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Radiometry, Throughput and Photon Detection

Follow optical power from a source through an aperture, a lens and a filter to a detector. Every loss is a row in the power budget, the collection cone sets both the collected power and the resolution, and the detector turns watts into photoelectrons with shot noise, dark counts and read noise.

Sliders have number boxes for exact values • Presets are reproducible experiments • Links reproduce the settings
NA — Object-side NA
Φc — Collected power
Φd — Power on detector
N — Mean signal e⁻
SNR — Signal-to-noise
δ — 0.61 λ₀/NA

Source → aperture → lens → detector thin lens, on-axis

Distances along the axis are to scale; heights are exaggerated by the stated factor. The shaded cone is the geometric collection cone (half-angle θ₀) from the source centre to the aperture edge; the image-side cone converges at half-angle θᵢ. Power labels follow the budget table below.

Image distance sᵢ
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Magnification m
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Image-side NAᵢ
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Resolution (object) 0.61λ₀/NA
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Airy radius at detector
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Image / spot radius
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Responsivity ηqλ₀/(hc)
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Photocurrent
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Power budget monochromatic, incoherent losses

Each row multiplies the power by its factor. Photon rate = P λ₀/(hc).
Stage Factor Power Photons or e⁻ per s How

Aperture NA: collected power and resolution

Top: fraction of the emitted power that enters the aperture (log scale); solid = exact, dashed = small-angle model. Bottom: Rayleigh resolution 0.61 λ₀/NA on the object side. Click, drag or use ←/→ on the plot to change the aperture.

Irradiance on the aperture plane numerical angular integral

Filled curve: E(ρ) at the lens plane, absolute units. For the Lambertian disk each point is a numerical integral of L cos θ dΩ over the directions that hit the source; the gold dot at ρ = 0 is the analytic on-axis value πL sin²θ. The cyan dashed line marks the aperture edge D/2 (only ρ < D/2 is collected); the pink dotted line marks the source radius a.

Analytic vs numerical cross-checks

Quantity Analytic Numerical / alternative Relative difference

Radiometric quantities SI

Quantity Symbol, unit Meaning Here

Radiance along the axial ray and étendue

Where n L L/n² (basic)
Étendue object n₀²A π sin²θ₀
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Étendue image nᵢ²A′ π sin²θᵢ
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Image E′ (radiance theorem)
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Image E′ (budget / area)
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Counts per exposure seeded MC

Filled cyan steps: histogram of N simulated exposures (Poisson signal + dark, clipped at the full well, plus Gaussian read noise). Gold dots: theoretical probability per bin (exact Poisson sums; a normal approximation above 2×10⁴ e⁻, flagged in the title). Gold dashed line μ: theoretical mean. Violet dotted lines: ±2 standard errors of a bin count, i.e. the scatter expected from finite N alone.

SNR regimes

Solid cyan: SNR = S/√(S + D + σ_r²). Gold dashed: shot-noise limit √S. Pink dotted: read-noise limit S/σ_r. Vertical markers: the current setting, the crossover S = σ_r² + D, and the full well, where the SNR line stops.

Physical noise vs Monte Carlo sampling error

Statistic Theory (physics) Monte Carlo estimate ± standard error Deviation / SE

Photometry (optional) CIE 1924 V(λ)

V(λ₀)
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Luminous efficacy 683 V
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Luminous flux on detector
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Collected luminous flux
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Φ_v = 683 lm/W × V(λ₀) × Φ for monochromatic light. V(λ) is the CIE 1924 photopic function tabulated at 10 nm (plus 555 nm) and interpolated linearly; it is 0 outside 380–780 nm, so infrared light has no luminous flux however many watts it carries.

Related tools

💡 How to use

Learn with this tool

Learning objectives

  • Build a power budget from radiance, intensity or beam power, with the geometric collection and every transmission loss written out, and predict how collected power and resolution scale with NA.
  • Convert optical power into expected photoelectrons with N = η P t λ₀/(hc), and say whether a measurement is shot-noise limited or read-noise limited.
  • Tell physical noise (the spread of real exposures) apart from Monte Carlo sampling error (the uncertainty of an estimate made from finite N).

Prerequisites

Model

Power leaving a surface element dA into solid angle dΩ at angle θ from its normal is d²Φ = L cos θ dA dΩ. A Lambertian source has the same radiance L in every direction, so its on-axis irradiance at a receiver that sees it within half-angle θ is E = πL sin²θ, and it emits πLA into the hemisphere. An isotropic point source delivers Φ = I Ω with Ω = 2π(1 − cos θ). A TEM₀₀ beam passes a centred aperture of radius R with fraction 1 − exp(−2R²/w²). The collected power is then multiplied by each transmission (two Fresnel surfaces, a filter, the fraction of the image that lands on the detector). The detector counts N = η Φ t λ₀/(hc) photoelectrons on average, plus dark electrons i_d t; the readout adds Gaussian read noise σ_r, and SNR = S/√(S + D + σ_r²).

