1. Introduction

Bragg solitons have been theoretically analyzed and experimentally observed in periodic dielectric media, specifically in optical fiber gratings [1–4]. The large second order dispersion at wavelengths near the photonic bandgap associated with the grating is compensated in the Bragg soliton regime by the third order optical nonlinear phase shifts in the fiber. Soliton formation is found even though there are strong higher order dispersion terms that one might initially think would disrupt soliton formation. The solitons that propagate near the photonic bandgap have group velocities substantially less than the speed of light in fiber. Experimentally we have measured velocities as low as c/2n [3], where c is the velocity of light and n is the refractive index for silica glass. Even lower velocities are predicted as the incident pulse wavelength approaches the edge of the photonic bandgap. However the pulse spectral width and imperfections in the grating which result in near-gap reflections eventually limit the velocity reduction. Reflections near the bandgap reduce the effective optical fields that propagate into the grating so that higher incident intensities are required for soliton formation. As the incident intensity is increased we find that for pulse widths in the 100 picosecond range the incident intensity is limited by damage at levels near 20 GW/cm². This damage limit prevents us from exploring solitons that are predicted to propagate with wavelength components very near or within the bandgap. For solitons within the gap, propagation velocities can vary over a wide range including stationary solitons. Ultra-low soliton velocities have not been observed although evidence for nonlinear propagation wavelengths primarily within the gap for a portion of the distance in the grating has been reported using longer pulse lengths in the nanosecond regime [4].

 

Fig. 1. A schematic diagram of the photonic bandgap and the optical pulse as a function of position and wavelength. Wavelength components are represented as a rainbow of colors. A representative set of three wavelength component in the optical pulse are shown at different time positions due to group velocity dispersion in the wavelength regime near the photonic bandgap.

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In the experiments reported here we study the nonlinear propagation of pulses with durations in the 100 ps regime in a chirped grating where the grating period varies along the axis of the grating [5]. Chirped gratings are interesting for nonlinear evolution studies since the dispersion varies as the pulse propagates along the grating. Bragg solitons are expected to evolve in a manner similar to the predictions of the nonlinear Schroedinger equation for slowly varying grating periods [3]. We are interested in approaching the bandgap as closely as possible with the pulse spectral components. This can be achieved in the chirped case since the grating can be tuned and chirped with decreasing period along the grating so that the pulse reflects from the grating at some point after propagating for a distance long enough for significant nonlinear evolution. The pulse is expected to slow as it approaches the gap edge. We hope to be able to trap these slow Bragg solitons in defect potentials formed by amplitude variations or phase shifts in the grating. Before studying trapping we need to understand the reflection process in the nonlinear regime where the reflected pulse shape depends on the incident pulse intensity.

We study nonlinear pulse evolution in the dispersion gradient of the chirped grating and the pulse dynamics as it reflects from the grating with nonlinear pulse reshaping. We use apodized gratings in order to reduce reflection from the ends of the grating and thus enhance the coupling of the pulse into the grating. A schematic diagram of the resulting photonic bandgap is shown in Figure 1. The rainbow shaded region is the linearly forbidden region for light propagation. The bandgap width increases gradually to the full width in the apodized regions at both ends of the grating. The linear chirp in the period of the grating tilts the gap. Our experiments use wavelengths near the short wavelength edge of the bandgap where the group velocity dispersion is anomalous. Anomalous dispersion counteracts nonlinear self-phase modulation and thus favors soliton formation. The light pulse in Figure 1 is shown as a cartoon with three wavelength components being dispersed by the grating. The cartoon is roughly to scale with the position axis spanning the 6 cm length of the grating and the spatial extent of the 100 ps pulse being roughly 1.5 cm. As shown schematically by the wavelength spread of the pulse, the shorter wavelengths propagate faster than the longer wavelengths up to the reflection point due to the anomalous grating group velocity dispersion. At the reflection point the chirped grating reflects the longer wavelength components first, thus partially compensating for the anomalous group velocity dispersion encountered during propagation before reflection. We will show that dominant feature of the nonlinearly reflected pulse is a breakup into two well separated short pulses that evolve from the dispersively broadened pulse that is observed for linear reflection from the chirped grating. This behavior is in agreement with numerical simulations and persists over a broad range of grating parameters.

2. Experiment

The fiber gratings used in the experiments were formed by UV exposure of a fiber through a phase mask [6]. They were apodized in order to minimize reflection losses near the photonic bandgap edge. The two gratings studied in the present experiments had bandgap widths of 0.17 and 0.2 nm, corresponding the index modulation amplitudes, Δn/n = 0.00017 and 0.0002. The gratings were 6 cm in length with 0.75 cm of gradually increasing index modulation on each end. The average index was kept approximately constant from the bare fiber through the grating region by exposing the fiber to uniform UV radiation. A small amount of chirp remained in these gratings due to an incomplete compensation of the average index.

