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Abstract
We demonstrate a modulation approach that relaxes the limitations imposed by stimulated Brillouin scattering (SBS) on amplification and propagation of narrow-linewidth light in fibers. By imposing synchronous amplitude and phase modulation on an input field, the optical spectrum after high-power fiber transmission is compressed using nonlinear self-phase modulation. This effectively reduces the SBS interaction length and increases the SBS threshold, enabling narrower linewidths. Using this technique, we demonstrate >2 × increase in SBS-limited spectral brightness from a kW-class amplifier. We show that SBS suppression becomes more effective for higher powers and longer fibers.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Narrow-linewidth, multi-kW Yb-doped fiber amplifiers (YDFAs) are of interest for parallel scaling of laser systems to higher powers with near diffraction-limited beam quality using coherent or spectral beam combining (CBC or SBC) methods [1]. For SBC laser systems in particular, the power and spectral linewidth of the underlying YDFA sources limits the attainable brightness of the combined beam. With spectrally broad YDFAs, the beam quality of the combined output can be degraded by angular dispersion from the combining optics [2]. For this reason it is of interest to scale kW-class YDFAs to higher peak optical power spectral density (PSD).
Stimulated Brillouin scattering (SBS) is a nonlinear impairment that limits the peak PSD attainable from >kW YDFAs [3]. SBS can be particularly limiting for beam combined system architectures, which often require multi-meter lengths of delivery fiber following each YDFA to enable termination in a common, close-packed array [4]. Since delivery fibers experience the full amplified power, YDFAs terminated with delivery fibers exhibit significantly increased nonlinear length and SBS gain compared to un-terminated YDFAs.
To increase the threshold power for SBS, it is common practice to broaden the input seed spectrum to the multi-GHz domain using phase modulation, or equivalently frequency modulation (FM). This reduces the optical coherence length and therefore reduces SBS gain. As YDFA powers increase to the multi-kW level, or as delivery fiber lengths increase, broader FM linewidths are needed to suppress SBS [5–11]. Consequently, peak PSD tends to stay constant or even decrease once linewidths become comparable to the Brillouin Stokes shift (~16 GHz in SiO2 fibers), at which point SBS can build up from resonant backscattered signal power rather than from noise [12–14].
A complementary approach to suppressing SBS is to spectrally shift some fraction of the laser power away from the final output spectrum (which we define as the “carrier” band). After traversing some or all of the fiber length, the spectrally shifted power is returned into the carrier spectral band. With a sufficiently large spectral shift, the SBS gain of the shifted power is independent from the SBS gain of the carrier. The net effect is to reduce the SBS gain of the carrier by reducing its effective nonlinear interaction length.
One example of this spectral shifting approach is the use of laser gain competition to transfer laser energy from a blue to red carrier band over a relatively short length of fiber [15]. This approach has been shown to provide ~2 × increase in spectral brightness at the kW level. However, implementing laser gain competition constrains the output wavelength to the red edge of the YDFA gain bandwidth, so it is of limited use in SBC systems whose channels typically span the accessible YDFA tuning range. Moreover, this approach is limited to actively lasing gain fibers, so it cannot reduce SBS arising from high power transmission through passive delivery fibers.
Another example of spectral shifting to reduce SBS is the use of a free space phase electro-optic modulator (EOM) at the output end of the amplifier to demodulate FM applied at the input end, thus recovering a narrow-line carrier [16–18]. Owing to the limited power handling, modulation depth, and modulation frequency of free space EOMs, it is challenging to extend this approach to kW-class fibers, particularly for beam combining applications that typically require fiber termination in close-packed arrays.
In this work, we demonstrate a novel spectral shifting approach that is conceptually similar to [16], but that uses the intrinsic fiber Kerr nonlinearity rather than a free space EOM for phase demodulation and carrier recovery [19]. This physical mechanism of fiber self-phase modulation (SPM) is widely used for nonlinear spectral compression of negatively chirped picosecond pulses [20,21]. By applying this to the continuous-wave (cw) domain, we demonstrate up to ~2.5 × increase in the SBS-limited peak PSD attainable from a kW-class fiber amplifier with a highly nonlinear delivery fiber.
