Abstract

For small-bend radii, the waveguide condition for total internal reflection is violated in a large angular spread of incident angles at the interface of the fiber core. To account for this, we derived an improved semianalytic bend-loss model that allows for the propagation of radiated fields outside the plane of the fiber bend. This new model is applied to large-mode-area helical-core fibers (which require small-bend radii) for use as high-power fiber lasers and amplifiers. In particular, the limits of scaling the mode area while maintaining good beam quality are explored. Single-mode operation is expected for a 40μm core with a 0.10 numerical aperture, and scaling to a 100μm core diameter is shown to be possible. Additionally, helical fibers are shown to readily perform where conventional coiled fibers are not appropriate to operate long term.

© 2006 Optical Society of America

Full Article  |  PDF Article

Corrections

Z. Jiang and J. R. Marciante, "Mode-area scaling of helical-core, dual-clad fiber lasers and amplifiers using an improved bend-loss model: erratum," J. Opt. Soc. Am. B 25, 1105-1105 (2008)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-25-7-1105

References

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    [CrossRef]
  7. C. C. Renaud, R. J. Selvas-Aguilar, J. Nilsson, P. W. Turner, and A. B. Grudinin, "Compact high-energy Q-switched cladding-pumped fiber laser with a tuning range over 40 nm," IEEE Photon. Technol. Lett. 11, 976-978 (1999).
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  8. M. Hofer, M. E. Fermann, A. Galvanauskas, D. Harter, and R. S. Windeler, "Low-noise amplification of high-power pulses in multimode fibers," IEEE Photon. Technol. Lett. 11, 650-652 (1999).
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  23. K. S. Kaufman, R. Terras, and R. F. Mathis, "Curvature loss in multimode optical fibers," J. Opt. Soc. Am. 71, 1513-1518 (1981).
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2006 (2)

2005 (1)

M. Lehtonen, G. Genty, and H. Ludvigsen, "Tapered microstructured fibers for efficient coupling to optical waveguides: a numerical study," Appl. Phys. B 81, 295-300 (2005).
[CrossRef]

2000 (2)

1999 (4)

U. Griebner and H. Schönnagel, "Laser operation with nearly diffraction-limited output from a Yb:YAG multimode channel waveguide," Opt. Lett. 24, 750-752 (1999).
[CrossRef]

J. M. Sousa and O. G. Okhotnikov, "Multimode Er-doped fiber for single-transverse-mode amplification," Appl. Phys. Lett. 74, 1528-1530 (1999).
[CrossRef]

C. C. Renaud, R. J. Selvas-Aguilar, J. Nilsson, P. W. Turner, and A. B. Grudinin, "Compact high-energy Q-switched cladding-pumped fiber laser with a tuning range over 40 nm," IEEE Photon. Technol. Lett. 11, 976-978 (1999).
[CrossRef]

M. Hofer, M. E. Fermann, A. Galvanauskas, D. Harter, and R. S. Windeler, "Low-noise amplification of high-power pulses in multimode fibers," IEEE Photon. Technol. Lett. 11, 650-652 (1999).
[CrossRef]

1998 (1)

1997 (1)

1996 (1)

1993 (1)

D. Marcuse, "Bend loss of slab and fiber modes computed with diffraction theory," IEEE J. Quantum Electron. 29, 2957-2961 (1993).
[CrossRef]

1990 (1)

A. Altintas and J. D. Love, "Effective cut-offs for modes on helical fibers," Opt. Quantum Electron. 22, 213-226 (1990).
[CrossRef]

1987 (2)

1986 (1)

A. Harris and P. Castle, "Bend loss measurements on high numerical aperture single-mode fibers as a function of wavelength and bend radius," J. Lightwave Technol. 4, 34-40 (1986).
[CrossRef]

1982 (1)

J. I. Sakai and T. Kimura, "Analytical bending loss formula of optical fibers with field deformation," Radio Sci. 17, 21-29 (1982).
[CrossRef]

1981 (1)

1976 (3)

1972 (1)

D. Gloge, "Optical power flow in multimode fibers," Bell Syst. Tech. J. 51, 1767-1783 (1972).

Abrmczyk, J.

