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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 8 — Apr. 16, 2007
  • pp: 4647–4662
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Four-wave mixing in nanosecond pulsed fiber amplifiers

Jean-Philippe Fève, Paul E. Schrader, Roger L. Farrow, and Dahv A.V. Kliner  »View Author Affiliations


Optics Express, Vol. 15, Issue 8, pp. 4647-4662 (2007)
http://dx.doi.org/10.1364/OE.15.004647


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Abstract

We present an experimental and theoretical analysis of four-wave mixing in nanosecond pulsed amplifiers based on double-clad ytterbium-doped fibers. This process leads to saturation of the amplified pulse energy at 1064 nm and to distortion of the spectral and temporal profiles. These behaviours are well described by a simple model considering both Raman and four-wave-mixing contributions. The role of seed laser polarization in birefringent fibers is also presented. These results point out the critical parameters and possible tradeoffs for optimization.

© 2007 Optical Society of America

1. Introduction

Pulsed fiber amplifiers have recently gained considerable interest for practical applications, such as materials processing, printing, and lidar, because of their ability to deliver nanosecond pulses with very high average and peak powers and diffraction-limited beam quality in highly reliable devices. Recent developments in double-clad, Yb-doped, large-mode-area (LMA) fibers have led to a record combination of average and peak output powers at 1064 nm [1–4

1. F. Di Teodoro and C. Brooks, “Multistage Yb-doped fiber amplifier generating megawatt peak-power, subnanosecond pulses,” Opt. Lett. 30, 3299–3301 (2005). [CrossRef]

]. Due to the very high peak irradiances of the amplified pulses, nonlinear effects limit the extracted pulse energy. The threshold power for nonlinear processes is increased by increasing the effective mode area and decreasing the fiber length. The mode field diameter of single-mode, low-numerical-aperture, LMA fibers is limited by increased sensitivity to bend-losses as the core diameter is increased and the numerical aperture (NA) is correspondingly decreased. Different techniques have successfully been implemented to preserve single-mode output in large-core fibers: selective bend-loss in coiled multimode (MM) fibers [5

5. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25, 442–444 (2000). [CrossRef]

], control of the seed conditions of MM fibers [6

6. M. Ferman, “Single-mode excitation of multimode fibers with ultrashort pulses,” Opt. Lett. 23, 52–54 (1998). [CrossRef]

], use of photonic crystal fibers [1

1. F. Di Teodoro and C. Brooks, “Multistage Yb-doped fiber amplifier generating megawatt peak-power, subnanosecond pulses,” Opt. Lett. 30, 3299–3301 (2005). [CrossRef]

,7

7. J. Limpert, A. Liem, M. Reich, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “Low-nonlinearity single-transverse-mode ytterbium-doped photonic crystal fiber amplifier,” Opt. Express 12, 1313–1319 (2004). [CrossRef] [PubMed]

], design of radial index and dopant profiles in MM fibers [8

8. H. Offerhaus, N. Broderick, D. Richardson, R. Sammuk, J. Caplen, and L. Dong, “High-energy single-transverse- mode Q-switched fiber laser based on a multimode large-mode-area erbium-doped fiber,” Opt. Lett. 23, 1683–1685 (1998). [CrossRef]

,9

9. M. Hotoleanu, M. Söderlund, D. Kliner, J. Koplow, S. Tammela, and V. Philipov, “Higher-order modes suppression in large mode area active fibers by controlling the radial distribution of the rare earth dopant,” Proc. SPIE 6102, 61021T (2006). [CrossRef]

], and selective bend-loss in helical-core MM fibers [10

10. P. Wang, L. Cooper, J. Sahu, and W. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber amplifier,” Opt. Lett. 31, 226–228 (2006). [CrossRef] [PubMed]

]. Despite significant advances based on these approaches, nonlinear processes remain the limiting factor for power scaling because of practical limitations on the core diameter and the very high peak irradiances generated in the fiber (>440 GW/cm2 [3

3. R. L. Farrow, D. A. V. Kliner, P. E. Schrader, A. A. Hoops, S. W. Moore, G. R. Hadley, and R. L. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” Proc. SPIE 6102, 61020L (2006). [CrossRef]

]). The limiting nonlinear effects depend on the pulse temporal and spectral characteristics and have been thoroughly studied in many configurations. For long pulses (>2 ns) with a transform-limited linewidth, stimulated Brillouin scattering (SBS) is the most limiting factor. For short pulses (<0.5 ns), self-phase modulation induces very large distortion of the spectrum. Recent results have suggested that a pulse duration of ~1 ns is suitable for minimizing spectral and temporal distortion in high-energy pulsed fiber amplifiers [3

3. R. L. Farrow, D. A. V. Kliner, P. E. Schrader, A. A. Hoops, S. W. Moore, G. R. Hadley, and R. L. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” Proc. SPIE 6102, 61020L (2006). [CrossRef]

,11–13

11. F. Di Teodoro, J. P. Koplow, S. W. Moore, and D. A. V. Kliner, “Diffraction-limited, 300-kW peak-power pulses from a coiled multimode fiber amplifier,” Opt. Lett. 27, 518–520 (2002). [CrossRef]

]; in this case, stimulated Raman scattering (SRS) generally becomes the most limiting effect [1

1. F. Di Teodoro and C. Brooks, “Multistage Yb-doped fiber amplifier generating megawatt peak-power, subnanosecond pulses,” Opt. Lett. 30, 3299–3301 (2005). [CrossRef]

,5

5. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25, 442–444 (2000). [CrossRef]

,14

14. G. Agrawal, Nonlinear Fiber Optics, 3rd ed., Optics and Photonics Series (Academic, San Diego, Calif., (2001).

].

Combined effects of SRS and FWM have been widely studied. For example, they play a key role in the generation of supercontinuum [18

18. A. Proulx, J. M. Ménard, N. H̑, J. Daniel, R. Vallée, and C. Paré, “Intensity and polarization dependences of the supercontinuum generation in birefringent highly nonlinear fibers,” Opt. Express 11, 3338–3345 (2003). [CrossRef] [PubMed]

,19

19. W. Wadsworth, N. Joly, J. Knight, T. Birks, F. Biancalana, and P. Russell, “Supercontinuum and four-wave mixing with Q-switched pulses in endlessly single-mode photonic crystal fibres,” Opt. Express 12, 299–309 (2004). [CrossRef] [PubMed]

] and in polarization attraction [16

16. S. Pitois, A. Sauter, and G. Millot, “Simultaneous achievement of polarization attraction and Raman amplification in isotropic optical fibers,” Opt. Lett. 29, 599–601 (2004). [CrossRef] [PubMed]

]. One process can enhance [20

20. J. Chee and J. Liu, “Raman-assisted parametric frequency and polarization conversion in a birefringent fiber,” Opt. Lett. 14, 820–822 (1989). [CrossRef] [PubMed]

