Abstract

Prospects for the development of high-order parametric amplifiers as an alternative to the existing inversion-based soft-x-ray lasers are analyzed. We show how the problem of phase matching can be solved. The possibility of gain transfer from a low-order parametric process to the high-order process is discussed. Initial encouraging experimental results toward the demonstration of high-order parametric amplifiers are reported.

© 1999 Optical Society of America

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  1. S. Meyer, H. Eichmann, T. Menzel, S. Nolte, B. Wellegehausen, B. N. Chichkov, and C. Momma, “Phase-matched high-order difference-frequency mixing in plasmas,” Phys. Rev. Lett. 76, 3336–3339 (1996).
    [CrossRef] [PubMed]
  2. A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280, 1412–1415 (1998).
    [CrossRef] [PubMed]
  3. Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Kapteyn, “Generation of coherent soft x rays at 2.7 nm using high harmonics,” Phys. Rev. Lett. 79, 2967–2970 (1997).
    [CrossRef]
  4. Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent x-rays in the water window using 5-femtosecond laser pulses,” Science 278, 661–664 (1997).
    [CrossRef]
  5. See the rest of this feature and previous features on optical parametric processes and devices: J. Opt. Soc. Am. B 10, 1657–1791 (1993); 10, 2147–2243 (1993); 11, 2083–2322 (1995).
  6. A. Tünnermann, K. Mossavi, and B. Wellegehausen, “Nonlinear-optical processes in the near-resonant two-photon excitation of xenon by femtosecond KrF-excimer-laser pulses,” Phys. Rev. A 46, 2707–2717 (1992).
    [CrossRef] [PubMed]
  7. S. Babin, U. Hinze, E. Tiemann, and B. Wellegehausen, “Continuous resonant four-wave mixing in double-Λ level configuration of Na2,” Opt. Lett. 21, 1186–1188 (1996); A. Apolonskii, S. Baluschev, U. Hinze, and B. Wellegehausen, “Continuous frequency upconversion in a double-Λ scheme of Na2,” Appl. Phys. B 64, 435–442 (1997).
    [CrossRef] [PubMed]
  8. P. L. Shkolnikov, A. E. Kaplan, and A. Lago, “Phase matching for large-scale frequency upconversion in plasma,” Opt. Lett. 18, 1700–1702 (1993).
    [CrossRef] [PubMed]
  9. S. Nolte, C. Momma, H. Jacobs, A. Tünnermann, B. N. Chichkov, B. Wellegehausen, and H. Welling, “Ablation of metals by ultrashort laser pulses,” J. Opt. Soc. Am. B 14, 2716–2722 (1997).
    [CrossRef]

1998 (1)

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280, 1412–1415 (1998).
[CrossRef] [PubMed]

1997 (3)

Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Kapteyn, “Generation of coherent soft x rays at 2.7 nm using high harmonics,” Phys. Rev. Lett. 79, 2967–2970 (1997).
[CrossRef]

Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent x-rays in the water window using 5-femtosecond laser pulses,” Science 278, 661–664 (1997).
[CrossRef]

S. Nolte, C. Momma, H. Jacobs, A. Tünnermann, B. N. Chichkov, B. Wellegehausen, and H. Welling, “Ablation of metals by ultrashort laser pulses,” J. Opt. Soc. Am. B 14, 2716–2722 (1997).
[CrossRef]

1996 (1)

S. Meyer, H. Eichmann, T. Menzel, S. Nolte, B. Wellegehausen, B. N. Chichkov, and C. Momma, “Phase-matched high-order difference-frequency mixing in plasmas,” Phys. Rev. Lett. 76, 3336–3339 (1996).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

A. Tünnermann, K. Mossavi, and B. Wellegehausen, “Nonlinear-optical processes in the near-resonant two-photon excitation of xenon by femtosecond KrF-excimer-laser pulses,” Phys. Rev. A 46, 2707–2717 (1992).
[CrossRef] [PubMed]

Backus, S.

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280, 1412–1415 (1998).
[CrossRef] [PubMed]

Burnett, N. H.

Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent x-rays in the water window using 5-femtosecond laser pulses,” Science 278, 661–664 (1997).
[CrossRef]

Chang, Z.

