Abstract

In this paper scaling laws governing loss in hollow core tube lattice fibers are numerically investigated and discussed. Moreover, by starting from the analysis of the obtained numerical results, empirical formulas for the estimation of the minimum values of confinement loss, absorption loss, and surface scattering loss inside the transmission band are obtained. The proposed formulas show a good accuracy for fibers designed for applications ranging from THz to ultra violet band.

© 2016 Optical Society of America

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References

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  1. V. Setti, L. Vincetti, and A. Argyros, “Flexible tube lattice fibers for terahertz applications,” Opt. Express 21(3), 3388–3399 (2013).
    [Crossref] [PubMed]
  2. W. Lu and A. Argyros, “Terahertz spectroscopy and imaging with flexible tube-lattice fiber probe,” J. Lightwave Technol. 29, 4621–4627 (2014).
  3. 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]
  4. A. F. Kosolapov, A. D. Pryamikov, A. S. Biriukov, V. S. Shiryaev, M. S. Astapovich, G. E. Snopatin, V. G. Plotnichenko, M. F. Churbanov, and E. M. Dianov, “Demonstration of CO2-laser power delivery through chalcogenide-glass fiber with negative-curvature hollow core,” Opt. Express 19(25), 25723–25728 (2011).
    [Crossref] [PubMed]
  5. 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]
  6. W. Belardi and J. C. Knight, “Hollow antiresonant fibers with low bending loss,” Opt. Express 22(8), 10091–10096 (2014).
    [Crossref] [PubMed]
  7. 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]
  8. 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]
  9. B. Debord, A. Amsanpally, M. Alharbi, L. Vincetti, J. Blondy, F. Gérôme, and F. Benabid, “Ultra-large core size hypocycloid-shape inhibited coupling kagome fibers for high-energy laser beam handling,” J. Lightwave Technol. 33(17), 3630–3634 (2015).
    [Crossref]
  10. A. Urich, R. R. Maier, F. Yu, J. C. Knight, D. P. Hand, and J. D. Shephard, “Flexible delivery of Er:YAG radiation at 2.94 µm with negative curvature silica glass fibers: a new solution for minimally invasive surgical procedures,” Biomed. Opt. Express 4(2), 193–205 (2013).
    [Crossref] [PubMed]
  11. B. Debord, M. Alharbi, L. Vincetti, A. Husakou, C. Fourcade-Dutin, C. Hoenninger, E. Mottay, F. Gérôme, and F. Benabid, “Multi-meter fiber-delivery and pulse self-compression of milli-Joule femtosecond laser and fiber-aided laser-micromachining,” Opt. Express 22(9), 10735–10746 (2014).
    [Crossref] [PubMed]
  12. P. Jaworski, F. Yu, R. R. Maier, W. J. Wadsworth, J. C. Knight, J. D. Shephard, and D. P. Hand, “Picosecond and nanosecond pulse delivery through a hollow-core Negative Curvature Fiber for micro-machining applications,” Opt. Express 21(19), 22742–22753 (2013).
    [Crossref] [PubMed]
  13. 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. St. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13(1), 236-244 (2005).
    [Crossref] [PubMed]
  14. E. N. Fokoua, F. Poletti, and D. J. Richardson, “Analysis of light scattering from surface roughness in hollow-core photonic bandgap fibers,” Opt. Express 20(19), 20980–20991 (2012).
    [Crossref] [PubMed]
  15. E. N. Fokoua, S. R. Sandoghchi, Y. Chen, G. T. Jasion, N. V. Wheeler, N. K. Baddela, J. R. Hayes, M. N. Petrovich, D. J. Richardson, and F. Poletti, “Accurate modelling of fabricated hollow-core photonic bandgap fibers,” Opt. Express 23(18), 23117–23132 (2015).
    [Crossref] [PubMed]
  16. F. Poletti, “Nested antiresonant nodeless hollow core fiber,” Opt. Express 22(20), 23807–23828 (2014).
    [Crossref] [PubMed]
  17. 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), 359–371 (2001).
    [Crossref]
  18. L. Vincetti and V. Setti, “Waveguiding mechanism in tube lattice fibers,” Opt. Express 18(22), 23133–23146 (2010).
    [Crossref] [PubMed]
  19. L. Vincetti, V. Setti, and M. Zoboli, “Terahertz tube lattice fibers with octagonal symmetry,” IEEE Photonics Technol. Lett. 22(13), 972–974 (2010).
    [Crossref]
  20. M. Kharadly and J. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEEE 116, 214–224 (1969).
  21. G. K. Alagashev, A. D. Pryamikov, A. F. Kosolapov, A. N. Kolyadin, A. Y. Lukovkin, and A. S. Biriukov, “Impact of geometrical parameters on the optical properties of negative curvature hollow-core fibers,” Laser Phys. 25(5), 055101 (2015).
    [Crossref]
  22. M. Masruri, A. Cucinotta, and L. Vincetti, “Scaling laws in tube lattice fibers,” in CLEO:2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper STu1N.8.
  23. E. A. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
    [Crossref]
  24. D. S. Wu, A. Argyros, and S. G. Leon-Saval, “Reducing the size of hollow terahertz waveguides,” J. Lightwave Technol. 29(1), 97–103 (2011).
    [Crossref]

