1. Introduction

Spatiotemporal dynamics in optical systems is an area of active research, as outlined below, with a long and rich history. In this paper, we describe a relatively simple numerical model that gives insight into the recently observed stochastic spatiotemporal dynamics of Stimulated Brillouin Scattering (SBS) in a two-mode fiber [1]. In these experiments, intensity fluctuations were observed along the transverse and longitudinal dimensions of the scattered waveform, and the degree of correlation between the stochastic intensity dynamics of distinct transverse points varied dramatically from shot-to-shot of the pump laser. Our model reproduces the observed fluctuations in intensity correlations and gives insight into the interplay of thermal and pump fluctuations in a multimode waveguide. We also introduce a method of parameter optimization for nonlinear stochastic optical systems that we believe will be of interest in the analysis of more complex systems showing nonlinear spatiotemporal dynamics.

A review of the last few years of optical physics research reveals a wide variety of projects focused on spatiotemporal dynamics. Here we give a brief overview of some of this work. In high power fiber lasers, thermally induced mode instabilities led to unwanted spatiotemporal dynamics in the output of the laser [2–4]. The maximum output power of fiber lasers is also limited by the onset of Brillouin scattering, prompting recent work to optimize the output power with respect to the dual constraints of Brillouin scattering and mode instability [4]. By way of contrast, space-division multiplexing exploits the orthogonality of different transverse modes to increase the capacity of fiber-optic communication links by creating, in effect, deterministic spatiotemporal dynamics in which different transverse intensity patterns are encoded with different bit streams [5–7]. There has even been work in which independent bit streams were encoded on the orbital angular momentum states of an optical beam in a free-space optical communication system [7]. Trebino and associates recently demonstrated a measurement technique for completely characterizing the spatiotemporal dynamics of ultrashort pulses in a single-shot measurement. The method generated an array of interference patterns, each of which represented the overlap of a particular spectral component of the pulse under test with a reference pulse of known spatial and temporal characteristics. This information could then be inverted to obtain the spatiotemporal characteristics of the test pulse [8]. Finally, Wabnitz recently reported a numerical study of the intensity and phase dynamics in a fiber laser in which the resulting patterns were plotted against a “slow time” related to the round trip time and a “fast time” related to the retarded time. These plots captured the complex nonlinear dynamics of intensity and phase as they evolved inside the cavity producing patterns which were conceptually analogous to turbulence [9].

Looking further back into the history of spatiotemporal dynamics, there was extensive early work on transverse intensity dynamics in lasers and in nonlinear optics in bulk media and waveguides. We list here a few examples to set the broader context for our own work. Tredicce et. al. observed instabilities in a CO₂ laser by translating intracavity lenses to vary the number and interaction of transverse modes. Under appropriate conditions, this could result in transverse intensity patterns that evolved in time [10]. Bespalov et al. observed transverse intensity dynamics in SBS in an acetone cell. Specifically, the temporal dynamics of the whole scattered beam was not the same as the temporal dynamics of light passing through a small aperture that spatially sampled part of the scattered beam [11]. Lim et al. observed spatiotemporal chaos in CW laser light undergoing SBS in a multimode fiber with optical feedback. They also looked at correlations in the intensity fluctuations at different transverse points. For short time scales the correlation between transverse points was very high, but it dropped off for longer time scales [12]. Finally, optical phase conjugation as used for beam cleanup can be thought of as an example of three-dimensional intensity dynamics since the input beam evolves through various transverse states as it propagates through the phase conjugate medium. Brillouin scattering in optical fiber can be used in these applications, but it requires highly multimode fiber to synthesize complex wavefronts [13–15].

Our work is focused on stochastic spatiotemporal dynamics in pulsed SBS in an optical fiber that supports only two transverse modes and results from the combined effects of thermal density fluctuations and transverse pump noise. While the physical system we consider is relatively simple, the three-dimensional nature of its stochastic dynamics can be thought of as conceptually analogous to turbulence in a fluid [16], and we hope that our success in constructing a phenomenological model will yield insight for tackling more complex systems that display full-fledged turbulent spatiotemporal dynamics. We are aware of no analogous work in the literature.

