Abstract

We have developed a statistical nonlinear model to explain an anomalous intensity saturation observed in the amplification of intense broadband incoherent pulses on neodynium-doped glass power chains. The physics behind this model is basically self-phase modulation creating new wavelengths scattered in the tail of the gain profile. The theory shows qualitative agreement with the experimental results.

© 1997 Optical Society of America

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References

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  1. O. Kinrot, I. S. Averbukh, and Y. Prior, “Measuring coherence while observing noise,” Phys. Rev. Lett. 75, 3822–3825 (1995).
    [CrossRef] [PubMed]
  2. D. De Beer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
    [CrossRef]
  3. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [CrossRef] [PubMed]
  4. R. H. Lehmberg and S. P. Obenschain, “Use of induced spatial incoherency for uniform illumination,” Opt. Commun. 46, 27–31 (1983).
    [CrossRef]
  5. A. M. Weiner, J. P. Heritage, and J. A. Saleti, “Encoding and decoding of femtosecond pulses,” Opt. Lett. 13, 300–302 (1988).
    [CrossRef] [PubMed]
  6. D. Véron, H. Ayral, C. Gouédard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, “Improved laser-beam uniformity using the angular dispersion of frequency-modulatedlight,” Opt. Commun. 65, 42–45 (1988); D. Véron, G. Thiell, and C. Gouédard, “Optical smoothing of the high power Phebus Nd-glass laser using themultimode optical fiber technique,” Opt. Commun. 97, 259–271 (1993).
    [CrossRef]
  7. D. Kopf, F. X. Kartner, K. J. Weingarten, and U. Keller, “Pulse shortening in a Nd:glass laser by gain reshaping and solitonformation,” Opt. Lett. 19, 2146–2148 (1994).
    [CrossRef] [PubMed]
  8. A. E. Siegman, Lasers (University ScienceBooks, Mill Valley, Calif., 1986).
  9. J. Garnier and J. P. Fouque, “Amplification of incoherentlight,” in Third International Conference on Mathematicaland Numerical Aspects of Wave Propagation, G. Cohen, ed. (Society forIndustrial and Applied Mathematics, Philadelphia, 1995), pp. 584–593.
  10. J. T. Manassah, “Self-phase modulation of incoherent light revisited,” Opt. Lett. 16, 1438–1441 (1991).
    [CrossRef]
  11. P. Donnat, C. Gouédard, D. Véron, O. Bonville, C. Sauteret, and A. Migus, “Induced spatial incoherence and nonlinear effects in Nd-glass amplifiers,” Opt. Lett. 17, 331–333 (1992).
    [CrossRef] [PubMed]
  12. D. Middleton, Introduction to Statistical CommunicationTheory (McGraw-Hill, New York, 1960), p. 141.
  13. P. Donnat, C. Treimany, and O. Morice, “MiròV2.0, guide utilisateur et manuel de référence,” noteCEA 2818 (Commissariat a l'Energie Atomique, Limeil-Valenton, France, 1997).
  14. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulationof waves in nonlinear medium,” Sov. Phys. JETP 34, 62–69 (1972).
  15. S. A. Gredeskul and Y. U. Kivshar, “Propagation and scattering of nonlinear waves in disordered systems,” Phys. Rep. 216, 1–61 (1992).
    [CrossRef]

1995 (1)

O. Kinrot, I. S. Averbukh, and Y. Prior, “Measuring coherence while observing noise,” Phys. Rev. Lett. 75, 3822–3825 (1995).
[CrossRef] [PubMed]

1994 (1)

1992 (2)

1991 (2)

J. T. Manassah, “Self-phase modulation of incoherent light revisited,” Opt. Lett. 16, 1438–1441 (1991).
[CrossRef]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

1988 (1)

1986 (1)

D. De Beer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef]

1983 (1)

R. H. Lehmberg and S. P. Obenschain, “Use of induced spatial incoherency for uniform illumination,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulationof waves in nonlinear medium,” Sov. Phys. JETP 34, 62–69 (1972).

Averbukh, I. S.

O. Kinrot, I. S. Averbukh, and Y. Prior, “Measuring coherence while observing noise,” Phys. Rev. Lett. 75, 3822–3825 (1995).
[CrossRef] [PubMed]

Beach, R.

D. De Beer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef]

Bonville, O.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

De Beer, D.

D. De Beer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef]

Donnat, P.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Gouédard, C.

