A Theoretical and Experimental Analysis of the Time-Domain Characteristics of a PRBS Phase-Modulated Laser System

Abstract

Pseudo-Random Binary Sequence (PRBS) phase modulation is an effective method for suppressing the stimulated Brillouin scattering (SBS) effect generated by narrow-linewidth fiber lasers during amplification. We noticed that backward time-domain pulses were generated when using PRBS modulation signals in fiber amplification. In this paper, the time-domain dynamic characteristics of the forward output laser and the backward Stokes light after PRBS phase modulation were studied theoretically. Through analyzing the transient SBS three-wave coupling theory and combining it with the SBS accumulation time constant, we knew that the forward and backward high-intensity pulses were caused by the long dwell time of the PRBS. For this purpose, we provided a new method for suppressing high-intensity pulses caused by a long dwell time; namely, we modified the maximum length sequence (MLS) of PRBS signals to eliminate the long dwell time, took the PRBS-9 signal at 1 GHz as an example, and then used MLS1 modulation and MLS2 modulation to compare them with unoptimized PRBS modulation. The output laser peaks of the MLS1 and MLS2 signals were reduced from ±55% to ±25% and ±10% relative to the original PRBSs, respectively, and the peaks of Stokes light were reduced from 39% to 19% and 11%, respectively. Additionally, we experimentally verified that the rational optimization of the sequence did not reduce the SBS threshold. The results provided a new method for suppressing high-intensity pulses during the amplification of a PRBS phase-modulated laser, which played an important role in the output stability of high-power narrow-linewidth fiber amplifiers.

1. Introduction

SBS is a nonlinear effect with the lowest threshold in narrow-linewidth CW fiber systems [1], which limits power scaling. At present, SBS suppression methods can be classified as follows: using large-mode fibers [2], using high-doped gain fibers [3], reducing the overlap between the light field and the sound field [4], broadening the SBS gain spectrum [5], and using phase modulation to broaden the signal light [6]. The phase broadening of the signal laser and a reduction in power spectral density are currently the most effective solutions to increase the SBS threshold [7,8,9]. At present, the mainstream phase modulation signals include the Sine signal, WNS (white noise source), and PRBS. Among these, Sine signal generation is the simplest of the several signals; however, it makes the average power of the seed source spectrum higher, which is due to its completely separated spectrum, resulting in its poor inhibition of SBS. The spectrum of WNS is inherently continuous, but the randomness of its time-domain sequence causes random spikes, which reduce the maximum output power of the laser. A PRBS is a digital signal that separates spectral lines and has a certain periodic nature, which produces an envelope distribution similar to a WNS, but the spectrum can be precisely controlled by adjusting the clock rate so that the SBS effect can be better suppressed. Therefore, the PRBS phase modulation scheme is one of the most effective schemes for suppressing SBS [10,11,12,13,14,15,16]. However, the PRBS phase modulation scheme is mainly focused on the selection of the pattern, the optimization of the ratio of the filter’s cut-off frequency to the clock rate, and the exploration of the optimal value of the modulation depth, but the time-domain characteristics of the forward and backward output have not been studied enough. In 2013, Melo et al. [17] reported that when a pulsed signal was amplified in a fiber amplifier, a very high peak power random pulse was generated backwards. These backward-generated high-peak-power (kW-level) pulses could damage the components in the upstream amplifier and main oscillator stage. In 2017, Hanzard et al. [18] also discovered kW-level pulses in self-pulsed fiber lasers, which were all attributed to SBS in fiber lasers. In 2020, Chu et al. [19] investigated the high-intensity pulses caused by SBS after WNS phase modulation in high-power fiber lasers and increased the SBS threshold by cascading the WNS. It could be seen that the instability of the output laser and the backward Stokes light caused by the SBS effect would have extreme consequences. Thus, it is important to study the time-domain characteristics of the output light in PRBS phase-modulated fiber lasers.

2. Theoretical Analysis

2.1. SBS Process Simulating

The PRBS phase-modulated monolithic laser system we considered is shown in Figure 1. The PRBS signal was amplified by an RF amplifier and driven by an electro-optic phase modulator (EOPM), and the modulated lightwave entered a passive fiber with a length of L.

