# Physics Formulae/Equations of Light

Lead Article: Tables of Physics Formulae

This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of photonics.

## Geometric Optics

### Definitions, Quantities

Definitions

For conveinece in the table below, "r-surface" refers to reflecting/refracting surface. This is not a standard abbreviation.

Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Refractive Index of substance n ${\displaystyle n={\frac {c}{v}}\,\!}$

${\displaystyle n={\sqrt {\frac {\epsilon \mu }{\epsilon _{0}\mu _{0}}}}={\sqrt {\epsilon _{r}\mu _{r}}}\,\!}$

dimensionless dimensionless
Object Distance s m [L]
Image Distance s' m [L]
Focal Length f m [L]
Optcal Power P ${\displaystyle P={\frac {1}{f}}={\frac {2}{r}}\,\!}$ D (Dipotres) = m-1 [L]-1

of r-surface

f m [L]
Lateral Magnification m ${\displaystyle m={\frac {h'}{h}}=-{\frac {s'}{s}}\,\!}$

m and h negative when upside down

dimensionless dimensionless
Angular Magnification m ${\displaystyle m={\frac {\theta '}{\theta }}={\frac {f}{f'}}\,\!}$ dimensionless dimensionless
Dispersive Power ω ${\displaystyle \omega ={\frac {n_{\mathrm {blue} }-n_{\mathrm {red} }}{n_{\mathrm {yellow} }-1}}\,\!}$

The refractive indicies are determined

by the frequencies of the Fraunhöfer lines.

dimensionless dimensionless

Sign Conventions and Implications

There are different sign conventions which can be used, perhaps the the simplist to understand and recall is the one below[1].

The general pattern is the following:

Distances for real rays of light actually traversed are positve

Distances for apparent (i.e. virtual) rays of light not actually traversed are negative.

Distances are measured to the the apex of the r-surface on the optic axis.

Quantity + -
s Object in front of r-surface Object behind r-surface
s' Real image Virtual image
f, P Converging r-surface Diverging r-surface
r r-surface centre of curvature

on same side as object

r-surface centre of curvature

on opposite side as object

### Laws of Geomtric Optics

 Law of Reflection ${\displaystyle \theta _{1}=\theta _{2}\,\!}$ Snell's Law of Refraction, Angles of Refraction ${\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}\,\!}$

Mirrors

 Image distance in a Plane Mirror ${\displaystyle s=-s'\,\!}$ Image distance in a Spherical Mirror ${\displaystyle {\frac {n_{1}}{s}}+{\frac {n_{2}}{s'}}={\frac {n_{2}-n_{1}}{r}}\,\!}$ Spherical Mirror Focal Length ${\displaystyle f=r/2\,\!}$ Spherical Mirror ${\displaystyle {\frac {1}{s}}+{\frac {1}{s'}}={\frac {1}{f}}={\frac {2}{r}}\,\!}$

General Media

 Critical Angle of Total Internal Reflection ${\displaystyle \sin \theta _{c}={\frac {n_{2}}{n_{1}}}\,\!}$

Lenses

 Thin Lens, Focal Length ${\displaystyle {\frac {1}{s}}+{\frac {1}{s'}}={\frac {1}{f}}\,\!}$ ${\displaystyle {\frac {1}{f}}={\frac {n_{\mathrm {lens} }}{n_{\mathrm {med} }-1}}\left({\frac {1}{r_{1}}}-{\frac {1}{r_{2}}}\right)\,\!}$ Newton's Formula ${\displaystyle xy=f^{2}\,\!}$ ${\displaystyle x=s-f\,\!}$ ${\displaystyle y=s'-f\,\!}$

Prisms

 Minimum Deviation Angle A = Prism Angle D = Deviation Angle ${\displaystyle n_{\mathrm {prism} }={\frac {\sin \left({\frac {A+D_{\mathrm {min} }}{2}}\right)}{\sin {\frac {A}{2}}}}\,\!}$ ${\displaystyle A\leq \theta _{c}\,\!}$

Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Unit Dimension
Radiant Power Q J = [M] [L]2 [T]-2
Radiant Intensity I W sr-1 [M] [L]2 [T]-3
Radiance, Radiant Intensity L W sr-1 m-2

Intensity incident on a surface

E, I W sr-1 m-2
Radiant Exitance, Radiant Emittance M W m-2

reflected Intensity leaving a surface

J, Jλ W m-2
Spectral Radiance Lλ, Lν W sr-1 m-3 = W sr-1 Hz-2
Spectral Irradiance Eλ, Eν W m-3 = W m-2 Hz-1

## Photometry

Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Luminous energy Qv J = lm s [M] [L]2 [T]-2
Luminous flux, luminous power F, Φv cd sr = lm = J s-1 [Φ]
Luminous intensity Iv cd = lm sr-1 [Φ]
Luminance Lv cd m-2 [Φ] [L]-2
Illuminance (light incident on a surface) Ev lx = lm m-2 [Φ] [L]-2
Luminous Emittance (light emitted from a surface Mv lx = lm m2 [Φ] [L]-2
Luminous efficacy ${\displaystyle {\frac {\Phi _{v}}{\Phi _{\lambda }}}\,\!}$ lm W-1 [Φ] [T]2 [M]-1 [L]-2

