## Abstract

We present a detailed theoretical investigation of claddingpumped Raman fiber amplification in an unexplored parameter space of high conversion efficiency (> 60%) and high brightness enhancement (> 1000). Fibers with large clad-to-core diameter ratios can provide a promising means for Raman-based brightness enhancement of diode pump sources. Unfortunately, the diameter ratio cannot be extended indefinitely since the intensity generated in the core can greatly exceed that in the cladding long before the pump is fully depleted. If left uncontrolled, this leads to the generation of parasitic second-order Stokes wavelengths in the core, limiting the conversion efficiency and as we will show, clamping the achievable brightness enhancement. Using a coupled-wave formalism, we present the upper limit on brightness enhancement as a function of diameter ratio for conventionally guided fibers. We further present strategies for overcoming this limit based upon depressed well core designs. We consider two configurations: (1) pulsed cladding-pumped Raman fiber amplifier (CPRFA) and (2) cw cladding-pumped Raman fiber laser (CPRFL).

©2010 Optical Society of America

## 1. Introduction

High average power fiber amplifiers and fiber laser oscillators are important candidates for directed energy laser sources, front-end drive laser sources for Inertial Confinement Fusion (ICF), short pulse, compact high flux X-ray sources, and machining. Generally, these are pumped by diode lasers, which are very efficient devices for converting electrical energy to optical energy. But at high powers, pump beam quality becomes difficult to achieve. Cladding-pumped fiber lasers provide a means for converting that low brightness energy into a diffraction-limited beam in a process referred to as brightness enhancement. To date, Yb^{3+} fiber lasers have proven to be the best systems for achieving brightness enhancement. However, rare-earth doped fibers emit in fixed spectral bands and require that the diode laser emission spectrum overlap with their absorption band. In this paper, we investigate a promising alternative method for converting low brightness pump diode laser energy into a high brightness, diffraction-limited beam with high efficiency using cladding-pumped Raman fiber amplification. Experimental demonstrations of this technique have shown high efficiency [1–3] although with modest brightness enhancement and high brightness enhancement although with limited efficiency [4]. In this article, we explore the parameter space over which the simultaneous achievement of both can be possible.

While not competitive at lower powers and pulse energies, at very high powers and/or energies, cladding-pumped Raman fiber amplifiers [5] possess multiple advantages when compared to rare-earth doped fiber laser amplifiers [6, 7]. These advantages include (1) the potential for higher quantum-defect-limited conversion efficiency, (2) lower heat dissipation, (3) a wider range of operating wavelengths, since the Raman gain bandwidth is determined by a fixed frequency offset from the pump source and spans many THz, (4) scaling to wide bandwidths by means of multiple spectrally-staggered pumps [8] (5) simplicity of fabrication, since the Raman process is non-resonant and hence does not depend on the addition of dopants beyond those required to define the waveguide; with benefits including the avoidance of photodarkening, operation at higher bulk damage thresholds, scaling to larger single-moded core diameters since the elimination of rare-earth dopants permits larger core sizes to be employed with good or even single mode beam quality due to superior refractive index control in non rare-earth doped fibers, and (6) Multi-stage scaling, where a high-power cladding-pumped Raman fiber laser could provide a means of combining the outputs of multiple lower-power cladding-pumped Raman fiber lasers in a simple and robust manner.

We present methods for designing and optimizing cladding-pumped Raman fiber amplifiers and laser oscillators. We show that the use of differential optical waveguide loss can suppress higher-order Stokes conversion and greatly enhances the utility of a cladding-pumped Raman fiber amplifier or laser. It does so by significantly raising the clad-to-core diameter ratio at which the process can be efficient. This in turn greatly increases the allowable brightness enhancement of the laser or amplifier. We envision that this scheme will allow for the leveraging of commercial, off-the-shelf low brightness laser diodes for the construction of high brightness and high efficiency fiber lasers at high energies and/or high powers.

## 2. Coupled Wave Formulation for Single Pass Amplification

We first consider the case of a single-pass cladding-pumped Raman amplifier. The coupledwave equations governing the interaction of pump *P _{p}*, desired first-order Stokes

*P*

_{s1}, and undesired second-order Stokes

*P*

_{s2}powers are given as [9, 10]:

where *A*
_{clad} and *A*
_{core} are the respective cladding and core effective areas, *ν*
_{j} are the interacting carrier frequencies, *α _{j}* are the losses and

*g*is the Raman gain coefficient. Compared to corepumped Raman amplifiers, gain at the first-order Stokes wave is weaker by the ratio of the clad-to-core area. However, the subsequent conversion of light in the core to higher Stokes orders progresses at the higher gains. A race then ensues between pump depletion and secondorder Stokes conversion, Except for very small clad-to-core areas, this race is usually won by the higher gain at the second-order Stokes wiping out the first-order Stokes before the pump has had a chance to fully deplete. This incomplete conversion is the primary drawback for claddingpumped Raman fiber amplification. A solution of these equations is plotted in Fig. 1 for a pulsed pump source with 50 kW of peak power. We do not explicitly solve the time dependence but rather assume a pump energy of 500 microJoules in a pulse profile that is 10 ns flat in time.

