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	$n = \frac{c}{v} \,\!$ $n = \sqrt{\frac{\epsilon \mu}{\epsilon_0 \mu_0}} = \sqrt{\epsilon_r \mu_r} \,\!$	dimensionless	dimensionless
Object Distance	s		m	[L]
Image Distance	s'		m	[L]
Focal Length	f		m	[L]
Optcal Power	P	$P = \frac{1}{f} = \frac{2}{r} \,\!$	D (Dipotres) = m^-1	[L]^-1
Radius of Curvature of r-surface	f		m	[L]
Lateral Magnification	m	$m = \frac{h'}{h} = -\frac{s'}{s} \,\!$ m and h negative when upside down	dimensionless	dimensionless
Angular Magnification	m	$m = \frac{\theta'}{\theta} = \frac{f}{f'} \,\!$	dimensionless	dimensionless
Dispersive Power	ω	$\omega = \frac{n_\mathrm{blue}-n_\mathrm{red}}{n_\mathrm{yellow} - 1}\,\!$ 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	$\theta_1=\theta_2\,\!$
Snell's Law of Refraction, Angles of Refraction	$n_1\sin\theta_1 = n_2\sin\theta_2\,\!$

Mirrors

Image distance in a Plane Mirror	$s = -s' \,\!$
Image distance in a Spherical Mirror	$\frac{n_1}{s} + \frac{n_2}{s'} = \frac{n_2 - n_1}{r} \,\!$
Spherical Mirror Focal Length	$f =r/2\,\!$
Spherical Mirror	$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f} = \frac{2}{r} \,\!$

General Media

Critical Angle of Total Internal Reflection	$\sin \theta_c = \frac{n_2}{n_1}\,\!$

Lenses

Thin Lens, Focal Length	$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f} \,\!$ $\frac{1}{f} = \frac{n_\mathrm{lens}}{n_\mathrm{med}-1} \left ( \frac{1}{r_1} - \frac{1}{r_2} \right )\,\!$
Newton's Formula	$xy = f^2 \,\!$ $x = s - f \,\!$ $y = s' - f \,\!$

Prisms

Minimum Deviation Angle A = Prism Angle D = Deviation Angle	$n_\mathrm{prism} = \frac{\sin\left ( \frac{A + D_\mathrm{min}}{2} \right ) }{\sin\frac{A}{2}} \,\!$ $A \le \theta_c \,\!$

Radiometry

Quantity (Common Name/s)	(Common) Symbol/s	SI Unit	Dimension
Radiant Power	Q	J =	[M] [L]² [T]^-2
Radiant Flux, Radiant Power	Φ	W
Radiant Intensity	I	W sr^-1	[M] [L]² [T]^-3
Radiance, Radiant Intensity	L	W sr^-1 m^-2
Irradiance, Incident Intensity, Intensity incident on a surface	E, I	W sr^-1 m^-2
Radiant Exitance, Radiant Emittance	M	W m^-2
Radiosity (heat transfer), Radiosity, emitted plus reflected Intensity leaving a surface	J, J_λ	W m^-2
Spectral Radiance	L_λ, L_ν	W sr^-1 m^-3 = W sr^-1 Hz^-2
Spectral Irradiance	E_λ, E_ν	W m^-3 = W m^-2 Hz^-1

Photometry

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Luminous energy	Q_v		J = lm s	[M] [L]² [T]^-2
Luminous flux, luminous power	F, Φ_v		cd sr = lm = J s^-1	[Φ]
Luminous intensity	I_v		cd = lm sr^-1	[Φ]
Luminance	L_v		cd m_-2	[Φ] [L]^-2
Illuminance (light incident on a surface)	E_v		lx = lm m^-2	[Φ] [L]^-2
Luminous Emittance (light emitted from a surface	M_v		lx = lm m²	[Φ] [L]^-2
Luminous efficacy		$\frac{\Phi_v}{\Phi_\lambda} \,\!$	lm W^-1	[Φ] [T]² [M]^-1 [L]^-2

