Abstract

We describe a family of dispersion-free and diffraction-free optical beams consisting in two-dimensional wave packets with a spatiotemporal Bessel (STB) profile propagating in media with anomalous dispersion. We also describe quasi-invariant optical beams with a spatiotemporal Bessel-Gauss (STBG) profile; these wave packets have finite dimensions and energy, conditions to be representative of physical beams. The paper provides a detailed account of the properties of STB and STBG beams, including their spatially resolved frequency spectrum, their far-field behaviour and a comparison of the propagation of STBG beams with that of Gaussian wave packets. An experimental setup based on a folded pulse shaper has allowed to generate STBG beams using the ultrashort pulses from a Ti:sapphire laser. The analysis of the spatially resolved frequency spectrum and of the spatial and temporal profiles obtained experimentally shows good agreement with theory.

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References

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  1. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23(3), 142–144 (1973).
    [CrossRef]
  2. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980).
    [CrossRef]
  3. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987).
    [CrossRef]
  4. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
    [CrossRef] [PubMed]
  5. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
    [CrossRef]
  6. M. A. Porras, R. Borghi, and M. Santarsiero, “Few-optical-cycle bessel-gauss pulsed beams in free space,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(44 Pt B), 5729–5737 (2000).
    [CrossRef] [PubMed]
  7. B. Lü and Z. Liu, “Propagation properties of ultrashort pulsed Bessel beams in dispersive media,” J. Opt. Soc. Am. A 20(3), 582–587 (2003).
    [CrossRef]
  8. C. A. Dartora, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Properties of localized pulses through the analysis of temporal modulation effects in Bessel beams and the convolution theorem,” Opt. Commun. 229(1-6), 99–107 (2004).
    [CrossRef]
  9. M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersive broadening of light with Bessel-Gauss beams,” Opt. Commun. 206(4-6), 235–241 (2002).
    [CrossRef]
  10. 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(5), 310–312 (1997).
    [CrossRef] [PubMed]
  11. M. A. Porras and I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217(1-6), 257–264 (2003).
    [CrossRef]
  12. D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, “Bessel X waves in two- and three-dimensional bidispersive optical systems,” Opt. Lett. 29(13), 1446–1448 (2004).
    [CrossRef] [PubMed]
  13. D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Three-dimensional arrays of optical bottle beams,” Opt. Commun. 225(4-6), 215–222 (2003).
    [CrossRef]
  14. A. G. Sedukhin, “Periodically focused propagation-invariant beams with sharp central peak,” Opt. Commun. 228(4-6), 231–247 (2003).
    [CrossRef]
  15. C. Paterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124(1-2), 131–140 (1996).
    [CrossRef]
  16. S. Longhi, “Localized subluminal envelope pulses in dispersive media,” Opt. Lett. 29(2), 147–149 (2004).
    [CrossRef] [PubMed]
  17. M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066606 (2004).
    [CrossRef] [PubMed]
  18. M. Dallaire, M. Piché, and N. McCarthy, “Spatiotemporal Bessel Beams,” Proc. SPIE 6796, 67963O (2007).
    [CrossRef]
  19. S. Malaguti, G. Bellanca, and S. Trillo, “Two-dimensional envelope localized waves in the anomalous dispersion regime,” Opt. Lett. 33(10), 1117–1119 (2008).
    [CrossRef] [PubMed]
  20. A. E. Siegman, Lasers, (University Science Books, Mill Valley, California, 1986). See p. 277, Eq. (32).
  21. P. M. Morse, and L. Feshbach, Methods of Theoretical Physics, (McGraw-Hill, New York, 1953).
  22. G. Arfken, Mathematical Methods for Physicists, third edition. (Academic Press, New York, 1985). See p.797.
  23. I. S. Gradshteyn, and I. M. Ryzhik, Tables of Integrals, Series, and Products, fourth edition,( Academic Press, New York, 1980).

