Abstract

We propose a criterion to predict the relative value of the stimulated Brillouin scattering (SBS) threshold in single-mode optical fibers with different refractive index profiles. We confirm our results by several representative measurements. We show that with the proper profile design one can achieve more than 3 dB increase in the SBS threshold compared to the standard single-mode optical fiber.

©2005 Optical Society of America

1. Introduction

SBS is a nonlinear effect caused by interaction between optical and acoustic waves (for an in-depth description see, e.g., [1]). For many fiber-optic applications it has a negative impact. For example, SBS has been shown to restrict performance of soliton-based long-haul transmission systems [2] and analog CATV systems [3]. In addition, SBS is an undesirable effect in the optical power delivery, high-power fiber lasers, and unrepeatered transmission where high input optical powers are required. Once the input power reaches some critical value, the amount of backscattered power increases quickly with the input power. The onset of SBS is determined by that critical input power which is called the SBS threshold (SBST) [4]. An increase in the SBST power which is highly desirable for many applications can be accomplished in several ways. For example, one can broaden the spectrum of the input signal using phase modulation to reduce its overlap with the narrow Brillouin gain spectrum [5] or impose a variation of fiber parameters such as stress [6], temperature [7], the core radius [8], or the Brillouin shift frequency [9, 10] along the fiber length. A more practical solution, however, is to control the Brillouin gain by the fiber design because fibers with different index profiles have different Brillouin gain spectra and therefore different SBS thresholds [11, 12, 13, 14].

In this paper we describe an approach to estimate the SBST power of a single-mode optical fiber from its index profile. To study the acousto-optic interaction we use the approach of Refs. [1, 15]. We present a simple reasoning that allows us to deduce a single parameter, the acousto-optic effective area, that quantifies the SBST depending on the radial distribution of the core dopant concentration. We confirm our theoretical results by measurement of the SBST in several GeO2-doped silica fibers. The paper is organized as follows. In Section 2 we derive the expression for the acousto-optic effective area which unlike the commonly used optical effective area determines the strength of the SBS process in optical fibers. We then calculate profiles of acoustic modes that are needed to evaluate the acousto-optic effective area (Section 3) and describe the calculation of SBST when input optical wave interacts with several acoustic modes (Section 4).

2. Key design parameter for fibers with enhanced SBS threshold

Similarly to bulk media [1], the equation for the acousto-optic interaction in an optical fiber can be written as [15]

2ρt2Γ2ρtvl2(r)2ρ=γ22E2,

where ρ is the material density fluctuation around its mean value ρ 0, Γ=η 11/ρ 0 is the damping factor, vl2 (r)=Y(r)/ρ(r) is the squared longitudinal sound velocity that depends on the transverse radial coordinate r due to silica doping with GeO2, Y(r) is the Young’s modulus, γ=n 4 ε 0 p 12 is the electrostriction constant, n is the glass refraction index, η 11 and p 12 are the respective components of the viscosity and electrostriction tensors, and ε 0 is the vacuum permittivity. Unlike Ref. [15], our approach takes into account the radial variation of mechanical properties of glass and therefore addresses acoustic guiding by the core region. The electric field E in the right-hand side of (1) is represented as a superposition of forward and backward propagating electro-magnetic waves

E(r,z,t)=12f(r)[A1(z,t)ei(ω1tβ1z)u1+A2(z,t)ei(ω2t+β2z)u2]+c.c.,

where f(r) is the dimensionless fundamental optical mode profile, Aj (z,t), j=1, 2, are the slowly varying envelopes of the optical field, u j are unit polarization vectors of the forward- and backward-propagating waves, ωj and β j are, respectively, frequencies and propagation constants of optical waves, and “c.c.” stands for complex-conjugate.

A rigorous treatment of the acousto-optic interaction requires the solution of the equation for the material displacement (see, e.g., Appendix I in Ref. [15]). But since the acoustic wave is predominantly longitudinal [14, 16], we neglect the transversal component of the acoustic wave for the SBST estimation and adopt the scalar approximation [1, 15] that takes the form of eqn. (1) where

2E212f2(r)A1(z,t)A2*(z,t)q2ei(Ωtqz)+c.c.,

Ω=ω 1-ω 2 is the acoustic frequency and q=β 1+β 2≈2ω 1 n eff/c≈4πn/λ with n eff, λ, and c being the effective refractive index of the optical mode, the input signal wavelength, and the speed of light, respectively.

We look for the solution of (1) in the form

ρ(z,t,r,θ)=12m=1Mρ¯m(z,t)ξm(r,θ)ei(Ωtqz)+c.c.,

where M is the number of acoustic modes ξm (r,θ) which are solutions of the unperturbed (with zero right-hand side and Γ=0) equation (1) and thus satisfy the eigenvalue equation

2ξm(r,θ)+[Ωm2vl2(r)q2]ξm(r,θ)=0,

where 2= 2/ r 2+(1/r)/ r+(1/r 2)∂2/ θ 2 is the transverse Laplacian operator in cylindrical coordinates. In what follows, we consider acoustic modes without axial variation (/ θ=0) since only those modes interact efficiently with the axially symmetric optical mode f(r). Upon substitution of (3), (4), and (5) in (1) we find after multiplying both sides of the resulting equation with ξ m (r) and integrating over the transverse plane

ρ¯m(z,t)=γq2A1(z,t)A2*(z,t)2(Ωm2Ω2+iΩΓq2)ξm(r)f2(r)ξm2(r),

where we have denoted integrated quantities by angled brackets 〈ζ(r)〉=2π0 ζ(r)rdr and used the fact that for relevant modal solutions of (5) 〈ξm (r)ξl (r)〉/〈ξm2 (r)〉≈10-8 (ml) for all studied fiber index profiles. Since each acoustic mode interacts independently with the optical field, from (4) and (6) the full material density variation is

ρ(z,t,r)=14m=1M[ξm(r)ξm(r)f2(r)ξm2(r)γq2A1(z,t)A2*(z,t)Ωm2Ω2+iΩΓq2ei(Ωtqz)]+c.c.

Following the perturbative approach for the derivation of the nonlinear pulse propagation in an optical fiber [17, 18] one can obtain the propagation equation for the optical envelopes A 1,2. Details of the derivation are given in the Appendix. Finally, using the normalization Pj =|Aj |2 ε 0 cnf 2〉/2 so that Pj has the dimension of the optical power in Watts we obtain the ordinary differential equation for the back-reflected (Stokes) power in the stationary regime

dP2dzαP2+gBAmao𝓛(ν)P1P2=0.