We used the transient SBS three-wave coupling equation to simulate the time course and obtained the output waveforms of the laser and Stokes light. The equations can be written as follows [20,21]:

$${\displaystyle \frac{c}{n}}{\displaystyle \frac{\partial A_L}{\partial z}}+{\displaystyle \frac{\partial A_L}{\partial t}}+\left[{\displaystyle \frac{c}{n}}{\displaystyle \frac{l_s}{2}}+{\displaystyle \frac{i{\mathsf{\omega }}_{\mathrm{s}}{\mathrm{n}}_2}{{nA}_{eff}}}\left({\left|A_s\right|}^2+2{\left|A_b\right|}^2\right)\right]A_L=\mathit{-}{\displaystyle \frac{{\gamma }_e\omega }{2{\rho }_0{n}^2}}{A}_{S}Q$$

(1)

$${\displaystyle \frac{c}{n}}{\displaystyle \frac{\partial A_S}{\partial z}}+{\displaystyle \frac{\partial A_S}{\partial t}}+\left[{\displaystyle \frac{c}{n}}{\displaystyle \frac{l_s}{2}}+{\displaystyle \frac{i{\mathsf{\omega }}_{\mathrm{s}}{\mathrm{n}}_2}{{nA}_{eff}}}\left({\left|A_b\right|}^2+2{\left|A_s\right|}^2\right)\right]A_S=-{\displaystyle \frac{{\gamma }_e\omega }{2{\rho }_0{n}^2}}{A}_L{Q}^{*}$$

(2)

$${\displaystyle \frac{\partial Q}{\partial t}}+\left({\displaystyle \frac{1}{2}}{\mathsf{\Gamma }}_B-j\left({\mathsf{\Omega }}_B-\mathsf{\Omega }\right)\right)Q=-{\displaystyle \frac{i{\gamma }_e\omega }{2c^2{v}_A{A}_{ao}}}{A}_L{A_S}^{*}+f$$

(3)

Here, $A_L$, $A_S$, and $Q$ denote the amplitudes of the laser, Stokes wave, and acoustic waves, respectively. $c$ represents the speed of light in a vacuum, while n is the fiber’s refractive index. The nonlinear refractive index is denoted by ${\mathrm{n}}_2$, and $l_s$ indicates the fiber’s background loss. The angular frequencies are $\omega$ ($\omega ={\mathsf{\omega }}_{\mathrm{L}}\simeq {\mathsf{\omega }}_{\mathrm{S}}$) for the laser and ${\mathsf{\Omega }}_B$ for the acoustic wave, and $\mathsf{\Omega }$ is the Brillouin scattering shift. The electrostrictive constant ${\gamma }_e$ connects electrostrictive pressure to the electric field, and ${\rho }_0$ is the fiber medium’s background density. Additionally, $A_{eff}$ is the effective mode field area, $v_A$ is the velocity of the acoustic fundamental mode in the fiber core, and $A_{ao}$ is the interaction area between the acoustic wave and the signal laser. The sound wave’s damping rate is ${\mathsf{\Gamma }}_B=2\pi \times w_{BGS}$, where $w_{BGS}$ is the Brillouin gain spectrum, and $f$ represents Langevin noise.

Table 1. Parameters and the values used for simulation.

Symbol	Quantity	Symbol	Quantity
T	293 K	${\rho }_0$	2201 kg/m3
n	1.45	$A_{ao}$	2.67 × 10−10 m2
$l_s$	15 dB/km	${\nu }_A$	5900 m/s
${\lambda }_s$	1064 nm	${\Omega }_B-\Omega$	set to 0
$n_2$	2.6 × 10−20	$w_{BGS}$	57.1 MHz
$A_{eff}$	2.6 × 10−10 m2	L	10 m
${\gamma }_e$	1.95

summarizes the key parameters and their values used in our simulations.

In passive fibers, the SBS setup time constant at the fiber inlet can be expressed as follows [22]:

$${\tau }_C={\displaystyle \frac{2n{A}_{eff}}{gPc}}$$

(4)

where P is the laser power, and $g$ is the SBS gain coefficient.

