Abstract

This paper presents a model of a subwavelength diameter adiabatic microfiber taper (nanotaper), which allows an asymptotically accurate solution of the wave equation. The evanescent field near the nanotaper is expressed through a Gaussian beam having a singularity at the nanotaper axis. For certain values of parameters of the nanotaper, when it has a swell in the middle and narrows down to zero at the infinity, the nanotaper is lossless. For other values, when the nanotaper has a biconical shape, it exhibits an exponentially small radiation loss, which is determined as a tunneling rate through an effective parabolic potential barrier. The latter case represents an exceptional example of the radiation loss being distributed along the length of an adiabatic nanotaper rather than being localized near focal circumferences in the evanescent field region.

© 2007 Optical Society of America

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References

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  1. L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  9. A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).
  10. W. L. Kath and G A. Kriegsmann, "Optical tunnelling: radiation losses in bent fibre-optic waveguides," IMA J. Appl. Math. 41, 85-103 (1988).
    [CrossRef]
  11. M. Sumetsky, "How thin can a microfiber be and still guide light?" Opt. Lett. 31, 870-872 (2006).
    [CrossRef] [PubMed]
  12. M. Sumetsky, "How thin can a microfiber be and still guide light? Errata," Opt. Lett. 31, 3577-3578 (2006).
    [CrossRef]
  13. M. Sumetsky, "Optics of tunneling from adiabatic nanotapers," Opt. Lett. 31, 3420-3422 (2006).
    [CrossRef] [PubMed]
  14. L. Tong, L. Hu, J. Zhang, J. Qiu, Q. Yang, J. Lou, Y. Shen, J. He, and Z. Ye, "Photonic nanowires directly drawn from bulk glasses," Opt. Express 14, 82-87 (2006).
    [CrossRef] [PubMed]
  15. A. D. Capobianco, M. Midrio, C. G. Someda, and S. Curtarolo, "Lossless tapers, Gaussian beams, free-space modes: Standing waves versus through-flowing waves," Opt. Quantum Electron. 32, 1161-1173 (2000).
    [CrossRef]
  16. V. M. Babič and V. S. Buldyrev, Short-wavelength diffraction theory (Springer, Berlin, 1991).
  17. M. Sumetskii, "Tunnel effect at the boundary of state stability: optical cavity and highly excited hydrogen atom in a magnetic field," Sov. Phys. JETP,  67, 49-59 (1988).
  18. J. Heading, Phase Integral Methods (New York, Wiley, 1962).
  19. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1965, 2nd edition).

2006 (6)

2004 (3)

2003 (2)

K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

2000 (1)

A. D. Capobianco, M. Midrio, C. G. Someda, and S. Curtarolo, "Lossless tapers, Gaussian beams, free-space modes: Standing waves versus through-flowing waves," Opt. Quantum Electron. 32, 1161-1173 (2000).
[CrossRef]

1988 (2)

M. Sumetskii, "Tunnel effect at the boundary of state stability: optical cavity and highly excited hydrogen atom in a magnetic field," Sov. Phys. JETP,  67, 49-59 (1988).

W. L. Kath and G A. Kriegsmann, "Optical tunnelling: radiation losses in bent fibre-optic waveguides," IMA J. Appl. Math. 41, 85-103 (1988).
[CrossRef]

Ashcom, J. B.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Birks, T. A.

Brambilla, G.

Capobianco, A. D.

A. D. Capobianco, M. Midrio, C. G. Someda, and S. Curtarolo, "Lossless tapers, Gaussian beams, free-space modes: Standing waves versus through-flowing waves," Opt. Quantum Electron. 32, 1161-1173 (2000).
[CrossRef]

Curtarolo, S.

A. D. Capobianco, M. Midrio, C. G. Someda, and S. Curtarolo, "Lossless tapers, Gaussian beams, free-space modes: Standing waves versus through-flowing waves," Opt. Quantum Electron. 32, 1161-1173 (2000).
[CrossRef]

DiGiovanni, D. J.

Dulashko, Y.

Finazzi, V.

Fini, J. M.

Gattass, R. R.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Hale, A.

He, J.

He, S.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Hu, L.

Kath, W. L.

W. L. Kath and G A. Kriegsmann, "Optical tunnelling: radiation losses in bent fibre-optic waveguides," IMA J. Appl. Math. 41, 85-103 (1988).
[CrossRef]

Kriegsmann, G A.

