Abstract

A scheme to enhance the fundamental-mode third harmonic generation efficiency in microfibers is presented. By introducing an appropriate counter-propagating pulse train, large propagation constant mismatch is partly overcome and nonlinear phase shifts could be corrected for, thus quasi-phase matching between the fundamental pump mode and the fundamental third harmonic mode is achieved, enabling the harmonic power to grow along the direction of propagation. Depending on the microfiber and pulse parameters, phase matching can enhance the conversion efficiency by several orders of magnitude with respect to the non-phase matched case. This scheme offers an alternative approach for harmonic generation and could potentially be applied to other small core waveguides.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  6. K. Tarnowski, B. Kibler, C. Finot, and W. Urbanczyk, “Quasi-phase-matched third harmonic generation in optical fibers using refractive-index gratings,” IEEE J. Quantum Elect. 47(5), 622–629 (2011).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  16. M. G. Moebius, F. Herrera, S. Griesse-Nascimento, O. Reshef, C. C. Evans, G. G. Guerreschi, A. Aspuru-Guzik, and E. Mazur, “Efficient photon triplet generation in integrated nanophotonic waveguides,” Opt. Express 24(9), 9932–9954(2016).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  27. D. Mao, X. Liu, Z. Sun, H. Lu, D. Han, G. Wang, and F. Wang, “Flexible high-repetition-rate ultrafast fiber laser,” Sci. Rep. 3, 3223(2013).
    [Crossref] [PubMed]
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    [Crossref]

2016 (3)

2015 (2)

2014 (1)

2013 (2)

C. Lecaplain and P. Grelu, “Multi-gigahertz repetition-rate-selectable passive harmonic mode locking of a fiber laser,” Opt. Express 21(9), 10897–10902 (2013).
[Crossref] [PubMed]

D. Mao, X. Liu, Z. Sun, H. Lu, D. Han, G. Wang, and F. Wang, “Flexible high-repetition-rate ultrafast fiber laser,” Sci. Rep. 3, 3223(2013).
[Crossref] [PubMed]

2012 (3)

2011 (2)

A. Lin, A. Ryasnyanskiy, and J. Toulouse, “Tunable third-harmonic generation in a solid-core tellurite glass fiber,” Opt. Lett. 36(17), 3437–3439 (2011).
[Crossref] [PubMed]

K. Tarnowski, B. Kibler, C. Finot, and W. Urbanczyk, “Quasi-phase-matched third harmonic generation in optical fibers using refractive-index gratings,” IEEE J. Quantum Elect. 47(5), 622–629 (2011).
[Crossref]

2010 (2)

2008 (2)

A. Bahabad, O. Cohen, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching and dispersion characterization of harmonic generation in the perturbative regime using counterpropagating beams,” Opt. Express 16(20), 15923–15931 (2008).
[Crossref] [PubMed]

B. Kibler, R. Fischer, G. Genty, D. N. Neshev, and J. M. Dudley, “Simultaneous fs pulse spectral broadening and third harmonic generation in highly nonlinear fibre: experiments and simulations,” Appl. Phys. B 91(2), 349–352(2008).
[Crossref]

2007 (2)

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic UV generation using low-order modes in a glass microfiber,” Opt. Commun. 274(2), 447–450 (2007).
[Crossref]

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[Crossref]

2005 (1)

2004 (1)

2003 (1)

1997 (1)

1993 (1)

D. Nicácio, E. Gouveia, N. Borges, and A. Gouveia-Neto, “Third-harmonic generation in GeO2-doped silica single-mode optical fibers,” Appl. Phys. Lett. 62(18), 2179–2181 (1993).
[Crossref]

1983 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2012), Chap. 1, Chap. 2, Chap. 3, Chap. 12.

Aspuru-Guzik, A.

Bahabad, A.

Bencheikh, K.

Borges, N.

