Stimulated Brillouin scattering in single-frequency fiber amplifiers with delivery fibers

Anping Liu

doi:10.1364/OE.17.015201

1. Introduction

Fiber lasers have recently demonstrated multi-kW output with diffraction-limited beam quality [1]. Their superior performance has created great opportunities to replace conventional solid-state lasers and CO₂ lasers in many application areas [2,3]. This leads to an increasing demand for highly reliable industrial-class fiber lasers as well as new development of dedicated high power fiber components. Fortunately, many of these components and subassemblies [4–6] can be developed with similar technologies developed by the telecom industry. These high quality fiber-based components, also called all-fiber-based components, can be easily connected together by fusion splicing to produce all-fiber-based fiber lasers or amplifiers. The use of all-fiber-based components significantly simplifies system configuration and thus makes the fiber lasers considerably smaller and more robust than the free-space-based devices. In addition, the all-fiber connections also reduce reflection and scattering losses from optical surfaces and greatly improve high power laser efficiency.

One of the key components enabling the all-fiber-based fiber laser configuration is a fiber pump combiner. It consists of a fiber bundle for coupling the pump light into the pump cladding of the gain fiber, a signal fiber which acts as either an input port for launching the signal into the core or an output port for signal emission from the core. It is manufactured with a bundle of specialty fibers using sophisticated processes at a high temperature. Once it is formed, it has a signal input/output port and a number of pump input ports on one side and a combiner port on the other side. When it is used to construct a fiber laser or amplifier, the combiner port is connected to the gain fiber. This all-fiber-based combiner has extremely low losses for both the signal and pump light and is capable of achieving over 1 kW output [7].

It is well-known that a counter-pumped amplifier provides great advantages over other pumping configurations. In a counter-pumped fiber amplifier, the seed signal is launched from one end of the amplifier while the pump and signal output are through the combiner connected to the gain fiber on the other end. To allow the high power signal to emit from the pump combiner, an additional passive fiber has to be used to construct the pump combiner. This un-doped fiber also acts as a power delivery fiber to delivery high power laser beam into the work station. However, due to the high intensity over the entire fiber, it is vulnerable to the onset of nonlinear effects such as stimulated Raman scattering (SRS), stimulated Brillouin scattering (SBS), four-wave mixing (FWM), and self-phase modulation (SPM). In a single-frequency amplifier, the onset of SBS can considerably undermine amplifier performance. For example, excessive SBS can generate huge signal spikes and cause catastrophic optical damage (COD) to many expensive components and even the entire device. Therefore, it is strongly desirable to understand SBS characteristics in such fiber amplifiers and how to mitigate its impact on amplifier performance.

In this paper, a rigorous model based on the rate-equations developed for designing high power single-frequency amplifier [8] is modified to analyze counter-pumped amplifiers with a delivery fiber operating at single-frequency. The impact of fiber length, temperature, Brillouin gain, and core size on system performance are evaluated. It is expected that these results provide guidance for designing the counter-pumped amplifier with a delivery fiber using all-fiber-based configuration.

2. Theory

Considering a counter-pumped fiber amplifier shown in Fig. 1 , which is seeded from the left end and pumped from the right end, the seed signal is first pre-amplified and then power amplified by a double-clad fiber amplifier. The high intensity signal propagating in the core of the double-clad fiber from the left to the right may induce SBS which propagates in an opposite direction. Based on the rate-equations combining with a number of discrete Brillouin Stokes at slightly different frequencies over the Brillouin gain bandwidth [8], the signal, backward pumps, and Brillouin Stokes in the gain fiber then can be described as

\frac{d P_{s}}{d z} = (N_{2} σ_{s}^{e} - N_{1} σ_{s}^{a}) Γ_{s} P_{s} - α_{s}^{a 0} P_{s} - P_{s} \sum_{i = 1}^{n} g_{S B S}^{i} P_{S B S}^{i} / A

\frac{d P_{b}}{d z} = - (N_{2} σ_{p}^{e} - N_{1} σ_{p}^{a}) Γ_{p} P_{b} + α_{p}^{a 0} P_{b}

\frac{d P_{S B S}^{i}}{d z} = - g_{S B S}^{i} P_{s} P_{S B S}^{i} / A + α_{s}^{a 0} P_{S B S}^{i} - (N_{2} σ_{s}^{e} - N_{1} σ_{s}^{a}) Γ_{s} P_{S B S}^{i}

where

P_{s}

is the signal power,

P_{b}

is the backward pump power (also called counter-propagating pump power). z is the location along the fiber. The seed is launched at

z = 0

while the amplified signal output at

z = L

, where L is the fiber length.

