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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 2 — Jan. 16, 2012
  • pp: 1790–1797
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Importance of residual stresses in the Brillouin gain spectrum of single mode optical fibers

Y. Sikali Mamdem, E. Burov, L-A. de Montmorillon, Y. Jaouën, G. Moreau, R. Gabet, and F. Taillade  »View Author Affiliations


Optics Express, Vol. 20, Issue 2, pp. 1790-1797 (2012)
http://dx.doi.org/10.1364/OE.20.001790


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Abstract

Residual stresses inside optical fibers can impact significantly on Brillouin spectrum properties. We have analyzed the importance of internal stresses on the Brillouin Gain Spectrum (BGS) for a conventional G.652 fiber and compared modeling results to measurements. Then the residual internal stresses have been investigated for a set of trench-assisted fibers: fibers are coming from a single preform with different draw tensions. Numerical modeling based on measured internal stresses profiles are compared with corresponding BGS experimental results. Clearly, Brillouin spectrum is shifted linearly versus draw tension with a coefficient of −20MHz/100g and its linewidth increases.

© 2012 OSA

1. Introduction

Fabrication of preforms and optical fibers always generates intrinsic residual stresses. The difference of doping compositions between core and cladding results in a mismatch of thermal expansion coefficients and viscosities. Thermal internal stresses are created during the cooling from fusion temperature to room temperature. Moreover, drawing conditions can modify significantly the internal frozen-stresses [1

1. A. D. Yablon, “Optical and mechanical effects of frozen-in stresses and strains in optical fibers,” IEEE J. Sel. Top. Quantum Electron. 10(2), 300–311 (2004). [CrossRef]

]. These stresses introduce changes in the refractive optical index profile and the acoustic velocity profile. So, Brillouin frequency shift is modified. While the dependence of the Brillouin Gain Spectrum (BGS) with the doping composition profile is already well-known [2

2. A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express 13(14), 5338–5346 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-14-5338. [CrossRef] [PubMed]

,3

3. Y. Koyamada, S. Sato, S. Nakamura, H. Sotobayashi, and W. Chujo, “Simulating and designing Brillouin gain spectrum in single-mode fibers,” J. Lightwave Technol. 22(2), 631–639 (2004). [CrossRef]

], only recently the influence of internal frozen- stresses in the BGS has been investigated [4

4. W. Zou, Z. He, A. D. Yablon, and K. Hotate, “Dependence of Brillouin frequency shift in optical fibers on draw-induced residual elastic and inelastic strains,” IEEE Photon. Technol. Lett. 19(18), 1389–1391 (2007). [CrossRef]

,5

5. W. Zou, Z. He, and K. Hotate, “Two-dimensional finite-element modal analysis of Brillouin gain spectra in optical fibers,” IEEE Photon. Technol. Lett. 18(23), 2487–2489 (2006). [CrossRef]

].

In this paper, we propose a 2D-FEM model to predict BGS of optical fibers considering the corresponding measured frozen-stresses profiles. Optical and acoustic properties are calculated from measured doping profiles of fibers [6

6. Y. Sikali Mamdem, X. Pheron, F. Taillade, Y. Jaouën, R. Gabet, V. Lanticq, G. Moreau, A. Boukenter, Y. Ouerdane, S. Lesoille, and J. Bertrand, “Two-dimensional FEM analysis of Brillouin Gain Spectra in acoustic guiding and antiguiding single mode optical fibers,” in Proceedings of COMSOL Multiphysics Conference (session Acoustic II, Paris, 2010), 111–124.

,7

7. Y. Sikali Mamdem, E. Burov, L.-A de Montmorillon, F. Taillade, Y. Jaouën, G. Moreau, and R. Gabet, “Importance of residual stresses in the Brillouin gain spectrum of single mode optical fibers,” ECOC 2011, paper We.10.P1.16, Geneva, Sept. 2011.

