1. Introduction

Optical diffraction is a universal phenomenon in nature. It originates from the different phase accumulation rates during the evolution of a beam with different spatial frequencies. Diffraction is also a geometrical effect and depends weakly on the medium through its refractive index. There are a variety of diffraction-free beams, e.g., Bessel beams, Airy Beams. However, such beams have theoretically infinite spatial size and thus infinity energy. In experiments, these beams inevitably suffer a diffraction since their size and energy is actually finite. Eisenberg et al. proposed an effective way to control the diffraction of a broad beam by launching it into zigzag waveguide arrays [1]. The reduced, canceled, and reversed diffraction was observed experimentally. Following this pioneer work, many efforts are devoted to the study of diffraction management, including diffraction reversal [2], self-collimation effects [2, 3], the discrete Talbot effect [4], diffraction suppression via Bloch oscillations [5], dynamic localization [6], astigmatic diffraction control [7], diffraction-managed discrete solitons [8], to name a few. For a review of early works, see e.g., [9] and references therein. Note that most studies focused on the suppression of diffraction for broad beams having a narrow spatial spectrum, with only few exceptions, e.g., self-collimation of white light in curved waveguides [3], beam evolution in frequency-chirped lattices [10]. Additionally, beam path is no longer straight except for [10].

Recently, beam evolution in $\mathrm{\mathbb{P}}\mathbb{T}$ symmetric potentials has attracted considerable interest due to their fundamental physics and potential applications in various physical fields [11–19]. The crucial reason is that a non-Hermitian complex potential can exhibit entirely real spectra, provided that the imaginary part of the potential is an odd function and the real part is an even function of x. Another necessary condition for the real spectra of $\mathrm{\mathbb{P}}\mathbb{T}$ potentials is that the gain-loss strength is below a certain phase-transition point or symmetric-breaking point [11–19]. In linear case, double refraction (for broad beam) and nonreciprocal diffraction (for narrow beam) occur in periodic $\mathrm{\mathbb{P}}\mathbb{T}$ lattices [14]. Various types of spatial solitons and their unusual properties have been explored theoretically and observed experimentally in diverse nonlinear settings [20–36].

The amplitude or frequency modulation of transversal photonic lattices provides new opportunities for controllable soliton steering and routing [37]. Chirped lattices change the existence conditions and stability properties of power thresholdless surface waves [38]. Recalling the fact that properly designed lattice chirp can be utilized to tune the strength and sign of beam diffraction in the broad band of spatial frequencies [10], one naturally asks: whether the beam diffraction can be managed in $\mathrm{\mathbb{P}}\mathbb{T}$ lattices with a suitable modulation? Another question is: what is the dynamics of solitons in chirped $\mathrm{\mathbb{P}}\mathbb{T}$ lattices?

To answer them, we study beam evolution in $\mathrm{\mathbb{P}}\mathbb{T}$ lattices with a quadratic frequency modulation. In linear regime, diffraction control of beams with a broad band of spatial frequencies can be realized by properly adjusting the lattice depth, chirp rate, and strength of gain-loss. Double refraction and nonreciprocal diffraction usually occurred in regular $\mathrm{\mathbb{P}}\mathbb{T}$ lattices for broad and narrow beams can be suppressed in chirped lattices, provided that the lattice depth is at a critical value and the gain-loss component remains odd symmetric. In focusing Kerr media, with the decrease of power, the delocalization rate of solitons in chirped lattices is far slower than that of solitons in periodic (unchirped) lattices. A new family of “bright” solitons are found in defocusing Kerr media modulated by chirped $\mathrm{\mathbb{P}}\mathbb{T}$ lattices. Such solitons are stable in their entire existence domain, in contrast to the stability of gap solitons in periodic $\mathrm{\mathbb{P}}\mathbb{T}$ lattices [25].

