## Formation of X-pulses in periodically gain/loss modulated materials |

Optics Express, Vol. 20, Issue 10, pp. 11271-11276 (2012)

http://dx.doi.org/10.1364/OE.20.011271

Acrobat PDF (1270 KB)

### Abstract

We propose a versatile and efficient technique for the formation of X-pulses in materials with a periodical gain/loss modulation on the wavelength scale. We show that in such materials the strong wave-vector anisotropy of amplification/attenuation of the Bloch modes enables the shaping of ultra-short light pulses around the edges of the first Brillouin zone. X-pulses generation is numerically demonstrated and the optimum conditions are derived; specific characteristics of X-pulses can be tailored by appropriate selection of the geometry and modulation depth.

© 2012 OSA

## 1. Introduction

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. Ferroelectr. Freq. Control **39**(1), 19–31 (1992). [CrossRef] [PubMed]

3. J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control **39**(3), 441–446 (1992). [CrossRef] [PubMed]

4. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. **79**(21), 4135–4138 (1997). [CrossRef]

5. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**(4), 651–654 (1987). [CrossRef]

6. J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**(15), 1499–1501 (1987). [CrossRef] [PubMed]

7. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. **46**(1), 15–28 (2005). [CrossRef]

8. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**(3), 264–267 (1979). [CrossRef]

9. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef] [PubMed]

10. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics **4**(2), 103–106 (2010). [CrossRef]

11. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. **105**(25), 253901 (2010). [CrossRef] [PubMed]

12. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. **44**(8), 592–597 (1954). [CrossRef]

13. S. Fujiwara, “Optical properties of conic surfaces: I. Reflecting cone,” J. Opt. Soc. Am. **52**(3), 287–292 (1962). [CrossRef]

8. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**(3), 264–267 (1979). [CrossRef]

9. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef] [PubMed]

14. H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. **22**(5), 310–312 (1997). [CrossRef] [PubMed]

15. 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**(9), 093904 (2003). [CrossRef] [PubMed]

16. O. Jedrkiewicz, M. Clerici, E. Rubino, and P. Di Trapani, “Generation and control of phase-locked conical wave packets in type-I seeded optical parametric amplification,” Phys. Rev. A **80**(3), 033813 (2009). [CrossRef]

17. A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **73**(1), 016608 (2006). [CrossRef] [PubMed]

18. D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. **96**(19), 193901 (2006). [CrossRef] [PubMed]

19. D. Faccio, A. Averchi, A. Dubietis, P. Polesana, A. Piskarskas, P. D. Trapani, and A. Couairon, “Stimulated Raman X waves in ultrashort optical pulse filamentation,” Opt. Lett. **32**(2), 184–186 (2007). [CrossRef] [PubMed]

*k*,

*ω*) space is mapped into the spatio-temporal pulse spectrum [20

20. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. **90**(17), 170406 (2003). [CrossRef] [PubMed]

21. K. Staliunas, R. Herrero, and R. Vilaseca, “Subdiffraction and spatial filtering due to periodic spatial modulation of the gain/loss profile,” Phys. Rev. A **80**(1), 013821 (2009). [CrossRef]

22. M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain/loss modulation on a wavelength scale,” Phys. Rev. A **82**(1), 013828 (2010). [CrossRef]

*ω*(

*k*), of the Bloch modes are modified in GLMMs in a different way than in PhCs. Nevertheless, some beam propagation effects reported for PhCs, such as self-collimation or flat lens focusing [23

23. R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. **34**(12), 1589–1617 (1987). [CrossRef]

21. K. Staliunas, R. Herrero, and R. Vilaseca, “Subdiffraction and spatial filtering due to periodic spatial modulation of the gain/loss profile,” Phys. Rev. A **80**(1), 013821 (2009). [CrossRef]

21. K. Staliunas, R. Herrero, and R. Vilaseca, “Subdiffraction and spatial filtering due to periodic spatial modulation of the gain/loss profile,” Phys. Rev. A **80**(1), 013821 (2009). [CrossRef]

24. C. Conti and S. Trillo, “Nonspreading wave packets in three dimensions formed by an ultracold bose gas in an optical lattice,” Phys. Rev. Lett. **92**(12), 120404 (2004). [CrossRef] [PubMed]

27. A. Di Falco, C. Conti, and S. Trillo, “Tunneling mediated by 2D+1 conical waves in a 1D lattice,” Phys. Rev. Lett. **101**(1), 013601 (2008). [CrossRef] [PubMed]

22. M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain/loss modulation on a wavelength scale,” Phys. Rev. A **82**(1), 013828 (2010). [CrossRef]

