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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 2 — Jan. 22, 2007
  • pp: 577–582
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Phase-matching and mitigation of four-wave mixing in fibers with positive gain

Jean-Philippe Fève  »View Author Affiliations


Optics Express, Vol. 15, Issue 2, pp. 577-582 (2007)
http://dx.doi.org/10.1364/OE.15.000577


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Abstract

We present a theoretical study of four-wave mixing interactions in fibers in the presence of gain. In contrast to passive fibers, positive gain at the pump wavelength leads to constructive generation of the signal and idler waves, even in the case of large phase-mismatch, so that FWM processes can be very efficient even in isotropic single-mode fibers with normal dispersion. We also propose simple ways to mitigate these parametric interactions by applying a controlled variation of the phase-mismatch along the fiber. These concepts apply to all optical amplifiers.

© 2007 Optical Society of America

1. Introduction

Pulsed fiber amplifiers have gained considerable interest because they are able to deliver nanosecond pulses with very high average and peak powers and diffraction-limited beam quality in highly reliable devices, which are major advantages for practical applications such as material processing and LIDAR. Recent advances in double-clad, Yb-doped, large-mode-area (LMA) fibers has led to a record combination of average and peak output powers at 1064nm [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]

, 2

2. R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley, and R. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,138–148 (2006).

]. Different techniques have been implemented to ensure single-transverse-mode output while increasing the effective mode area of the fiber [3–6

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

]. These approaches decrease the peak irradiance and the fiber length, thereby raising the threshold for nonlinear process and increasing the attainable output power and pulse energy.

In this paper, we present what is to our knowledge the first study of the phase-matching properties of FWM in fibers in the presence of gain. We point out the existence of a unique and previously unrecognized phenomenon, “gain-induced phase-matching”, which allows efficient FWM even for an interaction that is nominally highly phase-mismatched in isotropic single-mode fibers with normal dispersion. We show that FWM can become the limiting factor in the nanosecond temporal regime. The solution of the wave equation allows us to propose simple approaches to efficiently mitigating this detrimental nonlinear process by applying a controlled longitudinal variation of the phase-mismatch along the fiber.

2. FWM in nanosecond amplifiers

2.1. Nonlinear equations

{E1z=jγ[E12+(2ρ)(E32+E42)]E1gR2AeffE32E1+g2E1+jγE1*E3E4ejΔkzE3z=jγ[E32+(2ρ)(E12+E42)]E3+gR2AeffE12E3+jγE4*E1E1ejΔkzE4z=jγ[E42+(2ρ)(E12+E32)]E4+jγE3*E1E1ejΔkz
(1)

Assuming non-depletion of the pump field during FWM (E 3, 4«E 1z), the first equation is solved to yield E1(z)=P0exp(gz/2)exp(jγP0egz/g) with P 0 =|E 1(0)|2the seed peak power. Equations for E 3,4 are transformed by setting B 3 = E 3 exp(- j2γP 0 egz/g). The net phase-mismatch and the signal field obey:

κ(z)=Δkz+2γP0gegz
(2)
d2B3dz2+[j(Δk+2γP0egz)g]dB3dzγ2P02e2gzB3=0
(3)

In the limit of passive fibers, g=0, the above relations are similar to equations of Ref. [7

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

]. The signal field is efficiently generated along the fiber only if 0>Δk>-4γP 0 ; in the case of normal dispersion, the net phase-mismatch exhibits a linear variation along the propagation, and FWM is not efficient whatever the value of the pump field P 0.

A major consequence of positive gain g is that the coherence length of FWM varies along the fiber (L c(z)≈2π/[Δk+2γP 0 egz]). In order to derive analytical solutions for E 3,4(z), we note that the variation of the pump field along the fiber is much slower than the oscillation of the fields due to non-phase-matched FWM (i.e. since g«δk+2γP 0 egz, for typical values of the parameters the characteristic length for amplification L g~l/g~lm is much longer than the coherence length L c~lcm), so we approximate P 0 egz to a constant. This approximation is validated a posteriori by the agreement between analytical relations with the exact numerical solutions of Eq. (1). The generated signal power deduced from Eq. (3) is then of the form:

P3(z)=[Aexp(u+z)+Bexp(uz)]2
(4)

2.2. Asymptotic solutions, analysis of the generated fields

In the limits of large and small mismatches:

  • Δk»2γP 0 egz. In this case, g«Δk+2γP 0 egzz, u±jΔk/2±[γP0exp(gz)]2(Δk/2)2 u± is purely imaginary, and in the limit of very large mismatch Eq. (4) becomes:

    P3(z)sin2(Δkz2)
    (5)

    which is the usual evolution of the signal in the case of non-phase-matched interaction in media without gain. The signal power oscillates (the period is constant in that case, L c~2π/δk) and is not efficiently constructed along the propagation.