2. Principle of operation
Figure 1(a) shows the optical block diagram for the nonlinear phase demodulation concept. Light from a cw single-frequency master oscillator (MO) is modulated via FM and amplitude modulation (AM) in three functionally distinct stages. Figure 1 and the discussion below depict these stages as occurring in serial modulators both for pedagogical purposes and to align with our experimental setup. However, with an appropriate electrical driver all three modulations could be imposed at once using a single Mach-Zehnder EOM with independent drive inputs for each leg [22,23].
Fig. 1 Conceptual diagram for nonlinear phase demodulation. (a) Block diagram. (b) Notional PSD at different locations in the optical circuit. (c) Notional amplitude modulation at different locations.
The first modulation stage defines a carrier spectrum by imposing a time-dependent phase ϕ(t). Setting the MO field amplitude to unity and its optical frequency to zero for simplicity, the carrier field is
The notional spectrum shown in Fig. 1(b) illustrates the broadened PSD of the carrier, where is the Fourier transform of . The second stage modulates the laser amplitude, so that the normalized power is , and the corresponding field isFor , as illustrated in Figs. 1(b) and 1(c), this AM has little impact on the optical PSD. This field is further spectrally broadened in a third stage by applying a phase that is synchronized to a common waveform u(t) with the AM, with modulation depths m and β for the AM and FM respectively: The field after this final modulation stage isThis field seeds a high power YDFA or fiber transmission line characterized by a cw nonlinear phase shift, or B-integral, B. Neglecting dispersion, after the nonlinear transmission line the normalized output is:where is self-phase modulation driven by . We note that this treatment based on SPM automatically includes all four-wave mixing terms through intensity modulation as long as the field is polarized and all frequencies are included.From Eq. (6), and from the corresponding spectra shown in Fig. 1(b), one can see that spectral broadening due to can be gradually reduced as the signal propagates and accumulates nonlinear phase. The broadening is reversed at the output when the SPM phase exactly cancels the synchronous FM applied at the input:
For any given nonlinearity B, Eq. (7) corresponds to selection of the modulation depth parameters to satisfyIn the limit of small, this recovers the original carrier spectrum at the amplifier output. By spectrally shifting power out of the carrier band for much of the nonlinear interaction length, the SBS gain is reduced without increasing the output linewidth.3. Experimental implementation
This concept was demonstrated using the experimental configuration shown in Fig. 2. The master oscillator was a Nd:YAG nonplanar ring oscillator (NPRO) that emitted a single frequency signal at λ = 1064 nm. The carrier phase modulation was imposed using an amplified radio frequency (RF) thermal white noise source (WNS) to drive a carrier EOM. By adjusting the WNS RF amplifier, the carrier full-width-half-maximum (FWHM) optical linewidth could be tuned continuously over a range of 3 – 30 GHz.
Fig. 2 Experimental high power test layout. Tests were performed with and without the components with cross-hatched backgrounds (Dispersion compensation and Delivery fiber).