U. H. Manyam, B. Samson, V. Khitrov, D. P. Machewirth, J. Abrmczyk, N. Jacobson, J. Farroni, D. Guertin, A. Carter, and K. Tankala, "Laser fibers designed for single polarization output," in 2004 OSA Topical Meeting on Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004), pp. 118-122.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1989).

Altintas, A.

A. Altintas and J. D. Love, "Effective cut-offs for modes on helical fibers," Opt. Quantum Electron. 22, 213-226 (1990).
[CrossRef]

Alvarez-Chavez, J. A.

Alvarez-Estrada, R. F.

Birch, R. D.

R. D. Birch, "Fabrication and characterisation of circularly birefringent helical fibers," Electron. Lett. 23, 50-52 (1987).
[CrossRef]

Birks, T. A.

Broderick, N. G.

Calvo, M. L.

Caplen, J.

Carter, A.

U. H. Manyam, B. Samson, V. Khitrov, D. P. Machewirth, J. Abrmczyk, N. Jacobson, J. Farroni, D. Guertin, A. Carter, and K. Tankala, "Laser fibers designed for single polarization output," in 2004 OSA Topical Meeting on Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004), pp. 118-122.

Castle, P.

A. Harris and P. Castle, "Bend loss measurements on high numerical aperture single-mode fibers as a function of wavelength and bend radius," J. Lightwave Technol. 4, 34-40 (1986).
[CrossRef]

Clarkson, W. A.

Cooper, L. J.

Dong, L.

Ermeneux, S.

O. Schmidt, F. Roser, S. Linke, T. Schreiber, J. Limpert, S. Ermeneux, P. Yvernault, F. Salin, and A. Tunnermann, "High energy and high average power Q-switched photonic crystal fiber laser," in Advanced Solid-State Photonics 2006 (Optical Society of America, 2006).

Farroni, J.

U. H. Manyam, B. Samson, V. Khitrov, D. P. Machewirth, J. Abrmczyk, N. Jacobson, J. Farroni, D. Guertin, A. Carter, and K. Tankala, "Laser fibers designed for single polarization output," in 2004 OSA Topical Meeting on Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004), pp. 118-122.

Farrow, R. L.

G. R. Hadley, R. L. Farrow, and A. V. Smith, "Bent-waveguide modeling of large-mode-area double-clad fibers for high-power lasers," in Fiber Lasers III, A.J.Brown, J.Nilsson, D.J.Harter, and A.Tünnermann, eds., Proc. SPIE 61026102152006).

Fermann, M. E.

M. Hofer, M. E. Fermann, A. Galvanauskas, D. Harter, and R. S. Windeler, "Low-noise amplification of high-power pulses in multimode fibers," IEEE Photon. Technol. Lett. 11, 650-652 (1999).
[CrossRef]

Fini, J. M.

Galvanauskas, A.

M. Hofer, M. E. Fermann, A. Galvanauskas, D. Harter, and R. S. Windeler, "Low-noise amplification of high-power pulses in multimode fibers," IEEE Photon. Technol. Lett. 11, 650-652 (1999).
[CrossRef]

Genty, G.

M. Lehtonen, G. Genty, and H. Ludvigsen, "Tapered microstructured fibers for efficient coupling to optical waveguides: a numerical study," Appl. Phys. B 81, 295-300 (2005).
[CrossRef]

Gloge, D.

D. Gloge, "Optical power flow in multimode fibers," Bell Syst. Tech. J. 51, 1767-1783 (1972).

Goldberg, L.

Griebner, U.

Grudinin, A. B.

C. C. Renaud, R. J. Selvas-Aguilar, J. Nilsson, P. W. Turner, and A. B. Grudinin, "Compact high-energy Q-switched cladding-pumped fiber laser with a tuning range over 40 nm," IEEE Photon. Technol. Lett. 11, 976-978 (1999).
[CrossRef]

Grunwald, R.

Guertin, D.

U. H. Manyam, B. Samson, V. Khitrov, D. P. Machewirth, J. Abrmczyk, N. Jacobson, J. Farroni, D. Guertin, A. Carter, and K. Tankala, "Laser fibers designed for single polarization output," in 2004 OSA Topical Meeting on Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004), pp. 118-122.

