The numerical study of acoustic modal properties in w-shaped optical fibers with high-delta germanium-doped core and F-doped inner cladding (F-HDF) is demonstrated. The cutoff conditions of acoustic modes in the F-HDF show opposite behaviors in contrast with those of optical ones because F-doped inner cladding contributes differently to acoustic and optical waveguides. The acoustic dispersion characteristics vary to a great extent with respect to the location of the acoustic modes in the fiber’s core or in the fiber’s inner cladding. The resonance frequency spacing between neighboring acoustic modes is theoretically and experimentally found to have a quadratic relation to the core’s germanium concentration. We also investigate the critical conditions to move high-order acoustic modes into the F-doped inner cladding and validate the optimal feasibility of employing L 01 and L 03 acoustic modes to fiber-optic Brillouin-based discriminative sensing of strain and temperature.
©2008 Optical Society of America
Stimulated Brillouin scattering (SBS) phenomenon in optical fibers is generated from the nonlinear interaction among counter-propagating optical waves and thermally-initialized acoustic waves that are intensified by the optical waves’ interference via the electro-striction effect. Meanwhile, SBS phenomenon can also happen when only a strong optical wave is launched into optical fibers with optical power beyond the so-called threshold value. The SBS phenomenon in optical fibers induces a big trouble to optical fiber communications [1–4] but provides a good opportunity to build up novel fiber-optic sensors [5, 6].
The counter-propagating optical waves have frequency deviations from each other that are determined by the resonance frequencies of the existing acoustic modes in optical fibers. The resonance frequency of the main-peak Brillouin gain spectrum (BGS) generated from the fundamental L 01 acoustic mode, that is, Brillouin frequency shift (BFS), has a linear relationship to applied strain or temperature change. The linear relationship has been used to build a type of fully-distributed fiber-optic Brillouin sensors [5, 6] for smart materials and structures. However, Brillouin-based sensors encounter a physical difficulty in discriminating the response to strain from that to temperature. To solve this problem, a w-shaped triple-layer single-mode optical fiber structure with a high-delta GeO2-doped core and F-doped inner cladding (F-HDF) was recently proposed and investigated . Owing to the possible variance of F-doped silica from GeO2-doped silica, four resonance frequencies in that F-HDF from various L 0l longitudinal acoustic modes hold different dependence behaviors to strain and temperature. The combination of 1st-order L 01 together with 4th-order L 04 modes provided a lowest discriminative error. This is because the L 04 acoustic mode locates nearest to the F-doped inner cladding although still in the GeO2-doped core. It was expected that the discriminative sensing performance could be enhanced if high-order acoustic mode, such as L 02 or L 03 mode, was moved into the F-doped inner cladding with comparable Brillouin gain in contrast with the fundamental L 01 acoustic mode that is always confined in the GeO2-doped core.
In this paper, we demonstrate the theoretical analysis and control of acoustic modal properties in the F-HDF. The cutoff conditions of acoustic modes and the acoustic dispersion characteristics are thoroughly studied, which clarify the recently observed abnormal Brillouin scattering phenomenon in an F-HDF . We also investigate the critical conditions to move high-order acoustic modes into the F-doped inner cladding. The optimal feasibility of utilizing L 01 and L 03 acoustic modes for discriminative sensing is verified by setting fiber parameters to let L 03 mode below its critical condition while just close to its cutoff condition.
2. Mathematical principles
2.1. Optical refractive index and longitudinal acoustic index
The F-HDF includes triple layers: GeO2-doped core (r≤a 1), F-doped inner cladding (a 1≤r≤a 2) and pure-silica outer cladding (a 2≤r≤b 0) where r is a radial coordinate in the cylindrical optical fiber, a 1 the core radius, a 2 the outer radius of the inner cladding and b 0 the outer radius of the outer cladding (i.e., ~62.5 µm). The optical properties of F-HDF structure such as the optical cutoff condition and the modified optical dispersion properties have been comprehensively reported by Monerie  and the fabrication of the F-HDF has been demonstrated in .
