Abstract

We study intermodal four-wave mixing (FWM) in few-mode fibers in the presence of birefringence fluctuations and random linear mode coupling. Two different intermodal FWM processes are investigated by including all nonlinear contributions to the phase-matching condition and FWM bandwidth. We find that one of the FWM processes has a much larger bandwidth than the other. We include random linear mode coupling among fiber modes using three different models based on an analysis of the impact of random coupling on differences of propagation constants between modes. We find that random coupling always reduces the FWM efficiency relative to its vale in the absence of linear coupling. The reduction factor is relatively small (about 3 dB) when only a few modes are linearly coupled but can become very large (> 40 dB) when all modes couple strongly. In the limit of a coupling length much shorter than the nonlinear length, intermodal FWM efficiency becomes vanishingly small. These results should prove useful in the context of space-division multiplexing with few-mode and multimode fibers.

© 2014 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  9. J. Demas, P. Steinvurzel, B. Tai, Y. Chen, and S. Ramachandran, “Two octaves of frequency generation by cascaded intermodal nonlinear mixing in solid optical fiber,” Proc. Conf. on Lasers and Electro-optics (2013). Paper CTu2E.5.
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  30. P. Harremoës, “Maximum entropy on compact groups,” Entropy 11, 222–237 (2009).
    [Crossref]
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2013 (4)

2012 (7)

2011 (1)

C. Koebele, M. Salsi, G. Charlet, and S. Bigo, “Nonlinear effects in mode division multiplexed transmission over few-mode optical fiber,” IEEE Photon. Technol. Lett. 23, 1316–1318 (2011).
[Crossref]

2009 (1)

P. Harremoës, “Maximum entropy on compact groups,” Entropy 11, 222–237 (2009).
[Crossref]

2008 (1)

2004 (1)

2001 (1)

M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett. 13, 836–838 (2001).
[Crossref]

1993 (1)

1986 (1)

C. Poole and R. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibres,” Electronics Letters 22, 1029–1030 (1986).
[Crossref]

1983 (1)

S. Rashleigh, “Origins and control of polarization effects in single-mode fibers,” J. Lightwave Technol. 1, 312–331 (1983).
[Crossref]

1981 (1)

1975 (2)

R. Olshansky, “Mode coupling effects in graded-index optical fibers,” Appl. Opt. 14, 935–945 (1975).
[Crossref] [PubMed]

R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. 11, 100–103 (1975).
[Crossref]

1974 (1)

R. Stolen, J. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24, 308–310 (1974).
[Crossref]

1971 (1)

D. Gloge and et al., “Weakly guiding fibers,” Appl. Opt 10, 2252–2258 (1971).
[Crossref] [PubMed]

Abedin, K. S.

Agrawal, G. P.

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

S. Mumtaz, R.-J. Essiambre, and G. P. Agrawal, “Reduction of nonlinear penalties due to linear coupling in multicore optical fibers,” IEEE Photon. Technol. Lett. 24, 1574–1576 (2012).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic Press, New York, 2012).

Antonelli, C.

Ashkin, A.

R. Stolen, J. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24, 308–310 (1974).
[Crossref]

Bigo, S.

C. Koebele, M. Salsi, G. Charlet, and S. Bigo, “Nonlinear effects in mode division multiplexed transmission over few-mode optical fiber,” IEEE Photon. Technol. Lett. 23, 1316–1318 (2011).
[Crossref]

Bjorkholm, J.

R. Stolen, J. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24, 308–310 (1974).
[Crossref]

Blondel, M.

M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett. 13, 836–838 (2001).
[Crossref]

Bolle, C.

Bösch, M.

Boyd, R. W.

R. W. Boyd, Nonlinear optics (Academic Press, 1992).

Breuillard, E.

E. Breuillard, “Random walks on lie groups,” Survey, http://www.math.u-psud.fr/breuilla/part0gb.pdf .

Burrows, E. C.

Charlet, G.