  • Δk«2γP 0 egz. In this case, u±g/2jγP0egz±(γP0egz)2+(j2γP0egzg)2/4 ; for typical parameters, g«2γP 0 egz for all z, and Eq. (4) is approximated by:

    P3(z)exp(gz)
    (6)

This case is unique to the fiber with positive gain: the net mismatch (2) and the coherence length (L c(z)~π/(γP 0 egz) in this limit) vary nonlinearly with distance. Thus, the power flow from the pump to the signal and idler accumulated over the first part of one coherence length is no longer equal to the power flowing in the reverse way in the second part. As a consequence, the signal field never goes back to its initial value, but is progressively constructed as it propagates along the fiber, even with non-zero phase-mismatch in single-mode isotropic fibers with normal dispersion. It should also be noted that Raman amplification (not taken into account in the above expression) will enhance this effect for signal wavelengths close to the Stokes peak.

The above analytical expressions provide useful understanding of the important parameters; however they were derived through important assumptions and only apply in limit cases. In order to illustrate more accurately the effect of “gain-induced phase-matching,” Fig. 1 shows the evolution of the signal power at λ3=1094nm along the fiber, calculated from the numerical integration of Eq. (1) with the above parameters (input signal and idler are one photon each, Δk=41.5 m-1). It demonstrates the critical impact of gain on the generation of the signal: the field at λ3 is progressively constructed along the fiber, and the signal power reaches the same order of magnitude as the amplified pump power (dashed line) after a few meters. From this exact calculation, the variation of P 3(z) is more complicated than analytical Eq. (4)–Eq. (6). Close to the fiber input, Δk>2γP 0 egz, so the curve has a pronounced oscillatory behaviour as in Eq. (5). The end of the fiber corresponds to the limit Δk<2γP 0 egz so that the behaviour is close to an exponential shape, as expected from Eq.(6). Maximum amplification of the pump (12.2dB) is reached at z=5.65 m. Beyond that position, the undepleted-pump approximation does not hold. For comparison, the case of the same fiber with no gain (g=0, g R=0) is also shown on the same graph; in that case, the signal field exhibits the usual sinusoidal oscillation, as expected from Eq. (5). Figure 1(b) shows more clearly the departure of the signal power from purely oscillatory behaviour: constructive generation of signal occurs from the entrance of the fiber as soon as the gain is non-zero, even with weak pump input. In contrast, in a passive fiber (g=0) the signal field at λ3 keeps an oscillatory behavior even if the input pump power P 0 is 10 times larger. After 3m in the active fiber (g=0.5m-1), the signal is 6.3 times larger while the pump at λ1 is lower (58% compared to the passive fiber with stronger input). From these curves, we conclude that gain is required to achieve “phase-matching” in the case of normal dispersion, while large pump power alone is not sufficient [7

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

].

Fig. 1. Peak powers along propagation in the fiber: (a) Fixed seed energy; (b) Zoom towards fiber entrance. Solid curves are signal (λ3=1094nm), green dashed curve is pump (λ1=1064nm). Different values of seed energy, gain and Raman coefficient: red {E in=5μJ, g=0.5m-1, g R=5.10-14m2/W} ; magenta {E in=5μJ, g=0m-1, g R=0m2/W} ; blue {E in=5μJ, g=0m-1, g R=5.10-14m2/W} ; black {E in=50μJ, g=0m-1, g R=0m2/W}.

Calculations at other signal wavelengths show similar qualitative behavior. For input signal and idler fixed at one noise photon each, the spectral dependence of FWM arises from the relative importance of Raman gain g R and the phase-mismatch Δk; the signal generated along the fiber varies as in Fig. 1, but depletion occurs at different fiber length or pulse energy (peak power). A detailed study of these effects is beyond the scope of this paper; experimental data and calculated spectra are reported in Ref. [10

10. J. P. Féve, P. Schrader, R. Farrow, and D. Kliner, “Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers,” in Fiber Lasers IV: Technology, Systems and Applications, Proc. SPIE6453 (2007).

].

From the above analysis, a characteristic length for “gain-induced phase-matching” is:

LPM=1gln(Δk2γP0)=1gln(Δkλ1Aeff4πn2P0)
(7)

If the fiber length (L) is «L PM, gain has no noticeable impact on phase-matching. In contrast, if LL PM gain has a major effect on FWM. Using the parameters listed above and Δk=100 m-1 , (λ1=1064 nm and λ3~1100 nm) then L PM=6.9 m and the corresponding gain at λ1 is only 15dB. This example shows that FWM cannot be neglected in typical fiber amplifiers.

3. Mitigation of “gain-induced phase-matching” of FWM

From Eq. (7), “gain-induced phase-matching” remains negligible only if the output amplified power obeys P out«Δkλ1 A eff/4πn 2, which limits the power that can be extracted from the amplifier without distortions by the FWM process. As expected, this maximum power increases with the effective mode area (and hence the fiber core diameter) and with decreasing nonlinearity. Less intuitive, the maximum power scales with the phase-mismatch. Careful design of the fiber effective area and dispersion might then help mitigating FWM. However, current fiber-fabrication processes, practical handling considerations and non-availability of fused-fiber pump combiners with matched core diameters set limits for implementation in reliable all-fiber systems.

