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  • Vol. 17, Iss. 7 — Mar. 30, 2009
  • pp: 5083–5088
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All-optical self-switching in optimized phase-shifted fiber Bragg grating

Irina V. Kabakova, Bill Corcoran, Jeremy A. Bolger, C. Martijn de Sterke, and Benjamin J. Eggleton  »View Author Affiliations


Optics Express, Vol. 17, Issue 7, pp. 5083-5088 (2009)
http://dx.doi.org/10.1364/OE.17.005083


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Abstract

We experimentally demonstrate all-optical self-switching based on sub nanosecond pulse propagation through an optimized fiber Bragg grating with a π phase-jump. The jump acts as a cavity leading to an intensity enhancement by factor 19. At pulse peak powers of 1.5 kW we observe 4.2 dB nonlinear change in transmission. Experimental results are consistent with numerical simulations.

© 2009 Optical Society of America

1. Introduction

Fiber Bragg gratings (FBGs) are well studied periodic structures with a one-dimensional bandgap where light is forbidden to propagate. Most FBG applications, such as narrowband filtering [1

1. K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978). [CrossRef]

] and optical fiber sensing [2

2. R. Kashyap, Fiber Bragg Gratings (San Diego, CA: Academic, 1999).

], exploit their linear properties. However, phenomena such as bistability [3

3. H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979). [CrossRef]

], soliton formation [4

4. B. J. Eggleton and C. M. de Sterke, “Nonlinear pulse propagation in Bragg grating,” J. Opt. Soc. Am. B 14, 2980–2993 (1997). [CrossRef]

, 5

5. J. T. Mok, I. C. M. Littler, E. Tsoy, and B. J. Eggleton, “Soliton compression and pulse-train generation by use of microchip Q-switched pulses in Bragg gratings,” Opt. Lett. 30, 2457–2459 (2005). [CrossRef] [PubMed]

] and the optical push broom [6

6. N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Experimental observation of the nonlinear pulse compression in nonuniform Bragg gratings,” Opt. Lett. 22, 1837–1839 (1997). [CrossRef]

] show that they also have the potential to carry out nonlinear functions.

The principle of all-optical switching in a nonlinear periodic structure relies on the intensity-dependence of its transmission. This can result in a change from reflection at low intensities when the pulse wavelength is inside the photonic bandgap and close to one of the band edges, to transmission in the nonlinear, high intensity regime so the pulse is tuned out of the bandgap. Since most dielectric materials have a weak nonlinearity it takes large pulse energies to achieve the required nonlinear phase shift. In the experiment reported by Broderick et al., a uniform 20 cm long FBG was used to produce 20 dB switching with 5.5 ns pulses at a peak power of 1.8 kW, but with the maximum transmission of -15 dB [7

7. N. G. R. Broderick, D. J. Richardson, and M. Ibsen, “Nonlinear switching in 20-cm-long fiber Bragg grating,” Opt. Lett. 25, 536–538 (2000). [CrossRef]

]. In a pump-probe experiment, Larochelle et al. demonstrated a 6% transmission change in a FBG for YAG pump pulses at 4.5 kW peak power [8

8. S. Larochelle, Y. Hibino, V. Mizrahi, and G.I. Stegeman, “All-optical switching of grating transmission using cross-phase modulation in optical fibers,” Electron. Lett. 26, 1459–1460 (1990). [CrossRef]

].

The steeper the transmission feature, the smaller the nonlinear shift required, so the lower the peak power needed for all-optical switching. One way to achieve this is to introduce a defect, e.g. a phase jump, into the FBG. The phase jump acts as a cavity with associated field enhancement at resonance, which appears as a narrow transmission peak inside the bandgap. In addition, for a perfect structure the transmission of a phase-shifted FBG reaches 100 % on resonance. This is a very attractive feature and contrasts with the transmission of uniform gratings. Agrawal et al. numerically studied nonlinear pulse propagation through both uniform and phase-shifted FBGs [9

9. H. Lee and G. P. Agrawal, “Nonlinear switching of optical pulses in fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 508–515 (2003). [CrossRef]

