Optimization and visualization of phase modulation with filtered and amplified maximal-length sequence for SBS suppression in a short fiber system: a theoretical treatment

We use a model to investigate both the temporal and spectral characteristics of a signal lightwave which has been spectrally broadened through phase modulation with a maximal-length sequence (MLS), which is a common type of pseudo-random bit sequence. The enhancement of the stimulated Brillouin scattering (SBS) threshold of the modulated lightwave in a fiber system is evaluated by numerically simulating the coupled three-wave SBS interaction equations. We find that SBS can build up on a nanosecond-level time scale in a short fiber, which can reduce the SBS suppressing capability of MLS modulation waveforms with GHz-level clock rate, if the sub-sequence ("run") lengths with the same symbol (zero or one) of the MLS extend over several nanoseconds. To ensure the SBS buildup is perturbed and thus suppressed also during these long sub-sequences, we introduce a low-pass filter to average the signal over several bits so that the modulation waveform changes gradually even during long runs and amplify the RF modulation waveforms to the level required for sufficient spectral broadening and carrier suppression of the optical signal. We find that the SBS suppression depends non-monotonically on the parameters of the filtered and amplified MLS waveform such as pattern length, modulation depth, and the ratio of low-pass filter cutoff frequency to clock rate for maximum SBS mitigation. We optimize the SBS suppression through numerical simulations and discuss it in terms of the temporal and spectral characteristics of the lightwave and modulation waveform using derived analytical expressions and numerical simulations. The simulations indicate that the normalized SBS threshold reaches a maximum for a RMS modulation depth of 0.56π and a ratio of filter cutoff frequency to clock rate of 0.54 and that MLS9 is superior to other investigated patterns. © 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
Yb-doped fiber amplifiers (YDFAs) at wavelengths around 1 µm are well-established as reliable laser sources for high power and brightness [1,2]. Still, they are limited by nonlinearities, optical damage, and thermal problems [3][4][5][6]. Several beam combining schemes aiming to overcome their limitations were demonstrated in recent years [7][8][9]. Both spectral and coherent beam combining benefit from monolithic fiber amplifiers with narrow linewidth and good beam quality [10][11][12][13]. However, for those beam combining schemes, the nonlinear effect of stimulated Brillouin scattering (SBS) limits power scaling, and suppression techniques are therefore applied [14][15][16]. SBS is a spectrally narrow process that involves scattering off an acoustic wave. External phase modulation of a single-frequency seed laser with a temporal pseudo-random bit sequence (PRBS) modulation and thus spectral broadening of the lightwave. We then consider the SBS dynamic in this passive fiber. Although SBS could also occur in the amplifier, this is not considered. The optical waves propagating in the fiber core can be represented by the electric field A L of the forward propagating laser and the electric field A S of the backward propagating Brillouin Stokes wave [17]. We do not consider the SBS seeding effect [21] that can occur when the linewidth exceeds the Brillouin frequency shift (∼16 GHz for 1064 nm), because the spectral broadening in this work is small compared to that, and because it is possible to fine-tune the clock rate so that the spectral lines of the signal avoid the Brillouin gain peak [24]. Using the slowly varying envelope approximation and ignoring the group velocity dispersion and the background propagation losses (negligible in a short fiber), the temporal and spatial evolution of the laser, Stokes and acoustic fields in a fiber are determined by the following coupled system (corresponding to a Lorentzian Brillouin line): c n Here σ = ωγ e /2n 2 ρ 0 , χ = ϵ 0 γ e k 2 q /2Ω B , α = Γ B /Ω B , respectively. The phonon decay rate Γ B = 2π/17.5 ns −1 [17] = 2π × 57.1 × 10 6 s −1 . This is also the full-width at half-maximum (FWHM) of the Brillouin line in angular frequency. Thus, in "regular" frequency, the FWHM Brillouin linewidth ∆ν B becomes 57.1 MHz. The phonon lifetime τ p is the inverse of the decay rate and becomes 2.79 ns. Furthermore, ω = 2πc/λ and Ω B = 2nv A ω/c is the laser and resonant acoustic angular frequency, respectively. The acoustic wave number k q = 4π/λ. The definition of some other parameters are like this: γ e is the electrostrictive constant, ρ 0 is the mean density of the fiber medium, n is the core refractive index, c is the velocity of light in vacuum, ϵ 0 is the dielectric constant, v A is the speed of the acoustic wave, and ρ is the amplitudes