σ_{s}^{a}

: cross-section, the upper index represents absorption (a) or emission (e), the lower index represents the signal (s), or pump (p).

A = π R^{2}

is the effective area of the Yb-doped core.

Γ_{i}

are the overlap factors between the light-field modes and the Yb distribution. If the dopant distribution is uniform with a radius of R and the light field waist is ω, then

Γ = 1 - \exp (- R^{2} / ω^{2})

.

P_{S B S}^{i}

and

g_{S B S}^{i}

are the i-th Brillouin wave power and gain coefficient at frequency

ν_{S B S}^{i}

.

α_{s}^{a 0}

, and

α_{p}^{a 0}

are possible intrinsic background losses in the fiber for the signal and pump.

N_{1}

and

N_{2}

are the populations of the lower and upper states, respectively.

Fig. 1 Configuration of counter-pumped amplifier with delivery fiber. 1. seed source; 2. pre-amplifier; 3. isolator; 4. pump source; 5. pump combiner; 6. gain fiber; 7. delivery fiber.

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When the temperature induced Brillouin gain broadening is considered [8], the Brillouin gain can be expressed as:

g (ν_{S B S}^{i}) = g_{B} \frac{Γ_{0} / 2}{F_{0} - F_{c}} \times [\tan^{- 1} (\frac{F_{0} - ν_{S B S}^{i} + T_{c} \cdot C_{T}}{Γ_{0} / 2}) - \tan^{- 1} (\frac{F_{c} - ν_{S B S}^{i} + T_{c} \cdot C_{T}}{Γ_{0} / 2})]

For a typical all-fiber based amplifier shown in Fig. 1, the pump light is launched into the amplifier through the pump combiner and thus no pump light actually passes the delivery fiber. Only the signal and SBS are propagated in the delivery fiber. To describe their characteristics, one needs a different set of differential equations, which have different boundary conditions. This adds complexity to the modeling. To avoid this, a special treatment is implemented. It assumes that the pump light is launched from the delivery fiber but does not induce population inversion. In other words, the delivery fiber just passively passes the pump light into the gain fiber. With these treatments, the system does not need to add an additional boundary point at the joint between the gain fiber and comber for the pump light. The three boundary condition problem reduce to two boundary conditions. The two boundary points are the left end of the gain fiber and the right end of the delivery fiber. As a result, both the gain fiber and the delivery fiber can use the same differential Eqs. (1)~(3). The initial value for the signal is the pre-amplified signal power at the left end of the gain fiber; while for SBS and the pump, they are the noise power and pump power at the right end of the delivery fiber, respectively. During numerical computing, the fiber is divided into two sections with a special location pointer to distinguish the delivery fiber and gain fiber. For the delivery fiber, the signal and SBS are computed as normal except that $N_{1}$ and $N_{2}$ are set to be zero to reflect that no gain is induced by the pump. For the gain fiber, no special treatment is needed for $N_{1}$ and $N_{2}$ . This approach considerably simplifies the computing process and makes it possible to use a universal model to treat a complicated system. In the model, the gain fiber length refers to the length between the isolator (i.e., seed input port) and pump combiner, while the delivery fiber is from the combiner to its output port. As detailed in Ref [8], the model allows independent temperature distribution inputs for the gain fiber as well as the delivery fiber. The fiber parameters used in this paper are the same as detailed in Ref [8]. if they are not specially mentioned. The gain fiber core and cladding diameters are 25 μm and 400 μm, respectively.

3. Impact of delivery fiber on amplifier performance

To numerically solve differential Eqs. (1)~(3), a separate data file consisting of fiber dimension parameters, temperature distributions, and a location pointer is used to provide independent inputs to the model so it can deal with a variety of cases. The fiber dimension parameters include the core and cladding diameters as well as their numerical apertures (NA); while the temperature distributions for the gain fiber and delivery fiber are independent inputs. The location pointer indicates whether the fiber is the gain fiber or delivery fiber. Once the model detect the delivery fiber it sets $N_{1}$ and $N_{2}$ to be zero. This approach allows the model to stimulate a variety of situations without requiring additional modification on the code. In the following section, the impacts of delivery fiber temperature, length, diameter, and Brillouin gain coefficient on the system performance are analyzed.

