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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 25 — Dec. 11, 2006
  • pp: 11987–11996
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Delay-tunable gap-soliton-based slow-light system

Joe T. Mok, C. Martijn de Sterke, and Benjamin J. Eggleton  »View Author Affiliations


Optics Express, Vol. 14, Issue 25, pp. 11987-11996 (2006)
http://dx.doi.org/10.1364/OE.14.011987


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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 Δn=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 Δn 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

1. Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chenb, “80-micron Interaction Length Silicon Photonic Crystal Waveguide Modulator,” Appl. Phys. Lett. 87, 221,105 (2005). [CrossRef]

]. Enhanced nonlinearity that arises from the slow light effect has been observed in a photonic crystal waveguide [2

2. R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained Silicon as a New Electro-Optic Material,” Nature 441, 199–202 (2006). [CrossRef] [PubMed]

]. 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

3. R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Application of Slow Light in Telecommunications,” Opt. Photon. News 17, 18–23 (2006). [CrossRef]

]. 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

4. R. Boyd and D. Gauthier, “Slow and Fast Light,” in Prog. in Optics, E. Wolf, ed., vol. 43, pp. 497–530 (Elsevier, Amsterdam, 2002).

, 5

5. J. B. Khurgin, “Optical Buffers Based on Slow Light in Electromagnetically Induced Transparent Media and Coupled Resonator Structures: Comparative Analysis,” J. Opt. Soc. Am. B 22, 1062–1074 (2005). [CrossRef]

]. 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

6. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical Delay Lines Based on Optical Filters,” J. Quantum Electron. 37, 525–530 (2001). [CrossRef]

]. 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

7. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg Grating Solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996). [CrossRef] [PubMed]

, 8

8. D. Taverner, N. G. R. Broderick, D. J. Richardson, R. I. Laming, and M. Ibsen, “Nonlinear Self-Switching and Multiple Gap-Soliton Formation in a Fiber Bragg Grating,” Opt. Lett. 23, 328–330 (1998). [CrossRef]

, 9

9. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg Solitons in the Nonlinear Schrodinger Limit: Experiment and Theory,” J. Opt. Soc. Am. B 16, 587–599 (1999). [CrossRef]

, 10

10. N. G. R. Broderick, D. J. Richardson, and M. Ibsen, “Nonlinear Switching in a 20-Cm-Long Fiber Bragg Grating,” Opt. Lett. 25, 536–538 (2000). [CrossRef]

], leading to the generation of slow gap solitons [11

11. D. N. Christodoulides and R. I. Joseph, “Slow Bragg Solitons in Nonlinear Periodic Structures,” Phys. Rev. Lett. 62, 1746–1749 (1989). [CrossRef] [PubMed]

, 12

12. A. Aceves and S. Wabnitz, “Self-Induced Transparency Solitons in Nonlinear Refractive Periodic Media,” Phys. Lett. A 141, 37–42 (1989). [CrossRef]

, 13

13. C. M. de Sterke and J. E. Sipe, “Gap Solitons,” in Prog. in Optics, E. Wolf, ed., vol. 33, pp. 203–259 (Elsevier, Amsterdam, 1994).

, 19

19. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless Slow Light Using Gap Solitons,” Nature Phys. 2, 775–780 (2006). [CrossRef]

]. 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 Δn, 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 Δn 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

λB=2nΛ,
(1)

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

n=n0+n2I,
(2)

where n 0 is the refractive index at low intensities and n 2, taken to be positive, is the nonlinear coefficient, thus shifting λB 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

13. C. M. de Sterke and J. E. Sipe, “Gap Solitons,” in Prog. in Optics, E. Wolf, ed., vol. 33, pp. 203–259 (Elsevier, Amsterdam, 1994).

]. It is the possibility of the very low gap soliton velocities that gives rise to potentially large temporal delays.

Fig. 1. Illustration of (a) nonlinear bandgap shift, and the launching and evolution of (b) low- and (c) high-intensity pulses within the bandgap of a Bragg grating spanning 0<z<0.1 m. Contour lines representing the intensity are equally spaced on a logarithmic scale. At a sufficiently high launch intensity, the pulse propagates in the Bragg grating, as shown in (c), at a velocity less than c/n and emerges with a pulse delay.