Figure 1 schematically illustrates the refractive index ni profile (solid curve) and the longitudinal acoustic velocity Vli profile (dashed curve) of the F-HDF where the subscripts i=1, 2, 0 correspond to the core, inner cladding and outer cladding, respectively. In this study, we introduce a longitudinal acoustic index Ni, as also plotted in Fig. 1 (dotted curve), which is defined as the ratio of the longitudinal acoustic velocity of pure silica (V l0, 5944 m/s ) to that of GeO2-doped or F-doped silica (Vli):
From Fig. 1, one can qualitatively know that the F-doped inner-cladding layer depresses the optical waveguide but enhances the acoustic waveguide, which has been preliminarily clarified in . The ni and Ni are dependent on the ith-layer dopant concentrations:
where w 1 and w 2 are the concentrations of GeO2 and F in units of molecular percent (mol%) and weight percent (wt%), respectively. It is worth noting that the above relationships are basically referred to  but the unit of GeO2 concentration w 1 is thought to be mol% rather than wt% according to Ref.  and therein references in Ref. . We also note that the influence of the residual stress during fiber fabrication  is not taken into account in this study although SBS properties could be modified to some extent by the residual stress . The indices n 0=1.444 in 1.55-µm region  and N 0=1.0 correspond to pure silica.
2.2. Optical scalar-wave equation
We redefine the physical core and the physical inner cladding of an F-HDF as the calculated core region and the physical outer cladding as the calculated cladding region for the purpose of two-dimensional finite-element-method (2D-FEM) modal analysis of its BGS. The optical mode in the redefined calculated core region satisfies the following normalized scalar-wave equation:
where ∇2 a=∂2/∂x′2+∂2/∂y′2 is the transverse Laplacian in a coordinate normalized to the outer radius of the inner cladding a 2 (i.e., x′=x/a 2,y′=y/a 2); Ei, the transverse electric field of the fundamental LP 01 optical mode [i.e., Ex(x′,y′) for HEx 11 mode and Ey(x′,y′) for HEy 11 mode]. vop, Pop, and Wop are the optical normalized frequency, the normalized refractive index distribution, and the normalized optical transverse wave number, respectively, which are given by
where k 0 (=2π/λ 0) is the free-space optical wave number with λ 0 (=1.549 µm in this study) the optical wavelength in vacuum, ni(r′) with r′=r/a 2 is the radial refractive index distribution in the normalized coordinates, and neff is the optical effective refractive index that determines the effective propagation constant βop=k 0 neff.
The optical boundary condition at the interface between the calculated core region and the calculated cladding region can be expressed as :
where n⃗ is the outward pointing normal on the boundary and Km(Wop) is the mth-order second-kind modified Bessel function.
2.3. Acoustic scalar-wave equation
where ∇2=∂2/∂x 2+∂2/∂y 2, βac is the propagation constant of the longitudinal acoustic modes decided by the Bragg-like phase-matching condition as βac=2βop, and ω (l) ac is the lth-order angular resonance frequency that is relative to βac and the effective lth-order longitudinal acoustic velocity V (l) a as w (l) ac=V (l) a βac.
Taken into account the defined acoustic indices in Eq. (1), the above acoustic scalar-wave equation in the calculated core region can be modified into the similar scalar-wave equation to Eq. (4) in the normalized coordinates:
where vac, Pac, and Wac denote the normalized acoustic frequency, the normalized acoustic index distribution, and the normalized acoustic transverse wavenumber, respectively, which are defined by:
where Ni(r′) is the longitudinal acoustic index distribution and N (l) eff is the effective acoustic index of the longitudinal lth-order L 0l acoustic mode that determines the L 0l mode’s effective acoustic velocity as V (l) a=V l0/N (l) eff. As follows, the lth-order resonance frequency ν (l) ac=βac·V (l) a/2π in the F-HDF’s BGS can be expressed as:
The boundary condition of the longitudinal L 0l acoustic mode also satisfies the Eq. (6) except for the substitution of the acoustic eigenvalue Wac for the optical eigenvalue Wop.
