Abstract

We demonstrate efficient four-wave mixing among different spatial modes in a 1-km long two-mode fiber at telecommunication wavelengths. Two pumps excite the LP01 and LP11 modes, respectively, while the probe signal excites the LP01 mode, and the phase conjugation (PC) and Bragg scattering (BS) idlers are generated in the LP11 mode. For these processes we experimentally characterize their phase matching efficiency and bandwidth and find that they depend critically on the wavelength separation of the two pumps, in good agreement with the numerical study we carried out. We also confirm experimentally that BS has a larger bandwidth than PC for the optimum choice of the pump wavelength separation.

© 2016 Optical Society of America

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References

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2016 (2)

2015 (1)

2014 (1)

2013 (3)

S. M. M. Friis, K. Rottwitt, and C. J. McKinstrie, “Raman and loss induced quantum noise in depleted fiber optical parametric amplifiers,” Opt. Express 21, 29320–29331 (2013).
[Crossref]

R.-J. 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, 535–538 (2013).
[Crossref]

R.-J. 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, 539–542 (2013).
[Crossref]

2012 (5)

2011 (1)

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

2010 (3)

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Raman-induced asymmetric pump noise transfer in fiber-optical parametric amplifiers,” IEEE Photon. Technol. Lett. 22, 386–388 (2010).
[Crossref]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref] [PubMed]

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18, 2884–2893 (2010).
[Crossref] [PubMed]

2009 (1)

M. Liscidini, J. E. Sipe, and L. G. Helt, “Continuous wave photon pair generation in silicon-on-insultator waveguides and ring resonators,” Opt. Express 24, 16558–16570 (2009).

2008 (2)

S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. 2, 498–513 (2008).
[Crossref]

F. Poletti and P. Horak, “Description of ultrashort pulse propagation in multimode optical fibers,” J. Opt. Soc. Am. B 25, 1645–1654 (2008).
[Crossref]

2005 (1)

2002 (2)

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 (2002).
[Crossref]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

1974 (1)

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

1973 (1)

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[Crossref]

Agha, I.

Agrawal, G. P.

Y. Xiao, R.-J. Essiambre, M. Desgroseilliers, A. M. Tulino, R. Ryf, S. Mumtaz, and G. P. Agrawal, “Theory of intermodal four-wave mixing with random linear mode coupling in few-mode fibers”, Opt. Express 22, 32039–32059 (2014).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

F. Yaman, Q. Lin, and G. P. Agrawal, “Fiber optic parametric amplifiers for lightwave systems,” in Guided Wave Optical Components and Devices, B. P. Pal, ed. (Academic, 2006).
[Crossref]

Andrekson, P. A.

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Raman-induced asymmetric pump noise transfer in fiber-optical parametric amplifiers,” IEEE Photon. Technol. Lett. 22, 386–388 (2010).
[Crossref]

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18, 2884–2893 (2010).
[Crossref] [PubMed]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

Ashkin, A.

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

Begleris, I.

S. M. M. Friis, Y. Jung, I. Begleris, P. Horak, K. Rottwitt, P. Petropoulos, D. J. Richardson, and F. Parmigiani, “Detailed phase matching characterization of inter-modal four-wave mixing in a two-mode fiber,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2016), paper JTu5A.49.

F. Parmigiani, Y. Jung, S. M. M. Friis, Q. Kang, I. Begleris, P. Horak, P. Petropoulos, K. Rottwitt, and D. J. Richardson, “Study of inter-modal four wave mixing in two few-mode fibres with different phase matching properties,” Proceedings of European Conference on Optical Communication, Tu2D.4 (2016).

Bell, B. A.

Bjorkholm, J. E.

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

Bogris, A.

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Raman-induced asymmetric pump noise transfer in fiber-optical parametric amplifiers,” IEEE Photon. Technol. Lett. 22, 386–388 (2010).
[Crossref]

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18, 2884–2893 (2010).
[Crossref] [PubMed]

Brinkmann, M.

Cheng, J.

Choi, D.-Y.

