Abstract

An important design problem for dense WDM optical networks is the placement of optical amplifiers along the transmission lines connecting optical add–drop multiplexers. We present a dynamic programming algorithm that chooses the minimum-cost amplifier placement subject to bounds on introduced nonlinear phase shift and noise. The algorithm is in current use and is effective over a wide range of design scenarios.

© 2009 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. New York, NY: Academic, 2001.
  2. A. Aho, J. Hopcroft, J. Ullman, The Design and Analysis of Computer Algorithms. Reading, MA: Addison-Wesley, 1974.
  3. M. Garey, D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. New York, NY: W. H. Freeman, 1979.
  4. D. S. Hochbaum, “Various notions of approximations: good, better, best, and more,” in Approximation Algorithms for NP-Hard Problems, D. S. Hochbaum, ed., Boston, MA: PWS, 1997.
  5. M. Alicherry, H. Nagesh, V. Poosala, “Optimal configuration of optical line system under various constraints,” in 28th European Conf. on Optical Communication, 2002. ECOC 2002, Copenhagen, Denmark, Sept. 8–12, 2002, vol. 3, pp. 1–2.
  6. C. Chekuri, W. Lee, L. Zhang, “Optical line system configuration via dynamic programming,” in Proc. of the Int. Network Optimization Conf. (INOC), Paris, France, October 2003, pp. 156–162.
  7. R.-J. Essiambre, W. Lee, P. Claisse, P. Winzer, E. Burrows, A. Chraplyvy, “Optimum power evolution in heterogeneous optical networks,” Bell Laboratories Technical Memorandum, July 2005.
  8. W. Lee, R.-J. Essiambre, P. Claisse, “An analytic solution for the optimum power evolution in optical networks,” Bell Laboratories Technical Memorandum, July 2005.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. New York, NY: Academic, 2001.

Aho, A.

A. Aho, J. Hopcroft, J. Ullman, The Design and Analysis of Computer Algorithms. Reading, MA: Addison-Wesley, 1974.

Alicherry, M.

M. Alicherry, H. Nagesh, V. Poosala, “Optimal configuration of optical line system under various constraints,” in 28th European Conf. on Optical Communication, 2002. ECOC 2002, Copenhagen, Denmark, Sept. 8–12, 2002, vol. 3, pp. 1–2.

Burrows, E.

R.-J. Essiambre, W. Lee, P. Claisse, P. Winzer, E. Burrows, A. Chraplyvy, “Optimum power evolution in heterogeneous optical networks,” Bell Laboratories Technical Memorandum, July 2005.

Chekuri, C.

C. Chekuri, W. Lee, L. Zhang, “Optical line system configuration via dynamic programming,” in Proc. of the Int. Network Optimization Conf. (INOC), Paris, France, October 2003, pp. 156–162.

Chraplyvy, A.

R.-J. Essiambre, W. Lee, P. Claisse, P. Winzer, E. Burrows, A. Chraplyvy, “Optimum power evolution in heterogeneous optical networks,” Bell Laboratories Technical Memorandum, July 2005.

Claisse, P.

R.-J. Essiambre, W. Lee, P. Claisse, P. Winzer, E. Burrows, A. Chraplyvy, “Optimum power evolution in heterogeneous optical networks,” Bell Laboratories Technical Memorandum, July 2005.

W. Lee, R.-J. Essiambre, P. Claisse, “An analytic solution for the optimum power evolution in optical networks,” Bell Laboratories Technical Memorandum, July 2005.

Essiambre, R.-J.

W. Lee, R.-J. Essiambre, P. Claisse, “An analytic solution for the optimum power evolution in optical networks,” Bell Laboratories Technical Memorandum, July 2005.

R.-J. Essiambre, W. Lee, P. Claisse, P. Winzer, E. Burrows, A. Chraplyvy, “Optimum power evolution in heterogeneous optical networks,” Bell Laboratories Technical Memorandum, July 2005.

Garey, M.

M. Garey, D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. New York, NY: W. H. Freeman, 1979.

Hochbaum, D. S.

D. S. Hochbaum, “Various notions of approximations: good, better, best, and more,” in Approximation Algorithms for NP-Hard Problems, D. S. Hochbaum, ed., Boston, MA: PWS, 1997.

Hopcroft, J.

A. Aho, J. Hopcroft, J. Ullman, The Design and Analysis of Computer Algorithms. Reading, MA: Addison-Wesley, 1974.

Johnson, D.

M. Garey, D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. New York, NY: W. H. Freeman, 1979.

Lee, W.

R.-J. Essiambre, W. Lee, P. Claisse, P. Winzer, E. Burrows, A. Chraplyvy, “Optimum power evolution in heterogeneous optical networks,” Bell Laboratories Technical Memorandum, July 2005.

C. Chekuri, W. Lee, L. Zhang, “Optical line system configuration via dynamic programming,” in Proc. of the Int. Network Optimization Conf. (INOC), Paris, France, October 2003, pp. 156–162.

W. Lee, R.-J. Essiambre, P. Claisse, “An analytic solution for the optimum power evolution in optical networks,” Bell Laboratories Technical Memorandum, July 2005.

