Abstract

Using a recently developed technique (SEA TADPOLE) for easily measuring the complete spatiotemporal electric field of light pulses with micrometer spatial and femtosecond temporal resolution, we directly demonstrate the formation of theo-called boundary diffraction wave and Arago’s spot after an aperture, as well as the superluminal propagation of the spot. Our spatiotemporally resolved measurements beautifully confirm the time-domain treatment of diffraction. Also they prove very useful for modern physical optics, especially in micro- and meso-optics, and also significantly aid in the understanding of diffraction phenomena in general.

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References

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  1. G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in mezzo isotropo,” Ann. di Mat IIa,” 16, 21–48 (1888).
  2. A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180(4578), 160–162 (1957).
    [CrossRef]
  3. g. See, monograph M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1987, 6th ed) and references therein.
  4. Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026601 (2001).
    [CrossRef] [PubMed]
  5. P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral interferometry: a simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in real time,” Opt. Express 14(24), 11892–11900 (2006) (and references therein).
    [CrossRef] [PubMed]
  6. Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239(4-6), 243–250 (2004).
    [CrossRef]
  7. D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe velocity in Young-type experiments,” Phys. Lett. A 295(2-3), 78–80 (2002).
    [CrossRef]
  8. M. Vasnetsov, V. Pas’ko, A. Khoroshun, V. Slyusar, and M. Soskin, “Observation of superluminal wave-front propagation at the shadow area behind an opaque disk,” Opt. Lett. 32(13), 1830–1832 (2007).
    [CrossRef] [PubMed]
  9. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
    [CrossRef]
  10. P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34(15), 2276–2278 (2009).
    [CrossRef] [PubMed]
  11. H. Valtna-Lukner, P. Bowlan, M. Lõhmus, P. Piksarv, R. Trebino, and P. Saari, “Direct spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets,” Opt. Express 17(17), 14948–14955 (2009).
    [CrossRef] [PubMed]
  12. P. Piksarv, MSc thesis, University of Tartu (2009).
  13. P. Bowlan, P. Gabolde, and R. Trebino, “Directly measuring the spatio-temporal electric field of focusing ultrashort pulses,” Opt. Express 15(16), 10219–10230 (2007).
    [CrossRef] [PubMed]
  14. P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express 16(18), 13663–13675 (2008).
    [CrossRef] [PubMed]
  15. If viewing the plots without magnification in a computer screen the Moiré effect may obscure the actual small period of the intensity oscillations and increase of the period.

2009

2008

2007

2006

2004

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239(4-6), 243–250 (2004).
[CrossRef]

2002

D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe velocity in Young-type experiments,” Phys. Lett. A 295(2-3), 78–80 (2002).
[CrossRef]

2001

Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026601 (2001).
[CrossRef] [PubMed]

1997

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[CrossRef]

1957

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180(4578), 160–162 (1957).
[CrossRef]

1888

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in mezzo isotropo,” Ann. di Mat IIa,” 16, 21–48 (1888).

Akturk, S.

Bor, Z.

Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026601 (2001).
[CrossRef] [PubMed]

Bowlan, P.

Brunel, M.

D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe velocity in Young-type experiments,” Phys. Lett. A 295(2-3), 78–80 (2002).
[CrossRef]

Chauvat, D.

D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe velocity in Young-type experiments,” Phys. Lett. A 295(2-3), 78–80 (2002).
[CrossRef]

Emile, O.

D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe velocity in Young-type experiments,” Phys. Lett. A 295(2-3), 78–80 (2002).
[CrossRef]

Fuchs, U.

Gabolde, P.

Horváth, Z. L.

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239(4-6), 243–250 (2004).
[CrossRef]

Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026601 (2001).
[CrossRef] [PubMed]

Khoroshun, A.

Klebniczki, J.

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239(4-6), 243–250 (2004).
[CrossRef]

Kovács, A.

