1. Introduction

During the last three decades, nondiffracting beams and spatiotemporally-localized wave packets (SLWPs)—often called light bullets—have attracted much attention [1–5]. Two well-known examples are Besssel beams [1] and X-waves [2–4]. The Bessel beam is a free-space, beamlike, exact solution of the wave equation, free of transverse broadening. It has sharply defined field distribution in the form of the Bessel function in the transverse plane. The X-wave is a superposition of cylindrically symmetric Bessel beams with multiple frequencies. It is nondiffracting in both transverse and longitudinal directions. After being proposed and demonstrated with acoustic waves [2], its demonstration in the optical regime followed [3, 4]. Very recently, a Airy-Bessel wave packet [5] was proposed, which is a combination of a Bessel beam in the transverse plane with a Airy pulse in the longitudinal (or temporal) direction. In nonlinear media, SLWPs are called spatiotemporal solitons [6–8]. These result from the balance of diffraction and dispersion by self-focusing and nonlinear refraction, respectively. The requirement for high intensity in nonlinear media may be an obstacle in many real applications.

Photonic crystals (PCs) are periodic dielectric structures that have band gaps, frequency ranges over which wave propagation is forbidden. For waves with frequencies outside band gaps, PCs can modify their diffractive and dispersive properties due to complicated band structures. One of the phenomena is self-collimation [9–16], by which the transverse broadening of waves is counteracted with the diffractive properties of PCs. Many photonic devices based on the self-collimation of monochromatic beams have been demonstrated [14]. However, to transmit and process information in photonic devices, one needs SLWPs—not just monochromatic collimated beams—with a small amount of both diffraction and dispersion. Recently, a hybrid photonic crystal that supports broadband super-collimation was proposed [15]. It was pointed out that the hybrid structure has the potential importance for polychromatic as well as monochromatic light. Although SLWPs in free space or nonlinear media have been studied for decades, it is not long ago that those in periodic media [17–19] or photonic crystals [20–25] gained attention. Most of the works except for [24, 25] have been done analytically, based on specific models or ad hoc approximations. Those works provide some physical insights into the propagation of SLWPs. To utilize SLWPs in realistic PCs, one needs to understand the characteristics of their propagation along with detailed knowledge of physical parameters. Here we report the propagation of SLWPs—self-collimated ultrashort Gaussian pulses—in a hybrid photonic crystal structure by numerical computations.

2. Properties of a Hybrid Photonic Crystal

The photonic crystal we consider here is a hybrid structure proposed in [15]. It consists of a square lattice of circular rods and a waveguide array, and is shown schematically in Fig. 1(a). The hybrid structure is claimed to inherit useful properties from two constituent structures [15]. The waveguide array brings the flatness of equi-frequency contours (EFCs) over an extended frequency range. The array of circular rods breaks the continuous translational symmetry along the waveguides’ direction, resulting in some EFCs with sign flip of their curvature. These two properties facilitate broadband super-collimation.

 

Fig. 1 (a) Schematic of the refractive index distribution of a hybrid photonic crystal structure containing rods of radius r = 0.16a and waveguides of thickness t = 0.2a, both with refractive index n = 3.5 surrounded by air of n = 1. The red square is a unit cell of lattice constant a. Waves propagate along the positive x direction. (b) Equi-frequency contour plot for the fourth TM band. The numbers on the curves represent frequencies in units of 2πc/a with c being the speed of light in air.

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We compute the EFCs shown in Fig. 1(b) by the MPB package [26] based on the plane wave expansion method. The EFCs provide information on the direction of group velocity of a wave propagating inside a PC, which is incident with a certain angle at a single frequency [9, 11]. As shown similarly in [15], the EFCs are flat over an extended range of k_y for many different values of frequencies. Thus a narrow beam propagating in the x direction with a large range of k_y would not diffract in the y direction.

To analyze temporal dispersion of pulses, one needs the values of group velocity dispersion (GVD) and third-order dispersion (TOD) [25]. GVD is the derivative of an inverse of group velocity with respect to frequency, and is responsible for temporal broadening. TOD is the derivative of GVD with respect to frequency. It affects the shape of a ultrashort pulse with a large frequency spectrum even when GVD at its center frequency is zero. We compute GVD and TOD as a function of frequency by MPB as follows. Given a list of frequencies and the direction of a wavevector, MPB calculates the corresponding list of magnitudes of the wavevector, and then computes group velocities for the set of the eigenstates with wavevector-frequency pairs [26]. It is straightforward to compute GVD and TOD. The computed GVD (β ₂) and TOD (β ₃) are shown together in Fig. 2. We see that β ₂ ≈ 0 and β ₃ ≈ 90 at ω = 0.473(2πc/a), the frequency of interest for zero GVD. Another interesting frequency is ω = 0.495(2πc/a) for optimized super-collimation studied in [15]. At that frequency, β ₂ ≈ 1 and β ₃ ≈ 23, which is much smaller.

 

Fig. 2 Group velocity dispersion and third-order dispersion as a function of frequency.

