Roughness-induced radiation losses in optical micro or nanofibers

Gaoye Zhai; Limin Tong

doi:10.1364/OE.15.013805

1. Introduction

Surface roughness induced scattering loss is one of the most important properties of an optical waveguide, especially when the width or diameter of the waveguide is close to or smaller than the wavelength of the guided light, in which sidewall scattering is usually the dominant loss mechanism [1–3]. Recently, optical micro- or nanofibers (MNFs) have been fabricated by taper drawing of molten glasses [4–6]. Because of their low optical loss, strong evanescent field and low dimension, these fibers have been used as building blocks for a variety of microphotonic devices such as resonators [7,8], couplers [9], sensors [10–13], filters [14] and lasers [15], as well as for other applications in nonlinear optics [16–18] and atom trapping or manipulation [19–21]. Previous research has shown that, during the solidification of the glass, the thermally excited surface capillary waves froze onto the surface at the glass transition temperature [22], which will inevitably contribute to surface roughness of a molten-drawn glass fiber and lead to a certain degree of scattering (or radiation) loss as has been investigated in photonic crystal fibers [23,24]. In the past years, much effort had been devoted to the theoretical analysis of scattering losses of slab or planar waveguides and weakly guiding optical fibers [25–29]; however, there has been little research on surface roughness induced scattering loss in MNFs. Recently some work has been done on the MNF loss using the theory of nonadiabatic intermode transitions [30, 31], in which the loss of MNF due to the tapering of the fiber is studied. However, the surface roughness, which is another dominant loss mechanism, remains uninvestigated. Here we investigated the surface roughness induced scattering loss of a MNF using an induced-current model, in which we use an induced-current representation of nonuniformities [32], and treat the perturbation as induced current sources on the surface of an unperturbed MNF. Since the diameter fluctuation of a taper-drawn MNF can go down to 10^-7 [33], treating optical loss of an MNF solely based on sidewall roughness should represent one of the typical situations as has been applied in photonic crystal fibers [23, 24].

2. Theoretical analysis

Generally, when we draw MNFs from molten glasses, capillary wave frozes on the surface at glass transition temperature, resulting in sidewall roughness of the fiber. To model the roughness, we assume that the capillary wave on a MNF surface is sine deformed along its length L, as shown in Fig. 1(a). This does not lose generality since any function can be expressed in sine waves through Fourier transformation.

Assume the refractive-index profile of a real (with surface perturbation as shown in Fig. 1(a)) and an ideal (without perturbation as shown in Fig. 1(b)) MNF to be n(r,ϕ,z) and n̅(r,ϕ,z) respectively, the sinusoidal perturbation can be represented as current sources induced by an electric field, with an induced current density given by [32]

J = i \sqrt{\frac{ε_{0}}{μ_{0}}} k ({\bar{n}}^{2} - n^{2}) E,

where ε ₀ and μ ₀ are permittivity and magnetic permeability in vacuum, k=2π/λ, and E(r,ϕ,z) is the total electric field of the perturbed MNF.

Fig. 1. Refractive-index profiles of (a) a real MNF and (b) an ideal MNF with induced currents on the surface. The length of the MNF is L, and the refractive indices of the core and cladding of the MNF are n_co and n_cl, respectively.

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For taper-drawn MNFs, the sidewall roughness is usually lower than 0.5 nm [5, 6, 34], while the diameter of the fiber used in visible and near-infrared band is larger than 100 nm, the perturbation is very small. Therefore, in Eq. (1) it is reasonable to assume that E(r,ϕ,z)≅E̅ (r,ϕ,z), where E̅ (r,ϕ,z) is the electric field of an unperturbed fiber, as has been suggested in Ref. [32].

In most applications, MNFs are single-mode (or equivalently used as single-mode) optical waveguides, so we only consider the fundamental mode in the following discussions.

For a slightly perturbed MNF, the induced current can be well approximated by [32]

J = i \sqrt{\frac{ε_{0}}{μ_{0}}} k ({\bar{n}}^{2} - n^{2}) \bar{E} = i \sqrt{\frac{ε_{0}}{μ_{0}}} k ({\bar{n}}^{2} - n^{2}) {\bar{a}}_{1} e_{1} \exp (iβz),

where E̅ = a̅₁ e ₁ exp(iβz) is the electric field of the propagating HE₁₁ mode of an unperturbed MNF, a̅₁ and β are the amplitude and propagation constant of the fundamental mode, respectively.

With a sine deformed surface, the radius ρ(z) of a real MNF (see Fig. 1(a)) can be expressed as

ρ (z) = ρ_{0} + ξ \sin ωz,

where ρ₀ is the radius of an unperturbed MNF, ξ the amplitude of the surface roughness, and ω the spatial frequency of the perturbation. Since ξ≪ρ₀, we assume that the currents are localized on the surface of the ideal fiber, as shown in Fig. 1(b). Therefore, the item n̅² -n ² in Eq. (2) can be approximated as [32]

{\bar{n}}^{2} - n^{2} ≅ - (n_{co}^{2} - n_{cl}^{2}) ξ \sin (ωz) \bar{δ} (r - ρ_{0}),

where δ̅(r-ρ ₀) is the Dirac delta function, n_co and n_cl are refractive indices of the core and cladding of the MNF, respectively .

