The theoretical analysis of stimulated Raman scattering (SRS) in crystalline amplifiers with a tightly-focused pump geometry is presented. We predict the minimum Stokes seed power required for an efficient Raman power amplifier and verify this result experimentally. Conversion of a pump to a Stokes beam in a single-pass diamond amplifier is demonstrated using nanosecond pulses with gains of 5.8 from a 1.2-kW peak-power Stokes seed beam. The results demonstrate the possibility of amplifying and combining high-power continuous-wave lasers using current diamond Raman laser technology.
© 2015 Optical Society of America
Optical amplifiers based on the stimulated Raman scattering (SRS) process have proven to be a versatile and convenient approach for amplifying optical signals in traditionally hard-to-reach spectral regions. Early gas Raman amplifiers were motivated by the demand for high energy UV pulses with good beam quality in laser fusion [1–5]. The pulsed-pump peak powers tended to be in the gigawatt range with SRS occurring in meter-long gas-filled light guides. This geometry was dictated by the need for near-collimated pumping for combining multiple pump beams, and so necessitating guiding to maintain overlap over extended distances to reach a reasonable gain.
SRS amplification of pulsed lasers in Raman-active crystals such as silicon , barium nitrate , barium tungstate  and calcium tungstate , can easily reach single-stage amplification factors of several thousand. Indeed Raman generators have gains of order e25 to generate Stokes output from spontaneous Raman photons (e.g., ). Crystalline Raman materials have much stronger non-linearities than gases, and so very high gains are achieved without the need for guiding, or even for strong focusing of the pump pulses, with amplifiers often employing collimated beam geometries with Rayleigh ranges many times longer than the Raman-active media.
Continuous-wave Raman amplifiers are of great technological importance to the telecommunication industry. The lower peak powers of cw pump lasers, generally of the order of watts to kilowatts, require more care to achieve reasonable amplification. Fiber Raman amplifiers use guided interaction lengths that may extend over kilometers, with applications in telecoms  and laser guide stars . Silicon waveguide Raman amplifiers have also demonstrated low (∼ 0.4 W) cw thresholds with modest (∼ 6%) amplification factors . Operating at higher (kilowatt) cw powers in waveguides is limited by optical damage and parasitic nonlinear effects.
To the best of our knowledge, a cw Raman amplifier using an unguided geometry has not been demonstrated. Diamond is a promising candidate for such Raman amplification. Diamond has thermal properties that far exceed all other optical materials, as well as high Raman gain and optical damage thresholds well above the expected power densities in kilowatt amplifiers with tightly-focused beams .
In this work, we consider the design of a cw Raman power amplifier in which the Raman amplifier efficiently transfers incident pump power to a Stokes seed beam of similar power. We focus on two main questions which address key design parameters for power amplifiers: how to maximize gain in bulk unguided devices, and what is the resulting minimum Stokes power required for an efficient Raman power amplifier. We show that tight-focusing of the pump beams maximizes the single pass gain, and we present results with nanosecond lasers that suggest that efficient cw amplifiers should be feasible at power levels of order one kilowatt. Such an amplifier would have significant potential for power scaling and beam combination of kilowatt CW laser technology.
2. Efficient power transfer in Raman amplifiers
We first set out to calculate the minimum Stokes power required for a efficient Raman power amplifier based on a set of analytical equations.
It is well understood that single pass Raman gain is maximized for tight focusing, such that the Rayleigh range of the beams are much shorter than the length of the Raman medium [15,16]. Indeed, provided this criterion is satisfied, the Raman gain is independent of the length, and only depends on the properties of the pump beam (power, wavelength, and beam quality) and properties of the Raman medium (Raman gain and refractive index). The adaption of this theory has been developed in recent years by the present authors for crystalline Raman lasers [17, 18], and here we present an analysis for crystalline Raman power amplifiers.
The evolution of the Stokes and pump fields, govern by two time-dependent rate-equations in a Raman amplifier, assuming the paraxial approximations, are given as
We wish to design an efficient power amplifier such that the input pump power is efficiently transferred to the amplified beam. For conventional laser amplifiers, this is characterized by the saturation intensity required to deplete the gain. In a Raman amplifier, we can similarly calculate the Stokes saturation power—that is the minimum Stokes power required to guarantee significant depletion of the pump power at the exit of the amplifier. Such efficient power extraction is most challenging in a low-gain amplifier in which the Stokes power is much greater than the pump power; in this case the seed Stokes power is approximately constant through the amplifier.