of the acoustic wave, respectively. Since the second derivative of the acoustic field has negligible influence on the simulation results with low optical linewidths, we omit this term for simplicity. The quantity f j,k = √︁ ncQ/(∆t) 2 S j,k in RHS of Eq. (3) represents the initiation of the SBS process from Langevinian noise [25], where Q = 2k B T c Γ B ρ 0 /Aν 2 , k B is Boltzmann's constant, T c is the temperature in the fiber core, and A is the fiber effective mode area, respectively. The quantity S j,k denotes a complex random function with Gaussian distribution of zero mean and unit variance. The indices j and k enumerate the spatial and temporal points of the numerical grid along the fiber and in time. The symbols ∆z and ∆t denote the spatial and temporal grid spacing, respectively. The MLS coded phase modulation can be introduced by A L (0, t) = A L (0, 0)e iφ(t) , where ϕ(t) is the optical phase coded by MLS waveform and A L (0, 0) is the input laser amplitude at location z = 0. Given the noise seeding of SBS, the Stokes wave is often pulsing. The SBS level is therefore quantified by the time-averaged reflectivity R = ⟨|A S (0, t)| 2 ⟩/⟨|A L (0, t)| 2 ⟩, where the brackets indicate time-averaging over several transit times (20 fiber transit times in this work). The SBS threshold is defined as the laser power that results in R = 1%. The single-pass gain G, in units of nepers (Np), is defined by G = g p PL/A, where P is the laser power at z = 0, and g p = γ 2 e ω 2 /ρ 0 nc 3 v A Γ B is the peak value of the Brillouin gain. The three-wave equations Eqs. (1)-(3) are solved numerically using the Euler method along the characteristic lines dz/dt = ±c/n [17]. We use the normalized SBS threshold to quantify the SBS suppression (or the SBS threshold enhancement factor). This is the ratio of the SBS threshold with phase modulation to the unmodulated SBS threshold. It is also equivalent to the multiplicative inverse for the normalized SBS gain introduced in Ref. [23], i.e., one is the inverse of the other. Parameters and the values we have used are shown in Table 1. The fiber length of 7.4 m leads to a single-pass transit time of 35.8 ns. First of all, we use this numerical model to calculate the SBS threshold thus defined with no phase modulation. The dependence of the SBS reflectivity on the single-pass gain G is shown in Fig. 2(a). The fluctuation of the reflectivity is caused by the stochastic nature of the noise source and indicates the uncertainty resulting from the finite duration of the averaging. The threshold for 0.01 reflectivity is around G = 9 Np (∼5 W) for the parameters above. We also examine the temporal evolution of the Stokes power at its output (at the signal input end of the passive fiber at z = 0) and investigate the details about the SBS buildup time at this location. The SBS buildup time can be defined as the delay between the start of the Stokes and that of the laser [26]. Here in our simulation, according to the relaxation oscillations in SBS dynamics, we evaluate the buildup process in terms of the time constant in the time before the reflectivity reaches its maximum value, during which the temporal evolution of the Stokes power shows an approximately exponential form. For this we introduce an ON-OFF-ON-like phase modulation with a MLS waveform with parameters such that SBS is well suppressed when the MLS is ON. The switch-OFF moment of the MLS waveform represents the starting point of the SBS buildup. The modulating scheme and the SBS reflectivity as a function of time for G = 30 Np and 150 Np are shown in Fig. 2(b), as calculated by ensemble averaging over 100 times of the SBS temporal evolution to reduce noise fluctuations. It shows a rapid growth of the Stokes power in the first several nanoseconds for large G (such as G = 150 Np), and then relaxation oscillations appear, which agrees with the statement in Ref. [27]. To further investigate the dependence of time constant on G, we also examine the SBS buildup kinetics in a modulation switch-OFF time period of 10 ns, which is shown in Fig. 2(c). The dashed curves show the exponential fitting curves. The time constants for different G calculated by the curve fitting are shown in Fig. 2(d). It is clear that the time constant is inversely proportional to G and in the nanosecond time scale. To be certain, SBS in a short fiber system can build up within several nanoseconds at high power, with a time constant that is considerably shorter than the transit time of the optical waves, and shorter than the phonon lifetime. The smallest time constant becomes 0.9 ns in Fig. 2(d), which occurs for the highest considered single-pass gain of 200 Np, i.e., around 22 times the unbroadened threshold. Note that narrow-line fiber systems may well be broadened to a higher enhancement factor than this, e.g., 53 in [22], and we would then expect an even shorter time constant.