By substituting Eq. (4) into Eq. (2), we obtain the current on the surface of the MNF in Fig. 1(b) as

J = - i \sqrt{\frac{ε_{0}}{μ_{0}}} k (n_{co}^{2} - n_{cl}^{2}) \bar{δ} (r - ρ_{0}) ξ (\sin ωz) {\bar{a}}_{1} e_{1} \exp (iβz) .

The amplitudes of the radiation modes excited by the surface current are then expressed as [32]:

For forward-propagating radiation mode (i.e., z≥L)

a_{j}^{r} (Q) = - \frac{1}{4 N_{j}^{r} (Q)} \int_{0}^{L} \int_{A_{\infty}} e_{j}^{* r} (Q) • J \exp (- iβ (Q) z) dAdz

For backward-propagating radiation mode (i.e., z≤0)

a_{- j}^{r} (Q) = - \frac{1}{4 N_{j}^{r} (Q)} \int_{0}^{L} \int_{A_{\infty}} e_{- j}^{* r} (Q) • J \exp (iβ (Q) z) dAdz

where e ^r _j(Q) and e ^r _-j(Q) are electric fields of forward- and backward-propagating radiation modes, A _∞, is the infinite cross-section, ^* denotes complex conjugate, β(Q) is the propagation constant of the radiation mode, $N_{j}^{r} (Q) = \frac{1}{2} ∣ \int_{A_{\infty}} e_{j}^{r} (Q) \times h_{j}^{* r} (Q) • \hat{z} dA ∣$ is the normalization factor of the power of the radiation mode, in which ẑ is the unit vector parallel to the waveguide axis.

In right-hand parts of Eq. (6), all modes with j≠1 vanish after the spatial integration. Therefore the radiation field of the perturbed MNF is [32]

E_{rad} = a_{1}^{r (ITE)} (Q) e_{1}^{r (ITE)} (Q) \exp (- iβ (Q) z) + a_{- 1}^{r (ITE)} (Q) e_{- 1}^{r (ITE)} (Q) \exp (iβ (Q) z)

where superscripts ITE and ITM denote the ITE (free-space TE) and ITM (free-space TM) radiation modes respectively, and the positive and negative subscripts denote the forward- and backward-propagating modes respectively.

Therefore, the total power radiated by the surface current is [32]

P_{rad} = \frac{1}{2} Re {\int_{A_{\infty}} E_{rad} \times H_{rad}^{*} • \hat{z} dA}

The loss coefficient is then obtained as

α = 10 \lg (\frac{\bar{P} - P_{rad}}{\bar{P}}) / L,

where p̅ = |E̅|² = |a̅₁|² N is the incident power of the MNF, and $N = \frac{1}{2} ∣ \int_{A_{\infty}} e_{1} \times h_{1}^{*} • \hat{z} dA ∣$ is the normalization factor of the power of the HE₁₁ mode, in which h ₁ is the magnetic field of the HE₁₁ mode, ẑ is the unit vector parallel to the waveguide axis.

3. Numerical results and discussions

In this section, we’ll investigate the roughness-induced loss of MNFs numerically based on the theoretical analysis presented in section 2, which shows that the roughness-induced optical loss of a real MNF depends on the amplitude and the spatial frequency of the surface roughness, the radius and index of the MNF, and the wavelength of the guided light.

We first consider the dependence of the loss coefficient α on the perturbation period Γ (Γ=2π/ω). The amplitude of the surface roughness ξ is assumed to be 0.2 nm. Three types of MNFs, silica (ρ₀=350 nm, n_co=1.44 at 1550-nm wavelength), phosphate (ρ₀=350 nm, n_co=1.54 at 1550-nm wavelength) and tellurite (ρ₀=300 nm, n_co=2.05 at 1550-nm wavelength) MNFs, are assumed to be operated at 1550-nm wavelength. For reference, the loss behavior of a silicon MNF (ρ₀=200 nm, n_co=3.48 at 1550-nm wavelength), operating at 1550-nm wavelength, is also provided.

It should be noticed that in Eq. (6), when the sinusoidal function (i.e., sin(ωz)) is negative, we use the inside-fiber (core) electric fields; and when sin(ωz) is positive, we use the outside-fiber (cladding) electric fields (see appendix for detailed expressions of these fields).

Calculated Γ-dependent α is shown in Fig. 2. With the increasing of Γ, the loss coefficient α shows an oscillation dependence with a series of minima (α_min). This means that, when the period of the perturbation goes to some specific values (denoted as Γ_min), the roughness-induced loss is very weak. The oscillation behavior can be explained by destructive interference between radiation waves originated at opposite side of the sine deformed surface (similar behavior has been reported in weakly guiding waveguides [28, 35]), in which a MNF with higher effective index (that is, larger propagation constant) provides higher spatial frequency (that is, smaller spatial period) for offering more minima with a given length. For example, for a 700-nm-diameter phosphate MNF, the maximum α is about 5.35 dB/mm with Γ=16 μm; when Γ increases to 47.8 μm, α decreases to about 0.00043 dB/mm (over 4 orders of magnitude lower). Both the maximum and the minimum α decrease with the increasing of Γ. For relatively small Γ, radiation loss increases with the increasing of the refractive index of the MNF, i.e., α_silicon>α_tellurite >α_phosphate >α_silica (see inset of Fig. 2), which can be explained as stronger radiation loss existed in MNFs with higher core-cladding index contrast. It also shows that, for MNF with a given diameter, the first Γ_min required for the first α_min of a higher index MNF (e.g., the 350-nm-radius phosphate MNF with n_co=1.54 at 1550-nm wavelength) is smaller than that of a lower index MNF (e.g., the 350-nm-radius silica MNF with n_co=1.44 at 1550-nm wavelength), which can be explained as larger propagation constant in a higher-index MNF.

