1. Introduction

Recent technological advances have made possible fiber lasers with outputs well beyond the kilowatt level. However, the highest output powers are characterized by broad spectra making them unsuitable for a range of applications including coherent beam combination for directed energy purposes, harmonic generation, lidar, and gravitational wave detection. A major limitation for high power CW narrow linewidth fiber amplifiers is the onset of stimulated Brillouin Scattering (SBS) which is the lowest threshold nonlinear process for such amplifiers. To reduce SBS, a number of approaches - such as the use of thermal gradients [1], polarization effects [2], large flattened mode area fibers [3], and fibers designed to reduce the overlap between the incident light and the resulting acoustic wave or to induce a negative acoustic lens effect [4,5,6] - have been either demonstrated or suggested. This paper presents theoretical verification of another technique that increases the SBS threshold in a fiber amplifier using two narrow linewidth master oscillators, or ‘tones’, oscillating at wavelengths λ₁ and λ₂.

In this paper we investigate two variants of this ‘two-tone’ amplification, which are based on the frequency difference between the master oscillators. In one case, Δλ is large enough to seriously mitigate four-wave mixing (FWM) which is the next lowest-threshold nonlinear effect. It has been thought previously that while an approximately 100% increase in total power can be obtained, enhancement of either of the two laser signals is not possible [7]. We will show that for Δλ >10 nm, appreciable power enhancement of one of the tones is possible through power transfer between the two tones and an overall reduction of the integrated SBS gain. Another two-tone technique which we investigate was previously demonstrated experimentally by Wessels et al. [7] in a 72 m long Nd-doped double clad fiber. In this case, a fiber amplifier was constructed with two seed lasers at a Δλ twice the Brillouin shift. This enabled roughly a one-fold improvement in the amplified power of one of the seed signals through additional interactions among the input and Stokes signals, but also transferred large amount of power into numerous FWM-generated sidebands. This broadening of the optical power spectrum precludes the application of this method to fiber laser applications that require well-defined spectra such as electronically phased coherent arrays [8,9,10] and spectral beam combination [11,12]. Due to its suppression of FWM, the large wavelength separation technique (which we henceforth identify as any separation larger than twice the Brilloin shift) is therefore much more suited for use in coherent beam combination applications. Specifically, a technique based on “Locking of Optical Coherence by Single-Detector Electronic-Frequency Tagging” (LOCSET) [9] to coherently combine the output of several two-tone amplifiers will be discussed in future work. Both variants of two-tone amplification have the added benefit that they can be used in conjunction with the aforementioned SBS mitigation techniques. This paper presents the first extensive modeling results of these two-tone high power Yb-doped fiber amplifiers. A system of 8x8 coupled nonlinear differential equations describing SBS, FWM, and laser gain is obtained and solved numerically. The derivation of this model for the large e Δλ case is discussed in Section 2 along with the simulations and results. In Section 3, we investigate the technique proposed by Wessels et al.

2. Large wavelength separation

2.1 Derivation of the two-tone model

The two-tone method discussed here has been specifically designed to increase the SBS threshold and avoid FWM. Both effects are third-order nonlinear effects. SBS describes the interaction among the laser signal, its Stokes signal, and an acoustic signal that beats at the difference frequency of the optical fields due to electrostriction. In a fiber amplifier the Stokes photons and the acoustic phonons are initiated from noise. By assuming steady-state conditions and strong damping of the phonon field, a coupled 2x2 system of the laser and Stokes intensities is typically obtained. The effects of FWM are well-known; in the case of two input frequency components oscillating at ω ₁ and ω ₂ the interaction of the waves is mediated through the third-order susceptibility of the mediumχ ⁽³⁾. Two sidebands oscillating at ω ₃=ω ₁-Δω and ω ₄=ω ₁+2Δω, where Δω=ω ₂-ω ₁, are generated.

In order to formulate a self-consistent model laser gain, SBS, and FWM are all incorporated in our work into a set of 8x8 coupled nonlinear differential equations that describe the evolution of 7 optical fields and the population density of one of the atomic levels of a two-level laser system.

The evolution of the electric field of each frequency component can be derived from the nonlinear wave equation

where the subscript i represents the frequency component of the electric field and P_i ^(nl) is the nonlinear polarization. The electric field is expressed as

where j represents the mode, A_i,_j (z) is the amplitude, β _i,_j is the propagation constant, and ϕ_i,_j (x,y) is the transverse profile.

