Stopping of Light by the Dynamic Tuning of Photonic Crystal Slow Light Device

: We propose a simple technique of stopping light pulses using a slow-light device based on photonic crystal coupled waveguide (PCCW). Dynamically tuning the material index chirp in the PCCW adiabatically transforms slow-light pulses into stopped ones. We demonstrate this in finite-difference time-domain simulation assuming ideal and actual tuning of the index chirp. In the ideal case, the group velocity of the almost stopped pulse is reduced to 190 times smaller than that of simple slow light pulse. The smallest limit is affected by the timing error of the tuning between wavelengths. Re-ordering and stopping of a pulse train are possible by optimizing the device length and timing. As a practical tuning method, we discuss carrier effects induced by photo-excitation. Taking into account carrier distribution and free carrier absorption, the actual behaviors of stopped light are estimated. We define and evaluate an effective delay-bandwidth product, which is affected by free carrier absorption.


Introduction
The slowing and stopping of light pulses have been studied extensively for optical buffer memory and advanced time-domain signal processing in future photonic networks, interconnects, and instrumentations. Photonic nanostructures, which show large first-order dispersion dk/dω (k: wave number, ω: angular frequency), generate on-chip slow light with a group velocity υ g of typically a hundred times smaller than the light velocity c in vacuum [1]. So far, coupled-resonator waveguides [2,3], all pass filters [2,4], photonic crystal (PC) waveguides [5][6][7][8][9], and metamaterials [10,11] have been exploited as device structures. Those except for metamaterials are made of transparent dielectric media and can be low loss devices. The delay-bandwidth product (DBP) is evaluated as an essential performance factor for these devices [1]. It indicates that large dk/dω is obtained only in a narrow frequency bandwidth, and it constrains the maximum buffering capacity, i.e. the number of pulses buffered in the device, when the pulses are not broadened by group velocity dispersion (GVD). We have succeeded in experimentally demonstrating a high DBP of 110 in PC coupled waveguide (PCCW) of 800 µm length [8]. However, it is difficult to further enhance the DBP without increasing the length. The stopping of light, obtained by dynamically tuning photonic nanostructures, relaxes this constraint [12][13][14]. Here, dk/dω is enhanced instantaneously by locally changing the material index of the device in the presence of light pulses. Then, the device and pulses simultaneously narrow their bandwidths, and consequently the pulses almost stop. Some studies experimentally demonstrated similar operation by controlling the Q factor in a single cavity or coupled cavities [15][16][17][18].
In this paper, we propose a simple dynamic tuning in chirped PCCW. In a uniform PCCW, the photonic band of a waveguide mode becomes flat at a single frequency and dk/dω diverges to infinity, resulting in zero υ g [19]. When some structural parameters are gradually changed along the device (i.e. chirped structure), the flat band frequency is shifted so that the slow light effect is averaged over a finite bandwidth and slow light pulses can transmit with a moderate delay. The large DBP mentioned above is obtained in such a chirped PCCW. If the chirp range is changed instantaneously from an initial value to zero, slow light pulses with a wide spectrum are converted to stopped ones at a single frequency. Such dynamic tuning is possible by some practical methods. In comparison with other methods, it downsizes the device footprint because incident pulses are pre-compressed in space by the slow light effect before the tuning. Also, it allows subsequent pulses to pass through and stop in the device, which enable the re-ordering and stopping of a pulse train, respectively. This paper first explains the principle of the tuning process in detail in Section 2. Then, the stopping of a single pulse is demonstrated in finite-difference time-domain (FDTD) simulation in Section 3, assuming ideal tuning of material index. Here, we discuss some fundamental properties and limiting factors. In Sections 4 and 5, the re-ordering and stopping of a pulse train are demonstrated, respectively. Finally in Section 6, a real operation is discussed, assuming photo-excited carriers to change the chirp in a semiconductor device. Here, we take into account nonlinear carrier distribution, carrier plasma and band filling effects, and free carrier absorption (FCA), and discuss an effective DBP restricted by the FCA. Figure 1(a) shows a schematic of the PCCW consisting of two line defects