In the sections that follow, we will give a brief overview of our recent experiments to make it clear what is being modeled and then develop a phenomenological model for the transverse intensity dynamics that we observe. We also introduce useful techniques for systematically optimizing adjustable parameters in nonlinear optics simulations that give insight into the level of sensitivity of simulation results on adjustable parameters.

2. Overview of experiments

Before describing the numerical model, we will briefly review the main features of the experiments, highlighting those results that pertain directly to what we wish to understand using the numerical model. For a detailed discussion of the experiments and their results the reader is referred to [1]. A schematic of the experimental setup is shown in Fig. 1. Single-mode, Q-switched pulses from an Nd:YAG laser are coupled into a 10.1 m length of standard communication fiber with a core diameter of approximately 8 µm, which allowed the two lowest order transverse modes of the fiber to propagate at the operating wavelength of 1064 nm. Mode bending losses were also included which suppressed, but did not eliminate, the higher order mode; consequently, we expect that the fundamental fiber mode was dominant, as indicated by the model optimization with respect to mode fraction described in Section 4. Some of the incident pump light is backscattered by stimulated Brillouin scattering (SBS) and part of this light is sampled by the first beamsplitter near the fiber input and directed to a second beamsplitter that produces two copies of the scattered beam. The two arms of the apparatus each contain a 1 mm aperture that can be translated in the transverse plane so that the intensity dynamics of distinct transverse points can be monitored by the two fast photodiodes. The transverse dimension of the scattered beam was roughly 4 mm. For the results of experiments here reported, the two apertures sampled transverse points that were 2.5 mm apart and symmetrically placed about the intensity maximum of the scattered beam. Two sample waveform pairs, each pair associated with scattering from a single input pump pulse, are shown in Fig. 2. The peak power coupled into the fiber was 87 W, and the average amount of pump depletion due to SBS was approximately 20%. In the top frame of Fig. 2, the intensity fluctuations at the two transverse points are very similar, while in the bottom frame the intensity fluctuations are quite different. Both waveform pairs correspond to the same aperture separation of 2.5 mm and to the same input pump power.

 

Fig. 1 Setup for collecting SBS waveform ensembles: BSC, beam splitter cube; OBJ, microscope objective; AP, aperture; NDF, neutral density filter; VNDF, variable neutral density filter; FPD, fast photodiode.

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Fig. 2 Sample SBS waveform pairs at distinct transverse points for an input pump power of 87 W peak. (a) An example of highly correlated intensity fluctuations. (b) An example of essentially uncorrelated intensity fluctuations.

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If the normalized two-point intensity correlation coefficient, defined in Eq. (1) below [17], is calculated for an ensemble of successive waveform pairs then large-scale shot-to-shot fluctuations are observed.

 

Fig. 3 Intensity correlation coefficients for an ensemble of 1000 waveform pairs showing large scale shot-to-shot fluctuations for the same input pump power as Fig. 2.

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A few key features of the experiments are important to note as we consider the development of a phenomenological model for the “turbulent” intensity dynamics. First, large scale intensity correlation fluctuations were only observed in multimode fiber. For single mode fiber the intensity dynamics at distinct transverse points were always highly correlated [1]. Second, when there were no bending losses and the higher-order mode was dominant we observed that the symmetry of the LP₁₁ mode was broken, as illustrated by the contour plot in Fig. 4, which shows the backscattered Stokes beam when strong Brillouin scattering was occurring in the fiber. Note that the intensities of the two lobes differ significantly. A similar asymmetry was observed in the transmitted pump beam. We believe that the nonlinearity broke the symmetry of the higher order mode predicted for linear propagation. This suggests that the mixture of fundamental LP₀₁ and higher order LP₁₁ modes at symmetric transverse points lying on a common diameter might be different. Finally, it was discovered that the input pump pulses contained transverse fluctuations so that symmetric transverse points did not always have identical temporal profiles [1]. All three of these factors will be accounted for in the phenomenological model that we construct.

 

Fig. 4 Mode profile of the backscattered Stokes beam without mode stripping bending losses showing an asymmetric intensity pattern.

