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 for the case with and without postprocessing in the electrical regime. It is shown how the results are related to the traditional shot-noise formula and under which (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

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  1. N. R. Campbell, “The study of discontinuous phenomena,” Proc. Cambridge Philos. Soc. 8, 117 (1909); “Discontinuities in light emission,” Proc. Cambridge Philos. Soc. 15, 310 (1909).
  2. B. M. Oliver, “Thermal and quantum noise,” Proc. IEEE 53, 436–454 (1965).
    [CrossRef]
  3. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
  4. T. M. Niebauer, R. Schilling, K. Danzmann, A. Ruediger, and W. Winkler, “Nonstationary shot noise and its effect on the sensitivity of interferometers,” Phys. Rev. A 43, 5022–5029 (1991).
    [CrossRef] [PubMed]
  5. B. J. Meers and K. A. Strain, “Modulation, signal, and quantum noise in interferometers,” Phys. Rev. A 44, 4693–4703 (1991).
    [CrossRef] [PubMed]
  6. A more detailed description can be found in, e.g., Refs. 3, 7, 8, and 9.
  7. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, England, 1995).
  8. B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, New York, 1978).
  9. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  10. The expression O(Δt) stands for all terms in the series converging to zero faster than Δt.
  11. This quantity is also known as flux density; its magnitude is proportional to the squared magnitude of the complex envelope of any field quantity fulfilling the wave equation and being of dimension W/m2.
  12. H. A. Steinberg and J. T. La Tourette, “A note on photon-electron converters with reference to a paper by A. Arcese,” Appl. Opt. 3, 902 (1964).
    [CrossRef]
  13. E. A. Lee and D. G. Messerschmitt, Digital Communication, 2nd ed. (Kluwer Academic, Boston, Mass., 1994).
  14. The weight of Dirac impulses is denoted by a hat [e.g., Σˆ(0)]; the spectrum Σ(jω) thus equals Σˆ(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 fulfilling the 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) stands for 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 spectrum is even and the phase is odd; this means |H(0)|=H(0).
  18. Note that I0 is defined to be the direct current before the filter f(t) according to Eq. (12).
  19. This definition is not the only reasonable one; another well-motivated definition could be   SNR=〈s(t)〉e2σi2(t)¯, which does not yield the same results as Eq. (20), however.
  20. This is the case in a heterodyne detection arrangement, for instance, if the detector works well below its cutoff frequency, resulting in |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 first glance. However, it is seen in the main text that it in fact unifies the derivations for time-limited and time-unlimited signals.

1991 (2)

T. M. Niebauer, R. Schilling, K. Danzmann, A. Ruediger, and W. Winkler, “Nonstationary shot noise and its effect on the sensitivity of interferometers,” Phys. Rev. A 43, 5022–5029 (1991).
[CrossRef] [PubMed]

B. J. Meers and K. A. Strain, “Modulation, signal, and quantum noise in interferometers,” Phys. Rev. A 44, 4693–4703 (1991).
[CrossRef] [PubMed]

1965 (1)

B. M. Oliver, “Thermal and quantum noise,” Proc. IEEE 53, 436–454 (1965).
[CrossRef]

1964 (1)

1909 (1)

N. R. Campbell, “The study of discontinuous phenomena,” Proc. Cambridge Philos. Soc. 8, 117 (1909); “Discontinuities in light emission,” Proc. Cambridge Philos. Soc. 15, 310 (1909).

Campbell, N. R.

N. R. Campbell, “The study of discontinuous phenomena,” Proc. Cambridge Philos. Soc. 8, 117 (1909); “Discontinuities in light emission,” Proc. Cambridge Philos. Soc. 15, 310 (1909).

Danzmann, K.

T. M. Niebauer, R. Schilling, K. Danzmann, A. Ruediger, and W. Winkler, “Nonstationary shot noise and its effect on the sensitivity of interferometers,” Phys. Rev. A 43, 5022–5029 (1991).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Heywang, W.

R. Müller and W. Heywang, Rauschen (Springer-Verlag, New York, 1989).

La Tourette, J. T.

Lee, E. A.

E. A. Lee and D. G. Messerschmitt, Digital Communication, 2nd ed. (Kluwer Academic, Boston, Mass., 1994).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, England, 1995).

Meers, B. J.

B. J. Meers and K. A. Strain, “Modulation, signal, and quantum noise in interferometers,” Phys. Rev. A 44, 4693–4703 (1991).
[CrossRef] [PubMed]

Messerschmitt, D. G.

E. A. Lee and D. G. Messerschmitt, Digital Communication, 2nd ed. (Kluwer Academic, Boston, Mass., 1994).

Müller, R.

