Abstract

Interest in the nonlinear properties of multi-mode optical waveguides has seen a recent resurgence on account of the large dimensionality afforded by the platform. The large volume of modes in these waveguides provides a new spatial degree of freedom for phase matching nonlinear optical processes. However, this spatial dimension is quantized, which narrows the conversion bandwidths of intermodal processes and constrains spectral and temporal tailoring of the light. Here we show that by engineering the relative group velocity within the spatial dimension, we can tailor the phase-matching bandwidth of intermodal parametric nonlinearities. We demonstrate group-velocity-tailored parametric nonlinear mixing between higher-order modes in a multi-mode fiber with gain bandwidths that are more than an order of magnitude larger than that previously thought possible for intermodal four-wave mixing. As evidence of the technological utility of this methodology, we seed this process to generate the first high-peak-power wavelength-tunable all-fiber quasi-CW laser in the Ti:sapphire wavelength regime. More generally, with the combination of intermodal interactions, which dramatically expand the phase-matching degrees of freedom for nonlinear optics, and intermodal group-velocity engineering, which enables tailoring of the bandwidth of such interactions, we showcase a platform for nonlinear optics that can be broadband while being wavelength agnostic.

© 2018 Chinese Laser Press

Full Article  |  PDF Article
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References

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2018 (1)

K. Rottwitt, J. G. Koefoed, and E. N. Christensen, “Photon-pair sources based on intermodal four-wave mixing in few-mode fibers,” Fibers 6, 32–34 (2018).
[Crossref]

2017 (2)

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
[Crossref]

J. Demas, G. Prabhakar, T. He, and S. Ramachandran, “Wavelength-agile high-power sources via four-wave mixing in higher-order modes,” Opt. Express 25, 7455–7464 (2017).
[Crossref]

2016 (4)

D. Cruz-Delgado, R. Ramirez-Alarcon, E. Ortiz-Ricardo, J. Monroy-Ruz, F. Dominguez-Serna, H. Cruz-Ramirez, K. Garay-Palmett, and A. B. U’Ren, “Fiber-based photon-pair source capable of hybrid entanglement in frequency and transverse mode, controllably scalable to higher dimensions,” Sci. Rep. 6, 27377 (2016).
[Crossref]

M. Schnack, T. Hellwig, and K. Fallnich, “Ultrafast, all-optical control of modal phases in a few-mode fiber for all-optical switching,” Opt. Lett. 41, 5588–5591 (2016).
[Crossref]

M. Ma and L. R. Chen, “Harnessing mode-selective nonlinear optics for on-chip multi-channel all-optical signal processing,” APL Photon. 1, 086104 (2016).
[Crossref]

S. M. M. Friis, I. Begleris, Y. Jung, K. Rottwitt, P. Petropoulos, D. J. Richardson, P. Horak, and F. Parmigiani, “Inter-modal four-wave mixing study in a two-mode fiber,” Opt. Express 24, 30338–30348 (2016).
[Crossref]

2015 (4)

2014 (2)

2013 (5)

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

B. Zwan, S. Legge, J. Holdsworth, and B. King, “Spatio-spectral analysis of supercontinuum generation in higher order electromagnetic modes of photonic crystal fiber,” Opt. Express 21, 834–839 (2013).
[Crossref]

Y. Chen, Z. Chen, W. J. Wadsworth, and T. A. Birks, “Nonlinear optics in the LP02 higher-order mode of a fiber,” Opt. Express 21, 17786–17799 (2013).
[Crossref]

R. J. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photon. J. 5, 0701307 (2013).
[Crossref]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

2012 (3)

J. W. Nicholson, J. M. Fini, A. M. DeSantolo, X. Liu, K. Feder, P. S. Westbrook, V. R. Supradeepa, E. Monberg, F. DiMarcello, R. Ortiz, C. Headley, and D. J. DiGiovanni, “Scaling the effective area of higher-order-mode erbium-doped fiber amplifiers,” Opt. Express 20, 24575–24584 (2012).
[Crossref]

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
[Crossref]

J. Cheng, M. E. V. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101, 161106 (2012).
[Crossref]

2009 (2)