Gredeskul, S. A.

S. A. Gredeskul and Y. U. Kivshar, “Propagation and scattering of nonlinear waves in disordered systems,” Phys. Rep. 216, 1–61 (1992).
[CrossRef]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hartmann, S. R.

D. De Beer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Heritage, J. P.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Kartner, F. X.

Keller, U.

Kinrot, O.

O. Kinrot, I. S. Averbukh, and Y. Prior, “Measuring coherence while observing noise,” Phys. Rev. Lett. 75, 3822–3825 (1995).
[CrossRef] [PubMed]

Kivshar, Y. U.

S. A. Gredeskul and Y. U. Kivshar, “Propagation and scattering of nonlinear waves in disordered systems,” Phys. Rep. 216, 1–61 (1992).
[CrossRef]

Kopf, D.

Lehmberg, R. H.

R. H. Lehmberg and S. P. Obenschain, “Use of induced spatial incoherency for uniform illumination,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Manassah, J. T.

J. T. Manassah, “Self-phase modulation of incoherent light revisited,” Opt. Lett. 16, 1438–1441 (1991).
[CrossRef]

Migus, A.

Obenschain, S. P.

R. H. Lehmberg and S. P. Obenschain, “Use of induced spatial incoherency for uniform illumination,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

Prior, Y.

O. Kinrot, I. S. Averbukh, and Y. Prior, “Measuring coherence while observing noise,” Phys. Rev. Lett. 75, 3822–3825 (1995).
[CrossRef] [PubMed]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Saleti, J. A.

Sauteret, C.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulationof waves in nonlinear medium,” Sov. Phys. JETP 34, 62–69 (1972).

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Van Wagenen, L. G.

D. De Beer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef]

Véron, D.

Weiner, A. M.

Weingarten, K. J.

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulationof waves in nonlinear medium,” Sov. Phys. JETP 34, 62–69 (1972).

Opt. Commun. (1)

R. H. Lehmberg and S. P. Obenschain, “Use of induced spatial incoherency for uniform illumination,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

Opt. Lett. (4)

Phys. Rep. (1)

S. A. Gredeskul and Y. U. Kivshar, “Propagation and scattering of nonlinear waves in disordered systems,” Phys. Rep. 216, 1–61 (1992).
[CrossRef]

Phys. Rev. Lett. (2)

O. Kinrot, I. S. Averbukh, and Y. Prior, “Measuring coherence while observing noise,” Phys. Rev. Lett. 75, 3822–3825 (1995).
[CrossRef] [PubMed]

D. De Beer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulationof waves in nonlinear medium,” Sov. Phys. JETP 34, 62–69 (1972).

Other (5)

A. E. Siegman, Lasers (University ScienceBooks, Mill Valley, Calif., 1986).

J. Garnier and J. P. Fouque, “Amplification of incoherentlight,” in Third International Conference on Mathematicaland Numerical Aspects of Wave Propagation, G. Cohen, ed. (Society forIndustrial and Applied Mathematics, Philadelphia, 1995), pp. 584–593.

D. Véron, H. Ayral, C. Gouédard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, “Improved laser-beam uniformity using the angular dispersion of frequency-modulatedlight,” Opt. Commun. 65, 42–45 (1988); D. Véron, G. Thiell, and C. Gouédard, “Optical smoothing of the high power Phebus Nd-glass laser using themultimode optical fiber technique,” Opt. Commun. 97, 259–271 (1993).
[CrossRef]

D. Middleton, Introduction to Statistical CommunicationTheory (McGraw-Hill, New York, 1960), p. 141.

P. Donnat, C. Treimany, and O. Morice, “MiròV2.0, guide utilisateur et manuel de référence,” noteCEA 2818 (Commissariat a l'Energie Atomique, Limeil-Valenton, France, 1997).

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Figures (8)

Fig. 1
Fig. 1

Theoretical correlation function in the limit T2=0 as a function of the reduced time T expressed in units of Tc for different values of the B integral. For each B value we have normalized the maximum of the function to 1. The dashed, solid, dotted–dashed, and double-dotted–dashed curves plot cases B=0, 1, 2, 3, respectively.

Fig. 2
Fig. 2

Experimental energy output as a function of midchain energy input for a monochromatic pulse and for an incoherent pulse of bandwidth (FWHM) 1.2 nm.

