Abstract

The impact of dark 1/f noise on fundamental signal sensitivity in direct low optical signal detection is an understudied issue. In this theoretical manuscript, we study the limitations of an idealized detector with a combination of white noise and 1/f noise, operating in detector dark noise limited mode. In contrast to white noise limited detection schemes, for which there is no fundamental minimum signal sensitivity limit, we find that the 1/f noise characteristics, including the noise exponent factor and the relative amplitudes of white and 1/f noise, set a fundamental limit on the minimum signal that such a detector can detect.

© 2008 Optical Society of America

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References

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  1. S. O. Flyckt and C. Marmonier, Photomultiplier tubes: Principles and applications (Photonis, 2002).
  2. M. C. Teich, K. Matsuo, and B. E. A. Saleh, "Excess noise factors for conventional and superlattice avalanche photodiodes and photomultiplier tubes," IEEE J. Quantum Electron. 22, 1184-1193 (1986).
    [CrossRef]
  3. E. J. McDowell, X. Cui, Y. Yaqoob, and C. Yang, "A generalized noise variance analysis model and its application to the characterization of 1/f noise," Opt. Express 15, (2007).
    [CrossRef] [PubMed]
  4. P. Dutta and P. M. Horn, "Low-frequency fluctuations in solids - 1/f noise," Rev. Mod. Phys. 53, 497-516 (1981).
    [CrossRef]
  5. W. H. Press, "Flicker noises in astronomy and elsewhere," Comments Astrophys. 7, 103-119 (1978).
  6. M. B. Weissman, "1/f noise and other slow, nonexponential kinetics in condensed matter," Rev. Mod. Phys. 60, 537-571 (1988).
    [CrossRef]
  7. E. Milotti, "1/f noise: A pedagogical review," invited talk to E-GLEA-2 (2001).
  8. D. W. Allan, "Statistics of atomic frequency standards," Proc. Inst. Electr. Eng. 54, 221-230 (1966).
  9. J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuniga, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971).
    [CrossRef]
  10. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

2007 (1)

E. J. McDowell, X. Cui, Y. Yaqoob, and C. Yang, "A generalized noise variance analysis model and its application to the characterization of 1/f noise," Opt. Express 15, (2007).
[CrossRef] [PubMed]

1988 (1)

M. B. Weissman, "1/f noise and other slow, nonexponential kinetics in condensed matter," Rev. Mod. Phys. 60, 537-571 (1988).
[CrossRef]

1986 (1)

M. C. Teich, K. Matsuo, and B. E. A. Saleh, "Excess noise factors for conventional and superlattice avalanche photodiodes and photomultiplier tubes," IEEE J. Quantum Electron. 22, 1184-1193 (1986).
[CrossRef]

1981 (1)

P. Dutta and P. M. Horn, "Low-frequency fluctuations in solids - 1/f noise," Rev. Mod. Phys. 53, 497-516 (1981).
[CrossRef]

1978 (1)

W. H. Press, "Flicker noises in astronomy and elsewhere," Comments Astrophys. 7, 103-119 (1978).

1971 (1)

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuniga, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971).
[CrossRef]

1966 (1)

D. W. Allan, "Statistics of atomic frequency standards," Proc. Inst. Electr. Eng. 54, 221-230 (1966).

Allan, D. W.

D. W. Allan, "Statistics of atomic frequency standards," Proc. Inst. Electr. Eng. 54, 221-230 (1966).

Barnes, J. A.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuniga, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971).
[CrossRef]

Chi, A. R.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuniga, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971).
[CrossRef]

Cui, X.

E. J. McDowell, X. Cui, Y. Yaqoob, and C. Yang, "A generalized noise variance analysis model and its application to the characterization of 1/f noise," Opt. Express 15, (2007).
[CrossRef] [PubMed]

Cutler, L. S.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuniga, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971).
[CrossRef]

Dutta, P.

P. Dutta and P. M. Horn, "Low-frequency fluctuations in solids - 1/f noise," Rev. Mod. Phys. 53, 497-516 (1981).
[CrossRef]

Healey, D. J.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuniga, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971).
[CrossRef]

Horn, P. M.

P. Dutta and P. M. Horn, "Low-frequency fluctuations in solids - 1/f noise," Rev. Mod. Phys. 53, 497-516 (1981).
[CrossRef]

Leeson, D. B.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuniga, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971).
[CrossRef]

Matsuo, K.

M. C. Teich, K. Matsuo, and B. E. A. Saleh, "Excess noise factors for conventional and superlattice avalanche photodiodes and photomultiplier tubes," IEEE J. Quantum Electron. 22, 1184-1193 (1986).
[CrossRef]

McDowell, E. J.

