Abstract

For optical channels transmitting signals of weak power, several information criteria, such as the channel capacity, the cutoff rate and the rate distortion, are analyzed. These performances are calculated for various formats of modulation and for systems that include optical amplifiers and utilize photon-counting receivers. Comparison with the results that are derived from the Gaussian approximation of the intensity distribution is made.

© 2001 Optical Society of America

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References

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  1. N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7, 1071–1082 (1989).
    [CrossRef]
  2. D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. 8, 1816–1823 (1990).
    [CrossRef]
  3. P. A. Humblet, M. Azizoglu, “On the bit error rate of lightwave systems with optical amplifier,” J. Lightwave Technol. 9, 1576–1582 (1991).
    [CrossRef]
  4. B. Chan, J. Conradi, “On the non-Gaussian noise in erbium-doped fiber amplifiers,” J. Lightwave Technol. 15, 680–687 (1997).
    [CrossRef]
  5. C. W. Gardiner, Quantum Noise, Vol. 56 of Springer Series in Synergetics (Springer-Verlag, Heidelberg, Germany, 1992).
  6. C. Bendjaballah, G. Oliver, “Detection of coherent signal after nonlinear amplification,” IEEE Trans. Aerosp. Electron. Syst. AES-17, 620–625 (1981).
    [CrossRef]
  7. J. R. Pierce, E. C. Posner, E. R. Rodemich, “The capacity of the photon counting channel,” IEEE Trans. Inf. Theory IT-27, 61–77 (1981).
    [CrossRef]
  8. J. L. Massey, “Capacity, cutoff rate, and coding for a direct detection optical channel,” IEEE Trans. Commun. COM-29, 1615–1621 (1981).
    [CrossRef]
  9. C. Bendjaballah, Introduction to Photon Communication, Vol. 29 of Springer Lecture Notes in Physics (Springer-Verlag, Heidelberg, Germany, 1995), p. 168.
  10. M. Charbit, C. Bendjaballah, “Efficient capacity for a PPM weak noisy photon-counting channel,” Opt. Quantum Electron. 18, 49–55 (1986).
    [CrossRef]
  11. M. Charbit, C. Bendjaballah, “Cutoff rate for PPM noisy photon counting channel with soft decision,” IEEE Trans. Commun. COM-35, 122–125 (1987).
    [CrossRef]
  12. M. R. Bell, S. M. Tseng, “Capacity of the low-rate direct-detection optical pulse-position-modulation channel in the presence of noise photons,” Appl. Opt. 39, 1776–1782 (2000).
    [CrossRef]
  13. C. Bendjaballah, “Information rate for photon communication,” J. Opt. Commun. 19, 169–172 (1998).
    [CrossRef]
  14. C. Bendjaballah, “An approach of the rate distortion for photon communication,” J. Opt. Commun. 20, 50–53 (1999).
    [CrossRef]
  15. C. Bendjaballah, “Information rates in optical channels,” Opt. Commun. 17, 55–58 (1976).
    [CrossRef]
  16. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), pp. 1037–1039.
  17. J. Per̆ina, Coherence of Light (Van Nostrand Reinhold, London, 1971), p. 282.
  18. K. Kiasaleh, “Turbo-coded optical PPM communication systems,” J. Lightwave Technol. 16, 18–26 (1998).
    [CrossRef]
  19. T. Berger, Rate Distortion Theory: A Mathematical Basis for Data Compression (Prentice-Hall, New York, 1971).
  20. C. N. Georghiades, “Modulation and coding for throughput-efficient optical systems,” IEEE Trans. Inf. Theory IT-40, 1313–1326 (1994).
    [CrossRef]
  21. R. Jodoin, L. Mandel, “Information rate in an optical communication channel,” J. Opt. Soc. Am. 61, 191–198 (1971).
    [CrossRef]

2000

1999

C. Bendjaballah, “An approach of the rate distortion for photon communication,” J. Opt. Commun. 20, 50–53 (1999).
[CrossRef]

1998

C. Bendjaballah, “Information rate for photon communication,” J. Opt. Commun. 19, 169–172 (1998).
[CrossRef]

K. Kiasaleh, “Turbo-coded optical PPM communication systems,” J. Lightwave Technol. 16, 18–26 (1998).
[CrossRef]

1997

B. Chan, J. Conradi, “On the non-Gaussian noise in erbium-doped fiber amplifiers,” J. Lightwave Technol. 15, 680–687 (1997).
[CrossRef]

1994

C. N. Georghiades, “Modulation and coding for throughput-efficient optical systems,” IEEE Trans. Inf. Theory IT-40, 1313–1326 (1994).
[CrossRef]

