1. INTRODUCTION

Temporal optical solitons are solitary waves that can propagate over long distances with relatively robust immunity against distortions that may arise from nonlinearity and dispersion. Whereas the fundamental solitons maintain their shapes while traveling in media, high-order solitons undergo periodic changes in shape during propagation, which lead to an initial stage of temporal compression [1,2], prior to restoration of their original pulse shape [3]. Conventional temporal optical soliton formation is driven by (1) anomalous group velocity dispersion (GVD), which provides a defocusing effect, temporally advancing (retarding) the blue (red) frequencies; and (2) self-phase modulation (SPM), which induces a frequency chirp in the pulse, which when combined with anomalous GVD leads to compression. In principle, the defocusing effect from GVD may be derived from higher-order dispersion, in which case other types of solitons may manifest. Anomalous quartic dispersion (fourth-order dispersion) can have similar qualitative impacts as anomalous GVD as both exhibit the same symmetry, with the group velocity increasing with frequency for both. Therefore, it can be inferred that quartic dispersion may generate soliton formation with similar underlying mechanisms [4,5], referred to as quartic solitons, which rely on the interaction between SPM and negative GVD in the presence of strong fourth-order dispersion (FOD) [6].

Pure-quartic solitons are fundamentally driven by the interaction between FOD and SPM with negligible effects from GVD, in contrast with the above-mentioned quartic solitons that rely on GVD for soliton formation albeit in the presence of strong FOD. Recent studies have introduced pure-quartic solitons where anomalous FOD is the dominant dispersion term interacting with SPM while GVD and third-order dispersion (TOD) are negligible [4,7]. Since then, theoretical advancements have unveiled that higher orders of dispersion, even beyond the fourth order, may support soliton formation [8]. Solitons can be stable in the presence of negative FOD and used to reduce the pulse duration requirement to achieve a given pulse energy in short-pulse lasers with minimal GVD and TOD [9–11]. Pure-quartic solitons therefore have a favorable energy scaling, making it possible to attain higher pulse energies than conventional GVD solitons [12,13]. Recent advances in the area of pure-quartic solitons have focused on their existence in optical fiber [7,12] as well as their stationary and dynamical properties [13]. While there has been considerable theoretical progress on pure-quartic solitons, the challenge lies in producing the necessary dispersion profile in a platform that combines high nonlinearity and low nonlinear loss, resulting in limited experimental demonstrations of pure-quartic solitons [4,7].

Bragg solitons are a class of solitons that exist in one-dimensional periodic structures. Since their first observations made in optical fiber [14,15], studies into Bragg solitons have seen a recent resurgence in on-chip platforms [16–18]. Early work on Bragg solitons in fiber Bragg gratings leveraged the blue side of the grating stopband for its strong anomalous GVD [14,15], revealing soliton dynamics that are similar to conventional waveguide or optical fiber solitons, in that they form as a result of a balance between SPM and GVD in nonlinear media. Whereas waveguide and optical fiber solitons form from the waveguide dispersion, Bragg solitons form as a result of the strong Bragg grating induced GVD that forms as a result of interactions between the forward and backward propagation waves at frequencies near the stopband edge.

In this work, we report experimental observations of a new class of Bragg soliton, the pure- quartic Bragg soliton (PQBS), in a low-loss, integrated nonlinear Bragg grating on the ultra-silicon-rich nitride (USRN) platform. The USRN platform possesses a high Kerr nonlinearity of ${n_{2}} = 2.8 \times {{10}^{- 13}}\;{{\rm cm}^2}/{\rm W}$ and negligible nonlinear loss near 1550 nm of wavelength [19–21], making it highly advantageous for soliton-based applications [16,17,22]. In particular, USRN Bragg gratings can be engineered to have large anomalous GVD, three orders of magnitude larger than conventional waveguides [17], and can be designed to provide the specific dispersion properties required to support the formation of PQBSs on the red side of the photonic stopband. In contrast to conventional Bragg solitons, which are known to reside on the blue side of the grating stopband where GVD is large and anomalous, our grating design yields a newly discovered regime on the red side of the grating stopband where the interaction between the large grating induced FOD and SPM supports PQBS formation. Negative FOD is observed 7 nm to 8 nm away from the red side of the stopband edge, while GVD and TOD are negligible, the ideal conditions for the formation of PQBS. We demonstrate PQBS formation using USRN Bragg gratings, with the experimentally observed temporal and spectral behavior agreeing well with theoretical calculations. We further experimentally observe that even though input pulses have a hyperbolic secant profile, the PQBS formed at the output of the nonlinear Bragg grating has a profile close to a Gaussian shape, a signature of pure-quartic solitons. In addition to signatures of pure-quartic solitons, we demonstrate the strongest pure-quartic soliton-effect temporal compression to date in the pure-quartic soliton regime. We further confirm, through simulations, the effective energy-width scaling of $E \propto T_0^{\,- 3}$ for the PQBS, whereas conventional solitons present energy-width scaling of $E \propto T_0^{\,- 1}$, indicating that higher pulse energy can be achieved for the same short pulse duration for PQBS with the same pulse width. This development paves the way for future implementations of PQBS lasers and frequency combs, promising significant advances in ultrafast optics and optical communication technologies.