,21

21. T. Sylvestre, H. Maillotte, E. Lantz, and P. Tchofo Dinda, “Raman-assisted parametric frequency conversion in a normally dispersive single-mode fiber,” Opt. Lett. 24, 1561–1563 (1999). [CrossRef]

] or suppress [22

22. P. Tchofo Dinda, G. Millot, and S. Wabnitz, “Polarization switching and suppression of stimulated Raman scattering in birefringent optical fibers,” J. Opt. Soc. Am. B 15, 1433–1441 (1998). [CrossRef]

] the other. In particular, SRS can induce efficient FWM in regimes where phase-matching is normally impossible [23

23. P. Tchofo, E. Sève, G. Millot, T. Sylvestre, H. Maillote, and E. Lantz, “Raman-assisted three-wave-mixing of non-phase-matched waves in optical fibres: application to wide range frequency conversion,” Opt. Commun. 192, 107–121 (2001) [CrossRef]

,24

24. S. Coen, D. A. Wardle, and J. D. Harvey, “Observation of non-phase-matched parametric amplification in resonant nonlinear optics,” Phys. Rev. Lett. 89, 273901 (2002) [CrossRef]

]. A few studies have examined the influence of pump intensity on phase-matching of FWM; the depletion of the pump beam has been recognized to have important consequences for the dynamics of phase-matched FWM processes [25

25. G. Cappellini and S. Trillo, “Energy conversion in degenerate four-photon mixing in birefringent fibers,” Opt. Lett. 16, 895–897 (1991). [CrossRef] [PubMed]

,26

26. S. Trillo, G. Millot, E. Sève, and S. Wabnitz, “Failure of phase-matching concept in large-signal parametric frequency conversion,” Appl. Phys. Lett. 72, 150–152 (1998). [CrossRef]

]. The case of amplification gain and non-phase- matched FWM, however, have not been considered, to our knowledge. In a recent paper [27

27. J.-P. Fève, “Phase-matching and mitigation of four-wave mixing in fibers with positive gain,” Opt. Express 15, 577–582 (2007). [CrossRef] [PubMed]

], we pointed out the existence of “gain-induced phase-matching”, which allows efficient interaction even for highly mismatched FWM in isotropic single-mode fibers with normal dispersion. We showed that FWM can become the most limiting nonlinear process in fiber amplifiers operating in the ~1 ns temporal regime [27

27. J.-P. Fève, “Phase-matching and mitigation of four-wave mixing in fibers with positive gain,” Opt. Express 15, 577–582 (2007). [CrossRef] [PubMed]

].

The present work reports a detailed experimental and theoretical study of FWM in a nanosecond pulsed fiber amplifier operating at a wavelength of 1064 nm. We confirm that, in this specific regime, FWM is the most limiting nonlinear effect. It sets an upper limit to the “in-band” pulse energy at 1064 nm, beyond which all photons are converted to “out-of-band” wavelengths. FWM strongly impacts the spectrum and the temporal profile of the output pulses, so that a careful characterization is required, depending on the target application. A simple numerical model, taking into account the relevant effects for the specific regime of operation, allows us to explain the most important features for the considered regime of operation: it predicts the observed spectral and temporal shapes of the output pulses, as well as the impact of polarization on FWM. Based on these calculations, we propose possible approaches to mitigating FWM, and we examine tradeoffs for optimization.

2. Description of experiments and model

2.1. Experimental apparatus

The experimental setup used in the present study has been detailed previously [13

13. P. E. Schrader, R. L. Farrow, D. A. V. Kliner, J.-P. Fève, and N. Landru, “High power fiber amplifier with tunable repetition rate, fixed pulse duration, and multiple output wavelengths,” Opt. Express 14, 11528–11537 (2006). [CrossRef] [PubMed]

]. A fiber amplifier was seeded at 1064 nm by a passively Q-switched Nd:YAG/Cr:YAG microchip laser. For most of the present work, the seed laser was operated at its maximum repetition rate (f) of 27 kHz and with a pulse duration of 1.0 ± 0.1 ns and pulse energy of 3.2 ± 0.3 μJ. The output of the microchip laser was optically isolated and launched into the core of a double-clad, Yb-doped, polarization-maintaining (PM) fiber amplifier. The energy coupled in the fiber was typically 1.5 μJ. The fiber (Nufern) had a core diameter of 30 μm, core NA of 0.06, inner-cladding diameter of 250 μm, and length of 3.52 m, unless otherwise specified. The fiber was end-pumped with the ~976 nm output of a fiber-coupled diode bar (Apollo) with a measured pump-coupling efficiency of 85%. The fiber was coiled on orthogonal spools with a bend radius of 2.9 cm to suppress high-order modes via bend-loss-induced mode filtering [5

5. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25, 442–444 (2000). [CrossRef]

]. The amplifier output pulses were characterized spectrally, temporally, and spatially, as described in Ref. [3

3. R. L. Farrow, D. A. V. Kliner, P. E. Schrader, A. A. Hoops, S. W. Moore, G. R. Hadley, and R. L. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” Proc. SPIE 6102, 61020L (2006). [CrossRef]

].

2.2. Numerical model

{E1z=jγ[E12+(2ρ)(E32+E42)]E1gR2AeffE32E1+g2E1+jγE1*E3E4ejΔkzE3z=jγ[E32+(2ρ)(E12+E42)]E3gR2AeffE12E3+jγE4*E1E1ejΔkzE4z=jγ[E42+(2ρ)(E12+E32)]E4+jγE3*E1E1ejΔkz
(1)

The above approximations were employed to yield equations that are computationally tractable but capture the key physical processes necessary to describe high-peak-power, nanosecond-duration, pulsed fiber amplifiers. Complete modeling of pulsed fiber amplifiers, taking into account all effects, is very challenging, and such a comprehensive model is not currently available [32

32. G. R. Hadley, R. L. Farrow, and A. V. Smith, “Three-dimensional time-dependent modeling of high-power fiber amplifiers,” Proc. SPIE 6453, 64531B (2007) in press. [CrossRef]

]. By considering only the relevant phenomena in the specific regime of operation, we obtain a simple model that provides a very good qualitative and semi-quantitative description of all observed behaviors and that is applicable to peak powers and pulse energies of practical interest. Advantages of this approach are that it allows parametric studies for optimization of pulsed fiber amplifiers with modest computational effort, and it provides significant physical insight into the processes that influence fiber-amplifier performance.

3. Evidence and importance of four-wave mixing

3.1. Experimental demonstration

We first investigated the performance of the fiber amplifier at f = 26.9 kHz and with the seed polarization parallel to the slow axis of the fiber. Figure 1(a) shows the measured output spectrum at four output pulse energies. Figure 1(b) shows both the total output pulse energy (E out) and the pulse energy that is transmitted through a narrow band-pass filter with peak transmission at 1064 nm and 4 nm FWHM bandwidth (E 1064). Integrating the out-of-band power in the spectral measurements of Fig. 1(a) produces similar results to those obtained with the band-pass filter. At the highest pulse energy, ~50% of the energy was shifted to out-of- band wavelengths [Fig. 1(b)].

