Abstract

Space–time (ST) wave packets are coherent pulsed beams that propagate diffraction free and dispersion free by virtue of tight correlations introduced between their spatial and temporal spectral degrees of freedom. Less is known of the behavior of incoherent ST fields that maintain the spatio–temporal spectral structure of their coherent wave-packet counterparts while losing all purely spatial or temporal coherence. We show here that structuring the spatio–temporal spectrum of an incoherent field produces broadband incoherent ST fields that are diffraction free. The intensity profile of these fields consists of a narrow spatial feature atop a constant background. Spatio–temporal spectral engineering allows controlling the width of this spatial feature, tuning it from a bright to a dark diffraction-free feature, and varying its amplitude relative to the background. These results pave the way to new opportunities in the experimental investigation of optical coherence of fields jointly structured in space and time by exploiting the techniques usually associated with ultrafast optics.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (6)

B. Bhaduri, M. Yessenov, D. Reyes, J. Pena, M. Meem, S. R. Fairchild, R. Menon, M. C. Richardson, and A. F. Abouraddy, “Broadband space-time wave packets propagating 70  m,” Opt. Lett. 44, 2073–2076 (2019).

M. Yessenov, B. Bhaduri, L. Mach, D. Mardani, H. E. Kondakci, M. A. Alonso, G. A. Atia, and A. F. Abouraddy, “What is the maximum differential group delay achievable by a space-time wave packet in free space?” Opt. Express 27, 12443–12457 (2019).

H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets of arbitrary group velocity in free space,” Nat. Commun. 10, 929 (2019).
[Crossref]

B. Bhaduri, M. Yessenov, and A. F. Abouraddy, “Space-time wave packets that travel in optical materials at the speed of light in vacuum,” Optica 6, 139–146 (2019).
[Crossref]

H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-pulse-front space-time wave packets,” ACS Photon. 6, 475–481 (2019).
[Crossref]

M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Classification of propagation-invariant space-time light-sheets in free space: theory and experiments,” Phys. Rev. A 99, 023856 (2019).
[Crossref]

2018 (4)

2017 (2)

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time beams,” Nat. Photonics 11, 733–740 (2017).
[Crossref]

L. J. Wong and I. Kaminer, “Ultrashort tilted-pulsefront pulses and nonparaxial tilted-phase-front beams,” ACS Photon. 4, 2257–2264 (2017).
[Crossref]

2016 (3)

2015 (2)

Y. Lumer, Y. Liang, R. Schley, I. Kaminer, E. Greenfield, D. Song, X. Zhang, J. Xu, Z. Chen, and M. Segev, “Incoherent self-accelerating beams,” Optica 2, 886–892 (2015).
[Crossref]

P. Wang, J. A. Dominguez-Caballero, D. J. Friedman, and R. Menon, “A new class of multi-bandgap high-efficiency photovoltaics enabled by broadband diffractive optics,” Prog. Photovolt. 23, 1073–1079 (2015).
[Crossref]

2010 (2)

2009 (2)

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

J. E. Morris, M. Mazilu, J. Baumgartl, T. Čižmár, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express 17, 13236–13245 (2009).
[Crossref]

2008 (1)

2007 (2)

2006 (1)

P. Fischer, H. Little, R. L. Smith, C. Lopez-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Wavelength dependent propagation and reconstruction of white light Bessel beams,” J. Opt. A 8, 477–482 (2006).
[Crossref]

2005 (1)

2004 (2)

S. Longhi, “Gaussian pulsed beams with arbitrary speed,” Opt. Express 12, 935–940 (2004).
[Crossref]

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
[Crossref]

2002 (2)

Z. Bouchal and J. Peřina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49, 1673–1689 (2002).
[Crossref]

K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 056611 (2002).
[Crossref]

1997 (1)

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

1993 (1)

R. Donnelly and R. Ziolkowski, “Designing localized waves,” Proc. R. Soc. Lond. A 440, 541–565 (1993).
[Crossref]

1992 (1)

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. Ferroelec. Freq. Control 39, 19–31 (1992).
[Crossref]

1991 (2)

J. Turenen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
[Crossref]

A. T. Friberg, A. Vasara, and J. Turenen, “Partially coherent propagation-invariant beams: passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
[Crossref]

1987 (1)

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

1983 (1)

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[Crossref]

1979 (1)

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

1978 (1)

L. Mackinnon, “A nondispersive de Broglie wave packet,” Found. Phys. 8, 157–176 (1978).
[Crossref]

1977 (1)

C. J. R. Sheppard, “The use of lenses with annular aperture in scanning optical microscopy,” Optik 48, 329–334 (1977).

1976 (1)

1961 (1)

1872 (1)

J. W. Strutt, “On the diffraction of object glasses,” Mon. Not. R. Astron. Soc. 33, 59–63 (1872).
[Crossref]

Abouraddy, A. F.

