Abstract

A theory of light propagation along adiabatic photonic nanowire tapers (nanotapers) having diameters significantly less than the radiation wavelength λ1μm is developed. The fundamental mode of a nanotaper primarily consists of an evanescent field, which propagates in the ambient medium and is very sensitive to the nanotaper shape. General analytical expressions for the evanescent field and the radiation loss of adiabatic nanotapers are obtained and applied to the investigation of the optics of tunneling from a nanotaper of a characteristic shape. The radiation loss of this nanotaper occurs locally near a focal circumference of the evanescent field, representing an intersection of a complex caustic surface with real space, where the fundamental mode splits into the radiating and guiding components. The interference of these components gives rise to a sequence of circumferences with zero electromagnetic field.

© 2006 Optical Society of America

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References

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  1. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).
  2. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1991).
  3. M. Sumetsky, Opt. Lett. 31, 870 (2006).
    [CrossRef] [PubMed]
  4. V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics (Reidel, 1981).
    [CrossRef]
  5. To arrive at Eqs. 4 5, the semiclassical solution is seen in the form (Ref. 4) U(ρ,z)=f(γ)[∂ρ(z,γ)/∂γ]−1/2exp[ikz−i(γ2/2k)z]∣γ=γ(ρ,z). Here, the arbitrary function f(γ) is determined so that this solution coincides with Eq. 1 for γρ⪢1 when K0(γρ)≈[π/(2γρ)]1/2exp(−γρ).
  6. E. A. Solov'ev, Sov. Phys. JETP 43, 453 (1976).
  7. The commercial RSOFT BeamPROP, version 4, software was used.
  8. Yu. N. Demkov and V. N. Ostrovskii, Zero-Range Potentials and their Applications in Atomic Physics (Plenum, 1988).
  9. A. Z. Devdariani, Theor. Math. Phys. 11, 460 (1972).
    [CrossRef]
  10. M. V. Berry, Proc. R. Soc. London, Ser. A 460, 2629 (2004).
    [CrossRef]
  11. It occurs in the case γ(0)(z)=γ0[1−(z/L)2]−1/2 that will be considered elsewhere.
  12. W. L. Kath and G. A. Kriegsmann, IMA J. Appl. Math. 41, 85 (1988).
    [CrossRef]

2006

2004

M. V. Berry, Proc. R. Soc. London, Ser. A 460, 2629 (2004).
[CrossRef]

1988

W. L. Kath and G. A. Kriegsmann, IMA J. Appl. Math. 41, 85 (1988).
[CrossRef]

1976

E. A. Solov'ev, Sov. Phys. JETP 43, 453 (1976).

1972

A. Z. Devdariani, Theor. Math. Phys. 11, 460 (1972).
[CrossRef]

Berry, M. V.

M. V. Berry, Proc. R. Soc. London, Ser. A 460, 2629 (2004).
[CrossRef]

Demkov, Yu. N.

Yu. N. Demkov and V. N. Ostrovskii, Zero-Range Potentials and their Applications in Atomic Physics (Plenum, 1988).

Devdariani, A. Z.

A. Z. Devdariani, Theor. Math. Phys. 11, 460 (1972).
[CrossRef]

Fedoriuk, M. V.

V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics (Reidel, 1981).
[CrossRef]

Kath, W. L.

W. L. Kath and G. A. Kriegsmann, IMA J. Appl. Math. 41, 85 (1988).
[CrossRef]

Kriegsmann, G. A.

W. L. Kath and G. A. Kriegsmann, IMA J. Appl. Math. 41, 85 (1988).
[CrossRef]

Love, J. D.

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

Marcuse, D.

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

Maslov, V. P.

V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics (Reidel, 1981).
[CrossRef]

Ostrovskii, V. N.

Yu. N. Demkov and V. N. Ostrovskii, Zero-Range Potentials and their Applications in Atomic Physics (Plenum, 1988).

Snyder, A. W.

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

Solov'ev, E. A.

E. A. Solov'ev, Sov. Phys. JETP 43, 453 (1976).

Sumetsky, M.

IMA J. Appl. Math.

W. L. Kath and G. A. Kriegsmann, IMA J. Appl. Math. 41, 85 (1988).
[CrossRef]

Opt. Lett.