C. Koebele, M. Salsi, G. Charlet, and S. Bigo, “Nonlinear effects in mode division multiplexed transmission over few-mode optical fiber,” IEEE Photon. Technol. Lett. 23, 1316–1318 (2011).
[Crossref]

Chen, X. S.

X. S. Chen, W. Li, and W. W. Xu, “Perturbation analysis of the eigenvector matrix and singular vector matrices,” Taiwanese J. Math. 16, 179–194 (2012).

Chen, Y.

J. Demas, P. Steinvurzel, B. Tai, Y. Chen, and S. Ramachandran, “Two octaves of frequency generation by cascaded intermodal nonlinear mixing in solid optical fiber,” Proc. Conf. on Lasers and Electro-optics (2013). Paper CTu2E.5.

Chraplyvy, A. R.

R.-J. Essiambre, R. Ryf, M. A. Mestre, 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, 539–541 (2013).
[Crossref]

Defosse, Y.

M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett. 13, 836–838 (2001).
[Crossref]

Demas, J.

J. Demas, P. Steinvurzel, B. Tai, Y. Chen, and S. Ramachandran, “Two octaves of frequency generation by cascaded intermodal nonlinear mixing in solid optical fiber,” Proc. Conf. on Lasers and Electro-optics (2013). Paper CTu2E.5.

Ellis, A. D.

Esmaeelpour, M.

Essiambre, R.-J.

R.-J. Essiambre, R. Ryf, M. A. Mestre, 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, 539–541 (2013).
[Crossref]

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

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[Crossref]

S. Mumtaz, R.-J. Essiambre, and G. P. Agrawal, “Reduction of nonlinear penalties due to linear coupling in multicore optical fibers,” IEEE Photon. Technol. Lett. 24, 1574–1576 (2012).
[Crossref]

R.-J. Essiambre, R. W. Tkach, and R. Ryf, “Fiber nonlinearity and capacity: Single-mode and multimode fibers,” in “Optical Fiber Telecommunications VI B,”, I. Kaminow, T. Li, and A. E. Willner, eds. (Academic Press, 2013), chap. 1, pp. 1–37.
[Crossref]

Fini, J. M.

Galtarossa, A.

Gloge, D.

D. Gloge and et al., “Weakly guiding fibers,” Appl. Opt 10, 2252–2258 (1971).
[Crossref] [PubMed]

Gnauck, A. H.

R.-J. Essiambre, R. Ryf, M. A. Mestre, 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, 539–541 (2013).
[Crossref]

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[Crossref]

guang Sun, J.

G. W. Stewart and J. guang Sun, Matrix Perturbation Theory, Computer Science and Scientific computing (Academic Press, 1990).

Gunning, F. C.

Harremoës, P.

P. Harremoës, “Maximum entropy on compact groups,” Entropy 11, 222–237 (2009).
[Crossref]

Horak, P.

Jiang, X.

R.-J. Essiambre, R. Ryf, M. A. Mestre, 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, 539–541 (2013).
[Crossref]

Kato, T.

T. Kato, Perturbation theory for linear operators, vol. 132 of Grundlehren der mathematischen Wissenschaften (Springer, 1995).

Koebele, C.

C. Koebele, M. Salsi, G. Charlet, and S. Bigo, “Nonlinear effects in mode division multiplexed transmission over few-mode optical fiber,” IEEE Photon. Technol. Lett. 23, 1316–1318 (2011).
[Crossref]

Li, W.

X. S. Chen, W. Li, and W. W. Xu, “Perturbation analysis of the eigenvector matrix and singular vector matrices,” Taiwanese J. Math. 16, 179–194 (2012).

Lin, C.

Lingle, R.

R.-J. Essiambre, R. Ryf, M. A. Mestre, 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, 539–541 (2013).
[Crossref]

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[Crossref]

Mac Suibhne, N.

Marcuse, D.

D. Marcuse, Theory of dielectric optical waveguides (Academic Press, 1991).

Marhic, M. E.

M. E. Marhic, Fiber optical parametric amplifiers, oscillators and related devices (Cambridge university press, 2008).

McCurdy, A.

