Abstract

Four-wave mixing (FWM) is one the limiting factors for existing and future wavelength division multiplexed optical networks. A semianalytical method based on Monte Carlo and Extreme Value theory is proposed and applied to study the influence of the FWM noise on the performance of WDM systems. The statistical behavior of the FWM noise is investigated while the Bit-Error rate is calculated for various combinations of the design parameters and for both single and multiple span WDM systems. The semianalytical method is also compared to the Multicanonical Monte Carlo (MCMC) method showing the same efficiency and accuracy with the former providing however the advantage of deriving closed-form approximations for the cumulative distribution functions of the photocurrents in the mark and the space state and the BER.

© 2013 Optical Society of America

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References

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  1. B. Mukherjee, Optical WDM Networks (Springer, 2006).
  2. C.-H. Lee, S.-M. Lee, K.-M. Choi, J.-H. Moon, S.-G. Mun, K.-T. Jeong, J. H. Kim, and B. Kim, “WDM-PON experiences in Korea [Invited],” J. Opt. Netw.6(5), 451–464 (2007).
    [CrossRef]
  3. C. H. Lee, W. V. Sorin, and B. Y. Kim, “Fiber to the Home Using a PON Infrastructure,” Lightwave Technology, Journalism24, 4568–4583 (2006).
  4. I. Neokosmidis, T. Kamalakis, A. Chipouras, and T. Sphicopoulos, “New techniques for the suppression of the four-wave mixing-induced distortion in nonzero dispersion fiber WDM systems,” J. Lightwave Technol.23, 1137–1144 (2005).
  5. M. Eiselt, “Limits on WDM Systems Due to Four-Wave Mixing: A Statistical Approach,” J. Lightwave Technol.17(11), 2261–2267 (1999).
    [CrossRef]
  6. I. Neokosmidis, T. Kamalakis, A. Chipouras, and T. Sphicopoulos, “Evaluation by Monte Carlo Simulations of the Power Limits and Bit-Error Rate Degradation in Wavelength-Division Multiplexing Networks Caused by Four-Wave Mixing,” Appl. Opt.43(26), 5023–5032 (2004).
    [CrossRef] [PubMed]
  7. I. Neokosmidis, T. Kamalakis, A. Chipouras, and T. Sphicopoulos, “Estimation of the four-wave mixing noise probability-density function by the multicanonical Monte Carlo method,” Opt. Lett.30(1), 11–13 (2005).
    [CrossRef] [PubMed]
  8. R. Holzlöhner and C. R. Menyuk, “Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems,” Opt. Lett.28(20), 1894–1896 (2003).
    [CrossRef] [PubMed]
  9. E. Castillo, A. S. Hadi, N. Balakrishman, and J. M. Sarabia, Extreme value and related models with applications in engineering and science (John Wiley and Sons, 2005).
  10. S. Savory, F. Payne, and A. Hadjifotiou, “Estimating Outages Due to Polarization Mode Dispersion Using Extreme Value Statistics,” J. Lightwave Technol.24(11), 3907–3913 (2006).
    [CrossRef]
  11. K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” Lightwave Technology, Journalism12, 1423–1439 (1994).
  12. K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett.17(11), 801–803 (1992).
    [CrossRef] [PubMed]
  13. G. H. Einarsson, Principles of Lightwave Communications (John Wiley & Sons, 1996).

2007 (1)

2006 (2)

C. H. Lee, W. V. Sorin, and B. Y. Kim, “Fiber to the Home Using a PON Infrastructure,” Lightwave Technology, Journalism24, 4568–4583 (2006).

S. Savory, F. Payne, and A. Hadjifotiou, “Estimating Outages Due to Polarization Mode Dispersion Using Extreme Value Statistics,” J. Lightwave Technol.24(11), 3907–3913 (2006).
[CrossRef]

2005 (2)

2004 (1)

2003 (1)

1999 (1)

1994 (1)

K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” Lightwave Technology, Journalism12, 1423–1439 (1994).

1992 (1)

Chipouras, A.

Choi, K.-M.

Eiselt, M.

Hadjifotiou, A.

Holzlöhner, R.

Inoue, K.

K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” Lightwave Technology, Journalism12, 1423–1439 (1994).

K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett.17(11), 801–803 (1992).
[CrossRef] [PubMed]

Jeong, K.-T.