2.4. Numerical method
The 2D-FEM modal analysis  is used to solve Eq. (4) for optical modes and Eq. (8) for acoustic modes in the F-HDF, respectively, both based on the boundary equation of Eq. (6). Consequently, the lth-order L 0l acoustic resonance frequency in Eq. (10) is quantified; the lth-order Brillouin gain (gl) is evaluated which is inversely proportional to the acousto-optic effective area A ao(l) eff  as
where the inner product <f>=∫∫f(x′,y′)dx′dy′ is over all the calculated core and cladding regions.
3. Modal analysis and modal control
3.1. Cutoff conditions
Our numerical result, as illustrated in Fig. 2, shows that vac is 5.37 to 6.00 times of vop, which means that the longitudinal acoustic modes sense better confinement than the optical mode. This characteristic is responsible for the existence of multiple-peak BGS in a GeO2-doped single-mode fiber (SMF) and even in the standard step-index SMF , and also responsible for the greatest acousto-optic overlapping efficiency (i.e., 100%) of the fundamental L 01 acoustic mode among the multiple-peak BGS.
The optical cutoff normalized frequencies v0op for LP 01 and LP 11 modes are quantified at different concentration ratio R=w 2/w 1 and radius ratio S=a 2/a 1. To do that, the optical cutoff condition of Wop=0 (i.e., neff=n 0) in Eq. (5) is preset for the boundary condition of Eq. (6) as follows:
respectively, for LP 01 or LP 11.
The optical cutoff characteristics after our numerical analysis are depicted in Fig. 3(a). The v0op is increased when the concentration ratio R or radius ratio S is increased, and further the fundamental LP 01 optical mode has a nonzero cutoff v0op. Our results are in good agreement with previously-reported ones . The physical reason is because the F-doped inner cladding depresses the optical waveguide as mentioned above in Fig. 1.
Similarly, we evaluate the longitudinal acoustic cutoff properties that are summarized in Fig. 3(b) for L 02 or L 03 acoustic mode. The cutoff v0ac becomes reductive when the concentration ratio R or radius ratio S increases, which coincides with our previous estimation that F-doped inner cladding acts as an enhanced waveguide layer for acoustic modes [see Fig. 1].
3.2. Acoustic dispersion properties
We numerically solve the Eq. (8) for different vac at different concentration ratio R or radius ratio S to evaluate the dispersion curves of L 0l acoustic modes in the F-HDF which denote the relations between Wac/S and vac.
The dispersion curves of L 01 (blue solid curves), L 02 (green dashed curves) and L 03 (red dotted curves) acoustic modes for the concentration ratio R=0.05 and the GeO2 concentration w 1=10 mol% are plotted in Fig. 4(a) where a cluster of curves in the same color corresponds to different radius ratios S (=1.0, 2.0, 3.5, and 5.0 from right side to left side). Similarly, Fig. 4(b) denotes the corresponding acoustic dispersion curves at the same parameters as those of Fig. 4(a) except for R=0.10. In Figs. 4(a) or (b), a black solid linear curve as a based line corresponds to the case when Neff=N 2, which means that the acoustic mode is just cutoff at the interface between the physical core and the physical inner cladding.
A crossing point (Al with l=1, 2 or 3) between lth-order acoustic dispersion curve and the based line in Fig. 4 determines a critical value of v(l) c-ac. If vac is just below the v(l) c-ac then N (l) eff<N 2, which means that the lth-order acoustic mode enters into the F-doped physical inner cladding; vice versa, the lth-order acoustic mode still exists in the GeO2-doped physical core. For example, when w 1=10 mol%, R=0.05 and S=1.5, v(3) c-ac=8.12 for the 3rd-order L 03 acoustic mode. Figure 5 illustrates our evaluated field distributions of the fundamental LP 01 optical mode and all L 0l acoustic modes in the calculated core region for the above fiber parameters of w 1, R and S but at different acoustic vac. It can be clearly seen that when vac becomes smaller than v(3) c-ac the L 03 field enters more into the F-doped physical inner cladding.
