Abstract

We analyze the impact of Differential Mode Delay (DMD) Management on the nonlinear impairments in mode-division multiplexed transmission systems. It is found out that DMD Management can lead to a degraded performance, due to enhanced intermodal nonlinear interaction. This can be attributed to an increased correlation of co-propagating channels, similar to the effects that show up in dispersion managed single-mode systems.

© 2015 Optical Society of America

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References

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  1. R. J. Essiambre, G. Kramer, P.J. Winzer, G. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010).
    [Crossref]
  2. T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
    [Crossref]
  3. B. Inan, B. Spinnler, F. Ferreira, D. van den Borne, A. Lobato, S. Adhikari, V. a J. M. Sleiffer, M. Kuschnerov, N. Hanik, and S. L. Jansen, “DSP complexity of mode-division multiplexed receivers,” Opt. Express 20, 10859–10869 (2012).
    [Crossref] [PubMed]
  4. S. Randel, R. Ryf, and A. Gnauck, “Mode-Multiplexed 620-GBd QPSK Transmission over 1200-km DGD-Compensated Few-Mode Fiber,” Proc. Opt. Fiber Commun. Conf., Paper PDP5C.5 (2012).
    [Crossref]
  5. R. Essiambre, M. Mestre, R. Ryf, and A. Gnauck, “Experimental Investigation of Inter-Modal Four-Wave Mixing in Multimode Fibers,” IEEE Photon. Technol. Lett. 25, 539–542 (2013).
    [Crossref]
  6. 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, 1929–1932 (2012).
    [Crossref]
  7. H.-G. Unger, “Planar optical waveguides and fibres,” Clarendon Pr., pp. 465 ff. (1977).
  8. K.-P. Ho and J. M. Kahn, “Mode-Dependent Loss and Gain: Statistics and Effect on Mode-Division Multiplexing,” Opt. Express 19, 16612–16635 (2011).
    [Crossref] [PubMed]
  9. F. Ye, S. Warm, and K. Petermann, “Differential Mode Delay Management in Spliced Multimode Fiber Transmission Systems,” Proc. Opt. Fiber Commun. Conf., Paper OM3B.3 (2013).
    [Crossref]
  10. S. Warm and K. Petermann, “DMD Management in Few-Mode Fiber MDM Transmission Systems with Mode Coupling,” Proc. IEEE Summer Topicals, pp. 158–159 (2014).
  11. K. Petermann, “Nonlinear distortions and noise in optical communication systems due to fiber connectors,” IEEE J. Quantum Electron. 16, 761–770 (1980).
    [Crossref]
  12. A. Juarez, E. Krune, S. Warm, C. Bunge, and K. Petermann, “Modeling of Mode Coupling in Multimode Fibers With Respect to Bandwidth and Loss,” J. Lightwave Technol. 32, 1549–1558 (2014).
    [Crossref]
  13. S. Mumtaz, R. Essiambre, and G. Agrawal, “Nonlinear Propagation in Multimode and Multicore Fibers: Generalization of the Manakov Equations,” J. Lightwave Technol. 31, 398–406 (2013).
    [Crossref]

2014 (1)

2013 (2)

S. Mumtaz, R. Essiambre, and G. Agrawal, “Nonlinear Propagation in Multimode and Multicore Fibers: Generalization of the Manakov Equations,” J. Lightwave Technol. 31, 398–406 (2013).
[Crossref]

R. Essiambre, M. Mestre, R. Ryf, and A. Gnauck, “Experimental Investigation of Inter-Modal Four-Wave Mixing in Multimode Fibers,” IEEE Photon. Technol. Lett. 25, 539–542 (2013).
[Crossref]

2012 (2)

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, 1929–1932 (2012).
[Crossref]

B. Inan, B. Spinnler, F. Ferreira, D. van den Borne, A. Lobato, S. Adhikari, V. a J. M. Sleiffer, M. Kuschnerov, N. Hanik, and S. L. Jansen, “DSP complexity of mode-division multiplexed receivers,” Opt. Express 20, 10859–10869 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (1)

1980 (1)

K. Petermann, “Nonlinear distortions and noise in optical communication systems due to fiber connectors,” IEEE J. Quantum Electron. 16, 761–770 (1980).
[Crossref]

Abe, Y.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Adhikari, S.

Agrawal, G.

Bunge, C.

Essiambre, R.

