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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 10 — May. 7, 2012
  • pp: 11271–11276
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Formation of X-pulses in periodically gain/loss modulated materials

Yurii Loiko, Muriel Botey, Ramon Herrero, and Kestutis Staliunas  »View Author Affiliations


Optics Express, Vol. 20, Issue 10, pp. 11271-11276 (2012)
http://dx.doi.org/10.1364/OE.20.011271


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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

Propagation of pulses with non-gaussian spatio-temporal envelopes in diffractive and dispersive media can be peculiar. Perhaps the best known example of such exotic pulses is the so called X-pulse, characterized by an X-like shape of its spatio-temporal spectrum [1

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]

,2

2. H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (John Wiley and Sons, 2008).

]. In an ideal case, pulses with an infinitely narrow X-spectrum do not experience any distortion in propagation through homogeneous media with a quadratic dispersion. In a realistic case, i.e. for pulses of a finite spectral width and/or when higher dispersion orders are not negligible, “imperfect” X-pulses preserve their shapes for anomalously long propagation distances [3

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

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

].

X-pulses are three-dimensional (3D) analogs of non-diffractive Bessel beams in 2D, proposed by Durnin [5

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

,6

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

], and widely used in many areas of optics [7

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

]. Whereas Bessel beams are monochromatic beams that propagate without any diffractive spreading, X-pulses can be envisaged as a superposition of Bessel beams of different frequencies. Apart from X-pulses in 3D and Bessel beams in 2D, there exist several other exotic waveforms, such as Airy beams in 2D [8

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

,9

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]

] and Airy pulses in 3D, which owe an interesting property of self-bending, i.e. can propagate “behind” an obstacle [10

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

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]

].

Experimentally, the generation of monochromatic 2D either Bessel or Airy beams is relatively simple, since any arbitrary spatial distribution can be obtained by linear means, i.e. by using different phase and amplitude masks (axicon lenses for Bessel beams [12

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

,13

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

], or holograms for Airy beams [8

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

,9

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]

]). In contrast, the formation of exotic 3D pulses is more involved, as their envelopes cannot be factorized by spatial and temporal distributions. Therefore, the angular (space) and frequency (temporal) modulation have to be performed simultaneously. In principle, X-pulses can be generated by linear means [14

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]

], however the known linear techniques are not commonly used. In practice, nonlinear techniques for the spontaneous formation of X-pulses have been developed. Such nonlinear techniques are based on second harmonics generation [15

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]

], parametric amplification [16

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]

], filamentation [17

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

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]

], and Raman scattering [19

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]

]. In these techniques a shape of phase-matching function of nonlinear process in (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]

]. However, nonlinear techniques usually require high input light intensities.

Here we propose a relatively simple, versatile and direct linear technique for the formation of X-pulses by propagating them through Gain/Loss Modulated Materials (GLMM) [21

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

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]

], the materials where the gain/loss is periodically modulated on the wavelength scale. Hence, they differ from the well-known Photonic Crystals (PhCs), since not the real part of the refractive index but rather its imaginary part is modulated in space. The spatial dispersion curves represented by the isofrequencies, 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]

], are also possible in GLMMs [21

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]

]. In addition, GLMMs show novel effects: such as a strong dependence of the Bloch mode amplification/ absorption on its wave-vector, i.e. strong gain/ loss anisotropy [21

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]

]. The latter property is exploited here to provide a new and efficient technique for spatio-temporal shaping of ultra-short light pulses, and in particular for efficient formation of X-pulses. We emphasize that the proposed technique is based on anisotropic gain rather than on the modification of spatial dispersion reported in index-modulated materials [24

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

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]

].

