In this paper, slow and fast light in optical fiber through tilted fiber Bragg grating (TFBG) are reported. The experimental results show the capability of TFBGs to enable group velocity control of an optical pulse in optical fiber, due to the anomalous dispersion features induced by the coupling between the propagating core mode and each counter-propagating coupling cladding mode. In particular, superluminal propagation of a pulse train has been observed at optical communication wavelengths with time advancements in the picoseconds time scale in 1cm long TFBG and group velocity as large as about two times the speed of light in optical fiber (≈1.3∙c0). Very good agreement has been obtained comparing the measured group delay of the TFBG with the one retrieved from the amplitude response through Hilbert transform. Finally, tunable slow and fast light has also been reported, demonstrating the possibility to control the group velocity at single wavelength through both fluidic and thermal actuation.
©2009 Optical Society of America
In recent years, a renewed interest has been shown by the scientific community towards methods and technologies enabling the active control of the speed of a light signal, in virtue of their enormous potential applicability in fiber optic communication networks, optical processing and quantum computing . Many different mechanisms and systems have been proposed which can slow and speed up light, such as electromagnetically induced transparency [2,3], coherent population oscillations  and photonic crystal structures . In addition, complementary studies have demonstrated the possibility of overcoming the delay-bandwidth limit, affecting these systems, by means of ultrafast tuning of the optical structures . Also double resonance systems have been used to obtain very large pulse bandwidths, large group delays and small broadening .
Previously, Longhi et al., envisaging the great advantages to be obtained through light signals speed control within an optical fiber, in terms of direct integration in communication optical systems, demonstrated superluminal optical pulse propagation through Fiber Bragg Gratings (FBGs) . More recently, Thevenaz et al., demonstrated both time advancement and delay in optical fibers through stimulated Brillouin scattering [9,10]. The system, while demonstrating its effectiveness in producing large tunable delay and advancement, still suffers from the intrinsic tradeoff between long path and high pump power . Eggleton et al. also obtained considerable delays by launching powerful optical pulses at the edge of the rejection band of the FBG in transmission and using the Kerr effect to modify the delay via a shift of the FBG . The several slow and fast light structures proposed in the last decade, while present remarkable performances in terms of delay-bandwidth product and efficient tuning mechanisms (such as that based on pumping [2–4,9–11]), are still far from real applications especially in relation with the complexity of the overall system. The challenge ahead is to achieve a large time delay/advancement tunability over a large bandwidth and without dispersion by means of a low-cost and short length device.
In this work, the group delay of a 5° Tilted Fiber Bragg Grating (TFBG) has been experimentally measured in order to verify the possibility of exploiting these types of gratings as building blocks to design low-cost, all-in-fiber and multi-wavelength light signals speed controlling systems operating at room temperature. The speed-up and slow-down capability of TFBGs has never been investigated so far.
A TFBG consists of a refractive index modulation “tilted” with respect to the fiber axis that enables the coupling between the forward propagating core mode and circularly and non-circularly symmetric counter-propagating cladding modes . This multiple coupling confers unique and interesting spectral features in the TFBG transmission spectrum which presents numerous absorption resonances. The wavelength and amplitude of each resonance are governed respectively by the phase matching condition, involving the propagation constants of the coupling modes, the grating pitch and the tilt angle and by the overlap coefficient between the fields of the coupling modes. The main application of these devices is in the sensing field as multi-parameter sensors involving measurements of refractive index, strain and temperature .
2.2 Tilted fiber Bragg grating fabrication
The investigated TFBG was manufactured into hydrogen loaded Corning® single-mode fiber by means of frequency-doubled Argon ion laser emitting at 244 nm. A 1060-nm uniform phase mask was mounted on a rotating stage in order to apply a tilt in the plane perpendicular to the incident laser beam. The mean optical power was kept constant—equal to 60mW—and the grating was inscribed with a single sweep of the UV laser along the phase mask with a speed of 10 µm/s. After the writing process, the grating was annealed at 100°C for 24 h. Following this procedure, a 1-cm-long grating with tilt angle of 5° was fabricated.