Mecozzi, A.

Mégret, P.

M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett. 13, 836–838 (2001).
[Crossref]

Mestre, M. A.

R.-J. Essiambre, R. Ryf, M. A. Mestre, 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, 539–541 (2013).
[Crossref]

Mumtaz, S.

Olshansky, R.

Palmieri, L.

A. Galtarossa and L. Palmieri, “Spatially resolved PMD measurements,” J. Lightwave Technol. 22, 1103–1115 (2004).
[Crossref]

L. Palmieri, “Coupling mechanism in multimode fibers,” in Proc. SPIE 9009, Next-Generation Optical Communication: Components, Sub-Systems, and Systems III (2013). Paper 90090G.

Peckham, D. W.

Poletti, F.

Poole, C.

C. Poole and R. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibres,” Electronics Letters 22, 1029–1030 (1986).
[Crossref]

Ramachandran, S.

J. Demas, P. Steinvurzel, B. Tai, Y. Chen, and S. Ramachandran, “Two octaves of frequency generation by cascaded intermodal nonlinear mixing in solid optical fiber,” Proc. Conf. on Lasers and Electro-optics (2013). Paper CTu2E.5.

Randel, S.

Rashleigh, S.

S. Rashleigh, “Origins and control of polarization effects in single-mode fibers,” J. Lightwave Technol. 1, 312–331 (1983).
[Crossref]

Rogers, A. J.

M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett. 13, 836–838 (2001).
[Crossref]

Ryf, R.

R.-J. Essiambre, R. Ryf, M. A. Mestre, 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, 539–541 (2013).
[Crossref]

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[Crossref]

R.-J. Essiambre, R. W. Tkach, and R. Ryf, “Fiber nonlinearity and capacity: Single-mode and multimode fibers,” in “Optical Fiber Telecommunications VI B,”, I. Kaminow, T. Li, and A. E. Willner, eds. (Academic Press, 2013), chap. 1, pp. 1–37.
[Crossref]

Salsi, M.

C. Koebele, M. Salsi, G. Charlet, and S. Bigo, “Nonlinear effects in mode division multiplexed transmission over few-mode optical fiber,” IEEE Photon. Technol. Lett. 23, 1316–1318 (2011).
[Crossref]

Shtaif, M.

Sibbett, W.

Sierra, A.

Steinvurzel, P.

J. Demas, P. Steinvurzel, B. Tai, Y. Chen, and S. Ramachandran, “Two octaves of frequency generation by cascaded intermodal nonlinear mixing in solid optical fiber,” Proc. Conf. on Lasers and Electro-optics (2013). Paper CTu2E.5.

Stewart, G. W.

G. W. Stewart and J. guang Sun, Matrix Perturbation Theory, Computer Science and Scientific computing (Academic Press, 1990).

Stolen, R.

R. Stolen, M. Bösch, and C. Lin, “Phase matching in birefringent fibers,” Opt. Lett. 6, 213–215 (1981).
[Crossref] [PubMed]

R. Stolen, J. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24, 308–310 (1974).
[Crossref]

Stolen, R. H.

R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. 11, 100–103 (1975).
[Crossref]

Su, Z.

Sun, Y.

R.-J. Essiambre, R. Ryf, M. A. Mestre, 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, 539–541 (2013).
[Crossref]

Sygletos, S.

Tai, B.

J. Demas, P. Steinvurzel, B. Tai, Y. Chen, and S. Ramachandran, “Two octaves of frequency generation by cascaded intermodal nonlinear mixing in solid optical fiber,” Proc. Conf. on Lasers and Electro-optics (2013). Paper CTu2E.5.

Taunay, T. F.

Tkach, R. W.

R.-J. Essiambre, R. Ryf, M. A. Mestre, 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, 539–541 (2013).
[Crossref]

R.-J. Essiambre, R. W. Tkach, and R. Ryf, “Fiber nonlinearity and capacity: Single-mode and multimode fibers,” in “Optical Fiber Telecommunications VI B,”, I. Kaminow, T. Li, and A. E. Willner, eds. (Academic Press, 2013), chap. 1, pp. 1–37.
[Crossref]

Wagner, R.

