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Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Vol. 14, Iss. 10 — Oct. 1, 1997
  • pp: 2424–2429

Shot-noise formula for time-varying photon rates: a general derivation

Peter J. Winzer  »View Author Affiliations


JOSA B, Vol. 14, Issue 10, pp. 2424-2429 (1997)
http://dx.doi.org/10.1364/JOSAB.14.002424


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Abstract

An expression describing the variance of shot noise is derived for deterministic, time-varying photon rates on a semiclassical basis. This is done both forthe case with and without postprocessing in the electrical regime. It is shownhow the results are related to the traditional shot-noise formula and underwhich (restricted) conditions the well-known formula may be applied. This becomes significant when signal-to-noise ratios have to be calculated. An example related to optical heterodyning demonstrates similarities and differences between the traditional formula and the new results.

© 1997 Optical Society of America

Citation
Peter J. Winzer, "Shot-noise formula for time-varying photon rates: a general derivation," J. Opt. Soc. Am. B 14, 2424-2429 (1997)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-14-10-2424


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References

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  10. The expression O(Δt) standsfor all terms in the series converging to zero faster than Δt.
  11. This quantity is also known as flux density; its magnitudeis proportional to the squared magnitude of the complex envelope of any fieldquantity fulfilling the wave equation and being of dimension W/m2.
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  14. The weight of Dirac impulses is denoted by a hat [e.g., Σ⁁(0)]; the spectrum Σ(jω)thus equals Σ⁁(0)δ(ω)+ Σ ˜(jω).
  15. The duration and bandwidth of a signal x(t)can be defined in various ways. We use the following definitions fulfillingthe time–bandwidth product relation BxTx≥1/2: Bx=−∞ ∞|X(jν)|2dν2|X(jν)|max2, Tx=−∞ ∞|x(t)|2dt|x(t)|max2. In these relations, X(jν) standsfor the Fourier transform of x(t).
  16. R. Müller and W. Heywang, Rauschen(Springer-Verlag, New York, 1989).
  17. As h(t) is real, the magnitude of its spectrumis even and the phase is odd; this means |H(0)|=H(0).
  18. Note that I0 is defined to bethe direct current before the filter f(t) according toEq. (12).
  19. This definition is not the only reasonable one; another well-motivateddefinition could be SNR=〈s(t)〉e2σi2(t), which does not yield thesame results as Eq. (20), however.
  20. This is the case in a heterodyne detection arrangement, forinstance, if the detector works well below its cutoff frequency, resultingin |H(jω)|≈e, and if |F(jω)|, representing a fairly narrow bandpass filter, equals unity in the passband.
  21. These quantities are of dimension W/m2.
  22. The procedure of making the signal x˜(t)periodic for the definition of a time average may look strange at a firstglance. However, it is seen in the main text that it in fact unifies the derivationsfor time-limited and time-unlimited signals.

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