## Radiation losses in optical nanofibers with random rough surface

Optics Express, Vol. 16, Issue 8, pp. 5797-5806 (2008)

http://dx.doi.org/10.1364/OE.16.005797

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### 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

## 1. Introduction

1. 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 fibers,” Opt. Express **13**, 236–244 (2005). [CrossRef] [PubMed]

1. 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 fibers,” Opt. Express **13**, 236–244 (2005). [CrossRef] [PubMed]

2. P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Loss in solid-core photonic crystal fibers due to interface roughness scattering,” Opt. Express **13**, 7779–7793 (2005). [CrossRef] [PubMed]

4. 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]

5. L. M. Tong, L. L. Hu, J. J. Zhang, J. R. Qiu, Q. Yang, J. Y. Lou, Y. H. Shen, J. L. He, and Z. Z. Ye, “Photonic nanowires directly drawn from bulk glasses,” Opt. Express **14**, 82–87 (2006). [CrossRef] [PubMed]

6. G. Zhai and L. Tong, “Roughness-induced radiation losses in optical micro or nanofibers,” Opt. Express **15**, 13805–13816 (2007). [CrossRef] [PubMed]

^{-3}dB/mm and substantially depend on the perturbation period and refractive index contrast. Besides, the paper contains the conclusion that the model of sinusoidal perturbation can be generalized for all kinds of the surface deformation by means of Fourier analysis. This statement is controversial because direct generalization of the represented model assumes linear relationship between the amplitude of perturbation and the power of light scattered by this perturbation. Meanwhile, the radiated power is expressed as the square of linear functional of perturbation amplitude and thus can not be represented as a sum of independent contributions of perturbation’s Fourier components. This problem is solved with a help of statistical approach, when the perturbed surface is treated as a random field and the radiated power is determined with averaging over perturbation ensemble.

1. 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 fibers,” Opt. Express **13**, 236–244 (2005). [CrossRef] [PubMed]

2. P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Loss in solid-core photonic crystal fibers due to interface roughness scattering,” Opt. Express **13**, 7779–7793 (2005). [CrossRef] [PubMed]

3. J. Jäckle and K. Kawasaki, “Intrinsic roughness of glass surfaces,” J. Phys.: Condens. Matter **7**, 4351–4358 (1995). [CrossRef]

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

## 2. Radiation losses in approximation of weakly perturbed fiber surface

6. G. Zhai and L. Tong, “Roughness-induced radiation losses in optical micro or nanofibers,” Opt. Express **15**, 13805–13816 (2007). [CrossRef] [PubMed]

_{0}is the radius of unperturbed waveguide, the function ξ(

*z*, φ) describes the perturbation (Fig. 1).

_{z}=

*z*

_{2}-

*z*

_{1}, Δ

_{φ}=φ

_{2}-φ

_{1}

^{2}

_{ξ}is the roughness variance, γ

_{ξ}denotes normalized correlation. We suppose the perturbation magnitude ξ being small. The small order of ξ means that the amplitude of electric field of both propagating and radiation modes can be considered constant within the perturbed region. As in nanofibers the perturbation is less in scale than 1 nm [4

4. 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]

_{11}mode with electric component

**E**

_{11}=

**e**

_{1}(

*r*, φ) exp(

*i*β

_{1}

*z*). The scattered wave is expressed as a superposition of radiation ITE and ITM modes

**E**

^{r}_{ν}=

**e**

^{r}

_{ν}(

*Q*;

*r*, φ)exp(±

*i*β(

*Q*)

*z*), where

*Q*

^{2}=ρ

^{2}

_{0}(

*k*

^{2}

*n*

^{2}

_{c1}-β

^{2}), the signs +/- correspond to the forward and backward propagating modes, respectively. We choose the modal fields in the form with exponential dependence on the azimuthal coordinate [8], (Appendix)

**h**denotes the magnetic component of the modal field.

*L*, is given by [6

6. G. Zhai and L. Tong, “Roughness-induced radiation losses in optical micro or nanofibers,” Opt. Express **15**, 13805–13816 (2007). [CrossRef] [PubMed]

*ā*is the amplitude of the propagating HE

_{11}mode and the following designations are introduced

*f*

_{ν}(

*Q*;ξ) appears as a transform of the random perturbation ξ. Therefore, the amplitude of radiation mode is a random variable with statistical properties dependent on the perturbation statistics. Losses for radiation of the definite mode are given by the mean square of the proper modal coefficient

*P̅*=

*ā*

^{2}is the power of the fundamental mode.