Large mode area (LMA) fiber amplifiers are typically coiled to operate in the lowest-order mode. In the analysis that follows we will focus on the nonlinear coupling of the lowest-order modes of the various optical waves. We henceforth drop from our equations the subscript j and it is understood that our equations describe the lowest-order mode for each frequency component. Furthermore, since the frequency separation among the waves is much smaller than the optical wavelength, the modal profiles of all waves are set to be equal. In the limit of the slowly-varying envelope approximation, d ² A_i/dz ² << 2iβ_i(dA_i/dz) << β ² _iA_i, and using cβ ₁≈n ₁ ω ₁, where n ₁ is the linear index of refraction at ω₁, one can reduce through coupled mode theory the wave equation for the wave oscillating at ω ₁ to:

where the nonlinear index of refraction is related to χ ⁽³⁾ by n ⁽²⁾ ₁=3χ ⁽³⁾/8n ₁ and where g ₁ and g_B,₁ are the laser gain and the SBS gain coefficients, respectively due to the input seed oscillating at ω ₁. A _1S is the Stokes wave due to ω ₁ and n _{1 S} is the linear index of refraction corresponding to it. The third and fourth terms on the right hand side of the equation above are FWM terms and contribute to the phase mismatch as they represent self-(SPM) and cross-phase modulation (XPM), respectively, while terms (5-7) are also FWM terms responsible for energy transfer among the various waves. The linear phase mismatch terms Δβ ₁, Δβ ₂ and Δβ ₃ characterizing this energy transfer are given by:

where we used a Taylor’s expansion to relate these terms to β ⁽²⁾ the group-velocity dispersion (GVD) parameter. The overlap integral for the SPM and XPM, κ_pm, and the overlap integral for the acoustic and optical wave interaction, κ_ao, are given by:

Note that in obtaining κ_ao we assumed the transverse acoustic profile is described by |ϕ|² as can be inferred from the form of the electrostrictive force. If one were to subscribe to the notion of guided acoustic modes [4], then _ao κ will be of the form:

${\kappa }_{\mathit{ao}}=\frac{{\mid \iint {\mid \varphi \mid }^2{\psi }^{*}\mathit{dx}\mathit{dy}\mid }^2}{\iint {\mid \varphi \mid }^2\mathit{dx}\mathit{dy}·\iint {\mid \psi \mid }^2\mathit{dx}\mathit{dy}}$

where ψ represents the acoustic mode.

We model the gain medium as a two-level system. The laser Yb-gain coefficient, g ₁, is given by

where N ₂ and N ₁ are the population densities of the upper and lower energy levels, respectively, and where σ ^(e) ₁ and σ ^(a) ₁ a represent the emission and absorption cross sections for the seed frequency ω ₁, respectively. The integration in the numerator of Eq. (6) is carried out within the core.

The Stokes wave, A _{1 S}, is initiated from noise at the opposite end of the fiber and travels in the backward direction. The evolution of its amplitude along its direction of propagation is given by

where g _1S is the laser gain coefficient at the Stokes wavelength and has a similar form to that in Eq. (6) except that the emission and absorption cross sections correspond to the Stokes wavelength. Note that the noise contribution is incorporated into Eq. (7) as we employ a localized source model as proposed by Smith [15] or, alternatively, Zel’dovich et al. [16].

Similar equations can be derived for the seed wave oscillating at the input frequency ω ₂, and its associated Stokes waves. These equations are given by:

where g ₂ and g _{2 S} have similar expressions to Eq. (6). We neglect the SBS interaction for the FWM sidebands ω ₃ and ω ₄. This is justified as long as their amplitudes are much smaller than the laser signals. The evolution of their amplitudes along z is given by:

Where again g ₃ and g ₄ have similar expressions to Eq. (6). The population inversion equations are obtained from the rate equations at steady state. They are related to the absorption and emission cross sections, and the intensities of the various waves through the following equations:

where N _ο represents the density of Yb ions in the fiber core, τ is the lifetime of the upper laser level, and the subscripted I’s represent the intensities of the various waves. The intensity of the pump which in our simulation is taken to propagate in the same direction evolves according to:

where d_core and d_clad are the diameters of the core and the cladding, respectively.