Temporal and frequency characteristics of MLS phase modulation
The maximal-length sequence is a type of PRBS which possesses all three PRBS randomness criteria simultaneously [28]. An MLS is sometimes called an m-sequence or n-sequence, and can also be denoted as PRBSn or MLSn. It is commonly used in spread spectrum systems, bit error testing, and many other areas, and can be easily produced by a Linear Feedback Shift Register (LFSR) of length n. Mathematically, the maximal-length sequences are described by irreducible and primitive polynomials [29]. For MLS3, the polynomial is x 3 + x + 1, or abbreviated as (3,1,0). Correspondingly, there are (5,2,0) for MLS5, (7,1,0) for MLS7, (9,4,0) for MLS9, (11,2,0) for MLS11, (13,4,3,1,0) for MLS13, and (17,3,0) for MLS17, respectively. A MLS consists of an aperiodic base sequence of length N = 2 n − 1, which is then periodically repeated to form a continuous infinite sequence, and since it is periodically repeated, its bit sequence (or pattern) has a discrete spectrum with frequency spacing equal to the inverse of the period. The choice of the pattern length in MLS phase modulation in fiber system has always been a dilemma. If the pattern of the MLS is too short then the line spacing can become excessive and the number of spectral lines small, and the power per spectral line thus too high for efficient SBS suppression. On the other hand, the phase for a MLS phase modulated lightwave stays constant during the uninterrupted sub-sequences ("runs"), which could also lead to significant SBS in a fiber system if these sub-sequences are too long (large n). This suggests that there is a length that optimizes the trade-off between these conflicting considerations. In addition, with low-pass filtering, the MLS waveform is averaged over several bits due to significant inter-symbol interference (ISI) when the filter cutoff frequency is small compared to the clock rate. Thus, the phase can vary continuously, and SBS suppressed, even during long runs. Although averaging also reduces the amplitude of the modulation, this can be restored through amplification of the low-pass filtered RF wave. This control of the RF power (i.e., modulation amplitude) further extends the control of the optical spectrum. The temporal and frequency characteristics of unfiltered and filtered and amplified MLS phase modulation are important for the understanding of the SBS suppression capability.
In this Section, we investigate the temporal characteristics of the MLS waveform (Section 3.1), the optical spectra of MLS phase modulation (Section 3.2), and the optical spectra of filtered and amplified MLS phase modulation (Section 3.3).

Temporal characteristics of a MLS waveform
The properties of a MLS waveform are controlled by the sequence The quantity {a j } is a variable with a period of N that takes on discrete values -1 (if the bit is symbol 0) and +1 (if the bit is symbol 1). If an element from the sequence is chosen at random, then the probabilities of the different values become Pr +1 = (1 + 1/N)/2 and Pr −1 = (1 − 1/N)/2, respectively, where of course Pr +1 + Pr −1 = 1. To form a continuous waveform (e.g., for modulation) from the sequence, a single period x N (t) of the continuously repeated train x(t) can be written as where T is the bit duration (equal to the inverse of the clock rate f cr ) and * denotes convolution. The base shape p(t) of a single symbol is assumed to be the same for both types of symbols. It is often taken to be a rectangular function, constant throughout the duration T of the symbol slot and zero outside, and can thus be written as The intermediate aperiodic signal x N (t) can then be extended in time by using a comb of delta functions with a time spacing of T N = NT. This operation can be formulated as convolution resulting in the periodic train where j, k ∈ Z are used to specify discrete instants in time. period T N for sequences with N = 31, 127, 511, and 2047 bits (n = 5, 7, 9, 11) with a clock rate of 6.5 GHz, shown in Fig. 3(a). The signal filtered by a low-pass filter with a cutoff frequency of 2.2 GHz is shown in Fig. 3(b). We choose a low-pass filter with the amplitude characteristics of a 6 th -order Butterworth filter in this paper, because we have found that to agree well with the filtering in our previous experimental work [22]. However, the filter is of the zero-phase type, i.e., with flat phase characteristics. Such a filter is not causal but the resulting signal can still be generated by an arbitrary waveform generator (AWG) with pre-calculated sample values. We use Matlab's "filtfilt" function to implement zero-phase filtering [30]. It is clear in Fig. 3(b) and Fig. 3(d) that the normalized phase spreads beyond the range (-1, 1) and also attains intermediate values. We next consider the temporal properties of MLS and compare them to the SBS buildup time. A MLSn sequence contains 2 n−1 runs, including one single "1" run of length n (the "dwell time" in Ref. [17]), which is the longest run, and 2 n−1−i runs of length i, where i = 1, 2, 3, . . ., n-1.