One approach to mitigating “gain-induced phase-matching” in standard LMA fibers is to introduce a longitudinal variation in the phase-mismatch, in the form Δk eff(z)=Δk+f(z). An analytical solution of Eq. (1) is obtained in the form of Eq. (4) in the limit of slowly varying f(z). If f(z) verifies Δk + f(z) » 2γP 0egz ∀ z ∊ [0,L], the signal power is given by a relation similar to Eq. (5), and is then not efficiently generated.

In order to validate this concept, Fig. 2 shows the signal power deduced from numerical integration of Eq. (1) in fiber with controlled longitudinal variation of the phase-mismatch and different functions f(z).

This approach efficiently reduces the generated signal: in particular, the simplest case of a linear gradient of the phase-mismatch decreases the generated signal by up to 5 orders of magnitude, which has a major impact on the ability to extract more amplified power from the fiber. No depletion of the pump is observed for the considered fiber length, and the maximum amplification at λ1 is 14.8dB at z=6.85 m; in this case, the limiting nonlinear process is no longer FWM but SRS, because the same maximum is found with n 2=0. Linear variation of Δk is thus sufficient to effectively eliminate the impact of FWM.

Fig. 2. Signal power along the fiber with longitudinally controlled phase-mismatch. Red: constant Δk=41.5m-1. Magenta: linear increase of Δk, slope a= 4pm-1. Blue: exponential increase of δk , a=b=1.4m-1. Inset: zoom towards fiber exit.

4. Conclusion

We demonstrate, for the first time to our knowledge, the very unique behaviour of four-wave mixing in fiber amplifiers: positive gain at the pump wavelength leads to constructive generation of the signal and idler waves, even in the case of large phase-mismatch. We conclude that FWM processes can be very efficient even in isotropic single-mode fibers with normal dispersion. Practical examples show that this effect can be the most limiting one in pulsed fiber amplifiers operating in the nanosecond range. A detailed experimental study of these limitations will be reported in a separate paper.

Based on our theoretical analysis, we also propose simple ways to efficiently mitigate FWM by applying a controlled variation of the phase-mismatch along the fiber.

Finally, it is important to note that the equations are not specific to fibers, so that these concepts are generic to all optical amplifiers, including waveguides or bulk, or to fiber lasers.

Acknowledgments

The author thanks D. Kliner, R. Farrow and P. Schrader (Sandia National Laboratories, Livermore) and N. Landru (Teem Photonics) for stimulating discussions.

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.

R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley, and R. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,138–148 (2006).

3.

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

4.

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]

5.

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,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,425–432 (2006).

6.

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]

7.

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

8.

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]

9.

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

10.

J. P. Féve, P. Schrader, R. Farrow, and D. Kliner, “Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers,” in Fiber Lasers IV: Technology, Systems and Applications, Proc. SPIE6453 (2007).

11.

S. Murdoch, M. Thomson, R. Leonhardt, and J. Harvey, “Quasi-phase-matched modulation instability in birefringent fibers,” Opt. Lett. 22,682 (1997) [CrossRef] [PubMed]

OCIS Codes
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(140.3510) Lasers and laser optics : Lasers, fiber
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

ToC Category:
Nonlinear Optics

History
Original Manuscript: November 6, 2006
Revised Manuscript: January 9, 2007
Manuscript Accepted: January 9, 2007
Published: January 22, 2007

Citation
Jean-Philippe Fève, "Phase-matching and mitigation of four-wave mixing in fibers with positive gain," Opt. Express 15, 577-582 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-2-577


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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. R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley and R. Schmitt, "High-peak-power (>1.2MW) pulsed fiber amplifier," Proc. SPIE 6102, 138 -148 (2006).
  3. J. Koplow, D. Kliner and L. Goldberg, "Single-mode operation of a coiled multimode fiber amplifier," Opt. Lett. 25, 442-444 (2000). [CrossRef]
  4. 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]
  5. 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, 425-432 (2006).
  6. 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]
  7. G. Agrawal, Nonlinear Fiber Optics, 3rd ed., Optics and Photonics Series (Academic, San Diego, Calif., 2001).
  8. 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]
  9. Complete calculation of modal propagation shows waveguide contribution to phase-mismatch is less than 3%, so that the single-mode approximation is justified (R. Farrow, personal communication).
  10. J. P. Fève, P. Schrader, R. Farrow and D. Kliner, "Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers," Proc. SPIE 6453, (2007).
  11. S. Murdoch, M. Thomson, R. Leonhardt and J. Harvey, "Quasi-phase-matched modulation instability in birefringent fibers," Opt. Lett. 22, 682 (1997) [CrossRef] [PubMed]

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