2.2 ‘Two-tone’ simulations and results

The large wavelength separation case of ‘two-tone’ modeling is completed for a gain fiber with parameters based on Nufern’s 25/400µm LMA Yb-doped fiber. The dopant level N _ο is taken to be 1.2×10²⁶/m ³. The cross sections are taken from experimental values [13] and the lifetime τ of the upper level is 8.0×10^-4 s. The numerical aperture (NA) for this fiber is 0.06 and, as previously mentioned, the LMA fiber is assumed to be coiled so that the power in the higher-order fiber modes can be neglected. Since dispersion effects in the nonlinear index of refraction are negligible, for all waves we use n ^(2)I=3×10^-20 m ²/W, where n ^(2)I is related to the non-linear index of refraction n ⁽²⁾ used in Section 2 by n ^(2)I=(2µ _ο c·n ⁽²⁾)/n. We take a value of the group velocity dispersion, β⁽²⁾, of 15 ps²/km as computed from the fitted experimental measured dispersion of bulk-fused silica at 1.0 µm region [14]. This is done to estimate the values of the phase matching terms in our coupled system. The 8x8 coupled system of equations as described by equations (3), (7-11), (13) and (14) is solved numerically. To be precise, the nonlinear coupled system actually comprises 7 differential equations and one algebraic equation. It describes a two-point boundary value problem with the laser signals and the co-propagating pump known at z=0, while the two Stokes signals are initiated from noise at z=L. The two sidebands signals are set to zero at z=0. We subjected our model to several tests in order to verify both its numerical accuracy and to confirm that it captures the correct physics. One set of tests simulated a passive fiber with the SBS gain neglected. In this case, our simulations agreed with the expected results with regards to FWM, including the square sinusoidal behavior with distance in the case of phase mismatch and the quadratic behavior in the case of perfect phase-matching at low seed powers (low conversion efficiency). Furthermore, the total energy of the two seeds and the two sidebands was shown to be numerically conserved to better than one part in 10⁶. Another set of tests simulated a passive fiber with SBS gain but no FWM. Calculations of SBS thresholds compared well to those reported in the literature and to our own experimental tests. Furthermore, for a two seed input, we verified that the difference in optical powers, P ₁+P ₂-P _1S-P _2S, was constant along the length of the fiber to better than one part in 10⁶. After the model was verified, numerous runs were completed to validate the power improvement available for ‘two-tone’ amplification in the case of large Δλ. For the first test, 6.5 meters fiber segment was seeded with a 1.2 W 1064 nm signal and a 0.8 W 1068 nm signal. This particular ratio of seed powers was selected to optimize total output power. If the ratio of the input powers is not chosen carefully, one of the seeds can become more quickly amplified; this ‘robs’ the other input signal of its gain and ensures that the intensity of the quickly amplified signal will increase enough to reach the SBS threshold well below the maximum possible total output power. Figure (1) shows that the amount of SBS at constant pumping level near threshold can vary dramatically with the power ratio of the seed inputs.

 

Fig. 1. The sensitivity of SBS to the ratio of the input signals in ‘two-tone’ amplification. The x-axis shows the ratio of power in one input signal (1068nm) to the total input power. The y-axis represents percentage of Stokes power.

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The case of a single 2.0 W 1064 nm seed was also modeled for comparison. In these model runs, the power from a 977nm pump was increased for each seed method until the SBS threshold was reached. For clarity, we used a working definition of the SBS threshold as the point where the generated SBS power from a signal, that is, from a single 1064. nm seed or either of the ‘two-tone’ seeds, reached 1% of the output power for that amplified signal. The results of the model runs at this defined SBS threshold are shown in Figs. (2)-(4).

 

Fig. 2. The output powers of the two amplified signals in a simulated ‘two-tone’ amplifier and a typical single seed fiber amplifier versus propagation distance along fiber

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Fig. 3. Power of backward travelling Stokes light signals corresponding to 1068 nm and 1064 nm.

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The typical single seed amplification reached SBS threshold at 33 W of pump power for an amplified power of approximately 28 W; the two-tone case reached its SBS threshold at 62 W of pump. Here the output power of the 1064nm signal was approximately 24.5 W and 30 W for the 1068 nm signal for a total exceeding 54 W. Thus, the total amplified power from the two-tone amplifier was almost twice the single seed case with minimal FWM while maintaining the overall optical efficiency and suppressing SBS. Note from Fig. (3) and (4), FWM is an order of magnitude lower than SBS. Furthermore, it can be inferred that the coherence lengths of the two FWM sidebands are different. This is due to laser gain in the sidebands as well as SPM and XPM effects. Referring to Fig. (2), it is worthwhile to mention here that at some point along the fiber, the 1064 nm light will experience negative gain. We will elaborate on this point further below.