An arbitrary waveform generator (AWG) was used to generate the synchronous drive waveform u(t). While the nonlinear demodulation concept is general to any arbitrary waveform, for experimental simplicity we chose a single-tone sinusoid:
with modulation frequency = 32 GHz. This waveform was synchronously output to separate RF amplifiers to drive the AM EOM and FM EOM, with independently adjustable amplitudes (modulation depths) and relative RF phase. The AM EOM was biased in quadrature (50% transmission), so that the optical AM approximately followed the RF waveform for .Inserting Eqs. (3), (4), and (9) into Eq. (5), the modulated seed field using this single tone drive waveform is:
The FM term can be expanded using the Jacobi-Anger identity:where the Jn are Bessel functions of the first kind. Transforming Eq. (11) to the spectral domain, the seed field can be written as the convolution of the AM-modulated carrier and a frequency comb at harmonics of the modulation frequency :where is the convolution operator, and δ(ω) is the Dirac delta function. From Eq. (12), one can see the effect of the single-tone FM EOM is to generate replicas of the carrier PSD at sidebands corresponding to harmonics of ωmod. We chose a value ωmod/2π = 32 GHz to be approximately twice the 16-GHz SBS Stokes shift. This allowed increasing the carrier spectral FWHM to be on the order of the Stokes shift while avoiding self-seeding driven by spectral overlap between the Stokes-shifted return power and any modulation sidebands.In addition to the FM sidebands, some power is also shifted into spectral sidebands owing to AM. One can estimate the magnitude of this AM-shifted power in the limit by expanding the field amplitude to second order in m:
From Eq. (13), one can see the approximate DC carrier amplitude is 1 – m2/16, and therefore the carrier power is (1 – m2/16)2 ≈1 – m2/8. This represents the highest possible AM-limited compression efficiency; i.e., even with perfect phase demodulation, a power fraction m2/8 will remain in spectral sidebands due to AM that remains imprinted on the output.The fully modulated seed was pre-amplified to ~30 mW in a semiconductor optical amplifier (SOA) to seed a commercial 3-stage YDFA chain (Nufern, final stage 20/400 μm core/clad diameter, co-pumped, 13 m fiber coiled for single mode output) characterized by a B-integral B = 4.3 rad at the maximum 1.34 kW output. Since the YDFA fiber was non-polarization maintaining (non-PM), we inserted a manual polarization adjuster to enable control of the launch state of polarization (SOP). To assess the impact of higher nonlinearity, tests were conducted with and without an additional 20-m length of mode-matched passive delivery fiber spliced to the YDFA output as indicated in Fig. 2. The high-power output fiber was angle-cleaved to minimize backscattered light, which was monitored from a return power tap located upstream of the final amplifier stage. A sample of the collimated free space output beam was fiber-coupled into an optical spectrum analyzer (OSA) with ~3 GHz resolution.
Downstream of the modulators, the total fiber length was L ~100 m due to patch cords, pigtailed components, multiple YDFA stages, and delivery fiber. With this length of fiber, dispersion arising from the relatively narrowband carrier could be neglected, but dispersion of the 32 GHz sidebands had a significant impact on δP(t) due to FM-to-AM conversion [24]. While a full treatment of dispersion in the nonlinear regime requires propagation using the nonlinear Schrodinger equation, a simple approximation can be obtained by a single split step partitioning of dispersion followed by nonlinearity. This approximation is reasonably accurate for the experimental layout without delivery fiber, since the signal power was low until midway through the final YDFA stage, and nearly all the SPM arose from a relatively short fiber length (~10 m) near the output, over which dispersion was small and could be neglected. With this approximation, at the input to the nonlinear section of the fiber train, the dispersed field is:
Here the spectral phase shift due to dispersion is [25]:where Dλ = –30 ps/nm/km is the fiber dispersion coefficient at λ = 1064 nm, and c is the speed of light.Figure 3 uses Eq. (14) to illustrate the effect of dispersion on the time domain power for different input modulation depths m and β. Even with no AM applied at the input [Fig. 3(a), m = 0], dispersion can result in substantial AM at the nonlinear section that is mostly sinusoidal at . Figure 3(b) shows that since this dispersive AM is synchronous and in-phase with the AM applied at the input, its effect is primarily to increase the AM level at the nonlinear section. To account for dispersive FM-to-AM in the tests with no delivery fiber attached, we directly measured the AM at the YDFA output using a high speed photoreceiver (Newport 1474-A) and electronic spectrum analyzer.
Fig. 3 Calculated dispersive FM-to-AM conversion for L = 100 m. (a) m = 0, no AM applied at the modulator. (b) Applied modulator AM m = –0.4.