M. J. Collins, A. S. Clark, J. He, D.-Y. Choi, R. J. Williams, A. C. Judge, S. J. Madden, M. J. Withford, M. J. Steel, B. Luther-Davies, C. Xiong, and B. J. Eggleton, “Low Raman-noise correlated photon-pair generation in a dispersion-engineered chalcogenide As2S3 planar waveguide‘,” Opt. Lett. 37,. 3393–3395 (2012).
[Crossref]

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Chraplyvy, A. R.

R.-J. 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, 535–538 (2013).
[Crossref]

R.-J. 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, 539–542 (2013).
[Crossref]

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 (2002).
[Crossref]

Clark, A. S.

M. J. Collins, A. S. Clark, J. He, D.-Y. Choi, R. J. Williams, A. C. Judge, S. J. Madden, M. J. Withford, M. J. Steel, B. Luther-Davies, C. Xiong, and B. J. Eggleton, “Low Raman-noise correlated photon-pair generation in a dispersion-engineered chalcogenide As2S3 planar waveguide‘,” Opt. Lett. 37,. 3393–3395 (2012).
[Crossref]

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Clemmen, S.

A. Farsi, S. Clemmen, S. Ramelow, and A. L. Gaeta, “Low-Noise Quantum Frequency Translation of Single Photons,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2015), paper FM3A.4.

Collins, M. J.

Davanço, M.

Desgroseilliers, M.

Dorenbos, S. N.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Eggleton, B. J.

B. A. Bell, J. He, C. Xiong, and B. J. Eggleton, “Frequency conversion in silicon in the single photon regime,” Opt. Express 24, 5235–5242 (2016).
[Crossref]

M. J. Collins, A. S. Clark, J. He, D.-Y. Choi, R. J. Williams, A. C. Judge, S. J. Madden, M. J. Withford, M. J. Steel, B. Luther-Davies, C. Xiong, and B. J. Eggleton, “Low Raman-noise correlated photon-pair generation in a dispersion-engineered chalcogenide As2S3 planar waveguide‘,” Opt. Lett. 37,. 3393–3395 (2012).
[Crossref]

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Essiambre, R.-J.

Y. Xiao, R.-J. Essiambre, M. Desgroseilliers, A. M. Tulino, R. Ryf, S. Mumtaz, and G. P. Agrawal, “Theory of intermodal four-wave mixing with random linear mode coupling in few-mode fibers”, Opt. Express 22, 32039–32059 (2014).
[Crossref]

R.-J. 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, 539–542 (2013).
[Crossref]

R.-J. 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, 535–538 (2013).
[Crossref]

Fallnich, C.

Farsi, A.

A. Farsi, S. Clemmen, S. Ramelow, and A. L. Gaeta, “Low-Noise Quantum Frequency Translation of Single Photons,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2015), paper FM3A.4.

Friis, S. M. M.

S. M. M. Friis, K. Rottwitt, and C. J. McKinstrie, “Raman and loss induced quantum noise in depleted fiber optical parametric amplifiers,” Opt. Express 21, 29320–29331 (2013).
[Crossref]

F. Parmigiani, Y. Jung, S. M. M. Friis, Q. Kang, I. Begleris, P. Horak, P. Petropoulos, K. Rottwitt, and D. J. Richardson, “Study of inter-modal four wave mixing in two few-mode fibres with different phase matching properties,” Proceedings of European Conference on Optical Communication, Tu2D.4 (2016).

S. M. M. Friis, Y. Jung, I. Begleris, P. Horak, K. Rottwitt, P. Petropoulos, D. J. Richardson, and F. Parmigiani, “Detailed phase matching characterization of inter-modal four-wave mixing in a two-mode fiber,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2016), paper JTu5A.49.

Gaeta, A. L.

A. Farsi, S. Clemmen, S. Ramelow, and A. L. Gaeta, “Low-Noise Quantum Frequency Translation of Single Photons,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2015), paper FM3A.4.

Gnauck, A. H.

R.-J. 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, 535–538 (2013).
[Crossref]

R.-J. 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, 539–542 (2013).
[Crossref]

Grüner-Nielsen, L.

