A quasi-mode interpretation of acoustic radiation modes for analyzing Brillouin gain spectra of acoustically antiguiding optical fibers

Kyoungyoon Park; Yoonchan Jeong

doi:10.1364/OE.22.007932

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 [1–6]. 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 [7–18]. In particular, the use of acoustically antiguiding optical fibers (AAOFs) [10–13] 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 [13–17]. 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 [10–13]. 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, 20] 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 [11–13], a rigorous theoretical study on AAOFs has first been reported in [10], 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].

Recently, a different numerical method for analyzing the consequential effects of the CP-ACMs has been reported in [20] via revisiting the concept of leaky acoustic modes (LAMs) [21, 22] with an unbounded cladding, in which an acoustic mode solver, capable of finding both guided and leaky modes for an arbitrary circular acoustic waveguide, was developed and utilized to visualize the dispersion relations of acoustic modes. This method was followed by another similar approach considering higher-order acoustic modes (HOAMs) [23], which simplified the LAM method, removing the radiated acoustic power in the unbounded cladding from the normalization. While both the aforementioned methods are capable of providing with a numerical tool for analyzing AAOFs with significantly reduced number of acoustic modes, they still require solving complex-valued, transcendental characteristic equations, dealing with appropriate acoustic boundary conditions, which indeed makes it hard for one to intuitively capture the physical origin of the leaky modal behaviors of the corresponding acoustic modes.

Thus, considering the immediate importance of the comprehensive understanding of the leaky modal behavior of acoustic waves in AAOFs, the development of a novel concept to deal with NCG acoustics modes, such as CP-ACMs or equivalently LAMs, is still in demand. If so, such a concept must be capable of presenting physical insight into the detailed modal behaviors of acoustic modes and their interrelationships resulting in SBS. This should also lead to a significantly simplified numerical methodology to get rid of too elaborate numerical procedures normally required in the previous approaches.

Here, we propose a quasi-mode-interpretation (QMI) method capable of considering NCG acoustic modes as acoustic radiation modes (ARMs) that can be simplified into discrete acoustic quasi-modes (AQMs) [24], on which a preliminary discussion has been presented in [25]. We emphasize that the QMI method is methodologically simple and physically intuitive in terms of dealing with NCG acoustic modes in AAOFs, thereby helping one understand the detailed physics behind the acoustic modal characteristics in AAOFs. Based on AQMs, we fully analyze an AAOF that has typically been used for SBS suppression [10], and discuss its modal characteristics and BGS. We finally verify and discuss the validity and effectiveness of the QMI method by comparing our result with the experimental one reported in [10].

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 [26–28], 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.

We define an optical pump wave and the corresponding backscattered Stokes wave as $E_{p} = A_{p} ψ_{p} (r, ϕ) e^{i (ω_{p} t - β_{p} z)}$ and $E_{s} = A_{s} ψ_{s} (r, ϕ) e^{i (ω_{s} t + β_{s} z)}$ , respectively. In general, the pump and Stokes waves are represented by the optical fundamental mode of the given fiber [10]. It is noteworthy that we only consider axially symmetric acoustic waves throughout out our discussion, assuming that they are only driven by the optical fundamental mode of the given fiber that is, presumably, axially symmetric, too. Then, the acoustic waves produced via an SBS process are expressed by the density variation, i.e., $ρ^{'} (r, t) = ρ (r, t) - ρ_{0} (r)$ , which is governed by the following material equation [29]:

\frac{\partial^{2} ρ^{'}}{\partial t^{2}} - Γ^{'} \nabla^{2} \frac{\partial ρ^{'}}{\partial t} - v^{2} \nabla^{2} ρ^{'} = ε_{0} γ_{e} q^{2} A_{p} A_{s}^{*} ψ_{p} ψ_{s}^{*} e^{i (Ω t - q z)}

where

v

is the phase velocity of the acoustic longitudinal wave,

Γ^{'}

the acoustic damping parameter, and

γ_{e}

the electrostrictive constant, respectively. It is worth noting that Eq. (1) implicitly includes the energy and momentum conservation conditions given by