In (8), α is the fiber loss coefficient and

gB=4πn8p122λ3ρ0cwνB

is the Brillouin gain coefficient with wq 2/(2π) being the FWHM of the Brillouin gain spectrum. The latter has a Lorentzian shape

𝓛(ν)=(w2)2(νν1+νB)2+(w2)2,

with the peak shifted from the signal frequency ν 1 by νBm /(2π). A typical value of νB for most germania-doped fibers is ~11 GHz. It varies only slightly (by ~0.5 GHz) for different acoustic modes and different fibers. Hence, without loss of accuracy we take equal Brillouin shifts νB for all acoustic modes. Additional scaling factors can be introduced in (9) to account for polarization effects [19] and the finite spectral line width of the input signal [12]. We do not use these factors here though because, as will be discussed in Section 4, we will calculate the SBST power relative to a measured value for some reference fiber. Finally,

Amao=[f2(r)ξm(r)f2(r)]2ξm2(r),

is the acousto-optic effective area (in µm2) associated with mth acoustic mode. Unlike the optical effective area defined as [17, 18]

Aeff=f2(r)2f4(r)

and conventionally used to describe SBS in optical fibers, the quantity given by (11) actually determines the total Brillouin gain. As we will see in the following section, the acousto-optic effective area approximately equals A eff for fiber profiles which are close to step-index. This explains why approximating A ao with A eff gives a good agreement with numerous experimental data when step-index, standard single-mode fiber is used. However, if the fiber profile differs from the step-index, counter-intuitive results (higher SBS threshold for fibers with smaller effective area) can be obtained [11].

Although it was mentioned before that the Brillouin gain coefficient relates to the overlap integral between acoustic and squared optical modal profiles [14, 15], no such quantity as Amao was used to predict the SBS threshold in optical fibers. Another advantage of the definition (11) is that it does not require normalization of the mode profiles. Indeed, the dimensionless functions f(r) and ξm (r) appear to the same power in the numerator and denominator of (11). As a final comment on the Brillouin gain coefficient we mention that the same expression for the peak gain (9) was obtained when the guiding nature of acoustic waves was not accounted for, i.e., using the plane wave approximation for acoustic waves [1, 17]. Hence, the peak value of the SBS gain is the same in bulk and waveguide geometries. However, an important difference in the total Brillouin gain coefficient in the equation for the optical power evolution (8) is due to the modal overlap factor determined by the acousto-optic effective area (11).

In what follows we show that the acousto-optic effective area is a good measure for the SBST power of an optical fiber and that only several modes with the smallest Amao (i.e., the largest Brillouin gain) efficiently contribute to the back-reflected power. Therefore only those modes can be taken into account when calculating the SBST.

3. Computation of acoustic mode profiles

Even though the optical mode profile might change only slightly from one single-mode fiber profile to another, acoustic modes and corresponding acousto-optic effective areas can vary significantly. The SBS threshold in turn will depend on the fiber index profile. To calculate the acousto-optic effective area we numerically solve (5) for different refractive index profiles defined as Δ%(r)=100[n(r)nSiO2]nSiO2. We used the following relation for the longitudinal sound velocity (in m/s)

vl(r)=5944[10.078Δ%(r)]

derived from measured data [20, 21].

Fig. 1 shows the index profile together with calculated optical and acoustic mode profiles for two single-mode optical fibers. We only show the acoustic modes with the smallest acoustooptic effective areas whose values are given in Table 1 where we summarize results for five single-mode germania-doped optical fibers. All studied fibers have approximately the same attenuation coefficient α=0.2 dB/km.

The last two columns of Table 1 give calculated and measured values of the SBST power for 20 km long fibers. As can be seen from Fig. 1, the first acoustic mode of fiber 2 is more localized that results in a weaker overlap with the optical mode and hence a larger value of A1ao than in fiber 1. It leads to a smaller Brillouin gain coefficient and consequently higher SBST power of fiber 2 despite its smaller optical effective area compared to fiber 1 (see Table 1).

The SBST for fiber 1 in Table 1 is representative of commonly cited “standard single-mode fiber” (e.g., Corning® SMF-28e® Fiber and OFS/Furukawa Allwave® Fiber) and serves as a baseline for SBST comparison consistent with common industry practice. Fiber 2 is Corning® LEAF® Fiber compliant with the International Telecommunication Union (ITU) G.655 standard. SBST for fiber 5 (Corning® NexCor fiber [22]) is indicative of capability realizable with application of concepts in Section 2 to a fiber with parameters otherwise consistent with “standard single-mode fiber” (ITU G.652 compliant) [23, 24]. Below we explain how the calculated values of the SBST were obtained.

 figure: Fig. 1.

Fig. 1. Normalized index profile, fundamental optical mode f(r), and calculated from (5), (13) acoustic modes ξm (r) with the three smallest Amao for fiber 1 (a) and fiber 2 (b).

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4. Calculation of the SBS threshold power

To calculate the SBST from the found Amao we have used the following approach. The SBST is reached when the total Stokes power P S at the fiber input z=0 equals some predetermined fraction η of the input signal power P 1(0) [4, 25, 26]. From the solution of (8) obtained with the undepleted signal approximation P 1(z)≈P 1(0)exp(-αz) this condition can be written as [27]

PS(0)=m=1M2κTνB+νGm(ν)dν=ηP1(0)

where T is the fiber temperature and κ is the Boltzmann constant. From the short fiber approximation (fiber length L<50 km, see Ref. [27] Section 2C)

Gm(ν)=exp[gBP1(0)Amaoα𝓛(ν)(1eαL)αL].
Tables Icon

Table 1. Three smallest Amao in µm2, optical effective area Aeff [µm2] calculated from (12) and calculated and measured SBS threshold powers Pth in dBm for various GeO2-doped fibers

Substitution of (15) into (14) with the subsequent integration by the steepest descent method (see, e.g., Ref. [28], pp. 477–481) results in the following transcendental equation for the SBST

eαL1eαLm=1Mexp[rmkx(1eαL)]rmk=x32B.

Here

B=ηA11aoανBλπκTgBcw

is a constant and the second index in the acousto-optic effective area denotes the fiber number k from Table 1, while the first index m denotes, as before, the number of the acoustic mode, rmk =A11ao/Amkao (r mk ≠0 assumed). The solution of (16) for x gives the SBST power as

Pthcalc=xA11aoαgB.

Because accurate values of gB is difficult to obtain theoretically, we rely on relative calculations of the threshold. From the measured value of P th for a reference fiber 1 we obtain gB from (16) with x=P thgB /(αA11ao). Then we calculate the constant B from (17) and solve (16) for x for other fibers listed in Table 1. The corresponding threshold power is then obtained from (18). We therefore assume that the relative strength of the SBS interaction due to each acoustic mode is given by the ratio rmk , i.e., the widths of the Brillouin gain spectral lines w near the threshold are approximately equal for the lowest acoustic modes and that they do not change significantly from fiber to fiber. For calculations we have taken w=12 MHz, νB =11 GHz, M=3, η=0.01 [25, 26], λ=1.55 µm, and T=300 K. The exact values of these parameters are not important, since the solution of (16) depends on B very weakly [4]. We have also accounted for the connector loss by adding 0.1 dB to calculated values of the SBST.

