Abstract

In this Letter, we describe the optical field associated with transmittances characterized by a slit-shaped curve. The influence of the curvature is that the diffraction field generates focusing regions. The focusing geometry corresponds to the geometry of the transmittance curve, except for scaling, rotations or translations. A relevant point is that the changes in the morphology of the diffraction field are bounded by the focusing regions. Our experimental and computational results are in good agreement with the theoretical predictions.

© 2012 Optical Society of America

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References

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  1. M. V. Berry and C. Upstill, Vol. 42 of Elsevier Serieson Progress in Optics, E. Wolf, ed. (North Holland, 1980), 257–346.
  2. M. S. Sosskin and M. V. Vasnetsov, Vol. 42 of Elsevier Series on Progress in Optics, E. Wolf, ed. (North Holland, 2001), 219–276.
  3. G. Martinez-Niconoff, J. Carranza, and A. C. Rodriguez, in Elsevier Series on Optics Communications (1995), Vol. 114, pp. 194–198.
    [CrossRef]
  4. Y. S. Kivshar and E. A. Ostrovskaya, Opt. Photon. News 12(4), 24 (2001).
    [CrossRef]
  5. D. P. Rhodes, G. P. T. Lancaster, J. Livesey, D. Mcgloin, J. Arlt, and K. Dholakia, Opt. Commun. 214, 247 (2002).
    [CrossRef]
  6. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, Opt. Lett. 36, 1164 (2011).
    [CrossRef]
  7. K. Dholakia and T. Cizmar, Nat. Photonics 5, 335 (2011).
    [CrossRef]
  8. A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, Opt. Express 19, 5760 (2011).
    [CrossRef]
  9. W. C. Graustein, Differential Geometry (Dover, 2006).
  10. D. J. Struik, Lectures on Classical Differential Geometry (Dover, 1988).
  11. V. I. Arnold, Singularities of Caustics and Wave Fronts (Springer, 1990).
  12. P. R. Garabedian, Partial Differential Equations (Wiley, 1967).
  13. J. B. Keller, J. Opt. Soc. Am. 52, 116 (1962).
    [CrossRef]
  14. L. Elsgoltz, Differential Equations and Variational Calculus (MIR, 1977).

2011 (3)

2002 (1)

D. P. Rhodes, G. P. T. Lancaster, J. Livesey, D. Mcgloin, J. Arlt, and K. Dholakia, Opt. Commun. 214, 247 (2002).
[CrossRef]

2001 (1)

Y. S. Kivshar and E. A. Ostrovskaya, Opt. Photon. News 12(4), 24 (2001).
[CrossRef]

1962 (1)

Arlt, J.

D. P. Rhodes, G. P. T. Lancaster, J. Livesey, D. Mcgloin, J. Arlt, and K. Dholakia, Opt. Commun. 214, 247 (2002).
[CrossRef]

Arnold, V. I.

V. I. Arnold, Singularities of Caustics and Wave Fronts (Springer, 1990).

Baumgartl, J.

Berry, M. V.

M. V. Berry and C. Upstill, Vol. 42 of Elsevier Serieson Progress in Optics, E. Wolf, ed. (North Holland, 1980), 257–346.

Carranza, J.

G. Martinez-Niconoff, J. Carranza, and A. C. Rodriguez, in Elsevier Series on Optics Communications (1995), Vol. 114, pp. 194–198.
[CrossRef]

Cizmar, T.

K. Dholakia and T. Cizmar, Nat. Photonics 5, 335 (2011).
[CrossRef]

Dholakia, K.

K. Dholakia and T. Cizmar, Nat. Photonics 5, 335 (2011).
[CrossRef]

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, Opt. Express 19, 5760 (2011).
[CrossRef]

D. P. Rhodes, G. P. T. Lancaster, J. Livesey, D. Mcgloin, J. Arlt, and K. Dholakia, Opt. Commun. 214, 247 (2002).
[CrossRef]

Elsgoltz, L.

L. Elsgoltz, Differential Equations and Variational Calculus (MIR, 1977).

Garabedian, P. R.

P. R. Garabedian, Partial Differential Equations (Wiley, 1967).

Graustein, W. C.

W. C. Graustein, Differential Geometry (Dover, 2006).

Keller, J. B.

Kivshar, Y. S.

Lancaster, G. P. T.

D. P. Rhodes, G. P. T. Lancaster, J. Livesey, D. Mcgloin, J. Arlt, and K. Dholakia, Opt. Commun. 214, 247 (2002).
[CrossRef]

Liu, W.

Livesey, J.

D. P. Rhodes, G. P. T. Lancaster, J. Livesey, D. Mcgloin, J. Arlt, and K. Dholakia, Opt. Commun. 214, 247 (2002).
[CrossRef]

Martinez-Niconoff, G.

G. Martinez-Niconoff, J. Carranza, and A. C. Rodriguez, in Elsevier Series on Optics Communications (1995), Vol. 114, pp. 194–198.
[CrossRef]

Mcgloin, D.