Hadley, G. R.

G. R. Hadley, R. L. Farrow, and A. V. Smith, "Bent-waveguide modeling of large-mode-area double-clad fibers for high-power lasers," in Fiber Lasers III, A.J.Brown, J.Nilsson, D.J.Harter, and A.Tünnermann, eds., Proc. SPIE 61026102152006).

Harris, A.

A. Harris and P. Castle, "Bend loss measurements on high numerical aperture single-mode fibers as a function of wavelength and bend radius," J. Lightwave Technol. 4, 34-40 (1986).
[CrossRef]

Harter, D.

M. Hofer, M. E. Fermann, A. Galvanauskas, D. Harter, and R. S. Windeler, "Low-noise amplification of high-power pulses in multimode fibers," IEEE Photon. Technol. Lett. 11, 650-652 (1999).
[CrossRef]

Hofer, M.

M. Hofer, M. E. Fermann, A. Galvanauskas, D. Harter, and R. S. Windeler, "Low-noise amplification of high-power pulses in multimode fibers," IEEE Photon. Technol. Lett. 11, 650-652 (1999).
[CrossRef]

Jacobson, N.

U. H. Manyam, B. Samson, V. Khitrov, D. P. Machewirth, J. Abrmczyk, N. Jacobson, J. Farroni, D. Guertin, A. Carter, and K. Tankala, "Laser fibers designed for single polarization output," in 2004 OSA Topical Meeting on Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004), pp. 118-122.

Kaufman, K. S.

Khitrov, V.

U. H. Manyam, B. Samson, V. Khitrov, D. P. Machewirth, J. Abrmczyk, N. Jacobson, J. Farroni, D. Guertin, A. Carter, and K. Tankala, "Laser fibers designed for single polarization output," in 2004 OSA Topical Meeting on Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004), pp. 118-122.

Kimura, T.

J. I. Sakai and T. Kimura, "Analytical bending loss formula of optical fibers with field deformation," Radio Sci. 17, 21-29 (1982).
[CrossRef]

Kliner, D. A. V.

Knight, J. C.

Koch, R.

Koplow, P.

Lehtonen, M.

M. Lehtonen, G. Genty, and H. Ludvigsen, "Tapered microstructured fibers for efficient coupling to optical waveguides: a numerical study," Appl. Phys. B 81, 295-300 (2005).
[CrossRef]

Limpert, J.

O. Schmidt, F. Roser, S. Linke, T. Schreiber, J. Limpert, S. Ermeneux, P. Yvernault, F. Salin, and A. Tunnermann, "High energy and high average power Q-switched photonic crystal fiber laser," in Advanced Solid-State Photonics 2006 (Optical Society of America, 2006).

Linke, S.

O. Schmidt, F. Roser, S. Linke, T. Schreiber, J. Limpert, S. Ermeneux, P. Yvernault, F. Salin, and A. Tunnermann, "High energy and high average power Q-switched photonic crystal fiber laser," in Advanced Solid-State Photonics 2006 (Optical Society of America, 2006).

Love, J. D.

A. Altintas and J. D. Love, "Effective cut-offs for modes on helical fibers," Opt. Quantum Electron. 22, 213-226 (1990).
[CrossRef]

Ludvigsen, H.

M. Lehtonen, G. Genty, and H. Ludvigsen, "Tapered microstructured fibers for efficient coupling to optical waveguides: a numerical study," Appl. Phys. B 81, 295-300 (2005).
[CrossRef]

Machewirth, D. P.

U. H. Manyam, B. Samson, V. Khitrov, D. P. Machewirth, J. Abrmczyk, N. Jacobson, J. Farroni, D. Guertin, A. Carter, and K. Tankala, "Laser fibers designed for single polarization output," in 2004 OSA Topical Meeting on Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004), pp. 118-122.

Manyam, U. H.

U. H. Manyam, B. Samson, V. Khitrov, D. P. Machewirth, J. Abrmczyk, N. Jacobson, J. Farroni, D. Guertin, A. Carter, and K. Tankala, "Laser fibers designed for single polarization output," in 2004 OSA Topical Meeting on Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004), pp. 118-122.

Marciante, J. R.