]. This model is able to predict the BGS’s behaviors of any optical fiber with mono- or multi-doping composition, considering any residual stresses profile. In order to validate the inclusion of residual stresses in our model, we consider fibers with different stresses profiles and we compare our Brillouin gain simulations to experimental measurements. We present the BGS of a conventional G.652 step-index fiber including residual internal stresses profile. Then, we analyze a set of G.657 single-trench-assisted Bend Insensitive SMFs (BI-SMF). All these fibers are coming from a single preform but are drawn with different tensions. The calculated BGS spectra show very good agreement with corresponding measurements. These results confirm the influence of internal composition and residual frozen-in stresses on the BGS properties.

2. Theoretical background and 2D-FEM model of simulation

BGS calculation requires a rigorous determination of the acoustic-optical interaction (i.e. overlap integral between optical and acoustic modes). The modeling of optical and longitudinal acoustic waves was performed, neglecting the contribution of torsional modes in the BGS [8

8. A. Safaai-Jazi, C.-K. Jen, and G. W. Farnell, “Analysis of weakly guiding fiber acoustic waveguide,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 33(1), 59–68 (1986). [CrossRef] [PubMed]

]. The properties of the optical mode E(x,y) and the longitudinal acoustic mth-order L0m mode um(x,y)are determined by solving the 2D scalar-wave propagation Eqs. (1) and (2) respectively [6

6. Y. Sikali Mamdem, X. Pheron, F. Taillade, Y. Jaouën, R. Gabet, V. Lanticq, G. Moreau, A. Boukenter, Y. Ouerdane, S. Lesoille, and J. Bertrand, “Two-dimensional FEM analysis of Brillouin Gain Spectra in acoustic guiding and antiguiding single mode optical fibers,” in Proceedings of COMSOL Multiphysics Conference (session Acoustic II, Paris, 2010), 111–124.

,9

9. L. Tartara, C. Codemard, J.-N. Maran, R. Cherif, and M. Zghal, “Full modal analysis of the Brillouin gain spectrum of an optical fiber,” Opt. Commun. 282(12), 2431–2436 (2009). [CrossRef]

,10

10. S. Dasgupta, F. Poletti, S. Liu, P. Petropoulos, D. J. Richardson, L. Grüner-Nielsen, and S. Herstrøm, “Modeling Brillouin Gain Spectrum of solid and microstructured optical fibers using a finite element method,” J. Lightwave Technol. 29(1), 22–30 (2011). [CrossRef]

]:
Δt2E+(2πλ0)2(n2neff2)E=0
(1)
Δt2um+(Ωm2VL2βacoust2)um=0
(2)
where Δt is the transverse Laplacian operator, E(x,y) is the spatial distribution of the optical field, neff is the effective index of the fundamental LP01 optical mode, βacoust is the acoustic propagation constant, um(x,y) is the longitudinal displacement field of the L0m acoustic mode and Ωm is the acoustic Brillouin resonance angular frequency. Assuming that Ωm is much smaller than the optical frequency, the phase matching (i.e. Bragg condition) with the optical wave leads to the relation βacoust=2βopt where βopt=2πneff/λ is the propagation constant of the optical mode. The Brillouin frequency shift is then Ωm=4πneffVeff/λwhere Veffis the effective velocity of the longitudinal acoustic mode.

It is well-known that the refractive index n and the longitudinal acoustic velocity VL vary on the cross-section of the fiber according to the type and concentration of doping materials [11

11. C. A. S. de Oliveira, C. K. Jen, A. Shang, and C. Saravanos, “Stimulated Brillouin scattering in cascaded fibers of different Brillouin frequency shifts,” J. Opt. Soc. Am. B 10(6), 969–971 (1993). [CrossRef]

]. The refractive index profile is measured on optical fibers with a near-field measurement technique (EXFO NR-9200). The relative contributions of the different dopants are measured by chemical analysis on the preform. The fractional dependence for unit doping concentration on the optical and acoustic properties of pure silica is listed in Table 1

Table 1. Influence of Doping Concentrations on Optical and Acoustic Properties of Fibers Parameters [10]

table-icon
View This Table
for used dopants. Propagation equations were solved using COMSOL Multiphysics, a FEM-2D commercial solver. The model and the parameters of resolution are precisely described in [6

6. Y. Sikali Mamdem, X. Pheron, F. Taillade, Y. Jaouën, R. Gabet, V. Lanticq, G. Moreau, A. Boukenter, Y. Ouerdane, S. Lesoille, and J. Bertrand, “Two-dimensional FEM analysis of Brillouin Gain Spectra in acoustic guiding and antiguiding single mode optical fibers,” in Proceedings of COMSOL Multiphysics Conference (session Acoustic II, Paris, 2010), 111–124.

