This article summarizes equations used in optics , including geometric optics , physical optics , radiometry , diffraction , and interferometry .
Definitions
General fundamental quantities Quantity (common name/s) (Common) symbol/s SI units Dimension Object distance x, s, d, u, x 1 , s 1 , d 1 , u 1 m [L] Image distance x', s', d', v, x 2 , s 2 , d 2 , v 2 m [L] Object height y, h, y 1 , h 1 m [L] Image height y', h', H, y 2 , h 2 , H 2 m [L] Angle subtended by object θ, θo , θ 1 rad dimensionless Angle subtended by image θ', θi , θ 2 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 {\displaystyle P=1/f\,\!} m−1 = D (dioptre) [L]−1 Lateral magnification m m = − x 2 / x 1 = y 2 / y 1 {\displaystyle m=-x_{2}/x_{1}=y_{2}/y_{1}\,\!} dimensionless dimensionless Angular magnification m m = θ 2 / θ 1 {\displaystyle m=\theta _{2}/\theta _{1}\,\!} dimensionless dimensionless
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 = 1 μ 0 E × B = E × H {\displaystyle \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 Φ N = ∫ S N ⋅ d S {\displaystyle \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 r m s = ⟨ E 2 ⟩ = E / 2 {\displaystyle 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 E M = U / c {\displaystyle p_{EM}=U/c\,\!} J s m−1 [M][L][T]−1 Radiation pressure Pr , pr , PEM P E M = I / c = p E M / A t {\displaystyle P_{EM}=I/c=p_{EM}/At\,\!} W m−2 [M][T]−3
Radiometry Visulization of flux through differential area and solid angle. As always n ^ {\displaystyle \mathbf {\hat {n}} \,\!} d A = n ^ d A {\displaystyle \mathrm {d} \mathbf {A} =\mathbf {\hat {n}} \mathrm {d} A\,\!} e ^ ∠ {\displaystyle \mathbf {\hat {e}} _{\angle }\,\!} θ is the angle between them. The factor n ^ ⋅ e ^ ∠ d A = e ^ ∠ ⋅ d A = cos θ d A {\displaystyle \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\,\!} 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 = d Q / ( e ^ ∠ ⋅ d A ) {\displaystyle 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 ω e = d Q / d V {\displaystyle \omega _{e}=\mathrm {d} Q/\mathrm {d} V\,\!} J m−3 [M][L]−3 Radiant flux , radiant power Φ, Φe Q = ∫ Φ d t {\displaystyle Q=\int \Phi \mathrm {d} t} W [M][L]2 [T]−3 Radiant intensity I, Ie Φ = I d Ω {\displaystyle \Phi =I\mathrm {d} \Omega \,\!} W sr−1 [M][L]2 [T]−3 Radiance , intensity L, Le Φ = ∬ L ( e ^ ∠ ⋅ d A ) d Ω {\displaystyle \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 Φ = ∫ E ( e ^ ∠ ⋅ d A ) {\displaystyle \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 Φ = ∫ M ( e ^ ∠ ⋅ d A ) {\displaystyle \Phi =\int M\left(\mathbf {\hat {e}} _{\angle }\cdot \mathrm {d} \mathbf {A} \right)} W m−2 [M][T]−3 Radiosity J, Jν , Je, Jeν J = E + M {\displaystyle J=E+M\,\!} W m−2 [M][T]−3 Spectral radiant flux, spectral radiant power Φλ , Φν , Φeλ , Φeν Q = ∬ Φ λ d λ d t {\displaystyle Q=\iint \Phi _{\lambda }{\mathrm {d} \lambda \mathrm {d} t}} Q = ∬ Φ ν d ν d t {\displaystyle Q=\iint \Phi _{\nu }\mathrm {d} \nu \mathrm {d} t}
W m−1 (Φ λ ) −1 = J (Φ ν ) [M][L]−3 [T]−3 (Φ λ ) −2 [T]−2 (Φ ν ) Spectral radiant intensity Iλ , Iν , Ieλ , Ieν Φ = ∬ I λ d λ d Ω {\displaystyle \Phi =\iint I_{\lambda }\mathrm {d} \lambda \mathrm {d} \Omega } Φ = ∬ I ν d ν d Ω {\displaystyle \Phi =\iint I_{\nu }\mathrm {d} \nu \mathrm {d} \Omega }
W sr−1 m−1 (Iλ ) −1 Hz−1 (Iν ) [M][L]−3 [T]−3 (Iλ )2 [T]−2 (Iν ) Spectral radiance Lλ , Lν , Leλ , Leν Φ = ∭ L λ d λ ( e ^ ∠ ⋅ d A ) d Ω {\displaystyle \Phi =\iiint L_{\lambda }\mathrm {d} \lambda \left(\mathbf {\hat {e}} _{\angle }\cdot \mathrm {d} \mathbf {A} \right)\mathrm {d} \Omega } Φ = ∭ L ν d ν ( e ^ ∠ ⋅ d A ) d Ω {\displaystyle \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 )−1 m−2 Hz−1 (L ν ) [M][L]−1 [T]−3 (L λ )−2 [T]−2 (L ν ) Spectral irradiance Eλ , Eν , Eeλ , Eeν Φ = ∬ E λ d λ ( e ^ ∠ ⋅ d A ) {\displaystyle \Phi =\iint E_{\lambda }\mathrm {d} \lambda \left(\mathbf {\hat {e}} _{\angle }\cdot \mathrm {d} \mathbf {A} \right)} Φ = ∬ E ν d ν ( e ^ ∠ ⋅ d A ) {\displaystyle \Phi =\iint E_{\nu }\mathrm {d} \nu \left(\mathbf {\hat {e}} _{\angle }\cdot \mathrm {d} \mathbf {A} \right)}
