## Abstract

We numerically and analytically evaluate the delay of solitons propagating slowly, and without broadening, in an apodized Bragg grating. Simulations indicate that a 100 mm Bragg grating with Δ*n*=10^{-3} can delay sub-nanosecond pulses by nearly 20 pulse widths without any change in the output pulse width. Delay tunability is achieved by simultaneously adjusting the launch power and detuning. A simple analytic model is developed to describe the monotonic dependence of delay on Δ*n* and compared with simulations. As the intensity may be greatly enhanced due to a reduced velocity, a procedure for improving the delay while avoiding material damage is outlined.

© 2006 Optical Society of America

## 1. Introduction

Slow light is a phenomenon of fundamental interests that has already found use in a number of areas. For instances, a Mach-Zehnder intensity modulator having the benefit of requiring a reduced path length from a lower group velocity has been demonstrated [1]. Enhanced nonlinearity that arises from the slow light effect has been observed in a photonic crystal waveguide [2]. On the other hand, slow light systems have been suggested as candidates for optical re-timing devices such as optical buffers and optical delay lines, or functions such as data synchronization and jitter correction [3]. Ideally, such slow light systems should provide a potentially large and tunable delay without any pulse distortion. However, the largest delay in slow light systems reported to date is only a few pulse widths, and the slow pulse is always hampered by dispersive broadening [4, 5]. This broadening is a direct result of the Kramers-Kronig relations, which fundamentally limit all linear systems to have a finite bandwidth over which the low group velocity stays constant [6]. Consequently, the maximum length, and hence the maximum delay, of a slow light system is limited by how much broadening is acceptable in the application.

In this paper we investigate a nonlinear slow light medium, in which light pulses propagate as solitons, thus avoiding the broadening effects of dispersion–this slow light medium can thus be made arbitrarily long without dispersion penalty. The particular geometry considered here are Bragg gratings at high pulse intensities [7, 8, 9, 10], leading to the generation of slow gap solitons [11, 12, 13, 19]. Using numerical simulations, we evaluate the delay and tunability in this geometry, by incorporating the coupling and propagation of gap solitons. We show that the pulse delay increases with the grating’s refractive index contrast Δ*n*, while tunability can be achieved by varying the launch power and detuning. A simple model that relates the delay of the slow light system to Δ*n* is proposed and compared to simulations. A way to avoid material damage due to potentially large peak intensity enhancement is also outlined.

## 2. Device principles

Because of the periodic refractive index distribution, Bragg gratings exhibit a (one-dimensional) photonic bandgap (see Fig.1(a)) centered at the Bragg wavelength

where Λ is the spatial period and *n* is the average refractive index. Thus, a weak pulse with a spectrum lying inside this bandgap, that is launched into a Bragg grating is strongly reflected, as illustrated in Fig. 1(b), showing an intensity contour plot of time vs position. In contrast, an intense pulse with intensity *I*, through the Kerr nonlinear response of the material, can raise the refractive index according to

where *n*
_{0} is the refractive index at low intensities and *n*
_{2}, taken to be positive, is the nonlinear coefficient, thus shifting *λ _{B}* and the bandgap to a longer wavelength, allowing the pulse to be transmitted through the grating. This is illustrated in Fig.1(c), which shows that the pulse travels at a velocity, given by the inverse slope of the contour, less than that in a uniform medium. Such a pulse, a gap soliton, can propagate, in principle, at any velocity between 0 and

*c*/

*n*, where

*c*is the speed of light in vacuum, without broadening [13]. It is the possibility of the very low gap soliton velocities that gives rise to potentially large temporal delays.

As the gap soliton propagation is a nonlinear process and the Bragg grating has a strong spectral dependence of phase, the velocity of the gap solition depends on both its amplitude and frequency. As a result, tunable delay can be achieved by controlling the launch pulse power [14] and the detuning, which is proportional to the difference between the pulse wavelength and *λ _{B}*. The launch power can be adjusted using a variable optical attenuator, while the detuning can be adjusted by either varying Λ or

*n*[15], for example via the electro-optic effect, which is inherently fast.

