Abstract

We describe a robust shooting algorithm to model backward-pumped Raman amplifiers. This algorithm uses a continuation method and a Jacobi weight in conjunction with the shooting algorithm. We compare this algorithm to the commonly used relaxation algorithm. We find that the shooting algorithm is more flexible, in that it can be applied to amplifiers in which one fixes the gain, in contrast to the standard relaxation algorithm, which can be applied only to two-point boundary-value problems. However, it is less efficient when applied to two-point boundary-value problems, in that it requires more computer time.

© 2005 Optical Society of America

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  1. S. Namiki and Y. Emori, "Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelength-division-multiplexed high-power laser diodes," IEEE J. Sel. Top. Quantum Electron. 7, 3-16 (2001).
    [CrossRef]
  2. H. B. Keller, Numerical Methods for Two-Point Boundary-Value Problems (Ginn and Blaisdell, 1968).
  3. B. Min, W. J. Lee, and N. Park, "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photonics Technol. Lett. 12, 1486-1488 (2000).
    [CrossRef]
  4. H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijanona, "Pump interactions in a 100-nm bandwidth Raman amplifier," IEEE Photonics Technol. Lett. 11, 530-532 (1999).
    [CrossRef]
  5. V. Perlin and H. Winful, "Optimal design of flat-gain wide-band fiber Raman amplifiers," J. Lightwave Technol. 20, 250-254 (2002).
    [CrossRef]
  6. V. Perlin and H. Winful, "On distributed Raman amplification for ultrabroadband long-haul WDM system," J. Lightwave Technol. 20, 409-416 (2002).
    [CrossRef]
  7. J. Hu, B. S. Marks, and C. R. Menyuk, "Flat-gain fiber Raman amplifiers using equally spaced pumps," J. Lightwave Technol. 22, 1519-1522 (2004).
    [CrossRef]
  8. D. D. Morrison, J. D. Riley, and J. F. Zancanaro, "Multiple shooting method for two-point boundary value problems," Commun. ACM 5, 613-614 (1962).
    [CrossRef]
  9. X. Liu and B. Lee, "Effective shooting algorithm and its application to fiber amplifiers," Opt. Express 11, 1452-1461 (2003).
    [CrossRef] [PubMed]
  10. X. Liu and B. Lee, "A fast and stable method for Raman amplifier propagation equations," Opt. Express 11, 2163-2176 (2003).
    [CrossRef] [PubMed]
  11. X. Liu, "Powerful solution for simulating nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers," Opt. Express 12, 545-550 (2004).
    [CrossRef] [PubMed]
  12. S. A. E. Lewis, S. V. Chernikov, and J. R. Taylor, "Temperature-dependent gain and noise in fiber Raman amplifiers," Opt. Lett. 24, 1823-1825 (1999).
    [CrossRef]
  13. K. Rottwitt, M. Nissov, and F. Kerfoot, "Detailed analysis of Raman amplifiers for long-haul transmission," in Optical Fiber Communication Conference (Optical Society of America, 1998), paper TuG1.
  14. S. E. Miller and A. G. Chynoweth, Optical Fiber Telecommunications (Academic, 1979).
  15. J. Bromage, "Raman amplification for fiber communication systems," J. Lightwave Technol. 22, 79-93 (2002).
    [CrossRef]
  16. X. Liu, J. Chen, C. Lu, and X. Zhou, "Optimizing gain profile and noise performance for distributed fiber Raman amplifiers," Opt. Express 12, 6053-6066 (2004).
    [CrossRef] [PubMed]
  17. A. H. Hartog and M. P. Gold, "On the theory of backscattering in single-mode optical fibers," J. Lightwave Technol. LT-2, 76-82 (1984).
    [CrossRef]
  18. P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, "Rayleigh scattering limitations in distributed Raman pre-amplifiers," IEEE Photonics Technol. Lett. 10, 159-161 (1998).
    [CrossRef]
  19. W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 6th ed. (Wiley, 1997).
  20. G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt College Publishers, 1988).
  21. S. M. Roberts and J. S. Shipman, Two-Point Boundary Value Problems: Shooting Methods (Elsevier, 1972).
  22. G. E. Tudury, J. Hu, B. S. Marks, G. M. Carter, and C. R. Menyuk, "Spectral gain characteristics of an amplified hybrid Raman/EDFA 210 km link," in Conference on Laser and Electro Optics (Optical Society of America, 2003), paper CThM52.