S. Mumtaz, R. Essiambre, and G. Agrawal, “Nonlinear Propagation in Multimode and Multicore Fibers: Generalization of the Manakov Equations,” J. Lightwave Technol. 31, 398–406 (2013).
[Crossref]

R. Essiambre, M. Mestre, R. Ryf, and A. Gnauck, “Experimental Investigation of Inter-Modal Four-Wave Mixing in Multimode Fibers,” IEEE Photon. Technol. Lett. 25, 539–542 (2013).
[Crossref]

Essiambre, R. J.

Ferreira, F.

Foschini, G.

Gnauck, A.

R. Essiambre, M. Mestre, R. Ryf, and A. Gnauck, “Experimental Investigation of Inter-Modal Four-Wave Mixing in Multimode Fibers,” IEEE Photon. Technol. Lett. 25, 539–542 (2013).
[Crossref]

S. Randel, R. Ryf, and A. Gnauck, “Mode-Multiplexed 620-GBd QPSK Transmission over 1200-km DGD-Compensated Few-Mode Fiber,” Proc. Opt. Fiber Commun. Conf., Paper PDP5C.5 (2012).
[Crossref]

Goebel, B.

Goh, T.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Hanik, N.

Ho, K.-P.

Inan, B.

Ishida, I.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Jansen, S. L.

Juarez, A.

Kahn, J. M.

Kawakami, H.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Kobayashi, T.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Kramer, G.

Krune, E.

Kuschnerov, M.

Lobato, A.

Matsuo, S.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Mestre, M.

R. Essiambre, M. Mestre, R. Ryf, and A. Gnauck, “Experimental Investigation of Inter-Modal Four-Wave Mixing in Multimode Fibers,” IEEE Photon. Technol. Lett. 25, 539–542 (2013).
[Crossref]

Miyamoto, Y.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Mizuno, T.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Morioka, T.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Mumtaz, S.

Nakagawa, T.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Oguma, M.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Petermann, K.

A. Juarez, E. Krune, S. Warm, C. Bunge, and K. Petermann, “Modeling of Mode Coupling in Multimode Fibers With Respect to Bandwidth and Loss,” J. Lightwave Technol. 32, 1549–1558 (2014).
[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, 1929–1932 (2012).
[Crossref]

K. Petermann, “Nonlinear distortions and noise in optical communication systems due to fiber connectors,” IEEE J. Quantum Electron. 16, 761–770 (1980).
[Crossref]

F. Ye, S. Warm, and K. Petermann, “Differential Mode Delay Management in Spliced Multimode Fiber Transmission Systems,” Proc. Opt. Fiber Commun. Conf., Paper OM3B.3 (2013).
[Crossref]

S. Warm and K. Petermann, “DMD Management in Few-Mode Fiber MDM Transmission Systems with Mode Coupling,” Proc. IEEE Summer Topicals, pp. 158–159 (2014).

Rademacher, G.

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, 1929–1932 (2012).
[Crossref]

Randel, S.

S. Randel, R. Ryf, and A. Gnauck, “Mode-Multiplexed 620-GBd QPSK Transmission over 1200-km DGD-Compensated Few-Mode Fiber,” Proc. Opt. Fiber Commun. Conf., Paper PDP5C.5 (2012).
[Crossref]

Ryf, R.

R. Essiambre, M. Mestre, R. Ryf, and A. Gnauck, “Experimental Investigation of Inter-Modal Four-Wave Mixing in Multimode Fibers,” IEEE Photon. Technol. Lett. 25, 539–542 (2013).
[Crossref]

S. Randel, R. Ryf, and A. Gnauck, “Mode-Multiplexed 620-GBd QPSK Transmission over 1200-km DGD-Compensated Few-Mode Fiber,” Proc. Opt. Fiber Commun. Conf., Paper PDP5C.5 (2012).
[Crossref]

Saitoh, K.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Sakamoto, T.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Sano, A.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Sasaki, Y.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Sleiffer, V. a J. M.

Spinnler, B.

Takara, H.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Takenaga, K.

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

Unger, H.-G.

H.-G. Unger, “Planar optical waveguides and fibres,” Clarendon Pr., pp. 465 ff. (1977).

van den Borne, D.

Warm, S.

A. Juarez, E. Krune, S. Warm, C. Bunge, and K. Petermann, “Modeling of Mode Coupling in Multimode Fibers With Respect to Bandwidth and Loss,” J. Lightwave Technol. 32, 1549–1558 (2014).
[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, 1929–1932 (2012).
[Crossref]

F. Ye, S. Warm, and K. Petermann, “Differential Mode Delay Management in Spliced Multimode Fiber Transmission Systems,” Proc. Opt. Fiber Commun. Conf., Paper OM3B.3 (2013).
[Crossref]

S. Warm and K. Petermann, “DMD Management in Few-Mode Fiber MDM Transmission Systems with Mode Coupling,” Proc. IEEE Summer Topicals, pp. 158–159 (2014).