The slow light scheme considered here is well suited to integrated geometries using highly nonlinear materials, such as chalcogenide waveguides [16] which has *n*
_{2} of the order of 10^{-13} cm^{2}/W and *n*
_{0} of roughly 2.5. The large nonlinearity and the large core-cladding index difference, leading to a smaller effective mode area, substantially lower the required optical power. Fabrication of chalcogenide-based waveguide gratings via photosensitivity producing a large photo-induced Δ*n* has also been reported [17]. Here, without loss in generality, we first consider silica fiber gratings in our analysis, and then indicate how it may be applied to chalcogenide waveguide, in which much stronger gratings can be written than in silica fibers.

## 3. Method

We evaluate the propagation delay of the output pulse of the Bragg grating slow-light system by numerically solving the governing coupled mode equations

$$-i\frac{\partial {A}_{-}}{\partial z}+i\frac{n}{c}\frac{\partial {A}_{-}}{\partial t}+\kappa {A}_{+}+\gamma \left({\mid {A}_{+}\mid}^{2}+2{\mid {A}_{-}\mid}^{2}{A}_{-}\right)=0$$

for the electric field using a collocation method [18]. *A*
_{±}=|*A*
_{±}(*z*,*t*)|*e ^{iδ′}*(

*c*/

*n*)

*t*is the complex forward and backward propagating envelops with

*δ*′=(

*ω*′-

*ω*)/(

_{B}*c*/

*n*), where

*ω*′ is the pulse frequency,

*ω*is the Bragg frequency, located in the center of the photonic bandgap,

_{B}*γ*is the nonlinear coefficient and

*κ*=

*ηπ*Δ

*n*/

*λ*is the coupling strength, where

*η*≈0.8 in silica fibers represents the mode-field grating overlap. For our simulations we focus on parameters corresponding to our recent experiment [19], and take gratings designed to work near wavelengths

*λ*≈1

*µ*m, in the core of a silica fiber with typical parameters:

*γ*=6.4 /(W km) and Δ

*n*≤10

^{-3}, as appropriate for silica glass. The results can however be generalised to other materials and geometries. The grating is apodized, with Δ

*n*varying as [1-cos(

*πz*/

*l*)]/2 for 0<

_{a}*z*<

*l*and [1+cos(

_{a}*π*[

*z*-(

*L*-

*l*)]/

_{a}*l*)]/2 for

_{a}*L*-

*l*<

_{a}*z*<

*L*, with

*l*=15 mm, and Δ

_{a}*n*constant in between the two regions. We choose an apodized grating because it is the simplest design in which relatively efficient coupling can be achieved. This leaves us the following parameters: (1) index contrast Δ

*n*, (2) grating length

*L*, (3) launch power

*P*, (4) pulse detuning

*δ*′, and (5) pulse width Δ

*τ*. We note that in terms of the detuning

*δ*=(

*ω*-

*ω*)/(

_{B}*c*/

*n*), where

*ω*is frequency, the photonic bandgap extends from

*δ*=-

*κ*to

*δ*=+

*κ*. The definitions of various spectral parameters are illustrated in Fig. 2(a). Assuming a lossless grating, where the gap soliton travels at a constant velocity, the delay simply scales linearly with

*L*. We therefore fix

*L*=100 mm, a typical grating length. Since the gap solitons’ group velocity and width both vary with the launch peak power, of which the effects are illustrated in Fig. 2(b), we choose to launch at the power

*P*=

*P*that results in an output pulse having the same pulse width as the input. The timing of the output pulse under this condition is then used to evaluate the delay.

_{0}The delay is given by *τ _{d}*-

*τ*

_{d0}, where

*τ*is the pulse propagation delay when the device is on, and

_{d}*τ*

_{d0}is that when it is off. The device is considered to be off when the bandgap is detuned far away from the pulse frequency, in which case the pulse propagates unaffected by the grating and has a delay of

*τ*

_{d0}=

*nL*/

*c*. The device is considered on when the pulse is tuned to lie inside the bandgap. In practice, the detuning can be adjusted by varying the strain that is applied to the grating, and thus the device can be mechanically switched between the on and off states.

Our slow light mechanism requires the bandgap frequency to be shifted with respect to the pulse spectrum by nonlinear effects. Since we want to operate at the lowest possible switching powers that are consistent for gap soliton formation, this spectral shift should be as small as possible and correspond to no more than a few pulse spectral widths. Furthermore, the regime where the pulse is outside the bandgap has been studied both experimentally and numerically in Ref. [9]. For these reasons, we limit the range of *δ*′ such that 1.2*δ _{p}*<

*κ*-

*δ*′<3

*δ*, where

_{p}*δ*is the 3 dB spectral width of the pulse.

