Polychromatic photonic Floquet-Bloch oscillations

Zhen Zhang; Yuan Li; Changhong Chen; Qi Yu; Xiankai Sun; Xiankai Sun; Xuewen Shu; Xuewen Shu

doi:10.1364/OE.519007

1. Introduction

As one of the most fundamental phenomena in wave optics, the self-imaging effect also known as the Talbot effect, is characterized by periodic repetition of planar field distributions. The self-imaging effect was first discovered with a diffraction grating by Talbot in 1836 [1] and then theoretically explained by Rayleigh in 1881 [2]. Beyond the direct result of Fresnel diffraction, the self-imaging effect has been extended to a broader range of phenomena exhibiting wave packet revivals in both spatial and temporal domains. Abundant interesting phenomena involving the self-imaging effect have been observed in various continuous systems, such as Bose–Einstein condensates [3], plasmons [4], quantum optics [5], metamaterials [6], photonic lattices [7], and conformal waveguides [8]. To date, the exploration of the self-imaging effect in discrete systems has drawn tremendous attention because discrete wave dynamics present exotic characteristics that have no analogue in continuous systems. One typical discrete system to implement the self-imaging effect is evanescently coupled waveguide arrays [9–28]. In straight waveguide arrays, the discrete self-imaging effect would appear only for particular field distributions [9]. Nevertheless, periodic self-imaging with arbitrary field distributions can also be realized by properly engineering the waveguide arrays. For example, an artificial gauge field can be introduced through the curved trajectory of waveguides, offering new possibilities for light manipulations and giving rise to a wide range of intriguing self-imaging phenomena, such as Bloch oscillations (BOs) [16–20], dynamic localization (DL) [21–26], Floquet–Bloch oscillations (FBOs) [27], and subwavelength self-imaging [28].

Self-imaging effect in waveguide arrays plays a key role in both current imaging technology and on-chip integrated photonics [28–30]. For broadband applications, achieving approximate self-imaging over a broad spectral range has drawn tremendous attention. However, due to the wavelength sensitivity of coupling among waveguides [31], the majority of self-imaging effects in waveguide arrays are constrained to a narrow spectral range. In the case of DL [21,22], periodic self-imaging only occurs for the light at a specific wavelength that satisfies the resonance condition, whereas light with other wavelengths experiences residual diffraction. Surprisingly, by mixing the first and the second resonances of DL, polychromatic dynamic localization, i.e., an approximate dynamic localization over a broad spectral range, has been theoretically proposed by Garanovich et al. in 2006 [23] and experimentally observed in femtosecond-laser-written waveguide arrays by Szameit et al. in 2009 [24]. Subsequently, several approaches to achieve polychromatic self-imaging have been reported, such as polychromatic optical Bloch oscillations [17], polychromatic dynamic localization with long-range interaction [25], generalized exact dynamic localization [26], nonlinearity-induced polychromatic self-imaging [32], and quasi-Bloch oscillations [33]. However, for fulfilling the demand for the new generation of broadband devices in integrated optics, exploring new self-imaging mechanisms to improve the comprehensive performance of polychromatic self-imaging is imperative. Recently, photonic Floquet–Bloch oscillations have been visually observed as a novel self-imaging phenomenon in femtosecond-laser-written waveguide arrays [27]. The visual observation of photonic FBOs shows that the amplitude and functional form of the Floquet modulation significantly affect the light evolution without altering the self-imaging period, suggesting the adjustability of the self-imaging effect. Nevertheless, the spectral range of the approximate self-imaging based on the configuration of the Floquet–Bloch oscillations remains largely unexplored, and the feasibility of achieving polychromatic self-imaging remains elusive.

Here, we investigate the spectral range of approximate Floquet–Bloch oscillations in arrays of evanescently coupled optical waveguides and show the realization of approximate self-imaging over the entire visible spectral range (450–750 nm), which we refer to as “polychromatic photonic Floquet–Bloch oscillations.” The polychromatic photonic FBOs exhibit as an efficient suppression of polychromatic beam diffraction, which is achieved by properly choosing the Floquet modulation amplitude. Moreover, by designing the Floquet modulation function form of FBOs, we can further improve the performance of polychromatic self-imaging based on photonic FBOs with the cascade of two polychromatic photonic FBOs.

2. Theory

We start our analysis with a one-dimensional (1D) curved photonic lattice of evanescently coupled waveguides. As shown in Fig. 1, the paraxial propagation of light along the z-axis in the photonic lattice is described by the Schrödinger-type equation

(1)$${i}\frac{{\partial \mathrm{\psi }}}{{\partial {z}}}{\; = \; {-}\; }\frac{{1}}{{{2k}}}{\nabla ^{2}}\mathrm{\psi \;\ {-}\;\ }\frac{{k}}{{{n0}}}{\Delta n(x,\;\ y,\;\ z)\psi ,\;\ }$$

where ψ(x, y, z) is the envelope of the electric field E(x, y, z) = ψ(x, y, z)exp(ikz − i2πωt), k = 2πn₀/λ is the wavenumber in the ambient medium with refractive index n₀, ω = c/λ is the optical frequency, c and λ are respectively the velocity and wavelength of light in vacuum, ${\nabla ^{2}}=\partial _{x}^{2}+\partial _{y}^{2}$ is the Laplacian operator in the x–y plane, and the function Δn(x, y, z) = n(x, y, z) − n₀ describes the refractive index distribution of the whole photonic lattice. The waveguide array is composed of identical curved waveguides with waveguide spacing d, which means that Δn(x, y, z) is a periodic function in x-coordinate with period d. Each period in the x-coordinate consists of a single waveguide following a gradually curved trajectory x₀(z). By making a transformation of the reference frame ${\tilde{x}\;\ =\ \;\ x\;\ +\ \;\ }{{x}_{0}}{(z)}$ such that the waveguides are straight in z-coordinate, Eq. (1) can be rewritten as