D. Nodop, C. Jauregui, D. Schimpf, J. Limpert, and A. Tünnermann, “Efficient high-power generation of visible and mid-infrared light by degenerate four-wave-mixing in a large-mode-area photonic-crystal fiber,” Opt. Lett. 34, 3499–3501 (2009).
[Crossref]

C.-S. Bres, A. O. J. Wiberg, B. P.-P. Kuo, N. Alic, and S. Radic, “Wavelength multicasting of 320-Gb/s channel in self-seeded parametric amplifier,” IEEE Photon. Technol. Lett. 21, 1002–1004 (2009).
[Crossref]

2008 (1)

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[Crossref]

2004 (1)

2002 (1)

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

2000 (1)

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in a photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[Crossref]

1975 (1)

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. 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]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Elsevier, 2005).

Alic, N.

C.-S. Bres, A. O. J. Wiberg, B. P.-P. Kuo, N. Alic, and S. Radic, “Wavelength multicasting of 320-Gb/s channel in self-seeded parametric amplifier,” IEEE Photon. Technol. Lett. 21, 1002–1004 (2009).
[Crossref]

Andrekson, P. A.

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

Arriaga, J.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in a photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[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]

Barthélémy, A.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
[Crossref]

Begleris, I.

Birks, T. A.

Y. Chen, Z. Chen, W. J. Wadsworth, and T. A. Birks, “Nonlinear optics in the LP02 higher-order mode of a fiber,” Opt. Express 21, 17786–17799 (2013).
[Crossref]

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in a photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[Crossref]

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]

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Bres, C.-S.

C.-S. Bres, A. O. J. Wiberg, B. P.-P. Kuo, N. Alic, and S. Radic, “Wavelength multicasting of 320-Gb/s channel in self-seeded parametric amplifier,” IEEE Photon. Technol. Lett. 21, 1002–1004 (2009).
[Crossref]

Charan, K.

J. Cheng, M. E. V. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101, 161106 (2012).
[Crossref]

Chen, L. R.

M. Ma and L. R. Chen, “Harnessing mode-selective nonlinear optics for on-chip multi-channel all-optical signal processing,” APL Photon. 1, 086104 (2016).
[Crossref]

Chen, Y.

Chen, Z.

Cheng, J.

J. Cheng, M. E. V. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101, 161106 (2012).
[Crossref]

Chraplyvy, A. R.

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

Christensen, E. N.

K. Rottwitt, J. G. Koefoed, and E. N. Christensen, “Photon-pair sources based on intermodal four-wave mixing in few-mode fibers,” Fibers 6, 32–34 (2018).
[Crossref]

Christodoulides, D. N.

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9, 306–310 (2015).
[Crossref]

Cižmár, T.

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
[Crossref]

Cohen, O.

Couderc, V.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
[Crossref]

Cruz-Delgado, D.

D. Cruz-Delgado, R. Ramirez-Alarcon, E. Ortiz-Ricardo, J. Monroy-Ruz, F. Dominguez-Serna, H. Cruz-Ramirez, K. Garay-Palmett, and A. B. U’Ren, “Fiber-based photon-pair source capable of hybrid entanglement in frequency and transverse mode, controllably scalable to higher dimensions,” Sci. Rep. 6, 27377 (2016).
[Crossref]

Cruz-Ramirez, H.

D. Cruz-Delgado, R. Ramirez-Alarcon, E. Ortiz-Ricardo, J. Monroy-Ruz, F. Dominguez-Serna, H. Cruz-Ramirez, K. Garay-Palmett, and A. B. U’Ren, “Fiber-based photon-pair source capable of hybrid entanglement in frequency and transverse mode, controllably scalable to higher dimensions,” Sci. Rep. 6, 27377 (2016).
[Crossref]

Demas, J.

DeSantolo, A. M.

Dholakia, K.

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
[Crossref]

DiGiovanni, D. J.

DiMarcello, F.

Dominguez-Serna, F.

D. Cruz-Delgado, R. Ramirez-Alarcon, E. Ortiz-Ricardo, J. Monroy-Ruz, F. Dominguez-Serna, H. Cruz-Ramirez, K. Garay-Palmett, and A. B. U’Ren, “Fiber-based photon-pair source capable of hybrid entanglement in frequency and transverse mode, controllably scalable to higher dimensions,” Sci. Rep. 6, 27377 (2016).
[Crossref]

Essiambre, R.

