Abstract

Diffraction places a fundamental limitation on the distance an optical beam propagates before its size increases and spatial details blur. We show here that imposing a judicious correlation between spatial and spectral degrees of freedom of a pulsed beam can render its transverse spatial profile independent of location along the propagation axis, thereby arresting the spread of the time-averaged beam. Such correlation introduced into a beam with arbitrary spatial profile enables spatio-temporal dispersion to compensate for purely spatial dispersion that underlies diffraction. As a result, the spatio-temporal profile in the local time-frame of the pulsed beam remains invariant at all positions along the propagation axis. One-dimensional diffraction-free space-time beams are described – including non-accelerating Airy beams, despite the well-known fact that cosine waves and accelerating Airy beams are the only one-dimensional diffraction-free solutions to the monochromatic Helmholtz equation.

© 2016 Optical Society of America

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2016 (3)

U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” Prog. Opt. 61, 237–281 (2016).
[Crossref]

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space–time characterization of ultra-intense femtosecond laser beams,” Nat. Photon. 10, 547–553 (2016).
[Crossref]

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photon. 8, 200–227 (2016).
[Crossref]

2015 (4)

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref] [PubMed]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

E. DelRe, F. Di Mei, J. Parravicini, G. Parravicini, A. J. Agranat, and C. Conti, “Subwavelength anti-diffracting beams propagating over more than 1,000 Rayleigh lengths,” Nat. Photon. 9, 228–232 (2015).

2013 (3)

M. A. Bandres and B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New J. Phys. 15, 013054 (2013).
[Crossref]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

D. Dan, M. Lei, B. Yao, W. Wang, M. Winterhalder, A. Zumbusch, Y. Qi, L. Xia, S. Yan, Y. Yang, P. Gao, T. Ye, and W. Zhao, “DMD-based LED-illumination super-resolution and optical sectioning microscopy,” Sci. Rep. 3, 1116 (2013).
[Crossref] [PubMed]

2012 (2)

M. Duocastella and C. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[Crossref]

A. Dogariu, S. Sukhov, and J. Sáenz, “Optically induced ‘negative forces’,” Nat. Photon. 7, 24–27 (2012).
[Crossref]

2011 (3)

2010 (4)

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spinâǍŞorbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4, 103–106 (2010).
[Crossref]

B. M. Rodríguez-Lara, “Normalization of optical Weber waves and Weber-Gauss beams,” J. Opt. Soc. Am. A 27, 327–332 (2010).
[Crossref]

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Prog. Opt. 54, 1–88 (2010).
[Crossref]

2009 (2)

F. Frei, A. Galler, and T. Feurer, “Space-time coupling in femtosecond pulse shaping and its effects on coherent control,” J. Chem. Phys. 130, 034302 (2009).
[Crossref] [PubMed]

A. Luis, “Coherence, polarization, and entanglement for classical light fields,” Opt. Commun. 282, 3665–3670 (2009).
[Crossref]

2008 (3)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon. 2, 675–678 (2008).
[Crossref]

B. J. Sussman, R. Lausten, and A. Stolow, “Focusing of light following a 4-f pulse shaper: Considerations for quantum control,” Phys. Rev. A 77, 043416 (2008).
[Crossref]

C.-B. Huang, Z. Jiang, D. Leaird, J. Caraquitena, and A. Weiner, “Spectral line-by-line shaping for optical and microwave arbitrary waveform generations,” Laser Photon. Rev. 2, 227–248 (2008).
[Crossref]

2007 (5)

2005 (4)

2004 (3)

2003 (3)

J. C. Gutierrez-Vega, R. M. Rodriguez-Dagnino, M. A. Meneses-Nava, and S. Chavez-Cerda, “Mathieu functions, a visual approach,” Am. J. Phys. 71, 233–242 (2003).
[Crossref]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

J. C. Vaughan, T. Feurer, and K. A. Nelson, “Automated spatiotemporal diffraction of ultrashort laser pulses,” Opt. Lett. 28, 2408–2410 (2003).
[Crossref] [PubMed]

2002 (4)

2001 (1)

2000 (2)

R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A 17, 294–303 (2000).
[Crossref]

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929–1960 (2000).
[Crossref]

1998 (3)

R. M. Koehl, T. Hattori, and K. A. Nelson, “Automated spatial and temporal shaping of femtosecond pulses,” Opt. Commun. 157, 57–61 (1998).
[Crossref]

R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
[Crossref]

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

1997 (1)

1992 (1)

1990 (1)

1988 (1)

1987 (3)

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

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

1979 (1)

M. V. Berry, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

Abouraddy, A. F.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref] [PubMed]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

Agranat, A. J.

