Abstract

We introduce an illumination configuration which is a spatiotemporal analog of a non-diffracting X-wave. By interfering multiple ultrashort converging plane waves, we generate a tight central spot at which a transform limited ultrashort pulse is formed. Outside this tight focus a spatiotemporal speckle field with longer duration and reduced peak power is created. We investigate this spatiotemporal X-wave configuration analytically, numerically, and experimentally demonstrate the effect using two photon excitation fluorescence.

© 2009 Optical Society of America

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  1. J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  2. J. Lu and J. F. Greenleaf, "Nondiffracting X Waves-Exact Solutions to Free-Space Scalar Wave Equation and Their Finite Aperture Realizations," IEEE Trans. Ultrasonics Ferroelectrics and Frequency Control 39, 19-31 (1992).
    [CrossRef]
  3. J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
    [CrossRef]
  4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  5. J. Durnin, J. J. MicelliJr., and J. H. Eberly, "Diffraction free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  6. T. Wulle and S. Herminghaus, "Nonlinear optics of Bessel beams," Phys. Rev. Lett. 70, 1401-1404 (1993).
    [CrossRef] [PubMed]
  7. O. Manela, M. Segev, and D. N. Christodoulides, "Nondiffracting beams in periodic media," Opt. Lett. 30, 2611-2613 (2005).
    [CrossRef] [PubMed]
  8. Y. Lahini, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
    [CrossRef] [PubMed]
  9. C. J. R. Sheppard, "Generalized Bessel pulse beams," J. Opt. Soc. Am. A 19, 2218-2222 (2002).
    [CrossRef]
  10. O. E. Martinez, "3000 times grating compressor with positive group-velocity dispersion - application to fiber compensation in 1.3-1.6 m region," IEEE J. Quantum Electron. 23, 59 (1987).
    [CrossRef]
  11. A. M. Shaarawi, I. M. Besieris, and T. M. Said, "Temporal focusing by use of composite X waves," J. Opt. Soc. Am. A 20, 1658-1665 (2003).
    [CrossRef]
  12. D. Oron, E. Tal, and Y. Silberberg, "Scanningless depth resolved microscopy," Opt. Express 13, 1468-1476 (2005).
    [CrossRef] [PubMed]
  13. E. Tal and Y. Silberberg, "Transformation from an ultrashort pulse to spatiotemporal speckle by a thin scattering surface," Opt. Lett. 31, 3529-3531 (2006).
    [CrossRef] [PubMed]
  14. E. Tal, D. Oron, and Y. Silberberg, "Improved depth resolution in video-rate line-scanning multiphoton microscopy using temporal focusing," Opt. Lett. 30, 1686-1688 (2005).
    [CrossRef] [PubMed]
  15. G. Zhu, J. van Howe, M. Durst, W. Zipfel, and C. Xu, "Simultaneous spatial and temporal focusing of femtosecond pulses," Opt. Express 13, 2153-2159 (2005).
    [CrossRef] [PubMed]
  16. J. W. Goodman, Speckle Phenomena in Optics: Theory and application, (Roberts and Company Publishers, Greenwood Village, 2007).
  17. J. C Dainty "The statistics of speckle patterns," in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, (1977) Vol. 14, pp 1-44.

2007

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Y. Lahini, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

2006

2005

2003

2002

1996

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

1993

T. Wulle and S. Herminghaus, "Nonlinear optics of Bessel beams," Phys. Rev. Lett. 70, 1401-1404 (1993).
[CrossRef] [PubMed]

1992

J. Lu and J. F. Greenleaf, "Nondiffracting X Waves-Exact Solutions to Free-Space Scalar Wave Equation and Their Finite Aperture Realizations," IEEE Trans. Ultrasonics Ferroelectrics and Frequency Control 39, 19-31 (1992).
[CrossRef]

1987

J. Durnin, J. J. MicelliJr., and J. H. Eberly, "Diffraction free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

O. E. Martinez, "3000 times grating compressor with positive group-velocity dispersion - application to fiber compensation in 1.3-1.6 m region," IEEE J. Quantum Electron. 23, 59 (1987).
[CrossRef]

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

Besieris, I. M.