].Due to the exponential decay of acoustic wave magnitude, the spontaneous Brillouin spectrum of each acoustic mode has a Lorentzian shape. As each acoustic mode contribution adds up in an incoherent way, the BGS is then computed by adding Lorentzian curves centered at each mode’s Brillouin frequency shift νBm=Ωm/2π weighted by the acousto-optic overlap integral Imao [2

2. A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express 13(14), 5338–5346 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-14-5338. [CrossRef] [PubMed]

] (see Eq. (3) and Eq. (4)). In this work, the full width at half maximum (FWHM), noted Γ, of Lorentzian curves is assumed to be identical for all the modes and depends of the fiber-under-test.

BGS(ν)=mImao(Γ/2)2(Γ/2)2+(ννBm)2
(3)
Imao=(|E|2umdxdy)2|E|4dxdy.|um|2dxdy
(4)

The residual internal stresses in an optical fiber not only induce changes in optical refractive index [12

12. G. W. Scherer, “Stress-induced index profile distortion in optical waveguides,” Appl. Opt. 19(12), 2000–2006 (1980). [CrossRef] [PubMed]

] but also modify the acoustic velocity behavior. The acoustic velocity of a homogenous and isotropic infinite material subjected to a uniaxial longitudinal stresses can be expressed as [13

13. S. Chaki and G. Bourse, “Guided ultrasonic waves for non-destructive monitoring of the stress levels in prestressed steel strands,” Ultrasonics 49(2), 162–171 (2009). [CrossRef] [PubMed]

]:
VL=VL0(1+KLσ)
(5)
where VL0(x,y)is the longitudinal acoustic velocity profile on the fiber cross-section without stresses, σ(x,y)is the total axial stresses (both thermal and draw-induced stresses), varying on the fiber cross-section. The parameterKLis the acousto-elastic coefficient and represents the velocity variation of the acoustic wave with stresses. Second-order expression ofKL (Eq. (6)) can be found in [13

13. S. Chaki and G. Bourse, “Guided ultrasonic waves for non-destructive monitoring of the stress levels in prestressed steel strands,” Ultrasonics 49(2), 162–171 (2009). [CrossRef] [PubMed]

]. The Lame’s constants λ(Eq. (7)) andμ(Eq. (8)) are calculated from the profiles of the stress-free longitudinal and shear wave velocities VL0and VT0 respectively.
KL=12(λ+2μ)(3λ+2μ)(λ+μμ(4λ+10μ)+λ)
(6)
λ=ρ[(VL0)22(VT0)2]
(7)
μ=ρ(VT0)2
(8)
The stress free velocities VL0and VT0are computed considering the doping composition [11

11. C. A. S. de Oliveira, C. K. Jen, A. Shang, and C. Saravanos, “Stimulated Brillouin scattering in cascaded fibers of different Brillouin frequency shifts,” J. Opt. Soc. Am. B 10(6), 969–971 (1993). [CrossRef]

] and the acoustic velocities of pure fused silica VLsilicaand VTsilica. Note, that glass material’s has been measured on used preforms in the [5940m/s - 6000m/s] range depending of the fictive temperature during the drawing process [14

14. R. Le Parc, “Diffusion de rayonnement et relaxation structurale dans les verres de silice et les préformes de fibres optiques,” PhD thesis (Claude Bernard University, Lyon-1, 2002).

