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Why the Kelpie ORT coincidence background is a triangle

Note on the clock. This document is written for the 10 MHz reference clock, where the oscillator period P is 100 ns. P is one period of the reference clock, so it is 20 ns at 50 MHz, 200 ns at 5 MHz and 500 ns at 2 MHz, and the triangle's base, the CAR penalty T/P and the number of folded cycles all move with it. Every formula below is in terms of P; only the numbers assume 100 ns. In code, pass open_shutter_time_s (or full_scale_s) — nothing assumes 100 ns any more.

TL;DR — The Kelpie ORT timestamp records only where in a 100 ns oscillator cycle a photon arrived, not which cycle inside the ~9 µs frame. That single fact — losing the absolute timestamp and keeping only the phase — turns the flat accidental background of a normal time‑tagger into a triangle, and concentrates the accidentals under the peak by a factor T_frame / T_phase ≈ 9 µs / 100 ns ≈ 90. It is not an analysis bug; it is a direct consequence of folding 9 µs into 100 ns.

1. Two ways to timestamp the same photon

Consider a photon that arrives at absolute time t inside one acquisition frame of length T = 9 µs. The ring oscillator has period P = 100 ns, so we can write

t = k·P + φ ,     k = 0,1,…,89   (which 100 ns cycle),   φ ∈ [0, P)  (phase in the cycle)
  • Reference camera (absolute tagger). Stores the full t (100 ps resolution over the whole frame). Both k and φ are kept.
  • Kelpie ORT. Stores only φ — the phase within the current 100 ns cycle — and only one value per pixel per frame. The cycle index k is thrown away.

Everything below follows from that one difference.

2. A coincidence is a difference of two timestamps

For a signal photon (pixel A) and an idler photon (pixel B) we histogram

Δ = t_B − t_A          (absolute tagger)
Δ = φ_B − φ_A          (Kelpie ORT: phases only)
  • A true SPDC pair is simultaneous: t_A = t_B (to within the ~100 ps jitter). Then Δ = 0 in both schemes — same cycle, same phase. The peak at Δ = 0 survives the folding. This is why you still see a peak.
  • An accidental pair is two independent photons. Their Δ distribution is what changes shape — and that is the whole story.

3. The math: difference of two uniform times is a triangle

Photon arrival times within a frame are (to good approximation) uniform. Two independent uniform variables, differenced, give a triangular distribution — this is the autocorrelation of a boxcar:

If X, Y ~ Uniform(0, L) independently, then D = Y − X has density

        L − |D|
f(D) =  ───────── ,   |D| ≤ L ,   0 otherwise
          L²

a triangle peaking at D = 0, falling linearly to zero at |D| = L, with

peak density at 0 :  f(0) = 1 / L

Absolute tagger → base L = T

Absolute times are uniform over the whole frame, Uniform(0, T). So accidentals form a triangle with base ±T = ±9 µs:

f_abs(0) = 1 / T = 1 / 9000 ns   (accidentals per ns, per accidental pair)

Over a normal ±ns coincidence window, that triangle is essentially flat and almost empty — the background you expect. (Left panel of the figure.)

Kelpie ORT → base L = P

Here is the key step. Taking a uniform time modulo the period,

φ = t mod P

maps Uniform(0, T) onto Uniform(0, P) exactly when T is a whole number of periods (9000 = 90 × 100). Every one of the 90 cycles folds onto the same [0, 100 ns) interval. So the phase is uniform on [0, P), and the accidental difference φ_B − φ_A is a triangle with base ±P = ±100 ns:

f_phase(0) = 1 / P = 1 / 100 ns

4. The folding concentrates accidentals by T / P

Compare the accidental density right under the peak (at Δ = 0) in the two schemes:

f_phase(0)     1/P      T       9 µs
──────────  =  ───   =  ─   =  ──────  =  90
 f_abs(0)      1/T      P      100 ns

Folding 9 µs into 100 ns piles 90 frames' worth of accidental combinations on top of each other at Δ = 0. The number of true pairs at Δ = 0 is unchanged (they were already at 0). So the coincidence‑to‑accidental ratio drops by the same factor:

CAR_phase        P      100 ns      1
─────────   =    ─   =  ──────  =  ──── ≈ 0.011      →   ~90× worse contrast
 CAR_abs         T       9 µs        90

at the same singles rate, same resolution, same source.

