Abstract

A simplified closed form expression for the noise power due to four-wave mixing in coherent OFDM systems is derived. The proposed model is in very good agreement with the exact model. The derived analytical expressions can be used in performance evaluation of systems employing CO-OFDM with any number of subcarriers and/or as an integral part of physical layer aware routing algorithms.

© 2014 Optical Society of America

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References

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  1. Q. Yang, A. Al Amin, X. Chen, Y. Ma, S. Chen, and W. Shieh, “428-Gb/s single-channel coherent optical OFDM transmission over 960-km SSMF with constellation expansion and LDPC coding,” Opt. Express 18(16), 16883–16889 (2010).
    [Crossref] [PubMed]
  2. B. Zhu, S. Chandrasekhar, X. Liu, and D. W. Peckham, “Transmission performance of a 485-Gb/s CO-OFDM superchannel with PDM-16QAM subcarriers over ULAF and SSMF-based links,” IEEE Photon. Technol. Lett. 23(19), 1400–1402 (2011).
    [Crossref]
  3. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express 18(18), 19039–19054 (2010).
    [Crossref] [PubMed]
  4. 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]
  5. X. Zhu and S. Kumar, “Nonlinear phase noise in coherent optical OFDM transmission systems,” Opt. Express 18(7), 7347–7360 (2010).
    [Crossref] [PubMed]
  6. 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]
  7. C. T. Politi, V. Anagnostopoulos, and A. Stavdas, “PLI-aware routing in regenerated mesh topology optical networks,” J. Lightwave Technol. 30(12), 1960–1970 (2012).
    [Crossref]

2012 (1)

2011 (1)

B. Zhu, S. Chandrasekhar, X. Liu, and D. W. Peckham, “Transmission performance of a 485-Gb/s CO-OFDM superchannel with PDM-16QAM subcarriers over ULAF and SSMF-based links,” IEEE Photon. Technol. Lett. 23(19), 1400–1402 (2011).
[Crossref]

2010 (3)

2007 (1)

1992 (1)

Al Amin, A.

Anagnostopoulos, V.

Chandrasekhar, S.

B. Zhu, S. Chandrasekhar, X. Liu, and D. W. Peckham, “Transmission performance of a 485-Gb/s CO-OFDM superchannel with PDM-16QAM subcarriers over ULAF and SSMF-based links,” IEEE Photon. Technol. Lett. 23(19), 1400–1402 (2011).
[Crossref]

Chen, S.

Chen, X.

Inoue, K.

Kumar, S.

Liu, X.

B. Zhu, S. Chandrasekhar, X. Liu, and D. W. Peckham, “Transmission performance of a 485-Gb/s CO-OFDM superchannel with PDM-16QAM subcarriers over ULAF and SSMF-based links,” IEEE Photon. Technol. Lett. 23(19), 1400–1402 (2011).
[Crossref]

Lowery, A. J.

Ma, Y.

Peckham, D. W.

B. Zhu, S. Chandrasekhar, X. Liu, and D. W. Peckham, “Transmission performance of a 485-Gb/s CO-OFDM superchannel with PDM-16QAM subcarriers over ULAF and SSMF-based links,” IEEE Photon. Technol. Lett. 23(19), 1400–1402 (2011).
[Crossref]

Politi, C. T.

Premaratne, M.

Shieh, W.

Stavdas, A.

Wang, S.

Yang, Q.

Zhu, B.

B. Zhu, S. Chandrasekhar, X. Liu, and D. W. Peckham, “Transmission performance of a 485-Gb/s CO-OFDM superchannel with PDM-16QAM subcarriers over ULAF and SSMF-based links,” IEEE Photon. Technol. Lett. 23(19), 1400–1402 (2011).
[Crossref]

Zhu, X.