2015 (3)

2014 (4)

2013 (5)

2012 (1)

2011 (3)

2010 (2)

L. Vincetti and V. Setti, “Waveguiding mechanism in tube lattice fibers,” Opt. Express 18(22), 23133–23146 (2010).
[Crossref] [PubMed]

L. Vincetti, V. Setti, and M. Zoboli, “Terahertz tube lattice fibers with octagonal symmetry,” IEEE Photonics Technol. Lett. 22(13), 972–974 (2010).
[Crossref]

2007 (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]

2005 (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), 359–371 (2001).
[Crossref]

1969 (1)

M. Kharadly and J. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEEE 116, 214–224 (1969).

1964 (1)

E. A. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

Alagashev, G. K.

G. K. Alagashev, A. D. Pryamikov, A. F. Kosolapov, A. N. Kolyadin, A. Y. Lukovkin, and A. S. Biriukov, “Impact of geometrical parameters on the optical properties of negative curvature hollow-core fibers,” Laser Phys. 25(5), 055101 (2015).
[Crossref]

Alharbi, M.

Amsanpally, A.

Argyros, A.

Astapovich, M. S.

Baddela, N. K.

Belardi, W.

Benabid, F.

Biriukov, A. S.

Birks, T. A.

Blondy, J.

Bradley, T.

Chen, Y.

Churbanov, M. F.

Couny, F.

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), 359–371 (2001).
[Crossref]

Debord, B.

Dianov, E. M.

Farr, L.

Fokoua, E. N.

Fourcade-Dutin, C.

Gérôme, F.

Hand, D. P.

Hayes, J. R.

Hoenninger, C.

Husakou, A.

Jasion, G. T.

Jaworski, P.

Kharadly, M.

M. Kharadly and J. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEEE 116, 214–224 (1969).

Knight, J. C.

Kolyadin, A. N.

G. K. Alagashev, A. D. Pryamikov, A. F. Kosolapov, A. N. Kolyadin, A. Y. Lukovkin, and A. S. Biriukov, “Impact of geometrical parameters on the optical properties of negative curvature hollow-core fibers,” Laser Phys. 25(5), 055101 (2015).
[Crossref]

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]

Kosolapov, A. F.

Leon-Saval, S. G.

Lewis, J.

M. Kharadly and J. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEEE 116, 214–224 (1969).

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]

Lu, W.

W. Lu and A. Argyros, “Terahertz spectroscopy and imaging with flexible tube-lattice fiber probe,” J. Lightwave Technol. 29, 4621–4627 (2014).

Lukovkin, A. Y.

G. K. Alagashev, A. D. Pryamikov, A. F. Kosolapov, A. N. Kolyadin, A. Y. Lukovkin, and A. S. Biriukov, “Impact of geometrical parameters on the optical properties of negative curvature hollow-core fibers,” Laser Phys. 25(5), 055101 (2015).
[Crossref]

Maier, R. R.

Mangan, B. J.

Marcatili, E. A.

E. A. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

Mason, M. W.

Mottay, E.

Petrovich, M. N.

Plotnichenko, V. G.

Poletti, F.

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]

Richardson, D. J.

Roberts, P. J.

Sabert, H.

Sandoghchi, S. R.

Schmeltzer, R. A.

E. A. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

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), 359–371 (2001).
[Crossref]

Semjonov, S. L.

Setti, V.

Shephard, J. D.

Shiryaev, V. S.

Snopatin, G. E.

St. J. Russell, P.

Tomlinson, A.

Urich, A.

Vincetti, L.