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

Essiambre, R. J.

R. J. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photon. J. 5, 0701307 (2013).
[Crossref]

Fabert, M.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
[Crossref]

Fallnich, K.

Fang, B.

Feder, K.

Fini, J. M.

J. W. Nicholson, J. M. Fini, A. M. DeSantolo, X. Liu, K. Feder, P. S. Westbrook, V. R. Supradeepa, E. Monberg, F. DiMarcello, R. Ortiz, C. Headley, and D. J. DiGiovanni, “Scaling the effective area of higher-order-mode erbium-doped fiber amplifiers,” Opt. Express 20, 24575–24584 (2012).
[Crossref]

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[Crossref]

Fontaine, N. K.

R. J. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photon. J. 5, 0701307 (2013).
[Crossref]

Friis, S. M. M.

Garay-Palmett, K.

D. Cruz-Delgado, R. Ramirez-Alarcon, E. Ortiz-Ricardo, J. Monroy-Ruz, F. Dominguez-Serna, H. Cruz-Ramirez, K. Garay-Palmett, and A. B. U’Ren, “Fiber-based photon-pair source capable of hybrid entanglement in frequency and transverse mode, controllably scalable to higher dimensions,” Sci. Rep. 6, 27377 (2016).
[Crossref]

Ghalmi, S.

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[Crossref]

Gnauck, A. H.

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

Grüner-Nielsen, L.

J. Cheng, M. E. V. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101, 161106 (2012).
[Crossref]

Hansryd, J.

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

He, T.

J. Demas, G. Prabhakar, T. He, and S. Ramachandran, “Wavelength-agile high-power sources via four-wave mixing in higher-order modes,” Opt. Express 25, 7455–7464 (2017).
[Crossref]

J. Demas, G. Prabhakar, T. He, and S. Ramachandran, “Broadband and Wideband parametric gain via intermodal four-wave mixing in optical fiber,” in Conference on Lasers and Electro-Optics (CLEO) (2017), paper SM3M.1.

Headley, C.

Hedekvist, P. O.

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

Hellwig, T.

Holdsworth, J.

Horak, P.

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Jakobsen, D.

J. Cheng, M. E. V. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101, 161106 (2012).
[Crossref]

Jauregui, C.

Jiang, X.

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

Jung, Y.

King, B.

Knight, J. C.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in a photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[Crossref]

Koefoed, J. G.

K. Rottwitt, J. G. Koefoed, and E. N. Christensen, “Photon-pair sources based on intermodal four-wave mixing in few-mode fibers,” Fibers 6, 32–34 (2018).
[Crossref]

Konorov, S. O.

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

L. Rishøj, B. Tai, P. Kristensen, and S. Ramachandran, “Discovery of soliton self-mode conversion in multimode optical fibers,” arXiv:1805.06037.

Krupa, K.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
[Crossref]

Kuo, B. P.-P.

C.-S. Bres, A. O. J. Wiberg, B. P.-P. Kuo, N. Alic, and S. Radic, “Wavelength multicasting of 320-Gb/s channel in self-seeded parametric amplifier,” IEEE Photon. Technol. Lett. 21, 1002–1004 (2009).
[Crossref]

Legge, S.

Li, J.

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

Limpert, J.

Lingle, R.

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

Liscidini, M.

Liu, X.

J. W. Nicholson, J. M. Fini, A. M. DeSantolo, X. Liu, K. Feder, P. S. Westbrook, V. R. Supradeepa, E. Monberg, F. DiMarcello, R. Ortiz, C. Headley, and D. J. DiGiovanni, “Scaling the effective area of higher-order-mode erbium-doped fiber amplifiers,” Opt. Express 20, 24575–24584 (2012).
[Crossref]

J. Demas, L. Rishøj, X. Liu, G. Prabhakar, and S. Ramachandran, “High-power, wavelength-tunable NIR all-fiber lasers via intermodal four-wave mixing,” in Conference on Lasers and Electro-Optics (CLEO) (2017), paper JTh5A.8.

Lorenz, V. O.

Ma, M.

M. Ma and L. R. Chen, “Harnessing mode-selective nonlinear optics for on-chip multi-channel all-optical signal processing,” APL Photon. 1, 086104 (2016).
[Crossref]

Mafi, A.

