Abstract

Recently mode-division-multiplexing (MDM) has been widely investigated to enhance fiber optics capacity, in which modes or mode groups in few-mode fiber (FMF) or multi-mode fiber (MMF) are exploited as different spatial channels for data transmission. For short-reach applications, significantly reducing inter-spatial-channel crosstalk to avoid coherent detection and multiple-input-multiple-output (MIMO) equalization is preferred. Currently most studies focus on the design of weakly-coupled FMFs and mode (de)multiplexers. Alternatively, in this work, a wavelength-interleaved (WI) scheme is proposed to mitigate inter-spatial-channel crosstalk by optimizing the design of direct detection (DD) MDM and wavelength-division-multiplexing (WDM) system. In weakly-coupled MDM systems, crosstalk mainly comes from the adjacent spatial channels, and the signal-to-crosstalk beat interference (SCBI) constitutes main crosstalk impairment after square-law detection. The WI scheme interleaves the WDM grids in adjacent spatial channels by half WDM channel spacing and uses an electrical low-pass filtering (ELPF) to remove out-of-band SCBI. The effectiveness of SCBI suppression is theoretically analyzed. The feasibility of WI scheme is experimentally verified by 3-mode 3-wavelength MDM-WDM transmission over 500-m OM3 MMF. Enabled by WI scheme, record 120-km 10G-per-channel MDM-WDM transmission over 2-mode FMF without MIMO equalization is successfully demonstrated. The WI scheme is promising to enhance the performance of short reach or even metro MDM optics.

© 2017 Optical Society of America

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References

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2016 (3)

2015 (2)

A. Grieco, G. Porter, and Y. Fainman, “Integrated Space-Division Multiplexer for Application to Data Center Networks,” IEEE J. Quantum Electron. 22(6), 8200106 (2015).

F. Ren, J. Li, T. Hu, R. Tang, J. Yu, Q. Mo, Y. He, Z. Chen, and Z. Li, “Cascaded mode-division-multiplexing and time-division-multiplexing passive optical network based on low mode-crosstalk FMF and mode MUX/DEMUX,” Photon. Journal 7(5), 7903509 (2015).

2014 (2)

2012 (1)

2009 (1)

1981 (1)

K. Petermann and E. Weidel, “Semiconductor Laser Noise in an Interferometer System,” IEEE J. Quantum Electron. 17(7), 1251–1256 (1981).
[Crossref]

1973 (1)

D. Marcuse, “Losses and impulse response of a parabolic index fiber with random bends,” Bell Syst. Tech. J. 52(8), 1423–1437 (1973).
[Crossref]

1972 (1)

D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. 51(1), 229–237 (1972).
[Crossref]

Amezcua-Correa, R.

Antonio-Lopez, J.

Bland-Hawthorn, J.

Bolle, C.

Burrows, E.

Chand, N.

Chen, Z.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of Bidirectional PON Based on Mode Division Multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201–1204 (2016).
[Crossref]

Z. Wu, J. Li, D. Ge, F. Ren, P. Zhu, Q. Mo, Z. Li, Z. Chen, and Y. He, “Demonstration of all-optical MDM/WDM switching for short-reach networks,” Opt. Express 24(19), 21609–21618 (2016).
[Crossref] [PubMed]

F. Ren, J. Li, T. Hu, R. Tang, J. Yu, Q. Mo, Y. He, Z. Chen, and Z. Li, “Cascaded mode-division-multiplexing and time-division-multiplexing passive optical network based on low mode-crosstalk FMF and mode MUX/DEMUX,” Photon. Journal 7(5), 7903509 (2015).

Du, C.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of Bidirectional PON Based on Mode Division Multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201–1204 (2016).
[Crossref]

Effenberger, F.

Ercan, B.

Esmaeelpour, M.

Essiambre, R.

Fainman, Y.