G. T. Moore and J. R. Marciante, "Cladding-pumped fiber with helical rare-earth-doped core for fiber lasers and amplifiers," U.S. patent 6,650,664 (November 18, 2003).

Marcuse, D.

Mathis, R. F.

Moore, G. T.

G. T. Moore and J. R. Marciante, "Cladding-pumped fiber with helical rare-earth-doped core for fiber lasers and amplifiers," U.S. patent 6,650,664 (November 18, 2003).

Nilsson, J.

J. A. Alvarez-Chavez, H. L. Offerhaus, J. Nilsson, P. W. Turner, W. A. Clarkson, and D. J. Richardson, "High-energy, high-power ytterbium-doped Q-switched fiber laser," Opt. Lett. 25, 37-39 (2000).
[CrossRef]

C. C. Renaud, R. J. Selvas-Aguilar, J. Nilsson, P. W. Turner, and A. B. Grudinin, "Compact high-energy Q-switched cladding-pumped fiber laser with a tuning range over 40 nm," IEEE Photon. Technol. Lett. 11, 976-978 (1999).
[CrossRef]

Offerhaus, H. L.

Okhotnikov, O. G.

J. M. Sousa and O. G. Okhotnikov, "Multimode Er-doped fiber for single-transverse-mode amplification," Appl. Phys. Lett. 74, 1528-1530 (1999).
[CrossRef]

Renaud, C. C.

C. C. Renaud, R. J. Selvas-Aguilar, J. Nilsson, P. W. Turner, and A. B. Grudinin, "Compact high-energy Q-switched cladding-pumped fiber laser with a tuning range over 40 nm," IEEE Photon. Technol. Lett. 11, 976-978 (1999).
[CrossRef]

Richardson, D. J.

Roser, F.

O. Schmidt, F. Roser, S. Linke, T. Schreiber, J. Limpert, S. Ermeneux, P. Yvernault, F. Salin, and A. Tunnermann, "High energy and high average power Q-switched photonic crystal fiber laser," in Advanced Solid-State Photonics 2006 (Optical Society of America, 2006).

Russell, P. St. J.

Sahu, J. K.

Sakai, J. I.

J. I. Sakai and T. Kimura, "Analytical bending loss formula of optical fibers with field deformation," Radio Sci. 17, 21-29 (1982).
[CrossRef]

Salin, F.

O. Schmidt, F. Roser, S. Linke, T. Schreiber, J. Limpert, S. Ermeneux, P. Yvernault, F. Salin, and A. Tunnermann, "High energy and high average power Q-switched photonic crystal fiber laser," in Advanced Solid-State Photonics 2006 (Optical Society of America, 2006).

Sammut, R.

Samson, B.

U. H. Manyam, B. Samson, V. Khitrov, D. P. Machewirth, J. Abrmczyk, N. Jacobson, J. Farroni, D. Guertin, A. Carter, and K. Tankala, "Laser fibers designed for single polarization output," in 2004 OSA Topical Meeting on Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004), pp. 118-122.

Schmidt, O.

O. Schmidt, F. Roser, S. Linke, T. Schreiber, J. Limpert, S. Ermeneux, P. Yvernault, F. Salin, and A. Tunnermann, "High energy and high average power Q-switched photonic crystal fiber laser," in Advanced Solid-State Photonics 2006 (Optical Society of America, 2006).

Schönnagel, H.

Schreiber, T.

O. Schmidt, F. Roser, S. Linke, T. Schreiber, J. Limpert, S. Ermeneux, P. Yvernault, F. Salin, and A. Tunnermann, "High energy and high average power Q-switched photonic crystal fiber laser," in Advanced Solid-State Photonics 2006 (Optical Society of America, 2006).

Selvas-Aguilar, R. J.

C. C. Renaud, R. J. Selvas-Aguilar, J. Nilsson, P. W. Turner, and A. B. Grudinin, "Compact high-energy Q-switched cladding-pumped fiber laser with a tuning range over 40 nm," IEEE Photon. Technol. Lett. 11, 976-978 (1999).
[CrossRef]

Smith, A. V.

