Abstract

Einstein’s theory of relativity establishes the speed of light in vacuum, c, as a fundamental constant. However, the speed of light pulses can be altered significantly in dispersive materials. While significant control can be exerted over the speed of light in such media, to our knowledge no experimental demonstration of altered light speeds has hitherto been achieved in vacuum for “twisted” optical beams. We show that “twisted” light pulses exhibit subluminal velocities in vacuum, being 0.1% slowed relative to c. This work does not challenge relativity theory but experimentally supports a body of theoretical work on the counterintuitive vacuum group velocities of twisted pulses. These results are particularly important given recent interest in applications of twisted light to quantum information, communication, and quantum key distribution.

© 2016 Optical Society of America

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References

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2015 (2)

D. Giovannini, J. Romero, V. Potoček, G. Ferenczi, F. Speirits, S. M. Barnett, D. Faccio, and M. J. Padgett, Science 347, 857 (2015).
[Crossref]

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, Sci. Adv. 1, e1500087 (2015).
[Crossref]

2014 (1)

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, New J. Phys. 16, 113028 (2014).
[Crossref]

2013 (3)

2012 (1)

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[Crossref]

2010 (1)

T. Visser and E. Wolf, Opt. Commun. 283, 3371 (2010).
[Crossref]

2009 (3)

N. V. Budko, Phys. Rev. Lett. 102, 020401 (2009).
[Crossref]

K. B. Kuntz, B. Braverman, S. H. Youn, M. Lobino, E. M. Pessina, and A. I. Lvovsky, Phys. Rev. A 79, 043802 (2009).
[Crossref]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photonics 3, 720 (2009).
[Crossref]

2008 (1)

M. Berry and K. McDonald, J. Opt. A 10, 035005 (2008).
[Crossref]

2007 (2)

2006 (1)

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, Science 312, 895 (2006).
[Crossref]

2005 (1)

2004 (2)

2003 (2)

M. A. Porras, I. Gonzalo, and A. Mondello, Phys. Rev. E 67, 066604 (2003).
[Crossref]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200 (2003).
[Crossref]

2002 (1)

N.-H. Liu, S.-Y. Zhu, H. Chen, and X. Wu, Phys. Rev. E 65, 046607 (2002).
[Crossref]

2001 (2)

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, Phys. Rev. Lett. 87, 253902 (2001).
[Crossref]

A. Dogariu, A. Kuzmich, and L. Wang, Phys. Rev. A 63, 053806 (2001).
[Crossref]

1999 (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[Crossref]

1997 (2)

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
[Crossref]

J. Courtial, K. Dholakia, L. Allen, and M. Padgett, Phys. Rev. A 56, 4193 (1997).
[Crossref]

1996 (1)

Z. Horváth, J. Vinkó, Z. Bor, and D. Von der Linde, Appl. Phys. B 63, 481 (1996).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[Crossref]

Ahmed, N.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[Crossref]

Allen, L.

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57(5), 35 (2004).
[Crossref]

J. Courtial, K. Dholakia, L. Allen, and M. Padgett, Phys. Rev. A 56, 4193 (1997).
[Crossref]

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[Crossref]

Altucci, C.

Barnett, S. M.

D. Giovannini, J. Romero, V. Potoček, G. Ferenczi, F. Speirits, S. M. Barnett, D. Faccio, and M. J. Padgett, Science 347, 857 (2015).
[Crossref]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[Crossref]

Barsi, C.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, Science 312, 895 (2006).
[Crossref]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[Crossref]

Berry, M.

M. Berry and K. McDonald, J. Opt. A 10, 035005 (2008).
[Crossref]

Bigelow, M. S.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200 (2003).
[Crossref]

Bor, Z.

Z. Horváth, J. Vinkó, Z. Bor, and D. Von der Linde, Appl. Phys. B 63, 481 (1996).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), Chap. 1.

Boyd, R. W.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, Sci. Adv. 1, e1500087 (2015).
[Crossref]

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, Science 312, 895 (2006).
[Crossref]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200 (2003).
[Crossref]

Braverman, B.

K. B. Kuntz, B. Braverman, S. H. Youn, M. Lobino, E. M. Pessina, and A. I. Lvovsky, Phys. Rev. A 79, 043802 (2009).
[Crossref]

Budko, N. V.

N. V. Budko, Phys. Rev. Lett. 102, 020401 (2009).
[Crossref]

Cardano, F.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, Sci. Adv. 1, e1500087 (2015).
[Crossref]

Chen, H.

N.-H. Liu, S.-Y. Zhu, H. Chen, and X. Wu, Phys. Rev. E 65, 046607 (2002).
[Crossref]

Courtial, J.

