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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 21 — Oct. 17, 2007
  • pp: 13805–13816
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Roughness-induced radiation losses in optical micro or nanofibers

Gaoye Zhai and Limin Tong  »View Author Affiliations


Optics Express, Vol. 15, Issue 21, pp. 13805-13816 (2007)
http://dx.doi.org/10.1364/OE.15.013805


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Abstract

Roughness-induced radiation losses in optical micro or nanofibers (MNFs) are investigated using an induced-current model. Loss coefficients of silica, phosphate, tellurite and silicon MNFs with sinusoidal deformations on their surfaces are numerically calculated with respect to typical parameters of the guiding system. Interesting phenomena such as the existence of the loss minima at specific perturbation periods are observed. Results presented in this work may be generalized to all kinds of surface deformation and may provide useful guidelines for both estimating and tailoring waveguiding properties of MNFs.

© 2007 Optical Society of America

1. Introduction

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

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

]

J=iε0μ0k(n¯2n2)E,
(1)

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 nco and ncl, respectively.

For taper-drawn MNFs, the sidewall roughness is usually lower than 0.5 nm [5

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

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]

, 34

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]

], 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

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

].

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

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

]

J=iε0μ0k(n¯2n2)E¯=iε0μ0k(n¯2n2)a¯ 1e1exp(iβz),
(2)

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)=ρ0+ξsinωz,
(3)

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 ξρ0, 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

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

]

n¯ 2n2(nco2ncl2)ξsin(ωz)δ¯ (rρ0),
(4)

where δ̅(r-ρ 0) is the Dirac delta function, nco and ncl 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=iε0μ0k(nco2ncl2)δ¯(rρ0)ξ(sinωz)a¯1e1exp(iβz).
(5)

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

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

]:

For forward-propagating radiation mode (i.e., zL)

ajr(Q)=14Njr(Q)0LAej*r(Q)Jexp((Q)z)dAdz
=ika¯4Njr(Q)(ε0μ0)12(nco2ncl2)
×0LAξsin(ωz)δ¯ (rρ0)e1ej*r(Q)exp(i(β1β(Q))z)dAdz,
(6a)

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

ajr(Q)=14Njr(Q)0LAej*r(Q)Jexp((Q)z)dAdz
=ika¯4Njr(Q)(ε0μ0)12(nco2ncl2)
×0LAξsin(ωz)δ¯(rρ0)e1ej*r(Q)exp(i(β1+β(Q))z)dAdz,
(6b)

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, Njr(Q)=12Aejr(Q)×hj*r(Q)ẑdA 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

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

]

Erad=a1r(ITE)(Q)e1r(ITE)(Q)exp((Q)z)+a1r(ITE)(Q)e1r(ITE)(Q)exp((Q)z)
+a1r(ITM)(Q)e1r(ITM)(Q)exp((Q)z)+a1r(ITM)(Q)e1r(ITM)(Q)exp((Q)z),
(7)

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

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

]

Prad=12Re{AErad×Hrad*ẑdA}
=0ncl(a1r(ITE)(Q)2+a1r(ITE)(Q)2)N1r(ITE)(Q)dQ
+0ncl(a1r(ITM)(Q)2+a1r(ITM)(Q)2)N1r(ITM)(Q)dQ.
(8)

The loss coefficient is then obtained as

α=10lg(P¯ PradP¯ )/L,
(9)

where p̅ = |E̅|2 = |a̅1|2 N is the incident power of the MNF, and N=12Ae1×h1*ẑdA 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, nco=1.44 at 1550-nm wavelength), phosphate (ρ0=350 nm, nco=1.54 at 1550-nm wavelength) and tellurite (ρ0=300 nm, nco=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, nco=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 (αmin). This means that, when the period of the perturbation goes to some specific values (denoted as Γmin), 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

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

, 35

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

]), 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., αsilicon>αtellurite >αphosphate >αsilica (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 Γmin required for the first αmin of a higher index MNF (e.g., the 350-nm-radius phosphate MNF with nco=1.54 at 1550-nm wavelength) is smaller than that of a lower index MNF (e.g., the 350-nm-radius silica MNF with nco=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.

We then investigate the influence of the amplitude of the surface roughness ξ on the Γmin. 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 Γmin 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.

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.

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 Γmin, 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.

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 Γmin 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.

In addition, since supercontinuum generation in MNF is of great interest [16

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

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]

, 36

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

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]

], 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.

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

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

]

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

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Where R=r/ρ0, ρ0 is the core radius, U=ρ0(k 2 nco 22)1/2, V=ρ0(k 2 nco 2-k 2 ncl 2)1/2, W=ρ0(β 2-k 2 ncl 2)1/2, ∆=(nco 2-ncl 2)/2nco 2, k=2π/λ, core and cladding refractive indices are nco and ncl.

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

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where U=ρ0(k 2 nco 2-β(Q)2)1/2, Q=ρ0(β(Q)2-k 2 ncl 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.

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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]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(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: May 3, 2007
Revised Manuscript: August 19, 2007
Manuscript Accepted: August 22, 2007
Published: October 5, 2007

Citation
Gaoye Zhai and Limin Tong, "Roughness-induced radiation losses in optical micro or nanofibers," Opt. Express 15, 13805-13816 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-13805


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References

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