], and showed a significantly lower power threshold for all-optical switching in phase-shifted gratings. Brown et al. demonstrated optical switching in phase-shifted metal-semiconductor-metal Bragg reflectors exploiting the nonlinearity due to free carriers[10

10. A. E. Bieber, T. G. Brown, and R. C. Tiberio, “Optical switching in phase-shifted metal-semiconductor-metal Bragg reflectors,” Opt. Lett. 20, 2216–2218(1995). [CrossRef] [PubMed]

]. In a pump-probe experiment in a FBG with a phase jump, Melloni et al observed a 6 dB change in transmission using 1 kW, 7–16 ns pulses [11

11. A. Melloni, M. Chinello, and M. Martinelli, “All-optical switching in phase-shifted fiber Bragg grating,” IEEE Photon. Technol.Lett. 12, 42–44 (2000). [CrossRef]

]. However, to our knowledge self-switching in phase-shifted FBGs in silica fiber has not yet been experimentally demonstrated.

Here we present the experimental observation of all-optical self-switching based on 0.68 ns Nd:YAG laser pulses propagating through a phase-shifted FBG in a conventional silica fiber. The grating is designed to maximize optical field at resonance giving an intensity enhancement of 19. At a peak power of 1.5 kW we demonstrate a 4.2 dB change in the grating transmission due to 10 pm nonlinear red-shift of the transmission feature, with the maximum transmission close to 80%. We analyze the pulse evolution in both transmission and reflection, and show that the ultrafast Kerr nonlinearity is responsible for the switching. We compare experimental results with numerical simulations and discuss a potential of the FBG design with a phase-jump for all-optical switching applications.

2. Grating design, optimization and fabrication

Figure 1(a) shows a schematic of the FBG. The refractive index n(z) = n 0 + Δn cos (2πz/d + ϕ(z)) varies around the average value n 0 with the index modulation amplitude Δn, period d and phase ϕ(z), which here is piecewise constant, with a nominal π jump in the middle of the grating. The transmission spectrum exhibits a resonance peak (Fig. 1(b)) which at low intensities is at the bandgap center, i.e. at the Bragg wavelength λB = 2n 0 d. The normalized full width at half maximum (FWHM) of resonance can be derived from Coupled Mode Theory [12

12. D. Marcuse, Theory of dielectric optical waveguides (Academic Press, 1991).

] as

ΔλFWHMλB=π(sinhκL2)2,
(1)

where L is the grating length and κπΔn/λB is the coupling coefficient between forward and backward propagating modes with corresponding envelops E + (z), E - (z). The optical field inside the phase-shifted FBG (Fig. 1(a)) grows and decays exponentially, with the maximum at the position of the phase shift z = 0 as follows: E +(z) = E 0cosh(κ(L/2 - ∣z∣)), E -(z) = E 0 sinh(if(κ/2 - ∣z∣)), where E 0 = E +(0) is an incident field amplitude and the grating extends over - L/2 ≤ zL/2. Hence the total field enhancement, given by

E+(0)2+E(0)2E02=cosh(κL),
(2)

increases exponentially with the grating strength (κL). The higher the intensity enhancement, the narrower transmission notch and the lower the pulse power needed for switching. On the other hand, the grating resonance width needs to satisfy Δλ FWHM > 2 pm, since the microchip Q-switched Nd:YAG laser used in our experiments, emits pulses close to transform limited with the spectral width of approximately 2 pm. Thus, for a fixed Δλ FWHM, Eq. (1) shows that the field enhancement increases when the grating gets shorter and stronger. Our phase-shifted grating should therefore be as strong as possible (the photoinduced refractive index modulation amplitude in silica fiber is typically limited by Δn = 10-3 [2

2. R. Kashyap, Fiber Bragg Gratings (San Diego, CA: Academic, 1999).

]), with the length adjusted to give the required spectral width of the defect state.

Fig. 1. (a) Experimental setup. The phase-shifted FBG is shown schematically, together with the field intensity at resonance. (b) Calculated transmission spectrum exhibits a narrow resonance peak at Bragg wavelength λ 0 = λB (dotted line), which shifts to a longer wavelength λ 1 at high intensities (solid line) due to the Kerr effect.