A. Grieco, G. Porter, and Y. Fainman, “Integrated Space-Division Multiplexer for Application to Data Center Networks,” IEEE J. Quantum Electron. 22(6), 8200106 (2015).

Fontaine, N. K.

Ge, D.

Gnauck, A.

Grieco, A.

A. Grieco, G. Porter, and Y. Fainman, “Integrated Space-Division Multiplexer for Application to Data Center Networks,” IEEE J. Quantum Electron. 22(6), 8200106 (2015).

He, Y.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of Bidirectional PON Based on Mode Division Multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201–1204 (2016).
[Crossref]

Z. Wu, J. Li, D. Ge, F. Ren, P. Zhu, Q. Mo, Z. Li, Z. Chen, and Y. He, “Demonstration of all-optical MDM/WDM switching for short-reach networks,” Opt. Express 24(19), 21609–21618 (2016).
[Crossref] [PubMed]

F. Ren, J. Li, T. Hu, R. Tang, J. Yu, Q. Mo, Y. He, Z. Chen, and Z. Li, “Cascaded mode-division-multiplexing and time-division-multiplexing passive optical network based on low mode-crosstalk FMF and mode MUX/DEMUX,” Photon. Journal 7(5), 7903509 (2015).

Hu, T.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of Bidirectional PON Based on Mode Division Multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201–1204 (2016).
[Crossref]

F. Ren, J. Li, T. Hu, R. Tang, J. Yu, Q. Mo, Y. He, Z. Chen, and Z. Li, “Cascaded mode-division-multiplexing and time-division-multiplexing passive optical network based on low mode-crosstalk FMF and mode MUX/DEMUX,” Photon. Journal 7(5), 7903509 (2015).

Huang, B.

Kahn, J.

Ke, Y.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of Bidirectional PON Based on Mode Division Multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201–1204 (2016).
[Crossref]

Leon-Saval, S. G.

Li, G.

Li, J.

Z. Wu, J. Li, D. Ge, F. Ren, P. Zhu, Q. Mo, Z. Li, Z. Chen, and Y. He, “Demonstration of all-optical MDM/WDM switching for short-reach networks,” Opt. Express 24(19), 21609–21618 (2016).
[Crossref] [PubMed]

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of Bidirectional PON Based on Mode Division Multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201–1204 (2016).
[Crossref]

F. Ren, J. Li, T. Hu, R. Tang, J. Yu, Q. Mo, Y. He, Z. Chen, and Z. Li, “Cascaded mode-division-multiplexing and time-division-multiplexing passive optical network based on low mode-crosstalk FMF and mode MUX/DEMUX,” Photon. Journal 7(5), 7903509 (2015).

Li, Z.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of Bidirectional PON Based on Mode Division Multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201–1204 (2016).
[Crossref]

Z. Wu, J. Li, D. Ge, F. Ren, P. Zhu, Q. Mo, Z. Li, Z. Chen, and Y. He, “Demonstration of all-optical MDM/WDM switching for short-reach networks,” Opt. Express 24(19), 21609–21618 (2016).
[Crossref] [PubMed]

F. Ren, J. Li, T. Hu, R. Tang, J. Yu, Q. Mo, Y. He, Z. Chen, and Z. Li, “Cascaded mode-division-multiplexing and time-division-multiplexing passive optical network based on low mode-crosstalk FMF and mode MUX/DEMUX,” Photon. Journal 7(5), 7903509 (2015).

Lingle, R.

Liu, H.

Liu, X.

Liu, Z.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of Bidirectional PON Based on Mode Division Multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201–1204 (2016).
[Crossref]

Mao, W.

Marcuse, D.

D. Marcuse, “Losses and impulse response of a parabolic index fiber with random bends,” Bell Syst. Tech. J. 52(8), 1423–1437 (1973).
[Crossref]

D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. 51(1), 229–237 (1972).
[Crossref]

McCurdy, A.

Mo, Q.