(2)$$\; {i}\frac{{\partial {\tilde{\psi }}}}{{\partial {z}}}{\; = \; {-}\; }\frac{{1}}{{{2k}}}\tilde{\nabla }{\tilde{\psi }\;\ {-}\;\ }\frac{{k}}{{{n0}}}{[\Delta n(\tilde{x},\;\ y)\;\ +\ \;\ F(z)\tilde{x}]}\widetilde {{\psi ,}}$$

where ${\ \tilde{\psi }\;\ =\ \;\ \psi (\tilde{x},\;\ y,\;\ z)}$ is the consequently transformed state [20,22], ${\Delta n(\tilde{x},\;\ y)}$ describes the transverse refractive index profile, and ${F(z)\; = \; {-}}{{n}_0}\partial _{z}^{2}{x_{0}}{(z)}$. Equation (2) is analogous to the Schrödinger equation describing an electron in a one-dimensional lattice driven by an external electric field force F [see Fig. 1(a)]. The curvature of the waveguides is perceived as an effective electric field force F(z) acting on light waves. For well-confined single-mode waveguides, in the nearest-neighbor tight-binding (NNTB) approximation and assuming that only the lowest Bloch band of the array is excited, Eq. (2) can be derived to the following set of coupled-mode equations:

(3)$${i}\frac{{\partial {{a}_{m}}}}{{\partial {z}}}{\;\ =\ \;\ {-}\kappa (\lambda )(}{{a}_{{m} - {1}}}{\; + \; }{{a}_{{m + 1}}}{)\; {-}\; }\frac{{{mF(z)kd}}}{{{n0}}}{{a}_{m}}{,\; }$$

where a_m is the amplitude of the guided mode $\left| m \right\rangle $ in the m^th waveguide and κ(λ) is the coupling coefficient between the nearest-neighbor waveguides. Substituting a plane wave ansatz a_m(z) = a₀exp(imk_xd)exp(iβz) into Eq. (2) and assuming F(z) = 0, we can obtain the dispersion relation for a straight waveguide array: β = 2κcos(k_xd), where k_x and β are the transverse Bloch momentum and the longitudinal propagation constant of the photonic lattice, respectively. A plot of this dispersion relation or band structure is depicted in Fig. 1(b). The effective electric field force F(z) induces the z-dependent Bloch momentum k_x(z) = k_x(0) + γ(z), where k_x(0) is the initial Bloch momentum and ${\ \gamma (z)\;\ =\ \;\ }\frac{{k}}{{{{n}_{0}}}}\mathop \smallint \nolimits_{0}^{z} {F(\tau )d\tau }$ denotes the shift of Bloch momentum [34]. The shift of k_x causes a z-dependent propagation constant, leading to the oscillation of wave packets in the real space. By engineering the format of the force, versatile transport dynamics in the lattice can be realized, such as optical Bloch oscillations under a constant effective force [16–19], dynamic localization under a periodic effective force [21–26], and photonic Floquet–Bloch oscillations under a combination of constant and periodic effective force [27].

Fig. 1. (a) Schematic of a one-dimensional lattice under an external electric field force F. (b) The lowest band structure of the lattice in (a), where the Bloch momentum shifts under the driving of F according to the generalized acceleration theory. (c) Schematic of a 1D array composed of circularly curved optical waveguides with a cosine modulation. (d) z-dependent shift of the transverse Bloch momentum in structure (c), where the photonic Floquet–Bloch oscillations occur with period 2$\Lambda$_FL.