D. Nicácio, E. Gouveia, N. Borges, and A. Gouveia-Neto, “Third-harmonic generation in GeO2-doped silica single-mode optical fibers,” Appl. Phys. Lett. 62(18), 2179–2181 (1993).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 1992), Chap. 2.

Brambilla, G.

Broderick, N. G. R.

Cavanna, A.

Chekhova, M. V.

Chemnitz, M.

Cheng, T.

Codemard, C. A.

Cohen, O.

A. Bahabad, O. Cohen, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching and dispersion characterization of harmonic generation in the perturbative regime using counterpropagating beams,” Opt. Express 16(20), 15923–15931 (2008).
[Crossref] [PubMed]

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[Crossref]

Coillet, A.

A. Coillet and P. Grelu, “Third-harmonic generation in optical microfibers: from silica experiments to highly nonlinear glass prospects,” Opt. Commun. 285(16), 3493–3497 (2012).
[Crossref]

Deng, D.

Ding, M.

Duan, Z.

Dudley, J. M.

B. Kibler, R. Fischer, G. Genty, D. N. Neshev, and J. M. Dudley, “Simultaneous fs pulse spectral broadening and third harmonic generation in highly nonlinear fibre: experiments and simulations,” Appl. Phys. B 91(2), 349–352(2008).
[Crossref]

Ebendorff-heidepriem, H.

Efimov, A.

Evans, C. C.

Feinberg, J.

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic UV generation using low-order modes in a glass microfiber,” Opt. Commun. 274(2), 447–450 (2007).
[Crossref]

Finot, C.

K. Tarnowski, B. Kibler, C. Finot, and W. Urbanczyk, “Quasi-phase-matched third harmonic generation in optical fibers using refractive-index gratings,” IEEE J. Quantum Elect. 47(5), 622–629 (2011).
[Crossref]

Fischer, R.

B. Kibler, R. Fischer, G. Genty, D. N. Neshev, and J. M. Dudley, “Simultaneous fs pulse spectral broadening and third harmonic generation in highly nonlinear fibre: experiments and simulations,” Appl. Phys. B 91(2), 349–352(2008).
[Crossref]

Gabriagues, J. M.

Gao, W.

Genty, G.

B. Kibler, R. Fischer, G. Genty, D. N. Neshev, and J. M. Dudley, “Simultaneous fs pulse spectral broadening and third harmonic generation in highly nonlinear fibre: experiments and simulations,” Appl. Phys. B 91(2), 349–352(2008).
[Crossref]

Gooijer, F.

Gouveia, E.

D. Nicácio, E. Gouveia, N. Borges, and A. Gouveia-Neto, “Third-harmonic generation in GeO2-doped silica single-mode optical fibers,” Appl. Phys. Lett. 62(18), 2179–2181 (1993).
[Crossref]

Gouveia-Neto, A.

D. Nicácio, E. Gouveia, N. Borges, and A. Gouveia-Neto, “Third-harmonic generation in GeO2-doped silica single-mode optical fibers,” Appl. Phys. Lett. 62(18), 2179–2181 (1993).
[Crossref]

Grelu, P.

C. Lecaplain and P. Grelu, “Multi-gigahertz repetition-rate-selectable passive harmonic mode locking of a fiber laser,” Opt. Express 21(9), 10897–10902 (2013).
[Crossref] [PubMed]

A. Coillet and P. Grelu, “Third-harmonic generation in optical microfibers: from silica experiments to highly nonlinear glass prospects,” Opt. Commun. 285(16), 3493–3497 (2012).
[Crossref]

Griesse-Nascimento, S.

Grubsky, V.

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic UV generation using low-order modes in a glass microfiber,” Opt. Commun. 274(2), 447–450 (2007).
[Crossref]

V. Grubsky and A. Savchenko, “Glass micro-fibers for efficient third harmonic generation,” Opt. Express 13(18), 6798–6806 (2005).
[Crossref] [PubMed]

Guerreschi, G. G.