The obtained value of Pthcalc are within 0.5 dB (~12%) from the measured values (Table 1). We attribute this difference to a slight variation of the widths of Brillouin gain spectra in different fibers and the approximate nature of eqn. (1) discussed above. In most cases the impact of the second and higher-order acoustic modes on the SBS threshold is weak and the increase in the SBST of the kth fiber compared to the reference fiber can be roughly evaluated as

ΔPth[dB]10log10(A1kaoA11ao).

This approximation works especially well when A1kaoA2kao but tends to underestimate the threshold difference for index profiles with comparable effective acousto-optic areas for the first and higher-order modes. Since no acoustic core modes exist for pure silica core and F-doped cladding fibers [14], one has to account for cladding modes that have maximum overlap with the optical mode to calculate relevant acousto-optic effective areas.

As a final remark, we emphasize that apart from the strength of the acousto-optic interaction, change in the index profile also affects other important fiber parameters such as dispersion, loss, bending sensitivity, etc. Thus, a smart fiber profile design requires manipulation with acoustic properties of the fiber without compromising its the key optical properties [22].

5. Conclusions

In conclusion, we have theoretically and experimentally studied SBS in optical fibers with different index profiles. We have introduced the acousto-optic effective area which can be used for prediction of the SBS threshold power in optical fiber with various index profiles. Large acousto-optic effective area is a prerequisite for enhanced SBS threshold of the optical fiber.

Appendix: Derivation of evolutional equations for signal and Stokes waves

From Maxwell’s equations we obtain the following equation for the electric field [17]

2EεLc22Et2μ02𝓟NLt2=0

where εL and µ 0 are the linear part of the dielectric constant and the magnetic permeability, respectively, while the nonlinear polarization induced by the acousto-optic interaction is [1, 15]

𝓟NL=γρ0ρE=ε0εNLE

where εNL=γρ/(ε 0 ρ 0). Then (A1) can be rewritten as

2E=εtotc22Et2,

where ε totLNL=n 2(r)-inαc/ωNL with α being the loss coefficient of the optical fiber. In obtaining (A2) we made several assumptions which are similar to those in the standard approach for derivation of the nonlinear pulse propagation in an optical fiber [17, 18]. It was assumed that the nonlinear polarization 𝒫NL can be treated as a small perturbation, i.e., 2NL E)/∂ t 2≈εNL 2 E/∂ t 2, the optical field maintains its polarization so that the scalar approximation is valid, optical field is quasi-monochromatic, the nonlinear response is instantaneous, and the weakly guidance (or the small index gradient) condition ∇ε/ε≪1 is satisfied. In the linear part of the total dielectric constant εtot we have neglected both differences in the material properties at close frequencies ω=ω 1ω 2 and the spatial dependence of n in the linear attenuation term because 𝒯(ε L)≪ℜ(ε L). Assuming independent interaction between the optical mode and each acoustic mode we can write the nonlinear part of the total dielectric constant as

εNL=ξm[UA1A2*ei(ω1ω2)ti(β1+β2)z+c.c.]

where

U=γ2q24ρ0ε0(Ωm2Ω2+iΩΓq2)ξmf2ξm2.

Using the slowly varying envelope approximation [17, 18] we obtain relations for the derivatives of the total electric field (2), substitute them together with (A3) and (A4) in (A2) and group the resulting terms by the exponential factors ei(ω1tβ1z) and ei(ω2t+β2z) to obtain two equations for forward and backward propagating optical waves

Aj2f(r)f(r)(2iβiAjz±βj2Aj)
=[n2(r)inαcω]f(r)c2Ajω2U˜ξm(r)f(r)c2ω2AjA3j2,

where the upper and lower sign is for j=1 and j=2, respectively; Ũ=U for j=1, while Ũ=U* for j=2.

Neglecting the dependence of the optical modal profile on nonlinear effects [18] we can substitute the optical modal equation

2f(r)+[ω2n2(r)c2βj2]f(r)=0

in (A5), multiply its both sides with f(r) and integrate the result over the transverse crosssection to finally obtain the coupled ordinary differential equations

dA1dz=α2A1iσA1A22,,

dA2dz=α2A2+iσ*A12A2

where

σ=Uω2cnξmf2f2=n9ε0p122ω32ρ0c3(Ωm2Ω2+iΩΓq2)ξmf22ξm2f2..

Acknowledgments

The authors gratefully acknowledge discussions with A. Boskovic, K. Emig, S. Darmanyan, T. Hanson, M. Li, C. Mazzali, S. Ten, R. Vodhanel, and A. Woodfin.

*currently with the Department of Electrical and Computer Engineering, McMaster University Hamilton ON, Canada.

References and links

1. R. W. Boyd, Nonlinear Optics, 2nd edition, Academic Press, New York2003, Chapter 9.

2. E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. N. Starodumov, “Electrostriction mechanism of soliton interaction in optical fibers,” Opt. Lett. 15, 314–316 (1990). [CrossRef]   [PubMed]  

3. X. P. Mao, G. E. Bodeep, R. W. Tkach, A. R. Chraplyvy, T. E. Darcie, and R. M. Derosier, “Brillouin scattering in externally modulated lightwave AM-VSB CATV transmission systems,” IEEE Photon. Technol. Lett. 4, 287–289 (1992). [CrossRef]  

4. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2489–2494 (1972). [CrossRef]   [PubMed]  

5. F. W. Willems and W. Muys, “Suppression of interferometric noise in externally modulated lightwave AM-CATV systems by phase modulation,” Electron. Lett. 29, 2062–2063 (1993). [CrossRef]  

6. N. Yoshizawa and T. Imai, “Stimulated Brillouin scattering suppression by means of applying strain distribution to fiber with cabling,” IEEE J. Lightwave Technol. 11, 1518–1522 (1993). [CrossRef]  

7. J. Hansryd, F. Dross, M. Westlund, P. A. Andrekson, and S. N. Knudsen “Increase in the SBS threshold in a short highly nonlinear fiber by applying a temperature distribution,” IEEE J. Lightwave Technol. 19, 1691–1697 (2001). [CrossRef]  

8. K. Shiraki, M. Ohashi, and M. Tateda, “Suppression of stimulated Brillouin scattering in a fibre by changing the core radius,” Electron. Lett. 31, 668–669 (1995). [CrossRef]  

9. C. A. S. de Oliveira, C. K. Jen, A. Shang, and C. Saravanos, “Stimulated Brillouin scattering in cascaded fibers of different Brillouin frequency shift,” J. Opt. Soc. Am B 10, 969–972 (1993). [CrossRef]  

10. A. Kobyakov, M. Sauer, and J. E. Hurley, “SBS threshold of segmented fibers,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference on CD-ROM (Optical Society of America, Washington, DC, 2005), paper OME5. [PubMed]  

11. C. C. Lee and S. Chi, “Measurement of stimulated Brillouin scattering threshold for various types of fibers using Brillouin optical time-domain reflectometer,” IEEE Photon. Technol. Lett. 12, 672–674 (2000). [CrossRef]  

12. A. Yeniay, J. -M. Delavaux, and J. Toulouse, “Spontaneous and stimulated Brillouin scattering gain spectra in optical fibers,” IEEE J. Lightwave Technol. 20, 1425–1432 (2002). [CrossRef]  

13. J. Yu, I.-B. Kwon, and K. Oh, “Analysis of Brillouin frequency shift and longitudinal acoustic wave in a silica optical fiber with a triple-layered structure,” IEEE J. Lightwave Technol. 21, 1779–1786 (2003). [CrossRef]  

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

15. E. Peral and A. Yariv, “Degradation of modulation and noise characteristics of semiconductor lasers after propagation in optical fiber due to shift induced by stimulated Brillouin scattering,” IEEE J. Quantum Electron. 35, 1185–1195 (1999). [CrossRef]  

16. C.-K. Jen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guided fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. UFFC-33, 634–643 (1986).