2.3 Experimental setup
In order to demonstrate fast and slow light in optical fiber, the transmittance and the group delay of the 5° TFBG in the wavelength range 1534.2-1540.2nm were simultaneously retrieved by means of direct time domain measurements. For a narrow-band optical pulse, the time needed to cross the grating structure can be assumed equal to the grating group delay [8,15]. The experimental setup used to perform the measurements is schematically shown in Fig. 1 . It employs as source a tunable laser (wavelength range 1520-1620nm) modulated by a Mach-Zehnder Modulator (33GHz – 3dB electro-optical bandwidth). The modulator is driven by a RF pulse train, supplied by a pulse pattern generator and characterized by 300MHz repetition rate and 1.7ns pulse duration. The modulated signal is amplified through an Erbium Doped Fiber Amplifier (EDFA) and then it is split by an optical coupler in order to provide a time delay reference arm to compare with the optical signal going through the TFBG. The obtained signals, after the opto-electric conversion provided by fast photodiodes (25GHz bandwidth), were stored in an oscilloscope (20GSample/sec) and the time difference between the reference arm and the TFBG output has been evaluated for each wavelength in the investigated spectral range by comparing the respective pulses traces. The arrival time of each pulse has been evaluated as the mean between the time corresponding to the 50% amplitude level on the rise front and the fall front respectively, in order not to take into account the effect of pulse broadening. It is noteworthy that the RF frequency was chosen low enough to yield a modulated signal bandwidth narrower than the bandwidth of the transmittance resonances [8,15], in spite of the ideally square wave shape of the RF waveform. In particular, the Full Width Half Maximum (FWHM) bandwidths of each transmission resonance, range from 6GHz (for the peak around 1536.9nm) up to 20GHz (for the peak around 1534.7nm), while the bandwidth of the modulated signal is 3GHz, if up to the fifth harmonic (95% of the power) is taken into account. It should also be noted that the reference arm guarantees that the components of the optical chain before the optical coupler does not affect the measured time delay. In addition, in order to make negligible the effect of the dispersion of the optical fibers (nominal dispersion ≈17ps/(nm∙Km)) in the time delay measurements, both arms have been chosen with approximately the same length (about ten centimeters difference). Furthermore, the measured time delay has been referenced to the arrival time of the pulse out of TFBG bandwidth (at 1540.2nm) thus representing the zero-delay. In this experimental configuration, the investigated spectral range has not been extended to deeper resonances of the TFBG spectrum because of the limitations imposed by the minimum detectable signal by the oscilloscope and by the saturation power of the employed photodiodes.
3. Experimental Results and Discussion
3.1 Fast and slow light through tilted fiber Bragg grating
In Figs. 2(a)-2(b), the TFBG transmittance and group delay (solid circles) are reported in the investigated wavelength range 1534.2-1540.2nm with 5pm steps. In the Fig. 2(a), the correspondence between the absorption peaks and the coupled cladding modes (LP0n and LP1(n-1) with n>1) is also highlighted. As can be seen by comparing Figs. 2(a) and 2(b), for each resonance in the transmittance, a strong variation in the group delay occurs. In a similar way to an adsorptive medium exhibiting anomalous dispersion, here, the coupling between the propagating core mode and each counter-propagating cladding mode creates a strong change in the effective group index of the core mode. As a consequence, the core-cladding modes coupling modifies the delay of an optical pulse travelling through the TFBG with respect to the time delay that an optical pulse would employ in optical fiber in absence of grating. In particular, a temporal advancement of the transmitted pulse peak occurs in the central part of the resonances, while the pulse slows down at the edges of the resonance.
In order to exemplify the fast and slow light capability of the investigated TFBG, in Fig. 3(a) the normalized amplitudes of the square pulses with largest delay (at 1535.305nm) and advancement (at 1535.205nm) for the peak around 1535.2nm are shown as functions of time. In the same graph also the square pulse corresponding to 1540.2nm (out of the grating bandwidth), representing the zero-delay, is reported. In Figs. 3(b)-3(c) the rising and falling fronts of the same pulses have been shown. A slight pulse broadening can be appreciated as well as in the whole investigated spectral range. The maximum broadening (up to about 1% of the FWHM pulse duration) occurs in the central part of the resonances and it is attributable to the third order spectral dispersion of the TFBG itself .