Ω = ω_{p} - ω_{s}

and

q = β_{p} + β_{s}

, respectively. The acoustic vibration is driven at the Brillouin frequency shift

Ω

incurred by the frequency difference between the pump and Stokes waves, which is expressed as a source term in the right side of Eq. (1). Then, it is obvious that as the driving frequency

Ω

approaches to the center frequency

Ω_{m}

of an acoustic normal mode, SBS strongly occurs, owing to the resonance between the acoustic wave and the driving electrostriction. It is also worth noting that the acoustic normal mode is determined assuming that there are no acoustic damping and source terms. Thus, without loss of generality, one can readily obtain that the general solution of Eq. (1) is given by a linear combination of the all possible acoustic normal modes as the following [20]:

ρ^{'} (r, t) = [\sum_{m} ρ_{0 m} ξ_{m} (r, ϕ)] e^{i (Ω_{m} t - q z)}

, where

ξ_{m} (r, ϕ)

is the normalized acoustic field pattern in the transverse plane. Then, the total Brillouin gain is obtained as [20]:

G (Ω) = G_{0} \sum_{m} \frac{1}{σ_{m}^{a o} Γ_{m}} \frac{{(Γ_{m} / 2)}^{2}}{{(Ω - Ω_{m})}^{2} + {(Γ_{m} / 2)}^{2}}

where

G_{0}

denotes a proportional coefficient related with the medium, which is independent of modal characteristics,

Γ_{m}

the Brillouin linewidth for the m-th acoustic normal mode that is related with the modal propagation loss in the fiber, including both the material loss and waveguide loss, and

σ_{m}^{a o}

the corresponding acousto-optic effective mode area in the transverse plane of S, which is given by [20]

σ_{m}^{a o} = \frac{{(\int {| ψ |}^{2} d S)}^{2} {\int | ξ_{m} |}^{2} d S}{{| \int ψ^{*} ξ_{m} ψ d S |}^{2}} .

It is worth noting that the acousto-optic effective mode area is inversely proportional to the acousto-optic mode overlap factor. In addition, the Brillouin linewidth

Γ_{m}

in Eq. (2) can be expressed as

Γ_{m} = 2 q_{m i} v_{m}

[20], where

q_{m i}

is the imaginary part of the complex propagation constant given by

q_{m} = q_{m r} + i q_{m i}

, and

v_{m}

is the phase velocity of the m-th acoustic longitudinal normal mode.

In fact, Eq. (2) indicates that the entire Brillouin gain is given by the simple summation of individual Lorentzian functions of the mode center frequency $Ω_{m}$ and the Brillouin linewidth $Γ_{m}$ , the peak value of which is proportional to the mode overlap factor between the m-th acoustic normal mode and the optical fundamental mode [see Eq. (3)]. This is due to the fact that the contributions of individual modes to SBS are statistically independent [30]. In addition, as shown in Eq. (2), (3) and the expression of the Brillouin linewidth, each Lorentzian function is determined by the mode center frequency, complex propagation constant, and radial field patterns of the corresponding acoustic normal mode. Thus, in order to quantify the BGS properly, these three parameters must be determined accurately. For example, based on the ACM method for an AAOF, the BGS is formed by the superposition of all the Lorentzian functions for ACMs, such as 143 modes in total [10], which need to be determined individually. However, this extensive numerical procedure can readily be simplified by the QMI method to be discussed because the contributions from a large number of CP-ACMs can be quantified by a few of dominant AQMs that represent the whole ARMs in the given AAOF [24, 25]. While some previous numerical models could also represent the effect of CP-ACMs with a few discrete modes [20, 23], they instead needed to deal with complex-valued, transcendental characteristic equations, which is simply unnecessary for the QMI method.

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 $u = u_{r} {\hat{a}}_{r} + u_{ϕ} {\hat{a}}_{ϕ} + u_{z} {\hat{a}}_{z}$ in the cylindrical coordinate system [31]. In optical fibers the longitudinal component of the displacement vector, $u_{z}$ , 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]. Thus, one can only take the longitudinal component of the displacement vector into account to represent the density variation [10, 21].