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Here we focus on the photonic Floquet–Bloch oscillations under a combination of constant and periodic forces. The force can be introduced by a combined curved trajectory of waveguides according to x₀(z) = x_FL(z) + x_BO(z), where x_FL(z) is the periodically curved term following an arbitrary periodic function x_FL(z) = x_FL(z + $\Lambda$_FL) with the modulation period $\Lambda$_FL, and x_BO(z) is the circularly curved term with radius R (R >> z) [see Fig. 1(c)]. The periodically curved trajectory introduces a periodic electric field force ${{F}_{{\textrm {FL}}}}{\; = \; {-}}{{n}_0}\partial _{z}^{2}{x_{{\textrm {FL}}}}{(z)}$, which serves as the Floquet modulation with period $\Lambda$_FL. The circularly curved trajectory introduces a constant electric field force F_BO = n₀/R, which is responsible for optical BOs with period $\Lambda$_BO(λ) = 2πR/k(λ)d. The phenomenon of photonic FBOs corresponds to periodic self-imaging with a period $\Lambda$_FBO of extended least common multiple (LCM) of $\Lambda$_BO and $\Lambda$_FL, i.e., $\Lambda$_FBO = LCM($\Lambda$_BO, $\Lambda$_FL) [27]. In the NNTB approximation, the condition for photonic FBOs is $\mathop \smallint \nolimits_{0}^{z} {\textrm{exp}[{-}i {\gamma} (\tau )d]d\tau \;\ =\ \;\ 0}$ [21,27]. When $\Lambda$_FL ≠ $N\Lambda$_BO (N is a positive integer), the condition is always satisfied at a propagation length LCM($\Lambda$_BO, $\Lambda$_FL), indicating that the photonic FBO period $\Lambda$_FBO can be any integer multiples of the modulation period $\Lambda$_FL, i.e., 2$\Lambda$_FL, 3$\Lambda$_FL, 4$\Lambda$_FL…. We then analyze the photonic Floquet–Bloch oscillations with the minimum self-imaging period of 2$\Lambda$_FL. The corresponding z-dependent shift of the transverse Bloch momentum is shown in Fig. 1(d). The effective coupling strength after one self-imaging period of 2$\Lambda$_FL along the z-coordinate can be computed using averaging methods and is expressed as (see Appendix)

(4)$${{\kappa }_{{\textrm {eff}}}}{(\lambda )\;\ =\ \;\ }\frac{{{\kappa (\lambda )}}}{{{2}\varLambda_{\textrm {FL}}}}\left|{\mathop \int \nolimits_{0}^{{2}\varLambda_{\textrm {FL}}} {\textrm {exp}[{-}i {\gamma} (z)d]dz}} \right|{\;\ =\ \;\ \kappa (\lambda )\Gamma [k(\lambda )],\;\ }$$

with the effective force modulation ${\Gamma} [k(\lambda )]=|\frac{{{\sin(2\pi k/}{{k}_0}{)}}}{{{2\sin(\pi k/}{{k}_0}{)}}}\times\frac{{1}}{{{2\pi }}}\mathop \smallint \nolimits_{0}^{{2\pi }} { {\textrm {exp}}}\{ {{-}i}\frac{{k}}{{{{k}_{0}}}}[\theta +\frac{{{2\pi R}}}{{\varLambda_{\textrm {FL}}}}$ ${\partial_{z}}{x_{{\textrm {FL}}}}{(z)}|_{0}^{\frac{{\varLambda_{\textrm {FL}\theta }}}{{{2\pi }}}} ] \}{d\theta } |$ and the unit wavenumber k₀ = 2πR/$\Lambda$_FLd that satisfies the resonant modulation $\Lambda$_BO(k₀) = $\Lambda$_FL. The self-imaging effect at propagation distance 2$\Lambda$_FL can be achieved with κ_eff(λ) = 0, which means a complete suppression of discrete diffraction. Similarly, the approximate self-imaging at propagation distance 2$\Lambda$_FL can be achieved with κ_eff(λ) ≈ 0, which means an efficient suppression of discrete diffraction. When k = (N − 0.5)k₀ (N is a positive integer), Г[k(λ)] = 0 is attained since $\frac{{{\sin(2\pi k/}{{k}_0}{)}}}{{{2\sin(\pi k/}{{k}_0}{)}}}{\; = \; 0}$ is always satisfied. In this case, κ_eff(λ) = 0, which results in the occurrence of the photonic FBOs. Here we define the photonic FBOs at the wavenumber components k_FBO = (N − 0.5)k₀ as the Nth-order Floquet–Bloch oscillations. When k ≠ (N − 0.5)k₀, the condition Г = 0 caused by $\frac{{{\sin(2\pi k/}{{k}_0}{)}}}{{{2\sin(\pi k/}{{k}_0}{)}}}{\; = \; 0}$ is destroyed, resulting in the termination of self-imaging at propagation distance 2$\Lambda$_FL. Then we focus on the approximate self-imaging at propagation distance 2$\Lambda$_FL. Obviously, when the wavenumber k is sufficiently close to k_FBO, Г ≈ 0 can be achieved since the term $\frac{{{\sin(2\pi k/}{{k}_0}{)}}}{{{2\sin(\pi k/}{{k}_0}{)}}}$ is sufficiently close to zero. Consequently, there indeed exists an allowed spectral range Δk centered around k_FBO for the approximate self-imaging. For a certain k_FBO, the spectral range Δk of the approximate self-imaging is determined by the integral term of Г[k(λ)], i.e., $\frac{{1}}{{{2\pi }}}\mathop \smallint \nolimits_{0}^{{2\pi }} { {\textrm{exp}}}\left\{ {{{-}i}\frac{{k}}{{{{k}_{0}}}}\left[ {{\theta \;\ +\ \;\ }\frac{{{2\pi R}}}{{\varLambda_{\textrm {FL}}}}{\partial_{z}}{x_{{\textrm {FL}}}}{(z)}|_{0}^{\frac{{\varLambda_{\textrm {FL}\theta }}}{{{2\pi }}}}} \right]} \right\}{d\theta }$, which depends on the periodically curved trajectory x_FL(z). By properly engineering x_FL(z), it is possible to achieve Г[k(λ)] ≈ 0 over a relatively broad spectral range Δk centered around k_FBO, giving rise to the polychromatic self-imaging, i.e., polychromatic photonic Floquet–Bloch oscillations.

