## Broadband time-reversal of optical pulses using a switchable photonic-crystal mirror |

Optics Express, Vol. 19, Issue 15, pp. 14502-14507 (2011)

http://dx.doi.org/10.1364/OE.19.014502

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### Abstract

Recently, Chumak *et al.* have demonstrated experimentally the time-reversal of microwave spin pulses based on non-adiabatically tuning the wave speed in a spatially-periodic manner [Nat. Comm. 1, 141 (2010)]. Here, we solve the associated wave equations analytically, and give an explicit formula for the reversal efficiency. We discuss the implementation for short optical electromagnetic pulses and show that the new scheme may lead to their accurate time-reversal with efficiency higher than before.

© 2011 OSA

2. M. Fink, “Acoustic time-reversal mirrors: Imaging of complex media with acoustic and seismic waves,” Topics in Applied Physics **84**, 17–43 (2002). [CrossRef]

3. J. Aullbach, B. Gjonaj, P. M. Johnson, A. M. Mosk, and A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. **106**, 103901 (2011). [CrossRef]

4. O. Katz, Y. Bromberg, E. Small, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Phot. **5**, 372–377 (2011). [CrossRef]

2. M. Fink, “Acoustic time-reversal mirrors: Imaging of complex media with acoustic and seismic waves,” Topics in Applied Physics **84**, 17–43 (2002). [CrossRef]

6. J. B. Pendry, “Time-reversal and negative refraction,” Science **322**, 71–73 (2008). [CrossRef] [PubMed]

7. X. Li and M. I. Stockman, “Highly efficient spatio-temporal coherent control in nanoplasmonics on a nanometer-femtosecond scale by time-reversal,” Phys. Rev. B **77**, 195109 (2008). [CrossRef]

8. Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Phot. **2**, 110–115 (2008). [CrossRef]

9. Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, “Transmission of electrical signals by spin-wave interconversion in a magnetic insulator,” Nature **464**, 262–266 (2010). [CrossRef] [PubMed]

10. F. M. Cucchietti, “Time-reversal in an optical lattice,” J. Opt. Soc. Am. B **27**, 30–35 (2010). [CrossRef]

2. M. Fink, “Acoustic time-reversal mirrors: Imaging of complex media with acoustic and seismic waves,” Topics in Applied Physics **84**, 17–43 (2002). [CrossRef]

11. A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Qu. Electron. **28**, 2251–2261 (1992). [CrossRef]

12. D. Marom, D. Panasenko, R. Rokitski, P.-C. Sun, and Y. Fainman, “Time-reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves,” Opt. Lett. **25**, 132–134 (2000). [CrossRef]

13. O. Kuzucu, Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Spectral phase conjugation via temporal imaging,” Opt. Exp. **17**, 20605–20614 (2009). [CrossRef]

14. M. F. Yanik and S. Fan, “Time-reversal of light with linear optics and modulators,” Phys. Rev. Lett. **93**, 173903 (2004). [CrossRef] [PubMed]

15. S. Longhi, “Stopping and time-reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E **75**, 026606 (2007). [CrossRef]

16. A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time-reversal by a dynamic artificial crystal,” Nat. Comm. **1**, 141 (2010). [CrossRef]

16. A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time-reversal by a dynamic artificial crystal,” Nat. Comm. **1**, 141 (2010). [CrossRef]

22. Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , **106**, 193902 (2011). [CrossRef] [PubMed]

23. Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

16. A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time-reversal by a dynamic artificial crystal,” Nat. Comm. **1**, 141 (2010). [CrossRef]

## 1. Principles of time-reversal using a switchable mirror

**1**, 141 (2010). [CrossRef]

22. Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , **106**, 193902 (2011). [CrossRef] [PubMed]

23. Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

*U-turn*, i.e., the leading edge remains the leading edge etc.. Now, imagine that one could change of direction of the pulse propagation

*at all points in space and at the same time*. Then, obviously, the leading edge will become the trailing edge, and vice versa, i.e., the pulse is (time-) reversed.

*a spectral band as wide as possible*. Possibly the simplest way to do that would be to open a frequency bandgap by periodically modulating the material properties. Heuristically, when the bandgap is turned on, the wave cannot propagate in any direction. Instead, the forward waves are then repeatedly converted to backward waves, then back to forward waves and vice versa. If one re-establishes the transmissivity once most of the energy of the forward wave has been converted to a backward wave, then a time-reversed pulse is released backwards. In a sense, this procedure transforms a perfectly transmitting medium into a “volume” mirror. Accordingly, in what follows we refer to these schems as switchable mirror (SM) -based reversal schemes.