E. DelRe, F. Di Mei, J. Parravicini, G. Parravicini, A. J. Agranat, and C. Conti, “Subwavelength anti-diffracting beams propagating over more than 1,000 Rayleigh lengths,” Nat. Photon. 9, 228–232 (2015).

Aiello, A.

Alonso,

K. J. Parker, Alonso, and M. A., “The longitudinal iso-phase condition and needle pulses,” Opt. Express, (in press) 2016

Arnold, C.

M. Duocastella and C. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[Crossref]

Bandres, M. A.

Banzer, P.

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon. 2, 675–678 (2008).
[Crossref]

Berg-Johansen, S.

Bernet, S.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photon. Rev. 5, 81–101 (2011).
[Crossref]

Berry, M. V.

M. V. Berry, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

Borges, C. V. S.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spinâǍŞorbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

Borot, A.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space–time characterization of ultra-intense femtosecond laser beams,” Nat. Photon. 10, 547–553 (2016).
[Crossref]

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]

Caraquitena, J.

C.-B. Huang, Z. Jiang, D. Leaird, J. Caraquitena, and A. Weiner, “Spectral line-by-line shaping for optical and microwave arbitrary waveform generations,” Laser Photon. Rev. 2, 227–248 (2008).
[Crossref]

Chavez-Cerda, S.

J. C. Gutierrez-Vega, R. M. Rodriguez-Dagnino, M. A. Meneses-Nava, and S. Chavez-Cerda, “Mathieu functions, a visual approach,” Am. J. Phys. 71, 233–242 (2003).
[Crossref]

Chávez-Cerda, S.

Chichkov, B.

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4, 103–106 (2010).
[Crossref]

Christodoulides, D. N.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4, 103–106 (2010).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[Crossref] [PubMed]

Conti, C.

E. DelRe, F. Di Mei, J. Parravicini, G. Parravicini, A. J. Agranat, and C. Conti, “Subwavelength anti-diffracting beams propagating over more than 1,000 Rayleigh lengths,” Nat. Photon. 9, 228–232 (2015).

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

Dan, D.

D. Dan, M. Lei, B. Yao, W. Wang, M. Winterhalder, A. Zumbusch, Y. Qi, L. Xia, S. Yan, Y. Yang, P. Gao, T. Ye, and W. Zhao, “DMD-based LED-illumination super-resolution and optical sectioning microscopy,” Sci. Rep. 3, 1116 (2013).
[Crossref] [PubMed]

DelRe, E.

E. DelRe, F. Di Mei, J. Parravicini, G. Parravicini, A. J. Agranat, and C. Conti, “Subwavelength anti-diffracting beams propagating over more than 1,000 Rayleigh lengths,” Nat. Photon. 9, 228–232 (2015).

Derevyanko, S.

U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” Prog. Opt. 61, 237–281 (2016).
[Crossref]

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon. 2, 675–678 (2008).
[Crossref]

Di Giuseppe, G.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photon. 7, 72–78 (2013).
[Crossref]

Di Mei, F.

E. DelRe, F. Di Mei, J. Parravicini, G. Parravicini, A. J. Agranat, and C. Conti, “Subwavelength anti-diffracting beams propagating over more than 1,000 Rayleigh lengths,” Nat. Photon. 9, 228–232 (2015).

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J. Opt. Soc. Am. A (7)

J. Opt. Soc. Am. B (4)

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Nat. Photon. (6)

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[Crossref]

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New J. Phys. (2)

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Figures (5)