Fig. 2. Loss coefficients α of air-cladding MNFs as a function of the perturbation periods Γ. The four MNFs are assumed to have the same roughness amplitude ξ of 0.2 nm, and be operated at 1550-nm wavelength. Inset, Γ-dependent α with relatively small Γ.

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We then investigate the influence of the amplitude of the surface roughness ξ on the Γ_min. Fig. 3 shows Γ-dependent α of 600-nm-diameter tellurite MNFs with ξ of 0.1, 0.2 and 0.4 nm respectively. It is reasonable to see that, for a given Γ, larger ξ leads to higher loss (larger α); while Γ_min is almost independent on ξ, since the propagation constant is almost independent on ξ when ξ is small.

Fig. 3. Loss coefficient α of 600-nm-diameter tellurite MNFs as a function of the perturbation period Γ with λ=1550 nm.

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To further study the ξ-dependent loss, we calculate α of silica, phosphate, tellurite, and silicon MNFs as a function of ξ, with a given perturbation period Γ = 10 nm as shown in Fig. 4. It shows that, for a given MNF, α increases monotonously with ξ. With a given ξ, the higher the refractive index of the MNF, the larger the radiation loss. For example, with ξ=0.2 nm, α of a 700-nm-diameter phosphate MNF (0.011 dB/mm) is over 2 times higher than that of a 700-nm-diameter silica MNF (0.0051 dB/mm). This is reasonable as larger index-contrast causes higher radiation loss.

Fig. 4. Loss coefficient α of MNFs as a function of the roughness amplitude ξ, with λ=1550 nm and Γ=10 nm.

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With a given sidewall roughness, it is also interesting to compare the loss coefficients α of MNFs with different diameters. Figure 5 shows Γ-dependent a of silica MNFs with diameters of 700, 800 and 900 nm respectively. The three MNFs are assumed to bear the same roughness ξ of 0.2 nm and be operated at 1550-nm wavelength. The results show that, the smaller the diameter, the higher the radiation loss. For example, at ξ=10 μm, α of a 700-nm-diameter MNF (2.41 dB/mm) is over 4 times higher than that of a 900-nm-diameter one (0.58 dB/mm). This can be attributed to stronger interaction between the guided light and the surface in the thinner MNF due to higher field intensity on the surface. It also shows that, thicker MNFs provide smaller Γ_min, which can be explained as the larger propagation constant in a thicker MNF.

Fig. 5. Γ-dependent α of silica MNFs of different diameters, with λ=1550 nm and ξ=0.2 nm.

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Compared with the results shown in Fig. 5, an equivalent behavior obtained is Γ-dependent α of a given MNF operating at different wavelengths. As shown in Fig. 6, α of an 800-nm-diameter silica MNF are given at 1200-, 1400- and 1600-nm wavelengths, respectively. The surface roughness ξ is assumed to be 0.2 nm. It shows that, α increases with the increasing wavelength, as light with larger wavelength distributes higher fraction of guided power around the surface of the MNF; and light with shorter wavelength encounters the smaller first Γ_min due to its larger propagation constant.

Fig. 6. Γ-dependent α of a 800-nm-diameter silica MNF operating at different wavelengths with ξ=0.2 nm.

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In addition, since supercontinuum generation in MNF is of great interest [16, 17, 36, 37], the surface roughness induced loss within a broad spectral range is also investigated. Figure 7 shows wavelength-dependent α of silica MNFs with diameters of 300 and 500 nm, respectively. The two MNFs are assumed to have same roughness parameters with amplitude ξ of 0.2 nm and period Γ of 100 nm. The results show that, α increases with the increasing wavelength, and the thicker MNF presents much lower loss than the thinner one, which are in good agreement with those shown in Figs. 5 and 6. For example, α of the 500-nm-diameter MNF increased from 0.08 dB/mm at 600-nm wavelength to 0.2 dB/mm at 900-nm wavelength; at 600-nm wavelength, the loss of the 500-nm-diameter MNF (about 0.08 dB/mm) is about 20 times lower than that of a 300-nm-diameter MNF (about 1.6 dB/mm). In addition, when the ratio of the wavelength and the fiber diameter λ/(2ρ₀) exceeds a certain value (here is about 2, e.g., 600-nm wavelength for the 300-nm-diameter MNF and 1000-nm wavelength for the 500-nm-diameter MNF), the increasing of α becomes very slow, which can be explained as follows: when λ/(2ρ₀) >2, the intensity of the electromagnetic field at the MNF surface increases very slowly with the increasing of the wavelength.

Fig. 7. Wavelength-dependent α of silica MNFs with diameters of 300 and 500 nm, respectively. The two MNFs are assumed to have same roughness parameters with amplitude ξ of 0.2 nm and period Γ of 100 nm.

