Imaging SNR Calculator

01 Imaging Beam

Atom & Imaging Transition

Atom species

Detuning Δ/2π (MHz) — negative = red-detuned 0 MHz

Saturation parameter s = I/I_sat: 1.0

0.01×I_sat1×I_sat100×I_sat

Scattering rate

—

photons / s

— photons / ms

Saturation parameter

—

I / I_sat

—

Branching / linewidth

—

Γ/2π (MHz)

τ = 1/Γ

R_max at saturation

—

photons / s (s→∞, Δ=0)

= Γ/2

Two-level scattering rate

On resonance (Δ=0) a two-level atom saturates at Γ/2. Off resonance the Lorentzian denominator reduces the rate:

$$R_{\rm sc} = \frac{\Gamma}{2} \cdot \frac{s}{1 + s + (2\Delta/\Gamma)^2}$$ $s = I/I_{\rm sat}$, $\Gamma$ = natural linewidth, $\Delta$ = laser detuning

02 Collection & Camera

Objective & Optics

Numerical Aperture (NA): 0.50

NA 0.10NA 0.50NA 0.95

Camera quantum efficiency (%): 80%

Optical transmission T (%): 70%

Exposure time (ms): 5.0 ms

0.1 ms1 ms10 ms100 ms

Camera Noise Model

Camera type

EM gain (EMCCD only): 300×

1×100×1000×

Read noise σ_read (e⁻ RMS): 60 e⁻

01 e⁻ (sCMOS)60 e⁻ (EMCCD CIC)

Dark count rate (e⁻/px/s): 0.001

Background photon rate (photons/s collected): 100

03 Results

Signal-to-Noise Ratio —

15102050100+

Collection efficiency η

—

η = η_geo × QE × T

η_geo = —

Photons scattered (total)

—

in exposure time

Photons detected (signal)

—

N_sig = R_sc × t × η

—

Detection fidelity

—

Min exposure for SNR=10

target SNR = 10

—

Min exposure for SNR=50

target SNR = 50

Noise Budget

Source	σ² (photons²)	Fraction	Contribution
Set parameters above to compute noise budget.

04 SNR vs Exposure Time

SNR as a function of imaging duration

Current settings SNR = 10 target SNR = 50 target

05 Physics & Formulas

Collection geometry — solid angle ▶

Geometric collection fraction η_geo

A high-NA objective collects photons into a cone defined by the half-angle θ_max = arcsin(NA/n), where n is the refractive index of the medium (n=1 in air/vacuum).

$$\eta_{\rm geo} = \frac{1 - \sqrt{1 - {\rm NA}^2}}{2} \quad \text{[exact, for } n=1\text{]}$$ $$\approx \frac{{\rm NA}^2}{4} \quad \text{[paraxial, NA} \ll 1\text{]}$$ Example: NA $= 0.6 \Rightarrow \eta_{\rm geo} = 10\%$; NA $= 0.8 \Rightarrow \eta_{\rm geo} = 20\%$

Note that isotropic emission into 4π steradians means even NA=0.95 captures only ~26% of photons geometrically. High-NA objectives (0.5–0.85) are typical for tweezer experiments.

Camera noise model — EMCCD vs sCMOS ▶

Noise variance per pixel per frame

Each camera type has a distinct noise model. The total variance σ²_total determines the denominator of SNR = N_sig / σ_total.

EMCCD (excess noise factor $F = \sqrt{2}$): $$\sigma^2 = F^2(N_{\rm sig} + N_{\rm bg} + N_{\rm dark}) + (\sigma_{\rm read}/G)^2 = 2(N_{\rm sig} + N_{\rm bg} + N_{\rm dark}) + (\sigma_{\rm read}/G)^2$$ sCMOS (no multiplication noise): $$\sigma^2 = N_{\rm sig} + N_{\rm bg} + N_{\rm dark} + \sigma_{\rm read}^2$$ Ideal: $\sigma^2 = N_{\rm sig} + N_{\rm bg} + N_{\rm dark}$
where $G$ = EM gain, $\sigma_{\rm read}$ = read noise (e$^-$ RMS)

For EMCCD the read noise is divided by EM gain G, making it negligible at high gain (G ≳ 50). The penalty is the √2 excess noise factor that increases shot noise variance by 2×. For sCMOS cameras (1–2 e⁻ read noise), read noise is the dominant term at low photon counts.

Detection fidelity — Poisson statistics ▶

Binary atom detection

We distinguish "atom present" (Poisson mean μ_sig) from "no atom" (Poisson mean μ_bg) by choosing a threshold θ. The optimal threshold minimises total error probability.

For Gaussian approximation ($\mu \gg 1$): $$P_{\rm miss} = \Phi\!\left(\frac{-(\mu_{\rm sig} - \theta)}{\sqrt{\sigma^2_{\rm sig}}}\right) \quad \text{[false negative]}$$ $$P_{\rm false} = 1 - \Phi\!\left(\frac{\theta - \mu_{\rm bg}}{\sqrt{\sigma^2_{\rm bg}}}\right) \quad \text{[false positive]}$$ Optimal $\theta \approx (\mu_{\rm sig} + \mu_{\rm bg})/2$
Detection fidelity: $\mathcal{F} = 1 - (P_{\rm miss} + P_{\rm false})/2 \approx \Phi({\rm SNR}/(2\sqrt{2}))$
where $\Phi$ is the standard normal CDF, ${\rm SNR} = (\mu_{\rm sig} - \mu_{\rm bg})/\sigma$

For SNR ≥ 10, fidelity exceeds 99.9%. The key insight is that fidelity scales as erfc(SNR) — doubling SNR dramatically reduces error. Background suppression (e.g. dark-field or EIT imaging) improves fidelity by reducing μ_bg without reducing μ_sig.

Recoil heating & imaging limits ▶

Photon recoil vs trap depth

Every scattered photon imparts a recoil kick ℏk. At scattering rate R_sc the atom heats at rate dE/dt = E_rec × R_sc (in a σ⁺/σ⁻ Sisyphus beam, this is partially cancelled; in a σ beam near resonance it accumulates).

$$E_{\rm rec} = \frac{\hbar^2 k^2}{2m}$$ $^{87}$Rb D2: $E_{\rm rec}/k_{\rm B} = 362$ nK $= 3.77\times10^{-30}$ J
Heating rate: $\dot{T} \approx 2E_{\rm rec} R_{\rm sc}/k_{\rm B}$ [3D, no cooling]
Max imaging time ($U_0$ = trap depth): $t_{\rm max} \sim U_0/(E_{\rm rec} R_{\rm sc})$
Example: $U_0/k_{\rm B} = 1$ mK, $R_{\rm sc} = 50$ kHz $\Rightarrow t_{\rm max} \approx 55$ ms

In practice, tweezer imaging uses repump + imaging beams in a gray molasses or Λ-enhanced dark SPOT configuration to scatter >1000 photons without losing the atom. This pushes fidelity above 99.5% (Bergamini 2004; Liu 2019).

06 References & Further Reading

📄 Bergamini et al. (2004) — Single-atom detection with EMCCD, Science 📄 Liu & Bhatt (2019) — High-fidelity single-atom detection in optical tweezers 📄 Hutzler et al. (2017) — Eliminating light shifts for single-atom trapping 📊 Steck — Alkali Data Sheets (D1/D2 line data for all species) 📄 Kaufman & Greiner (2021) — Quantum science with optical tweezers (Rev. Mod. Phys.)