Abstract

We report an analytic model for quantitatively calculating the transmission attenuation of single-wall hollow-core anti-resonant fibers. Our calculations unveil the light leakage dependences on azimuthal angle, polarization, and geometrical shape and have been examined in a variety of fiber geometries. Based on our model, a simple and clear picture about light guidance in hollow-core lattice fibers is presented. Formation of equiphase surface at fiber’s outermost boundary and light emission ruled by Helmholtz equation in transverse plane constitute the basis of this picture. Using this picture, we explain how the geometrical shape of a single-wall hollow-core fiber influences its transmission properties.

© 2014 Optical Society of America

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References

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  1. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multi-octave optical-frequency combs,” Science 318(5853), 1118–1121 (2007).
    [Crossref] [PubMed]
  2. F. Luan, J. C. Knight, P. St. J. Russell, S. Campbell, D. Xiao, D. T. Reid, B. J. Mangan, D. P. Williams, and P. J. Roberts, “Femtosecond soliton pulse delivery at 800nm wavelength in hollow-core photonic bandgap fibers,” Opt. Express 12(5), 835–840 (2004).
    [Crossref] [PubMed]
  3. Y. Y. Wang, X. Peng, M. Alharbi, C. F. Dutin, T. D. Bradley, F. Gérôme, M. Mielke, T. Booth, and F. Benabid, “Design and fabrication of hollow-core photonic crystal fibers for high-power ultrashort pulse transportation and pulse compression,” Opt. Lett. 37(15), 3111–3113 (2012).
    [Crossref] [PubMed]
  4. M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
    [Crossref]
  5. J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
    [Crossref] [PubMed]
  6. W. Belardi and J. C. Knight, “Effect of core boundary curvature on the confinement losses of hollow antiresonant fibers,” Opt. Express 21(19), 21912–21917 (2013).
    [Crossref] [PubMed]
  7. F. Couny, F. Benabid, and P. S. Light, “Large-pitch kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. 31(24), 3574–3576 (2006).
    [Crossref] [PubMed]
  8. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27(18), 1592–1594 (2002).
    [Crossref] [PubMed]
  9. P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13(1), 236–244 (2005).
    [Crossref] [PubMed]
  10. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
    [Crossref] [PubMed]
  11. T. D. Hedley, D. M. Bird, F. Benabid, J. C. Knight, and P. S. J. Russell, “Modelling of a novel hollow-core photonic crystal fibre,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper QTuL4.
  12. G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. S. J. Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express 15(20), 12680–12685 (2007).
    [Crossref] [PubMed]
  13. S. Février, B. Beaudou, and P. Viale, “Understanding origin of loss in large pitch hollow-core photonic crystal fibers and their design simplification,” Opt. Express 18(5), 5142–5150 (2010).
    [Crossref] [PubMed]
  14. M. Alharbi, T. Bradley, B. Debord, C. Fourcade-Dutin, D. Ghosh, L. Vincetti, F. Gérôme, and F. Benabid, “Hypocycloid-shaped hollow-core photonic crystal fiber Part II: Cladding effect on confinement and bend loss,” Opt. Express 21(23), 28609–28616 (2013).
    [Crossref] [PubMed]
  15. Y. Y. Wang, N. V. Wheeler, F. Couny, P. J. Roberts, and F. Benabid, “Low loss broadband transmission in hypocycloid-core Kagome hollow-core photonic crystal fiber,” Opt. Lett. 36(5), 669–671 (2011).
    [Crossref] [PubMed]
  16. A. D. Pryamikov, A. S. Biriukov, A. F. Kosolapov, V. G. Plotnichenko, S. L. Semjonov, and E. M. Dianov, “Demonstration of a waveguide regime for a silica hollow-core microstructured optical fiber with a negative curvature of the core boundary in the spectral region > 3.5 μm,” Opt. Express 19(2), 1441–1448 (2011).
    [Crossref] [PubMed]
  17. B. Debord, M. Alharbi, T. Bradley, C. Fourcade-Dutin, Y. Y. Wang, L. Vincetti, F. Gérôme, and F. Benabid, “Hypocycloid-shaped hollow-core photonic crystal fiber Part I: Arc curvature effect on confinement loss,” Opt. Express 21(23), 28597–28608 (2013).
    [Crossref] [PubMed]
  18. J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photon. 1(1), 58–106 (2009).
    [Crossref]
  19. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
    [Crossref]
  20. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd Edition, (Academic, 1991).
  21. A. W. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall, 1983).
  22. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68(9), 1196–1201 (1978).
    [Crossref]
  23. S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33(4/5), 359–371 (2001).
    [Crossref]
  24. C. A. Balanis, Antenna Theory: Analysis and Design, 3rd Edition, (John Wiley & Sons, 2005).
  25. A. N. Kolyadin, A. F. Kosolapov, A. D. Pryamikov, A. S. Biriukov, V. G. Plotnichenko, and E. M. Dianov, “Light transmission in negative curvature hollow core fiber in extremely high material loss region,” Opt. Express 21(8), 9514–9519 (2013).
    [Crossref] [PubMed]
  26. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th Edition, (Cambridge University Press, 1999).