IEEE Photon. Technol. Lett. (1)

B. Zhu, S. Chandrasekhar, X. Liu, and D. W. Peckham, “Transmission performance of a 485-Gb/s CO-OFDM superchannel with PDM-16QAM subcarriers over ULAF and SSMF-based links,” IEEE Photon. Technol. Lett. 23(19), 1400–1402 (2011).
[Crossref]

J. Lightwave Technol. (1)

Opt. Express (4)

Opt. Lett. (1)

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Figures (5)

Fig. 1
Fig. 1 Statistical distribution of produced FWM terms for 256 subcarriers, Δf = 100MHz and L = 100 km.
Fig. 2
Fig. 2 FWM efficiency against Eq. (4).
Fig. 3
Fig. 3 The red curve shows the inner sum of Eq. (11) and the green curve shows the function inside the integral of Eq. (16), both plotted for various values of k.
Fig. 4
Fig. 4 Comparison of the Absolute Approximation Error (dB) of Eq. (2) and model in [3] against Eq. (1) for various number of subcarriers.
Fig. 5
Fig. 5 Comparison of our model against Inoue’s model with fractional spans [6, Eq. (10)] for 10 spans and 100 MHz and 200 MHz subcarrier spacing.

Equations (26)

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P N L i = D i j k 2 18 γ 2 L e f f 2 e a L P i j = N s u b / 2 N s u b / 2 k = N s u b / 2 N s u b / 2 P j P k 1 1 + ( 2 π λ 2 a c Δ f 2 ( i k ) ( j k ) D ) 2 sin 2 { N s 2 π λ 2 c Δ f 2 ( i k ) ( j k ) D L / 2 } sin 2 { 2 π λ 2 c Δ f 2 ( i k ) ( j k ) D L / 2 } ( 1 + 4 e a L sin 2 { 2 π λ 2 c Δ f 2 ( i k ) ( j k ) D L / 2 } ( 1 e a L ) 2 )
P N L = { D i j k 2 18 γ 2 L e f f 2 P 3 N s 2 N s u b 2 , N s u b 2 N s π λ 2 Δ f 2 D L 4 a 1 c < 1 D i j k 2 18 γ 2 L e f f 2 P 3 N s a 1 c λ 2 Δ f 2 D L ( 1 L o g [ 4 a 1 c N s u b 2 N s π λ 2 Δ f 2 D L ] ) , 1 N s u b 2 N s π λ 2 Δ f 2 D L 4 a 1 c π N s u b 4 D i j k 2 18 γ 2 L e f f 2 P 3 N s 2 ( N s u b a 1 c N s λ 2 Δ f 2 D L L o g [ 4 N s u b π ] ) , N s u b 2 N s π λ 2 Δ f 2 D L 4 a 1 c π N s u b 4
sin 2 { N s π λ 2 Δ f 2 ( i k ) ( j k ) D L / c } sin 2 { π λ 2 Δ f 2 ( i k ) ( j k ) D L / c }
π λ 2 Δ f 2 ( i k ) ( j k ) D L / c
1 1 + ( 2 π λ 2 a c Δ f 2 ( i k ) ( j k ) D ) 2
sin 2 { π λ 2 Δ f 2 ( i k ) ( j k ) D L / c } ( π λ 2 Δ f 2 ( i k ) ( j k ) D L / c ) 2
N s 2 sin c 2 { N s π λ 2 Δ f 2 ( i k ) ( j k ) D L / c }