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

14. M. De Sario, C. Conti, and G. Assanto, “Optically Controlled Delay Lines by Pulse Self-Trapping in Parametric Waveguides with Distributed Feedback,” IEEE J. Quant. Elect. 36, 931–943 (2000). [CrossRef]

] and the detuning, which is proportional to the difference between the pulse wavelength and λB. The launch power can be adjusted using a variable optical attenuator, while the detuning can be adjusted by either varying Λ or n [15

15. M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulos, “Slow-Light, Band-Edge Waveguides for Tunable Time Delays,” Opt. Express 13, 7145–7159 (2005). [CrossRef] [PubMed]

], 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

16. V. G. Ta’eed, M. Shokooh-Saremi, L. Fu, D. J. Moss, M. Rochette, I. C. M. Littler, B. J. Eggleton, Y. Ruan, and B. Luther-Davies, “Integrated All-Optical Pulse Regenerator in Chalcogenide Waveguides,” Opt. Lett. 30, 2900–2902 (2005). [CrossRef] [PubMed]

] which has n 2 of the order of 10-13 cm2/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 Δn has also been reported [17

17. M. Shokooh-Saremi, V. G. Taeed, N. J. Baker, I. C.M. Littler, D. J. Moss, B. J. Eggleton, Y. Ruan, and B. Luther- Davies, “High-Performance Bragg Gratings in Chalcogenide Rib Waveguides Written with a Modified Sagnac Interferometer,” J. Opt. Soc. Am. B 23, 1323–1331 (2006). [CrossRef]

]. 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

iA+z+incA+t+κA+γ(A+2+2A2A+)=0
iAz+incAt+κA++γ(A+2+2A2A)=0
(3)

for the electric field using a collocation method [18

18. C. M. de Sterke, K. R. Jackson, and B. D. Robert, “Nonlinear Coupled Mode Equations on a Finite Interval: A Numerical Procedure,” J. Opt. Soc. Am. B 8, 403–412 (1991). [CrossRef]

]. A ±=|A ±(z,t)|eiδ′(c/n)t is the complex forward and backward propagating envelops with δ′=(ω′-ωB)/(c/n), where ω′ is the pulse frequency, ωB is the Bragg frequency, located in the center of the photonic bandgap, γ is the nonlinear coefficient and κ=ηπΔn/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

19. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless Slow Light Using Gap Solitons,” Nature Phys. 2, 775–780 (2006). [CrossRef]

], and take gratings designed to work near wavelengths λ≈1 µm, in the core of a silica fiber with typical parameters: γ=6.4 /(W km) and Δn≤10-3, as appropriate for silica glass. The results can however be generalised to other materials and geometries. The grating is apodized, with Δn varying as [1-cos(πz/la)]/2 for 0<z<la and [1+cos(π[z-(L-la)]/la)]/2 for L-la<z<L, with la=15 mm, and Δn 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 Δn, (2) grating length L, (3) launch power P, (4) pulse detuning δ′, and (5) pulse width Δτ. We note that in terms of the detuning δ=(ω-ωB)/(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=P0 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.

Fig. 2. (a) Definitions of κ, δ′ and δp. Solid line represents the Bragg grating transmission spectrum as a function of δ. Dotted line represents the pulse spectrum. (b) Typical effects of different launch peak powers P on the output pulse width and delay. P0 is the power that results in the same input and output pulse widths Δτ, and τd is the propagation delay under this condition. Output pulses (solid) are drawn to have the same pulse energy as the input (dashed).

The delay is given by τd-τ d0, where τd is the pulse propagation delay when the device is on, and 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

9. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg Solitons in the Nonlinear Schrodinger Limit: Experiment and Theory,” J. Opt. Soc. Am. B 16, 587–599 (1999). [CrossRef]

]. For these reasons, we limit the range of δ′ such that 1.2δp<κ-δ′<3δp, where δp is the 3 dB spectral width of the pulse.

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

19. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless Slow Light Using Gap Solitons,” Nature Phys. 2, 775–780 (2006). [CrossRef]

]. 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 Δτ=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 Δn 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-1.2δp, κ-1.5δp, κ-2.0δp, κ-2.5δp and κ-3.0δp where δp=14.09 m-1 is the spectral fwhm of the 680 ps pulse with a constant temporal phase.

Figure 3 shows the pulse delay τd-τ d0 and the required launch peak power P 0 as a function of Δn for the different detunings. The right axis of Fig. 3(a) shows the scale corresponding to the fractional delay, F=(τd-τ d0)/Δτ, corresponding to the grating’s delay capacity. Note from Fig. 3 that as Δn 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 Δn 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 δn 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/cm2). 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 Δn, 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 Δn=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 τd. It is found that the output pulse delay can be sensitive to launch power. Generally, obtaining the accuracy in τd shown by the error bars requires 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.