2013 (4)

2012 (1)

2011 (2)

2010 (1)

2009 (1)

2007 (3)

G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. S. J. Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express 15(20), 12680–12685 (2007).
[Crossref] [PubMed]

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multi-octave optical-frequency combs,” Science 318(5853), 1118–1121 (2007).
[Crossref] [PubMed]

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
[Crossref]

2006 (1)

2005 (1)

2004 (1)

2002 (1)

2001 (1)

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33(4/5), 359–371 (2001).
[Crossref]

1999 (1)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[Crossref] [PubMed]

1998 (1)

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[Crossref] [PubMed]

1986 (1)

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[Crossref]

1978 (1)

Abeeluck, A. K.

Alharbi, M.

Allan, D. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[Crossref] [PubMed]

Beaudou, B.

Belardi, W.

Benabid, F.

Biriukov, A. S.

Birks, T. A.

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13(1), 236–244 (2005).
[Crossref] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[Crossref] [PubMed]

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[Crossref] [PubMed]

Booth, T.

Bradley, T.

Bradley, T. D.

Broeng, J.

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[Crossref] [PubMed]

Burger, S.

Campbell, S.

Couny, F.

Cregan, R. F.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[Crossref] [PubMed]

Cucinotta, A.

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33(4/5), 359–371 (2001).
[Crossref]

Debord, B.

Dianov, E. M.

Duguay, M. A.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[Crossref]

Dutin, C. F.

Eggleton, B. J.

Farr, L.

Février, S.

Fourcade-Dutin, C.

Gérôme, F.

Ghosh, D.

Headley, C.

Hu, J.

Knight, J. C.

Koch, T. L.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[Crossref]

Kokubun, Y.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[Crossref]

Kolyadin, A. N.

Kosolapov, A. F.

Light, P. S.

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multi-octave optical-frequency combs,” Science 318(5853), 1118–1121 (2007).
[Crossref] [PubMed]

F. Couny, F. Benabid, and P. S. Light, “Large-pitch kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. 31(24), 3574–3576 (2006).
[Crossref] [PubMed]

Litchinitser, N. M.

Luan, F.

Mangan, B. J.

Marom, E.

Mason, M. W.

Menyuk, C. R.

Mielke, M.

Pearce, G. J.

Peng, X.

Pfeiffer, L.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[Crossref]

Plotnichenko, V. G.

Poulton, C. G.

Pryamikov, A. D.

Raymer, M. G.

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multi-octave optical-frequency combs,” Science 318(5853), 1118–1121 (2007).
[Crossref] [PubMed]

Reid, D. T.

Roberts, P. J.

Russell, P. S. J.

G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. S. J. Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express 15(20), 12680–12685 (2007).
[Crossref] [PubMed]

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13(1), 236–244 (2005).
[Crossref] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[Crossref] [PubMed]

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[Crossref] [PubMed]

Russell, P. St. J.

Sabert, H.

Selleri, S.

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33(4/5), 359–371 (2001).
[Crossref]

Semjonov, S. L.

Tomlinson, A.

Tonouchi, M.

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
[Crossref]

Viale, P.

Vincetti, L.

Wang, Y. Y.

Wheeler, N. V.

Wiederhecker, G. S.

Williams, D. P.

Xiao, D.

Yariv, A.

Yeh, P.

Zoboli, M.