Hadfield, R. H.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Hansryd, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

He, J.

Hedekvist, P.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

Hellwig, T.

Helt, L. G.

M. Liscidini, J. E. Sipe, and L. G. Helt, “Continuous wave photon pair generation in silicon-on-insultator waveguides and ring resonators,” Opt. Express 24, 16558–16570 (2009).

Horak, P.

F. Poletti and P. Horak, “Description of ultrashort pulse propagation in multimode optical fibers,” J. Opt. Soc. Am. B 25, 1645–1654 (2008).
[Crossref]

F. Parmigiani, Y. Jung, S. M. M. Friis, Q. Kang, I. Begleris, P. Horak, P. Petropoulos, K. Rottwitt, and D. J. Richardson, “Study of inter-modal four wave mixing in two few-mode fibres with different phase matching properties,” Proceedings of European Conference on Optical Communication, Tu2D.4 (2016).

S. M. M. Friis, Y. Jung, I. Begleris, P. Horak, K. Rottwitt, P. Petropoulos, D. J. Richardson, and F. Parmigiani, “Detailed phase matching characterization of inter-modal four-wave mixing in a two-mode fiber,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2016), paper JTu5A.49.

Inoue, K.

Ippen, E. P.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[Crossref]

Jakobsen, D.

Jespersen, K. G.

Jiang, X.

R.-J. 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, 539–542 (2013).
[Crossref]

R.-J. 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, 535–538 (2013).
[Crossref]

Judge, A. C.

Jung, Y.

F. Parmigiani, Y. Jung, S. M. M. Friis, Q. Kang, I. Begleris, P. Horak, P. Petropoulos, K. Rottwitt, and D. J. Richardson, “Study of inter-modal four wave mixing in two few-mode fibres with different phase matching properties,” Proceedings of European Conference on Optical Communication, Tu2D.4 (2016).

S. M. M. Friis, Y. Jung, I. Begleris, P. Horak, K. Rottwitt, P. Petropoulos, D. J. Richardson, and F. Parmigiani, “Detailed phase matching characterization of inter-modal four-wave mixing in a two-mode fiber,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2016), paper JTu5A.49.

Kaminow, I.

I. Kaminow and T. Li, Optical Fiber Telecommunications IV-B: Systems and Impairments (Elsevier Science, 2002).

Kang, Q.

F. Parmigiani, Y. Jung, S. M. M. Friis, Q. Kang, I. Begleris, P. Horak, P. Petropoulos, K. Rottwitt, and D. J. Richardson, “Study of inter-modal four wave mixing in two few-mode fibres with different phase matching properties,” Proceedings of European Conference on Optical Communication, Tu2D.4 (2016).

Karlsson, M.

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18, 2884–2893 (2010).
[Crossref] [PubMed]

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Raman-induced asymmetric pump noise transfer in fiber-optical parametric amplifiers,” IEEE Photon. Technol. Lett. 22, 386–388 (2010).
[Crossref]

Li, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

Li, T.

I. Kaminow and T. Li, Optical Fiber Telecommunications IV-B: Systems and Impairments (Elsevier Science, 2002).

Lin, Q.

F. Yaman, Q. Lin, and G. P. Agrawal, “Fiber optic parametric amplifiers for lightwave systems,” in Guided Wave Optical Components and Devices, B. P. Pal, ed. (Academic, 2006).
[Crossref]

Lingle, R.

R.-J. 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, 535–538 (2013).
[Crossref]

R.-J. 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, 539–542 (2013).
[Crossref]

L. Grüner-Nielsen, Y. Sun, J. W. Nicholson, D. Jakobsen, K. G. Jespersen, R. Lingle, and B. Pálsdóttir, “Few mode transmission fiber with low DGD, low mode coupling, and low loss,” J. Lightwave Technol. 30, 3693–3698 (2012).
[Crossref]

Liscidini, M.

M. Liscidini, J. E. Sipe, and L. G. Helt, “Continuous wave photon pair generation in silicon-on-insultator waveguides and ring resonators,” Opt. Express 24, 16558–16570 (2009).