22. M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain/loss modulation on a wavelength scale,” Phys. Rev. A **82**(1), 013828 (2010). [CrossRef]

## 2. Model and methods

*μ*= 1) material, Maxwell's equations in 2D read: where

*E*,

_{z}*D*and,

_{z}*H*,

_{x,y}*B*are the electric and magnetic field and displacement components, and

_{x,y}*ε*

_{0}and

*μ*

_{0}are the electric and magnetic constants in vacuum. In Eq. (2) the refractive index is modulated by the spatial dependence of electric susceptibility,

*ε*(

*x,y*), whereas the spatial variation of conductivity

*σ*(

*x*,

*y*), describes the gain/loss modulation. Note that in paraxial models the gain/loss is usually accounted by an effective imaginary part of the complex refractive index.

*ε*(

*x,y*) = 1.

*N*= 2 non-collinear vectors

*x*-axis along the bisection between the two [see Fig. 2 ], i.e.

*α*is the angle between the vectors

*and L*

_{x}*denote the longitudinal (along the light propagation direction) and transverse lattice periods, respectively. For*

_{y}*N*= 3 lattices the third vector can be chosen as

*α*= 90° and 2

*α*= 60°(120°), respectively, whereas other values of

*α*correspond to rhombic lattices. Figures 2(b) and 2(c) show rhombic lattices formed by either two or three spatial harmonics, being of square- or hexagonal-like type. Light propagates along the

*x*-axis, i.e. along the lattice cell diagonal, which corresponds to ΓM direction in square lattices and ΓK (ΓM) crystallographic directions in rhombic lattices with L

_{x}< L

_{y}(L

_{x}> L

_{y}).

## 3. X-pulse formation

^{2}widths of 4 periods and 4 wavelengths [see Fig. 3(a) ]. The pulse enters on the front face (

*x*= 0) of a 20λ-long GLMM structure and propagates along the cell diagonal (

*x*-axis) [see Fig. 2]. This length is sufficient for X-pulse formation, but we have checked that the generated pulses propagate keeping their X-shape in GLMM for a distance at least of 100λ.The spatial and spectral profiles of the output pulse are characterized in free space just behind the structure. In all simulations presented below the input pulse is maintained while we scan longitudinal and transverse modulation periods.

*α*= 90°, we choose the gain-loss profile amplitude corresponding to an absorption/gain coefficient of 5 × 10

^{3}cm

^{−1}, which is a typical value for semiconductors. For simplicity the refractive index is assumed to be equal to unity, allowing us to investigate the effects caused by the pure gain/loss modulation. Figures 3(b-d) show the excitation of X-pulses in square GLMMs with different lattice periods. As expected, the most efficient X-pulse excitation, i.e. the highest amplitude of the output X-pulse, occurs when the carrier frequency of the input pulse coincides with the corner (M-point) of the first BZ, where the effective gain is maximum and X-shaped, see Fig. 3(b).

_{x}/L

_{y}, that characterizes the lattice, corresponds to a given 2

*α*angle between both reciprocal lattice vectors. Spatial profiles of the pulses behind the rhombic GLMMs are presented in Figs. 3(e,f) as obtained by PSTD simulations. Bright white points in the corresponding spatial spectra represent the enhanced light amplification at the edge of the BZ. In Figs. 3(e) and 3(f) the corner of the BZ corresponds to the central wavelength of the input pulse, resulting in an effective formation of the X-shaped output pulse. The apex angle of the output pulse [see Fig. 3(e)] is defined by the reciprocal lattice angle

*α*, making possible the generation of nondiffracting X-pulse for materials with different temporal dispersions.

*N*= 2, rhombic lattices, such condition is given by the crossing of the dispersion circles of the fundamental and two nearest spatial harmonics centered at

_{y}/λ > 1.

## 4. Conclusions

## Acknowledgments

## References and links

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. Ferroelectr. Freq. Control |

2. | H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, eds., |

3. | J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control |

4. | P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. |

5. | J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

6. | J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

7. | D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. |

8. | M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. |

9. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. |

10. | A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics |

11. | D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. |

12. | J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. |

13. | S. Fujiwara, “Optical properties of conic surfaces: I. Reflecting cone,” J. Opt. Soc. Am. |

14. | H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. |

15. | 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. |

16. | O. Jedrkiewicz, M. Clerici, E. Rubino, and P. Di Trapani, “Generation and control of phase-locked conical wave packets in type-I seeded optical parametric amplification,” Phys. Rev. A |

17. | A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

18. | D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. |

19. | D. Faccio, A. Averchi, A. Dubietis, P. Polesana, A. Piskarskas, P. D. Trapani, and A. Couairon, “Stimulated Raman X waves in ultrashort optical pulse filamentation,” Opt. Lett. |