G. R. Hadley, R. L. Farrow, and A. V. Smith, "Bent-waveguide modeling of large-mode-area double-clad fibers for high-power lasers," in Fiber Lasers III, A.J.Brown, J.Nilsson, D.J.Harter, and A.Tünnermann, eds., Proc. SPIE 61026102152006).

Sousa, J. M.

J. M. Sousa and O. G. Okhotnikov, "Multimode Er-doped fiber for single-transverse-mode amplification," Appl. Phys. Lett. 74, 1528-1530 (1999).
[CrossRef]

Tankala, K.

U. H. Manyam, B. Samson, V. Khitrov, D. P. Machewirth, J. Abrmczyk, N. Jacobson, J. Farroni, D. Guertin, A. Carter, and K. Tankala, "Laser fibers designed for single polarization output," in 2004 OSA Topical Meeting on Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004), pp. 118-122.

Terras, R.

Tunnermann, A.

O. Schmidt, F. Roser, S. Linke, T. Schreiber, J. Limpert, S. Ermeneux, P. Yvernault, F. Salin, and A. Tunnermann, "High energy and high average power Q-switched photonic crystal fiber laser," in Advanced Solid-State Photonics 2006 (Optical Society of America, 2006).

Turner, P. W.

J. A. Alvarez-Chavez, H. L. Offerhaus, J. Nilsson, P. W. Turner, W. A. Clarkson, and D. J. Richardson, "High-energy, high-power ytterbium-doped Q-switched fiber laser," Opt. Lett. 25, 37-39 (2000).
[CrossRef]

C. C. Renaud, R. J. Selvas-Aguilar, J. Nilsson, P. W. Turner, and A. B. Grudinin, "Compact high-energy Q-switched cladding-pumped fiber laser with a tuning range over 40 nm," IEEE Photon. Technol. Lett. 11, 976-978 (1999).
[CrossRef]

Wang, P.

Windeler, R. S.

M. Hofer, M. E. Fermann, A. Galvanauskas, D. Harter, and R. S. Windeler, "Low-noise amplification of high-power pulses in multimode fibers," IEEE Photon. Technol. Lett. 11, 650-652 (1999).
[CrossRef]

Yvernault, P.

O. Schmidt, F. Roser, S. Linke, T. Schreiber, J. Limpert, S. Ermeneux, P. Yvernault, F. Salin, and A. Tunnermann, "High energy and high average power Q-switched photonic crystal fiber laser," in Advanced Solid-State Photonics 2006 (Optical Society of America, 2006).

Appl. Phys. B (1)

M. Lehtonen, G. Genty, and H. Ludvigsen, "Tapered microstructured fibers for efficient coupling to optical waveguides: a numerical study," Appl. Phys. B 81, 295-300 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

J. M. Sousa and O. G. Okhotnikov, "Multimode Er-doped fiber for single-transverse-mode amplification," Appl. Phys. Lett. 74, 1528-1530 (1999).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Gloge, "Optical power flow in multimode fibers," Bell Syst. Tech. J. 51, 1767-1783 (1972).

Electron. Lett. (1)

R. D. Birch, "Fabrication and characterisation of circularly birefringent helical fibers," Electron. Lett. 23, 50-52 (1987).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Marcuse, "Bend loss of slab and fiber modes computed with diffraction theory," IEEE J. Quantum Electron. 29, 2957-2961 (1993).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

C. C. Renaud, R. J. Selvas-Aguilar, J. Nilsson, P. W. Turner, and A. B. Grudinin, "Compact high-energy Q-switched cladding-pumped fiber laser with a tuning range over 40 nm," IEEE Photon. Technol. Lett. 11, 976-978 (1999).
[CrossRef]

M. Hofer, M. E. Fermann, A. Galvanauskas, D. Harter, and R. S. Windeler, "Low-noise amplification of high-power pulses in multimode fibers," IEEE Photon. Technol. Lett. 11, 650-652 (1999).
[CrossRef]

J. Lightwave Technol. (1)

A. Harris and P. Castle, "Bend loss measurements on high numerical aperture single-mode fibers as a function of wavelength and bend radius," J. Lightwave Technol. 4, 34-40 (1986).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Opt. Express (1)

Opt. Lett. (7)

Opt. Quantum Electron. (1)