With the assumption that IS is constant through the amplifier, we examine only (1). Firstly we convert this equation to fundamental powers PF by integrating over the transverse coordinates such that
These equations are similar to those recently used by Spence  for intra-cavity Raman lasers and by Kitzler et al.  for extra-cavity Raman lasers, and follow the general form of those derived by Boyd et al.. Integrating with respect to z for a crystal of length l we find for the fundamental power:
These expressions (5, 6) describe the pump power depletion as a function of experimental parameters. The average effective area given by (6) can be calculated for arbitrary beam profiles, including the effects of diffraction through the dependence of the profiles on z. There are simple analytical results for the case that both beams are Gaussian in space. A particularly simple form for results for beams that have matched Rayleigh ranges with , so that the depletion coefficient αd becomes
For Rayleigh ranges much longer than the crystal length, the beams are effectively collimated and (7) reduces to αd = −g0PSl/ηAeff as required, with the depletion coefficient proportional to the crystal length. As we focus tighter, the depletion coefficient increases. In the limit of tight focusing, such that the Rayleigh range is now much shorter than the length of the Raman medium, the tan−1 term in (7) approaches π/2 and depletion coefficient is maximized to
The maximum depletion is therefore independent of crystal length in this tight-focusing limit.
We can calculate the Stokes saturation power required for an efficient single-pass power amplifier to be(9) is that required in the most challenging case.
The saturation Stokes power is illustrated by the 50% contour in Fig. 1. This plot shows the percentage of pump power depleted by a diamond Raman amplifier as a function of focusing strength and Stokes seed power, using the following other parameters: n = 2.41; g0 = 10 cm/GW; λF = 1064 nm; and λS = 1240 nm. It highlights that the Rayleigh range needs to be surprisingly short compared to crystal length to maximize gain. By double passing the pump and Stokes beams, a four-fold reduction of the saturation power is possible, owing to the contributions of both forwards- and backwards-SRS on each of the two passes.
The conclusion is that a diamond amplifier in the strong-focusing regime can in principle achieve efficient Raman amplification for Stokes powers at the kilowatt level, or just a few hundred watts for a double pass design. These power levels are well within reach for current cw laser technology, and so open the way for amplification and beam combination of these lasers to reach much higher power levels.
In order to experimentally investigate the depletion in a Raman amplifier at high power levels we used kilowatt pump pulses of duration 10 nanoseconds from a commercial 1064 nm Q-switched laser (ESKPLA; model NL-220). These pulses are sufficiently long to investigate steady-state Raman interaction (but not steady-state thermal conditions). The output of the pump laser, after passing through a Faraday isolator, was split into two paths using a dichroic beam splitter. One beam path was used to pump the seed laser: a diamond Raman laser which was similar to our previous work , and formed using a 20 cm radius-of-curvature input mirror, a planar output mirror, and an 8-mm long CVD-grown single-crystal diamond (“Type IIIa”; Element Six, UK). The mirrors were dielectric-coated with spectral characteristics selected primarily for first Stokes generation at 1240 nm and double-pass pumping. The Stokes output was filtered to remove residual 1064 nm and any second Stokes at 1485 nm. A telescope was used to closely match the waist size and divergence of the second pump beam in an second 9.5 mm-long diamond which acts as the Raman amplifier. The experimental arrangement is illustrated in Fig. 2. A short delay path was added to the second pump beam so that the peak intensities of the pump and Stokes seed signals were aligned temporally and to avoid any correlations between the pump and Stokes beams in the amplifier. Both 1064-nm pump and 1240-nm seed beams were combined on a dichroic mirror and passed through a Glan-laser polarizer which polarized both beams to the high-gain ⟨111⟩ axis of the diamond amplifier . The beams were focused into the middle of the diamond amplifier using a 75-mm focal length doublet lens. The pump waist diameter was approximately 28 μm with zR ≈ 1.27 mm in diamond. The seed beam was slightly larger with a 33 μm diameter waist at focus. Both beams had measured beam propagation factors of M2 < 1.2. The beam waists were measured using a single-lens magnifying system and an Ophir SP620 camera. The pump and seed beam propagation directions were along the ⟨110⟩ in the diamond and end-faces were AR coated for 1240 nm, with the crystal fixed onto a copper heat sink using a small amount of silver-impregnated thermal paste.
3. Results and discussion
Before we present results showing strong depletion of the pump in an efficient power amplifier, we first present the more conventional data to measure the gain in the opposite limit of a weak Stokes seed pulse. In this limit, the pump power is constant through the amplifier, and the Stokes gain obeys:
In Fig. 3, the gain as a function of peak pump power is shown in black for pump beam with a Rayleigh range approximately 6.8 times shorter than the amplifier crystal, satisfying the strong focusing conditions. For comparison, we also show the gain, in red, for “collimated” pump and probe beams with a confocal parameter approximately equal to the crystal length. The effect of tighter focusing is clearly shown with a factor of two higher exponential gain for similar peak pump powers.