 

Fig. 4. The first two FWM-generated sidebands in the case of ‘two-tone’ amplification of a 1064nm seed and 1068nm seed as described above. Note that the power in these sidebands is one order of magnitude lower than the SBS power.

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Since there has been recently considerable effort devoted to building high power narrow linewidth amplifiers [4,5,6], we examined the possible use of this technique in conjunction with other SBS-suppressing techniques. We simulated an SBS suppressing fiber in order to test regimes where FWM was the lowest threshold nonlinear effect and thus would impose an upper limit on achievable power. The two-tone amplifier (with a 4nm wavelength separation) was modeled with 350 W and 550 W of pump power corresponding to an approximately SBS gain reduction factor of 6 and 10, respectively. This mirrors the possible suppression available with fibers constructed to specific acoustic guiding properties [4,5,6]. The laser efficiencies of these model runs were approximately equal to our previous simulations. We found that the FWM was comparable to SBS at 350 W, but exceeded it at 550 W. Note from Fig. (5a) and (5b) that the coherence length of the FWM sidebands has decreased from our previous simulations due to the increased SPM and XPM effects. All of this suggests that at a wavelength separation of 4 nm, this technique can be applied to amplifiers with outputs exceeding 100 W while keeping FWM to reasonably low levels.

 

Fig. 5. The FWM behavior of SBS-suppressed two-tone amplifiers with a) 350 W (top) and b) 550 W (bottom) of pump power. The wavelength separation in this case is 4 nm. The additional pumping has altered the coherence period due to SPM and XPM.

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Going back to our initial amplifier configuration (with the SBS gain suppressing factor turned off), the wavelength separation of the two input signals was increased and decreased. For these tests, the pump power was held constant at approximately 60W which was near our defined SBS threshold for Δλ=4.0 nm. For Δλ<3nm the FWM increased considerably and became comparable to the Stokes light. For Δλ>10nm, the FWM was extremely small. Most remarkably for this wavelength separation, considerable enhancement in the power output of the lower laser frequency was obtained. For example, at Δλ=14nm and input seeds with wavelengths 1064 nm and 1050 nm, 46 W of output power was obtained for the 1064 nm light. The power ratio needed to obtain this output was approximately 9:1 with the 1050 nm having the higher input power. The 46 W output power represented a 64% enhancement over a 1064 nm single tone amplifier as shown in Fig. (6). Higher power output in one of the tones is possible to the point where almost all the output power would be in a single frequency. This can be achieved through an optimal ratio of seed and wavelength separation, or by selecting a more suitable fiber configuration; the details will be discussed in a future publication.

 

Fig. 6. The evolution of two laser signals along the direction of propagation. For comparison, the case of single tone seeding is plotted.

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Note that the 1050 nm signal reaches its maximum value at a distance shorter than the midway point of the fiber. This is due to the higher pump power, the skewed input seed power ratio, and the higher emission cross section of the 1050 nm light as compared to that of 1064 nm light. As the two laser signals propagate down the fiber, the population density of the upper level state, N ₂, decreases. Immediately past the point where the maximum power for the 1050 nm light is obtained, the population inversion is such that the 1050 nm light will experience negative gain. The 1064 nm light which has an appreciably lower absorption cross section will, however, continue to experience positive laser gain. As a consequence, power transfer occurs from the 1050 nm light and into the 1064 nm light. The SBS threshold is raised because the spatially integrated Stokes light gain for the two-tone 1064 nm light will be close to the 1064 nm single tone case even though more 1064 nm output power is obtained in the former. This is made possible because, for a significant portion of the fiber, the power in the 1064 nm light for the two tone case is less than that for the single tone case as can be seen from Fig. (6). We examined the total gain for the electric field amplitude of the Stokes light as a function of position. Referring to Eq. (7) in Section 2, this total gain is due to the total of the laser and Brillouin gain. It is worthwhile to point out here that the amplitude gain is half that of the intensity or power gain. Figure (7) represents a comparison of the amplitude gain for the two tone case pumped such that the power output at 1064 nm is equal to the power output in a single tone amplifier at threshold. Note that the spatially integrated Stokes gain for two-tone is reduced significantly, thus allowing for higher pumping power and consequently higher output at 1064 nm. As mentioned in the Section 1, this power enhancement was not thought to be possible. Thus, this is a novel way to increase power in CW narrow linewidth Yb-doped amplifiers.