Lobino, M.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Lombardo, D.

Luther-Davies, B.

M. J. Collins, A. S. Clark, J. He, D.-Y. Choi, R. J. Williams, A. C. Judge, S. J. Madden, M. J. Withford, M. J. Steel, B. Luther-Davies, C. Xiong, and B. J. Eggleton, “Low Raman-noise correlated photon-pair generation in a dispersion-engineered chalcogenide As2S3 planar waveguide‘,” Opt. Lett. 37,. 3393–3395 (2012).
[Crossref]

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Madden, S. J.

M. J. Collins, A. S. Clark, J. He, D.-Y. Choi, R. J. Williams, A. C. Judge, S. J. Madden, M. J. Withford, M. J. Steel, B. Luther-Davies, C. Xiong, and B. J. Eggleton, “Low Raman-noise correlated photon-pair generation in a dispersion-engineered chalcogenide As2S3 planar waveguide‘,” Opt. Lett. 37,. 3393–3395 (2012).
[Crossref]

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Marshall, G. D.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Mathews, J.

McGuinness, H. J.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref] [PubMed]

McKinstrie, C. J.

S. M. M. Friis, K. Rottwitt, and C. J. McKinstrie, “Raman and loss induced quantum noise in depleted fiber optical parametric amplifiers,” Opt. Express 21, 29320–29331 (2013).
[Crossref]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A,  85, 53829 (2012).
[Crossref]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref] [PubMed]

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 (2002).
[Crossref]

Mejling, L.

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A,  85, 53829 (2012).
[Crossref]

Mestre, M. A.

R.-J. 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, 539–542 (2013).
[Crossref]

R.-J. 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, 535–538 (2013).
[Crossref]

Mumtaz, S.

Natarajan, C. M.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Nicholson, J. W.

O’Brien, J. L.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Pálsdóttir, B.

Parmigiani, F.

S. M. M. Friis, Y. Jung, I. Begleris, P. Horak, K. Rottwitt, P. Petropoulos, D. J. Richardson, and F. Parmigiani, “Detailed phase matching characterization of inter-modal four-wave mixing in a two-mode fiber,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2016), paper JTu5A.49.

F. Parmigiani, Y. Jung, S. M. M. Friis, Q. Kang, I. Begleris, P. Horak, P. Petropoulos, K. Rottwitt, and D. J. Richardson, “Study of inter-modal four wave mixing in two few-mode fibres with different phase matching properties,” Proceedings of European Conference on Optical Communication, Tu2D.4 (2016).

Pedersen, M. E. V.

Peruzzo, A.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Petropoulos, P.

S. M. M. Friis, Y. Jung, I. Begleris, P. Horak, K. Rottwitt, P. Petropoulos, D. J. Richardson, and F. Parmigiani, “Detailed phase matching characterization of inter-modal four-wave mixing in a two-mode fiber,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2016), paper JTu5A.49.

F. Parmigiani, Y. Jung, S. M. M. Friis, Q. Kang, I. Begleris, P. Horak, P. Petropoulos, K. Rottwitt, and D. J. Richardson, “Study of inter-modal four wave mixing in two few-mode fibres with different phase matching properties,” Proceedings of European Conference on Optical Communication, Tu2D.4 (2016).

Poletti, F.

Radic, S.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref] [PubMed]

S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. 2, 498–513 (2008).
[Crossref]

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 (2002).
[Crossref]

Ramelow, S.

A. Farsi, S. Clemmen, S. Ramelow, and A. L. Gaeta, “Low-Noise Quantum Frequency Translation of Single Photons,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2015), paper FM3A.4.

Rarity, J. G.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Raymer, M. G.

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A,  85, 53829 (2012).
[Crossref]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref] [PubMed]

Richardson, D. J.

F. Parmigiani, Y. Jung, S. M. M. Friis, Q. Kang, I. Begleris, P. Horak, P. Petropoulos, K. Rottwitt, and D. J. Richardson, “Study of inter-modal four wave mixing in two few-mode fibres with different phase matching properties,” Proceedings of European Conference on Optical Communication, Tu2D.4 (2016).

