List of photonics equations

This article summarizes equations in the theory of photonics, including geometric optics, physical optics, radiometry, diffraction, and interferometry.

## Contents

* 1 Definitions
* 1.1 Geometric optics (luminal rays)
* 1.1.1 General fundamental quantities
* 1.2 Physical optics (EM luminal waves)
* 1.3 Radiometry
* 2 Equations
* 2.1 Luminal electromagnetic waves
* 2.2 Geometric optics
* 2.3 Polarization
* 2.4 Diffraction and interference
* 3 Astrophysics definitions
* 4 See also
* 5 Sources
* 6 Further reading

## Definitions

### Geometric optics (luminal rays)

Main article: Physics: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=−x2/x1=y2/y1 | dimensionless  | dimensionless
Angular magnification  | m | m=θ2/θ1 | dimensionless  | dimensionless

### Physical optics (EM luminal waves)

Main article: Physics: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 | 𝐍=1μ0𝐄×𝐁=𝐄×𝐇 | W m−2 | [M][T]−3
Poynting flux, EM field power flow  | ΦS, ΦN | ΦN=∫S𝐍⋅d𝐒 | W  | [M][L]2[T]−3
RMS Electric field of Light  | Erms | Erms=⟨E2⟩=E/2 | N C−1 = V m−1 | [M][L][T]−3[I]−1
Radiation momentum  | p, pEM, pr | pEM=U/c | J s m−1 | [M][L][T]−1
Radiation pressure | Pr, pr, PEM | PEM=I/c=pEM/At | W m−2 | [M][T]−3

### Radiometry

Main article: Astronomy:Radiometry

Visulization of flux through differential area and solid angle. As always n^ is the unit normal to the incident surface A, d𝐀=n^dA, and e^∠ is a unit vector in the direction of incident flux on the area element, θ is the angle between them. The factor n^⋅e^∠dA=e^∠⋅d𝐀=cos⁡θdA 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 | He=dQ/(e^∠⋅d𝐀) | J m−2 | [M][T]−3
Radiant energy density  | ωe | ωe=dQ/dV | J m−3 | [M][L]−3
Radiant flux, radiant power  | Φ, Φe | Q=∫Φdt | W  | [M][L]2[T]−3
Radiant intensity | I, Ie | Φ=IdΩ | W sr−1 | [M][L]2[T]−3
Radiance, intensity  | L, Le | Φ=∬L(e^∠⋅d𝐀)dΩ | W sr−1 m−2 | [M][T]−3
Irradiance | E, I, Ee, Ie | Φ=∫E(e^∠⋅d𝐀) | W m−2 | [M][T]−3
Radiant exitance, radiant emittance  | M, Me | Φ=∫M(e^∠⋅d𝐀) | W m−2 | [M][T]−3
Radiosity  | J, Jν, Je, Jeν | J=E+M | W m−2 | [M][T]−3
Spectral radiant flux, spectral radiant power  | Φλ, Φν, Φeλ, Φeν | Q=∬Φλdλdt

Q=∬Φνdνdt

| 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 | Φ=∬IλdλdΩ

Φ=∬IνdνdΩ

| 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 | Φ=∭Lλdλ(e^∠⋅d𝐀)dΩ

Φ=∭Lνdν(e^∠⋅d𝐀)dΩ

| 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 | Φ=∬Eλdλ(e^∠⋅d𝐀)

Φ=∬Eνdν(e^∠⋅d𝐀)

| 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  |

* ⟨u⟩ = mean energy density

| For a dielectric:

* ⟨u⟩=12(ϵ𝐄2+μ𝐁2)

Kinetic and potential momenta (non-standard terms in use)  |  | Potential momentum:

𝐩p=q𝐀

Kinetic momentum: 𝐩k=m𝐯

Cononical momentum: 𝐩=m𝐯+q𝐀

Irradiance, light intensity  |

* ⟨𝐒⟩ = time averaged poynting vector
* I = irradiance
* I0 = intensity of source
* P0 = power of point source
* Ω = solid angle
* r = radial position from source

| I=⟨𝐒⟩=Erms2/cμ0

At a spherical surface: I=P0Ω|r|2

Doppler effect for light (relativistic)  |  | λ=λ0c−vc+v

v=|Δλ|c/λ0

Cherenkov radiation, cone angle  |

* n = refractive index
* v = speed of particle
* θ = cone angle

| cos⁡θ=cnv=1vϵμ
Electric and magnetic amplitudes  |

* E = electric field
* H = magnetic field strength

| For a dielectric

|𝐄|=ϵμ|𝐇|

EM wave components  |  | Electric

𝐄=𝐄0sin⁡(kx−ωt)

Magnetic

𝐁=𝐁0sin⁡(kx−ωt)

### 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⁡θc=n2n1
Thin lens equation  |

* f = lens focal length
* x1 = object length
* x2 = image length
* r1 = incident curvature radius
* r2 = refracted curvature radius

| 1x1+1x2=1f

Lens focal length from refraction indices
1f=(nlensnmed−1)(1r1−1r2)

Image distance in a plane mirror  |  | x2=−x1
Spherical mirror  |

* r = curvature radius of mirror

| Spherical mirror equation

1x1+1x2=1f=2r

Image distance in a spherical mirror n1x1+n2x2=(n2−n1)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:

n1n2=v2v1=λ2λ1=ϵ1μ1ϵ2μ2

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  |

* θB = Reflective polarization angle, Brewster's angle

| tan⁡θB=n2/n1
intensity from polarized light, Malus's law  |

* I0 = Initial intensity,
* I = Transmitted intensity,
* θ = Polarization angle between polarizer transmission axes and electric field vector

| I=I0cos2θ

### 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: Nλ/n2
* Maxima:2L=(N+1/2)λ/n2

The grating equation  |

* a = width of aperture, slit width
* α = incident angle to the normal of the grating plane

| δ2πλ=a(sin⁡θ+sin⁡α)
Rayleigh's criterion  |  | θR=1.22λ/d
Bragg's law (solid state diffraction)  |

* d = lattice spacing
* δ = phase difference between two waves

| δ2πλ=2dsin⁡θ

* For constructive interference: δ/2π=n
* For destructive interference: δ/2π=n/2

where n∈𝐍

Single slit diffraction intensity  |

* I0 = source intensity
* Wave phase through apertures

ϕ=2πaλsin⁡θ

| I=I0[sin⁡(ϕ/2)(ϕ/2)]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

δ=2πdλsin⁡θ

| I=I0[sin⁡(Nδ/2)sin⁡(δ/2)]2

N-slit diffraction (all N)  |  | I=I0[sin⁡(ϕ/2)(ϕ/2)sin⁡(Nδ/2)sin⁡(δ/2)]2
Circular aperture intensity  |

* a = radius of the circular aperture
* J1 is a Bessel function

| I=I0(2J1(kasin⁡θ)kasin⁡θ)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(𝐫)∝∬apertureEinc(𝐫′)eik|𝐫−𝐫′|4π|𝐫−𝐫′|dx′dy′

Far-field (Fraunhofer) A(𝐫)∝eikr4πr∬apertureEinc(𝐫′)e−ik[sin⁡θ(cos⁡ϕx′+sin⁡ϕy′)]dx′dy′

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
* n^ = unit normal vector to the aperture
* 𝐫0⋅n^=|𝐫0|cos⁡α0
* 𝐫⋅n^=|𝐫|cos⁡α
* |𝐫||𝐫0|≪λ

| A(𝐫)=−i2λ∬apertureei𝐤⋅(𝐫+𝐫0)|𝐫||𝐫0|[cos⁡α0−cos⁡α]dS
Kirchhoff's diffraction formula |  | A(𝐫)=−14π∬apertureei𝐤⋅𝐫0|𝐫0|[i|𝐤|U0(𝐫0)cos⁡α+∂A0(𝐫0)∂n]dS

## 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 | DL=L4πF | pc (parsecs)  | [L]
Apparent magnitude in band j (UV, visible and IR parts of EM spectrum) (Bolometric)  | m | mj=−52log10|FjFj0| | dimensionless  | dimensionless
Absolute magnitude