1991

P. A. Humblet, M. Azizoglu, “On the bit error rate of lightwave systems with optical amplifier,” J. Lightwave Technol. 9, 1576–1582 (1991).
[CrossRef]

1990

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. 8, 1816–1823 (1990).
[CrossRef]

1989

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7, 1071–1082 (1989).
[CrossRef]

1987

M. Charbit, C. Bendjaballah, “Cutoff rate for PPM noisy photon counting channel with soft decision,” IEEE Trans. Commun. COM-35, 122–125 (1987).
[CrossRef]

1986

M. Charbit, C. Bendjaballah, “Efficient capacity for a PPM weak noisy photon-counting channel,” Opt. Quantum Electron. 18, 49–55 (1986).
[CrossRef]

1981

C. Bendjaballah, G. Oliver, “Detection of coherent signal after nonlinear amplification,” IEEE Trans. Aerosp. Electron. Syst. AES-17, 620–625 (1981).
[CrossRef]

J. R. Pierce, E. C. Posner, E. R. Rodemich, “The capacity of the photon counting channel,” IEEE Trans. Inf. Theory IT-27, 61–77 (1981).
[CrossRef]

J. L. Massey, “Capacity, cutoff rate, and coding for a direct detection optical channel,” IEEE Trans. Commun. COM-29, 1615–1621 (1981).
[CrossRef]

1976

C. Bendjaballah, “Information rates in optical channels,” Opt. Commun. 17, 55–58 (1976).
[CrossRef]

1971

Azizoglu, M.

P. A. Humblet, M. Azizoglu, “On the bit error rate of lightwave systems with optical amplifier,” J. Lightwave Technol. 9, 1576–1582 (1991).
[CrossRef]

Bell, M. R.

Bendjaballah, C.

C. Bendjaballah, “An approach of the rate distortion for photon communication,” J. Opt. Commun. 20, 50–53 (1999).
[CrossRef]

C. Bendjaballah, “Information rate for photon communication,” J. Opt. Commun. 19, 169–172 (1998).
[CrossRef]

M. Charbit, C. Bendjaballah, “Cutoff rate for PPM noisy photon counting channel with soft decision,” IEEE Trans. Commun. COM-35, 122–125 (1987).
[CrossRef]

M. Charbit, C. Bendjaballah, “Efficient capacity for a PPM weak noisy photon-counting channel,” Opt. Quantum Electron. 18, 49–55 (1986).
[CrossRef]

C. Bendjaballah, G. Oliver, “Detection of coherent signal after nonlinear amplification,” IEEE Trans. Aerosp. Electron. Syst. AES-17, 620–625 (1981).
[CrossRef]

C. Bendjaballah, “Information rates in optical channels,” Opt. Commun. 17, 55–58 (1976).
[CrossRef]

C. Bendjaballah, Introduction to Photon Communication, Vol. 29 of Springer Lecture Notes in Physics (Springer-Verlag, Heidelberg, Germany, 1995), p. 168.

Berger, T.

T. Berger, Rate Distortion Theory: A Mathematical Basis for Data Compression (Prentice-Hall, New York, 1971).

Chan, B.

B. Chan, J. Conradi, “On the non-Gaussian noise in erbium-doped fiber amplifiers,” J. Lightwave Technol. 15, 680–687 (1997).
[CrossRef]

Charbit, M.

M. Charbit, C. Bendjaballah, “Cutoff rate for PPM noisy photon counting channel with soft decision,” IEEE Trans. Commun. COM-35, 122–125 (1987).
[CrossRef]

M. Charbit, C. Bendjaballah, “Efficient capacity for a PPM weak noisy photon-counting channel,” Opt. Quantum Electron. 18, 49–55 (1986).
[CrossRef]

Conradi, J.

B. Chan, J. Conradi, “On the non-Gaussian noise in erbium-doped fiber amplifiers,” J. Lightwave Technol. 15, 680–687 (1997).
[CrossRef]

Gardiner, C. W.

C. W. Gardiner, Quantum Noise, Vol. 56 of Springer Series in Synergetics (Springer-Verlag, Heidelberg, Germany, 1992).

Georghiades, C. N.

C. N. Georghiades, “Modulation and coding for throughput-efficient optical systems,” IEEE Trans. Inf. Theory IT-40, 1313–1326 (1994).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), pp. 1037–1039.

Humblet, P. A.

P. A. Humblet, M. Azizoglu, “On the bit error rate of lightwave systems with optical amplifier,” J. Lightwave Technol. 9, 1576–1582 (1991).
[CrossRef]

Jodoin, R.

Kiasaleh, K.

Mandel, L.

Marcuse, D.

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. 8, 1816–1823 (1990).
[CrossRef]

Massey, J. L.