S. M. M. Friis, Y. Jung, I. Begleris, P. Horak, K. Rottwitt, P. Petropoulos, D. J. Richardson, and F. Parmigiani, “Detailed phase matching characterization of inter-modal four-wave mixing in a two-mode fiber,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2016), paper JTu5A.49.

Rottwitt, K.

S. M. M. Friis, K. Rottwitt, and C. J. McKinstrie, “Raman and loss induced quantum noise in depleted fiber optical parametric amplifiers,” Opt. Express 21, 29320–29331 (2013).
[Crossref]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A,  85, 53829 (2012).
[Crossref]

S. M. M. Friis, Y. Jung, I. Begleris, P. Horak, K. Rottwitt, P. Petropoulos, D. J. Richardson, and F. Parmigiani, “Detailed phase matching characterization of inter-modal four-wave mixing in a two-mode fiber,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2016), paper JTu5A.49.

F. Parmigiani, Y. Jung, S. M. M. Friis, Q. Kang, I. Begleris, P. Horak, P. Petropoulos, K. Rottwitt, and D. J. Richardson, “Study of inter-modal four wave mixing in two few-mode fibres with different phase matching properties,” Proceedings of European Conference on Optical Communication, Tu2D.4 (2016).

Ryf, R.

Y. Xiao, R.-J. Essiambre, M. Desgroseilliers, A. M. Tulino, R. Ryf, S. Mumtaz, and G. P. Agrawal, “Theory of intermodal four-wave mixing with random linear mode coupling in few-mode fibers”, Opt. Express 22, 32039–32059 (2014).
[Crossref]

R.-J. 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, 539–542 (2013).
[Crossref]

R.-J. 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, 535–538 (2013).
[Crossref]

Schnack, M.

Sipe, J. E.

M. Liscidini, J. E. Sipe, and L. G. Helt, “Continuous wave photon pair generation in silicon-on-insultator waveguides and ring resonators,” Opt. Express 24, 16558–16570 (2009).

Srinivasan, K.

Steel, M. J.

M. J. Collins, A. S. Clark, J. He, D.-Y. Choi, R. J. Williams, A. C. Judge, S. J. Madden, M. J. Withford, M. J. Steel, B. Luther-Davies, C. Xiong, and B. J. Eggleton, “Low Raman-noise correlated photon-pair generation in a dispersion-engineered chalcogenide As2S3 planar waveguide‘,” Opt. Lett. 37,. 3393–3395 (2012).
[Crossref]

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Stolen, R. H.

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

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[Crossref]

Sun, Y.

R.-J. 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, 535–538 (2013).
[Crossref]

R.-J. 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, 539–542 (2013).
[Crossref]

L. Grüner-Nielsen, Y. Sun, J. W. Nicholson, D. Jakobsen, K. G. Jespersen, R. Lingle, and B. Pálsdóttir, “Few mode transmission fiber with low DGD, low mode coupling, and low loss,” J. Lightwave Technol. 30, 3693–3698 (2012).
[Crossref]

Takesue, H.

Tanner, M. G.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Thompson, M. G.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Thurston, B.

Tkach, R. W.

R.-J. 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, 539–542 (2013).
[Crossref]

R.-J. 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, 535–538 (2013).
[Crossref]

Tong, Z.

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Raman-induced asymmetric pump noise transfer in fiber-optical parametric amplifiers,” IEEE Photon. Technol. Lett. 22, 386–388 (2010).
[Crossref]

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18, 2884–2893 (2010).
[Crossref] [PubMed]

Tulino, A. M.

Wang, K.

Westlund, M.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

Williams, R. J.

Withford, M. J.

Xiao, Y.

Xiong, C.