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

u_{z} = [- i q A_{a} X_{p} (k_{l} r) - k_{s} B_{a} Z_{p} (k_{s} r)] \cos (p ϕ) e^{i (Ω t - q z)},

where

A_{a}

and

B_{a}

are proportional constants,

X_{p}

and

Z_{p}

are linear combinations of the 1st and 2nd kind Bessel functions with

k_{l}^{2} = \frac{Ω^{2}}{v_{l}^{2}} - q^{2} = \frac{ρ_{0} Ω^{2}}{λ + 2 μ} - q^{2}, and k_{s}^{2} = \frac{Ω^{2}}{v_{s}} - q^{2} = \frac{ρ_{0} Ω^{2}}{μ} - q^{2},

where

v_{l}

and

v_{s}

are acoustic velocities of longitudinal and shear waves, respectively. The Lamé constants

λ

and

μ

are determined by acoustic material properties,

v_{l}

,

v_{s}

and

ρ

as

λ = ρ_{0} (v_{l}^{2} - 2 v_{s}^{2})

and

μ = ρ_{0} v_{s}^{2}

, respectively. In particular, we have

p = 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 with

k_{l}

) and shear waves (represented by the second term with

k_{s}

) are coupled into

u_{z}

. Thus, it seems that one should consider both waves simultaneously to describe

u_{z}

. However, in general the velocities of longitudinal waves are significantly larger than those of shear waves [10]. As a result, we normally have

k_{l}^{2} < k_{s}^{2}

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]. 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

u_{z} = - i q A_{a} X_{0} (k_{l} r) e^{i (Ω t - q z)} .

In addition, from the acoustic continuity equation [33], one can see that the density variation $ρ^{'}$ is proportional to $\nabla \cdot u$ , so that $ρ^{'}$ becomes proportional to $u_{z}$ as well, because we assume that density variation $ρ^{'}$ depends dominantly on $u_{z}$ . 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., $ξ_{m}$ .

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) and 1(b), simply modified from the fiber 2 given in [10] 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.

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.

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On the other hand, since each term of Eq. (6) is a solution to the homogeneous acoustic Helmholtz equation, one can define an acoustic radiation mode, i.e., an ARM, as a mode whose effective mode index is lower than the lowest of the waveguide materials, which is guided not by TIR but by FR. It is worth noting that acoustic refractive index is defined as $n_{a} = v_{l}^{c o r e} / v_{l}$ . In fact, the concept of ARMs is considered the same as that of electromagnetic radiation modes [24, 34]. While ARMs are conceptually equivalent to CP-ACMs [10], with ARMs the second boundary placed at the inner-cladding/outer-cladding interface can be ignored similarly to the way the previous numerical approaches with LAMs or HOAMs did [20, 23], because the phonon lifetime of such a mode in silica is normally much shorter than the phonon’s “time of flight” from the core to the cladding edge, which means there is nearly no bounce-back effect by the inner-cladding edge [20, 23]. In fact, this can significantly simplify the numerical procedure in dealing with NCG acoustic modes, such as CP-ACMs.

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]. Since the power density of the ARM confined in the core of the given fiber is approximately represented by $A_{a}^{2}$ as shown in Fig. 2, we define $A_{a}^{2}$ as the power density factor of the corresponding ARM. (See Appendix for more details of the representation of ARMs and the evaluation of $A_{a}^{2}$ ). Unlike guide modes, the solution for the power density factor $A_{a}^{2}$ forms a continuum, composed of repeating, Lorentzian-like peaks in the propagation-constant domain as shown in Fig. 2(a).

Fig. 2 The normalized power density factor $A_{a}^{2}$ for the step-index AAOF shown in Fig. 1, where AQM_m denotes the m-th acoustic longitudinal quasi-modes. (b) The normalized power density factor $A_{a}^{2}$ for AQM₁ in (a) and its Lorentzian fit.