A. Altintas and J. D. Love, "Effective cut-offs for modes on helical fibers," Opt. Quantum Electron. 22, 213-226 (1990).
[CrossRef]

Radio Sci. (1)

J. I. Sakai and T. Kimura, "Analytical bending loss formula of optical fibers with field deformation," Radio Sci. 17, 21-29 (1982).
[CrossRef]

Other (6)

G. R. Hadley, R. L. Farrow, and A. V. Smith, "Bent-waveguide modeling of large-mode-area double-clad fibers for high-power lasers," in Fiber Lasers III, A.J.Brown, J.Nilsson, D.J.Harter, and A.Tünnermann, eds., Proc. SPIE 61026102152006).

Optical Fiber Cabling Components Standard ANSI/TIA/EIA 568B.3.

G. T. Moore and J. R. Marciante, "Cladding-pumped fiber with helical rare-earth-doped core for fiber lasers and amplifiers," U.S. patent 6,650,664 (November 18, 2003).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1989).

O. Schmidt, F. Roser, S. Linke, T. Schreiber, J. Limpert, S. Ermeneux, P. Yvernault, F. Salin, and A. Tunnermann, "High energy and high average power Q-switched photonic crystal fiber laser," in Advanced Solid-State Photonics 2006 (Optical Society of America, 2006).

U. H. Manyam, B. Samson, V. Khitrov, D. P. Machewirth, J. Abrmczyk, N. Jacobson, J. Farroni, D. Guertin, A. Carter, and K. Tankala, "Laser fibers designed for single polarization output," in 2004 OSA Topical Meeting on Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004), pp. 118-122.

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Figures (8)

Fig. 1
Fig. 1

Depiction of a helical-core, dual-clad fiber. The helix has pitch P and offset Q.

Fig. 2
Fig. 2

Fiber is bent into a torus with a radius of curvature R B . The dotted surface indicates the cylindrical matching surface that is tangential to the torus at radius r = R B + a .

Fig. 3
Fig. 3

Bending loss versus bend radius for LP 01 and LP 11 modes for different models. The wavelength, core diameter, and NA are 1053 nm , 60 μ m , and 0.1, respectively.

Fig. 4
Fig. 4

Bending loss versus helix pitch for LP 01 (solid curve) and LP 11 (dashed curve) modes for different core-offset values. The wavelength, core diameter, and NA are 1053 nm , 60 μ m , and 0.1, respectively.

Fig. 5
Fig. 5

Modal discrimination ( LP 11 bending loss minus LP 01 bending loss) versus LP 01 bending loss for different core-diameter values. The wavelength and NA are 1053 nm and 0.1, respectively.

Fig. 6
Fig. 6

Modal discrimination ( LP 11 bending loss minus LP 01 bending loss) versus NA for different core-diameter values. The wavelength and LP 01 bending loss are 1053 nm and 1 dB m , respectively.

Fig. 7
Fig. 7

Helix pitch (left) and bend radius (right) versus core diameter for different NA values. The wavelength, core offset, and modal discrimination are 1053 nm , 100 μ m , and 5 dB m , respectively.

Fig. 8
Fig. 8

(a) Fundamental mode loss and (b) modal discrimination versus wavelength for different fundamental modes-loss values at a 1053 nm wavelength. The core diameter and NA are 40 μ m and 0.1, respectively.

Equations (28)