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4. Conclusion

In conclusion, we’ve investigated the surface roughness induced radiation (scattering) loss of a MNF using an induced-current model, in which we assume a sinusoidal fluctuation on the MNF surface. Loss behaviors with regard to typical parameters of the guiding system, including the amplitude and the perturbation period of the roughness, the diameter and the index of the MNF, and the operating wavelength, are investigated by numerical calculations. Interesting phenomena such as the existence of a series of loss minima at specific perturbation periods is observed.

Since any function can be expressed in sinusoidal waves through Fourier transformation, the approach presented in this work may be generalized to all kinds of surface deformation on a MNF providing that the amplitude of the deformation is small compared to the diameter, and therefore can be used for estimating roughness-induced losses of MNFs in a variety of applications.

In addition, the precise dependence of the radiation loss on the surface deformation may provide opportunities for tailoring the MNF (e.g., intentionally forming deformations on the surface) with desired guiding properties.

Appendix

Electric field e ₁ used in Eq. (5), e ^r _j(Q) (with j=1) used in Eq. (6) and mode normalization N used in Eq. (9) are expressed as following [32]

Table 1. Electric components e ₁ of HE₁₁ mode of the step-profile fiber

View Table | View all tables in this article

Where R=r/ρ₀, ρ⁰ is the core radius, U=ρ₀(k ² n_co ²-β²)^1/2, V=ρ₀(k ² n_co ²-k ² n_cl ²)^1/2, W=ρ₀(β ²-k ² n_cl ²)^1/2, ∆=(n_co ²-n_cl ²)/2n_co ², k=2π/λ, core and cladding refractive indices are n_co and n_cl.

Table 2. Electric field e ^r_j(Q) of radiation mode (with j=l) of a step-profile fiber

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where U=ρ₀(k ₂ n_co ²-β(Q)²)^1/2, Q=ρ₀(β(Q)²-k ² n_cl ²)^1/2.

Table 3. The mode normalization N of the HE₁₁ mode in the core and cladding

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Acknowledgment

This work is supported by the National Natural Science Foundation of China (No. 60425517) and the National Basic Research Program (973) of China (2007CB307003). The authors thank Yuhang Li and Zhe Ma for helpful discussions.

References and links

1. F. Ladouceur, “Roughness, inhomogeneity, and integrated optics,” J. Lightwave Technol. 15, 1020–1025 (1997). [CrossRef]

2. K. K. Lee, D. R. Lim, H. C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO₂ waveguide: experiments and model,” Appl. Phys. Lett. 77, 1617–1619 (2000). [CrossRef]

3. F. Grillot, L. Vivien, S. Laval, and E. Cassan, “Propagation Loss in Single-Mode Ultrasmall Square Silicon-on-Insulator Optical Waveguides,” J. Lightwave. Technol , 24, 891–896 (2006). [CrossRef]

4. J. Bures and R. Ghosh, “Power density of the evanescent field in the vicinity of a tapered fiber,” J. Opt. Soc. Am. A 16, 1992–1996 (1999). [CrossRef]

5. L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelengthdiameter silica wires for low-loss optical wave guiding,” Nature 426, 816–818 (2003). [CrossRef] [PubMed]

6. 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]

7. M. Sumetsky, Y. Dulashko, J. M. Fini, and A. Hale, “Optical microfiber loop resonator,” Appl. Phys. Lett. 86, 161108 (2005). [CrossRef]

8. X. S. Jiang, L. M. Tong, G. Vienne, X. Guo, Q. Yang, A. Tsao, and D. R. Yang, “Demonstration of optical microfiber knot resonators,” Appl. Phys. Lett. 88, 223501 (2006). [CrossRef]

9. L. M. Tong, J. Y. Lou, R. R. Gattass, S. L. He, X. W. Chen, L. Liu, and E. Mazur, “Assembly of silica nanowires on silica aerogels for microphotonics devices,” Nano. Lett. 5, 259–262 (2005). [CrossRef] [PubMed]

10. J. Y. Lou, L. M. Tong, and Z. Z. Ye, “Modeling of silica nanowires for optical sensing,” Opt. Express 13, 2135–2140 (2005). [CrossRef] [PubMed]

11. P. Polynkin, A. Polynkin, N. Peyghambarian, and M. Mansuripur, “Evanescent field-based optical fiber sensing device for measuring the refractive index of liquids in microfluidic channels,” Opt. Lett. 30, 1273–1275 (2005). [CrossRef] [PubMed]

12. J. Villatoro and D. Monzón-Hernández, “Fast detection of hydrogen with nano fiber tapers coated with ultra thin palladium layers,” Opt. Express 13, 5087–5092 (2005). [CrossRef] [PubMed]

13. W. Liang, Y. Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86, 151122 (2005). [CrossRef]

14. X. S. Jiang, Y. Chen, G. Vienne, and L. M. Tong, “All-fiber add-drop filters based on microfiber knot resonators,” Opt. Lett. 32, 1710–1712 (2007). [CrossRef] [PubMed]

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

16. S. Leon-Saval, T. Birks, W. Wadsworth, P. St. J. Russell, and M. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express 12, 2864–2869 (2004). [CrossRef] [PubMed]

17. R. R. Gattass, G. T. Svacha, L. M. Tong, and E. Mazur, “Supercontinuum generation in submicrometer diameter silica fibers,” Opt. Express 14, 9408–9414 (2006). [CrossRef] [PubMed]