H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets of arbitrary group velocity in free space,” Nat. Commun. 10, 929 (2019).
[Crossref]

H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-pulse-front space-time wave packets,” ACS Photon. 6, 475–481 (2019).
[Crossref]

M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Classification of propagation-invariant space-time light-sheets in free space: theory and experiments,” Phys. Rev. A 99, 023856 (2019).
[Crossref]

B. Bhaduri, M. Yessenov, and A. F. Abouraddy, “Space-time wave packets that travel in optical materials at the speed of light in vacuum,” Optica 6, 139–146 (2019).
[Crossref]

B. Bhaduri, M. Yessenov, D. Reyes, J. Pena, M. Meem, S. R. Fairchild, R. Menon, M. C. Richardson, and A. F. Abouraddy, “Broadband space-time wave packets propagating 70  m,” Opt. Lett. 44, 2073–2076 (2019).

M. Yessenov, B. Bhaduri, L. Mach, D. Mardani, H. E. Kondakci, M. A. Alonso, G. A. Atia, and A. F. Abouraddy, “What is the maximum differential group delay achievable by a space-time wave packet in free space?” Opt. Express 27, 12443–12457 (2019).

H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26, 13628–13638 (2018).
[Crossref]

B. Bhaduri, M. Yessenov, and A. F. Abouraddy, “Meters-long propagation of diffraction-free space-time light sheets,” Opt. Express 26, 20111–20121 (2018).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Self-healing of space-time light sheets,” Opt. Lett. 43, 3830–3833 (2018).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Airy wavepackets accelerating in space-time,” Phys. Rev. Lett. 120, 163901 (2018).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time beams,” Nat. Photonics 11, 733–740 (2017).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24, 28659–28668 (2016).
[Crossref]

H. E. Kondakci, M. A. Alonso, and A. F. Abouraddy, “Classical entanglement underpins the propagation invariance of space-time wave packets,” arXiv:1812.10566 (2018).

Alonso, M. A.

Atia, G. A.

Balazs, N. L.

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

Baumgartl, J.

Berry, M. V.

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

Bhaduri, B.

Bouchal, Z.

Z. Bouchal and J. Peřina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49, 1673–1689 (2002).
[Crossref]

Brittingham, J. N.

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[Crossref]

Brown, C. T. A.

P. Fischer, H. Little, R. L. Smith, C. Lopez-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Wavelength dependent propagation and reconstruction of white light Bessel beams,” J. Opt. A 8, 477–482 (2006).
[Crossref]

P. Fischer, C. T. A. Brown, J. E. Morris, C. López-Marscal, E. M. Wright, W. Sibbett, and K. Dholakia, “White light propagation invariant beams,” Opt. Express 13, 6657–6666 (2005).
[Crossref]

Cada, M.

Chen, Z.

Christodoulides, D. N.

H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-pulse-front space-time wave packets,” ACS Photon. 6, 475–481 (2019).
[Crossref]

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

Cižmár, T.

Derevyanko, S.

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

Dholakia, K.

Dominguez-Caballero, J. A.

P. Wang, J. A. Dominguez-Caballero, D. J. Friedman, and R. Menon, “A new class of multi-bandgap high-efficiency photovoltaics enabled by broadband diffractive optics,” Prog. Photovolt. 23, 1073–1079 (2015).
[Crossref]

Donnelly, R.

R. Donnelly and R. Ziolkowski, “Designing localized waves,” Proc. R. Soc. Lond. A 440, 541–565 (1993).
[Crossref]

Durnin, J.

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

Eberly, J. H.

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

Fairchild, S. R.

Fischer, P.

P. Fischer, H. Little, R. L. Smith, C. Lopez-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Wavelength dependent propagation and reconstruction of white light Bessel beams,” J. Opt. A 8, 477–482 (2006).
[Crossref]

P. Fischer, C. T. A. Brown, J. E. Morris, C. López-Marscal, E. M. Wright, W. Sibbett, and K. Dholakia, “White light propagation invariant beams,” Opt. Express 13, 6657–6666 (2005).
[Crossref]

Friberg, A. T.

