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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 7 — Apr. 7, 2014
  • pp: 7932–7946
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A quasi-mode interpretation of acoustic radiation modes for analyzing Brillouin gain spectra of acoustically antiguiding optical fibers

Kyoungyoon Park and Yoonchan Jeong  »View Author Affiliations


Optics Express, Vol. 22, Issue 7, pp. 7932-7946 (2014)
http://dx.doi.org/10.1364/OE.22.007932


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Abstract

We propose a novel quasi-mode interpretation (QMI) method to represent acoustic radiation modes in acoustically antiguiding optical fibers (AAOFs) in terms of discrete quasi-modes. The QMI method readily enables one to obtain the full quasi-modal properties of AAOFs, including the complex propagation constants, mode center frequencies, and field distributions in an intuitive and much simplified way, compared to other previous methods. We apply the QMI method to analyze the Brillouin gain spectrum of an AAOF that has typically been used to mitigate stimulated Brillouin scattering of optical waves. The result based on the QMI method is in good agreement with the numerical and experimental results for the same fiber structure previously reported in the literature. Considering the effectiveness and simplicity of its numerical procedure, we expect the use of the QMI method can further be extended to even more complicated numerical analyses with acoustic radiation modes, which include the acoustically antiguiding, large-core optical fibers in multi-mode regimes.

© 2014 Optical Society of America

1. Introduction

Stimulated Brillouin scattering (SBS) is a nonlinear optical scattering process resonated between optical waves and acoustic waves. While SBS is sometimes actively utilized to generate specific Stokes signals, it can also severely limit the power scaling of optical fiber sources if they are of high-power as well as of narrow linewidths [1

1. A. R. Chraplyvy, “Limitation on lightwave communication imposed by optical-fiber nonlinearities,” J. Lightwave Technol. 8(10), 1548–1557 (1990). [CrossRef]

6

6. Q. Fang, W. Shi, K. Kieu, E. Petersen, A. Chavez-Pirson, and N. Peyghambarian, “High power and high energy monolithic single frequency 2 μm nanosecond pulsed fiber laser by using large core Tm-doped germanate fibers: experiment and modeling,” Opt. Express 20(15), 16410–16420 (2012). [CrossRef]

]. Consequently, the mitigation of SBS has become an important issue with high-power, narrow-linewidth fiber sources, so that in recent years various SBS mitigation methods have been investigated via exploiting novel optical fiber designs and configurations [7

7. 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(6), 969–972 (1993). [CrossRef]

18

18. P. D. Dragic, “Novel dual-Brillouin-frequency optical fiber for distributed temperature sensing,” Proc. SPIE 7197, 719710 (2009). [CrossRef]

]. In particular, the use of acoustically antiguiding optical fibers (AAOFs) [10

10. 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]

13

13. P. D. Dragic, “Brillouin suppression by fiber design,” in Photonics Society Summer Topical Meeting Series, (IEEE, 2010), Paper TuC3.2.

] is one of the most efficient tactics to alleviate SBS while other acoustically tailored optical fibers, such as modified acoustically guiding optical fibers, can also be considered alternatives to AAOFs [13

13. P. D. Dragic, “Brillouin suppression by fiber design,” in Photonics Society Summer Topical Meeting Series, (IEEE, 2010), Paper TuC3.2.

17

17. S. Yoo, C. A. Codemard, Y. Jeong, J. K. Sahu, and J. Nilsson, “Analysis and optimization of acoustic speed profiles with large transverse variations for mitigation of stimulated Brillouin scattering in optical fibers,” Appl. Opt. 49(8), 1388–1399 (2010). [CrossRef] [PubMed]

]. In fact, AAOFs lead to low field overlaps between optical waves and acoustic waves in the fiber core, thereby significantly reducing the “effective” Brillouin gain coefficient for the optical waves [10

10. 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]

13

13. P. D. Dragic, “Brillouin suppression by fiber design,” in Photonics Society Summer Topical Meeting Series, (IEEE, 2010), Paper TuC3.2.

]. From a viewpoint of geometrical acoustics, acoustic modes existing in the core region of AAOFs are guided not by total internal reflection (TIR) but by Fresnel reflection (FR) at the core-cladding interface because the acoustic refractive index of the core is lower than that of the cladding. As a result, in the core of an AAOF, non-core-guided (NCG) acoustic modes result in dominant acousto-optic interaction with optical modes [19

19. L. Dong, “Limits of stimulated Brillouin scattering suppression in optical fibers with transverse acoustic waveguide designs,” J. Lightwave Technol. 21, 3156–3161 (2010).

, 20

20. L. Dong, “Formulation of a complex mode solver for arbitrary circular acoustic wave guides,” J. Lightwave Technol. 21, 3162–3175 (2010).

] although they are not totally bounded in the core. Therefore, an accurate analysis of NCG acoustic modes is crucial in determining or designing the Brillouin gain spectrum (BGS) of an AAOF for SBS mitigation.

