Abstract

We theoretically propose a temporal cloaking scheme based on accelerating wave packets. A part of a monochromatic light wave is endowed with a discontinuous nonlinear frequency chirp, so that two opposite accelerating caustics are created in space–time as the different frequency components propagate in the presence of dispersion. The two caustics open a biconvex time gap that contains negligible optical energy, thus concealing the enclosed events. In contrast to previous temporal cloaking schemes, where light propagates successively through two different media with opposite dispersions, accelerating wave packets open and close the cloaked time window continuously in a single dispersive medium. In addition, biconvex time gaps can be tailored into arbitrary shapes and offer a larger suppression of intensity compared with their rhombic counterparts.

© 2014 Optical Society of America

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
    [CrossRef]
  2. U. Leonhardt, Science 312, 1777 (2006).
    [CrossRef]
  3. A. Greenleaf, M. Lassas, and G. Uhlmann, Physiol. Meas. 24, 413 (2003).
    [CrossRef]
  4. D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, Science 314, 977 (2006).
    [CrossRef]
  5. W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, Nat. Photonics 1, 224 (2007).
    [CrossRef]
  6. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
    [CrossRef]
  7. L. Gabrielli, J. Cardenas, C. Poitras, and M. Lipson, Nat. Photonics 3, 461 (2009).
    [CrossRef]
  8. S. Zhang, in Transformation Electromagnetics and Metamaterials, D. H. Werner and D.-H. Kwon, eds. (Springer, 2014), pp. 289–314.
  9. M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, J. Opt. 13, 024003 (2011).
    [CrossRef]
  10. P. Kinsler and M. W. McCall, Ann. Phys. 526, 51 (2014).
    [CrossRef]
  11. M. Fridman, A. Farsi, Y. Okawachi, and A. Gaeta, Nature 481, 62 (2012).
    [CrossRef]
  12. R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, Opt. Lett. 33, 1047 (2008).
    [CrossRef]
  13. J. Lukens, D. Leaird, and A. Weiner, Nature 498, 205 (2013).
    [CrossRef]
  14. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
    [CrossRef]
  15. N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010).
    [CrossRef]
  16. I. Chremmos, P. Zhang, J. Prakash, N. Efremidis, D. Christodoulides, and Z. Chen, Opt. Lett. 36, 3675 (2011).
    [CrossRef]
  17. I. Chremmos, Z. Chen, D. Christodoulides, and N. Efremidis, Phys. Rev. A 85, 023828 (2012).
    [CrossRef]
  18. R. B. Li, L. Deng, E. W. Hagley, J. C. Bienfang, M. G. Payne, and M.-L. Ge, Phys. Rev. A 87, 023839 (2013).
    [CrossRef]
  19. I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, Opt. Lett. 36, 1890 (2011).
    [CrossRef]
  20. G. Agrawal, Nonlinear Fiber Optics, 5th ed., Series in Optics and Photonics (Academic, 2012).
  21. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, IEEE Press Series on Electromagnetic Wave Theory (Wiley, 1994).
  22. A. M. Weiner, Rev. Sci. Instrum. 71, 1929 (2000).
    [CrossRef]
  23. I. Besieris and A. Shaarawi, Phys. Rev. E 78, 046605 (2008).
    [CrossRef]
  24. R. Driben, Y. Hu, Z. Chen, B. A. Malomed, and R. Morandotti, Opt. Lett. 38, 2499 (2013).
    [CrossRef]
  25. A. V. Yulin, R. Driben, B. A. Malomed, and D. V. Skryabin, Opt. Express 21, 14481 (2013).
    [CrossRef]
  26. A. Chong, W. Renninger, D. Christodoulides, and F. Wise, Nat. Photonics 4, 103 (2010).
    [CrossRef]
  27. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
    [CrossRef]
  28. A. Farsi, M. Fridman, and A. L. Gaeta, in CLEO (Optical Society of America, 2013), paper CM3L.1.
  29. C. Ament, P. Polynkin, and J. V. Moloney, Phys. Rev. Lett. 107, 243901 (2011).
    [CrossRef]

