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List of optics equations
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This article summarizes equations used in optics, including geometric optics, physical optics, radiometry, diffraction, and interferometry.
Definitions
Geometric optics (luminal rays)
Main article: Geometrical optics
General fundamental quantities
| Quantity (common name/s) | (Common) symbol/s | SI units | Dimension | Object distance | Image distance | Object height | Image height | Angle subtended by object | Angle subtended by image | Curvature radius of lens/mirror | Focal length |
|---|---|---|---|---|---|---|---|---|---|---|---|
| *x, s, d, u,* *x*1, *s*1, *d*1, *u*1 | m | [L] | |||||||||
| *x', s', d', v,* *x*2, *s*2, *d*2, *v*2 | m | [L] | |||||||||
| *y, h,* *y*1, *h*1 | m | [L] | |||||||||
| *y', h', H,* *y*2, *h*2, *H*2 | m | [L] | |||||||||
| *θ, θo,* *θ*1 | rad | dimensionless | |||||||||
| *θ', θi,* *θ*2 | rad | dimensionless | |||||||||
| *r, R* | m | [L] | |||||||||
| *f* | m | [L] |
| Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension | Lens power | Lateral magnification | Angular magnification |
|---|---|---|---|---|---|---|---|
| *P* | P = 1/f \,\! | m−1 = D (dioptre) | [L]−1 | ||||
| *m* | m = - x_2/x_1 = y_2/y_1 \,\! | dimensionless | dimensionless | ||||
| *m* | m = \theta_2/\theta_1 \,\! | dimensionless | dimensionless |
Physical optics (EM luminal waves)
Main article: Physical optics
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 | Poynting flux, EM field power flow | RMS Electric field of Light | Radiation momentum | Radiation pressure |
|---|---|---|---|---|---|---|---|---|---|
| **S**, **N** | \mathbf{N} = \frac{1}{\mu_0}\mathbf{E}\times\mathbf{B} = \mathbf{E}\times\mathbf{H} \,\! | W m−2 | [M][T]−3 | ||||||
| *ΦS*, *ΦN* | \Phi_N = \int_S \mathbf{N} \cdot \mathrm{d}\mathbf{S} \,\! | W | [M][L]2[T]−3 | ||||||
| *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 | ||||||
| *p, pEM, pr* | p_{EM} = U/c\,\! | J s m−1 | [M][L][T]−1 | ||||||
| *Pr, pr, PEM* | P_{EM} = I/c = p_{EM}/At \,\! | W m−2 | [M][T]−3 |
Radiometry
Main article: 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 | Radiant exposure | Radiant energy density | Radiant flux, radiant power | Radiant intensity | Radiance, intensity | Irradiance | Radiant exitance, radiant emittance | Radiosity | Spectral radiant flux, spectral radiant power | Spectral radiant intensity | Spectral radiance | Spectral irradiance |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| *Q, E, Qe, Ee* | J | [M][L]2[T]−2 | |||||||||||||||
| *He* | H_e = \mathrm{d} Q/\left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right ) \,\! | J m−2 | [M][T]−3 | ||||||||||||||
| *ωe* | \omega_e = \mathrm{d} Q/\mathrm{d}V \,\! | J m−3 | [M][L]−3 | ||||||||||||||
| *Φ, Φe* | Q = \int \Phi \mathrm{d} t | W | [M][L]2[T]−3 | ||||||||||||||
| *I, Ie* | \Phi = I \mathrm{d} \Omega \,\! | W sr−1 | [M][L]2[T]−3 | ||||||||||||||
| *L, Le* | \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 | ||||||||||||||
| *E, I, Ee, Ie* | \Phi = \int E \left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right ) | W m−2 | [M][T]−3 | ||||||||||||||
| *M, Me* | \Phi = \int M \left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right ) | W m−2 | [M][T]−3 | ||||||||||||||
| *J, Jν, Je, Jeν* | J = E + M \,\! | W m−2 | [M][T]−3 | ||||||||||||||
| *Φλ, Φν, Φeλ, Φeν* | Q=\iint\Phi_\lambda{\mathrm{d} \lambda \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) | |||||||||||||||||
| *Iλ, Iν, Ieλ, Ieν* | \Phi = \iint I_\lambda \mathrm{d} \lambda \mathrm{d} \Omega | W sr−1 m−1 (*Iλ*) | |||||||||||||||
| W sr−1 Hz−1 (*Iν*) | [M][L]−3[T]−3 (*Iλ*) | ||||||||||||||||
| [M][L]2[T]−2 (*Iν*) | |||||||||||||||||
| *Lλ, Lν, Leλ, Leν* | \Phi = \iiint L_\lambda \mathrm{d} \lambda \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) | |||||||||||||||||