]. The typical value of 3750m/s has been taken for VTsilica [15

15. F. Terki, C. Levelut, M. Boissier, and J. Pelous, “Low-frequency dynamics and medium-range order in vitreous silica,” Phys. Rev. B Condens. Matter 53(5), 2411–2418 (1996). [CrossRef] [PubMed]

]. Lame’s silica constants are typically λ = 16GPa and μ = 31GPa, that results KL = 3.4 10−11Pa−1.

The calculated BGS is compared to experimental results that have been obtained using a ~1km fiber. The BGS measurement has been performed using the well-known self-heterodyne technique [16

16. V. Lanticq, S. Jiang, R. Gabet, Y. Jaouën, F. Taillade, G. Moreau, and G. P. Agrawal, “Self-referenced and single-ended method to measure Brillouin gain in monomode optical fibers,” Opt. Lett. 34(7), 1018–1020 (2009). [CrossRef] [PubMed]

] (see experimental set-up in Fig. 1
Fig. 1 Experimental set-up for the measurement of Brillouin spectrum.
). A 1560nm DFB laser is used as input signal. The heterodyne detection of both the Brillouin frequency shift back-propagated Stokes wave in the fiber-under-test and the injected input signal allows to direct measurement of the BGS. A polarization scrambler is used for polarization-insensitive measurement. The laser linewidth is of the order of 1MHz, compared to 30-40MHz for the acoustic modes, so the detected electrical spectrum corresponds directly to the spontaneous BGS.

3. Validation of stresses profile impact for a step-index fiber

4. Draw tension impact for trench-assisted profile

Single-trench-assisted BI-SMFs fibers [17

17. L.-A. de Montmorillon, P. Matthijsse, F. Gooijer, D. Molin, F. Achten, X. Meersseman, and C. Legrand, “Next Generation SMF with Reduced Bend Sensitivity for FTTH Networks,” in Proceedings of ECOC Conference (Cannes, France, paper Mo.3.3.2, 2006).

] corresponding to the recent G.657 fibers designed for small bending radius applications. These properties let BI-SMFs be also interesting for sensing applications especially for distributed stress and temperature sensors based on Brillouin effect. For our knowledge, the impact of the trench to the Brillouin shift had never been evaluated. A schematic trench-assisted step index profile shape is represented in Fig. 5
Fig. 5 Trench-assisted step index structure.
.

During fabrication process, with or without applied external forces, stresses develop inside the fibers during cooling at room temperature. The draw-induced stresses are due to a difference of the viscoelastic properties in the core and the cladding respectively [18

18. P. K. Bachmann, W. Hermann, H. Wehr, and D. U. Wiechert, “Stress in optical waveguides. 2: Fibers,” Appl. Opt. 26(7), 1175–1182 (1987). [CrossRef] [PubMed]

]. So the amplitude of residual stresses depends on the doping composition of fibers and the drawing tensions. In order to investigate the impact of draw-induced stresses on the BGS, we used optical fibers coming from a single preform (i.e. same doping composition and geometric parameters), with same melting point, but drawn with different tensions.

The draw tensions are going from very low values (20g, 40g) to very high values (120g, 160g). The internal stresses profiles have been measured based on a polarimetric technique [19

19. F. Dürr, H. G. Limberger, R. P. Salathé, F. Hindle, M. Douay, E. Fertein, and C. Przygodzki, “Tomographic measurement of femtosecond-laser induced stress changes in optical fibers,” Appl. Phys. Lett. 84(24), 4983–4985 (2004). [CrossRef]

]. The core stresses have negative values, corresponding to a compressive state, and decrease with increasing draw tensions. Acoustic velocities profiles have been calculated according to the measured index and stresses profiles.