This includes the second-longest run, which comprises n-1 of "0". For shorter runs, there are the same number of zero-runs and one-runs. Take MLS9 for example, there are a total of 256 runs, which include a single run of ones of length 9, a single run of zeros of length 8, 2 runs of length 7, 4 runs of length 6, 8 runs of length 5, 16 runs of length 4, 32 runs of length 3, 64 runs of length 2, and 128 runs of length 1. We plot the longest run of MLS9 (1.38 ns) and MLS11 (1.69 ns) before (Fig. 3(c)) and after Fig. 3(d) filtering. The dwell time nT as a function of clock rate for n = 3, 5, 7, 9, 11, 13, 17, and 31 is shown in Fig. 4(a). For clock rates ranging from 3 GHz to 10 GHz, the dwell times for patterns considered in this work are within several nanoseconds, which are comparable to that of the SBS buildup time constant in the 7.4-m of passive fiber that we consider. Such dwell times may then lead to significant SBS, which will be demonstrated in Section 4. The amplitude of the low-pass filtered MLS waveform no longer stays constant ( Fig. 4(b), which may then allow for better SBS suppression, although a small bandwidth may lead to excessive averaging. Also, the dependence of the period NT on clock rate for different MLS patterns is shown in Fig. 4(b). As seen, the period is much longer than the SBS buildup time (except for MLS3). In many cases, it is also longer than the 35.8 ns fiber transit time.

Optical spectra of MLS phase modulation
In this section, we discuss the optical spectra of lightwave phase-modulated with a MLS. We consider a linearly polarized single-frequency laser electric field whereĒ L is the constant amplitude of the laser field, which we ignore in the calculation of the power spectral density for simplicity. For convenience, we shift the optical spectrum by the carrier frequency ω c to be centered around ω = 0. According to the expression given by Eq. (7), the MLS coded phase ϕ(t) can be represented as where k p is the peak-to-peak phase modulation amplitude, which is defined as π times the ratio of the peak-to-peak modulation voltage to the half-wave voltage V π of the EOPM. Here, the laser electric field E(t) is an infinitely repeated periodic sequence with a period of N which takes two distinctive values exp(ik p /2) and exp(−ik p /2) (i.e., the phase ϕ(t) takes two values k p /2 and −k p /2). Based on these properties, we can then calculate the autocorrelation function (ACF) of E(t) as phase modulated by the MLS waveform. Recalling the periodic ACF (i.e., PACF) of a complex function (i.e. E(t) in our case) at discrete times τ = kT is defined by where x j x j+τ is the product of one bit in the base sequence with a corresponding bit shifted by kT, where j and k are integers. Figure 5 illustrates the evaluation of the PACF R E,E of E(t) phase modulated by MLS3 for k = 0, 1, and 2, according to Eq. (10). The calculating procedure is shown in Fig. 5. The first period of R E,E can be expressed as The minimum value of R E,E varies in the range of (−1/N, 1) according to k p , which is shown in Fig. 6. The power spectral density (PSD) of the laser field can then be given by the Fourier transform of the PACF according to the Wiener-Khinchin theorem, that is Through the use of elementary Fourier transform pairs and arithmetic, we can substitute Eq. (11) into Eq. (12), whereby Eq. (12) becomes Equation (13) is the normalized optical spectrum of the MLS phase modulation scheme, which is a discrete spectrum. The line spacing is given by f N = 1/NT = f cr /N and the clock rate f cr and the modulation amplitude k p determine the bandwidth. With a clock rate of 6.5 GHz, the line spacing becomes 928.57 MHz, 209.68 MHz, 51.18 MHz, 12.72 MHz, and 3.18 MHz for n = 3, 5, 7, 9, and 11, respectively. Spectral nulls occur at integer multiples of clock rate. The first part in RHS of Eq. (13) corresponds to the intensity (i.e., optical power) of the carrier while the second part corresponds to the intensity of the sidebands. The dependence of the intensities of the carrier and the adjacent sideband components on k p is shown in Fig. 7(a) and 7(b). When k p = (2m + 1)π (m ∈ Z), Eq. (13) becomes the PSD of the MLS waveform [28], viz At this point, the carrier reaches its lowest intensity 1/N 2 and the adjacent sideband components reach their highest intensity (N + 1)/N 2 . If there is dominating spectral line with more power than in other lines then it is well known that the SBS threshold is dictated mainly by this line, when the optical sidebands act independently (approximately when f N >Γ B /π) [17]. It is clear from Fig. 7(a) and 7(b) that the carrier can be such a dominant component. Most importantly, this must not be excessive, but nor should it be depleted of power since this would mean that the other lines carry more power than necessary. Thus, a value of k p that leads to the same carrier component intensity as in the adjacent (or the strongest) sideband components may be best for SBS suppression at large line spacing. components are compared in Fig. 7(c). We find that the values of k p that lead to the same intensity in the carrier and the adjacent sideband components (I c = I s ) are 0.8869π for MLS5, 0.9437π for MLS7, 0.9719π for MLS9, and 0.9859π for MLS11, respectively, as shown in Fig. 7(d).
