A nonuniform large mode area fiber having an larger core at most locations to reduce power intensity and having a relatively smaller core at certain locations to reduce bending sensitivity is proposed for suppression of stimulated Brillouin scattering (SBS) in a high power fiber amplifier. A comprehensive model taking into account fiber nonuniformity, profile and temperature gradient shows that the fiber can achieve kilowatt output at a temperature gradient of >250 °C. Compared with a conventional large mode area fiber, a total of 7 dB SBS suppression can be achieved using this unique fiber.
©2007 Optical Society of America
High power, single frequency fiber lasers are desirable for many applications such as laser radar and imaging, frequency conversion for generation of visible or UV light, and as modules for spectral and coherent beam combining. However, the maximum power of a CW, single frequency fiber amplifier is believed to be limited to several hundred watts due to stimulated Brillouin scattering (SBS) [1,2] even using a large mode area (LMA) fiber with a core diameter of 20~30 μm and a core NA of 0.04~0.06. Increasing the core diameter further to suppress SBS requires a reduction in the core NA in order to preserve diffraction limited beam quality. In practice, reducing the NA of a LMA fiber to less than 0.04 using conventional fiber fabrication processes can also be challenging. The use of photonic crystal fibers enables an ultra-low NA fiber (i.e. NA ≤ 0.03) to be made routinely by precisely controlling air-hole to glass ratio, but the fiber becomes very sensitive to bending and must be either kept straight or wound with a large diameter. As a result, the devices based on this kind of fiber are not only more bulky with more complex packaging requirements but also sensitive to environmental variation as well as vibration.
In this paper, a nonuniform fiber which not only has a large core but also is insensitive to bending is proposed to suppress SBS in a high power fiber amplifier. The fiber has a large core with a low NA at most locations to reduce nonlinear effects and preserve diffraction limited beam quality while at certain locations the fiber has a relatively smaller core with a higher NA. Compared with an ultra-low NA fiber requiring a bending diameter of >500 mm, the nonuniform fiber can be bent at the higher NA locations like a conventional LMA fiber with a bending diameter of <200 mm and kept relatively straight at the low NA locations.
The nonuniform fiber can be made with the well developed technique: thermally expanded core (TEC) , which uses high temperature heating sources to heat up a conventional LMA fiber at multiple locations and allow some dopants such as Ge or F to be diffused to the adjacent areas to modify the refractive index profile. The process is capable of expanding the model-field-diameter (MFD) of a conventional fiber by several times. The expansion ratio and tapering profile can be controlled by varying heating time at different locations along the fiber. With an appropriately chosen tapering profile (i.e. taper ratio and length), the taper transition loss can be negligible . Once the desirable core profile is achieved, the fiber is recoated to have similar protection and strength as conventional fibers. It should be noted that for a double-clad nonuniform fiber the pump cladding diameter keeps the same along the fiber and the heating process does not affect optical properties of the pump cladding, such as NA and propagation loss as the pure silica cladding is insensitive to moderate heating. Therefore, the nonuniform fiber can be pumped with either free-space optics or fiber couplers and has the same power handling capability as a conventional LMA fiber for achieving efficient operation at high power.
During the past the dependence of SBS gain on the temperature, stress and fiber profile [5, 10, 11] has been investigated, but most of them have only concentrated on passive transmission fibers in which the signal along the fiber varies slightly and can be described with a simplified model . Meanwhile, the analytical analysis of SBS in a conventional fiber amplifier  is clearly not suitable for such a fiber because the population inversion along the fiber is strongly location dependent and is too difficult to be described with a simple function. In this paper, a comprehensive model is developed to help understand SBS characteristics in the amplifier and design the nonuniform fiber with a low SBS gain. The model takes into account fiber nonuniformity, profile and temperature gradient and uses a new approach to obtain detailed SBS spectral information. Unlike most SBS models where the SBS gain spectrum was assumed to be flat, the model described here assumes that all facts mentioned above may lead to a SBS spectral change. To systematically model SBS characteristics, it is worth to divide the SBS wave into a series of individual SBS waves so that each SBS wave experiences different gain and linewidth broadening. As a result, a detailed high resolution SBS spectrum can be obtained and used for designing the SBS suppression fiber.