2. Theoretical model

We begin our analysis by considering optical wave propagation along the z axis in cubic nonlinear media with a linear transverse refractive index modulation. Evolution of a beam is thus governed by the dimensionless nonlinear Schrödinger equation:

The desired real part of lattice can be realized by several ways, e.g., Fourier transform synthesis technique, computer generated holograms, direct technological fabrication. The gain-loss component may be achieved by employing concatenated semiconductor optical amplifier and semiconductor-doped two-photon absorber sections [20]. In Eq. (1), transverse and longitudinal coordinates are scaled to the beam width a and diffraction length L_diff = kn₀a², respectively. k = 2π/λ and n₀ is the background refractive index. Lattice depth p is thus scaled to 1/(k²n₀a²). For an input beam with λ = 1μm, a = 10μm and n₀ = 3, unit length corresponds to ∼ 10μm in transverse direction and ∼ 1.9mm in propagation direction. p = 1 corresponds to a maximum variation of refractive-index ∼ 8.5 × 10⁻⁵.

3. Diffraction management in linear regime

Before we discuss the diffraction management, it is instructive to understand the formation of beam diffraction. After propagating over a distance z, each plane wave with wavenumber k adds a phase ϕ(k,z) = b(k)z [1], where k and b is the transverse wave number (spatial frequency) and longitudinal wave number (propagation constant), respectively. The components originally centered at k shifts transversely by a value $\mathrm{\Delta }x=\frac{d\varphi }{dk}=\frac{db}{dk}z$. The propagation direction of k component can be expressed as $\theta ={\tan }^{-1}\left(\frac{db}{dk}\right)$. The different θ for different k leads to the different propagation direction. In vacuum or uniform media, the divergence between different displacements $S=\frac{1}{z}\frac{d^2\varphi }{d{k}^2}=\frac{d^{2}b}{d{k}^2}$ at different k broadens the beam, which is usually termed as diffraction. Within the paraxial approximation, the Poynting vector of each k component is perpendicular to the diffraction curve.

The above analysis indicates that the sign and strength of beam diffraction can be controlled by changing the sign and value of S for different spatial frequencies. If S > 0, beam experiences normal diffraction and broadens during propagation. Nondiffraction beam forms when S = 0 and reversed diffraction can occur for S < 0. It gives us a helpful hint that, by applying suitable linear refractive index modulation, diffraction management is possible for a beam with a broad band of spatial frequencies.

By applying the Fourier transform on a complex $\mathrm{\mathbb{P}}\mathbb{T}$ lattice with a quadratic frequency modulation in both real and imaginary parts, the lattice spectrum obeys the following relation:

Now, we consider beam evolution in the absence of nonlinearity by setting σ = 0 in Eq. (1). By applying the Fourier transform to Eq. (1), we obtain the following equation for the spatial Fourier spectrum of the beam:

The term −k²/2 represents the normal diffraction with a convex phasefront (negative sign) in uniform media, the second term in the right side of Eq. (4) denotes a phase shift independent on k, and the last one describes the phase shift induced by the Bragg-type scattering of a lattice.

According to Eq. (2), we plot the spatial spectra of chirped $\mathrm{\mathbb{P}}\mathbb{T}$ lattices with different chirp rates in Fig. 1(a). When α ≪ 1, there is a local maximum of R_k in the vicinity of k = Ω. It decreases and shifts towards the high-frequency region for increasing α. This indicates that the width of a spatial frequency band [−k_m k_m] (k_m corresponding to R_kmax) can be enlarged by the increase of chirp rate α. We stress here again that the gain-loss component of a $\mathrm{\mathbb{P}}\mathbb{T}$ lattice plays the same role as the refractive index modulation on the lattice spectrum distribution. We also should note that although there is a term dependent on Ω + k in real lattices [10], its contribution to the spectrum distribution can be neglected because it is too small. In the region k ∈ [−k_m k_m], R_k curve is approximatively parabolic, which is crucial and makes it possible for the realization of diffraction management. By adjusting the lattice depth p and coefficient of gain-loss component W₀, it is possible to get a flat dispersion curve, which means that a beam can propagate diffraction-freely.

 

Fig. 1 (a) Spatial spectra of chirped $\mathrm{\mathbb{P}}\mathbb{T}$ lattices. Dispersion relation at α = 0.1 (b) and 0.3 (c). (d) Critical lattice depth versus lattice frequency. Threshold depth at Ω = 2 are marked by circles. (e) Critical lattice depth versus k at W₀ = 0.5,Ω = 1.8, and α = 0.3. Ω = 2 except for (d, e). In all panels, V₀ = 1, solid curves stand for W₀ = 0.5 and dashed ones denote W₀ = 0.