B. Debord, A. Amsanpally, M. Alharbi, L. Vincetti, J. Blondy, F. Gérôme, and F. Benabid, “Ultra-large core size hypocycloid-shape inhibited coupling kagome fibers for high-energy laser beam handling,” J. Lightwave Technol. 33(17), 3630–3634 (2015).
[Crossref]

B. Debord, M. Alharbi, L. Vincetti, A. Husakou, C. Fourcade-Dutin, C. Hoenninger, E. Mottay, F. Gérôme, and F. Benabid, “Multi-meter fiber-delivery and pulse self-compression of milli-Joule femtosecond laser and fiber-aided laser-micromachining,” Opt. Express 22(9), 10735–10746 (2014).
[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]

V. Setti, L. Vincetti, and A. Argyros, “Flexible tube lattice fibers for terahertz applications,” Opt. Express 21(3), 3388–3399 (2013).
[Crossref] [PubMed]

L. Vincetti and V. Setti, “Waveguiding mechanism in tube lattice fibers,” Opt. Express 18(22), 23133–23146 (2010).
[Crossref] [PubMed]

L. Vincetti, V. Setti, and M. Zoboli, “Terahertz tube lattice fibers with octagonal symmetry,” IEEE Photonics Technol. Lett. 22(13), 972–974 (2010).
[Crossref]

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), 359–371 (2001).
[Crossref]

Wadsworth, W. J.

Wang, Y. Y.

Wheeler, N. V.

Williams, D. P.

Wu, D. S.

Yu, F.

Zoboli, M.

L. Vincetti, V. Setti, and M. Zoboli, “Terahertz tube lattice fibers with octagonal symmetry,” IEEE Photonics Technol. Lett. 22(13), 972–974 (2010).
[Crossref]

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), 359–371 (2001).
[Crossref]

Bell Syst. Tech. J. (1)

E. A. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

Biomed. Opt. Express (1)

IEEE Photonics Technol. Lett. (1)

L. Vincetti, V. Setti, and M. Zoboli, “Terahertz tube lattice fibers with octagonal symmetry,” IEEE Photonics Technol. Lett. 22(13), 972–974 (2010).
[Crossref]

J. Lightwave Technol. (3)

Laser Phys. (1)

G. K. Alagashev, A. D. Pryamikov, A. F. Kosolapov, A. N. Kolyadin, A. Y. Lukovkin, and A. S. Biriukov, “Impact of geometrical parameters on the optical properties of negative curvature hollow-core fibers,” Laser Phys. 25(5), 055101 (2015).
[Crossref]

Opt. Express (13)

L. Vincetti and V. Setti, “Waveguiding mechanism in tube lattice fibers,” Opt. Express 18(22), 23133–23146 (2010).
[Crossref] [PubMed]

V. Setti, L. Vincetti, and A. Argyros, “Flexible tube lattice fibers for terahertz applications,” Opt. Express 21(3), 3388–3399 (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]

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. F. Kosolapov, A. D. Pryamikov, A. S. Biriukov, V. S. Shiryaev, M. S. Astapovich, G. E. Snopatin, V. G. Plotnichenko, M. F. Churbanov, and E. M. Dianov, “Demonstration of CO2-laser power delivery through chalcogenide-glass fiber with negative-curvature hollow core,” Opt. Express 19(25), 25723–25728 (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, “Hollow antiresonant fibers with low bending loss,” Opt. Express 22(8), 10091–10096 (2014).
[Crossref] [PubMed]

B. Debord, M. Alharbi, L. Vincetti, A. Husakou, C. Fourcade-Dutin, C. Hoenninger, E. Mottay, F. Gérôme, and F. Benabid, “Multi-meter fiber-delivery and pulse self-compression of milli-Joule femtosecond laser and fiber-aided laser-micromachining,” Opt. Express 22(9), 10735–10746 (2014).
[Crossref] [PubMed]

P. Jaworski, F. Yu, R. R. Maier, W. J. Wadsworth, J. C. Knight, J. D. Shephard, and D. P. Hand, “Picosecond and nanosecond pulse delivery through a hollow-core Negative Curvature Fiber for micro-machining applications,” Opt. Express 21(19), 22742–22753 (2013).
[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. St. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13(1), 236-244 (2005).
[Crossref] [PubMed]

E. N. Fokoua, F. Poletti, and D. J. Richardson, “Analysis of light scattering from surface roughness in hollow-core photonic bandgap fibers,” Opt. Express 20(19), 20980–20991 (2012).
[Crossref] [PubMed]

E. N. Fokoua, S. R. Sandoghchi, Y. Chen, G. T. Jasion, N. V. Wheeler, N. K. Baddela, J. R. Hayes, M. N. Petrovich, D. J. Richardson, and F. Poletti, “Accurate modelling of fabricated hollow-core photonic bandgap fibers,” Opt. Express 23(18), 23117–23132 (2015).
[Crossref] [PubMed]

F. Poletti, “Nested antiresonant nodeless hollow core fiber,” Opt. Express 22(20), 23807–23828 (2014).
[Crossref] [PubMed]

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), 359–371 (2001).
[Crossref]

Proc. IEEE (1)

M. Kharadly and J. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEEE 116, 214–224 (1969).

Science (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]

Other (1)

M. Masruri, A. Cucinotta, and L. Vincetti, “Scaling laws in tube lattice fibers,” in CLEO:2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper STu1N.8.