As a proof of concept, we consider a cosine curved trajectory as the periodically curved term, i.e., x_FL(z) = Acos(2πz/$\Lambda$_FL), where A is the modulation amplitude. The cosine trajectory introduces a well-defined effective electric field force with a simple mathematical form F_FL = F₀cos(2πz/$\Lambda$_FL), where F₀(A) = 4π²An₀/$\Lambda$_FL² is the amplitude of the effective periodic force. In this case, the effective force modulation Г[k(λ)] is reduced to

(5)$${\;\ \Gamma \;\ =\ \;\ }\left|{\frac{{{\sin(2\pi k/}{{k}_0}{)}}}{{{2\sin(\pi k/}{{k}_0}{)}}}{\;\ \times \;\ }\frac{{1}}{{{2\pi }}}\mathop \int \nolimits_{0}^{{2\pi }} { {\textrm{exp}}}\left\{ {{{-}i}\frac{{k}}{{{{k}_{0}}}}\left[ {{\theta \;\ +\ \;\ }\frac{{A}}{{{{A}_{0}}}}{\sin(\theta )}} \right]} \right\}{d\theta }} \right|{,\; }$$

where A₀ = $\Lambda$_FL²/4π²R is the unit modulation amplitude that satisfies F₀(A₀) = F_BO. Figure 2(a) shows the effective force modulation Г[k(λ)] with respect to k and A. When k = k_FBO, the condition Г = 0 is always satisfied with arbitrary amplitude A, corresponding to the photonic FBOs [the green dashed lines in Fig. 2(a)]. This phenomenon indicates that the occurrence of photonic FBOs does not depend on the modulation amplitude A. However, the allowed spectral range Δk centered around k_FBO of Г ≈ 0 varies with the modulation amplitude A, suggesting the dependence of approximate FBOs on modulation amplitude A. Especially for the even-order FBOs, a relatively broad spectral range Δk of Г ≈ 0 can be attained with a small modulation amplitude A. Taking the 2nd-order FBOs (k_FBO = 1.5k₀) as an example, Fig. 2(b) shows the effective force modulation Г as a function of k for different A. For A = 0.4A₀, Г < 0.01 can be attained in the spectral range 1.29–1.77k₀ (the gray dashed line represents Г = 0.01), indicating that the effective coupling strength κ_eff(λ) is suppressed to less than 1% of the coupling strength κ(λ) in the spectral range. In this case, the spectral range of Г < 0.01 far exceeds the spectral range with A = 0 (BOs). Moreover, the spectral range satisfying Г < 0.01 of the 2nd-order FBOs can also be smaller than that of BOs, e.g., the 2nd-order FBOs with A = A₀. The results suggest the adjustability of the spectral range of the approximate 2nd-order photonic FBOs and the achievement of polychromatic 2nd-order photonic FBOs. In a similar vein, the polychromatic Floquet–Bloch oscillations can be achieved for any even-order FBOs with the cosine curved trajectory x_FL(z).

Fig. 2. (a) Effective force modulation Г as a function of wavenumber k (in unit of k₀ = 2πR/$\Lambda$_FLd) and modulation amplitude A (in unit of A₀ = $\Lambda$_FL²/4π²R). The green dashed lines correspond to the Nth-order photonic Floquet–Bloch oscillations. (b) Effective force modulation Г as a function of k for different A. The gray dashed line represents Г = 0.01, meaning that the effective coupling strength κ_eff is suppressed to 1% of the coupling coefficient κ.

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3. Numerical simulation and analysis

Next, we verify the aforementioned theoretical findings through numerical simulations of polychromatic light (visible spectral range 450–750 nm) evolution in femtosecond-laser-written waveguide arrays [24]. The simulations were performed by solving the paraxial wave equation Eq. (1) with a single-site excitation. The single-site excitation in the real space corresponds to a broad excitation of Bloch modes in the reciprocal space. Therefore, it is enough to investigate the self-imaging effect in waveguide arrays by employing a single-site excitation. In the simulations, we employed 60-mm-long femtosecond-laser-written waveguide arrays in fused silica substrates (n₀ ∼ 1.46 in the visible spectral range). For describing the femtosecond-laser-written waveguide arrays in a fused silica substrate, the refractive index profile in the transverse cross-section we employed here follows the hyper-Gaussian function Δn(x, y) = p∑_mexp(−{[(x − md)/w_x]² + (y/w_y)²}³), with the refractive index modulation depth p = 7 × 10⁻⁴, the minor axis radius w_x = 2 µm, the major axis radius w_y = 5.5 µm, and the waveguide spacing d = 18 µm. The simulation parameters are derived from the reported works that utilize femtosecond-laser-written waveguide arrays in fused silica substrates as experimental platforms [24,27,35,36], ensuring the reliability and practicality of our results. For polychromatic light excitation, we used a superposition of 300 wavelength components with an equidistant flat spectrum from 450 to 750 nm as input. The propagation pattern and spectrally resolved output in an array composed of 41 straight waveguides are displayed in Figs. 3(a)–3(c), respectively. Note that the beam broadening becomes significantly stronger at longer wavelengths, indicating a sharp rise in the coupling strength with increasing wavelength [see Fig. 3(d)].

