Slow and Stopped Light in Coupled Resonator Systems

Chapter 7 Slow and Stopped Light in Coupled Resonator Systems Shanhui Fan, Sunil Sandhu, Clayton R. Otey, and Michelle L. Povinelli Abstract In the first part of this chapter, a theoretical overview is presented on the different approaches to the use of dynamic tuning for coherent optical pulse stopping and storage in coupled resonator systems, which are amenable to fabrication in on-chip devices such as photonic crystals. The use of such dynamic tuning overcomes the delay-bandwidth constraint of slow-light structures. The second part of this chapter presents a discussion on recent experimental work that has demonstrated the possibility of such dynamic tuning in on-chip systems. 7.1 Introduction This chapter describes how coupled resonator systems can be used to stop light – that is, to controllably trap and release light pulses in localized, standing wave modes. The inspiration for this work lies in previous research on stopped light in atomic gasses using electromagnetically induced transparency (EIT) [1], in which light is captured in “dark states” of the atomic system via adiabatic tuning [2–4]. However, such atomic systems are severely constrained to operate only at particular wavelengths corresponding to available atomic resonances and have only very limited bandwidth. The coupled resonator systems described in this chapter are amenable to fabrication in on-chip devices such as photonic crystals (PCs) [5–9] or microring resonators [10]. As such, the operating wavelength and other operating parameters can be engineered to meet flexible specifications, such as for optical communications applications. The idea of using dynamic tuning in a coupled resonator system is to modulate the properties of the resonators (e.g., the resonator frequencies) while a light pulse is in the system. In so doing, the spectrum of the pulse can be molded almost arbitrarily, leading to highly non-trivial information processing capabilities. In past S. Fan (B) Ginzton Laboratory, Stanford University, Stanford, CA, 94305, USA e-mail: shanhui@stanford.edu I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_7,  C Springer Science+Business Media, LLC 2010 165 166 S. Fan et al. work [11], it was shown that dynamic tuning can be used for time reversal of pulses. This chapter focuses on approaches to stopping light [12–17]. A wide variety of work has been done on slow-light structures employing coupled resonators [18–28]. However, in all such systems, the maximum achievable time delay scales inversely with the operating bandwidth [21]. As will be seen below, the use of dynamic tuning overcomes this constraint by manipulation of the photon spectrum in time. This chapter starts with an overview of theoretical work on light stopping in dynamically tuned coupled resonator systems. This is followed by a discussion on quite recent experimental results that have demonstrated the possibility of adiabatic tuning in on-chip systems and a review of the growing body of work inspired by dynamic optical modulation ideas. 7.2 Theory 7.2.1 Tuning the Spectrum of Light Here a simple example is provided to show how the spectrum of an electromagnetic wave can be modified by a dynamic photonic structure. Consider a linearly polarized electromagnetic wave in one dimension. The wave equation for the electric field is ∂ 2E ∂ 2E − (ε0 + ε (t)) μ0 2 = 0. 2 ∂x ∂t (7.1) Here ε (t) represents the dielectric modulation and ε0 is the background dielectric constant. Both ε0 and ε (t) are assumed to be independent of position. Hence different wave vector components do not mix in the modulation process. For a specific wave vector component at k0 , with electric field described by E (t) = √ f (t) ei(ω0 t−k0 x) , where ω0 = k0 / μ0 ε0 , we have −k02 f − [ε0 + ε (t)] μ0   ∂ 2f ∂f 2 − ω + 2iω f = 0. 0 0 ∂t ∂t2 (7.2) By using a slowly varying envelope approximation, i.e., ignoring the ∂ 2 f /∂t2 term, and by further assuming that the index modulations are weak, i.e., ε(t) << εo , (7.2) simplifies to i ε (t) ω0 ∂f ε (t) ω0 = f, f ≈ ∂t 2 [ε (t) + ε0 ] 2ε0 (7.3) which has an exact analytic solution: ⎡ f (t) = f (t0 ) exp ⎣−iω0 t t0   ⎤ ε t′ ′ dt ⎦ , 2ε0 (7.4) 7 Slow and Stopped Light in Coupled Resonator Systems 167 where t0 is the starting time of the modulation. Thus the “instantaneous frequency” of the electric field for this wave vector component is ω (t) = ω0 1 − ε (t) . 