3.1 Delivery fiber length and temperature

Nonuniform temperature distribution along the gain fiber is beneficial for SBS suppressing. According to Eq. (4), the nonuniform temperature distribution leads to a broadening of the Brillouin gain spectrum and thus reduces its peak amplitude. For a passively cooled gain fiber, the temperature gradient along the fiber is mainly induced by the nonuniform pump distribution due to the nature of fiber absorption. In a counter-pumped amplifier, the regions close to the pump end absorb more power than other regions and thus normally generate more heat to raise their local temperatures. The local temperatures can also be affected by many factors, such as cooling condition, ambience temperature, and fiber material, but in general it is fair to assume that they are linearly proportional to the absorbed powers. Under this assumption, the amplifier temperature distribution is estimated using the pump distribution when the maximum temperatures are given. For the delivery fiber, the temperature distribution is set to be uniform (i.e., flat) because of no pump absorption. It should be noted that the temperatures mentioned in this paper are the relative temperatures (i.e., the ambience temperature is subtracted) and can differ from the actual temperatures. As the impact of temperature gradient on SBS is due to temperature differences at different locations, using the relative temperatures instead of actual temperature can make the samples more universal and should not affect modeling results.

Figure 2 shows typical signal, pump, and SBS distributions along the fiber. In this case the gain and delivery fibers are 10 m and 2 m long, respectively. It can be seen that the signal starts to grow exponentially after propagating in the gain fiber for about 6 m and reaches its maximum output at the end of the gain fiber, in which the delivery fiber is connected via the pump combiner. After the amplified signal propagates through the delivery fiber, it loses its power slightly due to propagation loss of the delivery fiber. Meanwhile, SBS starts from noise at the delivery fiber output end and propagates backward to the signal input end. For the pump light, it has no change when it propagates in the delivery fiber. This reflects the fact that it just passively passes the delivery fiber into the gain fiber to generate signal amplification.

Fig. 2 Signal, pump, and SBS power distribution along the fiber. Gain fiber length: 10 m; delivery fiber length: 2 m. Fiber core diameter: 25 μm.

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Temperature distribution plays a critical role on SBS. To evaluate the impact of the delivery fiber temperature on amplifier performance, two delivery fiber temperature distributions are chosen in the model as shown in Fig. 3 . In each case, the gain fiber temperature and other parameters are kept unchanged. Figure 3a shows a temperature profile which is likely to be achieved by applying additional heating while Fig. 3b is very close to a real situation in which the heat generated from the delivery fiber is extremely low and is negligible. The SBS thresholds as a function of the delivery fiber lengths are plotted in Fig. 4 . To include the impact of gain fiber Brillouin gain coefficient on system performance, the delivery fibers with four Brillouin gain coefficients are used.

Fig. 3 Fiber temperature distribution of an amplifier with 10 m gain fiber and 2 m delivery fiber; a. (upper), delivery fiber temperature is 100 °C; b. (lower), delivery fiber temperature is 0 °C. Gain fiber temperatures are the same.

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Fig. 4 SBS threshold as a function of delivery fiber length for systems with different delivery fiber temperatures and Brillouin gain coefficients. g_B shown in figures are delivery fiber Brillouin gain coefficients and is in m/W.

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Overall the delivery fiber length makes the most significant impact on the amplifier performance. For example, adding a 1 m commonly used delivery fiber with g_B = 3 × 10⁻¹¹ m/W, reduces amplifier output to less than ~220 W which is more than a 40% reduction from its maximum 416 W output without a delivery fiber. To mitigate the impact, a specially designed SBS suppressing fiber should be used as a delivery fiber [9]. The delivery fiber length should also be shortened. For example, when a 0.5 m delivery fiber with an ultra-low g_B = 1 × 10⁻¹¹ m/W is used, it reduces the maximum power for approximately 10 W, which is much better than a conventional delivery fiber. The results for different operating conditions suggest that using a delivery fiber longer than 1 m can considerably undermine amplifier performance and reduce its maximum power in the range of 10 to 50%.