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Christodoulides, D. N.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Y. Lahini, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

O. Manela, M. Segev, and D. N. Christodoulides, "Nondiffracting beams in periodic media," Opt. Lett. 30, 2611-2613 (2005).
[CrossRef] [PubMed]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Droulias, S.

Y. Lahini, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Durnin, J.

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

J. Durnin, J. J. MicelliJr., and J. H. Eberly, "Diffraction free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Durst, M.

Eberly, J. H.

J. Durnin, J. J. MicelliJr., and J. H. Eberly, "Diffraction free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Fagerholm, J.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Friberg, A. T.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Greenleaf, J. F.

J. Lu and J. F. Greenleaf, "Nondiffracting X Waves-Exact Solutions to Free-Space Scalar Wave Equation and Their Finite Aperture Realizations," IEEE Trans. Ultrasonics Ferroelectrics and Frequency Control 39, 19-31 (1992).
[CrossRef]

Herminghaus, S.

T. Wulle and S. Herminghaus, "Nonlinear optics of Bessel beams," Phys. Rev. Lett. 70, 1401-1404 (1993).
[CrossRef] [PubMed]

Hizanidis, K.

Y. Lahini, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Huttunen, J.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Lahini, Y.

Y. Lahini, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Lu, J.

J. Lu and J. F. Greenleaf, "Nondiffracting X Waves-Exact Solutions to Free-Space Scalar Wave Equation and Their Finite Aperture Realizations," IEEE Trans. Ultrasonics Ferroelectrics and Frequency Control 39, 19-31 (1992).
[CrossRef]

Manela, O.

Martinez, O. E.

O. E. Martinez, "3000 times grating compressor with positive group-velocity dispersion - application to fiber compensation in 1.3-1.6 m region," IEEE J. Quantum Electron. 23, 59 (1987).
[CrossRef]

Micelli, J. J.

J. Durnin, J. J. MicelliJr., and J. H. Eberly, "Diffraction free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Morandotti, R.

Y. Lahini, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Morgan, D. P.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Oron, D.

Said, T. M.

Salomaa, M. M.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Segev, M.

Shaarawi, A. M.

Sheppard, C. J. R.

Silberberg, Y.

Siviloglou, G. A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Tal, E.

van Howe, J.

Wulle, T.

T. Wulle and S. Herminghaus, "Nonlinear optics of Bessel beams," Phys. Rev. Lett. 70, 1401-1404 (1993).
[CrossRef] [PubMed]

Xu, C.

Zhu, G.

Zipfel, W.

IEEE J. Quantum Electron.

O. E. Martinez, "3000 times grating compressor with positive group-velocity dispersion - application to fiber compensation in 1.3-1.6 m region," IEEE J. Quantum Electron. 23, 59 (1987).
[CrossRef]

IEEE Trans. Ultrasonics Ferroelectrics and Frequency Control

J. Lu and J. F. Greenleaf, "Nondiffracting X Waves-Exact Solutions to Free-Space Scalar Wave Equation and Their Finite Aperture Realizations," IEEE Trans. Ultrasonics Ferroelectrics and Frequency Control 39, 19-31 (1992).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Phys. Rev. E

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, "Angular-spectrum representation of nondiffracting X waves," Phys. Rev. E 54, 4347-4352 (1996).
[CrossRef]

Phys. Rev. Lett.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

J. Durnin, J. J. MicelliJr., and J. H. Eberly, "Diffraction free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

T. Wulle and S. Herminghaus, "Nonlinear optics of Bessel beams," Phys. Rev. Lett. 70, 1401-1404 (1993).
[CrossRef] [PubMed]

Y. Lahini, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Other

J. W. Goodman, Speckle Phenomena in Optics: Theory and application, (Roberts and Company Publishers, Greenwood Village, 2007).