There are also equivalent points for k p slightly larger than π. However, the power in the strongest spectral line varies only slowly near k p = π for every n (see Fig. 7(c)). Therefore, we take k p = π for the SBS suppression optimization for unfiltered MLS phase modulation in this work. Although this leads to marginally higher power in the strongest lines than necessary, this does not necessarily translate to a lower SBS threshold for small line spacings. Furthermore, we note that for most values of k p , the carrier is the dominant line and needs to be controlled precisely if it is to match the highest sideband peak. The slow variation of the power in the carrier around k p = π reduces the sensitivity to errors in k p . To verify the derivation, we construct a series of MLS waveforms with n = 5, 7, 9, 11, and calculate numerically the PSD of the MLS phase modulated laser field using the Wiener-Khinchin theorem. The clock rate is 6.5 GHz, and the modulation depth assumes the values π, 1.12π, 1.3π, and 2π. The calculated results are shown in Fig. 8. The intensity of both carrier and sidebands are in good agreement with the derivation. The results confirm that when k p = π, the carrier nearly vanishes and the sidebands reach their highest intensity. From these results we expect that in this scheme, errors in k p can easily result in a large carrier and thus drastically degraded SBS threshold.

Optical spectra of filtered and amplified MLS phase modulation
So far, the spectra of the phase modulated optical field are easy to find analytically because the unfiltered MLS is a sum of rectangular pulses. However, the spectra are sinc-shaped with slow decaying sidelobes, which is often undesirable, as is the distinctive behavior of the carrier. On the other hand, the bandwidth of rectangular pulses is infinite, so they will be distorted by the limited bandwidth of the different parts of a real system, in particular if there is a filter. As before, we will assume that a low-pass filter akin to a 6 th -order Butterworth filter but with spectrally flat phase characteristics determines the overall response. All other components are ideal, with bandwidth significantly larger than that of the filter, but other linear filter would be straightforward to implement. With reference to Fig. 1, the complex envelope at the RF input of the EOPM may be written as ϕ(t) * h(t), where h(t) is the impulse response of the filter. The frequency-domain response for a Butterworth low-pass filter is 1/ √︁ 1 + (f /f co ) 2m , where f co is the cutoff frequency (-3dB) and m is the order. The electric field of the lightwave phase modulated by this filtered signal can be described as We have not been able to obtain an analytical expression for the spectrum of a lightwave phase modulated according to Eq. (15). Instead, we calculated spectra numerically. The modulation depth of the low-pass filtered and amplified MLS is defined as k σ = πV σ /V π , where V σ is the standard deviation of the voltage of the modulation waveform. This is related to the RMS voltage where k AVG is the average of the modulation depth. Thus, the RMS value and the standard deviation are equal if k AVG = 0. We note that for an unfiltered MLS phase modulation with a peak-to-peak modulation amplitude of k p , the modulation depth k RMS = k p /2 when MLS waveform takes the values of −π/2 or π/2. Although k AVG ≠ 0, the resulting difference is small, especially for a long MLS, and therefore we neglect it. Optical spectra for n = 3, 5, 7, 9, 11, and 13 with f cr = 6.5 GHz are calculated with the ratio of the low-pass filter cutoff frequency to clock rate f co /f cr = 0.4, and k σ = 0.5π, π, 1.2π, 1.5π, and are shown in Fig. 9. The spectra become wider as k σ increases, and whereas the temporal periodicity means the spectra remain discrete, the envelopes are no longer sinc 2 shaped. There are no clear sidelobes, and, significantly, the power in the carrier does not depend critically on the modulation depth, at least for k σ >π/2. Still, the carrier is in many cases the strongest component by a considerable margin, which may lead to SBS. Next, we will investigate the SBS suppression capability of unfiltered and filtered MLS phase modulation scheme numerically, including the impact of the spectral spikes. We start with the unfiltered case in the next section. Fig. 9. Optical spectra as phase modulated by filtered and amplified MLS waveform with f cr = 6.5 GHz, f co /f cr = 0.4, and n = 3, 5, 7, 9, 11, and 13 for k σ equal to (a) 0.5π, (b) π, (c) 1.2π, and (d) 1.5π.