2008 (1)

2007 (1)

M. Dallaire, M. Piché, and N. McCarthy, “Spatiotemporal Bessel Beams,” Proc. SPIE 6796, 67963O (2007).
[CrossRef]

2004 (4)

S. Longhi, “Localized subluminal envelope pulses in dispersive media,” Opt. Lett. 29(2), 147–149 (2004).
[CrossRef] [PubMed]

M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066606 (2004).
[CrossRef] [PubMed]

D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, “Bessel X waves in two- and three-dimensional bidispersive optical systems,” Opt. Lett. 29(13), 1446–1448 (2004).
[CrossRef] [PubMed]

C. A. Dartora, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Properties of localized pulses through the analysis of temporal modulation effects in Bessel beams and the convolution theorem,” Opt. Commun. 229(1-6), 99–107 (2004).
[CrossRef]

2003 (4)

B. Lü and Z. Liu, “Propagation properties of ultrashort pulsed Bessel beams in dispersive media,” J. Opt. Soc. Am. A 20(3), 582–587 (2003).
[CrossRef]

D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Three-dimensional arrays of optical bottle beams,” Opt. Commun. 225(4-6), 215–222 (2003).
[CrossRef]

A. G. Sedukhin, “Periodically focused propagation-invariant beams with sharp central peak,” Opt. Commun. 228(4-6), 231–247 (2003).
[CrossRef]

M. A. Porras and I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217(1-6), 257–264 (2003).
[CrossRef]

2002 (1)

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersive broadening of light with Bessel-Gauss beams,” Opt. Commun. 206(4-6), 235–241 (2002).
[CrossRef]

2000 (1)

M. A. Porras, R. Borghi, and M. Santarsiero, “Few-optical-cycle bessel-gauss pulsed beams in free space,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(44 Pt B), 5729–5737 (2000).
[CrossRef] [PubMed]

1997 (1)

1996 (1)

C. Paterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124(1-2), 131–140 (1996).
[CrossRef]

1987 (3)

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

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980).
[CrossRef]

1973 (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23(3), 142–144 (1973).
[CrossRef]

Bellanca, G.

Borghi, R.

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersive broadening of light with Bessel-Gauss beams,” Opt. Commun. 206(4-6), 235–241 (2002).
[CrossRef]

M. A. Porras, R. Borghi, and M. Santarsiero, “Few-optical-cycle bessel-gauss pulsed beams in free space,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(44 Pt B), 5729–5737 (2000).
[CrossRef] [PubMed]

Christodoulides, D. N.

Dallaire, M.

M. Dallaire, M. Piché, and N. McCarthy, “Spatiotemporal Bessel Beams,” Proc. SPIE 6796, 67963O (2007).
[CrossRef]

Dartora, C. A.

C. A. Dartora, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Properties of localized pulses through the analysis of temporal modulation effects in Bessel beams and the convolution theorem,” Opt. Commun. 229(1-6), 99–107 (2004).
[CrossRef]

Dholakia, K.

D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Three-dimensional arrays of optical bottle beams,” Opt. Commun. 225(4-6), 215–222 (2003).
[CrossRef]

Di Trapani, P.

M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066606 (2004).
[CrossRef] [PubMed]

D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, “Bessel X waves in two- and three-dimensional bidispersive optical systems,” Opt. Lett. 29(13), 1446–1448 (2004).
[CrossRef] [PubMed]

Durnin, J.

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

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Efremidis, N. K.

Gonzalo, I.

M. A. Porras and I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217(1-6), 257–264 (2003).
[CrossRef]

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980).
[CrossRef]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23(3), 142–144 (1973).
[CrossRef]

Hernández-Figueroa, H. E.

C. A. Dartora, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Properties of localized pulses through the analysis of temporal modulation effects in Bessel beams and the convolution theorem,” Opt. Commun. 229(1-6), 99–107 (2004).
[CrossRef]

Liu, Z.

Longhi, S.

Lü, B.

Malaguti, S.

Malomed, B. A.

McCarthy, N.

M. Dallaire, M. Piché, and N. McCarthy, “Spatiotemporal Bessel Beams,” Proc. SPIE 6796, 67963O (2007).
[CrossRef]

McGloin, D.

D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Three-dimensional arrays of optical bottle beams,” Opt. Commun. 225(4-6), 215–222 (2003).
[CrossRef]

Melville, H.

D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Three-dimensional arrays of optical bottle beams,” Opt. Commun. 225(4-6), 215–222 (2003).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980).
[CrossRef]

Nóbrega, K. Z.

C. A. Dartora, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Properties of localized pulses through the analysis of temporal modulation effects in Bessel beams and the convolution theorem,” Opt. Commun. 229(1-6), 99–107 (2004).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

Paterson, C.

C. Paterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124(1-2), 131–140 (1996).
[CrossRef]

Piché, M.

M. Dallaire, M. Piché, and N. McCarthy, “Spatiotemporal Bessel Beams,” Proc. SPIE 6796, 67963O (2007).
[CrossRef]

Porras, M. A.

