Theoretical studies of modulation instability, Fermi–Pasta–Ulam recurrence and pattern formation in an ultra-silicon-rich-nitride Bragg grating

2.1 Tailorable refractive index

Silicon-rich nitride has recently gained attention as a promising material for integrated photonic platforms [50]. Its refractive index can be tuned between 2.1 and 3.1 by tailoring the silicon content, allowing optimized designs for photonic devices like waveguides, resonators, and modulators [51]. This tunability is essential for meeting specific optical requirements, such as phase matching in nonlinear processes or minimizing losses.

The Kramers–Kronig relation provides a foundation for silicon-rich nitride-based material design, showing that the refractive index increases as the bandgap decreases, while absorption drops with a larger bandgap [61], [62]. These principles, related to Miller’s rule, indicate that nonlinear polarizability (χ ³) scales with the refractive index n [63]. Carefully balancing bandgap, refractive index, and absorption is crucial for selecting silicon-rich nitride materials for nonlinear applications. Minimizing two-photon absorption at telecom wavelengths (1,550 nm), while maintaining a high nonlinear coefficient, is a key objective for effective material design.

2.2 Compatibility with CMOS fabrication

Silicon has advanced as a waveguide material and serves many applications [50], but it faces challenges as a nonlinear material. Its indirect bandgap (1.1 eV at room temperature) reduces two-photon absorption at 1,550 nm, but it can still occur at high intensities, converting optical energy into heat and limiting nonlinear effects like four-wave mixing and supercontinuum generation [64]. Additionally, free carriers cause extra losses through absorption and affect dispersion, degrading the performance of nonlinear optical devices [65], [66], [67].

To address these challenges, CMOS-compatible silicon nitride has emerged as a promising alternative. Beyond mitigating the drawbacks of silicon, silicon nitride offers excellent compatibility with CMOS processes, enabling scalable integration with application-specific integrated circuits. This compatibility has driven advancements in key technologies, such as on-chip frequency combs, high-density optical communications, and quantum entanglement systems. Furthermore, the ability to tune the silicon-to-nitrogen ratio in silicon nitride provides precise control over its nonlinear and dispersion properties, paving the way for the exploration of novel nonlinear phenomena at the chip scale [29], [55], [68].

Building on these developments, the USRN Bragg grating represents a next-generation platform that capitalizes on the benefits of silicon nitride while further enhancing nonlinear interactions. The USRN Bragg grating is constructed on a silicon-rich nitride platform with a silicon-to-nitrogen ratio of 7:3, resulting in a high silicon content that yields a large nonlinear refractive index of 2.9 × 10⁻¹⁷ m²/W and a linear refractive index of 3.1 at 1,550 nm. Fabricated using inductively coupled chemical vapor deposition at 250 °C, this low deposition temperature makes the platform ideal for CMOS backend processing [34].

While the USRN platform avoids two-photon absorption at 1,550 nm, it does exhibit three-photon absorption at the Urbach tail [69]. However, three-photon absorption impacts nonlinear efficiency far less than two-photon absorption. The high nonlinear parameter of USRN enhances nonlinear interactions, optimizing conversion efficiency in processes like four-wave mixing and optical parametric gain, which is particularly advantageous for achieving high conversion at low input powers [3].

2.3 Grating design

Figure 1 illustrates the design of a USRN Bragg grating. In Figure 1(a), an SEM image shows the 6 mm grating structure, while Figure 1(b) presents its transmission band. The shaded region near the stop-band edge, spanning wavelengths from 1,539 nm to 1,546 nm, demonstrates strong dispersion due to multiple reflections at the band edge [30]. A 3D schematic of the grating is provided in Figure 1(c). The design achieves high-value dispersion by incorporating cladding-modulated Bragg gratings on both sides of the central waveguide L ₁, which measures 6 mm × 600 nm × 300 nm. Precise dispersion engineering is enabled by adjusting parameters such as the pillar radius r, gap G ₁ (distance from the waveguide), and grating pitch Λ _B . To ensure smooth optical field interaction with the waveguide, the input and output gratings are apodized, resulting in a gradual effective index change and minimizing abrupt field transitions. A comprehensive description of the grating design and its parameters is provided in [55]. The detailed description of material properties, fabrication conditions, and geometric parameters provided in this section ensures the completeness and reproducibility of the device configuration used in this study.

4.1 Noise-induced MI in the grating: continuous output spectrum

At the edge of the stopband, the grating’s dispersion profile becomes highly complex, with higher-order dispersion terms playing a significant role. This complexity makes predicting the grating’s response to variations in the input wavelength challenging. Figure 2(a) presents the GVD and nonlinear parameter values, while Figure 2(b) displays the TOD and FOD parameters. In the wavelength range of 1,539–1,546 nm, there are 4,999 pump wavelengths spaced approximately 176 MHz apart. Each pump is characterized by specific values for the nonlinear parameter γ and dispersion parameters β ₂, β ₃, and β ₄. These parameters, combined with the initial condition given in Eq. (4), were used to solve Eq. (1). The results are shown in Figure 2(c), where the x-axis represents the pump wavelengths, and the y-axis displays the output spectrum.