The anisotropy of gain/loss has been studied in detail in [22

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]

], where the complex valued spatial dispersions have been obtained with the Plane Wave Expansion (PWE) method. A common description of PhC's properties is based on isolines of the real-valued frequency, which are calculated and plotted in wave-vector space. It has been shown [22

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]

] that the imaginary part of the frequency, which corresponds to the amplification/absorption of the propagating Bloch modes, displays sharp peaks and narrow lines in spatial Fourier domain. The enhanced gain areas are situated at the edges of the first Brillouin Zone (BZ). Figure 1
Fig. 1 (a) Gain/loss response of a GLMM with a square lattice in wave-vector space, as obtained by PWE. (b) Formation of an X-shaped wave: convolution of the GLMM response and a Gaussian input pulse (dashed circle) with carrier frequency centered at the corner of the BZ. (c) Gain/loss response of a GLMM structure with rhombic lattice. Brighter regions correspond to higher gain (the imaginary part of the frequency), in adimensional Lx/c units; where c is the speed of light in vacuum and L, Lx,y denote the lattice parameters.
presents a PWE calculation of the imaginary part of the frequencies in the first band to illustrate the gain profile.

Inspection of Fig. 1(a) suggests the possibility of pulse formation with particular spatio-temporal spectra, for instance, by injecting a Gaussian pulse with spatio-temporal spectrum centered at the corner of the first BZ, as indicated by the dashed circle in Fig. 1(b). This basic idea is numerically demonstrated in this letter by performing calculations of pulse propagation in GLMMs on the basis of Maxwell’s equations. The idea can be generalized to provide X-pulses with variable apex angle in GLMM with rhombic lattice symmetries, see Fig. 1(c).

2. Model and methods

Light propagation in a planar structure can be modeled by considering a 2D geometry. For TM polarization in a non-magnetic (μ = 1) material, Maxwell's equations in 2D read:
tDz=xHyyHx,tBx=yEz,tBy=xEz,
(1)
tDz=ε0ε(x,y)tEz+σ(x,y)Ez,Bx,y=μ0Hx,y,
(2)
where Ez, Dz and, Hx,y, Bx,y are the electric and magnetic field and displacement components, and ε0 and μ0 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.

GLMMs with specific parameters can be fabricated e.g. from semiconductor materials. We consider smooth spatial profiles for the conductivity, because carrier diffusion in semiconductors usually deteriorates steep spatial profile of the gain and loss areas. Moreover, this also eliminates numerical instabilities arising from mathematical discontinuities. In the simplest case, square, rhombic and hexagonal lattice symmetries can be imprinted by superimposing two or three spatial harmonics. The conductivity of such 2D GLMMs with enhanced gain areas embedded in a background of loss can be written as follows:
σ(x,y)=1Nσ0m=1Ncos(qmr),
(3)
where r=(x,y) denotes position in space and we assume ε(x,y) = 1.

For N = 2 non-collinear vectors q1,2 form basis of reciprocal lattice and we choose the x-axis along the bisection between the two [see Fig. 2
Fig. 2 Square (a), rhombic (b,c) and hexagonal (d) GLMM lattices formed by two (a,b) and three (c,d) spatial harmonics. Rhombic lattices could be envisaged as squeezed square and hexagonal lattices, respectively. Open circles in the insets represent direct lattice, while q1,2,3 denote the reciprocal lattice vectors.
], i.e. q1,2=(qx,±qy), where qx=qcosα=2π/Lx, qy=qsinα=2π/Ly, and 2α is the angle between the vectors q1,2. Lx and Ly denote the longitudinal (along the light propagation direction) and transverse lattice periods, respectively. For N = 3 lattices the third vector can be chosen as q1+q2=q3=(2qx,0). Square and hexagonal lattices correspond to 2α = 90° and 2α = 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 Lx < Ly (Lx > Ly).

3. X-pulse formation

In order to efficiently exploit the response of the GLMM structures around the edges of the first BZ, we apply an ultra-short and narrow input Gaussian pulse (λ = 800nm)-with temporal and spatial 1/e2 widths of 4 periods and 4 wavelengths [see Fig. 3(a)
Fig. 3 Spatial profiles and spatial spectra (insets) of the electric field (Ez) of light pulses propagated in free space (a), in GLMM with a square (b-d) and rhombic lattice with fixed Ly /λ = 1.075 (e,f). (g) Optimum conditions for X-pulses generation, as a function of longitudinal, Lx, and transverse, Ly, lattice periods. Symbols represent numerical PSTD results, whereas the solid black curve is the analytic solution from Eq. (4). Dashed lines refer to square and hexagonal lattices.
]. 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.