Figure 3 shows that as dispersive FM-to-AM increases, the power modulation is no longer purely sinusoidal, and it would be impossible to exactly satisfy Eq. (7) for SPM to cancel the FM [26]. Hence, for configurations with high β or for long delivery fibers, there is a need to compensate dispersion to optimize the spectral compression efficiency. Accordingly, for the configuration with the 20-m delivery fiber, we inserted a dispersion compensation module in the seed path. The module was tuned to pre-compensate dispersion by setting m = 0 at the modulator and minimizing AM measured at the fiber output.
4. Results
4.1 Nonlinear spectral compression
Figure 4 shows the modulation parameters measured at the YDFA output for different amplifier power levels with no delivery fiber attached. Here, the AM EOM modulation depth was kept fixed at the YDFA input. Both the launch SOP and the FM peak-to-peak modulation depth 2β were adjusted at each amplifier power for optimum spectral compression efficiency. Figure 4 also shows the values of the B-integral inferred from these optimized modulation parameters using Eq. (8). Due to dispersion, the AM depth 2|m| (i.e., the peak-to-peak power variation) measured at the YDFA output increased from 80% to 92% as 2β increased from ~1.5 to 4 rad.
Fig. 4 Measured modulation parameters optimized for spectral compression efficiency at each power level with no delivery fiber, and the corresponding inferred B-integrals.
Figure 5 shows the seed and amplified spectra with the modulation parameters m and β optimized for 1.34 kW, and with the carrier linewidth set to 12 GHz FWHM. Figure 5(a) shows that with no synchronous AM or FM, the seed and amplified spectra appeared identical. Figure 5(b) shows that engaging only the AM caused little change to the seed spectrum, but caused large sidebands to appear in the 1.3 kW output due to SPM. Figure 5(c) shows that engaging only the FM caused sidebands to appear in both the seed and output that were very similar to those due to SPM. Finally, Fig. 5(d) shows that with both AM and FM engaged synchronously, the FM and SPM nearly cancelled each other, and the original carrier spectrum was recovered.
Fig. 5 Measured seed spectra and 1.3 kW amplified spectra with synchronous amplitude and frequency modulations cycled on (m = –0.46, β = 2 rad) and off (m, β = 0) in different combinations, with no delivery fiber. (a) Both AM and FM off. (b) AM only on. (c) FM only on. (d) Both AM and FM on.
By comparing the 1.3 kW optical spectra with the synchronous AM/FM on and off (Fig. 6), one can see that 95% of the total power was compressed into the spectral band of the carrier, with ~5% remaining in spectral sidebands. Due to the sinusoidal AM that remained imprinted on the output, the expected residual sideband power was m2/8 ≈3% with m = –0.46. The excess 2% sideband power was likely due to imperfections in the RF drivers and EOMs, as well as polarization-related impairments arising from the non-PM amplifier components. Work is ongoing to better understand root causes and mitigations.
Fig. 6 Comparison of 1.3 kW optical PSD with synchronous AM + FM engaged (red) and disengaged (blue), with no delivery fiber.
Figure 7 shows the amplified spectral power dependence with the modulation parameters fixed at their optimum values for the full 1.34 kW output power (m = –0.46 and β = 2 rad), and with the carrier linewidth set to 7.6 GHz FWHM. The measured spectral compression behavior [Fig. 7(a)] is in good agreement with the calculated optical PSD of the output field using Eq. (6) [Fig. 7(b)]. As the power (nonlinearity) increased, the sideband power gradually shifted into the original carrier band (zero frequency shift) due to demodulation of the input phase by SPM. Figure 7 is also qualitatively representative of the expected spectral evolution down the amplifier fiber length at full power as SPM accumulates.
Fig. 7 Power evolution showing nonlinear linewidth narrowing of (a) measured and (b) modeled amplified optical PSDs.
