Abstract

We determine the threshold power for self-focusing collapse both in a bulk medium and in a hollow-core waveguide for various spatial profiles. We find that the threshold power for collapse in the waveguide is always equal to the lower-bound prediction for a bulk medium.

© 2000 Optical Society of America

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References

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  1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, Opt. Lett. 20, 73 (1995); E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysyrowicz, Opt. Lett. 21, 62 (1996); O. Kosareva, V. P. Kandidov, A. Brodeur, C. Y. Chien, and S. L. Chin, Opt. Lett. 22, 1332 (1997); L. Wöste, C. Wedeknd, H. Wille, P. Rairoux, B. Stein, S. Nikolov, Ch. Werner, St. Niedermeier, F. Ronneberger, H. Schillinger, and R. Sauerbrey, Laser Optoelectron. 29, 51 (1997).
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    [CrossRef]
  3. M. Nisoli, S. De Silvestri, and O. Svelto, Appl. Phys. Lett. 68, 2793 (1996); Opt. Lett. 22, 522 (1997); M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, S. Sartania, Z. Cheng, M. Lenzner, Ch. Spielmann, and F. Krausz, Appl. Phys. B 65, 189 (1997).
    [CrossRef]
  4. R. Chiao, E. Garmire, and C. Townes, Phys. Rev. Lett. 13, 479 (1964); P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965); E. Dawes and J. Marburger, Phys. Rev. 179, 862 (1969); P. Chernev and V. Petrov, Opt. Lett. 17, 172 (1992).
    [CrossRef]
  5. This expression is the same in both MKS and cgs units because n2 has units of inverse intensity.
  6. R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992), Sec. 6.2.
  7. M. Weinstein, Commun. Math. Phys. 87, 567 (1983).
    [CrossRef]
  8. G. Tempea and T. Brabec, Opt. Lett. 23, 762, 1286 (1998).
  9. G. Fibich and G. Papanicolaou, SIAM J. Applied Math 60, 183 (1999).
    [CrossRef]
  10. J. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
    [CrossRef]
  11. V. Malkin, Physica D 64, 251 (1993).
    [CrossRef]
  12. G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).
    [CrossRef] [PubMed]
  13. G. Fibich and G. Papanicolaou, Phys. Lett. A 239, 167 (1998).
    [CrossRef]
  14. E. A. J. Marcatili and R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).
    [CrossRef]
  15. G. Fibich, “Self-focusing in the damped nonlinear Schrödinger equation,” preprint (available at www.math.tau.ac.il/~fibich ) .

1999 (1)

G. Fibich and G. Papanicolaou, SIAM J. Applied Math 60, 183 (1999).
[CrossRef]

1998 (2)

G. Tempea and T. Brabec, Opt. Lett. 23, 762, 1286 (1998).

G. Fibich and G. Papanicolaou, Phys. Lett. A 239, 167 (1998).
[CrossRef]

1997 (1)

1996 (2)

M. Nisoli, S. De Silvestri, and O. Svelto, Appl. Phys. Lett. 68, 2793 (1996); Opt. Lett. 22, 522 (1997); M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, S. Sartania, Z. Cheng, M. Lenzner, Ch. Spielmann, and F. Krausz, Appl. Phys. B 65, 189 (1997).
[CrossRef]

G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).
[CrossRef] [PubMed]

1995 (1)

1993 (1)

V. Malkin, Physica D 64, 251 (1993).
[CrossRef]

1983 (1)

M. Weinstein, Commun. Math. Phys. 87, 567 (1983).
[CrossRef]

1974 (1)

J. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

1964 (2)

R. Chiao, E. Garmire, and C. Townes, Phys. Rev. Lett. 13, 479 (1964); P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965); E. Dawes and J. Marburger, Phys. Rev. 179, 862 (1969); P. Chernev and V. Petrov, Opt. Lett. 17, 172 (1992).
[CrossRef]

E. A. J. Marcatili and R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).
[CrossRef]

Ashkin, A.

J. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

Backus, S.

Bjorkholm, J.

J. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992), Sec. 6.2.

Brabec, T.

G. Tempea and T. Brabec, Opt. Lett. 23, 762, 1286 (1998).

Braun, A.

Chiao, R.

R. Chiao, E. Garmire, and C. Townes, Phys. Rev. Lett. 13, 479 (1964); P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965); E. Dawes and J. Marburger, Phys. Rev. 179, 862 (1969); P. Chernev and V. Petrov, Opt. Lett. 17, 172 (1992).
[CrossRef]

De Silvestri, S.

M. Nisoli, S. De Silvestri, and O. Svelto, Appl. Phys. Lett. 68, 2793 (1996); Opt. Lett. 22, 522 (1997); M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, S. Sartania, Z. Cheng, M. Lenzner, Ch. Spielmann, and F. Krausz, Appl. Phys. B 65, 189 (1997).
[CrossRef]

Du, D.

Durfee, C. G.

Fibich, G.

G. Fibich and G. Papanicolaou, SIAM J. Applied Math 60, 183 (1999).
[CrossRef]

G. Fibich and G. Papanicolaou, Phys. Lett. A 239, 167 (1998).
[CrossRef]

G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).
[CrossRef] [PubMed]

G. Fibich, “Self-focusing in the damped nonlinear Schrödinger equation,” preprint (available at www.math.tau.ac.il/~fibich ) .