2ε0 (7.5) Note that the frequency change is proportional to the magnitude of the refractive index shift alone. Thus, the process defined here differs in a fundamental way from traditional nonlinear optical processes. For example, in a conventional sum frequency conversion process, in order to convert the frequency of light from ω1 to ω2 , modulations at a frequency ω2 − ω1 need to be provided. In contrast, regardless of how slow the modulation is in the process described here, as long as light is in the system, the frequency shift can always be accomplished. Below, some spectacular consequences of such frequency shifts will be demonstrated, in particular when applied to stopping light pulses all optically in dynamic PC systems. The existence of the frequency shift in dynamic PC structures [29] and in laser resonators [30, 31] was also pointed out in a number of previous works. In practical optoelectronic or nonlinear optical devices, the achievable refractive index shift is generally quite small. Thus, in most practical situations the effect of dynamics is prominent only in structures in which the spectral feature is sensitive to small refractive index modulations. This motivates the design of Fano interference schemes described below, which are employed to enhance the sensitivity of photonic structures to small index modulations. 7.2.2 General Conditions for Stopping Light The aim of stopping light is to reduce the group velocity of a light pulse to zero, while completely preserving all the coherent information encoded in the pulse. Such ability holds the key to the ultimate control of light and has profound implications for optical communications and quantum information processing. There has been extensive work attempting to control the speed of light using optical resonances in static PC structures. Group velocities as low as 10−2 c have been experimentally observed at waveguide band edges [32, 33] or with coupled resonator optical waveguides (CROWs) [34–37]. Nevertheless, such structures are fundamentally limited by the delay-bandwidth product constraint – the group delay from an optical resonance is inversely proportional to the bandwidth within which the delay occurs. Therefore, for a given optical pulse with a certain temporal duration and corresponding frequency bandwidth, the minimum group velocity achievable is limited. In a CROW waveguide structure, for example, the minimum group velocity that can be accomplished for pulses at 10 Gbit/s rate at a wavelength of 1.55 μm is no smaller than 10−2 c. For this reason, static photonic structures can not be used to stop light. To stop light, it is necessary to use a dynamic system. The general condition for stopping light [12] is illustrated in Fig. 7.1. Imagine a dynamic PC system, with an 168 S. Fan et al. Fig. 7.1 The general conditions for stopping a light pulse. (a) The large-bandwidth state that is used to accommodate an incident light pulse. (b) The narrow-bandwidth state that is used to hold the light pulse. An adiabatic transition between these two states stops a light pulse inside the system [63]. Reprinted with permission. Copyright 2006 IEEE initial band structure possessing a sufficiently wide bandwidth. Such a state is used to accommodate an incident pulse, for which each frequency component occupies a unique wave vector component. After the pulse has entered the system, one can then stop the pulse by flattening the dispersion relation of the crystal adiabatically, while preserving the translational invariance. In doing so, the spectrum of the pulse is compressed, and its group velocity is reduced. In the meantime, since the translational symmetry is still preserved, the wave vector components of the pulse remain unchanged, and thus one actually preserves the dimensionality of the phase space. This is crucial in preserving all the coherent information encoded in the original pulse during the dynamic process. 7.2.3 Tunable Fano Resonance To create a dynamic PC, one needs to adjust its properties as a function of time. This can be accomplished by modulating the refractive index, either with electro-optic or with nonlinear optic means. However, the amount of refractive index tuning that can be achieved with standard optoelectronics technology is generally quite small, with a fractional change typically on the order of δn/n ≈ 10−4 . Therefore, Fano interference schemes are employed in which a small refractive index modulation leads to a very large change of the bandwidth of the system. The essence of a Fano interference scheme is the presence of multi-path interference, where at least one of the paths includes a resonant tunneling process [38]. Such interference can be used to greatly enhance the sensitivity of resonant devices to small refractive index modulation [14, 39, 40]. Here a waveguide side-coupled to two cavities is considered [41]. The cavities have resonant frequencies ωa,b ≡ ω0 ± δω/2. (This system represents an all-optical analogue of atomic systems exhibiting EIT [1]. Each optical resonance here is analogous to the polarization between the energy levels in the EIT system [26].) For simplicity, it is assumed that the cavities coupled to the waveguide with an equal coupling rate of γ , and the direct coupling between the side-cavities is ignored. 