Abstract

Gordon and Mollenauer [ Opt. Lett. 15, 1351 ( 1990)] have shown that the nonlinear Kerr effect limits the range of coherent communications systems using laser amplifiers. We show that parametric amplifiers avoid this limitation. Our method is novel in that we use quantum-optical master equations to model the communications systems. These are solved numerically for systems with either laser amplifiers or parametric amplifiers, with and without the nonlinear Kerr effect. Parametric amplifiers perform better because they preserve the signal-to-noise ratio and decrease the phase noise.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001 (1986).
    [CrossRef]
  2. Y. Yamamoto and T. Mukai, Opt. Quantum Electron. 21, S1 (1989).
    [CrossRef]
  3. J. P. Gordon and L. F. Mollenauer, Opt. Lett. 15, 1351 (1990).
    [CrossRef] [PubMed]
  4. R. Loudon, IEEE J. Quantum Electron. QE-21, 766 (1985).
    [CrossRef]
  5. R. E. Slusher and B. Yurke, IEEE J. Lightwave Technol. 8, 466 (1990).
    [CrossRef]
  6. K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd, Phys. Rev. A 42, 4102 (1990), and references therein.
    [CrossRef] [PubMed]
  7. I. R. Senitzky, Phys. Rev. 119, 670 (1960); Phys. Rev. 124, 642 (1961).
    [CrossRef]
  8. H. Haken, in Handbuch der Physik, L. Genzel, ed. (Springer-Verlag, Berlin, 1970), Vol. XXXV/2c.
  9. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
  10. H. J. Carmichael, Quantum Statistical Methods in Quantum Optics (Springer-Verlag, Berlin, to be published).
  11. P. D. Drummond and D. F. Walls, J. Phys. A 13, 725 (1980).
    [CrossRef]
  12. G. J. Milburn and C. A. Holmes, Phys. Rev. Lett. 56, 2237 (1986).
    [CrossRef] [PubMed]
  13. V. Perinova and A. Luks, Phys. Rev. A 41, 414 (1990).
    [CrossRef]
  14. R. J. Glauber, in Group Theoretical Methods in Theoretical Physics, M. A. Markov, V. I. Man’ko, and A. E. Shabad, eds. (Harwood Academic, New York, 1985), Vol. 1, p. 137.
  15. S. Tarzi, J. Phys. A 21, 3105 (1988).
    [CrossRef]
  16. D. F. Walls, Nature 306, 141 (1983), and references therein.
    [CrossRef]
  17. C. M. Savage and H. J. Carmichael, IEEE J. Quantum Electron. 24, 1495 (1988).
    [CrossRef]
  18. C. M. Savage, J. Mod. Opt. 37, 1711 (1990).
    [CrossRef]
  19. E. B. Davies, Quantum Theory of Open Systems (Academic, New York, 1976), Sec. 3.4.
  20. G. J. Milburn, Phys. Rev. A 33, 674 (1986).
    [CrossRef] [PubMed]
  21. S. Saito, T. Imai, T. Sugie, N. Ohakawa, Y. Ichihashi, and T. Ito, in Optical Fiber Communication, Vol. 1 of the OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper PD2.

1990 (5)

J. P. Gordon and L. F. Mollenauer, Opt. Lett. 15, 1351 (1990).
[CrossRef] [PubMed]

R. E. Slusher and B. Yurke, IEEE J. Lightwave Technol. 8, 466 (1990).
[CrossRef]

K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd, Phys. Rev. A 42, 4102 (1990), and references therein.
[CrossRef] [PubMed]

V. Perinova and A. Luks, Phys. Rev. A 41, 414 (1990).
[CrossRef]

C. M. Savage, J. Mod. Opt. 37, 1711 (1990).
[CrossRef]

1989 (1)

Y. Yamamoto and T. Mukai, Opt. Quantum Electron. 21, S1 (1989).
[CrossRef]

1988 (2)

S. Tarzi, J. Phys. A 21, 3105 (1988).
[CrossRef]

C. M. Savage and H. J. Carmichael, IEEE J. Quantum Electron. 24, 1495 (1988).
[CrossRef]

1986 (3)

G. J. Milburn, Phys. Rev. A 33, 674 (1986).
[CrossRef] [PubMed]

Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001 (1986).
[CrossRef]

G. J. Milburn and C. A. Holmes, Phys. Rev. Lett. 56, 2237 (1986).
[CrossRef] [PubMed]

1985 (1)

R. Loudon, IEEE J. Quantum Electron. QE-21, 766 (1985).
[CrossRef]

1983 (1)

D. F. Walls, Nature 306, 141 (1983), and references therein.
[CrossRef]

1980 (1)

P. D. Drummond and D. F. Walls, J. Phys. A 13, 725 (1980).
[CrossRef]

1960 (1)

I. R. Senitzky, Phys. Rev. 119, 670 (1960); Phys. Rev. 124, 642 (1961).
[CrossRef]

Blow, K. J.

