Abstract

There has been a trend of migration to high spectral efficiency transmission in optical fiber communications for which the frequency guard band between neighboring wavelength channels continues to shrink. In this paper, we derive closed-form analytical expressions for nonlinear system performance of densely spaced coherent optical OFDM (CO-OFDM) systems. The closed-form solutions include the results for the achievable Q factor, optimum launch power density, nonlinear threshold of launch power density, and information spectral efficiency limit. These analytical results clearly identify their dependence on system parameters including fiber dispersion, number of spans, dispersion compensation ratio, and overall bandwidth. The closed-form solution is further substantiated by numerical simulations using distributed nonlinear Schrödinger equation.

© 2010 OSA

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    [CrossRef]
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    [CrossRef]
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2010

2009

2008

2007

G. Goldfarb, G. F. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007).
[CrossRef]

A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express 15(20), 13282–13287 (2007).
[CrossRef] [PubMed]

2006

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–589 (2006).
[CrossRef]

2001

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

1995

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13(5), 841–849 (1995).
[CrossRef]

1992

1948

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

Al Amin, A.

Athaudage, C.

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–589 (2006).
[CrossRef]

Bao, H.

Buchali, F.

Chen, S.

Cho, P.

Chraplyvy, A. R.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13(5), 841–849 (1995).
[CrossRef]

Derosier, R. M.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13(5), 841–849 (1995).
[CrossRef]

Forghieri, F.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13(5), 841–849 (1995).
[CrossRef]

Gnauck, A. H.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13(5), 841–849 (1995).
[CrossRef]

Goldfarb, G.

G. Goldfarb, G. F. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007).
[CrossRef]

Haunstein, H.

Inoue, K.

Jansen, S. L.

Karagodsky, V.

Khurgin, J.

Li, G. F.

G. Goldfarb, G. F. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007).
[CrossRef]

Liu, X.

Lowery, A. J.

Ma, Y.

Mayrock, M.

Meiman, Y.

Mitra, P. P.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

Morita, I.

Nazarathy, M.

Noe, R.

Premaratne, M.

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

Shieh, W.

Shpantzer, I.

Stark, J. B.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

Takahashi, H.

Tanaka, H.

Tang, Y.

Taylor, M. G.

G. Goldfarb, G. F. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007).
[CrossRef]

Tkach, R. W.

X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol. 27(16), 3632–3640 (2009).
[CrossRef]

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13(5), 841–849 (1995).
[CrossRef]

Wang, S.

Weidenfeld, R.

Yang, Q.

Bell Syst. Tech. J.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

Electron. Lett.

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–589 (2006).
[CrossRef]

IEEE Photon. Technol. Lett.

G. Goldfarb, G. F. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007).
[CrossRef]

J. Lightwave Technol.

Nature

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Other

S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “Transmission of a 1.2-Tb/s 24-carrier no-guard-interval coherent OFDM superchannel over 7200-km of ultra-large-area fiber,” Eur. Conf. Optical Commun.,Vienna, Austria (2009), post-deadline Paper PD2.6.

R. Dischler, and F. Buchali, “Transmission of 1.2 Tb/s continuous waveband PDM-OFDM-FDM signal with spectral efficiency of 3.3 bit/s/Hz over 400 km of SSMF,” Optical Fiber Communication Conference, paper PDP C2, San Diego, USA (2009).

E. Yamada, A. Sano, H. Masuda, E. Yamazaki, T. Kobayashi, E. Yoshida, K. Yonenaga, Y. Miyamoto, K. Ishihara, Y. Takatori, T. Yamada, and H. Yamazaki, “1Tb/s (111Gb/s/ch × 10ch) no-guard-interval CO-OFDM transmission over 2100 km DSF,” Opto-Electronics Communications Conference/Australian Conference on Optical Fiber Technology, paper PDP6, Sydney, Australia (2008).

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Figures (5)

Fig. 1
Fig. 1

Conceptual diagram of densely spaced OFDM (DS-OFDM) systems: (a) with a frequency guard band Δ B that is much smaller than the wavelength channel bandwidth W, and (b) continuous without any frequency guard band.