Fig. 1. (a). Output spectrum at four values of E out. (b) Total output energy (integrated over the whole spectrum, open squares), energy transmitted through a band-pass filter at 1064 nm (filled circles) and out-of-band energy (stars) vs. diode pump power coupled into the fiber.

The results shown in Fig. 1 provide clear evidence of the major impact of FWM at high pulse energy (high peak power): FWM generates broad spectral peaks above a threshold energy (~150 μJ for our experimental conditions), and the out-of-band energy increases significantly with increasing output energy above this threshold. Even though the peak out-of- band intensity was at least 25 dB below the amplified peak at 1064 nm [Fig. 1(a)], at the highest energy, ~50% of the pulse energy was shifted to out-of-band wavelengths [Fig. 1(b)]. FWM effectively clamped the 1064 nm pulse energy to ~170 μJ, which was reached at a pump power of 13 W; further increases in diode pump power increased the total output energy, but all of this increase occurred at out-of-band wavelengths.

These results are in good agreement with predictions based on analytical integration of the system of Eq. (1), as detailed in [27

27. J.-P. Fève, “Phase-matching and mitigation of four-wave mixing in fibers with positive gain,” Opt. Express 15, 577–582 (2007). [CrossRef] [PubMed]

]. In particular, this analysis showed that above a certain power level (which depends on the fiber parameters), the 1064 nm pump beam is efficiently converted to out-of-band wavelengths, effectively clamping the 1064 nm pulse energy. This efficient conversion is caused by “gain-induced phase matching” of the FWM interaction. Although evidence of FWM in the output spectra of high-energy fiber amplifiers had been reported [12

12. C. Brooks and F. Di Teodoro, “1-mJ energy, 1-MW peak-power, 10 W-average power, spectrally-narrow, diffraction-limited pulses from a photonic-crystal fiber amplifier,” Opt. Express 13, 8999–9002 (2005). [CrossRef] [PubMed]

], to our knowledge its limiting effect had not been recognized. We present below additional consequences of FWM for the performance of pulsed fiber amplifiers operating in the ~1 ns and ~1 μm wavelength regime.

3.2 Pump depletion and temporal effects

To further understand the limitations due to FWM on the ability to extract energy from a pulsed fiber amplifier, Fig. 2 shows the calculated pump and signal peak powers as they propagate along the fiber, deduced from the numerical integration of Eq. (1). The calculations employed parameters corresponding to those of the experiments: a fiber length of 3.5 m and a 1064 nm seed pulse with a Gaussian temporal shape, 1 ns duration (FWHM), and 1.5 μJ pulse energy. The signal and idler beams were each seeded with one noise photon.

Fig. 2. Calculated peak powers along the fiber for the pump (λ 1 = 1064 nm, solid) and signal (λ 3 = 1084 nm, dotted) for two values of E out, 313 μJ (blue) and 576 μJ (black).

This example considers a signal wavelength of λ 3 = 1084 nm (so λ 4 = 1044.7 nm), which is far from the Raman gain peak (~1115 nm). The calculated phase mismatch is Δk = 18.8 m-1, and the corresponding coherence length for FWM is 33.4 cm. However, gain-induced phase-matching leads to progressive generation of the signal field along the fiber [27

27. J.-P. Fève, “Phase-matching and mitigation of four-wave mixing in fibers with positive gain,” Opt. Express 15, 577–582 (2007). [CrossRef] [PubMed]

]. In the case of large amplification, this process is so efficient that the pump field at λ 1 is fully depleted at the output of the fiber, and all photons are converted to signal and idler fields, as shown in Fig. 2 for an output energy of 576 μJ. Calculations at other wavelengths show qualitatively similar behavior, but depletion occurs at lower pulse energy (peak power) because 1084 nm lies close to the FWM minimum, as seen in Fig. 1(a).

An important consequence of the depletion of the pump field is observed in the pulse temporal profiles. Figure 3(a) shows the temporal profile of the total output pulse, the beam transmitted through the band-pass filter (i.e., the portion of the pulse at 1064 nm), and the beam reflected from the filter (i.e., the out-of-band portion of the pulse) at five values of E out. At low pulse energy, all of the energy is contained within the 1064 nm portion of the pulse. As E out increases, FWM becomes important, so that depletion of the pump and transfer to other wavelengths occurs. The effect of FWM is most pronounced near the center of the pulse, i.e., at the highest peak power. At E out = 184 μJ, a small dip is observed at the center of the pulse. As the pulse energy is further increased, depletion becomes more important, as expected from Fig. 2. Full depletion at the center of the peak is observed at the maximum E out value, with almost all photons at 1064 nm being converted to signal and idler fields for this peak power. Figure 3(b) shows corresponding calculated temporal profiles obtained by numerically solving Eq. (1). We assumed an idler wavelength of λ 3 = 1115 nm, corresponding to the peak of the FWM spectrum [Fig. 1(a)], and Δk = 124 m-1 in this case. The value of E out was varied through the gain coefficient g. As seen in Fig. 3, there is a very good agreement between this basic model and the experiments. The non-symmetric pulse shape observed in the experimental in-band data at high output energy is due to a time-dependent phase shift (frequency chirp) caused by SPM, which is not included in the calculations, and possibly by gain depletion. Nonetheless, Eq. (1) is sufficient to provide a useful understanding of the critical effects for the considered temporal and power regime.

Fig. 3. (a). Measured temporal profiles of the amplified pulses at five values of E out. Dashed: unfiltered pulses; solid: transmitted by band-pass filter at 1064 nm; diamonds: reflected from filter. (b). Calculated normalized temporal profiles.

Figure 4 shows calculated temporal profiles without normalization for increasing values of E out. There exists a maximum peak power for the pump field, beyond which depletion occurs. The existence of such a maximum extractable peak power was inferred from the analytical integration of Eq. (1): for a given A eff and nonlinear coefficient of the fiber, the maximum peak power increases linearly with the phase mismatch [27

27. J.-P. Fève, “Phase-matching and mitigation of four-wave mixing in fibers with positive gain,” Opt. Express 15, 577–582 (2007). [CrossRef] [PubMed]

]. When E out increases, the maximum power is reached earlier in the pulse, so that generation of signal and idler photons starts earlier. This phenomenon explains the broadening of the out-of-band pulse shape with increasing E out, which was also observed in the measured temporal profiles [Fig. 3(a)]. The saturation of the pump power in Fig. 4 also agrees with the observation of a saturation of the output energy at 1064 nm observed in Fig. 1(b); the calculated maximum peak power of 180 kW corresponds to a pulse energy of 190 μJ (for 1 ns pulses), which is comparable to the measured maximum energy at 1064 nm of ~170 μJ.