Physical Optics

Luminal EM Waves

Electric Field Component		$\mathbf{E} = \mathbf{E}_0 \sin(kx-\omega t)\,\!$
Magnetic Field Component		$\mathbf{B} = \mathbf{B}_0 \sin(kx-\omega t)\,\!$
Luminal Speed in Meduim		$c = \frac{1}{\sqrt{\mu \epsilon}} = \frac{\left \| \mathbf{E} \right \|}{\left \| \mathbf{B} \right \|} \,\!$
Poynting Vector	$Y_0 \,\!$ = Admittance of Free Space $Z_0 \,\!$ = Impedance of Free Space $Y_0 = \frac{1}{Z_0} = \sqrt{\frac{\mu_0}{\epsilon_0}} \,\!$	$\mathbf{S} = \frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}\,\!$ $\mathbf{S} = \mathbf{E}\times\mathbf{H}\,\!$ $\mathbf{S} = \epsilon_0 \mathbf{D}\times\mathbf{H}\,\!$ $\mathbf{S} = \frac{\epsilon_0}{\mu_0} \mathbf{D}\times\mathbf{B} = Y_0 \mathbf{D}\times\mathbf{B} \,\!$
Poynting Vector Magnitude		$\left \| \mathbf{S} \right \| = \frac{\left \| \mathbf{E} \right \| \left \| \mathbf{B} \right \|}{\mu_0} = \frac{\left \| \mathbf{E} \right \|^2}{c\mu_0} \,\!$
Root Mean Square Electric Field of Light		$\mathbf{E}_\mathrm{rms} = \frac{\mathbf{E}}{\sqrt{2}}\,\!$
Irradiance, Light Intensity		$I = \frac{\langle \left \| \mathbf{E} \right \|^2 \rangle}{c\mu_0} \,\!$
Irradiance, Light Intensity due to a Point Source	$\Omega\,\!$ = solid angle $r\,\!$ = position from source	$I = \frac{P_0}{\Omega r^2}\,\!$
Radiation Momentum, Total Absorption (Inelastic)		$\Delta p = \frac{\Delta U}{c}\,\!$
Radiation Momentum, Total Reflection (Elastic)		$\Delta p = {2 \Delta U}{c}\,\!$
Radiation Pressure, Total Absorption (Inelastic)		$p_r = I/c\,\!$
Radiation Pressure, Total Reflection (Elastic)		$p_r = 2I/c\,\!$
Intensity Unpolarized Light		$I = I_0/2\,\!$
Malus' Law, Plane Polarized Light		$I = I_0\cos^2\theta\,\!$
Brewster's Law of Total Reflective Polarisation, Brewster's Angle		$\tan\theta_B = \frac{n_2}{n_1} \,\!$

Diffraction/Interferance

Diffraction

Path Length Difference	$\Delta x = d \sin\theta\,\!$
Diffraction Grating Equation	$d \sin\theta = n\lambda\,\!$ Minima $n=m \,\!$ Maxima $n=m+\frac{1}{2} \,\!$ $m \in \mathbf{Z} \,\!$
Diffraction Grating Half-Width	$\Delta\theta_{hw} = \lambda/Nd\cos\theta\,\!$
Diffraction Grating Dispersion	$D=N/d \cos\theta\,\!$
Diffraction Grating resolving power	$R=Nn\,\!$
X-Ray Molecular Lattice Diffraction, Bragg's law, Lattice Distance	$2d\sin\theta = N\lambda\,\!$
Double-Slit Interference Intensity	$I = 4 I_0 \cos ^2 \left ( \frac{\pi d}{\lambda}\sin\theta \right )\,\!$
Thin-Film Optics	Air Minima $2L = \left ( N + \frac{1}{2} \right )\frac{\lambda}{n_2}\,\!$ Air Maxima $2L = N \frac{\lambda}{n_2}\,\!$
Single-Slit Intensity	$I \left ( \theta \right ) = I_0 \left ( \frac{\sin\alpha}{\alpha} \right )^2\,\!$
Double Slit Intensity	$I \left ( \theta \right ) = I_0 \left ( \cos^2 \beta \right ) \left ( \frac{\sin\alpha }{\alpha} \right )^2\,\!$ $\alpha = \frac{\pi a }{\lambda} \sin\theta \,\!$
Multiple-Slit Intensity	$I \left ( \theta \right ) = I_0 \left [ \dfrac{\sin \left ( \dfrac{N \pi a }{\lambda} \sin\theta \right ) }{ \sin \left ( \dfrac{\pi a }{\lambda} \sin\theta \right ) } \right ]^2\,\!$
Circular Aperture First Minimum	$\sin\theta = 1.22\frac{\lambda}{d}\,\!$
Rayleigh's Criterion	$\theta_R = 1.22\frac{\lambda}{d} \,\!$

External Links

Poynting Vector

EM Waves

Photometry

Radiometry

Interferometry

Referances

↑ Essential Principles of Physics, M.J. Hodgson and P.M. Whelan, John Murray 2nd Edition, 1978, ISBN 0-7195-3382-1

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