18. L. Shi, X. Chen, H. Liu, Y. Chen, Z. Ye, W. Liao, and Y. Xia, “Fabrication of submicron-diameter silica fibers using electric strip heater,” Opt. Express 14, 5055–5060 (2006). [CrossRef] [PubMed]

19. V. I. Balykin, K. Hakuta, Fam Le Kien, J. Q. Liang, and M. Morinaga, “Atom trapping and guiding with a subwavelength-diameter optical fiber,” Phys. Rev. A 70, 011401 (2004). [CrossRef]

20. F. Kien, V. I. Balykin, and K. Hakuta, “Atom trap and waveguide using a two-color evanescent light field around a subwavelength-diameter optical fiber,” Phys. Rev. A 70, 063403 (2004). [CrossRef]

21. K. P. Nayak, P. N. Melentiev, M. Morinaga, F. L. Kien, V. I. Balykin, and K. Hakuta, “Optical nanofiber as an efficient tool for manipulating and probing atomic fluorescence,” Opt. Express 15, 5431–5438 (2007). [CrossRef] [PubMed]

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

23. 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]

24. 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]

25. D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48, 3187–3215 (1969).

26. D. Marcuse, “Mode conversion caused by diameter changes of round dielectric waveguide,” Bell Syst. Tech. J. 48, 3217–3233 (1969).

27. F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Opt. Quantum. Electron. 26, 977–986 (1994). [CrossRef]

28. E. G. Rawson, “Analysis of scattering from fiber waveguides with irregular core surface,” Appl. Opt 13, 2370–2377 (1974). [CrossRef] [PubMed]

29. D. Marcuse, “Radiation losses of the HE₁₁ mode of a fiber with sinusoidally perturbed core boundary,” Appl. Opt. 14, 3021–3025 (1975). [CrossRef] [PubMed]

30. M. Sumetsky, “How thin can a microfiber be and still guide light?,” Opt. Lett. 31, 870–872 (2006). Errata, Opt. Lett. 31, 3577 (2006). [CrossRef] [PubMed]

31. M. Sumetsky, “Thinnest optical waveguide: experimental test,” Opt. Lett. 32, 754–756 (2007). [CrossRef] [PubMed]

32. A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, New York, NY1983).

33. G. Brambilla, V. Finazzi, and D. J. Richardson, “Ultra-low-loss optical fiber nanotapers,” Opt. Express 12, 2258–2263 (2004). [CrossRef] [PubMed]

34. L. M. Tong, J. Y. Lou, Z. Z. Ye, G. T. Svacha, and E. Mazur, “Self-modulated taper drawing of silica nanowires,” Nanotechnology 16, 1445–1448 (2005). [CrossRef]

35. D. Marcuse, Theory of dielectric optical waveguides (Academic Press, New York, NY1974).

36. M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004). [CrossRef]

37. M. A. Foster, J.M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino R., and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B 81, 363–367 (2005). [CrossRef]

Component	Core	Cladding
e_r	$- \frac{a_{1} J_{0} (UR) + a_{2} J_{2} (UR)}{J_{1} (U)} f_{1} (ϕ)$	$- \frac{U}{W} \frac{a_{1} K_{0} (WR) - a_{2} K_{2} (WR)}{K_{1} (W)} f_{1} (ϕ)$
e_ϕ	$- \frac{a_{1} J_{0} (UR) - a_{2} J_{2} (UR)}{J_{1} (U)} g_{1} (ϕ)$	$- \frac{U}{W} \frac{a_{1} K_{0} (WR) + a_{2} K_{2} (WR)}{K_{1} (W)} g_{1} (ϕ)$
e_z	$- \frac{iU}{ρ_{0} β} \frac{J_{1} (UR)}{J_{1} (UR)} f_{1} (ϕ)$	$- \frac{iU}{ρ_{0} β} \frac{K_{1} (WR)}{K_{1} (W)} f_{1} (ϕ)$
$f_{v} (ϕ) = e^{i (βz - ωt)} {\begin{matrix} \cos (νϕ) \\ \sin (νϕ) \end{matrix}; g_{v} (ϕ) = e^{i (βz - ωt)} {\begin{matrix} - \sin (νϕ) even \\ \cos (νϕ) odd \end{matrix}$
$a_{1} = \frac{(F_{2} - 1)}{2}; a_{3} = \frac{(F_{1} - 1)}{2}; a_{5} = \frac{(F_{1} - 1 + 2 Δ)}{2}$
$a_{2} = \frac{(F_{2} + 1)}{2}; a_{4} = \frac{(F_{1} + 1)}{2}; a_{6} = \frac{(F_{1} + 1 - 2 Δ)}{2}$
$b_{1} = \frac{1}{2 U} {\frac{J_{ν - 1} (U)}{J_{ν} (U)} - \frac{J_{ν + 1} (U)}{J_{ν} (U)};} b_{2} = - \frac{1}{2 W} {\frac{K_{ν - 1} (W)}{K_{ν} (W)} + \frac{K_{ν + 1} (W)}{K_{ν} (W)}}$
$F_{1} = {(\frac{UW}{V})}^{2} \frac{b_{1} + (1 - 2 Δ) b_{2}}{ν}; F_{2} = {(\frac{V}{UW})}^{2} \frac{ν}{b_{1} + b_{2}}$