_{p}## 4. Results

We consider launching gap solitons into a Bragg grating using slightly asymmetric pulses of width 680 ps, which matches our earlier experiment [19]. Their electric field envelope is given by sech(*t*/*a*
_{±}), with *a*-=257 ps for *t*<0 and a_{+}=514 ps for *t*>0, leading to a full width at half maximum (fwhm) of the intensity of Δ*τ*=680 ps. We obtain the pulse delay and required launch peak power by adjusting the launch power until the fwhm of the output pulse matches that of the input pulse (680 ps). This is repeated for Δ*n* from 10^{-4} to 10^{-3}, a typical range for silica gratings, while keeping the detuning to be at a constant value from the edge of the photonic bandgap (i.e. *κ*-*δ*′ is a constant). Five such detunings are chosen: *δ*′=*κ*-1.2*δ _{p}*,

*κ*-1.5

*δ*,

_{p}*κ*-2.0

*δ*,

_{p}*κ*-2.5

*δ*and

_{p}*κ*-3.0

*δ*where

_{p}*δ*=14.09 m

_{p}^{-1}is the spectral fwhm of the 680 ps pulse with a constant temporal phase.

Figure 3 shows the pulse delay *τ _{d}*-

*τ*

_{d0}and the required launch peak power

*P*

_{0}as a function of Δ

*n*for the different detunings. The right axis of Fig. 3(a) shows the scale corresponding to the fractional delay,

*F*=(

*τ*-

_{d}*τ*

_{d0})/Δ

*τ*, corresponding to the grating’s delay capacity. Note from Fig. 3 that as Δ

*n*increases, the delay increases sub-linearly, while the required launch power decreases. The increased delay is a result of stronger interaction between the forward and backward propagating waves, creating an effectively longer path length. The required launch power decreases with Δ

*n*as a result of intensity enhancement due to the slow light effect. Less incident power is required to maintain the intensity for the same bandgap shift to be achieved.

As seen from Fig. 3(a) and (b), with the maximum achievable δ*n* in silica limited to ~10 ^{-3}, the fractional delay for 680 ps pulses in a 100 mm grating, depending on the detuning, can be up to nearly 20 pulse widths, with the required launch power of the order of a few kW (1 kW corresponds to 3.65 GW/cm^{2}). Since all data points in Fig. 3 represent delayed pulses of 680 ps wide, they imply delay tunability without any change in pulse width. To achieve this for a given grating, namely for a fixed Δ*n*, the delay is tuned by varying both the detuning and the launch power appropriately. The inset in Fig. 3(b) shows the input pulse (dotted line) and output pulses (solid lines) through a Bragg grating with Δ*n*=10^{-3}, at launch powers and detunings indicated by the grey box in Fig. 3(b). It illustrates a continuously tunable delay from ~8 to ~12.5 ns, or group velocities ranging from 0.037 to 0.057 *c*/*n*. This corresponds to a tuning range of approximately 7 pulse widths, with a constant output pulse width of 680 ps. Transmission of the delayed pulses ranges from 20% to 28%, as illustrated in the inset of Fig. 3(a), with the loss associated with the input (output) coupling and velocity mismatch between the incident (transmitted) pulse and the gap soliton. Transmission decreases as the delay increases because of an increased velocity mismatch. For the same Δn, the delay increases with decreasing detuning, at the expense of increasing required launch power.

The error bars in the delay values in Fig. 3(a) arise from simulation granularity in *P*
_{0}. Recall that *P*
_{0} is the required launch power so that the output pulse width equals that of the input pulse. Since in the simulation we only have control over the launch power and not the output pulse width, the pulse widths are matched by trying different values of *P*
_{0}. Similarly, since pulse delay is a function of launch power, the same iterative process of finding P0 gives us a range that contains the delay *τ _{d}*. It is found that the output pulse delay can be sensitive to launch power. Generally, obtaining the accuracy in

*τ*shown by the error bars requires

_{d}*P*

_{0}to be accurate to within′ 0.1%, which is the reason why the error bars in

*P*

_{0}in Fig. 3(b) are virtually invisible on the scale of the plots. This fact also makes comparison of delay between experiments and simulations difficult, since uncertainties in power measurements are usually large (≳10%). A better grating design may be able to lower the sensitivity of delay on launch power.