Figure 2:

Noise-induced MI driven by the device’s dispersion and nonlinear parameters: (a) and (b) present the experimentally measured effective nonlinear parameter (γ) and dispersion parameters (β ₂, β ₃, β ₄) across the wavelength range of 1,539 nm–1,546 nm which are used in the numerical simulation. Using this range as the pump, (c) shows how noise-induced MI generates a new spectrum. The shaded regions in (a) and (b) correspond to specific parameter values that define three distinct regions in (c): region 1, where no new wavelengths are generated, and regions 2 and 3, where new spectral components are formed. The bottom panels, (d), (e), (f), and (g), illustrate the spectral evolution at four selected wavelengths indicated by black arrows in (c). In these panels, dashed white lines mark the pump wavelength, red dotted lines indicate the newly generated spectral components and the transmission bands are shown in blue lines.

The grating’s resonance response shows distinct features in regions labeled 1, 2, and 3 in Figure 2(c), corresponding to the dispersion and nonlinear parameter values within the shaded areas of Figure 2(a) and (b). In region 1, the narrow bandwidth and negligible frequency excitation result from the minimal values of β ₂ and β ₄ in the corresponding shaded area. The growth rate g(ω) from Eq. (6) indicates that when β ₂ = β ₄ ≈ 0, the MI gain bandwidth is highly restricted, resulting in almost no amplification of new frequency components and a vanishing growth rate. Although nonlinear effects γ and TOD may induce weak self-phase modulation and MI [3], their contribution remains insignificant.

We observe an abrupt widening of the output spectrum at both edges of the region 1, around 1,540 nm and 1,540.6 nm, driven by variations in β ₄. These broadenings are indicated by the white arrows in Figure 2(c) and the corresponding β ₄ values indicated with black arrows in Figure 2(b). The sign of β ₄ plays a crucial role: β ₄ > 0 but ≪ 1 enhances MI, resulting in spectral broadening, while β ₄ < 0 or β ₄ ≫ 1 suppresses MI, confining spectral components near the pump [82], [85]. Around 1,540 nm, for a narrow range of pump wavelengths with small positive values of β ₄, generate multiple MI gain bands, producing new frequencies farther from the pump and briefly widening the output spectrum. Within the 1,540.10–1,540.58 nm range, nonlinear interactions are significantly reduced because β ₄ has positive higher values, which also reduce the spectral bandwidth. In this range, the nonlinear parameter γ and dispersion parameters β ₂ and β ₃ remain nearly constant. Another widening of the spectrum is observed around 1,540.6 nm for a narrow range of pump wavelengths, as shown in region 1 of Figure 2(c). Here, the broad spectral components also arise due to β ₄ > 0 but β ₄ ≪ 1 values. It is important to note that for enhanced seeding and spectral amplification, FOD must remain positive, but with β ₄ ≪ 1 [82].

This behavior highlights FOD as a key control parameter for determining how the grating generates new wavelengths. This phenomenon is well-explained by the analytic gain expression in Eq. (6), a sixth-order polynomial in frequency ω. When β ₄ > 0, the condition g(ω)′ = 0 (where the prime denotes the first derivative with respect to ω) yields four MI gain bands symmetrically positioned on both sides of the pump [82], [85]. As the noisy CW pump evolves, frequencies within these gain bands experience strong phase-matching, amplifying spectral components away from the pump. In contrast, when β ₄ < 0, g(ω)′ = 0 produces only two symmetrical gain bands centered around the pump, without additional MI sub-bands. This narrows the gain spectrum and confines the instability region near the pump frequency, limiting the extent of spectral broadening.

To further examine the grating’s response within the narrow bandwidth region labeled 1 in Figure 2(c), we selected a pump wavelength of 1,540.3 nm, marked by the black arrow at d. The simulated result, shown in the bottom panel in Figure 2(d), reveals no spectral broadening along the grating. The developing MI experiences little to no growth for this range of wavelengths and no development of ABs in this region. In the upper panel of Figure 2(d), the black curve indicates minimal spectral change at the grating’s output. The blue curve above shows the grating’s transmission band, providing a comparison with the output spectrum.

The region 2 in Figure 2(c) displays intriguing spectral features corresponding to the shaded area 2 in Figure 2(a) and (b), covering the wavelength range from approximately 1,543.4 nm to 1,543.8 nm. Within this range, both symmetric and asymmetric spectral wings emerge, marked by arrows e, f, and g. For the asymmetric spectral broadening, pump wavelengths in this region exhibit positive TOD values, as shown by the peak of the blue curve in Figure 2(b) within the shaded area 2. These TOD values dominate over the effects of FOD, enhancing MI and amplifying the asymmetric spectral wings. In contrast, symmetric spectral broadening arises where the FOD curve crosses the zero line and β ₄ > 0. Small positive β ₄ values generate two symmetric resonance wings.

Alternatively, all downward black arrows in Figure 2(b) indicate the positions of small β ₄ values, which contribute to the symmetric broad spectrum in Figure 2(c), marked by downward white arrows. Similarly, all the upward black arrows in Figure 2(b) correspond to the positive TOD values responsible for the asymmetric spectrum in Figure 2(c), highlighted by upward white arrows. The symmetric and asymmetric spectral features are associated with pump wavelengths at 1,543.48 nm, 1,543.57 nm, and 1,543.65 nm. Their corresponding spectral evolutions are presented in Figure 2(e)–(g), respectively, in the bottom panel.