Nagesh, H.

M. Alicherry, H. Nagesh, V. Poosala, “Optimal configuration of optical line system under various constraints,” in 28th European Conf. on Optical Communication, 2002. ECOC 2002, Copenhagen, Denmark, Sept. 8–12, 2002, vol. 3, pp. 1–2.

Poosala, V.

M. Alicherry, H. Nagesh, V. Poosala, “Optimal configuration of optical line system under various constraints,” in 28th European Conf. on Optical Communication, 2002. ECOC 2002, Copenhagen, Denmark, Sept. 8–12, 2002, vol. 3, pp. 1–2.

Ullman, J.

A. Aho, J. Hopcroft, J. Ullman, The Design and Analysis of Computer Algorithms. Reading, MA: Addison-Wesley, 1974.

Winzer, P.

R.-J. Essiambre, W. Lee, P. Claisse, P. Winzer, E. Burrows, A. Chraplyvy, “Optimum power evolution in heterogeneous optical networks,” Bell Laboratories Technical Memorandum, July 2005.

Zhang, L.

C. Chekuri, W. Lee, L. Zhang, “Optical line system configuration via dynamic programming,” in Proc. of the Int. Network Optimization Conf. (INOC), Paris, France, October 2003, pp. 156–162.

Other (8)

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. New York, NY: Academic, 2001.

A. Aho, J. Hopcroft, J. Ullman, The Design and Analysis of Computer Algorithms. Reading, MA: Addison-Wesley, 1974.

M. Garey, D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. New York, NY: W. H. Freeman, 1979.

D. S. Hochbaum, “Various notions of approximations: good, better, best, and more,” in Approximation Algorithms for NP-Hard Problems, D. S. Hochbaum, ed., Boston, MA: PWS, 1997.

M. Alicherry, H. Nagesh, V. Poosala, “Optimal configuration of optical line system under various constraints,” in 28th European Conf. on Optical Communication, 2002. ECOC 2002, Copenhagen, Denmark, Sept. 8–12, 2002, vol. 3, pp. 1–2.

C. Chekuri, W. Lee, L. Zhang, “Optical line system configuration via dynamic programming,” in Proc. of the Int. Network Optimization Conf. (INOC), Paris, France, October 2003, pp. 156–162.

R.-J. Essiambre, W. Lee, P. Claisse, P. Winzer, E. Burrows, A. Chraplyvy, “Optimum power evolution in heterogeneous optical networks,” Bell Laboratories Technical Memorandum, July 2005.

W. Lee, R.-J. Essiambre, P. Claisse, “An analytic solution for the optimum power evolution in optical networks,” Bell Laboratories Technical Memorandum, July 2005.

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Figures (11)

Fig. 1
Fig. 1

Trade-off between noise and nonlinearities. Each dot represents a configuration (i.e., choice of optical amplifiers) for the transmission line in Fig. 11 below. Larger dots are more expensive configurations.

Fig. 2
Fig. 2

Unidirectional transmission line; triangles a 0 , , a n are amplifiers; s 1 , , s n are spans.

Fig. 3
Fig. 3

Noise-to-signal ratio for amplifiers L and H.

Fig. 4
Fig. 4

Reach as a function of amplifier spacing, considering noise-to-signal ratio and nonlinear phase shift in isolation.

Fig. 5
Fig. 5

Top, instance of span engineering; middle and bottom, eastbound and westbound configurations. In each configuration, locations l 1 and l 2 have glassthroughs, so segments g 1 , g 2 , and g 3 together form a span.

Fig. 6
Fig. 6

Dynamic programming algorithm for unidirectional span engineering. C j and D j are sets of configurations. Subroutine prune is described in the text.

Fig. 7
Fig. 7

Dynamic programming algorithm for bidirectional span engineering.

Fig. 8
Fig. 8

Heuristic algorithm bse for bidirectional span engineering; prune is discussed in the text.

Fig. 9
Fig. 9

Average cost of ten instances as a function of sample size and deletion rules.

Fig. 10
Fig. 10

This instance consists of six 20 km segments. The top configuration results from a liberal noise-to-signal target, the middle from a tight target, and the bottom from a moderate target. L and H indicate low- and high-power amplifiers (Section 2).

Fig. 11
Fig. 11

This instance consists of fifty 10 km segments. Five eastbound configurations are shown, with allowed margins decreasing from top to bottom. Westbound configurations are very similar.

Equations (6)

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g i = P i ( P i 1 T i ) .
σ i = B h ν N ( g i ) g i , i = 0 , 1 , , n ,
φ i = γ i 0 l i P i ( z ) d z = γ i α i P i ( 1 T i ) , i = 1 , , n ,
σ = i = 0 n σ i , φ = i = 1 n φ i .
[ ( i 0 , a 0 ) , ( i 1 , a 1 ) , , ( i t , a t ) ] ,
[ ( i 0 , a 0 , a 0 ) , ( i 1 , a 1 , a 1 ) , , ( i t , a t , a t ) ] ,