, cone angle | n = refractive index v = speed of particle θ = cone angle | $\cos \theta ={\frac {c}{nv}}={\frac {1}{v{\sqrt {\varepsilon \mu }}}}\,\!$ |
| Electric and magnetic amplitudes | E = electric field H = magnetic field strength | For a dielectric $\left|\mathbf {E} \right|={\sqrt {\varepsilon \over \mu }}\left|\mathbf {H} \right|\,\!$ |
| EM wave components | | Electric $\mathbf {E} =\mathbf {E} _{0}\sin(kx-\omega t)\,\!$ Magnetic $\mathbf {B} =\mathbf {B} _{0}\sin(kx-\omega t)\,\!$ |

### Geometric optics

| Physical situation | Nomenclature | Equations |
| Critical angle (optics) | n$_1$ = refractive index of initial medium n$_2$ = refractive index of final medium θ$_c$ = critical angle | $\sin \theta _{c}={\frac {n_{2}}{n_{1}}}\,\!$ |
| Thin lens equation | f = lens focal length x$_1$ = object distance x$_2$ = image distance r$_1$ = incident curvature radius r$_2$ = refracted curvature radius | ${\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}={\frac {1}{f}}\,\!$ Lens focal length from refraction indices ${\frac {1}{f}}=\left({\frac {n_{\mathrm {lens} }}{{n}_{\mathrm {med} }}}-1\right)\left({\frac {1}{r_{1}}}-{\frac {1}{r_{2}}}\right)\,\!$ |
| Image distance in a plane mirror | | $x_{2}=-x_{1}\,\!$ |
| Spherical mirror | r = curvature radius of mirror | Spherical mirror equation ${\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}={\frac {1}{f}}={\frac {2}{r}}\,\!$ Image distance in a spherical mirror ${\frac {n_{1}}{x_{1}}}+{\frac {n_{2}}{x_{2}}}={\frac {\left(n_{2}-n_{1}\right)}{r}}\,\!$ |
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:
${\frac {n_{1}}{n_{2}}}={\frac {v_{2}}{v_{1}}}={\frac {\lambda _{2}}{\lambda _{1}}}={\sqrt {\frac {\varepsilon _{1}\mu _{1}}{\varepsilon _{2}\mu _{2}}}}\,\!$
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 | $\tan \theta _{B}=n_{2}/n_{1}\,\!$ |
| intensity from polarized light, Malus's law | I$_0$ = Initial intensity, I = Transmitted intensity, θ = Polarization angle between polarizer transmission axes and electric field vector | $I=I_{0}\cos ^{2}\theta \,\!$ |

