Abstract

We show that in twisted microstructured optical fibers (MOFs) the coupling between the core and cladding modes can be obtained for helix pitch much greater than previously considered. We provide an analytical model describing scaling properties of the twisted MOFs, which relates coupling conditions to dimensionless ratios between the wavelength, the lattice pitch and the helix pitch of the twisted fiber. Furthermore, we verify our model using a rigorous numerical method based on the transformation optics formalism and study its limitations. The obtained results show that for appropriately designed twisted MOFs, distinct, high loss resonance peaks can be obtained in a broad wavelength range already for the fiber with 9 mm helix pitch, thus allowing for fabrication of coupling based devices using a less demanding method involving preform spinning.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref] [PubMed]
  5. V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  13. X. M. Xi, G. K. L. Wong, M. H. Frosz, F. Babic, G. Ahmed, X. Jiang, T. G. Euser, and P. S. J. Russell, “Orbital-angular-momentum-preserving helical Bloch modes in twisted photonic crystal fiber,” Optica 1(3), 165–169 (2014).
    [Crossref]
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    [Crossref] [PubMed]
  15. M. Napiorkowski and W. Urbanczyk, “Role of symmetry in mode coupling in twisted microstructured optical fibers,” Opt. Lett. 43(3), 395–398 (2018).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  23. M. Bass, Handbook of Optics, Vol. 4, 3rd ed. (McGraw-Hill, 2009).
  24. A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
    [Crossref]
  25. A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
    [Crossref]
  26. Z. Zhang, Y. Shi, B. Bian, and J. Lu, “Dependence of leaky mode coupling on loss in photonic crystal fiber with hybrid cladding,” Opt. Express 16(3), 1915–1922 (2008).
    [Crossref] [PubMed]
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2018 (1)

2017 (1)

P. S. J. Russell, R. Beravat, and G. K. L. Wong, “Helically twisted photonic crystal fibres,” Philos Trans A Math Phys Eng Sci 375(2087), 20150440 (2017).
[Crossref] [PubMed]

2014 (3)

2013 (2)

C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013).
[Crossref]

X. Xi, G. K. L. Wong, T. Weiss, and P. S. J. Russell, “Measuring mechanical strain and twist using helical photonic crystal fiber,” Opt. Lett. 38(24), 5401–5404 (2013).
[Crossref] [PubMed]

2012 (1)

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

2011 (1)

2009 (1)

W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009).
[Crossref]

2008 (2)

C. N. Alexeyev and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period helical core optical fibers,” Phys. Rev. A 78(4), 043828 (2008).
[Crossref]

Z. Zhang, Y. Shi, B. Bian, and J. Lu, “Dependence of leaky mode coupling on loss in photonic crystal fiber with hybrid cladding,” Opt. Express 16(3), 1915–1922 (2008).
[Crossref] [PubMed]

2007 (2)

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

V. I. Kopp, V. M. Churikov, G. Zhang, J. Singer, C. W. Draper, N. Chao, D. Neugroschl, and A. Z. Genack, “Single- and double-helix chiral fiber sensors,” J. Opt. Soc. Am. B 24(10), A48–A52 (2007).
[Crossref]

2006 (3)

2005 (1)

2004 (3)

A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
[Crossref]

M. Fuochi, J. Hayes, K. Furusawa, W. Belardi, J. Baggett, T. Monro, and D. Richardson, “Polarization mode dispersion reduction in spun large mode area silica holey fibres,” Opt. Express 12(9), 1972–1977 (2004).
[Crossref] [PubMed]

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004).
[Crossref] [PubMed]

2002 (1)

2000 (1)

1987 (1)

R. D. Birch, “Fabrication and Characterization of Circularly Birefringent Helical Fibers,” Electron. Lett. 23(1), 50–52 (1987).
[Crossref]

Agha, Y. O.

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

Ahmed, G.

Alexeyev, A. N.

C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013).
[Crossref]

Alexeyev, C. N.

C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013).
[Crossref]

C. N. Alexeyev and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period helical core optical fibers,” Phys. Rev. A 78(4), 043828 (2008).
[Crossref]

Andrés, P.

Babic, F.

Baggett, J.

Belardi, W.

Beravat, R.

P. S. J. Russell, R. Beravat, and G. K. L. Wong, “Helically twisted photonic crystal fibres,” Philos Trans A Math Phys Eng Sci 375(2087), 20150440 (2017).
[Crossref] [PubMed]

Bian, B.

Biancalana, F.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Birch, R. D.

R. D. Birch, “Fabrication and Characterization of Circularly Birefringent Helical Fibers,” Electron. Lett. 23(1), 50–52 (1987).
[Crossref]

Chamorovskii, Y.

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Chang, G.

Chao, N.

Churikov, V. M.

Clarkson, W. A.

Conti, C.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Cooper, L. J.

de Sterke, C.

Draper, C. W.

Euser, T. G.

Ferrando, A.

Frosz, M. H.

Fuochi, M.

Furusawa, K.

Galvanauskas, A.