2014

P. Kinsler and M. W. McCall, Ann. Phys. 526, 51 (2014).
[CrossRef]

2013

J. Lukens, D. Leaird, and A. Weiner, Nature 498, 205 (2013).
[CrossRef]

R. B. Li, L. Deng, E. W. Hagley, J. C. Bienfang, M. G. Payne, and M.-L. Ge, Phys. Rev. A 87, 023839 (2013).
[CrossRef]

R. Driben, Y. Hu, Z. Chen, B. A. Malomed, and R. Morandotti, Opt. Lett. 38, 2499 (2013).
[CrossRef]

A. V. Yulin, R. Driben, B. A. Malomed, and D. V. Skryabin, Opt. Express 21, 14481 (2013).
[CrossRef]

2012

I. Chremmos, Z. Chen, D. Christodoulides, and N. Efremidis, Phys. Rev. A 85, 023828 (2012).
[CrossRef]

M. Fridman, A. Farsi, Y. Okawachi, and A. Gaeta, Nature 481, 62 (2012).
[CrossRef]

2011

I. Chremmos, P. Zhang, J. Prakash, N. Efremidis, D. Christodoulides, and Z. Chen, Opt. Lett. 36, 3675 (2011).
[CrossRef]

I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, Opt. Lett. 36, 1890 (2011).
[CrossRef]

M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, J. Opt. 13, 024003 (2011).
[CrossRef]

C. Ament, P. Polynkin, and J. V. Moloney, Phys. Rev. Lett. 107, 243901 (2011).
[CrossRef]

2010

A. Chong, W. Renninger, D. Christodoulides, and F. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010).
[CrossRef]

2009

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
[CrossRef]

L. Gabrielli, J. Cardenas, C. Poitras, and M. Lipson, Nat. Photonics 3, 461 (2009).
[CrossRef]

2008

2007

W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, Nat. Photonics 1, 224 (2007).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef]

2006

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, Science 314, 977 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef]

U. Leonhardt, Science 312, 1777 (2006).
[CrossRef]

2003

A. Greenleaf, M. Lassas, and G. Uhlmann, Physiol. Meas. 24, 413 (2003).
[CrossRef]

2000

A. M. Weiner, Rev. Sci. Instrum. 71, 1929 (2000).
[CrossRef]

Abdollahpour, D.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 5th ed., Series in Optics and Photonics (Academic, 2012).

Ament, C.

C. Ament, P. Polynkin, and J. V. Moloney, Phys. Rev. Lett. 107, 243901 (2011).
[CrossRef]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
[CrossRef]

Besieris, I.

I. Besieris and A. Shaarawi, Phys. Rev. E 78, 046605 (2008).
[CrossRef]

Bienfang, J. C.

R. B. Li, L. Deng, E. W. Hagley, J. C. Bienfang, M. G. Payne, and M.-L. Ge, Phys. Rev. A 87, 023839 (2013).
[CrossRef]

Boardman, A.

M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, J. Opt. 13, 024003 (2011).
[CrossRef]

Cai, W.

W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, Nat. Photonics 1, 224 (2007).
[CrossRef]

Cardenas, J.

L. Gabrielli, J. Cardenas, C. Poitras, and M. Lipson, Nat. Photonics 3, 461 (2009).
[CrossRef]

Chen, Z.

Chettiar, U.

W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, Nat. Photonics 1, 224 (2007).
[CrossRef]

Chong, A.

A. Chong, W. Renninger, D. Christodoulides, and F. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Chremmos, I.

Christodoulides, D.

I. Chremmos, Z. Chen, D. Christodoulides, and N. Efremidis, Phys. Rev. A 85, 023828 (2012).
[CrossRef]

I. Chremmos, P. Zhang, J. Prakash, N. Efremidis, D. Christodoulides, and Z. Chen, Opt. Lett. 36, 3675 (2011).
[CrossRef]

A. Chong, W. Renninger, D. Christodoulides, and F. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Christodoulides, D. N.