Fig. 3. Propagation patterns, spectrally resolved output, and effective coupling strength in waveguide arrays with single-site excitations. (a)–(d) Diffraction and the corresponding coupling strength in a straight waveguide array. (e)–(h) Diffraction and the corresponding effective coupling strength for the case of BOs. (i)–(l) Diffraction and the corresponding effective coupling strength for the case of polychromatic photonic FBOs. (m)–(p) Diffraction and the corresponding effective coupling strength for the case of DL. (q) Transmission spectra of the center waveguide for the above four cases.

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To realize the 2nd-order FBOs at propagation distance L = 60 mm, waveguides following the combined curved trajectories with circularly curved radius R = 876 mm (corresponding to $\Lambda$_BO = 20 mm at λ = 600 nm) and cosine curved period $\Lambda$_FL = 30 mm were employed to meet the 2nd-order FBOs condition (k_FBO = 1.5k₀) at the resonant wavelength of λ_FBO = 600 nm. As shown in Figs. 3(e)–3(p), we constructed three sets of waveguide arrays with the same parameters except for different bending profiles, which correspond to BOs (R = 876 mm, A = 0), 2nd-order FBOs (R = 876 mm, A = 10 µm), and 1st-order DL (R = ∞, A = 42 µm), respectively. The curvatures of the waveguides are sufficiently small, thus allowing for the adiabatic assumption of guided modes. For the BOs case under a constant modulation, the monochromatic beam with λ = 600 nm experiences self-imaging with period $\Lambda$_BO = 20 mm [see Fig. 3(e)]. Unfortunately, the light with different wavelength components exhibits a shifted BO period, limiting the approximate self-imaging in a narrow spectral range around λ = 600 nm [see Figs. 3(f) and 3(g)]. As displayed in Fig. 3(h), the effective coupling strength in this structure is not sufficiently suppressed, especially for the red wavelength components, which experience significant diffraction [see Fig. 3(g)]. When the Floquet modulation is introduced to the BOs case, the photonic Floquet–Bloch oscillations occur with adjustable approximate self-imaging spectral range as expected, where the modulation amplitude A makes a significant difference. For the FBOs case with A = 10 µm (corresponding to A = 0.4A₀), it can be clearly observed that almost all the wavelength components in the visible range return to the input waveguide at the output [see Figs. 3(i)–3(k)], resulting in approximate self-imaging over an extremely broad spectral range, i.e., the polychromatic photonic FBOs. The corresponding effective coupling strength is much less than that in the structure without Floquet modulation [see Fig. 3(l)]. Especially in the spectral range 450–706 nm, more than 99% of light is localized in a single waveguide at the output [see Fig. 3(q)], indicating the sufficient suppression of coupling strength. When the Floquet modulation is introduced to a straight waveguide array, the phenomenon of DL can occur [see Figs. 3(m)–3(p)]. In this case, the approximate self-imaging is limited to a narrow spectral range, which is much smaller than that of polychromatic photonic FBOs. As shown in Fig. 3(q), the spectral bandwidths of 99% transmission at the output of the center waveguide are 256 nm for polychromatic photonic FBOs, 20 nm for BOs, and 17 nm for DL. The results indicate that both the constant modulation and the Floquet modulation have a significant impact on the spectral range of approximate self-imaging. With the constant modulation, it is possible to achieve near-zero Г[k(λ)] over an extremely broad spectral range by properly choosing the Floquet modulation amplitude A, so as to sufficiently suppress the coupling strength and achieve polychromatic photonic FBOs.

The performance of polychromatic self-imaging based on photonic FBOs can be further improved by fine-tuning the cosine functional form of the periodically curved trajectory x_FL(z). Inspired by the reported works that employ hybrid structures with a periodic bending profile for compensation of coupling dispersion in waveguide arrays [23–26], we consider the periodically curved trajectory x_FL(z) following a piecewise function [see Fig. 4(a)], which is expressed as

(6)$${{x}_{{\textrm {FL}}}}{(z)\; = \; }\left\{ \begin{array}{{c}} {{A\cos}\left( {\frac{{{2\pi z}}}{{{\varLambda_{{\textrm{FL}}}}}}} \right){,\; \;{N}}{{\varLambda}_{{\textrm {FL}}}}{\; } \le {\;\ z\;\ < \;\ N}{{\varLambda}_{{\textrm {FL}}}}{\; + \; }\frac{{{{\varLambda}_{{\textrm {FL}}}}}}{{4}}{,\; \; \; \; \; \; \; \; \; \; \; \; \; }}\\ {{{A}_{1}}{\cos}\left( {\frac{{{2\pi z}}}{{{\varLambda_{{\textrm {FL}}}}}}} \right){,\; \; N}{{\varLambda}_{{\textrm {FL}}}}{\; + \; }\frac{{{{\varLambda}_{{\textrm {FL}}}}}}{{4}}{\; } \le {\;\ z\;\ < \;\ N}{{\varLambda}_{{\textrm {FL}}}}{\; + \; }\frac{{{3}{{\varLambda }_{{\textrm {FL}}}}}}{{4}}{,}}\\ {{A\cos}\left( {\frac{{{2\pi z}}}{{{\varLambda _{{\textrm {FL}}}}}}} \right){,\; \;{N}}{{\varLambda }_{{\textrm {FL}}}}{\; + \; }\frac{{{3}{{\varLambda }_{{\textrm {FL}}}}}}{{4}}{\; } \le {\;\ z\;\ < \;\ N}{{\varLambda }_{{\textrm {FL}}}}{\; + \; }{{\varLambda }_{{\textrm {FL}}}}{.\; \;}} \end{array} \right.$$