Genack, A. Z.

Gubin, V.

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Guenneau, S.

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
[Crossref]

Hayes, J.

Hu, I. N.

Isaev, V.

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Jiang, X.

Kang, M. S.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Kaplan, A.

Kopp, V. I.

Kuhlmey, B.

Lapin, B. P.

C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013).
[Crossref]

Lee, H. W.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Lee, Y. L.

W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009).
[Crossref]

Liu, C. H.

Lu, J.

Ma, X.

Maystre, D.

McPhedran, R.

Milione, G.

C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013).
[Crossref]

Miret, J. J.

Monro, T.

Morshnev, S.

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Napiorkowski, M.

Neugroschl, D.

Nicolet, A.

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
[Crossref]

Noh, Y.

W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009).
[Crossref]

Oh, K.

W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009).
[Crossref]

Olszewski, J.

Park, J.

Pendry, J. B.

Renversez, G.

Richardson, D.

Robinson, P.

Russell, P. S. J.

P. S. J. Russell, R. Beravat, and G. K. L. Wong, “Helically twisted photonic crystal fibres,” Philos Trans A Math Phys Eng Sci 375(2087), 20150440 (2017).
[Crossref] [PubMed]

X. M. Xi, G. K. L. Wong, M. H. Frosz, F. Babic, G. Ahmed, X. Jiang, T. G. Euser, and P. S. J. Russell, “Orbital-angular-momentum-preserving helical Bloch modes in twisted photonic crystal fiber,” Optica 1(3), 165–169 (2014).
[Crossref]

X. Xi, G. K. L. Wong, T. Weiss, and P. S. J. Russell, “Measuring mechanical strain and twist using helical photonic crystal fiber,” Opt. Lett. 38(24), 5401–5404 (2013).
[Crossref] [PubMed]

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Sahu, J. K.

Sazonov, A.

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Schurig, D.

Shi, Y.

Shin, W.

W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009).
[Crossref]

Silvestre, E.

Singer, J.

Smith, D. R.

Starostin, N.

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Szpulak, M.

Urbanczyk, W.

Usov, A.

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Wang, P.

Weiss, T.

X. Xi, G. K. L. Wong, T. Weiss, and P. S. J. Russell, “Measuring mechanical strain and twist using helical photonic crystal fiber,” Opt. Lett. 38(24), 5401–5404 (2013).
[Crossref] [PubMed]

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Wlodawski, M.

Wong, G. K. L.

P. S. J. Russell, R. Beravat, and G. K. L. Wong, “Helically twisted photonic crystal fibres,” Philos Trans A Math Phys Eng Sci 375(2087), 20150440 (2017).
[Crossref] [PubMed]

X. M. Xi, G. K. L. Wong, M. H. Frosz, F. Babic, G. Ahmed, X. Jiang, T. G. Euser, and P. S. J. Russell, “Orbital-angular-momentum-preserving helical Bloch modes in twisted photonic crystal fiber,” Optica 1(3), 165–169 (2014).
[Crossref]

X. Xi, G. K. L. Wong, T. Weiss, and P. S. J. Russell, “Measuring mechanical strain and twist using helical photonic crystal fiber,” Opt. Lett. 38(24), 5401–5404 (2013).
[Crossref] [PubMed]

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Xi, X.

Xi, X. M.

Yavorsky, M. A.

C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013).
[Crossref]

C. N. Alexeyev and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period helical core optical fibers,” Phys. Rev. A 78(4), 043828 (2008).
[Crossref]

Yu, B.

W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009).
[Crossref]

Zhang, G.

Zhang, Z.

Zhu, C.

Zolla, F.

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
[Crossref]

Electron. Lett. (1)

R. D. Birch, “Fabrication and Characterization of Circularly Birefringent Helical Fibers,” Electron. Lett. 23(1), 50–52 (1987).
[Crossref]

Eur. Phys. J. Appl. Phys. (1)

A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009).
[Crossref]

Opt. Express (7)

Opt. Lett. (4)

Optica (1)

Philos Trans A Math Phys Eng Sci (1)

P. S. J. Russell, R. Beravat, and G. K. L. Wong, “Helically twisted photonic crystal fibres,” Philos Trans A Math Phys Eng Sci 375(2087), 20150440 (2017).
[Crossref] [PubMed]

Phys. Rev. A (2)

C. N. Alexeyev and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period helical core optical fibers,” Phys. Rev. A 78(4), 043828 (2008).
[Crossref]

C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013).
[Crossref]

Quantum Electron. (1)

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Science (2)

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004).
[Crossref] [PubMed]

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Wave Random Complex (1)

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

Other (3)

B. T. Kuhlmey, Theoretical and Numerical Investigation of the Physics of Microstructured Optical Fibres, PhD thesis, (Université Aix-Marseille III and University of Sydney, 2004).

M. Bass, Handbook of Optics, Vol. 4, 3rd ed. (McGraw-Hill, 2009).

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, A. Argyros, and S. Leon-Saval, Foundations of Photonic Crystal Fibers, 2nd ed. (Imperial College, 2012), Chap. 8.