	ITE	ITM
Core	$e_{r} = \frac{i ρ_{0}}{2 U} {(β (Q) a_{1}^{s} - k b_{1}^{s} \sqrt{\frac{μ_{0}}{ε_{0}}}) J_{0} (UR) - (β (Q) a_{1}^{s} + k b_{1}^{s} \sqrt{\frac{μ_{0}}{ε_{0}}}) J_{2} (UR)} f_{1} (ϕ)$
	$e_{ϕ} = \frac{i ρ_{0}}{2 U} {(β (Q) a_{1}^{s} - k b_{1}^{s} \sqrt{\frac{μ_{0}}{ε_{0}}}) J_{0} (UR) + (β (Q) a_{1}^{s} + k b_{1}^{s} \sqrt{\frac{μ_{0}}{ε_{0}}}) J_{2} (UR)} g_{1} (ϕ)$
	$e_{z} = a_{1} J_{1} (UR) f_{1} (ϕ)$
Cladding	$e_{r} = \frac{i ρ_{0}^{2}}{Q^{2}} {\begin{matrix} β (Q) (\frac{Q}{2 ρ_{0}} c_{1}^{f} (J_{0} (QR) - J_{2} (QR)) + \frac{Q}{2 ρ_{0}} c_{1}^{s} (H_{0}^{(1)} (QR) - H_{2}^{(1)} (QR))) \\ - \sqrt{\frac{μ_{0}}{ε_{0}}} \frac{k}{r} (d_{1}^{f} J_{1} (QR) + d_{1}^{s} H_{1}^{(1)} (QR)) \end{matrix}} f_{1} (ϕ)$
	$e_{ϕ} = \frac{i ρ_{0}^{2}}{Q^{2}} {\begin{matrix} \frac{β (Q)}{r} (c_{1}^{f} J_{1} (QR) + c_{1}^{s} H_{1}^{(1)} (QR)) \\ - k \sqrt{\frac{μ_{0}}{ε_{0}}} \frac{Q}{2 ρ_{0}} (d_{1}^{f} (J_{0} (QR) - J_{2} (QR)) + d_{1}^{s} (H_{0}^{(1)} (QR) - H_{2}^{(1)} (QR))) \end{matrix}} g_{1} (ϕ)$
	$e_{z} = {c_{ν}^{f} J_{ν} (UR) + c_{ν}^{s} H_{ν}^{(1)} (UR)} f_{ν} (ϕ)$
$f_{1} (ϕ) = e^{i (β (Q) z - ωt)} {\begin{matrix} \cos (ϕ) \\ \sin (ϕ) \end{matrix}; g_{1} (ϕ) = e^{i (β (Q) z - ωt)} {\begin{matrix} - \sin (ϕ) even \\ \cos (ϕ) odd \end{matrix}$
a^s ₁	$- \frac{4}{π} \frac{β (Q)}{k n_{1}^{2}} \frac{1}{k ρ_{0}} \frac{V^{2}}{U^{2} Q^{3}} \frac{1}{J_{1} (U) H_{1}^{(1)} (Q) M_{1}}$	$- \frac{4}{π} {(\frac{n_{cl}}{n_{co}})}^{2} \frac{1}{kβ (Q) ρ_{0}} \frac{F_{1}}{J_{1} (U) H_{1}^{(1)} (Q) M_{1}}$
b^s ₁	$- \frac{4}{π} \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{1}{Qk ρ_{0}} \frac{G_{1}}{J_{1} (U) H_{1}^{(1)} (Q) M_{1}}$	$- \frac{4}{π} \frac{n_{cl}^{2}}{n_{co}^{2}} \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{1}{k ρ_{0}} \frac{V^{2}}{U^{2} Q^{3}} \frac{1}{J_{1} (U) H_{1}^{(1)} (Q) M_{1}}$
c^f ₁	0	$2 i \frac{Q}{β (Q) ρ_{0}}$
c^s ₁	$- \frac{4}{π} \frac{β (Q)}{k n_{co}^{2}} \frac{1}{k ρ_{0}} \frac{V^{2}}{U^{2} Q^{3}} \frac{1}{{H_{1}^{(1)} (Q)}^{2} M_{1}}$	$- 2 i \frac{Q}{β (Q) ρ_{0}} \frac{J_{1} (Q)}{H_{1}^{(1)} (Q)} \frac{A_{1}}{M_{1}}$
d^f ₁	$2 i \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{Q}{ρ_{0} k}$	0
d^s ₁	$- 2 i \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{Q}{ρ_{0} k} \frac{J_{1} (Q)}{H_{1}^{(1)} (Q)} \frac{B_{1}}{M_{1}}$	$- \frac{4}{π} \frac{n_{cl}^{2}}{n_{co}^{2}} \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{1}{k ρ_{0}} \frac{V^{2}}{U^{2} Q^{3}} \frac{1}{{H_{1}^{(1)} (Q)}^{2} M_{1}}$
N^r ₁	$\frac{2 π ρ_{0}^{2}}{Q} \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{β (Q)}{k}$	$\frac{2 π ρ_{0}^{2}}{Q} \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{k n_{cl}^{2}}{β (Q)}$
$F_{1} = \frac{{J'}_{1} (U)}{U J_{1} (U)} - \frac{H_{1}^{(1)'} (Q)}{Q H_{1}^{(1)} (Q)} G_{1} = \frac{{J'}_{1} (U)}{U J_{1} (U)} - \frac{n_{cl}^{2}}{n_{co}^{2}} \frac{H_{1}^{(1)'} (Q)}{Q H_{1}^{(1)} (Q)}$
${A_{1} = M}_{1} - \frac{2 i}{π} \frac{n_{cl}^{2}}{n_{co}^{2}} \frac{F_{1}}{Q^{2} H_{1}^{(1)} (Q) J_{1} (Q)} {B_{1} = M}_{1} - \frac{2 i}{π} \frac{G_{1}}{Q^{2} H_{1}^{(1)} (Q) J_{1} (Q)}$
$M_{1} = {(\frac{β}{k n_{cl}})}^{2} {(\frac{V}{UQ})}^{2} - F_{1} G_{1}$

