Bending light via adiabatic optical transition in longitudinally modulated photonic lattices

Bending light in a controllable way is desired in various applications such as beam steering, navigating and cloaking. Different from the conventional way to bend light by refractive index gradient, transformation optics or special beams through wavefront design such as Airy beams and surface plasmons, we proposed a mechanism to bend light via resonant adiabatic optical transition between Floquet-Bloch (FB) modes from different FB bands in longitudinally modulated photonic lattices. The band structure of longitudinally modulated photonic lattices was calculated by employing the concept of quasi-energy based on the Floquet-Bloch theory, showing the existence of band discontinuities at specific resonant points which cannot be revealed by the coupled-mode theory. Interestingly, different FB bands can be seamlessly connected at these resonant points in longitudinally modulated photonic lattices driven by adiabatically varying the longitudinal modulation period along the propagation direction, which stimulates the adiabatic FB mode transition between different FB bands.


Results
Quasi-energy and Floquet-Bloch theory in longitudinally modulated photonic lattices. The evolution of a light beam with a slowly varying electric field amplitude ψ(x, z) propagating along the z direction of a longitudinally modulated photonic lattice can be described by 20 For simplicity but without loss of generality, the refractive index distribution of the photonic lattice is set to be n = n 0 + δncos(ω x x)cos(ωz), which is periodic in both x and z directions with ω x and ω being the transverse and longitudinal angular frequencies of the refractive index modulation of the lattice, and the corresponding transverse and longitudinal lattice periods are D = 2π/ω x and T = 2π/ω, respectively. n 0 is the background refractive index and δn is the refractive index modulation depth. Therefore, the optical potential in Eq. (1) can be expressed as where k is the wave number of light in background. Note that the longitudinal refractive index modulation depth of the lattice is the same as that of the transverse one, therefore, it cannot be simply treated as a weak perturbation as done in the literature 20,21,27 . Such a strong longitudinal periodic modulation in the refractive index of the lattice, which is an optical analogy to a strong temporally periodic driving field in a quantum system 26 , is expected to have dramatic effects on the band structure of the lattice and therefore the beam evolution behaviors in the lattice. In addition, the longitudinal modulation period T is another important parameter that will have significant influence on the band structure of the lattice and also the dynamic propagation behavior of light in the lattice, as we will show in the following.
According to the Floquet-Bloch theory, the solution of Eq. (1) is of the form x z e e x z 2 Where β(k x ) is the propagation constant, which will be in the form of a series of bands β n (k x ) with band index n = 1, 2, 3... for a photonic lattice 40 , k x is the transverse wave vector. The FB modes φ ( , ) β, x z k x are periodic in both x and z directions, satisfying φ φ φ ( , ) = ( + , ) = ( , + ) , , Here the eigen propagation constant β(k x ), the analogy of quasi-energy in a quantum system driven by a temporally periodic field 26,36 , is real-valued and z-independent although the Hamiltonian , with q being an integer q = 0, ± 1, ± 2,..., are also solutions of Eq. (1) but with shifted eigenvalue β q (k x ) = β(k x ) − qω. Such shifted solutions are originated from the longitudinal periodicity of the optical potential V(x, z). Therefore, similar to the Brillouin zone due to the transverse periodicity of a photonic lattice, one can also restrict the eigenvalue bands β q (k x ) into a unit region of a longitudinal reciprocal lattice vector ω.