The slope of the exponential gain also produces a measure of the stationary Raman gain coefficient, g0, in the limit of a non-depleting pump. The Raman gain coefficient is usually determined based on experimental observations from, for example, the performance of a Raman laser , or from the small-signal analysis of pump-probe measurements [22–24]. Pump-probe measurements typically use collimated beams and assume no depletion of the pump. In our experiments a non-depleting pump was achieved by using a seed beam with peak intensity more than two orders of magnitude smaller than the saturation power given by (9). With strong focusing the Raman gain was g0 = 9.9 cm/GW, and for the collimated beams was g0 = 10.5 cm/GW with an estimated experimental error of approximately 15% based on repeated experiments. The gain was calculated using the theory of section 2. The experimentally-measured gain is reduced because of the uncorrelated phase noise on the pump and Stokes fields. We must account for this effect because the linewidths of the pump and Stokes (ΔωF and ΔωS ≈ 0.83 cm−1) are significant compared to the Raman linewidth (ΔωR = 1.5 cm−1) . We thus find the narrow band Raman gain coefficient by dividing the experimental gain coefficient by the correction factor ΔωR/(ΔωR + ΔωF + ΔωS) = 0.46 [13, 18]. We note that for gas amplifiers with very narrow Raman linewidths, this gain reduction factor would be prohibitive, so dictating the use of correlated pump and Stokes beams in such amplifiers [1–4]. The experimentally measured Raman gain also included experimentally determined effective beam area and crystal lengths. Our measured values of g0 agree with previously reported values in the range of 8 to 17 cm/GW for single-crystal diamond at wavelengths around 1-μm . This agreement validates the strong-focusing theory presented above.
We now look at experiments demonstrating strong depletion of the pump beam. Figure 4 shows the the depletion of three pump pulses with initial peak powers of 2.9, 6.0 and 10.4 kW (left to right respectively) by a ∼ 1.2 kW Stokes beam, i.e. with peak power approximately 9% greater than the calculated saturation power of 1.1 kW. Stokes pulses were measured at the location indicated in Fig. 2 with the pump blocked (initial seed) or with the pump on (amplified seed). Additionally, pump pulses were also measured with the seed blocked (initial pump) or with the seed present (depleted pump). Each pulse was averaged over 16 traces and calibrated against average power measurements using a sensitive power meter for a range of power levels. Each pump beam is depleted by more than 50%. The depletion is greater for the stronger pumps, and greater than that predicted in Fig. 1 because we are not in the weak pump limit. The depleted pump power is transferred with near quantum-defect-limited efficiency (∼79% c.f. η = 85.8%) into the Stokes beam. The expected depletion from a 1.2-kW Stokes beam is 53% for weak pump beams. The amplification, G in Fig. 4, increases from a modest value of 1.88 from a 2.9 kW pump beam to more than 5.8 from a 10.4 kW pump beam. Correspondingly the depletion increases from approximately 58% to more than 66% at maximum pump power.
These results validate the theory above, showing that for Stokes powers of order 1 kW, there is efficient power transfer from a pump beam onto a Stokes beam. Cascading amplifier stages would allow sequential amplification to more powerful Stokes beams that, in principle, could be used to combine the power output of a large number of pump lasers onto a single Stokes beam. The simplest arrangement consists of a chain of single-pass diamond amplifiers, each similar to that described in this paper. The pump and Stokes fields were not correlated in the present experiments, and so uncorrelated pump beams can equally be used in subsequent stages in a cascaded Raman amplifier. Due to the unparalleled thermal properties of diamond we calculate that powers in kilowatt diamond Raman amplifiers are well below that for thermal fracture (estimated at 1 MW) . Likewise power densities are well below the coated surface optical damage threshold which exceeds 220 MW/cm2 in diamond [25, 26]. The first stage of such a cascaded amplifier is the most challenging because the minimum Stokes power needed to efficiently deplete the pump is of order 1 kW, although this power requirement can be reduced by using a double-pass pre-amplifier stage. A recently demonstrated 380 W diamond Raman laser , for example, could achieve 61.6% pump depletion even in that first challenging stage using a double-pass configuration, whereas a single-pass would achieve only 21.3% depletion.
In summary, we have presented a theoretical and experimental study of stimulated Raman scattering in crystalline Raman amplifiers with a tightly-focused pump geometry. We have validated the theory of tightly-focused SRS with a conventional measurement of the Raman gain of diamond.
Using nanosecond pulses as a model for continuous-wave laser systems, we have demonstrated an efficient Raman power amplifier in diamond using cw beam powers at less that a kilowatt, opening the way for sequential beam-combination of kW-scale conventional lasers onto a single Stokes laser beam.
This research was sponsored by the Australian Research Council Future Fellowship ( FT0990622) and Discovery Grant ( DP130103799) Schemes, and the US Air Force Research Laboratory under agreement number FA2386-12-1-4055.
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