### Diffraction and interference

| Property or effect | Nomenclature | Equation |
| Thin film in air | n$_1$ = refractive index of initial medium (before film interference) n$_2$ = refractive index of final medium (after film interference) | Min: $N\lambda /n_{2}\,\!$ Max:$2L=(N+1/2)\lambda /n_{2}\,\!$ |
| The grating equation | a = width of aperture, slit width α = incident angle to the normal of the grating plane | ${\frac {\delta }{2\pi }}\lambda =a\left(\sin \theta +\sin \alpha \right)\,\!$ |
| Rayleigh's criterion | | $\theta _{R}=1.22\lambda /\,\!d$ |
| Bragg's law (solid state diffraction) | d = lattice spacing δ = phase difference between two waves | ${\frac {\delta }{2\pi }}\lambda =2d\sin \theta \,\!$ For constructive interference: $\delta /2\pi =n\,\!$ For destructive interference: $\delta /2\pi =n/2\,\!$ where $n\in \mathbf {N} \,\!$ |
| Single slit diffraction intensity | I$_0$ = source intensity Wave phase through apertures $\phi ={\frac {2\pi a}{\lambda }}\sin \theta \,\!$ | $I=I_{0}\left[{\frac {\sin \left(\phi /2\right)}{\left(\phi /2\right)}}\right]^{2}\,\!$ |
| N-slit diffraction (N ≥ 2) | d = centre-to-centre separation of slits N = number of slits Phase between N waves emerging from each slit $\delta ={\frac {2\pi d}{\lambda }}\sin \theta \,\!$ | $I=I_{0}\left[{\frac {\sin \left(N\delta /2\right)}{\sin \left(\delta /2\right)}}\right]^{2}\,\!$ |
| N-slit diffraction (all N) | | $I=I_{0}\left[{\frac {\sin \left(\phi /2\right)}{\left(\phi /2\right)}}{\frac {\sin \left(N\delta /2\right)}{\sin \left(\delta /2\right)}}\right]^{2}\,\!$ |
| Circular aperture intensity | a = radius of the circular aperture J$_1$ is a Bessel function | $I=I_{0}\left({\frac {2J_{1}(ka\sin \theta )}{ka\sin \theta }}\right)^{2}$ |
| 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 (Fresnel) $A\left(\mathbf {r} \right)\propto \iint _{\mathrm {aperture} }E_{\mathrm {inc} }\left(\mathbf {r} '\right)~{\frac {e^{ik\left|\mathbf {r} -\mathbf {r} '\right|}}{4\pi \left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} x'\mathrm {d} y'$ Far-field (Fraunhofer) $A\left(\mathbf {r} \right)\propto {\frac {e^{ikr}}{4\pi r}}\iint _{\mathrm {aperture} }E_{\mathrm {inc} }\left(\mathbf {r} '\right)e^{-ik\left[\sin \theta \left(\cos \phi x'+\sin \phi y'\right)\right]}\mathrm {d} x'\mathrm {d} y'$ |
| Huygens–Fresnel–Kirchhoff principle | r$_0$ = position from source to aperture, incident on it r = position from aperture diffracted from it to a point α$_0$ = 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 $\mathbf {\hat {n}} \,\!$ = unit normal vector to the aperture $\mathbf {r} _{0}\cdot \mathbf {\hat {n}} =\left|\mathbf {r} _{0}\right|\cos \alpha _{0}\,\!$ $\mathbf {r} \cdot \mathbf {\hat {n}} =\left|\mathbf {r} \right|\cos \alpha \,\!$ $\left|\mathbf {r} \right|\left|\mathbf {r} _{0}\right|\ll \lambda \,\!$ | $A\mathbf { \mathbf {r} )={\frac {-i}{2\lambda }}\iint _{\mathrm {aperture} }{\frac {e^{i\mathbf {k} \cdot \left(\mathbf {r} +\mathbf {r} _{0}\right)}}{\left|\mathbf {r} \right|\left|\mathbf {r} _{0}\right|}}\left[\cos \alpha _{0}-\cos \alpha \right]\mathrm {d} S\,\!$ |
| Kirchhoff's diffraction formula | | $A\left(\mathbf {r} \right)=-{\frac {1}{4\pi }}\iint _{\mathrm {aperture} }{\frac {e^{i\mathbf {k} \cdot \mathbf {r} _{0}}}{\left|\mathbf {r} _{0}\right|}}\left[i\left|\mathbf {k} \right|U_{0}\left(\mathbf {r} _{0}\right)\cos {\alpha }+{\frac {\partial A_{0}\left(\mathbf {r} _{0}\right)}{\partial n}}\right]\mathrm {d} S$ |

## 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 | D$_M$ | | pc (parsecs) | [L] |
| Luminosity distance | D$_L$ | $D_{L}={\sqrt {\frac {L}{4\pi F}}}\,$ | pc (parsecs) | [L] |
| Apparent magnitude in band j (UV, visible and IR parts of EM spectrum) (Bolometric) | m | $m_{j}=-{\frac {5}{2}}\log _{10}\left|{\frac {F_{j}}{F_{j}^{0}}}\right|\,$ | dimensionless | dimensionless |
| Absolute magnitude (Bolometric) | M | $M=m-5\left[\left(\log _{10}{D_{L}}\right)-1\right]\!\,$ | dimensionless | dimensionless |
| Distance modulus | μ | $\mu =m-M\!\,$ | dimensionless | dimensionless |
| Colour indices | (No standard symbols) | $U-B=M_{U}-M_{B}\!\,$ $B-V=M_{B}-M_{V}\!\,$ | dimensionless | dimensionless |
| Bolometric correction | C$_{bol}$ (No standard symbol) | ${\begin{aligned}C_{\mathrm {bol} }&=m_{\mathrm {bol} }-V\\&=M_{\mathrm {bol} }-M_{V}\end{aligned}}\!\,$ | dimensionless | dimensionless |