W. L. Kath and G A. Kriegsmann, "Optical tunnelling: radiation losses in bent fibre-optic waveguides," IMA J. Appl. Math. 41, 85-103 (1988).
[CrossRef]

Leon-Saval, S. G.

Lou, J.

L. Tong, L. Hu, J. Zhang, J. Qiu, Q. Yang, J. Lou, Y. Shen, J. He, and Z. Ye, "Photonic nanowires directly drawn from bulk glasses," Opt. Express 14, 82-87 (2006).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Maxwell, I.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Mazur, E.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Midrio, M.

A. D. Capobianco, M. Midrio, C. G. Someda, and S. Curtarolo, "Lossless tapers, Gaussian beams, free-space modes: Standing waves versus through-flowing waves," Opt. Quantum Electron. 32, 1161-1173 (2000).
[CrossRef]

Nicholson, J. W.

Qiu, J.

Richardson, D. J.

Russell, P. St. J.

Shen, M.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Shen, Y.

Someda, C. G.

A. D. Capobianco, M. Midrio, C. G. Someda, and S. Curtarolo, "Lossless tapers, Gaussian beams, free-space modes: Standing waves versus through-flowing waves," Opt. Quantum Electron. 32, 1161-1173 (2000).
[CrossRef]

Sumetskii, M.

M. Sumetskii, "Tunnel effect at the boundary of state stability: optical cavity and highly excited hydrogen atom in a magnetic field," Sov. Phys. JETP,  67, 49-59 (1988).

Sumetsky, M.

Tong, L.

L. Tong, L. Hu, J. Zhang, J. Qiu, Q. Yang, J. Lou, Y. Shen, J. He, and Z. Ye, "Photonic nanowires directly drawn from bulk glasses," Opt. Express 14, 82-87 (2006).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Vahala, K. J.

K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003).
[CrossRef] [PubMed]

Wadsworth, W. J.

Yang, Q.

Ye, Z.

Zhang, J.

IMA J. Appl. Math. (1)

W. L. Kath and G A. Kriegsmann, "Optical tunnelling: radiation losses in bent fibre-optic waveguides," IMA J. Appl. Math. 41, 85-103 (1988).
[CrossRef]

J. Lightwave Technol. (1)

Nature (2)

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Lett. (4)

Opt. Quantum Electron. (1)

A. D. Capobianco, M. Midrio, C. G. Someda, and S. Curtarolo, "Lossless tapers, Gaussian beams, free-space modes: Standing waves versus through-flowing waves," Opt. Quantum Electron. 32, 1161-1173 (2000).
[CrossRef]

Sov. Phys. JETP (1)

M. Sumetskii, "Tunnel effect at the boundary of state stability: optical cavity and highly excited hydrogen atom in a magnetic field," Sov. Phys. JETP,  67, 49-59 (1988).

Other (5)

J. Heading, Phase Integral Methods (New York, Wiley, 1962).

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1965, 2nd edition).

X. Jiang, Q. Yang, G. Vienne, Y. Li, L. Tong, J. Zhang, L. Hu, "Demonstration of microfiber knot laser," Appl. Phys. Lett. 89, Art. 143513 (2006)
[CrossRef]

A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).

V. M. Babič and V. S. Buldyrev, Short-wavelength diffraction theory (Springer, Berlin, 1991).

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Figures (3)

Fig. 1.
Fig. 1.

Illustration of (a) a nanotaper and (b) a nanoswell.

Fig. 2.
Fig. 2.

(a) and (b) – Illustration of the distribution of the electromagnetic field density near a bent microfiber and near a NT r -(z), respectively. (a1) and (b1) – Effective transversal dielectric constant for a bent microfiber and for a NT r -(z), respectively. (a2) and (b2) – Transversal behavior of the electromagnetic field density near a bent microfiber and near a NT r -(z), respectively. Waved arrows indicate the classically allowed region.

Fig. 3.
Fig. 3.

(a) and (b) – Illustration of the distribution of the electromagnetic field density for a Gaussian beam and a nanoswell r +(z), respectively. (a1) and (b1) – Effective transversal dielectric constant for Gaussian beam and for a nanoswell r +(z), respectively. (a2) and (b2) – Transversal behavior of the electromagnetic field density near a Gaussian beam and near a nanoswell r +(z), respectively.