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Before proceeding to the modeling, we will describe how the transverse pump fluctuations that were incorporated into the model where measured. The correlation apparatus shown in Fig. 1 was used with the first beamsplitter replaced by a mirror that directed the input pump pulses into the correlation apparatus. The 1 mm apertures in each arm of the apparatus were translated so that they sampled two different transverse points of the pump pulses separated by 2.5 mm, symmetrically located about the intensity maximum of the beam. In this manner, an ensemble of 1000 pairs of pump pulse temporal shapes was recorded for the two transverse points.

3. Model for scattering along fiber length

The equations which form the basis of our phenomenological model are the equations for the pump field $E_p$, scattered field $E_s$, and density fluctuations $\rho$ that are appropriate for Brillouin scattering along the fiber length that is initiated by thermal noise [18–20]. Although these equations do not make any explicit reference to the transverse dimensions of the material, information about the transverse structure of the beam does enter into the model through the mode effective areas of the two fiber modes. This effective area determines the average intensity of the pump and Stokes fields inside the fiber and, as shown in the Appendix, the strength of the thermal density fluctuation term driving the spontaneous scattering. We mention this in passing now, but these ideas will be further developed in the next section.

These equations will be solved using the split-step method [20]. In particular, the linear portions of Eq. (2)–Eq. (4) have exact solutions that are used to propagate the fields each step, and these fields are then used in the nonlinear and stochastic terms which are solved using the midpoint method. This process is then repeated for each time step to calculate the complex field amplitudes at every point along the fiber as a function of time.

The modeling of nonlinear-stochastic processes is complicated by the presence of parameters whose values are often difficult to measure and can vary depending on the details of the experiments. It is not uncommon to start with typical values from the literature which are then varied to optimize the fit between experiment and model. This is an example of an “inverse problem” in applied mathematics [21]. In this paper, we solve Eq. (2)–Eq. (4) and optimize the agreement of simulations with experiments using formal techniques of solving inverse problems, which we describe below. Banks and Tran have an excellent introduction to the role of inverse problems within mathematical modeling with several examples in physics and biology in [21].

We have chosen to treat two parameters as adjustable in our model for scattering along the fiber length, the Brillouin gain coefficient $g_o$ and the phonon damping rate $\Gamma$. The remaining parameters in the model were things for which we had reasonable estimates from our own measurements or calculations, or they were published values that were not expected to vary significantly for our experimental setup. The nonlinear coupling coefficients in the SBS equations depend on both of the adjustable parameters, and we believe they are sensible choices because the gain coefficient is directly related to the scattering strength and the phonon damping rate to the timescale of intensity fluctuations of the scattered light [22–24].

At this stage of the model development, our primary aim is to choose good parameter values for Eq. (2)–Eq. (4) that will be used when we construct the phenomenological model of transverse scattering dynamics in the following section. To accomplish this, we will consider an ensemble of 1000 waveforms collected with the 1 mm aperture aligned with the intensity maximum of the scattered beam and average these waveforms to obtain an average shape for the scattered waveform in the experiments, as shown in Fig. 5. We then generate ensembles of waveforms in the simulations for different choices of $g_o$ and $\Gamma$ to compare the average scattered waveform shape from the simulations with the experiments.

 

Fig. 5 Ensemble of 1000 SBS waveforms as measured in the lab for an aperture position that samples the intensity maximum near the beam center. The bold curve is the ensemble average waveform shape. The input peak power is 87 W.

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To this end, denote the vector of adjustable parameters by $q=[g_0\text{,}\Gamma ]$. Denote the ensemble averaged experimental waveform intensity, sampled at m times $t_i$, by $〈I_{\exp }(t_i)〉$. The ensemble average waveform intensity predicted by the model is dependent on the values of the adjustable parameters, and we will denote it by $〈I_{\mathrm{mod}}(t_i;q)〉$. We calculate the mean square error, MSE(q), which is a function of the parameters by

 

Fig. 6 MSE landscape for the ensemble average waveform shape as a function of the Brillouin gain coefficient and phonon damping rate. Lower values represent better agreement between experiment and simulations.