Then we investigate the influence of A₁ on the function of effective force modulation Г with respect to wavenumber k and modulation amplitude A. As shown in Figs. 4(b)–4(e), A depends strongly on A₁ for achieving polychromatic 3rd-order FBOs (k_FBO = 2.5k₀), but is rarely affected by A₁ for achieving polychromatic 2nd-order FBOs (k_FBO = 1.5k₀). Interestingly, when A₁ = 0.4A [see Fig. 4(d)], both the polychromatic 2nd-order and 3rd-order photonic FBOs can be achieved with A ≈ 0.75A₀. In this case, the two spectral ranges of polychromatic photonic FBOs are connected, and can therefore be considered as an extremely broad spectral range of polychromatic self-imaging effect. Next, we constructed waveguides following the combined trajectories with circularly curved radius R = 701 mm and piecewise curved period $\Lambda$_FL = 30 mm. The parameters were set to simultaneously meet the 2nd-order FBOs condition (k_FBO = 1.5k₀) at the resonant wavelength of λ_FBO = 750 nm and the 3rd-order FBOs condition (k_FBO = 2.5k₀) at the resonant wavelength of λ_FBO = 450 nm. Figure 4(f) displays the effective coupling strengths in the waveguide arrays with modulation amplitude A = 23, 24, 25, and 26 µm, which corresponds to A ≈ 0.71A₀, 0.74A₀, 0.77A₀, and 0.80A₀, respectively. We observe that, in all the four structures, the effective coupling strengths are suppressed to less than 0.016 cm⁻¹ for all the wavelength components over the visible wavelength range. In these scenarios, the spectral ranges of polychromatic self-imaging are larger than that of polychromatic photonic FBOs shown in Fig. 3(l). Compared with cosine modulation, the piecewise periodic modulation provides a further suppression of coupling strength by cascading the 2nd-order and 3rd-order polychromatic photonic FBOs. Moreover, the sufficient suppression of waveguide coupling in all the four structures with different amplitude A also indicates robustness against variations in modulation amplitude A. These results show the potential of periodically curved trajectory designs in polychromatic self-imaging in waveguide arrays.

Fig. 4. Further optimization of polychromatic self-imaging based on photonic FBOs by designing the functional form of the periodically curved trajectory x_FL(z). (a) Periodically curved trajectory x_FL(z) with $\Lambda$_FL = 30 mm. (b)–(e) Effective force modulation Г as a function of the wavenumber k (in unit of k₀ = 2πR/$\Lambda$_FLd) and modulation amplitude A (in unit of A₀ = $\Lambda$_FL²/4π²R) with A₁ = 1.0A, 0.7A, 0.4A, and 0.1A, respectively. The green dashed lines correspond to the Nth-order photonic Floquet–Bloch oscillations. The gray dashed line represents A = 0.75A₀. (f) Effective coupling strength in the waveguide arrays with periodically curved amplitude A = 23, 24, 25, and 26 µm, corresponding to A = 0.71A₀, 0.74A₀, 0.77A₀, and 0.80A₀, respectively.

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4. Conclusion

In summary, we have developed a novel method to control the spectral range of approximate self-imaging in discrete systems, drawing inspiration from the photonic Floquet–Bloch oscillations. By properly engineering the Floquet modulation of the photonic FBOs, we can sufficiently suppress the effective coupling strength in an extremely broad spectral range, giving rise to polychromatic self-imaging. Further performance enhancement of polychromatic self-imaging based on photonic FBOs can be achieved by optimizing the functional form of the periodically curved trajectory, and machine learning may be a promising approach to achieve this goal. Beyond polychromatic self-imaging, our proposed method exhibits a remarkable capability for engineering effective coupling strength in a broad spectral range, suggesting the potential to achieve dispersionless coupling [37] and broadband diffraction management. Furthermore, as discrete diffraction phenomena, polychromatic photonic FBOs in waveguide arrays can be extended to synthetic dimensions of time [38,39] and frequency [40], providing new insights into signal processing and frequency manipulations.

Appendix: Derivation of the effective coupling strength

In this appendix, we present the detailed derivation of effective coupling strength κ_eff(λ) at propagation distance z = 2$\Lambda$_FL. In the high-frequency limit, the effective coupling coefficient ${\kappa }_{{\textrm {eff}}}^{^{\prime}}{(\lambda )}$ can be averaged as [21,37]