17. K. Okamoto, Fundamentals of Optical Waveguides, Academic Press, New York2000.

18. G. P. Agrawal, Nonlinear Fiber Optics, 3d editionAcademic Press, New York2001.

19. M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” IEEE J. Lightwave Technol. 12, 585–590 (1994). [CrossRef]  

20. N. Lagakos, J. A. Bucaro, and R. Hughes, “Acoustic sensitivity predictions of single-mode optical fibers using Brillouin scattering,” Appl. Opt. 19, 3668–3670 (1980). [CrossRef]   [PubMed]  

21. S. T. Gulati and J. D. Helfinstine, “Fatigue behavior of GeO2-SiO2 glasses,” Mat. Res. Soc. Symp. Proc. 531, 133–138 (1998). [CrossRef]  

22. http://www.corning.com/opticalfiber/products applications/products/nexcor.aspx

23. D. Chowdhury, A. Kobyakov, S. Kumar, B. Ruffin, and S. Bickham, “Application of doped optical glass for optical communication,” in Proceedings of XX International Congress on Glass (The Ceramic Society of Japan, Kyoto, Japan, 2004) paper I-01-005.

24. Patent application US Publication No. 2004/0218882.

25. P. Bayvel and P. M. Radmore, “Solutions of the SBS equations in single mode optical fibers and implications for fiber transmission systems,” Electron. Lett. 26, 434–436 (1990). [CrossRef]  

26. R. D. Esman and K. J. Williams, “Brillouin scattering: beyond threshold,” in Optical Fiber Communication Conference (Optical Society of America, Washington, DC, 1996) 227–228, paper ThF5.

27. A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. 20, 1635–1643 (2002). [CrossRef]  

28. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th edition, Academic Press, New York2001.

References

  • View by:

  1. R. W. Boyd, Nonlinear Optics, 2nd edition, Academic Press, New York2003, Chapter 9.
  2. E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. N. Starodumov, “Electrostriction mechanism of soliton interaction in optical fibers,” Opt. Lett. 15, 314–316 (1990).
    [Crossref] [PubMed]
  3. X. P. Mao, G. E. Bodeep, R. W. Tkach, A. R. Chraplyvy, T. E. Darcie, and R. M. Derosier, “Brillouin scattering in externally modulated lightwave AM-VSB CATV transmission systems,” IEEE Photon. Technol. Lett. 4, 287–289 (1992).
    [Crossref]
  4. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2489–2494 (1972).
    [Crossref] [PubMed]
  5. F. W. Willems and W. Muys, “Suppression of interferometric noise in externally modulated lightwave AM-CATV systems by phase modulation,” Electron. Lett. 29, 2062–2063 (1993).
    [Crossref]
  6. N. Yoshizawa and T. Imai, “Stimulated Brillouin scattering suppression by means of applying strain distribution to fiber with cabling,” IEEE J. Lightwave Technol. 11, 1518–1522 (1993).
    [Crossref]
  7. J. Hansryd, F. Dross, M. Westlund, P. A. Andrekson, and S. N. Knudsen “Increase in the SBS threshold in a short highly nonlinear fiber by applying a temperature distribution,” IEEE J. Lightwave Technol. 19, 1691–1697 (2001).
    [Crossref]
  8. K. Shiraki, M. Ohashi, and M. Tateda, “Suppression of stimulated Brillouin scattering in a fibre by changing the core radius,” Electron. Lett. 31, 668–669 (1995).
    [Crossref]
  9. C. A. S. de Oliveira, C. K. Jen, A. Shang, and C. Saravanos, “Stimulated Brillouin scattering in cascaded fibers of different Brillouin frequency shift,” J. Opt. Soc. Am B 10, 969–972 (1993).
    [Crossref]
  10. A. Kobyakov, M. Sauer, and J. E. Hurley, “SBS threshold of segmented fibers,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference on CD-ROM (Optical Society of America, Washington, DC, 2005), paper OME5.
    [PubMed]
  11. C. C. Lee and S. Chi, “Measurement of stimulated Brillouin scattering threshold for various types of fibers using Brillouin optical time-domain reflectometer,” IEEE Photon. Technol. Lett. 12, 672–674 (2000).
    [Crossref]
  12. A. Yeniay, J. -M. Delavaux, and J. Toulouse, “Spontaneous and stimulated Brillouin scattering gain spectra in optical fibers,” IEEE J. Lightwave Technol. 20, 1425–1432 (2002).
    [Crossref]
  13. J. Yu, I.-B. Kwon, and K. Oh, “Analysis of Brillouin frequency shift and longitudinal acoustic wave in a silica optical fiber with a triple-layered structure,” IEEE J. Lightwave Technol. 21, 1779–1786 (2003).
    [Crossref]
  14. Y. Koyamada, S. Sato, S. Nakamura, H. Sotobayashi, and W. Chujo, “Simulating and designing Brillouin gain spectrum in single-mode fibers,” IEEE J. Lightwave Technol. 22, 631–639 (2004).
    [Crossref]
  15. E. Peral and A. Yariv, “Degradation of modulation and noise characteristics of semiconductor lasers after propagation in optical fiber due to shift induced by stimulated Brillouin scattering,” IEEE J. Quantum Electron. 35, 1185–1195 (1999).
    [Crossref]
  16. C.-K. Jen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guided fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. UFFC-33, 634–643 (1986).
  17. K. Okamoto, Fundamentals of Optical Waveguides, Academic Press, New York2000.
  18. G. P. Agrawal, Nonlinear Fiber Optics, 3d editionAcademic Press, New York2001.
  19. M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” IEEE J. Lightwave Technol. 12, 585–590 (1994).
    [Crossref]
  20. N. Lagakos, J. A. Bucaro, and R. Hughes, “Acoustic sensitivity predictions of single-mode optical fibers using Brillouin scattering,” Appl. Opt. 19, 3668–3670 (1980).
    [Crossref] [PubMed]
  21. S. T. Gulati and J. D. Helfinstine, “Fatigue behavior of GeO2-SiO2 glasses,” Mat. Res. Soc. Symp. Proc. 531, 133–138 (1998).
    [Crossref]
  22. http://www.corning.com/opticalfiber/products applications/products/nexcor.aspx
  23. D. Chowdhury, A. Kobyakov, S. Kumar, B. Ruffin, and S. Bickham, “Application of doped optical glass for optical communication,” in Proceedings of XX International Congress on Glass (The Ceramic Society of Japan, Kyoto, Japan, 2004) paper I-01-005.
  24. Patent application US Publication No. 2004/0218882.
  25. P. Bayvel and P. M. Radmore, “Solutions of the SBS equations in single mode optical fibers and implications for fiber transmission systems,” Electron. Lett. 26, 434–436 (1990).
    [Crossref]
  26. R. D. Esman and K. J. Williams, “Brillouin scattering: beyond threshold,” in Optical Fiber Communication Conference (Optical Society of America, Washington, DC, 1996) 227–228, paper ThF5.
  27. A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. 20, 1635–1643 (2002).
    [Crossref]
  28. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th edition, Academic Press, New York2001.

