Abstract

We introduce a new family of (2+1)D light beams with pre-engineered abruptly autofocusing properties. These beams have a circularly symmetric input profile that develops outward of a dark disk and oscillates radially as a sublinear-chirp signal, creating a series of concentric intensity rings with gradually decreasing width. The light rays involved in this process form a caustic surface of revolution that bends toward the beam axis at an acceleration rate that is determined by the radial chirp itself. The collapse of the caustic on the axis leads to a large intensity buildup right before the intended focus. This ray-optics interpretation provides valuable insight into the dynamics of abruptly autofocusing waves.

© 2011 Optical Society of America

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References

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  1. N. Efremidis and D. Christodoulides, Opt. Lett. 35, 4045 (2010).
    [Crossref] [PubMed]
  2. G. Siviloglou and D. Christodoulides, Opt. Lett. 32, 979 (2007).
    [Crossref] [PubMed]
  3. G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
    [Crossref]
  4. Y. Kaganovsky and E. Heyman, Opt. Express 18, 8440 (2010).
    [Crossref] [PubMed]
  5. S. Vo, K. Fuerschbach, K. Thompson, M. Alonso, and J. Rolland, J. Opt. Soc. Am. A 27, 2574 (2010).
    [Crossref]
  6. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Wiley-IEEE, 1994).
    [Crossref]

2010 (3)

2007 (2)

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

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

Alonso, M.

Broky, J.

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

Christodoulides, D.

Dogariu, A.

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

Efremidis, N.

Felsen, L.

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Wiley-IEEE, 1994).
[Crossref]

Fuerschbach, K.

Heyman, E.

Kaganovsky, Y.

Marcuvitz, N.

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Wiley-IEEE, 1994).
[Crossref]

Rolland, J.

Siviloglou, G.

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

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

Thompson, K.

Vo, S.

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

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

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

Other (1)

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Wiley-IEEE, 1994).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1

Ray-optics schematic: The rays emerg ing from the circle with radius ρ c meet on axis at z = z c , exactly where the caustic surface collapses.

Fig. 2
Fig. 2

(a) Radial input amplitude, (b) intensity evolution (in logarithimic scale), and (c) intensity contrast versus propagation distance for a beam with C = π , β = 3 / 2 , r 0 = 4 , and A ( r ) = exp [ 0.2 ( 4 r ) ] . The red detail in (b) is the paraboloid caustic of Eq. (5).

Fig. 3
Fig. 3

Similar to Fig. 2 but with β = 5 / 3 .

Fig. 4
Fig. 4

Maximum intensity contrast versus r 0 for the beams of Fig. 2 (solid blue curve) and Fig. 3 (dashed red curve).

Fig. 5
Fig. 5

Similar to Fig. 3 with an envelope function A ( r ) = 1 for 4 r 4 + 80 3 / 5 and zero elsewhere.

Equations (7)

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2 i u z + u r r + r 1 u r + r 2 u φ φ = 0 ,
u ( r , z = 0 ) = A ( r ) sin [ q ( r ) ] ,
q ( r ) = { C ( r r 0 ) β , r r 0 0 , r < r 0 ,
u ( r , φ , z ) = 0 2 π 0 u ( ρ , θ , 0 ) 2 π i z e i ρ 2 + r 2 2 ρ r cos ( φ θ ) 2 z ρ d ρ d θ .
r = r 0 [ C β ( β 1 ) z ] ν ν 1 ,
u ( r = 0 , z c ) π ρ c A ( ρ c ) Ai ( 0 ) z c [ q ( ρ c ) / 2 ] 1 / 3 e i [ ρ c 2 / 2 z c q ( ρ c ) ] .
z f z c + 1 σ ( z c ) = [ ( ν 1 ) r 0 ] 1 / ν C β ( β 1 ) + 1 σ ( z c ) .

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