Optics & Beam Calculations
Gaussian beam propagation, collimation, focusing, NA, telescope magnification.
Fiber → Collimated → Focused Spot
Given the fiber MFD and collimating lens focal length, computes the collimated beam radius and (optionally) the focused spot after a second lens.
$$w_{\rm col} = \frac{f_1\,\lambda}{\pi\cdot{\rm MFR}} \qquad w_{\rm foc} = \frac{f_2\,\lambda}{\pi\,w_{\rm col}} \quad \text{[optional focus]}$$
$$z_R = \frac{\pi\,w_{\rm foc}^2}{\lambda}$$
nm
µm
mm
mm
NA ↔ Gaussian Beam Waist
For a Gaussian beam, NA = λ/(πw₀). Also gives the Abbe resolution limit.
$$w_0 = \frac{\lambda}{\pi\cdot{\rm NA}} \quad \text{[Gaussian]} \qquad d_{\rm Abbe} = \frac{\lambda}{2\cdot{\rm NA}} \quad \text{[Abbe resolution]}$$
$${\rm NA} = \frac{\lambda}{\pi\,w_0}$$
nm
Rayleigh Range & Divergence
Gaussian beam propagation from waist w₀ and wavelength.
$$z_R = \frac{\pi w_0^2}{\lambda} \qquad \theta_{\rm div} = \frac{\lambda}{\pi w_0} \quad \text{[far-field half-angle]}$$
nm
µm
Beam Telescope Magnification
2-lens Galilean / Keplerian expander. Output beam waist and divergence.
$$M = \frac{f_2}{f_1} \qquad w_{0,{\rm out}} = M\,w_{0,{\rm in}} \qquad \theta_{\rm out} = \frac{\theta_{\rm in}}{M}$$
mm
mm
mm
f-number → Spot Size
For a Gaussian beam: w₀ ≈ λf/(πw_in). For plane wave (overfilled): d_Airy = 2.44 λ (f/D).
$$w_0 = \frac{\lambda f}{\pi w_{\rm in}} \quad \text{[Gaussian]} \qquad d_{\rm Airy} = 2.44\,\lambda\,(f/D) \quad \text{[overfilled]}$$
nm
mm
mm
Power, RF & AOM Calculators
mW / dBm, gain/loss chains, AOM Bragg angles, shot noise.
mW ↔ dBm Conversion
1 mW = 0 dBm. Used everywhere: AOM drivers, amplifiers, photodetectors, VCOs.
$$P\,[\text{dBm}] = 10\cdot\log_{10}(P\,[\text{mW}]) \qquad P\,[\text{mW}] = 10^{P[\text{dBm}]/10}$$
mW
dB Gain / Loss Chain
Enter input power and gains/losses in dB (negative = loss). E.g. AOM −10 dB, amplifier +30 dB.
$$P_{\rm out}\,[\text{dBm}] = P_{\rm in}\,[\text{dBm}] + \sum \text{gains}\,[\text{dB}]$$
dBm
AOM / AOD Deflection
Bragg diffraction angle for a given optical wavelength and AOM drive frequency.
$$\Lambda = \frac{v_s}{f_{\rm AOM}} \quad \text{[acoustic wavelength]}$$
$$\theta_B = \frac{\lambda_{\rm opt}}{2\Lambda} \quad \text{[Bragg angle]} \qquad \theta_{\rm def} = \frac{n\,\lambda\,f_{\rm AOM}}{v_s} \quad \text{[}n\text{-th order]}$$
nm
MHz
Shot Noise on Photodetector
Photon shot noise sets the quantum noise floor for a given detection bandwidth.
$$I_{\rm dc} = \frac{\eta e P}{h\nu} \qquad i_{\rm shot} = \sqrt{2e\,I_{\rm dc}\,{\rm BW}} \quad [{\rm A}_{\rm rms}]$$
$${\rm NEP} = \frac{i_{\rm shot}}{R} \quad [{\rm W}]$$
nm
µW
Hz
Atomic Physics Calculators
Recoil, Doppler, de Broglie, Zeeman, saturation intensity, Maxwell–Boltzmann velocities.