Assuming conversion from a multi-mode cladding with diameter *D*
_{clad} and numerical aperture *NA*
_{clad} to a single-mode core with a beam quality parameter of *M*
^{2} = 1, the brightness enhancement is defined as [4]:

Note that this expression depends on the conversion efficiency, η, and the limitations imposed by radiance considerations. Increasing the clad-to-core diameter ratio can initially improve this quantity, but it then becomes increasingly difficult to attain a high level of conversion efficiency before the onset of second-order Stokes. To quantify this, we derive a simple analytic expression that predicts the attainable conversion efficiency as a function of clad-to-core diameter ratio.

We define the conversion efficiency as *η*(*z*)=*P*
_{s1}(*z*)/*P _{p}*(0). Assuming the generated secondorder Stokes is not appreciably large at the peak of first-order Stokes generation, and further assuming all other loss mechanisms are negligible, by energy conservation we have:

the peak conversion efficiency can thus be estimated from the following relation:

But this relation is not directly useful unless we further link the peak signal power to the secondorder Stokes seed which ultimately arrests the achievable conversion. This is complicated because near the peak, the signal intensity is being amplified by the pump while it is in turn amplifying the second-order Stokes wave. In order to simplify this dependence, we approximate the super-exponentially growing second-order Stokes power as simple exponential growth with an effective *gIL* product.

where the effective *gIL* product [11] can be approximated as:

Combining Eqs. (7) and (10) results in an expression linking the peak signal power to the second-order Stokes seed:

Finally, we make use of the fact that the rate of first-order Stokes generation goes to zero at peak conversion. For simplicity, we assume that the quantum defect is small (*ν*
_{s1} ≈ *ν*
_{s2}) and that the signal absorption is much weaker than the Raman signal gain *α*
_{s1} << *gI _{p}*. This yields an expression connecting the second-order Stokes power to the depleted pump power:

Substituting Eqs. (11) and (13) into Eq. (6) yields an analytic expression for the peak achievable conversion efficiency:

Increasing the pump and decreasing the fiber length does not help resulting in an upper bound of *I*
_{s1} ≈ 8*I _{p}* at 90% conversion [10]. In the limit of large clad-to-core diameter ratio, and for typical assumptions regarding the second-order Stokes seed, the conversion efficiency can be approximated as:

Because the conversion efficiency is inversely proportional to the clad-to-core area, it will directly cancel the other component of the brightness enhancement, which is limited by radiance considerations (directly proportional to the clad-to-core area). Combining Eqs. (4) and (15) the maximum achievable brightness enhancement is clamped at:

This result implies that making *NA*
_{clad} high and operating in a large core *D*
_{core} (with a low *NA*
_{core} to maintain single-moded operation) optimizes the achievable brightness enhancement. Practical considerations limit this brightness enhancement to approximately 1000. Figure 2 plots the brightness enhancement maximum and the attainable value when limited by incomplete pump depletion resulting from second-order Stokes generation in the core. Here, high peak pump powers are attained with nanosecond pulses as in [4]. Note that beyond a clad-tocore diameter ratio of around 2–3, the brightness enhancement reaches a limit.

But the potential for much higher levels of brightness enhancement exists. It is possible to suppress the onset of the Raman cascade by using a custom designed fiber waveguide (depressed well core) that provides a differential loss between the first and second-order Stokes waves of the Raman process [12]. This custom fiber design removes the restriction of low cladto-core area ratios of 8 to beyond 40. This in turn permits brightness enhancement factors well in excess of 1000. Figure 3 demonstrates how implementing loss at the second-order Stokes wavelength can increase the conversion efficiency to near the quantum-defect limit and hence the the achievable brightness enhancement to near the theoretical maximum governed by radiance considerations.

## 3. Oscillator Formulation

We next consider the case of a cw cladding-pumped Raman fiber laser (CPRFL). Here, lower pump powers are employed, but feedback ensures a longer interaction length. The model [13] consists of 5 coupled-wave equations for the pump, forward and backward traveling waves at the first-order Stokes and forward and backward traveling waves at the second-order Stokes.We assume that the end mirrors are reflective only at the first-order Stokes wavelength (selective dichroic mirrors or Bragg gratings) with a 100% reflector on one end and an output coupler with reflectivity R on the other end.

These equations can be solved using a fourth order Runge-Kutta ODE solver. Self-consistent solutions are found by shooting the Stokes waves (forward and backward) away from the 100% coupler and determining the reflectivity based on the ratio of the forward and backward going waves at the output coupler. A global optimizer can further be used to optimize output coupler reflectivity in order to maximize the output power in the first-order Stokes. When an optimum is found it is generally the case that decreased reflectivity results in inefficient pump depletion while increased reflectivity results in greater intra-cavity losses. It is desirable to keep the cladding diameter small enough to maintain high signal gain but large enough to accommodate the low brightness of the pump source used. Assuming a pump diode brightness at 0.03 W/(*µ*m^{2}-sr) mapped into in a 80 *µ*m, 0.45 NA cladding we first consider a pump source of ~ 100 W. Using this as a pump for a 12 *µ*m MFD core, in Fig. 4 we map out the parameter space of required attenuation on the first and second-order Stokes waves for efficient conversion in a 100 m fiber oscillator.