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280, 1412–1415 (1998).
[CrossRef] [PubMed]

Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Kapteyn, “Generation of coherent soft x rays at 2.7 nm using high harmonics,” Phys. Rev. Lett. 79, 2967–2970 (1997).
[CrossRef]

Chichkov, B. N.

S. Nolte, C. Momma, H. Jacobs, A. Tünnermann, B. N. Chichkov, B. Wellegehausen, and H. Welling, “Ablation of metals by ultrashort laser pulses,” J. Opt. Soc. Am. B 14, 2716–2722 (1997).
[CrossRef]

S. Meyer, H. Eichmann, T. Menzel, S. Nolte, B. Wellegehausen, B. N. Chichkov, and C. Momma, “Phase-matched high-order difference-frequency mixing in plasmas,” Phys. Rev. Lett. 76, 3336–3339 (1996).
[CrossRef] [PubMed]

Durfee III, C. G.

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280, 1412–1415 (1998).
[CrossRef] [PubMed]

Eichmann, H.

S. Meyer, H. Eichmann, T. Menzel, S. Nolte, B. Wellegehausen, B. N. Chichkov, and C. Momma, “Phase-matched high-order difference-frequency mixing in plasmas,” Phys. Rev. Lett. 76, 3336–3339 (1996).
[CrossRef] [PubMed]

Herne, C.

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280, 1412–1415 (1998).
[CrossRef] [PubMed]

Jacobs, H.

Kan, C.

Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent x-rays in the water window using 5-femtosecond laser pulses,” Science 278, 661–664 (1997).
[CrossRef]

Kaplan, A. E.

Kapteyn, H. C.

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280, 1412–1415 (1998).
[CrossRef] [PubMed]

Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Kapteyn, “Generation of coherent soft x rays at 2.7 nm using high harmonics,” Phys. Rev. Lett. 79, 2967–2970 (1997).
[CrossRef]

Koppitsch, R.

Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent x-rays in the water window using 5-femtosecond laser pulses,” Science 278, 661–664 (1997).
[CrossRef]

Krausz, F.

Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent x-rays in the water window using 5-femtosecond laser pulses,” Science 278, 661–664 (1997).
[CrossRef]

Lago, A.

Lenzner, M.

Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent x-rays in the water window using 5-femtosecond laser pulses,” Science 278, 661–664 (1997).
[CrossRef]

Menzel, T.

S. Meyer, H. Eichmann, T. Menzel, S. Nolte, B. Wellegehausen, B. N. Chichkov, and C. Momma, “Phase-matched high-order difference-frequency mixing in plasmas,” Phys. Rev. Lett. 76, 3336–3339 (1996).
[CrossRef] [PubMed]

Meyer, S.

S. Meyer, H. Eichmann, T. Menzel, S. Nolte, B. Wellegehausen, B. N. Chichkov, and C. Momma, “Phase-matched high-order difference-frequency mixing in plasmas,” Phys. Rev. Lett. 76, 3336–3339 (1996).
[CrossRef] [PubMed]

Momma, C.

S. Nolte, C. Momma, H. Jacobs, A. Tünnermann, B. N. Chichkov, B. Wellegehausen, and H. Welling, “Ablation of metals by ultrashort laser pulses,” J. Opt. Soc. Am. B 14, 2716–2722 (1997).
[CrossRef]

S. Meyer, H. Eichmann, T. Menzel, S. Nolte, B. Wellegehausen, B. N. Chichkov, and C. Momma, “Phase-matched high-order difference-frequency mixing in plasmas,” Phys. Rev. Lett. 76, 3336–3339 (1996).
[CrossRef] [PubMed]

Mossavi, K.

A. Tünnermann, K. Mossavi, and B. Wellegehausen, “Nonlinear-optical processes in the near-resonant two-photon excitation of xenon by femtosecond KrF-excimer-laser pulses,” Phys. Rev. A 46, 2707–2717 (1992).
[CrossRef] [PubMed]

Murnane, M. M.

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280, 1412–1415 (1998).
[CrossRef] [PubMed]

Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Kapteyn, “Generation of coherent soft x rays at 2.7 nm using high harmonics,” Phys. Rev. Lett. 79, 2967–2970 (1997).
[CrossRef]

Nolte, S.