Time advancements and delays increase with the absorption peak depth as shown in Fig. 4(a) for LP0n coupling cladding modes. A temporal advancement as high as ≈25ps and delay of ≈15ps have been obtained in correspondence of the highest order LP0n cladding mode (peak at 1534.4nm) offering a FWHM bandwidth of 13.8GHz. The maximum observed time advancement corresponds to a group velocity as large as about two times the speed of light in optical fiber (≈1.3∙c0). Figure 4(b) shows the 5°-TFBG capability to speed up and slow down the group velocity in terms of pulse advancement (delay) – FWHM bandwidth product for each LP0n resonance versus the peak depth. As can be seen, when the peak depth increases, also the advancement (delay) - bandwidth product increases, reaching the maxima values in correspondence of the higher order LP0n cladding modes in the investigated wavelength range. The same data are not reported for LP1(n-1) resonant modes because the evaluated FWHM bandwidths in the transmittance correspond to double peaks in the group delay (i.e. see the highest LP1(n-1) mode).
The achieved time delays and advancements in optical fiber by means of TFBG, while still comparable to results obtained in Ref , are significantly lower than ref [9–12], especially in terms of fractional delay (in this case about 0.02). As a matter of fact, the pulse train at 300MHz driving the modulator, while it has been suitable for 5°-TFBG group delay characterization, led to a modulated signal bandwidth of 3GHz against a pulse duration of 1.7ns and hence to a severely low fractional delay. However, since the experimentally measured group delay is a TFBG’s property, smoother and shorter pulses will experience the same time advancement/delay thus increasing the fractional delay, provided that the pulse train bandwidth remains narrower than the TFBG attenuation bandwidths. The theoretical limitation on the fractional delay, therefore, is ruled by the delay-bandwidth product reported in Fig.e 4(b) that prevents the fractional delay to reach values greater than unity. Nevertheless, in the present work, only lower order modes have been considered which in turn have weaker resonances when compared with high order modes at lower wavelengths, this means that larger product advancement (delay)-bandwidth can be obtained (according to Fig. 4(b)) by acting on the wavelength range or the tilt angles to couple light with higher order modes. In turn, higher delay-bandwidth products can be reached at the expenses of higher insertion losses, in a similar fashion of other slow and fast light structures intrinsically limited and ruled by their delay-bandwidth product  (see Section 3.2).
3.2 Group-delay reconstruction for tilted fiber Bragg gratings in transmission
The experimental measured group delay has also been compared with the theoretically predicted group delay in transmission. Although it is not possible to retrieve a general solution in closed form describing the transmission spectrum for a TFBG, it is possible to demonstrate that the group delay in transmission of TFBGs is uniquely determined by its transmittance, paralleling the analysis by Poladian in . The analysis, based on the linear coupled mode theory, is valid only under the hypothesis of coupling of two discrete modes, thus excluding radiating modes and simultaneous multiple coupling, beyond the classical assumptions of the coupled mode theory. Under these assumptions, the transmission spectrum of a TFBG can be modeled by the coupled modes equations that describe the interaction between the propagating core mode with the counter-propagating cladding mode. Note that the method developed by Poladian deals with counter-propagating modes without dependence on their nature (core or cladding modes). Hence, the group delay of a TFBG in transmission can be uniquely determined by its transmittance through Hilbert Transform in the same way as uniform FBG .
A minimum phase reconstruction algorithm based on the “Cepstrum” technique  has been implemented to retrieve numerically the group delay spectrum from the experimental transmittance data. The obtained results, shown in Fig. 2(b), demonstrate a good agreement between the reconstructed and measured group delay spectra. In addition the agreement between the group delay obtained through Hilbert Transform and the experimental spectrum also allows us to infer the capability of the TFBG to provide strong light advancements in correspondence of deeper and narrower resonances not experimentally investigated here.
Basically, we demonstrated that in TFBGs a relationship between amplitude and group delay exists, thus proving theoretically and experimentally the capability of this class of devices to speed up and slow down the light group velocity. Consequently, TFBGs share the same potential advantages and disadvantages (in terms of delay-bandwidth product, dispersion, insertion losses, fractional delay…) of all systems in which the amplitude and phase response are ruled by a Hilbert Transform . Nevertheless, some remarkable peculiarities related to the TFBGs capability to control in optical fiber light signals speed, pave the way for a wide range of applications.