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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] 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 $A_{a}^{2}$ if the specific distribution eventually yields the locally highest probability in terms of the field confinement in the fiber by Cauchy-Schwarz inequality [24].

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 $A_{a}^{2}$ if the ARMs form a well-defined AQM. A more detailed mathematical derivation and proof can be found in [24].

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 $A_{a}^{2}$ . 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 ( $q_{m}$ ) of the corresponding AQM by reading the peak position (resulting in $q_{m r}$ the real part of $q_{m}$ ), and the half width at the half maximum (HWHM) of the selected Lorentzian peak (resulting in $q_{m i}$ the imaginary part of $q_{m}$ ). In a similar manner, one can obtain all other higher-order AQMs, finding $q_{m}$ ’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.

Thus, once all AQMs of interest have been found, one can readily obtain the dispersion relations between q and Ω as depicted in Fig. 3. In particular, one can notice that there are a couple of interesting features in the dispersion relations. One is that the dispersion relations are basically similar to those of guided modes. In other words, for lower-order modes or higher frequencies, the acoustic power flux tends to be concentrated in the core region. While AQMs are partially guided in the core by FR, one can still see the typical modal characteristics of guided modes by TIR. The other is that one can readily determine the individual mode center frequencies to construct the BGS given in Eq. (2), utilizing the dispersion relation curves shown in Fig. 3(a) [20]. For example, the long dashed line in Fig. 3(a) denotes the dispersion relation curve of the first-order AQM (AQM₁), and the momentum conservation line indicates the phase-matching condition for SBS, i.e., $q = 2 β_{p}$ . Thus, the intersection point between the dispersion relation curve and the momentum conservation line denotes the phase-matching frequency at which Brillouin scattering is resonated. That is, at the intersection point the phase- and frequency-matching conditions are satisfied at the same time, thereby leading to a strong acousto-optic interaction between AQM₁ and optical waves. It is worth noting that in Fig. 3(a) the dispersion relation curves exhibit monotonic variations which are typical for well-defined normal modes since the number of peaks of the field variation for $r < a$ is maintained as long as the modal order is fixed [35], which is a bit different from what one can see with the LAM model given in [20]. In addition, the attenuation coefficients $q_{m i}$ ’s are shown in Fig. 3(b). One can see that as frequency increases, the attenuation coefficients tend to decrease, while they increase with the order of the mode. This implies that the phonon lifetime reduction due to FR becomes more significant for higher-order AQMs.

Fig. 3 (a) Dispersion relations and (b) attenuation coefficients of the three lowest-order AQMs.

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With the evaluated complex propagation constant at the mode center frequency of each AQM, the field pattern of the corresponding AQM in the transverse plane is readily obtained as

ξ_{q m} = ξ_{q m 0} η_{m} (r) e^{i (Ω t - q_{m r} z)} e^{- q_{m i} z},

where

ξ_{q m 0}

is a proportional constant and

η_{m} (r)

is given by

η_{m} (r) = \int F (γ) η_{a} (r, γ) d γ

[see Appendix for

F (γ)

and

η_{a} (r, γ)

]. That is, an individual AQM is defined by a linear combination of ARMs with a specific distribution, i.e., a specific weighting function

F (γ)

, which eventually maximizes the power flow of the resultant field within the fiber. In addition, it should be noted that the variable

γ

denotes the deviation of the propagation constant

q

from a specific modal propagation constant

q_{m r}

.

Based on Eq. (7) we obtain the radial field patterns for the three lowest-order AQMs as shown in Fig. 4(a), 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]. The undamped, oscillatory field patterns for $r > a$ indicates an outward radiating nature [34].

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.

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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] 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].