2. Modeling SBS in fiber amplifiers
Considering a general fiber amplifier seeded at one end and pumped from both ends, the seeded signal is amplified during propagating along the fiber amplifier while SBS propagates in an opposite direction. The SBS is assumed to include a series of discrete Brillouin waves at different frequencies and each Brillouin wave interacts with the intense signal along the fiber. For the sake of simplicity, the three-level Yb system is reduced to an effective two-level system so that the rate equations involve only the total population density of multiplets 1 and 2 . By adding the SBS term into the well known rate-equations, the signal, forward and backward pumps, and Brillouin waves can be described as
where Ps is the signal power, Pf and Pb are the forward and backward pump power, respectively. z is the fiber location. σas are the cross-sections and the upper index represents absorption (a) or emission (e) while the lower index represents the signal (s) or pump (p). A is the effective area of the Yb-doped core. Γi are the overlap factors between the light-field modes and the Yb distribution. PiSBS and giSBS are the i -th Brillouin wave power and gain coefficient at frequency νiSBS . α a0 s and α a0 p are intrinsic background losses in the fiber for the signal and pump. N 2 and N 1 are the ion population density in the upper and lower levels.
When the waveguide-induced inhomogeneous spectral broadening of Brillouin waves is taken into account, the Brillouin gain coefficient is given by 
where g 0 is the SBS gain coefficient for bulk silica and Γ0 is the homogeneous width of Brillouin spectral line (FWHM). F 0 and Fc are the Stokes frequency shifts related to NA and can be expressed as:F 0=2∙n∙ν/λs and Fc = 2∙n∙ν[1-(NA/n)2]1/ 2/λs, here ν is the velocity of sound, n is the refractive index of the core, λs is the signal wavelength.
The experimental results have shown that Brillouin frequency shift is directly proportional to fiber temperature while Brillouin gain shape is insensitive to the temperature change [5, 9, 10, 11]. Based on that, the temperature term can be simply added to Eq. (5) and thus the Brillouin gain is expressed as:
where Tc is the fiber core temperature which is directly proportional to the pump distribution, and CT is the temperature slope coefficient given in Tab.1. The spectral linewidth of the signal is assumed to be 60 kHz, which is significantly narrower than the SBS gain bandwidth.
The system of Eqs. (1)~(4) is a two-point-boundary-value problem with initial conditions for Eqs. (1)~(2) at z = 0 and remaining boundary conditions for Eqs. (3)~(4) at z = L. It should be noted that Eq. (4) actually represents a series of differential equations for individual Brillouin waves. To have a high spectral resolution result obtained within a reasonable computing time, a total of 97 Brillouin waves, corresponded to a total of 100 differential equations were numerically solved with the modified relaxation method . The fiber parameters are listed in Tab.1.
It is well known that a counter-pumping scheme (i.e. the pump light is launched into the fiber from the signal output side) can be more efficient with a higher SBS threshold compared with a co-pumping scheme (i.e. both the signal and pump are launched from the same end) and thus is commonly used to generate high power output . For this reason the model focuses on the counter-pumping configuration and thus the boundary conditions for the pumps are simplified as Pf≡0, Pb(L)=pump power. Assuming the fiber core temperature proportional to the pump power, the temperature gradient values mentioned below are the core temperature difference at two special locations along the fiber where the pump power is either maximum or minimum. Obviously, for a counter-pumped amplifier the gradient is the temperature difference at the two fiber ends. The pump cladding diameter and NA are 600 μm and 0.46 respectively. For such a large cladding fiber, the core temperature distribution should be fairly uniform along the cross-section and a radial temperature gradient is negligible.