*f*

_{ν}(

*Q*;ξ)

*G*in the azimuthal coordinate

_{f}*G*(Δ

_{f}_{φ}, Δ

_{z})=

*G*(Δ

_{f}_{φ}±2π, Δ

_{z}). Also, it is assumed that

*G*takes significant values only when Δ

_{f}_{z}≪

*L*. If the last condition is violated, the integral for dΔ

_{z}should be taken in limits -

*L*≤Δ

_{z}≤

*L*with the window

*w*(Δ

_{z})=1-|Δ

_{z}|

*L*

^{-1}.

*f*

_{ν}(6), the mean square of the amplitude is given by Fourier transform of the correlation

*G*. In addition, it follows from (11) that radiation losses are linear in

_{f}*L*.

## 3. Dependence of radiated power on roughness correlation

*f*

_{ν}(6). The multipliers

*u*

_{co},

*u*

_{cl}(7) are the scalar products of electric fields of fundamental and radiation modes in core and cladding. As the radial component of the electric field has discontinuity in core/cladding interface, the expression (6) defines sectionally linear mapping ξ→

*f*

_{ν}. The functional dependence of correlation

*G*on the roughness correlation is established with Price’s theorem as follows

_{f}_{1}∓β(

*Q*)>0, because |β(

*Q*)|<

*kn*

_{cl}<β

_{1}[13]. Therefore, the constant term from (17) does not contribute to the power

*p*

_{ν}(

*Q*) and can be eliminated. The function

_{z},Δ

_{φ}) is well approximated with a square law

^{2}

_{ξ}(Fig. 2). Assuming such approximation, the power of radiation mode is expressed through the roughness power spectrum

*S*

_{µ}(β) by

_{ξ}. Note that both terms are of the same order of smallness in roughness variance σ

^{2}

_{ξ}, but are different in dependence on the refractive index contrast because Γ

_{L}~(

*n*

^{2}

_{co}-

*n*

^{2}

_{cl})

^{2}, Γ

_{N}~(

*n*

^{2}

_{co}-

*n*

^{2}

_{cl})

^{4}. Therefore, the quadratic term in (20) is negligibly small for weakly guiding fibers, but it becomes significant in the particular case of nanofibers, in which

*n*

_{co}and

*n*

_{cl}are considerably different in value.

**e**equals to unperturbed field of the fundamental mode in the core if ξ<0, and it is equal to the field in the cladding otherwise:

*z*;Ω)=ξ

_{Ω}sinΩ

*z*. Since

*r*- component of the electric field of the fundamental mode is discontinuous in core/cladding transition, the correspondent component of the field

**e**is of the form of raised meander function with the period Ω

^{-1}, and can be expanded in Fourier series:

*e*=

_{r}*c*

_{0}+

*c*

_{1}sin(Ω

*z*)+

*c*

_{2}sin(2Ω

*z*)+…. Therefore, the induced current (23) is expressed as superposition of harmonic currents:

**J**

_{1}and

**J**

_{2}respectively.

**J**

_{m}exp(

*i*(β

_{1}-

*m*Ω)

*z*) radiates in resonance the plain waves in directions, which subtend an angle θ with

*Oz*axis. This angle satisfies the relation:

*kn*

_{cl}cos θ=(β

_{1}-

*m*Ω). As β

_{1}>

*kn*

_{cl}, the losses are caused with currents

**J**

_{m}with

*m*≥1. It is easy to show, that

**J**

_{1}~(

*n*

^{2}

_{co}-

*n*

^{2}

_{cl}),

**J**

_{m}~(

*n*

^{2}

_{co}-

*n*

^{2}

_{cl})

^{2},

*m*=2,3…. Therefore, in weakly guiding fibers, the induced currents

**J**

_{2},

**J**

_{3}… could be neglected. Then, the minimal frequency of perturbation, which still satisfies the resonance condition, equals Ω

_{min}=β

_{1}-

*kn*

_{cl}. The lower frequencies of the perturbation cause no radiation losses. When refractive index contrast is high, the currents

**J**

_{1}and

**J**

_{2}have the same order of smallness, and produce two meaningful components in the expression for total radiation losses. It should be noticed that the minimal frequency of the resonant perturbation is halved, when the current

**J**

_{2}is taken into account: Ω

_{min}=(β

_{1}-

*kn*

_{cl})/2.