An example of stresses impact on the BGS is plotted in Fig. 6(a)
Fig. 6 BGS comparisons: (a) Measured and simulated BGS with and without stress of fiber drawn with 90g, (b) Measured and simulated (with stress) BGS for different values of draw tension.
. As for the step-index G.652 fiber, the L01 mode has the most significant contribution to BGS in the trench-assisted fibers. Pure Silica used for the preform has a refractive index n = 1,444 and a velocity = 5990m/s. The BGS are plotted in Fig. 6(b) for three different tensions. A very good agreement between BGS calculation and measurements is obtained when the stresses are taken into account. Brillouin spectra are moving towards decreasing frequencies when draw tension increases, so when absolute value of core stresses increase. The evolution of L01 mode Brillouin frequency shift is plotted in Fig. 7(a)
Fig. 7 Evolution with draw tension of: (a) Measured Brillouin frequency shift versus draw tension, (b) BGS FWHM versus draw tension.
according to draw tension. The results are given with an uncertainty of 2MHz for the Brillouin frequency shifts. We have also plotted the evolution of the ratio R corresponding to core stresses values (σ(r)|r = 0), the highest draw tension (160g) is chosen as reference (Eq. (9)). We observe that in absolute value, the core stresses increase with draw tension. Due to the uncertainty in the draw tension ( ± 10g) and the difficulty of the drawing process stabilization at low tension, we have a quite significant scattering in the data. Moreover the uncertainty of stresses measurement, in the order of 1MPa, contributes to this scattering. Amplitude of the core stresses varies linearly with draw tension with a slope of (−0.48 ± 0.08)MPa/g.

R=σ(r)|r=0σ(r)|r=0160g
(9)

The obtained experimental results show that the Brillouin shift decreases with draw tension. A linear approximation give a −20MHz/100g dependence (accuracy ± 2.4MHz/100g.). Note that this value is about half the value found in [4

4. W. Zou, Z. He, A. D. Yablon, and K. Hotate, “Dependence of Brillouin frequency shift in optical fibers on draw-induced residual elastic and inelastic strains,” IEEE Photon. Technol. Lett. 19(18), 1389–1391 (2007). [CrossRef]

]. This difference can be explained by the higher GeO2 concentration and the multi-doping composition of the fibers. Moreover, the BI-SMF index profiles are different from [4

4. W. Zou, Z. He, A. D. Yablon, and K. Hotate, “Dependence of Brillouin frequency shift in optical fibers on draw-induced residual elastic and inelastic strains,” IEEE Photon. Technol. Lett. 19(18), 1389–1391 (2007). [CrossRef]

].

The FWHM Brillouin linewidth of L01 mode is shown in Fig. 7(b) for the different draw tensions. The measurement of Brillouin linewidth is estimated to ~1MHz. We have observed that the Brillouin linewidth increases significantly with draw tension. It can be probably explained by the structural order/disorder changes in the silica glass due to the increase of stresses level.

6. Conclusion

References and links

1.

A. D. Yablon, “Optical and mechanical effects of frozen-in stresses and strains in optical fibers,” IEEE J. Sel. Top. Quantum Electron. 10(2), 300–311 (2004). [CrossRef]

2.

A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express 13(14), 5338–5346 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-14-5338. [CrossRef] [PubMed]

3.

Y. Koyamada, S. Sato, S. Nakamura, H. Sotobayashi, and W. Chujo, “Simulating and designing Brillouin gain spectrum in single-mode fibers,” J. Lightwave Technol. 22(2), 631–639 (2004). [CrossRef]

4.

W. Zou, Z. He, A. D. Yablon, and K. Hotate, “Dependence of Brillouin frequency shift in optical fibers on draw-induced residual elastic and inelastic strains,” IEEE Photon. Technol. Lett. 19(18), 1389–1391 (2007). [CrossRef]

5.

W. Zou, Z. He, and K. Hotate, “Two-dimensional finite-element modal analysis of Brillouin gain spectra in optical fibers,” IEEE Photon. Technol. Lett. 18(23), 2487–2489 (2006). [CrossRef]

6.

Y. Sikali Mamdem, X. Pheron, F. Taillade, Y. Jaouën, R. Gabet, V. Lanticq, G. Moreau, A. Boukenter, Y. Ouerdane, S. Lesoille, and J. Bertrand, “Two-dimensional FEM analysis of Brillouin Gain Spectra in acoustic guiding and antiguiding single mode optical fibers,” in Proceedings of COMSOL Multiphysics Conference (session Acoustic II, Paris, 2010), 111–124.

7.