Cummer, S.

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, Science 314, 977 (2006).
[CrossRef]

Deng, L.

R. B. Li, L. Deng, E. W. Hagley, J. C. Bienfang, M. G. Payne, and M.-L. Ge, Phys. Rev. A 87, 023839 (2013).
[CrossRef]

Driben, R.

Efremidis, N.

I. Chremmos, Z. Chen, D. Christodoulides, and N. Efremidis, Phys. Rev. A 85, 023828 (2012).
[CrossRef]

I. Chremmos, P. Zhang, J. Prakash, N. Efremidis, D. Christodoulides, and Z. Chen, Opt. Lett. 36, 3675 (2011).
[CrossRef]

Efremidis, N. K.

Farsi, A.

M. Fridman, A. Farsi, Y. Okawachi, and A. Gaeta, Nature 481, 62 (2012).
[CrossRef]

A. Farsi, M. Fridman, and A. L. Gaeta, in CLEO (Optical Society of America, 2013), paper CM3L.1.

Favaro, A.

M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, J. Opt. 13, 024003 (2011).
[CrossRef]

Felsen, L.

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, IEEE Press Series on Electromagnetic Wave Theory (Wiley, 1994).

Foster, M. A.

Fridman, M.

M. Fridman, A. Farsi, Y. Okawachi, and A. Gaeta, Nature 481, 62 (2012).
[CrossRef]

A. Farsi, M. Fridman, and A. L. Gaeta, in CLEO (Optical Society of America, 2013), paper CM3L.1.

Gabrielli, L.

L. Gabrielli, J. Cardenas, C. Poitras, and M. Lipson, Nat. Photonics 3, 461 (2009).
[CrossRef]

Gaeta, A.

M. Fridman, A. Farsi, Y. Okawachi, and A. Gaeta, Nature 481, 62 (2012).
[CrossRef]

Gaeta, A. L.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, Opt. Lett. 33, 1047 (2008).
[CrossRef]

A. Farsi, M. Fridman, and A. L. Gaeta, in CLEO (Optical Society of America, 2013), paper CM3L.1.

Ge, M.-L.

R. B. Li, L. Deng, E. W. Hagley, J. C. Bienfang, M. G. Payne, and M.-L. Ge, Phys. Rev. A 87, 023839 (2013).
[CrossRef]

Geraghty, D. F.

Greenleaf, A.

A. Greenleaf, M. Lassas, and G. Uhlmann, Physiol. Meas. 24, 413 (2003).
[CrossRef]

Hagley, E. W.

R. B. Li, L. Deng, E. W. Hagley, J. C. Bienfang, M. G. Payne, and M.-L. Ge, Phys. Rev. A 87, 023839 (2013).
[CrossRef]

Hu, Y.

Justice, B.

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, Science 314, 977 (2006).
[CrossRef]

Kildishev, A.

W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, Nat. Photonics 1, 224 (2007).
[CrossRef]

Kinsler, P.

P. Kinsler and M. W. McCall, Ann. Phys. 526, 51 (2014).
[CrossRef]

M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, J. Opt. 13, 024003 (2011).
[CrossRef]

Lassas, M.

A. Greenleaf, M. Lassas, and G. Uhlmann, Physiol. Meas. 24, 413 (2003).
[CrossRef]

Leaird, D.

J. Lukens, D. Leaird, and A. Weiner, Nature 498, 205 (2013).
[CrossRef]

Leonhardt, U.

U. Leonhardt, Science 312, 1777 (2006).
[CrossRef]

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
[CrossRef]

Li, R. B.

R. B. Li, L. Deng, E. W. Hagley, J. C. Bienfang, M. G. Payne, and M.-L. Ge, Phys. Rev. A 87, 023839 (2013).
[CrossRef]

Lipson, M.