7 Slow and Stopped Light in Coupled Resonator Systems 169 Fig. 7.2 (a) Transmission spectrum through a waveguide side-coupled to a single-mode cavity. (b) and (c) Transmission spectra through a waveguide side-coupled to two cavities. The parameters for the cavities are ω0 = 2π c/L, γ = 0.05ω0 . The waveguide satisfies a dispersion relation β (ω) = ω/c, where c is the speed of light in the waveguide and L is the distance between the cavities. In (b), ωa,b = ω0 ± 1.5γ . In (c), ωa,b = ω0 ± 0.2γ [63]. Reprinted with permission. Copyright 2006 IEEE Consider a mode in the waveguide passing through the cavities. The transmission and reflection coefficients for a single side cavity can be derived using the Green’s function method [42] and are used to calculate the two-cavity transmission spectrum via the transfer matrix method [41]. The transmission spectra of one- and two-cavity structures are plotted in Fig. 7.2. In the case of one-cavity structure, the transmission features a dip in the vicinity of the resonant frequency, with the width of the dip controlled by the strength of waveguide-cavity coupling (Fig. 7.2a). With two cavities, when the condition 2β (ω0 ) L = 2nπ (7.6) is satisfied, the transmission spectrum features a peak centered at ω0 . The width of the peak is highly sensitive to the frequency spacing between the resonances δω. When the cavities are lossless, the center peak can be tuned from a wide peak when δω is large (Fig. 7.2b), to a peak that is arbitrarily narrow with δω→0 (Fig. 7.2c). The two-cavity structure, appropriately designed, therefore behaves as a tunable bandwidth filter (as well as a tunable delay element with delay proportional to the inverse peak width [26]), in which the bandwidth can in principle be adjusted by any order of magnitude with very small refractive index modulation. 7.2.4 From Tunable Bandwidth Filter to Light-Stopping System By cascading the tunable bandwidth filter structure described in the previous section, one can configure a structure that is capable of stopping light (Fig. 7.3a). In 170 S. Fan et al. Fig. 7.3 (a) Schematic of a coupled-cavity structure used to stop light. (b) and (c) Band structures for the system shown in (a), as the frequency separation between the cavities are varied, using the same waveguide and cavity parameters as in Fig. 7.2b and c, with the additional parameter L2 = 0.7L1 . The thicker lines highlight the middle band that will be used to stop a light pulse [63]. Reprinted with permission. Copyright 2006 IEEE such a light-stopping device, the photonic band diagram becomes highly sensitive to small refractive index modulation. The photonic bands for the structure in Fig. 7.3a can be calculated using a transmission matrix method [13]. The band diagrams are shown in Fig. 7.3, in which the waveguide and cavity parameters are the same as those used to generate the transmission spectrum in Fig. 7.2. In the vicinity of the resonances, the system supports three photonic bands, with two gaps occurring around ωa and ωb . The width of the middle band depends strongly on the resonant frequencies ωa , ωb . By modulating the frequency spacing between the cavities, one goes from a system with a large bandwidth (Fig. 7.3b), to a system with a very narrow bandwidth (Fig. 7.3c). In fact, it can be analytically proven that the system supports a band that is completely flat in the entire first Brillouin zone [13], allowing a light pulse to be frozen inside the structure with the group velocity reduced to zero. Moreover, the gaps surrounding the middle band have sizes on the order of the cavity-waveguide coupling rate γ and are approximately independent of the slope of the middle band. Thus, by increasing the waveguide-cavity coupling rate, this gap can be made large, which is important for preserving the coherent information during the dynamic bandwidth compression process [12]. 7.2.5 Numerical Demonstration in a Photonic Crystal The system presented above can be implemented in a PC of a square lattice of dielectric rods n = 3.5 with a radius of 0.2a (a is the lattice constant) embedded in air n = 1 [13] (Fig. 7.4). The photonic crystal possesses a band gap for TM modes with electric field parallel to the rod axis. Removing one row of rods along the pulse propagation direction generates a single-mode waveguide. Decreasing the radius of 7 Slow and Stopped Light in Coupled Resonator Systems 171 Fig. 7.4 Light-stopping process in a PC simulated using the FDTD method. The crystal consists of a waveguide side-coupled to 100 cavity pairs. Fragments of the PC are shown in part b. The three fragments correspond to unit cells 12–13, 55–56, 97–98. The dots indicate the positions of the dielectric rods. The black dots represent the cavities. (a) The middle dashed horizontal and vertical lines (with “pulsed released” and “pulse stopped” points indicated) represent the variation of ωa and ωb as a function of time, respectively. The far left pulse (solid line) is the incident pulse as recorded at the beginning of the waveguide. The pulse centered at ∼1.65. tpass (dotted line) and the far-right pulse (solid line) are both the output pulses at the end of the waveguide, in the absence and in the presence of modulation, respectively. tpass is the passage time of the pulse in the absence of modulation. (b) Snapshots of the electric field distributions in the PC at the indicated times [13]. Reprinted with permission. Copyright 2004 American Physical Society a rod