Equations (30)

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

β ( z ) = β 0 2 + γ ( z ) 2 β 0 + γ 2 ( z ) 2 β 0 .
γ ( z ) = 1.273 r ( z ) exp [ η 8 λ f 2 r 2 ( z ) ] , λ f = 1 β 0 η + 1 η ( η 1 ) , η = n 1 2 n 2 2 .
d 2 Λ dv 2 + 1 v d Λ dv + ( a bv 2 ) Λ = 0
v ( ρ , z ) = ρ σ ( z )
σ ( z ) = ( c + b 0 2 z 2 ) 1 2
γ ± ( z ) = γ 0 1 ± ( z L ) 2
L 2 = b c 2 β 0 2 , γ 0 = ( a c ) 1 2
α ( z ) = 2 π γ 2 ( z ) β 0 exp ( π γ 0 2 L 2 β 0 )
ΔΨ + β 0 2 Ψ = 0 , β 0 = 2 πn 2 λ , ρ r ( z ) .
β ( z ) = β 0 2 + γ ( z ) 2 β 0 + γ 2 ( z ) 2 β 0 , γ ( z ) β 0 .
γ ( z ) = 1.655 r ( z ) exp ( 0.0713 λ 2 r 2 ( z ) )
r ( z ) λ f ln [ λ f γ ( z ) ] 1 2
Ψ ( 0 ) ( ρ , z ) = π 1 2 γ ( z ) K 0 ( γ ( z ) ρ ) exp ( i z β ( z ) dz ) , 2 π 0 Ψ ( 0 ) ( ρ , z ) 2 ρdρ = 1 ,
ψ ( 0 ) ρ z = ( γ ( z ) 2 ρ ) 1 2 exp ( γ ( z ) ρ + i z β ( z ) dz ) .
Ψ ρ φ z = exp ( imφ ) ρ Λ ( ρ σ ( z ) ) exp [ ikz ia 2 k z dz σ 2 ( z ) + ik 2 σ ( z ) ( z ) dz ρ 2 ] .
σ ( z ) = ( A 11 + 2 A 12 z + A 22 z 2 ) 1 2 ,
d 2 Λ ( v ) dv 2 + 1 v d Λ ( v ) dv + ( a bv 2 m 2 v 2 ) Λ ( v ) = 0 ,
A 11 A 22 A 12 2 = b β 0 2
Λ ( v ) Cv 1 2 ( bv 2 a ) 1 4 exp ( 0 v dv ( bv 2 a ) 1 2 )
Ψ ρ z C ( ρσ ( z ) ) 1 2 ( bv 2 a ) 1 4 exp ( 0 v dv ( bv 2 a ) 1 2 ) × exp [ ikz ia 2 k z dz σ 2 ( z ) + ik 2 σ ( z ) ( z ) dz ρ 2 ] .
γ ( z ) = ( a ) 1 2 σ ( z ) , C = ( a 2 ) 1 2
γ ( z ) = 1 g 11 + 2 g 12 z + g 22 z 2 .
g 11 g 22 g 12 2 = b a 2 k 2
Ψ ρ z C ( ρσ ( z ) ) 1 2 ( a bv 2 ) 1 4 exp ( 0 v turn dv ( bv 2 a ) 1 2 + i v turn v dv ( a bv 2 ) 1 2 + 4 ) × exp [ ikz ia 2 k z dz σ 2 ( z ) + ik 2 σ ( z ) ( z ) dz ρ 2 ] , v > v turn = ( a b ) 1 2 .
I ( z ) = 2 π 0 Ψ ρ z 2 ρdρ
I ( z ) = I ( z 0 ) exp ( z 0 z α ( z ) dz ) ,
α ( z ) = 1 I ( z ) dI ( z ) dz
k dI ( z ) dz = U ρ z U * ρ z ρ U * ρ z U ρ z ρ ,
U ρ z ( 2 π ) 1 2 C σ ( z ) 1 2 ( a bv 2 ) 1 4 exp ( πa 4 ( b ) 1 2 + i ( a b ) 1 2 v dv ( a bv 2 ) 1 2 )
α ( z ) = 2 πa β 0 σ 2 ( z ) exp ( πa 2 ( b ) 1 2 )

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