Mermelstein, M.

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[Crossref]

Mestre, M. A.

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

Millot, G.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
[Crossref]

Monberg, E.

Monroy-Ruz, J.

D. Cruz-Delgado, R. Ramirez-Alarcon, E. Ortiz-Ricardo, J. Monroy-Ruz, F. Dominguez-Serna, H. Cruz-Ramirez, K. Garay-Palmett, and A. B. U’Ren, “Fiber-based photon-pair source capable of hybrid entanglement in frequency and transverse mode, controllably scalable to higher dimensions,” Sci. Rep. 6, 27377 (2016).
[Crossref]

Nazemosadat, E.

Nicholson, J. W.

J. W. Nicholson, J. M. Fini, A. M. DeSantolo, X. Liu, K. Feder, P. S. Westbrook, V. R. Supradeepa, E. Monberg, F. DiMarcello, R. Ortiz, C. Headley, and D. J. DiGiovanni, “Scaling the effective area of higher-order-mode erbium-doped fiber amplifiers,” Opt. Express 20, 24575–24584 (2012).
[Crossref]

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[Crossref]

Nodop, D.

Ortigosa-Blanch, A.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in a photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[Crossref]

Ortiz, R.

Ortiz-Ricardo, E.

D. Cruz-Delgado, R. Ramirez-Alarcon, E. Ortiz-Ricardo, J. Monroy-Ruz, F. Dominguez-Serna, H. Cruz-Ramirez, K. Garay-Palmett, and A. B. U’Ren, “Fiber-based photon-pair source capable of hybrid entanglement in frequency and transverse mode, controllably scalable to higher dimensions,” Sci. Rep. 6, 27377 (2016).
[Crossref]

Parmigiani, F.

Pedersen, M. E. V.

J. Cheng, M. E. V. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101, 161106 (2012).
[Crossref]

Petropoulos, P.

Pourbeyram, H.

Prabhakar, G.

J. Demas, G. Prabhakar, T. He, and S. Ramachandran, “Wavelength-agile high-power sources via four-wave mixing in higher-order modes,” Opt. Express 25, 7455–7464 (2017).
[Crossref]

J. Demas, G. Prabhakar, T. He, and S. Ramachandran, “Broadband and Wideband parametric gain via intermodal four-wave mixing in optical fiber,” in Conference on Lasers and Electro-Optics (CLEO) (2017), paper SM3M.1.

J. Demas, L. Rishøj, X. Liu, G. Prabhakar, and S. Ramachandran, “High-power, wavelength-tunable NIR all-fiber lasers via intermodal four-wave mixing,” in Conference on Lasers and Electro-Optics (CLEO) (2017), paper JTh5A.8.

Radic, S.

C.-S. Bres, A. O. J. Wiberg, B. P.-P. Kuo, N. Alic, and S. Radic, “Wavelength multicasting of 320-Gb/s channel in self-seeded parametric amplifier,” IEEE Photon. Technol. Lett. 21, 1002–1004 (2009).
[Crossref]

Ramachandran, S.

J. Demas, G. Prabhakar, T. He, and S. Ramachandran, “Wavelength-agile high-power sources via four-wave mixing in higher-order modes,” Opt. Express 25, 7455–7464 (2017).
[Crossref]

J. Demas, L. Rishøj, and S. Ramachandran, “Free-space beam shaping for precise control and conversion of modes in optical fiber,” Opt. Express 23, 28531–28545 (2015).
[Crossref]

J. Demas, P. Steinvurzel, B. Tai, L. Rishøj, Y. Chen, and S. Ramachandran, “Intermodal nonlinear mixing with Bessel beams in optical fiber,” Optica 2, 14–17 (2015).
[Crossref]

P. Steinvurzel, J. Demas, B. Tai, Y. Chen, L. Yan, and S. Ramachandran, “Broadband parametric wavelength conversion at 1 μm with large mode area fibers,” Opt. Lett. 39, 743–746 (2014).
[Crossref]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[Crossref]

L. Rishøj, B. Tai, P. Kristensen, and S. Ramachandran, “Discovery of soliton self-mode conversion in multimode optical fibers,” arXiv:1805.06037.