The grating is fabricated, using an interferometric phase-mask scanning technique, in photosensitive, hydrogenated Nufern PS1060 silica fiber with an effective mode area of approximately 30 μm2. It is nominally L = 3 mm long and apodized from both ends to suppress side-lobes. The Gaussian apodization profile and the π phase-jump are implemented using a piezoelectric dither in one arm of the interferometer. The phase-shifted FBG spectrum, as measured by an Optical Spectrum Analyzer, has a minimum transmission of -21 dB inside the bandgap. However, this instrument was not further used in our work since the resolution of 10 pm is insufficient for our purposes.

3. Fiber grating characterization

Figure 1(a) shows the setup for the grating characterization. A Q-switched Nd:YAG laser emits 0.68 ns pulses at a repetition rate of 6.6 kHz. Since the laser operates at the fixed wavelength of 1064.28 nm, we have to find a way to tune the grating resonance close to the laser pulse central wavelength. We do this by mounting one end of the sample with the FBG on a translation stage. By stretching the sample on the stage we can change the spectral position of the grating bandgap [1

1. K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978). [CrossRef]

] in a range of 0.5 nm. We measure the low intensity grating transmission with 134 W peak power YAG pulses using a power meter. The measured spectrum (solid curve in Fig. 2) has a weak asymmetry which is probably due to a small deviation of the phase-jump from the nominal value of π. Our grating is slightly weaker than ideal but still has a steep, Δλ exp FWHM = 17.7 pm wide transmission notch that almost reaches full transmission at resonance. The asymmetry in the grating spectrum and the apodized FBG amplitude profile make it difficult to calculate the coupling coefficient κ accurately. By matching the measured reflection spectrum around the notch to Coupled Mode Theory, we deduce κ = 1350 m-1, corresponding to a refractive index modulation amplitude of Δn = 4.57 × 10-4, and L = 2.7 mm. At the Bragg wavelength of λB = 1064.28 nm, these give a resonance width of Δλ th FWHM = 18.4 pm. We note that the value for L is slightly smaller than the physical length of the grating, accounting for the grating’s apodization. From Eq. (2) we then estimate a field enhancement of approximately 19.

Fig. 2. Measured transmission spectrum at low intensities (solid curve) and at 1.5 kW peak power (dotted curve). Inset shows a close-up of the resonance region; δ 1,2,3 = -2,7,18 pm correspond to detunings from the Bragg wavelength for which detailed results are shown.

To minimize thermal effects [13

13. I. C. M. Littler, T. Grujic, and B. J. Eggleton, “Photothermal effects in fiber Bragg gratings,” Appl. Opt. 45, 4679–4685 (2006). [CrossRef] [PubMed]

] that can affect the grating properties at high intensities, we cover the sample with highly conductive thermopaste. We also ensure equal average power in the high and low-intensity measurements by reducing the repetition rate at high intensities through optical chopping. The dotted line in Fig. 2 shows the high-power transmission spectrum of our FBG, measured with 1.5 kW peak power pulses. Note the 10 pm shift of the transmission notch to longer wavelengths. This shift can be explained if we take the medium to have a Kerr nonlinearity with nonlinear coefficient γ = 4.7 W-1km-1, corresponding to n 2 = 2.4 × 10-20 m2/W. The resonance frequency depends on the average refractive index of the grating, which depends on the pulse peak intensity I via the Kerr law n = n 0 + n 2 I. Based on Coupled Mode Theory [12

12. D. Marcuse, Theory of dielectric optical waveguides (Academic Press, 1991).

] we estimate a nonlinear shift of the transmission notch for a π phase-shifted FBG to be δλ NL = 9.77 pm, close to the experimental result.

