Abstract

We develop an analytical theory which allows us to identify the information spectral density limits of multimode optical fiber transmission systems. Our approach takes into account the Kerr-effect induced interactions of the propagating spatial modes and derives closed-form expressions for the spectral density of the corresponding nonlinear distortion. Experimental characterization results have confirmed the accuracy of the proposed models. Application of our theory in different FMF transmission scenarios has predicted a ~10% variation in total system throughput due to changes associated with inter-mode nonlinear interactions, in agreement with an observed 3dB increase in nonlinear noise power spectral density for a graded index four LP mode fiber.

© 2013 OSA

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2013

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. N. Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavík, and D. J. Richardson, “Towards high-capacity fibre optic communications at the speed of light in vacuum,” Nat. Photonics7(4), 279–284 (2013).
[CrossRef]

R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental observation of inter-modal cross-phase modulation in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 535–538 (2013).
[CrossRef]

R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 539–542 (2013).
[CrossRef]

S. Mumtaz, R.-J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalisation of the Manakov equations,” J. Lightwave Technol.31(3), 398–406 (2013).
[CrossRef]

D. Rafique, S. Sygletos, and A. D. Ellis, “Impact of power allocation strategies in long-haul few-mode fiber transmission systems,” Opt. Express21(9), 10801–10809 (2013).
[CrossRef] [PubMed]

D. Rafique, S. Sygletos, and A. D. Ellis, “Impact of power allocation strategies in long-haul few-mode fiber transmission systems,” Opt. Express21(9), 10801–10809 (2013).
[CrossRef] [PubMed]

2012

F. Ferreira, S. Jansen, P. Monteiro, and H. Silva, “Nonlinear semi-analytical model for simulation of few-mode fiber transmission,” IEEE Photon. Technol. Lett.24(4), 240–242 (2012).
[CrossRef]

Y. Yang, Y. Yan, N. Ahmed, Y. Jeng-Yuan, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J.4(2), 535–543 (2012).
[CrossRef]

P. J. Winzer, “Optical Networking Beyond WDM,” IEEE Photon. J.4(2), 647–651 (2012).
[CrossRef]

G. Rademacher, S. Warm, and K. Petermann, “Analytical description of cross modal nonlinear interaction in mode multiplexed multi-mode fibers,” IEEE Photon. Technol. Lett.24(21), 1929–1932 (2012).
[CrossRef]

N. Bai, E. Ip, Y.-K. Huang, E. Mateo, F. Yaman, M. J. Li, S. Bickham, S. Ten, J. Liñares, C. Montero, V. Moreno, X. Prieto, V. Tse, K. Man Chung, A. P. Lau, H. Y. Tam, C. Lu, Y. Luo, G. D. Peng, G. Li, and T. Wang, “Mode-division multiplexed transmission with inline few-mode fiber amplifier,” Opt. Express20(3), 2668–2680 (2012).
[CrossRef] [PubMed]

S. Kilmurray, T. Fehenberger, P. Bayvel, and R. I. Killey, “Comparison of the nonlinear transmission performance of quasi-Nyquist WDM and reduced guard interval OFDM,” Opt. Express20(4), 4198–4205 (2012).
[CrossRef] [PubMed]

B. Inan, B. Spinnler, F. Ferreira, D. van den Borne, A. Lobato, S. Adhikari, V. A. Sleiffer, M. Kuschnerov, N. Hanik, and S. L. Jansen, “DSP complexity of mode-division multiplexed receivers,” Opt. Express20(10), 10859–10869 (2012).
[CrossRef] [PubMed]

A. Mecozzi, C. Antonelli, and M. Shtaif, “Coupled Manakov equations in multimode fibers with strongly coupled groups of modes,” Opt. Express20(21), 23436–23441 (2012).
[CrossRef] [PubMed]

V. A. J. M. Sleiffer, Y. Jung, V. Veljanovski, R. G. H. van Uden, M. Kuschnerov, H. Chen, B. Inan, L. G. Nielsen, Y. Sun, D. J. Richardson, S. U. Alam, F. Poletti, J. K. Sahu, A. Dhar, A. M. J. Koonen, B. Corbett, R. Winfield, A. D. Ellis, and H. de Waardt, “73.7 Tb/s (96 x 3 x 256-Gb/s) mode-division-multiplexed DP-16QAM transmission with inline MM-EDFA,” Opt. Express20(26), B428–B438 (2012).
[CrossRef] [PubMed]