Z. Wu, J. Li, D. Ge, F. Ren, P. Zhu, Q. Mo, Z. Li, Z. Chen, and Y. He, “Demonstration of all-optical MDM/WDM switching for short-reach networks,” Opt. Express 24(19), 21609–21618 (2016).
[Crossref] [PubMed]

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of Bidirectional PON Based on Mode Division Multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201–1204 (2016).
[Crossref]

F. Ren, J. Li, T. Hu, R. Tang, J. Yu, Q. Mo, Y. He, Z. Chen, and Z. Li, “Cascaded mode-division-multiplexing and time-division-multiplexing passive optical network based on low mode-crosstalk FMF and mode MUX/DEMUX,” Photon. Journal 7(5), 7903509 (2015).

Mumtaz, S.

Panicker, R.

Peckham, D.

Petermann, K.

K. Petermann and E. Weidel, “Semiconductor Laser Noise in an Interferometer System,” IEEE J. Quantum Electron. 17(7), 1251–1256 (1981).
[Crossref]

Porter, G.

A. Grieco, G. Porter, and Y. Fainman, “Integrated Space-Division Multiplexer for Application to Data Center Networks,” IEEE J. Quantum Electron. 22(6), 8200106 (2015).

Randel, S.

Ren, F.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of Bidirectional PON Based on Mode Division Multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201–1204 (2016).
[Crossref]

Z. Wu, J. Li, D. Ge, F. Ren, P. Zhu, Q. Mo, Z. Li, Z. Chen, and Y. He, “Demonstration of all-optical MDM/WDM switching for short-reach networks,” Opt. Express 24(19), 21609–21618 (2016).
[Crossref] [PubMed]

F. Ren, J. Li, T. Hu, R. Tang, J. Yu, Q. Mo, Y. He, Z. Chen, and Z. Li, “Cascaded mode-division-multiplexing and time-division-multiplexing passive optical network based on low mode-crosstalk FMF and mode MUX/DEMUX,” Photon. Journal 7(5), 7903509 (2015).

Ryf, R.

Salazar-Gil, J. R.

Shemirani, M.

Sierra, A.

Sillard, P.

Tang, R.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of Bidirectional PON Based on Mode Division Multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201–1204 (2016).
[Crossref]

F. Ren, J. Li, T. Hu, R. Tang, J. Yu, Q. Mo, Y. He, Z. Chen, and Z. Li, “Cascaded mode-division-multiplexing and time-division-multiplexing passive optical network based on low mode-crosstalk FMF and mode MUX/DEMUX,” Photon. Journal 7(5), 7903509 (2015).

Velázquez-Benítez, A.

Weidel, E.

K. Petermann and E. Weidel, “Semiconductor Laser Noise in an Interferometer System,” IEEE J. Quantum Electron. 17(7), 1251–1256 (1981).
[Crossref]

Wen, H.

Winzer, P.

Wu, Z.

Xia, C.

Yu, J.

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of Bidirectional PON Based on Mode Division Multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201–1204 (2016).
[Crossref]

F. Ren, J. Li, T. Hu, R. Tang, J. Yu, Q. Mo, Y. He, Z. Chen, and Z. Li, “Cascaded mode-division-multiplexing and time-division-multiplexing passive optical network based on low mode-crosstalk FMF and mode MUX/DEMUX,” Photon. Journal 7(5), 7903509 (2015).

Zheng, H.

Zhu, P.

Bell Syst. Tech. J. (2)

D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. 51(1), 229–237 (1972).
[Crossref]

D. Marcuse, “Losses and impulse response of a parabolic index fiber with random bends,” Bell Syst. Tech. J. 52(8), 1423–1437 (1973).
[Crossref]

IEEE J. Quantum Electron. (2)

K. Petermann and E. Weidel, “Semiconductor Laser Noise in an Interferometer System,” IEEE J. Quantum Electron. 17(7), 1251–1256 (1981).
[Crossref]

A. Grieco, G. Porter, and Y. Fainman, “Integrated Space-Division Multiplexer for Application to Data Center Networks,” IEEE J. Quantum Electron. 22(6), 8200106 (2015).