in a triangle lattice PC slab, with lattice constant a and background airhole diameter 2r. The diameter 2r' of the center row of airholes and the position of other airholes are modified. Figure 1(b) shows the corresponding photonic band calculated by two-dimensional (2D) FDTD method with the periodic and absorbing boundary condition. We assume an equivalent slab index n eq of 2.917, normalized hole diameters 2r/a = 0.59, 2r'/a = 0.20, normalized shifts s 1 /a = 0.10 and s 2 /a = 0.20, and a polarization parallel to the 2D plane. Two coupled modes appear, and the even mode exhibits the target flat band at a normalized frequency ωa/2πc ≡ a/λ = 0.257 (λ: wavelength in vacuum). Here we define the flat band as that sandwiched by two white dots in Fig. 1(b), which are separated by the spectral resolution in this calculation i.e. ∆ω res a/2πc = 6.7 × 10 −5 . The band shifts to higher frequencies as n eq decreases along the device. When a light pulse is incident on the device, each frequency component of the pulse reaches the slow light condition of the flat band at a different position. This means that, under the slow light condition, the pulse profile is transformed into a Fourier spectral distribution. Before and after passing through this condition, it passes GVD(1) and GVD(2) in Fig. 1(b), respectively. Thus, the total GVD is compensated and the profile of the incident pulse is recovered at the output end. The bandwidth of slow light is proportional to the chirp range. On the other hand, the effective length of slow light for each frequency component is inversely proportional to the chirp range. In consequence, the delay and bandwidth are constrained by the DBP [1,14]. Now we propose the dynamic tuning, as shown in Fig. 1(c). Here, the initial, fixed chirp is formed so that n eq decreases along the device. A control pulse is incident on the device behind the signal pulse with an appropriate delay and velocity. The control pulse chases the signal pulse, and successively catches up with each frequency component that is slowing down. By means of index change induced by the pump pulse, (such as carrier effects, optical Kerr effect), n eq at this position is instantaneously reduced. The amount of index change ∆n eq ' slopes along the device because the control pulse decays due to some propagation losses. In the ideal case, such dynamic chirp ∆n eq ' cancels the initial fixed chirp ∆n eq , and all frequency components of the situation when light at the flat-band frequency enters a chirp-less PCCW. Therefore, the delay is extended dramatically. In the reverse process, ∆n eq ' is removed from the input to output side, and the initial pulse profile is recovered.

Simulation
In this section, we demonstrate the stopping of light using 2D FDTD simulation. Here, the model in Fig. 1(a) is used, and n eq decreasing linearly from 2.963 to 2.870 (∆n eq = 0.093) is assumed as the initial fixed chirp for a device length L = 125a. This leads to a band shift of ∆ω c a/2πc = 8.3 × 10 −3 , depicted by the gray region in Fig. 1(b). The signal pulse is launched in the even mode, at the input end of two waveguides of the PCCW, by in-phase Gaussian excitation with ω p a/2πc centered at 0.263. Normalized time and spectral full widths at half maximum (FWHM) of the pulse are set at c∆t p /a = 290 and ∆ω p a/2πc = 1.53 × 10 −3 , respectively. This spectral FWHM is sufficiently covered by the band shift. These values correspond to ∆t p = 0.39 ps and wavelength FWHM ∆λ p = 8.7 nm for a = 0.40 µm and λ p = 1.53 µm, which are often seen in experiments. The square control pulse forming the ideal dynamic chirp is incident with a normalized delay c∆t d /a = 720 and a velocity of c/3.25, which equals the modal group velocity at ∆ωa/2πc = 0.01 higher than the flat band in Fig.  1(b).
Figures 2(a)-(d) compare the propagation of slow light pulse without and with the dynamic tuning. ((i)-(l) are their animations, respectively.) Without tuning (a), the pulse enters the slow light region and pauses at ct/a = 300, and exit at 1200. With the tuning at the best timing (c), the slow light pulse transforms into a stopped pulse, almost maintaining its profile at least until ct/a = 3000. We notice through careful observation that a small amount of light escapes from the stopped pulse in the forward direction. This is dues to slight timing error of the tuning between different wavelengths against the constant tuning velocity. When the tuning is early (b), the pulse propagates without pause because the tuning takes place before the light (particularly high frequency components) reaches the slow light condition. With late tuning (d), the pulse propagates with dispersion before the dispersion compensation of GVD(1) and GVD(2) is completed.