Apart from experimental demonstrations of AAOFs [11

11. M. D. Mermelstein, M. J. Andrejco, J. Fini, A. Yablon, C. Headley III, D. J. DiGiovanni, and A. H. McCurdy, “11.2 dB SBS gain suppression in a large mode area Yb-doped optical fiber,” Proc. SPIE 6873, U63–U69 (2008). [CrossRef]

13

13. P. D. Dragic, “Brillouin suppression by fiber design,” in Photonics Society Summer Topical Meeting Series, (IEEE, 2010), Paper TuC3.2.

], a rigorous theoretical study on AAOFs has first been reported in [10

10. 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]

], including its comparison with experimental results. The BGS was obtained through extensive numerical calculations based on a complete set of acoustic cladding modes (ACMs), i.e., 143 modes in total, with a presumptive boundary condition that the outer cladding is stress-free. In fact, these ACMs consist of two different types of NCG acoustic modes: One is a set of modes bounded mainly in the inner-cladding region and forbidden or evanescent in the core region (defined by “core-evanescent acoustic cladding modes: CE-ACMs”), such that their displacement vectors are represented by normal Bessel functions in the inner cladding and modified Bessel functions in the core, respectively. The other is a set of modes bounded across both inner cladding and core regions, such that their displacement vectors are represented by normal Bessel functions in both inner cladding and core (defined by “core-passing acoustic cladding modes: CP-ACMs”). In fact, the main difference between CE-ACMs and CP-ACMs is noted by whether the radial oscillation of acoustic displacement vectors or the radial phonon propagation is allowed or forbidden in the core region. In this approach, the influence of the phonon lifetime of acoustic modes is considered separately [10

10. 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]

].

2. Stimulated Brillouin scattering and acoustic modal analysis

In general, SBS in optical fibers is understood in a way that an optical wave is backscattered via Bragg reflection caused by the density variation slowly propagating in the forward direction that is basically induced by photo-elastically produced acoustic waves. From a viewpoint of quantum mechanics, SBS can also be regarded as a scattering process between photons and acoustic phonons. Thus, it can readily be deduced that the strength of Brillouin scattering is proportional to the field overlap factors between optical waves and acoustic waves [26

26. 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]

28

28. P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19(10), 4986–4998 (1979). [CrossRef]

], which are, consequently, of great importance in properly analyzing SBS phenomena in optical fibers. Thus, one can start with the analysis, deriving the field overlap factors between optical waves and acoustic waves as the following.

3. Theory of acoustic quasi-modes

3.1. Acoustic waves in an optical fiber

In general, one can equivalently express an acoustic wave, i.e., its density variation, in terms of a displacement vector u, which is given by u=ura^r+uϕa^ϕ+uza^z in the cylindrical coordinate system [31

31. R. A. Waldron, “Some problems in the theory of guided microsonic waves,” IEEE Trans. Microw. Theory Tech. 17(11), 893–904 (1969). [CrossRef]

]. In optical fibers the longitudinal component of the displacement vector, uz, dominantly contributes to the density variation and thereby to SBS because the Stokes waves are resonantly backscattered by the longitudinally distributed, optical refractive index variations, regardless of their acoustic waveguiding properties whether they are acoustically guiding or antiguiding [32

32. N. Shibata, K. Okamoto, and Y. Azuma, “Longitudinal acoustic modes and Brillouin-gain spectra for GeO2-doped-core single mode fibers,” J. Opt. Soc. Am. B 6(6), 1167–1174 (1989). [CrossRef]

]. Thus, one can only take the longitudinal component of the displacement vector into account to represent the density variation [10

10. 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]

, 21

21. K. J. Chen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guiding fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 33(6), 634–643 (1986). [CrossRef] [PubMed]

].

Here, we define the longitudinal component of displacement vector that satisfies the acoustic wave equation as [31

31. R. A. Waldron, “Some problems in the theory of guided microsonic waves,” IEEE Trans. Microw. Theory Tech. 17(11), 893–904 (1969). [CrossRef]

]
uz=[iqAaXp(klr)ksBaZp(ksr)]cos(pϕ)ei(Ωtqz),
(4)
where Aa and Ba are proportional constants, Xpand Zp are linear combinations of the 1st and 2nd kind Bessel functions with
kl2=Ω2vl2q2=ρ0Ω2λ+2μq2,andks2=Ω2vsq2=ρ0Ω2μq2,
(5)
where vl and vs are acoustic velocities of longitudinal and shear waves, respectively. The Lamé constants λ and μare determined by acoustic material properties, vl, vsand ρas λ=ρ0(vl22vs2) and μ=ρ0vs2, respectively. In particular, we havep=0 since we assume that there is no azimuthal variation in acoustic modes. In Eq. (4), in fact, both acoustic longitudinal waves (represented by the first term withkl) and shear waves (represented by the second term with ks) are coupled into uz. Thus, it seems that one should consider both waves simultaneously to describe uz. However, in general the velocities of longitudinal waves are significantly larger than those of shear waves [10

10. 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]