C. Poole and R. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibres,” Electronics Letters 22, 1029–1030 (1986).
[Crossref]

Winzer, P. J.

Wuilpart, M.

M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett. 13, 836–838 (2001).
[Crossref]

Xu, W. W.

X. S. Chen, W. Li, and W. W. Xu, “Perturbation analysis of the eigenvector matrix and singular vector matrices,” Taiwanese J. Math. 16, 179–194 (2012).

Yan, M. F.

Zhu, B.

Zhu, X.

Appl. Opt (1)

D. Gloge and et al., “Weakly guiding fibers,” Appl. Opt 10, 2252–2258 (1971).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. Stolen, J. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24, 308–310 (1974).
[Crossref]

Electronics Letters (1)

C. Poole and R. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibres,” Electronics Letters 22, 1029–1030 (1986).
[Crossref]

Entropy (1)

P. Harremoës, “Maximum entropy on compact groups,” Entropy 11, 222–237 (2009).
[Crossref]

IEEE J. Quantum Electron. (1)

R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. 11, 100–103 (1975).
[Crossref]

IEEE Photon. Technol. Lett. (4)

C. Koebele, M. Salsi, G. Charlet, and S. Bigo, “Nonlinear effects in mode division multiplexed transmission over few-mode optical fiber,” IEEE Photon. Technol. Lett. 23, 1316–1318 (2011).
[Crossref]

S. Mumtaz, R.-J. Essiambre, and G. P. Agrawal, “Reduction of nonlinear penalties due to linear coupling in multicore optical fibers,” IEEE Photon. Technol. Lett. 24, 1574–1576 (2012).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Wave configurations and notations for two non-degenerate IM-FWM processes: (a) PROC1 and (b) PROC2.
Fig. 2
Fig. 2 Generation of the idler through PROC1 in a 4.7-km-long FMF supporting three spatial modes. Comparison of input (solid green) and output (dotted red) spectra indicates that the idler is created at the expected wavelength of 1538 nm. The idler is generated in the LP11a mode.
Fig. 3
Fig. 3 Same as in Fig. 2 but for PROC2. The wavelength of the idler is λidler = 1536 nm. A second idler is generated in the LP01 mode at λ = 1540nm through cascaded IM-FWM (see main text).
Fig. 4
Fig. 4 (a) Efficiency as a function of λprobe and λpump2 for PROC1 when λpump1 = 1530 nm. (b) Efficiency curve for λpump2 = 1554 nm (full green curve) from Eq. (22). Numerical result (input power Pp1 = 4, Pp2 = 6, PB = − 6 dBm, dotted red curve) agrees well with the analytic approach.
Fig. 5
Fig. 5 Same as Fig. 4 but for PROC2 and with λpump2 = 1546 nm from Eq. (23). Notice the difference in scale of the λprobe axis and difference in bandwidth of the process. The analytical formula agrees well with the numerical results.
Fig. 6
Fig. 6 Efficiency curves all shift to shorter wavelengths as Pp1 or Pp2 increase for (a) PROC1 while they shift in opposite directions for (b) PROC2 depending on whether Pp1 or Pp2 is increased. Powers vector is defined as P = (Pp1, Pp2, PB).
Fig. 7
Fig. 7 Gray-coded values of the elements of the matrix T with white and black representing 0 and 1, respectively. The values of p with p = σ are shown above each plot. Top row N = 1; bottom row N = 10.
Fig. 8
Fig. 8 Gray-coded values of the elements of the matrix T for N = 10 when σ = 1 is fixed and p decreases from 0.1 to 10−4.

Tables (3)

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Table 1 Dispersion parameters for the LP01 and LP11 spatial modes for the 6 mode FMF.

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Table 2 Power of the idler (in dBm) generated by the IM-FWM PROC2 for five different realizations of the random linear mode coupling for the three models considered.

