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Light beams undergoing optical Bloch oscillations (OBOs) in two-dimensional (2D) photonic lattices suffer from severe diffraction along the perpendicular direction to the oscillation plane. In this paper, we propose and demonstrate that such diffraction could be suppressed in hexagonal photonic lattices via sophisticated managements of the discrete diffraction. By positioning the Fourier spectrum of the beam to a special region in the Brillouin zone, the light driven by the OBO experiences normal and anomalous diffractions alternatively, leading to a non-diffracting propagation for a long distance. We show that non-diffracting OBOs can be implemented not only for Gaussian beam but also for other complex 2D beams including self-accelerating Airy beams and vortex beams. Our results provide novel insights into the diffraction or dispersion engineering of waves in periodic structures.
© 2017 Optical Society of America
1. Introduction
A finite sized light beam is inevitably subject to the diffractive distortion in free space during its propagation. To manage such a linear diffraction, optical wave packages associated with special functions, named as diffraction-free beams, have been successively developed including Bessel beams [1,2], Airy beams [3,4], Mathieu beams [5,6]. Meanwhile optical periodical structures (e.g. photonic lattices) have been serving as important tools for diffraction management of light owing to their unique photonic bandgap and symmetry [7–13], and also become a platform for producing diffraction-free beams [14,15]. In one-dimensional (1D) photonic lattices, it has been demonstrated that light diffraction can be suppressed by optical Bloch oscillations (OBOs) [16–22]. Resembling the oscillating behaviors of a charged particle in a biased semiconductor superlattice, OBOs have been theoretically predicted and experimentally observed in photonic lattices with transverse index gradients. Undergoing the OBOs, the light beam is forced to follow a cosine-like path with a non-diffracting feature. Recently we have demonstrated that finite-energy 1D Airy beams exhibit non-spreading and periodically reviving during long distance propagations in the presence of the OBOs [22]. However, for a two-dimensional (2D) case, since the OBO only happens along the direction of the index gradient, the diffraction of the optical field would never be suppressed along the perpendicular direction to the oscillation plane [23–25]. As a result, 2D light beams would be distorted during the OBOs, posing a challenge to achieve non-diffracting OBOs in 2D photonic lattices. A possible solution is to carefully design the symmetry of the 2D beam to match with that of the lattice structure, as the OBO-enabled recovery of 2D Airy beams in square lattices being demonstrated recently [26]. Nevertheless, how to maintain the characteristics of the 2D light beams undergoing the OBOs in more general conditions is still an open challenge.
In this paper, we provide a solution for suppressing the diffraction of 2D light beams along the perpendicular direction to the OBOs in hexagonal photonic lattices (HPLs). By launching the input beam at a proper direction, the OBO can drive the Fourier spectrum of the beam crossing through different diffraction regions inside the 1st Brillouin zone. Consequently, the light beam is alternatively subjected to normal and anomalous diffractions, leading to a non-diffracting propagation along with the beam restoration. This prevents the dramatic collapses of the 2D beams during the OBOs. Our method is not only valid for Gaussian beams, but also for complex beams including Airy beams and vortex beams. The results provide novel insights into the diffraction or dispersion managements with periodic structures in general.
2. Theoretical model
Within the paraxial limit, the propagation of a monochromatic light beam B traveling along the z-axis in a nonuniform medium can be described by the normalized Schrödinger equation
where B denotes the complex amplitude of the optical field, and Δn is the modulation of refractive index. The dimensionless parameters (x, y, z, Δn) are related to the physical parameters (x', y', z', Δn') by the expressions (x, y) = k0 (Δn0n)1/2(x', y'), z = k0Δn0z', and Δn = Δn'/Δn0, where k0 is the wave number of light in vacuum, Δn0 is the normalized parameter of refractive index, and n is the index of the medium [10,11].The HPL can be optically induced in a biased photorefractive crystal by a light pattern of hexagonal array [8–12], of which the intensity can be expressed as
where A is the normalized amplitude, g = 4π/3d, k1 = (1,0), k2 = (−1,)/2, k3 = (−1,-)/2, r = (x,y) denotes the coordinate vector, and d is the lattice spacing. The vector kn (n = 1,2,3) refers to the lattice symmetry in the reciprocal space, as illustrated in Fig. 1(b), where the hexagon denotes the 1st Brillouin zone of the HPL, and the three high-symmetry points are marked as Γ, X and M. The index distribution of a typical HPL, proportional to Il/(1 + Il), is shown in the inset of Fig. 1(a).