S. Nolte, C. Momma, H. Jacobs, A. Tünnermann, B. N. Chichkov, B. Wellegehausen, and H. Welling, “Ablation of metals by ultrashort laser pulses,” J. Opt. Soc. Am. B 14, 2716–2722 (1997).
[CrossRef]

S. Meyer, H. Eichmann, T. Menzel, S. Nolte, B. Wellegehausen, B. N. Chichkov, and C. Momma, “Phase-matched high-order difference-frequency mixing in plasmas,” Phys. Rev. Lett. 76, 3336–3339 (1996).
[CrossRef] [PubMed]

Rundquist, A.

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280, 1412–1415 (1998).
[CrossRef] [PubMed]

Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Kapteyn, “Generation of coherent soft x rays at 2.7 nm using high harmonics,” Phys. Rev. Lett. 79, 2967–2970 (1997).
[CrossRef]

Sartania, S.

Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent x-rays in the water window using 5-femtosecond laser pulses,” Science 278, 661–664 (1997).
[CrossRef]

Schnürer, M.

Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent x-rays in the water window using 5-femtosecond laser pulses,” Science 278, 661–664 (1997).
[CrossRef]

Shkolnikov, P. L.

Spielmann, Ch.

Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent x-rays in the water window using 5-femtosecond laser pulses,” Science 278, 661–664 (1997).
[CrossRef]

Tünnermann, A.

S. Nolte, C. Momma, H. Jacobs, A. Tünnermann, B. N. Chichkov, B. Wellegehausen, and H. Welling, “Ablation of metals by ultrashort laser pulses,” J. Opt. Soc. Am. B 14, 2716–2722 (1997).
[CrossRef]

A. Tünnermann, K. Mossavi, and B. Wellegehausen, “Nonlinear-optical processes in the near-resonant two-photon excitation of xenon by femtosecond KrF-excimer-laser pulses,” Phys. Rev. A 46, 2707–2717 (1992).
[CrossRef] [PubMed]

Wang, H.

Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Kapteyn, “Generation of coherent soft x rays at 2.7 nm using high harmonics,” Phys. Rev. Lett. 79, 2967–2970 (1997).
[CrossRef]

Wellegehausen, B.

S. Nolte, C. Momma, H. Jacobs, A. Tünnermann, B. N. Chichkov, B. Wellegehausen, and H. Welling, “Ablation of metals by ultrashort laser pulses,” J. Opt. Soc. Am. B 14, 2716–2722 (1997).
[CrossRef]

S. Meyer, H. Eichmann, T. Menzel, S. Nolte, B. Wellegehausen, B. N. Chichkov, and C. Momma, “Phase-matched high-order difference-frequency mixing in plasmas,” Phys. Rev. Lett. 76, 3336–3339 (1996).
[CrossRef] [PubMed]

A. Tünnermann, K. Mossavi, and B. Wellegehausen, “Nonlinear-optical processes in the near-resonant two-photon excitation of xenon by femtosecond KrF-excimer-laser pulses,” Phys. Rev. A 46, 2707–2717 (1992).
[CrossRef] [PubMed]

Welling, H.

Wobrauschek, P.

Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent x-rays in the water window using 5-femtosecond laser pulses,” Science 278, 661–664 (1997).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Phys. Rev. A (1)

A. Tünnermann, K. Mossavi, and B. Wellegehausen, “Nonlinear-optical processes in the near-resonant two-photon excitation of xenon by femtosecond KrF-excimer-laser pulses,” Phys. Rev. A 46, 2707–2717 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

S. Meyer, H. Eichmann, T. Menzel, S. Nolte, B. Wellegehausen, B. N. Chichkov, and C. Momma, “Phase-matched high-order difference-frequency mixing in plasmas,” Phys. Rev. Lett. 76, 3336–3339 (1996).
[CrossRef] [PubMed]

Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Kapteyn, “Generation of coherent soft x rays at 2.7 nm using high harmonics,” Phys. Rev. Lett. 79, 2967–2970 (1997).
[CrossRef]

Science (2)

Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent x-rays in the water window using 5-femtosecond laser pulses,” Science 278, 661–664 (1997).
[CrossRef]

A. Rundquist, C. G. Durfee III, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280, 1412–1415 (1998).
[CrossRef] [PubMed]

Other (2)

See the rest of this feature and previous features on optical parametric processes and devices: J. Opt. Soc. Am. B 10, 1657–1791 (1993); 10, 2147–2243 (1993); 11, 2083–2322 (1995).