T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator with coherent detection,” Opt. Express20(27), 28779–28785 (2012).
[CrossRef] [PubMed]

2011

2010

2001

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

1992

D. A. Cleland, A. D. Ellis, and C. H. F. Sturrock, “Precise modelling of four wave mixing products over 400km of step index fibre,” Electron. Lett.28(12), 1171–1172 (1992).
[CrossRef]

1984

1965

P. D. Maker and R. W. Terhune, “Study of the optical effects due to an induced polarisation third order in the electric field strength,” Phys. Rev.137(3A), A801–A818 (1965).
[CrossRef]

Adhikari, S.

Agrawal, G. P.

Ahmed, N.

Y. Yang, Y. Yan, N. Ahmed, Y. Jeng-Yuan, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J.4(2), 535–543 (2012).
[CrossRef]

Alam, S. U.

Antonelli, C.

Baddela, N.

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. N. Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavík, and D. J. Richardson, “Towards high-capacity fibre optic communications at the speed of light in vacuum,” Nat. Photonics7(4), 279–284 (2013).
[CrossRef]

Bai, N.

Bayvel, P.

Bickham, S.

Birnbaum, K. M.

Y. Yang, Y. Yan, N. Ahmed, Y. Jeng-Yuan, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J.4(2), 535–543 (2012).
[CrossRef]

Bosco, G.

G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Opt. Express19(26), B440–B449 (2011).
[CrossRef] [PubMed]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical Modeling of Non-Linear Propagation in Uncompensated Optical Transmission Links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011).
[CrossRef]

Carena, A.

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical Modeling of Non-Linear Propagation in Uncompensated Optical Transmission Links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011).
[CrossRef]

G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Opt. Express19(26), B440–B449 (2011).
[CrossRef] [PubMed]

Chen, H.

Chen, X.

Chraplyvy, A. R.

R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 539–542 (2013).
[CrossRef]

R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental observation of inter-modal cross-phase modulation in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 535–538 (2013).
[CrossRef]

Cleland, D. A.

D. A. Cleland, A. D. Ellis, and C. H. F. Sturrock, “Precise modelling of four wave mixing products over 400km of step index fibre,” Electron. Lett.28(12), 1171–1172 (1992).
[CrossRef]

Corbett, B.

Cotter, D.

Curri, V.

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical Modeling of Non-Linear Propagation in Uncompensated Optical Transmission Links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011).
[CrossRef]

G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Opt. Express19(26), B440–B449 (2011).
[CrossRef] [PubMed]

de Waardt, H.

Dhar, A.

Dolinar, S.

Y. Yang, Y. Yan, N. Ahmed, Y. Jeng-Yuan, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J.4(2), 535–543 (2012).
[CrossRef]

Ellis, A. D.

Erkmen, B. I.

Y. Yang, Y. Yan, N. Ahmed, Y. Jeng-Yuan, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J.4(2), 535–543 (2012).
[CrossRef]

Essiambre, R.

R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 539–542 (2013).
[CrossRef]

R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental observation of inter-modal cross-phase modulation in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 535–538 (2013).
[CrossRef]

R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol.28(4), 662–701 (2010).
[CrossRef]

Essiambre, R.-J.

Fehenberger, T.

Ferreira, F.

B. Inan, B. Spinnler, F. Ferreira, D. van den Borne, A. Lobato, S. Adhikari, V. A. Sleiffer, M. Kuschnerov, N. Hanik, and S. L. Jansen, “DSP complexity of mode-division multiplexed receivers,” Opt. Express20(10), 10859–10869 (2012).
[CrossRef] [PubMed]

F. Ferreira, S. Jansen, P. Monteiro, and H. Silva, “Nonlinear semi-analytical model for simulation of few-mode fiber transmission,” IEEE Photon. Technol. Lett.24(4), 240–242 (2012).
[CrossRef]

Fischer, J. K.

Fokoua, E. N.

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. N. Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavík, and D. J. Richardson, “Towards high-capacity fibre optic communications at the speed of light in vacuum,” Nat. Photonics7(4), 279–284 (2013).
[CrossRef]

Forghieri, F.

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical Modeling of Non-Linear Propagation in Uncompensated Optical Transmission Links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011).
[CrossRef]

G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Opt. Express19(26), B440–B449 (2011).
[CrossRef] [PubMed]

Foschini, G. J.