]. As a result, we normally have kl2<ks2 for an optical fiber of an acoustic antiguide structure. This implies that shear waves result in highly oscillating field patterns in the radial direction while longitudinal waves result in relatively slowly oscillating radial field patterns. Consequently, shear waves’ contributions to the field overlap factors with optical waves would be substantially smaller than those of longitudinal waves [17

17. S. Yoo, C. A. Codemard, Y. Jeong, J. K. Sahu, and J. Nilsson, “Analysis and optimization of acoustic speed profiles with large transverse variations for mitigation of stimulated Brillouin scattering in optical fibers,” Appl. Opt. 49(8), 1388–1399 (2010). [CrossRef] [PubMed]

]. Therefore, the shear waves’ contributions in Eq. (4) are obviously limited. Based on the fact, the longitudinal component of the displacement vector is now simplified into

uz=iqAaX0(klr)ei(Ωtqz).
(6)

In addition, from the acoustic continuity equation [33

33. L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of acoustics, 4th ed. (Wiley, 2010), Chap. 5.

], one can see that the density variation ρis proportional to u, so that ρbecomes proportional to uz as well, because we assume that density variation ρ depends dominantly on uz. Thus, one can evaluate the BGS using the longitudinal component of the displacement vector instead of using the density variation. It is not too difficult to check that Eq. (6) also satisfies the homogeneous acoustic Helmholtz equation having no acoustic damping and driving terms in Eq. (1). Then, transverse field pattern of Eq. (6) becomes the same as the corresponding field pattern of the density variation, i.e., ξm.

3.2. Quasi-mode interpretation of acoustic radiation modes

To explain the QMI theory for analyzing acoustic modes in an AAOF with simplicity, we adopt a typical acoustic step-index antiguide structure as shown in Fig. 1(a)
Fig. 1 (a) Acoustic step-index antiguide structure: An ARM is guided by FR because the acoustic effective refractive index is lower than the lowest index of the fiber materials. (b) Acoustical and optical material parameters.
and 1(b), simply modified from the fiber 2 given in [10

10. 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]

] although it is not limited to a specific design of AAOF. The core radius a is given by 4.3 μm, and the optical wavelength is assumed to be 1.55 μm.

Let us consider an acoustic wave as illustrated in Fig. 1(a). While a part of the acoustic wave propagates in the core, being guided by FR, the other part radiates from the core into the unbounded cladding region. As a result, the total power flow in the core region decreases with z because of the consecutive FRs between the core and cladding boundary. In other words, ARMs are partially guided modes with attenuation, so that they become leaky from a view point of the power flow within a finite cross-section of the given fiber [24

24. Y. Jeong, B. Lee, J. Nilsson, and D. J. Richardson, “A quasi-mode interpretation of radiation modes in long-period fiber gratings,” IEEE J. Quantum Electron. 39(9), 1135–1142 (2003). [CrossRef]

]. Since the power density of the ARM confined in the core of the given fiber is approximately represented by Aa2 as shown in Fig. 2
Fig. 2 The normalized power density factor Aa2 for the step-index AAOF shown in Fig. 1, where AQMm denotes the m-th acoustic longitudinal quasi-modes. (b) The normalized power density factor Aa2 for AQM1 in (a) and its Lorentzian fit.
, we define Aa2 as the power density factor of the corresponding ARM. (See Appendix for more details of the representation of ARMs and the evaluation of Aa2). Unlike guide modes, the solution for the power density factor Aa2forms a continuum, composed of repeating, Lorentzian-like peaks in the propagation-constant domain as shown in Fig. 2(a).

While it is assumed that all ARMs are normalized to carry the same acoustic power (see Appendix for details), the power densities confined within the fiber crucially depends on the propagation constant of the mode as shown in Fig. 2(a). That is, at certain positions in terms of the propagation constant, the power density factor has local maxima. We emphasize that this feature observed in Fig. 2(a) is in a similar form with those of electromagnetic radiation modes discussed in [24

24. Y. Jeong, B. Lee, J. Nilsson, and D. J. Richardson, “A quasi-mode interpretation of radiation modes in long-period fiber gratings,” IEEE J. Quantum Electron. 39(9), 1135–1142 (2003). [CrossRef]

] where the concept of the QMI of optical radiation modes was first proposed and discussed. In fact, the homogeneous Helmholtz equation for longitudinal acoustic waves is basically equivalent to that for electromagnetic waves [see Eq. (1)]. Thus, one is sufficiently guaranteed to utilize the QMI method for the analysis of longitudinal ARMs as well.

Since a general expression for an NCG acoustic wave can be given by a linear combination of ARMs [as to be shown in Eq. (7)], we define an acoustic “quasi-mode” (AQM) as a group of localized ARMs with a specific distribution centered by the local maximum of Aa2 if the specific distribution eventually yields the locally highest probability in terms of the field confinement in the fiber by Cauchy-Schwarz inequality [24

24. Y. Jeong, B. Lee, J. Nilsson, and D. J. Richardson, “A quasi-mode interpretation of radiation modes in long-period fiber gratings,” IEEE J. Quantum Electron. 39(9), 1135–1142 (2003). [CrossRef]

].