W m−3 (E λ ) −2 Hz−1 (E ν ) [M][L]−1 [T]−3 (E λ ) −2 [T]−2 (E ν )
Equations
Luminal electromagnetic waves Physical situation Nomenclature Equations Energy density in an EM wave ⟨ u ⟩ {\displaystyle \langle u\rangle \,\!} For a dielectric: ⟨ u ⟩ = 1 2 ( ε E 2 + B 2 μ ) {\displaystyle \langle u\rangle ={\frac {1}{2}}\left(\varepsilon \mathbf {E} ^{2}+{\mathbf {B} ^{2} \over \mu }\right)\,\!} Kinetic and potential momenta (non-standard terms in use) Potential momentum: p p = q A {\displaystyle \mathbf {p} _{\mathrm {p} }=q\mathbf {A} \,\!}
Kinetic momentum: p k = m v {\displaystyle \mathbf {p} _{\mathrm {k} }=m\mathbf {v} \,\!}
Canonical momentum: p = m v + q A {\displaystyle \mathbf {p} =m\mathbf {v} +q\mathbf {A} \,\!}
Irradiance , light intensity ⟨ S ⟩ {\displaystyle \langle \mathbf {S} \rangle \,\!} time averaged poynting vector I = irradiance I 0 = intensity of sourceP 0 = power of point sourceΩ = solid angle r = radial position from source I = ⟨ S ⟩ = E r m s 2 / c μ 0 {\displaystyle I=\langle \mathbf {S} \rangle =E_{\mathrm {rms} }^{2}/c\mu _{0}\,\!} At a spherical surface: I = P 0 Ω | r | 2 {\displaystyle I={\frac {P_{0}}{\Omega \left|r\right|^{2}}}\,\!}
Doppler effect for light (relativistic) λ = λ 0 c − v c + v {\displaystyle \lambda =\lambda _{0}{\sqrt {\frac {c-v}{c+v}}}\,\!} v = | Δ λ | c / λ 0 {\displaystyle v=|\Delta \lambda |c/\lambda _{0}\,\!}
Cherenkov radiation , cone angle cos θ = c n v = 1 v ε μ {\displaystyle \cos \theta ={\frac {c}{nv}}={\frac {1}{v{\sqrt {\varepsilon \mu }}}}\,\!} Electric and magnetic amplitudes E = electric fieldH = magnetic field strength For a dielectric | E | = ε μ | H | {\displaystyle \left|\mathbf {E} \right|={\sqrt {\varepsilon \over \mu }}\left|\mathbf {H} \right|\,\!}
EM wave components Electric E = E 0 sin ( k x − ω t ) {\displaystyle \mathbf {E} =\mathbf {E} _{0}\sin(kx-\omega t)\,\!}
Magnetic
B = B 0 sin ( k x − ω t ) {\displaystyle \mathbf {B} =\mathbf {B} _{0}\sin(kx-\omega t)\,\!}
Geometric optics 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:
n 1 n 2 = v 2 v 1 = λ 2 λ 1 = ε 1 μ 1 ε 2 μ 2 {\displaystyle {\frac {n_{1}}{n_{2}}}={\frac {v_{2}}{v_{1}}}={\frac {\lambda _{2}}{\lambda _{1}}}={\sqrt {\frac {\varepsilon _{1}\mu _{1}}{\varepsilon _{2}\mu _{2}}}}\,\!}
where:
Polarization Physical situation Nomenclature Equations Angle of total polarisation θB = Reflective polarization angle, Brewster's angle tan θ B = n 2 / n 1 {\displaystyle \tan \theta _{B}=n_{2}/n_{1}\,\!} intensity from polarized light, Malus's law I 0 = Initial intensity,I = Transmitted intensity,θ = Polarization angle between polarizer transmission axes and electric field vector I = I 0 cos 2 θ {\displaystyle I=I_{0}\cos ^{2}\theta \,\!}
Diffraction and interference Property or effect Nomenclature Equation Thin film in air n 1 = refractive index of initial medium (before film interference)n 2 = refractive index of final medium (after film interference) Min: N λ / n 2 {\displaystyle N\lambda /n_{2}\,\!} Max: 2 L = ( N + 1 / 2 ) λ / n 2 {\displaystyle 2L=(N+1/2)\lambda /n_{2}\,\!} The grating equation a = width of aperture, slit widthα = incident angle to the normal of the grating plane δ 2 π λ = a ( sin θ + sin α ) {\displaystyle {\frac {\delta }{2\pi }}\lambda =a\left(\sin \theta +\sin \alpha \right)\,\!} Rayleigh's criterion θ R = 1.22 λ / d {\displaystyle \theta _{R}=1.22\lambda /\,\!d} Bragg's law (solid state diffraction) d = lattice spacingδ = phase difference between two waves δ 2 π λ = 2 d sin θ {\displaystyle {\frac {\delta }{2\pi }}\lambda =2d\sin \theta \,\!} For constructive interference: δ / 2 π = n {\displaystyle \delta /2\pi =n\,\!} For destructive interference: δ / 2 π = n / 2 {\displaystyle \delta /2\pi =n/2\,\!} where n ∈ N {\displaystyle n\in \mathbf {N} \,\!}