(Bolometric)

| M | M=m−5[(log10DL)−1] | dimensionless  | dimensionless
Distance modulus | μ | μ=m−M | dimensionless  | dimensionless
Colour indices | (No standard symbols)  | U−B=MU−MB

B−V=MB−MV

| dimensionless  | dimensionless
Bolometric correction | Cbol (No standard symbol)  | Cbol=mbol−V=Mbol−MV | dimensionless  | dimensionless

## See also

* Defining equation
* Defining equation (physical chemistry)
* List of electromagnetism equations
* List of equations in classical mechanics
* List of equations in gravitation
* List of equations in nuclear and particle physics
* List of equations in quantum mechanics
* List of equations in wave theory
* List of relativistic equations

## Sources

* P.M. Whelan; M.J. Hodgeson (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 0-7195-3382-1.
* G. Woan (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. ISBN 978-0-521-57507-2. https://archive.org/details/cambridgehandboo0000woan.
* A. Halpern (1988). 3000 Solved Problems in Physics, Schaum Series. Mc Graw Hill. ISBN 978-0-07-025734-4.
* R.G. Lerner; G.L. Trigg (2005). Encyclopaedia of Physics (2nd ed.). VHC Publishers, Hans Warlimont, Springer. pp. 12–13. ISBN 978-0-07-025734-4.
* C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. ISBN 0-07-051400-3. https://archive.org/details/mcgrawhillencycl1993park.
* P.A. Tipler; G. Mosca (2008). Physics for Scientists and Engineers: With Modern Physics (6th ed.). W.H. Freeman and Co. ISBN 978-1-4292-0265-7.
* L.N. Hand; J.D. Finch (2008). Analytical Mechanics. Cambridge University Press. ISBN 978-0-521-57572-0.
* T.B. Arkill; C.J. Millar (1974). Mechanics, Vibrations and Waves. John Murray. ISBN 0-7195-2882-8.
* H.J. Pain (1983). The Physics of Vibrations and Waves (3rd ed.). John Wiley & Sons. ISBN 0-471-90182-2.
* J.R. Forshaw; A.G. Smith (2009). Dynamics and Relativity. Wiley. ISBN 978-0-470-01460-8.
* G.A.G. Bennet (1974). Electricity and Modern Physics (2nd ed.). Edward Arnold (UK). ISBN 0-7131-2459-8.
* I.S. Grant; W.R. Phillips; Manchester Physics (2008). Electromagnetism (2nd Edition). John Wiley & Sons. ISBN 978-0-471-92712-9.
* D.J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley. ISBN 81-7758-293-3.

## Further reading

* L.H. Greenberg (1978). Physics with Modern Applications. Holt-Saunders International W.B. Saunders and Co. ISBN 0-7216-4247-0. https://archive.org/details/physicswithmoder0000gree.
* J.B. Marion; W.F. Hornyak (1984). Principles of Physics. Holt-Saunders International Saunders College. ISBN 4-8337-0195-2.
* A. Beiser (1987). Concepts of Modern Physics (4th ed.). McGraw-Hill (International). ISBN 0-07-100144-1.
* H.D. Young; R.A. Freedman (2008). University Physics – With Modern Physics (12th ed.). Addison-Wesley (Pearson International). ISBN 0-321-50130-6.

* v
* t
* e

SI units

Base units|

* ampere
* candela
* kelvin
* kilogram
* {{{sp}}}}}|us|er|re}}
* mole
* second

Derived units
with special names|

* becquerel
* coulomb
* degree Celsius
* farad
* gray
* henry
* hertz
* joule
* katal
* lumen
* lux
* newton
* ohm
* pascal
* radian
* siemens
* sievert
* steradian
* tesla
* volt
* watt
* weber

Other accepted units|

* astronomical unit
* dalton
* day
* decibel
* degree of arc
* electronvolt
* hectare
* hour
* {{{sp}}}}}|us|er|re}}
* minute
* minute and second of arc
* neper
* tonne

See also|

* Conversion of units
* Metric prefixes
* 2005–2019 definition
* 2019 redefinition
* Systems of measurement

* Book
* Category

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