A notable feature in Figure 2(e) is the delayed emergence of the phase-matched wavelength, which becomes evident around z ≈ 2.5 mm, as indicated by the red dashed line. The high loss within the grating initially causes an exponential power reduction, suppressing MI and delaying the appearance of phase-matched frequencies. Additionally, at this wavelength, the TOD values correspond to small negative FOD values, which further suppress MI. As the evolution progresses, the exponential growth of MI eventually overcomes this suppression, resulting in a modest spectral gain at the grating’s output. This is reflected in the upper spectral wings along the black arrow e in Figure 2(c).

In contrast, Figure 2(f) and (g), corresponding to arrows f and g in Figure 2(c), show regular early MI development without delay. While linear stability analysis (Eq. (6)) predicts that β ₃ (TOD) does not directly contribute to MI development, this analysis reveals asymmetric spectral features driven by TOD. Further investigation is needed to understand how TOD influences MI in the presence of FOD.

In regions 1 and 3, where broader spectral output occurs for a narrow range of pump wavelengths, the spectral evolution resembles that shown in Figure 2(f) with a symmetric profile, though with varying spectral bandwidths. Additional details on the temporal and spectral evolution near these pump wavelengths are provided in the Supplementary material (see Figure 1).

While Figure 2(c) demonstrates the combined effects of nonlinear and dispersion parameters on the output spectrum, analyzing the individual contributions of these parameters to MI can provide valuable insights for optimizing grating performance through precise dispersion engineering. In Figure 3, we identify the ranges of γ, β ₂, β ₃, and β ₄ that drive the generation of new spectral components. Notably, the region 2 in Figure 2(c), characterized by multiple spectral wings, corresponds to rapidly changing sections of the dispersion and nonlinear parameter curves. In Figure 3(a)–(d), the nonlinear and dispersion values within the white boxes align with region 2 in Figure 2(a)–(c), revealing their role in producing multiple wide spectral regions.

Figure 3:

Individual contributions of nonlinear and dispersion parameters: (a) shows the contribution of the nonlinear parameter (γ), while (b), (c), and (d) illustrate the contributions of the dispersion parameters β ₂, β ₃, and β ₄, respectively. Key features are highlighted with white arrows, and enlarged views of these regions are presented in A, B, C, and D in the bottom panel.

The data presented in Figure 3 are derived from experimentally measured dispersion and nonlinearity parameters obtained across a dense set of discrete wavelengths in the 1,539–1,546 nm range (see Methods). For the purpose of numerical modeling and visualization, these curves were interpolated using a reduced number of wavelength points. Since the effective nonlinearity γ and second-order dispersion β ₂ exhibit nearly linear dependence over the measured range, their interpolated profiles appear smooth and monotonic in Figure 3(a) and (b). In contrast, the higher-order dispersion terms β ₃ and β ₄ exhibit strong nonlinear dependence on wavelength, with inflection points and curvature. Consequently, approximating these curves with fewer points leads to moderate point-to-point variations, resulting in the less monotonic trends observed in Figure 3(c) and (d).

The spectral shape and orientation within these white boxes vary significantly, as seen in Figure 2(c). However, further investigation is needed to uncover the intricate relationships among these parameters and how they shape the resulting spectral content.

Additionally, we observe a gap in region 1 of Figure 2(c), where nonlinear interactions are minimal. This gap reappears in Figure 3(a) and (b) when analyzing γ and β ₂, marked by white arrows A and B. These values correspond precisely with shaded region 1 in Figure 2(a), indicating that γ and β ₂ are the primary contributors to nonlinear interactions in this part of the grating. Conversely, the absence of similar gaps in Figure 3(c) and (d) suggests that β ₃ and β ₄ play a lesser role in these interactions. Enlarged views of key features are presented in the bottom panels A− D.

Furthermore, we observe a symmetric, wider spectral content at γ = 1.67 and β ₂ = −1.8 in Figure 3(a) and (b), respectively. These correspond to the points where the β ₄ curve intersects the zero line (red-dotted) in region 3 of Figure 2(b). These specific parameter values are directly responsible for generating the corresponding wide spectrum in Figure 2(c), highlighting the crucial interplay between these parameters in shaping the output spectral characteristics.

A notable general feature arises from the contribution of β ₄. In Figure 2(b), every instance where the β ₄ curve intersects the zero line results in a wider spectral regime in Figure 2(c), regardless of the behavior of other parameters (see all the downward black arrows). This underscores the pivotal role of β ₄ in determining the overall spectral behavior of the grating.

4.2 MI dynamics using linear stability analysis

MI in the grating can also be analyzed using linear stability analysis [3]. One may wonder: if MI can be studied analytically through linear stability, why use computationally expensive numerical methods? The answer lies in the limitations of linear stability analysis. The MI expression, Eq. (6), is derived without including the third-order dispersion (TOD) term β ₃. Using this expression, we generate the MI structure for the grating’s nonlinear and dispersion parameters, as shown in Figure 4(a). However, the numerical analysis incorporates all dispersion and nonlinear terms, producing the MI structure in Figure 2(c). Comparing Figure 4(a) with Figure 2(c), it is evident that several spectral wings, missing in the analytical result, appear in the numerical MI structure. These additional spectral components, which arise from TOD, are crucial for a comprehensive understanding of MI and can only be captured through numerical methods.