(A1)$${\kappa }_{{\textrm {eff}}}^{^{\prime}}{(\lambda )\;\ =\ \;\ }\frac{{{\kappa (\lambda )}}}{{{2}\varLambda_{\textrm {FL}}}}\mathop \int \nolimits_{0}^{{2}\varLambda_{\textrm {FL}}} {\textrm{exp}[{-}i {\gamma} (z)d]dz\;\ =\ \;\ }{{\kappa }_{{\textrm {eff}}}}{(\lambda )}{{e}^{{i\zeta }}}{,\; }$$

with the effective coupling strength

(A2)$${{\kappa }_{{\textrm {eff}}}}{(\lambda )\;\ =\ \;\ }\frac{{{\kappa (\lambda )}}}{{{2}\varLambda_{\textrm {FL}}}}\left|{\mathop \int \nolimits_{0}^{{2}\varLambda_{\textrm {FL}}} {\textrm{exp}[{-}i {\gamma} (z)d]dz}} \right|{,\; }$$

and the phase of the coupling

(A3)$${\;\ \zeta \;\ =\ \;\ ta}{{n}^{{ - 1}}}\left\{ {\frac{{\mathop \int \nolimits_{0}^{{2}\varLambda_{\textrm {FL}}} {\sin[{-}\gamma (z)d]dz}}}{{\mathop \int \nolimits_{0}^{{2}\varLambda_{\textrm {FL}}} {\cos[{-}\gamma (z)d]dz}}}} \right\}{.\; }$$

Note that the phase of coupling will not affect the evolution of light intensity. Therefore, we would like to analyze the effective coupling strength. The effective coupling strength is significantly affected by the effective force modulation Г(λ), which is expressed as

(A4)$$\begin{aligned} \Gamma(\lambda) & =\frac{\kappa_{\mathrm{eff}}(\lambda)}{\kappa(\lambda)}=\frac{1}{2 \varLambda_{\mathrm{FL}}}\left|\int_0^{2 \varLambda_{\mathrm{FL}}} \exp [-i \gamma(z) d] \mathrm{d} z\right| \\ & =\frac{1}{2 \varLambda_{\mathrm{FL}}}\left[\left(\int_0^{2 \varLambda_{\mathrm{FL}}} \cos [\gamma(z) d] \mathrm{d} z\right)^2+\left(\int_0^{2 \varLambda_{\mathrm{FL}}} \sin [\gamma(\tau) d] \mathrm{d} z\right)^2\right]^{1 / 2}.\end{aligned}$$

For photonic FBOs under a combination of constant and periodic force, the shift of Bloch momentum γ(z) is derived as

(A5)$${\;\ \gamma (z)\;\ =\ \;\ }\frac{{k}}{{{{n}_{0}}}}\mathop \int \nolimits_{0}^{z} {F(\tau )d\tau \;\ =\ \;\ }\frac{{k}}{{{{n}_{0}}}}\mathop \int \nolimits_{0}^{z} \frac{{{{n}_{0}}}}{{R}}{\; {-}\; }{{n}_0}\partial _{z}^{2}{x_{{\textrm {FL}}}}{(\tau )d\tau \;\ =\ \;\ }\frac{{{kz}}}{{R}}{\; {-}\; k}{\partial _{z}}{x_{{\textrm {FL}}}}{(\tau )}|_{0}^{z}{.}$$

Substituting γ(z) into the integral term of Eq. (A4), one arrives at the expressions:

(A6)$$\begin{aligned} & \int_0^{2 \varLambda_{\mathrm{FL}}} \cos [\gamma(z) d] \mathrm{d} z \\ & =\int_0^{\varLambda_{\mathrm{FL}}} \cos \left[\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}-\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^z\right] \mathrm{d} z+\int_{\varLambda_{\mathrm{FL}}}^{2 \varLambda_{\mathrm{FL}}} \cos \left[\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}-\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^z\right] \mathrm{d} z \\ & =\int_0^{\varLambda_{\mathrm{FL}}} \cos \left[\frac{2 \pi \varLambda_{\mathrm{FL}}}{\varLambda_{\mathrm{BO}}}-\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}-\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^{-z}\right] d z \\ & +\int_0^{\varLambda \mathrm{FL}} \cos \left[\frac{4 \pi \varLambda_{\mathrm{FL}}}{\varLambda_{\mathrm{BO}}}-\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}-\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^{-z}\right] d z \\ & =\frac{\sin \left(2 \pi \varLambda_{\mathrm{FL}} / \varLambda_{\mathrm{BO}}\right)}{\sin \left(\pi \varLambda_{\mathrm{FL}} / \varLambda_{\mathrm{BO}}\right)} \times\left\{\cos \left(\frac{3 \pi \varLambda_{\mathrm{FL}}}{\varLambda_{\mathrm{BO}}}\right) \int_0^{\varLambda_{\mathrm{FL}}} \cos \left[\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}+\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^{-z}\right] d z\right. \\ & \left.+\sin \left(\frac{3 \pi \varLambda_{\mathrm{FL}}}{\varLambda_{\mathrm{BO}}}\right) \int_0^{\varLambda \mathrm{FL}} \sin \left[\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}+\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^{-z}\right] d z\right\}, \\ & \end{aligned}$$