**1**, 141 (2010). [CrossRef]

23. Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

*although the wave-front is reversed, it is not conjugated*. Thus, the scheme can lead to perfect time-reversal only if it is complemented by a consequent step of phase-conjugation, e.g., via nearly-degenerate 4WM [22

22. Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , **106**, 193902 (2011). [CrossRef] [PubMed]

**1**, 141 (2010). [CrossRef]

**106**, 193902 (2011). [CrossRef] [PubMed]

19. A. B. Matsko, Y. V. Rostovtsev, O. Kocharovskaya, A. S. Zibrov, and M. O. Scully, “Nonadiabatic approach to quantum optical information storage,” Phys. Rev. A **64**, 043809 (2001). [CrossRef]

10. F. M. Cucchietti, “Time-reversal in an optical lattice,” J. Opt. Soc. Am. B **27**, 30–35 (2010). [CrossRef]

**106**, 193902 (2011). [CrossRef] [PubMed]

**1**, 141 (2010). [CrossRef]

## 2. Analysis

**106**, 193902 (2011). [CrossRef] [PubMed]

*x*-direction in a homogeneous medium which is time-modulated in the following manner Here,

*n*

_{0}is the average refractive index, the modulation is spatially periodic, i.e., and time-localized around

*t*

_{0}with

*max*[

*m*(

*t – t*

_{0})] =

*m*(

*t*

_{0}) = 1, i.e., the modulation essentially turns on a periodic grating perpendicular to the direction of propagation of the pulse. In this case, the Maxwell equations reduce to the 1D wave equation

**106**, 193902 (2011). [CrossRef] [PubMed]

25. C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E **54**, 1969–1989 (1996). [CrossRef]

**106**, 193902 (2011). [CrossRef] [PubMed]

*E*

^{±}represent the Slowly-Varying Envelopes (SVEs) of the forward and backward field compoenets, respectively;

*ω*

_{0}=

*ck*

_{0},

*k*

_{0}= 2

*πn*

_{0}/

*λ*and

_{v}*n*

_{0}are the carrier frequency, wavevector and refractive index of the medium, respectively, with

*λ*being the vacuum wavelength. Substituting the ansatz (4) into Eq. (3), neglecting the c.c. terms and the second order derivative terms, and removing the factor

_{v}*e*

^{−iω0t}gives

*ɛ*(

_{m}*x*) couples the forward and backward field components. When

*k*

_{0}is close to the first bandgap, i.e., when

*k*

_{0}=

*k*

^{(g)}+

*δk*(with

*k*

^{(g)}≡

*π*/

*d*or equivalently,

*j*= ±1 components of the grating are close to the phase mismatch between the forward and backward field components, so that the coupling becomes most efficient. Following [24], we now expand the spatial part of the modulation as a Fourier series as follows For a weak grating, Δ

*ɛ*≪ 1, it is justified to take only the

*j*= ±1 components of the grating [24]; this is equivalent to setting Substituting Eq. (7) in Eq. (5), neglecting all the fast-oscillating terms and separating into two sets of equations gives

**106**, 193902 (2011). [CrossRef] [PubMed]

*x*

^{(f,b)}≡

*x*∓

*vt*. We then assume that the coupling is weak and neglect the coupling term on the RHS of Eq. (10). In this case, the solution of Eq. (10) is simply where

**106**, 193902 (2011). [CrossRef] [PubMed]

*wave-front reversal*rather than a complete time-reversal (which requires also the

*conjugation*of the envelope).

*t*

_{0}= 0 and

*δk*= 0, the wave-front of the reversed component is given by where

*t*=

*t*

_{0}= 0), the width of the gap opened by the modulation is given approximately by

17. D. A. B Miller, “Time reversal of optical pulses by four-wave mixing,” Opt. Lett. **5**, 300–302 (1980). [CrossRef] [PubMed]

18. M. Tsang and D. Psaltis, “Spectral phase conjugation with cross-phase modulation compensation,” Opt. Exp. **12**, 2207–2219 (2004). [CrossRef]

14. M. F. Yanik and S. Fan, “Time-reversal of light with linear optics and modulators,” Phys. Rev. Lett. **93**, 173903 (2004). [CrossRef] [PubMed]