Single slit diffraction intensity I 0 = source intensityWave phase through apertures ϕ = 2 π a λ sin θ {\displaystyle \phi ={\frac {2\pi a}{\lambda }}\sin \theta \,\!}
I = I 0 [ sin ( ϕ / 2 ) ( ϕ / 2 ) ] 2 {\displaystyle 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 slitsN = number of slitsPhase between N waves emerging from each slit δ = 2 π d λ sin θ {\displaystyle \delta ={\frac {2\pi d}{\lambda }}\sin \theta \,\!}
I = I 0 [ sin ( N δ / 2 ) sin ( δ / 2 ) ] 2 {\displaystyle 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 [ sin ( ϕ / 2 ) ( ϕ / 2 ) sin ( N δ / 2 ) sin ( δ / 2 ) ] 2 {\displaystyle 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 ( 2 J 1 ( k a sin θ ) k a sin θ ) 2 {\displaystyle 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 apertureE inc , magnitude of incident electric field at aperture Near-field (Fresnel) A ( r ) ∝ ∬ a p e r t u r e E i n c ( r ′ ) e i k | r − r ′ | 4 π | r − r ′ | d x ′ d y ′ {\displaystyle 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 ( r ) ∝ e i k r 4 π r ∬ a p e r t u r e E i n c ( r ′ ) e − i k [ sin θ ( cos ϕ x ′ + sin ϕ y ′ ) ] d x ′ d y ′ {\displaystyle 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 r 0 = position from source to aperture, incident on itr = position from aperture diffracted from it to a pointα0 = 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 ^ {\displaystyle \mathbf {\hat {n}} \,\!} r 0 ⋅ n ^ = | r 0 | cos α 0 {\displaystyle \mathbf {r} _{0}\cdot \mathbf {\hat {n}} =\left|\mathbf {r} _{0}\right|\cos \alpha _{0}\,\!} r ⋅ n ^ = | r | cos α {\displaystyle \mathbf {r} \cdot \mathbf {\hat {n}} =\left|\mathbf {r} \right|\cos \alpha \,\!} | r | | r 0 | ≪ λ {\displaystyle \left|\mathbf {r} \right|\left|\mathbf {r} _{0}\right|\ll \lambda \,\!} A ( r ) = − i 2 λ ∬ a p e r t u r e e i k ⋅ ( r + r 0 ) | r | | r 0 | [ cos α 0 − cos α ] d S {\displaystyle 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 ( r ) = − 1 4 π ∬ a p e r t u r e e i k ⋅ r 0 | r 0 | [ i | k | U 0 ( r 0 ) cos α + ∂ A 0 ( r 0 ) ∂ n ] d S {\displaystyle 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.
See also
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 ISBN 978-0-521-57507-2 . 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 ISBN 0-07-051400-3 . 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 ed.). John Wiley & Sons. ISBN 978-0-471-92712-9 . D.J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley. ISBN 978-81-7758-293-2 .
Further reading L.H. Greenberg (1978). Physics with Modern Applications ISBN 0-7216-4247-0 . 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 978-0-321-50130-1 . 
![{\displaystyle I=I_{0}\left[{\frac {\sin \left(\phi /2\right)}{\left(\phi /2\right)}}\right]^{2}\,\!}](./_assets_/dbb8da44a507fe22f2d2ad8a28b2f2043ea21706.svg)
![{\displaystyle I=I_{0}\left[{\frac {\sin \left(N\delta /2\right)}{\sin \left(\delta /2\right)}}\right]^{2}\,\!}](./_assets_/a402d3363d129f2369af83440da669ac9e18b0ee.svg)
![{\displaystyle 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}\,\!}](./_assets_/6289a9a8239bf8d151515dab46b5348d30a11c95.svg)
![{\displaystyle 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'}](./_assets_/d352a6eb78726b4e7afaafcca8fd4c18985283be.svg)
![{\displaystyle 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\,\!}](./_assets_/40eca54a607f6968c482166ae98b9a6323e067a6.svg)
![{\displaystyle 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}](./_assets_/f4b73ff83fbd0b08f6c393e5daef916fb7d53c9f.svg)
![{\displaystyle M=m-5\left[\left(\log _{10}{D_{L}}\right)-1\right]\!\,}](./_assets_/d7c00dabee47ab3dc1627bea2d6cdd701918b865.svg)