Winzer, P.J.

Ye, F.

F. Ye, S. Warm, and K. Petermann, “Differential Mode Delay Management in Spliced Multimode Fiber Transmission Systems,” Proc. Opt. Fiber Commun. Conf., Paper OM3B.3 (2013).
[Crossref]

IEEE J. Quantum Electron. (1)

K. Petermann, “Nonlinear distortions and noise in optical communication systems due to fiber connectors,” IEEE J. Quantum Electron. 16, 761–770 (1980).
[Crossref]

IEEE Photon. Technol. Lett. (2)

R. Essiambre, M. Mestre, R. Ryf, and A. Gnauck, “Experimental Investigation of Inter-Modal Four-Wave Mixing in Multimode Fibers,” IEEE Photon. Technol. Lett. 25, 539–542 (2013).
[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, 1929–1932 (2012).
[Crossref]

J. Lightwave Technol. (3)

Opt. Express (2)

Other (5)

F. Ye, S. Warm, and K. Petermann, “Differential Mode Delay Management in Spliced Multimode Fiber Transmission Systems,” Proc. Opt. Fiber Commun. Conf., Paper OM3B.3 (2013).
[Crossref]

S. Warm and K. Petermann, “DMD Management in Few-Mode Fiber MDM Transmission Systems with Mode Coupling,” Proc. IEEE Summer Topicals, pp. 158–159 (2014).

H.-G. Unger, “Planar optical waveguides and fibres,” Clarendon Pr., pp. 465 ff. (1977).

S. Randel, R. Ryf, and A. Gnauck, “Mode-Multiplexed 620-GBd QPSK Transmission over 1200-km DGD-Compensated Few-Mode Fiber,” Proc. Opt. Fiber Commun. Conf., Paper PDP5C.5 (2012).
[Crossref]

T. Mizuno, T. Kobayashi, H. Takara, A. Sano, H. Kawakami, T. Nakagawa, Y. Miyamoto, Y. Abe, T. Goh, M. Oguma, T. Sakamoto, Y. Sasaki, I. Ishida, K. Takenaga, S. Matsuo, K. Saitoh, and T. Morioka, “12-core × 3-mode Dense Space Division Multiplexed Transmission over 40 km Employing Multt-carrier Signals with Parallel MIMO Equalization,” Proc. Opt. Fiber Commun. Conf., Paper Th5B.2 (2014).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Link configuration for a DMD managed link. The two different fibers are designed to fully compensate the delay spread between pulses that travel in different fiber modes by exchanging the fastest mode of fiber 1 the slowest mode of fiber 2.
Fig. 2
Fig. 2 Schematic representation of the delay spread between pulses that travel in different fiber modes. (a) without DMD management for no coupling (linear increase) and random unitary coupling (square root dependance). (b) with DMD management for no coupling (periodic compensation of the delay spread) and random unitary coupling (square root dependance).
Fig. 3
Fig. 3 Setup of the simulated transmission system. Each of the three fiber modes carries one QPSK signal that is ideally multiplex to the fiber eigenmodes. Each of 10 spans consists of 80 km DMD-compensated few mode fiber while different numbers of DMD-compensated segments are simulated. An ideal amplifier recovers all losses, including splice-induced mode dependent losses. Chromatic dispersion compensation, Noise loading and MIMO processing as well as error counting are performed after an ideal mode de-multiplexer.
Fig. 4
Fig. 4 Required OSNR to reach a Bit Error Rate (BER) of 10−3 as a function of the number of DMD-compensated segments per span for the signal in (a) the LP01 and (b) the LP11a mode. The simulation is performed for different input-powers in order to indicate the nonlinear origin of the error. Markers indicate the simulated transmission setups.
Fig. 5
Fig. 5 Required OSNR for a Bit Error Rate (BER) of 10−3 as function of the number of filter tabs for different DMD management scenarios with splice-induced mode coupling. (a) represents the signal that was launched into the LP01 and (b) LP11a mode.

Equations (2)

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A i z = α 2 A i + i β 0 i A i β 1 i A i t i β 2 i 2 2 A i t 2 + i ( 8 9 γ i i | A i | 2 + 4 3 j i γ i j | A j | 2 ) A i
f i j = | F i | 2 | F j | 2 d x d y | F i | 2 d x d y | F j | 2 d x d y .

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