Absorption Imaging Lab

The standard OD formula $-\ln(I_{\rm out}/I_{\rm in})$ underestimates atom number when $I/I_{\rm sat} \gtrsim 0.1$, because saturation reduces the absorption cross section. The corrected formula adds back the "consumed photons" term:

This is exact within the two-level model. In practice, use $I/I_{\rm sat} \sim 0.1$–$0.5$ to balance photon count (SNR) against saturation distortion. At $I = I_{\rm sat}$ uncorrected OD can be badly biased unless the saturation term is included; at $I = 0.1\,I_{\rm sat}$ the correction is usually small but still worth tracking. If you can't apply the correction (e.g., unknown $I_{\rm sat}$ for mixed states), use $I \ll I_{\rm sat}$.

For high-field imaging (Paschen-Back regime, B > 100 G), the cycling transition breaks down and the effective cross section must be computed from the actual magnetic sublevel populations. Imaging at the stretched states $|F, m_F=\pm F\rangle$ restores a near-cycling cross section.

At large probe photon counts (N_ph > 1000/pixel), shot noise $\delta\text{OD} = \sqrt{2/N_{\rm ph}}$ becomes negligible — but OD noise floors around 0.01–0.1 remain due to imaging fringes: interference patterns from dust, optical surfaces, or air currents that shift slightly between the probe and reference frames.

PCA / eigenface removal: take a library of reference images (no atoms), decompose into principal components, and subtract the best fit from each probe image. This suppresses common fringe modes, reducing effective noise to ~0.002–0.01 OD in well-optimized setups. Implementation: ref_lib @ pinv(ref_lib) @ probe gives the optimal projection.

Practical tips: take probe and reference within the same camera frame (jitter-limited by vibration, not drift), enclose beam path, use antireflection-coated optics, keep the probe pulse < 50 μs to freeze air fluctuations, and take the reference frame immediately (within 1 ms) after the probe frame.

For single atoms in tweezers, absorption imaging is typically not used — the OD of one atom in a ~μm² area is OD ~ 10⁻⁴ to 10⁻³, well below any noise floor. Instead, fluorescence imaging (collecting photons scattered by the atom during a detection pulse) is the standard, giving ~50–200 detected photons per atom and a detection fidelity > 99%.

Absorption imaging is most powerful for ensemble experiments: MOTs (10⁶–10⁹ atoms), BECs (10³–10⁶ atoms), and Fermi gases, where the column density OD is 0.1–5 across a ~10–100 μm cloud. For atom numbers below ~500 in a MOT-scale trap, fluorescence wins even for ensembles.

Everything above destroys the sample: resonant photons that make the shadow also scatter the atoms out of the trap, and the calculator's saturation correction only pushes the OD ceiling higher, it doesn't remove it. Phase-contrast (dispersive) imaging sidesteps both limits by probing far off resonance and detecting the atoms' effect on the light's phase rather than its amplitude, so almost no photons scatter and the cloud survives the shot.

A phase plate (or, in modern setups, a spatial light modulator) at the imaging system's Fourier plane converts this phase shift into an intensity signal a camera can record, the same trick used in Zernike phase-contrast microscopy. Because $\phi$ scales as $1/\Delta$ while photon scattering (and saturation) fall off much faster with detuning, there is a wide detuning window where the phase signal is still easily measurable but the sample is essentially unperturbed.

Andrews et al. (1996) introduced this to cold-atom physics to solve a problem absorption imaging cannot: watching the same Bose-Einstein condensate evolve over time. A single absorption shot destroys the cloud, so any dynamics study needed a fresh cloud per data point; phase-contrast imaging let them take repeated, non-destructive snapshots of one condensate.

Frometa Fernandez et al. (2025) push the technique to the opposite regime this calculator's OD sweet-spot warns you about: very dense clouds. Using a circular SLM phase mask (phase shift $\pi/3$, far detunings of −298 and −1010 MHz), they validated phase-contrast imaging up to resonant optical depth OD ≈ 64, densities well above $\lambda^{-3}$, where absorption imaging's two-level model (and this page's saturation correction) has already broken down from collective and many-body effects.