Kamalakis, T.

Kim, B.

Kim, B. Y.

C. H. Lee, W. V. Sorin, and B. Y. Kim, “Fiber to the Home Using a PON Infrastructure,” Lightwave Technology, Journalism24, 4568–4583 (2006).

Kim, J. H.

Lee, C. H.

C. H. Lee, W. V. Sorin, and B. Y. Kim, “Fiber to the Home Using a PON Infrastructure,” Lightwave Technology, Journalism24, 4568–4583 (2006).

Lee, C.-H.

Lee, S.-M.

Menyuk, C. R.

Moon, J.-H.

Mun, S.-G.

Nakanishi, K.

K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” Lightwave Technology, Journalism12, 1423–1439 (1994).

Neokosmidis, I.

Oda, K.

K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” Lightwave Technology, Journalism12, 1423–1439 (1994).

Payne, F.

Savory, S.

Sorin, W. V.

C. H. Lee, W. V. Sorin, and B. Y. Kim, “Fiber to the Home Using a PON Infrastructure,” Lightwave Technology, Journalism24, 4568–4583 (2006).

Sphicopoulos, T.

Toba, H.

K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” Lightwave Technology, Journalism12, 1423–1439 (1994).

Appl. Opt. (1)

J. Lightwave Technol. (3)

J. Opt. Netw. (1)

Lightwave Technology, Journalism (2)

C. H. Lee, W. V. Sorin, and B. Y. Kim, “Fiber to the Home Using a PON Infrastructure,” Lightwave Technology, Journalism24, 4568–4583 (2006).

K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” Lightwave Technology, Journalism12, 1423–1439 (1994).

Opt. Lett. (3)

Other (3)

E. Castillo, A. S. Hadi, N. Balakrishman, and J. M. Sarabia, Extreme value and related models with applications in engineering and science (John Wiley and Sons, 2005).

G. H. Einarsson, Principles of Lightwave Communications (John Wiley & Sons, 1996).

B. Mukherjee, Optical WDM Networks (Springer, 2006).

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Figures (5)

Fig. 1
Fig. 1

Comparison between the proposed hybrid method and the MCMC method for Nch = 16 channels.

Fig. 2
Fig. 2

Normalized CDF of the photocurrent for Nch = 16 channels, chromatic dispersion D = 5ps/km/nm and channel spacing Δf = 25GHz.

Fig. 3
Fig. 3

Bit Error Rate as a function of the input peak power Pin for a) Nch = 8, b) Nch = 16 and c) Nch = 32 channels. D1 = 2ps/km/nm and D2 = 5ps/km/nm.

Fig. 4
Fig. 4

CDF of multispan WDM systems with Nch = 32 channels, input peak power Pin = 6dBm, chromatic dispersion coefficient D = 2ps/km/nm and channel spacing Δf = 50GHz.

Fig. 5
Fig. 5

Bit Error Rate as a function of the input peak power Pin of multispan WDM systems with Nch = 32 channels, chromatic dispersion coefficient D = 2ps/km/nm, channel spacing Δf = 50GHz and various values of the number of spans.

Equations (32)