15. S. Longhi, “Stopping and time-reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E **75**, 026606 (2007). [CrossRef]

**1**, 141 (2010). [CrossRef]

## 3. Implementation

*LiNbO*

_{3}which is spatially-modulated in a periodic manner. A 100% reversal efficiency can be easily obtained using index modulations on the scale of 10

^{−3}[22

**106**, 193902 (2011). [CrossRef] [PubMed]

*x*direction only is to confine the probe (signal) into a thin waveguide and send a much wider and much shorter intensity-modulated pump at right angles to the waveguide (see e.g., [17

17. D. A. B Miller, “Time reversal of optical pulses by four-wave mixing,” Opt. Lett. **5**, 300–302 (1980). [CrossRef] [PubMed]

18. M. Tsang and D. Psaltis, “Spectral phase conjugation with cross-phase modulation compensation,” Opt. Exp. **12**, 2207–2219 (2004). [CrossRef]

27. G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D **217**, 31–57 (2006). [CrossRef]

28. Y. Sivan, G. Fibich, and M. I. Weinstein, “Waves in nonlinear lattices: ultrashort optical pulses and bose-einstein condensates,” Phys. Rev. Lett. **97**, 193902 (2006). [CrossRef] [PubMed]

29. B. A. Malomed, Y. V. Kartashov, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. **83**, 247–306 (2011). [CrossRef]

## References and links

1. | M. Fink, “Time-reversed acoustics,” Scientific American |

2. | M. Fink, “Acoustic time-reversal mirrors: Imaging of complex media with acoustic and seismic waves,” Topics in Applied Physics |

3. | J. Aullbach, B. Gjonaj, P. M. Johnson, A. M. Mosk, and A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. |

4. | O. Katz, Y. Bromberg, E. Small, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Phot. |

5. | D. M. Pepper, |

6. | J. B. Pendry, “Time-reversal and negative refraction,” Science |

7. | X. Li and M. I. Stockman, “Highly efficient spatio-temporal coherent control in nanoplasmonics on a nanometer-femtosecond scale by time-reversal,” Phys. Rev. B |

8. | Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Phot. |

9. | Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, “Transmission of electrical signals by spin-wave interconversion in a magnetic insulator,” Nature |

10. | F. M. Cucchietti, “Time-reversal in an optical lattice,” J. Opt. Soc. Am. B |

11. | A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Qu. Electron. |

12. | D. Marom, D. Panasenko, R. Rokitski, P.-C. Sun, and Y. Fainman, “Time-reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves,” Opt. Lett. |

13. | O. Kuzucu, Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Spectral phase conjugation via temporal imaging,” Opt. Exp. |

14. | M. F. Yanik and S. Fan, “Time-reversal of light with linear optics and modulators,” Phys. Rev. Lett. |

15. | S. Longhi, “Stopping and time-reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E |

16. | A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time-reversal by a dynamic artificial crystal,” Nat. Comm. |

17. | D. A. B Miller, “Time reversal of optical pulses by four-wave mixing,” Opt. Lett. |

18. | M. Tsang and D. Psaltis, “Spectral phase conjugation with cross-phase modulation compensation,” Opt. Exp. |

19. | A. B. Matsko, Y. V. Rostovtsev, O. Kocharovskaya, A. S. Zibrov, and M. O. Scully, “Nonadiabatic approach to quantum optical information storage,” Phys. Rev. A |

20. | G. A. Melkov, A. A. Serga, V. S. Tiberkevich, A. N. Oliynyk, and A. N. Slavin, “Wave Front Reversal of a Dipolar Spin Wave Pulse in a Non-Stationary Three-Wave Parametric Interaction,” Phys. Rev. Lett. |

21. | L. Tkeshelashvili and K. Busch, “Nonlinear three-wave interaction in photonic crystals,” Appl. Phys. B |

22. | Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , |

23. | Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583. |

24. | C. M. de Sterke and J. E. Sipe, in |

25. | C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E |

26. | P. Yeh, |

27. | G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D |

28. | Y. Sivan, G. Fibich, and M. I. Weinstein, “Waves in nonlinear lattices: ultrashort optical pulses and bose-einstein condensates,” Phys. Rev. Lett. |

29. | B. A. Malomed, Y. V. Kartashov, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. |

**OCIS Codes**

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(190.2055) Nonlinear optics : Dynamic gratings