Fig. 3
Fig. 3

Experimental chain output spectra. The preamplifier output spectrum (dashed curve) has a FWHM of 1.2 nm. The chain output spectrum (solid curve) has a FWHM of 2.4 nm and corresponds to an output energy of 1.1 kJ.

Fig. 4
Fig. 4

Experimental spectral broadening as a function of energy output with an incoherent pulse of bandwidth (FWHM) 1.2 nm.

Fig. 5
Fig. 5

Theoretical amplification efficiency, defined as the ratio r of the average output intensity given by relation (7) over the expected output intensity I0eg. The efficiency is expressed as a percentage function of the B integral. The small signal-gain is taken to be g=3.5 and the ratio is δ=0.09.

Fig. 6
Fig. 6

Theoretical chain output spectrum versus the preamplifier spectrum (dashed curve) for B=1.4 (solid curve). The small-signal gain is taken to be g=3.5, and the ratio is δ =0.09.

Fig. 7
Fig. 7

Variation of the spectral broadening Δλoutput/Δλinput with the B integral. We compare the approximation that is valid for a coherent Gaussian pulse (dashed curve) with the theoretical chain output spectral broadening for incoherent light for δ=0 (dotted–dashed curve), for g=3.5 and δ=0.09 (solid curve), and for g=6 and δ=0.09 (double-dotted–dashed curve).

Fig. 8
Fig. 8

Experimental B integral as a function of energy output with an incoherent pulse of bandwidth (FWHM) 1.2 nm. The values of the B integrals are calculated from the experimental values of the spectral broadenings given in Fig. 4 by the theoretical formula that connects the spectral broadening Δλoutput/ Δλinput with the B integral and that is plotted in Fig. 7.

Equations (29)

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E0(t)E0*(t+T)=I0 exp(-T2/2Tc2),
i Ez+σ 2Et2+k0n22n0|E|2E=12P,
T2 Pt+P=iγE,
σ=ω(k0)2ω(k0)3,
σγTc21.
C(T, B)|T2=0=egI0 exp(-T2/2Tc2){1+B2[1-exp(-T2/Tc2)]}2,
I(g, B)egI0[1-δ2g-2δ2B2f2(g)+O(δ3)].
C(T, g, B)egI0 exp(-T2/2Tc2){1+B2[1-exp(-T2/Tc2)]}2×[1+δ2h1(g, T/Tc)+2δ2B2h2(g, T/Tc)+O(δ3)].
i Ez+σ 2Et2+k0n22n0|E|2E=iγ2E.
c2=|E|4-|E|22|E|22,
Et2=-Re 2CT2T=0.
0<σγTc21,
Et2I0egTc21+4B21+44BσγTc2.
H:=σEt2-k0n24n0|E|4,
Hz=-γk0n24n0|E|4+γH.
|E|4|E0|4e2g1+16BσγTc2+528B2σ2γ2Tc4.
c21+32BσγTc2+1056B2σ2γ2Tc4,
I(g, B)egI01-δ2g-2δ2B21+88Bσ3γTc2+O(δ3).
a(τ, z)=a˜0(τ, z)+δa˜1(τ, z)+δ2a˜2(τ, z)+,
ϕ(τ, z)=ϕ˜0(τ, z)+δϕ˜1(τ, z)+δ2ϕ˜2(τ, z)+,
i Ez+σTc22Eτ2+k0n22n0|E|2E=P2,
δ Pτ+P=iγE.
I(g)ega02-δ2(B/I0)2f2(g)ega04a02-δ2geg(a02+a02ϕ02)+O(δ3),
A(g, I0)=f3(g)a04a04+f4(g)a02a06,
A¯(g)=-1763+176ge-g-352ge-2g+528e-2g-24g2e-3g-176ge-3g-14083e-3g,
ΔC(τ, g)δ2egg2k0n2(eg-1)2n0γ2a02a0(0)a02a0(τ)×exp{i[ϕ0(0)+ϕNL(0)-ϕ0(τ)-ϕNL(τ)]}.
ΔC(τ, g)2δ2egg2B2I0×exp(-τ2/2)[1-τ2-exp(-τ2)]{1+B2[1-exp(-τ2)]}2+negligibleterms,
δ2 I0 exp(-τ2/2)h(g, B, τ){1+B2[1-exp(-τ2)]}j,
I(g, B)egI01-δ2g-2δ2B2f2(g)+l=3kfl(g, B)δl+O(δk+1),

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