20. | C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. |

21. | K. Staliunas, R. Herrero, and R. Vilaseca, “Subdiffraction and spatial filtering due to periodic spatial modulation of the gain/loss profile,” Phys. Rev. A |

22. | M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain/loss modulation on a wavelength scale,” Phys. Rev. A |

23. | R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. |

24. | C. Conti and S. Trillo, “Nonspreading wave packets in three dimensions formed by an ultracold bose gas in an optical lattice,” Phys. Rev. Lett. |

25. | S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B |

26. | S. Longhi, “Localized and nonspreading spatiotemporal Wannier wave packets in photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

27. | A. Di Falco, C. Conti, and S. Trillo, “Tunneling mediated by 2D+1 conical waves in a 1D lattice,” Phys. Rev. Lett. |

28. | B. Fornberg, |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(050.1940) Diffraction and gratings : Diffraction

(070.6110) Fourier optics and signal processing : Spatial filtering

(320.5540) Ultrafast optics : Pulse shaping

(320.5550) Ultrafast optics : Pulses

(070.3185) Fourier optics and signal processing : Invariant optical fields

(050.5298) Diffraction and gratings : Photonic crystals

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: February 28, 2012

Revised Manuscript: April 13, 2012

Manuscript Accepted: April 17, 2012

Published: May 2, 2012

**Citation**

Yurii Loiko, Muriel Botey, Ramon Herrero, and Kestutis Staliunas, "Formation of X-pulses in periodically gain/loss modulated materials," Opt. Express **20**, 11271-11276 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-10-11271

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### References

- 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. Ferroelectr. Freq. Control39(1), 19–31 (1992). [CrossRef] [PubMed]
- H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (John Wiley and Sons, 2008).
- J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control39(3), 441–446 (1992). [CrossRef] [PubMed]
- P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett.79(21), 4135–4138 (1997). [CrossRef]
- J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A4(4), 651–654 (1987). [CrossRef]
- J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987). [CrossRef] [PubMed]
- D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys.46(1), 15–28 (2005). [CrossRef]
- M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264–267 (1979). [CrossRef]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007). [CrossRef] [PubMed]
- A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4(2), 103–106 (2010). [CrossRef]
- D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett.105(25), 253901 (2010). [CrossRef] [PubMed]
- J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am.44(8), 592–597 (1954). [CrossRef]
- S. Fujiwara, “Optical properties of conic surfaces: I. Reflecting cone,” J. Opt. Soc. Am.52(3), 287–292 (1962). [CrossRef]
- H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett.22(5), 310–312 (1997). [CrossRef] [PubMed]
- 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(9), 093904 (2003). [CrossRef] [PubMed]
- O. Jedrkiewicz, M. Clerici, E. Rubino, and P. Di Trapani, “Generation and control of phase-locked conical wave packets in type-I seeded optical parametric amplification,” Phys. Rev. A80(3), 033813 (2009). [CrossRef]
- A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.73(1), 016608 (2006). [CrossRef] [PubMed]
- D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett.96(19), 193901 (2006). [CrossRef] [PubMed]
- D. Faccio, A. Averchi, A. Dubietis, P. Polesana, A. Piskarskas, P. D. Trapani, and A. Couairon, “Stimulated Raman X waves in ultrashort optical pulse filamentation,” Opt. Lett.32(2), 184–186 (2007). [CrossRef] [PubMed]
- C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett.90(17), 170406 (2003). [CrossRef] [PubMed]
- K. Staliunas, R. Herrero, and R. Vilaseca, “Subdiffraction and spatial filtering due to periodic spatial modulation of the gain/loss profile,” Phys. Rev. A80(1), 013821 (2009). [CrossRef]
- M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain/loss modulation on a wavelength scale,” Phys. Rev. A82(1), 013828 (2010). [CrossRef]
- R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt.34(12), 1589–1617 (1987). [CrossRef]
- C. Conti and S. Trillo, “Nonspreading wave packets in three dimensions formed by an ultracold bose gas in an optical lattice,” Phys. Rev. Lett.92(12), 120404 (2004). [CrossRef] [PubMed]
- S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B70(23), 235123 (2004). [CrossRef]
- S. Longhi, “Localized and nonspreading spatiotemporal Wannier wave packets in photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(1), 016603 (2005). [CrossRef] [PubMed]
- A. Di Falco, C. Conti, and S. Trillo, “Tunneling mediated by 2D+1 conical waves in a 1D lattice,” Phys. Rev. Lett.101(1), 013601 (2008). [CrossRef] [PubMed]
- B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge University Press, 1996).

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