Han, D.

D. Mao, X. Liu, Z. Sun, H. Lu, D. Han, G. Wang, and F. Wang, “Flexible high-repetition-rate ultrafast fiber laser,” Sci. Rep. 3, 3223(2013).
[Crossref] [PubMed]

Herrera, F.

Hölzer, P.

Horak, P.

Ishaaya, A. A.

Jiang, X.

Joly, N. Y.

Jung, Y.

Just, F.

Kapteyn, H. C.

A. Bahabad, O. Cohen, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching and dispersion characterization of harmonic generation in the perturbative regime using counterpropagating beams,” Opt. Express 16(20), 15923–15931 (2008).
[Crossref] [PubMed]

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[Crossref]

Khudus, M. I. M. A.

Kibler, B.

K. Tarnowski, B. Kibler, C. Finot, and W. Urbanczyk, “Quasi-phase-matched third harmonic generation in optical fibers using refractive-index gratings,” IEEE J. Quantum Elect. 47(5), 622–629 (2011).
[Crossref]

B. Kibler, R. Fischer, G. Genty, D. N. Neshev, and J. M. Dudley, “Simultaneous fs pulse spectral broadening and third harmonic generation in highly nonlinear fibre: experiments and simulations,” Appl. Phys. B 91(2), 349–352(2008).
[Crossref]

Knight, J. C.

Kostecki, R.

Krabshuis, G.

Lecaplain, C.

Lee, T.

Leuchs, G.

Levenson, J. A.

Liao, M.

Lin, A.

Liu, X.

D. Mao, X. Liu, Z. Sun, H. Lu, D. Han, G. Wang, and F. Wang, “Flexible high-repetition-rate ultrafast fiber laser,” Sci. Rep. 3, 3223(2013).
[Crossref] [PubMed]

Lou, J. Y.

Lu, H.

D. Mao, X. Liu, Z. Sun, H. Lu, D. Han, G. Wang, and F. Wang, “Flexible high-repetition-rate ultrafast fiber laser,” Sci. Rep. 3, 3223(2013).
[Crossref] [PubMed]

Lytle, A. L.

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[Crossref]

Mao, D.

D. Mao, X. Liu, Z. Sun, H. Lu, D. Han, G. Wang, and F. Wang, “Flexible high-repetition-rate ultrafast fiber laser,” Sci. Rep. 3, 3223(2013).
[Crossref] [PubMed]

Matsumoto, M.

Mazur, E.

Mélin, G.

Misumi, T.

Moebius, M. G.

Monro, T. M.

Montz, Z.

Murnane, M. M.

A. Bahabad, O. Cohen, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching and dispersion characterization of harmonic generation in the perturbative regime using counterpropagating beams,” Opt. Express 16(20), 15923–15931 (2008).
[Crossref] [PubMed]

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[Crossref]

Nazarkin, A.

Neshev, D. N.

B. Kibler, R. Fischer, G. Genty, D. N. Neshev, and J. M. Dudley, “Simultaneous fs pulse spectral broadening and third harmonic generation in highly nonlinear fibre: experiments and simulations,” Appl. Phys. B 91(2), 349–352(2008).
[Crossref]

Nicácio, D.

D. Nicácio, E. Gouveia, N. Borges, and A. Gouveia-Neto, “Third-harmonic generation in GeO2-doped silica single-mode optical fibers,” Appl. Phys. Lett. 62(18), 2179–2181 (1993).
[Crossref]

Nold, J.

Ohishi, Y.

Omenetto, F. G.

Peatross, J.

Podlipensky, A.

Popmintchev, T.

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[Crossref]

Prokopovich, I.

Reshef, O.

Richard, S.

Russell, P. St. J.

Ryasnyanskiy, A.

Savchenko, A.

Scharrer, M.

Schmidt, M. A.

Sun, Z.