D. P. Rhodes, G. P. T. Lancaster, J. Livesey, D. Mcgloin, J. Arlt, and K. Dholakia, Opt. Commun. 214, 247 (2002).
[CrossRef]

Miroshnichenko, A. E.

Mourka, A.

Neshev, D. N.

Ostrovskaya, E. A.

Y. S. Kivshar and E. A. Ostrovskaya, Opt. Photon. News 12(4), 24 (2001).
[CrossRef]

Rhodes, D. P.

D. P. Rhodes, G. P. T. Lancaster, J. Livesey, D. Mcgloin, J. Arlt, and K. Dholakia, Opt. Commun. 214, 247 (2002).
[CrossRef]

Rodriguez, A. C.

G. Martinez-Niconoff, J. Carranza, and A. C. Rodriguez, in Elsevier Series on Optics Communications (1995), Vol. 114, pp. 194–198.
[CrossRef]

Shadrivov, I. V.

Shanor, C.

Sosskin, M. S.

M. S. Sosskin and M. V. Vasnetsov, Vol. 42 of Elsevier Series on Progress in Optics, E. Wolf, ed. (North Holland, 2001), 219–276.

Struik, D. J.

D. J. Struik, Lectures on Classical Differential Geometry (Dover, 1988).

Upstill, C.

M. V. Berry and C. Upstill, Vol. 42 of Elsevier Serieson Progress in Optics, E. Wolf, ed. (North Holland, 1980), 257–346.

Vasnetsov, M. V.

M. S. Sosskin and M. V. Vasnetsov, Vol. 42 of Elsevier Series on Progress in Optics, E. Wolf, ed. (North Holland, 2001), 219–276.

Wright, E. M.

J. Opt. Soc. Am. (1)

Nat. Photonics (1)

K. Dholakia and T. Cizmar, Nat. Photonics 5, 335 (2011).
[CrossRef]

Opt. Commun. (1)

D. P. Rhodes, G. P. T. Lancaster, J. Livesey, D. Mcgloin, J. Arlt, and K. Dholakia, Opt. Commun. 214, 247 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Photon. News (1)

Y. S. Kivshar and E. A. Ostrovskaya, Opt. Photon. News 12(4), 24 (2001).
[CrossRef]

Other (8)

L. Elsgoltz, Differential Equations and Variational Calculus (MIR, 1977).

M. V. Berry and C. Upstill, Vol. 42 of Elsevier Serieson Progress in Optics, E. Wolf, ed. (North Holland, 1980), 257–346.

M. S. Sosskin and M. V. Vasnetsov, Vol. 42 of Elsevier Series on Progress in Optics, E. Wolf, ed. (North Holland, 2001), 219–276.

G. Martinez-Niconoff, J. Carranza, and A. C. Rodriguez, in Elsevier Series on Optics Communications (1995), Vol. 114, pp. 194–198.
[CrossRef]

W. C. Graustein, Differential Geometry (Dover, 2006).

D. J. Struik, Lectures on Classical Differential Geometry (Dover, 1988).

V. I. Arnold, Singularities of Caustics and Wave Fronts (Springer, 1990).

P. R. Garabedian, Partial Differential Equations (Wiley, 1967).

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Figures (4)

Fig. 1.
Fig. 1.

(Left) Transmittance plane containing a slit curve; the curvature centers generate a focusing region that is the separatrix for regions with different orders. (Right) Transmittance containing a cycloid shape curve and its evolute. For this case, the higher order points are in the neighborhood of the cusped regions.

Fig. 2.
Fig. 2.

Experimental results for the diffraction field on two planes separated 15 cm. The transmittance consists in a cycloid slit. It was recorded on high resolution plate and it is contained in a square of 0.5 cm per side. The width of the slit is approximately 0.2 mm. The wavelength used is 623.8 nm.

Fig. 3.
Fig. 3.

Experimental results on two propagation planes for the diffraction of screens containing an epicycloid slit with n=3, 4, 6, respectively. The curves were recorded on high resolution plate of 5 mm per side.

Fig. 4.
Fig. 4.

Computer simulation for the interference between a plane wave with a diffraction field that presents a focusing region with four and six cusps. Bifurcation of the phase function occurs in the neighborhood of the cusped regions where kind speckle pattern is observed.

Equations (8)

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φ(P)=12πSeikrr{(ik1r)Ar⃗A}·ndl.
φB(P)=ik2πSeikrrAr⃗·ndlik2πeikRRB(θ),
φB(P)=ik2πSeikrrAr⃗·ndliNik2πeikRiRiBi(θ),
α=xy(x2+y2)xyxy;β=y+x(x2+y2)xyxy,
x=a(tsint);y=a(1cost).
α=a(tsint);β=a(1cost)2a.
x=(RR0)cost+R0cos(RR0R0t),y=(RR0)sint+R0sin(RR0R0t),
α=(R2R0)R[(RR0)costR0cos(RR0R0t)]β=1R[(RR0)(R+2R0)sint+(3R2R0)R0sin(RR0R0t)].

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