4. Switching results

For the switching experiments we use the same setup as before, but additionally monitor the waveform of the transmitted and reflected pulses with photodiode and a sampling oscilloscope, with an overall impulse response of 22 ps. We characterize the nonlinear pulse propagation through the phase-shifted FBG at three different detunings of the input pulse δ1,2,3 = λ pulse - λB = -2,7,18 pm indicated in the inset of Fig. 2 by vertical lines. These are chosen to show the most distinct switching dynamics. We also model the input pulse evolution by numerically solving a set of nonlinear coupled-mode equations which describe nonlinear pulse propagation in gratings. [3

3. H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979). [CrossRef]

, 12

12. D. Marcuse, Theory of dielectric optical waveguides (Academic Press, 1991).

, 14

14. 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]

] Fig. 3 shows the experimental results (square, triangle and diamond markers) compared with the corresponding numerical simulations (solid, dotted and dash dotted lines). The average output versus coupled peak power is shown in Fig. 3(a) as power transfer curves.

Fig. 3. (a) Power transfer curves: the markers (square, triangle and diamond) represent measured data for detunings δ 1,2,3 = -2,7,18 pm from the resonance. The curves (solid, dot, dash dot) are corresponding numerical results. (b) Transmittance at the different de-tunings.

The transmittance, the ratio of the average power transmitted by the grating to the input average power coupled into the sample, is shown in Fig. 3(b). At the detuning δ 3 = 18 pm, the transmittance increases by 4.2 dB from T = 24 % at low powers to T = 62 % at 1.5 kW peak power. We understand this since the transmission spectrum shifts to longer wavelengths (red) around the resonance at high peak powers (dotted curve in Fig. 2). When the grating is strained so that the pulse is tuned slightly to the blue of the notch (left in the inset of Fig. 2), i.e. δ 1 = -2 pm, the transmittance decreases by 2 dB with increasing peak power for the same reason. Finally, the transmittance does not change appreciably for pulses at the detuning δ 2 = 7 pm since the low power transmission is the same as the high power transmission due to the intersection of spectral curves in Fig. 2. Thus, the switching results shown in Fig. 3(b) are consistent with the FBG spectra. We find satisfactory agreement between the experimental results and numerical simulations, except at the highest intensities for δ 3 = 18 pm.

Figure 4 shows (a) experimental, and (b) numerical results for the incident, reflected and and transmitted pulse profiles at 1.5 kW peak power and the detuning δ 3 = 18 pm. At this detuning we observed the largest transmittance change, making it most suited for all-optical switching. A significant waveform change in reflection is observed both in experiments and in simulations: the double-hump shape is due to the intensity threshold of the nonlinear switching. It means that only the central part of the input pulse is intense enough to change the cavity and switch the grating transmission, but not wings. The asymmetry in the reflected waveform, which is stronger on the trailing edge, occurs since the cavity is empty at the start of the pulse, but is filled with energy at the end.

Fig. 4. Pulse profiles in reflection (r) and transmission (t) with the reference input signal (i) at 1.5 kW peak power and detuning δ 3 = 18 pm off the resonance: (a) experimental results, (b) numerical simulations.

5. Discussion and conclusion

In conclusion, we present the experimental demonstration of nonlinear all-optical switching in a FBG with 0.68 ns pulses at a wavelength λ = 1.064 µm. By operating at this wavelength we make use of the convenient availability of high-power laser sources based on Nd:YAG. The FBG has a π phase shift in the middle and is carefully optimized. The phase shift acts as a cavity and leads to a 17.7 pm wide, high-transmission resonant feature (T ≈ 80%), in a strongly reflective background with reflection as large as 21 dB. The cavity enhancement is estimated to be 19. At 1.5 kW peak power, we observe a 10 pm nonlinear shift in the spectrum, leading to a maximum transmission change of 74 %, all in good agreement with simulations. These results represent significant improvements over earlier results [7

7. N. G. R. Broderick, D. J. Richardson, and M. Ibsen, “Nonlinear switching in 20-cm-long fiber Bragg grating,” Opt. Lett. 25, 536–538 (2000). [CrossRef]

, 8

8. S. Larochelle, Y. Hibino, V. Mizrahi, and G.I. Stegeman, “All-optical switching of grating transmission using cross-phase modulation in optical fibers,” Electron. Lett. 26, 1459–1460 (1990). [CrossRef]

] with a uniform FBG.