It is worth noting that a prefix “quasi” is used because an AQM is not directly obtained as an eigenstate from the characteristic equation but is obtained by appropriately combining eigenstates (ARMs), thereby indicating its “mode-like” behavior as a whole. In addition, based on the Fourier analysis, it is not too difficult for one to deduce that the specific distribution should result in a Lorentzian function for the power flow function of the resultant AQM if one assumes that it exponentially attenuates with propagation. In other words, the Fourier transform of a Lorentzian function of the propagation constant (q) into the spatial domain (z) results in an exponentially decaying function. Thus, it turns out that a Lorentzian function eventually yields the best fit to a localized, individual peak of the power density factor Aa2 if the ARMs form a well-defined AQM. A more detailed mathematical derivation and proof can be found in [24

24. Y. Jeong, B. Lee, J. Nilsson, and D. J. Richardson, “A quasi-mode interpretation of radiation modes in long-period fiber gratings,” IEEE J. Quantum Electron. 39(9), 1135–1142 (2003). [CrossRef]

].

As a typical example, we zoom in on one localized peak that is the rightmost one in Fig. 2(a) together with a Lorentzian fitting function, which are now shown in Fig. 2(b). We denote |F(q)|2 as the normalized function of Aa2. While in Fig. 2(b) there are small discrepancies between the Lorentzian fitting function and |F(q)|2, particularly, in the edge regions, their impacts are limited as long as the contribution from the central part of the Lorentzian distribution is significantly dominant compared to those from the edges. Then, it is straightforward to obtain the complex propagation constant (qm) of the corresponding AQM by reading the peak position (resulting in qmr the real part of qm), and the half width at the half maximum (HWHM) of the selected Lorentzian peak (resulting in qmithe imaginary part ofqm). In a similar manner, one can obtain all other higher-order AQMs, finding qm’s matched with the corresponding peaks. It is noteworthy that under the QMI method, it is not necessary to solve complicated, complex-valued transcendental equations to determine modes, which eventually leads to a considerable simplification of the whole numerical procedure.

Based on Eq. (7) we obtain the radial field patterns for the three lowest-order AQMs as shown in Fig. 4(a)
Fig. 4 (a) Radial field patterns of AQMs based on the QMI method, neglecting the contributions of shear waves. (b) Radial field patterns of CP-ACMs obtained by the ACM method given in [10], considering both longitudinal waves and shear waves.
, which actually satisfy the phase- and frequency-matching conditions shown in Fig. 3(a). One can see that the field patterns follow the same trend as those of guided modes for r<a, by noting that the number of peaks of the field variation for r<a is proportional to the order of the mode as typical as well-defined normal modes [35

35. K. F. Graff, Wave Motion in Elastic Solids (Dover Publications, 1991), Chap. 8.

]. The undamped, oscillatory field patterns for r>a indicates an outward radiating nature [34

34. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic Press, 1991), Chap. 1–2.

].

The field patterns of AQMs based on the QMI method, shown Fig. 4(a), are generally in a good agreement with the field patterns of CP-ACMs based on the ACM method in [10

10. 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]

] separately shown in Fig. 4(b). However, it should be noted that with the ACM method, we had to go through elaborate numerical calculations to determine 186 ACMs for the given fiber [see Fig. 1(a)], likewise the use of 143 acoustic modes in [10

10. 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]

].

3.3. Quasi-mode interpretation in the frequency domain

In fact, the QMI method applied in the frequency domain essentially leads to the identical outcome, which must be obvious as both results shown in Fig. 5 are obtained based on the same phase- and frequency-matching conditions. In fact, Fig. 5(b) is a snapshot of Fig. 5(a) when the latter satisfies the phase-matching conditions. Consequently, the QMI method applied in the frequency domain offers a great convenience to directly determining both mode center frequencies and Brillouin linewidths for the individual AQMs at the same time, skipping the redundant calculation steps required to obtain the dispersion curves shown in Fig. 3(a) although they do have physical meanings.

4. Verification the QMI method to the Brillouin gain analysis of an acoustically antiguiding optical fiber

To verify our acoustic quasi-mode theory, we analyze the BGS of fiber 2 previously investigated in [10

10. 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]

] by means of the QMI method. We think that this comparative verification is important because fiber 2 has a very typical structure frequently utilized in AAOFs and the experimentally measured BGS of fiber 2 is also well presented. In fact, the fiber has a core made of pure silica, and its cladding is doped with fluorine to increase its acoustical refractive index while decreasing its optical refractive index. The core radius is given by 4.3 μm, the optical wavelength 1.55 μm, and the fluorine concentrations of the core and the cladding 0 and 1.3 wt%, respectively. The optical refractive index profile (ORIP) and the acoustical refractive index profile (ARIP) are illustrated in Fig. 6(a)
Fig. 6 (a) Acoustical and optical refractive index profiles of an AAOF following fiber 2 in [10]. (b) Dispersion relations (top figure) and the power density factor Aa2(bottom figure), indicating the mode center frequencies for individual AQMs. (c) The radial field patterns of the three lowest-order AQMs.
.