N_co	$\frac{π ρ_{0}^{2}}{2} \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{k n_{1}^{2}}{β J_{1}^{2} (U)} [\begin{matrix} a_{1} a_{3} (J_{0}^{2} (U) - J_{1} (U) J_{- 1} (U)) \\ + a_{2} a_{4} (J_{2}^{2} (U) - J_{1} (U) J_{3} (U)) \end{matrix}]$
N_cl	$\frac{- π ρ_{0}^{2}}{2} \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{k n_{2}^{2}}{β K_{1}^{2} (W)} \frac{U^{2}}{W^{2}} [\begin{matrix} a_{1} a_{5} (K_{0}^{2} (W) - K_{1} (W) K_{- 1} (W)) \\ + a_{2} a_{6} (K_{2}^{2} (W) - K_{1} (W) K_{3} (W)) \end{matrix}]$

Component	Core	Cladding
e_r	$- \frac{a_{1} J_{0} (UR) + a_{2} J_{2} (UR)}{J_{1} (U)} f_{1} (ϕ)$	$- \frac{U}{W} \frac{a_{1} K_{0} (WR) - a_{2} K_{2} (WR)}{K_{1} (W)} f_{1} (ϕ)$
e_ϕ	$- \frac{a_{1} J_{0} (UR) - a_{2} J_{2} (UR)}{J_{1} (U)} g_{1} (ϕ)$	$- \frac{U}{W} \frac{a_{1} K_{0} (WR) + a_{2} K_{2} (WR)}{K_{1} (W)} g_{1} (ϕ)$
e_z	$- \frac{iU}{ρ_{0} β} \frac{J_{1} (UR)}{J_{1} (UR)} f_{1} (ϕ)$	$- \frac{iU}{ρ_{0} β} \frac{K_{1} (WR)}{K_{1} (W)} f_{1} (ϕ)$
$f_{v} (ϕ) = e^{i (βz - ωt)} {\begin{matrix} \cos (νϕ) \\ \sin (νϕ) \end{matrix}; g_{v} (ϕ) = e^{i (βz - ωt)} {\begin{matrix} - \sin (νϕ) even \\ \cos (νϕ) odd \end{matrix}$
$a_{1} = \frac{(F_{2} - 1)}{2}; a_{3} = \frac{(F_{1} - 1)}{2}; a_{5} = \frac{(F_{1} - 1 + 2 Δ)}{2}$
$a_{2} = \frac{(F_{2} + 1)}{2}; a_{4} = \frac{(F_{1} + 1)}{2}; a_{6} = \frac{(F_{1} + 1 - 2 Δ)}{2}$
$b_{1} = \frac{1}{2 U} {\frac{J_{ν - 1} (U)}{J_{ν} (U)} - \frac{J_{ν + 1} (U)}{J_{ν} (U)};} b_{2} = - \frac{1}{2 W} {\frac{K_{ν - 1} (W)}{K_{ν} (W)} + \frac{K_{ν + 1} (W)}{K_{ν} (W)}}$
$F_{1} = {(\frac{UW}{V})}^{2} \frac{b_{1} + (1 - 2 Δ) b_{2}}{ν}; F_{2} = {(\frac{V}{UW})}^{2} \frac{ν}{b_{1} + b_{2}}$