As in most nonlinear experiments that rely on the Kerr effect, the switching threshold is inversely proportional to the strength of the nonlinearity. In addition, as discussed in Section 2, in this experiment the switching threshold also depends exponentially on the grating strength. Thus the peak power threshold can be reduced by using guided-wave structures, made, for example, from a soft glass such as chalcogenide. This glass exhibits third-order Kerr nonlinearity up to 1000 times that of silica [15

15. M. Asobe, “Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching,” Opt. Laser Technol. 3, 142–148 (1997).

], and has the additional advantages of having a high refractive index (n 0 ≈ 2.4), so the light can be strongly confined, and of being more strongly photosensitive than silica. Recent achievements in waveguides fabrication [16

16. M. Shokooh-Saremi, V. G. Ta’eed, N. J. Baker, I. C. M. Littler, D. J. Moss, and B. J. Eggleton, “High-performance Bragg gratings in chalcogenide rib waveguides written with a modified Sagnac interferometer,” J. Opt. Soc. Am. B 23, 1323–1331 (2006). [CrossRef]

, 17

17. H. C. Hong, D. -I. Yeom, E. C. Mägi, L. B. Fu, B. T. Kuhlmey, C. Martijn de Sterke, and B. J. Eggleton,“Nonlinear switching using long-period gratings in As2S3 chalcogenide fiber,” J. Opt. Soc. Am. B 25, 1393–1401 (2008). [CrossRef]

] in chalcogenides make these glasses highly promising for all-optical grating-based switching technology.

Acknowledgments

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

References and links

1.

K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978). [CrossRef]

2.

R. Kashyap, Fiber Bragg Gratings (San Diego, CA: Academic, 1999).

3.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979). [CrossRef]

4.

B. J. Eggleton and C. M. de Sterke, “Nonlinear pulse propagation in Bragg grating,” J. Opt. Soc. Am. B 14, 2980–2993 (1997). [CrossRef]

5.

J. T. Mok, I. C. M. Littler, E. Tsoy, and B. J. Eggleton, “Soliton compression and pulse-train generation by use of microchip Q-switched pulses in Bragg gratings,” Opt. Lett. 30, 2457–2459 (2005). [CrossRef] [PubMed]

6.

N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Experimental observation of the nonlinear pulse compression in nonuniform Bragg gratings,” Opt. Lett. 22, 1837–1839 (1997). [CrossRef]

7.

N. G. R. Broderick, D. J. Richardson, and M. Ibsen, “Nonlinear switching in 20-cm-long fiber Bragg grating,” Opt. Lett. 25, 536–538 (2000). [CrossRef]

8.

S. Larochelle, Y. Hibino, V. Mizrahi, and G.I. Stegeman, “All-optical switching of grating transmission using cross-phase modulation in optical fibers,” Electron. Lett. 26, 1459–1460 (1990). [CrossRef]

9.

H. Lee and G. P. Agrawal, “Nonlinear switching of optical pulses in fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 508–515 (2003). [CrossRef]

10.

A. E. Bieber, T. G. Brown, and R. C. Tiberio, “Optical switching in phase-shifted metal-semiconductor-metal Bragg reflectors,” Opt. Lett. 20, 2216–2218(1995). [CrossRef] [PubMed]

11.

A. Melloni, M. Chinello, and M. Martinelli, “All-optical switching in phase-shifted fiber Bragg grating,” IEEE Photon. Technol.Lett. 12, 42–44 (2000). [CrossRef]

12.

D. Marcuse, Theory of dielectric optical waveguides (Academic Press, 1991).

13.

I. C. M. Littler, T. Grujic, and B. J. Eggleton, “Photothermal effects in fiber Bragg gratings,” Appl. Opt. 45, 4679–4685 (2006). [CrossRef] [PubMed]

14.

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]

15.

M. Asobe, “Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching,” Opt. Laser Technol. 3, 142–148 (1997).

16.

M. Shokooh-Saremi, V. G. Ta’eed, N. J. Baker, I. C. M. Littler, D. J. Moss, and B. J. Eggleton, “High-performance Bragg gratings in chalcogenide rib waveguides written with a modified Sagnac interferometer,” J. Opt. Soc. Am. B 23, 1323–1331 (2006). [CrossRef]

17.