J. Demas, L. Rishøj, X. Liu, G. Prabhakar, and S. Ramachandran, “High-power, wavelength-tunable NIR all-fiber lasers via intermodal four-wave mixing,” in Conference on Lasers and Electro-Optics (CLEO) (2017), paper JTh5A.8.

J. Demas, G. Prabhakar, T. He, and S. Ramachandran, “Broadband and Wideband parametric gain via intermodal four-wave mixing in optical fiber,” in Conference on Lasers and Electro-Optics (CLEO) (2017), paper SM3M.1.

Ramirez-Alarcon, R.

D. Cruz-Delgado, R. Ramirez-Alarcon, E. Ortiz-Ricardo, J. Monroy-Ruz, F. Dominguez-Serna, H. Cruz-Ramirez, K. Garay-Palmett, and A. B. U’Ren, “Fiber-based photon-pair source capable of hybrid entanglement in frequency and transverse mode, controllably scalable to higher dimensions,” Sci. Rep. 6, 27377 (2016).
[Crossref]

Randel, S.

R. J. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photon. J. 5, 0701307 (2013).
[Crossref]

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Richardson, D. J.

Rishøj, L.

J. Demas, L. Rishøj, and S. Ramachandran, “Free-space beam shaping for precise control and conversion of modes in optical fiber,” Opt. Express 23, 28531–28545 (2015).
[Crossref]

J. Demas, P. Steinvurzel, B. Tai, L. Rishøj, Y. Chen, and S. Ramachandran, “Intermodal nonlinear mixing with Bessel beams in optical fiber,” Optica 2, 14–17 (2015).
[Crossref]

L. Rishøj, B. Tai, P. Kristensen, and S. Ramachandran, “Discovery of soliton self-mode conversion in multimode optical fibers,” arXiv:1805.06037.

J. Demas, L. Rishøj, X. Liu, G. Prabhakar, and S. Ramachandran, “High-power, wavelength-tunable NIR all-fiber lasers via intermodal four-wave mixing,” in Conference on Lasers and Electro-Optics (CLEO) (2017), paper JTh5A.8.

Rottwitt, K.

K. Rottwitt, J. G. Koefoed, and E. N. Christensen, “Photon-pair sources based on intermodal four-wave mixing in few-mode fibers,” Fibers 6, 32–34 (2018).
[Crossref]

S. M. M. Friis, I. Begleris, Y. Jung, K. Rottwitt, P. Petropoulos, D. J. Richardson, P. Horak, and F. Parmigiani, “Inter-modal four-wave mixing study in a two-mode fiber,” Opt. Express 24, 30338–30348 (2016).
[Crossref]

Russell, P. St. J.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in a photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[Crossref]

Ryf, R.

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

R. J. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photon. J. 5, 0701307 (2013).
[Crossref]

Schimpf, D.

Schnack, M.

Serebrannikov, E. E.

Shalaby, B. M.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
[Crossref]

Sipe, J. E.

Steinvurzel, P.

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]

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]

Sun, Y.

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

Supradeepa, V. R.

Tai, B.

Tarasevitch, A.

Tkach, R. W.

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

Tonello, A.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
[Crossref]

Tünnermann, A.

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

U’Ren, A. B.

D. Cruz-Delgado, R. Ramirez-Alarcon, E. Ortiz-Ricardo, J. Monroy-Ruz, F. Dominguez-Serna, H. Cruz-Ramirez, K. Garay-Palmett, and A. B. U’Ren, “Fiber-based photon-pair source capable of hybrid entanglement in frequency and transverse mode, controllably scalable to higher dimensions,” Sci. Rep. 6, 27377 (2016).
[Crossref]

von der Linde, D.

Wabnitz, S.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
[Crossref]

Wadsworth, W. J.

Y. Chen, Z. Chen, W. J. Wadsworth, and T. A. Birks, “Nonlinear optics in the LP02 higher-order mode of a fiber,” Opt. Express 21, 17786–17799 (2013).
[Crossref]

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in a photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[Crossref]

Wang, K.

J. Cheng, M. E. V. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101, 161106 (2012).
[Crossref]

Westbrook, P. S.

Westlund, M.

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

Wiberg, A. O. J.