M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066606 (2004).
[CrossRef] [PubMed]

M. A. Porras and I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217(1-6), 257–264 (2003).
[CrossRef]

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersive broadening of light with Bessel-Gauss beams,” Opt. Commun. 206(4-6), 235–241 (2002).
[CrossRef]

M. A. Porras, R. Borghi, and M. Santarsiero, “Few-optical-cycle bessel-gauss pulsed beams in free space,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(44 Pt B), 5729–5737 (2000).
[CrossRef] [PubMed]

Rätsep, M.

Recami, E.

C. A. Dartora, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Properties of localized pulses through the analysis of temporal modulation effects in Bessel beams and the convolution theorem,” Opt. Commun. 229(1-6), 99–107 (2004).
[CrossRef]

Saari, P.

Santarsiero, M.

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersive broadening of light with Bessel-Gauss beams,” Opt. Commun. 206(4-6), 235–241 (2002).
[CrossRef]

M. A. Porras, R. Borghi, and M. Santarsiero, “Few-optical-cycle bessel-gauss pulsed beams in free space,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(44 Pt B), 5729–5737 (2000).
[CrossRef] [PubMed]

Sedukhin, A. G.

A. G. Sedukhin, “Periodically focused propagation-invariant beams with sharp central peak,” Opt. Commun. 228(4-6), 231–247 (2003).
[CrossRef]

Sibbett, W.

D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Three-dimensional arrays of optical bottle beams,” Opt. Commun. 225(4-6), 215–222 (2003).
[CrossRef]

Smith, R.

C. Paterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124(1-2), 131–140 (1996).
[CrossRef]

Sõnajalg, H.

Spalding, G. C.

D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Three-dimensional arrays of optical bottle beams,” Opt. Commun. 225(4-6), 215–222 (2003).
[CrossRef]

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23(3), 142–144 (1973).
[CrossRef]

Trillo, S.

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23(3), 142–144 (1973).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (7)

C. A. Dartora, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Properties of localized pulses through the analysis of temporal modulation effects in Bessel beams and the convolution theorem,” Opt. Commun. 229(1-6), 99–107 (2004).
[CrossRef]

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersive broadening of light with Bessel-Gauss beams,” Opt. Commun. 206(4-6), 235–241 (2002).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

M. A. Porras and I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217(1-6), 257–264 (2003).
[CrossRef]

D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Three-dimensional arrays of optical bottle beams,” Opt. Commun. 225(4-6), 215–222 (2003).
[CrossRef]

A. G. Sedukhin, “Periodically focused propagation-invariant beams with sharp central peak,” Opt. Commun. 228(4-6), 231–247 (2003).
[CrossRef]

C. Paterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124(1-2), 131–140 (1996).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066606 (2004).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

M. A. Porras, R. Borghi, and M. Santarsiero, “Few-optical-cycle bessel-gauss pulsed beams in free space,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(44 Pt B), 5729–5737 (2000).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980).
[CrossRef]

Proc. SPIE (1)

M. Dallaire, M. Piché, and N. McCarthy, “Spatiotemporal Bessel Beams,” Proc. SPIE 6796, 67963O (2007).
[CrossRef]

Other (4)

A. E. Siegman, Lasers, (University Science Books, Mill Valley, California, 1986). See p. 277, Eq. (32).

P. M. Morse, and L. Feshbach, Methods of Theoretical Physics, (McGraw-Hill, New York, 1953).

G. Arfken, Mathematical Methods for Physicists, third edition. (Academic Press, New York, 1985). See p.797.

I. S. Gradshteyn, and I. M. Ryzhik, Tables of Integrals, Series, and Products, fourth edition,( Academic Press, New York, 1980).

Supplementary Material (3)

» Media 1: MPG (414 KB)     
» Media 2: MPG (474 KB)     
» Media 3: MPG (1513 KB)     

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Figures (13)

Fig. 1
Fig. 1

Relative intensity distribution of a spatiotemporal Bessel beam of azimuthal order m = 0 along normalized space and time axes.

Fig. 2
Fig. 2

Schematic representation of the propagation of an ideal STB beam (see Media1).

Fig. 3
Fig. 3

Relative intensity profile at z = 0 of an STB beam and an STBG beam of order m = 0 as a function of the normalized space-time position. The size wo of the Gaussian envelope has been set to 14.4 in normalized units (a value equivalent to 6ΔX or 6ΔT).

Fig. 4
Fig. 4

(Media 2) illustrates the evolution of an STBG beam during propagation.