SBS suppression with unfiltered MLS phase modulation
As mentioned previously, phase modulation by an unfiltered MLS can provide an effective way of suppressing the SBS gain. We examine SBS suppression with phase modulation by a MLS with k p = π (approximate to k σ = π/2) using the simulation method introduced in Section 2. This modulation amplitude avoids strong spectral lines in the optical spectrum (see Fig. 8). The normalized SBS threshold (i.e., the enhancement factor) as a function of clock rate for different bit patterns is shown in Fig. 10(a). It should be noted that, even though the normalized SBS threshold is calculated by time averaging over 20 transit times, there is still some residual randomness in the simulation results, due to the Langevinian noise introduced in every temporal and spatial grid point. The increase in enhancement factor with clock rate saturates for MLS3 and MLS5, for a clock rate of ∼1.7 GHz (line spacing ∼243 MHz) and ∼7.5 GHz (line spacing ∼242 MHz), respectively. Also, a rollover point of about 30 GHz can be obtained for MLS7 (line spacing ∼236 MHz), which is shown in Fig. 10(b). The spectral line separation is more than twice the spontaneous Brillouin linewidth ∆ν B of 57.1 MHz in this paper at these points. For clock rates higher than these points, the spectral lines act independently and the SBS threshold no longer grows [17]. For high SBS threshold, the power should then be distributed over a large number of spectral lines, without excessive power in any line. Thus, in that regime, spectral considerations suggest a long MLS with smaller line spacing is preferable. However, once the spectral line spacing becomes sufficiently small for cross-interactions (beating) between adjacent spectral lines to contribute to the SBS gain, the SBS threshold decreases. Fig. 10. Normalized SBS threshold with unfiltered MLS and k p = π vs. clock rate for (a) n = 3, 5, 7, 9, 11, and 17 for clock rates from 0 GHz to 10 GHz, and (b) n = 7 for clock rates from 0 GHz to 30 GHz.
It is shown in Fig. 10(a) that the SBS threshold enhancement with MLS11 is smaller than with MLS9 and that with MLS17 is smaller than with MLS9 as well as with MLS11. Thus, the shorter sequences are better, and we note that all of these sequences have line spacings smaller than the Brillouin linewidth for the range of clock rates plotted in Fig. 10(a). The largest spacing becomes 10 GHz / 511 = 19.6 MHz. This is often explained as a result of the smaller line spacing of the longer sequences (= f cr /N), but in the simulations in [16], smaller line spacing, down to 12.5 MHz (the smallest considered) led to better SBS suppression for optimized, non-MLS, waveforms. However, this is not always the case, and a temporal-domain description may provide more insight into why one waveform should be better than the other, than the line spacing does. This includes MLS sequences, for the situation that the clock rate is sufficiently low relative to the rollover point. Then we believe that the dwell time for the MLS signal plays an increasingly important role in the SBS buildup kinetics, and for the same clock rate, the dwell time for MLS11 is longer than MLS9 (shown in Fig. 4(a)). For example, the dwell time for MLS9 and MLS11 with a clock rate of 1.5 GHz is 5.99 ns and 7.33 ns, respectively. Given that the Stokes power grows exponentially when the signal is unmodulated (Fig. 2(c)), for the case of G = 160 Np (corresponding to an enhancement factor of ∼18 relative to the unbroadened threshold gain of 9 Np), the reflectivity can grow up to 6.13×10 −5 for 5.99 ns and 1.91 × 10 −4 for 7.33 ns, which means the reflectivity of the MLS11 case is 3 times than that of the MLS9 case. It is clear that there can be a significant penalty for the longer dwell time of MLS11. For MLS17, the dwell time is much longer still, and its SBS suppressing capability thus reduced.

Optimization of filtered and amplified MLS phase modulation
In this section, we numerically optimize a filtered and amplified MLS waveform used to drive a phase modulator for the best SBS mitigation. Still, we use a low-pass filter similar to a 6 th -order Butterworth filter but with zero phase. As a starting point, we investigate the SBS suppression capability for phase modulation with low-pass-filtered MLS (no RF amplification). A plot of the normalized SBS threshold vs. clock rate for different f co /f cr and k p = π for MLS9 is shown in Fig. 11(a). It is shown that the SBS threshold increases with f co /f cr for k p = π. The highest threshold is reached for the unfiltered case (with infinite f co ). The reason for this is that the low-pass filtering is equivalent to temporal averaging, whereby the modulation depth k σ becomes too small, leaving too much power in the carrier for effective SBS suppression (shown in Fig. 9).