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Figures (11)

Fig. 1
Fig. 1 (a) Cross section of a HC-TLF with non-touching tubes and N = 8. Black is air with unitary refractive index and white the dielectric with refractive index nd; (b) fundamental core mode. (c) holes cladding mode; (d)-(e) dielectric cladding modes with fast and slow spatial oscillation respectively.
Fig. 2
Fig. 2 Left: Examples of the electric field modulus distribution of some dielectric modes of a single tube fiber. Right: Normalized cut-off frequencies versus ρ parameter, for two different dielectric refractive indices: nd = 1.44, 2.42. Different colors show curves of modes with different azimuthal number μ from 0 to 4. Empty and filled circles refer to HE and EH modes respectively.
Fig. 3
Fig. 3 Confinement loss (left) and dielectric overlap (right) of four fibers with t = 1um, N = 8, δ = 0, and different values of ρ and nd.
Fig. 4
Fig. 4 CL (left column) and CL Rco44.5 (right column) of fibers THz with t = 131um, rext = 1746um, nd = 1.521 (top), MIR with t = 1.0um, rext = 10um, nd = 2.42 (middle), and NIR with t = 0.5um, rext = 5um, nd = 1.44 (bottom) having different number of tubes N and thus different Rco. All fibers have δ = 0.
Fig. 5
Fig. 5 CL Rco44.5 (left) and CL Rco44.5ρ12 (right) of fibers VUV having different ρ parameter, different number of tubes N, and Rco = 8um, t = 0.2um, nd = 1.5. All fibers have δ = 0.
Fig. 6
Fig. 6 CL Rco44.5ρ12 (left) and CL R co 4 / λ 4.5 ρ 12 / n 2 1 (right) of MIS fibers having different nd. and t. All fibers have δ = 0, Rco = 25.5um, and ρ = 0.92.
Fig. 7
Fig. 7 CL (a), CL R co 4 / λ 4.5 ρ 12 / n 2 1 (b), and NLC (c) spectra of the fibers: VUV, NIR, MIR, and THz, described in the Table 1 with δ = 0.
Fig. 8
Fig. 8 CL (top), and NCL (bottom) of NIR and THz with different values of N and δ.
Fig. 9
Fig. 9 DO (left) and NDO (right) of the fibers described in the table and with δ=0 (top) and δ0 (bottom).
Fig. 10
Fig. 10 EI (left) and NEI (right) parameters for some of the fibers described in the table and figure labels.
Fig. 11
Fig. 11 Trends of CL (left) and DO (right) with respect to number of tubes N (top), tube-tube spacing k (middle), and normalized tube size rext/t (bottom).

Tables (2)

Tables Icon

Table 1 Sets of fibers considered in the analysis.

Tables Icon

Table 2 HC-TLFs scaling laws.

Equations (15)

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R co =[ ( k/ sin( π N ) )1 ] r ext ,
CL(λ)=8.686Im[γ(λ)] [dB/m],
AL(λ)= α d (λ)DO(λ),
DO(λ)= s d p z dS / s p z dS ,
SS L min (λ)=ηEI(λ) ( λ 0 λ ) 3 ,
EI= ε 0 μ 0 l d | E | 2 dl / S p z dS ,
F= 2t λ n 2 1 .
NCL=CL R co 4 λ 4.5 t ρ 12 n 2 1 r ext λ e 2 λ r ext ( n 2 1) .
C L min =5 10 4 λ 4.5 R co 4 ( 1 t r ext ) 12 n 2 1 t r ext e 2λ r ext ( n 2 1) =5 10 4 ( k sin( π N ) 1 ) 4 ( λ r ext ) 4.5 ( 1 t r ext ) 12 n 2 1 t e 2λ r ext ( n 2 1) [dB/m].
NDO=DO [ ( k/ sin( π N ) )1 ] 2.93 ( r ext t ) 0.93 ( r ext λ ) 2 n.
D O min = 2.4 10 1 n ( λ R co ) 2 ( t R co ) 0.93 = 2.4 10 1 n ( k sin( π N ) 1 ) 2.93 ( t r ext ) 0.93 ( λ r ext ) 2 .
A L min =2.4 10 1 α d (λ) n ( λ R co ) 2 ( t R co ) 0.93 =2.4 10 1 α d (λ) n ( k sin( π N ) 1 ) 2.93 ( t r ext ) 0.93 ( λ r ext ) 2 ,
NEI=EI R co 3 λ 2 =EI [ ( k/ sin( π N ) )1 ] 3 r ext 3 λ 2 .
E I min =0.63 ( λ R co ) 2 1 R co =0.63 ( k sin( π N ) 1 ) 3 λ 2 r ext 3 .
SS L min (λ)=0.63η λ 0 3 1 R co 3 1 λ .

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