S. Babin, U. Hinze, E. Tiemann, and B. Wellegehausen, “Continuous resonant four-wave mixing in double-Λ level configuration of Na2,” Opt. Lett. 21, 1186–1188 (1996); A. Apolonskii, S. Baluschev, U. Hinze, and B. Wellegehausen, “Continuous frequency upconversion in a double-Λ scheme of Na2,” Appl. Phys. B 64, 435–442 (1997).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

High-order parametric amplification scheme.

Fig. 2
Fig. 2

Phase-matched schemes for HOPA’s in low-charged plasmas.

Fig. 3
Fig. 3

Schematic illustration of the possibility of gain transfer.

Fig. 4
Fig. 4

Experimental setup used in preliminary experiments.

Fig. 5
Fig. 5

Dependence of the 7ω emission signal on the medium length. As a nonlinear medium, Xe gas at a backing pressure of 2 bar is used.

Equations (27)

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dAidz=iπωicni(q+1)2q-1Nχ(q+1)(ωi) ApqAs* exp(iΔkz),
dAsdz=iπωscns(q+1)2q-1Nχ(q+1)(ωs) ApqAi* exp(iΔkz),
Is=ωsωiγ2G2Ii0 sinh2(Gz),Ii=Ii01+γ2G2sinh2(Gz),
G=[γ2-(Δk/2)2]1/2,
γ=πcωiωsnins1/2(q+1)2q-1Nχ(q+1)|Ap|q.
n(ω)=na(ω)+ni(ω)+nel(ω)-2
na,i(ω)=1+2πe2mNa,i(t)Rkf0kω0k2-ω2+iΓ0kω,
nel(ω)1-2πe2mNel(t)1ω2=1-ωpl22ω2,
qωp=ωs+ωiqnpωp=ωs+niωiqωp(np-1)=ωi(ni-1).
d Aidz=iπωicniN n(n+1)2n-1×χ(n+1)(ωi)ApnAsn* exp(iΔknz),
dAidz=iπωicniNn=2,q(n+1)2n-1χ(n+1)(ωi)×ApnAsn* exp(iΔkn z),
dAs2dz=i3πωs22cns2×Nχ(3)(ωs2)Ap2Ai* exp(iΔk2 z),
dAsqdz=iπωsqcnsq(q+1)2q-1×Nχ(q+1)(ωsq)ApqAi* exp(iΔkq z),
d2Aidz2=XAi+iΔkdAidz,
X=n=2,q(n+1)2π2ωiωsn4n-1c2ninsn×N2[χ(n+1)(ωi)χ(n+1)*(ωsn)]|Ap|2n.
Isq=ωsqωiγq2G2Ii0 sinh2(Gz),
G=[γ22+γq2-(Δk/2)2]1/2,
Is2=ωs2ωiγ22G2Ii0 sinh2(Gz),
Ii=Ii01+γ22+γq2G2sinh2(Gz),
γ2=3π2cωiωs2nins21/2Nχ(3)|Ap|2,
γq=πcωiωsqninsq1/2(q+1)2q-1Nχ(q+1)|Ap|q.
Isq=ωsqωiγq2(G2+a2)2Ii0{[b sinh(Gz)+c sin(az)]2+c2[cosh(Gz)-cos(az)]2},
Is2=ωs2ωiγ22G2Ii0 sinh2(Gz),
Ii=Ii01+γ22G2sinh2(Gz),
G=[γ22-(Δk2/2)2]1/2,
Isq=ωsqωiγq2G2Ii0sinh2(Gz)+Δk2G2 sinh2(Gz/2).
Isq=ωsqωiγq2(G2+Δkq2)2Ii0{[G sinh(Gz)+Δkq sin(Δkqz)]2+Δkq2[cosh(Gz)-cos(Δkqz)]2}ωsqωiγq2G2Ii0 sinh2(Gz),

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