# List of photonics equations

This article summarizes equations in the theory of photonics, 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 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 $P = 1/f \,\!$ 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)

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 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 ΦS, ΦN $\Phi_N = \int_S \mathbf{N} \cdot \mathrm{d}\mathbf{S} \,\!$ W [M][L]2[T]−3
RMS Electric field of Light Erms $E_\mathrm{rms} = \sqrt{\langle E^2 \rangle} = E/\sqrt{2}\,\!$ N C−1 = V m−1 [M][L][T]−3[I]−1
Radiation momentum p, pEM, pr $p_{EM} = U/c\,\!$ J s m−1 [M][L][T]−1
Radiation pressure Pr, pr, PEM $P_{EM} = I/c = p_{EM}/At \,\!$ W m−2 [M][T]−3

Visulization of flux through differential area and solid angle. As always $\mathbf{\hat{n}} \,\!$ is the unit normal to the incidant surface A, $\mathrm{d} \mathbf{A} = \mathbf{\hat{n}}\mathrm{d}A \,\!$, and $\mathbf{\hat{e}}_{\angle} \,\!$ is a unit vector in the direction of incident flux on the area element, θ is the angle between them. The factor $\mathbf{\hat{n}} \cdot \mathbf{\hat{e}}_{\angle} \mathrm{d}A = \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} = \cos \theta \mathrm{d}A \,\!$ arises when the flux is not normal to the surface element, so the area normal to the flux is reduced.

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 $H_e = \mathrm{d} Q/\left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right ) \,\!$ J m−2 [M][T]−3
Radiant energy density ωe $\omega_e = \mathrm{d} Q/\mathrm{d}V \,\!$ J m−3 [M][L]−3
Radiant flux, radiant power Φ, Φe $Q = \int \Phi \mathrm{d} t$ W [M][L]2[T]−3
Radiant intensity I, Ie $\Phi = I \mathrm{d} \Omega \,\!$ W sr−1 [M][L]2[T]−3
Radiance, intensity 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
Irradiance E, I, Ee, Ie $\Phi = \int E \left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right )$ W m−2 [M][T]−3
Radiant exitance, radiant emittance M, Me $\Phi = \int M \left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right )$ W m−2 [M][T]−3
Radiosity J, Jν, Je, J $J = E + M \,\!$ W m−2 [M][T]−3
Spectral radiant flux, spectral radiant power Φλ, Φν, Φ, Φ $Q=\iint\Phi_\lambda{\mathrm{d} \lambda \mathrm{d} t}$

$Q = \iint \Phi_\nu \mathrm{d} \nu \mathrm{d} t$

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, I, I $\Phi = \iint I_\lambda \mathrm{d} \lambda \mathrm{d} \Omega$

$\Phi = \iint I_\nu \mathrm{d} \nu \mathrm{d} \Omega$

W sr−1 m−1 (Iλ)
W sr−1 Hz−1 (Iν)
[M][L]−3[T]−3 (Iλ)
[M][L]2[T]−2 (Iν)
Spectral radiance Lλ, Lν, L, L $\Phi = \iiint L_\lambda \mathrm{d} \lambda \left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right ) \mathrm{d} \Omega$

$\Phi = \iiint L_\nu \mathrm{d} \nu \left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right ) \mathrm{d} \Omega \,\!$

W sr−1 m−3 (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, E, E $\Phi = \iint E_\lambda \mathrm{d} \lambda \left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right )$

$\Phi = \iint E_\nu \mathrm{d} \nu \left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right )$

W m−3 (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 Equations
Energy density in an EM wave
• $\langle u \rangle \,\!$ = mean energy density
For a dielectric:
• $\langle u \rangle = \frac{1}{2} \left ( \epsilon \mathbf{E}^2 + \mu \mathbf{B}^2 \right ) \,\!$
Kinetic and potential momenta (non-standard terms in use) Potential momentum:

$\mathbf{p}_\mathrm{p} = q\mathbf{A} \,\!$

Kinetic momentum: $\mathbf{p}_\mathrm{k} = m\mathbf{v} \,\!$

Cononical momentum: $\mathbf{p} = m\mathbf{v} + q\mathbf{A} \,\!$

• $\langle \mathbf{S} \rangle \,\!$ = time averaged poynting vector
• I = irradiance
• I0 = intensity of source
• P0 = power of point source
• Ω = solid angle
• r = radial position from source
$I = \langle \mathbf{S} \rangle = E^2_\mathrm{rms}/c\mu_0\,\!$

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
$\cos \theta = \frac{c}{n v} = \frac{1}{v\sqrt{\epsilon\mu}} \,\!$
Electric and magnetic amplitudes
• E = electric field
• H = magnetic field strength
For a dielectric

$\left | \mathbf{E} \right | = \sqrt{\frac{\epsilon}{\mu}} \left | \mathbf{H} \right | \,\!$

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)
• n1 = refractive index of initial medium
• n2 = refractive index of final medium
• θc = critical angle
$\sin\theta_c = \frac{n_2}{n_1}\,\!$
Thin lens equation
• f = lens focal length
• x1 = object length
• x2 = image length
• r1 = incident curvature radius
• r2 = 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:

$\frac{n_1}{n_2} = \frac{v_2}{v_1} = \frac{\lambda_2}{\lambda_1} = \sqrt{\frac{\epsilon_1 \mu_1}{\epsilon_2 \mu_2}} \,\!$

where:

### Polarization

Physical situation Nomenclature Equations
Angle of total polarisation
$\tan \theta_B = n_2/n_1\,\!$
intensity from polarized light, Malus' law
• I0 = 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
• n1 = refractive index of initial medium (before film interference)
• n2 = refractive index of final medium (after film interference)
• Minima: $N \lambda/n_2\,\!$
• Maxima:$2L = (N + 1/2)\lambda/n_2\,\!$
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
• I0 = 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
$I = I_0 \left ( \frac{2 J_1(ka \sin \theta)}{ka \sin \theta} \right )^2$
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
• Einc, 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) $A \left ( \mathbf{r} \right ) \propto \frac{e^{ik r}}{4 \pi r} \iint_\mathrm{aperture} E_\mathrm{inc}\left ( \mathbf{r}' \right ) e^{-ik \left [ \sin \theta \left ( \cos \phi x' + \sin \phi y' \right ) \right ] } \mathrm{d}x'\mathrm{d}y'$

Huygen-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
• $\mathbf{\hat{n}}\,\!$ = 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 DM pc (parsecs) [L]
Luminosity distance DL $D_L = \sqrt{\frac{L}{4\pi F}} \,$ 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 Cbol (No standard symbol) \begin{align} C_\mathrm{bol} & = m_\mathrm{bol} - V \\ & = M_\mathrm{bol} - M_V \end{align} \!\, dimensionless dimensionless

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