C.-S. Bres, A. O. J. Wiberg, B. P.-P. Kuo, N. Alic, and S. Radic, “Wavelength multicasting of 320-Gb/s channel in self-seeded parametric amplifier,” IEEE Photon. Technol. Lett. 21, 1002–1004 (2009).
[Crossref]

Willner, A. E.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Wise, F. W.

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9, 306–310 (2015).
[Crossref]

Wright, L. G.

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9, 306–310 (2015).
[Crossref]

Xu, C.

J. Cheng, M. E. V. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101, 161106 (2012).
[Crossref]

Yan, L.

Yan, M. F.

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[Crossref]

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Zheltikov, A. M.

Zhou, P.

Zwan, B.

APL Photon. (1)

M. Ma and L. R. Chen, “Harnessing mode-selective nonlinear optics for on-chip multi-channel all-optical signal processing,” APL Photon. 1, 086104 (2016).
[Crossref]

Appl. Phys. Lett. (2)

J. Cheng, M. E. V. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101, 161106 (2012).
[Crossref]

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]

Fibers (1)

K. Rottwitt, J. G. Koefoed, and E. N. Christensen, “Photon-pair sources based on intermodal four-wave mixing in few-mode fibers,” Fibers 6, 32–34 (2018).
[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 J. Sel. Top. Quantum Electron. (1)

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

IEEE Photon. J. (1)

R. J. Essiambre, R. Ryf, N. K. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photon. J. 5, 0701307 (2013).
[Crossref]

IEEE Photon. Technol. Lett. (3)

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

C.-S. Bres, A. O. J. Wiberg, B. P.-P. Kuo, N. Alic, and S. Radic, “Wavelength multicasting of 320-Gb/s channel in self-seeded parametric amplifier,” IEEE Photon. Technol. Lett. 21, 1002–1004 (2009).
[Crossref]

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in a photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[Crossref]

Laser Photon. Rev. (1)

S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photon. Rev. 2, 429–448 (2008).
[Crossref]

Nat. Commun. (1)

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
[Crossref]

Nat. Photonics (2)

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9, 306–310 (2015).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
[Crossref]

Opt. Express (8)

J. Demas, G. Prabhakar, T. He, and S. Ramachandran, “Wavelength-agile high-power sources via four-wave mixing in higher-order modes,” Opt. Express 25, 7455–7464 (2017).
[Crossref]

S. M. M. Friis, I. Begleris, Y. Jung, K. Rottwitt, P. Petropoulos, D. J. Richardson, P. Horak, and F. Parmigiani, “Inter-modal four-wave mixing study in a two-mode fiber,” Opt. Express 24, 30338–30348 (2016).
[Crossref]

J. W. Nicholson, J. M. Fini, A. M. DeSantolo, X. Liu, K. Feder, P. S. Westbrook, V. R. Supradeepa, E. Monberg, F. DiMarcello, R. Ortiz, C. Headley, and D. J. DiGiovanni, “Scaling the effective area of higher-order-mode erbium-doped fiber amplifiers,” Opt. Express 20, 24575–24584 (2012).
[Crossref]

B. Zwan, S. Legge, J. Holdsworth, and B. King, “Spatio-spectral analysis of supercontinuum generation in higher order electromagnetic modes of photonic crystal fiber,” Opt. Express 21, 834–839 (2013).
[Crossref]

Y. Chen, Z. Chen, W. J. Wadsworth, and T. A. Birks, “Nonlinear optics in the LP02 higher-order mode of a fiber,” Opt. Express 21, 17786–17799 (2013).
[Crossref]

S. O. Konorov, E. E. Serebrannikov, A. M. Zheltikov, P. Zhou, A. Tarasevitch, and D. von der Linde, “Mode-controlled colors from microstructure fibers,” Opt. Express 12, 730–735 (2004).
[Crossref]

H. Pourbeyram, E. Nazemosadat, and A. Mafi, “Detailed investigation of intermodal four-wave mixing in SMF-28: blue-red generation from green,” Opt. Express 23, 14487–14500 (2015).
[Crossref]

J. Demas, L. Rishøj, and S. Ramachandran, “Free-space beam shaping for precise control and conversion of modes in optical fiber,” Opt. Express 23, 28531–28545 (2015).
[Crossref]

Opt. Lett. (3)

Optica (2)