Fig. 1 (a) Schematic drawing of the optical Bloch oscillation (OBO) of a two-dimensional (2D) beam in a hexagonal photonic lattice (HPL) with an index gradient. (b) Reciprocal space and the 1st Brillouin zone of the HPL. (c) OBO of a 2D Gaussian beam in the HPL (see Visualization 1); Top: side view of propagation process in the y-z plane; Bottom: intensity distributions at different propagation distances.
To realize the OBO of a light beam, a linear transverse index gradient is superimposed on the photonic lattice [23–26]. The index modulation is given by
where γ is a constant proportional to the bias voltage, ρ denotes the transverse index gradient along the y-axis. When a beam is launched into the HPL as shown in Fig. 1(a), the index gradient and Bragg reflection of the lattice drive the input beam oscillating along a cosine-like path in the y-z plane. Meanwhile, the Fourier spectrum of the input beam moves along the line through the two X points [see Fig. 1(b)] in the 1st Brillouin zone of the HPL. This evolution process of light beam in the HPL can be derived by numerically solving Eq. (1) with the beam propagation method (BPM).3. Diffraction during optical Bloch oscillation (OBO)
To excite the OBO in the HPL, a Gaussian beam B0 = exp(-r2/w02) is incident normally to the lattice, and its evolution is numerically simulated with the parameters of A = 0.2, d = 3, γ = 2, ρ = 0.025, and w0 = 10. The top panel of Fig. 1(c) displays the side view of the beam propagation in the y-z plane within a normalized propagation distance z = 300, while the bottom four panels depict the intensity distributions of the input (z = 0) and the output beams at z = 100, 200 and 300, respectively. It can be seen that the Gaussian beam experiences an oscillation of three periods within z = 300, without showing obvious width variation along the y-axis. However, along the x-axis, an apparent diffraction happens and broadens the beam substantially during the OBO. As a result, the circular Gaussian beam is distorted into an elliptical one. Such a diffraction effect represents a universal behavior of a 2D beam undergoing the OBO in periodic systems.
4. Diffraction suppression utilizing diffraction characteristics during the OBO
In order to achieve the diffraction suppression in the HPL, it is necessary to consider the diffraction characteristics of light beams. It has been demonstrated that the optical band-gap structures of photonic lattices, expressed as the relationship of the propagation constant β versus the transverse wave vector (kx, ky), can be used for describing the optical diffraction behaviors inside the lattice [7–11]. The band-gap structure β(kx,ky) of the HPL can be solved by substituting B = b(x,y)exp(iβz) in Eq. (1). Figure 2(a) displays the calculated first two transmission bands corresponding to the HPL in Fig. 1. Since the OBO mainly refers to the propagation modes on the 1st band, only the diffraction characteristics of the 1st band will be discussed below.
Fig. 2 Diffraction characteristics of a HPL. (a) 3D view of the 1st and 2nd transmission bands of the HPL. (b) Contour plot of the 1st band β(kx,ky) with the 1st Brillouin zone marked by a dotted hexagon. (c) Diffraction characteristics along the x-axis, where the dashed lines indicate locations with Dx = 0.
The 1st transmission band structure is displayed in Fig. 2(b), with the 1st Brillouin zone marked by a dotted hexagon. The diffractions characteristics of a light beam along the x- and y-axes in the photonic lattice are defined as Dx = ∂2β/∂kx2, Dy = ∂2β/∂ky2, respectively [7–11]. It is revealed that the value of |Dx| or |Dy| measures the light broadening, and the negative and positive signs of the Dx or Dy are respectively corresponding to the normal and anomalous diffractions, while the zero of Dx or Dy means no diffraction. The diffraction along the x-axis Dx is calculated and displayed in Fig. 2(c). The non-diffraction region (Dx = 0) is drawn with the dashed lines, marking the boundaries of normal and anomalous diffraction regions. Specifically, light beams with their spectra residing in the region enclosed by the dashed lines of the 1st Brillouin zone (Dx<0) experience light broadening, while beyond this region (Dx>0), light experiences convergence.