Gnauck, A. H.

R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 539–542 (2013).
[CrossRef]

R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental observation of inter-modal cross-phase modulation in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 535–538 (2013).
[CrossRef]

Goebel, B.

Gray, D. R.

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. N. Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavík, and D. J. Richardson, “Towards high-capacity fibre optic communications at the speed of light in vacuum,” Nat. Photonics7(4), 279–284 (2013).
[CrossRef]

Haider, A. F. M. Y.

Hanik, N.

Hayes, J. R.

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. N. Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavík, and D. J. Richardson, “Towards high-capacity fibre optic communications at the speed of light in vacuum,” Nat. Photonics7(4), 279–284 (2013).
[CrossRef]

Huang, H.

Y. Yang, Y. Yan, N. Ahmed, Y. Jeng-Yuan, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J.4(2), 535–543 (2012).
[CrossRef]

Huang, Y.-K.

Inan, B.

Ip, E.

Jansen, S.

F. Ferreira, S. Jansen, P. Monteiro, and H. Silva, “Nonlinear semi-analytical model for simulation of few-mode fiber transmission,” IEEE Photon. Technol. Lett.24(4), 240–242 (2012).
[CrossRef]

Jansen, S. L.

Jeng-Yuan, Y.

Y. Yang, Y. Yan, N. Ahmed, Y. Jeng-Yuan, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J.4(2), 535–543 (2012).
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[CrossRef]

R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental observation of inter-modal cross-phase modulation in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 535–538 (2013).
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Li, M. J.

Li, Z.

F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. N. Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavík, and D. J. Richardson, “Towards high-capacity fibre optic communications at the speed of light in vacuum,” Nat. Photonics7(4), 279–284 (2013).
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Liñares, J.

Lingle, R.

R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental observation of inter-modal cross-phase modulation in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 535–538 (2013).
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R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 539–542 (2013).
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Lu, C.

Luo, Y.

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P. D. Maker and R. W. Terhune, “Study of the optical effects due to an induced polarisation third order in the electric field strength,” Phys. Rev.137(3A), A801–A818 (1965).
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Man Chung, K.

Mateo, E.

Matera, M.

Mecozzi, A.

Mestre, M. A.

R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 539–542 (2013).
[CrossRef]

R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental observation of inter-modal cross-phase modulation in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 535–538 (2013).
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F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. N. Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavík, and D. J. Richardson, “Towards high-capacity fibre optic communications at the speed of light in vacuum,” Nat. Photonics7(4), 279–284 (2013).
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F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. N. Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavík, and D. J. Richardson, “Towards high-capacity fibre optic communications at the speed of light in vacuum,” Nat. Photonics7(4), 279–284 (2013).
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F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. N. Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavík, and D. J. Richardson, “Towards high-capacity fibre optic communications at the speed of light in vacuum,” Nat. Photonics7(4), 279–284 (2013).
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Figures (7)

Fig. 1
Fig. 1

(a) Calculated group delay versus wavelength for a four mode fiber showing (shaded boxes) example regions of velocity matching for intra mode (brown) and inter-mode (green) interactions. (b) Black line, a typical optical super channel spectrum; brown line FWM efficiency curves for intra-mode FWM; green line typical FWM efficiency curve for inter-mode FWM Curves are plotted from Eq. (6)for a 4 amplifier system, with an inter-mode velocity matching offset of 2.3 GHz and an exaggerated velocity-matching bandwidth of 200 GHz (for clarity).

Fig. 2
Fig. 2

Relative nonlinear noise power coefficient (in dB relative to SMF) for a step index 12-mode fiber (four LP modes) with a maximum differential mode delay of 880ps/km, 0.2 dB/km loss and assumed chromatic dispersion of 20 ps2/km (left) and with a maximum DMD of 110ps/km (right). Contours show 10log10ijkn)as a function of effective area and VMO for a WDM bandwidth of 5THz. Colored dots represent the calculated values of these parameters for nonlinear noise generated in the LP01(red), LP02(green), LP11(blue) and LP21 (purple) modes for different inter-mode interactions. Fiber parameters calculated using a commercial mode solver.

Fig. 3
Fig. 3

Nonlinear noise power normalized to LP01 mode at 100GHz bandwidth as a function of WDM signal bandwidth for the high (left) and low (right) DMD fibers of Fig. 2. LP01 (red), LP02 (green), LP11 (blue) and LP21 (purple).

