Abstract

We analyze the combined effect of small time dispersion and nonparaxiality on self-focusing and its ability to arrest the blowup of laser pulses by deriving reduced equations that depend on only the propagation distance and time. We calculate the pulse duration for which time dispersion dominates over nonparaxiality, or vice versa. We identify additional terms (shock term, group-velocity nonparaxiality, etc.)  that should be retained when time dispersion and nonparaxiality are of comparable magnitude. These additional terms lead to temporal asymmetry, and in the visible spectrum they can dominate over both time dispersion and nonparaxiality.

© 1997 Optical Society of America

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References

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1996 (3)

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

J. Ranka, R. Schirmer, and A. Gaeta, Phys. Rev. Lett. 77, 3783 (1996).
[CrossRef] [PubMed]

G. Fibich, Opt. Lett. 21, 1735 (1996).
[CrossRef] [PubMed]

1995 (2)

S. Chi and Q. Guo, Opt. Lett. 20, 1598 (1995).
[CrossRef] [PubMed]

G. Fibich, V. Malkin, and G. Papanicolaou, Phys. Rev. A 52, 4218 (1995).
[CrossRef] [PubMed]

1992 (1)

1988 (1)

1986 (1)

N. Zharova, A. G. Litvak, T. A. Petrova, A. M. Sergeev, and A. D. Yunakovsky, JETP Lett. 44, 13 (1986); J. E. Rothenberg, Opt. Lett. 17, 583 (1992); P. Chernev and V. Petrov, Opt. Lett. 17, 172 (1992); G. Luther, A. Newell, and J. Moloney, Phys. D 74, 59 (1994).
[CrossRef] [PubMed]

1985 (1)

1982 (1)

1981 (1)

N. Tzoar and M. Jain, Phys. Rev. A 23, 1266 (1981).
[CrossRef]

1965 (1)

Anderson, D.

Bor, Z.

Chi, S.

Feit, M.

Fibich, G.

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

G. Fibich, Opt. Lett. 21, 1735 (1996).
[CrossRef] [PubMed]

G. Fibich, V. Malkin, and G. Papanicolaou, Phys. Rev. A 52, 4218 (1995).
[CrossRef] [PubMed]

G. Fibich, “Self-focusing in the nonlinear Schrödinger equation for ultrashort laser–tissue interactions,” Ph.D. dissertation (Courant Institute, New York University, New York, New York, 1994).

G. Fibich and G. Papanicolaou, (University of California, Los Angeles, Los Angeles, Calif., 1997).

Fleck, J.

Gaeta, A.

J. Ranka, R. Schirmer, and A. Gaeta, Phys. Rev. Lett. 77, 3783 (1996).
[CrossRef] [PubMed]

Guo, Q.

Jain, M.

N. Tzoar and M. Jain, Phys. Rev. A 23, 1266 (1981).
[CrossRef]

Lisak, M.

Litvak, A. G.

N. Zharova, A. G. Litvak, T. A. Petrova, A. M. Sergeev, and A. D. Yunakovsky, JETP Lett. 44, 13 (1986); J. E. Rothenberg, Opt. Lett. 17, 583 (1992); P. Chernev and V. Petrov, Opt. Lett. 17, 172 (1992); G. Luther, A. Newell, and J. Moloney, Phys. D 74, 59 (1994).
[CrossRef] [PubMed]

Malitson, I.

Malkin, V.

G. Fibich, V. Malkin, and G. Papanicolaou, Phys. Rev. A 52, 4218 (1995).
[CrossRef] [PubMed]

Papanicolaou, G.

G. Fibich, V. Malkin, and G. Papanicolaou, Phys. Rev. A 52, 4218 (1995).
[CrossRef] [PubMed]

G. Fibich and G. Papanicolaou, (University of California, Los Angeles, Los Angeles, Calif., 1997).

Petrova, T. A.

N. Zharova, A. G. Litvak, T. A. Petrova, A. M. Sergeev, and A. D. Yunakovsky, JETP Lett. 44, 13 (1986); J. E. Rothenberg, Opt. Lett. 17, 583 (1992); P. Chernev and V. Petrov, Opt. Lett. 17, 172 (1992); G. Luther, A. Newell, and J. Moloney, Phys. D 74, 59 (1994).
[CrossRef] [PubMed]

Querry, M.

M. Querry, D. Wieliczka, and D. Segelstein, in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, San Diego, Calif., 1991), pp. 1059–1077.

Rácz, B.

Ranka, J.

J. Ranka, R. Schirmer, and A. Gaeta, Phys. Rev. Lett. 77, 3783 (1996).
[CrossRef] [PubMed]

Rothenberg, J.

Schirmer, R.

J. Ranka, R. Schirmer, and A. Gaeta, Phys. Rev. Lett. 77, 3783 (1996).
[CrossRef] [PubMed]

Segelstein, D.

M. Querry, D. Wieliczka, and D. Segelstein, in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, San Diego, Calif., 1991), pp. 1059–1077.

Sergeev, A. M.