During the OBO as shown in Fig. 1, the spectrum of the input beam moves along the line through the two X points in the normal diffraction region, as depicted by the white arrowhead in Fig. 2(c) (see also Visualization 1). This is the reason why the beam width gradually increases. Given the movement of the beam spectrum during the OBO, it is not possible to make the beam spectrum follow the curved non-diffraction line in the reciprocal space of the HPL [the dashed lines in Fig. 2(c)]. Then a natural question comes up: can the diffraction of beam be suppressed if the beam experiences alternately normal and anomalous diffractions during the OBO? Such regions are highlighted by the rectangles in Fig. 2(c). When the beam spectrum is set to be within those regions, it will be driven by the index gradient along the blue arrowhead, and eventually it moves through the boundaries of two diffraction regions alternately.
To test the performance of the above assumption, the incident Gaussian beam as same as that employed in Fig. 1 with the wave vector (kx, 0) is launched into the HPL, and the propagation process is simulated with the BPM. By scanning the wave vectors kx within the highlighted regions shown in Fig. 2(c), we find the best diffraction suppression effect happens at kx≈0.5g. Under such a condition, the beam spectrum, as also shown in Visualization 2, firstly moves from the normal diffraction region [i.e. (0.5g, 0)] to the anomalous one [point M (0.5g, 0.5g)], and finally goes back to the normal one after a Bragg reflection. The corresponding propagation results are shown in Fig. 3, where 3(a) reveals the movement of the light beam, 3(b)-3(e) depict the input (z = 0) and the outputs at z = 100, 200 and 300, respectively. Comparing with the results shown in Fig. 1(c), it is clearly seen that the light beam is scarcely broadened even after an oscillation of three periods, proving that the diffraction of light beam is effectively suppressed. Yet, due to the oblique incidence, the light beam continually moves towards the x-axis during oscillating along the y-axis [see Figs. 3(b)-3(e)]. For the convenience of visualization, we tilt the incident beam and the HPL as a whole [see Fig. 3(f)], and make the OBO happen in the y-z plane, as shown in Fig. 3(g). We can obtain the same outputs as that shown in Figs. 3(c)-3(e) at z = 100-300, but with the beam centers located at the origin. The evolution process is also shown in Visualization 2.
Fig. 3 Non-diffracting OBO of a Gaussian beam. (a) Numerically simulated OBO of a tilted Gaussian beam in a HPL, where the translucent planes correspond to y = 0, z = 0, 100, 200, and 300, respectively. (b)-(e) Intensity cross-sections at different propagation distances, where the coordinates of the beam centers (marked by the crosses) are labeled. (f) Exciting geometry of the input beam and the HPL, and (g) the corresponding propagation process in the y-z plane (see Visualization 2).
Figure 3 clearly indicates that by setting the beam spectrum to alternatively pass through the normal and anomalous diffraction regions of the HPL indeed offers diffraction suppressions of the OBOs. We point out that there still exists a bit divergence of the light beam after a long distance propagation, as shown in Fig. 3(e), even under the condition of diffraction suppression. This can be explained by the relatively wide beam spectrum, which covers a certain range inside the Brillouin zone. Light components associated with the spectrum located in different regions experience different extents of diffractions, leading to an asymmetric diffraction suppression. Therefore, the proposed method is more applicable to the beams with narrow spectra.
5. Non-diffracting OBOs of Airy and vortex beams
In corroboration of the diffraction suppression scheme with the configuration shown in Fig. 3(f), an Airy beam with a finite energy [see Fig. 4(a1)], expressed by B0 = Ai(x/6)Ai(y/6)exp[(x + y)/64], is obliquely incident into the tilted HPL with an index gradient along the y-axis. The corresponding results are shown in Fig. 4(d). The propagation processes of the same input Airy beam at the normal incidence in a uniform HPL and in a HPL with an index gradient are displayed in Figs. 4(b) and 4(c), respectively. In the uniform HPL, the finite-energy Airy beam essentially maintains its profile for a certain distance, and yet finally evolves into a wide distribution after a long-distance propagation (z = 300) due to the diffraction, as also shown in Fig. 4(b2). Under the OBO in the nonuniform HPL [see Fig. 4(c)], the Airy profile in the oscillation direction (y-axis) can be periodically revived. While in the perpendicular direction (x-axis), normal diffraction induces the divergence of the Airy profile. As a result, the 2D Airy beam is distorted after a long distance propagation, e.g. three periods of oscillation in Fig. 4(c2). By positioning the spatial spectrum of the Airy beam to the highlighted region in Fig. 2(c), i.e. under the diffraction suppression condition, the diffraction along the x-axis of the Airy beam is largely resisted, and the output field represents the characteristics of 2D Airy profile as shown in Fig. 4(d). Note that the Airy profile in x-axis is slightly broadened because of the wide spectrum, as shown in the bottom of Fig. 4(a2).