Although the results above correspond to a silica fiber grating, they can be generalised to other materials and geometries, such as chalcogenide-based waveguide gratings, which has a reported Δ*n* of 0.01 [17]. According to Fig. 3, a large Δ*n* means that the maximum delay can be greatly improved. With *n*
_{2} of chalcogenide up to ~1000× higher than that of silica [20], and a higher core-cladding index difference leading to a smaller effective mode area, the power requirement can be lowered to a few watts or possibly sub-watt levels. However, despite a lower power requirement, one has to consider material damage because of the potentially large intensity enhancement within the grating due to a reduced velocity. For example, at Δ*n*=10 ^{-3} and *δ*′=*κ*-1.5*δ _{p}*, the pulse slows down by a factor of 19 while the total internal peak intensity of the forward and backward waves reaches 12 times that of the incident pulse. Generally, the enhancement increases with the slow down factor since a slowed pulse is spatially compressed, though the exact degree of enhancement partly depends on the amount of incident pulse energy coupled into the grating, which is determined by the pulse shape and velocity mismatches between the incident pulse and the gap soliton. In the next section, we develop and verify an analytic model to predict the delay, particularly in chalcogenide-based gratings, then look at how the internal intensity varies with

*δ*′ and

*κ*, and suggest on a way avoid material damage.

## 5. Prediction of the delay

The delay results shown in Fig. 3 are obtained through full simulations of the coupled mode equations, which can be time-consuming. Here we develop a simplified analytic model to relate the delay of the gap solitons to typical grating parameters. The delay of the gap solitons arises from their low group velocity, which, in turn, follows from the (linear) dispersion relation of the grating. We therefore approximate the simulated delay results by treating the system as if it were a linear Bragg grating. The (linear) dispersion relation of a medium with a Bragg grating can be written as

where *q*=*k*-*k _{B}*, where

*k*is the wave number and

*k*=

_{B}*π*/

*d*is the wave number at the Bragg condition. The group velocity

*V*, ignoring end effects, can then be found from the dispersion relation, and can be shown to be [21]

_{g}The delay *τ _{d}*-

*τ*

_{d0}of a pulse is thus

We treat the gap soliton as having locally shifted the entire bandgap such that its centre frequency now lies outside the bandgap at an *apparent* detuning *δ*′+Δ*δ* with Δ*δ*>*κ*-*δ*′ the bandgap shift induced by the positive Kerr nonlinearity. The delay *τ _{d}* is then given by the linear dispersion relation of the Bragg grating at the apparent detuning, with

*δ*=

*δ*′+Δ

*δ*in Eq. (5) where Δ

*δ*can be found by fitting the numerical results. The dashed lines in Fig. 3(a) are fitted in such a way.

As shown by the fits in Fig. 3(a), although Eqs. (5) and (6) describe a linear property of the grating, they model the delay well even at high intensities, with the introduction of the free parameter Δ*δ*. Fig. 3(a) indicates that Δ*δ* does not depend on the individual values of *κ* (i.e. Δ*n*) or *δ*′, but depends on their difference *κ*-*δ*′. The fitted values of Δ*δ* for each detuning are summarised in Table 1. Using this model, we estimate that the maximum delay achievable in chalcogenide gratings, which has a reported Δ*n*≈0.01 [17], can be improved to over 60 pulse widths, with a range of tunable delay of more than 20 pulse widths.

## 6. Intensity enhancement and avoidance of material damage

It is perhaps surprising that the bandgap shift Δ*δ* is independent of *κ* as long as *κ*-*δ*′ is fixed, that is, if one always launch the gap soliton at a fixed detuning from the band-edge. As this shift is proportional to the intensity *I* through Eqs. (1) and (2), this suggests that we have inadvertently launched gap solitons having the same peak intensity regardless of *κ* (or Δ*n*). We found that this is the result of imposing a fixed output pulse width and a fixed pulse frequency (relative to the band-edge). By requiring the output pulse width to be fixed, the propagating gap soliton inside the grating also has a temporal fwhm Δ*τ*
_{GS} of approximately the same value. Δ*τ*
_{GS} can be shown from the analytic solutions for gap soliton [12], which is uniquely defined by two parameters, namely the normalised detuning $\tilde{\delta}$ and the normalised velocity *ν*=*V _{g}*/(

*c*/

*n*), to equal

*$\tilde{\gamma}$*=(1-

*v*

_{2})

^{-1/2}. In addition, for small

*v*(≲0.4), the two gap soliton parameters

*$\tilde{\delta}$*and

*v*can be related through the approximation

Using a fixed Δ*τ*
_{GS} together with known values of *δ*′ and *κ*, one can then solve numerically for *δ* and *v* using Eqs. (7) and (8), and thus determine the peak intensities of the forward- and backward-propagating waves *I*
_{±} comprising the gap soliton given by

where Δ^{4}=(1-*v*)/(1+*v*).