Fig. 1
Fig. 1 Representations of optical beams in (kx, kz) space. Each point uniquely identifies a monochromatic plane wave (kx, kz; ω), ω = c k x 2 + k z 2. A circle in this plane represents all possible iso-frequency plane waves. (a) A monochromatic beam of spatial bandwidth Δkx is represented by the red arc. This representation captures the plane waves needed in constructing such a beam but not its actual shape, which depends on the amplitudes F(kx). (b) A pulsed plane wave of spectral bandwidth Δω represented by the vertical red line. (c) A pulsed beam of spatial and spectral bandwidths Δkx and Δω, respectively, is represented by the red patch. The beam diffracts and the pulse waveform disperses. (d) A diffraction-free space-time beam represented by a horizontal red line corresponding to a constant k z = K. Both spatial and temporal profiles are z-independent.
Fig. 2
Fig. 2 (a) ST spectrum F(kx, λ) for a Gaussian beam of initial FWHM xo = 10 µm and Δλ = 0.01 nm. (b) ST intensity | f (x, 0; t)|2 at z = 0. Inset shows |f (x, z; t)|2 at z/zR = 20. (c) Beam profile | f (x, z; 0)|2 at z/zR = 0, 10, 20. The beam profiles correspond to the peaks of the pulses at each z. Inset shows the time-averaged profile ∫dt| f (x, z; t)|2 at the same z. (d) Pulse shape | f (0, z; t)|2 at the beam center for z/zR = 0, 10, 20. Inset is the spatially averaged pulse shape ∫dx| f (x, z; t)|2, which is independent of z. All color maps are normalized independently. (e)–(h) Correspond to (a)–(d) except that ST correlations are introduced between kx and λ; δλ =0.01 nm and Δλ =1 nm. (e) The ST spectrum F(kx, λ). (f) The ST intensity | f (x, 0; t)|2 is localized in x and t. The inset is removed since there is no axial variation. (g) Beam profile | f (x, z; 0)|2 at z/zR =0, 10, 20. The beam profile is z-invariant. (h) Pulse shape | f (0, z; t)|2 at the beam center for z/zR =0, 10, 20. Inset is the spatially averaged pulse shape ∫dx| f (x, z; t)|2, which is independent of z.
Fig. 3
Fig. 3 Diffraction-free beams of arbitrary spatial profile with ST correlations introduced between kx and λ; the correlation uncertainty is δλ = 0.01 nm and the bandwidth is Δλ =1 nm. (a) ST spectrum F(kx, λ) for a ’bottle-beam,’ where the phase along the spatio-temporal spectrum is phase-modulated to create an odd-parity beam with a minimum at its center. (b) ST intensity | f (x, 0; t)|2 at z = 0. The inset is removed since there is no axial variation. (c) Beam profile | f (x, z; 0)|2 at z/zR =0, 10, 20. The beam profiles correspond to the peaks of the pulses at each z. Inset shows the time-averaged profile ∫dt| f (x, z; t)|2 at the same z. (d) Pulse shape | f (0, z; t)|2 at the beam center for z/zR =0, 10, 20. Inset is the spatially averaged pulse shape ∫dx| f (x, z; t)|2, which is independent of z. All color maps are normalized independently. (e)–(h) Correspond to (a)–(d) except that the phase modulation is e i π ( k x x o ) 3, which is designed to produce a non-accelerating Airy beam. (e) Real part of the ST spectrum F(kx, λ), while the inset shows the imaginary part. (f) ST intensity | f (x, 0; t)|2 at z = 0. (g) Beam profile | f (x, z; 0)2 at z/zR =0, 10, 20. Inset shows the time-averaged profile ∫dt| f (x, z; t)|2 at the same z. (h) Pulse shape | f (0, z; t)|2 at the beam center for z/zR =0, 10, 20. Inset is the spatially averaged pulse shape ∫dx |f (x, z; t)|2, which is independent of z.
Fig. 4
Fig. 4 The impact of the ST correlation uncertainty δλ on the diffraction-free propagation length. (a) ST spectrum F(kx, λ) for a Gaussian beam with ST correlations introduced between kx and λ; δλ = 0.05 nm and Δλ = 1 nm. (b) ST intensity |f (x, 0; t)|2 at z = 0. Insets show | f (x, z; t)|2 at z/zR =10, 20. (c) Beam profile |f (x, z; 0)|2 at z/z = 0, 10, 20. The beam profiles correspond to the peaks of the pulses at each z. Inset shows the time-averaged profile ∫dt| f (x, z; t)|2 at the same z. (d) Pulse shape | f (0, z; t)|2 at the beam center for z/zR = 0, 10, 20. Inset is the spatially averaged pulse shape ∫dx| f (x, z; t)|2, which is independent of z. The pulse at the beam center |f (0, z; t)|2 undergoes z-dependent delay and deformation. All color maps are normalized independently.
Fig. 5
Fig. 5 Impact of ST correlation uncertainty on the diffraction-free (DF) range zc of ST beams compared to monochromatic Gaussian beams (Δλ =0.01 nm). Smaller δλ increases zc.

Equations (4)

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f ( x , z ; t ) = d k x d ω F ( k x , ω ) e i { k x x + k z ( k x , ω ) z ω t } ;
K 2 = ( ω / c ) 2 k x 2 .
f ( x , z ; t ) = e i K z d k x F ( k x ) e i ( k x x ω x t ) = e i K z g ( x ; t ) .
Δ ω = ω max ω min = ( c Δ k x ) 2 / 2 ω o ,

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