Fig. 4. Calculated pulse shapes for five values of E out: 100μJ, 152μJ, 216μJ, 267μJ and 329μJ (increasing along arrow). All parameters are similar to Fig. 3. (a) pump λ 1 = 1064 nm; (b) signal λ 3 = 1115 nm.

We performed similar experiments at a lower repetition rate, 8 kHz, and observed identical behavior for the pulse energies, spectra, and temporal profiles. We also checked the effect of 1064 nm pump depletion on the transverse beam profile of the output beam by imaging the fiber output face with and without the 1064 nm band-pass filter; both the 1064 nm and the total beam had the same nearly Gaussian transverse profile, even at the maximum E out (278 μJ at f = 26.9 kHz), where out-of-band energy represents ~50% of E out. This absence of mode distortion is consistent with the excellent beam quality, measured to be M2 < 1.15 without spectral filtering [13

13. P. E. Schrader, R. L. Farrow, D. A. V. Kliner, J.-P. Fève, and N. Landru, “High power fiber amplifier with tunable repetition rate, fixed pulse duration, and multiple output wavelengths,” Opt. Express 14, 11528–11537 (2006). [CrossRef] [PubMed]

].

Figures 1, 3, and 4 clearly illustrate that FWM can be the limiting process for energy extraction in nanosecond fiber amplifiers. As a consequence, practical devices will require detailed characterization of the output pulses. For applications in materials processing, outof- band power participates in the process similarly to the 1064 nm power, so all output energy is used, and FWM should not be highly detrimental. Alternatively, for applications demanding narrow linewidth (e.g., nonlinear frequency conversion or spectroscopy), FWM limits the useful energy. The temporal distortion of the 1064 nm pulse profile clearly has to be taken into account when designing laser systems for frequency conversion or lidar. Similarly, the distortion of the temporal profile may be detrimental for time-of-flight measurements (e.g., for ranging or altimetry).

Finally, it is important to note that the quantitative limits discussed above depend on the fiber parameters (primary A eff and length), and different results will be obtained for different operating conditions. For example, we have obtained higher peak powers and thus higher in-band pulse energies with shorter, more highly Yb-doped, fiber amplifiers.

3.3. Simulated output spectra

This section presents more detailed calculations in order to examine the critical parameters that determine the FWM spectra. Figure 5(a) shows normalized output spectra calculated from the numerical integration of Eq. (1) for different sets of wavelengths {λ 3, λ 4} and at four values of E out (corresponding to four values of the peak power). The input conditions were similar to those used in the above calculations, and the total output energy was again varied through the value of the gain coefficient g. The spectral dependence comes from two parameters, the Raman gain coefficient g R and the phase-mismatch Δk, whose respective variations with signal wavelength are shown in Fig. 5(b).

As discussed above, this calculation considers each set of signal and idler fields at a given wavelength independent from the others, which is no longer accurate when strong depletion of the pump at λ 1 takes place. Nonetheless, the calculations provide valuable insight into the FWM process in a fiber amplifier, and they reproduce the overall shape of the experimental spectra of Fig. 1, as well as the dependence of nonlinear effects on the output energy.

Fig. 5. (a). Calculated normalized peak power spectrum (t = 0) for four values of E out (L = 3.5 m, E seed = 1.5 μJ, Δt = 1 ns FWHM, Δk offset = 3 m-1). (b). Spectral dependence of the Raman gain g R (from [31]) and the phase-mismatch Δk for a pump wavelength of 1064 nm; the value of Δk offset used in the calculations is also shown.

The spectral shape arises from the relative importance of Δk and g R shown in Fig. 5(b). For signal wavelengths close to the pump wavelength of 1064 nm, the relatively small phase mismatch leads to efficient FWM, but this contribution decreases with increasing signal wavelength due to very limited Raman amplification. On the other hand, for wavelengths close to 1115 nm, large amplification from SRS leads to efficient generation of the signal field even though FWM is less efficient because of larger phase mismatch. Both the model and measurements show a broadening of the peak near the seed wavelength λ 1 when E out is increased. As was shown in Ref. [27

27. J.-P. Fève, “Phase-matching and mitigation of four-wave mixing in fibers with positive gain,” Opt. Express 15, 577–582 (2007). [CrossRef] [PubMed]

], higher gain (corresponding to higher E out in the calculations) leads to efficient FWM through gain-induced phase-matching, even for signal and idler wavelengths with a large phase mismatch. The experimental spectra of Fig. 1(a) exhibit slightly less signal near 1064 nm than near 1115 nm [in contrast to the calculations of Fig. 5(a)], which probably indicates that phase mismatch does not be completely cancel. Variations in the core dimensions, Yb concentration, birefringence, and temperature along the fiber may create an offset in Δk. For this reason, the calculations shown in Fig. 5(a) include a small offset to the phase mismatch, Δk offset (Section 2.2); as a representative example, we used Δk offset = 3 m-1, but the results are not highly sensitive to this choice. This term has negligible effect on the signal peak near 1115 nm (where Δk offset is less than 3% of Δk), but it reduces the signal amplitude for wavelengths very close to 1064 nm (Δk offset is 14% of Δk at 1084 nm). As discussed previously, the model tends to over-predict the FWM signal because we neglected the interaction between signal fields at different wavelengths. The idler contribution is also over-predicted compared to the experiments because we neglected losses in our calculations, and the Yb absorption increases at shorter wavelengths (typically α = 4 m-1 for λ 4 = 1015 nm for the fiber used in our experiments, corresponding to a 17 dB/m loss).

3.4. Optimization

Given the good agreement between the basic model and the experiments shown in Sections 3.2 and 3.3, we employed the model to propose possible tradeoffs for the mitigation of FWM in fiber amplifiers. In order to calculate the fraction of E out present in out-of-band wavelengths, we first integrated Eq. (1) with a time-dependent input power, similar to the calculations shown in Fig. 3(b); we repeated the calculation for different sets of wavelengths {λ 3, λ 4} and integrated over the whole spectrum. Figure 6 shows the relative contributions of the 1064 nm and out-of-band beams to E out. The simple model accounts very well for the observed saturation of the 1064 nm pulse energy at ~170 μJ, and it correctly predicts the threshold energy above which FWM begins to deplete the amplified 1064 nm beam.

Fig. 6. Amplified 1064 nm and out-of-band pulse energies vs. total output pulse energy. Points: experimental data from Fig. 1(b); dashed curves: calculated with parameters of Fig. 5.

This simple model allows us to propose approaches to mitigating the effects of FWM. As expected, the maximum extractable power scales with the effective mode area, so larger-core fibers are an attractive solution. When designing a practical system, however, the mode area is limited by several factors: sensitivity to bend-loss, availability of all-fiber components (e.g., fused-fiber pump combiners matched to the double-clad fiber, output delivery fiber), and ease of splicing and packaging. Furthermore, for a given mode-field diameter, many applications will desire the maximum output energy, so that an understanding of fundamental limits and mitigation of nonlinear effects will always remain of interest. Conversely, for a given target pulse energy, it is often desirable to employ the fiber with the smallest possible core diameter to minimize the pump threshold power (maximize the system efficiency) and to facilitate mode filtering [5

5. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25, 442–444 (2000). [CrossRef]

].