The bends in these curves occur at the magnitude of second-order Stokes losses that is sufficient to suppress the runaway Stokes cascade. For an oscillator to be immune from a runaway Stokes cascade, the imposed losses on the second-order Stokes need to be greater than the maximum possible Raman gain. Since the gain can occur from forward and backward traveling signal fields, this translates into the following relationship:

Figure 5 shows an example single-mode waveguide design showing how a core could be formed in an optical fiber [14, 15]. A 1480 nm laser diode array might be used as the pump source. Bend losses were calculated by numerically propagating the appropriately deformed mode in an index profile corresponding to a coiled fiber. The propagation for a given fiber is performed by the well known Beam Propagation Method (BPM) [16], a scalar calculation valid for weakly guiding fibers. Coiling of the fiber is treated as an exponential perturbation to the index profile in the plane of the coil [17], which for practical cases in indistinguishable from a linear tilt. Signal lost from the core is absorbed at the edges of the domain of the calculation, i.e., we neglect trapping of signal light by the cladding. The calculations match analytic predictions [18] of bend loss in weakly guiding step index fibers over a range of parameters spanning six orders of magnitude. Using this simulation, we predict that when coiled to a diameter of 475 mm, the induced losses at 1702 nm could be made as high as 1 dB/m while maintaining losses < 1 dB/km at 1583 nm. The spectral cutoff for the differential loss behavior can be engineered by changing the ratio of the diameter of the well to the diameter of the inner core. This permits the construction of a cladding-pumped Raman fiber laser that overcomes the Stokes limit over a broad range of wavelengths. A scaling analysis further shows that a 1 dB/m vs 1 dB/km 2nd to 1st order Stokes ratio can be achieved for core areas up to 260 microns squared albeit with larger coiled bend diameters.

A particular evolution at the losses realizable with this fiber geometry is plotted in Fig. 6. The output coupler reflectivity required is 88%. The reflectivity may be further increased to shorten the fiber length but this can result in a greater circulating intensity in the oscillator cavity. For high power, high conversion operation, it may be desirable to limit the reflectivity so as to practically extract the maximum amount of power and avoid damage near 20 W/*µ*m^{2}. Figure 7 summarizes these tradeoffs. There are clearly identifiable regimes on this plot. For low pump powers, high reflectivities can provide enough feedback to deplete the pump, but the attenuation of the signal becomes the limiting factor on efficiency. For short fiber lengths, incomplete pump depletion becomes the limiting factor on efficiency and damage can be an issue. For powers or reflectivities higher than the optimum, the generation of second-order Stokes clamps the output power.

Another particular solution at *α*
_{s1} =1 dB/km, *α*
_{s2} =1 dB/m, *P _{p}* =1 kW in a 350 m fiber is plotted in Fig. 8. In this aggressively pumped regime, less reflectivity is required and 63% conversion efficiency is predicted with a brightness enhancement > 1600. Higher levels of cw brightness enhancement may be possible with higher damage thresholds, lower signal attenuation, or larger cores.

Care must also be taken to ensure that signal light remains in the core and does not leak into the cladding where it can also be amplified, robbing pump power and spoiling the output mode quality. This can result from imperfect coupling with an external bulk dichroic mirror. A better solution would be to employ Bragg gratings written into the core.

Finally, we examine the relationship between input and output power. Figure 9 shows the curves for 100 m oscillators with 88% and 2.3% reflectivities for *α*
_{s2} = 1,5 dB/m. Because the oscillator is not resonant at second-order Stokes wavelengths, there is a clamping of output power. The power curves can be parameterized by a threshold, asymptotic slope efficiency and the clamped output power. The threshold is determined by the fiber length and reflectivity required to achieve oscillation:

Just above threshold, the slope efficiency can be larger than the asymptotic slope efficiency and even exceed unity. It however eventually approaches the slope efficiency limited by signal losses and quantum defect. Clamped output power is limited by the threshold for second-order Stokes in Eq. (22).

To summarize, proper oscillator design requires specification of at least 9 parameters (Pump power, clad diameter, clad NA, core diameter, core NA, signal loss, 2nd order Stokes loss, fiber length, output coupler reflectivity). The dependence of conversion efficiency and damage fraction on these parameters is complicated. In the spirit of simplifying the design process navigation through parameter space, we present a methodology in Fig. (10).

## 4. Conclusion

In conclusion, we have presented a detailed investigation of cladding-pumped Raman fiber amplification aimed at high conversion efficiency (> 60%) at high brightness enhancement (> 1000). A scheme for differential loss can be applied to both single-pass configurations appropriate for pulsed amplification and laser oscillator configurations applied to high average power cw source generation.

## Acknowledgments

This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

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