IEEE Photonics Technol. Lett. (1)

T. Hu, J. Li, F. Ren, R. Tang, J. Yu, Q. Mo, Y. Ke, C. Du, Z. Liu, Y. He, Z. Li, and Z. Chen, “Demonstration of Bidirectional PON Based on Mode Division Multiplexing,” IEEE Photonics Technol. Lett. 28(11), 1201–1204 (2016).
[Crossref]

J. Lightwave Technol. (3)

Nat. Photonics (1)

P. Winzer, “Making spatial multiplexing a reality,” Nat. Photonics 8(5), 345–348 (2014).
[Crossref]

Opt. Express (2)

Photon. Journal (1)

F. Ren, J. Li, T. Hu, R. Tang, J. Yu, Q. Mo, Y. He, Z. Chen, and Z. Li, “Cascaded mode-division-multiplexing and time-division-multiplexing passive optical network based on low mode-crosstalk FMF and mode MUX/DEMUX,” Photon. Journal 7(5), 7903509 (2015).

Other (15)

J. Luo, J. Li, Q. Sui, and Z. Li, “30 Gb/s 2×2 MDM-DD-OFDM Transmission over 200m Conventional MMF Link without MIMO Compensation,” in Asia Communications and Photonics Conference (OSA, 2015), paper AS4D. 2.
[Crossref]

J. Carpenter, B. Eggleton, and J. Schröder, “LCoS-based devices for MDM,” in Optical Fiber Communication Conference (OSA, 2015), paper W1A.1.
[Crossref]

C. Koebele, M. Salsi, L. Milord, R. Ryf, C. Bolle, P. Sillard, S. Bigo, and G. Charlet, “40km Transmission of Five Mode Division Multiplexed Data Streams at 100Gb/s with low MIMO-DSP Complexity,” in European Conference on Optical Communication (IEEE, 2011), paper Th.13.C.3.
[Crossref]

Cisco, “Cisco Global Cloud Index: Forecast and Methodology, 2015–2020,” Cisco White Paper, (2016).

G. Keiser, Optical Fiber Communication (McGraw-Hill, 2000), Chap. 7.

M. Fiorani, M. Tornatore, J. Chen, L. Wosinska, and B. Mukherjee, “Optical Spatial Division Multiplexing for Ultra-High-Capacity Modular Data Centers,” in Optical Fiber Communication Conference (OSA, 2016), paper Tu2H. 2.
[Crossref]

M. Li, “MMF for High Data Rate and Short Length Applications,” in Optical Fiber Communication Conference (OSA, 2014), paper M3F.1.
[Crossref]

Y. Li, N. Hua, X. Zheng, and G. Li, “CapEx advantages of few-mode fiber networks,” in Optical Fiber Communication Conference (OSA, 2015), paper Th2A.43.
[Crossref]

K. Ho and J. Kahn, “Mode Coupling and its Impact on Spatially Multiplexed Systems,” in Optical Fiber Telecommunications VI-B: Systems and Networks, I. Kaminow, T. Li, and A. Willner ed. (Academic Press, 2013).

H. Li, G. Gao, J. Zhang, C. Xu, M. Lan, A. Fei, and W. Gu, “2.5Gbit/s MDM Transmission over 180m Two-mode Fiber Using 1550nm SFP without MIMO Processing,” in International Conference on Optical Communications and Networks (IEEE, 2016), paper T3-O-05.