In addition, it is worth noting that when the field patterns based on the ACM method were calculated, both acoustic longitudinal and shear waves were considered at the same time to match the acoustic boundary condition [31]. Comparing the two results, one can also see that our assumption of neglecting the shear wave contribution to $u_{z}$ is valid and well supported. In addition, the fields denoted by ACM₁, ACM₂, and ACM₃ shown in Fig. 4(b), are actually the selected acoustic modes which yield the three largest acousto-optic field overlap factors among the 186 acoustic modes. It is worth noting that the mode center frequencies obtained by the QMI method are nearly the same as those by the ACM method, as given in the insets of Fig. 4(a) and 4(b). This is very intriguing feature of the QMI method and highlights that the QMI method intrinsically leads to a set of ARMs having a specific spectral distribution, with which it will result in the best confinements within the given fiber, thereby yielding the maximum modal field overlaps with the optical fundamental mode of the fiber [24].

3.3. Quasi-mode interpretation in the frequency domain

In the previous section, we have applied the QMI method in the propagation-constant domain, and obtained the complex propagation constant for each AQM. However, the mathematical analogy between space and time in wave equations hints that one can also apply the QMI method in the frequency domain, given that the propagation constant is fixed via the phase-matching condition, i.e., $q = 2 β_{p}$ . Then, it is not too difficult to represent the power density factor $A_{a}^{2}$ in terms of the frequency of acoustic waves for a fixed propagation constant $q$ , as shown in Fig. 5(b). Obviously, the peak positions of the power density factor $A_{a}^{2}$ coincide with the intersection points between the dispersion curves and the momentum conservation line as shown in Fig. 5(a). It should be noted that the peak position and full width at the half maximum (FWHM) of each Lorentzian lobe denote, respectively, the mode center frequency and linewidth of the individual BGS.

Fig. 5 Concurrence of mode center frequencies: (a) Dispersion relations of AQMs by the QMI method represented in the propagation-constant domain [the same figure as Fig. 3(a)]. (b) The frequency spectrum of $A_{a}^{2}$ by the QMI method represented in the frequency domain of. The 4th and 5th peaks in (b) are also matched with the intersection points from AQM₄ and AQM₅ although they are not shown in (a).

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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] 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 $A_{a}^{2}$ (bottom figure), indicating the mode center frequencies for individual AQMs. (c) The radial field patterns of the three lowest-order AQMs.

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

The evaluated BGS is shown in Fig. 7, where the contributions of AQMs and CE-ACMs to the total BGS are superposed as indicated. For comparison, the measured experimental data numerically sampled from Fig. 4 in [10] is presented at the same time. It is worth noting that the substantially higher peak at ~11.2 GHz is due mainly to AQMs while the peaks at the lower shoulder at ~10.8 GHz are due mainly to CE-ACMs. The detailed BGS parameters for the individual AQMs are summarized in Table 1. We emphasize that the BGS obtained by the QMI method is in good agreement with the experimental result presented in [10] as well as with the numerical results by other previous models [10, 20]. One can readily see that in the acoustic antiguide structure AQMs or ARMs play an important role in forming the BGS because they have significantly larger acousto-optic mode overlaps and narrower linewidths than those by CE-ACMs. Thus, an accurate evaluation of the contributions of AQMs or ARMs must be crucial for determining the BGS of AAOFs.

Fig. 7 The total BGS of the AAOF given in Fig. 6(a) via the QMI method together with the ACM method. The dotted graph denotes the measured experimental data numerically sampled from Fig. 4 in [10].

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Table 1. Summary of the Individual Brillouin Gain Spectra due to AQMs

View Table

5. Conclusion

We have proposed a novel numerical methodology for analyzing AAOFs in terms of ARMs and AQMs that can be an immediate alternative way to the previously existing ones based on CP-ACMs, LAMs, and HOAMs [10, 20, 23]. The proposed method substantially simplifies the extensive numerical procedures normally required from the previous methods, removing the necessity for solving complicated, complex-valued transcendental characteristic equations. This method also clarifies modal characteristics of acoustic modes in AAOFs, including complex propagation constants, dispersion relations and radial field patterns. It is worth noting that this describes in an intuitively physical and mathematical manner the leaky nature of the CP-ACMs or LAMs, which is, in fact, due to the radiating nature of the acoustic modes incurred by the partial FR between the core and cladding interface. In addition, the proposed QMI method enables one to consider a continuum of ARMs with specific distributions as discrete AQMs with propagation constants of complex values, which is a unique feature compared to conventional mode analyses that determine modes primarily relying on the transcendental characteristic equations.