2004 (3)

2003 (2)

2002 (3)

2001 (1)

S. Namiki and Y. Emori, "Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelength-division-multiplexed high-power laser diodes," IEEE J. Sel. Top. Quantum Electron. 7, 3-16 (2001).
[CrossRef]

2000 (1)

B. Min, W. J. Lee, and N. Park, "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photonics Technol. Lett. 12, 1486-1488 (2000).
[CrossRef]

1999 (2)

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijanona, "Pump interactions in a 100-nm bandwidth Raman amplifier," IEEE Photonics Technol. Lett. 11, 530-532 (1999).
[CrossRef]

S. A. E. Lewis, S. V. Chernikov, and J. R. Taylor, "Temperature-dependent gain and noise in fiber Raman amplifiers," Opt. Lett. 24, 1823-1825 (1999).
[CrossRef]

1998 (1)

P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, "Rayleigh scattering limitations in distributed Raman pre-amplifiers," IEEE Photonics Technol. Lett. 10, 159-161 (1998).
[CrossRef]

1984 (1)

A. H. Hartog and M. P. Gold, "On the theory of backscattering in single-mode optical fibers," J. Lightwave Technol. LT-2, 76-82 (1984).
[CrossRef]

1962 (1)

D. D. Morrison, J. D. Riley, and J. F. Zancanaro, "Multiple shooting method for two-point boundary value problems," Commun. ACM 5, 613-614 (1962).
[CrossRef]

Boyce, W. E.

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 6th ed. (Wiley, 1997).

Bromage, J.

Carter, G. M.

G. E. Tudury, J. Hu, B. S. Marks, G. M. Carter, and C. R. Menyuk, "Spectral gain characteristics of an amplified hybrid Raman/EDFA 210 km link," in Conference on Laser and Electro Optics (Optical Society of America, 2003), paper CThM52.

Chen, J.

Chernikov, S. V.

Chynoweth, A. G.

S. E. Miller and A. G. Chynoweth, Optical Fiber Telecommunications (Academic, 1979).

DeMarco, J. J.

P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, "Rayleigh scattering limitations in distributed Raman pre-amplifiers," IEEE Photonics Technol. Lett. 10, 159-161 (1998).
[CrossRef]

DiGiovanni, D. J.

P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, "Rayleigh scattering limitations in distributed Raman pre-amplifiers," IEEE Photonics Technol. Lett. 10, 159-161 (1998).
[CrossRef]

DiPrima, R. C.

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 6th ed. (Wiley, 1997).

Emori, Y.

S. Namiki and Y. Emori, "Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelength-division-multiplexed high-power laser diodes," IEEE J. Sel. Top. Quantum Electron. 7, 3-16 (2001).
[CrossRef]

Eskildsen, L.

P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, "Rayleigh scattering limitations in distributed Raman pre-amplifiers," IEEE Photonics Technol. Lett. 10, 159-161 (1998).
[CrossRef]

Gold, M. P.

A. H. Hartog and M. P. Gold, "On the theory of backscattering in single-mode optical fibers," J. Lightwave Technol. LT-2, 76-82 (1984).
[CrossRef]

Hansen, P. B.

P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, "Rayleigh scattering limitations in distributed Raman pre-amplifiers," IEEE Photonics Technol. Lett. 10, 159-161 (1998).
[CrossRef]

Hartog, A. H.

A. H. Hartog and M. P. Gold, "On the theory of backscattering in single-mode optical fibers," J. Lightwave Technol. LT-2, 76-82 (1984).
[CrossRef]

Hu, J.

J. Hu, B. S. Marks, and C. R. Menyuk, "Flat-gain fiber Raman amplifiers using equally spaced pumps," J. Lightwave Technol. 22, 1519-1522 (2004).
[CrossRef]

G. E. Tudury, J. Hu, B. S. Marks, G. M. Carter, and C. R. Menyuk, "Spectral gain characteristics of an amplified hybrid Raman/EDFA 210 km link," in Conference on Laser and Electro Optics (Optical Society of America, 2003), paper CThM52.

Judkins, J.