Figure 4:

MI using linear stability analysis: (a) shows the output MI frequencies (x-axis) as a function of input pump wavelengths (y-axis) at a pump power of P ₀ = 8 W, without accounting for loss in Eq. (6). The pump wavelengths correspond to the nonlinear and dispersion values shown in Figure 2(a) and (b). (b) presents the output frequencies for five selected pump wavelengths. (c) shows the same five examples as in (b), but with loss included at the same power level.

In Figure 4(a), the y-axis represents pump wavelengths, while the x-axis displays the corresponding MI frequency bandwidth. These pump wavelengths are associated with the nonlinear and dispersion parameters shown in Figure 2(a) and (b). For clarity, the analysis assumes zero loss and a pump power of P ₀ = 8 W. Regions 1, 2, and 3 in Figure 4(a) correspond to those in Figure 2(c). Note that Figure 4(a) highlights the MI frequencies responsible for initiating instabilities, in contrast to Figure 2(c), which presents the output spectrum.

To explore the impact of grating loss on MI dynamics, we analyze specific pump wavelengths marked by black arrows in Figure 4(a). Figure 4(b) shows the corresponding MI bandwidths without loss, with an expanded x-axis to display the full range of MI frequencies. In Figure 4(b), arrow 1 at 1,540.35 nm, located in a region of minimal nonlinear interaction, generates a narrow MI band with a lower growth rate, shown by the thick green curve. Pump wavelengths at arrows 2, 4, and 5 produce additional MI subbands (purple dot-dashed, brick-red dashed, and blue dotted lines) with varying bandwidths and growth rates, resulting in strong MI consistent with the wide spectral wings observed in Figure 2(c). Arrow 3, however, at 1,542.39 nm, does not generate MI subbands and shows a narrow MI bandwidth, represented by the solid yellow line.

When loss is introduced, the MI dynamics change significantly. Setting P ₀ = 8 W and a loss of α = 1.3 dB/mm, Figure 4(c) displays the MI structure at the end of the 3 mm grating. Loss reduces both the MI bandwidth and growth rate. As a result, narrower-band, lower-intensity spectral components dominate the output. Importantly, the location of the MI subbands remains unchanged despite the inclusion of loss.

Figure 5 illustrates how the MI band structure varies with power in the presence of loss. Figure 5(a) corresponds to the MI bands at 8 W with waveguide loss, showing reduced bandwidth compared to Figure 4(a). As the power decreases to 4 W in Figure 5(b), the MI bandwidth narrows further, and distinct features emerge. Specifically, the resonant structure begins to separate from the central MI band, as marked by the curved white arrows. At lower power levels, such as 500 mW in Figure 5(c), the upper MI resonance region is nearly eliminated, leaving the lower resonance region still connected to the central MI band. Separation from the main MI band becomes more pronounced as power decreases further, with additional details provided in the Supplementary material (see Figure 3).

Figure 5:

Power dependence of the grating’s MI: (a) shows the MI spectrum at P ₀ = 8 W with loss, corresponding to the example in Figure 4(c). (b) and (c) present the spectra at lower powers, P ₀ = 4 W and P ₀ = 500 mW, respectively, with loss included in all cases. The white arrows indicate the separation of the resonant MI sub-bands from the central MI band as the power decreases.

These power-dependent changes in the MI band structure are critical for understanding MI dynamics at the chip scale. On-chip gratings and waveguides often operate at power levels in the range of hundreds to tens of milliwatts. At these lower powers, the MI bandwidth becomes narrow, allowing only a limited number of MI frequencies to drive instability. However, the separation of MI subbands from the central band at lower powers enables phase-matched MI frequencies in the separated subbands, supporting strong MI even at milliwatt power levels.

4.3 Emergence of AB in the grating confirmed by inverse scattering transform (IST) analysis

The IST is a powerful and well-established method for deriving closed-form solutions to the NLSE [86], [87]. Beyond its original purpose, IST has been applied to solve practical problems in optical communications. In 1993, Hasegawa and colleagues proposed an efficient fiber communication system based on IST [88], and more recently, it has been used to encode, transmit, and process data by mitigating nonlinear channel cross-talk in optical fiber systems [89].

In nonlinear wave systems such as fibers, waveguides, and gratings, the evolution of CW fields often leads to the formation of localized structures, including solitons, ABs, and rogue waves [90]. Identifying these solutions within a chaotic wave field is challenging, but numerical techniques combined with statistical methods have been employed to address this [71], [91], [92]. Recently, IST has emerged as an effective tool for identifying and classifying localized nonlinear structures within chaotic wave fields [93], [94], [95]. Moreover, it has been applied to study how chaotic initial conditions in the NLSE evolve into solitons, ABs, and rogue wave solutions [96], [97].

While most IST analyses for communication and localized structure identification have been conducted in fiber systems, their application in on-chip data processing remains largely unexplored. Before adopting the AB model to explain MI and related nonlinear phenomena in these gratings, it is essential to confirm that they do emerge in this system. To achieve this, we employ the IST technique, as previously used to identify AB-type solutions in [94].