Tables Icon

Table 3 Same as Table 2 but for 3 dB lower powers for the two pumps and the probe.

Equations (41)

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ω l = ω i + ω j ω k ,
Δ β = β ( m ) ( ω i ) + β ( n ) ( ω j ) β ( o ) ( ω k ) β ( p ) ( ω l ) 0 ,
β ( r ) ( ω ) β 0 ( r ) + β 1 ( r ) Δ ω + β 2 ( r ) 2 ( Δ ω ) 2 + β 3 ( r ) 6 ( Δ ω ) 3 + ,
ω 4 = ω 1 ω 2 + ω 3 , ( PROC 1 )
ω 4 = ω 1 + ω 2 ω 3 . ( PROC 2 )
β 1 11 + β 2 11 2 ( Δ ω 1 + Δ ω 4 ) = β 1 01 + β 2 01 2 ( Δ ω 2 + Δ ω 3 ) ,
A z = α 2 A + i B 0 A B 1 A t i B 2 2 2 A t 2 + B 3 6 3 A t 3 + i QA + i γ 3 [ ( A T G ( 2 ) A ) G ( 2 ) * A * + 2 ( A H G ( 1 ) A ) G ( 1 ) A ] dxdy .
A = [ A LP 01 x , A LP 01 y , A LP 11 ax , A LP 11 ay , A LP 11 bx , A LP 11 by ] T .
G i j ( 1 ) = Γ i j F i * F j , G i j ( 2 ) = Γ i j F i F j .
Γ i j = δ [ mod ( i + j + 1 , 2 ) , 1 ] .
f l m n p = F l * F m F n F p * d x d y .
f l m n p = { 1 for 4 waves in LP 01 mode 0.747 for 4 waves in LP 11 a or LP 11 b 0.496 for 2 in LP 01 and 2 in LP 11 a or LP 11 b 0.249 for 2 in LP 11 a and 2 in LP 11 b 0 for all other cases
q i j ( z ) = Γ i j k 0 2 n eff Δ ε ( x , y , z ) F i ( x , y ) F j * ( x , y ) d x d y .
A 1 z = D ^ 1 A 1 + i γ f 1111 | A 1 | 2 A 1 + i γ f 1331 A 3 2 A 1 * + i γ ( f 3131 + f 3311 ) | A 3 | 2 A 1 ,
A 3 z = D ^ 3 A 3 + i γ f 3333 | A 3 | 2 A 3 + i γ f 3113 A 1 2 A 3 * + i γ ( f 1313 + f 1133 ) | A 1 | 2 A 3 ,
D ^ r = i β 0 ( r ) β 1 ( r ) t i β 2 ( r ) 2 2 t 2 + β 3 ( r ) 6 3 t 3 α 2 .
d A I d z i γ ( f 3131 + f 3311 ) A p 1 ( 0 ) A p 2 ( 0 ) A B * ( 0 ) e 3 α z / 2 ( α / 2 ) A I ,
η = [ sin ( Δ β L / 2 ) ( Δ β L / 2 ) ] 2 ,
Δ β = ( Δ ω 1 Δ ω 4 ) { β 1 11 β 1 01 + 1 2 β 2 11 ( Δ ω 1 + Δ ω 4 ) 1 2 β 2 01 ( Δ ω 2 + Δ ω 3 ) + 1 6 β 3 11 [ Δ ω 1 2 + Δ ω 1 Δ ω 4 + Δ ω 4 2 ] 1 6 β 3 01 [ Δ ω 2 2 + Δ ω 2 Δ ω 3 + Δ ω 3 2 ] } ,