To solve Eq. (3), one may expand the FB modes φ ( , ) β, x z k x on the basis of a set of orthogonal basis , ,   , the non-diagonal matrix element in Eq. (7) is non-zero only when (δ q,q′+1 + δ q,q′−1 )(δ n,n′−2 + δ n,n′+2 + δ n=1,n′=2 + δ n=2,n′=1 ) = 1 is satisfied. This means that only interactions among the nearest and the next-nearest bands have to be considered, which will dramatically simplify the band structure calculation. As an example while at the same time without loss of generality, let us consider a lattice with D = 10 μm, T = 1300 μm, δn = 5 × 10 −4 , and n 0 = 2.3, which may be fabricated by using the proton-exchange or Ti in-diffusion techniques in lithium niobate crystals 19,42,43 or the light-induction technique in photosensitive materials 28 . The wavelength of the incident light is set at λ = 0.6328 μm. The band structure of the lattice can be obtained by directly solving Eq. (6). Figure 1(a) shows a portion of the band structure β(k x ) for the case when V(x, z) = 0, where the red solid thin curve is the 1st-band with band index (n, q) = (1, 0) and the green and blue solid thin curves are the ones indexed by (2, 0) and (3, 0), while the dashed thin curves are the corresponding shifted bands, respectively. Note that, as a proof of principle and for clarity, other higher-order bands with band index n > 3 are not shown here. One notes that there are crossing points A1 and A2 between the two bands indexed by (1, 0) and (2, − 1), and B1 and B2 between the two bands indexed by (2, 0) and (1, 1), respectively, at the resonant transverse wave vector k x 0 denoted by the vertical black dashed lines in Fig. 1. These crossings, according to the von-Neumann-Wigner degeneracy theorem 41 , will be avoided when a non-zero periodic refractive index modulation δncos(ω x x)cos(ωz) is introduced, therefore, leading to band discontinuities at resonant points, as shown in Fig. 1(b). Note that the degeneracy is not lifted at other crossing points such as C1, C2 and those at the boundary of Brillouin zone because the crossing points belong to different symmetry groups of Hamiltonian H k x and they do not interact with each other 26,41 . Referring to the Brillouin zone in the transverse dimension, one could also restrict the extended FB bands into a reciprocal lattice primitive zone in the longitudinal dimension 36,44 , and the reduced band structure for the first two bands is shown by the bold red and green curves in the shadow region in Fig. 1(c). Here, higher-order bands with n ≥ 3 are not shown in the reduced shadow region for clarity. Typical FB modes with transverse wave vectors k x = 0 and π = .
/ k D 0 44 x 0 (at the resonant point) are shown in Fig. 2, which can be experimentally excited by two interfering plane waves or prism coupling method, or asymptotically implemented by simply launching a wide Gaussian beam with a well-defined central transverse wave vector matched with that of the FB mode 45 , as will be confirmed numerically in the following. Figure 1(d) shows a portion of the reduced band structure of a lattice with a longitudinal lattice period T = 500 μm, other lattice parameters are the same as those in Fig. 1(c). Again, only band indexed by n = 1 (band 1, the red bold solid curves) and band indexed by n = 3 (band 3, the blue bold solid curves) are shown in the reduced shadow region for clarity. In this case, crossing points between bands indexed by n = 1 and 3 are found to be avoided. Physically, such band discontinuity is originated from the introduction of periodic modulation on the refractive index in both the transverse and longitudinal dimensions. Note that the resonant point k x 0 , where the band discontinuity occurs, shifts when the longitudinal modulation period T changes. It can be easily confirmed that the band structure will return back exactly to the normal one with the band discontinuity located at the boundary of the first Brillouin zone (in this case, one calls it as band gap in general) when the longitudinal period T tends to be infinitely large. Therefore, the band structure of waveguide arrays without longitudinal modulation is only a special case in which the longitudinal modulation period T is infinitely large.
From the band structure in Fig. 1(c,d), one sees that, although there is band discontinuity in the same band, bands with different n-index will be smoothly connected at the resonant wave vector k x 0 . This is because the propagation constant, i.e., the quasi-energy in quantum mechanism, of the FB modes at the resonant point in both bands will be shifted by a same amount, as can be clearly seen in Eq. (7) where the off-diagonal matrix elements responsible for the band shift are exactly the same. One may also note that the field distribution of the FB modes at the resonant point, although in different bands, are the same (see Fig. 2(c-f)). Combining the fact that the resonant transverse wave vector k x 0 moves as the longitudinal lattice period changes, this provides a novel and efficient way to stimulate adiabatic optical transition between optical modes in different FB bands, which can be used to manipulate the flow of light in lattices, as we will discuss in the following section.