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57(5), 35 (2004).
[Crossref]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[Crossref]

J. Courtial, K. Dholakia, L. Allen, and M. Padgett, Phys. Rev. A 56, 4193 (1997).
[Crossref]

de Lisio, C.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, Sci. Adv. 1, e1500087 (2015).
[Crossref]

De Martini, F.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photonics 3, 720 (2009).
[Crossref]

DeLong, K. W.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
[Crossref]

Dholakia, K.

J. Courtial, K. Dholakia, L. Allen, and M. Padgett, Phys. Rev. A 56, 4193 (1997).
[Crossref]

Dogariu, A.

A. Dogariu, A. Kuzmich, and L. Wang, Phys. Rev. A 63, 053806 (2001).
[Crossref]

Dolinar, S.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[Crossref]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[Crossref]

Faccio, D.

D. Giovannini, J. Romero, V. Potoček, G. Ferenczi, F. Speirits, S. M. Barnett, D. Faccio, and M. J. Padgett, Science 347, 857 (2015).
[Crossref]

T. Roger, J. J. Heitz, E. M. Wright, and D. Faccio, Sci. Rep. 3, 3491 (2013).
[Crossref]

Fazal, I. M.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[Crossref]

Ferenczi, G.

D. Giovannini, J. Romero, V. Potoček, G. Ferenczi, F. Speirits, S. M. Barnett, D. Faccio, and M. J. Padgett, Science 347, 857 (2015).
[Crossref]

Fickler, R.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, New J. Phys. 16, 113028 (2014).
[Crossref]

Fink, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, New J. Phys. 16, 113028 (2014).
[Crossref]

Fittinghoff, D. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
[Crossref]

Franke-Arnold, S.

Gaeta, A.

Gehring, G. M.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, Science 312, 895 (2006).
[Crossref]

Gibson, G.

Giovannini, D.

D. Giovannini, J. Romero, V. Potoček, G. Ferenczi, F. Speirits, S. M. Barnett, D. Faccio, and M. J. Padgett, Science 347, 857 (2015).
[Crossref]

Gonzalo, I.

M. A. Porras, I. Gonzalo, and A. Mondello, Phys. Rev. E 67, 066604 (2003).
[Crossref]

Handsteiner, J.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, New J. Phys. 16, 113028 (2014).
[Crossref]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[Crossref]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[Crossref]

Heitz, J. J.

T. Roger, J. J. Heitz, E. M. Wright, and D. Faccio, Sci. Rep. 3, 3491 (2013).
[Crossref]

Horváth, Z.

Z. Horváth, J. Vinkó, Z. Bor, and D. Von der Linde, Appl. Phys. B 63, 481 (1996).
[Crossref]

Huang, H.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[Crossref]

Kane, D. J.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
[Crossref]

Karimi, E.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, Sci. Adv. 1, e1500087 (2015).
[Crossref]

E. Karimi, C. Altucci, V. Tosa, R. Velotta, and L. Marrucci, Opt. Express 21, 24991 (2013).
[Crossref]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photonics 3, 720 (2009).
[Crossref]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
[Crossref]

Kostinski, N.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, Science 312, 895 (2006).
[Crossref]

Krenn, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, New J. Phys. 16, 113028 (2014).
[Crossref]

Krumbügel, M. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
[Crossref]

Kuntz, K. B.

K. B. Kuntz, B. Braverman, S. H. Youn, M. Lobino, E. M. Pessina, and A. I. Lvovsky, Phys. Rev. A 79, 043802 (2009).
[Crossref]

Kuzmich, A.

A. Dogariu, A. Kuzmich, and L. Wang, Phys. Rev. A 63, 053806 (2001).
[Crossref]

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200 (2003).
[Crossref]

Liu, N.-H.

N.-H. Liu, S.-Y. Zhu, H. Chen, and X. Wu, Phys. Rev. E 65, 046607 (2002).
[Crossref]

Lobino, M.

K. B. Kuntz, B. Braverman, S. H. Youn, M. Lobino, E. M. Pessina, and A. I. Lvovsky, Phys. Rev. A 79, 043802 (2009).
[Crossref]

Lvovsky, A. I.

K. B. Kuntz, B. Braverman, S. H. Youn, M. Lobino, E. M. Pessina, and A. I. Lvovsky, Phys. Rev. A 79, 043802 (2009).
[Crossref]

Mahon, R. J.

Malik, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, New J. Phys. 16, 113028 (2014).
[Crossref]

Marrucci, L.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, Sci. Adv. 1, e1500087 (2015).
[Crossref]

E. Karimi, C. Altucci, V. Tosa, R. Velotta, and L. Marrucci, Opt. Express 21, 24991 (2013).
[Crossref]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photonics 3, 720 (2009).
[Crossref]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
[Crossref]

Massa, F.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, Sci. Adv. 1, e1500087 (2015).
[Crossref]

McDonald, K.

M. Berry and K. McDonald, J. Opt. A 10, 035005 (2008).
[Crossref]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[Crossref]

Mondello, A.