E. J. McDowell, X. Cui, Y. Yaqoob, and C. Yang, "A generalized noise variance analysis model and its application to the characterization of 1/f noise," Opt. Express 15, (2007).
[CrossRef] [PubMed]

McGuniga, T. E.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuniga, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971).
[CrossRef]

Mullen, J. A.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuniga, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971).
[CrossRef]

Press, W. H.

W. H. Press, "Flicker noises in astronomy and elsewhere," Comments Astrophys. 7, 103-119 (1978).

Saleh, B. E. A.

M. C. Teich, K. Matsuo, and B. E. A. Saleh, "Excess noise factors for conventional and superlattice avalanche photodiodes and photomultiplier tubes," IEEE J. Quantum Electron. 22, 1184-1193 (1986).
[CrossRef]

Smith, W. L.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuniga, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971).
[CrossRef]

Sydnor, R. L.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuniga, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971).
[CrossRef]

Teich, M. C.

M. C. Teich, K. Matsuo, and B. E. A. Saleh, "Excess noise factors for conventional and superlattice avalanche photodiodes and photomultiplier tubes," IEEE J. Quantum Electron. 22, 1184-1193 (1986).
[CrossRef]

Vessot, R. F. C.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuniga, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971).
[CrossRef]

Weissman, M. B.

M. B. Weissman, "1/f noise and other slow, nonexponential kinetics in condensed matter," Rev. Mod. Phys. 60, 537-571 (1988).
[CrossRef]

Winkler, G. M. R.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuniga, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971).
[CrossRef]

Yang, C.

E. J. McDowell, X. Cui, Y. Yaqoob, and C. Yang, "A generalized noise variance analysis model and its application to the characterization of 1/f noise," Opt. Express 15, (2007).
[CrossRef] [PubMed]

Yaqoob, Y.

E. J. McDowell, X. Cui, Y. Yaqoob, and C. Yang, "A generalized noise variance analysis model and its application to the characterization of 1/f noise," Opt. Express 15, (2007).
[CrossRef] [PubMed]

Comments Astrophys. (1)

W. H. Press, "Flicker noises in astronomy and elsewhere," Comments Astrophys. 7, 103-119 (1978).

IEEE J. Quantum Electron. (1)

M. C. Teich, K. Matsuo, and B. E. A. Saleh, "Excess noise factors for conventional and superlattice avalanche photodiodes and photomultiplier tubes," IEEE J. Quantum Electron. 22, 1184-1193 (1986).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuniga, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, "Characterization of frequency stability," IEEE Trans. Instrum. Meas. 20, 105-120 (1971).
[CrossRef]

Opt. Express (1)

E. J. McDowell, X. Cui, Y. Yaqoob, and C. Yang, "A generalized noise variance analysis model and its application to the characterization of 1/f noise," Opt. Express 15, (2007).
[CrossRef] [PubMed]

Proc. Inst. Electr. Eng. (1)

D. W. Allan, "Statistics of atomic frequency standards," Proc. Inst. Electr. Eng. 54, 221-230 (1966).

Rev. Mod. Phys. (2)

P. Dutta and P. M. Horn, "Low-frequency fluctuations in solids - 1/f noise," Rev. Mod. Phys. 53, 497-516 (1981).
[CrossRef]

M. B. Weissman, "1/f noise and other slow, nonexponential kinetics in condensed matter," Rev. Mod. Phys. 60, 537-571 (1988).
[CrossRef]

Other (3)

E. Milotti, "1/f noise: A pedagogical review," invited talk to E-GLEA-2 (2001).

S. O. Flyckt and C. Marmonier, Photomultiplier tubes: Principles and applications (Photonis, 2002).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

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Figures (4)

Fig. 1.
Fig. 1.

A comparison between the measurement schemes in Ref. [3] (Scenario A) and the current work (Scenario B). In Scenario A, an amplitude modulated message is transmitted in steps of duration τ over a total time T. In this work, we simply wish to confirm the presence of the light source in an experiment where both signal and noise are measured for equivalent time periods.

Fig. 2.
Fig. 2.

Power spectral density of the dark noise count of a photon counting APD. 1/f noise is visible at and below frequencies in the mHz range. This averaged trace displays an α value of 1.6.

Fig. 3.
Fig. 3.

SNR versus integration time for a combination of white noise and 1/f noise with α values ranging from 0.4 to 1.6 (A1/f/Awhite=0.01). The slope of the SNR trace decreases with increasing α. For α>1, the SNR reaches a peak value and begins to decrease with increasing integration time. The existence of a peak SNR value implies that there is a limit on the smallest signal that the system is capable of measuring.

Fig. 4.
Fig. 4.