To clarify the best timing quantitatively, we estimated the motion of the pulse from the band curve. In a linearly chirped structure, the band shift with position z can be considered as the frequency shift along a fixed band [19]. Then, z and ω are related as where the subscript 's' denotes the slow light condition. The time t is expressed with respect to z as [ ] Since k(ω) is equivalent to the photonic band, we can derive the light propagation by applying Fig. 1(b) and Eq. (1) to (2). The result is shown in Fig. 2(e)-(h). Here, colors represent different frequency components overlapping with each other at the input end to form the signal pulse. Light outside of the pulse duration is neglected as they cancel with each other by interference. In (e), different frequencies branch, and the earliest and latest parts of each frequency reach the slow light condition on upper dotted and dashed lines, and exit on lower dotted and dashed lines, respectively. Finally, the different colors overlap again to form the output pulse. Thus the light propagation in (a) is well explained by (e). The gray region shows the timing window, in which all the frequency components are lying under the slow light condition. If the thick blue line indicating the motion of the tuning overlaps with the gray region, velocities of all the frequencies are fixed by the tuning. If the dynamic chirp completely cancels the initial chirp under this condition, all the frequencies are blue-shifted to the same final frequency and the velocities become almost zero. Figure 2(g) shows a case close to the ideal one, but small errors occur at edge frequencies, which are caused by the mismatch between slopes of the blue line and gray region. This is the reason that light partly escapes from the stopped pulse in (c). At off-timing (f, h), velocities fixed after the tuning are disrupted, resulting in large dispersion of the pulse.   From (2), the delay due to slow light, ∆t s , is given as where ∆k s is the shift of k at the flat band (distance between white dots in Fig. 1(b)). To achieve the stopping of light, the gray region must be opened by the condition ∆t s -∆t p > 0. For a Gaussian pulse satisfying ∆t p (∆ω p /2π) = 0.44, this condition is rewritten as From Fig. 1(b) and parameters assumed above, ∆ω p /∆ω c = 0.18, ∆k s /2π = 0.028/a and L = 125a, which lead to (∆ω p /∆ω c )(∆k s /2π)L = 0.63 > 0.44. Here, the spectral efficiency ∆ω p /∆ω c will be enhanced up to 0.5 by narrowing the signal pulse and/or the chirp range. ∆k s increases slightly when the structure is optimized; the maximum ∆k s /2π would be limited to around 0.04/a [19]. Then, the shortest L satisfying (4) is derived as 22a, which is 9 µm for a = 0.40 µm. Figure 3 summarizes velocities at three different parts of the pulse. White circles show the velocity at the weighted center, which is equivalent to υ g . Blue and red circles show those at front and back ends whose intensity is 10% of the pulse peak. Dispersion can be evaluated from their difference. For example, dispersion is minimum and υ g = 4.2 × 10 −4 c (group index n g ≡ c/υ g = 2400) at c∆t d /a = 720 when the pulse looks to be almost stopped in Fig. 2(c). From this value, the normalized DBP, defined as n g (∆ω p /ω p ) [1], is evaluated to be 14. For slow light without tuning, it is evaluated from the delay inside the device in Fig. 2(a) and the spectral FWHM ∆ω p to be 0.075. Thus, the tuning provides 190-fold enhancement. Since the spectral FWHM ∆ω p is 5.4 times narrower than the slow light band ∆ω c in this simulation, the normalized DBP without tuning can be enhanced by expanding ∆ω p and/or narrowing ∆ω c . The spectra should be optimized similarly for the stopped pulse to maintain its advantage. However, if ∆ω p is comparable to ∆ω c , the timing error at low frequencies increases and the stopped pulse is dispersed more severely. If the tuning velocity is not constant but changes along the device so that the blue line in Fig. 2(e)-(h) is always included inside the gray region, the error will be neglected and υ g will be minimized closer to zero.

Re-ordering of pulses
If the dynamic tuning is performed with cavities, the transmission is limited inside the resonant spectrum. Therefore, when multiple pulses with the same spectrum are incident on the device, subsequent pulses cannot pass through the device after an earlier pulse is stopped by the tuning. In contrast, our method maintains continuous pass-bands around the frequency of stopped light, and so subsequent pulses can pass while earlier pulses are stopped. The order of pulses can be changed by releasing the earlier pulse after the subsequent pulses pass by, as demonstrated in Fig. 4. Here, four Gaussian pulses of c∆t p /a = 290 are successively launched with a peak-to-peak interval of c∆t i /a = 960. The tuning is performed for the second pulse with c∆t d /a = 720 and a duration equal to the interval. (b) shows a complicated light intensity profile due to overlap of pulses, but the corresponding animation (d) displays counterchange of the second and third pulses. The third pulse incident during the tuning propagates with a constant υ g . As observed in (c), the slow light pulses (first, last), the stopped pulse (second), and the non-stop pulse (third) almost maintain their initial shape at the output end. Such pulse re-ordering will be meaningful for some signal processing if each pulse has its own intensity, phase, quantum information, etc.