Fig. 2
Fig. 2

Comparison of closed-form theory and simulation results for (a) FWM power density, and (b) Q factor as a function of the launch power density. Theo.: Theory; Simu.: Simulation; CD: Chromatic dispersion with a unit of ps/nm/km; CR: (CD) compensation ratio. Both (a) and (b) assume 10x100 km single-polarization transmission systems.

Fig. 3
Fig. 3

(a) The maximum Q factor, and (b) the optimal launch power density versus number of spans with various dispersion maps. CD: chromatic dispersion. CR: (CD) compensation ratio

Fig. 4
Fig. 4

Information spectral efficiency as a function of the number spans for various dispersion maps. The total bandwidth B is assumed to be 40 nm. The other OFDM and link parameters are the same as those described at the beginning of this section.

Fig. 5
Fig. 5

Multi-span noise enhancement factor as a function of the dispersion compensation ratio with fiber span losses of 10 and 20 dB. The number of spans is maintained at 10. The link losses of 10 and 20 dB are obtained by setting the span length to 50 and 100 km respectively.

Equations (53)

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P g = D x 2 9 γ 2 P i P j P k e α L η
η = η 1 η 2
η 1 = | 1 e α L e j Δ β i j k L j Δ β i j k + α | 2 1 ( Δ β i j k ) 2 + α 2
η 2 = sin 2 { N s Δ β ˜ i j k / 2 } sin 2 Δ β ˜ i j k / 2 ,           Δ β ˜ i j k = Δ β i j k L + Δ β i j k , 1 L 1
Δ β i j k = 2 π λ 2 c ( f i f k ) ( f j f k ) D = 4 π 2 β 2 ( f i f k ) ( f j f k )             = 4 π 2 β 2 Δ f 2 ( i k ) ( j k ) ,               β 2 = λ 2 2 π c D
Δ β ˜ i j k = 2 π λ 2 c ( f i f k ) ( f j f k ) D r = 4 π 2 β 2 ( f i f k ) ( f j f k ) L ( 1 ρ )           = 4 π 2 β 2 Δ f 2 L ( 1 ρ ) ( i k ) ( j k ) D r = D L + D 1 L 1 = D L ( 1 ρ ) = D L ζ
P g = D x 2 9 γ 2 P i P j P k η
P N L i = 2 γ 2 P i k = N / 2 N / 2     j = N / 2 N / 2 P j P k η
                                                          P N L = 2 γ 2 P 3 k = N / 2 N / 2     j = N / 2 N / 2 η 1 η 2 η 1 = | sin ( N s j ( k j ) Δ f 2 / ( 2 f P A 2 ) ) sin ( j ( k j ) Δ f 2 / ( 2 f P A 2 ) ) | 2 ,       η 2 = 1 β 2 2 ( 2 π ) 4 1 Δ f 4 j 2 ( k j ) 2 + f W 4                                             f P A 1 2 π 1 | β 2 | L ζ ,               f W 1 2 π α | β 2 |
P N L = 2 γ 2 P 3 β 2 2 ( 2 π ) 4 m = N / 2 j N / 2 j     j = N / 2 N / 2 | sin ( N s j m Δ f 2 / ( 2 f P A 2 ) ) sin ( j m Δ f 2 / ( 2 f P A 2 ) ) | 2 1 Δ f 4 j 2 m 2 + f W 4