## 4. Losses from perturbation with inverse square-law power spectrum

3. J. Jäckle and K. Kawasaki, “Intrinsic roughness of glass surfaces,” J. Phys.: Condens. Matter **7**, 4351–4358 (1995). [CrossRef]

*S*(

**β**)

*T*is the glass transition temperature, α is the surface tension in transition,

**β**is bidirectional vector of spatial frequencies, bounded in the absolute value as β

_{low}<|

**β**|<β

_{high}. The lower cutoff frequency is conditioned with gravity

*d*is the glass density. Its value is small in comparison with the typical frequencies of the optical range. Thus, for α=0.1 J·m

^{-2}[3

3. J. Jäckle and K. Kawasaki, “Intrinsic roughness of glass surfaces,” J. Phys.: Condens. Matter **7**, 4351–4358 (1995). [CrossRef]

*d*=2500kg·m

^{-3}, the estimation gives β

_{low}≈0.5 mm

^{-1}. The high-frequency cutoff is specified with atomic structure of the matter.

_{cut}is not equivalent to β

_{low}of the unbounded plane surface, and is introduced into (26) phenomenologically in order to avoid singularity of

*S*

_{0}(β). With that, a numerical value of β

_{cut}remains uncertain. This uncertainty, nevertheless, presents no obstacle to estimation of the radiated power. Let us assume that the lower cutoff β

_{cut}is small enough against the inverse radius of the fiber ρ

^{-1}

_{0}. In such approximation, the roughness variance σ

^{2}

_{ξ}and the convolved spectrum

*S̃*

_{µ}(β) (22) are given by

*M*=2, if µ=0, and

*M*=1, otherwise. The power

*p*

_{ν}(

*Q*) (20) depends on the parts of the spectra (26), (28), which correspond to the spatial frequencies β=β

_{1}∓β(

*Q*)>0. Therefore, the points

*S*

_{µ}(0),

*S̃*

_{µ}(0) lie aside the integration boundaries (9), which gives an opportunity to calculate the radiation losses in the limit β

_{cut}→0.

*S*

_{0}(β

_{1}∓β(

*Q*)),

*S̃*

_{0}(β

_{1}∓β(

*Q*)) are considerably larger than the rest of

*S*

_{µ},

*S̃*

_{µ}. These components “are responsible” for excitation of the radiation modes, which have the same azimuthal number with the fundamental mode. So, the terms with ν=1 are the most significant in the expression for the total radiated power (9).

## 5. Numerical results, discussion and conclusion

4. 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]

5. L. M. Tong, L. L. Hu, J. J. Zhang, J. R. Qiu, Q. Yang, J. Y. Lou, Y. H. Shen, J. L. He, and Z. Z. Ye, “Photonic nanowires directly drawn from bulk glasses,” Opt. Express **14**, 82–87 (2006). [CrossRef] [PubMed]

*n*

_{co}=1.46,

*T*=1500K, α=0.3 J·m

^{-2}[1

**13**, 236–244 (2005). [CrossRef] [PubMed]

*S*

_{0}(β) approximation. As follows from these calculations, the total loss estimation increases essentially when the term, proportional to the convolved spectrum

*S̃*

_{0}(β), is taken into account. So, the surface induced radiation losses are considerably nonlinear in the perturbation spectrum.

_{cut}, demonstrate that the choice of this parameter makes effect only when the fiber diameter is especially small, in our case being less than 180 nm (Fig. 3(b)). The result, which fits qualitatively to the experiment [4

**426**, 816–819 (2003). [CrossRef] [PubMed]

5. L. M. Tong, L. L. Hu, J. J. Zhang, J. R. Qiu, Q. Yang, J. Y. Lou, Y. H. Shen, J. L. He, and Z. Z. Ye, “Photonic nanowires directly drawn from bulk glasses,” Opt. Express **14**, 82–87 (2006). [CrossRef] [PubMed]

_{cut}→0.

*D*<300 nm. For very small diameters

*D*<180 nm, the plot of losses becomes very sensitive to the choice of the lower cutoff frequency of the perturbation. To explain the plot’s behavior, we shall notice at first that the radiated power (20) is proportional to the power density

*w*

_{1}of electric component of the fundamental mode, taken in the perturbed area. Dependence of

*w*

_{1}on the waveguide diameter is nonmonotonic, and has the maximum at

*D*≈300 nm (Fig. 4(a)). We notice also that the radiation arises from the spectral components of the perturbation which have the spatial frequencies β exceeding Ω

_{min}: β>Ω

_{min}=(β

_{1}-

*kn*

_{cl})/2. As the gap (β

_{1}-

*kn*

_{cl}) decreases with the decrease of the waveguide diameter, the low frequency perturbation components add their contribution into the total radiation losses. The perturbation power spectrum is expected to have inverse-square low, therefore, the contribution of the low frequencies is very critical, and could be explained in terms of the effective depth of the perturbed layer Δ

_{ξ}.