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33(4/5), 359–371 (2001).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (1)

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[Crossref]

J. Opt. Soc. Am. (1)

Nat. Photonics (1)

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
[Crossref]

Opt. Express (9)

F. Luan, J. C. Knight, P. St. J. Russell, S. Campbell, D. Xiao, D. T. Reid, B. J. Mangan, D. P. Williams, and P. J. Roberts, “Femtosecond soliton pulse delivery at 800nm wavelength in hollow-core photonic bandgap fibers,” Opt. Express 12(5), 835–840 (2004).
[Crossref] [PubMed]

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13(1), 236–244 (2005).
[Crossref] [PubMed]

S. Février, B. Beaudou, and P. Viale, “Understanding origin of loss in large pitch hollow-core photonic crystal fibers and their design simplification,” Opt. Express 18(5), 5142–5150 (2010).
[Crossref] [PubMed]

A. D. Pryamikov, A. S. Biriukov, A. F. Kosolapov, V. G. Plotnichenko, S. L. Semjonov, and E. M. Dianov, “Demonstration of a waveguide regime for a silica hollow-core microstructured optical fiber with a negative curvature of the core boundary in the spectral region > 3.5 μm,” Opt. Express 19(2), 1441–1448 (2011).
[Crossref] [PubMed]

A. N. Kolyadin, A. F. Kosolapov, A. D. Pryamikov, A. S. Biriukov, V. G. Plotnichenko, and E. M. Dianov, “Light transmission in negative curvature hollow core fiber in extremely high material loss region,” Opt. Express 21(8), 9514–9519 (2013).
[Crossref] [PubMed]

W. Belardi and J. C. Knight, “Effect of core boundary curvature on the confinement losses of hollow antiresonant fibers,” Opt. Express 21(19), 21912–21917 (2013).
[Crossref] [PubMed]

B. Debord, M. Alharbi, T. Bradley, C. Fourcade-Dutin, Y. Y. Wang, L. Vincetti, F. Gérôme, and F. Benabid, “Hypocycloid-shaped hollow-core photonic crystal fiber Part I: Arc curvature effect on confinement loss,” Opt. Express 21(23), 28597–28608 (2013).
[Crossref] [PubMed]

M. Alharbi, T. Bradley, B. Debord, C. Fourcade-Dutin, D. Ghosh, L. Vincetti, F. Gérôme, and F. Benabid, “Hypocycloid-shaped hollow-core photonic crystal fiber Part II: Cladding effect on confinement and bend loss,” Opt. Express 21(23), 28609–28616 (2013).
[Crossref] [PubMed]

G. J. Pearce, G. S. Wiederhecker, C. G. Poulton, S. Burger, and P. S. J. Russell, “Models for guidance in kagome-structured hollow-core photonic crystal fibres,” Opt. Express 15(20), 12680–12685 (2007).
[Crossref] [PubMed]

Opt. Lett. (4)

Opt. Quantum Electron. (1)

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33(4/5), 359–371 (2001).
[Crossref]

Science (3)

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[Crossref] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285(5433), 1537–1539 (1999).
[Crossref] [PubMed]

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multi-octave optical-frequency combs,” Science 318(5853), 1118–1121 (2007).
[Crossref] [PubMed]

Other (5)

T. D. Hedley, D. M. Bird, F. Benabid, J. C. Knight, and P. S. J. Russell, “Modelling of a novel hollow-core photonic crystal fibre,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference, Technical Digest (Optical Society of America, 2003), paper QTuL4.

C. A. Balanis, Antenna Theory: Analysis and Design, 3rd Edition, (John Wiley & Sons, 2005).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th Edition, (Cambridge University Press, 1999).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd Edition, (Academic, 1991).