f P A > > Δ f
f W > > Δ f
P N L = 2 γ 2 β 2 2 ( 2 π ) 4 P 3 Δ f 2 B / 2 f 1 B / 2 f 1 B / 2 B / 2 η 1 ( f , f 1 ) η 2 ( f , f 1 ) d f 1 d f η 1 ( f , f 1 ) = | sin ( N s f 1 f / ( 2 f P A 2 ) ) sin ( f 1 f / ( 2 f P A 2 ) ) | 2 ,         η 2 ( f , f 1 ) = 1 ( f 1 f ) 2 + f W 4
I N L P N L Δ f ,                 I P Δ f
I N L = 2 γ 2 β 2 2 ( 2 π ) 4 I 3 B / 2 f 1 B / 2 f 1 B / 2 B / 2 η 1 ( f , f 1 ) η 2 ( f , f 1 ) d f 1 d f
I N L = 8 γ 2 β 2 2 ( 2 π ) 4 I 3 B 0 / 2 B / 2 0 η 1 ( f , f 1 ) η 2 ( f , f 1 ) d f 1 d f
B 0 = 2 f W 2 / B
B > > f W
I N L = I B 0 B / 2 i N L ( f ) d f ,       i N L ( f ) = 8 γ 2 β 2 2 ( 2 π ) 4 I 2 0 η 1 ( f , f 1 ) η 2 ( f , f 1 ) d f 1
i N L ( f ) = γ 2 I 2 2 π α | β 2 | ( ( N s 1 + e α ζ L N s N s e α ζ L ) e α ζ L ( e α ζ L 1 ) 2 + N 2 ) 1 f
I N L = γ 2 I 3 1 π α | β 2 | ( 2 ( N s 1 + e α ζ L N s N s e α ζ L ) e α ζ L ( e α ζ L 1 ) 2 + N s )           = γ 2 N s ln ( B / B 0 ) h e π α | β 2 | I 3
h e 2 ( N s 1 + e α ζ L N s N s e α ζ L ) e α ζ L N s ( e α ζ L 1 ) 2 + 1
I N L = ( I I 0 ) 2 I ,           I 0 1 γ π α | β 2 | N s h e ln ( B / B 0 )
I = I exp ( ( I / I 0 ) 2 ) I
n = n 0 + I ( 1 exp ( ( I / I 0 ) 2 ) ) ,         n 0 = N s ( G 1 ) n s p h υ 0.5 N s e α L h υ N F
S N R = I exp ( ( I / I 0 ) 2 ) n 0 + I ( 1 exp ( ( I / I 0 ) 2 ) )
S N R I n 0 + I ( I / I 0 ) 2
S = log 2 ( 1 + S N R ) = log 2 ( 1 + I exp ( ( I / I 0 ) 2 ) n 0 + I ( 1 exp ( ( I / I 0 ) 2 ) ) )       log 2 ( 1 + I n 0 + I ( I / I 0 ) 2 )
S o p t = log 2 ( 1 + 1 3 ( 2 I 0 / n 0 ) 2 / 3 )
Q = S N R = I exp ( ( I / I 0 ) 2 ) n 0 + I ( 1 exp ( ( I / I 0 ) 2 ) ) I n 0 + I ( I / I 0 ) 2
I o p t = ( n 0 I 0 2 / 2 ) 1 / 3 = ( n 0 π α | β 2 | 2 γ 2 N s h e ln ( B / B 0 ) ) 1 / 3
Q m a x = 1 3 ( 2 I 0 / n 0 ) 2 / 3 = ( 4 π α | β 2 | ) 1 / 3 3 ( n 0 2 γ 2 N s h e ln ( B / B 0 ) ) 1 / 3
I t h = I 0 q 0 = 1 q 0 γ π α | β 2 | N s h e ln ( B / B 0 )
S = log 2 ( 1 3 ( 2 I 0 / n 0 ) 2 / 3 ) = log 2 ( 1 3 ( 4 π α | β 2 | ) 1 / 3 ( γ 2 n 0 2 N s h e ln ( B / B 0 ) ) 1 / 3 )
I N L = 8 γ 2 I 3 β 2 2 ( 2 π ) 4 I N L 1 ,           I N L 1 = 1 4 B / 2 B / 2 B / 2 B / 2 η 1 ( f , f 1 ) η 2 ( f , f 1 ) d f 1 d f η 1 ( f , f 1 ) = | sin ( N s f 1 f / ( 2 f P A 2 ) ) sin ( f 1 f / ( 2 f P A 2 ) ) | 2 ,         η 2 ( f , f 1 ) = 1 ( f 1 f ) 2 + f W 4
I N L 1 = 0 B / 2 0 B / 2 η 1 ( f , f 1 ) η 2 ( f , f 1 ) d f 1 d f
I N L 1 = A 1 A 2 + A 3