Based on the proposed theoretical model on the QMI of ARMs, we analyzed a typical AAOF design [10] which is normally utilized to alleviate SBS in optical fiber. Our numerical result on the BGS of the given AAOF is in good agreement with the experimental result given in [10] as well as with the numerical results by other previous models [10, 20]. We emphasize that the QMI method is mathematically simple and physically intuitive. It can deal with the continuum of ARMs into discrete AQMs with much simplified numerical steps to obtain the mode center frequencies, Brillouin linewidths, and acousto-optic effective mode areas required for constructing the total BGS. In addition, it should be noted that the definition of AQMs is conceptually different from the conventional one of LAMs [19–22] while both are represented by complex-valued propagation constants: AQMs are formed by a continuum of ARMs having a particular distribution yielding the highest probability in terms of the field confinement in the fiber, which actually results in a Lorentzian distribution. In fact, the QMI method is not limited to the Brillouin gain analysis of AAOFs in optically single-mode regimes. Because of its numerical simplicity, its application can further be extended to the analysis of the BGS of AAOFs in optically multi-mode large-core regimes [2–6, 11] in which more complicated inter-modal features are expected to be involved. We highlight that the QMI method can be a good numerical basis for the optical fiber designs for SBS suppression.

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]:

u_{z} = - i q A_{a} X_{0} (k_{l} r) e^{i (Ω t - q z)} = - i q A_{a} U_{z} (r) e^{i (Ω t - q z)} = - i q η_{a} (r) e^{i (Ω t - q z)},

U_{z} (r) = {\begin{cases} J_{0} (k_{l 1} r), r \leq a_{1} \\ B_{1} J_{0} (k_{l 2} r) + B_{2} Y_{0} (k_{l 2} r), a_{1} < r \leq a_{2} \\ C_{1} J_{0} (k_{l 3} r) + C_{2} Y_{0} (k_{l 3} r), a_{2} < r \leq a_{3} \\ D_{1} J_{0} (k_{l 4} r) + D_{2} Y_{0} (k_{l 4} r), a_{3} < r \end{cases}

with

k_{l i} = \sqrt{{(Ω / v_{l i})}^{2} - q^{2}}

. The positions of

a_{1}

,

a_{2}

and

a_{3}

are depicted in Fig. 6(a). The radial component of the displacement vector is obtained as

u_{r} (r) = A_{a} \frac{d}{d r} U_{z} (r) e^{i (Ω t - q z)} = A_{a} U_{r} (r) e^{i (Ω t - q z)},

U_{r} (r) = {\begin{cases} k_{l 1} J_{0}^{'} (k_{l 1} r), r \leq a_{1} \\ B_{1} k_{l 2} J_{0}^{'} (k_{l 2} r) + B_{2} k_{l 2} Y_{0}^{'} (k_{l 2} r), a_{1} < r \leq a_{2} \\ C_{1} k_{l 3} J_{0}^{'} (k_{l 3} r) + C_{2} k_{l 3} Y_{0}^{'} (k_{l 3} r), a_{2} < r \leq a_{3} \\ D_{1} k_{l 4} J_{0}^{'} (k_{l 4} r) + D_{2} k_{l 4} Y_{0}^{'} (k_{l 4} r), a_{3} < r \end{cases} .

Then,

B_{i}, C_{i}, and D_{i} (i = 1, 2)

can be expressed in terms of

A_{a}

from the acoustic boundary conditions that both the displacement vector components (

u_{z}

and

u_{r}

) and the stress components (

T_{1}

and

T_{5}

) are continuous across the boundaries [31]. In addition, the stress components can also be determined by the method described in [31].