A. W. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall, 1983).

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Figures (10)

Fig. 1
Fig. 1 (a) Schematic diagram of an M-type slab waveguide. (b) Field amplitudes (logarithmic scale) and phases of the dominant electric field components for the s- and p-polarization waves as a function of the x coordinate. Parameters used in calculations include ain = 2 μm, t = 0.67 μm, n2 = 1.45, and λ0 = 0.938 μm.
Fig. 2
Fig. 2 Attenuation coefficient and phase of the field at the outermost boundary (the pink line in the insert) for the fundamental core mode in an M-type slab waveguide. The vertical gray lines represent the resonant wavelengths of the glass modes. ain = 2 μm, n2 = 1.45, and t = 0.67 μm.
Fig. 3
Fig. 3 (a) Equivalence of a circular ring fiber and a series of slab structures. (b) Attenuation coefficients and propagation constants (real parts of the effective modal indices) of a single-wall circular ring fiber as a function of the wavelength. The precisely calculated results (hollow squares) are from a transfer matrix approach [22]. The results of our model (red curves) are from Eqs. (4) and (5). ain = 9.76 μm, n2 = 1.45, and t = 0.67 μm.
Fig. 4
Fig. 4 (a) Geometry transformation from a triangular fiber to a series of slab structures. The green arrows represent the equivalent light leaking processes occurring in 2D and 1D. The pink arrow denotes the radiation direction of ξ. (b) Analytically modelled (red curves) and numerically simulated (hollow squares) attenuation coefficients and propagation constants of the fundamental core mode in this triangle fiber as a function of the working wavelength (logarithmic scale). Parameters used in calculations: ain = 9.76 μm, n2 = 1.45, t = 0.67 μm.
Fig. 5
Fig. 5 Analytically modelled (red curves) and numerically simulated (black curves) field amplitudes and phases at (b) the outermost boundary and (c) a far-distance circle R. The schematic diagrams (a) depict a triangle single-wall fiber (orange), its outermost boundary (red), a far-distance circle R (gray), and the light polarizations (double arrows). Under each polarization, only the dominant electric field component, either Ex or Ey, is plotted. All the field amplitudes have been normalized for the convenience of comparison. Parameters used in calculations: ain = 9.76 μm, n2 = 1.45, t = 0.67 μm, R = 60 μm and λ0 = 0.68 μm.
Fig. 6
Fig. 6 Simulated attenuation spectra of a single-wall square fiber (black squares) and its variant (green squares), comparing with the calculated result based on our analytic model (red curve). ain = 9.76 μm, n2 = 1.45, t = 0.67 μm, and rc = 2.5 μm.
Fig. 7
Fig. 7 Simulated (black and blue curves) and modelled (red and green curves) attenuation spectra of a modified square fiber under vertical (black and red curves) and horizontal (blue and green curves) polarizations. The cyan and the pink dashed curves represent the two normal square fibers having uniform thicknesses of 0.6 and 0.67 μm respectively. In the vertical polarization, the modified square fiber shows lower loss than the normal square fiber having t = 0.67 μm in the wavelength range marked by the reddish shaded area. In the horizontal polarization, the modified square fiber shows lower loss than the normal square fiber having t = 0.6 μm in the greenish shaded area. ain = 9.76 μm, and n2 = 1.45.
Fig. 8
Fig. 8 Simulated and modelled attenuation spectra of a regular triangle (black), square (red), hexagon (green), and octagon (blue) single-wall fiber. The gray dashed curves in both graphs represent a circular ring fiber. ain = 9.76 μm, n2 = 1.45, and t = 0.67 μm.
Fig. 9
Fig. 9 Simulated and modelled attenuation spectra of a square shape fiber (black curves) and a hypocycloidal square fiber (red curves). The hypocycloid shape fiber is schematically depicted in the insert. ain = 9.76 μm, n2 = 1.45, and t = 0.67 μm.
Fig. 10
Fig. 10 Energy flow in the transverse plane. Field integral is implemented along the outermost boundary (the pink line) of the circular ring fiber (the orange ring). The polarization directions of E(s,p) and Ex,y, and the two azimuthal angles (ϕ, ξ) are depicted.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