2004 (1)

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

2003 (1)

J. Yu, I.-B. Kwon, and K. Oh, “Analysis of Brillouin frequency shift and longitudinal acoustic wave in a silica optical fiber with a triple-layered structure,” IEEE J. Lightwave Technol. 21, 1779–1786 (2003).
[Crossref]

2002 (2)

A. Yeniay, J. -M. Delavaux, and J. Toulouse, “Spontaneous and stimulated Brillouin scattering gain spectra in optical fibers,” IEEE J. Lightwave Technol. 20, 1425–1432 (2002).
[Crossref]

A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. 20, 1635–1643 (2002).
[Crossref]

2001 (1)

J. Hansryd, F. Dross, M. Westlund, P. A. Andrekson, and S. N. Knudsen “Increase in the SBS threshold in a short highly nonlinear fiber by applying a temperature distribution,” IEEE J. Lightwave Technol. 19, 1691–1697 (2001).
[Crossref]

2000 (1)

C. C. Lee and S. Chi, “Measurement of stimulated Brillouin scattering threshold for various types of fibers using Brillouin optical time-domain reflectometer,” IEEE Photon. Technol. Lett. 12, 672–674 (2000).
[Crossref]

1999 (1)

E. Peral and A. Yariv, “Degradation of modulation and noise characteristics of semiconductor lasers after propagation in optical fiber due to shift induced by stimulated Brillouin scattering,” IEEE J. Quantum Electron. 35, 1185–1195 (1999).
[Crossref]

1998 (1)

S. T. Gulati and J. D. Helfinstine, “Fatigue behavior of GeO2-SiO2 glasses,” Mat. Res. Soc. Symp. Proc. 531, 133–138 (1998).
[Crossref]

1995 (1)

K. Shiraki, M. Ohashi, and M. Tateda, “Suppression of stimulated Brillouin scattering in a fibre by changing the core radius,” Electron. Lett. 31, 668–669 (1995).
[Crossref]

1994 (1)

M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” IEEE J. Lightwave Technol. 12, 585–590 (1994).
[Crossref]

1993 (3)

C. A. S. de Oliveira, C. K. Jen, A. Shang, and C. Saravanos, “Stimulated Brillouin scattering in cascaded fibers of different Brillouin frequency shift,” J. Opt. Soc. Am B 10, 969–972 (1993).
[Crossref]

F. W. Willems and W. Muys, “Suppression of interferometric noise in externally modulated lightwave AM-CATV systems by phase modulation,” Electron. Lett. 29, 2062–2063 (1993).
[Crossref]

N. Yoshizawa and T. Imai, “Stimulated Brillouin scattering suppression by means of applying strain distribution to fiber with cabling,” IEEE J. Lightwave Technol. 11, 1518–1522 (1993).
[Crossref]

1992 (1)

X. P. Mao, G. E. Bodeep, R. W. Tkach, A. R. Chraplyvy, T. E. Darcie, and R. M. Derosier, “Brillouin scattering in externally modulated lightwave AM-VSB CATV transmission systems,” IEEE Photon. Technol. Lett. 4, 287–289 (1992).
[Crossref]

1990 (2)

E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. N. Starodumov, “Electrostriction mechanism of soliton interaction in optical fibers,” Opt. Lett. 15, 314–316 (1990).
[Crossref] [PubMed]

P. Bayvel and P. M. Radmore, “Solutions of the SBS equations in single mode optical fibers and implications for fiber transmission systems,” Electron. Lett. 26, 434–436 (1990).
[Crossref]

1986 (1)

C.-K. Jen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guided fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. UFFC-33, 634–643 (1986).

1980 (1)

1972 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3d editionAcademic Press, New York2001.

Andrekson, P. A.

J. Hansryd, F. Dross, M. Westlund, P. A. Andrekson, and S. N. Knudsen “Increase in the SBS threshold in a short highly nonlinear fiber by applying a temperature distribution,” IEEE J. Lightwave Technol. 19, 1691–1697 (2001).
[Crossref]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th edition, Academic Press, New York2001.

Bayvel, P.

P. Bayvel and P. M. Radmore, “Solutions of the SBS equations in single mode optical fibers and implications for fiber transmission systems,” Electron. Lett. 26, 434–436 (1990).
[Crossref]

Bickham, S.

D. Chowdhury, A. Kobyakov, S. Kumar, B. Ruffin, and S. Bickham, “Application of doped optical glass for optical communication,” in Proceedings of XX International Congress on Glass (The Ceramic Society of Japan, Kyoto, Japan, 2004) paper I-01-005.

Bodeep, G. E.

X. P. Mao, G. E. Bodeep, R. W. Tkach, A. R. Chraplyvy, T. E. Darcie, and R. M. Derosier, “Brillouin scattering in externally modulated lightwave AM-VSB CATV transmission systems,” IEEE Photon. Technol. Lett. 4, 287–289 (1992).
[Crossref]

Boot, A. J.

M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” IEEE J. Lightwave Technol. 12, 585–590 (1994).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 2nd edition, Academic Press, New York2003, Chapter 9.

Bucaro, J. A.

Chi, S.

C. C. Lee and S. Chi, “Measurement of stimulated Brillouin scattering threshold for various types of fibers using Brillouin optical time-domain reflectometer,” IEEE Photon. Technol. Lett. 12, 672–674 (2000).
[Crossref]

Chowdhury, D.

D. Chowdhury, A. Kobyakov, S. Kumar, B. Ruffin, and S. Bickham, “Application of doped optical glass for optical communication,” in Proceedings of XX International Congress on Glass (The Ceramic Society of Japan, Kyoto, Japan, 2004) paper I-01-005.

Chraplyvy, A. R.