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

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

J. Turenen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
[Crossref]

A. T. Friberg, A. Vasara, and J. Turenen, “Partially coherent propagation-invariant beams: passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
[Crossref]

Friedman, D. J.

P. Wang, J. A. Dominguez-Caballero, D. J. Friedman, and R. Menon, “A new class of multi-bandgap high-efficiency photovoltaics enabled by broadband diffractive optics,” Prog. Photovolt. 23, 1073–1079 (2015).
[Crossref]

Greenfield, E.

Greenleaf, J. F.

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. Ferroelec. Freq. Control 39, 19–31 (1992).
[Crossref]

Hendrych, M.

Huang, W.

Kaminer, I.

L. J. Wong and I. Kaminer, “Ultrashort tilted-pulsefront pulses and nonparaxial tilted-phase-front beams,” ACS Photon. 4, 2257–2264 (2017).
[Crossref]

Y. Lumer, Y. Liang, R. Schley, I. Kaminer, E. Greenfield, D. Song, X. Zhang, J. Xu, Z. Chen, and M. Segev, “Incoherent self-accelerating beams,” Optica 2, 886–892 (2015).
[Crossref]

Kondakci, H. E.

M. Yessenov, B. Bhaduri, L. Mach, D. Mardani, H. E. Kondakci, M. A. Alonso, G. A. Atia, and A. F. Abouraddy, “What is the maximum differential group delay achievable by a space-time wave packet in free space?” Opt. Express 27, 12443–12457 (2019).

M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Classification of propagation-invariant space-time light-sheets in free space: theory and experiments,” Phys. Rev. A 99, 023856 (2019).
[Crossref]

H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-pulse-front space-time wave packets,” ACS Photon. 6, 475–481 (2019).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets of arbitrary group velocity in free space,” Nat. Commun. 10, 929 (2019).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Airy wavepackets accelerating in space-time,” Phys. Rev. Lett. 120, 163901 (2018).
[Crossref]

H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26, 13628–13638 (2018).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Self-healing of space-time light sheets,” Opt. Lett. 43, 3830–3833 (2018).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time beams,” Nat. Photonics 11, 733–740 (2017).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24, 28659–28668 (2016).
[Crossref]

H. E. Kondakci, M. A. Alonso, and A. F. Abouraddy, “Classical entanglement underpins the propagation invariance of space-time wave packets,” arXiv:1812.10566 (2018).

Levy, U.

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

Liang, Y.

Little, H.

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K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
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H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-pulse-front space-time wave packets,” ACS Photon. 6, 475–481 (2019).
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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. Ferroelec. Freq. Control 39, 19–31 (1992).
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P. Fischer, H. Little, R. L. Smith, C. Lopez-Mariscal, C. T. A. Brown, W. Sibbett, and K. Dholakia, “Wavelength dependent propagation and reconstruction of white light Bessel beams,” J. Opt. A 8, 477–482 (2006).
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J. Opt. Soc. Am. A (1)

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J. W. Strutt, “On the diffraction of object glasses,” Mon. Not. R. Astron. Soc. 33, 59–63 (1872).
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H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets of arbitrary group velocity in free space,” Nat. Commun. 10, 929 (2019).
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H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time beams,” Nat. Photonics 11, 733–740 (2017).
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Opt. Express (9)

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24, 28659–28668 (2016).
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K. J. Parker and M. A. Alonso, “The longitudinal iso-phase condition and needle pulses,” Opt. Express 24, 28669–28677 (2016).
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B. Bhaduri, M. Yessenov, and A. F. Abouraddy, “Meters-long propagation of diffraction-free space-time light sheets,” Opt. Express 26, 20111–20121 (2018).
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M. Yessenov, B. Bhaduri, L. Mach, D. Mardani, H. E. Kondakci, M. A. Alonso, G. A. Atia, and A. F. Abouraddy, “What is the maximum differential group delay achievable by a space-time wave packet in free space?” Opt. Express 27, 12443–12457 (2019).