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Fig. 7 Comparison of the ensemble average waveform of experiments and simulations for the optimum values of $g_0$ and Γ, which minimize the MSE.

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4. Phenomenological model of transverse dynamics

Developing a rigorous model for SBS in three spatial dimensions (fiber length and cross-section) would be a formidable task. We believe we can answer the most fundamental questions about the causes of the large-scale intensity correlation fluctuations by constructing a phenomenological model of the transverse dynamics using the basic equations, Eq. (2)–Eq. (4), for scattering along the fiber length. There are three key features of the experiments that will guide our efforts: 1) interesting transverse dynamics are only observed in multimode fiber; 2) the higher-order fiber mode is observed to be asymmetric in our experiments; and 3) the input pump pulses have transverse irregularities in their intensity. The first two features suggest that the fractions of fundamental and higher order modes may have differed for different transverse points, and the third feature suggests that different transverse points were subject to different pump fluctuations.

Differences between fiber modes enter into the SBS model in Eq. (2)–Eq. (4) through the effective mode area or effective mode radius [18–20]. The effective mode area directly affects the input pump intensity, and it also directly affects the thermal noise strength, as outlined in the Appendix. The higher order mode will have a larger “footprint” in the fiber cross-section and result in less scattering for the same input pump peak power. Given the fiber parameters, we can calculate effective mode areas for both modes and use this in the SBS equations to model multimode scattering. The effective mode area calculations are described in the appendix on model parameters. Now we are in a position to develop the phenomenological model for SBS at a pair of transverse points. We begin by rewriting Eq. (2)–Eq. (4) to explicitly indicate the modal dependence of the scattering.

We will model the scattering at each transverse point by splitting the available input pump power between the fundamental and higher order modes and solving Eq. (7)–Eq. (9) for each fiber mode, using the appropriate effective mode area and its fraction of the total input pump power. We also incorporate transverse pump noise as described below. A composite waveform ensemble for each transverse point is constructed by taking the incoherent sum of the waveforms for the two fiber modes at that point.

To account for the transverse pump noise, distinct input pump fluctuations, taken from the experiments described at the end of Section 2, are used for each transverse point. In particular, variations in the peak intensity and leading and trailing half-widths of the input pump pulses were taken from the measured ensemble of input pump pulses for different transverse points and incorporated in the model.

To summarize our algorithm for generating waveform ensembles, the factors that distinguish one transverse point from another are the fractions of $L{P}_{01}$ and $L{P}_{11}$ modes and the pump pulse fluctuations. The thermal density fluctuations that initiate the scattering are the same for both transverse points and both fiber modes, but they differ from one input pump pulse to the next to build up an ensemble of independent scattering events.

In this manner, two waveform ensembles representing scattering at distinct transverse points are constructed, and individual waveforms from the two ensembles can be paired to calculate intensity correlations between them. When this is done for an ensemble of many input pump pulses, the result is a series of fluctuating intensity correlation coefficients like what is observed in the experiments. This series of correlations can be used to estimate the probability density function of the fluctuations.

The probability density function for the two-point intensity correlation fluctuations from the experiments (shown in Fig. 3) was compared with the probability density function from simulations by calculating the MSE between the two curves in the same manner as described earlier. Simulations were done in which the fraction of available input pump power in the two fiber modes were varied systematically for the two transverse points, and a MSE landscape was generated. The result of these simulations is shown in Fig. 8. One can see that there are two valleys, and this is to be expected since swapping the mode power fractions between the two transverse points should yield the same MSE, all other things being equal. However, it turns out that the valleys aren’t identical because differences in input pump fluctuations at the two transverse points break the symmetry of the MSE landscape. There is an absolute minimum MSE corresponding to $L{P}_{01}$ mode fractions of 0.77 and 0.88 at the two transverse points, and Fig. 9 shows the probability density functions from the experiment and the simulations for this minimum. They agree remarkably well. Note also that the fundamental mode carries well over 50% of the available input pump power for the optimum fractions, and this is consistent with the fact that bending losses were introduced in the experiments to discriminate against the higher order mode.