X. P. Mao, G. E. Bodeep, R. W. Tkach, A. R. Chraplyvy, T. E. Darcie, and R. M. Derosier, “Brillouin scattering in externally modulated lightwave AM-VSB CATV transmission systems,” IEEE Photon. Technol. Lett. 4, 287–289 (1992).
[Crossref]

Chujo, W.

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

Darcie, T. E.

X. P. Mao, G. E. Bodeep, R. W. Tkach, A. R. Chraplyvy, T. E. Darcie, and R. M. Derosier, “Brillouin scattering in externally modulated lightwave AM-VSB CATV transmission systems,” IEEE Photon. Technol. Lett. 4, 287–289 (1992).
[Crossref]

de Oliveira, C. A. S.

C. A. S. de Oliveira, C. K. Jen, A. Shang, and C. Saravanos, “Stimulated Brillouin scattering in cascaded fibers of different Brillouin frequency shift,” J. Opt. Soc. Am B 10, 969–972 (1993).
[Crossref]

Delavaux, J. -M.

A. Yeniay, J. -M. Delavaux, and J. Toulouse, “Spontaneous and stimulated Brillouin scattering gain spectra in optical fibers,” IEEE J. Lightwave Technol. 20, 1425–1432 (2002).
[Crossref]

Derosier, R. M.

X. P. Mao, G. E. Bodeep, R. W. Tkach, A. R. Chraplyvy, T. E. Darcie, and R. M. Derosier, “Brillouin scattering in externally modulated lightwave AM-VSB CATV transmission systems,” IEEE Photon. Technol. Lett. 4, 287–289 (1992).
[Crossref]

Dianov, E. M.

Dross, F.

J. Hansryd, F. Dross, M. Westlund, P. A. Andrekson, and S. N. Knudsen “Increase in the SBS threshold in a short highly nonlinear fiber by applying a temperature distribution,” IEEE J. Lightwave Technol. 19, 1691–1697 (2001).
[Crossref]

Esman, R. D.

R. D. Esman and K. J. Williams, “Brillouin scattering: beyond threshold,” in Optical Fiber Communication Conference (Optical Society of America, Washington, DC, 1996) 227–228, paper ThF5.

Evans, A. F.

A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. 20, 1635–1643 (2002).
[Crossref]

Farnell, G. W.

C.-K. Jen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guided fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. UFFC-33, 634–643 (1986).

Gulati, S. T.

S. T. Gulati and J. D. Helfinstine, “Fatigue behavior of GeO2-SiO2 glasses,” Mat. Res. Soc. Symp. Proc. 531, 133–138 (1998).
[Crossref]

Hansryd, J.

J. Hansryd, F. Dross, M. Westlund, P. A. Andrekson, and S. N. Knudsen “Increase in the SBS threshold in a short highly nonlinear fiber by applying a temperature distribution,” IEEE J. Lightwave Technol. 19, 1691–1697 (2001).
[Crossref]

Helfinstine, J. D.

S. T. Gulati and J. D. Helfinstine, “Fatigue behavior of GeO2-SiO2 glasses,” Mat. Res. Soc. Symp. Proc. 531, 133–138 (1998).
[Crossref]

Hughes, R.

Hurley, J. E.

A. Kobyakov, M. Sauer, and J. E. Hurley, “SBS threshold of segmented fibers,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference on CD-ROM (Optical Society of America, Washington, DC, 2005), paper OME5.
[PubMed]

Imai, T.

N. Yoshizawa and T. Imai, “Stimulated Brillouin scattering suppression by means of applying strain distribution to fiber with cabling,” IEEE J. Lightwave Technol. 11, 1518–1522 (1993).
[Crossref]

Jen, C. K.

C. A. S. de Oliveira, C. K. Jen, A. Shang, and C. Saravanos, “Stimulated Brillouin scattering in cascaded fibers of different Brillouin frequency shift,” J. Opt. Soc. Am B 10, 969–972 (1993).
[Crossref]

Jen, C.-K.

C.-K. Jen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guided fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. UFFC-33, 634–643 (1986).

Knudsen, S. N.

J. Hansryd, F. Dross, M. Westlund, P. A. Andrekson, and S. N. Knudsen “Increase in the SBS threshold in a short highly nonlinear fiber by applying a temperature distribution,” IEEE J. Lightwave Technol. 19, 1691–1697 (2001).
[Crossref]

Kobyakov, A.

A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. 20, 1635–1643 (2002).
[Crossref]

D. Chowdhury, A. Kobyakov, S. Kumar, B. Ruffin, and S. Bickham, “Application of doped optical glass for optical communication,” in Proceedings of XX International Congress on Glass (The Ceramic Society of Japan, Kyoto, Japan, 2004) paper I-01-005.

A. Kobyakov, M. Sauer, and J. E. Hurley, “SBS threshold of segmented fibers,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference on CD-ROM (Optical Society of America, Washington, DC, 2005), paper OME5.
[PubMed]

Koyamada, Y.

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

Kumar, S.

D. Chowdhury, A. Kobyakov, S. Kumar, B. Ruffin, and S. Bickham, “Application of doped optical glass for optical communication,” in Proceedings of XX International Congress on Glass (The Ceramic Society of Japan, Kyoto, Japan, 2004) paper I-01-005.

Kwon, I.-B.

J. Yu, I.-B. Kwon, and K. Oh, “Analysis of Brillouin frequency shift and longitudinal acoustic wave in a silica optical fiber with a triple-layered structure,” IEEE J. Lightwave Technol. 21, 1779–1786 (2003).
[Crossref]

Lagakos, N.

Lee, C. C.

C. C. Lee and S. Chi, “Measurement of stimulated Brillouin scattering threshold for various types of fibers using Brillouin optical time-domain reflectometer,” IEEE Photon. Technol. Lett. 12, 672–674 (2000).
[Crossref]

Luchnikov, A. V.

Mao, X. P.

X. P. Mao, G. E. Bodeep, R. W. Tkach, A. R. Chraplyvy, T. E. Darcie, and R. M. Derosier, “Brillouin scattering in externally modulated lightwave AM-VSB CATV transmission systems,” IEEE Photon. Technol. Lett. 4, 287–289 (1992).
[Crossref]

Mehendale, M.

A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. 20, 1635–1643 (2002).
[Crossref]

Muys, W.

F. W. Willems and W. Muys, “Suppression of interferometric noise in externally modulated lightwave AM-CATV systems by phase modulation,” Electron. Lett. 29, 2062–2063 (1993).
[Crossref]

Nakamura, S.

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

Oh, K.

J. Yu, I.-B. Kwon, and K. Oh, “Analysis of Brillouin frequency shift and longitudinal acoustic wave in a silica optical fiber with a triple-layered structure,” IEEE J. Lightwave Technol. 21, 1779–1786 (2003).
[Crossref]

Ohashi, M.

K. Shiraki, M. Ohashi, and M. Tateda, “Suppression of stimulated Brillouin scattering in a fibre by changing the core radius,” Electron. Lett. 31, 668–669 (1995).
[Crossref]

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides, Academic Press, New York2000.