_{ξ}as:

_{ξ}on the waveguide diameter is demonstrated at Fig. 4(b). One can see that Δ

_{ξ}increases with decreasing diameter, and essentially depends on the choice of β

_{cut}when

*D*becomes less than 180 nm. Thus, the growth of losses with decreasing

*D*from 450 nm to 300 nm is explained with simultaneous growth of the power density of the fundamental mode in the perturbed layer and of the depth of this layer. In the range 180 nm<

*D*<300 nm, the trends of Δ

_{ξ}and

*w*

_{1}have the opposite character and compensate each other. For

*D*<180 nm, the loss estimation depends on the phenomenological parameter β

_{cut}. Passage to the limit β

_{cut}→0 gives the reasonable estimation of losses, but leads to the growth of the effective depth of the perturbation to the values which exceed the waveguide diameter. To our opinion, the extremely low frequency perturbations with large amplitudes should be understood as random waveguide bends. Naturally, there are no reasons to describe such perturbations with the inverse-square law spectrum. So, the estimation of radiation losses in the silica nanofibers with

*D*<180 nm has rather qualitative character.

## Appendix A

*r*, the second one depends on φ. A choice of the modal system is ambiguous because the modes are degenerate in propagation constant. In [6

**15**, 13805–13816 (2007). [CrossRef] [PubMed]

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]

**e**

_{11, even},

**e**

_{11, odd}are the fields of the even and the odd HE

_{11}mode,

**e**

^{r}

_{ν, even},

**e**

^{r}

_{ν, odd}are ITE or ITM modes,

*N*,

*N*

_{ν}(

*Q*) are the normalization factors [6

**15**, 13805–13816 (2007). [CrossRef] [PubMed]

## References and links

1. | 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 fibers,” Opt. Express |

2. | P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Loss in solid-core photonic crystal fibers due to interface roughness scattering,” Opt. Express |

3. | J. Jäckle and K. Kawasaki, “Intrinsic roughness of glass surfaces,” J. Phys.: Condens. Matter |

4. | 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 |

5. | L. M. Tong, L. L. Hu, J. J. Zhang, J. R. Qiu, Q. Yang, J. Y. Lou, Y. H. Shen, J. L. He, and Z. Z. Ye, “Photonic nanowires directly drawn from bulk glasses,” Opt. Express |

6. | G. Zhai and L. Tong, “Roughness-induced radiation losses in optical micro or nanofibers,” Opt. Express |

7. | D. Marcuse, “Radiation losses of dielectric waveguides in terms of the power spectrum of the wall distortion function,” Bell Syst. Tech. J. |

8. | D. Marcuse, |

9. | L. Mandel and E. Wolf, |

10. | 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 (Ukrainian Physical Journal) |

11. | R. Price, “A useful Theorem for Non-Linear devices having Gaussian Inputs,” IEEE Trans. Inf. Theory |

12. | A. Papoulis, |

13. | A. W. Snyder and J. D. Love, |

14. | L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express |

**OCIS Codes**

(060.2400) Fiber optics and optical communications : Fiber properties

(290.5880) Scattering : Scattering, rough surfaces

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: February 15, 2008

Revised Manuscript: April 3, 2008

Manuscript Accepted: April 6, 2008

Published: April 10, 2008

**Citation**

A. V. Kovalenko, V. N. Kurashov, and A. V. Kisil, "Radiation losses in optical nanofibers with random rough surface," Opt. Express **16**, 5797-5806 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5797

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### References

- P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, and A. Tomlinson, T. A. Birks, J. C. Knight and P. St. J. Russell, "Ultimate low loss of hollow-core photonic crystal fibers," Opt. Express 13, 236-244 (2005). [CrossRef] [PubMed]
- P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, "Loss in solid-core photonic crystal fibers due to interface roughness scattering," Opt. Express 13, 7779-7793 (2005). [CrossRef] [PubMed]
- J. Jäckle and K. Kawasaki, "Intrinsic roughness of glass surfaces," J. Phys.: Condens. Matter 7, 4351-4358 (1995). [CrossRef]
- 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]
- L. M. Tong, L. L. Hu, J. J. Zhang, J. R. Qiu, Q. Yang, J. Y. Lou, Y. H. Shen, J. L. He, and Z. Z. Ye, "Photonic nanowires directly drawn from bulk glasses," Opt. Express 14, 82-87 (2006). [CrossRef] [PubMed]
- G. Zhai and L. Tong, "Roughness-induced radiation losses in optical micro or nanofibers," Opt. Express 15, 13805-13816 (2007). [CrossRef] [PubMed]
- 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).
- 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).
- 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).
- R. Price, "A useful Theorem for Non-Linear devices having Gaussian Inputs," IEEE Trans. Inf. Theory 4, 69-72 (1958). [CrossRef]
- 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).
- 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]

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