Photon-Counting BER (OOK and DPSK)

Exact Poisson bit error rate of photon-counting OOK and DPSK receivers versus detected photons per bit and dark plus background counts.

Model: Exact

Inputs

Detected Signal Photons per Bit (average)

Dark + Background Counts per Bit Slot (per detector)

Modulation

Results

BER (OOK)1.00e-13

Photons per "1" pulse200

Decision threshold (ML)≥ 183 counts

Required for BER 1e-335.6 photons/bit

Required for BER 1e-658.7 photons/bit

Required for BER 1e-977.7 photons/bit

Model (exact for an ideal photon-counting receiver): Poisson signal and noise counts, equiprobable bits, no dead time or ISI.

OOK: a "1" carries 2n̄ photons. Decide "1" when k ≥ k_T = ⌊2n̄ / ln(1 + 2n̄/n_b)⌋ + 1 (maximum likelihood). BER = ½[P(K ≥ k_T | n_b) + P(K < k_T | 2n̄ + n_b)].

DPSK: delay-line interferometer with one counter per port. BER = P(K_d > K_c) + ½P(K_d = K_c), K_c ~ Poisson(n̄ + n_b), K_d ~ Poisson(n_b).

No noise (dotted): ½e^−2n̄ (OOK, 10 photons/bit at 10⁻⁹) and ½e^−n̄ (DPSK, 20 photons/bit). Ref: D. O. Caplan, in Free-Space Laser Communications (Springer, 2008), doi:10.1007/978-0-387-28677-8_4.

Model and references

Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers

Exact for an ideal photon-counting receiver: equiprobable bits, Poisson signal and noise counts. Not modelled: dead time, intersymbol interference, timing jitter or interferometer phase error.

Caplan D. O. (2008). Laser communication transmitter and receiver design. In Majumdar A. K., Ricklin J. C. (eds.), Free-Space Laser Communications, pp. 109–246. Springer. Photon-counting quantum limits of OOK and DPSK. doi:10.1007/978-0-387-28677-8_4

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