2.2. Optimized PRBS Signal

In a periodic binary sequence with period N, there are m ones and N-m zeros. If its autocorrelation function

$$C\left(v\right)={\sum }_{j = 0}^{N - 1} a_j{a}_{j + v}$$

(5)

has only two values

$$C\left(v\right)=\left\{\begin{array}{cc}m,\hfill & \mathrm{if}v\equiv 0\left(\mathrm{mod}N\right)\\ mc,\hfill & \mathrm{otherwise}\hfill \end{array}\right.\mathrm{and}c={\displaystyle \frac{m-1}{N-1}}$$

(6)

then this binary sequence becomes a PRBS. c is called the duty cycle of the PRBS. The following Figure 2 shows a sequence diagram of PRBS-9 with a clock rate $f_{cr}$ of 1 GHz.

It can be noticed in Figure 2 that the PRBS has a specific tour property, i.e., for a PRBS with pattern n, there is one single “1” run of length n and one single “0” run of length n − 1, 2ⁿ⁻²⁻ⁱ runs of length i “111…1”, and 2ⁿ⁻²⁻ⁱ runs of length i “000…0”, where i ≤ n − 2. For example, a PRBS-9 period of a certain run length is shown in Figure 2, including one longest sequence (MLS) (111111111), one second longest sequence (00000000), two third longest sequences (1111111) (0000000), four fourth longest sequences, eight fifth longest sequences, and so on.

In 2021, Liu Meizhong and collaborators [23] investigated how SBS responds instantaneously to binary sequence waveforms in optical fiber systems. They observed that long-period binary sequences, exhibiting damped oscillations akin to the SBS attenuation process, can effectively suppress SBS if the dwell time is less than the time constant required for SBS to develop and the sequence period is sufficiently long. In 2022, He Xuan and his team [24] found that the occurrence of high peak pulses in the backward direction was related to the residence time of a PRBS. Our study theoretically examined how extended dwell times impact these backward high peak pulses and proposed an optimization for the longest PRBSs to prevent the formation of giant pulses.

3. Simulation

3.1. Time-Domain Pulses of MLS Phase-Modulated Lasers

Based on the transient SBS three-wave coupling theory, we simulated the time-domain waveforms of the output laser and the backward Stokes light after the phase modulation of PRBS-9 at an $f_{cr}$ of 1 GHz by using MATLAB, as shown in Figure 3; the positions of the first and second longest sequences of the PRBS are shown in this figure. We observed that the output laser produced a strong pulse and a second strong pulse of the backward Stokes light during each PRBS period, which corresponded to the first longest dwell time (“111111111”) and the second longest dwell time (“000000000”) in the PRBS.

In reference to [23], we knew that the strong pulses in the output waveform were generated by the long dwell time in the PRBS and that the laser and Stokes light produced corresponding pulses during each PRBS period. This phenomenon was elucidated by establishing a time constant for SBS. The laser power at the passive fiber input is 6.4 W, which corresponded to a time constant of about 5.6 ns for SBS establishment. The duration of each bit of the PRBS-9 signal was T = $f_{cr}$/N = 1 ns; the first longest dwell time was nT = 9 ns, which exceeded the SBS establishment time constant; and the spike of Stokes intensity was gradually generated in the time window of the first longest dwell time. The second longest dwell time needed to run (n − 1)T = 8 ns, which also exceeded the SBS establishment time constant, but the Stokes intensity relaxation oscillation time decreased, and the pulses generated were relatively weak, and the forward output light and backward Stokes light produced second-intensity pulses. A deeper explanation revealed that when the dwell time was longer than the SBS establishment time constant, the laser was equivalent to an unmodulated state, which behaved as a single-frequency signal, which carried a power much higher than the threshold in the actual single-frequency case, resulting in a giant pulse. The third longest dwell time (“1111111” “0000000”) was (n − 2) T = 7 ns, which was close to the time constant of SBS establishment, and the pulses generated were smaller, and in other smaller dwell times, the pulses were not enough time to be established, and SBS could be better suppressed.

3.2. Eliminating Time-Domain Pulses by Simulating MLS

Based on the discussion in the previous section, we analyzed the reasons why the strong pulses in the output light were caused by the longest dwell time in the PRBS signal. This strong pulse greatly affected the stability of the output light in the actual fiber amplifier, and the strong pulse in the backward light would destroy the upstream amplifier. In order to suppress pulses caused by long dwell times, we used two models to adjust and optimize the PRBS: in model 1 (MLS1), the longest run (111111111) was replaced by “101010101”; in model 2 (MLS2), the second longest run (00000000) was replaced with “01010101”. Both $f_{cr}$ and the incident laser power remained constant.