The simplest approach to reduce FWM is to decrease the length of the fiber. The above equations show that the maximum extractable energy is inversely proportional to L. However, minimizing L requires fabrication of fibers with higher gain per unit length, and there are limitations to the attainable rare-earth-dopant concentration and hence pump-absorption coefficient. For a given fiber, the required length is minimized by pumping at the peak of the dopant absorption spectrum; in the case of Yb, this approach places demands on the spectrum and stability of the pump source (because the 976 nm transition is relatively sharp). For a given core design, the required length is minimized by decreasing the inner-cladding area, but this approach places demands on the brightness of the pump source and the pumping method.

Based on the analytical solution of Eq. (1), we previously proposed introducing a controlled variation of the phase mismatch along the fiber as an effective solution to mitigate gain-induced phase-matching [27

27. J.-P. Fève, “Phase-matching and mitigation of four-wave mixing in fibers with positive gain,” Opt. Express 15, 577–582 (2007). [CrossRef] [PubMed]

]. We showed that a linear gradient in Δk is sufficient to reduce FWM by up to five orders of magnitude, and we proposed several practical implementations.

As an illustration of the design tradeoffs entailed in amplifier optimization, Fig. 7 presents calculations of the maximum output energy as well as the energy at 1064 nm for different commercially available fibers. Both fibers have a 30 μm core diameter, 0.06 core NA, and 250 μm inner-cladding diameter, but Fiber A has a 2.4 × higher Yb concentration than the fiber employed in the above experiments (Fiber B). For a given fiber design and diode pump power, we calculated the saturated cw output power using Liekki Application Designer software. Assuming this output power would equal the average power of a similarly pumped, high-repetition-rate, pulsed amplifier, we converted to pulse energy at a repetition rate of 27 kHz. The contribution of the 1064 nm beam was deduced from integration of Eq. (1) with respect to time (as in Section 3.2) with the parameters λ 3 = 1115 nm, E seed = 1.5 μJ, Δt = 1 ns FWHM, Δk offset = 0 m-1. Figure 7 shows the dependence of output energy on fiber length, accounting for effective absorption at the diode wavelength of 976 nm (assuming a flat spectral profile with 6 nm bandwidth), inversion of active ions, and absorption losses. The calculations indicate that significantly higher values of E out and of the in-band pulse energy should be attainable from fibers with higher Yb concentrations because the higher pump absorption allows use of shorter fibers (optimum L = 1.8 m for Fiber A).

Fig.7. Total output energy and energy at 1064 nm vs. fiber length for different fiber designs. Calculations assume 21 W of launched diode pump power and λ 3 = 1115nm. Fiber parameters (core diameter in μm/inner-cladding diameter in μm / 976 nm pump-absorption coefficient for light propagating in the core in dB/m): A: 30/250/1200; B: 30/250/500 (corresponding to the fiber employed in the present experiments).

3.5. Birefringent fibers and the effect of seed polarization

All of the above experiments used a linearly polarized seed pulse with the polarization aligned parallel to the slow axis of the PM fiber, and our calculations thus considered only interactions between waves with identical polarizations. We have also investigated the case of mixed polarizations both experimentally and theoretically. Figure 8 shows experimental measurements of E out, E 1064, the polarization extinction ratio (PER), and the output spectrum as a function of the input polarization angle (α, defined with respect to the slow axis of the fiber). For a constant diode pump power of 20.5 W launched into the fiber, corresponding to the maximum output energy of 290 μJ in Figs. 1(b) and 6, rotation of the input polarization angle did not induce significant changes in E out. E 1064 radically changed with the seed polarization, however, increasing from only 50% of E out when the seed was polarized along the slow axis to 98% of E out at α = 45°. The output spectra of Fig. 8(b) show suppression of the out-of-band peaks to below the detection limit as the input polarization was rotated toward 45°; simultaneously, the PER decreased to almost 0 dB. Thus, rotating the seed polarization from α = 0° to α = 45° leads to a non-polarized output beam with twice the in-band output energy. While this effect provides no advantage for applications that require a polarized output beam, it might be very advantageous in some cases, especially given the simplicity of this approach to mitigating FWM.

Fig. 8. (a). Measured values of E out (squares), E 1064 (circles), and the PER (stars) vs. angle of the seed polarization. (b). Measured output spectra for different orientations of the seed polarization.

To further understand and quantify the influence of polarization on amplifier performance, we measured E out and E 1064 using both PM and non-PM fibers with the same design as the above fiber (core diameter = 30 μm, core NA = 0.06, inner-cladding diameter = 250 μm; Nufern). Both fibers were pumped with 20.5 W at ~976 nm and seeded by the microlaser operating with f = 27 kHz. Table 1 lists the experimental results. Although E out was comparable in all experiments, the out-of-band fraction was much larger with the non-PM than with the PM fiber. Therefore, using a PM fiber amplifier can be advantageous even for the delivery of a non-polarized beam, and control of the seed polarization with a PM gain fiber provides an additional approach for mitigating FWM.

Table 1. Influence of seed polarization on E out and E 1064.

table-icon
View This Table

In order to explain the above observations, we analyzed the case of mixed polarizations, which is described by a system of six coupled differential equations (one for each field E i,p at frequency i = 1,3,4 with polarization p = s,f). As an example, the pump field along the slow axis obeys:

dE1,sdz=jγ[E1,s2+23E1,f2+(2ρ)(E3,s2+E4,s2)+2ρ3(E3,f2+E4,f2)]E1,sgR,2AeffE3,s2E1,s
gR,2AeffE3,f2E1,s+g2E1,s+jγ{E1,s*E3,sE4,sexp(jΔkssssz)+13E1,s*E3,fE4,fexp(jΔkssffz)+13E1,f*E3,sE4,fexp(jΔksffsz)+13E1,f*E3,sE4,fexp(jΔksfsfz)}
(2)

and the other equations were derived by similarity with Eq. (1). The perpendicular Raman gain g R,˔ was taken from [31

31. D. J. Dougherty, F. X. Kärtner, H. A. Haus, and E. P. Ippen, “Measurement of the Raman gain spectrum of optical fibers,” Opt. Lett. 20, 31–33 (1995). [CrossRef] [PubMed]

]. There are six combinations of polarizations of the interacting waves associated with a nonzero element of the χ(3) tensor, namely (s,s,s,s), (f,f,f,f), (s,s,f,f), (f,f,s,s), (s,f,s,f) and (s,f,f,s) for (λ 1,λ 1,λ 3,λ 4). The respective phase-mismatches Δk ijkl were calculated from the material dispersion and birefringence of the fiber.

Figure 9 shows the calculated output spectrum for different orientations of the seed polarization. The model reproduces the observed suppression of the out-of-band peaks as the polarization is rotated from 0° to 45°.