Peral, E.

E. Peral and A. Yariv, “Degradation of modulation and noise characteristics of semiconductor lasers after propagation in optical fiber due to shift induced by stimulated Brillouin scattering,” IEEE J. Quantum Electron. 35, 1185–1195 (1999).
[Crossref]

Pilipetskii, A. N.

Radmore, P. M.

P. Bayvel and P. M. Radmore, “Solutions of the SBS equations in single mode optical fibers and implications for fiber transmission systems,” Electron. Lett. 26, 434–436 (1990).
[Crossref]

Ruffin, B.

D. Chowdhury, A. Kobyakov, S. Kumar, B. Ruffin, and S. Bickham, “Application of doped optical glass for optical communication,” in Proceedings of XX International Congress on Glass (The Ceramic Society of Japan, Kyoto, Japan, 2004) paper I-01-005.

Safaai-Jazi, A.

C.-K. Jen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guided fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. UFFC-33, 634–643 (1986).

Saravanos, C.

C. A. S. de Oliveira, C. K. Jen, A. Shang, and C. Saravanos, “Stimulated Brillouin scattering in cascaded fibers of different Brillouin frequency shift,” J. Opt. Soc. Am B 10, 969–972 (1993).
[Crossref]

Sato, S.

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

Sauer, M.

A. Kobyakov, M. Sauer, and J. E. Hurley, “SBS threshold of segmented fibers,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference on CD-ROM (Optical Society of America, Washington, DC, 2005), paper OME5.
[PubMed]

Shang, A.

C. A. S. de Oliveira, C. K. Jen, A. Shang, and C. Saravanos, “Stimulated Brillouin scattering in cascaded fibers of different Brillouin frequency shift,” J. Opt. Soc. Am B 10, 969–972 (1993).
[Crossref]

Shiraki, K.

K. Shiraki, M. Ohashi, and M. Tateda, “Suppression of stimulated Brillouin scattering in a fibre by changing the core radius,” Electron. Lett. 31, 668–669 (1995).
[Crossref]

Smith, R. G.

Sotobayashi, H.

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

Starodumov, A. N.

Tateda, M.

K. Shiraki, M. Ohashi, and M. Tateda, “Suppression of stimulated Brillouin scattering in a fibre by changing the core radius,” Electron. Lett. 31, 668–669 (1995).
[Crossref]

Tkach, R. W.

X. P. Mao, G. E. Bodeep, R. W. Tkach, A. R. Chraplyvy, T. E. Darcie, and R. M. Derosier, “Brillouin scattering in externally modulated lightwave AM-VSB CATV transmission systems,” IEEE Photon. Technol. Lett. 4, 287–289 (1992).
[Crossref]

Toulouse, J.

A. Yeniay, J. -M. Delavaux, and J. Toulouse, “Spontaneous and stimulated Brillouin scattering gain spectra in optical fibers,” IEEE J. Lightwave Technol. 20, 1425–1432 (2002).
[Crossref]

Tsuda, S.

A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. 20, 1635–1643 (2002).
[Crossref]

van Deventer, M. O.

M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” IEEE J. Lightwave Technol. 12, 585–590 (1994).
[Crossref]

Vasilyev, M.

A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. 20, 1635–1643 (2002).
[Crossref]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th edition, Academic Press, New York2001.

Westlund, M.

J. Hansryd, F. Dross, M. Westlund, P. A. Andrekson, and S. N. Knudsen “Increase in the SBS threshold in a short highly nonlinear fiber by applying a temperature distribution,” IEEE J. Lightwave Technol. 19, 1691–1697 (2001).
[Crossref]

Willems, F. W.

F. W. Willems and W. Muys, “Suppression of interferometric noise in externally modulated lightwave AM-CATV systems by phase modulation,” Electron. Lett. 29, 2062–2063 (1993).
[Crossref]

Williams, K. J.

R. D. Esman and K. J. Williams, “Brillouin scattering: beyond threshold,” in Optical Fiber Communication Conference (Optical Society of America, Washington, DC, 1996) 227–228, paper ThF5.

Yariv, A.

E. Peral and A. Yariv, “Degradation of modulation and noise characteristics of semiconductor lasers after propagation in optical fiber due to shift induced by stimulated Brillouin scattering,” IEEE J. Quantum Electron. 35, 1185–1195 (1999).
[Crossref]

Yeniay, A.

A. Yeniay, J. -M. Delavaux, and J. Toulouse, “Spontaneous and stimulated Brillouin scattering gain spectra in optical fibers,” IEEE J. Lightwave Technol. 20, 1425–1432 (2002).
[Crossref]

Yoshizawa, N.

N. Yoshizawa and T. Imai, “Stimulated Brillouin scattering suppression by means of applying strain distribution to fiber with cabling,” IEEE J. Lightwave Technol. 11, 1518–1522 (1993).
[Crossref]

Yu, J.

J. Yu, I.-B. Kwon, and K. Oh, “Analysis of Brillouin frequency shift and longitudinal acoustic wave in a silica optical fiber with a triple-layered structure,” IEEE J. Lightwave Technol. 21, 1779–1786 (2003).
[Crossref]

Appl. Opt. (2)

Electron. Lett. (3)

F. W. Willems and W. Muys, “Suppression of interferometric noise in externally modulated lightwave AM-CATV systems by phase modulation,” Electron. Lett. 29, 2062–2063 (1993).
[Crossref]

K. Shiraki, M. Ohashi, and M. Tateda, “Suppression of stimulated Brillouin scattering in a fibre by changing the core radius,” Electron. Lett. 31, 668–669 (1995).
[Crossref]

P. Bayvel and P. M. Radmore, “Solutions of the SBS equations in single mode optical fibers and implications for fiber transmission systems,” Electron. Lett. 26, 434–436 (1990).
[Crossref]

IEEE J. Lightwave Technol. (7)

A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. 20, 1635–1643 (2002).
[Crossref]

M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” IEEE J. Lightwave Technol. 12, 585–590 (1994).
[Crossref]

N. Yoshizawa and T. Imai, “Stimulated Brillouin scattering suppression by means of applying strain distribution to fiber with cabling,” IEEE J. Lightwave Technol. 11, 1518–1522 (1993).
[Crossref]

J. Hansryd, F. Dross, M. Westlund, P. A. Andrekson, and S. N. Knudsen “Increase in the SBS threshold in a short highly nonlinear fiber by applying a temperature distribution,” IEEE J. Lightwave Technol. 19, 1691–1697 (2001).
[Crossref]

A. Yeniay, J. -M. Delavaux, and J. Toulouse, “Spontaneous and stimulated Brillouin scattering gain spectra in optical fibers,” IEEE J. Lightwave Technol. 20, 1425–1432 (2002).
[Crossref]

