Abstract

We show that optical-cycle steepening in a nonlinear dielectric before focusing results in an arbitrarily large enhancement of the focused intensity and energy density. The focusing of an optical shock produces singular intensity and energy density at the focal point.

© 2001 Optical Society of America

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References

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  1. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1964), p. 44.
  2. T. T. Wu, J. Appl. Phys. 57, 2370 (1985).
    [CrossRef]
  3. E. M. Belenov and A. V. Nazarkin, J. Opt. Soc. Am. A 11, 168 (1994).
    [CrossRef]
  4. M. A. Porras, F. Salazar-Bloise, and L. Vazquez, Phys. Rev. Lett. 85, 2104 (2000).
    [CrossRef] [PubMed]
  5. G. Rosen, Phys. Rev. 139, A539 (1965).
    [CrossRef]
  6. R. G. Flesch, A. Pushkarev, and J. V. Moloney, Phys. Rev. Lett. 76, 2488 (1996).
    [CrossRef] [PubMed]
  7. L. Gilles, J. V. Moloney, and L. Vázquez, Phys. Rev. E 60, 1051 (1999).
    [CrossRef]
  8. Z. Bor and Z. L. Horváth, Opt. Commun. 94, 249 (1992).
    [CrossRef]
  9. M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, J. Opt. Soc. Am. B 9, 1158 (1992).
    [CrossRef]
  10. E. Ibragimov, Appl. Opt. 34, 7280 (1995).
    [CrossRef] [PubMed]
  11. J. C. Chanteloup, H. Baldis, A. Mingus, G. Mourou, B. Loiseaux, and J. P. Huignard, Opt. Lett. 23, 475 (1998).
    [CrossRef]
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  15. A. M. Weiner, S. Oudin, D. E. Leaird, and D. H. Reitze, J. Opt. Soc. Am. A 10, 1112 (1993).
    [CrossRef]
  16. See for example, J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, Berlin, 1983).
    [CrossRef]
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    [CrossRef]

2000 (3)

1999 (1)

L. Gilles, J. V. Moloney, and L. Vázquez, Phys. Rev. E 60, 1051 (1999).
[CrossRef]

1998 (1)

1996 (1)

R. G. Flesch, A. Pushkarev, and J. V. Moloney, Phys. Rev. Lett. 76, 2488 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (3)

1993 (1)

1992 (2)

1989 (1)

1985 (1)

T. T. Wu, J. Appl. Phys. 57, 2370 (1985).
[CrossRef]

1965 (1)

G. Rosen, Phys. Rev. 139, A539 (1965).
[CrossRef]

Andrés, P.

Baldis, H.

Belenov, E. M.

Bor, Z.

Z. Bor and Z. L. Horváth, Opt. Commun. 94, 249 (1992).
[CrossRef]

Chanteloup, J. C.

Davidson, D. B.

Ferrando, A.

Flesch, R. G.

R. G. Flesch, A. Pushkarev, and J. V. Moloney, Phys. Rev. Lett. 76, 2488 (1996).
[CrossRef] [PubMed]

Gilles, L.

L. Gilles, J. V. Moloney, and L. Vázquez, Phys. Rev. E 60, 1051 (1999).
[CrossRef]

Horváth, Z. L.

Z. Bor and Z. L. Horváth, Opt. Commun. 94, 249 (1992).
[CrossRef]

Huignard, J. P.

Ibragimov, E.

Kempe, M.

Leaird, D. E.

Loiseaux, B.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1964), p. 44.

Mingus, A.

Miret, J. J.

Moloney, J. V.

L. Gilles, J. V. Moloney, and L. Vázquez, Phys. Rev. E 60, 1051 (1999).
[CrossRef]

R. G. Flesch, A. Pushkarev, and J. V. Moloney, Phys. Rev. Lett. 76, 2488 (1996).
[CrossRef] [PubMed]

Mourou, G.

Nakajima, T.

Nazarkin, A. V.

Oudin, S.

Porras, M. A.

M. A. Porras, F. Salazar-Bloise, and L. Vazquez, Phys. Rev. Lett. 85, 2104 (2000).
[CrossRef] [PubMed]

Pushkarev, A.

R. G. Flesch, A. Pushkarev, and J. V. Moloney, Phys. Rev. Lett. 76, 2488 (1996).
[CrossRef] [PubMed]

Reitze, D. H.

Rosen, G.

G. Rosen, Phys. Rev. 139, A539 (1965).
[CrossRef]

Rudolph, W.

Salazar-Bloise, F.

M. A. Porras, F. Salazar-Bloise, and L. Vazquez, Phys. Rev. Lett. 85, 2104 (2000).
[CrossRef] [PubMed]

Sherman, G. C.

Silvestre, E.

Smoller, J.

See for example, J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, Berlin, 1983).
[CrossRef]

Stamm, U.

Vazquez, L.

M. A. Porras, F. Salazar-Bloise, and L. Vazquez, Phys. Rev. Lett. 85, 2104 (2000).
[CrossRef] [PubMed]

Vázquez, L.

L. Gilles, J. V. Moloney, and L. Vázquez, Phys. Rev. E 60, 1051 (1999).
[CrossRef]

Weiner, A. M.

Wilhelmi, B.

Wu, T. T.

T. T. Wu, J. Appl. Phys. 57, 2370 (1985).
[CrossRef]

Ziolkowski, R. W.

Appl. Opt. (1)

J. Appl. Phys. (1)

T. T. Wu, J. Appl. Phys. 57, 2370 (1985).
[CrossRef]

J. Opt. Soc. Am. A (4)

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

Opt. Commun. (1)

Z. Bor and Z. L. Horváth, Opt. Commun. 94, 249 (1992).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. (1)

G. Rosen, Phys. Rev. 139, A539 (1965).
[CrossRef]

Phys. Rev. E (1)

L. Gilles, J. V. Moloney, and L. Vázquez, Phys. Rev. E 60, 1051 (1999).
[CrossRef]

Phys. Rev. Lett. (2)

R. G. Flesch, A. Pushkarev, and J. V. Moloney, Phys. Rev. Lett. 76, 2488 (1996).
[CrossRef] [PubMed]

M. A. Porras, F. Salazar-Bloise, and L. Vazquez, Phys. Rev. Lett. 85, 2104 (2000).
[CrossRef] [PubMed]

Other (2)

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1964), p. 44.

See for example, J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, Berlin, 1983).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

(a) Steepening of the sinusoidal wave ft=Asinω0t (dashed curve) in a medium with PNL=ε0χ3f3. Numerical values are given in the text. (b) Amplitude and (c) instantaneous intensity of the same waves in the focal point of an ideal focusing system.

Fig. 2
Fig. 2

(a) Energy-density enhancement and (b) Ziolkowski’s factor Yint at the focal point as functions of distance zD of steepening in the nonlinear dielectric.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

E˜r,z,ω=f˜ω1+z/qexpiωr22cq+zexpωi-zc,
Eont=a22cRdftdt,
won=a22cR2-dftdt2dt.
f=f0τ,t=τ+στz,
zs=min-1/σττ=minσττ
t-ts-σττττsτ-τs3/6σττs.
fzs,tf0τs-fτ0τs6σττsσττττst-ts1/3.
wr=12a2cR20dωω2f˜ω2exp-2ωar2cR2.

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