K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd, Phys. Rev. A 42, 4102 (1990), and references therein.
[CrossRef] [PubMed]

Carmichael, H. J.

C. M. Savage and H. J. Carmichael, IEEE J. Quantum Electron. 24, 1495 (1988).
[CrossRef]

H. J. Carmichael, Quantum Statistical Methods in Quantum Optics (Springer-Verlag, Berlin, to be published).

Davies, E. B.

E. B. Davies, Quantum Theory of Open Systems (Academic, New York, 1976), Sec. 3.4.

Drummond, P. D.

P. D. Drummond and D. F. Walls, J. Phys. A 13, 725 (1980).
[CrossRef]

Glauber, R. J.

R. J. Glauber, in Group Theoretical Methods in Theoretical Physics, M. A. Markov, V. I. Man’ko, and A. E. Shabad, eds. (Harwood Academic, New York, 1985), Vol. 1, p. 137.

Gordon, J. P.

Haken, H.

H. Haken, in Handbuch der Physik, L. Genzel, ed. (Springer-Verlag, Berlin, 1970), Vol. XXXV/2c.

Haus, H. A.

Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001 (1986).
[CrossRef]

Holmes, C. A.

G. J. Milburn and C. A. Holmes, Phys. Rev. Lett. 56, 2237 (1986).
[CrossRef] [PubMed]

Ichihashi, Y.

S. Saito, T. Imai, T. Sugie, N. Ohakawa, Y. Ichihashi, and T. Ito, in Optical Fiber Communication, Vol. 1 of the OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper PD2.

Imai, T.

S. Saito, T. Imai, T. Sugie, N. Ohakawa, Y. Ichihashi, and T. Ito, in Optical Fiber Communication, Vol. 1 of the OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper PD2.

Ito, T.

S. Saito, T. Imai, T. Sugie, N. Ohakawa, Y. Ichihashi, and T. Ito, in Optical Fiber Communication, Vol. 1 of the OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper PD2.

Loudon, R.

K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd, Phys. Rev. A 42, 4102 (1990), and references therein.
[CrossRef] [PubMed]

R. Loudon, IEEE J. Quantum Electron. QE-21, 766 (1985).
[CrossRef]

Louisell, W. H.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).

Luks, A.

V. Perinova and A. Luks, Phys. Rev. A 41, 414 (1990).
[CrossRef]

Milburn, G. J.

G. J. Milburn and C. A. Holmes, Phys. Rev. Lett. 56, 2237 (1986).
[CrossRef] [PubMed]

G. J. Milburn, Phys. Rev. A 33, 674 (1986).
[CrossRef] [PubMed]

Mollenauer, L. F.

Mukai, T.

Y. Yamamoto and T. Mukai, Opt. Quantum Electron. 21, S1 (1989).
[CrossRef]

Ohakawa, N.

S. Saito, T. Imai, T. Sugie, N. Ohakawa, Y. Ichihashi, and T. Ito, in Optical Fiber Communication, Vol. 1 of the OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper PD2.

Perinova, V.

V. Perinova and A. Luks, Phys. Rev. A 41, 414 (1990).
[CrossRef]

Phoenix, S. J. D.

K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd, Phys. Rev. A 42, 4102 (1990), and references therein.
[CrossRef] [PubMed]

Saito, S.

S. Saito, T. Imai, T. Sugie, N. Ohakawa, Y. Ichihashi, and T. Ito, in Optical Fiber Communication, Vol. 1 of the OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper PD2.

Savage, C. M.

C. M. Savage, J. Mod. Opt. 37, 1711 (1990).
[CrossRef]

C. M. Savage and H. J. Carmichael, IEEE J. Quantum Electron. 24, 1495 (1988).
[CrossRef]

Senitzky, I. R.

I. R. Senitzky, Phys. Rev. 119, 670 (1960); Phys. Rev. 124, 642 (1961).
[CrossRef]

Shepherd, T. J.

K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd, Phys. Rev. A 42, 4102 (1990), and references therein.
[CrossRef] [PubMed]

Slusher, R. E.

R. E. Slusher and B. Yurke, IEEE J. Lightwave Technol. 8, 466 (1990).
[CrossRef]

Sugie, T.