R. Müller and W. Heywang, Rauschen (Springer-Verlag, New York, 1989).

Niebauer, T. M.

T. M. Niebauer, R. Schilling, K. Danzmann, A. Ruediger, and W. Winkler, “Nonstationary shot noise and its effect on the sensitivity of interferometers,” Phys. Rev. A 43, 5022–5029 (1991).
[CrossRef] [PubMed]

Oliver, B. M.

B. M. Oliver, “Thermal and quantum noise,” Proc. IEEE 53, 436–454 (1965).
[CrossRef]

Ruediger, A.

T. M. Niebauer, R. Schilling, K. Danzmann, A. Ruediger, and W. Winkler, “Nonstationary shot noise and its effect on the sensitivity of interferometers,” Phys. Rev. A 43, 5022–5029 (1991).
[CrossRef] [PubMed]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, New York, 1978).

Schilling, R.

T. M. Niebauer, R. Schilling, K. Danzmann, A. Ruediger, and W. Winkler, “Nonstationary shot noise and its effect on the sensitivity of interferometers,” Phys. Rev. A 43, 5022–5029 (1991).
[CrossRef] [PubMed]

Steinberg, H. A.

Strain, K. A.

B. J. Meers and K. A. Strain, “Modulation, signal, and quantum noise in interferometers,” Phys. Rev. A 44, 4693–4703 (1991).
[CrossRef] [PubMed]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Winkler, W.

T. M. Niebauer, R. Schilling, K. Danzmann, A. Ruediger, and W. Winkler, “Nonstationary shot noise and its effect on the sensitivity of interferometers,” Phys. Rev. A 43, 5022–5029 (1991).
[CrossRef] [PubMed]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, England, 1995).

Appl. Opt. (1)

Phys. Rev. A (2)

T. M. Niebauer, R. Schilling, K. Danzmann, A. Ruediger, and W. Winkler, “Nonstationary shot noise and its effect on the sensitivity of interferometers,” Phys. Rev. A 43, 5022–5029 (1991).
[CrossRef] [PubMed]

B. J. Meers and K. A. Strain, “Modulation, signal, and quantum noise in interferometers,” Phys. Rev. A 44, 4693–4703 (1991).
[CrossRef] [PubMed]

Proc. Cambridge Philos. Soc. (1)

N. R. Campbell, “The study of discontinuous phenomena,” Proc. Cambridge Philos. Soc. 8, 117 (1909); “Discontinuities in light emission,” Proc. Cambridge Philos. Soc. 15, 310 (1909).

Proc. IEEE (1)

B. M. Oliver, “Thermal and quantum noise,” Proc. IEEE 53, 436–454 (1965).
[CrossRef]

Other (17)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

A more detailed description can be found in, e.g., Refs. 3, 7, 8, and 9.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, England, 1995).

B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, New York, 1978).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

The expression O(Δt) stands for all terms in the series converging to zero faster than Δt.

This quantity is also known as flux density; its magnitude is proportional to the squared magnitude of the complex envelope of any field quantity fulfilling the wave equation and being of dimension W/m2.

E. A. Lee and D. G. Messerschmitt, Digital Communication, 2nd ed. (Kluwer Academic, Boston, Mass., 1994).

The weight of Dirac impulses is denoted by a hat [e.g., Σˆ(0)]; the spectrum Σ(jω) thus equals Σˆ(0)δ(ω)+Σ˜(jω).

The duration and bandwidth of a signal x(t) can be defined in various ways. We use the following definitions fulfilling the 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) stands for the Fourier transform of x(t).

R. Müller and W. Heywang, Rauschen (Springer-Verlag, New York, 1989).

As h(t) is real, the magnitude of its spectrum is even and the phase is odd; this means |H(0)|=H(0).

Note that I0 is defined to be the direct current before the filter f(t) according to Eq. (12).

This definition is not the only reasonable one; another well-motivated definition could be   SNR=〈s(t)〉e2σi2(t)¯, which does not yield the same results as Eq. (20), however.

This is the case in a heterodyne detection arrangement, for instance, if the detector works well below its cutoff frequency, resulting in |H(jω)|≈e, and if |F(jω)|, representing a fairly narrow bandpass filter, equals unity in the passband.

These quantities are of dimension W/m2.

The procedure of making the signal x˜(t) periodic for the definition of a time average may look strange at a first glance. However, it is seen in the main text that it in fact unifies the derivations for time-limited and time-unlimited signals.

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Figures (2)

Fig. 1
Fig. 1

Photodetector with impulse response h(t) is followed by an electrical filter of impulse response f(t). The photon rate λph(t) produces a current i(t) after the detector, which causes a dc component I0.

