All the key relations of Chapter 1 on one page. Each block links to the chapter where the formula is derived and used. SI units; engineering convention \(e^{j(\omega t-kz)}\) , \(N=n-jK\) . Use your browser’s Print → Save as PDF for a paper copy.
\(\lambda\) vacuum wavelength · \(k=2\pi/\lambda\) · \(\nu\) frequency · \(\omega=2\pi\nu\) · \(n\) refractive index · \(N=n-jK\) complex index · \(\varepsilon_r\) relative permittivity · \(c\) speed of light · \(\theta_i,\theta_t\) incidence and transmission angles (from the normal) · \(R,T\) reflectance and transmittance (power) · \(r,t\) amplitude coefficients
plane wave
\(E_x=E_0\cos(\omega t-kz+\phi_0)\)
wave vector form
\(E=E_0\cos(\omega t-\mathbf k\cdot\mathbf r+\phi_0)\)
phase velocity
\(v=\omega/k=\nu\lambda=c/n\)
phase change over \(\Delta z\)
\(\Delta\phi=k\,\Delta z=2\pi\Delta z/\lambda\)
in a medium
\(\lambda_\text{medium}=\lambda/n\) , \(\ k_\text{medium}=nk\)
photon energy
\(E=h\nu=hc/\lambda\) (\(1.24/\lambda[\mu\text{m}]\) eV)
spectral width
\(\Delta\lambda=\lambda^2\Delta\nu/c\) (100 GHz ↔︎ 0.8 nm at 1550 nm)
spherical wave
amplitude \(\propto1/r\) , irradiance \(\propto1/r^2\)
irradiance
\(I(r,z)=\dfrac{2P}{\pi w^2}e^{-2r^2/w^2}\)
beam radius
\(w(z)=w_0\sqrt{1+(z/z_0)^2}\)
Rayleigh range
\(z_0=\pi w_0^2n/\lambda\)
divergence
\(\theta=\lambda/(\pi w_0)\) ; real beams \(2\theta=M^2\dfrac{4\lambda}{\pi(2w_0)}\)
wavefront radius
\(R(z)=z[1+(z_0/z)^2]\)
power inside \(r=w\)
\(1-e^{-2}\approx86\,\%\)
ABCD law
\(q=z+jz_0\) , \(\ q'=\dfrac{Aq+B}{Cq+D}\)
Gaussian coupling
\(\eta=\left(\dfrac{2w_1w_2}{w_1^2+w_2^2}\right)^2e^{-2d^2/(w_1^2+w_2^2)}\)
magnetic field
\(B=E/v\)
energy density
\(u=\varepsilon_0\varepsilon_rE^2\)
Poynting vector
\(\mathbf S=v^2\varepsilon_0\varepsilon_r\,\mathbf E\times\mathbf B\)
average irradiance
\(I=\tfrac12c\,\varepsilon_0nE_0^2\)
Gaussian peak
\(I_0=2P/(\pi w_0^2)\)
isotropic source
\(I=P/(4\pi r^2)\)
Refractive index and absorption (Chapter 4
index
\(n=\sqrt{\varepsilon_r}\) (at the optical frequency)
Sellmeier
\(n^2=1+\sum_i\dfrac{A_i\lambda^2}{\lambda^2-\lambda_i^2}\)
Lorentz oscillator
\(\varepsilon_r=1+\dfrac{\omega_p^2}{\omega_0^2-\omega^2+j\gamma\omega}\) , \(\ \omega_p^2=\dfrac{Ne^2}{\varepsilon_0m}\)
Drude (metals)
Lorentz with \(\omega_0=0\)
complex index