Figure 4 shows the time-domain characteristics of the original PRBS, MLS1, and MLS2 phase-modulated laser output in passive fiber and Stokes light. We can see that these three sequences did not change the periodic characteristics of the pulses after modulation, and the output light fluctuated periodically during each PRBS-9 period. However, the intensity of the output light was not same. It could be seen that when MLS1 and MLS2 were employed, the time-domain pulses of the output laser and backward Stokes light were greatly reduced. The output laser peaks of the MLS1 and MLS2 signals were reduced from ±55% to ±25% and ±10% relative to the original PRBSs, respectively, and the peaks of Stokes light were reduced from 39% to 19% and 11%, respectively.

The decrease in pulse intensity also proved that the strong pulses of the output light were caused by the long dwell time. When the bit sequence was a combination of 1 or 0, it was equivalent to the unmodulated state. At this time, SBS would accumulate amplification, and the longer the dwell time, the stronger the generated SBS time-domain pulse; when the long dwell time was broken, it hindered the establishment time of SBS and then effectively inhibited the generation of SBS. Our scheme could be further optimized to eliminate smaller pulses in the time domain, e.g., by replacing the third longest run with “0101010”; however, such a modification gradually reduced the randomness of the sequence, and the spectrum of the signal gradually lost the shape of the Sinc² envelope, which might lower the SBS threshold. MLS2 might be a compromise between maintaining sequence randomness and removing long dwell times. In the actual system, an AWG could be used to optimize the coding of PRBS signals for phase modulation and improve the stability of the output light in high-power amplification experiments.

4. Experiment Setup and Results

4.1. Experiment Setup

We set up an experimental setup to measure the temporal behavior of SBS intensity, as shown in Figure 5. We employed a low-noise single-frequency laser (NKT, Koheras Basik Y10, Copenhagen, Denmark) with a linewidth of 20 kHz as the seed source, delivering an output power of 10 mW. The seed light was then amplified through three stages of pre-amplification, each stage comprising 1 m of polarization-maintaining ytterbium-doped fiber (PM YDF) with an absorption coefficient of 250 dB/m, and the pump light had a central wavelength of 980 nm, resulting in an amplified output power of 50 mW. Signal generation involved an arbitrary waveform generator (AWG, Tektronix 70001B, Portland, OR, USA) to produce both unoptimized and optimized MLS signals, but the signal emitted from the AWG was discrete. We accessed a low-pass filter (LPF) to flatten the signal so that the phase of the light could be kept on the π modulation, and the signal was then amplified by an RF amplifier and fed into an EOPM to broaden the phase of the pre-amplified seed light. The main amplification stage utilized a 4.5 m long ytterbium-doped fiber with an absorption coefficient of 4.95 dB/m, and the pump optical center wavelength was 980 nm. After the main amplifier, we connected it to the circulator, which could isolate the backward light and prevent the upstream amplification stage from being damaged, and also, it could be used to test the spectral characteristics and waveforms of the backward light. A 99/1 2 × 2 fiber splitter was then plugged in, with 99% of the ports used to output the laser and 1% of the rear ports used to test the return optical power. We used a 1000 m passive fiber as the gain fiber for the SBS effect, which was used to amplify the SBS effect, and finally pumped the stripping at the output with an angle of 8° to strip the pump light and prevent end face reflection.

4.2. Optimized Experiments for MLS Signal Inhibition of SBS

We measured the backward power at the back end of the beam splitter and defined the SBS threshold as the output power when the return power was 0.5‰. For the AWG output signal, we used the feature polynomial x⁹ + x⁴ + 1 to write the PRBS-9 sequence, PRBS-9 MLS1, PRBS-9 MLS2, and PRBS-9 MLS3 by ourselves. We knew that MLS3 may reduce the SBS threshold even though it could better reduce the pulse intensity of the output light, and we tested this guess in our experiments.

Figure 6 shows an $f_{cr}$ of 8.5 GHz, an LPF cut-off frequency of 4.4 GHz, and the phase modulation of the original sequence and the optimized sequence in the output and backward power after seed laser amplification.