Fig. 4
Fig. 4

(top) Experimental configuration used to measure few mode fibers (lenses and positioning stages at the ends of the few mode fiber are omitted for clarity). (bottom) Comparison of theoretical (solid lines) and experimental (dots) results for the LP01(red) and LP11 (blue) mode showing the normalized nonlinear power spectral density at the center of the band as a function of the bandwidth of an amplified spontaneous emission source with a 50GHz frequency notch at the center. Theoretical predictions are based on typical measured DMD from the same fiber draw as the fiber used in this experiment [17], also shows theoretical prediction neglecting the inter mode nonlinearity (dotted lines)and typical output spectra (inset for signal widths between 1.5 (red) and 3 (dark blue) THz).

Fig. 5
Fig. 5

(left) ISD versus signal power spectral density for different four-mode fibers (separate line for each mode) calculated according to Eq. (9) and (right) total capacity for the four FMF systems obtained by summing the ISDs of all modes at the optimum PSD. Both figures show ISD for SMF (black),high effective area step index FMF (green), high effective area graded index FMF (red), small effective area graded index FMF) (blue) (DMDs shown in Table 1). Left figure shows ISD per mode, right figure shows total ISD (of six fibers for SMF). Systems have 80km amplifier spacing (4.8dB noise figure), 100 channels (50GHz spacing, assuming OFDM or Nyquist signaling), 0.2dB loss and dispersion (β”) of 20ps2/km.

Fig. 6
Fig. 6

Predicted maximum total information spectral density (right axis) and information spectral density per mode (left axis) for trench assisted fiber designs with a core radius of 10.4 µm as index curvature is varied resulting in differences in DMD, plotted as a function of resultant DMD between LP01 and LP21 for positive (red filled circles) and negative values (blue open circles). Inset shows a similar fiber with a core radius of 30 µm (filled blue circles). Solid red line shows a logarithmic fit over points falling within 0.1 and 8 ns/km DMD as a guide to the eye. Both plots are for a 3,200km system with 80km amplifier spacing (4.8dB noise figure), 100 channels (50GHz spacing), 0.2dB loss and dispersion (β”) of 20ps2/km.

Fig. 7
Fig. 7

Predicted information spectral density per mode for step index fiber designs supporting four LP modes, plotted as a function of resultant LP01 effective area. System parameters correspond to a 3,200km system with 80km amplifier spacing (4.8dB noise figure), 100 channels (50GHz spacing), 0.2dB loss and dispersion (β”) of 20ps2/km.

Tables (1)

Tables Icon

Table 1 Effective areas and differential mode delays of fibers used to calculate ISD variations of Fig. 5

Equations (10)

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E n = ω 0 n 2 c A i j k n E i E j E k * 1 e α L e j Δ β i j k n L j Δ β i j k n + α sin ( N S Δ β i j k n L / 2 ) sin ( Δ β i j k n L / 2 )
1 A i j k n = E i ( r , θ ) E j ( r , θ ) E k ( r , θ ) E n ( r , θ ) d r d θ E i ( r , θ ) d r d θ . E j ( r , θ ) d r d θ . E k ( r , θ ) d r d θ . E n ( r , θ ) d r d θ
Δ β i j k n = β ' n + β ' k β ' i β ' j
Δ β i j k n , p q r s = β n ~ + β k ~ β i ~ β j ~ 4 π 2 β ' ' ( f i , s f k , r ) ( f j , q f k , r )
Δ f i j k n = β n ~ + β k ~ β i ~ β j ~ 2 π β ' '
η i j k n = ξ i j k n A i j k n 2 ω 0 2 n 2 2 N S . c 2 π α | β ' ' | { ln ( B 2 + 2 B Δ f i j k n 2. f w 2 ) + s . ln ( s B 2 2 B Δ f i j k n 2. f w 2 ) } s = S i g n ( B 2 Δ f ) f w = α 4 π 2 β ' '
a n = i , j , k η i j k n
I S D N L = n = 1 M log 2 ( 1 + P s P N + i , j , k η i j k n P s 3 )
I S D N L | M = M ( C B | 1 ) log 2 [ n = 1 M ( 2 3 P 1 P M + 1 3 a n P M 2 a 1 P 1 2 ) ]
M = i = 1 M 3 a i P M 3 ( P N + a i P M 3 )

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