Slow and Stopped Light in Coupled Resonator Systems

Theory

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

Here ε (t) represents the dielectric modulation and ε 0 is the background dielectric constant. Both ε 0 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 k 0 , with electric field described by E (t) = f (t) e i(ω 0 t−k 0 x) , where ω 0 = k 0 / √ μ 0 ε 0 , we have

By using a slowly varying envelope approximation, i.e., ignoring the ∂ 2 f /∂t 2 term, and by further assuming that the index modulations are weak, i.e., ε(t) << ε o , (7.2) simplifies to

3)

which has an exact analytic solution:

where t 0 is the starting time of the modulation. Thus the "instantaneous frequency" of the electric field for this wave vector component is ω (t) = ω 0 1 − ε (t) 2ε 0 . (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 ω 1 to ω 2 , modulations at a frequency ω 2 − ω 1 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.

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][35][36][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 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.

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 ω a,b ≡ ω 0 ± 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. 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

is satisfied, the transmission spectrum features a peak centered at ω 0 . 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.

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

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 ω a and ω b . The width of the middle band depends strongly on the resonant frequencies ω a , ω b . 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].

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 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 excitation reaches its peak at t = 0.8 t pass , where t pass 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 t pass ), and the pulse propagates at a relatively high speed of v g = 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 t pass ). 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 t pass , and the pulse is then released by repeating the same index modulation in reverse (Fig. 7.4b, t = 6.3 t pass ). 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.

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 k c as

where ω

It can be shown [16] that the output width of the pulse in time ( t out ) after a total delay time τ is given by

where it is assumed that v g (τ ) = v g (0). For a static system, this reduces to the result 9) and the pulse spreads with increasing delay. For the dynamic system, however, ω

k c (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,

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.

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 ω 1 = ω 2 = ω 0 and γ 1 = γ 2 = γ 0 , the system has an eigenstate, with eigenfrequency ω 0 − β and resonator amplitudes α 1 = −α 2 , which does not leak into the waveguide. Starting from this dark state, if the resonators are tuned to ω 1 = ω 2 , 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 ω c = ω 0 − 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 [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.

Experimental Progress

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].

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 × 10 6 in a silicon ring resonator [51] suggests that storage times of several nanoseconds may be possible.

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.

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 × 10 6 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.

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.

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].

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.