D. Mao, X. Liu, Z. Sun, H. Lu, D. Han, G. Wang, and F. Wang, “Flexible high-repetition-rate ultrafast fiber laser,” Sci. Rep. 3, 3223(2013).
[Crossref] [PubMed]

Suzuki, T.

Tarnowski, K.

K. Tarnowski, B. Kibler, C. Finot, and W. Urbanczyk, “Quasi-phase-matched third harmonic generation in optical fibers using refractive-index gratings,” IEEE J. Quantum Elect. 47(5), 622–629 (2011).
[Crossref]

Taylor, A. J.

Tong, L. M.

Toulouse, J.

Urbanczyk, W.

K. Tarnowski, B. Kibler, C. Finot, and W. Urbanczyk, “Quasi-phase-matched third harmonic generation in optical fibers using refractive-index gratings,” IEEE J. Quantum Elect. 47(5), 622–629 (2011).
[Crossref]

Voronov, S.

Wadsworth, W. J.

Wang, F.

D. Mao, X. Liu, Z. Sun, H. Lu, D. Han, G. Wang, and F. Wang, “Flexible high-repetition-rate ultrafast fiber laser,” Sci. Rep. 3, 3223(2013).
[Crossref] [PubMed]

Wang, G.

D. Mao, X. Liu, Z. Sun, H. Lu, D. Han, G. Wang, and F. Wang, “Flexible high-repetition-rate ultrafast fiber laser,” Sci. Rep. 3, 3223(2013).
[Crossref] [PubMed]

Warren-smith, S. C.

Wie, J.

Wong, G. K. L.

Zhang, X.

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[Crossref]

Zhou, X.

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[Crossref]

Appl. Phys. B (1)

B. Kibler, R. Fischer, G. Genty, D. N. Neshev, and J. M. Dudley, “Simultaneous fs pulse spectral broadening and third harmonic generation in highly nonlinear fibre: experiments and simulations,” Appl. Phys. B 91(2), 349–352(2008).
[Crossref]

Appl. Phys. Lett. (1)

D. Nicácio, E. Gouveia, N. Borges, and A. Gouveia-Neto, “Third-harmonic generation in GeO2-doped silica single-mode optical fibers,” Appl. Phys. Lett. 62(18), 2179–2181 (1993).
[Crossref]

IEEE J. Quantum Elect. (1)

K. Tarnowski, B. Kibler, C. Finot, and W. Urbanczyk, “Quasi-phase-matched third harmonic generation in optical fibers using refractive-index gratings,” IEEE J. Quantum Elect. 47(5), 622–629 (2011).
[Crossref]

J. Optics (1)

G. Brambilla, “Optical fibre nanowires and microwires: a review,” J. Optics 12(4), 043001 (2010).
[Crossref]

Nat. Phys. (1)

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[Crossref]

Opt. Commun. (2)

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic UV generation using low-order modes in a glass microfiber,” Opt. Commun. 274(2), 447–450 (2007).
[Crossref]

A. Coillet and P. Grelu, “Third-harmonic generation in optical microfibers: from silica experiments to highly nonlinear glass prospects,” Opt. Commun. 285(16), 3493–3497 (2012).
[Crossref]

Opt. Express (9)

J. Peatross, S. Voronov, and I. Prokopovich, “Selective zoning of high harmonic emission using counter-propagating light,” Opt. Express 1(5), 114–125 (1997).
[Crossref] [PubMed]

A. Efimov, A. J. Taylor, F. G. Omenetto, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Phase-matched third harmonic generation in microstructured fibers,” Opt. Express 11(20), 2567–2576 (2003).
[Crossref] [PubMed]

L. M. Tong, J. Y. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12(6), 1025–1035 (2004).
[Crossref] [PubMed]

V. Grubsky and A. Savchenko, “Glass micro-fibers for efficient third harmonic generation,” Opt. Express 13(18), 6798–6806 (2005).
[Crossref] [PubMed]