It is worth noting that all the simulation parameters are exactly the same as given in [10

10. 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]

], including dopant concentrations, optical refractive indices, acoustical longitudinal velocities, shear velocities, and glass densities, while there may be small discrepancies in terms of dealing with the materials’ phonon lifetimes and the detailed shape of the acoustic refractive index profile, which are not explicitly stated in the report. In our calculations the material’s intrinsic Brillouin linewidth is assumed to be 35 MHz.

Since the fiber is based on an acoustic antiguide structure, it supports ARMs or AQMs partially guided by FR in the core region while there exist some CE-ACMs when one considers the second boundary between the inner cladding and outer cladding. It should be noted that we do not discuss in detail the existence of CE-ACMs here, because the numerical procedure to obtain them were well described in [10

10. 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]

]. Applying the QMI method in the frequency domain, we readily obtain the mode center frequencies and Brillouin linewidths for the individual AQMs. For each mode, its radial field pattern is also given by Eq. (7). It should be noted that even though there are higher-order AQMs other than the lowest three AQMs evaluated in Fig. 6(c), the higher-order AQMs have substantially small contributions to the BGS because of their weak acousto-optic field overlaps and short photon lifetimes. Based on the field pattern, the acousto-optic effective mode area is determined by Eq. (3). Then, by simply substituting the center frequencies, Brillouin linewidths, and acousto-optic effective mode areas into Eq. (2), the total BGS is straightforwardly evaluated. In addition, the CE-ACMs can readily be found by the ACM method presented in [10

10. 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]

].

5. Conclusion

Appendix

Assuming that the acoustic waves are axially symmetric and that the contributions of shear waves to the displacement vector are negligible, one can derive the longitudinal component of the displacement vector for an ARM as follows [31

31. R. A. Waldron, “Some problems in the theory of guided microsonic waves,” IEEE Trans. Microw. Theory Tech. 17(11), 893–904 (1969). [CrossRef]

]:
uz=iqAaX0(klr)ei(Ωtqz)=iqAaUz(r)ei(Ωtqz)=iqηa(r)ei(Ωtqz),
(8)
Uz(r)={J0(kl1r),ra1B1J0(kl2r)+B2Y0(kl2r),a1<ra2C1J0(kl3r)+C2Y0(kl3r),a2<ra3D1J0(kl4r)+D2Y0(kl4r),a3<r
(9)
with kli=(Ω/vli)2q2. The positions of a1, a2 and a3are depicted in Fig. 6(a). The radial component of the displacement vector is obtained as
ur(r)=AaddrUz(r)ei(Ωtqz)=AaUr(r)ei(Ωtqz),
(10)
Ur(r)={kl1J0(kl1r),ra1B1kl2J0(kl2r)+B2kl2Y0(kl2r),a1<ra2C1kl3J0(kl3r)+C2kl3Y0(kl3r),a2<ra3D1kl4J0(kl4r)+D2kl4Y0(kl4r),a3<r.
(11)
Then, Bi,Ci,andDi(i=1,2) can be expressed in terms of Aa from the acoustic boundary conditions that both the displacement vector components (uzand ur) and the stress components (T1and T5) are continuous across the boundaries [31

31. R. A. Waldron, “Some problems in the theory of guided microsonic waves,” IEEE Trans. Microw. Theory Tech. 17(11), 893–904 (1969). [CrossRef]

]. In addition, the stress components can also be determined by the method described in [31

31. R. A. Waldron, “Some problems in the theory of guided microsonic waves,” IEEE Trans. Microw. Theory Tech. 17(11), 893–904 (1969). [CrossRef]

].

By the boundary conditions, one can have
[uruzT1T5]layer1r=a1=[uruzT1T5]layer2r=a1andQ1(r=a1)[10]=[uruzT1T5]layer1r=a1,
(12)
where Q1(r=a1) is a 4 × 2 matrix. It should be noted that one can obtain two more equations in the same form of Eq. (12) from the boundary conditions at r=a2 and r=a3. Plugging the coefficient matrices, Qi(r=aj)(i,j=1,2,3,4) derived in the same way as in Eq. (12) into the boundary conditions, one can obtain
[B1B2]=[Q2(r=a1)]1[Q1(r=a1)][10]=M1[10],[C1C2]=M2M1[10],and[D1D2]=M3M2M1[10],
(13)
where Mi=[Qi+1(r=ai)]1[Qi(r=ai)](i=1,2,3). Thus, one finally obtains the longitudinal acoustic fields in terms of Aa2.