Fig. 5
Fig. 5

Evolution of the relative intensity distribution during propagation in fused silica of a) an STBG beam and b) a spatiotemporal Gaussian wave packet matched to central lobe of the STBG beam. All curves are normalized such that their maximum is unity. The local intensity at center of both beams is compared in c) as a function of propagation distance; the gray lines indicate the distance where the intensity has fallen to 50% of its value at the waist.

Fig. 6
Fig. 6

Intensity distribution at distance z = 30 zc for an STBG beam having the following parameters: Δλ = 40 nm with a central wavelength λ o = 1550 nm (which gives a = 40 mm−2), index of refraction n0 = 1.444, β 2 = −27.95 ps2/km, Gaussian envelope sizes wo = 2.95 mm and wot = 1.19 ps.

Fig. 7
Fig. 7

Theoretical spatially resolved spectrum of a spatiotemporal Bessel beam. The vertical (spatial) scale goes to infinity.

Fig. 8
Fig. 8

Beam parameter a and pulse duration ΔT as a function of the half-width of the spectrum of an STB beam (λ0 = 1550 nm, propagation in fused silica).

Fig. 9
Fig. 9

Experimental setup based on a pulse shaper with a folded diffraction grating. The figure follows the path of a short laser pulse through the pulse shaper. The reflective mask selects the optical frequencies that are recombined on the diffraction grating. Cylindrical lens #2 produces the spatial Fourier transform of the mask. See also (Media 3).

Fig. 10
Fig. 10

Autocorrelation trace of the central part of an STBG beam generated with pulses having an 18-nm spectral bandwidth.

Fig. 11
Fig. 11

- a) Experimental and b) theoretical spatial profiles of an STBG beam. c) Comparison between the theoretical and experimental intensity distributions along X (at Y = 0).

Fig. 12
Fig. 12

a) Theoretical and b) experimental spatially resolved spectra of an STBG beam.

Fig. 13
Fig. 13

Experimental spatiotemporal profile of an STBG beam, retrieved from the spatially resolved spectrum of Fig. 12(b).

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

E(r,t)     =     Re{u˜(r,t)ej(ωotβoz)},
U˜(r,ω')z  =  j2β(ω)t2U˜(r,ω')  jΔβ(ω)U˜(r,ω').
β(ω)  =  βo + β1ω' +   β2ω'22 + ...  =  βo + Δβ(ω)
v˜(x,z,T)z  =  j2βo   2v˜(x,z,T)x2 + j2β22v˜(x,z,T)T2,
2x2      βoβ22T2    2ρ2 + 1ρ   ρ     +     1ρ2   2θ2.
v˜(ρ,θ,z)z  =  j2βo[2ρ2     +     1ρ   ρ     +     1ρ2   2θ2]v˜(ρ,θ,z).
E˜m(x,z,t)     =     AmJm(ax2      T2βoβ2)ejmθ   ejωo(tz/vp)   ,
ΔX=4.8a,
ΔT=4.8βoβ2a.
u˜m(ρ,θ,z=0)  =  Am   exp[   ρ2wo2]Jm(aρ)ejmθ,
u˜m(ρ,θ,z)=jm+12πλozexp[jπρ2λoz]          ×     0Jm(2πρρλoz)exp[jπρ2λoz]u˜m(ρ,θ,z=0)ρ'dρ.
E˜m(ρ,θ,z)     =     AmjmzRejΦ(z)z2+zR2   exp[awo2z4(z+jzR)]   ×   exp[jπρ2λo(z+jzR)]   Im(azRρz+jzR)     ejmθej(ωotβoz),
wot2=wo2×(βoβ2).
zBG/zc0.38aw0,
|E˜(ρ,z)|2     Po(z)exp[2(ρσ(z))2wf2(z)],
wf(z)=λozπwo,
σ(z)     =     aλoz2π,
Po(z)     =     |Ao|2wo22aλozσ(z).
Δtz=2zaβ2βo.
δ   =     arctan(a/βo)          a/βo.
F˜(x,z,ω)=A(ω)cos   (kxx)ejkzz,
kx=a+βoβ2(ωωo)2,
kz=βoa/2βo+β1(ωωo)
A(ω)=Aoβoβ2kx,
(ωoΔωHW)     <     ω     <     (ωo+ΔωHW),
ΔωHW=aβoβ2,
a=(β0β2)×ΔωHW2.
ΔT=4.8ΔωHW.
kx2+kz2=β2(ω).
a<kx<a.
f2=πdmaskλa=πdmaskλΔωHWβ0β2,

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