to 0.1a and the dielectric constant to n = 2.24 + provides a single-mode cavity with resonance frequency at ωc = 0.357 · (2π c a). The nearest neighbor cavities are separated by a distance of l1 = 2a along the propagation direction, and the unit cell periodicity is l = 8a. The waveguide-cavity coupling occurs through a barrier of one rod, with a coupling rate of γ = ωc /235.8. The resonant frequencies of the cavities are tuned by refractive index modulation of the cavity rods. The entire process of stopping light for N = 100 pairs of cavities is simulated with the FDTD method, which solves Maxwell’s equations without approximation [43]. The dynamic process for stopping light is shown in Fig. 7.4. A Gaussian pulse is generated in the waveguide (the process is independent of the pulse shape). The 172 S. Fan et al. excitation reaches its peak at t = 0.8 tpass , where tpass is the traversal time of the pulse through the static structure. During pulse generation, the cavities have a large frequency separation. The field is concentrated in both the waveguide and the cavities (Fig. 7.4b, t = 1.0 tpass ), and the pulse propagates at a relatively high speed of vg = 0.082c. After the pulse is generated, the frequency separation is gradually reduced to zero. During this process, the speed of light is drastically reduced to zero. As the bandwidth of the pulse is reduced, the field concentrates in the cavities (Fig. 7.4b, t = 5.2 tpass ). When zero group velocity is reached, the photon pulse can be kept in the system as a stationary waveform for any time duration. In this simulation, the pulse is stored for a time delay of 5.0 tpass , and the pulse is then released by repeating the same index modulation in reverse (Fig. 7.4b, t = 6.3 tpass ). The pulse intensity as a function of time at the right end of the waveguide is plotted in Fig. 7.4a and shows the same temporal shape as both the pulse that propagates through the unmodulated system, and the initial pulse recorded at the left end of the waveguide. 7.2.6 Dispersion Suppression Through Dynamic Tuning The dynamic tuning scheme largely eliminates the dispersive effects associated with static delay lines. The time-varying dispersion relation ω(k,t) can be expanded around a central wave vector kc as (2) ω(k,t) ≈ (1) ω(kc ,t) + ωkc (t)(k − kc ) + ωkc (t) 2 (k − kc )2 , (7.7) (n) where ωkc (t) ≡ dn ω(k,t)/dkn |k=kc . It can be shown [16] that the output width of the pulse in time ( tout ) after a total delay time τ is given by 2 tout = 2 tin + / τ 0 (2) ωkc (t′ )dt′ v2g (0) tin 2 , (7.8) where it is assumed that vg (τ ) = vg (0). For a static system, this reduces to the result 2 tout = 2 tin +  (2) ωkc (0)τ v2g (0) tin 2 , (7.9) and the pulse spreads with increasing delay. For the dynamic system, however, (2) ωkc (t) (and all higher order derivatives) are identically zero in the flat band state. If the bandwidth compression and decompression processes each occupy a duration T, 2 = tout ⎡ /T ⎤2 (2) ωkc (t′ )dt′ ⎥ ⎢2 ⎥ ⎢ 0 2 tin +⎢ 2 ⎥ . ⎣ vg (0) tin ⎦ (7.10) 7 Slow and Stopped Light in Coupled Resonator Systems 173 The pulse spreading is independent of the delay time τ , since it only occurs during spectrum compression and decompression. The delay can thus be increased arbitrarily without any additional increase in dispersion. 7.2.7 Capturing Light Pulses Using Few Dynamically Tuned Microresonators Instead of using the many resonators approach shown in Fig. 7.4, the capture and release of light pulses can also be performed using a dynamically tuned system with few resonators [44]. An example of such a system, shown in Fig. 7.5a, consists of two resonators coupled to a waveguide. The key feature to the pulse capturing/releasing process lies in the presence of a state that is decoupled from the waveguide, which we refer to as the dark state. When ω1 = ω2 = ω0 and γ1 = γ2 = γ0 , the system has an eigenstate, with eigenfrequency ω0 − β and resonator amplitudes α1 = −α2 , which does not leak into the waveguide. Starting from this dark state, if the resonators are tuned to ω1 = ω2 , the energy from the resonators leaks into the waveguide, generating a released pulse. Since the underlying physics of the system is time-reversal invariant, performing the time-reversed temporal detuning trajectory allows for the complete capture of the time-reversed pulse into the dark state. The entire pulse capture/release process is simulated using the system shown in Fig. 7.5b with the FDTD method [44]. The dynamic process for pulse capture/release is shown in Fig. 7.6. A Gaussian pulse is generated with carrier frequency ωc = ω0 − β = 200 THz and width T = 4 ps in the waveguide. These pulse parameters are for a system with lattice constant a = 370 nm. During the pulse generation, the resonators have zero detuning and the waveguide is decoupled from the resonators (dark state). As the pulse approaches the resonators are detuned by gradually tuning the dielectric constants within a region 1.25a around Fig. 7.5 Schematic of double resonator system used for pulse capture/release process: (a) On the left is the waveguide with mode amplitude awg coupled to two resonators with modal amplitudes a1,2 ; γ 1,2 are the coupling constants between the waveguide and the resonators, while β is the coupling constant between the two resonators. (b) Actual structure used in FDTD simulations [44]. Reprinted with permission. Copyright 2009 American Institute of Physics 174 S. Fan et al. Fig. 7.6 Dynamics of the pulse capture/release process simulated using the FDTD method. (a) The out-of-plane Hz field during the pulse capture process. (b) The Hz field in the dark state of the system. (c) The dielectric modulation curves for the two resonators, used in the simulation. ε0 is the dielectric constant in the absence of any dielectric modulation. (d) The pulse amplitude measured in the waveguide in a pulse release simulation from FDTD (solid line) and coupled mode theory (circles) [44]. Reprinted with permission. Copyright 2009 American Institute of Physics the resonators in order to couple the incident pulse energy into the resonators. Figure 7.6c shows how proper tuning of the resonators results in the near complete capture (99.61%) of the incident pulse energy by the resonators. Details of the generation of a resonators tuning trajectory, which results in negligible reflection of the incident pulse, are discussed in [45]. The pulse field during and after this pulse capture process are shown in Fig. 7.6a, b. At the end of the pulse capture process, the pulse energy is in the resonators and the system can be kept in this dark state for any duration. In addition, there is high spatial compression of the pulse energy in the two resonators and consequently, very high nonlinearity enhancement may be achieved in this pulse capture regime [45]. In order to release the pulse energy trapped in the resonators, the time-reverse of the tuning trajectory shown in Fig. 7.6c is used. This results in the release of the Gaussian pulse that is the time-reverse of the captured pulse, and the near complete transfer of energy from the resonators to the waveguide. Figure 7.6d shows the pulse amplitude measured in the waveguide at the end of the release process. Here a brief comment is given on the differences between the use of a resonator array or only two resonators. In the case when an array is used, the dynamic modulation process can start after the entire pulse is contained in the array. As a result, the temporal profile of the modulation is independent of the pulse format, as long as the modulation remains adiabatic. Moreover, there is no spatial compression of electromagnetic energy during the light-stopping process. In contrast, with the use of two cavities, in order to completely capture a pulse, the temporal profile of the modulation is strongly dependent upon the format of the pulse. 7 Slow and Stopped Light in Coupled Resonator Systems 175 7.3 Experimental Progress 7.3.1 General Requirements for Microresonators The numerical examples above have demonstrated the use of PC microresonators for slowing and stopping light. However, the phenomena described are quite general and apply to arbitrary coupled resonator systems. To be useful for stopping light, the particular resonator implementation should satisfy several criteria. First, the resonator should be highly tunable on the time scale of operation of the device. The resonance frequency can be tuned by changing the refractive index of the material via electro-optic methods. For a small refractive index shift of δn/n = 10−4 , achievable in practical optoelectronic devices [46], and assuming a carrier frequency of approximately 200 THz, as used in optical communications, the achievable bandwidths are on the order of 20 GHz, which is comparable to the bandwidth of a single wavelength channel in high-speed optical systems. Second, the intrinsic quality factor of the resonator should be as high as possible, since it limits the delay time. Light stopped for longer than the cavity lifetime will substantially decay. However, the optical loss might be counteracted with the use of gain media within or external to the cavities. Third, small size of the resonator is generally desirable, since shorter length devices tend to consume less power. Moreover, for fixed device length, decreasing the size of the resonator increases the storage capacity [24]. 7.3.2 Experiments with Microring Resonators Experiments with silicon microring resonators have demonstrated the use of a tunable Fano resonance in a double-resonator system [47] to controllably trap and release light pulses [48]. Initially, the frequencies of the two microring resonators are slightly detuned, as in Fig. 7.2b. In this state, input light couples into a “supermode” of the two resonators. The frequencies of the two resonators are then tuned into resonance with one another, as in Fig. 7.2c. In this state, the supermode is isolated from the input and output waveguides, and light is stored in and between the two resonators. After a given storage time, the resonator frequencies are again detuned to release the light. The resonators are tuned using the free-carrier dispersion effect in silicon [49] to blue-shift the resonant wavelength. In this experiment, an optical pump pulse at 415 nm was used to excite free carriers in the microrings. Electro-optic tuning of the ring resonances via built-in p-i-n junctions [50] should allow electrically controlled storage, with an expected bandwidth of over 10 GHz. In the experiment, the storage time was limited to < 100 ps by the intrinsic Q of the microresonators (Q = 143,000). However, the demonstration of Q ∼ 4.8 × 106 in a silicon ring resonator [51] suggests that storage times of several nanoseconds may be possible. 