Figure 4 also shows how the delivery fiber temperature affects SBS threshold. The delivery fibers operating at 0 °C actually produce less SBS power than that of operating at 100 °C. This surprising result implies that applying additional heating to the delivery fiber does not necessarily help improve amplifier performance. For example, in Fig. 4, using a delivery fiber with a uniform temperature at 100 °C actually degrades amplifier performance no matter which delivery fiber is used. This phenomenon may be due to the relative Brillouin spectrum shift over the entire fiber. For a delivery fiber operating at 100 °C, its Brillouin spectrum overlaps well with that of generated from the last section of the gain fiber due to the same temperature. This great overlap thus enhances SBS gain over the region near the combiner. In contrast, for a delivery fiber operating at 0 °C, the Brillouin spectrum generated from the delivery fiber is shifted from what is generated from the last section of the gain fiber due to a large difference in their temperatures. The reduced overlap between the two spectra leads to a lower overall SBS gain.

3.2 Delivery fiber core diameter

Increasing the core diameter to suppress SBS is common in high power amplifiers. The same concept can be applied to the delivery fiber. However, in general practice, a core significantly larger than that of the gain fiber requires a considerable reduction in NA, which leads to excessive bending loss. A delivery fiber with a high bending loss undermines the purpose of delivery fiber and thus is undesirable for high power applications. In addition, a large core diameter difference between the gain fiber and delivery fiber requires special treatments to achieve mode-field matching when fusion splicing them together. The process may increase fiber connection loss and leads to possible cross-talk between the signal and pump and risks the pump laser lifespan. These limitations must be factored in when choosing the delivery fiber core dimension. As a result, the delivery fiber core diameters are chosen from 20 μm to 30 μm in the modeling, while the gain fiber is the same as mentioned above.

Figure 5 shows performance differences among three delivery fiber sizes. It is interesting to see how a fiber size change can make a considerable difference on amplifier performance. An increase in the core diameter from 25 μm to 30 μm enables a more than 1 m long deliver fiber to achieve ~400 W output when the delivery fiber is an SBS suppressing fiber with an ultra-low g_B = 1 × 10⁻¹¹ m/W. To find whether the SBS threshold is directly proportional to the core area, a 20 μm delivery fiber is modeled. When the delivery fiber is 1 m long with g_B = 1 × 10⁻¹¹ m/W, the SBS thresholds for 20 μm, 25 μm, 30 μm are 343 W, 389 W, 400 W, respectively. The SBS threshold values do not scale with the core area ratio of 1:1.56:2.25. An increase in the delivery fiber core diameter by 1.5 times only increases the SBS threshold by 17%, which is far less than the increase of the core area. This is mainly because the delivery fiber is only a section of the whole fiber amplifier system. Another section, i.e., the gain fiber still plays a critical role in SBS. Nevertheless, the use of a large core delivery fiber enables a long delivery fiber.

Fig. 5 SBS threshold as a function of delivery fiber length for delivery fiber with core diameter a. (left) 20 μm; b. (middle) 25 μm; c. (right) 30 μm. Delivery fiber temperature is 0 °C. g_B is in m/W.

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3.3 Delivery fiber Brillouin gain coefficient and gain fiber length

So far, the modeling results indicate that the delivery fiber makes a significant impact on amplifier performance. No doubt, the delivery fiber is a critical component for constructing an all-fiber-based high power single-frequency amplifier. For a typical amplifier, the delivery fiber length is limited to about 1 m to avoid unwanted SBS. Now the question is “are there any other factors enabling a longer delivery fiber?” To answer the question, this section analyzes the amplifier SBS threshold verses Brillouin gain coefficient of the delivery fiber for amplifiers with different gain fiber lengths.

The goal of this modeling is to find whether it is possible to find a design which allows an amplifier to have a 2 m delivery fiber attached. The first attempt is to reduce the gain fiber length from its previous 10 m to 8 m while keep the delivery fiber 2 m long. The SBS threshold as a function of Brillouin gain coefficient for two fiber amplifiers is plotted in Fig. 6a . Compared to the 10 m amplifier, a reduction in the gain fiber length by 2 m marginally increases its SBS threshold. For a SBS suppressing delivery fiber, the SBS threshold increases to 306 W from its 288 W using a 10 m gain fiber. Compared to the output of 416 W without a delivery fiber, this is still a 25% penalty.