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According to the zero diffraction condition d²b/dk² = 0, we obtain the explicit expression of critical lattice depth for fixed W₀ and α: p_cr(k) = {4(V₀ + W₀)(3αΩ)^−4/3(Ω−2k) Ai[(3αΩ)^−1/3(Ω − 2k)]⁻¹. It indicates that the critical lattice depth is a function of k. However, p_cr(k) is approximately equal to p_cr(0) within a certain range of k values [Fig. 1(e)]. To suppress the diffraction of beams, it is useful to replace the p_cr(k) by its average. By setting a properly designed mask with a gradient distribution of light transmittance in front of the optically induced lattice, it is possible to eliminate the diffraction completely.

A beam experiences a naturally “positive” diffraction for p = 0. The diffraction rate is decreased if 0 < p < p_cr. When p = p_cr, the diffraction is suppressed and beyond which, the beam will suffer a “negative” or anomalous diffraction. That is to say, in the spatial domain, a qualitative change of phase front shape from convex to concave occurs when the lattice depth crosses p_cr. These features are summarized in Fig. 1(b) and 1(c).

The frequency of lattices is another important parameter for controlling the beam diffraction [Fig. 1(d)]. The critical lattice depth p_cr increases rapidly with the growth of lattice frequency Ω. When Ω is below a certain value, p_cr of the lattice with α = 0.1 is smaller than the lattice with α = 0.3. The reverse case occurs if Ω is large. The p_cr curve (blue solid) of a $\mathrm{\mathbb{P}}\mathbb{T}$ lattice is always lower than that of a real lattice (blue dashed). The difference between them enlarges with the growth of Ω, which indicates again that the introduction of odd symmetric gain-loss component can effectively reduce the critical lattice depth for nondiffraction beam.

The aforementioned analytical results are verified by the direct numerical integration of Eq. (1). We conduct the propagation simulations of beams by using the split-step Fourier method and some representative examples are illustrated in Fig. 2. We set the incidence beam in the form of $q\left(x,z=0\right)=A\exp \left(-x^2/X_0^2\right)$, where A and X₀ are the amplitude and width of the input Gaussian beam, respectively. In periodic $\mathrm{\mathbb{P}}\mathbb{T}$ lattices, beam experiences a discrete diffraction [Fig. 2(a)]. The energy is coupled into the neighboring lattice channels after a short propagation distance. If W₀ is close to the phase-transition point W_cr, an asymmetry diffraction is obvious because of the strong gain-loss component [14]. In the present scheme, since W₀ = 0.5 is far below W_cr = 1, the asymmetry diffraction due to the skewness of the Floquet-Bloch modes can be ignored [Fig. 2(a)]. When the lattice is chirped and the corresponding critical lattice depth is set, the diffraction of beam vanishes [Fig. 2(b)]. We stress that in periodic $\mathrm{\mathbb{P}}\mathbb{T}$ lattices, a narrow beam with width approximating a lattice channel experiences a strong discrete diffraction. In contrast to this, chirped lattices with appropriate depth can suppress the diffraction, even when the input beam is narrow and with a broad band of spatial frequencies. Note that a chirped real lattice also admits nondiffration beams [10]. Nevertheless, the required p_cr is obviously higher than that in $\mathrm{\mathbb{P}}\mathbb{T}$ lattices [Fig. 2(c)]. If the lattice depth is the same as that of chirped $\mathrm{\mathbb{P}}\mathbb{T}$ lattice, the beam experiences a power oscillation and the energy radiates away slowly during propagation [Fig. 2(d)].

 

Fig. 2 Beam propagations in linear media. (a) Unchirped $\mathrm{\mathbb{P}}\mathbb{T}$ lattice with α = 0. (b) Chirped $\mathrm{\mathbb{P}}\mathbb{T}$ lattice with α = 0.3. (c) Chirped real lattice with p = 4.65, (d) Chirped real lattice with p = 3.1. V₀ = 1, W₀ = 0.5, p = 2.1 in (a, b) and V₀ = 1, W₀ = 0, α = 0.3 in (c, d). In all panels Ω = 2.