3.3 Group velocity control through tilted fiber Bragg gratings
The most important reason in the employment of TFBGs to control the light signals speed with an all in fiber operation relies on the low cost manufacturing process of the gratings, on the low complexity of the system needed to obtain time delays and advancements, on the short device length, on the room temperature operation and above all on the multi-wavelength operation and wide tuning possibility that TFBGs offer. It can be envisaged, in fact, that the dependence of the attenuation bands’ wavelengths on the temperature (strain) through the thermo-optic (elasto-optic) effect would allow a fine control of the time delay experienced by a single wavelength signal by means of a fine wavelength shift of the resonances or by switching to adjacent resonances. At the same time, the presence of multiple “delaying” resonances in the same device does not require complex multiplexing strategies among different delay lines when differential time delays have to be provided among multiple signals.
Furthermore, the dependence of the attenuation bands’ wavelength and amplitudes on the surrounding refractive index (SRI), through respectively the phase matching condition and the overlap integral between the coupling modes, enables wide control on the time advancements and delays at single wavelength. This opens the way for multi-wavelength opto-fluidic light control systems, low-cost, all-in-fiber operating at room temperature.
As proofs of principle, in the following the group velocity of a single wavelength signal has been tuned by acting on the refractive index of the medium surrounding the TFBG as well as by using simple thermal control.
To this aim, the group delay associated to the absorption peak at about 1534.4nm has been measured when the TFBG was immersed in water at a controlled temperature of 26 ± 0.2°C. It is noteworthy, by comparing Figs. 2 and Fig. 5 , that the absorption peak becomes deeper from air to water, resulting in a larger maximum time delay and advancement in water than in air. Further SRI variations have been induced by immersing the TFBG in glycerin aqueous solutions at the same temperature (26 ± 0.2°C) and with different concentrations previously characterized with an Abbe refractometer at 589nm.
To exemplify the spectral behavior of the TFBGs with the SRI, in Figs. 5(a)-5(b), the experimentally measured transmittance and group delay of the selected absorption peak are shown for SRIs equal to 1.33, 1.4 and 1.449. From the figures can be observed that an increase of the SRI, far from the cut-off of the selected cladding mode, basically leads to a shift of the absorption peak towards higher wavelengths due to the corresponding increase in the cladding modes effective refractive index. The shift is also accompanied by a slight reshaping of the adsorption peak. Particularly, its depth slightly decreases with the SRI. Further SRI increases up to the glass refractive index would lead to additional shifts towards higher wavelengths until the higher order cladding modes are no longer guided in the cladding. This mechanism results in a progressive smoothing of the associated dips in the transmission spectrum up to their complete disappearance when the cut-off wavelengths of the respective cladding modes are approached . The experimental measured group delay spectra, shown in Fig. 5 (b), evolve similarly as could be predicted according to the Hilbert Transform.
By exploiting this dependence of the TFBG on the SRI, single wavelength time delay measurements have been performed at 1534.59nm, corresponding to the top of the rising front of the group delay spectrum in water, by changing the SRI from 1.33 to 1.4. The SRI range and operating wavelength have been chosen to guarantee a wide group velocity excursion. When the SRI is equal to 1.4 the cladding mode is still far from its cut-off point and the corresponding group delay peak approaches its minimum at 1534.59nm.
Figure 6(a) displays the time delay evolution at 1534.59nm with respect to the SRI in the range 1.33-1.4. As can be seen, the SRI increase led to a time advancement with respect to the starting point (SRI = 1.33) of about 35ps, revealing a tuning efficiency of 500ps/RIU.
Similarly, single wavelength time delay measurements have been carried out at 1534.56nm by simply changing the temperature of the water surrounding the TFBG from 24 to 29°C. A temperature variation leads to a wavelength shift of the TFBG transmittance as well as of the group delay. In particular, the selected resonance shifts with a wavelength sensitivity of 12pm/°C. No reshaping effect due to the thermo-optic effect is appreciable in the investigated temperature range. The obtained time delay variations are shown in Fig. 6 (b) in which the time advancements with respect to the 24°C starting point are reported. A tuning efficiency of 7.6ps/°C has been obtained for the absorption peak at 1534.5nm.
These results show how this simple all fiber approach allows to tune and control the light group velocity enabling compact all fiber devices and components like opto fluidics and thermal delay lines working in single or multi-wavelength configuration as well as advanced all fiber interferometers.