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3. Propagation of Self-Collimated Ultrashort Pulses

We use the Meep package [27] based on the finite-difference time-domain method to simulate the propagation of self-collimated ultrashort Gaussian pulses. The computational domain has a size of 600a × 30a with −300a ≤ x ≤ 300a and −15a ≤ y ≤ 15a, and is filled with the hybrid structure. The domain boundaries are surrounded by the perfectly matched layers as absorbing boundary conditions. The SLWP of a spatial Gaussian beam with temporal Gaussian profile is launched just near the left boundary in the positive x direction. We test the propagation of the SLWPs with different center frequencies ω ₀ to see if they retain their shapes over a long distance. We find that the SLWPs with ω ₀ = 0.473(2πc/a) and ω ₀ = 0.495(2πc/a) propagate with minimal dispersion, so we choose these two frequencies as typical values for further analysis. The full-width at half-maximum (FWHM) of the spectral width is given Δω ≈ 0.20(2πc/a), which yields the longitudinal FWHM length l ≈ 8a of the field distribution at a fixed time (See Fig. 3.). The transverse FWHM width is fixed as w ≈ 4a. These two values are chosen approximately from the minima after testing with many different values.

 

Fig. 3 Electric field distributions in portions of the computational domain at fixed instances of time (red for positive and blue for negative values): (a) and (b) for ω ₀ = 0.473(2πc/a); (c) and (d) for ω ₀ = 0.495(2πc/a).

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In Fig. 3, the enlarged electric field distributions are shown at two instances of time. The total elapsed time in the computation is t = 150(a/c) for Figs. 3(a) and 3(c), and t = 2000(a/c) for Figs. 3(b) and 3(d). It appears that our SLWPs nearly maintain their shapes after propagation over ∼ 575a, and that the distribution in Fig. 3(b) is not as good as that in Fig. 3(d) due to the larger value of TOD. To see variations of width and length with propagation distance, the field distributions at various instances of time are saved during the computation. We then calculate the values of width and length at the positions with the peak values of each field, and plot them as a function of position in Fig. 4.

 

Fig. 4 Variation of (a) width and (b) length with propagation distance. The black curves are for ω ₀ = 0.473(2πc/a), and the blue ones for ω ₀ = 0.495(2πc/a)

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It is clearly seen in Fig. 4(a) that the initial width 4a does not vary. This is because the EFCs in Fig. 1(b) are flat for a wide range of k_y. There is a jump to a lower value around x ≈ −230a for ω ₀ = 0.473(2πc/a), which is probably a calculation error. In Fig. 4(b), the initial length 8a increases slowly to 10a for ω ₀ = 0.473(2πc/a), whereas it remains unchanged over ∼ 300a and increases to 9a for ω ₀ = 0.495(2πc/a).

Finally, to compare the field distributions in Figs. 3(b) and 3(d), we plot the transverse field profiles in Fig. 5 and the longitudinal field profiles in Fig. 6, both at the positions with the peak field values. As expected from Fig. 4(a), the transverse profiles are nearly identical for Figs. 5(a) and 5(b). The longitudinal profile in Fig. 6(a) has a long tail, which is also seen in Figs. 3(b). The origin of the long tail is obviously the larger value of TOD.

 

Fig. 5 Transverse field profiles for: (a) Fig. 3(a) (red dotted line) and Fig. 3(b) (green solid line); (b) Fig. 3(c) (red dotted line) and Fig. 3(d) (blue solid line).

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Fig. 6 Longitudinal field profiles for: (a) Fig. 3(b) and (b) Fig. 3(d).

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We compare our SLWPs with similar ones in realistic PCs reported in [24, 25]. In [24], a SLWP with Gaussian profile in both transverse and longitudinal directions propagates in a 2D planar PC slab with a square lattice of air holes. Its initial transverse FWHM width is estimated to be ∼4.7a from its double transverse waist 2w _⊥ = 1.8 μm and lattice constant a = 320 nm. In Figs. 4(e) and 5(e) of [24], its longitudinal FWHM lengths are roughly estimated to be ∼ 12.9a and ∼ 11.1a after propagation over ∼ 160a, respectively, for TM and TE. Its width and length are larger than those of our SLWP. Moreover, its propagation distance is much smaller, and its longitudinal length increases as it propagates. In [25], a spatiotemporally Gaussian pulse propagates in a 2D PC with a square lattice of air holes. Its transverse FWHM width is given 16a, and its longitudinal FWHM length is estimated to be ∼48a from the inset in Fig. 2 of [25]. Although the pulse propagates without spatiotemporal broadening over 400a, it is much wider and significantly longer than ours.

4. Conclusion

We have reported the propagation of self-collimated ultrashort Gaussian pulses—our spatiotemporally-localized wave packets—in a hybrid photonic crystal structure. We have found that the wave packets propagate with very small spatiotemporal broadening over a long distance, and that they are narrower and shorter than similar wave packets in realistic photonic crystals reported previously. We believe that our results may help utilize spatiotemporally-localized wave packets in photonic devices based on periodic media or photonic crystals without nonlinearity.