The location of the peak SNR value is dependent on the relative amplitudes of white and 1/f noise. As A1/f/Awhite is increased (for fixed α=1.6), the location of the maximum SNR, denoted by stars, moves towards shorter integration times. Curve fitting to the data shown in Fig. 1 we find A1/f/Awhite=3.03×10-4 for the photon counting APD described. The dashed curve shows an SNR trace corresponding to this value, with an optimal integration time of ~50 seconds.

Tables (1)

Tables Icon

Table 1. A comparison of the important equations in both Ref. [3] (Scenario A) and the current study (Scenario B).

Equations (29)

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x ( t ) = x signal + Δ x ( t ) ,
X noise ( τ ) = 0 τ Δ x ( t ) dt ,
X signal + noise ( τ ) = τ 2 τ ( x signal + Δ x ( t ) ) dt .
E ( X signal + noise ( τ ) X noise ( τ ) ) > σ ( X signal + noise ( τ ) X noise ( τ ) ) .
σ 2 ( X signal + noise X noise ) = E [ ( τ 2 τ Δ x ( t ) dt 0 τ Δ x ( t ) dt ) 2 ] .
σ 2 ( X signal + noise X noise ) = E [ ( 0 τ ( Δ x ( t + τ ) Δ x ( t ) ) dt ) 2 ] .
σ 2 ( X signal + noise X noise )
= E [ 0 τ 0 τ Δ x ( t 1 + τ ) Δ x ( t 2 + τ ) dt 1 dt 2 + 0 τ 0 τ Δ x ( t 1 ) Δ x ( t 2 ) dt 1 dt 2 2 0 τ 0 τ Δ x ( t 1 + τ ) Δ x ( t 2 ) dt 1 dt 2 ]
= 0 τ 0 τ E ( Δ x ( t 1 + τ ) Δ x ( t 2 + τ ) ) dt 1 dt 2 + 0 τ 0 τ E ( Δ x ( t 1 ) Δ x ( t 2 ) ) dt 1 dt 2 2 0 τ 0 τ E ( Δ x ( t 1 + τ ) Δ x ( t 2 ) ) dt 1 dt 2
= 2 0 τ 0 τ R ( t 1 t 2 ) dt 1 dt 2 2 0 τ 0 τ R ( t 1 t 2 + τ ) dt 1 dt 2
σ 2 ( X signal + noise X noise )
= 2 Re [ 0 0 τ 0 τ S ( f ) e i 2 π f ( t 1 t 2 ) dfdt 1 dt 2 0 0 τ 0 τ S ( f ) e i 2 π f ( t 1 t 2 + τ ) dfdt 1 dt 2 ]
= 2 Re [ 0 0 τ 0 τ S ( f ) e i 2 π f ( t 1 t 2 ) ( 1 e i 2 π f τ ) dfdt 1 dt 2 ]
= 2 Re [ 0 S ( f ) ( 1 e i 2 π f τ ) ( 0 τ e i 2 π f t 1 dt 1 ) ( 0 τ e i 2 π f t 2 dt 2 ) df ]
= 2 Re [ 0 S ( f ) ( 1 e i 2 π f τ ) 1 ( 2 π f ) 2 ( e i 2 π f τ 1 ) ( e i 2 π f τ 1 ) df ]
= 2 Re [ 0 S ( f ) ( 1 e i 2 π f τ ) 1 ( π f ) 2 sin 2 ( π f τ ) df ]
= 2 Re [ 0 S ( f ) e i π f τ ( e i π f τ e i π f τ ) 1 ( π f ) 2 sin 2 ( π f τ ) df ]
= 2 Re [ 0 S ( f ) e i π f τ ( 2 i ) 1 ( π f ) 2 sin 3 ( π f τ ) df ]
= 2 Re [ 0 S ( f ) ( cos ( π f τ ) + i sin ( π f τ ) ) ( 2 i ) 1 ( π f ) 2 sin 3 ( π f τ ) df ]
= 4 0 S ( f ) ( π f ) 2 sin 4 ( π f τ ) df
σ white 2 ( τ ) = 0 4 A white π 2 f 2 sin 4 ( π f τ ) df = A white τ ,
SNR white = x signal τ A white .
σ 1 f 2 ( τ ) = 0 4 A 1 f π 2 f 2 + α sin 4 ( π f τ ) df .
σ 1 f 2 ( τ ) = A 1 f C τ 1 + α ,
C = ( 2 2 + α 2 1 + 2 α ) π α 1 Γ ( 1 α ) sin ( α π 2 ) α > 0 , α Z ,
SNR 1 f = x signal τ A 1 f C τ 1 + α = x signal τ ( 1 α ) 2 A 1 f C .
σ total 2 ( τ ) = A white τ + A 1 f C τ 1 + α ,
SNR = x signal τ A white τ + A 1 f C τ 1 + α = x signal τ A white + A 1 f C τ α .
τ opt = [ A 1 f C A white ( 1 + 2 α ) ] 1 α ,

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