In conclusion, fast and slow light propagation of a train pulse has been achieved at optical communication wavelengths with the observation of time advancement in the picoseconds time scale in 1cm long TFBG and group velocity as large as 1.3 times the light velocity in vacuum. The superluminal and subluminal propagation is caused by the anomalous dispersion, due to the coupling between the propagating core mode and the counter-propagating cladding modes. The multiple resonances, arising from this multiple coupling, lead to multiple peaks in TFBG group delay, enabling multi-wavelength fast and slow light operation in the same short device.
The experimental results agree well with the theoretical analysis that demonstrated that the group delay in transmission of TFBGs is uniquely determined by its transmittance through the Hilbert transform. On the other hand, the analysis highlights that TFBGs share the same potential advantages and disadvantages of all systems in which the amplitude and phase response are ruled by a Hilbert Transform. Particularly, the delay-bandwidth product limits the performances of these systems, unless an ultrafast tuning is provided as demonstrated in ref .
As a matter of fact, the achieved time delays and advancements in optical fiber by means of the TFBG are comparable to results obtained with uniform fiber Bragg gratings , but they are significantly lower than performances obtained in ref [9–12]. which in addition offer efficient tuning mechanisms, even if at the expenses of high complexity systems.
Nevertheless, optimization margins to obtain performances improvements with TFBGs-based fast and slow structures exist. Such optimizations can be achieved by acting on the basic building-block, the TFBG, tailoring its spectral features to enhance fast and slow light operation, and by acting on the optical system (i.e. by mitigating the insertion losses through optical amplification) with the aim to maintain anyway low the overall complexity of the optical system.
On this line of arguments and on the basis of the highlighted advantages and limitations of TFBGs as fast and slow light structures, their employment to control the light signals speed with an all in fiber operation is thus mainly justified by the low cost manufacturing process of the gratings, by the low complexity of the system needed to obtain time delays and advancements, by the short device length, by the room temperature operation and by the multi-wavelength operation and wide tuning possibility that TFBGs offer. In particular, two different methods to achieve tunable slow and fast light through controlling the surrounding refractive index and the temperature have been investigated in order to demonstrate that these systems can be used to dynamically control light group velocity for all-in-fiber optical systems and applied photonics.
References and links
2. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]
5. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008). [CrossRef]
6. Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys. 3(6), 406–410 (2007). [CrossRef]
7. R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A 73(6), 063812 (2006). [CrossRef]
8. S. Longhi, M. Marano, P. Laporta, and M. Belmonte, “Superluminal optical pulse propagation at 1.5 μm in periodic fiber Bragg gratings,” Phys. Rev. E64, 055602 (2001) [CrossRef]
9. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94(15), 153902 (2005). [CrossRef] [PubMed]
11. K. Y. Song, M. Herráez, and L. Thévenaz, “Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering,” Opt. Express 13(1), 82–88 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-1-82. [CrossRef] [PubMed]
12. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2(11), 775–780 (2006). [CrossRef]
13. T. Erdogan and J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13(2), 296–313 (1996), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-13-2-296. [CrossRef]
14. G. Laffont and P. Ferdinand, “Tilted short-period fibre-Bragg-grating induced coupling to cladding modes for accurate refractometry,” Meas. Sci. Technol. 12(7), 765–770 (2001). [CrossRef]
15. M. Ware, S. A. Glasgow, and J. Peatross, “Role of group velocity in tracking field energy in linear dielectrics,” Opt. Express 9(10), 506–518 (2001), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-9-10-506. [CrossRef] [PubMed]
16. M. Gonzalez Herraez and L. Thévenaz, “Physical limits to broadening compensation in a linear slow light system,” Opt. Express 17(6), 4732–4739 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-6-4732. [CrossRef] [PubMed]
17. L. Poladian, “Group-delay reconstruction for fiber Bragg gratings in ref lection and transmission,” Opt. Lett. 22(20), 1571–1573 (1997), http://www.opticsinfobase.org/abstract.cfm?URI=ol-22-20-1571. [CrossRef] [PubMed]
18. R. C. Kemerait and D. G. Childers, “Signal Detection and Extraction by Cepstrum Techniques,” Trans. Inf. Theory, 18 6, 745–759 (1972) [CrossRef]