Since we assume that acoustic waves are normalized to have the same average power flow in the z direction, regardless of modes, one can derive the acoustic power flow in the z direction, which satisfies the following orthogonal relation [36

36. B. A. Auld, Acoustic Fields and Waves in Solids Volume 1 (Wiley, 1973), Chap. 5.

]:
14{-vn*Tm-vm*Tn}a^zdS=P0δ(qmqn)
(14)
wherev and Tcan readily be derived from the acoustic fields we obtain from Eq. (9), (11) and (13). After some straightforward algebra, one can obtain [24

24. Y. Jeong, B. Lee, J. Nilsson, and D. J. Richardson, “A quasi-mode interpretation of radiation modes in long-period fiber gratings,” IEEE J. Quantum Electron. 39(9), 1135–1142 (2003). [CrossRef]

, 34

34. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic Press, 1991), Chap. 1–2.

]
Aa2=CA0D12+D22
(15)
where CA0 is a proportional constant depending on the power flow of the given acoustic mode. It should be noted that since Aacan be a function of either q or Ω, fixing one variable as constant, one can readily apply the QMI method either in the propagation-constant domain or in the frequency domain.

Acknowledgment

This work was supported in part by the Ministry of Trade, Industry and Energy (Project no. 10040429).

References and links

1.

A. R. Chraplyvy, “Limitation on lightwave communication imposed by optical-fiber nonlinearities,” J. Lightwave Technol. 8(10), 1548–1557 (1990). [CrossRef]

2.

Y. Jeong, J. K. Sahu, D. J. Richardson, and J. Nilsson, “Seeded erbium/ytterbium codoped fibre amplifier source with 87 W of single-frequency output power,” Electron. Lett. 39(24), 1717–1719 (2003). [CrossRef]

3.

Y. Jeong, J. K. Sahu, D. B. Soh, C. A. Codemard, and J. Nilsson, “High-power tunable single-frequency single-mode erbium:ytterbium codoped large-core fiber master-oscillator power amplifier source,” Opt. Lett. 30(22), 2997–2999 (2005). [CrossRef] [PubMed]

4.

Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. 30(5), 459–461 (2005). [CrossRef] [PubMed]

5.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency Ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007). [CrossRef]

6.

Q. Fang, W. Shi, K. Kieu, E. Petersen, A. Chavez-Pirson, and N. Peyghambarian, “High power and high energy monolithic single frequency 2 μm nanosecond pulsed fiber laser by using large core Tm-doped germanate fibers: experiment and modeling,” Opt. Express 20(15), 16410–16420 (2012). [CrossRef]

7.

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(6), 969–972 (1993). [CrossRef]

8.

A. Kobyakov, S. Kumar, D. Chowdhury, A. B. Ruffin, M. Sauer, S. R. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express 13(14), 5338–5346 (2005). [CrossRef] [PubMed]

9.

S. Gray, D. T. Walton, X. Chen, J. Wang, M.-J. Li, A. Liu, A. B. Ruffin, J. A. Demeritt, and L. A. Zenteno, “Optical fibers with tailored acoustic speed profiles for suppressing stimulated Brillouin scattering in high-power,” J. Lightwave Technol. 15, 37–46 (2009).

10.

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]

11.

M. D. Mermelstein, M. J. Andrejco, J. Fini, A. Yablon, C. Headley III, D. J. DiGiovanni, and A. H. McCurdy, “11.2 dB SBS gain suppression in a large mode area Yb-doped optical fiber,” Proc. SPIE 6873, U63–U69 (2008). [CrossRef]

12.

P. D. Dragic, “Ultra-flat Brillouin gain spectrum via linear combination of two acoustically anti-guiding optical fibers,” Electron. Lett. 48(23), 1492–1493 (2012). [CrossRef]

13.

P. D. Dragic, “Brillouin suppression by fiber design,” in Photonics Society Summer Topical Meeting Series, (IEEE, 2010), Paper TuC3.2.

14.

P. D. Dragic, C. H. Liu, G. C. Papen, and A. Galvanauskas, “Optical fiber with an acoustic guiding layer for stimulated Brillouin scattering suppression,” in Proceedings of the Conference on Lasers and Electro-optics, 2005 OSA Technical Digest Series (Optical Society of America, 2005), paper CThZ3. [CrossRef]

15.

S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. Ruffin, J. A. Demeritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17044–17050 (2007). [CrossRef] [PubMed]

16.

M. J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007). [CrossRef] [PubMed]

17.

S. Yoo, C. A. Codemard, Y. Jeong, J. K. Sahu, and J. Nilsson, “Analysis and optimization of acoustic speed profiles with large transverse variations for mitigation of stimulated Brillouin scattering in optical fibers,” Appl. Opt. 49(8), 1388–1399 (2010). [CrossRef] [PubMed]

18.

P. D. Dragic, “Novel dual-Brillouin-frequency optical fiber for distributed temperature sensing,” Proc. SPIE 7197, 719710 (2009). [CrossRef]

19.

L. Dong, “Limits of stimulated Brillouin scattering suppression in optical fibers with transverse acoustic waveguide designs,” J. Lightwave Technol. 21, 3156–3161 (2010).

20.

L. Dong, “Formulation of a complex mode solver for arbitrary circular acoustic wave guides,” J. Lightwave Technol. 21, 3162–3175 (2010).

21.

K. J. Chen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guiding fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 33(6), 634–643 (1986). [CrossRef] [PubMed]

22.