Fig. 2
Fig. 2

(a) Two optical fields are added by a beam combiner and detected by a photodetector. Postprocessing of the electrical signal is represented by the filter f(t), which is assumed to be a first-order low-pass filter, f1(t), or an ideal bandpass, f2(t). (b) The positive frequency parts of the magnitudes of the spectra of the filters shown in (a).

Equations (48)

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σi2=2eI0Bh.
P[onephotonin[t, t+Δt]]=λph(t)Δt+O(Δt),
λph(t)=1ωAI(r, t)dr,
P[onephotoelectronin[t, t+Δt]]=λ(t)Δt,
i(t)=limΔt0 k=-Xkh(t-kΔt),
i(t)e=(λ*h)(t),
σi2(t)=[i(t)-i(t)e]2e=(λ*h2)(t),
(x*y)(t)=-x(τ)y(t-τ)dτ.
H(jω)=12π-H(jω˜)H[j(ω-ω˜)]dω˜,
Σ(jω)=Λ(jω)H(jω).
σi2(t)¯=12πΛˆ(0)H(0)=Λˆ(0) 1(2π)2-|H(jω)|2dω.
σi2(t)¯=2|H(jω)|max2Bh 12πΛˆ(0)=2|H(0)|2Bh 12πΛˆ(0),
i(t)e¯=12πΛˆ(0)|H(0)|
I0i(t)e¯,
σi2(t)¯=2eI0Bh,
σi2(t)¯=(I0/e)-|H(jν)|2dν,
σi2(t)¯=2eI00|Hn(jν)|2dν.
σi2(t)¯=12πΛˆ(0)-|G(jν)|2dν.
σi2(t)¯=(I0/e)-|G(jν)|2dν,
σi2(t)¯=2I0Bg|G(jω)|max2/e.
SNR=s(t)e2¯σi2(t)¯,
SNR=s(t)e2¯2eI0Bh,
SNR=es(t)e2¯2I0Bg|H(jω)F(jω)|max2,
F1(jω)=F1(0)1+jω/ωf1,
F2(jω)=F2ωl<|ω|<ωu0elsewhere,
λ(t)=κ[V12+V22+2V1V2 cos(ωst)],
i(t)e=κ{(V12+V22)|H(0)|+2V1V2|H(jωs)|cos[ωst+ϕH(ωs)]},
σi2(t)=κ{(V12+V22)|H(0)|+2V1V2|H(jωs)|cos[ωst+ϕH(ωs)]},
i(t)e=κ{(V12+V22)|H(0)F1(0)|+2V1V2|H(jωs)F1(jωs)|×cos[ωst+ϕH(ωs)+ϕF1(ωs)]},
σi2(t)=κ(V12+V22)-|H(jω)F1(jω)|2dω+2V1V2-H(jω)F1(jω)H[j(ωs-ω)]×F1[j(ωs-ω)]dω×cos[ωst+ϕHF1(ωs)],
σi2(t)¯=2ei(t)e¯BgF1(0)=2eI0BgF12(0).
i(t)e=κ{2V1V2|H(jωs)F2(jωs)|×cos[ωst+ϕH(ωs)+ϕF2(ωs)]},
σi2(t)=κ(V12+V22)-|H(jω)F2(jω)|2dω+2V1V2-H(jω)F2(jω)H[j(ωs-ω)]×F2[j(ωs-ω)]dωcos[ωst+ϕHF2(ωs)].
σi2(t)¯=2BgI0 4e sin2(ωlTh/2)ωl2Th2F22,
λ(t)κV12.
i(t)e=(λ*h)(t)κeV12i(t)e¯,
σi2(t)=(λ*h2)(t)κV12-h2(t)dtσi2(t)¯,
x(t)t,=limT 1T-T/2T/2x(t)dt
x˜(t)t,T=1T-T/2T/2x˜(t)dt
x(t)t,=limT 1T-T/2T/2 12π-X(jω)exp(jωt)dωdt,
X(jω)=-x(t)exp(-jωt)dt.
x(t)t,=12πlimT -X(jω) sin(ωT/2)ωT/2dω,
x(t)t,=12πlimT -Xˆ(0)δ(ω) sin(ωT/2)ωT/2dω+12π-X˜(jω)limT sin(ωT/2)ωT/2dω=12πXˆ(0).
x(t)t,=12πXˆ(0)=limT 1T-T/2T/2x(t)dt=limN 1(2N+1)Tk=-NNkT(k+1)Tx˜(t)dt=x˜(t)t,T,
x(t)¯x(t)t,=12πXˆ(0)
Bx=-|X(jν)|2dν2|X(jν)|max2,
Tx=-|x(t)|2dt|x(t)|max2.
SNR=s(t)e2σi2(t)¯,

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