P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, "Rayleigh scattering limitations in distributed Raman pre-amplifiers," IEEE Photonics Technol. Lett. 10, 159-161 (1998).
[CrossRef]

Keller, H. B.

H. B. Keller, Numerical Methods for Two-Point Boundary-Value Problems (Ginn and Blaisdell, 1968).

Kerfoot, F.

K. Rottwitt, M. Nissov, and F. Kerfoot, "Detailed analysis of Raman amplifiers for long-haul transmission," in Optical Fiber Communication Conference (Optical Society of America, 1998), paper TuG1.

Kidorf, H.

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijanona, "Pump interactions in a 100-nm bandwidth Raman amplifier," IEEE Photonics Technol. Lett. 11, 530-532 (1999).
[CrossRef]

Lee, B.

Lee, W. J.

B. Min, W. J. Lee, and N. Park, "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photonics Technol. Lett. 12, 1486-1488 (2000).
[CrossRef]

Lewis, S. A.

Liu, X.

Lu, C.

Ma, M.

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijanona, "Pump interactions in a 100-nm bandwidth Raman amplifier," IEEE Photonics Technol. Lett. 11, 530-532 (1999).
[CrossRef]

Marks, B. S.

J. Hu, B. S. Marks, and C. R. Menyuk, "Flat-gain fiber Raman amplifiers using equally spaced pumps," J. Lightwave Technol. 22, 1519-1522 (2004).
[CrossRef]

G. E. Tudury, J. Hu, B. S. Marks, G. M. Carter, and C. R. Menyuk, "Spectral gain characteristics of an amplified hybrid Raman/EDFA 210 km link," in Conference on Laser and Electro Optics (Optical Society of America, 2003), paper CThM52.

Menyuk, C. R.

J. Hu, B. S. Marks, and C. R. Menyuk, "Flat-gain fiber Raman amplifiers using equally spaced pumps," J. Lightwave Technol. 22, 1519-1522 (2004).
[CrossRef]

G. E. Tudury, J. Hu, B. S. Marks, G. M. Carter, and C. R. Menyuk, "Spectral gain characteristics of an amplified hybrid Raman/EDFA 210 km link," in Conference on Laser and Electro Optics (Optical Society of America, 2003), paper CThM52.

Miller, S. E.

S. E. Miller and A. G. Chynoweth, Optical Fiber Telecommunications (Academic, 1979).

Min, B.

B. Min, W. J. Lee, and N. Park, "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photonics Technol. Lett. 12, 1486-1488 (2000).
[CrossRef]

Morrison, D. D.

D. D. Morrison, J. D. Riley, and J. F. Zancanaro, "Multiple shooting method for two-point boundary value problems," Commun. ACM 5, 613-614 (1962).
[CrossRef]

Namiki, S.

S. Namiki and Y. Emori, "Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelength-division-multiplexed high-power laser diodes," IEEE J. Sel. Top. Quantum Electron. 7, 3-16 (2001).
[CrossRef]

Nissov, M.

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijanona, "Pump interactions in a 100-nm bandwidth Raman amplifier," IEEE Photonics Technol. Lett. 11, 530-532 (1999).
[CrossRef]

K. Rottwitt, M. Nissov, and F. Kerfoot, "Detailed analysis of Raman amplifiers for long-haul transmission," in Optical Fiber Communication Conference (Optical Society of America, 1998), paper TuG1.

Park, N.

B. Min, W. J. Lee, and N. Park, "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photonics Technol. Lett. 12, 1486-1488 (2000).
[CrossRef]

Pedrazzani, R.

P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, "Rayleigh scattering limitations in distributed Raman pre-amplifiers," IEEE Photonics Technol. Lett. 10, 159-161 (1998).
[CrossRef]

Perlin, V.

Rabarijanona, E.

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijanona, "Pump interactions in a 100-nm bandwidth Raman amplifier," IEEE Photonics Technol. Lett. 11, 530-532 (1999).
[CrossRef]

Riley, J. D.

D. D. Morrison, J. D. Riley, and J. F. Zancanaro, "Multiple shooting method for two-point boundary value problems," Commun. ACM 5, 613-614 (1962).
[CrossRef]

Roberts, S. M.

S. M. Roberts and J. S. Shipman, Two-Point Boundary Value Problems: Shooting Methods (Elsevier, 1972).

Rottwitt, K.

