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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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  19. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1965, 2nd edition).

2006 (7)

2004 (3)

2003 (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]

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]

Babic, V. M.

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

Birks, T. A.

Brambilla, G.

Buldyrev, V. S.

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

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.

M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, “The Microfiber Loop Resonator: Theory, Experiment, and Application,” IEEE J. Lightwave Technol., 24,242–250 (2006).
[CrossRef]

Dulashko, Y.

M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, “The Microfiber Loop Resonator: Theory, Experiment, and Application,” IEEE J. Lightwave Technol., 24,242–250 (2006).
[CrossRef]

M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and J. W. Nicholson, “Probing optical microfiber nonuniformities at nanoscale,” Opt. Lett. 31,2393–2395 (2006).
[CrossRef] [PubMed]

Finazzi, V.

Fini, J. M.

M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, “The Microfiber Loop Resonator: Theory, Experiment, and Application,” IEEE J. Lightwave Technol., 24,242–250 (2006).
[CrossRef]

M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and J. W. Nicholson, “Probing optical microfiber nonuniformities at nanoscale,” Opt. Lett. 31,2393–2395 (2006).
[CrossRef] [PubMed]

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.

M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, “The Microfiber Loop Resonator: Theory, Experiment, and Application,” IEEE J. Lightwave Technol., 24,242–250 (2006).
[CrossRef]

M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and J. W. Nicholson, “Probing optical microfiber nonuniformities at nanoscale,” Opt. Lett. 31,2393–2395 (2006).
[CrossRef] [PubMed]

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]

Heading, J.

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

Hu, L.

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

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]

Jiang, X.

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

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]

Landau, L. D.

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

Leon-Saval, S. G.

Li, Y.

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

Lifshitz, E. M.

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

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]

Love, J. D.

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

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.

Snyder, A. W.

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

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.

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

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]

Vienne, G.

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

Wadsworth, W. J.

Yang, Q.

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]

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

Ye, Z.

Zhang, 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]

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

Appl. Phys. Lett. (1)

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

IEEE J. Lightwave Technol. (1)

M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, “The Microfiber Loop Resonator: Theory, Experiment, and Application,” IEEE J. Lightwave Technol., 24,242–250 (2006).
[CrossRef]

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]

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 (4)

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

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

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

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

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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)

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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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