J. Yu, I.-B. Kwon, and K. Oh, “Analysis of Brillouin frequency shift and longitudinal acoustic wave in a silica optical fiber with a triple-layered structure,” IEEE J. Lightwave Technol. 21, 1779–1786 (2003).
[Crossref]

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

IEEE J. Quantum Electron. (1)

E. Peral and A. Yariv, “Degradation of modulation and noise characteristics of semiconductor lasers after propagation in optical fiber due to shift induced by stimulated Brillouin scattering,” IEEE J. Quantum Electron. 35, 1185–1195 (1999).
[Crossref]

IEEE Photon. Technol. Lett. (2)

X. P. Mao, G. E. Bodeep, R. W. Tkach, A. R. Chraplyvy, T. E. Darcie, and R. M. Derosier, “Brillouin scattering in externally modulated lightwave AM-VSB CATV transmission systems,” IEEE Photon. Technol. Lett. 4, 287–289 (1992).
[Crossref]

C. C. Lee and S. Chi, “Measurement of stimulated Brillouin scattering threshold for various types of fibers using Brillouin optical time-domain reflectometer,” IEEE Photon. Technol. Lett. 12, 672–674 (2000).
[Crossref]

IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. (1)

C.-K. Jen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guided fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. UFFC-33, 634–643 (1986).

J. Opt. Soc. Am B (1)

C. A. S. de Oliveira, C. K. Jen, A. Shang, and C. Saravanos, “Stimulated Brillouin scattering in cascaded fibers of different Brillouin frequency shift,” J. Opt. Soc. Am B 10, 969–972 (1993).
[Crossref]

Mat. Res. Soc. Symp. Proc. (1)

S. T. Gulati and J. D. Helfinstine, “Fatigue behavior of GeO2-SiO2 glasses,” Mat. Res. Soc. Symp. Proc. 531, 133–138 (1998).
[Crossref]

Opt. Lett. (1)

Other (9)

R. W. Boyd, Nonlinear Optics, 2nd edition, Academic Press, New York2003, Chapter 9.

A. Kobyakov, M. Sauer, and J. E. Hurley, “SBS threshold of segmented fibers,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference on CD-ROM (Optical Society of America, Washington, DC, 2005), paper OME5.
[PubMed]

http://www.corning.com/opticalfiber/products applications/products/nexcor.aspx

D. Chowdhury, A. Kobyakov, S. Kumar, B. Ruffin, and S. Bickham, “Application of doped optical glass for optical communication,” in Proceedings of XX International Congress on Glass (The Ceramic Society of Japan, Kyoto, Japan, 2004) paper I-01-005.

Patent application US Publication No. 2004/0218882.

K. Okamoto, Fundamentals of Optical Waveguides, Academic Press, New York2000.

G. P. Agrawal, Nonlinear Fiber Optics, 3d editionAcademic Press, New York2001.

R. D. Esman and K. J. Williams, “Brillouin scattering: beyond threshold,” in Optical Fiber Communication Conference (Optical Society of America, Washington, DC, 1996) 227–228, paper ThF5.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th edition, Academic Press, New York2001.

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Figures (1)

Fig. 1.
Fig. 1. Normalized index profile, fundamental optical mode f(r), and calculated from (5), (13) acoustic modes ξm (r) with the three smallest Amao for fiber 1 (a) and fiber 2 (b).

Tables (1)

Tables Icon

Table 1. Three smallest Amao in µm2, optical effective area A eff [µm2] calculated from (12) and calculated and measured SBS threshold powers P th in dBm for various GeO2-doped fibers

Equations (25)

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2 ρ t 2 Γ 2 ρ t v l 2 ( r ) 2 ρ = γ 2 2 E 2 ,
E ( r , z , t ) = 1 2 f ( r ) [ A 1 ( z , t ) e i ( ω 1 t β 1 z ) u 1 + A 2 ( z , t ) e i ( ω 2 t + β 2 z ) u 2 ] + c . c . ,
2 E 2 1 2 f 2 ( r ) A 1 ( z , t ) A 2 * ( z , t ) q 2 e i ( Ω t qz ) + c . c . ,
ρ ( z , t , r , θ ) = 1 2 m = 1 M ρ ¯ m ( z , t ) ξ m ( r , θ ) e i ( Ω t qz ) + c . c . ,
2 ξ m ( r , θ ) + [ Ω m 2 v l 2 ( r ) q 2 ] ξ m ( r , θ ) = 0 ,
ρ ¯ m ( z , t ) = γ q 2 A 1 ( z , t ) A 2 * ( z , t ) 2 ( Ω m 2 Ω 2 + i Ω Γ q 2 ) ξ m ( r ) f 2 ( r ) ξ m 2 ( r ) ,
ρ ( z , t , r ) = 1 4 m = 1 M [ ξ m ( r ) ξ m ( r ) f 2 ( r ) ξ m 2 ( r ) γ q 2 A 1 ( z , t ) A 2 * ( z , t ) Ω m 2 Ω 2 + i Ω Γ q 2 e i ( Ω t qz ) ] + c . c .
d P 2 d z α P 2 + g B A m ao 𝓛 ( ν ) P 1 P 2 = 0 .
g B = 4 π n 8 p 12 2 λ 3 ρ 0 c w ν B
𝓛 ( ν ) = ( w 2 ) 2 ( ν ν 1 + ν B ) 2 + ( w 2 ) 2 ,
A m ao = [ f 2 ( r ) ξ m ( r ) f 2 ( r ) ] 2 ξ m 2 ( r ) ,
A eff = f 2 ( r ) 2 f 4 ( r )
v l ( r ) = 5944 [ 1 0.078 Δ % ( r ) ]
P S ( 0 ) = m = 1 M 2 κ T ν B + ν G m ( ν ) d ν = η P 1 ( 0 )
G m ( ν ) = exp [ g B P 1 ( 0 ) A m ao α 𝓛 ( ν ) ( 1 e α L ) α L ] .
e α L 1 e α L m = 1 M exp [ r mk x ( 1 e α L ) ] r mk = x 3 2 B .
B = η A 11 ao α ν B λ π κ T g B cw
P th calc = x A 11 ao α g B .
Δ P th [ dB ] 10 log 10 ( A 1 k ao A 11 ao ) .
2 E ε L c 2 2 E t 2 μ 0 2 𝓟 NL t 2 = 0
2 E = ε tot c 2 2 E t 2 ,
ε NL = ξ m [ U A 1 A 2 * e i ( ω 1 ω 2 ) t i ( β 1 + β 2 ) z + c . c . ]
U = γ 2 q 2 4 ρ 0 ε 0 ( Ω m 2 Ω 2 + i Ω Γ q 2 ) ξ m f 2 ξ m 2 .
A j 2 f ( r ) f ( r ) ( 2 i β i A j z ± β j 2 A j )
= [ n 2 ( r ) i n α c ω ] f ( r ) c 2 A j ω 2 U ˜ ξ m ( r ) f ( r ) c 2 ω 2 A j A 3 j 2 ,

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