2. RESULTS

A. Pure-Quartic Bragg Solitons and Dispersion Properties of the Bragg Grating

The fabrication of the USRN nonlinear Bragg grating involves several steps. Initially, 330 nm of USRN is deposited onto a 3 µm thermal oxide layer on silicon substrate using inductively coupled plasma chemical vapor deposition. This deposition process takes place at a low temperature of 250°C, which allows for back-end CMOS compatibility. Next, the device is patterned using electron-beam lithography and reactive ion etching. Finally, a 2-µm-thick ${{\rm SiO}_2}$ overcladding is deposited using plasma enhanced chemical vapor deposition.

The USRN platform was designed to have a strong optical nonlinearity while also having negligible nonlinear loss near the 1550 nm wavelength [19–21]. We utilize cladding modulated Bragg gratings (CMBGs), which implement periodic modulation of the effective index by placing cylinders at a distance ($G$) from a central waveguide as shown in Figs. 1(b) and 1(c) [16,17,23,24]. The CMBG is designed with a 3 mm central waveguide with a width, $W$, of 600 nm. Pillars with a radius ($r$) of 100 nm are placed periodically alongside. The 3-mm CMBG includes apodized sections (${L_A} = {600}\;\unicode{x00B5}{\rm m}$) on each end of the grating to ensure a smooth transmission with minimal ripple at the edges of the photonic bandgap. The apodization is performed by gradually reducing $G$ from the ends (${G_2} = {150}\;{\rm nm}$) to the center (${G_1} = {50}\;{\rm nm}$) of the grating. The grating pitch ($\Lambda$) is 339 nm. The CMBGs structure is chosen to provide greater control over the apodization because the coupling coefficient of the grating can be adjusted by changing the distance between the cylinders and the central waveguide. Effective apodization is crucial in reducing phase distortions at the edges of the grating band, enabling improved optical properties for nonlinear applications. The CMBGs use a 600 nm (width) by 300 nm (height) USRN core with ${{\rm SiO}_2}$ under- and over-cladding with the length of 3 mm.

 

Fig. 1. (a) Classical Bragg soliton versus pure-quartic Bragg soliton and their operating regimes. (b) Scanning electron micrograph of a CMBG used for demonstrating PQBS. (c) Schematic of the CMBG.