L
radiance in the medium where it is quoted, W m⁻² sr⁻¹; L/n² is the basic radiance
E
irradiance, W m⁻²
I
radiant intensity, W sr⁻¹
Φ
radiant flux (power), W
θ₀, NA
object-side collection half-angle; NA = n₀ sin θ₀
G
étendue n² A π sin²θ, m² sr
η
quantum efficiency (photoelectrons per incident photon)
λ₀
vacuum wavelength; photon energy hc/λ₀ is the same in every medium
S, D, σ_r
mean signal electrons, mean dark electrons, read noise (e⁻ rms)

Assumptions. Monochromatic, incoherent power addition; thin lens obeying n₀/sₒ + nᵢ/sᵢ = 1/f with no aberrations, vignetting or cos⁴ fall-off; on-axis detector; Fresnel losses at normal incidence without multiple reflections; a Lambertian image of uniform irradiance; a single integrating detector that is linear up to its full well.

Derivation: πL sin²θ, the n² law and the counting formula

Irradiance. On axis, directions within θ of the normal hit the disk, so E = ∫ L cos θ dΩ = 2πL ∫₀^θ cos θ′ sin θ′ dθ′ = πL sin²θ. The tool evaluates the same integral numerically for off-axis points: for each azimuth φ the directions that hit the disk form one interval in θ, so E(ρ) = (L/2) ∫ [sin²θ₂(φ) − sin²θ₁(φ)] dφ. Integrating E(ρ) over the aperture must reproduce the analytic disk-to-disk configuration factor.

Étendue and n². For a ray bundle crossing an interface, Snell's law n₁ sin θ₁ = n₂ sin θ₂ gives n₁² cos θ₁ dΩ₁ = n₂² cos θ₂ dΩ₂, so the étendue dG = n² dA cos θ dΩ is unchanged. Power dΦ = L dA cos θ dΩ = (L/n²) dG is also unchanged in a lossless system, so L/n² is conserved: radiance in glass is n² times that in air. Losses simply multiply: L₂/n₂² = T L₁/n₁². No passive optic can raise L/n², which is why the image irradiance is capped at E′ = T π (L/n₀²) NA′².

Counting. Photons arrive at rate Φ/(hc/λ₀); each is detected with probability η. Independent detections in time t give a Poisson count with mean S = η Φ t λ₀/(hc) and variance S. Units: W · s · m / (J s · m s⁻¹) = 1. Dark electrons are Poisson too, and read noise is added once per readout.

Exercise 1: Open the aperture

  1. Start from Lamp → lens. Predict how the collected power and the resolution change when D goes from 12.5 mm to 25 mm.
  2. Set D = 12.5 mm, then D = 25 mm.
  3. Read the “Collected by the aperture” row and the object-side resolution.
  4. Explain the two scalings with sin²θ₀ and 1/NA.
Show answer

At small angles sin²θ₀ ≈ (D/2sₒ)², so doubling D multiplies the collected power by ≈ 4.0 (3.99 here, because tan and sin differ slightly), and the resolution 0.61λ₀/NA halves from about 10.7 µm to 5.38 µm. The same cone controls both, which is why a fast lens is both brighter and sharper.

Exercise 2: Where does the noise come from?

  1. Load Photon-starved. The mean signal is about 20 e⁻ and σ_r = 5 e⁻. Predict the variance of the readout.
  2. Run with N = 20 000 exposures and a few seeds.
  3. Read the sample variance and its standard error in the Monte Carlo table.
  4. Which part of the spread is physics, and which part is the finite N?
Show answer

Var = S + D + σ_r² ≈ 20 + D + 25 ≈ 45 e⁻². The sample variance changes from seed to seed by about its standard error, which falls as 1/√N. The standard deviation √45 ≈ 6.7 e⁻ does not fall with N: it is the physical noise of one exposure. The SNR (about 3) is read-noise dominated because σ_r² exceeds S.

Exercise 3 (limiting cases): Short and long exposures

  1. Keep Photon-starved. Predict the slope of log SNR against log t for very short and for long exposures, and the crossover time.
  2. Look at the SNR plot, then move t across the crossover marker.
  3. Measure the slopes over one decade on each side.
  4. Explain in terms of which variance term dominates.
Show answer

For S ≪ σ_r², SNR ≈ S/σ_r ∝ t (slope 1). For S ≫ σ_r² + D, SNR ≈ √S ∝ t^½ (slope ½). They cross where S = σ_r² + D, i.e. t_c = σ_r²/(ηΦλ₀/(hc) − i_d). With σ_r = 0 (Ideal Poisson) the slope is ½ everywhere, and SNR = √S exactly.