To overcome this and achieve a high SBS threshold also with the low-pass filter, we use a RF amplifier to boost the modulation depth of the filtered MLS9 waveform. Then we investigate the dependence of normalized SBS threshold on the clock rate at f co /f cr = 0.4 for different modulation depth k σ , which is shown in Fig. 11(b). As EOPMs often have a maximum RF input power of around 28 dBm into 50 Ω impedance (thus RMS voltage of 5.6 V) and a half-wave voltage around 5 V, the modulation depth K σ in our calculations is chosen to be within π (corresponding to 27 dB). We note that the unfiltered case with k p = π (thus with k σ = π/2) approximately matches the filtered case with k σ = π. We plotted k σ /k p as a function of f co /f cr for different n, as shown in Fig. 11(c). For large values of filter cutoff frequency, k σ is close to k p /2.
Although Fig. 11 suggests that higher cutoff frequency and modulation are better, this is not generally the case. Figure 12(a) shows the dependence of the normalized SBS threshold on both f co /f cr and k σ . The value of f cr is chosen to be 10 GHz, which is a typical value. We note that within the parameter range investigated in Fig. 12(a), f co /f cr must be at least around 0.1. Furthermore, for large f co /f cr , good suppression is only achieved when k σ equals an integer multiple of π/2. From a design perspective for filtered and amplified MLS phase modulation with a high SBS suppression, the optical linewidth plays an important role in this optimization. Figure 12(b) shows corresponding contours of the RMS linewidth ∆ν RMS according to these two parameters. Often, a narrow linewidth is preferred, and it is then desirable to suppress SBS with the smallest possible linewidth broadening. It is often assumed that the SBS threshold increases linearly with linewidth, and, the ratio of the normalized SBS threshold to ∆ν RMS is used as a figure of merit. This ratio is plotted in Fig. 12(c). Generally, there is significant structure with large variations in the plot, and the local maxima for normalized SBS threshold follows approximately a k σ = [0.06/(f co /f cr − 0.1) + 0.5] × π behavior, where f co /f cr >0.1. We especially consider five local maxima in Fig. 12(c) (marked in red circles), and calculate the optical spectra using parameters in these zones, as shown in Fig. 13. It is interesting to note that there are no strong side bands in Fig. 13(a) and Fig. 13(b) (local maximum 1 and 2, respectively), whereas the other three local maxima (Fig. 13(c)-Fig. 13(e)) exhibit spectral "spikes" located at integer multiple of f cr . In addition, the carriers are suppressed in all these five local maxima. It may safe to draw the conclusion that the suppression of the carrier and the strong sidebands for the filtered and amplified MLS phase modulation is necessary for SBS threshold enhancement in a fiber system. If SBS is to be suppressed without regard to linewidth, Fig. 12(a) suggests it is best to use a large modulation depth, where suppression is good and the structure is reduced. However, as mentioned, modulation depths k σ larger than π are difficult to reach because of limitations on the half-wave voltage and the maximum RF input power of commonly used EOPMs. We therefore restrict further investigations to k σ below π, and use numerical simulations to find the dependence of the normalized SBS threshold on both f co /f cr and k σ for n = 3, 5, 7, 9, 11 with f cr = 6.5 GHz in this regime. The results are shown in Fig. 14(a)-14(e). For a large pattern power n, especially n = 9, a distinct curve for the local maximum of the normalized SBS threshold appears (see Fig. 14(d)). By comparing Fig. 14(a)-14(e), we find that the global maximum for MLS9 outperforms the other patterns for k σ <π, and reaches the maximum around f co /f cr = 0.54 and k σ = 0.56π within the investigated parameter range. The corresponding optical spectrum is shown in Fig. 13(e). Figure 14(a)-14(e) also indicate that a minimum value of f co /f cr of 0.3∼0.4 and of k σ of 0.5π is needed for high SBS suppression. To further distinguish the SBS suppression capability from different patterns, we plot the dependence of the normalized SBS threshold on the modulation depth k σ with f cr = 6.5 GHz and f co /f cr = 0.4 in Fig. 14(f). These corresponds to cuts through the Fig. 14(a)-14(e) at f co /f cr = 0.4. It is clear from this figure that the n = 9 pattern is superior to other patterns for k σ more than 0.5π. We also calculate the normalized SBS  threshold for MLS9 with clock rate of 1 GHz, 3 GHz, 4 GHz, 5 GHz, 6.5 GHz and 10 GHz, as shown in Fig. 15(a)-15(f). Figure 15(f) is a subset of Fig. 12(a). It is shown that the maximum of the normalized SBS threshold depends on f co /f cr but not the absolute value of f cr . Returning to Fig. 13(e), this contains spectra for the optimized parameters (f co /f cr = 0.54 and k σ = 0.56π) for MLS9 with clock rates of 1 GHz, 3 GHz, 5 GHz, 6.5 GHz, and 10 GHz. For all the clock rates, there are strong components at f cr . We calculate the RMS optical linewidth of this optimized