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijanona, "Pump interactions in a 100-nm bandwidth Raman amplifier," IEEE Photonics Technol. Lett. 11, 530-532 (1999).
[CrossRef]

K. Rottwitt, M. Nissov, and F. Kerfoot, "Detailed analysis of Raman amplifiers for long-haul transmission," in Optical Fiber Communication Conference (Optical Society of America, 1998), paper TuG1.

Shipman, J. S.

S. M. Roberts and J. S. Shipman, Two-Point Boundary Value Problems: Shooting Methods (Elsevier, 1972).

Stentz, A. J.

P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, "Rayleigh scattering limitations in distributed Raman pre-amplifiers," IEEE Photonics Technol. Lett. 10, 159-161 (1998).
[CrossRef]

Strang, G.

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt College Publishers, 1988).

Strasser, T. A.

P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, "Rayleigh scattering limitations in distributed Raman pre-amplifiers," IEEE Photonics Technol. Lett. 10, 159-161 (1998).
[CrossRef]

Taylor, J. R.

Tudury, G. E.

G. E. Tudury, J. Hu, B. S. Marks, G. M. Carter, and C. R. Menyuk, "Spectral gain characteristics of an amplified hybrid Raman/EDFA 210 km link," in Conference on Laser and Electro Optics (Optical Society of America, 2003), paper CThM52.

Winful, H.

Zancanaro, J. F.

D. D. Morrison, J. D. Riley, and J. F. Zancanaro, "Multiple shooting method for two-point boundary value problems," Commun. ACM 5, 613-614 (1962).
[CrossRef]

Zhou, X.

Commun. ACM (1)

D. D. Morrison, J. D. Riley, and J. F. Zancanaro, "Multiple shooting method for two-point boundary value problems," Commun. ACM 5, 613-614 (1962).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

S. Namiki and Y. Emori, "Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelength-division-multiplexed high-power laser diodes," IEEE J. Sel. Top. Quantum Electron. 7, 3-16 (2001).
[CrossRef]

IEEE Photonics Technol. Lett. (3)

P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, "Rayleigh scattering limitations in distributed Raman pre-amplifiers," IEEE Photonics Technol. Lett. 10, 159-161 (1998).
[CrossRef]

B. Min, W. J. Lee, and N. Park, "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photonics Technol. Lett. 12, 1486-1488 (2000).
[CrossRef]

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijanona, "Pump interactions in a 100-nm bandwidth Raman amplifier," IEEE Photonics Technol. Lett. 11, 530-532 (1999).
[CrossRef]

J. Lightwave Technol. (5)

Opt. Express (4)

Opt. Lett. (1)

Other (7)

K. Rottwitt, M. Nissov, and F. Kerfoot, "Detailed analysis of Raman amplifiers for long-haul transmission," in Optical Fiber Communication Conference (Optical Society of America, 1998), paper TuG1.

S. E. Miller and A. G. Chynoweth, Optical Fiber Telecommunications (Academic, 1979).

H. B. Keller, Numerical Methods for Two-Point Boundary-Value Problems (Ginn and Blaisdell, 1968).

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 6th ed. (Wiley, 1997).

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt College Publishers, 1988).

S. M. Roberts and J. S. Shipman, Two-Point Boundary Value Problems: Shooting Methods (Elsevier, 1972).

G. E. Tudury, J. Hu, B. S. Marks, G. M. Carter, and C. R. Menyuk, "Spectral gain characteristics of an amplified hybrid Raman/EDFA 210 km link," in Conference on Laser and Electro Optics (Optical Society of America, 2003), paper CThM52.

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Figures (7)

Fig. 1
Fig. 1

(a) Raman gain spectrum g R ( Δ ν ) of a typical silica fiber at the pump wavelength λ 0 = 1 μ m . (b) Loss profile for a typical silica fiber.

Fig. 2
Fig. 2

Simulation time as a function of the Δ z step size for different integration types with the threshold c = 5 × 10 4 . Squares denote the trapezoidal rule, and triangles denote the power-average method.

Fig. 3
Fig. 3

Convergence parameter c versus iteration number for different Jacobi weights ω.

Fig. 4
Fig. 4

Block diagram of the continuation method.

