Abstract

We introduce a new class of (2+1)D spatial and (3+1)D spatiotemporal waves that tend to autofocus in an abrupt fashion. While the maximum intensity of such a radial wave remains almost constant during propagation, it suddenly increases by orders of magnitude right before its focal point. These waves can be generated through the use of radially symmetric Airy waves or by appropriately superimposing Airy wave packets. Possible applications of such abruptly focusing beams are also discussed.

© 2010 Optical Society of America

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References

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  1. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).
  2. A. E. Seigman, Lasers (University Science, 1986).
  3. T. Juhasz, F. Loesel, R. Kurtz, C. Horvath, J. Bille, and G. Mourou, IEEE J. Sel. Top. Quantum Electron. 5, 902 (1999).
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    [CrossRef] [PubMed]
  7. G. S. He, in Vol. 53 of Progress in Optics (Elsevier, 2009), Chap. 4, pp. 201–292.
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    [CrossRef]
  10. M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
    [CrossRef]
  11. J. Baumgartl, T. Čižmár, M. Mazilu, V. C. Chan, A. E. Carruthers, B. A. Capron, W. McNeely, E. M. Wright, and K. Dholakia, Opt. Express 18, 17130 (2010).
    [CrossRef] [PubMed]

2010 (1)

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

2001 (1)

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

1999 (1)

T. Juhasz, F. Loesel, R. Kurtz, C. Horvath, J. Bille, and G. Mourou, IEEE J. Sel. Top. Quantum Electron. 5, 902 (1999).
[CrossRef]

1998 (1)

1996 (1)

1979 (1)

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Baumgartl, J.

Bergé, L.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Bille, J.

T. Juhasz, F. Loesel, R. Kurtz, C. Horvath, J. Bille, and G. Mourou, IEEE J. Sel. Top. Quantum Electron. 5, 902 (1999).
[CrossRef]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Capron, B. A.

Carruthers, A. E.

Chan, V. C.

Christodoulides, D. N.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef] [PubMed]

Cižmár, T.

Couairon, A.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

Davis, K. M.

Dholakia, K.

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Franco, M.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

He, G. S.

G. S. He, in Vol. 53 of Progress in Optics (Elsevier, 2009), Chap. 4, pp. 201–292.
[CrossRef]

Hirao, K.

Horvath, C.

T. Juhasz, F. Loesel, R. Kurtz, C. Horvath, J. Bille, and G. Mourou, IEEE J. Sel. Top. Quantum Electron. 5, 902 (1999).
[CrossRef]

Juhasz, T.

T. Juhasz, F. Loesel, R. Kurtz, C. Horvath, J. Bille, and G. Mourou, IEEE J. Sel. Top. Quantum Electron. 5, 902 (1999).
[CrossRef]

Kurtz, R.

T. Juhasz, F. Loesel, R. Kurtz, C. Horvath, J. Bille, and G. Mourou, IEEE J. Sel. Top. Quantum Electron. 5, 902 (1999).
[CrossRef]

Loesel, F.

T. Juhasz, F. Loesel, R. Kurtz, C. Horvath, J. Bille, and G. Mourou, IEEE J. Sel. Top. Quantum Electron. 5, 902 (1999).
[CrossRef]

Mazilu, M.

McNeely, W.

Miura, K.

Mlejnek, M.

Moloney, J. V.

Mourou, G.

T. Juhasz, F. Loesel, R. Kurtz, C. Horvath, J. Bille, and G. Mourou, IEEE J. Sel. Top. Quantum Electron. 5, 902 (1999).
[CrossRef]

Mysyrowicz, A.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

Prade, B.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Seigman, A. E.

A. E. Seigman, Lasers (University Science, 1986).

Siviloglou, G. A.

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Sugimoto, N.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Tzortzakis, S.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

Wright, E. M.

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

T. Juhasz, F. Loesel, R. Kurtz, C. Horvath, J. Bille, and G. Mourou, IEEE J. Sel. Top. Quantum Electron. 5, 902 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. Lett. (2)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

Other (3)

G. S. He, in Vol. 53 of Progress in Optics (Elsevier, 2009), Chap. 4, pp. 201–292.
[CrossRef]

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

A. E. Seigman, Lasers (University Science, 1986).

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Figures (3)

Fig. 1
Fig. 1

Dynamics of radially symmetric Airy beams for α = 0.05 , r 0 = 10 , and I max ( z = 0 ) = 1 . (a) Detailed plot of the central part of the propagation dynamics. (b) Maximum intensity as a function of z. (c) Maximum intensity that the Airy beam reaches during propagation for different values of the initial radius r 0 . (d) Hankel transform of the initial condition as given by Eq. (3).

Fig. 2
Fig. 2

Dynamics of the two-dimensional Airy superposition given by Eq. (5) for α = 0.05 , r 0 = 10 , and I max ( z = 0 ) = 1 . (a), (b), (c) Same as in Fig. 1. (d) Amplitude of the solution.

Fig. 3
Fig. 3

(a) Isointensity hemisphere of the Airy wave. (b), (c) Same as in Fig. 1 for the three-dimensional Airy solution given by Eq. (7) for α = 0.05 , r 0 = 15 , and I max ( z = 0 ) = 1 .

Equations (9)

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u z = ( i / 2 ) ( u x x + u y y ) = ( i / 2 ) ( u r / r + u r r ) ,
u ( r , z ) = 1 2 π 0 d k k u ˜ 0 ( k ) J 0 ( k r ) e i k 2 z / 2 ,
u ˜ 0 ( k ) = 2 π 0 d r r u 0 ( r ) J 0 ( k r ) .
g ( x , z ) = Ai ( x z 2 / 4 + i α z ) exp [ i ( 6 α 2 z 6 i α ( 2 x z 2 ) + 6 x z z 3 ) / 12 ] .
u ( r , r 0 , z ) = 0 2 π g ( x ( r , r 0 , θ 0 ) , z ) g ( y ( r , r 0 , θ 0 ) , z ) d θ 0 ,
u z = ( i / 2 ) ( u x x + u y y + u t t ) = ( i / 2 ) ( 2 u r / r + u r r ) ,
u ( r , z ) = [ g ( r 0 r , z ) g ( r 0 + r , z ) ] / r ,
u ( r , z ) 2 g r 0 ( r 0 , z ) g r 0 r 0 r 0 ( r 0 , z ) ( r 2 / 3 ) ,
u ( r = 0 , z ) = 2 g r 0 ( r 0 , z ) = e i Ψ ( r 0 , z ) [ ( 2 α + i z ) Ai ( ξ ( r 0 , z ) ) + 2 Ai ( ξ ( r 0 , z ) ) ] ,

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