s/p-Pol.: E y,x (z,x)=exp(iβz){ cos( k x1 | x |), (Core) A cl (s,p) cos( k x2 | x |+ ζ (s,p) ), (Cladding) A sur (s,p) exp[i k x1 (| x | a in t)], (Surrounding)
Im( n eff )= n 1 | E(x= a in +t+) | 2 n 1 2 [Re( n eff )] 2 2 k 0 Re( n eff ) 0 a in +t | E(x) | 2 dx n 1 | E(x= a in +t+) | 2 n 1 2 [Re( n eff )] 2 | E(x=0) | 2 k 0 a in Re( n eff )
| E (s,p) (at fiber's outer boundary) |= { | E 0 | 2 k 0 a in 2 Im[ n eff (s,p) (ϕ)]Re[ n eff (s,p) (ϕ)] 1Re [ n eff (s,p) (ϕ)] 2 } 1/2
Re( n eff )= 1 2π 0 2π {Re[ n eff (s) (ϕ)] cos 2 ϕ+Re[ n eff (p) (ϕ)] sin 2 ϕ}dϕ
α[ dB /m ]=8.69 k 0 Im( n eff )=8.69 k 0 | e Rad | 2 1Re ( n eff ) 2 / Re( n eff )
| e Rad | 2 1 8π a in 2 0 2π [ | e x (ξ) | 2 + | e y (ξ) | 2 ]dξ
{ e x (ξ)= 2 k 0 a in | E (p) || E (s) | | E 0 | cosϕsinϕ cos(ξϕ)+1 2 exp[i k T (cosξx+sinξy)]dl e y (ξ)= 2 k 0 a in | E (s) | cos 2 ϕ+| E (p) | sin 2 ϕ | E 0 | cos(ξϕ)+1 2 exp[i k T (cosξx+sinξy)]dl
| E (s,p) (at fiber's outer boundary) |= { | E 0 | 2 k 0 a in 2 Im[ n eff (s,p) (ϕ)]Re[ n eff (s,p) (ϕ)] 1Re [ n eff (s,p) (ϕ)] 2 } 1/2 κ(ϕ)
Re( n eff )= 1 2π 0 2π {Re[ n eff (s) (ϕ)] sin 2 θ+Re[ n eff (p) (ϕ)] cos 2 θ}dϕ
α[ dB /m ]=8.69 k 0 | e Rad | 2 1Re ( n eff ) 2 / Re( n eff )
| e Rad | 2 1 8π a in 2 0 2π [ | e x (ξ) | 2 + | e y (ξ) | 2 ]dξ
{ e x (ξ)= 2 k 0 a in | E (p) || E (s) | | E 0 | cosθsinθ sin(θξ)+1 2 e i k T (cosξx+sinξy) dL e y (ξ)= 2 k 0 a in | E (s) | sin 2 θ+| E (p) | cos 2 θ | E 0 | sin(θξ)+1 2 e i k T (cosξx+sinξy) dL
E x,y (r)= [ E x,y ( G / n )G( E x,y / n ) ]dl
E x,y (r) e 3πi /4 k T 2πr E x,y (r') cos( n ^ , s ^ )+1 2 exp(i k T s)dl
{ E x (r')=| E (p) |sinϕcosϕ| E (s) |cosϕsinϕ E y (r')=| E (s) | cos 2 ϕ+| E (p) | sin 2 ϕ
F T = k T Δz R | E | 2 dl = k T Δz k T 2π 0 2π [ | E x (ξ) | 2 + | E y (ξ) | 2 ]dξ { E x (ξ) [| E (p) || E (s) |]sinϕcosϕ cos( n ^ , s ^ )+1 2 exp(i k T s)dl E y (ξ) [| E (s) | cos 2 ϕ+| E (p) | sin 2 ϕ] cos( n ^ , s ^ )+1 2 exp(i k T s)dl
F L = k z ( Core | E | 2 ds | z Core | E | 2 ds | z+Δz )=[2 k 0 Im( n eff )Δz] k z Core | E(x,y) | 2 dA [2 k 0 Im( n eff )Δz] k 0 Re( n eff ) | E 0 | 2 2π a 2 2
Im( n eff ) 1Re ( n eff ) 2 8π a in 2 Re( n eff ) 0 2π [ | e x (ξ) | 2 + | e y (ξ) | 2 ]dξ { e x (ξ) ( e (p) e (s) )sinϕcosϕ cos( n ^ , s ^ )+1 2 exp(i k T s)dl e y (ξ) ( e (s) cos 2 ϕ+ e (p) sin 2 ϕ ) cos( n ^ , s ^ )+1 2 exp(i k T s)dl e (s,p) { Im[ n eff (s,p) (ϕ)]Re[ n eff (s,p) (ϕ)] 1Re [ n eff (s,p) (ϕ)] 2 } 1/2

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