filtered and amplified MLS phase modulation with f cr ∈ (0, 10) GHz, as shown in Fig. 16. The normalized SBS threshold for MLS9 with f co /f cr = 0.54 and k σ = 0.56π for f cr = 1 GHz, 3 GHz, 4 GHz, 5 GHz, 6.5 GHz, and 10 GHz (also the global maxima in Fig. 15) is also shown in Fig. 16 (red circles). It is shown that both the RMS linewidth of the optical spectra and the maximum normalized SBS threshold with the optimized parameters increase linearly with f cr , although a roll-off in threshold is expected when the line spacing (which stay below 19.6 MHz in Fig. 16) is sufficiently large, relative to the Brillouin line. The maximum RMS optical linewidth with the parameters considered here is calculated to be within 15 GHz, which indicates that the optimization in this work is suitable for both spectral and coherent beam combining. The calculations for Fig. 12(a), Fig. 12(c), Fig. 14(a)-14(e) and Fig. 15 are time consuming even for the 61 × 61 grid that we use, because the SBS threshold is found by numerical iteration with an increasing laser power, incremented in small laser power steps to ensure the accuracy. During the integration, the power increment is adapted to reduce the number of iterations to save computation time: first, we choose a relatively big power increment (20 W). Then the iteration goes quickly through the linear zone of the SBS reflectivity vs. G curve. The nonlinear zone of the reflectivity curve needs a smaller increment to achieve a high accuracy. When the calculated SBS reflectivity exceeds 1% (i.e., when the laser power is above our definition of the threshold), the iteration is repeated from a low power with a new increment which is the previous increment divided by 2. A solution is found when, the SBS reflectivity deviates from the threshold of 1% by less than 1×10 −6 (relative error < 10 −4 ). We use the Parallel Computing Toolbox T M in Matlab to split the execution of iterations over eight workers in a parallel pool. The execution time for each sub-figure in Fig. 12(a), Fig. 12(c), Fig. 14(a)-14(e) and Fig. 15 is less than 12 hours. Further optimization of the algorithm for threshold searching can improve the computational efficiency.

Conclusion
We have investigated SBS suppression in a 7.4-m-long passive fiber through phase-modulation of a single-frequency lightwave with a maximal-length sequence, assumed to be a RF wave. We considered ideal sequences comprising undistorted rectangular bits with π phase difference between symbols, as well as those that were low-pass-filtered and amplified to different levels of phase modulation. The temporal and spectral characteristics of the modulation waveform and the modulated lightwave were investigated analytically with expressions we derived, as

Research Article
Vol. 0, No. 0 / 00 00 0000 / Optics Express 22 well as numerically. The SBS threshold power was evaluated through numerical integration of a time-dependent three-wave coupled nonlinear system that describes the SBS dynamics of the phase-modulated lightwave, and subsequent time-averaging. The threshold was quantified relative to the threshold for the unbroadened lightwave (the so-called enhancement factor). For an enhancement factor of around 20, we found that the Brillouin Stokes wave can build up exponentially with a sub-nanosecond time constant in our 7.4-m fiber length. This can then reduce the SBS suppressing capability for MLS waveforms at GHz-level clock rate, insofar as these have uninterrupted single-symbol sub-sequences ("runs") extending over several nanoseconds. The combination of a low-pass filter and a RF amplifier can be used to distort the uninterrupted long sub-sequences and improve the phase distribution and SBS suppression with the MLS waveform. Aiming to accomplish the best SBS mitigation, parameters of the filtered and amplified MLS such as pattern length, modulation depth, and the ratio of filter cutoff frequency to clock rate are optimized numerically. The simulations indicate that for a RMS modulation depth of 0.56π and a ratio of filter cutoff frequency to clock rate of 0.54, the normalized SBS threshold reaches a maximum, and that MLS9 is superior to other investigated patterns (but note that MLS8 and MLS10 were not investigated). Unless the modulation is close to an integer multiple of ±π/2, a cutoff frequency larger than around 80% of the clock rate leads to lower SBS suppression in the cases we considered, which we attribute to inadequate suppression of the lightwave's carrier. By contrast, there is no upper limit on the modulation depth, beyond which SBS suppression is no longer effective. Nevertheless, the suppression does not increase monotonically with modulation depth, which necessitates a careful selection of waveform parameters. Similarly, the suppression vs. linewidth characteristics vary significantly, and the suppression can be poor even with large linewidths. Our results provide new insights into SBS and its dynamics, relevant for kilowatt-class fiber systems for spectral and coherent beam combining which need to balance the linewidth and the SBS.