Fig.9. Calculated normalized peak-power spectra for various polarizations of the seed pulse. Top: output power polarized along the slow axis; center: output power polarized along the fast axis; bottom: total output power. Parameters used in the calculations: L = 3.5 m, Δk offset = 3 m-1, E seed = 1.5 μJ, E out = 175 μJ.

The reduced FWM contribution when approaching α = 45° has two causes. First, the pump peak power at 1064 nm is effectively reduced because it is split between the two axes. As a consequence, gain-induced phase-matching is less important. Second, the field along each axis is now generated through several interactions, which are associated with different phase mismatches and different coherence lengths. This effect can be seen in Fig. 9 by comparing the fast- and slow-axis spectra at wavelengths close to 1064 nm, where the Δk term dominates Raman amplification. The combination of out-of-phase rapidly oscillating functions reduces the efficiency of FWM. This effect is similar to a scheme that was implemented for the suppression of non-phase-matched quadratic interactions simultaneous with four-wave processes [33

33. J.-P. Fève and B. Boulanger, “Suppression of quadratic cascading in four-photon interactions using periodically poled media,” Phys. Rev. A 65, 063814 1–l6 (2002). [CrossRef]

]. The use of a PM fiber and tuning of the seed polarization is thus a simple way to control the phase mismatch, offering an alternative approach to mitigating FWM.

4. Conclusions

We have reported the impact of FWM on the performance of a high-peak-power, nanosecond fiber amplifier based on a double-clad, Yb-doped fiber. In particular, we have shown, for the first time to our knowledge, that FWM induces substantial distortion of both the spectral and temporal profiles of the output pulses and that FWM is the most limiting nonlinear process in the considered temporal regime, where the other nonlinear effects (SPM, SRS and SBS) are minimized. Because of FWM, the pulse energy at the seed wavelength (1064 nm) reaches a maximum beyond which all photons are converted to out-of-band wavelengths. Depending on the target application, careful characterization of the output pulses of such fiber amplifiers is thus required.

We have also reported a simple numerical model that semi-quantitatively explains the experimental observations and identifies the most important parameters for the considered regime. Our results confirm that the process of “gain-induced phase-matching” is responsible for very efficient FWM in isotropic, single-mode, LMA fibers with normal dispersion, where conventional phase-matching of FWM is not possible. When designing an amplifier system, the limitations due to FWM must be taken into account in order to optimize the design (e.g, the fiber parameters and pump source).

Conversely, the very efficient FWM offers an alternative approach for the generation of new wavelengths: Watt-level out-of-band power can be delivered from a simple and rugged system with diffraction-limited beam quality and a smooth temporal profile. This approach could, for example, be used for generation of yellow light via second-harmonic generation, which is of great interest for various applications [34

34. S. Sinha, C. Langrock, M. Digonnet, M. Fejer, and R. Byer, “Efficient yellow-light generation by frequency doubling a narrow-linewidth 1150 nm ytterbium fiber oscillator,” Opt. Lett. 31, 347–349 (2006). [CrossRef] [PubMed]

]. FWM in pulsed fiber amplifiers also offers a new route for the generation of non-classical states of light, through pure χ(3) processes, which is attractive for quantum optics studies [35

35. J.-P. Fève, B. Boulanger, and J. Douady, “Specific properties of cubic optical parametric interactions compared to quadratic interactions,” Phys. Rev. A 66, 063817 1–11 (2002). [CrossRef]

].

Acknowledgment

Supported by Laboratory Directed Research and Development, Sandia National Laboratories, U.S. Department of Energy, under contract DE-AC04-94AL85000.

References and links

1.

F. Di Teodoro and C. Brooks, “Multistage Yb-doped fiber amplifier generating megawatt peak-power, subnanosecond pulses,” Opt. Lett. 30, 3299–3301 (2005). [CrossRef]

2.

M. Y. Cheng, Y. Chang, A. Galvanauskas, P. Mamidipudi, R. Chankatoti, and P. Gatchell, “High-energy and high-peak-power nanosecond pulse generation with beam quality control in 200-μm core highly multimode Yb-doped fiber amplifiers,” Opt. Lett. 30, 358–360 (2005). [CrossRef] [PubMed]

3.

R. L. Farrow, D. A. V. Kliner, P. E. Schrader, A. A. Hoops, S. W. Moore, G. R. Hadley, and R. L. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” Proc. SPIE 6102, 61020L (2006). [CrossRef]

4.

W. Torruellas, Y. Chen, B. McIntosh, J. Faroni, K. Tankala, S. Webster, D. Hagan, and M. Soileau, “High peak power Yb doped fiber amplifiers,” Proc. SPIE 6102, 61020N (2006). [CrossRef]

5.

J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25, 442–444 (2000). [CrossRef]

6.

M. Ferman, “Single-mode excitation of multimode fibers with ultrashort pulses,” Opt. Lett. 23, 52–54 (1998). [CrossRef]

7.

J. Limpert, A. Liem, M. Reich, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “Low-nonlinearity single-transverse-mode ytterbium-doped photonic crystal fiber amplifier,” Opt. Express 12, 1313–1319 (2004). [CrossRef] [PubMed]

8.

H. Offerhaus, N. Broderick, D. Richardson, R. Sammuk, J. Caplen, and L. Dong, “High-energy single-transverse- mode Q-switched fiber laser based on a multimode large-mode-area erbium-doped fiber,” Opt. Lett. 23, 1683–1685 (1998). [CrossRef]

9.

M. Hotoleanu, M. Söderlund, D. Kliner, J. Koplow, S. Tammela, and V. Philipov, “Higher-order modes suppression in large mode area active fibers by controlling the radial distribution of the rare earth dopant,” Proc. SPIE 6102, 61021T (2006). [CrossRef]

10.

P. Wang, L. Cooper, J. Sahu, and W. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber amplifier,” Opt. Lett. 31, 226–228 (2006). [CrossRef] [PubMed]

11.

F. Di Teodoro, J. P. Koplow, S. W. Moore, and D. A. V. Kliner, “Diffraction-limited, 300-kW peak-power pulses from a coiled multimode fiber amplifier,” Opt. Lett. 27, 518–520 (2002). [CrossRef]

12.

C. Brooks and F. Di Teodoro, “1-mJ energy, 1-MW peak-power, 10 W-average power, spectrally-narrow, diffraction-limited pulses from a photonic-crystal fiber amplifier,” Opt. Express 13, 8999–9002 (2005). [CrossRef] [PubMed]

13.

P. E. Schrader, R. L. Farrow, D. A. V. Kliner, J.-P. Fève, and N. Landru, “High power fiber amplifier with tunable repetition rate, fixed pulse duration, and multiple output wavelengths,” Opt. Express 14, 11528–11537 (2006). [CrossRef] [PubMed]

14.

G. Agrawal, Nonlinear Fiber Optics, 3rd ed., Optics and Photonics Series (Academic, San Diego, Calif., (2001).

15.