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]

23.

P. D. Dragic, P. C. Law, and Y. S. Liu, “Higher order modes in acoustically antiguiding optical fiber,” Microw. Opt. Technol. Lett. 54(10), 2347–2349 (2012). [CrossRef]

24.

Y. Jeong, B. Lee, J. Nilsson, and D. J. Richardson, “A quasi-mode interpretation of radiation modes in long-period fiber gratings,” IEEE J. Quantum Electron. 39(9), 1135–1142 (2003). [CrossRef]

25.

K. Park and Y. Jeong, “A quasi-mode interpretation of acoustic radiation modes for the analysis of acoustically antiguiding optical fibers,” in Advanced Solid-State Lasers, (Optical Society of America, Paris, 2013), Paper ATu3A.08.

26.

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]

27.

Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. 137(6A), A1787–A1805 (1965). [CrossRef]

28.

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19(10), 4986–4998 (1979). [CrossRef]

29.

R. W. Boyd, Nonlinear optics, 3rd ed. (Academic Press, 2008), Chap. 9.

30.

L. Tartara, C. Codemard, J. 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]

31.

R. A. Waldron, “Some problems in the theory of guided microsonic waves,” IEEE Trans. Microw. Theory Tech. 17(11), 893–904 (1969). [CrossRef]

32.

N. Shibata, K. Okamoto, and Y. Azuma, “Longitudinal acoustic modes and Brillouin-gain spectra for GeO2-doped-core single mode fibers,” J. Opt. Soc. Am. B 6(6), 1167–1174 (1989). [CrossRef]

33.

L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of acoustics, 4th ed. (Wiley, 2010), Chap. 5.

34.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic Press, 1991), Chap. 1–2.

35.

K. F. Graff, Wave Motion in Elastic Solids (Dover Publications, 1991), Chap. 8.

36.

B. A. Auld, Acoustic Fields and Waves in Solids Volume 1 (Wiley, 1973), Chap. 5.

OCIS Codes
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(290.5830) Scattering : Scattering, Brillouin

ToC Category:
Nonlinear Effects in Fibers

History
Original Manuscript: January 15, 2014
Revised Manuscript: March 3, 2014
Manuscript Accepted: March 18, 2014
Published: March 27, 2014

Virtual Issues
2013 Advanced Solid State Lasers (2013) Optics Express

Citation
Kyoungyoon Park and Yoonchan Jeong, "A quasi-mode interpretation of acoustic radiation modes for analyzing Brillouin gain spectra of acoustically antiguiding optical fibers," Opt. Express 22, 7932-7946 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-7932