Fig. 6 SBS threshold as a function of Brillouin gain coefficient of delivery fiber with core diameter; a. (left) 25 μm; b. (right) 35 μm. Delivery fiber length: 2 m, temperature: 0 °C. Gain fiber temperature difference: 100 °C, Brillouin gain coefficient: 3 × 10⁻¹¹ m/W.

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The next attempt is to find which kind of delivery fiber is required to meet the target. After many trials, it is found that the minimum core diameter of the delivery fiber is 35 μm with g_B = 1 × 10⁻¹¹ m/W. The use of a 35 μm core delivery fiber enables 400 W output from the amplifier even with a 2 m long delivery fiber. The SBS threshold as a function of Brillouin gain coefficient for two amplifier lengths is shown in Fig. 6b. Similar to Fig. 6a, the differences between the 10 m and 8 m gain fiber are marginal. The maximum difference is less than 20 W, which indicates that shortening the gain fiber length can only improve the SBS threshold by a few percent. Further shortening the gain fiber can be problematic since it not only considerably reduces pump absorption and leads to efficiency reduction, but also requires additional components to filter out unabsorbed pump at the signal input end.

4. Discussion

The model results clearly show limitations of all-fiber-based single-frequency fiber amplifier. Surprisingly, the bottleneck turns out to be the passive delivery fiber. Even with a very sophisticated SBS suppressing fiber, the SBS threshold is still around 400 W. This implies that the all-fiber-based single-frequency amplifier is limited to several hundred watts output until some key breakthroughs are made to mitigate SBS in the passive fiber. For applications which require a long delivery fiber (i.e., >2 m), an alternative is a pseudo all-fiber approach. In this case, a short piece of delivery fiber, for example less than 0.5 m, is used to construct the all-fiber-based amplifier, and then a high power isolator [11] is inserted between the short delivery fiber and an additional long delivery fiber. The isolator prevents the SBS generated in the delivery fiber from propagating back to the amplifier and thus suppresses the SBS buildup in the gain fiber. In this case, SBS characteristics of the delivery fiber has no impact on the amplifier performance and the threshold of the delivery fiber can be calculated using a well-known equation [10] $P_{c r} \approx 21 A / g_{B} L$ , where A and L are the effective core area and fiber length, respectively. The SBS threshold $P_{c r}$ as a function of delivery fiber length is plotted in Fig. 7 . With the help of an isolator, a 2 m long delivery fiber with a core diameter of 25 μm can deliver more than 500 W output.

Fig. 7 SBS threshold as a function of delivery fiber length for a device having an isolator. Delivery fiber core diameter is 25 μm. g_B is in m/W.

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5. Conclusion

A rate-equation based model has been adopted to evaluate SBS characteristics in single-frequency fiber amplifiers with a delivery fiber. By treating the delivery fiber as a passive fiber for the pump, a mid-point boundary is eliminated and the device can be modeled thoroughly. The model allows independent inputs of many critical parameters, such as the delivery fiber length, temperature distribution, core size, Brillouin gain coefficient, as well as the gain fiber length. The model reveals some interesting results and clearly shows that adding a short piece of delivery fiber can greatly affect SBS characteristics in the amplifier. Amongst many factors the delivery fiber lengths, core diameter, and Brillouin gain coefficient are the most influential parameters which make great impacts on amplifier performance. The delivery fiber temperature can also affect amplifier performance. To mitigate the effect, the delivery fiber temperature should be kept as low as possible. In addition, the gain fiber length also plays a role on SBS characteristics, but its impact is modest and is normally less than a few percent. To achieve more than 400 W output, many amplifier and delivery fiber parameters have to be optimized. For example, the delivery fiber should have a Brillouin gain coefficient <1 × 10⁻¹¹ m/W with a length of less than 2 m and a core diameter of greater than 35 μm.

References and links

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Stimulated Brillouin scattering in single-frequency fiber amplifiers with delivery fibers

Abstract

1. Introduction

2. Theory

3. Impact of delivery fiber on amplifier performance

3.1 Delivery fiber length and temperature

3.2 Delivery fiber core diameter

3.3 Delivery fiber Brillouin gain coefficient and gain fiber length

4. Discussion

5. Conclusion

References and links

Cited By

Figures (7)

Equations (4)

Optics Express