Y. Sikali Mamdem, E. Burov, L.-A de Montmorillon, F. Taillade, Y. Jaouën, G. Moreau, and R. Gabet, “Importance of residual stresses in the Brillouin gain spectrum of single mode optical fibers,” ECOC 2011, paper We.10.P1.16, Geneva, Sept. 2011.

8.

A. Safaai-Jazi, C.-K. Jen, and G. W. Farnell, “Analysis of weakly guiding fiber acoustic waveguide,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 33(1), 59–68 (1986). [CrossRef] [PubMed]

9.

L. Tartara, C. Codemard, J.-N. Maran, R. Cherif, and M. Zghal, “Full modal analysis of the Brillouin gain spectrum of an optical fiber,” Opt. Commun. 282(12), 2431–2436 (2009). [CrossRef]

10.

S. Dasgupta, F. Poletti, S. Liu, P. Petropoulos, D. J. Richardson, L. Grüner-Nielsen, and S. Herstrøm, “Modeling Brillouin Gain Spectrum of solid and microstructured optical fibers using a finite element method,” J. Lightwave Technol. 29(1), 22–30 (2011). [CrossRef]

11.

C. A. S. de Oliveira, C. K. Jen, A. Shang, and C. Saravanos, “Stimulated Brillouin scattering in cascaded fibers of different Brillouin frequency shifts,” J. Opt. Soc. Am. B 10(6), 969–971 (1993). [CrossRef]

12.

G. W. Scherer, “Stress-induced index profile distortion in optical waveguides,” Appl. Opt. 19(12), 2000–2006 (1980). [CrossRef] [PubMed]

13.

S. Chaki and G. Bourse, “Guided ultrasonic waves for non-destructive monitoring of the stress levels in prestressed steel strands,” Ultrasonics 49(2), 162–171 (2009). [CrossRef] [PubMed]

14.

R. Le Parc, “Diffusion de rayonnement et relaxation structurale dans les verres de silice et les préformes de fibres optiques,” PhD thesis (Claude Bernard University, Lyon-1, 2002).

15.

F. Terki, C. Levelut, M. Boissier, and J. Pelous, “Low-frequency dynamics and medium-range order in vitreous silica,” Phys. Rev. B Condens. Matter 53(5), 2411–2418 (1996). [CrossRef] [PubMed]

16.

V. Lanticq, S. Jiang, R. Gabet, Y. Jaouën, F. Taillade, G. Moreau, and G. P. Agrawal, “Self-referenced and single-ended method to measure Brillouin gain in monomode optical fibers,” Opt. Lett. 34(7), 1018–1020 (2009). [CrossRef] [PubMed]

17.

L.-A. de Montmorillon, P. Matthijsse, F. Gooijer, D. Molin, F. Achten, X. Meersseman, and C. Legrand, “Next Generation SMF with Reduced Bend Sensitivity for FTTH Networks,” in Proceedings of ECOC Conference (Cannes, France, paper Mo.3.3.2, 2006).

18.

P. K. Bachmann, W. Hermann, H. Wehr, and D. U. Wiechert, “Stress in optical waveguides. 2: Fibers,” Appl. Opt. 26(7), 1175–1182 (1987). [CrossRef] [PubMed]

19.

F. Dürr, H. G. Limberger, R. P. Salathé, F. Hindle, M. Douay, E. Fertein, and C. Przygodzki, “Tomographic measurement of femtosecond-laser induced stress changes in optical fibers,” Appl. Phys. Lett. 84(24), 4983–4985 (2004). [CrossRef]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(290.5830) Scattering : Scattering, Brillouin

ToC Category:
Fibers, Fiber Devices, and Amplifiers

History
Original Manuscript: October 3, 2011
Revised Manuscript: December 22, 2011
Manuscript Accepted: December 22, 2011
Published: January 12, 2012

Virtual Issues
European Conference on Optical Communication 2011 (2011) Optics Express