Stopping of pulse train
For optical buffering, it is particularly important to stop an entire pulse train simultaneously. When cavities are used for stopping pulses, it requires many cavities each assigned to one pulse, and complicated tuning process. Here, we show two different approaches to achieve this using our simple tuning process. As noted in Section 2, the pulse incident on chirped PCCW is expanded into a Fourier spectral distribution due to GVD(1). When a pulse train is incident, pulses do not pause separately but their spectral distributions overlap and slow down, keeping their initial time differences. If timing of the tuning is optimized for one pulse, the timing error occurs for other pulses, resulting in severe dispersion. The key is the blue line of the tuning overlapping with the gray region ∆t s -∆t p . In Fig. 2(e)-(h), the gray region is opened for the FWHM of one pulse. For densely-packed return-to-zero pattern of M pulses, ∆t p is extended to (2M−1)∆t p . Therefore, ∆t s must be extended similarly to keep the gray region still opening. Such a situation is obtainable by elongating the device (2M−1) times without changing the chirp range ∆n eq , so that the pulses decelerate and accelerate more slowly. For example, let us consider two pulses and a three-fold longer device, i.e. L = 375a. The second pulse is incident with an interval of c∆t i /a = 720. Then, the optimum delay of the tuning becomes c∆t d /a = 2520. Figure 5 shows the FDTD simulation of stopping two pulses. Incident pulses overlap and exhibit a complex interference pattern under the slow light condition, which is almost fixed and stopped after the tuning. It goes into action but does not separate into two pulses again after the tuning is removed. This is due to the incomplete dispersion compensation of the first structure. When the interval is slightly extended to c∆t i /a = 720, clear separation was confirmed in the same simulation. In Section 3, we discussed the shortest length of the device required for stopping one pulse with ideal parameters to be 22a. Therefore, M (>>1) pulses can be buffered in the device of approximately 44aM length, e.g. 1.8 mm with a = 0.4 µm for 100 pulses if the incomplete dispersion compensation is improved. The other method for stopping a pulse train is to divide areas, each of which stop one pulse. Figure 6 shows the case of stopping two pulses in a double-step chirped structure. Here, two chirped PCCWs are simply connected in series. Without tuning, pulses repeat slowing and moving in the PCCWs. If a moderate interval is set between the pulses, they slow simultaneously. If the tuning is performed at this moment, the pulses are equally stopped. Ideally, the tuning should cancel ∆n eq in each PCCW. But it is difficult to form a multi-step dynamic chirp. Here we use the single-step dynamic chirp with twice larger ∆n eq ', i.e. 0.186. Then, the index slope of each chirp is flattened although their indices after the tuning are not the same. In the FDTD simulation, delays of the second pulse and tuning are set to be c∆t d /a = 1440 and 2580, respectively. Figure 7(a) shows the light propagation in a two-step chirped structure with ∆n eq = 0.093. The tuning at the first chirping does not operate as expected; strong reflection occurs at the boundary with the index discontinuity. To confirm the expected operation, the same simulation was performed for a smaller ∆n eq of 0.0093, as shown in Fig.  7(b). Here, two pulses stop simultaneously although weak reflection still remains. After removing ∆n eq ', the two pulses appear at the output end. This approach can be applied to M pulses using M-step chirp. Two drawbacks are the reflection at the boundary particularly for pulses with a short interval and N-fold ∆n eq ' required.

Practical dynamic tuning using photo-excited carriers
So far, we presented the dynamic tuning with an ideal index change. To realize such index change, let us discuss carrier effects induced by photo-excitation, i.e. carrier plasma dispersion and band filling effects. As discussed in Section 2, pump light is incident on the device in the same manner as signal pulse. We can consider linear inter-band absorption and nonlinear two-photon absorption [20] of pump light at frequencies higher and lower than the bandgap frequency of the slab material, respectively. Let us discuss the linear absorption as it generates a carrier slope closer to a linear one, which is suitable for canceling the initial linear chirp. It should be noted that FCA cannot be neglected to estimate the real propagation of signal pulse in this approach. The totally absorbed power in a distance from 0 to z, P ab (z), is given by ab ab ex ( )