Quick-access calculations for the daily work of an AMO physicist, organized into four tabs: Optics (beam parameters, coupling, AOM shifts, shot noise floor), Atomic physics (single-photon recoil scale, Doppler limit, saturation intensity, Zeeman shifts), Trapping (tweezer frequency, Lamb-Dicke parameter, cavity FSR and finesse), and Clebsch-Gordan coefficients. These are the numbers you reach for dozens of times a week, keeping them in one place avoids the mental overhead of re-deriving units every time.
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
Gaussian Beam Profile
Side-view of beam propagation from the waist. Uses λ and w₀ from the Rayleigh Range inputs above.
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, single-photon recoil energy, and two-recoil scatter heating scale. Sets the energy scale for laser cooling.
$$v_{\rm rec} = \frac{\hbar k}{m} = \frac{h}{m\lambda} \qquad E_{\rm rec} = \frac{(\hbar k)^2}{2m}$$
$$T_{\rm rec} = \frac{E_{\rm rec}}{k_{\rm B}} \qquad T_{\rm scatter} \approx \frac{2E_{\rm rec}}{k_{\rm B}}$$
$$E_{\rm rec}/h = \frac{\hbar k^2}{4\pi m} \quad [\text{kHz}]$$
Common mistake, recoil factor-of-two.
\(E_{\rm rec}\) is the kinetic energy from one momentum kick \(\hbar k\).
A scatter event usually has absorption plus spontaneous-emission recoil, so the
heating scale is often near \(2E_{\rm rec}\), but the emitted photon projection
depends on geometry and dimensional averaging.
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)EOM / AOM Sideband Calculator
Bessel-function sideband depths for phase (PM) and amplitude (AM) modulation. AOM double-pass shift and efficiency.
EOM Sideband Depths — Bessel Functions Jn(β)
An EOM with modulation index β splits the carrier into sidebands at ±Ω, ±2Ω, … Each sideband carries
power fraction Jn(β)² relative to the input.
AM: sidebands are co-phased (real amplitude modulation).
PM/FM: sidebands have alternating phase (±1 sidebands are π out of phase) — no net AM on a PD.
$$P_n / P_0 = J_n(\beta)^2 \qquad \sum_{n=-\infty}^{\infty} J_n(\beta)^2 = 1$$
Spectrum — carrier + sidebands (height ∝ field amplitude)
Key β Values
Special modulation depths used in common lab applications.
| β | J₀(β)² | J₁(β)² | Use |
|---|---|---|---|
| 1.08 | 0.319 | 0.339 | PDH: max error signal ∝ J₀J₁ |
| 2.405 | ≈ 0 | 0.269 | Carrier suppression (STIRAP, EIT) |
| 5.520 | ≈ 0 | 0.204 | 2nd carrier null |
| 0.5 | 0.940 | 0.058 | Weak modulation (small sidebands) |
AOM Frequency Shift & Double-Pass Setup
AOMs shift laser frequency by ±f_AOM. A double-pass configuration (retroreflect through AOM a second time)
shifts by 2f_AOM, is polarization-maintaining, and makes the output direction independent of frequency —
essential for laser locking scan ranges without beam steering.
$$\Delta f_{\rm single} = \pm f_{\rm AOM} \qquad \Delta f_{\rm double} = 2 f_{\rm AOM} \qquad \eta_{\rm DP} = \eta_{\rm SP}^2$$
Double-pass alignment tip:
the cat's-eye retroreflector (λ/4 plate + mirror) must send the beam back through the AOM crystal,
not just the fiber coupler. Align by checking that the retro beam overlaps the AOM input beam on an iris,
then optimize the second-order extinction (should be >30 dB).
EOM Half-Wave Voltage Vπ
Converts RF power to modulation index β = π V_RF / V_π. Useful for comparing specs across EOMs.
$$\beta = \frac{\pi V_{\rm RF}}{V_\pi} = \pi\sqrt{\frac{2 R P_{\rm RF}}{1}}\cdot\frac{1}{V_\pi}$$
Hood Lab: We use a resonant EOM at 9.192 GHz to generate Cs hyperfine sidebands for
saturated absorption spectroscopy and a non-resonant broadband EOM at 20 MHz for PDH locking to
the 685 nm ULE cavity. The 9 GHz resonant EOM is particularly unforgiving — driving off resonance
by even 5 MHz drops the modulation depth by >10 dB; always match the drive frequency to the EOM's
resonant frequency with a VNA before use.
Unit Converter
Instant conversions for the units AMO physicists actually use. Enter any value and all equivalents update live.
Frequency / Energy
Hz ↔ kHz ↔ MHz ↔ GHz ↔ cm⁻¹ ↔ nm ↔ THz
Power — mW ↔ dBm
P[dBm] = 10 log₁₀(P[mW])
Magnetic Field
Gauss ↔ Tesla ↔ mT
Temperature — Cooling Scales
Kelvin ↔ mK ↔ μK ↔ nK ↔ μeV ↔ recoil units
Angle / Solid Angle
degrees ↔ mrad ↔ μrad
Trap Depth — μK ↔ mK ↔ E_r
Tweezer / optical lattice trap depth in practical units.