S. Saito, T. Imai, T. Sugie, N. Ohakawa, Y. Ichihashi, and T. Ito, in Optical Fiber Communication, Vol. 1 of the OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper PD2.

Tarzi, S.

S. Tarzi, J. Phys. A 21, 3105 (1988).
[CrossRef]

Walls, D. F.

D. F. Walls, Nature 306, 141 (1983), and references therein.
[CrossRef]

P. D. Drummond and D. F. Walls, J. Phys. A 13, 725 (1980).
[CrossRef]

Yamamoto, Y.

Y. Yamamoto and T. Mukai, Opt. Quantum Electron. 21, S1 (1989).
[CrossRef]

Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001 (1986).
[CrossRef]

Yurke, B.

R. E. Slusher and B. Yurke, IEEE J. Lightwave Technol. 8, 466 (1990).
[CrossRef]

IEEE J. Lightwave Technol. (1)

R. E. Slusher and B. Yurke, IEEE J. Lightwave Technol. 8, 466 (1990).
[CrossRef]

IEEE J. Quantum Electron. (2)

R. Loudon, IEEE J. Quantum Electron. QE-21, 766 (1985).
[CrossRef]

C. M. Savage and H. J. Carmichael, IEEE J. Quantum Electron. 24, 1495 (1988).
[CrossRef]

J. Mod. Opt. (1)

C. M. Savage, J. Mod. Opt. 37, 1711 (1990).
[CrossRef]

J. Phys. A (2)

P. D. Drummond and D. F. Walls, J. Phys. A 13, 725 (1980).
[CrossRef]

S. Tarzi, J. Phys. A 21, 3105 (1988).
[CrossRef]

Nature (1)

D. F. Walls, Nature 306, 141 (1983), and references therein.
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

Y. Yamamoto and T. Mukai, Opt. Quantum Electron. 21, S1 (1989).
[CrossRef]

Phys. Rev. (1)

I. R. Senitzky, Phys. Rev. 119, 670 (1960); Phys. Rev. 124, 642 (1961).
[CrossRef]

Phys. Rev. A (3)

V. Perinova and A. Luks, Phys. Rev. A 41, 414 (1990).
[CrossRef]

K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd, Phys. Rev. A 42, 4102 (1990), and references therein.
[CrossRef] [PubMed]

G. J. Milburn, Phys. Rev. A 33, 674 (1986).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

G. J. Milburn and C. A. Holmes, Phys. Rev. Lett. 56, 2237 (1986).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001 (1986).
[CrossRef]

Other (6)

H. Haken, in Handbuch der Physik, L. Genzel, ed. (Springer-Verlag, Berlin, 1970), Vol. XXXV/2c.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).

H. J. Carmichael, Quantum Statistical Methods in Quantum Optics (Springer-Verlag, Berlin, to be published).

R. J. Glauber, in Group Theoretical Methods in Theoretical Physics, M. A. Markov, V. I. Man’ko, and A. E. Shabad, eds. (Harwood Academic, New York, 1985), Vol. 1, p. 137.

S. Saito, T. Imai, T. Sugie, N. Ohakawa, Y. Ichihashi, and T. Ito, in Optical Fiber Communication, Vol. 1 of the OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper PD2.

E. B. Davies, Quantum Theory of Open Systems (Academic, New York, 1976), Sec. 3.4.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Schematic diagram of our model communications system. The graph above the laser shows the transmitted coherent states on the amplitude phase plane. The state representing 0 has a positive amplitude, while the state representing 1 has a negative amplitude. The area of the circles represents quantum noise in the amplitudes.

Fig. 2
Fig. 2

Schematic diagrams of the effect of amplification on a quantum-mechanical state. The axes define the complex amplitude phase plane. The arrows represent the signal amplitude, and the circles represent the quantum noise of the state. The latter may be interpreted as contours of the Q function discussed in Section 4. (a) Linear laser amplifier. The total noise after amplification is the amplified noise of the input state (striped) plus extra noise due to amplified spontaneous emission (outer circle). (b) Degenerate parametric amplifier. The amplitude noise is amplified by the same factor as the amplitude itself, and no extra noise is added. The phase noise is attenuated.

Fig. 3
Fig. 3

Signal-to-noise ratio [Eq. (3)] versus propagation distance for the communications system depicted in Fig. 1. The inset plots the amplitude quadrature variance versus propagation distance. The solid curve is for the parametric-amplifier-based systems. The parametric amplifier systems with and without the nonlinear Kerr effect are identical on this scale. The short-dashed curve is for the laser-amplifier-based system without the nonlinear Kerr effect. The long-dashed curve is for the laser-amplifier-based system with the nonlinear Kerr effect. The amplifiers are spaced 5.4 km apart, and their gain was chosen to compensate exactly the amplitude loss after propagation. Other parameters: χ = 10−3 rad (photons/pulse)−1 km−1, κ = 0.2 km−1.

