Abstract

Radiation losses of optical nanofibers are investigated in assumption of Gaussian statistics of distorted glass/air interface. Nonlinear relationship between the radiated power and roughness power spectrum is established. The losses in the single mode silica nanofibers are estimated for the case of inverse-square law of the roughness power spectrum.

© 2008 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  8. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, New York, 1974).
  9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge, 1995).
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  11. R. Price, "A useful Theorem for Non-Linear devices having Gaussian Inputs," IEEE Trans. Inf. Theory 4, 69-72 (1958).
    [CrossRef]
  12. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York 1984).
  13. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, NY 1983).
  14. L. Tong, J. Lou, and E. Mazur, "Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides," Opt. Express 12, 1025-1035 (2004).
    [CrossRef] [PubMed]

2007

2006

2005

2004

2003

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

1995

J. Jäckle and K. Kawasaki, "Intrinsic roughness of glass surfaces," J. Phys.: Condens. Matter 7, 4351-4358 (1995).
[CrossRef]

1991

O. I. Barchuk, A. V. Kovalenko, V. N. Kurashov, and A. I. Maschenko, "Statistical characteristics of fluctuations of dielectric constant in planar waveguide with rough walls," (Ukrainskiy Fizicheskiy Zhurnal) Ukr. Phys. J. 36, 612-617 (1991).

1969

D. Marcuse, "Radiation losses of dielectric waveguides in terms of the power spectrum of the wall distortion function," Bell Syst. Tech. J. 48, 3233-3242 (1969).

1958

R. Price, "A useful Theorem for Non-Linear devices having Gaussian Inputs," IEEE Trans. Inf. Theory 4, 69-72 (1958).
[CrossRef]

Bell Syst. Tech. J.

D. Marcuse, "Radiation losses of dielectric waveguides in terms of the power spectrum of the wall distortion function," Bell Syst. Tech. J. 48, 3233-3242 (1969).

IEEE Trans. Inf. Theory

R. Price, "A useful Theorem for Non-Linear devices having Gaussian Inputs," IEEE Trans. Inf. Theory 4, 69-72 (1958).
[CrossRef]

J. Phys.: Condens. Matter

J. Jäckle and K. Kawasaki, "Intrinsic roughness of glass surfaces," J. Phys.: Condens. Matter 7, 4351-4358 (1995).
[CrossRef]

Nature

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

Opt. Express

Ukrainskiy Fizicheskiy Zhurnal)

O. I. Barchuk, A. V. Kovalenko, V. N. Kurashov, and A. I. Maschenko, "Statistical characteristics of fluctuations of dielectric constant in planar waveguide with rough walls," (Ukrainskiy Fizicheskiy Zhurnal) Ukr. Phys. J. 36, 612-617 (1991).

Other

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York 1984).

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

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, New York, 1974).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge, 1995).

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Figures (4)

Fig. 1.
Fig. 1.

Perturbed fiber waveguide. n co, n cl, are the refractive indices of the core and the cladding (air), respectively.

Fig. 2.
Fig. 2.

Square law approximation of the nonlinear term of the correlation Gf ξ).

Fig. 3.
Fig. 3.

Dependence of the loss coefficient on the fiber diameter. (a) Linear approximation in S 0(β) (dash), nonlinear loss component (dot), aggregate losses (solid), βcut=0. (b) Losses for different cutoff frequencies of surface perturbation, βcut=1.0 mm-1 (blue), βcut=0.1 mm-1 (red), βcut=0 (black).

Fig. 4.
Fig. 4.

(a). Dependence of the power density of the electric component of the fundamental mode in the core/cladding interface on the fiber diameter. (b) Effective depth of the perturbed layer for different cutoff frequencies of surface perturbation, βcut=1.0 mm-1 (blue), βcut=0.1 mm-1 (red), βcut=0 (black).

Equations (37)