Fig. 5
Fig. 5

Convergence of the shooting algorithm with the constraint that the averaged pump power equals 20 mW for four pump waves with 50 signal waves. The pump waves are equally spaced between 1437 and 1462 nm. The signal waves are equally spaced between 1530 and 1570 nm with an input signal power of 3 dBm per channel. (a) Average input pump power as a function of iteration number. (b) Input pump power at z = L as a function of iteration number.

Fig. 6
Fig. 6

Convergence of the shooting algorithm with the constraint that the averaged pump power equals 20 mW for eight pump waves and 100 signal waves. The pump waves are equally spaced between 1430 and 1490 nm. The signal waves are equally spaced between 1525 and 1605 nm with an input power of 3 dBm per channel. (a) Average input pump power as a function of iteration number. (b) Input pump power at z = L as a function of iteration number.

Fig. 7
Fig. 7

Power evolution of the backward-propagating pumps. (a) Result using the relaxation algorithm. (b) Result using the shooting algorithm. The threshold of c = 5 × 10 4 was the convergence criterion used to stop the iteration for both algorithms.

Equations (21)

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

± d P k d z = α k P k + j = 1 m + n g j k P j P k ,
± d P ASE , k d z = α k P ASE , k + j = 1 m + n g j k P j ( P ASE , k + h ν k Δ ν F j k ) ,
d P SRB , k d z = α k P SRB , k + j = 1 m + n g j k P j P SRB , k + K P k ,
d P DRB , k d z = α k P DRB , k + j = 1 m + n g j k P j P DRB , k + K P SRB , k ,
d P d z = h ( P ) = [ A + G ( P ) ] P ,
P k ( 0 ) = P k 0 k = 1 , 2 , , m ,
P k ( L ) = P k 0 k = m + 1 , m + 2 , , m + n ,
d P ¯ d z = h ( P ¯ ) , P ¯ ( 0 ) = P ¯ 0 .
f [ P ¯ ( 0 ) , P ¯ ( L ) ]
[ P 1 ( 0 ) P ¯ 1 ( 0 ) P 2 ( 0 ) P ¯ 2 ( 0 ) P m ( 0 ) P ¯ m ( 0 ) P m + 1 ( L ) P ¯ m + 1 ( L ) P m + n ( L ) P ¯ m + n ( L ) ] = [ 0 0 0 P m + 1 ( L ) P ¯ m + 1 ( L ) P m + n ( L ) P ¯ m + n ( L ) ]
Q i j = h i P j ( z ) , M i j = f i P j ( 0 ) , N i j = f i P j ( L ) ,
d ( P ¯ + δ P ) d z = h ( P ¯ + δ P ) , f [ P ¯ ( 0 ) + δ P ( 0 ) , P ¯ ( L ) + δ P ( L ) ] = 0 .
d δ P d z = Q δ P , M δ P ( 0 ) + N δ P ( L ) = f [ P ¯ ( 0 ) , P ¯ ( L ) ] .
d U d z = QU U ( 0 ) = I ,
d P ¯ d z = h ( P ¯ ) P ¯ ( 0 ) = P ¯ 0 ,
f [ P ¯ ( 0 ) , I ¯ ] [ P 1 ( 0 ) P ¯ 1 ( 0 ) P 2 ( 0 ) P ¯ 2 ( 0 ) P m ( 0 ) P ¯ m ( 0 ) I m + 1 I ¯ m + 1 I m + n I ¯ m + n ] = [ 0 0 0 I m + 1 I ¯ m + 1 I m + n I ¯ m + n ] ,
Q i j = h i ( z ) P j ( z ) , M i j = f i P j ( 0 ) , N i j = f i I j ,
d ( P ¯ + δ P ) d z = h ( P ¯ + δ P ) , f [ P ¯ ( 0 ) + δ P ( 0 ) , I ¯ + δ I ] = 0 ,
d δ P d z = Q δ P , M δ P ( 0 ) + N δ I = f [ P ¯ ( 0 ) , I ¯ ] .
c = { 1 n l i = 1 n j = 1 l [ P i ( r ) ( z j ) P i ( r 1 ) ( z j ) P i ( r ) ( z j ) ] 2 } 1 2 ,
P 0 , J ( r + 1 ) = ( 1 ω ) P 0 ( r ) + ω P 0 ( r + 1 )

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