Fig. 4 Diffraction suppression of an Airy beam during the OBO in a HPL. (a) Input Airy beam and its spectrum, where the dashed hexagon marks the 1st Brillouin zone of the HPL. (b)-(d) Evolutions of the Airy beam in an uniform HPL, in a HPL with index gradient along the y-axis, and in a tilted HPL matching the diffraction suppression condition, respectively; the figures from left to right correspond to the propagation processes in the y-z plane and the output at z = 300, respectively. The crosses mark the initial beam positions, and the scales of (a1), (b2), (c2) and (d2) are the same.
To further manifest the diffraction suppression effect of the OBOs of Airy beams, a 2D finite-energy Airy beam accelerating in the y-axis with an amplitude of B0 = Ai[(y + x)/6]Ai[(y-x)/6]exp(y/32)is studied [see Fig. 5(a)]. Figures 5(b) and 5(c) exhibit the comparison of propagations of the Airy beam without and with imposing the condition of diffraction suppression in the x-axis during the OBO. Obviously, without diffraction suppression, only the Airy profile in the oscillation direction is inherited under the OBO, while in the perpendicular direction the beam collapses completely. The beam under diffraction suppression condition, by contrast, maintains the 2D Airy profile well for a long distance, with a rotation which is introduced by the residual diffraction caused again by the wide spectrum.
Fig. 5 Non-diffracting OBO of an Airy beam accelerating in the y-axis. (a) Input Airy beam and its spectrum. (b),(c) The OBOs of the Airy beam without and with imposing the condition of diffraction suppression, respectively; the figures from left to right correspond to the propagation processes in the y-z plane and the output at z = 300, respectively. The crosses marks the initial beam positions.
The efficient diffraction suppression of a vortex beam is also demonstrated. We select a singly charged vortex Gaussian beam, which is expressed by B0 = (r/15)2exp(-r2/152)exp(iφ), where (r,φ) denotes the polar coordinate. The propagation processes of the vortex beam without and with diffraction suppression in the x-axis during the OBO are shown in Figs. 6(b) and 6(c), respectively. The right two columns of Figs. 6(b) and 6(c) correspond to the output intensity and phase distributions at z = 300, respectively. It can be seen from the results that without the diffraction suppression, the vortex beam is deformed and evolves into a dipole-like mode with the action of the OBO along the y-axis and the normal diffraction parallel to the x-axis. An anticlockwise rotation can be observed during the OBO due to the orbital angular momentum carried by the input beam. While with diffraction suppression, the input vortex beam can hold its annular distribution of intensity and spiral phase, as clearly depicted in Fig. 6(c).
Fig. 6 Non-diffracting OBO of a vortex beam. (a) Intensity and phase distributions of the input beam. (b),(c) The OBOs of a vortex beam without and with diffraction suppression in the x-axis, respectively; the figures from left to right correspond to the propagation processes in the y-z plane, the output intensity and phase distributions at z = 300. The crosses marks the initial beam positions, and the scales of (a), (b2) and (c2) are the same.
6. Summary
In summary, we have proposed and demonstrated a method to achieve diffraction suppression of a 2D light beam during the OBO in a HPL. By examining the diffraction characteristics of the HPL, we have identified a special region inside the Brillouin zone of the lattice, where the beam spectrum alternatively passes through normal and anomalous diffraction regions during the OBO. By positioning the input beam spectrum within the special region, the diffraction of the light beam can be effectively suppressed. As examples, the non-diffracting OBOs of three typical 2D light beams including a Gaussian beam, an Airy beam and a vortex beam, have been demonstrated in a HPL. Our results revealed that the OBO perfectly revives the beam structure within the oscillation plane, while the anomalous diffraction mostly offsets the normal diffraction in the perpendicular direction. It has been demonstrated that our method can effectively prevent the broadening of a 2D light beam during the OBO, and essentially recovers the propagation characteristic of the beam periodically, even though slight distortions of the beams are inevitable for long distance propagations. Our findings provide novel insights into the diffraction or dispersion engineering with periodic systems.
Funding
National Natural Science Foundation (NSFC) (61675168, 11634010, 11304250, and 11574389); Joint Fund of the National Natural Science Foundation Committee of China Academy of Engineering Physics (NSAF) (U1630125); the Fundamental Research Funds for the Central Universities (3102015ZY057); the Natural Science Basic Research Plan in Shaanxi Province of China (2016JQ6004).
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