4.2 SBS suppression
SBS was suppressed in this scheme since the sideband powers located at harmonics of were widely spaced relative to both the carrier linewidth and the SBS Stokes shift, and hence contributed little to the SBS gain of the carrier. This can be seen from the data in Fig. 8, which shows a comparison of the nonlinearly demodulated output spectrum to the backwards power spectrum measured from the return tap coupler (Fig. 2). There is little spectral overlap in the return power spectrum between the backscattered light at the harmonic sidebands, and the Brillouin Stokes lines that are redshifted by –16 GHz from the sidebands.
Fig. 8 Nonlinearly demodulated output and return PSD. The return PSD shows SBS Stokes light at –16 GHz offsets (highlighted by the dashed red lines) from the 32 GHz modulation harmonics.
Proof that this concept suppresses SBS is shown in Fig. 9. At each YDFA power level, the SBS-limited linewidth was identified by the onset of random pulsing in the return direction [27,28] as the carrier linewidth was reduced. Upon engaging the synchronous AM and FM, the SBS-limited FWHM linewidth was reduced from 10.8 GHz to 6.5 GHz at 1.34 kW [Fig. 9(a)].
Fig. 9 (a) Measured FWHM SBS-limited linewidths with no delivery fiber. (b) Ratio of SBS linewidths with and without the synchronous AM/FM, compared to the calculated ratio of the nonlinear strengths of the carrier using Eq. (19).
Power that is spectrally shifted out of the carrier band should not contribute to SBS gain of the carrier. We estimated the SBS-limited linewidth as being proportional to the nonlinear strength of the carrier only; i.e., the integral of carrier band power over the fiber length. Using Eq. (6), we can write the nonlinearly demodulated field in a form analogous to Eq. (12):
The power fraction in the carrier is . We modeled the total signal power distribution P(z) as a function of length z in the co-pumped YDFA, and used this to calculate the B-integral distribution:
where n2 is the nonlinear index for SiO2, and Aeff is the effective mode field area in the fiber. Using this relationship, the integrated power-length product of the carrier iswhere the integral is over the length of the final YDFA stage (and delivery fiber, if applicable). The ratio of these integrated carrier power-length products without and with synchronous AM/FM applied is the expected reduction in SBS-limited linewidth:As shown in Fig. 9(b), Eq. (19) is in good agreement with the measured data.4.3 Scaling to higher nonlinearity
The data in Fig. 9(b) indicate the effectiveness of SBS suppression increases at higher power. This is more generally true for higher B, corresponding to any combination of higher YDFA powers and longer delivery fibers. There are two reasons for this which can be inferred from Eq. (8).
First, for any given AM depth m, a higher value of B enables higher FM modulation depth β, which corresponds to a broader input seed spectrum and consequently more SBS suppression. Figure 10 shows the estimated SBS suppression based on the reduction in nonlinear length of the carrier as described above. Factors well beyond 2 × appear feasible for β > 3, which is readily accessible with commercial EOMs and RF amplifiers.
Fig. 10 Estimated reduction in SBS-limited linewidths as FM depth increases, along with residual spectrally uncompressed sideband power for different values of B.
Second, a higher nonlinearity allows reduction in |m|. This improves the output power stability, and it also enables higher spectral compression efficiency. Figure 10 shows the calculated AM-limited residual sideband power (~m2/8) for different values of B. The case B = 4.3 rad corresponds to the measured 1.34 kW YDFA output. At higher B ≥ 10 rad, one can see the combination of high nonlinearity, high FM depth, and low AM depth enables SBS suppression (linewidth reduction) factors of >3 × with only a few percent of power remaining in spectral sidebands.