J. L. Massey, “Capacity, cutoff rate, and coding for a direct detection optical channel,” IEEE Trans. Commun. COM-29, 1615–1621 (1981).
[CrossRef]

Oliver, G.

C. Bendjaballah, G. Oliver, “Detection of coherent signal after nonlinear amplification,” IEEE Trans. Aerosp. Electron. Syst. AES-17, 620–625 (1981).
[CrossRef]

Olsson, N. A.

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7, 1071–1082 (1989).
[CrossRef]

Per?ina, J.

J. Per̆ina, Coherence of Light (Van Nostrand Reinhold, London, 1971), p. 282.

Pierce, J. R.

J. R. Pierce, E. C. Posner, E. R. Rodemich, “The capacity of the photon counting channel,” IEEE Trans. Inf. Theory IT-27, 61–77 (1981).
[CrossRef]

Posner, E. C.

J. R. Pierce, E. C. Posner, E. R. Rodemich, “The capacity of the photon counting channel,” IEEE Trans. Inf. Theory IT-27, 61–77 (1981).
[CrossRef]

Rodemich, E. R.

J. R. Pierce, E. C. Posner, E. R. Rodemich, “The capacity of the photon counting channel,” IEEE Trans. Inf. Theory IT-27, 61–77 (1981).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), pp. 1037–1039.

Tseng, S. M.

Appl. Opt.

IEEE Trans. Aerosp. Electron. Syst.

C. Bendjaballah, G. Oliver, “Detection of coherent signal after nonlinear amplification,” IEEE Trans. Aerosp. Electron. Syst. AES-17, 620–625 (1981).
[CrossRef]

IEEE Trans. Commun.

J. L. Massey, “Capacity, cutoff rate, and coding for a direct detection optical channel,” IEEE Trans. Commun. COM-29, 1615–1621 (1981).
[CrossRef]

M. Charbit, C. Bendjaballah, “Cutoff rate for PPM noisy photon counting channel with soft decision,” IEEE Trans. Commun. COM-35, 122–125 (1987).
[CrossRef]

IEEE Trans. Inf. Theory

C. N. Georghiades, “Modulation and coding for throughput-efficient optical systems,” IEEE Trans. Inf. Theory IT-40, 1313–1326 (1994).
[CrossRef]

J. R. Pierce, E. C. Posner, E. R. Rodemich, “The capacity of the photon counting channel,” IEEE Trans. Inf. Theory IT-27, 61–77 (1981).
[CrossRef]

J. Lightwave Technol.

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7, 1071–1082 (1989).
[CrossRef]

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. 8, 1816–1823 (1990).
[CrossRef]

P. A. Humblet, M. Azizoglu, “On the bit error rate of lightwave systems with optical amplifier,” J. Lightwave Technol. 9, 1576–1582 (1991).
[CrossRef]

B. Chan, J. Conradi, “On the non-Gaussian noise in erbium-doped fiber amplifiers,” J. Lightwave Technol. 15, 680–687 (1997).
[CrossRef]

K. Kiasaleh, “Turbo-coded optical PPM communication systems,” J. Lightwave Technol. 16, 18–26 (1998).
[CrossRef]

J. Opt. Commun.

C. Bendjaballah, “Information rate for photon communication,” J. Opt. Commun. 19, 169–172 (1998).
[CrossRef]

C. Bendjaballah, “An approach of the rate distortion for photon communication,” J. Opt. Commun. 20, 50–53 (1999).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

C. Bendjaballah, “Information rates in optical channels,” Opt. Commun. 17, 55–58 (1976).
[CrossRef]

Opt. Quantum Electron.

M. Charbit, C. Bendjaballah, “Efficient capacity for a PPM weak noisy photon-counting channel,” Opt. Quantum Electron. 18, 49–55 (1986).
[CrossRef]

Other

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), pp. 1037–1039.

J. Per̆ina, Coherence of Light (Van Nostrand Reinhold, London, 1971), p. 282.

T. Berger, Rate Distortion Theory: A Mathematical Basis for Data Compression (Prentice-Hall, New York, 1971).

C. W. Gardiner, Quantum Noise, Vol. 56 of Springer Series in Synergetics (Springer-Verlag, Heidelberg, Germany, 1992).

C. Bendjaballah, Introduction to Photon Communication, Vol. 29 of Springer Lecture Notes in Physics (Springer-Verlag, Heidelberg, Germany, 1995), p. 168.

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Figures (5)

Fig. 1
Fig. 1

Plot of the optimized length L c * of the PPM format versus the average photon number constrained capacity Γ* with M = 1, 3, 5, 7, 10 and N 0 = 0.02, the average value of the noise photon number. In dashed curve (a), MN 0= 0.02 and in dashed curve (b), MN 0 = 0.2; the curves correspond to the cases in which the noise and signal distributions are Poisson.

