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 [1,2]. 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 [3,4].

X-pulses are three-dimensional (3D) analogs of non-diffractive Bessel beams in 2D, proposed by Durnin [5,6], and widely used in many areas of optics [7]. 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 [8,9] and Airy pulses in 3D, which owe an interesting property of self-bending, i.e. can propagate “behind” an obstacle [10,11].

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 [12,13], or holograms for Airy beams [8,9]). 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], 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], parametric amplification [16], filamentation [17,18], and Raman scattering [19]. 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]. 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) [21,22], 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], are also possible in GLMMs [21]. 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]. 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 [24–27].

The anisotropy of gain/loss has been studied in detail in [22], 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] 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 presents a PWE calculation of the imaginary part of the frequencies in the first band to illustrate the gain profile.

 

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 L_x/c units; where c is the speed of light in vacuum and L, L_x,y denote the lattice parameters.

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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:

Equations (1,2) mathematically represent a linear problem of light propagation, and, hence, the possible dependence of ε(x,y) and σ(x,y) on the field intensity is neglected. In order to avoid any divergence related to the exponential unsaturated growth of amplifying Bloch modes, we fix the length of the GLMM structures to 20 λ. We solve Eqs. (1, 2) by pseudo-spectral time-domain (PSTD) method [28] using 20 points per wavelength.

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:

For N = 2 non-collinear vectors ${\overrightarrow{q}}_{1,2}$ form basis of reciprocal lattice and we choose the x-axis along the bisection between the two [see Fig. 2 ], i.e. ${\overrightarrow{q}}_{1,2}=\text{(}{q}_x,\pm q_y\text{)}$, where $q_x=q\text{cos}\alpha =2\pi /{\text{L}}_{\text{x}}$, $q_y=q\text{sin}\alpha =2\pi /{\text{L}}_y$, and 2α is the angle between the vectors ${\overrightarrow{q}}_{1,2}$. L_x and L_y denote the longitudinal (along the light propagation direction) and transverse lattice periods, respectively. For N = 3 lattices the third vector can be chosen as ${\overrightarrow{q}}_1+{\overrightarrow{q}}_2=-{\overrightarrow{q}}_3=\text{(2}{q}_x\text{,0)}$. 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 L_x < L_y (L_x > L_y).

 

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 q_1,2,3 denote the reciprocal lattice vectors.

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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) ]. 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.

 

Fig. 3 Spatial profiles and spatial spectra (insets) of the electric field (E_z) of light pulses propagated in free space (a), in GLMM with a square (b-d) and rhombic lattice with fixed L_y /λ = 1.075 (e,f). (g) Optimum conditions for X-pulses generation, as a function of longitudinal, L_x, and transverse, L_y, 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.

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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).

Next we generalize the results to rhombic lattices, by squeezing the square lattice, see Fig. 2(b). Each period ratio L_x/L_y, that characterizes the lattice, corresponds to a given 2α 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.

The most efficient generation of X-shaped pulses occurs for a certain ratio between the wavelength and the lattice periods. Figure 3(g) summarizes the search of the optimum condition, which occurs precisely when the pulse spectrum in centered at the corner of the BZ. For 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 ${\overrightarrow{q}}_{1,2}=\text{(}{q}_x,\pm q_y\text{)}$:

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. L_y /λ > 1.

Additionally, we mention that the X-shaped pulses can be found in hexagonal lattices with modulation periods of${\text{L}}_{\text{y}}\text{/λ}={\text{sin}}^{\text{-1}}\text{(}\pi /3\text{)}$ and either $2{\text{L}}_{\text{x}}\text{/λ}={\text{sin}}^{\text{-2}}\text{(}\pi \text{/6)}$or $2{\text{L}}_{\text{x}}\text{/λ}={\text{cos}}^{\text{-2}}\text{(}\pi \text{/6)}$ [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.