coupling of light, and after loading of a signal pulse it is switched to a high-Q
condition through ultrafast (that is, within the cavity photon lifetime) re- fractive index tuning due to free-carriers injection. This functionality is usually achieved by breaking the coupling to an access waveguide, and the signal pulse can be trapped for a time interval longer than the lifetime of the low-Q static system, therefore breaking the delay-bandwidth product limit. A second ultrafast tuning event restores the system in a low-Q condition, releasing the trapped pulse.
Such an approach was first demonstrated by Xu et al. [17] in a com- pact structure consisting of two ring resonators (Fig. 3.4b). Tuning was accomplished by injecting free carriers in the same region where light was stored, inducing very high loss and limiting the maximum achievable delay to less than 100 ps for a 14 ps pulse. Two very elegant alternatives were proposed, minimising the overlap of the tuning region with the storage re- gion. The first solution was implemented by Upham et al. [18, 151] in a photonic crystal cavity-waveguide-mirror system (Fig. 3.4c) whoseQ-factor depends on whether constructive or destructive interference occurs between light outputting the cavity and light reflected back from the mirror. The phase difference between the two beam paths is controlled by changing the refractive index of silicon in the waveguide section between the cavity and the mirror. So the overallQ-factor can be controlled without acting directly on the cavity, where most of the light is stored. The second solution was reported by Elshaari et al. [150] in a ring resonator system similar to that of Xu et al., but with the addition of a third ring to act as an input gate (Fig. 3.4d), which is to be tuned to trap light in the other two rings.
Note, however, that in such trap-and-release schemes once a pulse has been loaded no other pulse can be accepted until the release of the first one. Therefore, the maximum storage time defining the performance of the device represents also its intrinsic limitation on the hold-off time. Furthermore, it is still challenging to experimentally achieve a good balance between the initial low-Qcondition (bandwidth) for pulse loading and the high-Qcondition for pulse storage [150, 151].
3.2
Tunable delay based on wavelength conversion
and dispersion
An alternative approach for realising tunable delays is based on wavelength conversion and group velocity dispersion [19]: the input signal is shifted in frequency and then injected into a dispersive medium, where different frequencies propagate at different speed. The shifted signal will therefore accumulate a delay with respect to an unshifted one. The achievable delay is proportional to the wavelength shift and the group velocity dispersion and length of the dispersive medium [152]. The signal wavelength can then be
Figure 3.5: Examples of wavelength conversion processes in silicon. (a) Schematic of four-wave mixing: two pump photons are converted to a signal photon and an idler photon (from [22]. Reprinted by permission from Macmillan Publishers Ltd: Nature [22], copyright 2006). (b-c) Adiabatic wavelength conversion in a photonic crystal cavity (from [23]. Copyright 2006 by The American Physical Society) con- sisting of four missing air holes in a triangular lattice (b); the refractive index of the gray region is tuned. (c) FDTD calculations of wavelength spectra before (blue) and after (red) index tuning.
reconverted to its original value in a subsequent wavelength conversion step. Several demonstrations of continuously tunable delay based on this tech- nique have been reported, mainly using optical fibres as a dispersive medium, allowing for very high optical delays — up to few microseconds [153–155] — to be observed. Despite the large achievable delays, however, these solutions are extremely bulky, and not suitable for device integration. We may there- fore investigate the possibility of realising a tunable delay line based on wavelength conversion and dispersion in more compact photonic structures such as silicon photonic crystals.
3.2.1 Wavelength conversion in silicon waveguides
Wavelength conversion in silicon may be achieved with various effects. One important example is four-wave mixing [156], a third-order nonlinear pro- cess in which two pump photons are converted into a signal and idler pho- tons (Fig. 3.5a). The information carried by the signal can therefore be transferred also to the idler. Efficient conversion over a wavelength range of several tens of nanometres in a SOI channel waveguide was reported by Foster et al. [22]. Conversion efficiency and tunability range, however, come at a price: the need for very powerful pumps (11 W peak power in [22]), a minimum light travelling distance (6 to 17 mm in [22]), and phase- matching. At more realistic (continuous-wave) pump powers of around 100 mW, even in low-loss silicon nanowires the best conversion efficiencies range from −40 dB [61] to less than −10 dB [157], depending also on the waveguide length (almost 2 cm in Ref. [157]), whereas the best reported value to-date for slow light photonic crystals is −24 dB at ng ∼30 [61].