(A7)$$\begin{aligned} & \int_0^{2 A_{\mathrm{FL}}} \sin [\gamma(z) d] \mathrm{d} z \\ & =\int_0^{\varLambda_{\mathrm{FL}}} \sin \left[\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}-\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^z\right] \mathrm{d} z+\int_{\varLambda_{\mathrm{FL}}}^{2 \varLambda_{\mathrm{FL}}} \sin \left[\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}-\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^z\right] \mathrm{d} z \\ & =\int_0^{\varLambda \mathrm{FL}} \sin \left[\frac{2 \pi \varLambda_{\mathrm{FL}}}{\varLambda_{\mathrm{BO}}}-\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}-\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^{-z}\right] d z \\ & +\int_0^{\varLambda_{\mathrm{FL}}} \sin \left[\frac{4 \pi \varLambda_{\mathrm{FL}}}{\varLambda_{\mathrm{BO}}}-\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}-\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^{-z}\right] d z \\ & =\frac{\sin \left(2 \pi \varLambda_{\mathrm{FL}} / \varLambda_{\mathrm{BO}}\right)}{\sin \left(\pi \varLambda_{\mathrm{FL}} / \varLambda_{\mathrm{BO}}\right)} \times\left\{\sin \left(\frac{3 \pi \varLambda_{\mathrm{FL}}}{\varLambda_{\mathrm{BO}}}\right) \int_0^{\varLambda_{\mathrm{FL}}} \cos \left[\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}+\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^{-z}\right] d z\right. \\ & \left.-\cos \left(\frac{3 \pi \varLambda_{\mathrm{FL}}}{\varLambda_{\mathrm{BO}}}\right) \int_0^{\varLambda_{\mathrm{FL}}} \sin \left[\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}+\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^{-z}\right] d z\right\}. \\ & \end{aligned}$$

Substituting Eq. (A6) and (A7) into (A4), we can obtain

(A8)$$\begin{aligned} & \Gamma(\lambda)=\frac{1}{2 \varLambda_{\mathrm{FL}}}\left[\left(\int_0^{2 \varLambda_{\mathrm{FL}}} \cos [\gamma(z) d] \mathrm{d} z\right)^2+\left(\int_0^{2 \varLambda_{\mathrm{FL}}} \sin [\gamma(\tau) d] \mathrm{d} z\right)^2\right]^{1 / 2} \\ & =\frac{1}{2 \varLambda_{\mathrm{FL}}}\left|\frac{\sin \left(2 \pi \varLambda_{\mathrm{FL}} / \varLambda_{\mathrm{BO}}\right)}{\sin \left(\pi \varLambda_{\mathrm{FL}} / \varLambda_{\mathrm{BO}}\right)}\right| \times\left[\left(\int_0^{\varLambda_{\mathrm{FL}}} \cos \left[\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}+\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^{-z}\right] d z\right)^2\right. \\ & \left.+\left(\int_0^{\varLambda_{\mathrm{FL}}} \sin \left[\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}+\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^{-z}\right] d z\right)^2\right]^{1 / 2} \\ & =\left|\frac{\sin \left(2 \pi \varLambda_{\mathrm{FL}} / \varLambda_{\mathrm{BO}}\right)}{2 \sin \left(\pi \varLambda_{\mathrm{FL}} / \varLambda_{\mathrm{BO}}\right)} \times \frac{1}{\varLambda_{\mathrm{FL}}} \int_0^{\varLambda_{\mathrm{FL}}} \exp \left\{-i\left[\frac{2 \pi z}{\varLambda_{\mathrm{BO}}}+\left.\frac{2 \pi n_0 d}{\lambda} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^{-z}\right]\right\} d z\right|. \\ & \end{aligned}$$

Considering the change of variable z = $\Lambda$_FLθ/(2π), one arrives at the expression of Г(λ) as

(A9)$$\begin{aligned} \Gamma(\lambda) & =\left|\frac{\sin \left(2 \pi \varLambda_{\mathrm{FL}} / \varLambda_{\mathrm{BO}}\right)}{2 \sin \left(\pi \varLambda_{\mathrm{FL}} / \varLambda_{\mathrm{BO}}\right)} \times \frac{1}{2 \pi} \int_0^{2 \pi} \exp \left\{-i \frac{\varLambda_{\mathrm{FL}}}{\varLambda_{\mathrm{BO}}}\left[\theta+\left.\frac{2 \pi R}{\varLambda_{\mathrm{FL}}} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^{\frac{-\varLambda_{\mathrm{FL}} \theta}{2 \pi}}\right]\right\} d \theta\right| \\ & =\left|\frac{\sin \left(2 \pi k / k_0\right)}{2 \sin \left(\pi k / k_0\right)} \times \frac{1}{2 \pi} \int_0^{2 \pi} \exp \left\{-i \frac{k}{k_0}\left[\theta+\left.\frac{2 \pi R}{\varLambda_{\mathrm{FL}}} \partial_z x_{\mathrm{FL}}(\tau)\right|_0 ^{\frac{-\varLambda_{\mathrm{FL}} \theta}{2 \pi}}\right]\right\} d \theta\right|, \end{aligned}$$

with the wavenumber k(λ) = 2πn₀/λ = 2πR/$\Lambda$_BO(λ)d and the unit wavenumber k₀ = 2πR/$\Lambda$_FLd.

Funding

National Key Research and Development Program of China (2023YFE0105800); National Natural Science Foundation of China (62275093); Research Grants Council of Hong Kong (C4050-21E, N_CUHK479/23, RFS2324-4S03).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Polychromatic photonic Floquet-Bloch oscillations

Abstract

1. Introduction

2. Theory

3. Numerical simulation and analysis

4. Conclusion

Appendix: Derivation of the effective coupling strength

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Equations (15)

Optics Express