We first show the formation of X-pulses in GLMMs with the square lattice symmetry [see an example in Fig. 2(a)]. In Eq. (3) with 2α = 90°, we choose the gain-loss profile amplitude corresponding to an absorption/gain coefficient of 5 × 103 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).

(qyk)2+(qxk)2=2(qxk),Lxλ=11±1(λ/Ly)2.
(4)

This analytic solution, depicted by the solid curve in Fig. 3(g), is in good agreement with the numerical PSTD results indicated by crosses. Also from Fig. 3(g) it follows that the effective generation of X-shaped pulses is restricted to the domain where the transverse lattice period of GLMMs is larger than light wavelength, i.e. Ly /λ > 1.

Additionally, we mention that the X-shaped pulses can be found in hexagonal lattices with modulation periods ofLy=sin-1(π/3) and either 2Lx=sin-2(π/6)or 2Lx=cos-2(π/6) [see two open circles in Fig. 3(g)]. Moreover, the reported results remain unchanged for the “inverted” GLMM structures, i.e. when the cosine functions in Eq. (3) are substituted by the sine ones. The only difference is a stronger amplification by the sine-type structures, because in this latter case the loss areas are embedded in a background of gain areas (honeycomb gain profile) resulting in higher amplification of the Bloch modes.

In lattices formed by three spatial harmonics [see Figs. 2(c,d)] and with spatial discontinuities, i.e. when more spatial harmonics became involved, the corner of the BZ becomes more complex [see Fig. 1(c)]. Flat segments can appear in the gain/loss spectra of the GLMM response. These effects will be discussed elsewhere.

4. Conclusions

Propagation of light pulses through gain/loss modulated structures has been considered by numerical integration of Maxwell’s equations using pseudo-spectral time-domain method. We show that most efficiently the X-pulses can be formed close to the corners of the Brillouin zone, where the locking effects in GLMM provide enhancement of light amplification for particular wavelengths due to anisotropy. Optimum conditions for X-shaped pulse formation have also been derived.

In square GLMMs the X-shaped pulses are effectively generated along the ΓM direction when the lattice period along the diagonal matches the light wavelength. Apex angle of the X-pulse is equal to π/2 in this case, but can be tailored to arbitrary value in GLMMs of rhombic symmetry by an appropriate choice of the angle between the basis vectors of the reciprocal lattice, which means that GLMMs could be specially designed to generate X-pulses propagating invariantly in materials with a given temporal dispersion.

We have demonstrated the effect in 2D GLMMs, where, importantly, the gain/loss is modulated in both transverse and longitudinal directions. However, this procedure can be generalized to 3D GLMMs allowing the formation of full 3D light X-bullets.

Finally, we envisage that other exotic spatio-temporal spectra could be obtained by multiple imprinting, i.e. by serial propagation through two or more GLMMs with different symmetries, or also, by “parallel” propagation, i.e. embedding two different symmetries one into another, or equivalently by using quasi-periodic lattices.

Acknowledgments

This work is financially supported by Spanish Ministerio de Educación y Ciencia and European FEDER through project FIS2011-29734-C02-01. We acknowledge useful discussions with A. Dubietis.

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 39(1), 19–31 (1992). [CrossRef] [PubMed]

2.

H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (John Wiley and Sons, 2008).

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]

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]

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]

23.

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

25.

S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B 70(23), 235123 (2004). [CrossRef]

26.

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]

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]

28.

B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge University Press, 1996).

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

  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. Control39(1), 19–31 (1992). [CrossRef] [PubMed]
  2. H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (John Wiley and Sons, 2008).
  3. 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]
  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. A4(4), 651–654 (1987). [CrossRef]
  6. J. Durnin, J. J. Miceli, 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]
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