Proc. R. Soc. London, Ser. A

M. V. Berry, Proc. R. Soc. London, Ser. A 460, 2629 (2004).
[CrossRef]

Sov. Phys. JETP

E. A. Solov'ev, Sov. Phys. JETP 43, 453 (1976).

Theor. Math. Phys.

A. Z. Devdariani, Theor. Math. Phys. 11, 460 (1972).
[CrossRef]

Other

The commercial RSOFT BeamPROP, version 4, software was used.

Yu. N. Demkov and V. N. Ostrovskii, Zero-Range Potentials and their Applications in Atomic Physics (Plenum, 1988).

It occurs in the case γ(0)(z)=γ0[1−(z/L)2]−1/2 that will be considered elsewhere.

V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics (Reidel, 1981).
[CrossRef]

To arrive at Eqs. 4 5, the semiclassical solution is seen in the form (Ref. 4) U(ρ,z)=f(γ)[∂ρ(z,γ)/∂γ]−1/2exp[ikz−i(γ2/2k)z]∣γ=γ(ρ,z). Here, the arbitrary function f(γ) is determined so that this solution coincides with Eq. 1 for γρ⪢1 when K0(γρ)≈[π/(2γρ)]1/2exp(−γρ).

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

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

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Figures (2)

Fig. 1
Fig. 1

(a) Comparison of the NT transversal propagation constant variation defined by Eq. (7) and the corresponding NT diameter variation calculated using equations given in Ref. [1]. (b) Radiation loss of a NT as a function of its characteristic length L calculated with Eq. (8) (curves) compared with the loss computed with BPM (dots). Parameters of Eqs. (7, 8), are shown in the figure.

Fig. 2
Fig. 2

(a) Two-sheet surface plot of Re [ γ 1 ( ρ , z ) γ ] for γ 0 γ = 0.5 . (b) Logarithmic plot of the evanescent field amplitude for L = 0.5 mm , k = 4 μ m 1 , γ = 0.4 μ m 1 , and γ 0 = 0.2 μ m 1 .

Equations (14)

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U ( ρ , z ) γ ( 0 ) ( z ) ρ 1 ( 1 π k ) 1 2 γ ( 0 ) ( z ) K 0 [ γ ( 0 ) ( z ) ρ ] exp { i Z d z [ k γ ( 0 ) ( z ) 2 2 k ] } ,
ρ ( z , γ ) = ( γ k ) [ z z ( 0 ) ( γ ) ] ,
U j ( ρ , z ) γ j ( 0 ) ( z ) ρ 1 A j ( ρ , z ) exp [ i S j ( ρ , z ) ] ,
A j ( ρ , z ) = ( i γ 2 k ρ ) 1 2 { z z j ( 0 ) ( γ ) γ [ d z j ( 0 ) ( γ ) d γ ] 1 } 1 2 γ = γ j ( ρ , z ) ,
S j ( ρ , z ) = k z γ 2 2 k z + γ ρ + 1 k γ d γ γ z j ( 0 ) ( γ ) γ = γ j ( ρ , z ) .
P = π 3 2 4 k 1 2 d z ( 0 ) ( γ ) d γ γ = 0 Im [ z ( 0 ) ( 0 ) ] 3 2 exp [ 2 k Im γ ( 0 ) ( ) 0 d γ γ z ( 0 ) ( γ ) ] .
γ ( 0 ) ( z ) = γ + γ 0 γ 1 + ( z L ) 2 .
P = π 3 2 8 ( k γ 2 L ) 1 2 B ( δ ) exp [ L γ 2 4 k f ( δ ) ] ,
f ( δ ) = ( δ + 3 ) ( δ 1 ) ln ( 1 + δ 1 2 1 δ 1 2 ) + 2 δ 1 2 ( 3 δ ) ,
B ( δ ) = ( 1 δ ) δ 5 4 , δ = γ 0 γ .
P = ( π 3 2 8 ) Λ 1 2 exp ( 8 Λ 15 ) , Λ = L γ 0 5 2 ( k γ 1 2 ) .
γ 3 ( L 2 + z 2 ) γ 2 ( γ 0 L 2 + γ z 2 + 2 k ρ z ) + γ ( k 2 ρ 2 + 2 γ k ρ z ) γ k 2 ρ 2 = 0 ,
ρ b = L k γ Q + Q 2 δ 3 , z b = 0 ,
Q = 1 8 ( δ 2 + 18 δ 27 ) , δ = γ 0 γ .

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