Lukens, J.

J. Lukens, D. Leaird, and A. Weiner, Nature 498, 205 (2013).
[CrossRef]

Malomed, B. A.

Marcuvitz, N.

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, IEEE Press Series on Electromagnetic Wave Theory (Wiley, 1994).

McCall, M. W.

P. Kinsler and M. W. McCall, Ann. Phys. 526, 51 (2014).
[CrossRef]

M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, J. Opt. 13, 024003 (2011).
[CrossRef]

Mock, J.

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, Science 314, 977 (2006).
[CrossRef]

Moloney, J. V.

C. Ament, P. Polynkin, and J. V. Moloney, Phys. Rev. Lett. 107, 243901 (2011).
[CrossRef]

Morandotti, R.

Okawachi, Y.

M. Fridman, A. Farsi, Y. Okawachi, and A. Gaeta, Nature 481, 62 (2012).
[CrossRef]

Papazoglou, D. G.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Payne, M. G.

R. B. Li, L. Deng, E. W. Hagley, J. C. Bienfang, M. G. Payne, and M.-L. Ge, Phys. Rev. A 87, 023839 (2013).
[CrossRef]

Pendry, J.

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, Science 314, 977 (2006).
[CrossRef]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef]

Poitras, C.

L. Gabrielli, J. Cardenas, C. Poitras, and M. Lipson, Nat. Photonics 3, 461 (2009).
[CrossRef]

Polynkin, P.

C. Ament, P. Polynkin, and J. V. Moloney, Phys. Rev. Lett. 107, 243901 (2011).
[CrossRef]

Prakash, J.

Renninger, W.

A. Chong, W. Renninger, D. Christodoulides, and F. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Salem, R.

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef]

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, Science 314, 977 (2006).
[CrossRef]

Shaarawi, A.

I. Besieris and A. Shaarawi, Phys. Rev. E 78, 046605 (2008).
[CrossRef]

Shalaev, V.

W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, Nat. Photonics 1, 224 (2007).
[CrossRef]

Siviloglou, G. A.

Skryabin, D. V.

Smith, D.

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, Science 314, 977 (2006).
[CrossRef]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef]

Starr, A.

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, Science 314, 977 (2006).
[CrossRef]

Suntsov, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Turner, A. C.

Tzortzakis, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Uhlmann, G.

A. Greenleaf, M. Lassas, and G. Uhlmann, Physiol. Meas. 24, 413 (2003).
[CrossRef]

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
[CrossRef]

Weiner, A.

J. Lukens, D. Leaird, and A. Weiner, Nature 498, 205 (2013).
[CrossRef]

Weiner, A. M.

A. M. Weiner, Rev. Sci. Instrum. 71, 1929 (2000).
[CrossRef]

Wise, F.

A. Chong, W. Renninger, D. Christodoulides, and F. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Yulin, A. V.

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
[CrossRef]

Zhang, P.

Zhang, S.

S. Zhang, in Transformation Electromagnetics and Metamaterials, D. H. Werner and D.-H. Kwon, eds. (Springer, 2014), pp. 289–314.

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
[CrossRef]

Ann. Phys.

P. Kinsler and M. W. McCall, Ann. Phys. 526, 51 (2014).
[CrossRef]

J. Opt.

M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, J. Opt. 13, 024003 (2011).
[CrossRef]

Nat. Mater.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
[CrossRef]

Nat. Photonics

L. Gabrielli, J. Cardenas, C. Poitras, and M. Lipson, Nat. Photonics 3, 461 (2009).
[CrossRef]

W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, Nat. Photonics 1, 224 (2007).
[CrossRef]