Photon Recoil
Recoil velocity & temperature from one photon kick. Sets the energy scale for laser cooling.
$$v_{\rm rec} = \frac{\hbar k}{m} = \frac{h}{m\lambda} \qquad T_{\rm rec} = \frac{m v_{\rm rec}^2}{k_{\rm B}} = \frac{(\hbar k)^2}{m k_{\rm B}}$$
$$E_{\rm rec}/h = \frac{\hbar k^2}{4\pi m} \quad [\text{kHz}]$$
nm
amu
Doppler Shift
Frequency shift from atom/mirror velocity. Used to find which velocity class a detuned laser addresses.
$$\Delta f = \frac{v}{\lambda} \quad \text{[counter-propagating]} \qquad v = \Delta f \cdot \lambda$$
nm
m/s
Thermal de Broglie Wavelength
λ_dB ≫ inter-particle spacing → quantum degeneracy. Also for matter-wave interferometry.
$$\lambda_{\rm dB} = \frac{h}{\sqrt{2\pi m k_{\rm B} T}}$$
amu
µK
Zeeman Shift (Linear Regime)
Valid for |B| where Zeeman energy ≪ HF splitting (typically B ≪ few hundred Gauss for alkalis).
$$\Delta E = g_F\,m_F\,\mu_B\,B \qquad \Delta f = \frac{\Delta E}{h} \quad [\text{MHz}]$$
(Cs F=4: +1/4)
G
Saturation Intensity
I_sat for a dipole-allowed transition. Above I_sat, the transition is power-broadened.
$$I_{\rm sat} = \frac{\pi h c \Gamma}{3\lambda^3} \quad \text{[cycling transition]} \qquad \tau = \frac{1}{\Gamma/2\pi}$$
nm
MHz
Thermal Velocity Distribution
Maxwell–Boltzmann rms, most-probable, and mean speeds.
$$v_{\rm rms} = \sqrt{\frac{3k_{\rm B}T}{m}} \qquad v_{\rm prob} = \sqrt{\frac{2k_{\rm B}T}{m}} \quad \text{[most probable]}$$
$$v_{\rm mean} = \sqrt{\frac{8k_{\rm B}T}{\pi m}} \quad \text{[mean speed]}$$
amu
µK
Trap & Cavity Calculators
Optical tweezer frequencies, Lamb–Dicke parameter, Fabry–Pérot cavity, mode matching.
Optical Tweezer Trap Frequencies
Harmonic approximation near the trap minimum. Radial and axial frequencies from trap depth and beam waist.
$$\omega_r = \sqrt{\frac{4U_0}{m w_0^2}} \quad \text{[radial]} \qquad \omega_z = \sqrt{\frac{2U_0}{m z_R^2}} \quad \text{[axial]}$$
$$z_R = \frac{\pi w_0^2}{\lambda}$$
mK
µm
amu
nm
Lamb–Dicke Parameter
η = k x_zpf. Resolved sideband cooling requires η ≪ 1. x_zpf = √(ħ/2mω) is the zero-point motion.
$$x_{\rm zpf} = \sqrt{\frac{\hbar}{2m\omega_{\rm trap}}} \qquad \eta = k\cos\theta\cdot x_{\rm zpf}$$
$$\langle n\rangle_{\rm min} \approx \left(\frac{\Gamma}{2\omega}\right)^2 \quad \text{[RSB cooling limit]}$$
nm
amu
kHz
°
Optical Cavity / Fabry–Pérot
FSR, finesse, linewidth, round-trip time. Used for laser locking, ULE cavities, transfer cavities.
$${\rm FSR} = \frac{c}{2nL} \qquad \mathcal{F} = \frac{\pi\sqrt{R}}{1-R}$$
$$\delta\nu = \frac{{\rm FSR}}{\mathcal{F}} \quad \text{[linewidth FWHM]}$$
mm
Gaussian Mode Matching
Coupling efficiency into a fiber or cavity mode via overlap integral of two Gaussian modes.
$$\eta = \left(\frac{2w_1 w_2}{w_1^2+w_2^2}\right)^2 \quad [\Delta z=0]$$
Reduced by wavefront curvature for $\Delta z \neq 0$
µm
µm
µm
nm
References:
Foot (2005) Atomic Physics ·
Grimm et al. (2000) Adv. At. Mol. Opt. Phys. 42 ·
Yariv (1989) Quantum Electronics ·
Saleh & Teich (2019) Fundamentals of Photonics ·
NIST CODATA 2018 constants.