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4. Dynamics of solitons

In this section, we will discuss the properties of self-trapped nonlinear modes in chirped $\mathrm{\mathbb{P}}\mathbb{T}$ lattices. The stationary solutions of Eq. (1) are sought by assuming the light field obeying the form of plane-wave solution, i.e., q(x,z) = w(x)exp(ibz), where w is a complex function representing the soliton profile and b is the propagation constant. Substitution of the above expression into Eq. (1), we obtain: w_xx −bw+σw³ + pR(x)w = 0, which can be solved numerically by the relaxation iterative algorithm.

The stability of solitons can be analyzed by assuming the perturbed solutions of Eq. (1) as q(x,z) = {w(x) + [u(x) − v(x)]exp(λ z) + [u(x) + v(x)]^∗ exp(λ^∗z)} exp(ibz). The perturbations u, v can grow upon propagation with a complex growth rate λ. Linearization of Eq. (1) around the stationary solution w yields a linear eigenvalue problem:

In focusing Kerr media, the power of solitons defined as $U={\displaystyle {\int }_{-\infty }^{\infty }|q{|}^{2}dx}$ increases monotonously with the growth of propagation constant b [Fig. 3(a)]. The lower propagation constant cutoff of solitons in a $\mathrm{\mathbb{P}}\mathbb{T}$ lattice (b_cut = 0.53) is slightly smaller than that in a periodic real lattice (b_cut = 0.64) with the same chirp rate. With the growth of power, soliton becomes more localized and the ratio between the maxima of imaginary and real parts decreases [Fig. 3(b)], which leads to the decrease of difference between the power in $\mathrm{\mathbb{P}}\mathbb{T}$ and real lattices [Fig. 3(a)]. It also implies that the imaginary part of solitons is suppressed when the nonlinearity is strong. A unique feature of a chirped lattice is that it changes the existence domain of solitons [Fig. 3(c)]. Soliton solutions can be found only when b > b_cut. The existence domain expands slowly with the growth of chirp rate. A periodic lattice becomes a non-periodic one when α ≠ 0. The first band of unchirped lattice shrinks and eventually disappears with the increase of α.

The most important characteristic of lattices with a quadratic frequency modulation is that it can be used to control the beam diffraction in the linear regime. When nonlinearity is introduced, this property is also useful in the suppression of soliton width. For solitons with higher power, the width at α = 0 approximately equals to that at α ≠ 0 [Fig. 3(d)], due to that soliton resides mainly in the central lattice channel and the lattice chirp can be neglected [Fig. 3(b)]. However, soliton at lower power in unchirped lattices expands rapidly and covers many lattice channels, at this time, solitons tend to transform into linear Floquet-Bloch modes, in sharp contrast to the case in chirped lattices, where solitons are well localized. The soliton with zero power degenerates to a linear mode. Therefore, the zero-power soliton bridges the linear mode and thresholdless nonlinear mode with the same width. It demonstrates the role of chirped lattices in focusing nonlinear regime, i.e., for controlling the beam width. According to Eqs. (5), we perform the linear stability analysis on the whole family of soliton solutions and find no eigenvalues with a nonzero real part. Two typical propagation examples are displayed in Figs. 3(e) and 3(f). Either soliton with low power and wide profile or soliton with high power and narrow profile can propagate stably without any deformation.

 

Fig. 3 (a) U − b curves of solitons in lattices with α = 0.3,W₀ = 0.5 (upper) and 0 (lower). (b) Profiles of solitons marked in (a). (c) Propagation constant cutoff versus lattice chirp rate. (d) Dependence of soliton width on power. (e, f) Propagation examples of solitons at b = 0.53 and 4 marked in (c). White noises are added into the initial inputs. In all panels p = 2, V₀ = 1, Ω = 2. W₀ = 0.5 except for (a). Focusing medium.

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Fig. 4 (a) U − b curves of solitons. Profiles of solitons at b = 0.1 and 0.4 in lattices with α = 0 (b) and 0.3 (c). (d) Dependence of soliton width on power. In all panels p = 2, V₀ = 1, W₀ = 0.5, Ω = 2. Defocusing media.