( Δ β ) 1 = 1 2 δ ω ( Δ ω 2 Δ ω 3 + δ ω ) [ β 2 01 + β 2 11 + 1 3 β 3 11 ( 3 Δ ω 1 2 Δ ω 2 + 2 Δ ω 3 δ ω ) + 1 3 β 3 01 ( Δ ω 3 + 2 Δ ω 2 + δ ω ) ] .
( Δ β ) 2 = 1 2 δ ω ( Δ ω 2 Δ ω 3 δ ω ) [ β 2 11 β 2 01 + 1 3 β 3 11 ( 3 Δ ω 1 + 2 Δ ω 2 2 Δ ω 3 + δ ω ) 1 3 β 3 01 ( Δ ω 3 + 2 Δ ω 2 + δ ω ) ] .
( Δ β ) 1 1 2 δ ω ( Δ ω 2 Δ ω 3 ) M 1 ,
( Δ β ) 2 1 2 δ ω ( Δ ω 2 Δ ω 3 ) M 2 ,
M 1 = β 2 11 + 1 3 β 3 11 ( 3 Δ ω 1 2 Δ ω 2 + 2 Δ ω 3 ) + β 2 01 + 1 3 β 3 01 ( Δ ω 3 + 2 Δ ω 2 ) ,
M 2 = β 2 11 + 1 3 β 3 11 ( 3 Δ ω 1 + 2 Δ ω 2 2 Δ ω 3 ) β 2 01 1 3 β 3 01 ( Δ ω 3 + 2 Δ ω 2 ) .
( Δ β ) 1 N L = γ L ( f 3333 P p 1 + f 1111 P p 2 ) ,
( Δ β ) 2 N L = γ L ( f 3333 P p 1 f 1111 P p 2 ) .
A ( z ) z = i [ B 0 + Q ( z ) ] A ( z ) .
R ( z ) = [ r 11 p r 12 p 0 0 0 0 r 21 p r 22 p 0 0 0 0 0 0 r 11 l r 12 l 0 0 0 0 r 21 l r 22 l 0 0 0 0 0 0 r 11 m r 12 m 0 0 0 0 r 21 m r 22 m ] ,
T Δ ( z ) = R ( z ) exp [ i ( B 0 + Q ( z ) ) Δ z ] .
T 1 ( z ) = [ T 11 p T 12 p 0 0 0 0 T 21 p T 22 p 0 0 0 0 0 0 T 11 l T 12 l 0 0 0 0 T 21 l T 22 l 0 0 0 0 0 0 T 11 m T 12 m 0 0 0 0 T 21 m T 22 m ] .
T 2 ( z ) = [ T 11 p T 12 p 0 0 0 0 T 21 p T 22 p 0 0 0 0 0 0 T 11 l T 12 l T 13 l T 14 l 0 0 T 21 l T 22 l T 23 l T 24 l 0 0 T 31 l T 32 l T 33 l T 34 l 0 0 T 41 l T 42 l T 43 l T 44 l ] ,
T 3 ( z ) = [ T 11 T 12 T 13 T 14 T 15 T 16 T 21 T 22 T 23 T 24 T 25 T 26 T 31 T 32 T 33 T 34 T 35 T 36 T 41 T 42 T 43 T 44 T 45 T 46 T 51 T 52 T 53 T 54 T 55 T 56 T 61 T 62 T 63 T 64 T 65 T 66 ] .
B 0 = Δ B ^ Σ I , A = A ^ exp ( i Σ z ) ,
z = z ^ / Δ , Q = Δ σ Q ^ ,
A ^ ( z ^ ) z ^ = i [ B ^ + σ Q ^ ( z ^ ) ] A ^ ( z ^ ) ,
A ^ ( z ^ ) = T ^ ( z ^ ) A ^ ,
T ^ ( z ^ 2 ) = exp [ i ( B ^ + σ Q ^ ( z ^ ) ) ( z ^ 2 z ^ 1 ) ] T ^ ( z ^ 1 ) .
T = j = 1 N exp [ ( i / p ) ( B ^ + σ Q ^ j ) ] ,
ε = δ M F min i j | λ i λ j | , α = ( 1 + 1 1 / n ) M 2 min i j | λ i λ j | .
U ˜ U F 2 ε [ 1 2 α ε + 1 4 α ε ε 2 ] 1 / 2 .

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