C. Lin, W. Reed, A. Pearson, and H. Shang, “Phase matching in the minimum-chromatic-dispersion region of single-mode fibers for stimulated four-photon mixing,” Opt. Lett. 6, 493–495 (1981). [CrossRef] [PubMed]

16.

S. Pitois, A. Sauter, and G. Millot, “Simultaneous achievement of polarization attraction and Raman amplification in isotropic optical fibers,” Opt. Lett. 29, 599–601 (2004). [CrossRef] [PubMed]

17.

R. Stolen, M. Bösch, and C. Lin, “Phase matching in birefringent fibers,” Opt. Lett. 6, 213–215 (1981). [CrossRef] [PubMed]

18.

A. Proulx, J. M. Ménard, N. H̑, J. Daniel, R. Vallée, and C. Paré, “Intensity and polarization dependences of the supercontinuum generation in birefringent highly nonlinear fibers,” Opt. Express 11, 3338–3345 (2003). [CrossRef] [PubMed]

19.

W. Wadsworth, N. Joly, J. Knight, T. Birks, F. Biancalana, and P. Russell, “Supercontinuum and four-wave mixing with Q-switched pulses in endlessly single-mode photonic crystal fibres,” Opt. Express 12, 299–309 (2004). [CrossRef] [PubMed]

20.

J. Chee and J. Liu, “Raman-assisted parametric frequency and polarization conversion in a birefringent fiber,” Opt. Lett. 14, 820–822 (1989). [CrossRef] [PubMed]

21.

T. Sylvestre, H. Maillotte, E. Lantz, and P. Tchofo Dinda, “Raman-assisted parametric frequency conversion in a normally dispersive single-mode fiber,” Opt. Lett. 24, 1561–1563 (1999). [CrossRef]

22.

P. Tchofo Dinda, G. Millot, and S. Wabnitz, “Polarization switching and suppression of stimulated Raman scattering in birefringent optical fibers,” J. Opt. Soc. Am. B 15, 1433–1441 (1998). [CrossRef]

23.

P. Tchofo, E. Sève, G. Millot, T. Sylvestre, H. Maillote, and E. Lantz, “Raman-assisted three-wave-mixing of non-phase-matched waves in optical fibres: application to wide range frequency conversion,” Opt. Commun. 192, 107–121 (2001) [CrossRef]

24.

S. Coen, D. A. Wardle, and J. D. Harvey, “Observation of non-phase-matched parametric amplification in resonant nonlinear optics,” Phys. Rev. Lett. 89, 273901 (2002) [CrossRef]

25.

G. Cappellini and S. Trillo, “Energy conversion in degenerate four-photon mixing in birefringent fibers,” Opt. Lett. 16, 895–897 (1991). [CrossRef] [PubMed]

26.

S. Trillo, G. Millot, E. Sève, and S. Wabnitz, “Failure of phase-matching concept in large-signal parametric frequency conversion,” Appl. Phys. Lett. 72, 150–152 (1998). [CrossRef]

27.

J.-P. Fève, “Phase-matching and mitigation of four-wave mixing in fibers with positive gain,” Opt. Express 15, 577–582 (2007). [CrossRef] [PubMed]

28.

S. Trillo and S. Wabnitz, “Parametric and Raman amplification in birefringent fibers,” J. Opt. Soc. Am. B 9, 1061–1082 (1992). [CrossRef]

29.

Complete calculation of modal propagation shows waveguide contribution to phase mismatch is less than 3%, so that the single-mode approximation is justified.

30.

D. Milam, “Review and assessment of measured values of the nonlinear refractive-index coefficient of fused silica,” Appl. Opt. 37, 546 (1998). [PubMed]

31.

D. J. Dougherty, F. X. Kärtner, H. A. Haus, and E. P. Ippen, “Measurement of the Raman gain spectrum of optical fibers,” Opt. Lett. 20, 31–33 (1995). [CrossRef] [PubMed]

32.

G. R. Hadley, R. L. Farrow, and A. V. Smith, “Three-dimensional time-dependent modeling of high-power fiber amplifiers,” Proc. SPIE 6453, 64531B (2007) in press. [CrossRef]

33.

J.-P. Fève and B. Boulanger, “Suppression of quadratic cascading in four-photon interactions using periodically poled media,” Phys. Rev. A 65, 063814 1–l6 (2002). [CrossRef]

34.

S. Sinha, C. Langrock, M. Digonnet, M. Fejer, and R. Byer, “Efficient yellow-light generation by frequency doubling a narrow-linewidth 1150 nm ytterbium fiber oscillator,” Opt. Lett. 31, 347–349 (2006). [CrossRef] [PubMed]

35.

J.-P. Fève, B. Boulanger, and J. Douady, “Specific properties of cubic optical parametric interactions compared to quadratic interactions,” Phys. Rev. A 66, 063817 1–11 (2002). [CrossRef]

OCIS Codes
(060.2420) Fiber optics and optical communications : Fibers, polarization-maintaining
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(140.3280) Lasers and laser optics : Laser amplifiers
(140.3510) Lasers and laser optics : Lasers, fiber
(140.3540) Lasers and laser optics : Lasers, Q-switched
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: February 28, 2007
Manuscript Accepted: March 27, 2007
Published: April 3, 2007

Citation
Jean-Philippe Fève, Paul E. Schrader, Roger L. Farrow, and Dahv A. V. Kliner, "Four-wave mixing in nanosecond pulsed fiber amplifiers," Opt. Express 15, 4647-4662 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-4647