Fig. 2
Fig. 2

Plot of the optimized length L r * of the PPM format versus the average photon number constrained cutoff rate ρ* with M = 1, 3, 5, 7, 10 and N 0 = 0.02, the average value of the noise photon number. In dashed curve (a), MN 0= 0.02 and in dashed curve (b), MN 0 = 0.2; the curves correspond to the cases in which the noise and signal distributions are Poisson.

Fig. 3
Fig. 3

Comparison of the optimized length L* of the PPM format versus η, which is either the average photon number constrained capacity Γ* for curve C or the average photon number constrained cutoff rate ρ* for curve R 0, with M = 5 and the average value of the noise photon number N 0 = 0.02.

Fig. 4
Fig. 4

Information rate R N for the amplified coherent-source intensity modulated versus d, the average value of absolute difference fidelity criterion, using the photon number decoding for different values of M = 1, 5, 10, 20. The R N for the coherent source with the average value of the noise photon number N 0= 0 is plotted in a dashed curve. Curve R l is the Shannon lower bound.

Fig. 5
Fig. 5

Continuous modulation of the signal intensity. The average number of signal photons is N s = 5, and the average number of noise photons is N 0 = 0.05. Comparison of the channel capacity C and the cutoff rate R 0 versus M, the number of independent modes, with the results of the Gaussian approximation that are plotted in dashed curves.

Equations (39)

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e0ϑ=k=1M ck expiωkϑ.
p0J=1J0MJM-1M-1!exp-JJ0,
p1J=1J0JJsM-12 exp-J+JsJ0IM-12JJsJ0,
pin=1n!0dJ0T Jϑdϑn×exp-0T JϑdϑpiJ,  i=0, 1.
p0n=n+M-1!n!M-1!N0n1+N0n+M,  N0=J0T.
p1n=1n+M-1!N0n1+N0n+M×exp-Ns1+N0LnM-1-NsN01+N0.
iΔM=jl=1M1nl+μl! Lnlμlzl=1j+ΠM! LjΠMl=1M zl,ii L0M-1z=M-1!,   z,iii LnM-10=n+M-1!2n!M-1!,iv z,  2LnM-1-zn+M-1!n! zn,
p1nPn, n1n1nn!exp-n1,
p1J1/σJ2πexp-J2/2σJ2σJ0 δJ-J1
pn1, n2,, nL|l=p1nli=1,ili=L p0ni.
Γ=CβL=1βL1-logL1-+p logp+L-1q logq,
Ns=βL,=n=0 p1np0nL-1,p=n=1 p1nk=0n-1 p0kL-1,q=1-p-L-1.
log Lc*γ1+NsM1-λ/λ)-1γ1+λNs+Ns2M1-λ-1λ1expNs-1expNs-Ns-1,
Γ*1-γNslog Lc* N0=01-exp-Ns2Ns exp-NsexpNs-1-Ns.
Lc* Ns0expΓ*1+MN0.
R0=-logminPnΩNl=1L πlpn|l2,
ρ=R0sβL=1βLlog L-log1+L-1×n=0p0np1n2,
1+γL-11-γlogL1+L-1γ=1+γNsLL-11-γ1-LNsN0NsL1-LNsN0,
Lr*2Ns1+2MN0,  ρ*1-Ns+Ns24.
Lr* Ns021-ρ*1+2MN0,
INa, Ns=n01dxPx, nlogqn|x01dξPξ, n,
C=maxpax IN maxa IN,
R0maxa-logn01dxpaxqn|x2.
INa, NsNs0a=0Ns224,Ns0a0a3 Ns+faNs224,Nsa 0,
IN0, NsINaam, Ns,
IN0, Ns>INa>am, Ns,
Cmaxπ1π1Hp1+π0Hp0-HP,
R0R0*=-log(1-2π0π11-exp×-n1-n02).
R0=log21+n=0p1np0n1/2.
p1np0n1-Ns1-ν2+Ns21-ν28×n p0n1+znM+z2n-1n2MM+11/2,
R0Ns0Ns21-ν216MνNs216MN0,
INa, Ns=n01 qn|xpax×logqn|x01 qn|ξpaξdξdx,
dNa, Ns=n01 cx, nqn|xpaxdx,
cx, n=|x-xˆn|,
Qx|n=qn|xpaxrn,  rn=01qn|ξpa(ξ)dξ,
dNa, Ns=n01x-nNs+aqn|xpaxdx.
gs; x=exp-scxΩXexp-sczdz=exp-s|x|-exp-s|z|dz.
Rld=12logπσX22ed2  for 0dπσX22e1/2,
π24e1/2 exp-Cdπ24e1/2 exp-R0,

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