A Q-switched, mode-locked YLF laser generates sequences of 80 ps wide pulses at a 500 Hz rate [2]. The laser wavelength is 1053 nm. A single 80 ps pulse from each Q-switched pulse sequence is selected by an electro-optic modulator. This pulse is positioned at the beginning of each Q-switched sequence where self-phase modulation in the laser cavity [7] is at a minimum. Even for these initial pulses we measure a pulse chirp of 0.13π radians from the laser [8]. This input pulse chirp is small but non-negligible in the present experiments. For example, it results in as much as a 15% pulse narrowing as the pulse propagates linearly through the dispersive region near the photonic bandgap. The 500 Hz pulse train is coupled into the fiber with a microscope objective that also serves to collect the reflected light from the grating as shown in Figure 2. The pulse energy is measured by inserting a calibrated meter before the microscope objective. The light output of the fiber core is measured using a fast photodiode and sampling oscilloscope with a combined time response of 20 ps, adequate to temporally resolve a major portion of our experimental results.

The period of the grating is chosen so that the grating bandgap can be strain tuned to the laser wavelength. The fiber is cleaved so that there are only a few millimeters of grating free fiber at the incident face. This eliminates any spectral broadening due to self-phase modulation before propagation in the grating. The fiber is strain tuned by epoxying it to two optical mounts separated by approximately 25 cm and stretching it with a micrometer adjustment screw. The major grating chirp was controlled by embedding the grating region of the fiber with silicone grease in a 0.5mm x 0.5mm x 6cm stainless steel channel. A temperature gradient of up to 50 °C could be applied to this channel by heating one end and maintaining the other end at room temperature with a cooler as shown in Figure 2. The chirp obtained in this manner [9] is approximately 0.0065 nm/°C. The temperatures near both ends of the grating are held constant to within 0.1 °C by active feedback from temperature sensors. Grating spectra measured in various temperature gradients are shown in Figure 3. The chirp is not completely linear. Delays of the reflected pulse are used as an accurate measure of the actual grating period as a function of position.

 

Fig. 2. A schematic diagram of the experimental apparatus. The incidents pulse (horizontal red arrow) is focused into the fiber (orange line) and the reflected pulse (vertical arrow) is directed to the fast response photodiode (D) by the beam-splitter (BS). The photo-current is displayed on a sampling oscilloscope. The grating is chirped by a temperature gradient (shown shaded from a hot red to a cold blue), produced by a heater (H) and a cooler (C). The grating bandgap is tuned by applying a strain (S) with a micrometer positioner.

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Fig. 3. Transmission spectra for a chirped fiber grating in a series of temperature gradients applied as shown in Figure 2. One end of the fiber is kept at room temperature by the cooler and the other end is at 24 °C (black), 34 °C (red), 44 °C (green), 54 °C (blue), and 64 °C (magenta). Note that the photonic bandgap shift is toward longer wavelengths as the average temperature increases. The incident pulse for the nonlinear propagation experiments is positioned at the short wavelength edge of the grating as shown is Figure 1.

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2. Results

Delays of a linearly reflected pulse as a function of its wavelength relative to the photonic bandgap are obtained using the fixed laser wavelength and strain tuning the grating. The grating chirp is obtained with an applied temperature gradient near 20 °C over the 6 cm length of the grating, corresponding to total chirp near 0.13 nm. Large delays near 1ns are measured as the pulse reflects from the short wavelength end of the bandgap; approximately 6 cm from the front face of the grating (see Figure 1). These delays are measured relative to a small reflected pulse from the front cleaved face of the fiber. The measured delays as the grating is strain tuned correspond to a grating chirp that is approximately equal to that predicted using the thermal expansion and thermal refractive index changes for silica. Group velocity delays for tuning near the grating edge increase the measured delays from those expected for the bare fiber.

The nonlinear pulse evolution is studied for a grating strain such that the pulse reflects from the back of the grating, i.e. maximum delay, but with less than 10% transmission. The reflected pulse shapes in the linear and nonlinear regimes are shown in Figures 4 and 5 for two the different gratings. The linearly reflected pulses show dispersive broadening indicating that the group velocity dispersion is not completely compensated by the chirped grating reflection. The linearly dispersed pulse width and shape for the grating corresponding to the data in Figure 4 is in agreement with numerical simulations. The linearly reflected pulse shape in Figure 5 is also dispersed, but there is an additional modulation of the pulse shape with a period of approximately 150 ps, possibly due to grating imperfections introduced during fabrication.