The top panel of Figure 6 illustrates the IST analysis of an ideal periodic AB solution. Figure 6(a) shows the evolution of the AB from a dimensionless distance of −4 to 4, with three consecutive periods along the transverse dimension t. The solution is symmetric at both ends. For IST spectral analysis, intensity profiles at z = 0 and z = 1 are selected, shown in Figure 6(b) as thin and thick curves. The profile at z = 0 displays maximum intensity, which decreases significantly by z = 1. However, the IST spectra for both profiles, shown in Figure 6(c), remain identical, confirming that the IST spectra of an AB solution do not depend on the propagation distance [94], particularly near z = 0. Later, we confirm that this property also holds for ABs formed through CW propagation in the grating. Note, far from the AB’s peak intensity at z = 0, the IST spectrum resembles that of a plane wave [45].

Figure 6:

IST spectral analysis of the emergence of AB in the grating: [Ideal field] (a) shows the intensity profile of an ideal AB solution derived from the NLSE. Two intensity profiles, selected at z = 0 and z = 1 (indicated by white dashed lines), are presented in (b) as thin and thick blue curves, respectively. (c) displays their corresponding IST spectra. [Real field] (d) shows the evolution of a CW pump at 1,542 nm over a full 6 mm grating length. A temporal frame of intensity selected at z = 1.9 mm (marked by the white dashed line) is shown in (e). Six localized structures (i − vi) from the envelope are selected for IST spectral analysis, and their corresponding IST spectra are shown in (f) within the labeled boxes (i − vi). The genus (g) of each structure is indicated at the bottom. Note that (d) and (e) has the same x-axis.

The bottom panel of Figure 6 presents the IST analysis of a real optical field. Figure 6(d) shows the evolution of a CW in a 6 mm grating. Initially, strong nonlinear interactions and MI compress the CW, forming an AB-like wave profile around z ≈ 1.9. As the wave propagates, the grating loss parameter gradually reduces nonlinear interactions and MI, resulting in low-intensity localized structures at its output. With a loss of 1.3 dB/mm, the grating incurs a total loss of approximately 7.8 dB over its entire length, leaving only 17 % of the initial power at the output. Consequently, intense nonlinear interactions are primarily confined to the first ≈ 3 mm , with significantly reduced intensity thereafter.

We extract the transverse profile at z ≈ 1.9, where the AB achieves maximum compression, as marked by the white dashed line in Figure 6(d). The wave envelope at this point is shown in Figure 6(e). To verify that the localized formations within the envelope correspond to AB solutions, six structures labeled i through vi are highlighted in different colors. Their corresponding IST spectra, or “IST spectral portraits,” are presented in the bottom panel.

To confirm these spectral portraits correspond to AB-type solutions, we analyze them using finite-gap theory [84]. According to this theory, IST spectral data consists of spectral bands and gaps, characterized by ‘genus-g’ [93], [94]. The genus g of an NLSE solution is related to the number of gaps as g = N − 1, where N is the number of spectral bands [45]. For instance, a plane wave has g = 0, a soliton has g = 1, and an AB solution has g = 2. The IST spectra of the six localized structures show N = 3 spectral bands, confirming their genus g = 2 and verifying that they are AB-type solutions.

It is noteworthy that the wave envelope in Figure 6(d) does not perfectly align with the AB’s maximum compression point due to the highly modulated chaotic wave field that disrupts the symmetry observed in regular AB solutions in Figure 6(b). Structures in the pink-shaded areas (i, iii, v) fall slightly before or after this point, while those in the gray-shaded areas (ii, iv, vi) align more closely. Despite this, all structures share the same AB-type IST spectral portrait, illustrating that the eigenvalues of the Zakharov–Shabat problem remain independent of propagation distance [93], [94], as also emphasized in the ideal IST analysis in the upper panel. We analyzed and described the IST spectrum of the entire envelope in the Supplemental material (see Figure 4), highlighting that most of the localized structure within the envelop displays characteristic AB behavior.

4.4 Exciting AB in the grating: discrete output spectrum

An AB at its maximum compression point exhibits a discrete spectral pattern [98]. To excite an AB solution and generate such a spectrum within the grating, we employed the initial condition ψ ( z = 0 , t ) = P 0 1 + ϵ m o d ⁡ cos ( ω m o d t ) (Eq. (5)) to solve the GNLSE. Experimentally, this setup corresponds to a modulated CW laser at the grating input, a standard configuration for AB studies in optical fibers, particularly at the telecom wavelength of 1,550 nm [80], [99]. Simulations utilized the full range of nonlinear and dispersion values across the wavelengths shown in Figure 2(a), with a power of P ₀ = 8 W and modulation amplitude ϵ _mod = 10⁻⁴.

Figure 7(a) and (b) shows pump wavelengths along the x-axis, with each pump associated with unique nonlinear and dispersion values, while the y-axis in Figure 7(c) presents the output spectrum. Interestingly, the grating does not support AB excitation for a significant portion of the pump wavelengths. The central high-intensity white line in Figure 7(c) represents the output pump, while the two modulation sidebands remain visible but muted within the 1,539–1,543.30 nm range. AB formation begins only when the pump reaches 1,543.4 nm, producing new frequencies up to 1,545.6 nm.