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ρ ( z , φ ) = ρ 0 + ξ ( z , φ )
ξ * ( z 1 , φ 1 ) ξ ( z 2 , φ 2 ) = σ ξ 2 γ ξ ( Δ z , Δ φ )
e 1 ( r , φ ) = ψ 1 ( r ) exp ( i φ )
e v r ( Q ; r , φ ) = ψ v r ( Q ; r ) exp ( iv φ ) , v = 0 , ± 1 , ± 2
0 2 π d φ 0 + r d r e 1 × h 1 * · z = 1
0 2 π d φ 0 + r d r e v ( Q ) × h μ * ( Q ) · z = δ ( Q Q ) δ v μ
a v ( Q ) = ± a ¯ 0 L d z exp [ i ( β 1 β ( Q ) ) z ] 0 2 π exp [ i ( 1 v ) φ ] f v ( Q ; ξ )
f v ( Q ; ξ ) = { u co ξ , if ξ ( z , φ ) < 0 u cl ξ , if ξ ( z , φ ) > 0
u co = C ( Ψ v r * ( Q ; ρ 0 0 ) · Ψ 1 ( ρ 0 0 ) ) ,
u cl = C ( Ψ v r * ( Q ; ρ 0 + 0 ) · Ψ 1 ( ρ 0 + 0 ) ) ,
C = i k ρ 0 4 ( ε 0 μ 0 ) 1 2 ( n co 2 n cl 2 )
P v ( Q ) = a v ( Q ) 2
P rad = v 0 k ρ n cl p v ( + ITE ) ( Q ) d Q + v 0 k ρ n cl p v ( ITE ) ( Q ) d Q +
+ v 0 k ρ n cl p v ( + ITM ) ( Q ) d Q + v 0 k ρ n cl p v ( ITM ) ( Q ) d Q
η = 101 g ( 1 P rad P ¯ ) L
p v ( Q ) = 2 π L a ¯ 2 d Δ z π π d Δ φ G f ( Δ z , Δ φ ) exp [ i ( 1 v ) Δ φ ] exp [ i ( β 1 β ( Q ) ) Δ z ]
G f = f v * ( Q ; ξ 1 ) f v ( Q ; ξ 2 ) , ξ 1 ξ ( φ 1 , z 1 ) ) , ξ 2 ξ ( φ 2 , z 2 )
d 2 G f d γ ξ 2 = σ ξ 4 d 2 f * ( ξ 1 ) d ξ 1 2 d 2 f ( ξ 2 ) d ξ 2 2
d 2 G f d γ ξ 2 = σ ξ 2 u co u cl 2 2 π 1 γ ξ 2
G f ( γ ξ = 0 ) = f 2 = σ ξ 2 2 π u co u cl 2 ,
G f ( γ ξ = 1 ) = f 2 = σ ξ 2 2 ( u co 2 + u cl 2 )
G f ( γ ξ ) = σ ξ 2 4 u co + u cl 2 γ ξ + σ ξ 2 2 π u co u cl 2 ( γ ξ arcsin γ ξ + 1 γ ξ 2 )
G f ( γ ξ ) = Γ L ( Q ) σ ξ 2 γ ξ ( Δ z , Δ φ ) + Γ N ( Q ) σ ξ 2 γ ~ ( Δ z , Δ φ ) + const
Γ L ( Q ) = 1 4 u co + u cl 2 , Γ N ( Q ) = ( π 2 ) 4 π u co u cl 2
γ ~ ( Δ z , Δ φ ) = 2 π 2 ( γ ξ arcsin γ ξ + 1 γ ξ 2 1 )
p v ( Q ) 2 π L a 2 Γ L ( Q ) S 1 v ( β 1 β ( Q ) ) + Γ N ( Q ) S ˜ 1 v ( β 1 β ( Q ) )
S μ ( β ) = σ ξ 2 d Δ z π π d Δ φ γ ξ ( Δ z , Δ φ ) exp ( i β Δ z ) exp ( i μ Δ φ )
S ˜ μ ( β ) = S μ ( β ) * S μ ( β ) 4 π 2 σ ξ 2 1 4 π 2 σ ξ 2 m = + + d β S m ( β ) S μ m ( β β )
J = i ( ε 0 μ 0 ) 1 2 k ( n co 2 n cl 2 ) δ ( r ρ 0 ) ξ ( φ , z ) a ¯ e exp ( i β 1 z )
e = { e 1 ( ρ 0 0 ) , if ξ ( z ) < 0 e 1 ( ρ 0 + 0 ) , if ξ ( z ) > 0
S ( β ) = k B T α 1 β 2
S μ ( β ) = k B T α ρ 0 · 1 v μ 2 + β 2 , v μ = { μ ρ 0 1 , μ = ± 1 , ± 2 ± μ max β cut , μ = 0
σ ξ 2 = ( 2 π ) 2 μ + d β S μ ( β ) k B T 4 π α ρ 0 β cut
S ˜ μ ( β ) k B T α ρ 0 · M M 2 v μ 2 + β 2
Δ ξ 2 = Ω min + S 0 ( β ) d β
e ± 1 ( r , φ ) = N 1 2 ( e 11 , even ( r , φ ) ± i e 11 , odd ( r , φ ) ) = Ψ ± 1 ( r ) exp ( ± i φ )
e ± v r ( Q ; r , φ ) = N v ( Q ) 1 2 ( e v , even r ( Q ; r , φ ) ± i e v , odd r ( Q ; r , φ ) ) = Ψ ± v r ( Q ; r ) exp ( ± i v φ )

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