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P pqr = γ 2 9 d pqr 2 P p P q P r e αL L eff 2 η
η= α 2 α 2 + ( Δβ ) 2 { 1+ 4 e αL sin 2 ( ΔβL/2 ) [ 1 e αL ] 2 }
Δβ 2π λ 2 D c Δ f 2 ( pr )( qr )
S ( m ) =k | E ( m ) | 2 k P z exp( αL )+2kδ P z exp( αL ) I m
S ( s ) =k | E ( s ) | 2 k δ 2 I s
δ= γc 2π λ 2 DΔ f 2 P in 3/2 exp( αL/2 )
I m = 1 3 pqr B p B q B r d pqr | pz || qz | cos( θ pqr θ z )
I s = ( 1 3 pqr rz B p B q B r d pqr | pz || qz | cos( θ pqr ) ) 2 + ( 1 3 pqr rz B p B q B r d pqr | pz || qz | sin( θ pqr ) ) 2
η= α 2 α 2 + ( Δβ ) 2 sin 2 ( MΔβL/2 ) sin 2 ( ΔβL/2 ) { 1+ 4 e αL sin 2 ( ΔβL/2 ) [ 1 e αL ] 2 }
F n ( x )= P r ( M n x )= i=1 n F( x ) = [ F( x ) ] n
F n ' ( x )= P r ( M n ' x )=1 [ 1F( x ) ] n
lim n F n ( μ n +x σ n )=G( x ),x
G( x )=exp[ ( 1+ξx ) 1/ξ ],1+ξx>0
F n ( x )=exp[ ( 1+ξ( x μ n σ n ) ) 1/ξ ]
F n,G ( x )=exp{ exp[ a n ( x u n ) ] }
F n,G ' ( x )=1exp{ exp[ a n ' ( x u n ' ) ] }
BER= 1 2 [ Q f S ( m ) ( ξ )dξ + Q f S ( s ) ( ξ )dξ ]= = 1 2 [ P( S m Q )+P( S s Q ) ]= 1 2 { P( S m Q )+[ 1P( S s Q ) ] }
F ' ( S m )=P( S m s m )=1 [ 1 F n,G ' ( s m ) ] 1/n ( mark )
F( s s )=P( S s s s )= [ F n,G ( s s ) ] 1/n ( space )
BER= 1 2 { 1 [ 1 F n,G ' ( Q ) ] 1/n }+ 1 2 { 1 [ F n,G ( Q ) ] 1/n }= = 1 2 { 2 { exp[ exp( a n ' ( Q u n ' ) ) ] } 1/n { exp[ exp( a n ( Q u n ) ) ] } 1/n }
E ( m ) = P z exp( αL ) exp( j θ z )+ pqr B p B q B r P pqr exp( j θ pqr )
E ( s ) = pqr B p B q B r P pqr exp( j θ pqr )
S ( m ) =k | E ( m ) | 2 =k E ( m ) E ( m )* = =k( P z exp( αL ) exp( j θ z )+ pqr B p B q B r P pqr exp( j θ pqr ) )× ×( P z exp( αL ) exp( j θ z )+ pqr B p B q B r P pqr exp( j θ pqr ) )= =k[ P z exp( αL )+ + P z exp( αL ) ( pqr B p B q B r P pqr exp( j θ pqr )exp( j θ z ) + pqr B p B q B r P pqr exp( j θ pqr )exp( j θ z ) )+ + pqr B p B q B r P pqr exp( j θ pqr ) pqr B p B q B r P pqr exp( j θ pqr ) ]
S ( m ) =k P z exp( αL )+2k P z exp( αL ) pqr B p B q B r P pqr cos( θ pqr θ z )
P pqr = γ 2 9 d pqr 2 P p P q P r e αL L eff 2 η
η= α 2 α 2 + ( Δβ ) 2 { 1+ 4 e αL sin 2 ( ΔβL/2 ) [ 1 e αL ] 2 }
S ( m ) =k P z exp( αL )+ +2k P z exp( αL ) × × pqr B p B q B r γ 2 9 d pqr 2 P p P q P r exp( αL ) [ 1exp( αL ) ] 2 α 2 α 2 α 2 + ( Δβ ) 2 { 1+ 4exp( αL ) sin 2 ( ΔβL/2 ) [ 1exp( αL ) ] 2 } cos( θ pqr θ z )
Δβ 2π λ 2 D c Δ f 2 ( pr )( qr )
S ( m ) =k P z exp( αL )+2k P z exp( αL ) pqr B p B q B r γ 3 d pqr Δβ P p P q P r exp( αL ) cos( θ pqr θ z )
S ( m ) =k P z exp( αL )+ +2k P z exp( αL ) pqr B p B q B r γ 3 d pqr c 2π λ 2 DΔ f 2 ( pz )( qz ) P in 3 exp( αL ) cos( θ pqr θ z ) = =k P z exp( αL )+ 2k P z exp( αL ) γc 2π λ 2 DΔ f 2 P in 3/2 exp( αL/2 ) 1 3 pqr B p B q B r d pqr ( pz )( qz ) cos( θ pqr θ z )
η= α 2 α 2 + ( Δβ ) 2 sin 2 ( MΔβL/2 ) sin 2 ( ΔβL/2 ) { 1+ 4 e αL sin 2 ( ΔβL/2 ) [ 1 e αL ] 2 }
S ( s ) = ( pqr B p B q B r P pqr cos( θ pqr ) ) 2 + ( pqr B p B q B r P pqr sin( θ pqr ) ) 2

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