Garmire, E.

R. Chiao, E. Garmire, and C. Townes, Phys. Rev. Lett. 13, 479 (1964); P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965); E. Dawes and J. Marburger, Phys. Rev. 179, 862 (1969); P. Chernev and V. Petrov, Opt. Lett. 17, 172 (1992).
[CrossRef]

Kapteyn, H. C.

Korn, G.

Liu, X.

Malkin, V.

V. Malkin, Physica D 64, 251 (1993).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili and R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).
[CrossRef]

Mourou, G.

Murnane, M. M.

Nisoli, M.

M. Nisoli, S. De Silvestri, and O. Svelto, Appl. Phys. Lett. 68, 2793 (1996); Opt. Lett. 22, 522 (1997); M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, S. Sartania, Z. Cheng, M. Lenzner, Ch. Spielmann, and F. Krausz, Appl. Phys. B 65, 189 (1997).
[CrossRef]

Papanicolaou, G.

G. Fibich and G. Papanicolaou, SIAM J. Applied Math 60, 183 (1999).
[CrossRef]

G. Fibich and G. Papanicolaou, Phys. Lett. A 239, 167 (1998).
[CrossRef]

Schmeltzer, R. A.

E. A. J. Marcatili and R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).
[CrossRef]

Squier, J.

Svelto, O.

M. Nisoli, S. De Silvestri, and O. Svelto, Appl. Phys. Lett. 68, 2793 (1996); Opt. Lett. 22, 522 (1997); M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, S. Sartania, Z. Cheng, M. Lenzner, Ch. Spielmann, and F. Krausz, Appl. Phys. B 65, 189 (1997).
[CrossRef]

Tempea, G.

G. Tempea and T. Brabec, Opt. Lett. 23, 762, 1286 (1998).

Townes, C.

R. Chiao, E. Garmire, and C. Townes, Phys. Rev. Lett. 13, 479 (1964); P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965); E. Dawes and J. Marburger, Phys. Rev. 179, 862 (1969); P. Chernev and V. Petrov, Opt. Lett. 17, 172 (1992).
[CrossRef]

Weinstein, M.

M. Weinstein, Commun. Math. Phys. 87, 567 (1983).
[CrossRef]

Appl. Phys. Lett. (1)

M. Nisoli, S. De Silvestri, and O. Svelto, Appl. Phys. Lett. 68, 2793 (1996); Opt. Lett. 22, 522 (1997); M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, S. Sartania, Z. Cheng, M. Lenzner, Ch. Spielmann, and F. Krausz, Appl. Phys. B 65, 189 (1997).
[CrossRef]

Bell Syst. Tech. J. (1)

E. A. J. Marcatili and R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964).
[CrossRef]

Commun. Math. Phys. (1)

M. Weinstein, Commun. Math. Phys. 87, 567 (1983).
[CrossRef]

Opt. Lett. (3)

Phys. Lett. A (1)

G. Fibich and G. Papanicolaou, Phys. Lett. A 239, 167 (1998).
[CrossRef]

Phys. Rev. Lett. (3)

G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).
[CrossRef] [PubMed]

J. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

R. Chiao, E. Garmire, and C. Townes, Phys. Rev. Lett. 13, 479 (1964); P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965); E. Dawes and J. Marburger, Phys. Rev. 179, 862 (1969); P. Chernev and V. Petrov, Opt. Lett. 17, 172 (1992).
[CrossRef]

Physica D (1)

V. Malkin, Physica D 64, 251 (1993).
[CrossRef]

SIAM J. Applied Math (1)

G. Fibich and G. Papanicolaou, SIAM J. Applied Math 60, 183 (1999).
[CrossRef]

Other (3)

This expression is the same in both MKS and cgs units because n2 has units of inverse intensity.

R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992), Sec. 6.2.

G. Fibich, “Self-focusing in the damped nonlinear Schrödinger equation,” preprint (available at www.math.tau.ac.il/~fibich ) .

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Figures (2)

Fig. 1
Fig. 1

Townes profile Rr. Also shown is a Gaussian with the same power and on-axis intensity.

Fig. 2
Fig. 2

Plots of the peak intensity and the beam width inside the hollow waveguide as a function of propagation distance for P/Pcrlb=0.5.

Tables (1)

Tables Icon

Table 1 Comparison of Threshold Parameter α with Theoretical Lower Bound Ncr1.86225 and Upper Bound Gf [Eq. (8)] for Various Initial Profiles CA0r/r0a

Equations (14)

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Pcr=αλ2/4πn0n2,
2ik0Az+ΔA+40ck02n2A2A=0,  Ar,0=A0r,
P=4π0n0cA02rdr.
r˜=r/r0,  z˜=z/2Ldf,  ψ=2k0r0c0n2A,
iψz+Δψ+ψ2ψ=0,  ψr,0=ψ0r.
ψ=expizRr,
ΔR-R+R3=0,  R0=0,  R=0.
Ncr=R2rdr1.86225.
Pcrlb=Ncrλ24πn0n2.
H0=ψ02rdr-1/2ψ04rdr<0.
ψ02rdr>Gf,
Gf=2f2rdrf2rdrf4rdr.
Pcrub=Gfλ24πn0n2.
Ncr=αlbααub=Gf.

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