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The conditions for forming PQBS are satisfied when (1) FOD is anomalous and the dominant dispersion term driving interactions with SPM, and (2) GVD and TOD are negligible at the pump wavelength [4]. Figure 1(a) shows a comparison between the operating regimes for classical Bragg solitons compared to PQBS. For small/normal GVD, solitons can be generated by anomalous FOD when it simultaneously interacts with SPM induced by the larger effective nonlinearity of the Bragg grating. Figure 2(a) shows how PQBS is generated in the Bragg grating. The GVD is normal at the red side of photonic bandgap as shown in the blue curve of Fig. 2(a) whereas GVD is anomalous at the blue side of photonic bandgap. 7 nm to 8 nm away from the red side of the photonic bandgap, it is found that FOD is negative [Fig. 2(d)] and TOD vanishes [Fig. 2(c)] while normal GVD is unaffectedly small [Fig. 2(b)]. In other words, for a pump wavelength of 1558.8 nm, the conditions for PQBS formation are satisfied. GVD (${\beta _2}$) is obtained from the derivative of group index (${n_g}$) with respect to angular frequency ($\omega$). ${\beta _2} = \frac{1}{c}\frac{{\partial {n_g}}}{{\partial \omega}}$, where $c$ is the speed of light in vacuum and the group index, ${n_g}$, was measured using an interferometric component analyzer. TOD (${\beta _3}$) and FOD (${\beta _4}$) were obtained by fitting polynomials of ${\beta _3} = \frac{{\partial {\beta _2}}}{{\partial \omega}}$ and ${\beta _4} = \frac{{\partial {\beta _3}}}{{\partial \omega}}$, respectively. Ripples are increased in the plots for TOD and FOD as these are calculated through several rounds of derivatives of the ${n_g}$ versus $\omega$ data. However, the wavelength trend may still be clearly observed in Figs. 2(c) and 2(d). At a wavelength of 1558.8 nm, FOD is anomalous, possessing a value of ${-}{0.54}\;{{\rm ps}^4}/{\rm mm}$, TOD is negligible, and GVD is small/normal (${0.075}\;{{\rm ps}^2}/{\rm mm}$) as shown by the red arrows of Figs. 2(d), 2(c), and 2(b), respectively.

 

Fig. 2. (a) Linear transmission of the USRN Bragg grating and the GVD at the red side (red line) and blue side (blue line) of the band edge. (b) GVD, (c) TOD, and (d) FOD near the pump wavelength of 1558.8 nm (red side). The inset of (d) shows the FOD (${{\rm ps}^4}/{\rm mm}$) at the blue side of the band edge. The red arrows depict the dispersion at the pump wavelength of 1558.8 nm.

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B. Theoretical Analysis of Pure-Quartic Bragg Soliton Dynamics

The nonlinear Schrödinger equation (NLSE) effectively elucidates the pulse propagation dynamics of the quartic Bragg solitons [25]:

The parameter $A$ represents the pulse amplitude possessing the slowly varying envelope, while $z$ and $t$ denote the propagation and temporal coordinates, respectively. $\alpha$ is the linear loss coefficient of ${1.84}\;{{\rm cm}^{- 1}}$, which depicts the propagation loss of 8 dB/cm at the input wavelength of 1558.8 nm. The effective nonlinear parameter, ${\gamma _{{\rm eff}}} = \frac{{{\omega _0}{n_2}}}{{c{A_{{\rm eff}}}}}{\big(\frac{{{n_g}}}{{{n_0}}}\big)^2}$. Here, ${\omega _0}$ and ${A_{\rm{eff}}}$ refer to the input pulse’s angular frequency and effective mode area, respectively. ${n_2}$ is the Kerr nonlinear refractive index, and ${n_o}$ is the refractive index of the material. As ${A_{\rm{eff}}}$ and group index are ${0.25}\;\unicode{x00B5}{\rm m}^2$ and 3.85 respectively, ${\gamma _{\rm{eff}}}$ is calculated to ${700}\;{{\rm W}^{- 1}}/{\rm m}$. ${\beta _{2}} = {0.075}\;{{\rm ps}^2}/{\rm mm}$, ${\beta _{3}}\sim {0}\;{{\rm ps}^3}/{\rm mm}$, and ${\beta _{4}} = - {0.54}\;{{\rm ps}^4}/{\rm mm}$ at the pump wavelength of 1558.8 nm as shown in Figs. 2(d), 2(c), and 2(b), respectively.

The NLSE as an approximation for the nonlinear coupled mode equations has been well established in describing soliton pulse dynamics in Bragg gratings for the frequencies outside the stopband with two constraints on the field bandwidth and the light intensity [26]. The conditions for this approximation to be valid are (1) $c\kappa \tau \gg {1}$ and (2) ${n_2}I \ll \Delta n$ [15,27], respectively. ($\kappa$ is the grating’s coupling coefficient, $I$ is the light intensity, $\tau$ is the temporal width, and $\Delta n$ is the amplitude of the refractive index modulation in the Bragg grating). As $c\kappa \tau \sim {27}$ is much larger than 1 and ${n_2}I = {1.72} \times {{10}^{- 3}}$ is ${30} \times$ smaller magnitude than $\Delta n$, the NLSE model can be used to approximate the soliton pulse dynamics in place of the generalized nonlinear coupled mode equations [17].