J. Turunen, “Space-time coherence of polychromatic propagation-invariant fields,” Opt. Express 16, 20283–20294 (2008).
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P. Fischer, C. T. A. Brown, J. E. Morris, C. López-Marscal, E. M. Wright, W. Sibbett, and K. Dholakia, “White light propagation invariant beams,” Opt. Express 13, 6657–6666 (2005).
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J. E. Morris, M. Mazilu, J. Baumgartl, T. Čižmár, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express 17, 13236–13245 (2009).
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H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26, 13628–13638 (2018).
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Optica (2)

Optik (1)

C. J. R. Sheppard, “The use of lenses with annular aperture in scanning optical microscopy,” Optik 48, 329–334 (1977).

Phys. Rev. A (3)

M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Classification of propagation-invariant space-time light-sheets in free space: theory and experiments,” Phys. Rev. A 99, 023856 (2019).
[Crossref]

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[Crossref]

A. T. Friberg, A. Vasara, and J. Turenen, “Partially coherent propagation-invariant beams: passage through paraxial optical systems,” Phys. Rev. A 43, 7079–7082 (1991).
[Crossref]

Phys. Rev. E (2)

K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 056611 (2002).
[Crossref]

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
[Crossref]

Phys. Rev. Lett. (3)

H. E. Kondakci and A. F. Abouraddy, “Airy wavepackets accelerating in space-time,” Phys. Rev. Lett. 120, 163901 (2018).
[Crossref]

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

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

Proc. R. Soc. Lond. A (1)

R. Donnelly and R. Ziolkowski, “Designing localized waves,” Proc. R. Soc. Lond. A 440, 541–565 (1993).
[Crossref]

Prog. Opt. (2)

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

U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” Prog. Opt. 61, 237–281 (2016).
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Prog. Photovolt. (1)

P. Wang, J. A. Dominguez-Caballero, D. J. Friedman, and R. Menon, “A new class of multi-bandgap high-efficiency photovoltaics enabled by broadband diffractive optics,” Prog. Photovolt. 23, 1073–1079 (2015).
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Other (3)

H. E. Hernández-Figueroa, E. Recami, and M. Zamboni-Rached, eds., Non-Diffracting Waves (Wiley-VCH, 2014).

H. E. Kondakci, M. A. Alonso, and A. F. Abouraddy, “Classical entanglement underpins the propagation invariance of space-time wave packets,” arXiv:1812.10566 (2018).

B. E. A. Saleh and M. C. Teich, Principles of Photonics (Wiley, 2007).

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Figures (6)