Fig. 3. (a) Delay and (b) launch power as a function of Δn in a 100 mm Bragg grating for 680 ps pulses for five detunings (1 kW corresponds to 3.65 GW/cm2). Data points (with error bars) in (a) show the simulated delay, whereas each dashed line is an analytical fit using Eqs. (5) and (6), with the fit parameters summarised in Table 1. Inset(top): Transmission vs delay at Δn=10-3. Inset(bottom): Incident (dashed) and transmitted (solid) pulses at Δn=10-3 with the detuning and launch peak power indicated by the grey box: (i) δ′=κ-1.2δp, P 0=0.393 kW (ii) δ′=κ-1.5δp, P0=0.463 kW (iii) δ′=κ-2.0δp, P 0=0.577 kW (iv) δ′=κ-2.5δp, P 0=0.691 kW and (v) δ′=κ-3.0δp, P 0=0.805 kW.

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

q2=δ2κ2,
(4)

where q=k-kB, where k is the wave number and kB=π/d is the wave number at the Bragg condition. The group velocity Vg, ignoring end effects, can then be found from the dispersion relation, and can be shown to be [21

21. P. S. J. Russell, “Bloch Wave Analysis of Dispersion and Pulse Propagation in Pure Distributed Feedback Structures,” J. Mod. Opt. 38, 1599–1619 (1991). [CrossRef]

]

Vg=cn[1(κδ)2]12.
(5)

The delay τd-τ d0 of a pulse is thus

τdτd0=L(1Vgnc).
(6)

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′+Δδ with Δδ>κ-δ′ the bandgap shift induced by the positive Kerr nonlinearity. The delay τd is then given by the linear dispersion relation of the Bragg grating at the apparent detuning, with δ=δ′+Δδ in Eq. (5) where Δδ 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 Δδ. Fig. 3(a) indicates that Δδ does not depend on the individual values of κ (i.e. Δn) or δ′, but depends on their difference κ-δ′. The fitted values of Δδ 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 Δn≈0.01 [17

17. M. Shokooh-Saremi, V. G. Taeed, N. J. Baker, I. C.M. Littler, D. J. Moss, B. J. Eggleton, Y. Ruan, and B. Luther- Davies, “High-Performance Bragg Gratings in Chalcogenide Rib Waveguides Written with a Modified Sagnac Interferometer,” J. Opt. Soc. Am. B 23, 1323–1331 (2006). [CrossRef]

], 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

δγ~κcosδ~.
(8)

Using a fixed Δτ 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

I±=(λ2πn2)κΔ2sin2δ~(1v23v2)(21+cosδ~),
(9)

where Δ4=(1-v)/(1+v).

Following this approach, we set Δτ 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 Δn for the five different detunings. The result indicates that for each detuning I tot does stay almost constant (within ±0.5%) for the range 10-4n<10-2, and appears to approach an asymptotic value at large Δn. 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 Δn 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 Δn 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.

Fig. 4. Results obtained from the gap soliton solutions for Δτ GS=680 ps in a L=100 mm Bragg grating.(a) Calculated delay vs Δn for different detunings, and (b) calculated delay (×) and total peak intensity (˚) vs detuning at Δn=10-3.

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

22. R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, “Slow-Light Optical Buffers: Capabilities and Fundamental Limitations,” J. Lightwave Technol. 23, 4046–4066 (2005). [CrossRef]

, 19

19. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless Slow Light Using Gap Solitons,” Nature Phys. 2, 775–780 (2006). [CrossRef]

]. 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 Δn=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 Δn, 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.

References and links

1.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chenb, “80-micron Interaction Length Silicon Photonic Crystal Waveguide Modulator,” Appl. Phys. Lett. 87, 221,105 (2005). [CrossRef]

2.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained Silicon as a New Electro-Optic Material,” Nature 441, 199–202 (2006). [CrossRef] [PubMed]

3.

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Application of Slow Light in Telecommunications,” Opt. Photon. News 17, 18–23 (2006). [CrossRef]

4.

R. Boyd and D. Gauthier, “Slow and Fast Light,” in Prog. in Optics, E. Wolf, ed., vol. 43, pp. 497–530 (Elsevier, Amsterdam, 2002).

5.

J. B. Khurgin, “Optical Buffers Based on Slow Light in Electromagnetically Induced Transparent Media and Coupled Resonator Structures: Comparative Analysis,” J. Opt. Soc. Am. B 22, 1062–1074 (2005). [CrossRef]

6.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical Delay Lines Based on Optical Filters,” J. Quantum Electron. 37, 525–530 (2001). [CrossRef]

7.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg Grating Solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996). [CrossRef] [PubMed]

8.