 

Fig. 3. Temporal and spectral evolutions of PQBS at an input peak power of (a),(d) 1.12 W, (b),(e) 8.5 W, and (c),(f) 15.4 W.

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Fig. 4. Temporal and spectral outputs for an input peak power of (a),(e) 1.12 W, (b),(f) 1.5 W, (c),(g) 8.5 W, and (d),(h) 15.4 W. Experimental results in the time and spectral domains are black, navy, green, and pink empty dots (lines) for input peak power of 1.12 W, 1.5 W, 8.5 W, and 15.4 W, respectively. Theoretical results in the time/spectral domain are gray, blue, light green, and red lines for input peak power of 1.12 W, 1.5 W, 8.5 W, and 15.4 W, respectively. Black dashed lines in (a)–(d) depict the temporal traces of the input pulse.

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Theoretical calculations of the temporal and spectral evolution of the pure-quartic Bragg soliton along the propagation direction are performed for input coupled peak power (${P_{\rm{in}}}$) of 1.12 W, 8.5 W, and 15.4 W as shown in Fig. 3. The input pulse full width at half-maximum (FWHM) and the device length are 1.3 ps and 3 mm, respectively. To determine the conditions for fundamental PQBS formation, we performed NLSE simulations at a range of input powers. It is found that at ${P_{\rm{in}}} = {1.12}\;{\rm W}$, shape-preserving propagation occurs, in line with formation of the fundamental PQBS. For ${P_{\rm{in}}} = {1.12}\;{\rm W}$, we observe the fundamental PQBS exhibiting both temporal and spectral shape-preserving propagation as shown in Fig. 3(a) and Fig. 3(d), respectively. The pulse intensity is further observed to decrease along the propagation direction due to linear propagation losses. For ${P_{\rm{in}}} = {8.5}\;{\rm W}$, temporal pulses are observed to be compressed maximally at a propagation distance of 0.6–0.8 mm, indicating that nonlinear effects dominate compared to the FOD effect as the signature of high-order solitons where they experience an initial period of strong temporal narrowing, is observed in Fig. 3(b). Figure 3(e) shows the strong spectral broadening coinciding with where temporal compression occurs due to high-order solitons. For ${P_{\rm{in}}} = {15.4}\;{\rm W}$, periodic soliton evolution is observed, where three points of maximum temporal compression and spectral broadening are clearly seen in Figs. 3(c) and 3(f) at $z = {0.3}\;{\rm mm}$, 1.5 mm, and 2.7 mm. With larger ${P_{\rm{in}}}$, there are much stronger nonlinear effects compared to FOD, conducive to the formation of higher-order solitons. Temporal compression and spectral broadening observed in both the time domain and spectral domain simulations occur from $z = {2.7}\;{\rm mm}$ to $z = {3}\;{\rm mm}$. The results indicate that maximum compression and spectral broadening are achieved at the peak power of 15.4 W.

C. Experimental Results and the Comparison with the Theory

To experimentally study PQBS behavior in the USRN grating, we characterize both the temporal and spectral properties. A 1.3 ps laser emitting pulses at a 20 MHz repetition rate with a central wavelength of 1558.8 nm is utilized. The laser pulses are adjusted for TE polarization prior to coupling into the USRN CMBG using a tapered lensed fiber. The output’s spectral and temporal characteristics are then examined using an optical spectrum analyzer and autocorrelator, respectively.

Figure 4 shows the experimentally measured temporal and spectral outputs at the end of the propagation length as ${P_{\rm{in}}}$ is varied from 1.12 W, 1.5 W, 8.5 W, and 15.4 W. These are compared with theoretical simulations calculated by Eq. (1). First, we focus on the output temporal traces from Figs. 4(a)–4(d). The output autocorrelation traces are measured, and their time scales are obtained by applying the deconvolution factor (1.54 for ${{\rm sech}^2}$ pulse shape) to obtain the pulse FWHM of the temporal pulse [22]. For ${P_{\rm{in}}} = {1.12}\;{\rm W}$ [Fig. 4(a)], the output temporal shape is almost the same as the input temporal shape, which indicates that the fundamental quartic soliton is created preserving the temporal shape as it propagates. For ${P_{\rm{in}}} = {1.5}\;{\rm W}$ [Fig. 4(b)], there is only weak temporal compression, with the temporal profile being similar to that of the input pulse. For ${P_{\rm{in}}} = {8.5}\;{\rm W}$ [Fig. 4(c)], the pulses undergo temporal compression, which agrees well with theory. For ${P_{\rm{in}}} = {15.4}\;{\rm W}$ [Fig. 4(d)], stronger compression is achieved and the main peaks of the experiment match well the theory. The theory clearly shows pedestals in the pulse. We note that the experimental traces were not able to reveal these two features since an intensity autocorrelator was used, in which the second harmonic intensity between one and time-delayed outputs is measured. However, the pulse width can be estimated by the intensity autocorrelation as the width of intensity autocorrelation of a pulse is related to the intensity width by the deconvolution factor. The experimental traces’ pulse width match well theoretical values indicating that the experiment and theory are in good agreement with each other.

