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_col = f₁ · λ / (π · MFR)
[optional] w_foc = f₂ · λ / (π · w_col)
z_R = π w_foc² / λ
nm
µm
mm
mm
NA ↔ Gaussian Beam Waist
For a Gaussian beam, NA = λ/(πw₀). Also gives the Abbe resolution limit.
w₀ = λ / (π · NA) [Gaussian]
d_Abbe = λ / (2 · NA) [Abbe resolution]
NA = λ / (π · w₀)
nm
Rayleigh Range & Divergence
Gaussian beam propagation from waist w₀ and wavelength.
z_R = π w₀² / λ
θ_div = λ / (π w₀) (far-field half-angle)
nm
µm
Beam Telescope Magnification
2-lens Galilean / Keplerian expander. Output beam waist and divergence.
M = f₂ / f₁
w₀_out = M · w₀_in
θ_out = θ_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).
w₀ = λ f / (π w_in) [Gaussian]
d_Airy = 2.44 λ (f/D) [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 [dBm] = 10 · log₁₀(P [mW])
P [mW] = 10^(P [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_out [dBm] = P_in [dBm] + Σ gains [dB]
dBm
AOM / AOD Deflection
Bragg diffraction angle for a given optical wavelength and AOM drive frequency.
Λ = v_s / f_AOM (acoustic wavelength)
θ_B = λ_opt / (2Λ) (Bragg angle)
θ_def = n·λ·f_AOM / v_s (n-th order)
nm
MHz
Shot Noise on Photodetector
Photon shot noise sets the quantum noise floor for a given detection bandwidth.
I_dc = η · e · P / (hν)
i_shot = √(2 e I_dc BW) [A_rms]
NEP = i_shot / R [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_rec = ħk / m = h / (mλ)
T_rec = m·v_rec² / k_B = (ħk)² / (m·k_B)
E_rec/h = ħk² / (4πm) [kHz]
nm
amu
Doppler Shift
Frequency shift from atom/mirror velocity. Used to find which velocity class a detuned laser addresses.
Δf = v / λ (counter-propagating)
v = Δf · λ
nm
m/s
Thermal de Broglie Wavelength
λ_dB ≫ inter-particle spacing → quantum degeneracy. Also for matter-wave interferometry.
λ_dB = h / √(2π m k_B T)
amu
µK
Zeeman Shift (Linear Regime)
Valid for |B| where Zeeman energy ≪ HF splitting (typically B ≪ few hundred Gauss for alkalis).
ΔE = g_F · m_F · μ_B · B
Δf = ΔE / h [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_sat = π h c Γ / (3 λ³) (cycling transition)
τ = 1 / (Γ/2π)
nm
MHz
Thermal Velocity Distribution
Maxwell–Boltzmann rms, most-probable, and mean speeds.
v_rms = √(3k_BT/m)
v_prob = √(2k_BT/m) (most probable)
v_mean = √(8k_BT/πm) (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.
ω_r = √(4 U₀ / m w₀²) [radial]
ω_z = √(2 U₀ / m z_R²) [axial]
z_R = π w₀² / λ
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_zpf = √(ħ / 2mω_trap)
η = k · cos(θ) · x_zpf
⟨n⟩_min ≈ (Γ/2ω)² [RSB cooling limit]
nm
amu
kHz
°
Optical Cavity / Fabry–Pérot
FSR, finesse, linewidth, round-trip time. Used for laser locking, ULE cavities, transfer cavities.
FSR = c / (2nL)
F = π√R / (1−R)
δν = FSR / F (linewidth FWHM)
mm
Gaussian Mode Matching
Coupling efficiency into a fiber or cavity mode via overlap integral of two Gaussian modes.
η = (2w₁w₂/(w₁²+w₂²))² [Δz=0]
Reduced by wavefront curvature for Δz≠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.