1. Introduction

The manipulation of light wave is a fundamental and very important field in optics. There are a plenty of ways to control the light wave, such as classic optical effects (reflection, refraction and diffraction), nonlinear optical effects [1–3]. As well known to us, light can also be controlled in the photonic lattice (PL). It may appear anomalous refraction and discrete diffraction in the PL [2, 4, 5] due to the photonic band gap. The light wave could be localized and form a soliton, which can maintain its initial shape and energy while it travels, when the nonlinear effects balance the diffraction or dispersion [4, 6]. If the periodicity of the lattice is broken by introducing inhomogeneities, i.e., detuned waveguides or altered separation between particular waveguides, then the diffraction of the light wave will be modified [7–9]. Taking this factor into consideration, the papers on the reflectionless potential (RP) were reported recently in coupled-resonator structures and photonic lattice in which the coupling coefficient is a hyperbolic secant function [10, 11]. And one can control the coupling coefficient by changing the separation between adjacent waveguides if every waveguide are identical [10, 11, 15]. On the other hand, the modulated separation between waveguides is a kind of off-diagonal disorder [15]. It was shown that the structure with off-diagonal disorder supports a pairs of conjugated modes, e.g. the photonic lattice with RP which can fully transmit incident waves supports a pair of localized modes [10, 15]. However, the light behavior in the lattice with double off-diagonal disordered potential wells hasn’t been studied.

Considering the region between these double potentials can be regarded as a well which may be used for light trapping, in this paper we study numerically the light propagation in one-and two-dimensional PLs with four types of potential wells, respectively. We utilize nonlinear Schrödinger equation (NLSE) as the theoretical model. The potential wells in the waveguide arrays are described by the superposition of two negative or positive hyperbolic secant or rectangular functions, respectively. The propagation behaviors in the potential wells are studied systematically.

2. Theoretical analysis

In order to explain our approach, we analysis the theoretical basis for the separation-modulated waveguide arrays firstly. The light wave is assumed to linearly propagate along the Z axis and diffract in the transverse directions of X and Y axes. The evolution of such light wave can be described by a normalized NLSE for the spatial beam envelope u(x, y, z) [12], $i\frac{\partial u}{\partial z}+\frac{\lambda z_s}{4\pi n_0{x}_s^2}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)+\frac{2\pi z_s}{\lambda }V\left(x,y\right)u=0$, where x, y and z are the dimensionless transverse and longitudinal coordinates normalized to x_s=10μm and z_s=1mm, respectively. The wavelength of the light wave at vacuum is λ =532nm and the average refractive index of the waveguide array is n₀=2.35. V(x, y) is the refractive index profile used to describe the PLs which consists of one-or two-dimensional (1D or 2D) structures. For both these two structures, we assume that every waveguide are identical and each waveguide can support one mode.

In our simulation, V(x, y) can be approximately described by a series of Gaussian functions [12]. We assume that V(x, y) of the 2D waveguide arrays has the form of $V\left(x,y\right)=\xi {\sum }_{n=0}^N{e}^{\left[-{\left(x-nd\right)}^2-{\left(y-nd\right)}^2\right]/{\omega }^2}$, where ξ = 2.5 × 10⁻⁴ is the refractive index modulation depth, N = 30 is the total number of waveguides, n represents the waveguide number from 1 to N. d is the separation between the centers of two adjacent waveguides. We change d to modulate the coupling coefficient and d has four forms corresponding to four different potential wells.

In the case of homogeneous waveguide arrays (HWA), d = d₀ is a constant equal to 17μm, ω=5μm is the HWHM of each waveguide. Figure 1(a) depicts the waveguide spacing profiles of HWA. For the waveguide arrays with double negative or positive hyperbolic secant potentials (NHSP or PHSP), d = ±A × sech(m − n_a)/ω₂ ± A × sech(m − n_b)/ω₂ + d₀. When d is double negative or positive rectangular potentials (NRP and PRP), d are given by d = ±A × sech(m − n_a)¹⁶/ω₃ ± A × sech(m − n_b)¹⁶/ω₃ + d₀, respectively. The sign ± corresponds to positive and negative potential wells, respectively. A=5μm is the depth of the wells, m is the spacing number, n_a and n_b are the extremum positions of the potential wells, respectively. N is the total number of waveguides, ω₂ and ω₃ are the widths of the hyperbolic secant and rectangular functions, respectively. Figure 1(b, c) show the separation distributions for these four types of potential wells in X direction, respectively.

 

Fig. 1 The separation profiles of PLs with (a), 1D HWA, (b), double hyperbolic secant potential wells and (c), double rectangular potentials wells. The blue and red triangular lines in (b, c) indicate positive and negative potential wells, respectively. A = 5μm is the amplitude.

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3. 1D transverse separation modulated PLs

We perform numerical simulations of the light propagation in the 1D transverse separation modulated PLs. The light intensity distributions from top view are shown in Fig. 2. Figure 2(c, d) show the evolutions of a narrow Gaussian beam with single site excitation in the PHSP and PRP wells, respectively. The propagation behavior of the light waves will be strongly modified in these two types of potential wells. Strong reflections happen at the edges of these wells. Therefore the light waves are confined in these wells and the light wave could recur periodically in the Z axis, especially in the PHSP wells [Fig. 2(c)]. In Fig. 2(d), the refocusing effect is not very obvious since the light wave with different k⃗ vectors will be reflected simultaneously at the steep edges of the wells. These cases are actually periodic breathings.

 

Fig. 2 Two localized modes (a), (b) and their corresponding evolutions of Gaussian beams in PHSP wells (c) and PRP wells (d) in 1D PLs, respectively. (e), the reflectionless transmission through double NHSPs and (f), the light decaying in NRP wells.