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R B = Q sin 2 θ Q tan 2 θ = P 2 4 π 2 Q ,
E z = A K ν ( γ ν r ) cos ( ν ϕ ) e i β ν R B ϕ ,
E ϕ = i γ ν 2 n 2 k 0 A { K ν + 1 ( γ ν r ) sin [ ( ν + 1 ) ϕ ] K ν 1 ( γ ν r ) sin [ ( ν 1 ) ϕ ] } e i β ν R B ϕ ,
A = 2 γ ν V J ν ( K ν a ) K ν ( γ ν a ) [ Z 0 P k 0 e ν π β ν J ν 1 ( κ ν a ) J ν + 1 ( κ ν a ) ] 1 2 ,
E z = n k 0 n k 0 [ c 1 ( β ) ε z β 1 + c 2 ( β ) ε z β 2 ] d β ,
E ϕ = n k 0 n k 0 [ c 1 ( β ) ε ϕ β 1 + c 2 ( β ) ε ϕ β 2 ] d β ,
ε z β j = ε z j e i β z = B j H μ ( 2 ) ( ρ r ) e i μ ϕ e i β z ,
h z β j = h z j e i β z = B j F j H μ ( 2 ) ( ρ r ) e i μ ϕ e i β z ,
Γ 1 = H μ ( 2 ) [ ρ ( R B + a ) ] H μ ( 2 ) [ ρ ( R B + a ) ] ,
Γ 2 = H μ ( 2 ) [ ρ ( R B + a ) ] H μ ( 2 ) [ ρ ( R B + a ) ] .
ε ϕ β j = ε ϕ j e i β z = [ μ β ρ 2 r H μ ( 1 ) ( ρ r ) i μ 0 ω ρ F j H μ ( 1 ) ( ρ r ) ] B j e i μ ϕ e i β z ,
h ϕ β j = h ϕ j e i β z = [ μ β ρ 2 r F j H μ ( 1 ) ( ρ r ) i ε 0 n 2 ω ρ H μ ( 1 ) ( ρ r ) ] B j e i μ ϕ e i β z ,
ε r β j = ε r j e i β z = [ i β ρ H μ ( 1 ) ( ρ r ) μ 0 ω μ ρ 2 r F j H μ ( 1 ) ( ρ r ) ] B j e i μ ϕ e i β z ,
h r β j = h r j e i β z = [ i β ρ F j H μ ( 1 ) ( ρ r ) ε 0 n 2 ω μ ρ 2 r H μ ( 1 ) ( ρ r ) ] B j e i μ ϕ e i β z ,
d z [ ε ϕ β j h z β j * ε z β j h ϕ β j * ] = d z { [ μ β ρ 2 r H μ ( 1 ) ( ρ r ) i μ 0 ω ρ F j H μ ( 1 ) ( ρ r ) ] × B j e i μ ϕ e i β z B j * F j * H μ ( 1 ) * ( ρ r ) e i μ ϕ e i β z B j H μ ( 1 ) ( ρ r ) e i μ ϕ e i β z [ μ β ρ 2 r F j * H μ ( 1 ) * ( ρ r ) i ε 0 n 2 ω ρ H μ ( 1 ) * ( ρ r ) ] B j * e i μ ϕ e i β z } = ( 2 π ) δ ( β β ) N β β j j = N β β j j B j B j * ( ( β ρ 2 β ρ 2 ) H μ ( 1 ) [ ρ ( R B + a ) ] μ r F j * H μ ( 1 ) * [ ρ ( R B + a ) ] + i ε 0 n 2 ω { H μ ( 1 ) [ ρ ( R B + a ) ] H μ ( 1 ) * [ ρ ( R B + a ) ] ρ Γ j Γ j * H μ ( 1 ) [ ρ ( R B + a ) ] H μ ( 1 ) * [ ρ ( R B + a ) ] ρ } ) .
[ E ϕ h z β j * E z h ϕ β j * ] d z = { n k 0 n k 0 [ c 1 ( β ) ε ϕ β 1 + c 2 ( β ) ε ϕ β 2 ] h z β j * d β n k 0 n k 0 [ c 1 ( β ) ε z β 1 + c 2 ( β ) ε z β 2 ] h ϕ β j * d β } d z = j = 1 2 n k 0 n k 0 ( 2 π ) δ ( β β ) N β β j j c j ( β ) d β = ( 2 π ) B j 2 i ε 0 n 2 ω ρ H μ ( 1 ) H μ ( 1 ) * ( 1 Γ j * 2 ) c j ( β ) .