Fig. 1.
Fig. 1. Concept of diffraction-free coherent and incoherent space–time (ST) fields. Each panel depicts a spatio–temporal spectrum in the (kx,ω) plane. For coherent fields, the spectra should be interpreted as spectral densities |E˜(kx,ω)|2. For incoherent fields, the spectral coherence function G˜(kx,ω;kx,ω) in Eq. (8) is cast as G˜(kx,ω)δ(ωω)δ(|kx||kx|), and the spectra plotted here correspond to G˜(kx,ω). Insets show the spatio–temporal spectral configurations on the surface of the light-cone kx2+kz2=(ωc)2. (a) Typical spectrum separable with respect to the spatial and temporal degrees of freedom; Δkx and Δω are the spatial and temporal bandwidths, respectively. (b) Spatial filtering of the spectrum in (a) yields a plane wave with finite temporal bandwidth, corresponding to the intersection of the light-cone with the plane kx=0. (c) Temporal filtering of the spectrum in (a) yields a quasi-monochromatic field with finite spatial bandwidth, corresponding to the intersection of the light-cone with a plane ω=ωo. (d) Spatio–temporal filtering of the spectrum in (a) yields a curved spectral trajectory that we refer to as an ST field, corresponding to the intersection of the light-cone with a tilted plane parallel to the kx axis. If this trajectory satisfies the constraint in Eq. (1) with coherent light, then we have a propagation-invariant pulsed beam or wave packet; otherwise, we have a diffraction-free broadband incoherent ST field. Here, δω is the spectral uncertainty in the association between spatial and temporal frequencies.
Fig. 2.
Fig. 2. Optical setup for synthesizing incoherent ST fields. The arrangement consists of four sections delimited by the different-colored backgrounds: (i) spatial filtering via a 2f-system comprising spherical lenses (L2s and L3s) with a 200-μm-diameter pinhole P placed in the Fourier plane; (ii) field synthesis consisting of diffraction gratings DG1 and DG2, a transmissive phase plate, and cylindrical lenses (L4y and L5y) for collimation; (iii) field analysis in physical space via a two-lens telescope (L6x and L7x) and an axially scanning camera CCD1; (iv) spatio–temporal spectral analysis in the (kx,λ) domain via a 2f-system along the x axis (L8s) to CCD2. The gratings are used in reflection mode, but are depicted here in transmission mode for simplicity. d1=L1=L2=4cm. d2=L3=3.5cm. d3=d4=L4=L5=25cm. L6=20cm. d5=L7=5cm. d6=32.5cm. d7=L8=7.5cm. DG, grating; A, aperture; P, pinhole; M, mirror; BS, beam splitter; Ls, spherical lens; Ly, cylindrical lens along y; Lx, cylindrical lens along x.
Fig. 3.
Fig. 3. Measurements of the spatio–temporal spectrum and free-space propagation of broadband incoherent ST wave packets. The first column displays the spatio–temporal spectrum G˜(kx,λ), the second displays the same spectral density in the (kz,ω) plane, with kz normalized with respect to ko=2πλo [λo=830nm, except in (a), where λo=803nm] and ω is normalized with respect to ωo, and the third column displays the time-averaged intensity of the field along the z axis, I(x,z) with y=0. (a) Measurements carried out with a coherent mode-locked femtosecond Ti-Sapphire laser. (b)–(d) Measurements carried out with the incoherent field from a LED. From (b) to (d) we gradually increase the spatial bandwidth Δkx, which results also in an increase in the temporal bandwidth Δλ. The spatial width of the field in the transverse plane Δx accordingly decreases from (b) Δx=17.2μm to (d) 10 μm. The dashed white lines in (a) and (d) identify the plane z=z0 of the measurements in Figs. 5, 6, and 4(b).
Fig. 4.
Fig. 4. Measurement of the spectral bandwidth of ST field along propagation axis and intensity from fast detector at fixed axial plane. (a) FWHM of spectrum of ST field generated from coherent mode-locked femtosecond Ti-Sapphire laser (blue points with blue linear fit line) and broadband incoherent LED (red points with red linear fit line). (b) Normalized intensity of the coherent and incoherent ST fields.
Fig. 5.
Fig. 5. Controlling the ratio of the constant background G1(0) and the diffraction-free narrow spatial feature G2(2x) through spatio–temporal spectral filtering. The first column shows the portion of the phase distribution ϕ on the phase plate that is unblocked; the second column shows the spatio–temporal spectrum G˜(kx,ω), the third column the time-averaged transverse intensity I(x,y,z0) captured by CCD1 at the plane z=z0, which is identified by the dashed white line in the plot of I(x,z) in Fig. 3(d); and the last column shows the 1D propagation-invariant intensity distribution I(x,z0) resulting from integration over y.
Fig. 6.
Fig. 6. Bright and dark diffraction-free incoherent ST fields. The first column shows the transverse time-averaged intensity I(x,y,z0), and the second column shows a 1D intensity I(x,z0) resulting from integration over y. (a) A bright incoherent ST field is formed when the spatial spectral function G˜2(kx) is even. (b) An intermediate bright-dark incoherent ST field in which the relative phase between the positive and negative halves of the spatial spectrum is π2. (c) A dark incoherent ST field is formed when the spatial spectral function G˜2(kx) is odd, corresponding to a π-step in phase between the two halves of the spectrum.

Equations (13)

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ωc=ko+(kzko)tanθ,
E(x,z,t)=ei(kozωot)dkxψ˜(kx)eikxxei(ωωo)(tzccotθ),
I(x,z)=I(x,0)=dkx|ψ˜(kx)|2+dkxψ˜(kx)ψ˜*(kx)ei2kxx,
G(x1,z;x2,z)=dkxdkxG˜(kx,kx)ei(kxx1kxx2)ei(kzkz)z,
G˜(kx,kx)=G˜1(kx)δ(kxkx)+G˜2(kx)δ(kx+kx),
G(x1,x2)=G1(x1x2)+G2(x1+x2),
I(x)=G1(0)+G2(2x)=G1(0)+dkx|G˜2(kx)|cos(2kxx+ϕ(kx)).
G(x1,z1,t1;x2,z2,t2)=dkxdkxdωdωG˜(kx,ω;kx,ω)ei(kxx1kxx2)ei(kzz1kzz2)ei(ωt1ωt2),
G(x1,z1,t1;x2,z2,t2)=dkxdkxG˜(kx,kx)ei(kxx1kxx2)ei(ωωo)(t1z1ccotθ)ei(ωωo)(t2z2ccotθ),
G(x1,z,t1;x2,z,t2)=eiωo(t1t2)dkxdkxG˜(kx,kx)ei(kxx1kxx2)ei(ωt1ωt2)eizccotθ(ωω).
G(x1,z,t1;x2,z,t2)=G1(x1x2,t1t2)+G2(x1+x2,t1t2),
I(x)=G1(0,0)+G2(2x,0),
G(x,t;x,t+τ)=G1(2x,τ)+G2(0,τ).