 

Fig. 7. Total Stokes gain for the 1064 nm light for the two tone and one tone cases. For the two tone case the pump power is approximately 38 W which generated an output 1064 nm power of approximately 28 W (equal to the output at threshold in the single tone case).

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3. Wavelength separation of twice the Brillouin shift

We now investigate theoretically the two-tone suppressing technique implemented experimentally by Wessels et al. [7]. This technique relies on selecting the wavelength separation between the two input frequencies to be equal to twice the Brillouin shift ν_B, i.e. twice the frequency of the phonon field. In optical fibers, this value is approximately 34 GHz (Δλ≈0.1 nm). It is important to point out here if the frequency separation does not lie within the range defined by 2(ν_B±Δν_B, where Δν_B is the SBS gain linewidth, then there will be a one fold maximum increase in total amplifier output power as we described in Section 2. For Δν=2Δν_B, the equations describing the spatial evolution of A ₁ and A _{2 S} are modified to become:

As indicated by the equations above, The Stokes light generated by the input frequency ω ₂ is SBS-scattered into the input frequency ω ₁, thus effectively raising the SBS threshold for ω ₂. In order to gain maximum benefit from such a system, an optimal power ratio between the two input beams should be selected. To theoretically determine this ratio, we define r=P ₁/P ₂ and neglect laser gain and FWM. For small SBS signal gain, we can work in the undepleted pump limit to obtain from Eq. (7) and Eq. (16):

Where for simplicity we set g _B1=g _B2=g_B, and n ₁=n ₂=n. In order to achieve the largest suppression of SBS, the SBS small signal gain for each of the Stokes lights has to be approximately equal. Therefore r=1/2, i.e. the input power of ω ₂ is twice that of ω ₁. This is the power ratio used by Wessels et al. and our analysis above is further borne out by the numerical simulations presented below.

The fiber parameters used in our simulations were the same as described in Section 2.2; the wavelengths selected were close to 1068 nm. The amplified powers are shown in Fig. (8) with the single-seed case also shown for comparison. The maximum total power out of this two-tone amplifier was greater than 80 W, an approximately two-fold increase in total power over the single seed case. Furthermore, the total power output in one of the signals was twice that of the single tone case. This is close to the improvement noted in the experimental results of Reference [7]. The nonlinear effects in this amplifier are shown in Fig. (9). The FWM effects are more pronounced than in the case of the two tone amplifier with large wavelength separation, but they are still roughly a factor of 10 less than what was previously experimentally observed [7]. This is due mainly to the much shorter length of out Yb-doped amplifier.

 

Fig. 8. The power output of a two tone amplifier. The two tones are separated by twice the Brillouin shift allowing Stokes light generated by seed light to transfer its energy to the second seed. λ=1068 (2ν_B/ν) nm.

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Fig. 9. The FWM for the two tones separated by twice the Brillouin shift. Due to the small wavelength separation the FWM is the lowest threshold nonlinear process.

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We also tested additional aspects of this case of two-tone amplification, such as the optimal 2:1 ratio that we theoretically investigated above. Keeping track of the total output power, we varied this input ratio from 1:1 up to 3:1. For these set of simulations, we held the total seed power constant but varied the pump power until the SBS threshold was reached.

 

Fig. 10. The dependence of total output power for two-tone amplifier on seed power ratio. The two frequencies are separated by twice the Brillouin shift. Here, The seed power ratio is defined as the input power of the higher frequency wave divided by that of the lower frequency.

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As shown in Fig. (10), there is actually a broad range of input ratios spanning approximately 1.8 to 2.3 that effectively mitigate SBS and provide an output power that is within 5% of the maximum achievable power output.

In comparing these results with those obtained in Section 2 for Δλ=14 nm, we note that the total power for the latter is one third less, but that fairly comparable outputs are obtained at the wavelength possessing the higher power. Therefore, if certain applications require the use of single frequency, the case of large wavelength separation will have a higher efficiency.

4. Conclusion

We have rigorously formulated the problem of an Yb-doped amplifier seeded with two laser frequencies to account for stimulated Brillouin scattering and four-wave mixing. Two cases were considered and were shown to enhance the power output of one of the laser signals: 1) a large wavelength separation and 2) a wavelength separation corresponding to precisely the Brillouin shift. It was thought previously that power enhancement was not possible for the former case. Experimental implementation of this two-tone work as well as theoretical extension to multi-tone seeding is already underway.