\(N=n-jK\) , \(\ n^2-K^2=\varepsilon_r'\) , \(\ 2nK=\varepsilon_r''\)
absorption
\(\alpha=4\pi K/\lambda\) , \(\ I=I_0e^{-\alpha z}\)
reflectance
\(R=\dfrac{(n-1)^2+K^2}{(n+1)^2+K^2}\)
Kramers–Kronig
\(\varepsilon_r'-1=\dfrac2\pi\mathcal P\!\int_0^\infty\dfrac{\omega'\varepsilon_r''(\omega')}{\omega'^2-\omega^2}d\omega'\)
group velocity
\(v_g=d\omega/dk=c/N_g\)
group index
\(N_g=n-\lambda\,dn/d\lambda\)
beat of two waves
\(E=2E_0\cos(\delta\omega t-\delta kz)\cos(\omega t-kz)\)
group delay
\(\tau_g=LN_g/c\)
material dispersion
\(D=-\dfrac{\lambda}{c}\dfrac{d^2n}{d\lambda^2}\)
Snell’s law and TIR (Chapter 6
reflection
\(\theta_r=\theta_i\)
Snell
\(n_1\sin\theta_i=n_2\sin\theta_t\)
critical angle
\(\sin\theta_c=n_2/n_1\) (\(n_1>n_2\) )
lateral displacement
\(d=L\sin\theta_i\left[1-\dfrac{\cos\theta_i}{\sqrt{n^2-\sin^2\theta_i}}\right]\)
numerical aperture
\(\mathrm{NA}=n_0\sin\theta_a=\sqrt{n_1^2-n_2^2}\)
Fermat
the actual path makes the travel time stationary
Fresnel equations (Chapter 7
\(r_\perp\) \(\dfrac{\cos\theta_i-\sqrt{n^2-\sin^2\theta_i}}{\cos\theta_i+\sqrt{n^2-\sin^2\theta_i}}\) , \(\ n=n_2/n_1\)
\(r_\parallel\) \(\dfrac{\sqrt{n^2-\sin^2\theta_i}-n^2\cos\theta_i}{\sqrt{n^2-\sin^2\theta_i}+n^2\cos\theta_i}\)
\(t\) relations\(t_\perp=1+r_\perp\) , \(\ r_\parallel+nt_\parallel=1\)
normal incidence
\(R=\left(\dfrac{n_1-n_2}{n_1+n_2}\right)^2\) (glass: 4 %)
Brewster
\(\tan\theta_p=n_2/n_1\) (\(r_\parallel=0\) )
TIR phase ⊥
\(\tan(\phi_\perp/2)=\sqrt{\sin^2\theta_i-n^2}/\cos\theta_i\)
TIR phase ∥
\(\tan(\phi_\parallel/2+\pi/2)=\sqrt{\sin^2\theta_i-n^2}/(n^2\cos\theta_i)\)
evanescent decay
\(\alpha_2=\dfrac{2\pi n_2}{\lambda}\sqrt{\dfrac{n_1^2}{n_2^2}\sin^2\theta_i-1}\) , \(\ \delta=1/\alpha_2\)
Goos–Hänchen
\(\Delta z\approx2\delta\tan\theta_i\)
transmittance
\(T=\dfrac{n_2\cos\theta_t}{n_1\cos\theta_i}\lvert t\rvert^2=1-R\)
round-trip phase
\(\phi=2\cdot\dfrac{2\pi}{\lambda}n_2d\)
film reflection
\(r=\dfrac{r_{12}+r_{23}e^{-j\phi}}{1+r_{12}r_{23}e^{-j\phi}}\)
AR coating
\(d=m\dfrac{\lambda}{4n_2}\) (\(m\) odd), \(\ n_2=\sqrt{n_1n_3}\)
\(R_\text{min}\) (\(n_1<n_2<n_3\) )\(\left(\dfrac{n_1n_3-n_2^2}{n_1n_3+n_2^2}\right)^2\) at \(\phi=\) odd \(\pi\)
\(R_\text{max}\) (\(n_1<n_2<n_3\) )\(\left(\dfrac{n_3-n_1}{n_3+n_1}\right)^2\) at \(\phi=\) even \(\pi\)
incoherent plate
\(T=(1-R)/(1+R)\)
coherence time