Figure 7:

Generating ABs through CW wave modulation: (a) shows the nonlinear parameter (γ) and the group velocity dispersion (β ₂), while (b) presents the higher-order dispersion parameters (β ₃ and β ₄). (c) illustrates the output spectrum of ABs generated based on the nonlinear and dispersion values from (a) and (b). The shaded regions indicate phase-matched areas that result in new frequency generation, labeled as regions 1 and 2 in (c). The bottom panels, (d), (e), (f), and (g), depict the spectral evolution at selected pump wavelengths (highlighted in white text) indicated by their position by arrows in (c). A large portion of the grating’s dispersion profile, spanning from ≈ 1,539 nm to 1,543.4 nm, is identified as not supporting AB formation.

This absence of AB excitation can be explained using the AB parameters in Eq. (2). The modulation frequency ω _mod relates to the parameter a as 2 a = 1 − ( ω m o d / ω c ) 2 , where ω c 2 = 4 γ P 0 / | β 2 | . AB formation requires ω _c > ω _mod > 0, allowing MI to grow within the gain band defined by b = 8 a ( 1 − 2 a ) . However, in the grating, higher-order dispersion terms are significant. The MI gain expression for the standard AB solution, which includes only β ₂, does not fully apply. For AB excitation, the MI gain expression Eq. (6), incorporating β ₄, must hold. In the range 1,539–1,543.30 nm, large values of γ, β ₂, and β ₄ significantly narrow the MI gain band, pushing MI frequencies outside the band and preventing AB growth. Notably, β ₄ is largely negative in this range but briefly becomes positive near 1,540–1,540.6 nm, generating ABs and discrete spectra, as indicated by the white arrows in Figure 7(c).

The resonance frequencies or phase-matched new frequencies shown in Figure 7(c) align closely with Figure 2(c). Regions 1 and 2 in Figure 7(c) correspond to regions 2 and 3 in Figure 2(c). In Region 1, high loss reduces the visibility of resonance frequencies. Higher-order dispersion plays a critical role in generating discrete frequencies, with prominent spectral features near the stopband edge (1,543.4–1,545.6 nm) where β ₄ > 0. This demonstrates that the grating’s dispersion near the stopband edge is crucial for phase-matched spectral generation. We also examine the individual contributions of nonlinear (γ) and dispersion parameters (β ₂, β ₃, and β ₄) to the generation of discrete spectrum in the Supplementary material (see Figure 5).

Figure 7(d), representing a pump wavelength at 1,540.3 nm, shows no AB development. Here, modulation sidebands around the pump remain inactive, generating no new frequencies along the grating’s evolution. In contrast, Figure 7(e)–(g) demonstrates new frequency generation, highlighting the grating’s ability to amplify harmonic frequencies depending on phase-matching contributions from β ₃ or β ₄. Asymmetric amplification arises from β ₃, while symmetric amplification is driven by β ₄. These effects are marked by white arrows in the spectral profiles.

Note that the apparent difference in spectral resolution among Figure 7(d)–(g) arises from the dependence of the modulation frequency and time window on the second-order dispersion |β ₂|. A smaller |β ₂| results in a higher modulation frequency and thus a shorter time window t, which lowers the spectral resolution and causes the sidebands to appear broader. This effect is reflected in the initial spectral components shown in Figure 7(c). In contrast, larger |β ₂| values lead to a longer t, yielding higher spectral resolution and sharper, more distinct sideband structures. This trend is clearly observable in Figure 7(c), where the pump and the first two sidebands appear thicker at the beginning and progressively become narrower and more resolved along the dispersion profile.

In Figure 7(c), the central white line represents the output pump, while sidebands initially appear symmetrically around it. As pump wavelengths approach 1,546 nm, the sidebands converge. This behavior reflects the dispersion curves in Figure 7(a): lower dispersion values at shorter pump wavelengths support wideband MI, placing sidebands farther from the pump. As dispersion values increase, the MI band narrows, bringing sidebands closer to the pump and producing a converging spectral profile. Note that the black and white spikes appearing at various spectral positions, including along the central white pump line, are numerical artifacts and are not associated with any physical effects.

In the next sections, we explore FPU recurrence, pattern formation, and spectral enhancement in the grating within the AB framework, illustrating its versatility as a nonlinear platform.

4.5 FPU recurrence in the grating

The FPU recurrence is a fascinating nonlinear phenomenon with an equally intriguing history [100]. It describes how an active nonlinear system periodically returns to its initial state [23]. Since its discovery, this phenomenon has profoundly influenced nonlinear science, with applications ranging from wave dynamics to complex systems [101], [102], [103], [104]. The dynamics of the AB solution align closely with the behavior described by FPU recurrence [77], [104], [105]. In AB dynamics, a small modulation on a CW field causes exponential amplification of two sidebands. Through four-wave mixing, these sidebands generate new harmonic frequencies, drawing energy from the CW. This process continues until it reverses and returns to its initial state, illustrating the FPU recurrence. We show that the USRN Bragg grating, as a nonlinear platform, can also exhibit this cyclical process.

While most demonstrations of FPU recurrence within the AB framework have been conducted in weakly dispersive and nonlinear systems [102], there is limited research on how this phenomenon manifests in strongly dispersive and nonlinear systems, particularly on-chip optical devices. The USRN Bragg grating provides a unique platform to explore this behavior. In Figure 8, we excite an AB in the grating both without (left box) and with loss (right box) to observe FPU recurrence.