4 e a L sin 2 { π λ 2 Δ f 2 ( i k ) ( j k ) D L / c } ( 1 e a L ) 2
j = N s u b / 2 N s u b / 2 k = N s u b / 2 N s u b / 2 N s 2 ( e ( N s π λ 2 Δ f 2 ( i k ) ( j k ) D L / c ) 2 a 1 2 ) 2
j = N s u b / 2 N s u b / 2 k = N s u b / 2 N s u b / 2 N s 2 ( a 1 2 a 1 2 + ( N s π λ 2 Δ f 2 ( i k ) ( j k ) D L / c ) 2 ) 2
k = N s u b / 2 N s u b / 2 n = N s u b / 2 k N s u b / 2 k N s 2 ( a 1 2 a 1 2 + ( N s π λ 2 Δ f 2 k n D L / c ) 2 ) 2
N s u b / 2 N s u b / 2 N s u b / 2 k N s u b / 2 k N s 2 ( a 1 c N s π λ 2 Δ f 2 k D L ) 4 ( 1 ( a 1 c N s π λ 2 Δ f 2 k D L ) 2 + n 2 ) 2 d n d k
1 ( x 2 + m 2 ) k d x = x 2 m 2 ( k 1 ) ( x 2 + m 2 ) k 1 + 2 k 3 2 m 2 ( k 1 ) 1 ( x 2 + m 2 ) k 1 d x
1 ( x 2 + m 2 ) d x = 1 m A r c Tan [ x m ]
N s u b / 2 N s u b / 2 N s 2 ( a 1 2 2 ( N s u b / 2 k ( N s π λ 2 Δ f 2 k D L / c ) 2 ( N s u b / 2 k ) 2 + a 1 2 N s u b / 2 k ( N s π λ 2 Δ f 2 k D L / c ) 2 ( N s u b / 2 k ) 2 + a 1 2 ) + a 1 2 ( N s π λ 2 Δ f 2 k D L / c ) ( A r c Tan ( N s π λ 2 Δ f 2 k D L / c a 1 ( N s u b / 2 k ) ) A r c Tan ( N s π λ 2 Δ f 2 k D L / c a 1 ( N s u b / 2 k ) ) ) ) d k
N s u b / 2 N s u b / 2 N s 2 ( a 1 2 2 ( N s u b ( N s π λ 2 Δ f 2 k D L / c ) 2 ( N s u b / 2 ) 2 + a 1 2 ) + a 1 ( N s π λ 2 Δ f 2 k D L / c ) A r c Tan ( N s π λ 2 Δ f 2 k D L / c a 1 ( N s u b / 2 ) ) ) d k
( a 1 2 2 ( N s u b ( N s π λ 2 Δ f 2 k D L / c ) 2 ( N s u b / 2 ) 2 + a 1 2 ) + a 1 ( N s π λ 2 Δ f 2 k D L / c ) A r c Tan ( N s π λ 2 Δ f 2 k D L / c a 1 ( N s u b / 2 ) ) )
a 1 2 2 ( N s u b ( N s π λ 2 Δ f 2 k D L / c ) 2 ( N s u b / 2 ) 2 + a 1 2 ) + i a 1 2 ( N s π λ 2 Δ f 2 k D L / c ) ( L o g ( 1 i N s π λ 2 Δ f 2 k D L N s u b 2 a 1 c ) L o g ( 1 + i N s π λ 2 Δ f 2 k D L N s u b 2 a 1 c ) )
a 1 2 2 ( N s u b ( N s π λ 2 Δ f 2 k D L / c ) 2 ( N s u b / 2 ) 2 + a 1 2 ) + a 1 2 ( N s π λ 2 Δ f 2 k D L / c ) ( A r g ( 1 i N s π λ 2 Δ f 2 k D L N s u b 2 a 1 c ) A r g ( 1 + i N s π λ 2 Δ f 2 k D L N s u b 2 a 1 c ) )
a 1 2 2 ( N s u b ( N s π λ 2 Δ f 2 k D L / c ) 2 ( N s u b / 2 ) 2 + a 1 2 ) + a 1 π 2 ( N s π λ 2 Δ f 2 k D L / c )