Fig. 4
Fig. 4

Q functions of the transmitted and received density operators for the laser-amplifier-based system. The phase plane axes αr and αi are, respectively, the real and imaginary parts of the coherent-state amplitude forming the argument of the Q function. The phase plane origin is marked with a dot. The transmitted state is a coherent state and has a Gaussian Q function, labeled (a). The received state is labeled (b). (i) No nonlinear Kerr effect. (ii) Nonlinear Kerr effect. Note how the Q function wraps around the origin and that higher amplitudes have larger nonlinear rotations. For convenience the same phase plane has been used for both the transmitted and received states, and hence the effective rotation due to the signal tracking by the phase-locked loop detection system is not shown.

Fig. 5
Fig. 5

Q function of the transmitted and received density operators for the parametric-amplifier-based system. The phase plane origin is marked by a dot. The transmitted coherent state is labeled (a). The received state is labeled (b). (i) No nonlinear Kerr effect. (ii) Nonlinear Kerr effect. From this viewing angle we are looking at the broad side of the distribution. It is narrower from a viewpoint rotated by 90°.

Fig. 6
Fig. 6

Signal-to-noise ratio (SNR) [Eq. (3)] versus mean photon number of the transmitted coherent state. The dashed curve is the laser amplifier system, and the solid curve is the parametric amplifier system. In the laser system the trade-off between the linear and nonlinear noises results in an optimal SNR, corresponding to a nonlinear signal phase shift of ~1.8 rad. In the parametric amplifier system the SNR is an increasing function of photon number over the range considered. The amplifiers are spaced ~0.3 km apart, and their gain was chosen to compensate exactly the amplitude loss after propagation. Other parameters: χ = 0.033 rad (photons/pulse)−1 km−1, κ = 2 km−1.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

a 1 = 1 2 ( a + a ) ,             a 2 = 1 2 i ( a - a ) .
Δ a 1 2 = a 1 2 - a 1 2 ,             Δ a 2 2 = a 2 2 - a 2 2 .
SNR = a 1 2 / Δ a 1 2 .
b 1 = G 1 1 / 2 a 1 + N 1 ,             b 2 = G 2 1 / 2 a 2 + N 2 ,
[ N 1 , N 2 ] = i 2 [ 1 - ( G 1 G 2 ) 1 / 2 ] .
Δ N 1 2 Δ N 2 2 1 16 [ 1 - ( G 1 G 2 ) 1 / 2 ] 2 .
b 1 = G 1 1 / 2 a 1 ,             Δ b 1 2 = G 1 Δ a 1 2 + Δ N 1 2 .
SNR out = ( 1 + G 1 - 1 Δ N 1 2 Δ a 1 2 ) - 1 SNR in .
G 1 = G 2 = G ,             Δ N 1 2 = Δ N 2 2 = 1 4 ( G - 1 ) .
SNR out = ( 1 + 1 - G - 1 4 Δ a 1 2 ) - 1 SNR in .
SNR out = 1 2 SNR in .
b 1 = G 1 / 2 a 1 ,             b 2 = G - 1 / 2 a 2 .
b 1 = G 1 / 2 a 1 ,             Δ b 1 2 = G Δ a 1 2 ; b 2 = G - 1 / 2 a 2 ,             Δ b 2 2 = G - 1 Δ a 2 2 .
E ( + ) ( z ) = - B B b k exp ( i k c z ) d k ,
F n ( z ) = sin [ π B ( z - n / B ) ] π ( B z - n )
a n = 1 2 π - B B b k F ˜ n ( k ) d k ,
ρ t = - i χ [ ( a a ) 2 , ρ ] + κ ( 2 a ρ a - a a ρ - ρ a a ) ,
ρ t = K ( 2 a ρ a - a a ρ - ρ a a ) ,
ρ t = - i E [ a 2 - a 2 , ρ ] ,
d ρ d t = L ρ ,
ρ ( t ) = [ I + ( t / k ) L ] k ρ ( 0 )             ( k ) .
α x = - 2 i χ α 2 α - κ α ,             α ( x = 0 ) = α 0 .
α X = - 2 i α 2 α - α ,             α ( X = 0 ) = ( χ κ ) 1 / 2 α 0 .

Metrics