Different authors have therefore explored the possibility of realising wave- length conversion i) with high efficiency, ii) for weak light and iii) in compact
3.2 Tunable delay based on wavelength conversion and dispersion
photonic structures. Winn et al.[158] showed theoretically that simultane- ous spatial and temporal modulation of the refractive index promotes inter- band indirect transitions of photons, by imparting frequency and wavevector shifts. These transitions can be interpreted as the optical analogue to elec- tronic indirect transitions in a semiconductor, in which the absorption of a photon and a phonon causes an electron to change both its energy and momentum. Refractive index tuning of this kind, however, is not trivial to accomplish [159], and to-date the only experimental demonstration has been very recently reported by Liraet al. in a slotted waveguide [160].
A simpler wavelength conversion effect is instead based on adiabatic tuning, the same mechanism explored by Fan et al. [15] for dynamically stopping a light pulse (section 3.1.2). The results of Fan et al. suggest the possibility of dynamically controlling the wavelength of light through adiabatic tuning of the dispersion curve, but explicit numerical investigation of such controllability was first reported by Notomiet al.[23] for a photonic crystal cavity (Fig. 3.5b). The authors showed that if the resonant frequency of the cavity is tunedwhile light is stored into it, the wavelength of the light follows the changes in the cavity resonant wavelength (Fig. 3.5c). In fact, adiabatic wavelength conversion is the optical equivalent of the tuning of a classical resonator [75]: by plucking the string of a guitar, we generate a sound at a certain frequency, but if we slide the finger along the string before the sound dies away, the tone of the sound changes, following the resonant frequency of the string.
When compared to four-wave mixing, which typically allows for conver- sion efficiencies of less than−20 dB and over long waveguides, the adiabatic wavelength conversion has the big advantage of an efficiencyclose to unity, providing that losses from other sources are negligible [23]; furthermore, it does not depend on the initial light intensity nor on phase-matching condi- tions, and occurs over lengths on the order of the physical length of a light pulse. When compared to inter-band indirect photonic transitions [158,160], adiabatic tuning involves only one photonic state and the wavelength shift is proportional to the magnitude of the refractive index shift, not to the fre- quency of the modulation [146]. The tuning should maintain the structure periodicity in order to conserve the wavevector, and it should be adiabatic to avoid coupling to other modes [15].
Modulation of the material properties in the presence of photons requires the use of high-Qcavities or slow light media to hold the light during the tun- ing process. Adiabatic wavelength conversion was first observed by Preble
et al.[161] in a silicon ring resonator and by Tanabeet al.[162] in a photonic crystal cavity. It was then experimentally demonstrated also in the slow light regime of a silicon W1 photonic crystal waveguide by Kampfrathet al.[57]: a probe light pulse travelling through the waveguide is blue-shifted when the silicon refractive index is changed by illuminating the material with a pump pulse. This is shown in Fig. 3.6, where the output spectrum of the probe
Figure 3.6: Ultrafast adiabatic frequency shifting as reported by Kampfrathet al.
(from [57]. Copyright 2010 by The American Physical Society). (a) Experimentally determined and (b) calculated intensity spectra of a W1 waveguide response to a 1.3 ps-long input probe pulse at center frequency ωc/2π= 202 THz, for different pump-probe delaysτ. (c) Measured and calculated output spectra at τ = −4 ps and 1 ps.
pulse is reported as a function of the pump-probe delayτ. If the probe ar- rives before the pump (τ <−4 ps), it propagates in the unshifted waveguide. If the probe arrives after the pump (τ >1 ps), its frequency falls in the band gap of the blue-shifted band diagram and is reflected. Around τ = −1 ps pump and probe have maximum overlap, and the probe spectrum is adia- batically blue-shifted by 0.39 THz. With respect to configurations based on nanocavities, the use of slow light waveguide modes increases the conversion efficiency to more than 80%. This is due to the fact that in a waveguide the probe pulsepropagates through the tuned region rather than being held inside it, and is therefore affected by free-carrier absorption only in the time necessary for wavelength conversion.
On-the-fly wavelength conversion in a silicon photonic crystal waveguide was also reported by Uphamet al., [163], but the probe pulse was not prop- agating in a slow mode; as a result, the observed frequency shift was less than the induced refractive index change, due to the probe not being fully contained within the tuning spot during the entire index change. This high- lights the importance of the use of slow light for this type of effect.