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239(4-6), 243–250 (2004).
[CrossRef]

Kurdi, G.

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239(4-6), 243–250 (2004).
[CrossRef]

Le Floch, A.

D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe velocity in Young-type experiments,” Phys. Lett. A 295(2-3), 78–80 (2002).
[CrossRef]

Lõhmus, M.

Maggi, G. A.

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in mezzo isotropo,” Ann. di Mat IIa,” 16, 21–48 (1888).

McGresham, K.

Pas’ko, V.

Piksarv, P.

Reivelt, K.

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[CrossRef]

Rubinowicz, A.

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180(4578), 160–162 (1957).
[CrossRef]

Saari, P.

Shreenath, A.

Slyusar, V.

Soskin, M.

Trebino, R.

Valtna-Lukner, H.

Vasnetsov, M.

Zeitner, U. D.

Ann. di Mat

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in mezzo isotropo,” Ann. di Mat IIa,” 16, 21–48 (1888).

Nature

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180(4578), 160–162 (1957).
[CrossRef]

Opt. Commun.

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239(4-6), 243–250 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe velocity in Young-type experiments,” Phys. Lett. A 295(2-3), 78–80 (2002).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026601 (2001).
[CrossRef] [PubMed]

Phys. Rev. Lett.

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[CrossRef]

Other

g. See, monograph M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1987, 6th ed) and references therein.

If viewing the plots without magnification in a computer screen the Moiré effect may obscure the actual small period of the intensity oscillations and increase of the period.

P. Piksarv, MSc thesis, University of Tartu (2009).

Supplementary Material (2)

» Media 1: MOV (4013 KB)     
» Media 2: MOV (4013 KB)     

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Figures (4)

Fig. 1
Fig. 1

Schematic of the formation of the Arago spot in the case of illumination with ultrashort pulses. A pancake-shaped pulsed wave from the left illuminates a disk-shaped obstacle (D) with radius a. The obstacle removes its central region according to the shadow boundaries (horizontal dashed lines), forming the geometrical-wave (GW) component of the output field. In addition, the edges of the obstacle excite the boundary diffraction wave (BW), which expands from a ring torus shape trough a spindle-torus-like stage (cross-section depicted in the figure) into a spherical wave at infinity. On the axis, overlapping and interfering boundary waves form the Arago spot. Around the shadow boundary in the overlap regions (also indicated by ovals) of the BW and GW, the common interference rings appear. The Arago spot (AS) propagates along the axis behind the front (indicated by vertical dashed line) of the transmitted GW but catches up with the latter at infinity because its velocity is superluminal.

Fig. 2
Fig. 2

Formation and evolution of the Arago spot behind an opaque disk 4mm in diameter. The magnitude of the electric field E is shown at three different propagation distances z in pseudo-color code according to the color bar (white has been taken for the zero of the scale in order to better reveal areas of weak field).

Fig. 3
Fig. 3

Formation and evolution of the diffracted field behind a circular hole 4mm in diameter. The boundary waves interfere with each other and with the directly transmitted pulse, but the interference maximum on the axis (actually a temporally resolved spot of Arago) lags behind the direct pulse, and eventually catches up with it.

Fig. 4
Fig. 4

Videos showing simulations of the diffraction of a plane wave pulse from a circular disc of d = 1 mm diameter. The pulse parameters are the same as in Figs. 2 and 3. Color represents the normalized amplitude of the electric field. (a) The reference frame is fixed with respect to the disc at z = 0 (Media 1). The diffracted field is calculated for z > 0 mm. (b) Close-up of the evolution of the boundary wave pulse in a reference frame moving at the luminal velocity c, or with the incident plane wave pulse (Media 2). Note that the x-scale in (b) is finer than in (a) by two orders of magnitude.

Equations (1)

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Ψ B W ( r , z , t ) e i ω 0 t 0 π v ( t s c ) e i k 0 s r a cos φ a 2 ( z s ) s d φ ,

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