 

Fig. 4. Experimental (solid curves) reflected pulses from a grating with 0.2 nm bandgap and 0.15 nm chirp over the 6 cm length of the grating. The red curves are in the linear regime with intensities near 1 GW/cm². The blue curves are in the nonlinear regime with intensities near 10 GW/cm² for the experimental data. The simulation (dotted curves) in this case is for a uniform grating with a center wavelength of 1053.2 nm, a chirp of 0.15 nm and a bandgap of 0.3 nm. Both the experimental and simulated fiber in this case had a 3 cm bare fiber section before the grating. The incident pulse wavelength is 1052.8 nm and the intensity is 20GW/cm². Uniform gratings typically give similar results as an apodized grating except that the nonlinear regime is reached at nearly a factor of 2 higher intensity for uniform grating.

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Nonlinear evolution of the reflected pulse begins at intensities near 3 GW/cm². Between 1 and 3 GW/cm² there is a slight narrowing of the pulse and a small shift, of the order of 50 ps, toward longer delays. Above 3 GW/cm² the delays decrease and a second pulse evolves at smaller delays as shown in Figures 4 and 5. As the intensity increases the two pulses evolve to nearly equal intensities and then, at intensities just below the damage, a more complex multiple pulse structure is observed. The additional structure at high intensities in Figures 4 and 5 at times before and after the major pulse components is probably due to imperfections in the chirped grating formed during fabrication. This additional structure does not appear in the simulations.

 

Fig. 5. Experimental reflected pulses from a chirped fiber grating with a 0.17 nm bandgap width and a chirp of 0.13 nm over the 6 cm length of the grating. The red curve is for an incident intensity of 3 GW/cm², near the linear regime. The green curve is in the nonlinear regime with an intensity of 6 GW/cm² and the blue curve is further into the nonlinear regime at 9 GW/cm². The total delay of the reflected pulse sequence shown here is approximately 1 ns relative to a small reflection from the incident face of the fiber.

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Numerical simulations using the coupled mode equations [2] reveal pulse breakup into two pulses similar to the experimental data as shown in Figure 4. The numerical simulations use parameters similar to the experimental gratings and laser pulses. They include the effects of the apodized grating profiles. The pulse can be tuned very close to the edge of the photonic bandgap so that a significant fraction of the light is transmitted through the grating, e.g. 10 to 20% of the incident intensity for the linear regime. As the pulse intensity is increased by a factor of 2 above the range where reflected pulse break-up is observed, the transmitted pulse also evolves into two pulses, corresponding the formation of higher order solitons. However both experiment and simulations show that for incident intensities where the reflected pulse breaks into two pulses, the transmitted pulse is narrowed but remains a single pulse. Simulations of this partially transmitting case are shown in Figure 6.

The pulse breakup persists in both simulations and experiments when the chirp is increased or decreased by a factor of two. Experimentally the pulse breakup is not as pronounced when the pulse is tuned to reflect nearer the incident face of the grating and higher intensities (above 10 GW/cm²) are required to observe any nonlinear effects. When the pulse is tuned to reflect just at the front face of the grating the pulse broadens and the delay increases at intensities where the pulse breaks up at reflection from the back of the grating. When the pulse is tuned near midgap for reflection, only a slight nonlinear reshaping of the pulse for intensities below the damage threshold (near 20 GW/cm²) is observed experimentally in agreement with the numerical simulations.

4. Analysis

Self-phase modulation certainly plays an important role in the pulse pair formation. The nonlinear phase shift in the 6 cm of fiber before the reflection point is approximately 0.5π, for the incident intensity corresponding to the pulse pair formation. However the linear simulations for the chirped gratings parameters used in these experiments show that the peak pulse intensity in the grating is enhanced by the slower group velocity in the grating region by a factor in the range from 1.5 to 2 as the pulse narrows and reaches the reflection point (see the linear simulation movie in Figure7). At a nonlinear phase shift of 1.5 π the self-phase modulation is expected to split the spectrum of the pulse into two distinct frequency components [10]. These two components could then reflect from the chirped grating to form two distinct pulses in time. However there is also group velocity dispersion exerting a strong influence during the pulse evolution and self-phase modulation alone is a poor model. Simulations with 3 to 6cm sections of bare fiber before the grating do not show a strong influence on the pulse break-up. Since self-phase modulation is nearly doubled by adding these initial segments of fiber, these simulation results indicate that the simple model of self-phase modulation induced spectral splitting followed by chirped grating reflection is not the key to understanding the results.