 

Fig. 5. (a) Input ${{\rm sech}^2}$ pulses and output pulse shapes fitted with Gaussian for an input peak power of (b) 15.4 W and (c) 1.5 W in simulation. (a) Measured input pulse (dashed line) and the theoretical ${{\rm sech}^2}$ input pulse (orange line). (b) Theoretical output (${P_{\rm{in}}} = {15.4}\;{\rm W}$, red line) with the main peak fitted with Gaussian (green line) and ${{\rm sech}^2}$ shapes (purple line). (c) Theoretical output (${P_{\rm{in}}} = {1.5}\;{\rm W}$, brown line) fitted with Gaussian shape (pink line).

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Fig. 6. Numerically calculated energy-width scaling for theoretical quartic solitons (empty squares) fitted with pulse energy (red line) and experimentally obtained fundamental pure-quartic Bragg soliton for ${T_0} = {0.74}$ ps (green empty circle), $E \propto T_0^{\,- 3}$ compared with conventional solitons (blue line, GVD of ${-}{0.81}\;{{\rm ps}^2}/{\rm mm}$). All pertain to the case where the fundamental soliton is created.

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Second, output spectra in the experiment and theory are compared as shown in Figs. 4(e)–4(h). For ${P_{\rm{in}}} = {1.12}\;{\rm W}$ [Fig. 4(e)], the experimental spectral bandwidth has good agreement with the theoretically calculated fundamental soliton. It shows that the spectrum is broadened as ${P_{\rm{in}}}$ increases as the SPM effect is largely affected by ${P_{\rm{in}}}$ linked to $L_{\rm NL}(= 1/(\gamma_{\rm eff}\cdot {P_{\rm{in}}}))$. For ${P_{\rm{in}}} = {8.5}\;{\rm W}$ [Fig. 4(g)], the experimentally measured output spectrum is observed to possess small side lobes around the main peaks at a normalized power level of 0.5. For ${P_{\rm{in}}} = {15.4}\;{\rm W}$ [Fig. 4(h)], a larger SPM effect is achieved with wider spectral broadening splitting into three main peaks as ${P_{\rm{in}}}$ increases (${L_{\rm{NL}}}$ is smaller). The theory also shows similar spectral broadening as well as splitting into three peaks, which is in good agreement with the experiment.

We can further verify the PQBS and analyze the experiment by comparing ${L_{\rm{NL}}}$ and the dispersion and device lengths ($L$). The GVD length ($L_{\rm GVD} = {T_0}^2/| {{\beta _2}} |$) is 7.3 mm, TOD length ($L_{\rm TOD} = {T_0}^3/| {{\beta _3}} |$) is infinity, and ${L_{\rm{FOD}}} = {0.59}\;{\rm mm}$. It is confirmed that ${L_{\rm{GVD}}}$ is one order of magnitude larger than ${L_{\rm{FOD}}}$ indicating ${L_{\rm{GVD}}} = {12.4}\;{L_{\rm{FOD}}}$. As ${L_{FOD}} \lt L \lt {L_{\rm{GVD}}}$ with ${\beta _{4}} \lt {0}$ (${L_{\rm{GVD}}} = {2.43} L$ and $L = {5.08}{L_{\rm{FOD}}}$), FOD is the dominant dispersion order in the PQBS dynamics, far outweighing effects from GVD. ${L_{\rm{NL}}}$ for ${P_{\rm{in}}} = {1.12}\;{\rm W}$ and ${L_{\rm{FOD}}}$ are comparable, and therefore there is no soliton period observed as the fundamental soliton in Fig. 3(a). ${L_{\rm{FOD}}}/\;{L_{\rm{NL}}}$ for ${P_{\rm{in}}} = {15.4}\;{\rm W}$ is larger than ${L_{\rm{FOD}}}/ {L_{\rm{NL}}}$ for ${P_{\rm{in}}} = {1.5}\;{\rm W}$; the higher-order soliton is generated for higher input power. However, even though the trends appear similar to conventional solitons, the scaling of SPM and FOD with pulse width is distinct, indicating thag the conventional definition of soliton number and soliton period do not apply to pure-quartic solitons [4,13,28,29].