Exercise 4: Immersion and the n² law

  1. Load Immersion n² (n₀ = 1.515, nᵢ = 1, AR coated, no filter). Predict L/n² in image space relative to object space.
  2. Read the radiance table, then set the coating off.
  3. Compare the basic radiance in every row.
  4. Why does the image-space radiance L fall while L/n² only changes by the transmission?
Show answer

L/n² is conserved except for losses: coated, it drops by 0.9975² ≈ 0.995; uncoated, by the two Fresnel factors. The radiance in air is L(1/1.515)² ≈ 0.44 L: the same rays spread into a larger solid angle after refraction, so radiance falls although no power is lost.

Worked example: counting photons from a lamp

A Lambertian disk of radius a = 2 mm and radiance L = 1.00 mW m⁻² sr⁻¹ at λ₀ = 550 nm sits sₒ = 200 mm from an uncoated (n = 1.5) f = 50 mm lens stopped to D = 25 mm, followed by a 90 % filter. A detector of radius 1 mm, η = 0.7, dark current 100 e⁻/s and σ_r = 5 e⁻ integrates for 1 ms. This is the Lamp → lens preset.

  1. Collection: θ₀ = arctan(12.5/200) = 3.58°, NA = 0.0624. Emitted πL·πa² = 3.95×10⁻⁸ W. The disk-to-disk configuration factor is 0.003891 (sin²θ₀ = 0.003891), so Φ_c = 1.536×10⁻¹⁰ W.
  2. Losses: each surface transmits 1 − 0.04 = 0.96; with the filter, 0.96 × 0.96 × 0.9 = 0.829, so 1.274×10⁻¹⁰ W reaches the image.
  3. Imaging: sᵢ = 1/(1/50 − 1/200) = 66.7 mm and m = −1/3, so the image radius is 0.667 mm < 1 mm and the whole image lands on the detector.
  4. Counting: hc/λ₀ = 3.61×10⁻¹⁹ J, so 3.53×10⁸ photons/s and 2.47×10⁸ e⁻/s. In 1 ms, S = 2.47×10⁵ e⁻ and D = 0.1 e⁻.
  5. Noise: σ = √(246 918 + 0.1 + 25) = 497 e⁻, SNR = 497. √S = 496.9, so this is shot-noise limited: read noise changes the SNR by only 0.005 %.
  6. Resolution: 0.61 × 550 nm / 0.0624 = 5.38 µm at the source; at the detector the Airy radius is 0.61λ₀/NAᵢ = 1.82 µm, close to |m| × 5.38 µm = 1.79 µm (the paraxial thin lens does not obey the sine condition exactly; see “When the model fails”).

When the model fails

  • Large NA or off-axis points. The thin lens uses paraxial conjugates with tan-angles and does not satisfy the sine condition, so object and image étendues differ by (cos θᵢ/cos θ₀)². Aberrations, vignetting and cos⁴ fall-off are not modelled.
  • Coherent light. Fresnel losses are added as incoherent powers at normal incidence. Thin-film interference inside the lens or filter is ignored (see thin films).
  • Hard-aperture diffraction. A clipped laser beam gets diffraction rings and a larger focus; the model keeps the geometric spot |m|w₀ (see aperture propagation).
  • Non-Poisson light and detectors. Thermal light counted over times shorter than its coherence time is super-Poissonian (bunching). Avalanche or EM gain adds an excess-noise factor. Real sensors also have 1/f noise, ADC quantisation, non-linearity near the full well and pixel-to-pixel gain variation.
  • Real sources. Few sources are perfectly Lambertian or isotropic. Broadband sources need ∫ Φ_λ η(λ) λ/(hc) dλ, not a single wavelength.

References

  • R. W. Boyd, Radiometry and the Detection of Optical Radiation, Wiley (1983): radiance theorem, étendue, detector noise.
  • W. R. McCluney, Introduction to Radiometry and Photometry, 2nd ed., Artech House (2014): quantities, units and photometry.
  • J. R. Howell, M. P. Mengüç, K. Daun, R. Siegel, Thermal Radiation Heat Transfer, 7th ed., CRC (2020), and Howell's online catalogue of configuration factors: coaxial parallel disks.
  • M. Born and E. Wolf, Principles of Optics, 7th ed., §8.5: Airy pattern and encircled energy 1 − J₀² − J₁².
  • J. R. Janesick, Photon Transfer: DN → λ, SPIE Press (2007): shot, dark and read noise, full well.
  • CIE 018:2019, The Basis of Physical Photometry, and G. Wyszecki and W. S. Stiles, Color Science, 2nd ed., Table I(4.3.2): CIE 1924 V(λ). BIPM, The International System of Units, 9th ed. (2019): K_cd = 683 lm/W at 540 THz.