As we can see from Figure 6a, the output power reached the SBS threshold at 2.844 W when the original sequence modulation was used, and the backward power was 1.637 mW. Figure 6b–d present that when MLS1 modulation was applied, the output power could be increased to 2.847 W, and the backward power was 1.606 mW, and the threshold was not reduced compared with the sequence. When MLS2 modulation was applied, when the output power reached 2.873 W, the backward power was 1.600 mW, and compared with the original sequence, the SBS threshold was not reduced but slightly increased. When MLS3 modulation was applied, the output power could reach 2.833 W, and the backward power was 1.595 mW, which was a relatively slight decrease in output power. In order to demonstrate universal applicability, we also changed the modulation parameters of the PRBS and performed control experiments, the results of which are available in

Table 2. SBS thresholds under different modulations under different n, clock rate, and LPF frequency.

n	Clock Rate (GHz)	LPF Frequency (GHz)	Normal SBS Threshold (W)	MLS1 SBS Threshold (W)	MLS2 SBS Threshold (W)	MLS3 SBS Threshold (W)
7	3.75	2	1.408	1.459	1.461	1.367
	8.5	4.4	2.801	2.808	2.812	2.756
	18	9.5	2.783	2.772	2.791	2.767
9	3.75	2	1.448	1.516	1.670	1.440
	8.5	4.4	2.844	2.847	2.873	2.833
	18	9.5	2.895	2.898	2.902	2.858

.

The results showed that the reasonable optimization of MLS sequences not only did not reduce the SBS threshold but also had certain advantages in improving the SBS threshold. MLS3 had the opposite effect on the output power increase, because the modification of the sequence reduced its randomness, resulting in the power spectral density no longer being the shape of the rigorous Sinc² envelope, which was consistent with the speculation in Section 3.2. Moreover, the effect of pattern 9 is better than that of pattern 7, which also confirmed the conclusion that a long sequence period inhibits SBS better, as in Ref. [23].

Figure 7 presents the output spectrum with a pattern n of 9, an $f_{cr}$ of 8.5 GHz, an LPF cut-off frequency of 4.4 GHz, and an output power of 2.844 W. As can be seen from Figure 7a, the signal-to-noise ratio of the output spectrum was greater than 50 dB, and the spectrum contains only 1064 nm of signal light, indicating that the SBS effect was well suppressed by phase modulation, and Figure 7b shows a spectral pattern measured by an optical spectrum analyzer (OSA, HyperFine, HN-8995-2, Lightmachinery, Ottawa, ON, Canada) with a resolution of 0.8 pm, and the spectral half-height and width of 4.86 GHz were obtained.

The beam quality at an output power of 2.844 W was measured, as shown in Figure 8. The beam transmission factor in the X direction was $M_x^2=1.05$, and the beam transmission factor in the Y direction was $M_y^2=1.10$. The measurement results showed that the beam propagation factor M² of the laser was close to the diffraction limit of the fundamental mode Gaussian beam.

5. Conclusions

In our study, we investigated the temporal characteristics of the output laser in a fiber system modulated by a PRBS using the transient SBS three-wave coupling equation. During our analysis, we identified periodic high-intensity pulses occurring between the output laser and the backward Stokes light. These pulses were found to originate from the extended dwell times present in the PRBS signal, with particular emphasis on the longest and second longest dwell times. To address this issue, we optimized the PRBS signal while taking into account the SBS establishment time constant of 5.6 ns. The output laser peaks of the MLS1 and MLS2 signals were reduced from ±55% to ±25% and ±10% relative to the original PRBSs, respectively, and the peaks of Stokes light were reduced from 39% to 19% and 11%, respectively. This was essential to maintain the stability of the lasers’ output and protect the components in the upstream amplifier and main oscillator stage. The MLS2 signal modulation was particularly effective as a compromise solution, balancing the need to maintain the randomness of the sequence while also addressing the issue of long dwell times. Additionally, we experimentally verified that the rational optimization of the sequence does not reduce the SBS threshold, which also proved that our optimization method does not affect the output of lasers.

The theoretical and experimental results obtained from this study provide crucial guidance for the development of high-power fiber lasers with narrow linewidths, offering valuable insights for optimizing their performance.