One signature of pure-quartic solitons is their distinctive pulse shape [4,13,30]. Conventional temporal optical solitons have a ${{\rm sech}^2}$ shape. To further confirm the quartic nature of the PQBS, we analyze the output pulse shape as shown in Fig. 5. In the experiments and numerical calculations, hyperbolic secant pulses representative of the pulse shape from the optical source are used as shown in Fig. 5(a). Figure 5(b) provides a further analysis of the output pulse shapes in the numerical simulation (red line) for ${P_{\rm{in}}} = {15.4}\;{\rm W}$, comparing them with hyperbolic secant (purple line) and Gaussian profiles (green line). It shows that the main peak of theoretical output pulses is closer to a Gaussian profile rather than the ${{\rm sech}^2}$ profile. It can be confirmed again that the theoretical output for ${P_{\rm{in}}} = {1.5}\;{\rm W}$ (brown line) has a better fit with the Gaussian profile (pink line) as shown in Fig. 5(c). It may be clearly seen that the output pulses, despite having originated from ${{\rm sech}^2}$ pulses, have evolved such that they are now closer to a Gaussian profile.

We further analyze the impact of the PQBS pulse shape on the pulse properties. Figure 6 compares the energy-width scaling of the PQBS obtained using NLSE calculations [Eq. (1)], compared to conventional Bragg solitons as a function of the pulse width for fundamental solitons. For conventional solitons, ${\beta _2}$ used here is ${-}{0.81}\;{{\rm ps}^2}/{\rm mm}$ [17], a value typical for conventional Bragg solitons for the fundamental soliton with the same effective nonlinear parameter. The conventional soliton nature results in a $E \propto T_0^{\,- 1}$ relationship as shown as the blue line in Fig. 5. For PQBS, however, simulations reveal a $E \propto T_0^{\,- 3}$ relationship as shown as the red line in Fig. 5. The comparison between the conventional Bragg soliton and PQBS shows that larger pulse energy can be achieved at the short pulse duration of input pulse FWHM less than 500 fs. Therefore, it may be observed that PQBS, by virtue of their unique pulse shape, possess a different energy scaling, $E \propto T_0^{\,- 3}$, as opposed to $E \propto T_0^{\,- 1}$, particularly apparent at higher pulse energies [12]. These signatures of quartic solitons that we observe in PQBS provide confirmation of the quartic nature of the experimentally observed pulses generated in the nonlinear Bragg grating. The experimental pulse energy obtained is 1.46 pJ for ${T_0} = {0.74}\;{\rm ps}$ (${T_{\rm{FWHM}}} = 1.3\;{\rm ps}$) corresponding to the fundamental PQBS, which is in good agreement with theoretical calculations. It is confirmed through agreement between experimental characterization [Figs. 4(a) and 4(e)] and theoretical calculations [Figs. 3(a) and 3(d)] that the fundamental PQBS occurs at a peak power of 1.12 W.

 

Fig. 7. (a) Pulse FWHM, (b) compression factor, and (c) ratio of output peak power to input peak power as input peak power is varied. Empty squares and red lines depict experiment and the theory, respectively.

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Fig. 8. Normalized autocorrelation traces for input peak power of 4.22 W at a pump wavelength of 1552 nm. The inset shows the linear transmission (black line) and GVD (blue line) near the pump wavelength.