By carefully choosing the nonuniform fiber profile such as the taper slope and length as well as period, the model shows that a nonuniform fiber amplifier can achieve greater than 1 kW output without the onset of SBS. The nonuniform fiber is assumed to be made with the TEC technique based on a conventional 30 μm LMA fiber with a MFD of ~29 μm. The fiber core is expanded to a diameter of 40 μm at most locations with a MFD of ~39 μm as illustrated in Fig. 1. It should be noted that a very low taper ratio of 1.33 over a length of about 0.6 m is chosen here to avoid possible loss from the tapered core. According to the modeling results in Ref. 4, the fundamental mode should be transformed smoothly along the taper region without mode deformation and the taper transmission loss should be negligibly low. It should also note that all the nonuniform fibers modeled in this paper have the same profile for the nonuniform portion. The fibers with different lengths are made by shortening or extending the last section starting from the location ~4.8 m with a uniform core diameter of 40 μm.
The amplified signal and SBS power as a function of temperature gradient for four slightly different length fibers are shown in Fig. 2. It is found that with a total pump power of 1600 W, the signal output powers from the nonuniform fiber amplifiers strongly depend on the fiber temperature gradient. At a temperature gradient of 50 °C the amplifiers with slightly different fiber lengths generate less than 600 W of signal output with about 500 W of SBS output. When the temperature gradient increases to >250 °C all fiber amplifiers achieve >1000 W signal output with less than 70 W SBS power. The temperature gradient induced performance improvement reflects the nature of SBS and is caused by the dependence of Brillouin frequency shift on temperature. When the Brillouin waves propagate along a fiber with a nonuniform temperature distribution, they experience different gains at different frequencies. The higher the temperature gradient, the greater the gain difference and thus results in a lower SBS power.
The performance improvement of the nonuniform fiber amplifier can also be clearly seen when comparing with a conventional 30 μm LMA fiber amplifier. For two types of fibers, a length of longer than 4.83 m is chosen to achieve > 90% of pump absorption at 1600 W pump power. As shown in Fig. 3(a), at 50 °C both fiber amplifiers experience excessive SBS generation. However, an increase in temperature gradient noticeably differentiates the nonuniform fiber from the conventional 30 μm fiber. For example, increasing the temperature gradient from 50 °C to 300 °C suppresses the SBS gain by 12.5 dB for the nonuniform fiber while by only 6.3 dB for the 30 μm fiber. As a result, at a temperature gradient of 300 °C, the use of the nonuniform fiber reduces the SBS gain by as much as 7 dB.
Figure 3(b) shows how the two fiber amplifiers are affected by fiber lengths. In contrast with the 30 μm fiber amplifier with a longer fiber leading to an increase in the SBS gain, the nonuniform fiber amplifier with a longer fiber tends to reduce the SBS gain. It is believed that the difference is due to the unique fiber profile along the nonuniform fiber. To help qualitatively explain the performance difference, an average core diameter defined as the fiber volume divided by the fiber length can be induced. It is easy to understand that the larger the average core diameter, the lower the power intensity in the fiber and thus the less SBS is generated. For the given nonuniform profile as shown in Fig. 1, the use of a longer fiber actually increases the length of larger core section which is equivalent to an increase in the average core diameter. As a result, the longer fiber having a larger average core diameter helps suppress SBS. However, further extending the large core section does not necessarily suppress SBS further. For an extreme case in which the large core section is much longer than that of the nonuniform section, the nonuniform fiber essentially degenerates to a uniform fiber and thus the situation becomes similar to what occurs to the 30 μm fiber, in which the longer the fiber, the higher the SBS power is expected. It should also be noted that extending the large core section too long may also increase bending sensitivity of the last section and lead to an undesirable bulky device requiring complex packaging.