	ITE	ITM
Core	$e_{r} = \frac{i ρ_{0}}{2 U} {(β (Q) a_{1}^{s} - k b_{1}^{s} \sqrt{\frac{μ_{0}}{ε_{0}}}) J_{0} (UR) - (β (Q) a_{1}^{s} + k b_{1}^{s} \sqrt{\frac{μ_{0}}{ε_{0}}}) J_{2} (UR)} f_{1} (ϕ)$
	$e_{ϕ} = \frac{i ρ_{0}}{2 U} {(β (Q) a_{1}^{s} - k b_{1}^{s} \sqrt{\frac{μ_{0}}{ε_{0}}}) J_{0} (UR) + (β (Q) a_{1}^{s} + k b_{1}^{s} \sqrt{\frac{μ_{0}}{ε_{0}}}) J_{2} (UR)} g_{1} (ϕ)$
	$e_{z} = a_{1} J_{1} (UR) f_{1} (ϕ)$
Cladding	$e_{r} = \frac{i ρ_{0}^{2}}{Q^{2}} {\begin{matrix} β (Q) (\frac{Q}{2 ρ_{0}} c_{1}^{f} (J_{0} (QR) - J_{2} (QR)) + \frac{Q}{2 ρ_{0}} c_{1}^{s} (H_{0}^{(1)} (QR) - H_{2}^{(1)} (QR))) \\ - \sqrt{\frac{μ_{0}}{ε_{0}}} \frac{k}{r} (d_{1}^{f} J_{1} (QR) + d_{1}^{s} H_{1}^{(1)} (QR)) \end{matrix}} f_{1} (ϕ)$
	$e_{ϕ} = \frac{i ρ_{0}^{2}}{Q^{2}} {\begin{matrix} \frac{β (Q)}{r} (c_{1}^{f} J_{1} (QR) + c_{1}^{s} H_{1}^{(1)} (QR)) \\ - k \sqrt{\frac{μ_{0}}{ε_{0}}} \frac{Q}{2 ρ_{0}} (d_{1}^{f} (J_{0} (QR) - J_{2} (QR)) + d_{1}^{s} (H_{0}^{(1)} (QR) - H_{2}^{(1)} (QR))) \end{matrix}} g_{1} (ϕ)$
	$e_{z} = {c_{ν}^{f} J_{ν} (UR) + c_{ν}^{s} H_{ν}^{(1)} (UR)} f_{ν} (ϕ)$
$f_{1} (ϕ) = e^{i (β (Q) z - ωt)} {\begin{matrix} \cos (ϕ) \\ \sin (ϕ) \end{matrix}; g_{1} (ϕ) = e^{i (β (Q) z - ωt)} {\begin{matrix} - \sin (ϕ) even \\ \cos (ϕ) odd \end{matrix}$
a^s ₁	$- \frac{4}{π} \frac{β (Q)}{k n_{1}^{2}} \frac{1}{k ρ_{0}} \frac{V^{2}}{U^{2} Q^{3}} \frac{1}{J_{1} (U) H_{1}^{(1)} (Q) M_{1}}$	$- \frac{4}{π} {(\frac{n_{cl}}{n_{co}})}^{2} \frac{1}{kβ (Q) ρ_{0}} \frac{F_{1}}{J_{1} (U) H_{1}^{(1)} (Q) M_{1}}$
b^s ₁	$- \frac{4}{π} \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{1}{Qk ρ_{0}} \frac{G_{1}}{J_{1} (U) H_{1}^{(1)} (Q) M_{1}}$	$- \frac{4}{π} \frac{n_{cl}^{2}}{n_{co}^{2}} \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{1}{k ρ_{0}} \frac{V^{2}}{U^{2} Q^{3}} \frac{1}{J_{1} (U) H_{1}^{(1)} (Q) M_{1}}$
c^f ₁	0	$2 i \frac{Q}{β (Q) ρ_{0}}$
c^s ₁	$- \frac{4}{π} \frac{β (Q)}{k n_{co}^{2}} \frac{1}{k ρ_{0}} \frac{V^{2}}{U^{2} Q^{3}} \frac{1}{{H_{1}^{(1)} (Q)}^{2} M_{1}}$	$- 2 i \frac{Q}{β (Q) ρ_{0}} \frac{J_{1} (Q)}{H_{1}^{(1)} (Q)} \frac{A_{1}}{M_{1}}$
d^f ₁	$2 i \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{Q}{ρ_{0} k}$	0
d^s ₁	$- 2 i \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{Q}{ρ_{0} k} \frac{J_{1} (Q)}{H_{1}^{(1)} (Q)} \frac{B_{1}}{M_{1}}$	$- \frac{4}{π} \frac{n_{cl}^{2}}{n_{co}^{2}} \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{1}{k ρ_{0}} \frac{V^{2}}{U^{2} Q^{3}} \frac{1}{{H_{1}^{(1)} (Q)}^{2} M_{1}}$
N^r ₁	$\frac{2 π ρ_{0}^{2}}{Q} \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{β (Q)}{k}$	$\frac{2 π ρ_{0}^{2}}{Q} \sqrt{\frac{ε_{0}}{μ_{0}}} \frac{k n_{cl}^{2}}{β (Q)}$
$F_{1} = \frac{{J'}_{1} (U)}{U J_{1} (U)} - \frac{H_{1}^{(1)'} (Q)}{Q H_{1}^{(1)} (Q)} G_{1} = \frac{{J'}_{1} (U)}{U J_{1} (U)} - \frac{n_{cl}^{2}}{n_{co}^{2}} \frac{H_{1}^{(1)'} (Q)}{Q H_{1}^{(1)} (Q)}$
${A_{1} = M}_{1} - \frac{2 i}{π} \frac{n_{cl}^{2}}{n_{co}^{2}} \frac{F_{1}}{Q^{2} H_{1}^{(1)} (Q) J_{1} (Q)} {B_{1} = M}_{1} - \frac{2 i}{π} \frac{G_{1}}{Q^{2} H_{1}^{(1)} (Q) J_{1} (Q)}$
$M_{1} = {(\frac{β}{k n_{cl}})}^{2} {(\frac{V}{UQ})}^{2} - F_{1} G_{1}$

Roughness-induced radiation losses in optical micro or nanofibers

Abstract

1. Introduction

2. Theoretical analysis

3. Numerical results and discussions

4. Conclusion

Appendix

Acknowledgment

References and links

Cited By

Figures (7)

Tables (3)

Equations (17)

Optics Express