N. Zharova, A. G. Litvak, T. A. Petrova, A. M. Sergeev, and A. D. Yunakovsky, JETP Lett. 44, 13 (1986); J. E. Rothenberg, Opt. Lett. 17, 583 (1992); P. Chernev and V. Petrov, Opt. Lett. 17, 172 (1992); G. Luther, A. Newell, and J. Moloney, Phys. D 74, 59 (1994).
[CrossRef] [PubMed]

Tzoar, N.

N. Tzoar and M. Jain, Phys. Rev. A 23, 1266 (1981).
[CrossRef]

Wieliczka, D.

M. Querry, D. Wieliczka, and D. Segelstein, in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, San Diego, Calif., 1991), pp. 1059–1077.

Yunakovsky, A. D.

N. Zharova, A. G. Litvak, T. A. Petrova, A. M. Sergeev, and A. D. Yunakovsky, JETP Lett. 44, 13 (1986); J. E. Rothenberg, Opt. Lett. 17, 583 (1992); P. Chernev and V. Petrov, Opt. Lett. 17, 172 (1992); G. Luther, A. Newell, and J. Moloney, Phys. D 74, 59 (1994).
[CrossRef] [PubMed]

Zharova, N.

N. Zharova, A. G. Litvak, T. A. Petrova, A. M. Sergeev, and A. D. Yunakovsky, JETP Lett. 44, 13 (1986); J. E. Rothenberg, Opt. Lett. 17, 583 (1992); P. Chernev and V. Petrov, Opt. Lett. 17, 172 (1992); G. Luther, A. Newell, and J. Moloney, Phys. D 74, 59 (1994).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

JETP Lett. (1)

N. Zharova, A. G. Litvak, T. A. Petrova, A. M. Sergeev, and A. D. Yunakovsky, JETP Lett. 44, 13 (1986); J. E. Rothenberg, Opt. Lett. 17, 583 (1992); P. Chernev and V. Petrov, Opt. Lett. 17, 172 (1992); G. Luther, A. Newell, and J. Moloney, Phys. D 74, 59 (1994).
[CrossRef] [PubMed]

Opt. Lett. (4)

Phys. Rev. A (2)

N. Tzoar and M. Jain, Phys. Rev. A 23, 1266 (1981).
[CrossRef]

G. Fibich, V. Malkin, and G. Papanicolaou, Phys. Rev. A 52, 4218 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

J. Ranka, R. Schirmer, and A. Gaeta, Phys. Rev. Lett. 77, 3783 (1996).
[CrossRef] [PubMed]

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

Other (3)

M. Querry, D. Wieliczka, and D. Segelstein, in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, San Diego, Calif., 1991), pp. 1059–1077.

G. Fibich, “Self-focusing in the nonlinear Schrödinger equation for ultrashort laser–tissue interactions,” Ph.D. dissertation (Courant Institute, New York University, New York, New York, 1994).

G. Fibich and G. Papanicolaou, (University of California, Los Angeles, Los Angeles, Calif., 1997).

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Figures (1)

Fig. 1
Fig. 1

Evolution of the on-axis intensity 1/L versus time according to Eqs.  (6) and (7) at the propagation distances indicated.

Equations (20)

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iψz+Δψ+ψ2ψ=0, ψ0, r=ψ0r.
Ex, y, z, t=eAx, y, z, texpik0z-iω0t,
Azz+2ik0Az+ΔA+2ik0cgAt-1cg2+k0kωωAtt=2n2n0c2 expiω0tA2A exp-iω0ttt,
r˜=rr0, z˜=z2Ldiff, t˜=t-z/cgT, ψ=r0k02n2n0A,
iψz+Δψ+ψ2ψ+1ψzz+22in0cgcψ2ψt-ψzt-3ψtt=0,
1=14r02k02, 2=1cgk0T=1ω0Tcn0cg, 3=LdiffkωωT2.
22=13F, F=4cg2k0kωω.
-2ψzt-i2Δψt+ψ2ψt,
Tb=2Ldiffk0kωω=4FLdiffcg.
F2n0ωnω1,
iψz+Δψ+ψ2ψ+1ψzz+i2ψ2ψt-Δψt-3ψtt=0.
ψz, t, r1Lz, tRrLexpiζz, t+iLzr24L,
ζzz, t=1L2, Lzzz, t=-βz, tL3,
βzz, t=-γ11L2z-γ21L2t+γ3ζtt,
Lz, t=LZct-z, βz, t=βZct-z, ζz, t=ζZct-z.
βz=-γ11L2z+γ2Z·c1L2z+γ3-Z¨cζz+Z·c2ζzz, where .=ddt.
gss=sg+κg3, with g=L-1>0.
s=β0-γ3Z¨cζγ3Z¨c-2/3, β0β0, t, κ=-γ1-γ2Z·c-γ3Z·c2γ3Z¨c-2/3.
s0t:=sz=0, tβ0, tγ3Z¨c-2/3.
γ3Z·c2>γ1-γZ·c+2L20, tAi2s0γ3Z¨c2/3,

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