P. Sillard and D. Molin, “A Review of Few-Mode Fibers for Space-Division Multiplexed Transmissions,” in European Conference on Optical Communication (IEEE, 2013), paper Mo.3.A.1.
[Crossref]

L. Ma, S. Jiang, J. Du, C. Yang, W. Tong, and Z. He, “Ring-assisted 7-LP-mode Fiber with Ultra-low Inter-mode Crosstalk,” in Asia Communications and Photonics Conference (OSA, 2016), paper AS4A.5.
[Crossref]

G. Keiser, Optical Fiber Communication (McGraw-Hill, 2000), Chap.3.

Z. Wu, J. Li, Y. Tian, D. Ge, J. Zhu, Q. Mo, F. Ren, J. Yu, Z. Li, Z. Chen, and Y. He, “4-Mode MDM transmission over MMF with direct detection enabled by cascaded mode-selective couplers,” in Optical Fiber Communication Conference (OSA, 2017), paper Th2A.40.
[Crossref]

P. Sillard, “Few-Mode Fibers for Space Division multiplexing,” in Optical Fiber Communication Conference (OSA, 2015), paper Th1J. 1.

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Figures (13)

Fig. 1
Fig. 1 (a) Illustration of WA and WI MDM-WDM transmission scheme. (b) Principle of WI scheme for crosstalk mitigation
Fig. 2
Fig. 2 Illustration of power spectrum of SCBI and useful signal in an IM-DD MDM system where LP01, LP11 and LP21 are multiplexed. Symbols for simulation results considering mode dependent dispersion. Lines for theoretical results given by Eq. (8).
Fig. 3
Fig. 3 Experimental setup for 10-km 2 × 3 × 10-Gb/s MDM-WDM OOK transmission. VOA: variable optical attenuator.
Fig. 4
Fig. 4 BER performance under different relative CFOs for single wavelength 2 × 10-Gb/s MDM OOK signal after 10-km FMF transmission.
Fig. 5
Fig. 5 BER performance under different received powers for three different WDM channels after 10-km FMF transmission. Symbols represent experiment results. Lines represent fitting results using least squares method.
Fig. 6
Fig. 6 Experimental setup for 500-m 3 × 3 × 10-Gb/s MDM-WDM OOK transmission.
Fig. 7
Fig. 7 Transmitted optical spectra in (a) WA and (b) WI scheme. (c) Received optical spectra in WI scheme after 500-m MMF transmission.
Fig. 8
Fig. 8 Performance comparison between WA and WI scheme.
Fig. 9
Fig. 9 Setup for 2 × 3 × 10-Gb/s MDM-WDM OOK recirculating loop transmission.
Fig. 10
Fig. 10 (a) Transmitted optical spectra in WI scheme. (b) Received optical spectra in WI scheme after 10-km FMF transmission.
Fig. 11
Fig. 11 Q2-factors of WDM channels in (a) LP01 mode and (b) LP11 mode at different transmission distance. The purple dotted lines denote the FEC limit @ BER = 1e−3.
Fig. 12
Fig. 12 Electrical eye diagrams of 10-Gb/s channel (a) CH3 and (b) CH4 in WA and WI schemes.
Fig. 13
Fig. 13 Long-term measurement of Q2-factor.

Tables (1)

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Table 1 Parameters of weakly-coupled FMF

Equations (26)