\(\Delta t\approx1/\Delta\nu\)
coherence length
\(l_c=c\Delta t=\lambda^2/\Delta\lambda\)
Gaussian pulse
\(\Delta t\,\Delta\nu\approx0.44\) (FWHM)
two-beam interference
\(I=I_1+I_2+2\sqrt{I_1I_2}\cos\delta\)
phase difference
\(\delta=k(r_2-r_1)+(\phi_2-\phi_1)\)
visibility
\(V=\dfrac{2\sqrt{I_1I_2}}{I_1+I_2}\)
Mach–Zehnder
\(I\propto\cos^2(\Phi/2)\) ; quadrature at \(\Phi=\pi/2\)
unbalanced MZI
\(\mathrm{FSR}=\lambda^2/(N_g\Delta L)\)
etalon rings
\(2nL\cos\theta=m\lambda\)
Bragg condition
\(n_1d_1+n_2d_2=m\lambda/2\)
quarter-wave stack
\(d_1=\lambda/4n_1\) , \(\ d_2=\lambda/4n_2\)
peak reflectance
\(R_N=\left[\dfrac{n_0n_2^{2N}-n_3n_1^{2N}}{n_0n_2^{2N}+n_3n_1^{2N}}\right]^2\)
stop band
\(\Delta\lambda\approx\dfrac{4\lambda_0}{\pi}\arcsin\dfrac{n_1-n_2}{n_1+n_2}\)
penetration depth
\(L_\text{pen}=-\tfrac12\,d\phi/dk\approx\lambda/(4\Delta n)\)
1D band structure
\(\cos K\Lambda=\cos k_1d_1\cos k_2d_2-\tfrac12\left(\tfrac{n_1}{n_2}+\tfrac{n_2}{n_1}\right)\sin k_1d_1\sin k_2d_2\)
modes
\(m\lambda_m/2=nL\) , \(\ \nu_m=mc/(2nL)\)
free spectral range
\(\nu_f=c/(2nL)\) , \(\ \Delta\lambda=\lambda^2/(2nL)\)
finesse
\(\mathcal F=\dfrac{\pi\sqrt R}{1-R}\)
linewidth
\(\delta\nu_m=\nu_f/\mathcal F\)
quality factor
\(Q=\nu_m/\delta\nu_m=m\mathcal F\)
Airy transmission
\(T=\dfrac{(1-R)^2}{(1-R)^2+4R\sin^2(2\pi nL/\lambda)}\)
Fraunhofer regime
\(z\gg D\) and \(z\gg\pi D^2/\lambda\)
single slit
\(I=I_0\,\mathrm{sinc}^2\beta\) , \(\ \beta=\pi a\sin\theta/\lambda\) ; zeros \(\sin\theta=m\lambda/a\)
circular aperture
\(I=I_0[2J_1(\gamma)/\gamma]^2\) , \(\ \gamma=\tfrac12kD\sin\theta\)
Rayleigh criterion
\(\sin\Delta\theta_\text{min}=1.22\lambda/D\)
grating
\(d(\sin\theta_m-\sin\theta_i)=m\lambda\)
blaze
\(2\gamma=\theta_m-\theta_i\)
resolving power
\(\lambda/\Delta\lambda=mN\)
Rayleigh scattering
\(\alpha_R\propto\lambda^{-4}\) , pattern \(\propto1+\cos^2\theta\)
Brillouin shift
\(\nu_B=2nv_a/\lambda\)
Typical numbers worth remembering
glass–air reflectance (normal incidence)
≈ 4 % per surface
Si–air reflectance
≈ 30 %
Brewster angle air → glass
56.3°
critical angle glass → air
≈ 42°
silica at 1550 nm
\(n\approx1.444\) , \(N_g\approx1.463\)
optical period at 1550 nm
≈ 5.2 fs
He-Ne laser (multimode) coherence length
≈ 20 cm
LED (1550 nm, 150 nm wide) coherence length
≈ 16 µm
human eye angular resolution
≈ 1 arcmin