(250.4110) Optoelectronics : Modulators

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: May 31, 2011

Revised Manuscript: June 20, 2011

Manuscript Accepted: June 20, 2011

Published: July 13, 2011

**Citation**

Yonatan Sivan and John B. Pendry, "Broadband time-reversal of optical pulses using a switchable photonic-crystal mirror," Opt. Express **19**, 14502-14507 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14502

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### References

- M. Fink, “Time-reversed acoustics,” Scientific American 91, (November1999).
- M. Fink, “Acoustic time-reversal mirrors: Imaging of complex media with acoustic and seismic waves,” Topics in Applied Physics 84, 17–43 (2002). [CrossRef]
- J. Aullbach, B. Gjonaj, P. M. Johnson, A. M. Mosk, and A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. 106, 103901 (2011). [CrossRef]
- O. Katz, Y. Bromberg, E. Small, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Phot. 5, 372–377 (2011). [CrossRef]
- D. M. Pepper, Laser handbook , Vol. 4, (North-Holland Physics, Amsterdam1988).
- J. B. Pendry, “Time-reversal and negative refraction,” Science 322, 71–73 (2008). [CrossRef] [PubMed]
- X. Li and M. I. Stockman, “Highly efficient spatio-temporal coherent control in nanoplasmonics on a nanometer-femtosecond scale by time-reversal,” Phys. Rev. B 77, 195109 (2008). [CrossRef]
- Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Phot. 2, 110–115 (2008). [CrossRef]
- Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, “Transmission of electrical signals by spin-wave interconversion in a magnetic insulator,” Nature 464, 262–266 (2010). [CrossRef] [PubMed]
- F. M. Cucchietti, “Time-reversal in an optical lattice,” J. Opt. Soc. Am. B 27, 30–35 (2010). [CrossRef]
- A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Qu. Electron. 28, 2251–2261 (1992). [CrossRef]
- D. Marom, D. Panasenko, R. Rokitski, P.-C. Sun, and Y. Fainman, “Time-reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves,” Opt. Lett. 25, 132–134 (2000). [CrossRef]
- O. Kuzucu, Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Spectral phase conjugation via temporal imaging,” Opt. Exp. 17, 20605–20614 (2009). [CrossRef]
- M. F. Yanik and S. Fan, “Time-reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004). [CrossRef] [PubMed]
- S. Longhi, “Stopping and time-reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E 75, 026606 (2007). [CrossRef]
- A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time-reversal by a dynamic artificial crystal,” Nat. Comm. 1, 141 (2010). [CrossRef]
- D. A. B Miller, “Time reversal of optical pulses by four-wave mixing,” Opt. Lett. 5, 300–302 (1980). [CrossRef] [PubMed]
- M. Tsang and D. Psaltis, “Spectral phase conjugation with cross-phase modulation compensation,” Opt. Exp. 12, 2207–2219 (2004). [CrossRef]
- A. B. Matsko, Y. V. Rostovtsev, O. Kocharovskaya, A. S. Zibrov, and M. O. Scully, “Nonadiabatic approach to quantum optical information storage,” Phys. Rev. A 64, 043809 (2001). [CrossRef]
- G. A. Melkov, A. A. Serga, V. S. Tiberkevich, A. N. Oliynyk, and A. N. Slavin, “Wave Front Reversal of a Dipolar Spin Wave Pulse in a Non-Stationary Three-Wave Parametric Interaction,” Phys. Rev. Lett. 84, 3438–3441 (2000). [CrossRef] [PubMed]
- L. Tkeshelashvili and K. Busch, “Nonlinear three-wave interaction in photonic crystals,” Appl. Phys. B 81, 225–229 (2005). [CrossRef]
- Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]
- Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583 .
- C. M. de Sterke and J. E. Sipe, in Prog. in Opt. , Vol. XXXIV, (North-Holland, Amsterdam1994).
- C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996). [CrossRef]
- P. Yeh, Optical Waves in Layered Media , (Wiley-Interscience, 2nd edition2005).
- G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D 217, 31–57 (2006). [CrossRef]
- Y. Sivan, G. Fibich, and M. I. Weinstein, “Waves in nonlinear lattices: ultrashort optical pulses and bose-einstein condensates,” Phys. Rev. Lett. 97, 193902 (2006). [CrossRef] [PubMed]
- B. A. Malomed, Y. V. Kartashov, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–306 (2011). [CrossRef]

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