N s u b / 2 N s u b / 2 N s 2 ( 2 a 1 2 c 2 N s u b ( N s π λ 2 Δ f 2 D L ) 2 ( 1 ( 2 a 1 c N s u b N s π λ 2 Δ f 2 D L ) 2 + k 2 ) + a 1 c ( N s π λ 2 Δ f 2 k D L ) A r c Tan ( N s π λ 2 Δ f 2 k D L N s u b 2 a 1 c ) ) d k , N s u b 2 < a 1 c 2 N s λ 2 Δ f 2 D L a n d a 1 c 2 N s λ 2 Δ f 2 D L a 1 c 2 N s λ 2 Δ f 2 D L N s 2 ( 2 a 1 2 c 2 N s u b ( N s π λ 2 Δ f 2 D L ) 2 ( 1 ( 2 a 1 c N s u b N s π λ 2 Δ f 2 D L ) 2 + k 2 ) + a 1 c ( N s π λ 2 Δ f 2 k D L ) A r c Tan ( N s π λ 2 Δ f 2 k D L N s u b 2 a 1 c ) ) d k + 2 N s 2 ( N s u b 2 a 1 c 2 N s λ 2 Δ f 2 D L ) , N s u b 2 a 1 c 2 N s λ 2 Δ f 2 D L
x 1 x 1 1 x A r c Tan ( x y ) d x = i ( P o l y L o g ( 2 , i y x 1 ) P o l y L o g ( 2 , i y x 1 ) )
N s 2 ( a 1 c N s π λ 2 Δ f 2 D L ( A r c Tan ( N s π λ 2 Δ f 2 D L N s u b 2 4 a 1 c ) A r c Tan ( N s π λ 2 Δ f 2 D L N s u b 2 4 a 1 c ) ) + i a 1 c N s π λ 2 Δ f 2 D L ( P o l y L o g ( 2 , i N s π λ 2 Δ f 2 D L N s u b 2 4 a 1 c ) P o l y L o g ( 2 , i N s π λ 2 Δ f 2 D L N s u b 2 4 a 1 c ) ) , N s u b 2 < a 1 c 2 N s λ 2 Δ f 2 D L a n d N s 2 ( a 1 c N s π λ 2 Δ f 2 D L ( A r c Tan ( π N s u b 4 ) A r c Tan ( π N s u b 4 ) ) + i a 1 c N s π λ 2 Δ f 2 D L ( P o l y L o g ( 2 , i π N s u b 4 ) P o l y L o g ( 2 , i π N s u b 4 ) ) ) + 2 N s 2 ( N s u b 2 a 1 c 2 N s λ 2 Δ f 2 D L ) , N s u b 2 a 1 c 2 N s λ 2 Δ f 2 D L
N s 2 ( 2 a 1 c N s π λ 2 Δ f 2 D L A r c Tan ( N s π λ 2 Δ f 2 D L N s u b 2 4 a 1 c ) + i a 1 c N s π λ 2 Δ f 2 D L ( P o l y L o g ( 2 , i N s π λ 2 Δ f 2 D L N s u b 2 4 a 1 c ) P o l y L o g ( 2 , i N s π λ 2 Δ f 2 D L N s u b 2 4 a 1 c ) ) ) , N s u b 2 < a 1 c 2 N s λ 2 Δ f 2 D L a n d N s 2 ( 2 a 1 c N s π λ 2 Δ f 2 D L A r c Tan ( π N s u b 4 ) + i a 1 c N s π λ 2 Δ f 2 D L ( P o l y L o g ( 2 , i π N s u b 4 ) P o l y L o g ( 2 , i π N s u b 4 ) ) ) + 2 N s 2 ( N s u b 2 a 1 c 2 N s λ 2 Δ f 2 D L ) , N s u b 2 a 1 c 2 N s λ 2 Δ f 2 D L
N s 2 N s u b 2 , N s π λ 2 Δ f 2 D L N s u b 2 4 a 1 c < 1 N s a 1 c λ 2 Δ f 2 D L ( 1 L o g ( 4 a 1 c N s u b 2 N s π λ 2 Δ f 2 D L ) ) , N s π λ 2 Δ f 2 D L N s u b 2 4 a 1 c 1 N s 2 ( N s u b a 1 c N s λ 2 Δ f 2 D L L o g ( 4 N s u b π ) ) , N s u b N s λ 2 Δ f 2 D L a 1 c 1
L = L 1 + L 2 + ... + L N s N s

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