Adiabatic optical transition and bending light in longitudinally modulated photonic lattices. Figure 3(a) shows a typical spectrum of propagation constant β as a function of longitudinal modulation period T at a specific incident transverse wave vector k x = 0.7π/D. The lattice parameters are D = 10 μm, δn = 5 × 10 −4 , and n 0 = 2.3, respectively. One sees that the band indexed by n = 1 is smoothly connected to the band indexed by n = 3 at the resonant longitudinal period T 0 = 428 μm. Therefore, for a lattice with its longitudinal modulation period varying adiabatically from T = 600 μm to T = 300 μm (linearly here) along the propagation z direction, when one launches a FB mode with a transverse wave vector k x = 0.7π/D into the lattice at the T = 600-μm port (at point a in Fig. 3(a)), it will evolve along the upper spectral curve in Fig. 3(a) and transit adiabatically to the FB mode in band 3 at the resonant modulation period T 0 (at point b in Fig. 3(a)). This is an exact optical analogy to spin exchange related to a rapid adiabatic crossing of resonance in nuclear magnetic resonance (NMR) when it is driven by a temporally varying magnetic field with adiabatically increased frequency 26,44,46 . Interestingly, such adiabatic mode evolution and optical transition can be used to control the beam propagation trajectory in the lattice because different FB modes are of different transverse propagation velocity determined by ∂β/∂k x . Figure 3(b) shows the beam propagation trajectory of a wide input Gaussian beam in the lattice with the same lattice parameters as those in Fig. 3(a) when the light evolves along the upper spectral curve (a → b → c) in Fig. 3(a). Here, a wide input Gaussian beam with a waist width being 5D is used in order to define a narrow spatial spectral content of the excitation. The central incident transverse wave vector of the light at the T = 600-μm input port is set at k x = 0.7π/D. It can be easily confirmed numerically that over 80% of the input energy of the Gaussian beam will be converted into the target FB mode of lattice with its transverse wave vector k x = 0.7π/D. One sees that, along with the adiabatic mode evolution and optical transition, the beam trajectory bends during its propagation in lattice. It is worth mentioning that the adiabatic optical transition is reversible and the FB modes in band 3 can transit adiabatically to the FB modes in band 1 via the reversed spectral curve (c → b → a) in Fig. 3(a). This means that, by alternatively repeating the adiabatic optical transition from band 1 to band 3 and its reverse process along the propagation z direction, light can be designed to propagate in the lattice with a snake-like trajectory, as shown in Fig. 4(a). More flexibly, different adiabatic mode evolution and optical transition processes may be combined and followed one by another one, and therefore / k D 0 44 x 0 (c,d) and π = .
/ k D 0 46 x 0 (e,f), respectively. The longitudinal modulation period T is 1300 μm for (a-d) and 500 μm for (e,f), respectively, and the other lattice parameters are the same as those in Fig. 1. The vertical dashed blue lines denote position of the refractive index peak at the input lattice surface. complicated light propagation trajectory could be designed in principle. Figure 4(b) shows such an example, where a cascading adiabatic optical transition scheme (band 1 → band 3 → band 1 → band 2 → band 1) is designed, and a bending light with a designable large curvature trajectory is achieved. It is evident that any perturbation on the lattice parameter will influence the band structure of lattice, therefore the beam propagation trajectory. However, the practical tolerance on the deviation of lattice parameter is indeed very complicated and dependent on the requirement of specific applications, which is evidently deviated from the main topic of this paper and will not be discussed in detail here. Also, we suppose that the lattice is large enough so that the boundary effect is negligible.

Discussion
Bending the light propagation trajectory in a designable way is desired in many practical applications such as beam steering and switching, beam navigation and even cloaking, and it may be achieved by  designing the refractive index gradient along the light propagation trajectory 47 or through transformation optics 48,49 or by employing special beams with special wavefront structure such as Airy beams [50][51][52] or Airy plasmons [53][54][55][56][57][58] . Note that the effective average refractive index of our longitudinally modulated photonic lattice is a constant n 0 , and the mechanism to bend the light propagation trajectory in our case relies on the adiabatic optical mode evolution and optical transition of FB modes among different FB bands, which is completely different from those through refractive index gradient or wavefront design mentioned above.