A 1 = B 0 / 2 B / 2 0 η 1 η 2 d f 1 d f ,       A 2 = B 0 / 2 B / 2 B / 2 η 1 η 2 d f 1 d f ,       A 3 = 0 B 0 / 2 0 B / 2 η 1 η 2 d f 1 d f B 0 = 2 f W 2 / B
A 1 = π 2 f W 2 ln ( B / B 0 )
A 2 = B 0 / 2 B / 2 1 f f W 2 arctan ( f W 2 f B / 2 ) d f B 0 / 2 B / 2 1 f f W 2 f W 2 f B / 2 d f         = 1 B / 2 B 0 / 2 B / 2 1 f 2 d f = 1 B / 2 1 B 0 / 2 = 2 f W 2
A 3 = 0 B 0 0 B / 2 1 ( f 1 f ) 2 + f W 4 d f 1 d f         < 0 B 0 / 2 0 B / 2 1 f W 4 d f 1 d f = 1 f W 4 ( B 0 / 2 ) B / 2 = 1 2 f W 2
η 1 = | 1 + e j θ + e j 2 θ + ... + + e j ( N 1 ) θ | 2 = N + 2 n = 1 N 1 ( N n ) cos ( n θ )                                                                       θ = f f 1 / f W 2
A 3 = 0 B 0 / 2 0 B / 2 η 1 1 ( f 1 f ) 2 + f W 4 d f 1 d f 1 f W 4 0 B 0 / 2 0 B / 2 η 1 d f 1 d f       = 1 f W 4 0 B 0 / 2 0 B / 2 [ N + 2 n = 1 N S 1 ( N S n ) cos ( n f f 1 / f P A 2 ) ] d f 1 d f         = N s 2 f W 2 + 2 1 f W 4 n = 1 N 1 ( N S n ) 0 B 0 / 2 0 B / 2 cos ( n f f 1 / f P A 2 ) d f 1 d f         = N s 2 f W 2 + 2 1 f W 4 n = 1 N 1 ( N S n ) 0 B 0 / 2 f P A 2 n f sin ( n f B / ( 2 f P A 2 ) ) d f         = N s 2 f W 2 + 2 1 f W 4 n = 1 N 1 ( N S n ) f P A 2 n 0 n f W 2 4 f P A 2 1 f sin ( f ) d f
        A 3 N s f W 2 + π f P A 2 f W 4 n = 1 N 1 ( N S n ) n f P A 2           N s 2 f W 2 + π f P A 2 ( N s log N s N s + 1 ) f W 4 N s 2 f W 2
A 1 = π N s 2 f W 2 ln ( B / B 0 )
I N L = 8 γ 2 I 2 β 2 2 ( 2 π ) 4 I N L 1 ,             I N L 1 = B 0 / 2 B / 2 0 η 1 ( f , f 1 ) η 2 ( f , f 1 ) d f 1 d f
i N L ( f ) = 8 γ 2 I 2 β 2 2 ( 2 π ) 4 i N L 1 ( f ) ,     i N L 1 ( f ) = 0 η 1 ( f , f 1 ) η 2 ( f , f 1 ) d f 1
B 0 = max ( 2 f W 2 / B ,     2 B P E )
f ( x ) x 2 + a 2 d x = f ( x ) x 2 + a 2 d x = π a f ( i a )
i N L 1 ( f ) = 1 f 0 [ N + 2 n = 1 N S 1 ( N S n ) cos ( n f 2 / f P A 2 ) ] 1 f 2 2 + f W 4 d f 2                     = 1 2 f Re { [ N + 2 n = 1 N S 1 ( N S n ) exp ( j n f 2 / f P A 2 ) ] 1 f 2 2 + f W 4 d f 2 }
i N L 1 ( f ) = 1 2 f { N s + 2 n = 1 N S 1 ( N S n ) exp ( n f W 2 / f P A 2 ) }                     = π 2 f f W 2 { 2 ( N s 1 + e N s f W 2 / f P A 2 N s e f W 2 / f P A 2 ) e f W 2 / f P A 2 ( e f W 2 / f P A 2 1 ) 2 + N s }
i N L 1 ( f ) = 2 π 3 | β 2 | f α { 2 ( N s 1 + e N s α L ζ N s e α L ζ ) e α L ζ ( e α L ζ 1 ) 2 + N s }
i N L ( f ) = γ 2 π I 2 α | β 2 | { 2 ( N s 1 + e N s α L ζ N s e α L ζ ) e α L ζ ( e α L ζ 1 ) 2 + N s } 1 f

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