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s m ( t )= d m ( t ) e j[ ω 0 t+ Φ m ( t ) ] ,
Φ m ( t )=Δ ω m t+ ψ m ( t ).
r m ( t )= s m ( t )+ n[ 1,M ] nm Γ ¯ mn s n ( t ) ,
i m,SCBI ( t )=2 n[ 1,M ] nm | Γ ¯ mn | d m ( t ) d n ( t ) cos[ κ mn ( t ) ] ,
κ mn ( t )= Φ m ( t ) Φ n ( t ) θ mn .
P m.SCBI ( ω )= | F[ i m,SCBI ( t ) ] | 2
R m,SCBI ( τ )= i m,SCBI ( t+τ ) i m,SCBI ( t ) =2 n[ 1,M ] nm { | Γ ¯ mn | 2 R m,d ( τ ) R n,d ( τ ) e | τ |/ τ c cos[ ( Δ ω n Δ ω m )τ ] } ,
P m.SCBI ( ω )= 1 4 π 2 n[ 1,M ] nm { | Γ ¯ mn | 2 P m,d ( ω ) P n,d ( ω ) P laser ( ω ) [ δ( ω+Δ ω m Δ ω n )+δ( ω+Δ ω n Δ ω m ) ] }.
d A m ( z,ω ) dz = n[ 1,M ] nm K mn f( z ) A n ( z,ω ) e jΔ β nm ( ω )z .
Δ β nm ( ω )=Δ β nm ( 0 ) +Δ β nm ( 1 ) ω+Δ β nm ( 2 ) ω 2 = 2π λ Δ n eff,nm +Δ ρ nm ω+ Δ D nm λ 2 2πc ω 2 ,
Γ nn ( ω )= A n ( L,ω ) A n ( 0,ω ) 1,
Γ mn ( ω )= A m ( L,ω ) A n ( 0,ω ) K mn 0 L f( z ) e jΔ β nm ( ω )z dz = K mn 1 ( jΔ β nm ( ω ) ) 2 0 L Θ( z ) e jΔ β nm ( ω )z dz , mn,
Θ( z )= d 2 f( z ) d z 2
cos[ κ mn ( t+τ ) κ ml ( t ) ] =Re[ e j{ [ Φ m ( t+τ ) Φ m ( t ) ][ Φ n ( t+τ ) Φ l ( t ) ][ θ mn θ ml ] } ].
cos[ κ mn ( t+τ ) κ ml ( t ) ] =Re[ e j[ Φ m ( t+τ ) Φ m ( t ) ] e j[ Φ n ( t+τ ) Φ l ( t ) ] e j[ θ mn θ ml ] ].
ψ m ( t )= t ξ( u )du ,
e j[ ψ m ( t+τ ) ψ m ( t ) ] = e | τ | 2 τ c ,
e j[ ψ m ( t ) ] = e j[ ψ m ( t ) ψ m ( ) ] = e | t+ | 2 τ c =0.
e j[ Φ m ( t+τ ) Φ m ( t ) ] = e jΔ ω m τ e j[ ψ m ( t+τ ) ψ m ( t ) ] = e jΔ ω m τ e | τ | 2 τ c ,
e j[ Φ n ( t+τ ) Φ l ( t ) ] ={ e jΔ ω n ( t+τ ) e jΔ ω l t e j ψ n ( t+τ ) e j ψ l ( t ) =0, nl e jΔ ω n τ e | τ | 2 τ c , n=l = e jΔ ω n τ e | τ | 2 τ c δ( nl ).
cos[ κ mn ( t+τ ) κ ml ( t ) ] = e | τ | τ c cos[ ( Δ ω m Δ ω n )τ ]δ( nl ).
cos[ κ mn ( t+τ )+ κ ml ( t ) ] =Re[ e j[ Φ m ( t+τ )+ Φ m ( t ) ] e j[ Φ n ( t+τ )+ Φ l ( t ) ] e j[ θ mn + θ ml ] ].
e j[ Φ m ( t+τ )+ Φ m ( t ) ] = e jΔ ω m ( 2t+τ ) e j[ 2 ψ m ( t )+ ψ m ( t+τ ) ψ m ( t ) ] .
e j[ Φ m ( t+τ )+ Φ m ( t ) ] = e jΔ ω m ( 2t+τ ) e j2 ψ m ( t ) e j[ ψ m ( t+τ ) ψ m ( t ) ] =0.
cos[ κ mn ( t+τ )+ κ ml ( t ) ] =0.
cos[ κ mn ( t+τ ) ]cos[ κ ml ( t ) ] = 1 2 e | τ | τ c cos[ ( Δ ω n Δ ω m )τ ]δ( nl ).

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