A. Chong, W. Renninger, D. Christodoulides, and F. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Nature

M. Fridman, A. Farsi, Y. Okawachi, and A. Gaeta, Nature 481, 62 (2012).
[CrossRef]

J. Lukens, D. Leaird, and A. Weiner, Nature 498, 205 (2013).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

I. Chremmos, Z. Chen, D. Christodoulides, and N. Efremidis, Phys. Rev. A 85, 023828 (2012).
[CrossRef]

R. B. Li, L. Deng, E. W. Hagley, J. C. Bienfang, M. G. Payne, and M.-L. Ge, Phys. Rev. A 87, 023839 (2013).
[CrossRef]

Phys. Rev. E

I. Besieris and A. Shaarawi, Phys. Rev. E 78, 046605 (2008).
[CrossRef]

Phys. Rev. Lett.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

C. Ament, P. Polynkin, and J. V. Moloney, Phys. Rev. Lett. 107, 243901 (2011).
[CrossRef]

Physiol. Meas.

A. Greenleaf, M. Lassas, and G. Uhlmann, Physiol. Meas. 24, 413 (2003).
[CrossRef]

Rev. Sci. Instrum.

A. M. Weiner, Rev. Sci. Instrum. 71, 1929 (2000).
[CrossRef]

Science

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, Science 314, 977 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef]

U. Leonhardt, Science 312, 1777 (2006).
[CrossRef]

Other

S. Zhang, in Transformation Electromagnetics and Metamaterials, D. H. Werner and D.-H. Kwon, eds. (Springer, 2014), pp. 289–314.

G. Agrawal, Nonlinear Fiber Optics, 5th ed., Series in Optics and Photonics (Academic, 2012).

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, IEEE Press Series on Electromagnetic Wave Theory (Wiley, 1994).

A. Farsi, M. Fridman, and A. L. Gaeta, in CLEO (Optical Society of America, 2013), paper CM3L.1.

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Figures (3)

Fig. 1.
Fig. 1.

(a) Ray schematic of TC with accelerating wave packets. A section of a CW lightwave with frequency ω0 (green rays) is frequency-modulated by a time-lens. The different frequency components propagate under dispersion as colored rays with different slopes to form a biconvex bottle. A second time-lens demodulates the wave back to its CW form. Stars denote cloaked events. (b) Instantaneous frequency detuning δω(t)=ω(t)ω0 of the probe wave versus time at different distances. Time is measured in a frame that moves with the group velocity at ω0. (c) Ray schematic serving the analysis.

Fig. 2.
Fig. 2.

(a) TC with a wave packet with input phase q(τ)=4|τ|8|τ|3/2/(310) for |τ|40, corresponding to the caustic f(z)=4z0.4z2 (dashed curves). The input amplitude has been assumed uniform. (b) Phase of the wave packet at the input and output. The solid and dashed lines correspond to q(t,L) and q(t,0+), respectively. (c) and (d) Same information for the caustic f(z)=20/3(4/75)|z5|3. For the input amplitude, we have chosen a(τ)=0.5 in |τ>40| and a(τ)=1z/10 in |τ|<40, where z is obtained by solving numerically τ=f(z)zf(z). The input phase has also been computed numerically.

Fig. 3.
Fig. 3.

Comparison between TC with parabolic biconvex and rhombic time gaps. (a) Wave packet with input phase q(τ)=6|τ|4|τ|3/2/(15) for |τ|60 corresponding to the parabolic caustic f(z)=6z0.6z2. (b) Wave packet with parabolic input phase q(τ)=(|τ|15)2/10 in |τ|30 and constant elsewhere. (c) Intensities of (a) [solid blue line] and (b) [dashed red line] at z=L/2.

Equations (8)

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

u(t,z)=12πizu(τ,0+)ei(tτ)22zdτ,
t=τ+q(τ)z,
τc=f(zc)zcf(zc)q(τc)=f(zc),
u(tc,L)izcLzca(τc)eiq(τc)+i(tcτc)22L,
tc=f(zc)+(Lzc)f(zc),
q(τc)+q(tc)+L2(f(zc))2=π2,
q(tc)=f(zc).
f(zc)=f(Lzc),

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