Figure 8:

Observing FPU recurrence without and with waveguide loss: [Left box] (a) shows the temporal and (b) the spectral evolution of an AB at λ ₀ = 1,543.57 nm with no loss (α = 0). (c) depicts the temporal evolution of the AB’s amplitude on a complex plane, while (d) illustrates the energy exchange among spectral sidebands during evolution. [Right box] (e) and (f) show the temporal and spectral evolution, respectively, when the grating experiences a loss of 1.3 dB/mm. (g) demonstrates the effect of loss on the AB’s temporal evolution, and (h) shows how energy exchange among the spectral modes is altered in the presence of loss.

Without loss, as shown in Figure 8(a), the AB first emerges at z = 1 mm, with a second recurrence observed around z ≈ 2.5 mm. In Figure 8(b), the AB exhibits a narrow spectral bandwidth between recurrences, while the compression points display a comparatively broader bandwidth. This behavior arises from the grating’s strong dispersion, which reduces the MI bandwidth, limiting the number of excited sidebands [106]. The critical MI frequency parameter, ω c 2 = 4 γ P 0 / | β 2 | , highlights this effect, as large β ₂ values narrow the MI band. Including strong β ₄ further reduces the bandwidth, as predicted by Eq. (6). For this demonstration, we selected a pump wavelength of 1,543.57 nm within Region 1 of Figure 7(c), where the influence of higher-order dispersion on AB dynamics is minimal. This choice allows for a smooth observation of FPU recurrence.

In Figure 8(c), the AB’s amplitude trajectory is shown on a complex plane, with color representing the evolution distance z. The central trajectory corresponds to an undisturbed AB solution, while arrows indicate the flow of evolution. This is to differentiate how external perturbation impacts the FPU processes. The first half of the trajectory reflects the AB’s first emergence, and the second half represents its second recurrence. In an ideal NLSE system, subsequent AB recurrences would perfectly overlap this trajectory unless influenced by external dynamics. In contrast, the outer trajectory reflects the AB evolution from Figure 8(a), showing significant deviations due to strong nonlinearity and dispersion effects.

Figure 8(d) illustrates the energy exchange between the pump and discrete sidebands during the AB’s evolution. Taken from Figure 8(b), the plot highlights one side of the symmetric spectral profile. Downward black arrows indicate points where the pump’s energy is maximally transferred to the sidebands. The two nearest sidebands (in red and yellow) capture most of the power, while those below the black dashed line retain minimal energy, resulting in lower intensity.

In the right panel, we examine AB evolution with loss included. Figure 8(e) shows that loss delays the AB’s appearance, reduces its intensity, and introduces multiple low-intensity recurrences as the grating loses power. MI-driven growth is counteracted by the exponential loss term, ω _c = ω _c  exp(−α z/2), creating a dynamic interplay. This competition produces a rapid oscillatory recurrence pattern, where AB power diminishes with each cycle. Figure 8(f) reflects these spectral dynamics, showing energy concentrated near the pump due to loss suppressing MI-induced gain, thereby reducing the effectiveness of four-wave mixing.

Building on the complex plane evolution, Figure 8(g) illustrates four orbits of the AB trajectory corresponding to four recurrence events observed in Figure 8(e). Temporal overlap among the recurrences is indicated by crossing points, marked by inward black arrows. This reveals a critical distinction in FPU recurrence behavior between conservative and non-conservative systems. In the absence of loss, the grating acts as a conservative system, where energy is preserved and transferred equally during each successive recurrence. However, when loss is introduced, the system transitions to a non-conservative state. Each successive recurrence inherits progressively less energy from the previous one, resulting in diminished intensity along the grating’s length.

Figure 8(h) highlights energy transfer among spectral components in the presence of loss. As significant power is lost during each recurrence, four-wave mixing weakens, leading to fewer and less intense sidebands over time. The decreasing sideband intensity, both above and below the dashed line, is indicated by a downward trend. Black arrows on the pump denote progressively weaker energy exchange, emphasizing the impact of loss on spectral dynamics.

To further analyze the effects of loss, Figure 9 compares the AB’s spectral profile at its first maximum compression point. The blue solid line represents the loss-free case, while the red-dashed line shows the profile with loss. With loss, the shorter wavelength side of the spectrum exhibits a slight gain, but there is a significant reduction in spectral power on the longer wavelength side. This asymmetric spectral profile indicates that TOD dominates for the pump at λ ₀ = 1,543.56986 nm, exciting fewer modes compared to the loss-free case.

Figure 9:

Spectral comparison at AB’s first maxima: The spectra without loss (blue) and with loss included (red) are compared.

4.6 Pattern formation and spectral enhancement in the grating

In nonlinear systems, complex patterns arise from simple interactions, revealing structures and behaviors shaped by feedback loops and self-organization. Coherent and incoherent MI is crucial in forming patterns in instantaneous and non-instantaneous nonlinear systems, respectively [60], [107], [108]. In coherent MI, where the driving field is typically a monochromatic CW laser, the instability occurs due to the interaction between the medium’s nonlinearity and the small perturbations in the input field. This interaction leads to the exponential growth of sidebands around the input frequency, producing periodic intensity patterns within the beam. By leveraging this coherent MI, we demonstrate that the grating can generate complex patterns and induce parametric amplification in the spectral components of the driving field. In the left box of Figure 10, we present how the grating generates patterns and spectral amplification without loss, while the right box shows loss included.