Following this approach, we set Δ*τ*
_{GS} to 680 ps, and calculate the the total peak intensity *I*
_{tot}=*I*
_{+}+*I*
_{-} of the gap soliton inside the grating as function of Δ*n* for the five different detunings. The result indicates that for each detuning *I*
_{tot} does stay almost constant (within ±0.5%) for the range 10^{-4}<Δ*n*<10^{-2}, and appears to approach an asymptotic value at large Δ*n*. The asymptotic value of *I*
_{tot} for each detuning is shown in the last column of Table 1, which shows that *I*
_{tot} increases with decreasing detuning, since a larger bandgap shift is required (*c.f.*
Fig. 2(b)). Therefore, while the required launch power *P*
_{0} decreases with *κ* (c.f. Fig. 3(b)), the field enhancement due to the slow light effect ensures that the gap soliton propagating inside the grating has nearly the same intensity regardless of the value of *κ*. This calculation confirms that the bandgap shift is expected to be weakly dependent on *κ* if *κ*-*δ*′ is unchanged, and thus validates the analytic model.

To cross-check the calculations with the delay simulations shown in Fig. 3(a), we calculate the delay as a function of Δ*n* using the normalised velocity *v* obtained above and Eq. (6) with *L*=100 mm. The result is plotted in Fig. 4(a), which shows the same trend in the delay as a function of Δ*n* and *δ*′ as the simulation results in Fig. 3(a). Discrepancy exists between the two figures because the model does not take into account apodization or any effects of coupling in general.

Table 1 further suggests a general way to improve the delay without causing material damage due to the potentially large intensity enhancement. As shown, the peak intensity within the structure is independent of *κ*, but increases with *κ*-*δ*′. Given the maximum non-destructive intensity of the structure for a particular pulse width, one could first make sure the detuning is sufficiently large to prevent the peak intensity from reaching the damage threshold. Meanwhile, one can safely use materials exhibiting a large maximum index contrast, and thus large *κ*, to improve the delay without worrying about excessive intensity enhancement causing damage. This is so because the required launch power is lowered accordingly to maintain a relatively constant gap soliton intensity regardless of *κ*. The upper limit of delay is only restricted by *κ* (see Fig. 3(b) or Fig. 4(a)), and ultimately by loss [22, 19]. For lossless gratings at a fixed *κ*, the delay is the limited by the damage threshold. Figure 4(b) shows the calculated delay and total peak intensity as a function of detuning from the bandedge (*κ*-*δ*′) for Δ*n*=10^{-3} (*κ*=23.6 cm^{-1}). It can seen from Fig. 4(b) that both *I*
_{tot} and the delay increases with *κ*-*δ*′, indicating a trade-off between delay and high internal intensity that may cause material damage.

While the above procedure avoids damage in the uniform section of the grating, where the gap soliton solution applies, it is conceivable that pulse shaping effects in the apodization section could momentarily enhance the peak intensity further. For example, we observe such further enhancement, by a factor of 2, when the input pulse width becomes 70 ps, although no enhancement is observed for 680 ps pulses. This suggests that one should allow extra room for estimating the maximum intensity due to potential enhancement in the apodization section.

## 7. Conclusion

We investigated dispersionless slow light propagation based on gap soliton propagation in Bragg gratings. Simulations show that the delay increases sub-linearly with the refractive index contrast of the grating. A simple model based on the linear dispersion relation of the Bragg grating is developed to describe the sub-linear relationship between delay and index contrast. For any given grating with a fixed Δ*n*, simultaneous tuning of launch power and detuning leads to tunable delay without any change in the pulse width. Analysis on the gap soliton solution reveals that while the pulse intensity within the grating increases with the parameter *κ*-*δ*′, is independent of *κ*. To avoid material damage, a sufficiently large detuning should be chosen. Meanwhile, the delay can be safely improved by using materials with large maximum index contrast.

This work was produced with the assistance of the Australian Research Council (ARC) under the ARC Centres of Excellence program.

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