## Physical Optics

### Luminal EM Waves

 Electric Field Component ${\displaystyle \mathbf {E} =\mathbf {E} _{0}\sin(kx-\omega t)\,\!}$ Magnetic Field Component ${\displaystyle \mathbf {B} =\mathbf {B} _{0}\sin(kx-\omega t)\,\!}$ Luminal Speed in Meduim ${\displaystyle c={\frac {1}{\sqrt {\mu \epsilon }}}={\frac {\left|\mathbf {E} \right|}{\left|\mathbf {B} \right|}}\,\!}$ Poynting Vector ${\displaystyle Y_{0}\,\!}$ = Admittance of Free Space ${\displaystyle Z_{0}\,\!}$ = Impedance of Free Space ${\displaystyle Y_{0}={\frac {1}{Z_{0}}}={\sqrt {\frac {\mu _{0}}{\epsilon _{0}}}}\,\!}$ ${\displaystyle \mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} \,\!}$ ${\displaystyle \mathbf {S} =\mathbf {E} \times \mathbf {H} \,\!}$ ${\displaystyle \mathbf {S} =\epsilon _{0}\mathbf {D} \times \mathbf {H} \,\!}$ ${\displaystyle \mathbf {S} ={\frac {\epsilon _{0}}{\mu _{0}}}\mathbf {D} \times \mathbf {B} =Y_{0}\mathbf {D} \times \mathbf {B} \,\!}$ Poynting Vector Magnitude ${\displaystyle \left|\mathbf {S} \right|={\frac {\left|\mathbf {E} \right|\left|\mathbf {B} \right|}{\mu _{0}}}={\frac {\left|\mathbf {E} \right|^{2}}{c\mu _{0}}}\,\!}$ Root Mean Square Electric Field of Light ${\displaystyle \mathbf {E} _{\mathrm {rms} }={\frac {\mathbf {E} }{\sqrt {2}}}\,\!}$ Irradiance, Light Intensity ${\displaystyle I={\frac {\langle \left|\mathbf {E} \right|^{2}\rangle }{c\mu _{0}}}\,\!}$ Irradiance, Light Intensity due to a Point Source ${\displaystyle \Omega \,\!}$ = solid angle ${\displaystyle r\,\!}$ = position from source ${\displaystyle I={\frac {P_{0}}{\Omega r^{2}}}\,\!}$ Radiation Momentum, Total Absorption (Inelastic) ${\displaystyle \Delta p={\frac {\Delta U}{c}}\,\!}$ Radiation Momentum, Total Reflection (Elastic) ${\displaystyle \Delta p={2\Delta U}{c}\,\!}$ Radiation Pressure, Total Absorption (Inelastic) ${\displaystyle p_{r}=I/c\,\!}$ Radiation Pressure, Total Reflection (Elastic) ${\displaystyle p_{r}=2I/c\,\!}$ Intensity Unpolarized Light ${\displaystyle I=I_{0}/2\,\!}$ Malus' Law, Plane Polarized Light ${\displaystyle I=I_{0}\cos ^{2}\theta \,\!}$ Brewster's Law of Total Reflective Polarisation, Brewster's Angle ${\displaystyle \tan \theta _{B}={\frac {n_{2}}{n_{1}}}\,\!}$

## Diffraction/Interferance

### Diffraction

 Path Length Difference ${\displaystyle \Delta x=d\sin \theta \,\!}$ Diffraction Grating Equation ${\displaystyle d\sin \theta =n\lambda \,\!}$ Minima ${\displaystyle n=m\,\!}$ Maxima ${\displaystyle n=m+{\frac {1}{2}}\,\!}$ ${\displaystyle m\in \mathbf {Z} \,\!}$ Diffraction Grating Half-Width ${\displaystyle \Delta \theta _{hw}=\lambda /Nd\cos \theta \,\!}$ Diffraction Grating Dispersion ${\displaystyle D=N/d\cos \theta \,\!}$ Diffraction Grating resolving power ${\displaystyle R=Nn\,\!}$ X-Ray Molecular Lattice Diffraction, Bragg's law, Lattice Distance ${\displaystyle 2d\sin \theta =N\lambda \,\!}$ Double-Slit Interference Intensity ${\displaystyle I=4I_{0}\cos ^{2}\left({\frac {\pi d}{\lambda }}\sin \theta \right)\,\!}$ Thin-Film Optics Air Minima ${\displaystyle 2L=\left(N+{\frac {1}{2}}\right){\frac {\lambda }{n_{2}}}\,\!}$ Air Maxima ${\displaystyle 2L=N{\frac {\lambda }{n_{2}}}\,\!}$ Single-Slit Intensity ${\displaystyle I\left(\theta \right)=I_{0}\left({\frac {\sin \alpha }{\alpha }}\right)^{2}\,\!}$ Double Slit Intensity ${\displaystyle I\left(\theta \right)=I_{0}\left(\cos ^{2}\beta \right)\left({\frac {\sin \alpha }{\alpha }}\right)^{2}\,\!}$ ${\displaystyle \alpha ={\frac {\pi a}{\lambda }}\sin \theta \,\!}$ Multiple-Slit Intensity ${\displaystyle I\left(\theta \right)=I_{0}\left[{\dfrac {\sin \left({\dfrac {N\pi a}{\lambda }}\sin \theta \right)}{\sin \left({\dfrac {\pi a}{\lambda }}\sin \theta \right)}}\right]^{2}\,\!}$ Circular Aperture First Minimum ${\displaystyle \sin \theta =1.22{\frac {\lambda }{d}}\,\!}$ Rayleigh's Criterion ${\displaystyle \theta _{R}=1.22{\frac {\lambda }{d}}\,\!}$