One may note that curved beam trajectory can also be observed in other transversely or longitudinally modulated lattices, for example, through Bloch oscillation in waveguide arrays with equally spaced increasing propagation constant in the transverse dimension [10][11][12][13] and Rabi oscillation in waveguide arrays with weak homogeneous modulation in the longitudinal dimension 20,21 . However, the mechanisms leading to curved beam trajectories in Bloch oscillation and Rabi oscillation are totally different from that in our case. Bloch oscillations, which manifests themselves as transverse oscillations of the propagating light beam, is due to the excitation and beating of FB modes with equally spaced propagation constants 12 . For Rabi oscillation, the beam oscillates between two different FB modes from different bands, either directly or indirectly, depending on the momentum compensation of the transverse wave vector 20 . In our case, the longitudinal period of the photonic lattices changes linearly and adiabatically along the propagation direction, and the beam evolves continuously and adiabatically during its propagation, which experiences many FB modes in sequence in one band and then transits adiabatically into the FB modes in other band at the resonant point and evolves continuously and adiabatically again in the second band. Note that these phenomena are optical analogies of different quantum effects, i.e., Bloch oscillation, Rabi oscillation, and adiabatic electron spin exchange in NMR driven by a temporally varying magnetic field with adiabatically increased or decreased frequency, respectively. The mechanism in our case is also different from that based on the periodic longitudinal modulation on the coupling coefficient between neighboring waveguides through curved waveguide arrays 59 .
We would emphasize that the adiabatic optical transition shown in Fig. 3 is different from that achieved through Rabi oscillation reported in refs 20,21. Although both of them are driven by the longitudinal periodic modulation along the propagation z direction, the period of longitudinal modulation varies linearly and adiabatically along the propagation z direction in our case while it is kept to be the same in the Rabi oscillation case. In addition, two specific FB modes in different bands with different field distribution are involved in the Rabi oscillation, and the band structure of the lattice is invariant along the propagation dimension. While for our case, adiabatic mode transition occurs across the resonance point via band structure evolution and the two FB modes at the resonant point in two bands are of the same field distribution, as shown in Fig. 2(c-f). Note that the adiabatic optical transition is achieved by adiabatically scanning the longitudinal modulation period and the two involved FB modes are of the same field distribution, therefore it is always perfect with a 100%-conversion efficiency in our case. In contrast, the optical transition via Rabi oscillation is based on the parametric mixing between the FB modes from different bands, and the conversion efficiency may be relatively low, depending on the interacting bands 20,21 .
The resonant transition point is specially interesting (see the point R marked in Fig. 1(c)). Such resonant transition points have been extensively studied in quantum system driven by a temporally periodic field, many novel effects such as driven quantum tunneling and coherent destruction of tunneling have been observed 26,36,41 . For the longitudinally modulated photonic lattices here, besides the adiabatic optical transition between FB modes in different bands, one can also achieve effective negative refraction near this special resonant transition point because of the smooth connection between different FB bands at the resonant point. From the reduced band structure shown in Fig. 1(c), one sees that the FB modes of band 1 and band 2 in Fig. 1(c) with a resonant transverse wavevector k x 0 propagate in the same direction in the longitudinally modulated photonic lattice, leading to an effective negative refraction phenomenon, as shown in Fig. 5. The seemly fringe pattern along the beam propagation path in the modulated lattice is a discrete characteristic of the lattice, which occurs in all transversely modulated lattices 2,6,40 . Note that the observed negative refraction effect is determined by the band structure of the longitudinally modulated lattice, therefore, it occurs when the transverse wave vector k x of the exciting beam is matched, no matter the longitudinal modulation at the interface is switched on smoothly or abruptly. It can be numerically confirmed that a slight deviation of the incidence beam from the resonant point within Δ k x < 0.1π/D is acceptable for the observation of negative refraction. This is very different from that in a photonic lattice without longitudinal modulation, where the excited two FB modes in two different bands will propagate in different directions 1,40,60 .
In conclusion, based on the concept of quasi-energy and the Floquet-Bloch theory, we have succeeded in getting the accurate band structures of longitudinally modulated photonic lattices in which the perturbation approximation of the coupled-mode theory may not be applicable. Band discontinuity is observed due to the avoiding crossing effect at the resonant transverse wave vectors, where, on the other hand, different FB bands happened to be smoothly connected in the reduced band diagram, leading to interesting effects such as negative refraction at these resonant points. More interestingly, by adiabatically varying the longitudinal modulation period along the propagation direction, adiabatic optical transition between FB modes from different bands can be achieved with a perfect conversion efficiency, which can Scientific RepoRts | 5:15805 | DOi: 10.1038/srep15805 be used to bend the light propagation trajectory in the lattices, and designable snake-like beam propagation is achieved. This may have potential applications in beam steering, navigating and even cloaking.