By the boundary conditions, one can have

{[\begin{array}{l} u_{r} \\ u_{z} \\ T_{1} \\ T_{5} \end{array}]}_{\begin{array}{l} l a y e r 1 \\ r = a_{1} \end{array}} = {[\begin{array}{l} u_{r} \\ u_{z} \\ T_{1} \\ T_{5} \end{array}]}_{\begin{array}{l} l a y e r 2 \\ r = a_{1} \end{array}} and Q^{1} (r = a_{1}) [\begin{array}{l} 1 \\ 0 \end{array}] = {[\begin{array}{l} u_{r} \\ u_{z} \\ T_{1} \\ T_{5} \end{array}]}_{\begin{array}{l} l a y e r 1 \\ r = a_{1} \end{array}},

where

Q^{1} (r = a_{1})

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 = a_{2}

and

r = a_{3}

. Plugging the coefficient matrices,

Q^{i} (r = a_{j}) (i, j = 1, 2, 3, 4)

derived in the same way as in Eq. (12) into the boundary conditions, one can obtain

\begin{array}{l} [\begin{array}{l} B_{1} \\ B_{2} \end{array}] = [^{Q^{2} (r = a_{1})] - 1} [Q^{1} (r = a_{1})] [\begin{array}{l} 1 \\ 0 \end{array}] = M_{1} [\begin{array}{l} 1 \\ 0 \end{array}], \\ [\begin{array}{l} C_{1} \\ C_{2} \end{array}] = M_{2} M_{1} [\begin{array}{l} 1 \\ 0 \end{array}], and [\begin{array}{l} D_{1} \\ D_{2} \end{array}] = M_{3} M_{2} M_{1} [\begin{array}{l} 1 \\ 0 \end{array}], \end{array}

where

M_{i} = {[Q^{i + 1} (r = a_{i})]}^{- 1} [Q^{i} (r = a_{i})] (i = 1, 2, 3)

. Thus, one finally obtains the longitudinal acoustic fields in terms of

A_{a}^{2}

.

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]:

\frac{1}{4} \int_{\infty} {- v_{n}^{*} \cdot T_{m} - v_{m}^{*} \cdot T_{n}} \cdot {\hat{a}}_{z} d S = P_{0} δ (q_{m} - q_{n})

where

v

and

T

can readily be derived from the acoustic fields we obtain from Eq. (9), (11) and (13). After some straightforward algebra, one can obtain [24, 34]

A_{a}^{2} = \frac{C_{A 0}}{D_{1}^{2} + D_{2}^{2}}

where

C_{A 0}

is a proportional constant depending on the power flow of the given acoustic mode. It should be noted that since

A_{a}

can 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.

In addition, one can also define ${| F (γ) |}^{2}$ as a normalized function of $A_{a}^{2}$ for a specific Lorentzian-like peak, which can also be regarded as a normalized Lorentzian fitting function to $A_{a}^{2}$ as discussed in Sec. 3.2. The variable $γ$ denotes the deviation of the propagation constant $q$ from a specific modal propagation constant $q_{m r}$ . Thus, $η_{a} (r)$ given in Eq. (8) eventually becomes a function of $γ$ as well, so that we rewrite it down as $η_{a} (r, γ)$ .

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.

Order of quasi-mode	Peak position (GHz)	Relative peak strength	Brillouin line-width (MHz)
AQM₁	11.17	1.00	46.52
AQM₂	11.21	9.74 × 10⁻²	72.19
AQM₃	11.30	6.26 × 10⁻⁵	114.7

A quasi-mode interpretation of acoustic radiation modes for analyzing Brillouin gain spectra of acoustically antiguiding optical fibers

Abstract

1. Introduction

2. Stimulated Brillouin scattering and acoustic modal analysis

3. Theory of acoustic quasi-modes

3.1. Acoustic waves in an optical fiber

3.2. Quasi-mode interpretation of acoustic radiation modes

3.3. Quasi-mode interpretation in the frequency domain

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

5. Conclusion

Appendix

Acknowledgment

References and links

Cited By

Figures (7)

Tables (1)

Equations (15)

Optics Express