Due to the limited power and B-integral of the YDFA used in this work, we spliced a 20-m length of matched passive delivery fiber to the YDFA output. This increased the nonlinear strength from ~3 rad/kW to ~14 rad/kW. It also lowered the SBS power threshold. While the compositional differences between the active and passive fibers would be expected to slightly shift the SBS frequency, this effect is minor relative to our ~10 GHz class linewidths, although for very narrow (GHz-class) linewidth it could potentially be leveraged to reduce overall SBS gain [29]. Figure 11(a) displays the measured reduction in SBS-limited linewidth as FM modulation depth β was increased, and Fig. 11(b) shows the same data plotted as a function of output power. The SBS-limited linewidths were reduced by ~2.5 × at powers above 600 W.
Fig. 11 Reduction in SBS limited linewidths with 20-m delivery fiber as a function of (a) FM depth β; (b) amplifier power.
The solid curve in Fig. 11(a) shows the SBS linewidth reduction calculated using Eq. (19), which is in good agreement with the measured data for β < 2. But for values of β > 2, corresponding to powers > 600 W and carrier linewidths >12 GHz, the measured SBS reduction was greater than predicted by Eq. (19). This increased effectiveness of the SBS suppression scheme is correlated with the onset of self-seeding [12–14]. From the spectra measured at 700 W (Fig. 12), one can see the edge of the carrier spectrum with the AM/FM off (m = 0, β = 0, 14.6 GHz FWHM) extends past the −16 GHz SBS Stokes shift. This overlap was reduced by 10 dB when the AM/FM was enabled (m = –0.23, β = 2.3) and the carrier linewidth was reduced to 6.1 GHz. This suggests that once the carrier linewidth becomes comparable to the Stokes shift, by selecting a synchronous modulation waveform that avoids self-seeding one can achieve even higher SBS suppression than the estimates based on nonlinear strength shown in Figs. 9–11.
Fig. 12 SBS-limited amplified spectra measured with 20-m delivery fiber at 700 W output, with an estimated B = 10 rad.
For the data in Fig. 12, with 700 W emitted from the 20-m long delivery fiber, the nonlinear phase shift estimated using Eq. (8) was B = 10 rad due to SPM. With FM depth β = 2.3 rad, this enabled reduction of the AM depth to m = –0.23. As shown by the loss curve for B = 10 in Fig. 10, the estimated uncompressed sideband power with β = 2.3 rad is 0.5%, compared to ~3% measured. Comparing this against the 5% residual sideband power from the bare YDFA confirms the expected increase in spectral compression efficiency with higher nonlinearity (lower |m|).
5. Summary
We demonstrated up to 2.5 × narrowing of SBS-limited linewidths from a cw fiber amplifier by using nonlinear SPM to demodulate a spectrally spread seed source. SBS suppression was demonstrated to increase with higher B-integrals, which allows for higher initial FM (more spreading) and lower AM (higher compression efficiency). For this reason, the method becomes more effective with higher powers and longer fibers.
This nonlinear phase demodulation technique is relevant for active or passive transmission over fibers deep in the nonlinear regime (B > few radians), which is typical for kW-class fiber amplifiers. This also suggests the technique may be useful to generate higher spectral brightness from otherwise SBS-limited pulsed fiber amplifiers [18], which naturally operate in the high-nonlinearity regime with B-integrals in excess of 30 rad before reaching limits imposed by stimulated Raman scattering [30]. The technique appears complementary with carrier modulation formats that are more spectrally efficient than WNS, such as pseudo-random bit sequences [31,32], linear chirp [33], or other waveforms [34], potentially further reducing linewidths. In addition to enabling further power scaling of spectrally beam combined lasers due to the narrower linewidths, we anticipate this approach could also relax SBS limits on delivery fiber lengths, which could facilitate packaging of combined fiber sources on mobile platforms.
Funding
Directed Energy Joint Transition Office and Air Force Research Laboratory (FA9451-18-C-0101).
Acknowledgments
Approved for Public Release, Distribution Unlimited: 377ABW-2019-0006; NG18-2664. Portions of this work were presented at the 2018 Advanced Solid State Lasers Conference [19].
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