The same thing in the time domain: aliasing pulls random Δ toward zero

Write any pair's true separation as

Δ_true = t_B − t_A = (k_B − k_A)·P  +  (φ_B − φ_A)
                     └── global cycle ──┘   └ what ORT keeps ┘

ORT records only the second term, Δ_rec = φ_B − φ_A, so

Δ_rec = Δ_true − (k_B − k_A)·P

The discarded (k_B − k_A)·P — the global cycle offset — is the absolute‑time information. Dropping it means:

  • Random pairs are recorded with a shorter |Δ| than they truly have. A random pair genuinely 3.5 µs apart (k_B − k_A = 35) is stored at Δ_rec = φ_B − φ_A ∈ (−100, +100) ns — it masquerades as a near‑coincidence.
  • And they cluster at zero, not spread flat. Because the two phases are each uniform, Δ_rec is triangular — the aliasing preferentially yields small apparent separations, piling the fake near‑coincidences right under the true peak (density at 0 is 2× a flat spread of the same counts).

This is the physical content of the T/P concentration above: the long real delta‑ts of random pairs — which on an absolute tagger form the flat, out‑of‑the‑way background — are folded back onto Δ ≈ 0. True pairs already sit at Δ = 0, so only the background moves onto the peak. The lost cycle information does not just reduce contrast uniformly; it dumps the accidentals exactly where they do the most damage.

5. Demonstration (same photons, two timestamps)

absolute vs phase-folded

Both panels use the identical 3000 simultaneous pairs and 300 000 accidental pairs. Only the timestamp differs.

  • Left — absolute timestamps: accidentals spread over ±9 µs, so in a ±3 ns window the background is flat and nearly empty; the peak dominates (CAR ≈ 480). This is your reference camera.
  • Right — phase only (9 µs folded into 100 ns): every accidental collapses into the ±100 ns triangle; the peak is a thin spike on a large triangular pedestal (CAR ≈ 7). This is Kelpie ORT.

The real 2026‑07‑24 SPDC data reproduces the right panel exactly: a clean triangle from −100 ns to +100 ns with a sharp spike at 0.

6. Why this is not fixable in the analysis

The information that would flatten the background — the cycle index k, i.e. the absolute time within the frame — was never recorded. Verified against the MATLAB reference decoder (kelpie_data_ddr3.m): the timestamp field is only a 10‑bit coarse + 4‑bit fine value that wraps at P / 71.6 ps codes (~1397 at 10 MHz, ~6985 at 2 MHz; the field itself saturates at 8184 ≈ 586 ns), with no counter for k. You cannot un‑fold a triangle back into a flat background without k.

Two things also make it worse in practice:

  • One timestamp per pixel per frame (first photon only): at high occupancy the true pair's photon is often preempted by an earlier uncorrelated photon, suppressing the peak on top of the concentrated background.
  • Larger frames make it worse: the factor is T/P, so a longer T (the DCR‑consistency check hints T may be ~40 µs, not 9 µs) means an even bigger triangle.

7. What restores it

Only recovering absolute time, or shrinking the fold:

Lever Effect on T/P Notes
Absolute / start‑stop timestamp mode removes the fold entirely needs firmware/HW support (a "stopped" TDC vs the free‑running one); memory notes a run failing because "the TDC never stopped", implying such a mode exists
Shorter frame T linear improvement T = 900 ns → 10× better CAR
Better focusing / more statistics no change to CAR improves count rate and significance, not contrast — still worth doing, just not a fix for the triangle

8. Separate issue: peak width

The triangle explains the background shape and the contrast, but not the peak width (~500 ps measured vs ~100 ps expected). That is a timing‑ resolution question — code→time calibration (the flat ~71.6 ps/code is only an average; the density‑test LUT gives per‑code widths) and detector jitter — and is independent of everything in this document.