Sci. Rep. (1)

D. Cruz-Delgado, R. Ramirez-Alarcon, E. Ortiz-Ricardo, J. Monroy-Ruz, F. Dominguez-Serna, H. Cruz-Ramirez, K. Garay-Palmett, and A. B. U’Ren, “Fiber-based photon-pair source capable of hybrid entanglement in frequency and transverse mode, controllably scalable to higher dimensions,” Sci. Rep. 6, 27377 (2016).
[Crossref]

Science (1)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Other (4)

L. Rishøj, B. Tai, P. Kristensen, and S. Ramachandran, “Discovery of soliton self-mode conversion in multimode optical fibers,” arXiv:1805.06037.

G. P. Agrawal, Nonlinear Fiber Optics (Elsevier, 2005).

J. Demas, G. Prabhakar, T. He, and S. Ramachandran, “Broadband and Wideband parametric gain via intermodal four-wave mixing in optical fiber,” in Conference on Lasers and Electro-Optics (CLEO) (2017), paper SM3M.1.

J. Demas, L. Rishøj, X. Liu, G. Prabhakar, and S. Ramachandran, “High-power, wavelength-tunable NIR all-fiber lasers via intermodal four-wave mixing,” in Conference on Lasers and Electro-Optics (CLEO) (2017), paper JTh5A.8.

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Figures (5)

Fig. 1.
Fig. 1. (a) Intensity profiles of the pump modes LP0,4 and LP0,5. (b) Schematic representation of intermodal parametric frequency conversion of the pump (green pulse) to the anti-Stokes (blue pulse) and Stokes (red pulse) waves along the length of the fiber. Schematic representation of (c) energy conservation and (d) phase matching for four-wave mixing. (e) Phase matching in the effective refractive index picture: solutions exist where the straight, dashed lines intersect the neff curves for different modes (solid lines), where integer “m” denotes radial mode order. (f) Parametric gain for typical intermodal processes (black line) and a group-velocity-tailored process (purple line), offset for clarity; the purple line in (e) is tangent to the neff curves for an extended wavelength range leading to broadband parametric gain.
Fig. 2.
Fig. 2. (a) Setup schematic for intermodal four-wave mixing experiments. SLM, spatial light modulator; OSA, optical spectrum analyzer. (b) Simulated electric field profiles of the LP0,4 (blue line) and LP0,5 (red line) modes plotted alongside the phase profile of a LP0,5 binary phase plate stretched by 12% (green line, right axis). (c) Simulations of the relative power coupled to each mode of the test fiber for a phase plate stretched by 12%. (d) Simulated group velocity for the LP0,4 and LP0,5 modes. (e) Simulated parametric gain for group-velocity-matched four-wave mixing pumped with LP0,4 and LP0,5.
Fig. 3.
Fig. 3. (a) Spontaneous four-wave mixing spectra as a function of pump wavelength; each spectrum offset by 50 dB for clarity. (b) Zoom-in of the bandwidth-optimized spontaneous spectrum (λp=1047.6  nm) with mode images inset.
Fig. 4.
Fig. 4. Full experimental spectra as a function of seed wavelength (1535–1570 nm) showing conversion to the Ti:sapphire band (786–795 nm) with representative experimental mode images shown as inset; each spectrum offset by 70 dB for clarity.
Fig. 5.
Fig. 5. Experimentally measured pump profiles for the (a) pump (pump alone shown as a solid black line; pump combined with seed shown as a dotted purple line), (b) anti-Stokes, and (c) Stokes wavelengths for a 1545 nm wavelength seed. (d) Peak pump depletion (purple markers, left axis) and peak Stokes gain (brown markers, right axis) as a function of seed wavelength. (e) Peak power (anti-Stokes shown with blue markers; Stokes shown with red markers).

Tables (1)

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Table 1. Additional Simulated Broadband FWM Processes

Equations (3)

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Δβ=2πλpneff(j)(λp)+2πλpneff(k)(λp)2πλsneff(l)(λs)2πλasneff(m)(λas)0,
Δβ=βj(ωp)+βk(ωp)[βj(ωas)+Δωdβjdω|ωas][βk(ωs)Δωdβkdω|ωs].
Δβ=Δω(dβkdω|ωsdβjdω|ωas).