| *Eλ, Eν, Eeλ, Eeν* | \Phi = \iint E_\lambda \mathrm{d} \lambda \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 | Kinetic and potential momenta (non-standard terms in use) | Irradiance, light intensity | Doppler effect for light (relativistic) | Cherenkov radiation, cone angle | Electric and magnetic amplitudes | EM wave components |
|---|---|---|---|---|---|---|---|---|---|
| \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 ) \,\! | |||||||||
| Potential momentum: | |||||||||
| I = \langle \mathbf{S} \rangle = E^2_\mathrm{rms}/c\mu_0\,\! | |||||||||
| \lambda=\lambda_0\sqrt{\frac{c-v}{c+v}}\,\! | |||||||||
| \cos \theta = \frac{c}{n v} = \frac{1}{v\sqrt{\varepsilon\mu}} \,\! | |||||||||
| For a dielectric | |||||||||
| Electric |
Geometric optics
| Physical situation | Nomenclature | Equations | Critical angle (optics) | Thin lens equation | Image distance in a plane mirror | Spherical mirror |
|---|---|---|---|---|---|---|
| \sin\theta_c = \frac{n_2}{n_1}\,\! | ||||||
| \frac{1}{x_1} +\frac{1}{x_2} = \frac{1}{f} \,\! | ||||||
| x_2 = -x_1\,\! | ||||||
| *r* = curvature radius of mirror | Spherical mirror equation |
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 | intensity from polarized light, Malus's law |
|---|---|---|---|---|
| *θB* = Reflective polarization angle, Brewster's angle | \tan \theta_B = n_2/n_1\,\! | |||
| I = I_0\cos^2\theta\,\! |
Diffraction and interference
| Property or effect | Nomenclature | Equation | Thin film in air | The grating equation | Rayleigh's criterion | Bragg's law (solid state diffraction) | Young's double slit experiment | Single slit diffraction intensity | *N*-slit diffraction (*N* ≥ 2) | *N*-slit diffraction (all *N*) | Circular aperture intensity | Amplitude for a general planar aperture | Huygens–Fresnel–Kirchhoff principle | Kirchhoff's diffraction formula | |||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \frac{\delta}{2\pi}\lambda = a \left ( \sin\theta + \sin\alpha \right ) \,\! | |||||||||||||||||
| \theta_R = 1.22\lambda/\,\!d | |||||||||||||||||
| \frac{\delta}{2\pi} \lambda = 2d \sin\theta \,\! | |||||||||||||||||
| \Delta y = \lambda D / a | |||||||||||||||||
| I = I_0 \left [ \frac{ \sin \left( \phi/2 \right ) }{\left( \phi/2 \right )} \right ]^2 \,\! | |||||||||||||||||
| I = I_0 \left [ \frac{ \sin \left( N \delta/2 \right ) }{\sin \left( \delta/2 \right )} \right ]^2 \,\! | |||||||||||||||||
| 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 \,\! | |||||||||||||||||
| I = I_0 \left ( \frac{2 J_1(ka \sin \theta)}{ka \sin \theta} \right )^2 | |||||||||||||||||
| Cartesian and spherical polar coordinates are used, xy plane contains aperture | Near-field (Fresnel) | ||||||||||||||||
| \mathbf{r} \right | \left | \mathbf{r}_0 \right | } \left [ \cos \alpha_0 - \cos \alpha \right ] \mathrm{d}S \,\! | ||||||||||||||
| \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 | Luminosity distance | Apparent magnitude in band *j* (UV, visible and IR parts of EM spectrum) (Bolometric) | Absolute magnitude | Distance modulus | Colour indices | Bolometric correction | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| *DM* | pc (parsecs) | [L] | ||||||||||
| *DL* | D_L = \sqrt{\frac{L}{4\pi F}} \, | pc (parsecs) | [L] | |||||||||
| *m* | \frac {F_j}{F_j^0} \right | \, | dimensionless | dimensionless | ||||||||
| *M* | M = m - 5 \left [ \left ( \log_{10}{D_L} \right ) - 1 \right ]\!\, | dimensionless | dimensionless | |||||||||
| *μ* | \mu = m - M \!\, | dimensionless | dimensionless | |||||||||
| (No standard symbols) | U-B = M_U - M_B\!\, | |||||||||||
| dimensionless | dimensionless | |||||||||||
| *C*bol (No standard symbol) | \begin{align} C_\mathrm{bol} & = m_\mathrm{bol} - V \\ | dimensionless | dimensionless |
Notes
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