 

Fig. 8 MSE landscape for the intensity correlation coefficient probability distribution as a function of the fraction of $L{P}_{01}$ at each transverse point. Lower values represent better agreement between experiment and simulations.

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Fig. 9 Comparison of the intensity correlation coefficient probability distribution of experiments and simulations for the optimum choice of mode fractions at the two transverse points. Transverse pump noise is also included.

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It should be noted that leaving out the transverse pump fluctuations resulted in a very poor fit between simulations and experiments, as shown in Fig. 10. This indicates that transverse irregularities in the pump pulses are necessary to explain the large scale fluctuations in the two-point intensity correlation coefficient. Furthermore, if we choose a poor value of the mode fractions but incorporate transverse pump fluctuations then the agreement between simulations and experiments is also poor, as shown in Fig. 11. Thus, both factors play an important role in the model: transverse pump noise and differing mode fractions at the transverse points. Finally, Fig. 12 shows the series of intensity correlation fluctuations from the experiments (same as Fig. 3) and the fluctuations corresponding to Fig. 9, Fig. 10, and Fig. 11. As can be seen, the match between simulations and experiments is best for the optimum mode fractions and transverse pump noise.

 

Fig. 10 Comparison of the probability distributions for the optimum mode fractions but without transverse pump noise.

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Fig. 11 Comparison of the probability distributions with transverse pump noise but non-optimal mode fractions.

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Fig. 12 Comparison of the two-point intensity correlation coefficients for an ensemble of 1000 scattering events: (a) experiments (as in Fig. 3); (b) simulations using optimum mode fractions and transverse pump noise (as in Fig. 9); (c) simulations using optimum mode fractions but without transverse pump noise (as in Fig. 10); and (d) simulations with transverse pump noise but non-optimal mode fractions (as in Fig. 11).

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5. Conclusions

In this article we have presented a phenomenological model of stochastic transverse intensity dynamics in SBS in a multimode fiber, and we have also illustrated a systematic technique for parameter optimization. The physical system was quite simple, meeting the minimum requirements for spatiotemporal dynamics since the optical fiber supported only two transverse modes. Our success in identifying key system parameters and their optimum values should yield insight for the treatment of more complex systems that display turbulent spatiotemporal dynamics. A phenomenological approach of a similar character to the one we have illustrated could help identify key factors driving the dynamics and provide estimates of key system parameters. Information from the phenomenological model could then be incorporated into a more rigorous model, saving substantial computing time by first pursuing the simpler optimization process.

For the system we have considered, two sources of noise play a pivotal role in driving the spatiotemporal dynamics. First, thermal density fluctuations along the fiber length are a primary factor in determining the longitudinal structure of the stochastic intensity fluctuations. In concert with this, the transverse pump fluctuations drive the fluctuations over the cross-section of the fiber. If we could watch the evolution of the transverse structure of each SBS pulse in real time, we would see some shots where the intensity of different transverse points rise and fall in synchrony as the complex longitudinal waveforms streamed through the plane of observation. For other shots we would observe the intensity of different transverse points rise and fall in an uncorrelated manner. We can infer that the degree of correlation is determined in part by whether the driving pump pulse has a smooth, symmetric transverse intensity profile or an irregular transverse intensity profile.

The right mixture of $L{P}_{01}$ and $L{P}_{11}$ modes must also exist to match the experimentally observed distribution of intensity correlation fluctuations. If each fiber mode interacts with the density fluctuations in a distinct way then the structure of the scattered waveform will depend on the mixture of modes at each transverse point. Multimode scattering and the broken symmetry of the higher order mode most likely also play a role in the effectiveness of the transverse pump fluctuations since the fiber cannot act as an effective spatial filter for the pump beam.