The comprehensive model presented here also provides in-depth information about the SBS gain spectra at different locations to further help us understand how the nonuniform fiber differentiates from a conventional LMA fiber. The differences between a nonuniform fiber and a 30 μm LMA fiber with a same length of 5.33 m and at a temperature gradient of 250 °C are shown in Figs. 4(a) and 4(b). Overall, the SBS spectra of both fibers are locations dependent. Compared with the conventional LMA fiber having relatively smooth SBS spectral curves, the nonuniform fiber shows more features with multiple peaks. At the signal output end where backward propagating SBS starts from noise, both fibers have the same initial SBS spectrum. After the SBS propagating for a short distance at the location of ~4.7 m, the spectral difference is noticeable. The nonuniform fiber shows broader linewidth with a lower amplitude compared with that of the 30 μm fiber. At the location of ~4.2 m, the nonuniform fiber shows at least two times wider SBS linewidth than that of the 30 μm fiber. It is believed that the inhomogenous broadening induced by the nonuniform NA distribution contributes to adding the features to the SBS gain spectra. Meanwhile, the multiple peaks appear to be induced by the small core sections along the fiber in which SBS waves have slightly higher gain than that of in the adjacent locations with different frequency shifts due to the nonuniform temperature distribution. Nevertheless, the use of the nonuniform fiber reduces the SBS gain by 6.5 dB at a temperature gradient of 250 °C. As a result, the nonuniform fiber can achieve much higher signal output without generating excessive SBS by appropriately designing the fiber profile so that the SBS waves are spread out over a wide frequency range.
So far, the modeling results have clearly showed advantages of the nonuniform fiber as well as critical importance of the temperature gradient and fiber profile for SBS suppression. It should be noted that for practical reasons the model limits the maximum temperature gradient to 300 °C based on the thermal analysis of a kW fiber laser  indicating a possible core temperature of as high as ~350 °C. Given that, a temperature gradient of 300 °C seems reasonable to be achieved by proper thermal management, even though it may require a high temperature fiber coating , preferably, a glass over-cladding combined with a high temperature coating to avoid possible coating degradation or damage. Further increasing the temperature gradient can suppress SBS even more and scale up the signal output power, but in practice it can be very challenging and may require significant changes in fiber design and thermal management in order to handle such a high temperature.
As mentioned above, the main purpose of the paper is to design a compact fiber amplifier capable of achieving greater than 1000 W output without the onset of SBS. To make the concept more practical, it uses a conventional 30 μm fiber with a NA of 0.04 and a taper ratio of 1.33. To make a compact device, the fiber NA is limited to greater than 0.03. When these requirements are added, the fiber period and profile have to be compromised by performance. As a result, the samples given in the paper are not necessarily an optimal design for performance. When a larger package size is allowed, a nonuniform fiber with a longer large-core section (i.e. longer period and fewer necks), or with a larger core size can increase SBS threshold and improve performance further.
It should be noted that a total of 97 Brillouin waves are used in the simulation to provide sufficient resolution for these specific fibers as shown in Fig. 4. Doubling the number of Brillouin waves has not shown a noticeable difference on the modeling results but increases calculation time. However, an increase in the number of Brillouin waves may be needed for some fibers with more complex nonuniform profiles.
In addition, the model has also been used to analyze the experimental results described in Ref.  for verification purpose. With the provided fiber parameters, the model predicts that the fiber amplifier is capable of generating more than 300 W output at a temperature gradient of 120 °C. This result agrees well with both experimental data as well as the thermal modeling results in Ref.  where the fiber temperature gradient is predicted to be greater than 100 °C for a 300 W fiber laser.
In conclusion, a novel nonuniform Yb-doped fiber amplifier capable of achieving kW single frequency output has been proposed and analyzed with a comprehensive model taking into account the fiber nonuniformity, profile, and temperature gradient. It was found that by periodically expanding a conventional 30 μm LMA fiber and utilizing the temperature gradient, a total of 7 dB SBS suppression can be achieved and a substantially smaller fiber amplifier can be demonstrated.
References and links
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