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D. Temporal Pulse Properties as a Function of Input Peak Power

We further analyze the temporal properties of the pulse, obtaining the pulse FWHM and its compression factor defined as $\frac{{{\rm Input\;Pulse\;FWHM}}}{{{\rm Output\;Pulse\;FWHM}}}$ as shown in Figs. 7(a) and 7(b), respectively. The maximally compressed pulse FWHM is 0.55 ps, equivalent to a compression factor of ${2.4} \times$. That is the largest compression factor experimentally observed to date in the quartic soliton regime. The experimental trends of pulse FWHM and compression factor follow well the theoretical trends as ${P_{\rm{in}}}$ is varied.

Next, we examine the quality of the compressed pulse, defined as $\frac{{P{\rm peak},{\rm out}}}{{P{\rm peak},{\rm in}}}$, where ${P_{\rm peak,out(in)}}$ refer to the peak output (input) power of the pulse. The results are plotted as a function of ${P_{\rm{in}}}$ as shown in Fig. 7(c). The peak output power is determined using ${P_{{\rm peak},{\rm out}}} = \frac{{({f \times {P_{{\rm avg}}}})}}{{(R\int_{- \infty}^{+ \infty} P(t){\rm d}t)}}$, where $f$ is the proportion of the energy in the main lobe with respect to the total energy, ${P_{\rm{avg}}}$ is the measured average power, and $R$ is the repetition rate of input pulses. $P(t)$ is the normalized pulse intensity, which is assumed to be a ${{\rm sech}^2}$ pulse [22,31]. Both experiments and theoretical calculations show the same trend where $\frac{{P{\rm peak},{\rm out}}}{{P{\rm peak},{\rm in}}}$ increases with ${P_{\rm{in}}}$. The maximum $\frac{{P{\rm peak},{\rm out}}}{{P{\rm peak},{\rm in}}}$ is 1.47, achieved for ${P_{\rm{in}}} = {15.4}\;{\rm W}$. It shows that the increase in pulse FWHM and $\frac{{P{\rm peak},{\rm out\;}}}{{P{\rm peak},{\rm in}}}$ with increasing ${P_{\rm{in}}}$ is due to the increased nonlinear phase shift, which allows for sufficient compression. It is confirmed from Fig. 3(c) that a device length of 3 mm generates the strongest pulse compression at the output for ${P_{\rm{in}}} = {15.4}\;{\rm W}$.

 

Fig. 9. Calculated temporal phase at the end of the propagation length for an input peak power of (a) 1.5 W and (b) 15.4 W.

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E. Measured Temporal Pulse Profile in the Presence of Large Normal GVD

The observations of PQBS are confirmed using a control experiment by locating the input pulses at 1552 nm, a wavelength at the right side of the band edge as shown in Fig. 8. Note that the GVD on the red side of the grating band edge is typically large and normal. At this wavelength, GVD is large and normal, possessing a value of ${\beta _2} = 1.8\; {\rm ps}^2 / {\rm mm}$, which is 24 times larger than the GVD at pump wavelength of 1558.8 nm where a pure-quartic soliton is generated. In this regime where the conditions for formation of PQBS are not satisfied, pulses are observed to undergo temporal broadening rather than exhibit soliton dynamics.

For ${P_{\rm{in}}} = {4.22}\;{\rm W}$, ${L_{\rm{NL}}} = {0.34}\;{\rm mm}$, and ${L_{\rm{GVD}}} = {0.30}\;{\rm mm}$, which corresponds to a conventional soliton number, $N = {L_{\rm{GVD}}}/{L_{\rm{NL}}}\sim{1}$. For $N = {1}$ with anomalous GVD, the pulse preserves its shape, propagating unchanged as a fundamental soliton. However, for $N = {1}$ in the presence of normal GVD, temporal broadening as shown in Fig. 8 is observed. The phase change generated by SPM leads to the creation of new frequency components, with red-shifted frequencies near the leading edge of the pulse and blue-shifted frequencies near the trailing edge of the pulse. The faster propagation of the red-shifted components compared to the blue-shifted components in normal-dispersion in nonlinear media results in an increased rate of pulse broadening, compared to the pulse broadening expected from GVD alone [25]. This result shows normal GVD is maintained on the red side of photonic bandgap and the solitons generated in this work form as a result of anomalous FOD.

