List_of_optics_equations Knowpia

This article summarizes equations used in optics, including geometric optics, physical optics, radiometry, diffraction, and interferometry.

Definitions edit

Geometric optics (luminal rays) edit

General fundamental quantities edit

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, θ_o, θ₁	rad	dimensionless
Angle subtended by image	θ', θ_i, 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) edit

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 edit

Visulization of flux through differential area and solid angle. As always

\mathbf {\hat {n}} \,\!

is the unit normal to the incident 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, 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 edit

Luminal electromagnetic waves edit

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: $\mathbf {p} _{\mathrm {k} }=m\mathbf {v} \,\!$ Canonical momentum: $\mathbf {p} =m\mathbf {v} +q\mathbf {A} \,\!$
Irradiance, light intensity	$\langle \mathbf {S} \rangle \,\!$ = time averaged poynting vector I = irradiance I₀ = intensity of source P₀ = power of point source Ω = solid angle r = radial position from source	$I=\langle \mathbf {S} \rangle =E_{\mathrm {rms} }^{2}/c\mu _{0}\,\!$ 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	$\cos \theta ={\frac {c}{nv}}={\frac {1}{v{\sqrt {\epsilon \mu }}}}\,\!$
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 edit

Physical situation	Nomenclature	Equations
Critical angle (optics)	n₁ = refractive index of initial medium n₂ = refractive index of final medium θ_c = 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:

ε = permittivity of medium,
μ = permeability of medium,
λ = wavelength of light in medium,
v = speed of light in media.

Polarization edit

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	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 edit

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)	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	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	a = radius of the circular aperture J₁ 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	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 edit

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

Sources edit

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