A. Bahabad, O. Cohen, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching and dispersion characterization of harmonic generation in the perturbative regime using counterpropagating beams,” Opt. Express 16(20), 15923–15931 (2008).
[Crossref] [PubMed]

M. G. Moebius, F. Herrera, S. Griesse-Nascimento, O. Reshef, C. C. Evans, G. G. Guerreschi, A. Aspuru-Guzik, and E. Mazur, “Efficient photon triplet generation in integrated nanophotonic waveguides,” Opt. Express 24(9), 9932–9954(2016).
[Crossref] [PubMed]

S. C. Warren-smith, J. Wie, M. Chemnitz, R. Kostecki, H. Ebendorff-heidepriem, T. M. Monro, and M. A. Schmidt, “Third harmonic generation in exposed-core microstructured optical fibers,” Opt. Express 24(16),17860–17867 (2016).
[Crossref] [PubMed]

T. Lee, Y. Jung, C. A. Codemard, M. Ding, N. G. R. Broderick, and G. Brambilla, “Broadband third harmonic generation in tapered silica fibres,” Opt. Express 20(8), 8503–8511 (2012).
[Crossref] [PubMed]

C. Lecaplain and P. Grelu, “Multi-gigahertz repetition-rate-selectable passive harmonic mode locking of a fiber laser,” Opt. Express 21(9), 10897–10902 (2013).
[Crossref] [PubMed]

Opt. Lett. (7)

T. Cheng, W. Gao, M. Liao, Z. Duan, D. Deng, M. Matsumoto, T. Misumi, T. Suzuki, and Y. Ohishi, “Tunable third-harmonic generation in a chalcogenide-tellurite hybrid optical fiber with high refractive index difference,” Opt. Lett. 39(4), 1005–1007 (2014).
[Crossref] [PubMed]

Z. Montz and A. A. Ishaaya, “Dual-bandgap hollow-core photonic crystal fibers for third harmonic generation,” Opt. Lett. 40(1), 56–59 (2015).
[Crossref]

M. I. M. A. Khudus, T. Lee, P. Horak, and G. Brambilla, “Effect of intrinsic surface roughness on the efficiency of intermodal phase matching in silica optical nanofibers,” Opt. Lett. 40(7), 1318–1321 (2015).
[Crossref] [PubMed]

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Optica (1)

Sci. Rep. (1)

D. Mao, X. Liu, Z. Sun, H. Lu, D. Han, G. Wang, and F. Wang, “Flexible high-repetition-rate ultrafast fiber laser,” Sci. Rep. 3, 3223(2013).
[Crossref] [PubMed]

Other (3)

R. W. Boyd, Nonlinear Optics (Academic, 1992), Chap. 2.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2012), Chap. 1, Chap. 2, Chap. 3, Chap. 12.

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Figures (9)