Clebsch–Gordan Coefficients
⟨j₁m₁; j₂m₂ | JM⟩ via Racah formula — exact ±√(p/q) results, CG tables, and Wigner–Eckart reference.
📖 Background & Selection Rules
$$|J,M\rangle = \sum_{m_1,m_2} \langle j_1 m_1;\, j_2 m_2 \mid J M\rangle\; |j_1,m_1\rangle|j_2,m_2\rangle$$
Selection rules — coefficient is zero unless:
- M = m₁ + m₂ (z-component conservation)
- |j₁ − j₂| ≤ J ≤ j₁ + j₂ (triangle rule)
- |mᵢ| ≤ jᵢ for i = 1, 2
$J \in \{|j_1-j_2|,\;|j_1-j_2|+1,\;\ldots,\;j_1+j_2\}$
$$\sum_J (2J+1) = (2j_1+1)(2j_2+1) \quad \text{[dimension check]}$$ Exchange: $\langle j_1 m_1;j_2 m_2|JM\rangle = (-1)^{j_1+j_2-J}\langle j_2 m_2;j_1 m_1|JM\rangle$
Time-rev: $\langle j_1 m_1;j_2 m_2|JM\rangle = (-1)^{j_1+j_2-J}\langle j_1{-}m_1;j_2{-}m_2|J{-}M\rangle$
$$\sum_J (2J+1) = (2j_1+1)(2j_2+1) \quad \text{[dimension check]}$$ Exchange: $\langle j_1 m_1;j_2 m_2|JM\rangle = (-1)^{j_1+j_2-J}\langle j_2 m_2;j_1 m_1|JM\rangle$
Time-rev: $\langle j_1 m_1;j_2 m_2|JM\rangle = (-1)^{j_1+j_2-J}\langle j_1{-}m_1;j_2{-}m_2|J{-}M\rangle$
Compute ⟨j₁m₁; j₂m₂ | JM⟩
M is fixed automatically as m₁ + m₂. Results as exact ±√(p/q).
First angular momentum
Second angular momentum
Total angular momentum
M = —
Adjust inputs above to compute
CG Table for fixed j₁, j₂, J
Rows = m₁, columns = m₂. Cell: ⟨j₁m₁; j₂m₂ | J, m₁+m₂⟩. Keep j ≤ 4 for fast computation.
Wigner–Eckart Theorem
$$\langle \alpha', j', m' \mid T^{(k)}_q \mid \alpha, j, m\rangle = \langle j, m;\, k, q \mid j', m'\rangle \times \frac{\langle \alpha'j' \| T^{(k)} \| \alpha j\rangle}{\sqrt{2j'+1}}$$
The CG coefficient carries ALL $m$-dependence.
The reduced matrix element is independent of $m$, $m'$, $q$.
The reduced matrix element is independent of $m$, $m'$, $q$.
E1 ($k=1$): $\Delta j = 0,\pm1$ (no $0\to0$); $\Delta m = 0$ ($\pi$), $\pm1$ ($\sigma^\pm$)
M1 ($k=1$): Same $\Delta j$, $\Delta m$; $\Delta l = 0$
M1 ($k=1$): Same $\Delta j$, $\Delta m$; $\Delta l = 0$
E2 ($k=2$): $\Delta j = 0,\pm1,\pm2$; $\Delta m = 0,\pm1,\pm2$
Hyperfine: All $\langle F,m_F|T|F',m_{F'}\rangle$ reduce to one $\langle F\|T\|F'\rangle$ via Wigner–Eckart
Hyperfine: All $\langle F,m_F|T|F',m_{F'}\rangle$ reduce to one $\langle F\|T\|F'\rangle$ via Wigner–Eckart
Wigner 3j relation:
$$\langle j_1 m_1;\, j_2 m_2 \mid J M\rangle = (-1)^{j_1-j_2+M}\sqrt{2J+1}\begin{pmatrix}j_1 & j_2 & J \\ m_1 & m_2 & -M\end{pmatrix}$$
In Python:
sympy.physics.wigner.wigner_3j(j1, j2, j3, m1, m2, m3)