J. C Dainty "The statistics of speckle patterns," in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, (1977) Vol. 14, pp 1-44.

Supplementary Material (1)

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Figures (6)

Fig. 1.
Fig. 1.

(a) Scheme of the optical setup used for producing the spatio-temporal x-wave. A wide beam of ultrashort pulses (i) impinges upon a random diffuser placed at the back focal plane of a lens. The ultrashort pulse beam is scattered by the diffuser into multiple propagating spherical wavefronts (ii–iii). The spherical wavefronts are transformed by the lens into converging plane wavefronts, producing a spatio-temporal x-wave on the optical axis at the lens’ front focal plane (FFP) (iv). (b) Numerically simulated “snapshot” of the field intensity at the front focal plane of the lens at the x-wave formation time (see text). (c) “Zoom-in” on the center of the x-wave reveals its the spatiotemporal speckle structure.

Fig. 2.
Fig. 2.

A movie of the propagation of a pulse through the system. The pulse is longitudinal stretched in order to emphasize its behavior, therefore there seem to be lack of continuously when propagating through the lens. Media 1

Fig. 3.
Fig. 3.

Experimental linearly imaged speckle pattern at the lens’ FFP (Fourier-plane). (a) Ordinary speckle pattern measured using a monochromatic cw incident beam. (b) Radially smeared speckle pattern obtained using an ultrashort pulse incident beam. The radial smearing results from the superimposed re-scaled Fourier images of the wavelengths in the pulse’s broad spectrum. The sharp speckles at the center form the X-wave focus which is also temporally short.

Fig. 4.
Fig. 4.

(a) Linear imaging of the z=f X-wave formation plane averaged over 10 diffuser realizations. Scale bars indicates 1mm. (b) Two-photon image of the same plane, showing an order of magnitude reduced radii confirming the short temporal duration of pulses forming the spatiotemporal X-wave focus. (c) Radial cross-sections of the experimental and simulated signals: red - one photon signals; blue - two photons signals; solid line - experimental signals; dashed line - simulated signals. simulated result

Fig. 5.
Fig. 5.

(a) Experimental one- and two-photon radial cross-sections for planes at several axial (z) positions. Blue - focal plane, red - 10 mm away from the focal plane, green - 21 mm away from the focal plane. Solid line - two photons signal, dotted line - one photon signal. (b) Simulated one- and two-photon radial FWHM as function of axial (z). Red - one photon’s FWHM, Blue - Two photons’ FWHM. Black crosses mark the experimental FWHM in the measured places

Fig. 6.
Fig. 6.

(a),(c) Experimental one- and two-photon images for an isotropic diffuser and an elliptic (astigmatic) input beam. (b),(d) One- and two-photon images for a anisotropic diffuser and a circular incident beam.

Equations (9)

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Γλ (x1,y1;x2,y2,z=f)I(x1,y1)μλ(Δx,Δy)
Γλ(x1,y1;x2,y2,z=f)=1λfI(α2,β2)e2πiλf(Δxα2+Δyβ2)dα2dβ2×1λfµλ(∆α,∆β)e2πiλf(x1∆α+y1∆β)dαdβ
R(λ12Δλ)fR(λ+12Δλ)f <1σ
R<λΔλ(λfσ)
Γ(z)=1λ0λ1λ2Eλ0(x0,y0,0)Eλ1*(x1,y1,0)eiπλ0z(x02+y02)iπλ0z(x12+y12)dx0dy0dx1dy1
Γ(z)=1aλ02z2Eλ0(x0,y0,0)Eλ0*(x2a,y2a,0)eiπλ0z(x02+y02x22y22)dx0dy0dx2dx2
Γ(z)=4π2σ2d2aλ04z2f2[eiπλ0z(x02-x22)e12(2πdλ0f)2x02e12(2πσλ0f)2(x0x2a)2dx0dx2]2
ρ(z)=Γ(z)Iλ0(0,0,z)11+iπΔλσ2f2λ2z
Δ zxwaυe λ2f2πΔλσ2

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