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Now, we focus on the properties of solitons in chirped $\mathrm{\mathbb{P}}\mathbb{T}$ lattices imprinted in defocusing Kerr media. Unlike the solitons in focusing media, the power is a descending function of propagation constant [Fig. 4(a)]. The existence domain of solitons in chirped $\mathrm{\mathbb{P}}\mathbb{T}$ lattices is much narrower than that of solitons in unchirped lattices. Particularly, the upper propagation constant cutoff of solitons in a chirped lattice with α = 0.3 is b_cut = 0.53, which is approximately equal to the lower cutoff in focusing media [Fig. 3(c)]. Generally to say, the lower cutoffs in focusing media coincide with the upper cutoffs in defocusing media when α exceeds a certain value, below which there is an interval between them due to the emergence of periodicity of lattices.

In unchirped periodic lattices, gap soliton consists of a main peak in the central lattice channel and out-of-phase lobes in the neighboring lattice channels [Fig. 4(b)]. It can be seen as a higher-order nonlinear state since the dominant real part crosses the x axis several times. Interestingly, when the lattice is chirped, such soliton transforms into a fundamental one whose real part is totally above the x axis [Fig. 4(c)], just similar to the bright soliton in focusing media [Fig. 3(b)]. The competition between a reduced diffraction and defocusing nonlinearity accounts for the formation of such new type of solitons. Soliton in chirped lattices broadens with the growth of power [Fig. 4(d)], in contrary to the case in focusing media [Fig. 3(d)]. It is also different from the nonlinear mode in unchirped lattices, where the delocalization is strong either for solitons with low power or for solitons with high power.

It was demonstrated that in defocuing Kerr media with a periodic $\mathrm{\mathbb{P}}\mathbb{T}$ lattice, gap solitons with high power are usually unstable (see e.g., Fig. 5 in [25]). Here, we reveal another important application of lattice modulation, that is, a suitable chirp of lattice can be utilized to suppress the instability of gap solitons. While there is a wide instability window for gap solitons in a periodic $\mathrm{\mathbb{P}}\mathbb{T}$ lattice [Fig. 5(a)], solitons in a chirped $\mathrm{\mathbb{P}}\mathbb{T}$ lattice is stable in their whole existence region. To confirm the instability analysis results, we perform the propagation simulation of solitons exhaustively and several typical examples are plotted in Figs. 5(b)–5(d). These figures demonstrate clearly that the instability of solitons in unchirped lattices can be totally suppressed by an appropriate lattice modulation.

 

Fig. 5 (a) Instability growth rate versus propagation constant. (b) Unstable propagation of soliton at b = 0.1 in a unchirped $\mathrm{\mathbb{P}}\mathbb{T}$ lattice. Stable propagation of solitons at b = 0.1 (c) and 0.5 (d) in a chirped lattice with α = 0.3. In all panels p = 2, V₀ = 1, W₀ = 0.5, Ω = 2. Defocusing media.

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5. Conclusions

To summary, we investigated beam evolution in $\mathrm{\mathbb{P}}\mathbb{T}$ lattices with a quadratic frequency modulation. In linear regime, beam diffraction can be reduced, canceled, and reversed by properly adjusting the lattice depth, chirp rate, and strength of gain-loss. The gain-loss component, usually leading to double refraction and nonreciprocal diffraction in regular $\mathrm{\mathbb{P}}\mathbb{T}$ lattices, plays an important role on the suppression of beam diffraction. The lattice with high chirp rate favors to the formation of narrow nondiffraction beams with a broad frequency band which experience strongly asymmetric diffraction in harmonic $\mathrm{\mathbb{P}}\mathbb{T}$ lattices. In focusing media, the solitons with low power in chirped lattices are much narrower than the solitons in unchirped lattices. It implies that the reduction of diffraction by chirped lattices in linear cases still works in the nonlinear regime. Soliton remains localized even when its power approaches to zero. In defocusing media, we found a novel family of solitons corresponding to gap solitons in unchirped lattices, i.e., nonlinear localized modes similar to a bright fundamental soliton. Such solitons are completely stable in their existence domain, which manifests that the suitable modulation of lattices can be used to suppress the instability of solitons.