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References

  1. F. Di Teodoro and C. Brooks, "Multistage Yb-doped fiber amplifier generating megawatt peak-power, subnanosecond pulses," Opt. Lett. 30,3299-3301 (2005). [CrossRef]
  2. M. Y. Cheng, Y. Chang, A. Galvanauskas, P. Mamidipudi, R. Chankatoti and P. Gatchell, "High-energy and high-peak-power nanosecond pulse generation with beam quality control in 200-µm core highly multimode Yb-doped fiber amplifiers," Opt. Lett. 30,358-360 (2005). [CrossRef] [PubMed]
  3. R. L. Farrow, D. A. V. Kliner, P. E. Schrader, A. A. Hoops, S. W. Moore, G. R. Hadley and R. L. Schmitt, "High-peak-power (>1.2MW) pulsed fiber amplifier," Proc. SPIE 6102, 61020L (2006). [CrossRef]
  4. W. Torruellas, Y. Chen, B. McIntosh, J. Faroni, K. Tankala, S. Webster, D. Hagan and M. Soileau, "High peak power Yb doped fiber amplifiers," Proc. SPIE 6102, 61020N (2006). [CrossRef]
  5. J. P. Koplow, D. A. V. Kliner and L. Goldberg, "Single-mode operation of a coiled multimode fiber amplifier," Opt. Lett. 25, 442-444 (2000). [CrossRef]
  6. M. Ferman, "Single-mode excitation of multimode fibers with ultrashort pulses," Opt. Lett. 23, 52-54 (1998). [CrossRef]
  7. J. Limpert, A. Liem, M. Reich, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson and C. Jakobsen, "Low-nonlinearity single-transverse-mode ytterbium-doped photonic crystal fiber amplifier," Opt. Express 12, 1313-1319 (2004). [CrossRef] [PubMed]
  8. H. Offerhaus, N. Broderick, D. Richardson, R. Sammuk, J. Caplen and L. Dong, "High-energy single-transverse-mode Q-switched fiber laser based on a multimode large-mode-area erbium-doped fiber," Opt. Lett. 23, 1683-1685 (1998). [CrossRef]
  9. M. Hotoleanu, M. Söderlund, D. Kliner, J. Koplow, S. Tammela and V. Philipov, "Higher-order modes suppression in large mode area active fibers by controlling the radial distribution of the rare earth dopant," Proc. SPIE 6102, 61021T (2006). [CrossRef]
  10. P. Wang, L. Cooper, J. Sahu and W. Clarkson, "Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber amplifier," Opt. Lett. 31, 226-228 (2006). [CrossRef] [PubMed]
  11. F. Di Teodoro, J. P. Koplow, S. W. Moore and D. A. V. Kliner, "Diffraction-limited, 300-kW peak-power pulses from a coiled multimode fiber amplifier," Opt. Lett. 27, 518-520 (2002). [CrossRef]
  12. C. Brooks and F. Di Teodoro, "1-mJ energy, 1-MW peak-power, 10 W-average power, spectrally-narrow, diffraction-limited pulses from a photonic-crystal fiber amplifier," Opt. Express 13, 8999-9002 (2005). [CrossRef] [PubMed]
  13. P. E. Schrader, R. L. Farrow, D. A. V. Kliner, J.-P. Fève and N. Landru, "High power fiber amplifier with tunable repetition rate, fixed pulse duration, and multiple output wavelengths," Opt. Express 14, 11528-11537 (2006). [CrossRef] [PubMed]
  14. G. Agrawal, Nonlinear Fiber Optics, 3rd ed., Optics and Photonics Series (Academic, San Diego, Calif., 2001).
  15. C. Lin, W. Reed, A. Pearson and H. Shang, "Phase matching in the minimum-chromatic-dispersion region of single-mode fibers for stimulated four-photon mixing," Opt. Lett. 6, 493-495 (1981). [CrossRef] [PubMed]
  16. S. Pitois, A. Sauter and G. Millot, "Simultaneous achievement of polarization attraction and Raman amplification in isotropic optical fibers," Opt. Lett. 29, 599-601 (2004). [CrossRef] [PubMed]
  17. R. Stolen, M. Bösch and C. Lin, "Phase matching in birefringent fibers," Opt. Lett. 6, 213-215 (1981). [CrossRef] [PubMed]
  18. A. Proulx, J. M. Ménard, N. Hô, J. Daniel, R. Vallée and C. Paré, "Intensity and polarization dependences of the supercontinuum generation in birefringent highly nonlinear fibers," Opt. Express 11, 3338-3345 (2003). [CrossRef] [PubMed]
  19. W. Wadsworth, N. Joly, J. Knight, T. Birks, F. Biancalana and P. Russell, "Supercontinuum and four-wave mixing with Q-switched pulses in endlessly single-mode photonic crystal fibres," Opt. Express 12, 299-309 (2004). [CrossRef] [PubMed]
  20. J. Chee and J. Liu, "Raman-assisted parametric frequency and polarization conversion in a birefringent fiber," Opt. Lett. 14, 820-822 (1989). [CrossRef] [PubMed]
  21. T. Sylvestre, H. Maillotte, E. Lantz and P. Tchofo Dinda, "Raman-assisted parametric frequency conversion in a normally dispersive single-mode fiber," Opt. Lett. 24, 1561-1563 (1999). [CrossRef]
  22. P. Tchofo Dinda, G. Millot and S. Wabnitz, "Polarization switching and suppression of stimulated Raman scattering in birefringent optical fibers," J. Opt. Soc. Am. B 15, 1433-1441 (1998). [CrossRef]
  23. P. Tchofo, E. Sève, G. Millot, T. Sylvestre, H. Maillote and E. Lantz, "Raman-assisted three-wave-mixing of non-phase-matched waves in optical fibres: application to wide range frequency conversion," Opt. Commun. 192, 107-121 (2001) [CrossRef]
  24. S. Coen, D. A. Wardle and J. D. Harvey, "Observation of non-phase-matched parametric amplification in resonant nonlinear optics," Phys. Rev. Lett. 89, 273901 (2002) [CrossRef]
  25. G. Cappellini and S. Trillo, "Energy conversion in degenerate four-photon mixing in birefringent fibers," Opt. Lett. 16, 895-897 (1991). [CrossRef] [PubMed]
  26. S. Trillo, G. Millot, E. Sève and S. Wabnitz, "Failure of phase-matching concept in large-signal parametric frequency conversion," Appl. Phys. Lett. 72, 150-152 (1998). [CrossRef]
  27. J.-P. Fève, "Phase-matching and mitigation of four-wave mixing in fibers with positive gain," Opt. Express 15, 577-582 (2007). [CrossRef] [PubMed]
  28. S. Trillo and S. Wabnitz, "Parametric and Raman amplification in birefringent fibers," J. Opt. Soc. Am. B 9, 1061-1082 (1992). [CrossRef]
  29. Complete calculation of modal propagation shows waveguide contribution to phase mismatch is less than 3%, so that the single-mode approximation is justified.
  30. D. Milam, "Review and assessment of measured values of the nonlinear refractive-index coefficient of fused silica," Appl. Opt. 37, 546 (1998). [PubMed]
  31. D. J. Dougherty, F. X. Kärtner, H. A. Haus, and E. P. Ippen, "Measurement of the Raman gain spectrum of optical fibers," Opt. Lett. 20, 31-33 (1995). [CrossRef] [PubMed]
  32. G. R. Hadley, R. L. Farrow, and A. V. Smith, "Three-dimensional time-dependent modeling of high-power fiber amplifiers," Proc. SPIE 6453, 64531B (2007) in press. [CrossRef]
  33. J.-P. Fève and B. Boulanger, "Suppression of quadratic cascading in four-photon interactions using periodically poled media," Phys. Rev. A  65, 063814 1-6 (2002). [CrossRef]
  34. S. Sinha, C. Langrock, M. Digonnet, M. Fejer and R. Byer, "Efficient yellow-light generation by frequency doubling a narrow-linewidth 1150 nm ytterbium fiber oscillator," Opt. Lett. 31, 347-349 (2006). [CrossRef] [PubMed]
  35. J.-P. Fève, B. Boulanger and J. Douady, "Specific properties of cubic optical parametric interactions compared to quadratic interactions," Phys. Rev. A 66, 063817 1-11 (2002). [CrossRef]

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