Comparing Fig. 4(a) against Fig. 4(b), we know that the critical value v(l) c-ac is almost independent on the radius ratio S; however, it is sensitive to the concentration ratio R: when R is increased, it is correspondingly increased. These dependences can be further understood from Fig. 6 where the lth-order acoustic mode’s critical value v(l) c-ac and the (l+1)th-order acoustic mode’s cutoff value v(l+1) 0ac are plotted together as functions of the concentration ratio R for different radius ratio S.
In fact, a starting point of lth-order acoustic dispersion curve from the horizontal axis in Fig. 4 corresponds to its cutoff value v(l) 0ac. From it, we can clearly see that the cutoff value v(l) 0ac is reduced when the radius ratio S is increased, which means that the higher-order acoustic modes become more difficult to be cut off by the pure-silica outer cladding. Further, when the radius ratio S is increased, the parts of the dispersion curves below the based line become closer, which results in closer acoustic modes existing in the F-doped inner cladding. These analyzed properties can well explain the experimental observation in Ref.  where two groups of BGS were observed in that F-HDF: one group including two separate peaks is due to its location in the GeO2-doped core but the other group including three closer peaks due to its location in the F-doped region. Figure 7 depicts our simulated BGS of the F-HDF demonstrated by Yeniay et al. , in which a dashed line is used to mark the two disparate groups of BGS existing in the GeO2-doped core and the F-doped inner cladding, respectively.
Figure 8(a) illustrates the analyzed acoustic dispersion curves for two different w 1 of 3.65 mol% and 10 mol%. From it, we know that they match each other within less than 1% difference. This similarity further shows that the distinction of Wac/S between neighboring acoustic modes is almost independent on w 1 for a constant vac. From Eq. (10) together with Eq. (5) and Eq. (9), we can deduce
where b=W 2 op/v2 op is the normalized optical propagation constant. The resonance frequency spacing (Δv (l) ac≡v (l+1) ac-v (l) ac) between neighboring longitudinal acoustic modes can be further deduced to be
According to the definition of vac in Eq. (9), for a fixed vac meaning a fixed vop and b approximately, an increase of w 1 corresponds to a reduction of the core size a 1 resulting in a smaller resonance frequency v (l) ac [see Eq. (14)]. Furthermore, according to Eq. (15), a quadratic increase of the neighboring resonance frequency spacing arises from a increase of w 1 or a reduction of a 1. This is the reason why a high-delta optical fiber (HDF) with highly GeO2-doped core is preferred in this study since the enlarged frequency spacing is helpful to improve the measurement system performance in discriminative sensing of strain and temperature , and is useful to increase the sensitivity of higher-order resonance frequency to the change of the fiber parameters as will be described below.
The above theoretical prediction is experimentally confirmed by measuring the entire BGS in a 3.65-mol% step-index SMF and in a 17.0-mol% HDF via pump-probe SBS-based experimental configuration . The experimental results are depicted in Fig. 8(b) showing that the neighboring frequency spacing is ~50–60 MHz for the step-index SMF while ~700–720 MHz for the HDF.
4. Application for discriminative sensing
At first, we investigate the possibility of L 01 and L 02 acoustic modes for discriminative sensing, for which L 03 acoustic mode is cutoff by the pure-silica outer cladding and L 02 acoustic mode is moved into the F-doped inner cladding. Note that the fundamental L 01 acoustic mode is always located in the GeO2-doped core since its critical v(1) c-ac is extremely low, such as less than 1.9 as illustrated in Fig. 4, which corresponds to vop=~0.35 according to Fig. 2. The modal control conditions can be understood from Fig. 6(a) in which the crossing point decides a set of R and vac-value (vop-value correspondingly). A smaller S gives a greater vac-value or vop-value providing better waveguiding efficiency. In this case, our calculation shows that the vop is necessarily small, for instance, vac=~4.82 (or vop=~0.85) and R=0.05 when S=2 and w 1=10 mol%. The corresponding optical effective area defined by
is evaluated to be ~103 µm 2, which means that the optical waveguiding efficiency in the physical core is extremely weak. In fact, this fiber design is impractical even although the Brillouin gain of L 02 acoustic mode located in the F-doped inner cladding could be even greater than that of the fundamental L 01 acoustic mode located in the GeO2-doped core.