B. A. Bell, J. He, C. Xiong, and B. J. Eggleton, “Frequency conversion in silicon in the single photon regime,” Opt. Express 24, 5235–5242 (2016).
[Crossref]

M. J. Collins, A. S. Clark, J. He, D.-Y. Choi, R. J. Williams, A. C. Judge, S. J. Madden, M. J. Withford, M. J. Steel, B. Luther-Davies, C. Xiong, and B. J. Eggleton, “Low Raman-noise correlated photon-pair generation in a dispersion-engineered chalcogenide As2S3 planar waveguide‘,” Opt. Lett. 37,. 3393–3395 (2012).
[Crossref]

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Xu, C.

Yaman, F.

F. Yaman, Q. Lin, and G. P. Agrawal, “Fiber optic parametric amplifiers for lightwave systems,” in Guided Wave Optical Components and Devices, B. P. Pal, ed. (Academic, 2006).
[Crossref]

Zhao, Y.

Zijlstra, T.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Zwiller, V.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Appl. Phys. Lett. (3)

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

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

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Figures (8)

Fig. 1
Fig. 1 Different configurations of FWM using two pumps, p and q, a signal, s, and an idler, i; (a) modulation instability (MI), (b) phase conjugation (PC), (c) Bragg scattering (BS) and (d) the full spectrum. The arrows indicate gaining (up) and losing (down) energy.
Fig. 2
Fig. 2 (a) Principle of two-mode phase matching for distant BS in a RIGV vs. wavelength diagram; dots denote wave components and squares denote the average wavelength of the two waves in either mode. (b) RIGV measurements of LP01 and LP11 vs. wavelength; lines denote second order polynomial fits and the grey stars denote the inverse group velocity difference.
Fig. 3
Fig. 3 Conversion efficiency of the PC (left) and BS (right) for a variation of signal and pump q wavelengths. λp = 1549 nm and input powers are Pp = Pq = 27.5 dBm, Ps = 3.5 dBm.
Fig. 4
Fig. 4 Conversion efficiency of the PC (left) and BS (right) for selected λq shown in the legend. The paramaters are the same as in Fig. 3.
Fig. 5
Fig. 5 Sketch of the experimental setup. TLS, tunable laser source; MMUX, mode-multiplexer; EDFA, Erbium-doped fiber amplifier; 50/50, 3-dB non-polarization beam-splitter; PBS, polarization beam-splitter; PP, phase plate; TMF, two-mode fiber; MDMUX, mode-demultiplexer; OSA, optical spectrum analyzer.
Fig. 6
Fig. 6 Performance of the MDMUX; the LP01 port (left) extinguishes the LP11 mode by 25 dB, and the LP11 port (right) extinguishes the LP01 mode by 18 dB.
Fig. 7
Fig. 7 Measured conversion efficiency for PC vs. signal wavelength for cases (a)–(f) of pump q wavelengths corresponding to the values given in Table 1.
Fig. 8
Fig. 8 Measured conversion efficiency for BS vs. signal wavelength for cases (a)–(f) of pump q wavelengths corresponding to the values given in Table 1.

Tables (1)

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Table 1 Pump q wavelengths used in experiments; pump p is λp = 1549 nm in all cases.

Equations (4)

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Δ β = β ( ω s ) + β ( ω q ) β ( ω p ) β ( ω i ) [ β 1 ( 01 ) + β 2 ( 01 ) ( Δ ω s + Δ ω p 2 ) β 1 ( 11 ) β 2 ( 11 ) ( Δ ω q + Δ ω i 2 ) ] ( ω s ω p ) ,
A μ ( z , t ) z = 𝒟 μ ( z , t ) + 𝒩 μ ( z , t ) ,
𝒟 μ ( z , t ) = ( j ( β 0 ( μ ) β 0 ( 01 ) ) ( β 1 ( μ ) β 0 ( 01 ) ) t j β 2 ( μ ) 2 2 t 2 + β 3 ( μ ) 6 3 t 3 ) A μ ( z , t ) ,
𝒩 μ ( z , t ) = j n 2 ω ˜ c l , m , n ( 1 + j ω 0 t ) Q μ l m n A l ( z , t ) ( 3 ( 1 f R ) A m ( z , t ) A n * ( z , t ) + 3 f R h ( τ ) A m ( z , t τ ) A n * ( z , t τ ) d τ )

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