Fig. 1
Fig. 1 (a) Dependence of effective index on microfiber diameter for the fundamental pump mode HE11(ω1) (red) and for the third harmonic hybrid HEνm and EHνm modes with azimuthal order ν = 1 (solid blue) and ν = 3 (dashed blue). The pump wavelength is 1550nm. (b) Overlap integral J3 between the fundamental pump mode and the fundamental third harmonic mode for different microfiber diameters.
Fig. 2
Fig. 2 Schematic of third harmonic generation configuration when a counter-propagating pulse train is introduced.
Fig. 3
Fig. 3 (a) Magnitude and (b) phase of the backward-to-forward pump amplitude ratio r required for quasi-phase matching in different zones along the forward pump propagation. G = 0, p = 2, q = 1, |r| = 1.73, ϕr1 = −1.6434, Δϕr = 0.6166, and Lc = 0.8μm.
Fig. 4
Fig. 4 Third harmonic conversion against the propagation distance: (a) with only the forward pump; (b), (c) and (d) with both the forward pump and the counter-propagating pulses. Lc = 0.8μm.
Fig. 5
Fig. 5 (a) Magnitude and (b) phase of r 1 (solid green with square), r 1 (solid blue) and r 1 (dashed red with circle), three solutions to the first backward pulse required for quasi-phase matching with different enhancement coefficient G. p = 30000, q = 1001, Lc = 0.8μm.
Fig. 6
Fig. 6 Comparison between the analytical approximation and the numerical simulations with and without slowly-varying-amplitude approximation (SVAA). p = 2, q = 1, |r| = 1.73, ϕr1 = −1.6434, Δϕr = 0.6166, Lc = 0.8μm.
Fig. 7
Fig. 7 Third harmonic conversion against the propagation distance when the counter-propagating pulses are injected with stepped phase (Δϕr =−0.0145) or identical phase (Δϕr =0, ϕr = ϕr1). p = 3016, q = 1201, |r| = 8.99, ϕr1 = 1.3959, Lc = 0.8μm.
Fig. 8
Fig. 8 Third harmonic conversion against the propagation distance when SPM and XPM effects are corrected for with different ratio r. p = 3108, q = 1201, Lc = 0.8μm.
Fig. 9
Fig. 9 THG enhancement factor against the relative counter-propagating pulse amplitude with a microfiber 120mm long. T: pulse width, RR: pulse repetition rate.

Equations (10)

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E ( x , y , z , t ) = j = 1 , 3 E j ( x , y , z , t ) = 1 2 x j = 1 , 3 A j ( z , t ) F j ( x , y ) exp [ i ( β j z ω j t ) ] + c . c . ,
1 i 2 k 1 2 A 1 z 2 + A 1 z + 1 v g 1 A 1 t + i β 21 2 2 A 1 t 2 = i n ( 2 ) k 1 [ ( J 1 | A 1 | 2 + 2 J 2 | A 3 | 2 ) A 1 + J 3 A 1 * 2 A 3 exp ( i δ β z ) ] ,
1 i 6 k 1 2 A 3 z 2 + A 3 z + 1 v g 3 A 3 t + i β 23 2 2 A 3 t 2 = i n ( 2 ) k 1 [ ( 6 J 2 | A 1 | 2 + 3 J 5 | A 3 | 2 ) A 3 + J 3 * A 1 * 3 exp ( i δ β z ) ] ,
1 i 2 k 1 2 A 1 z 2 + A 1 z = i n ( 2 ) k 1 [ ( J 1 | A 1 | 2 + 2 J 2 | A 3 | 2 ) A 1 + J 3 A 1 * 2 A 3 exp ( i δ β z ) ] ,
1 i 6 k 1 2 A 3 z 2 + A 3 z = i n ( 2 ) k 1 [ ( 6 J 2 | A 1 | 2 + 3 J 5 | A 3 | 2 ) A 3 + J 3 * A 1 3 exp ( i δ β z ) ] .
E 1 ( x , y , z , t ) = E 1 + ( x , y , z , t ) + E 1 ( x , y , z , t ) = 1 2 x A 1 + ( z , t ) F 1 ( x , y ) exp [ i ( β 1 z ω 1 t ) ] + 1 2 x A 1 ( z , t ) F 1 ( x , y ) exp [ i ( β 1 z ω 1 t ) ] + c . c . ,
E 1 ( x , y , z , t ) = 1 2 x A 1 ( z , t ) F 1 ( x , y ) exp [ i ( β 1 z ω 1 t ) ] + c . c . ,
A 1 ( z , t ) = A 1 + [ 1 + r exp ( i 2 β 1 z ) ] .
A 3 ( z 1 z 2 ) C z 1 z 2 [ 1 + r exp ( i 2 β 1 z ) ] 3 exp ( i δ β z ) d z ,
A 3 p u l s e = C ( N p q ) L c N p L c [ 1 + r exp ( i 2 β 1 z ) ] 3 exp ( i δ β z ) d z = G A 3 0

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