Signal & quantificationFrom a spot of light to a number

A mologram concentrates diffracted light at a focus. The camera records intensity, which depends on the combined optical fields. Converting that intensity into coherent mass density (CMD) requires calibration and treatment of the initial field. Explore how focus size, field addition, and molecular arrangement affect the result.

Objective. Turn the focal spot into coherent surface mass density, and see what “coherent” leaves out. In the training: The MACS Matchmaker has three readouts.

Recommended first: the mologram and surface chemistry.

An ideal focus: the Airy disc

A circular, uniformly illuminated aperture produces an ideal Airy pattern: a bright central disc surrounded by faint rings. The lab uses that circular-aperture model. For an aperture of width D at focal distance f in air, the numerical aperture is NA ≈ D/(2f) and the first-zero radius is r ≈ 0.61λ/NA, so a wider mologram or a shorter focal distance gives a tighter spot. The lab uses an illustrative geometry, not the instrument's, and it is not an exact spot prediction for a square or nonuniformly illuminated mologram. The earlier research geometry used NA 0.33 and reported a 1.17 µm radius: Frutiger 2019, §IV.

The reader uses this shape. The focal-plane image is convolved with the expected Airy pattern before the peak intensity is read, which weights the pixels where the mologram's light should fall and suppresses single-pixel noise. The conversion from that peak to diffracted power depends on the mologram outline, square or round.

Intensity is the square of coherent mass

At the focus, the coherent contributions add as optical fields. Their phases depend on position within the pattern. A camera measures intensity, which is amplitude squared. With fixed illumination and no initial coherent field, doubling the added coherent mass quadruples intensity. An initial field changes this relation for newly bound mass.

Raw intensity is not generally proportional to newly bound mass. Background treatment removes light that does not come from the pattern. Optical normalization divides the diffracted power P_out by the power P_in that reaches the mologram, so CMD scales with √(P_out/P_in). P_in is read for each mologram from the coupling camera, which images the beam profile along the chip. Guided light loses power as it travels, so molograms far from the coupler receive less light; dividing by their own P_in removes that gradient together with changes in laser power or coupling. Calibration then converts the normalized amplitude to pg/mm² against a known mass loading. Square-root intensity alone does not identify the sign or molecular origin of a mass change.

Binding adds to an existing optical field

The initial field E₀ may include scattering from the ligand pattern, a deliberate bias, and coherent background. Binding adds another field E_b.

With optical factors held fixed, I ∝ |E₀ + E_b|². Expanding this gives |E₀|² + |E_b|² + 2|E₀||E_b| cos φ, where φ is their relative phase.

Fields with the same phase reinforce each other. Opposite-phase fields can partially cancel, so added mass can initially decrease intensity. Phase describes optical alignment, not a negative amount of protein.

Taking √I returns a magnitude, not the phase or the sign of a field change. Baseline intensity subtraction alone does not isolate binding intensity.

Why a trace can dip through zero and come back mirrored

CMD carries a sign, and the sign is a convention. The reported quantity is the contrast between ridges and grooves. Whichever side carries more material when the baseline is zeroed becomes the positive direction for the rest of the run. Mass itself is never negative; only the label of the heavier side can change.

If binding during a run reverses which side is heavier, the trace falls toward zero, passes through it, and continues as a mirror image of the true binding curve. In the lab above, set the phase to opposite and raise the added field past the initial one: intensity falls to zero and rises again, and √I traces a V. The software detects such a V near zero and can invert the trace from that point on, so that it reads on one consistent scale. A near-zero V supports an inferred sign change; it does not identify the cause. Excess backfilling is one possible cause.

Worked example. The polystyrene bias creates a nonzero initial field. The measured streptavidin response therefore starts above zero. See Frutiger et al. (2021), Figure 1.

Coherent mass density, and the mass you cannot see

The calibrated readout is coherent surface mass density, CMD, in picograms per square millimeter. It is defined as the amount of mass arranged perfectly in step with the grating that would give the measured signal. Real bound mass is only partly in step, so the total splits in two. The sensor sees the coherent part; an invisible part contributes no coherent amplitude at the selected detection location; it can still scatter elsewhere. How large the coherent part is, for a given amount of bound analyte, is the analyte efficiency.

Four quantities that answer different questions

Numerical example. Assume Γ_tot = 10 pg/mm² and a sinusoidal analyte distribution with analyte efficiency η = 0.5. Then Γ_coh = ηΓ_tot = 5 pg/mm².

If η changes to 0.25 while total bound mass stays fixed, CMD falls to 2.5 pg/mm². This change does not imply that half the molecules detached.

Neither result specifies occupancy or solution concentration. Under an ideal 1:1 equilibrium model, θ = c/(K_D + c); using it requires active capacity and the free analyte concentration.

This numerical example is synthetic and uses the same surface-area definition for both mass densities. It is not a reconstruction of the streptavidin experiment. Blickenstorfer et al. (2021) defines the coherent-mass framework.

Modulation shape: why arrangement matters

Take the same total bound mass and lay it down three ways: a gentle sinusoid, sharp rectangular ridges, or concentrated perfectly at the grating lines. Each puts a different fraction into the Fourier component at the grating period, which is what CMD measures. The selected Fourier component determines the coherent fraction. Edge sharpness alone does not determine that fraction; duty cycle and the full distribution also matter.