To ensure the optical waveguiding efficiency, we propose to utilize L 03 acoustic mode together with the fundamental L 01 acoustic mode for discriminative sensing, for which L 04 acoustic mode is cutoff by the outer cladding and L 03 acoustic mode is moved into the inner cladding while L 01 and L 02 acoustic modes are maintained in the core. Again, as depicted in Fig. 6(b), a smaller radius ratio S (e.g.,=1.5) is chosen to get a greater crossed vac-value of ~8.12 corresponding to vop-value of ~1.43 and a larger R=0.05 when w 1=10 mol%.
Figure 9(a) depicts the analyzed resonance frequencies changing as a function of vop from vop=1.43 to vop=1.15, which corresponds to the range from below the crossed vac-value (~8.12) to beyond the L 03 mode’s cutoff v(3) 0ac (~5.93). The resonance frequency change of the L 03 acoustic mode is ~7 times as that of the fundamental L 01 acoustic mode providing a higher sensitivity to the fiber parameter’s change. The Brillouin gain ratios (g 1/g 2 and g 1/g 3) and the optical effective area (Aeff) are also evaluated and thus plotted in Fig. 9(b), respectively. When vop (vac correspondingly) decreases, the optical effective area (Aeff) increases because the optical confinement is weakened as can be seen from Fig. 5. On the other hand, the Brillouin gain (g3) of the 3rd-order L 03 acoustic mode becomes more comparable to the fundamental L 01 acoustic mode when compared to the g 2 of the 2rd-order L 02 acoustic mode. This is because the displacement field of the 3rd-order L 03 acoustic mode enters more into the F-doped inner-cladding region for a smaller acoustic vac (<v(3) c-ac) while the field of L 02 acoustic mode dominantly confined in the GeO2-doped core has tiny change with vac (see Fig. 5). Consequently, comparing to L 02 acoustic mode, L 03 acoustic mode has greater acousto-optic overlapping efficiency with the weakly-guided LP 01 optical mode.
Furthermore, these analyzed results show a feasibility of utilizing L 01 and L 03 acoustic mode for discriminative sensing of strain and temperature by choosing vop=1.15 or vac=6.53. Firstly, Aeff can be kept to be as low as ~53 µm 2. Secondly, the simulated BGS of the optimized fiber design illustrated in Fig. 10 shows that the Brillouin gain of L 03 acoustic mode is only ~-5 dB lower than that of the fundamental L 01 acoustic mode. This is because the acoustic vac=6.53 is very close to the L 03 acoustic mode’s cutoff v(3) 0ac=~5.93, so that the existence of the L 03 acoustic field in the F-doped inner cladding is significantly enhanced (see Fig. 5(d)).
We have demonstrated the theoretical and numerical analysis of optical and acoustic modal properties in the F-HDF including cutoff conditions and acoustic dispersion curves. With appropriately designed parameters (radii and dopant concentrations), we can put the L 03 longitudinal acoustic mode into the F-doped inner cladding effectively with comparable Brillouin gain in contrast with the fundamental L 01 longitudinal acoustic mode and with more sensitive change (~6 times) of the resonance frequency than that of the fundamental L 01 mode. When the L 01 and L 03 resonance BGS in the optimally designed F-HDF are utilized for fiber-optic Brillouin sensors, an improvement of our preliminarily investigated accuracy of discriminative measurement of strain and temperature (e.g., strain error of 44 µε and temperature error of 1.8 °C ) is hopefully achievable.
This work was supported by the “Grant-in-Aid for Creative Scientific Research” and the “Global Center of Excellence Program (G-COE)” from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
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