c j ( β ) = [ E ϕ h z β j * E z h ϕ β j * ] d z N β j ,
N β j = ( 2 π ) B j 2 i ε 0 n 2 ω ρ H μ ( 1 ) [ ρ ( R B + a ) ] H μ ( 1 ) * [ ρ ( R B + a ) ] N β j ,
N β 1 = ( 1 Γ 1 * 2 ) ,
N β 2 = ( 1 Γ 2 2 ) .
r = ( a 2 + z 2 ) 1 2 ,
ϕ = arctan ( z a ) .
C j = 1 N β j ( E ϕ h z β j * E z h ϕ β j * ) d z = 1 N β j ( i γ ν 2 n k 0 A { K ν + 1 ( γ ν r ) sin [ ( ν + 1 ) ϕ ] K ν 1 ( γ ν r ) sin [ ( ν 1 ) ϕ ] } × e i β ν R B ϕ B j * F j * H μ ( 1 ) * e i μ ϕ e i β z A K ν ( γ ν r ) cos ( ν ϕ ) e i β ν R B ϕ × [ μ β ρ 2 r F j * H ν ( 1 ) * i ε 0 n 2 ω ρ H μ ( 1 ) * ] B j * e i μ ϕ e i β z ) d z = i ε 0 n 2 ω A B j * ρ 2 N β j ( [ ρ H μ ( 1 ) * i μ β r ε 0 n 2 ω F j * H μ ( 1 ) * ] × K ν [ γ ν ( a 2 + z 2 ) 1 2 ] cos [ ν arctan ( z a ) ] e i β z d z ρ 2 γ ν F j * H μ ( 1 ) * 2 k 0 ε 0 n 3 ω × { K ν + 1 [ γ ν ( a 2 + z 2 ) 1 2 ] sin [ ( ν + 1 ) arctan ( z a ) ] K ν 1 [ γ ν ( a 2 + z 2 ) 1 2 ] sin [ ( ν 1 ) arctan ( z a ) ] } e i β z d z ) .
{ K ν + 1 [ γ ν ( a 2 + z 2 ) 1 2 ] sin [ ( ν + 1 ) arctan ( z a ) ] K ν 1 [ γ ν ( a 2 + z 2 ) 1 2 ] sin [ ( ν 1 ) arctan ( z a ) ] } e i β z d z = π 2 i γ ν ν e a ( γ ν 2 + β 2 ) 1 2 ( γ ν 2 + β 2 ) 1 2 ( γ ν 1 { [ ( γ ν 2 + β 2 ) 1 2 + β ] ν + 1 [ ( γ ν 2 + β 2 ) 1 2 β ] ν + 1 } γ ν { [ ( γ ν 2 + β 2 ) 1 2 + β ] ν 1 [ ( γ ν 2 + β 2 ) 1 2 β ] ν 1 } ) .
K ν [ γ ν ( a 2 + z 2 ) 1 2 ] cos [ ν arctan ( z a ) ] e i β z d z = π 2 γ ν ν e a ( γ ν 2 + β 2 ) 1 2 ( γ ν 2 + β 2 ) 1 2 { [ ( γ ν 2 + β 2 ) 1 2 + β ] ν + [ ( γ ν 2 + β 2 ) 1 2 β ] ν } .
C j = A 4 ρ B j H μ ( 1 ) N β j I ν β j ,
I ν β j = { γ ν ν e a ( γ ν 2 + β 2 ) 1 2 ( γ ν 2 + β 2 ) 1 2 [ ( ρ i μ β n k 0 r Γ j * Γ 1 * ) { [ ( γ ν 2 + β 2 ) 1 2 + β ] ν + [ ( γ ν 2 + β 2 ) 1 2 β ] ν } + i ρ 2 γ ν Γ j * Γ 1 * 2 ( n k 0 ) 2 ( γ ν 1 { [ ( γ ν 2 + β 2 ) 1 2 + β ] ν + 1 [ ( γ ν 2 + β 2 ) 1 2 β ] ν + 1 } γ ν { [ ( γ ν 2 + β 2 ) 1 2 + β ] ν 1 [ ( γ ν 2 + β 2 ) 1 2 β ] ν 1 } ) ] } .
2 α = Δ P P L = Re [ r ̂ ( E × H * ) π r ] r d z P L = [ n γ ν a NA J ν ( κ ν a ) K ν ( γ ν a ) ] 2 2 π e ν R J ν 1 ( κ ν a ) J ν + 1 ( κ ν a ) 1 1 Re { j , j = 1 2 I ν β j I ν β j * ( 1 + Γ j Γ j * ) ( ρ a ) 4 H μ ( 1 ) [ ρ ( R + a ) ] 2 N β j N β j * } d x .

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