 

Fig. 6. Numerical simulations of the transmitted and reflected pulses from a chirped grating with parameters corresponding to a 0.3 nm bandgap and chirp of 0.1 nm at intensities in the linear regime a 1 GW/cm² (red curve is multiplied by 10 for comparison with 10 GW/cm²) and nonlinear regime at 10 GW/cm² (green curve) and 20 GW/cm² (blue curve). The incident pulse is shown for the 10 GW/cm² data at a time near 400 ps. Transmitted pulses appear between 900 and 1200 ps and reflected pulses are between 1200 and 1700 ps. The center wavelength for bandgap at the incident face (cool) of the grating is 1053.2 nm. The center wavelength of the incident pulse is 1052.95 nm.

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The numerical simulations show that as the pulse approaches the reflection point it narrows and intensifies, eventually reaching levels where a secondary peak evolves. The linear and nonlinear simulation results are shown in Figure 7 as a movie where each frame shows spatial profiles of the forward and backward propagating pulses at various times. The double pulse structure that evolves along the grating as the pulse propagates nonlinearly in the forward direction is broadened at reflection. After reflection the pulse pair is reconstituted in the backward propagating wave with a separation that is nearly 5 times the separation when the secondary peak appeared during the forward evolution. The temporal profile of the incident and reflected pulse corresponding to the simulation in Figure 7 is shown in Figure 8. These simulations are in reasonably good agreement with the experimental results in Figure 5 when grating fabrication errors and initial pulse chirp are taken into account.

 

Fig. 7. Numerical simulations in the form of snapshots of the spatial profiles of the forward (red for the nonlinear 15 GW/cm² case and blue for the linear 1 GW/cm² case) and backward (green for the nonlinear 15 GW/cm² case and magenta for the linear 1 GW/cm² case) propagating pulses in movie format (a single frame at a time of 692 ps is shown, open link for full movie). The nonlinear sequence is followed by the linear sequence. The incident pulse width is 80 ps. The snapshots are taken at times of 173, 346, 519, 692, 865, and 1038 ps, relative to the time when the pulse enters the grating. The center wavelength of the grating at the incident (hot) face is 1053.2 nm. The grating chirp is 0.15 nm over the 6 cm length. The center wavelength of the incident pulse is 1053 nm. [Media 1]

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The nonlinear Schroedinger equation model for Bragg solitons [3] predicts that the grating lengths required for complete evolution into a soliton for the parameters in these experiments are in the range of several centimeters, i.e. we cannot think of the pulse pair formation in terms of completely developed solitons. The low threshold for the pulse pair formation relative to the N=1 soliton formation in transmission appears to be associated with the pulse spectral components approaching the bandgap edge as closely as possible just before reflection. This leads to large field enhancements associated with the slow group velocities [3] for propagation near the bandgap edge, which in turn result in pulse break-up. The chirped reflection then accentuates the pulse splitting by increasing the temporal separation of the two pulses. The amplitudes for the two pulses appear to equilibrate during reflection process in the simulations.

 

Fig. 8. Time dependence of the nonlinearly reflected pulse (blue curve), linearly reflected pulse (red, multiplied by 15 in order to compare with nonlinearly reflected pulse) and incident pulse (green curve) for parameters as given in Figure 7. This simulation can be compared with the experimental results in Figure 5.

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5. Conclusions

In summary we have studied the nonlinear reflection of light pulses from chirped fiber gratings. We find that for incident pulse intensities in the regime that will form fundamental solitons in transmission, pulse pairs are formed in reflection. The pair formation is found to be associated with the strong fields produced as the pulse reflects from the chirped grating and the effects of the chirped reflection process. The pair formation persists over a broad range of chirped grating parameters. This reflection configuration optimizes the nonlinear multiple pulse evolution in the grating both because the pulse spectral components reach the bandgap edge resulting in enhanced fields and because of the doubled interaction length, compared to the transmission case, that contributes to enhanced nonlinear evolution.

In the future we hope to study nonlinear trapping of optical pulses at defects in the grating structure. The present studies of nonlinear reflection are important preliminary results for trapping since one possible configuration is to trap the pulse after reflection from a portion of the grating with high index modulation. The present experiments show that the pulse breakup regime should be avoided for trapping of single solitons. We are presently studying the frequency structure of the pulse sequences after they reflect in the nonlinear regime. For example, if the two backward propagating pulses have sufficiently distinct frequency components, trapping of a portion of the pulse sequence may be possible. Chirping of the grating may be useful in trapping studies in order to adjust the effective trapping potentials for nonlinear release or capture of the pulse. It will be interesting to study more complex chirped grating profiles than the simple linear profile studied in the present experiments. For example, reflection from an abrupt change in the grating period is of interest for nonlinear optical pulse trapping experiments. These studies of nonlinear reflection may also be important for the application of fiber gratings to pulsed fiber lasers where the optical fields can easily reach the nonlinear regime.