H. C. Hong, D. -I. Yeom, E. C. Mägi, L. B. Fu, B. T. Kuhlmey, C. Martijn de Sterke, and B. J. Eggleton,“Nonlinear switching using long-period gratings in As2S3 chalcogenide fiber,” J. Opt. Soc. Am. B 25, 1393–1401 (2008). [CrossRef]

OCIS Codes
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(060.3735) Fiber optics and optical communications : Fiber Bragg gratings

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: January 29, 2009
Revised Manuscript: March 3, 2009
Manuscript Accepted: March 3, 2009
Published: March 16, 2009

Citation
Irina V. Kabakova, Bill Corcoran, Jeremy A. Bolger, Martijn de Sterke, and Benjamin J. Eggleton, "All-optical self-switching in optimized phase-shifted fiber Bragg grating," Opt. Express 17, 5083-5088 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-5083


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References

  1. K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, "Photosensitivity in optical fiber waveguides: application to reflection filter fabrication," Appl. Phys. Lett. 32, 647-649 (1978). [CrossRef]
  2. R. Kashyap, Fiber Bragg Gratings (San Diego, CA: Academic, 1999).
  3. H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in nonlinear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979). [CrossRef]
  4. B. J. Eggleton and C. M. de Sterke,"Nonlinear pulse propagation in Bragg grating," J. Opt. Soc. Am. B 14, 2980-2993 (1997). [CrossRef]
  5. J. T. Mok, I. C. M. Littler, E. Tsoy, and B. J. Eggleton, "Soliton compression and pulse-train generation by use of microchip Q-switched pulses in Bragg gratings," Opt. Lett. 30, 2457-2459 (2005). [CrossRef] [PubMed]
  6. N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, "Experimental observation of the nonlinear pulse compression in nonuniform Bragg gratings," Opt. Lett. 22, 1837-1839 (1997). [CrossRef]
  7. N. G. R. Broderick, D. J. Richardson, and M. Ibsen, "Nonlinear switching in 20-cm-long fiber Bragg grating," Opt. Lett. 25, 536-538 (2000). [CrossRef]
  8. S. Larochelle, Y. Hibino, V. Mizrahi, and G.I. Stegeman, "All-optical switching of grating transmission using cross-phase modulation in optical fibers," Electron. Lett. 26, 1459-1460 (1990). [CrossRef]
  9. H. Lee, and G. P. Agrawal, "Nonlinear switching of optical pulses in fiber Bragg gratings," IEEE J. Quantum Electron. 39, 508-515 (2003). [CrossRef]
  10. A. E. Bieber, T. G. Brown, and R. C. Tiberio, "Optical switching in phase-shifted metal-semiconductor-metal Bragg reflectors," Opt. Lett. 20, 2216-2218(1995). [CrossRef] [PubMed]
  11. A. Melloni, M. Chinello, and M. Martinelli, "All-optical switching in phase-shifted fiber Bragg grating," IEEE Photon. Technol. Lett. 12, 42-44 (2000). [CrossRef]
  12. D. Marcuse, Theory of dielectric optical waveguides (Academic Press, 1991).
  13. I. C. M. Littler, T. Grujic, B. J. Eggleton, "Photothermal effects in fiber Bragg gratings," Appl. Opt. 45, 4679-4685 (2006). [CrossRef] [PubMed]
  14. 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]
  15. M. Asobe, "Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching," Opt. Laser Technol. 3, 142-148 (1997).
  16. M. Shokooh-Saremi, V. G. Ta’eed, N. J. Baker, I. C. M. Littler, D. J. Moss, and B. J. Eggleton, "Highperformance Bragg gratings in chalcogenide rib waveguides written with a modified Sagnac interferometer," J. Opt. Soc. Am. B 23, 1323-1331 (2006). [CrossRef]
  17. H. C. Hong, D. -I. Yeom, E. C. M¨agi, L. B. Fu, B. T. Kuhlmey, C. Martijn de Sterke, and B. J. Eggleton,"Nonlinear switching using long-period gratings in As2S3 chalcogenide fiber," J. Opt. Soc. Am. B 25, 1393-1401 (2008). [CrossRef]

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