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Figures (6)

Fig. 1
Fig. 1 Cross-section of the analyzed twisted microstructured optical fiber. Arrow shows the twist direction.
Fig. 2
Fig. 2 Waveguide loss calculated versus the wavelength for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOF with ΛL = 3 µm, d/ΛL = 0.4 and A = 4π mm−1 (green), A = 6π mm−1 (blue) or A = 8π mm−1 (red). Dashed line shows the loss of the fundamental modes in untwisted fiber. In (b) we show the loss for the same parameters versus rescaled wavelength λ′ = λ·(4π mm−1)/A. In this case, colors of dashed lines correspond to values of A.
Fig. 3
Fig. 3 Electric field amplitude |E| distribution in the cladding modes at selected resonances shown in Fig. 2.
Fig. 4
Fig. 4 Loss calculated versus the wavelength for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOF with A = 4π mm−1, d/ΛL = 0.4 and ΛL = 3 µm (green), ΛL = 3·1.50.5 µm (blue) or ΛL = 3·20.5 (red) (b). Dashed lines show the loss of the fundamental modes in untwisted fibers, colors of lines correspond to ΛL. In (b) we show the loss for the same parameters versus rescaled wavelength λ′′ = λ·(3 µm)2L.
Fig. 5
Fig. 5 Loss calculated versus wavelength for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOFs with d/ΛL = 0.4, ΛL = 3 µm (red), ΛL = 9 µm (blue) or ΛL = 18 µm (green). In (a) ΛH = 0.5 mm for ΛL = 3 µm, ΛH = 4.5 mm for ΛL = 9 µm and ΛH = 18 mm for ΛL = 18 µm. In (b) ΛH = 0.25 mm for ΛL = 3 µm, ΛH = 2.25 mm for ΛL = 9 µm and ΛH = 9 mm for ΛL = 18 µm. In (c) ΛH = 0.1(6) mm for ΛL = 3 µm, ΛH = 1.5 mm for ΛL = 9 µm and ΛH = 6 mm for ΛL = 18 µm.
Fig. 6
Fig. 6 In (a) we present the loss spectra for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOF with A = 4π mm−1 and ΛL = 3µm calculated using dispersive nM (red) and nM = 1.45 (blue). Dashed lines show the loss of the fundamental modes in untwisted fibers. In (b) we present the imaginary part of neff versus normalized wavelength for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOF with A = 8π mm−1 and ΛL = 3µm (red) or by A = 4π mm−1 and ΛL = 6µm (blue) calculated using dispersive nM.

Equations (25)

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λ res = 2π l n SM ρ 2 A,
x i' = x i Q ,
ε i'j' = | det( Λ i' i ) | 1 Λ i' i Λ j' j ε ij ,
μ i'j' = | det( Λ i' i ) | 1 Λ i' i Λ j' j μ ij ,
Λ i' i = δ x i' δ x i .
ε' ε = ε r ' ε r = μ' μ = μ r ' μ r =Q.
×( 1 μ r ( r ) × E ( r ) )= ( 2π λ ) 2 ε r ( r ) E ( r ),
×( 1 Q μ r ( r ) × E ( r ) )= ( 2π λ' ) 2 Q ε r ( r ) E ( r ).
×( 1 μ r ( r ) × E ( r ) )= ( 2π Q 1 λ' ) 2 ε r ( r ) E ( r ),
λ'=Qλ.
n ˜ eff ( λ,d, Λ L ) n mat ( λ )=const = n ˜ eff ( λ/ Λ L ,d/ Λ L ),
n ˜ eff ( λ,d, Λ L , Λ H ) n mat ( λ )=const = n ˜ eff ( λ/ Λ L ,d/ Λ L , Λ H / Λ L ),
L= 20 ln10 2π λ Im( n ˜ eff )[ dB m ].
n ' eff = n eff +J λ Λ H ,
J ( λ,d, Λ L , Λ H ) n mat ( λ )=const =J( λ/ Λ L ,d/ Λ L , Λ H / Λ L )
n ' eff ( λ,d, Λ L , Λ H ) n mat ( λ )=const =n ' eff ( λ/ Λ L ,d/ Λ L , Λ H / Λ L ),
n eff n M ( λ Λ L ) 2 q,
n ' eff = n eff +( λ Λ L )J Λ L Λ H n M ( λ Λ L ) 2 q+( λ Λ L )J Λ L Λ H
Δn ' eff =n ' eff,1 n ' eff,2 ( λ Λ L ) 2 ( q 2 q 1 )( λ Λ L )( J 2 J 1 ) Λ L Λ H
λ res Λ L =2 ΔJ Δq Λ L Λ H .
λ res = 1 π ΔJ Δq Λ L 2 A.
n M ( λ )= 1+ i=1 3 A i λ 2 λ 2 Z i 2 ,
α= sin 1 ( Aρ 1+ A 2 ρ 2 )Aρ
αA Λ L ,
α A Λ L 2 =const 1 Λ L ,

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