M. A. Porras, I. Gonzalo, and A. Mondello, Phys. Rev. E 67, 066604 (2003).
[Crossref]

Murphy, J. A.

Nagali, E.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photonics 3, 720 (2009).
[Crossref]

Notomi, M.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, Phys. Rev. Lett. 87, 253902 (2001).
[Crossref]

Okawachi, Y.

Padgett, M.

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57(5), 35 (2004).
[Crossref]

J. Courtial, K. Dholakia, L. Allen, and M. Padgett, Phys. Rev. A 56, 4193 (1997).
[Crossref]

Padgett, M. J.

D. Giovannini, J. Romero, V. Potoček, G. Ferenczi, F. Speirits, S. M. Barnett, D. Faccio, and M. J. Padgett, Science 347, 857 (2015).
[Crossref]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[Crossref]

Paparo, D.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, Sci. Adv. 1, e1500087 (2015).
[Crossref]

Pas’ko, V.

Pessina, E. M.

K. B. Kuntz, B. Braverman, S. H. Youn, M. Lobino, E. M. Pessina, and A. I. Lvovsky, Phys. Rev. A 79, 043802 (2009).
[Crossref]

Piccirillo, B.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photonics 3, 720 (2009).
[Crossref]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
[Crossref]

Porras, M. A.

M. A. Porras, I. Gonzalo, and A. Mondello, Phys. Rev. E 67, 066604 (2003).
[Crossref]

Potocek, V.

D. Giovannini, J. Romero, V. Potoček, G. Ferenczi, F. Speirits, S. M. Barnett, D. Faccio, and M. J. Padgett, Science 347, 857 (2015).
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Supplementary Material (1)

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Figures (3)

Fig. 1.
Fig. 1.

Group velocities of focused Laguerre–Gauss modes as a function of propagation distance. (a) Group velocities as a function of propagation distance for LG modes with = 0 and various values of p . These velocities pertain to the on-axis case, r = 0 . Three different propagation regions can be identified: a first subluminal zone z < z R , a superluminal zone | z | < z R , and a second subluminal zone z > z R , where z R = ω w 0 2 / ( 2 c ) is the Rayleigh range, at which all modes travel at speed c . (b) Propagation dependence of group velocities for LG beams characterized by p = 0 and different values of . These group velocities are calculated not along the beam axis but at the radial position corresponding to the beam intensity maximum r max = w ( z ) | | / 2 . r max defines the circle through which will pass all rays associated with a given LG mode in the ray tracing picture. With the exception of the = 0 case, all modes are found to exhibit subluminal behavior throughout propagation. These results are obtained from simulations of light at a wavelength of 795 nm with a beam waist of 2.5 mm, focused by a thin lens of focal length 400 mm.

Fig. 2.
Fig. 2.

Experimental setup for measuring subluminal speeds of twisted pulsed beams in vacuum. The test HyGG beam is generated using a spatial light modulator (SLM) onto which is displayed the hologram shown in the figure. Reference LG 0,0 and test HyGG beams are focused into a type-I beta-barium borate (BBO) crystal. Second-harmonic generation (SHG) output transverse intensity profiles of the test, cross, and reference beams are shown in parts (i)–(iii) of the rightmost inset. When the delay between the test and reference arms is on the order of the pulse duration, one photon from each can be upconverted to produce an SHG photon. By conservation of linear momentum, this photon will exit the BBO crystal at the bisector between the SHG beams produced by the test and reference beams alone. Conservation of OAM further requires that the SHG outputs (i)–(iii) carry OAM values of 2 , , and 0 [22,23]. The inset shows beam profiles obtained experimentally for the case = 1 . Note that the hologram shown here generates a pure mode with an infinite superposition of p modes, where most of the power resides in the fundamental p = 0 mode. Legend: BS-beam splitter, T-delay line (trombone), M-mirror, f-lens, I-iris, and D-detector.

Fig. 3.
Fig. 3.

Subluminal light propagation of HyGG modes in free space. (a) Normalized power obtained from the autocorrelation technique described in the text as a function of delay stage position, d , for HyGG beams with = 0 , 2 , 4 , 6 . Raw data are presented along with the fits used to estimate pulse peak positions. Power was measured at each stage position over a period of sufficient length to suppress Poissonian noise. The widths of the autocorrelation traces were found to fall in the range of 201 ± 6    μm , but in the interest of clarity, only the peaks of the autocorrelation traces are shown. (b) Arrival times of HyGG pulses relative to an LG 0,0 reference pulse extracted from peak positions determined directly from experimental data. A linear relationship between arrival time and is observed, and the experimental data (blue dots) are plotted along with theoretical data (red dots) obtained from the group velocity expression derived in Supplement 1.

Equations (1)

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Φ LG ( r , ω ) = ω c z + ω r 2 2 c R + φ ( 2 p + | | + 1 ) ζ ,

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