D. Taverner, N. G. R. Broderick, D. J. Richardson, R. I. Laming, and M. Ibsen, “Nonlinear Self-Switching and Multiple Gap-Soliton Formation in a Fiber Bragg Grating,” Opt. Lett. 23, 328–330 (1998). [CrossRef]

9.

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg Solitons in the Nonlinear Schrodinger Limit: Experiment and Theory,” J. Opt. Soc. Am. B 16, 587–599 (1999). [CrossRef]

10.

N. G. R. Broderick, D. J. Richardson, and M. Ibsen, “Nonlinear Switching in a 20-Cm-Long Fiber Bragg Grating,” Opt. Lett. 25, 536–538 (2000). [CrossRef]

11.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg Solitons in Nonlinear Periodic Structures,” Phys. Rev. Lett. 62, 1746–1749 (1989). [CrossRef] [PubMed]

12.

A. Aceves and S. Wabnitz, “Self-Induced Transparency Solitons in Nonlinear Refractive Periodic Media,” Phys. Lett. A 141, 37–42 (1989). [CrossRef]

13.

C. M. de Sterke and J. E. Sipe, “Gap Solitons,” in Prog. in Optics, E. Wolf, ed., vol. 33, pp. 203–259 (Elsevier, Amsterdam, 1994).

14.

M. De Sario, C. Conti, and G. Assanto, “Optically Controlled Delay Lines by Pulse Self-Trapping in Parametric Waveguides with Distributed Feedback,” IEEE J. Quant. Elect. 36, 931–943 (2000). [CrossRef]

15.

M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulos, “Slow-Light, Band-Edge Waveguides for Tunable Time Delays,” Opt. Express 13, 7145–7159 (2005). [CrossRef] [PubMed]

16.

V. G. Ta’eed, M. Shokooh-Saremi, L. Fu, D. J. Moss, M. Rochette, I. C. M. Littler, B. J. Eggleton, Y. Ruan, and B. Luther-Davies, “Integrated All-Optical Pulse Regenerator in Chalcogenide Waveguides,” Opt. Lett. 30, 2900–2902 (2005). [CrossRef] [PubMed]

17.

M. Shokooh-Saremi, V. G. Taeed, N. J. Baker, I. C.M. Littler, D. J. Moss, B. J. Eggleton, Y. Ruan, and B. Luther- Davies, “High-Performance Bragg Gratings in Chalcogenide Rib Waveguides Written with a Modified Sagnac Interferometer,” J. Opt. Soc. Am. B 23, 1323–1331 (2006). [CrossRef]

18.

C. M. de Sterke, K. R. Jackson, and B. D. Robert, “Nonlinear Coupled Mode Equations on a Finite Interval: A Numerical Procedure,” J. Opt. Soc. Am. B 8, 403–412 (1991). [CrossRef]

19.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless Slow Light Using Gap Solitons,” Nature Phys. 2, 775–780 (2006). [CrossRef]

20.

R. E. Slusher, G. Lenz, J. Hodelin, J. Sanghera, L. B. Shawand, and I. D. Aggarwal, “Large Raman Gain and Nonlinear Phase Shifts in High-Purity As2Se3 Chalcogenide Fibers,” J. Opt. Soc. Am. B 21, 1146–1155 (2004). [CrossRef]

21.

P. S. J. Russell, “Bloch Wave Analysis of Dispersion and Pulse Propagation in Pure Distributed Feedback Structures,” J. Mod. Opt. 38, 1599–1619 (1991). [CrossRef]

22.

R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, “Slow-Light Optical Buffers: Capabilities and Fundamental Limitations,” J. Lightwave Technol. 23, 4046–4066 (2005). [CrossRef]

OCIS Codes
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: September 26, 2006
Revised Manuscript: November 27, 2006
Manuscript Accepted: November 29, 2006
Published: December 11, 2006

Citation
Joe T. Mok, C. Martijn de Sterke, and Benjamin J. Eggleton, "Delay-tunable gap-soliton-based slow-light system," Opt. Express 14, 11987-11996 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-11987