176 S. Fan et al. The detailed theory for light pulse capturing in such double-resonator system has been discussed in [44, 45] and reproduced in a previous section in this chapter. One drawback of using this double-resonator system for pulse delay is that the pulse shape and spectrum are not preserved in the process. The information encoded in the shape of the original pulse can be retained using a cascaded multiresonator system [12, 13]. Nevertheless, this experiment represents a major first step toward the realization of the theoretical ideas for stopping light that was presented above. 7.3.3 Experiments with Photonic Crystals PC microcavities may represent the ultimate limit of miniaturization for resonator modes. Such microcavities have been demonstrated with Q up to 2 × 106 and modal volumes as small as a cubic wavelength [52]. A recent experiment has demonstrated the fundamental requirement for dynamic trapping and delay: the ability to tune between a supermode that is strongly coupled to an input waveguide, and one that is decoupled, or isolated [53]. The geometry used is shown schematically in Fig. 7.7. A single cavity is side-coupled to a waveguide that is terminated by a mirror. The coupling between the input waveguide and the supermode of the resonator–waveguide–mirror complex is determined by the reflection phase from the mirror. When the wave emitted from the cavity in the backward direction interferes constructively with the wave emitted from the cavity in the forward direction and reflected backwards by the mirror, light can easily couple from the supermode to the input waveguide. Conversely, when the waves interfere destructively, the coupling is reduced. We note that this structure is in fact conceptually very similar to the structure shown in Fig. 7.2b. The mirror, in essence, creates a mirror image of the first resonator. In the experiment, a pump pulse was used to dynamically tune the refractive index of the waveguide between the nanocavity and the mirror, adjusting the reflection phase. Pump-probe measurements of the power emitted from the cavity to free space show that the coupling properties of the supermode could be tuned on the picosecond timescale. Fig. 7.7 Schematic of system used for experiments on dynamic light trapping in PCs 7 Slow and Stopped Light in Coupled Resonator Systems 177 7.3.4 Aligning Microresonator Resonances Using Differential Thermal Tuning An important pre-requisite for the experimental demonstration of multiple microresonators in PC structures that have been proposed for slowing and stopping light is the ability to tune the different microresonators to a desired resonance frequency. Due to geometrical errors during fabrication, it is generally not possible to fabricate two PC resonators with identical resonances. Hence, practical resonant frequency tuning methods are important for removing slight fabrication differences in nominally identical microresonators, relaxing fabrication tolerances required to realize multiple microresonators in PC structures. One example of such a post-fabrication tuning method is differential thermal tuning [54], which does not require any extra materials or structures and, consequently, avoids the potential quality degradation of PC microresonators and excess fabrication complexity. The differential thermal tuning technique generally involves focusing the output of a pump laser in the vicinity of a microresonator, which as a result experiences a shift in resonance frequency due to the induced thermal gradient. In the experiments demonstrated in [54], an initial difference of 3.15 nm in the resonant wavelength of two closely separated micro-resonators in a Si PC slab was decreased to zero using this thermal tuning technique. 