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References

  1. A. R. Chraplyvy, “Limitation on lightwave communication imposed by optical-fiber nonlinearities,” J. Lightwave Technol. 8(10), 1548–1557 (1990). [CrossRef]
  2. Y. Jeong, J. K. Sahu, D. J. Richardson, J. Nilsson, “Seeded erbium/ytterbium codoped fibre amplifier source with 87 W of single-frequency output power,” Electron. Lett. 39(24), 1717–1719 (2003). [CrossRef]
  3. Y. Jeong, J. K. Sahu, D. B. Soh, C. A. Codemard, J. Nilsson, “High-power tunable single-frequency single-mode erbium:ytterbium codoped large-core fiber master-oscillator power amplifier source,” Opt. Lett. 30(22), 2997–2999 (2005). [CrossRef] [PubMed]
  4. Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. 30(5), 459–461 (2005). [CrossRef] [PubMed]
  5. Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, P. W. Turner, “Power scaling of single-frequency Ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13(3), 546–551 (2007). [CrossRef]
  6. Q. Fang, W. Shi, K. Kieu, E. Petersen, A. Chavez-Pirson, N. Peyghambarian, “High power and high energy monolithic single frequency 2 μm nanosecond pulsed fiber laser by using large core Tm-doped germanate fibers: experiment and modeling,” Opt. Express 20(15), 16410–16420 (2012). [CrossRef]
  7. C. A. S. de Oliveira, C. K. Jen, A. Shang, C. Saravanos, “Stimulated Brillouin scattering in cascaded fibers of different Brillouin frequency shift,” J. Opt. Soc. Am. B 10(6), 969–972 (1993). [CrossRef]
  8. A. Kobyakov, S. Kumar, D. Chowdhury, A. B. Ruffin, M. Sauer, S. R. Bickham, R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express 13(14), 5338–5346 (2005). [CrossRef] [PubMed]
  9. S. Gray, D. T. Walton, X. Chen, J. Wang, M.-J. Li, A. Liu, A. B. Ruffin, J. A. Demeritt, L. A. Zenteno, “Optical fibers with tailored acoustic speed profiles for suppressing stimulated Brillouin scattering in high-power,” J. Lightwave Technol. 15, 37–46 (2009).
  10. Y. Koyamada, S. Sato, S. Nakamura, H. Sotobayashi, W. Chujo, “Simulating and designing Brillouin gain spectrum in single mode fibers,” J. Lightwave Technol. 22(2), 631–639 (2004). [CrossRef]
  11. M. D. Mermelstein, M. J. Andrejco, J. Fini, A. Yablon, C. Headley, D. J. DiGiovanni, A. H. McCurdy, “11.2 dB SBS gain suppression in a large mode area Yb-doped optical fiber,” Proc. SPIE 6873, U63–U69 (2008). [CrossRef]
  12. P. D. Dragic, “Ultra-flat Brillouin gain spectrum via linear combination of two acoustically anti-guiding optical fibers,” Electron. Lett. 48(23), 1492–1493 (2012). [CrossRef]
  13. P. D. Dragic, “Brillouin suppression by fiber design,” in Photonics Society Summer Topical Meeting Series, (IEEE, 2010), Paper TuC3.2.
  14. P. D. Dragic, C. H. Liu, G. C. Papen, and A. Galvanauskas, “Optical fiber with an acoustic guiding layer for stimulated Brillouin scattering suppression,” in Proceedings of the Conference on Lasers and Electro-optics, 2005 OSA Technical Digest Series (Optical Society of America, 2005), paper CThZ3. [CrossRef]
  15. S. Gray, A. Liu, D. T. Walton, J. Wang, M.-J. Li, X. Chen, A. B. Ruffin, J. A. Demeritt, L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15(25), 17044–17050 (2007). [CrossRef] [PubMed]
  16. M. J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007). [CrossRef] [PubMed]
  17. S. Yoo, C. A. Codemard, Y. Jeong, J. K. Sahu, J. Nilsson, “Analysis and optimization of acoustic speed profiles with large transverse variations for mitigation of stimulated Brillouin scattering in optical fibers,” Appl. Opt. 49(8), 1388–1399 (2010). [CrossRef] [PubMed]
  18. P. D. Dragic, “Novel dual-Brillouin-frequency optical fiber for distributed temperature sensing,” Proc. SPIE 7197, 719710 (2009). [CrossRef]
  19. L. Dong, “Limits of stimulated Brillouin scattering suppression in optical fibers with transverse acoustic waveguide designs,” J. Lightwave Technol. 21, 3156–3161 (2010).
  20. L. Dong, “Formulation of a complex mode solver for arbitrary circular acoustic wave guides,” J. Lightwave Technol. 21, 3162–3175 (2010).
  21. K. J. Chen, A. Safaai-Jazi, G. W. Farnell, “Leaky modes in weakly guiding fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 33(6), 634–643 (1986). [CrossRef] [PubMed]
  22. A. Safaai-Jazi, C. K. Jen, G. W. Farnell, “Analysis of weakly guiding fiber acoustic waveguide,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 33(1), 59–68 (1986). [CrossRef] [PubMed]
  23. P. D. Dragic, P. C. Law, Y. S. Liu, “Higher order modes in acoustically antiguiding optical fiber,” Microw. Opt. Technol. Lett. 54(10), 2347–2349 (2012). [CrossRef]
  24. Y. Jeong, B. Lee, J. Nilsson, D. J. Richardson, “A quasi-mode interpretation of radiation modes in long-period fiber gratings,” IEEE J. Quantum Electron. 39(9), 1135–1142 (2003). [CrossRef]
  25. K. Park and Y. Jeong, “A quasi-mode interpretation of acoustic radiation modes for the analysis of acoustically antiguiding optical fibers,” in Advanced Solid-State Lasers, (Optical Society of America, Paris, 2013), Paper ATu3A.08.
  26. S. Dasgupta, F. Poletti, S. Liu, P. Petropoulos, D. J. Richardson, L. Grüner-Nielsen, 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]
  27. Y. R. Shen, N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. 137(6A), A1787–A1805 (1965). [CrossRef]
  28. P. J. Thomas, N. L. Rowell, H. M. van Driel, G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19(10), 4986–4998 (1979). [CrossRef]
  29. R. W. Boyd, Nonlinear optics, 3rd ed. (Academic Press, 2008), Chap. 9.
  30. L. Tartara, C. Codemard, J. Maran, R. Cherif, M. Zghal, “Full modal analysis of the Brillouin gain spectrum of an optical fiber,” Opt. Commun. 282(12), 2431–2436 (2009). [CrossRef]
  31. R. A. Waldron, “Some problems in the theory of guided microsonic waves,” IEEE Trans. Microw. Theory Tech. 17(11), 893–904 (1969). [CrossRef]
  32. N. Shibata, K. Okamoto, Y. Azuma, “Longitudinal acoustic modes and Brillouin-gain spectra for GeO2-doped-core single mode fibers,” J. Opt. Soc. Am. B 6(6), 1167–1174 (1989). [CrossRef]
  33. L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of acoustics, 4th ed. (Wiley, 2010), Chap. 5.
  34. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic Press, 1991), Chap. 1–2.
  35. K. F. Graff, Wave Motion in Elastic Solids (Dover Publications, 1991), Chap. 8.
  36. B. A. Auld, Acoustic Fields and Waves in Solids Volume 1 (Wiley, 1973), Chap. 5.

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