It is interesting to note that in both cases of parameter estimation, the MSE landscape did not have a sharply defined minimum but instead a valley where a range of parameter values would yield similar results. This implies that the model is not extremely sensitive to the particular choice of parameter values. It could be that the model is overdetermined with respect to the data being matched [21]. For example, it may be that it is the mathematical relationship between key parameters rather than their specific values that is most important in achieving a good match between experiment and simulations. A simple example of this would be the ratio of parameter values, although we suspect something more complex in our case due to the curvature of the valleys in the MSE landscapes. This flexibility with respect to parameter values may be a common feature of nonlinear stochastic optical systems as a consequence of their complexity, and it has the potential to make interpretation of the results of experiments more challenging when comparisons are made with a numerical model. It is important to let the results of experiments guide the choice of key features of the model, as we have done. It is also important to understand what conclusions are strongly supported by the experiments and modeling and what conclusions are more tentative in character. We hope that this example system will stimulate similar approaches to interpreting other nonlinear optics experiments with complex deterministic or stochastic spatiotemporal dynamics.

We close by briefly discussing, in general terms, the requirements for a more rigorous model for the stochastic, spatiotemporal dynamics we have observed. Recent experiments by Song et al have shown that there can be significant couplings between different transverse modes in SBS for pump and probe beams in different combinations of fiber modes [28]. This is obviously not incorporated into our simple phenomenological model. Work by Ward et al on modeling mode coupling in SBS uses a continuous wave model that neglects pump depletion and accounts for the coupling between modes averaged over the fiber cross-section [29]. A fully three-dimensional model will account for couplings between the electromagnetic and acoustic modes point-by-point in the cross-section of the fiber as they propagate along its length, and also include pump depletion and spontaneous initiation of SBS from thermal density fluctuations. As mentioned earlier, this is a formidable undertaking, but it would be necessary and worthwhile if the experiments used fast imaging to capture the full transverse intensity dynamics, producing a “movie” of the complex transverse dynamics for each scattering event, rather than simply sampling two distinct transverse points as we have done.

6 Appendix

In this appendix we discuss the details of determining model parameter values. We will start with the basic parameters that appear in Eq. (2)–Eq. (4), focusing on how the adjustable parameters enter into the modeling, and then discuss the calculation of effective mode areas and their incorporation into the phenomenological model for transverse intensity dynamics.

We begin by noting that Eq. (2)–Eq. (4) are expressed in the cgs system of units. The nonlinear coupling coefficients are given by the following relationships [18, 20]:

$$\kappa =\frac{\gamma \pi }{2{\lambda }_0{\rho }_{0}n}$$

and

$$\Lambda =\frac{\gamma n}{4V{\lambda }_0}$$

where $\gamma$ is the electrostrictive coupling constant, ${\rho }_0$ is the density of glass, V is the speed of sound in glass, n is the refractive index of the fiber core, and ${\lambda }_0$ is the free space wavelength of the pump laser. The values of the parameters that are not treated as adjustable are given in Table 1 below.

Table 1. Parameter values used in simulations

View Table

The dimensionless electrostrictive coupling constant has the adjustable parameters of the model embedded in it:

$${\gamma }^2=\frac{nc{\lambda }_0^2{\rho }_{0}V\Gamma g_0}{4{\pi }^2}$$

where Γ is the phonon damping rate and $g_0$ is the Brillouin gain coefficient, the two adjustable parameters in the basic model for SBS along the fiber length. Boyd and Gaeta list the values of these parameters in their experiments as 2.5 × 10⁻¹⁶ cm/(erg/s) for the Brillouin gain coefficient and 135 MHz for the phonon damping rate [19]. We vary these parameters about the values used by Boyd and Gaeta to optimize the agreement between experiment and simulation.

The one parameter in Eq. (2)–Eq. (4) that remains to be defined is the noise strength for the Longevin noise term:

$$Q=\frac{2k_{B}T{\rho }_0\Gamma }{V^2{A}_{eff}}$$

where $k_B$ is Boltzmann’s constant, T is the absolute temperature of the glass fiber, and $A_{eff}$ is the effective mode area [18, 20]. The calculation of the effective mode areas for the two lowest order modes of the fiber and their incorporation into the model will be discussed separately below.