7.4 Outlook and Concluding Remarks Beyond the work described above, the idea of using dynamic tuning of the refractive index for stopping, storing, and time-reversing pulses has sparked a wide range of research. For example, alternate dynamic tuning schemes that do not require translational invariance have recently been investigated [55]. Moreover, the generality of the physics governing coupled resonators has suggested the possibility of light stopping and time reversal in quite diverse physical systems. In semiconductor multiple quantum well structures, tuning of the excitonic resonance via the AC Stark effect can potentially flatten the photonic band structure to stop light pulses in a similar fashion as described here [56, 57]. In superconducting qubit systems, tuning of the qubit transition frequency can theoretically stop pulses on the single photon level [58]. Such an ability to manipulate single photons is of increasing interest for quantum information processing and quantum computing. The concept of using dynamic index tuning for frequency conversion is also being actively explored. Ideally, one could use a coupled resonator system to change the center frequency of a pulse while leaving its shape unchanged, a feat achieved via a uniform shift of the band structure [59]. While experiments are not yet feasible, a similar effect can be observed in single cavity systems. For a single cavity, changing the resonance frequency of the cavity mode on a timescale faster than the cavity decay time results in frequency conversion [60]. The frequency shift is linearly proportional to the index shift. The phenomenon has been demonstrated experimentally in both silicon microring resonators [61] and PC microcavities [62]. 178 S. Fan et al. In summary, dynamic tuning of coupled resonator systems opens the possibility for coherent optical pulse stopping and storage. More generally, dynamic processes in coupled resonator systems allow one to mold the spectrum of a photon pulse almost at will, while preserving coherent information in the optical domain. In the future, the use of dynamic photonic structures, as envisioned here, may provide a unifying platform for diverse optical information processing tasks. Acknowledgments The work is supported in part by NSF, DARPA, and the Lucile and Packard Foundation. The authors acknowledge the important contributions of Prof. Mehmet Fatih Yanik to the works presented here. References 1. Harris, S.E. Electromagnetically induced transparency. Phys. Today 50, 36–42 (1997) 2. Lukin, M.D., Yelin, S.F., et al. Entanglement of atomic ensembles by trapping correlated photon states. Phys. Rev. Lett. 84, 4232–4235 (2000) 3. Liu, C., Dutton, Z., et al. Observation of coherent optical information storage in an atomic medium using halted light pulses. Nature 409, 490–493 (2001) 4. Phillips, D.F., Fleischhauer, A., et al. Storage of light in an atomic vapor. Phys. Rev. Lett. 86, 783–786 (2001) 5. Joannopoulos, J.D., Meade, R.D., et al. Photonic Crystals: Molding the Flow of Light. Princeton University Press, Princeton (2008) 6. Joannopoulos, J.D., Villeneuve, P.R., et al. Photonic crystals: Putting a new twist on light. Nature. 386, 143–147 (1997) 7. Soukoulis, C. (ed.) Photonic crystals and light localization in the 21st century (NATO ASI Series). Kluwer, Netherlands (2001) 8. Johnson, S.G., Joannopoulos, J.D. Photonic crystals: The road from theory to practice. Kluwer, Boston (2002) 9. Inoue, K., Ohtaka, K. Photonic crystals. Springer-Verlag, Berlin (2004) 10. Haus, H.A., Popovic, M.A., et al. Optical resonators and filters. In: Vahala, K. (ed.) Optical Microcavities, pp. 1–38. World Scientific, Singapore, (2004) 11. Yanik, M.F., Fan, S. Time-reversal of light with linear optics and modulators. Phys. Rev. Lett. 93, 173903 (2004) 12. Yanik, M.F., Fan, S. Stopping light all-optically. Phys. Rev. Lett. 92, 083901 (2004) 13. Yanik, M.F., Suh, W., et al. Stopping light in a waveguide with an all-optical analogue of electromagnetically induced transparency. Phys. Rev. Lett. 93, 233903 (2004) 14. Yanik, M.F., Fan, S. Stopping and storing light coherently. Phys. Rev. A 71, 013803 (2005) 15. Yanik, M.F., Fan, S. Dynamic photonic structures: Stopping, storage, and time-reversal of light. Stud. in Appl. Math. 115, 233–254 (2005) 16. Sandhu, S., Povinelli, M.L., et al. Dynamically-tuned coupled resonator delay lines can be nearly dispersion free. Opt. Lett. 31, 1985–1987 (2006) 17. Sandhu, S., Povinelli, M.L., et al. Stopping and time-reversing a light pulse using dynamic loss-tuning of coupled-resonator delay lines. Opt. Lett. 32, 3333–3335 (2007) 18. Stefanou. N., Modinos, A. Impurity bands in photonic insulators. Phys. Rev. B 57, 12127–12133 (1998) 19. Yariv, A., Xu, Y., et al. Coupled-resonator optical waveguide: A proposal and analysis. Opt. Lett. 24, 711–713 (1999) 20. Bayindir, M., Temelkuran, B., et al. Tight-binding description of the coupled defect modes in three-dimensional photonic crystals. Phys. Rev. Lett. 84, 2140–2143 (2000) 21. Lenz, G., Eggleton, B.J., et al. Optical delay lines based on optical filters. IEEE J. of Quant. Electron. 37, 525–532 (2001) 7 Slow and Stopped Light in Coupled Resonator Systems 179 22. Heebner, J.E., Boyd, R.W., et al. Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide. Phys. Rev. E 65, 036619 (2002) 23. Heebner, J.E., Boyd, R.W., et al. SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides. J. Opt. Soc. Am. B 19, 722–731 (2002) 24. Wang, Z., Fan, S. Compact all-pass filters in photonic crystals as the building block for high capacity optical delay lines. Phys. Rev. E 68, 066616 (2003) 25. Smith, D.D., Chang, H., et al. Coupled resonator-induced transparency. Phys. Rev. A 69, 063804 (2004) 26. Maleki, L., Matsko, A.B., et al. Tunable delay line with interacting whispering-gallery-mode resonators. Opt. Lett. 29, 626–628 (2004) 27. Poon, J.K.S., Scheuer, J., et al. Designing coupled-resonator optical waveguide delay lines. J. Opt. Soc. Am. B 21, 1665–1673 (2004) 28. Khurgin, J.B. Expanding the bandwidth of slow-light photonic devices based on coupled resonators. Opt. Lett. 30, 513–515 (2005) 29. Reed, E.J., Soljacic, M., et al. Color of shock waves in photonic crystals. Phys. Rev. Lett. 91, 133901 (2003) 30. Yariv, A. Internal modulation in multimode laser oscillators. J. Appl. Phys. 36, 388–391 (1965) 31. Siegmann, A.E. Lasers. University Science Books, Sausalito (1986) 32. Notomi, M., Yamada, K., et al. Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs. Phys. Rev. Lett. 87, 253902 (2001) 33. Vlasov, Y.A., O’Boyle, M., et al. Active control of slow light on a chip with photonic crystal waveguides. Nature 438, 65–69 (2005) 34. Altug, H., Vuckovic, J. Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays. Appl. Phys. Lett. 86, 111102 (2005) 35. Xia, F., Sekaric, L., et al. Coupled resonator optical waveguides based on silicon-on-insulator photonic wires. Appl. Phys. Lett. 89, 041122 (2006) 36. Huang, S.C., Kato, M., et al. Time-domain and spectral-domain investigation of inflectionpoint slow-light modes in photonic crystal coupled waveguides. Opt. Exp. 15, 3543–3549 (2007) 37. O’Brien, D., Settle, M.D., et al. Coupled photonic crystal heterostructure cavities. Opt. Exp. 15, 1228–1233 (2007) 38. Fano, U. Effects of configuration interaction on intensities and phase shifts. Phys. Rev. 124, 1866–1878 (1961) 39. Fan, S. Sharp asymmetric lineshapes in side-coupled waveguide-cavity systems. Appl. Phys. Lett. 80, 910–912 (2002) 40. Fan, S., Suh, W., et al. Temporal coupled mode theory for Fano resonances in optical resonators. J. Opt. Soc. Am. A 20, 569–573 (2003) 41. Suh, W., Wang, Z., et al. Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multi-mode cavities. IEEE J. of Quantum Electron. 40, 1511–1518 (2004) 42. Fan, S., Villeneuve, P.R., et al. Theoretical investigation of channel drop tunneling processes. Phys. Rev. B 59, 15882–15892 (1999) 43. Taflove, A. Hagness, S.C. Computational electrodynamics: The finite-difference time-domain method. Artech House, Norwood (2005) 44. Otey, C., Povinelli, M.L., et al. Capturing light pulses into a pair of coupled photonic crystal cavities. Appl. Phys. Lett. 94, 231109 (2009) 45. Otey, C., Povinelli, M.L., et al. Completely capturing light pulses in a few dynamically tuned microcavities. IEEE J. Lightwave Technol. 26, 3784–3793 (2008) 46. Lipson, M. Guiding, modulating and emitting light on silicon- challenges and opportunities. IEEE J. Lightwave Technol. 23, 4222–4238 (2005) 47. Xu, Q., Sandhu, S., et al. Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency. Phys. Rev. Lett. 96, 123901 (2006) 180 S. Fan et al. 48. Xu, Q., Dong, P., et al. Breaking the delay-bandwidth limit in a photonic structure. Nature Phys. 3, 406–410 (2007) 49. Soref, R.A., Bennett, B.R. Electrooptical effects in silicon. IEEE J. of Quantum Electron. 23, 123–129 (1987) 50. Xu, Q., Schmidt, B., Pradhan, S., Lipson, M. Micrometre-scale silicon electro-optic modulator. Nature 435, 325–237 (2005) 51. Borselli, M. High-Q microresonators as lasing elements for silicon photonics (Thesis). California Institute of Technology, Pasadena (2006) 52. Noda, S., Fujita, M., et al. Spontaneous-emission control by photonic crystals and nanocavities. Nat. Photonics 1, 449–458 (2007) 53. Tanaka, Y., Upham, J., et al. Dynamic control of the Q factor in a photonic crystal nanocavity. Nat. Mater. 6, 862–865 (2007) 54. Pan, J., Huo, Y., et al. Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning. Appl. Phys. Lett. 92, 103114 (2008) 55. Longhi, S. Stopping and time reversal of light in dynamic photonic structures via Bloch oscillations. Phys. Rev. B 75, 026606 (2007) 56. Yang, Z.S., Kwong, N.H., et al. Distortionless light pulse delay in quantum-well Bragg structures. Opt. Lett. 30, 2790–2792 (2005) 57. Yang, Z.S., Kwong, N.H., et al. Stopping, storing, and releasing light in quantum-well Bragg structures. J. Opt. Soc. Am. B 22, 2144–2156 (2005) 58. Shen, J.T., Povinelli, M.L., et al. Stopping single photons in one-dimensional quantum electrodynamics systems. Phys. Rev. B 75, 035320 (2007) 59. Gaburro, Z., Ghulinyan, M., et al. Photon energy lifter. Opt. Exp. 14, 7270–7278 (2006) 60. Notomi, M., Mitsugi, S. Wavelength conversion via dynamic refractive index tuning of a cavity. Phys. Rev. A 73, 051803(R) (2006) 61. Preble, S.F., Xu, Q.F., et al. Changing the colour of light in a silicon resonator. Nat. Photonics 1, 293–296 (2007) 62. McCutcheon, M.W., Pattantyus-Abraham, A.G., et al. Emission spectrum of electromagnetic energy stored in a dynamically perturbed microcavity. Opt. Exp. 15, 11472–11480 (2007) 63. Shin, J., Shen, J.T., Catrysse, P.B., Fan, S. Cut-through metal slit array as an anisotropic metamaterial film. IEEE J. Sel. Top. Quantum Electron. 12, 1116–1122 (2006)