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The origin of such breathings is due to interference of the two simultaneously excited modes [10, 11]. Figure 2(e, f) are the calculated modes of Fig. 2(c, d), respectively. In our system, the waveguides are identical, so this modulated separation is a kind of off-diagonal disorder [13, 14]. The eigenvalues and eigenmodes are conjugated due to the off-diagonal disorder properties [Fig. 2(e, f)]. Such eigenmodes in pair with conjugated eigenvalues are localized modes with opposite signs of the propagation constants [15, 16].

In the structure with two NHSPs, the light tunnels through the potentials without any reflection and diffracts continuously in the homogeneous parts after these two potentials [Fig. 2(e)]. This is very similar to the light behavior in HWA where light distributes symmetrically in the two outer lobes with single site excitation [1]. In the lattice with double NRPs, such structure doesn’t support localized mode. The intensities of the light waves decay as they propagate after a very long distance for 300mm in Z axis [Fig. 2(f)].

4. 2D transverse separation modulated PLs

Considering 2D lattice is more general, we expanded our investigations from 1D to 2D lattices, where the transverse separation-modulation and coupling effects are much complicated than those in 1D case. As well known, a narrow Gaussian beam experiences discrete diffraction when it is launched into the 2D homogeneous waveguide arrays [3]. In our studies, we investigate the light behavior in the other four types of 2D transverse separation modulated PLs. The four potentials have the same profiles in the x or y directions as those in the 1D cases in Fig. 1(b, c), respectively. The top view of these four potentials look like a pound sign [Fig. 3(e), Fig. 4(e) and Fig. 5(e)]. This kind of pound sign structures form potential wells. The light beam is launched into the central waveguide in such potential wells, respectively. The linear diffraction in these four types of structures is studied numerically.

 

Fig. 3 The propagation dynamics of a Gaussian beam in 2D lattice with four PHSPs. (ad), the transverse intensity profiles at different z connected by the dash lines with (f), the longitudinal intensity profile. (e), the refractive index distribution of the lattice. The black arrow represents the input position. Δn, the potential depth. Site n, the waveguide numbers in the transverse directions.

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Fig. 4 The propagation dynamics of a Gaussian beam in 2D lattice with four PRPs. (ad), the transverse intensity profiles at different z connected by the dash lines with (f), the longitudinal intensity profile. (e), the refractive index distribution of the lattice. The black arrow represents the input position. Δn, the potential depth. Site n, the waveguide numbers in the transverse directions.

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Fig. 5 The propagation dynamics of a Gaussian beam in 2D lattice with four NHSPs. (ad), the transverse intensity profiles at different z connected by the dash lines with (f), the longitudinal intensity profile. (e), the refractive index distribution of the lattice. The black arrow represents the input position. Δn, the potential depth. Site n, the waveguide numbers in the transverse directions.

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Figure 3 show the propagation dynamics of a Gaussian beam in the lattices with PHSPs. The light evolution will be drastically changed. When a narrow Gaussian beam is launched into the central waveguide between the PHSP wells, the beam will experience linear diffraction firstly and then extend to about 7×7 waveguides [Fig. 3(b)]. With the propagation distance increasing, most of the outer lobes will go back into center by the reflection at the boundaries of such pound-sign PHSPs [Fig. 3(a–d) and (f)]. And there is almost no energy leak at the boundaries resulting in obvious refocusing and recurrence during its propagation since such structure could also support localized mode. This means the PHSP well could act as mirrors for the light wave of 532 nm at normal incidence. So this kind of PHSP well is a cavity which can trap light. This structure provides a new way for the design of optical cavity.

Figure 4 shows the case in the PRP well. A narrow Gaussian beam is launched at the normal incidence into a single waveguide at the center of such pound sign structure [Fig. 4(a, e)]. Figure 4(e) shows the input position of the Gaussian beam and the refractive index distribution of the lattice. The potential depth Δn is 2.5 × 10⁻⁴ and the waveguide numbers in the transverse directions are 29 × 29. It is obvious that this kind of potential well could also confine light and the light beam can propagate stably for a long distance [Fig. 4(f)]. However, there is also no very strong refocusing phenomenon similar to that in 1D lattice with PRP well [Fig. 4(f) and Fig. 2(d)].

Figure 5 shows the behavior of light wave in 2D lattice with NHSPs. The light wave can transmit through the NHSPs [Fig. 5(a–d) and (f)] and results in a larger diffraction scope than that in the 2D homogeneous PLs [3]. The intensity patterns are divided into nine parts by four dark regions due to the reflectionless effects. Although each part is pretty far away from the others, the whole intensity pattern is still very similar to that in 2D homogeneous PLs. This is coincident with the 1D case, mean such structure being a kind of reflectionless potential [11]. We also investigate the light behavior in NRP well in 2D PLs. Such structure could not support stable propagation since there is no localized mode. So the light intensity will decay with z due to the energy leak through the potentials.

5. Conclusion

In conclusion, we have studied the light propagation in four kinds of transverse separation modulated photonic lattices. Our results show that the light waves can be refocused and recur at the central waveguide in the lattices with double or pound-sign potentials obeying positive hyperbolic secant and positive rectangular functions, respectively. This is because such two kinds of off-diagonal disorder lattices can both support a pair of conjugated localized modes. Furthermore, the light waves can transmit with no reflection through double or pound-sign potentials obeying negative hyperbolic secant functions. However, the light waves decay in the negative rectangular potentials since such structure could not support localized mode. In a word, such transverse separation modulated photonic lattices can act as a kind of “cavity” which has important potential applications in the light manipulation and optical signal processing.