Citation
Y. Sikali Mamdem, E. Burov, L-A. de Montmorillon, Y. Jaouën, G. Moreau, R. Gabet, and F. Taillade, "Importance of residual stresses in the Brillouin gain spectrum of single mode optical fibers," Opt. Express 20, 1790-1797 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1790


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References

  1. A. D. Yablon, “Optical and mechanical effects of frozen-in stresses and strains in optical fibers,” IEEE J. Sel. Top. Quantum Electron.10(2), 300–311 (2004). [CrossRef]
  2. A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express13(14), 5338–5346 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-14-5338 . [CrossRef] [PubMed]
  3. Y. Koyamada, S. Sato, S. Nakamura, H. Sotobayashi, and W. Chujo, “Simulating and designing Brillouin gain spectrum in single-mode fibers,” J. Lightwave Technol.22(2), 631–639 (2004). [CrossRef]
  4. W. Zou, Z. He, A. D. Yablon, and K. Hotate, “Dependence of Brillouin frequency shift in optical fibers on draw-induced residual elastic and inelastic strains,” IEEE Photon. Technol. Lett.19(18), 1389–1391 (2007). [CrossRef]
  5. W. Zou, Z. He, and K. Hotate, “Two-dimensional finite-element modal analysis of Brillouin gain spectra in optical fibers,” IEEE Photon. Technol. Lett.18(23), 2487–2489 (2006). [CrossRef]
  6. Y. Sikali Mamdem, X. Pheron, F. Taillade, Y. Jaouën, R. Gabet, V. Lanticq, G. Moreau, A. Boukenter, Y. Ouerdane, S. Lesoille, and J. Bertrand, “Two-dimensional FEM analysis of Brillouin Gain Spectra in acoustic guiding and antiguiding single mode optical fibers,” in Proceedings of COMSOL Multiphysics Conference (session Acoustic II, Paris, 2010), 111–124.
  7. Y. Sikali Mamdem, E. Burov, L.-A de Montmorillon, F. Taillade, Y. Jaouën, G. Moreau, and R. Gabet, “Importance of residual stresses in the Brillouin gain spectrum of single mode optical fibers,” ECOC 2011, paper We.10.P1.16, Geneva, Sept. 2011.
  8. A. Safaai-Jazi, C.-K. Jen, and G. W. Farnell, “Analysis of weakly guiding fiber acoustic waveguide,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control33(1), 59–68 (1986). [CrossRef] [PubMed]
  9. L. Tartara, C. Codemard, J.-N. Maran, R. Cherif, and M. Zghal, “Full modal analysis of the Brillouin gain spectrum of an optical fiber,” Opt. Commun.282(12), 2431–2436 (2009). [CrossRef]
  10. S. Dasgupta, F. Poletti, S. Liu, P. Petropoulos, D. J. Richardson, L. Grüner-Nielsen, and S. Herstrøm, “Modeling Brillouin Gain Spectrum of solid and microstructured optical fibers using a finite element method,” J. Lightwave Technol.29(1), 22–30 (2011). [CrossRef]
  11. C. A. S. de Oliveira, C. K. Jen, A. Shang, and C. Saravanos, “Stimulated Brillouin scattering in cascaded fibers of different Brillouin frequency shifts,” J. Opt. Soc. Am. B10(6), 969–971 (1993). [CrossRef]
  12. G. W. Scherer, “Stress-induced index profile distortion in optical waveguides,” Appl. Opt.19(12), 2000–2006 (1980). [CrossRef] [PubMed]
  13. S. Chaki and G. Bourse, “Guided ultrasonic waves for non-destructive monitoring of the stress levels in prestressed steel strands,” Ultrasonics49(2), 162–171 (2009). [CrossRef] [PubMed]
  14. R. Le Parc, “Diffusion de rayonnement et relaxation structurale dans les verres de silice et les préformes de fibres optiques,” PhD thesis (Claude Bernard University, Lyon-1, 2002).
  15. F. Terki, C. Levelut, M. Boissier, and J. Pelous, “Low-frequency dynamics and medium-range order in vitreous silica,” Phys. Rev. B Condens. Matter53(5), 2411–2418 (1996). [CrossRef] [PubMed]
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