With these values in hand, the electrostrictive constant can be written as

$$\gamma =\sqrt{1.611\times {10}^7{g}_0\Gamma }.$$

For reference, $\gamma \cong 0.737$ for the values published in Boyd and Gaeta’s article [19]. The nonlinear coupling constants can now be expressed as multiples of the electrostrictive constant for the modeling:

$$\kappa =4628\gamma {\text{cm}}^{\text{2}}/\text{g}\text{and}\Lambda =5.804\times {10}^{-3}\gamma \text{s}/{\text{cm}}^{\text{2}}.$$

Note that the noise strength for the Longevin noise term also depends on the mode effective area, to which we now turn. For our simulations, we need the effective mode areas for the LP₀₁ and LP₁₁ modes of the fiber. The effective mode area was estimated by numerically evaluating the integrals in the expression below [26]:

$$A_{eff}=\frac{{\left[{\displaystyle \iint I_{\ell m}(x,y)dxdy}\right]}^2}{{\displaystyle \iint I_{\ell m}^2(x,y)dxdy}}$$

where $I_{\ell m}$ is the mode intensity of the $L{P}_{\ell m}$ mode of the fiber. The intensity of the $L{P}_{\ell m}$ modes for weakly guiding optical fibers is given by the expressions [27]

$$\begin{array}{l}{I}_{\ell m}(r,\varphi )=I_0{J}_{\ell }^2\left(\frac{ur}{a}\right){\cos }^2(\ell \varphi )\text{for}r\le a\\ I_{\ell m}(r,\varphi )=I_0{\left(\frac{J_{\ell }(u)}{K_{\ell }(w)}\right)}^2{K}_{\ell }^2\left(\frac{wr}{a}\right){\cos }^2(\ell \varphi )\text{for}r\ge a\end{array}$$

where $a$ is the core radius and $I_0$ is an overall constant determining the peak intensity of the mode. The $J_{\ell }$ and $K_{\ell }$ are Bessel functions and modified Bessel functions, respectively. The subscript $\ell$ denotes the number of nodal lines passing through the core’s center, and the subscript $m$ denotes the number of nodal circles enclosing the central axis of the fiber (the largest occurring at infinity). The values of $u$ and $w$ are determined from the eigenvalue equation [27]

$$u\frac{J_{\ell -1}(u)}{J_{\ell }(u)}+w\frac{K_{\ell -1}(w)}{K_{\ell }(w)}=0$$

where $u$ and $w$ are related by $V^2=u^2+w^2$ and $V$ is known as the “normalized frequency” given by the relationship

$$V=k_{0}a\sqrt{n_1^2-n_2^2}\cong \frac{2\pi }{{\lambda }_0}a\left(N.A.\right)$$

where $n_1$ is the core index, $n_2$ is the cladding index, and $N.A.$ is the numerical aperture of the optical fiber. The last equality assumes the fiber is immersed in air with an index of approximately one. For our calculations we used a core radius of a = 4.0 µm and a normalized frequency of V = 2.8, which allowed propagation of the two lowest $L{P}_{\ell m}$ modes. For these values, the effective mode area and radii for the two modes are

$$\begin{array}{l}{A}_{01}=50.7\mu {\text{m}}^{\text{2}}\text{or}{r}_{01}=4.0\mu \text{m}\text{and}\\ A_{11}=76.2\mu {\text{m}}^{\text{2}}\text{or}{r}_{11}=4.9\mu \text{m}\end{array}$$

We are now in a position to write an expression for the noise strength which depends on one of the adjustable parameters and on the effective mode area. For the two fiber modes we have

$$Q_{01}=1.035\times {10}^{-18}\Gamma {\text{g}}^{\text{2}}/\left({\text{cm}}^{\text{5}}\text{s}\right)\text{and}{Q}_{11}=6.897\times {10}^{-19}\Gamma {\text{g}}^{\text{2}}/\left({\text{cm}}^{\text{5}}\text{s}\right).$$

Finally, note that the pump intensity in the fiber will depend on the effective mode area; consequently, this also was accounted for in our model.

Acknowledgments

J. R. Thompson acknowledges partial support from an Ayres Faculty Development Leave at the Virginia Military Institute, and also very helpful guidance from Professor Christopher Goedde on implementing the split-step method to solve the SBS equations. It is also a pleasure to acknowledge helpful comments on the manuscript from Professor Rajarshi Roy.

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