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References

  1. Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chenb, "80-micron Interaction Length Silicon Photonic Crystal Waveguide Modulator," Appl. Phys. Lett. 87, 221,105 (2005). [CrossRef]
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  3. R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, "Application of Slow Light in Telecommunications," Opt. Photon. News 17, 18-23 (2006). [CrossRef]
  4. R. Boyd and D. Gauthier, "Slow and Fast Light," in Prog. in Optics, E.Wolf, ed., vol. 43, pp. 497-530 (Elsevier, Amsterdam, 2002).
  5. J. B. Khurgin, "Optical Buffers Based on Slow Light in Electromagnetically Induced Transparent Media and Coupled Resonator Structures: Comparative Analysis," J. Opt. Soc. Am. B 22, 1062-1074 (2005). [CrossRef]
  6. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical Delay Lines Based on Optical Filters," J. Quantum Electron. 37, 525-530 (2001). [CrossRef]
  7. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996). [CrossRef] [PubMed]
  8. D. Taverner, N. G. R. Broderick, D. J. Richardson, R. I. Laming, and M. Ibsen, "Nonlinear Self-Switching and Multiple Gap-Soliton Formation in a Fiber Bragg Grating," Opt. Lett. 23, 328-330 (1998). [CrossRef]
  9. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, "Bragg Solitons in the Nonlinear Schrodinger Limit: Experiment and Theory," J. Opt. Soc. Am. B 16, 587-599 (1999). [CrossRef]
  10. N. G. R. Broderick, D. J. Richardson, and M. Ibsen, "Nonlinear Switching in a 20-Cm-Long Fiber Bragg Grating," Opt. Lett. 25, 536-538 (2000). [CrossRef]
  11. D. N. Christodoulides and R. I. Joseph, "Slow Bragg Solitons in Nonlinear Periodic Structures," Phys. Rev. Lett. 62, 1746-1749 (1989). [CrossRef] [PubMed]
  12. A. Aceves and S. Wabnitz, "Self-Induced Transparency Solitons in Nonlinear Refractive Periodic Media," Phys. Lett. A 141, 37-42 (1989). [CrossRef]
  13. C. M. de Sterke and J. E. Sipe, "Gap Solitons," in Prog. in Optics, E. Wolf, ed., vol. 33, pp. 203-259 (Elsevier, Amsterdam, 1994).
  14. M. De Sario, C. Conti, and G. Assanto, "Optically Controlled Delay Lines by Pulse Self-Trapping in Parametric Waveguides with Distributed Feedback," IEEE J. Quant. Elect. 36, 931-943 (2000). [CrossRef]
  15. M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulos, "Slow-Light, Band-Edge Waveguides for Tunable Time Delays," Opt. Express 13, 7145-7159 (2005). [CrossRef] [PubMed]
  16. V. G. Ta’eed, M. Shokooh-Saremi, L. Fu, D. J. Moss, M. Rochette, I. C. M. Littler, B. J. Eggleton, Y. Ruan, and B. Luther-Davies, "Integrated All-Optical Pulse Regenerator in Chalcogenide Waveguides," Opt. Lett. 30, 2900-2902 (2005). [CrossRef] [PubMed]
  17. <jrn>. M. Shokooh-Saremi, V. G. Taeed, N. J. Baker, I. C.M. Littler, D. J. Moss, B. J. Eggleton, Y. Ruan, and B. Luther- Davies, "High-Performance Bragg Gratings in Chalcogenide Rib Waveguides Written with a Modified Sagnac Interferometer," J. Opt. Soc. Am. B 23, 1323-1331 (2006).</jrn> [CrossRef]
  18. C. M. de Sterke, K. R. Jackson, and B. D. Robert, "Nonlinear Coupled Mode Equations on a Finite Interval: A Numerical Procedure," J. Opt. Soc. Am. B 8, 403-412 (1991). [CrossRef]
  19. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless Slow Light Using Gap Solitons," Nature Phys. 2, 775-780 (2006). [CrossRef]
  20. R. E. Slusher, G. Lenz, J. Hodelin, J. Sanghera, L. B. Shawand, and I. D. Aggarwal, "Large Raman Gain and Nonlinear Phase Shifts in High-Purity As2Se3 Chalcogenide Fibers," J. Opt. Soc. Am. B 21, 1146-1155 (2004). [CrossRef]
  21. P. S. J. Russell, "Bloch Wave Analysis of Dispersion and Pulse Propagation in Pure Distributed Feedback Structures," J. Mod. Opt. 38, 1599-1619 (1991). [CrossRef]
  22. R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, "Slow-Light Optical Buffers: Capabilities and Fundamental Limitations," J. Lightwave Technol. 23, 4046-4066 (2005). [CrossRef]

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