• Reinaudi et al. (2007) — "Strong saturation absorption imaging of dense clouds of ultracold atoms." Opt. Lett. 32, 3143–3145. Derives the saturation-corrected OD formula used in this calculator. Essential reading before imaging any cloud with OD > 1 or I > 0.1 I_sat. DOI →
• Ockeloen et al. (2010) — "Detection of small atom numbers through image processing." Phys. Rev. A 82, 061606(R). Introduces PCA/eigenface fringe removal for absorption images. Demonstrates detection of ~10 atoms in a BEC with standard CCD cameras. Includes MATLAB code outline. DOI →
• Hueck et al. (2017) — "Calibrating high intensity absorption imaging of ultracold atoms." Rev. Sci. Instrum. 88, 123702. Rigorous treatment of saturation correction beyond the two-level model, including sublevel populations and polarization effects. Provides calibration procedure for I_sat in the actual experimental setup. DOI →
• Schmidutz et al. (2014) — "Quantum Joule-Thomson Effect in a Saturated Homogeneous Bose Gas." Phys. Rev. Lett. 112, 040403. Appendix contains a concise, pedagogical summary of the three-image absorption technique and noise sources — useful as a quick reference alongside this calculator. DOI →
• Ketterle, Durfee & Stamper-Kurn (1999) — "Making, probing and understanding Bose-Einstein condensates." In: Bose-Einstein Condensation in Atomic Gases, Proc. Int. School of Physics "Enrico Fermi." arXiv:cond-mat/9904034. Sections 3–5 give a complete treatment of absorption imaging, cross sections, and cloud reconstruction for BEC experiments. The standard reference for BEC groups. arXiv →
• Andrews et al. (1996) — "Direct, Nondestructive Observation of a Bose Condensate." Science 273, 84–87. Introduces phase-contrast imaging to cold-atom physics: the far-detuned, non-destructive alternative to everything above. Source for the new "Beyond Absorption" note in Practical Considerations. DOI →
• Frometa Fernandez et al. (2025) — "Phase-contrast imaging of a dense atomic cloud." Rev. Sci. Instrum. 96, 083203. Validates phase-contrast imaging up to resonant OD ≈ 64 using an SLM phase mask, well past where absorption imaging's two-level model breaks down. Free preprint below. arXiv →

• Steck atomic data pages (steck.us/alkalidata) — comprehensive tables of σ₀, I_sat, hyperfine A/B coefficients, transition dipole matrix elements, and Clebsch-Gordan factors for the cycling transition. Values used in this calculator are taken directly from these tables.
• Steck, "Rubidium-87 D Line Data" (2019) — steck.us/alkalidata/rubidium87numbers.pdf — Table 7 gives cycling-transition σ₀ = 2.907×10⁻¹³ m² and I_sat = 1.669 mW/cm² for the F=2→F'=3 D2 line used in this calculator.
• NIST Atomic Spectra Database (physics.nist.gov/asd) — Einstein A-coefficients and oscillator strengths for transitions not covered by Steck (e.g., Sr, Yb, Dy, Er).
• Loftus et al. (2004) — "Narrow line cooling and momentum-space crystals." Phys. Rev. A 70, 031401(R). Contains saturation intensity and cross section data for the Sr ¹S₀→³P₁ intercombination line, useful if imaging Sr on the narrow line. DOI →

• Metcalf & van der Straten, Laser Cooling and Trapping (1999) — Springer. Chapter 2 derives the resonant cross section σ₀ = 3λ²/2π from the oscillator model and discusses the role of degeneracy, polarization, and hyperfine pumping in real atoms.
• Foot, Atomic Physics (2005) — Oxford University Press. Chapter 8 "Doppler cooling" and Chapter 9 contain derivations of the scattering force, relevant for understanding the probe-atom interaction during the imaging pulse.
• Pethick & Smith, Bose-Einstein Condensation in Dilute Gases (2002, 2008) — Cambridge University Press. Appendix covers absorption imaging as a diagnostic tool for BEC density profiles.
• Inguscio & Fallani, Atomic Physics: Precise Measurements and Ultracold Matter (2013) — Oxford University Press. Chapter 4 covers time-of-flight expansion and absorption imaging together, useful for connecting this calculator to the TOF thermometry tool.