Figure 10:

Pattern formation: [Left box – without loss] (a) temporal and (b) spectral evolution of an AB at pump wavelength λ ₀ = 1,545.35 nm, where the FPU-recurrence phenomenon disappears after the first growth-return cycle. (c) presents the amplitude evolution on a complex plane, with each dot representing a temporal peak along the propagation direction (z). The color gradient (blue to red) indicates progression along z, and increased scattering reflects greater amplitude variations. (d) shows energy exchange between the pump and sidebands, with the black dashed line separating highly amplified from moderately amplified sidebands. The colored spectral lines highlighted by the long downward arrow represent the first 10 sidebands respectively on the right side of the pump in (b). [Right box – with loss] (e) temporal and (f) spectral evolution of the same AB under loss. (g) amplitude evolution on a complex plane, where peaks are more resolved, indicating reduced amplitude variations. (h) illustrates energy exchange among the pump and sidebands, with loss-reducing amplification slightly.

In Figure 10(a), the temporal evolution of the pump field λ ₀ = 1,545.34998 nm appeared as an AB at z ≈ 0.75 mm. The presence of small positive β ₄ at this wavelength symmetrically creates additional MI sub-bands. One example of such MI bands is presented in Figure 4(c) with dotted blue lines (legend 5). The MI frequencies from these sub-bands strongly seed the MI process, leading to an enhanced phase matching among the pump and the new generating sidebands. This facilitates an efficient and sustained energy transfer from the pump to the new excited frequency modes. This, in turn, in the temporal domain, translates to the emission of strong background dispersive waves from the base of the AB. The generated dispersive waves are strong enough to sweep across the wave field and prevent the formation of new AB recurrences as the field advances in the grating length [82].

As a result of strong phase matching, in Figure 10(b), we observe a high-intensity spectral profile with many discrete frequency modes. Because there is no further development of AB in the z direction, we do not observe any FPU recurrence both in the temporal and spectral domains. From start to end, the wideband spectral intensity nearly remains the same. Although the speckle filaments in the temporal domain appear chaotic to some extent, they are strongly phase-matched. To elucidate how the patterns form, we provide a detailed explanation using the phase portrait of the temporal evolution in Figure 6 in the Supplementary material.

The reason behind the disappearance of the AB is pronounced on the AB’s complex plane trajectory of amplitude in Figure 10(c). In the outer orbit, the trajectory starts smoothly at z = 0 with dark blue. After a while, the scattered blue dots are indicative of the sudden deviation of the trajectory from the AB state to the multipeak background radiation waves. The highly scattered dots strongly indicate that the background waves have a diverse range of peaks. However, because they still remain within a well-defined region of an orbit means, until the evolution ends, the peaks repeat themselves a number of times, creating a pattern in the spatiotemporal domain. Note that, for a standard AB, the amplitude of the periodic pulse train smoothly changes in z. As a result, we observe a near continuous trajectory of its amplitude, such as in Figure 8(c) and (g).

Figure 10(d) shows that the pump strongly couples with the spectral modes. Sidebands closer to or farther from the pump are separated by a black dashed line. The intensity of the sidebands nearest the pump (above the dashed line) remains nearly equal to that of the pump itself. Notably, there is no periodic energy exchange in the sidebands, both above and below the dashed line, indicating that the AB no longer participates in the reccurence dynamics. This observation highlights that energy transfer from the pump to the sidebands is steady and robust rather than oscillatory, maintaining a nearly constant spectral intensity along the entire grating length.

High loss in the grating significantly impacts these dynamics, as shown in the right box. In Figure 10(e), the nonlinear interactions in the temporal evolution are now subdued, with the AB completely dissipating after two recurrences due to exponential power decay. The intense radiation waves seen in Figure 10(a) are now much weaker, and the background patterns are less pronounced. Interestingly, MI remains strong despite high loss, as evident in the spectral domain in Figure 10(f). Although power loss reduces spectral intensity somewhat, the overall spectral width remains unchanged. Notably, the intensity of the spectral sidebands near the pump is still comparable to the loss-free case shown in Figure 10(b).

Further insights come from the AB’s amplitude evolution on the complex plane in Figure 10(g). Here, the peaks are more resolved than those in Figure 10(c). Starting from dark blue, the trajectory in the outer orbit approaches the inward direction, with points clustering closer together, indicating fewer peak developments and a less pronounced temporal pattern.

In the energy exchange dynamics shown in Figure 10(h), the pump and its neighboring sidebands (above the black dashed line) retain high intensity, appearing less affected by loss. The sidebands farther from the pump (below the dashed line) maintain a relatively lower power level compared to the loss-free case in Figure 10(d). Remarkably, MI can strongly counteract the high loss at this pump wavelength, producing high-intensity, wideband spectral content at the output.

Finally, Figure 11 compares the output spectral profiles for the loss-free case (solid blue) and with loss (red dashed). Without loss, the shorter wavelength side of the spectrum gains more intensity than the longer wavelength side, reflecting asymmetry due to negative TOD. As previously shown in Figure 10(h), the sidebands near the pump exhibit similar intensity in both cases. This demonstrates the USRN Bragg grating’s potential for robust parametric amplification of spectral intensity, even under high-loss conditions.

Figure 11:

Grating’s output spectrum in pattern formation: The output spectrum is shown for the cases without loss (solid blue) and with loss (dashed red).