F. Temporal Phase as a Function of Input Peak Power

The temporal phase for low and high peak power (${P_{\rm{in}}} = {1.5}\;{\rm W}$ and 15.4 W) pulses are numerically calculated to confirm that nearly non-perturbed solitons are created as a result of implementation on the USRN platform with negligible nonlinear loss at 1550 nm, as shown in Fig. 9. As the intensity autocorrelator we have used for temporal measurement cannot measure the phase variation in time, we can assume that input pulses have flat temporal phase with a value of 0 as the input pulses are transform-limited. For ${P_{\rm{in}}} = {1.5}\;{\rm W}$, the output pulse shape is minimally perturbed because the FOD length and nonlinear length are comparable to each other. This leads to the flat temporal phase across the pulse duration as shown in Fig. 9(a). Temporally compressed pulses are observed alongside for the high-order pure-quartic soliton (${P_{\rm{in}}} = {15.4}\;{\rm W})$ as shown in Figs. 3(c) and 4(d). The temporal phase is flat at the pulse center, as perturbations from nonlinear loss (eg. TPA or FCA) are absent due to their absence in USRN. This result is shown in Fig. 9(b). The range over which temporal phase is flat is also shorter for ${P_{\rm{in}}} = {15.4}\;{\rm W}$ compared to ${P_{\rm{in}}} = {1.5}\;{\rm W}$, commensurate with the stronger compression experienced by the input pulses as a signature of high-order solitons.

3. DISCUSSION

Optical solitons have a rich history in optical fiber, first theoretically predicted in 1973 by Hasegawa and Tappert [32] followed by their experimental observations in 1980 [1]. Their application to soliton lasers quickly followed [33]. It is therefore serendipitous that 50 years after their initial discovery, the study of optical solitons in chip-scale media has unveiled a new class of Bragg solitons—the PQBS.

Pure-quartic solitons, which are distinguished from quartic solitons by their reliance primarily on ${\beta _4}$, were first demonstrated in chip-scale photonic crystal waveguides [4]. The pure-quartic solitons experimentally demonstrated in silicon photonic crystal waveguides elucidated a maximum temporal compression factor of 1.63, which was limited by silicon’s two-photon absorption (TPA) and the related free-carrier absorption/dispersion at telecommunications wavelengths; the nonlinear loss results in an intensity plateau in the waveguide and a concomitant limit in achievable compression. Free-carrier dispersion was also reported to result in self-accelerated/asymmetric pulses, while free-carrier absorption led to an attenuation of the trailing edge of the pulse in the case of a high-order quartic soliton [4,34]. It was further reported in [9] that substantial propagation losses (${\sim}{70}\;{\rm dB/cm}$) led to the quartic soliton decaying after some extent of propagation as a result of the reduction in optical intensity. This leads to FOD dominating over the Kerr nonlinearity and dispersive broadening overwhelming the previously equilibrated quartic soliton-based propagation. More recently, with precise dispersion control using a free-space waveshaper, quartic soliton lasers in optical fiber were demonstrated [7].

The lower linear and nonlinear losses involved in the formation of PQBS are one key advantage compared to their implementation in photonic crystal waveguides. The first observations of PQBS in this work could potentially lead to integrated, compact PQBS lasers in the future, with the added advantage that the energy scaling of $E \propto T_0^{\,- 3}$ could provide pulses with higher output energies particularly at shorter pulse widths.

Pure-quartic solitons have also been theoretically analyzed to lead to frequency combs with better line-to-line power variation [35,36]. This property stems from the enhanced spectral flatness in quartic solitons. In the future, configuring the nonlinear Bragg grating described in this work in a cavity and pumping it at the PQBS regime could provide a pathway towards the implementation of PQBS frequency combs.

4. CONCLUSION

In conclusion, we first demonstrate a new class of Bragg solitons—the PQBS. Contrary to previously demonstrated classes of Bragg solitons, which reside either on the blue side of the grating stopband or within the stopband (as in the case of gap solitons) [37], PQBSs reside on the red edge of the grating stopband, which is broadly characterized as being in the normal dispersion regime. We demonstrate that under specific design conditions, pulses propagating on the red side of the stopband can satisfy the dispersion condition for pure-quartic soliton formation—negative FOD, negligible TOD, and GVD. We further confirm signatures of pure-quartic solitons including their profile being close to Gaussian rather than a ${{\rm sech}^2}$ shape, and a pulse energy scaling with $T_0^{\,- 3}$. Temporal compression of ${2.4} \times$, which is the largest observed experimentally in the quartic soliton regime, is experimentally demonstrated.