## Focused X-shaped pulses

JOSA A, Vol. 21, Issue 8, pp. 1564-1574 (2004)

http://dx.doi.org/10.1364/JOSAA.21.001564

Acrobat PDF (1027 KB)

### Abstract

The space–time focusing of a (continuous) succession of localized X-shaped pulses is obtained by suitably integrating over their speed, i.e., over their axicon angle, thus generalizing a previous (discrete) approach. New superluminal wave pulses are first constructed and then tailored so that they become temporally focused at a chosen spatial point, where the wave field can reach very high intensities for a short time. Results of this kind may find applications in many fields, besides electromagnetism and optics, including acoustics, gravitation, and elementary particle physics.

© 2004 Optical Society of America

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(060.4080) Fiber optics and optical communications : Modulation

(070.1060) Fourier optics and signal processing : Acousto-optical signal processing

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(140.3300) Lasers and laser optics : Laser beam shaping

(170.0170) Medical optics and biotechnology : Medical optics and biotechnology

(320.5540) Ultrafast optics : Pulse shaping

(320.5550) Ultrafast optics : Pulses

**Citation**

Michel Zamboni-Rached, Amr M. Shaarawi, and Erasmo Recami, "Focused X-shaped pulses," J. Opt. Soc. Am. A **21**, 1564-1574 (2004)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-8-1564

Sort: Year | Journal | Reset

### References

- H. Bateman, Electrical and Optical Wave Motion (Cambridge U. Press, Cambridge, UK, 1915).
- J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 356.
- R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1966), Vol. 2, p. 760.
- I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bi-directional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
- R. Donnelly and R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. London, Ser. A 440, 541–565 (1993). See also Ref. 30 below.
- J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X-waves: exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
- E. Recami, “On localized X-shaped superluminal solutions to Maxwell equations,” Phys. A 252, 586–610 (1998) and references therein.
- R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
- M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,” Eur. Phys. J. D 21, 217–228 (2002).
- For short review papers, see, for instance, E. Recami, “Superluminal motions? A bird’s-eye view of the experimental situation,” Found. Phys. 31, 1119–1135 (2001). Also see Ref. 11.
- E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
- See, e.g., J.-Y. Lu, H.-H. Zou, and J. F. Greenleaf, “Biomedical ultrasound beam forming,” Ultrasound Med. Biol. 20, 403–428 (1994).
- P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).
- M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” (e-print physics/0209101), Opt. Commun. 226, 15–23 (2003).
- C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (2003).
- M. A. Porras, S. Trillo, C. Conti, and P. Di Trapani, “Paraxial envelope X-waves,” Opt. Lett. 28, 1090–1092 (2003).
- A. M. Attiya, “Transverse (TE) electromagnetic X-waves: propagation, scattering, diffraction and generation problems,” Ph.D. thesis (Cairo University, Cairo, 2001).
- See E. Recami, “Classical tachyons and possible applications,” Riv. Nuovo Cimento 9 (6), 1–178 (1986) and references therein.
- A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
- 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).
- J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992). See also Ref. 22.
- In the case of Ref. 21, the beam speed is larger than the sound (not of the light) speed in the considered medium.
- P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
- D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000).
- P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneous formation of nonspreading X-shaped wavepackets” (e-print physics/0303083) in LANL Archives.
- M. Zamboni-Rached, E. Recami, and F. Fontana, “Superluminal localized solutions to Maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603 (2001).
- M. Zamboni-Rached, F. Fontana, and E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
- M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
- M. Zamboni-Rached and H. E. Hernández-Figueroa, “A rigorous analysis of localized wave propagation in optical fibers,” Opt. Commun. 191, 49–54 (2000).
- I. M. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions of the scalar wave equation,” Prog. Electromagn. Res. 19, 1–48 (1998).
- S. He and J. Y. Lu, “Sidelobe reduction of limited-diffraction beams with Chebyshev aperture apodization,” J. Acoust. Soc. Am. 107, 3556–3559 (2000).
- J.-Y. Lu and S. He, “High frame rate imaging with a small number of array elements,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 1416–1421 (1999).
- J.-Y. Lu, “Experimental study of high frame rate imaging with limited-diffraction beams,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 84–97 (1998).
- J.-Y. Lu, “Producing bowtie limited-diffraction beams with synthetic array experiments,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 893–900 (1996).
- J.-Y. Lu and J. F. Greenleaf, “Producing deep depth of field and depth-independent resolution in NDE with limited-diffraction beams,” Ultrason. Imaging 15, 134–149 (1993).
- A. A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, and M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2287–2295 (1998).
- M. Abdel-Rahman, I. M. Besieris, and A. M. Shaarawi, “A comparative study on the reconstruction of localized pulses,” in Proceedings of the IEEE Southeast Conference (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 113–117.
- 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).
- See, e.g., C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “A general formulation for the analysis of scalar limited-diffraction beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
- H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. 22, 310–312 (1997).
- J.-Y. Lu, “An X-wave transform,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47, 1472–1481 (2000).
- P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).
- J. Salo, A. T. Friberg, and M. Salomaa, “Orthogonal X-waves,” J. Phys. A. 34, 9319–9327 (2001).
- M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernandez, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
- I. S. Gradshteyn and I. M. Ryzhik, Integrals, Series and Products, 4th ed. (Academic, New York, 1965).
- A. T. Friberg, J. Fagerholm, and M. M. Salomaa, “Space-frequency analysis of non-diffracting pulses,” Opt. Commun. 136, 207–212 (1997).
- 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).
- P. Saari, “Superluminal localized waves of electromagnetic field in vacuo,” in Time’s Arrows, Quantum Measurements and Superluminal Behavior, D. Mugnai, A. Ranfagni, and L. S. Shulman, eds. (C.N.R., Rome, 2001), pp. 37–48.
- D. Mugnai, A. Ranfagni, and R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A 311, 77–81 (2003).
- G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426–435 (1952).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.