(a) Noise coefficient α_{Geom} , for the scenario of a Poisson-distributed signal that is amplified by multiplication, and (b) noise coefficient α_{GeomR} , for the scenario of a Poisson-distributed signal that is amplified by multiplication and subsequently corrupted by measurement noise. In both (a) and (b), the noise coefficient is shown as a function of the mean ν_{θ} of the signal, which ranges in value from 0.27 to 201.28. The signal is amplified through *N* = 536 stages of standard geometric multiplication, and the different curves correspond to mean gain values of *m*^{N} = 1.01 (*), 1.03 (◦), 1.06 (×), 1.31 (⋄), 1.71 (·), 4.98 (□), 14.49 (+), and 1015.46 (△). In (a), the red curve represents the noise coefficient α_{H} of the high gain approximation, while the blue and green curves represent the noise coefficient α_{Gauss} of the Gaussian approximation for mean gains of 1015.46 and 4.98, respectively. In (b), the measurement noise is Gaussian-distributed with mean η_{w} = 0 and standard deviation σ_{w} = 8, and as a reference, the red curve represents the noise coefficient α_{R} for the scenario where there is no signal amplification.

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