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We introduce recent advances in dynamic control over the Q factor of a photonic crystal nanocavity system. By carefully timing a rapid increase of the Q factor from 3800 to 22,000, we succeed in capturing a 4ps signal pulse within the nanocavity with a photon lifetime of 18ps. By performing an additional transition of the Q factor within the photon lifetime, the held light is once again ejected from of the system on demand.
©2008 Optical Society of America
1. Introduction
High quality (Q) factor photonic crystal (PC) nanocavities have been receiving a great deal of attention because of their ability to strongly confine photons in a tiny space [1–3]. Recent advances in their design are approaching theoretical Q values of 109 [4] and refinement of fabrication processes have resulted in experimental ultra-high Q factors of 2.5 million [5], all with modal volumes on the order of a cubic wavelength. The photon lifetimes of ultra-high Q factor nanocavities have reached the nanosecond regime experimentally. Unfortunately, their resulting ultra-narrow resonant spectra dictate that only optical pulses of temporal duration as long or longer than the cavity’s photon lifetime can be effectively coupled to them. However, if a method to efficiently capture relatively short pulses in high Q cavities for controllable periods of time was developed, several areas of active research would benefit significantly. These include nonlinear optics [6,7], where confining short pulses with high peak powers could lead to enhanced the nonlinear effects; slowing/stopping light [8,9] for all-optical communication; and cavity quantum electrodynamics [10–12]. In 2005, we proposed that this limitation could be overcome by introducing the concept of a dynamic change of the Q factor [13]. In this method the Q value is initially set at a very low value (allowing efficient coupling of a short pulse) then is rapidly increased to a high value (ensuring a photon lifetime greater than the pulse duration) as light is introduced into the cavity. We have recently demonstrated a technique to dynamically increase the Q factor of a PC nanocavity within picoseconds by manipulation of interference conditions [14]. Another method of control by the dynamic detuning of coupled ring resonators has also been demonstrated [15]. In this paper, we experimentally demonstrate the above concept with this technique: short pulses can be efficiently held within cavities with photon lifetimes significantly longer than their pulse duration. Moreover, we introduce an approach for on-demand, dynamic lowering the Q factor to eject light from the cavity when desired. This multi-step, dynamic control over Q is demonstrated theoretically and experimentally.
2. Sample fabrication and Q control technique
The photonic crystal device used in this study is shown in Fig. 1. The sample is made of an air-suspended 250 nm thick silicon slab with triangular lattice of air holes with lattice constant a=413.75 nm and air hole radius of 120 nm. A cavity consisting of three missing holes is placed next to a waveguide formed by a single row of missing holes. The air holes at the cavity edges are shifted by 0.15a to increase the vertical Q (Qv) [1]. The resulting nanocavity’s resonant wavelength is 1550 nm and Qv is 55,000. The coupling strength between the nanocavity and the waveguide can also be quantified by a Q factor of 3000 (Qin-orig). A hetero-interface, made by changing the lattice constant to 403.75 nm, is introduced 80a (~33 µm) away from the center of the nanocavity and acts as a perfectly reflecting mirror. This changes the strength of the in-plane coupling from the original value Qin-orig, which gives us the freedom to control the in-plane Q factor.
Light leaving the cavity travels in both the left and right directions in the waveguide. Right-moving light will reflect back off the hetero-interface and return to interfere with the left-moving light. The phase difference (θ) between these two beam paths determines the power of light which leaks through the waveguide. The renormalized in-plane Q factor of the cavity-waveguide-mirror system (defined as Qin) calculated by the coupled mode analysis [16,17] is given as:
Accordingly, the total Q factor of the system (Qsys) can then be expressed as:
Dynamic control of the Q factor is achieved by altering θ. Assuming Qv≫Q in-orig, the minimum (θ=0) and maximum (θ=π) values of this dynamic range will be Q in-orig/2 and Qv, respectively. Experimentally, we use a control light pulse which strikes and is absorbed by the waveguide between the cavity and the hetero-interface, generating free-carriers and lowering that region’s refractive index by the carrier plasma effect. A optical fiber based, tunable, passively mode-locked laser (operating wavelength 1535–1555 nm, pulse width ~4 ps) and a second harmonic generator are used as light sources. The fundamental light pulse is the light to be captured (signal pulse) and a frequency doubled pulse (~4 ps duration, center wavelength ~775 nm) is used as the control to be shone on the waveguide. The control pulse spot size that is produced by the current experimental set-up has a Gaussian distribution with a full width half maximum no smaller than ~10 µm. This limitation of the experimental system is what dictates that the separation between cavity and hetero-interface must be about 30µm, rather than any inherent property of the device itself. A variable optical delay line is used to change the timing between them.
3. Holding light by increasing Q
We have already proposed that dynamic control over the Q factor is essential to efficiently capture a short light pulse with a cavity lifetime longer than the pulse duration [13]. In order to explain the concept, the time evolution of the cavity energy (E cav) when a 4 ps optical pulse is injected into the system is calculated using coupled mode theory for three different cases and is shown in Fig. 2. The red and black lines show the behavior for example static Q values, low Q and high Q, respectively. For the black line, where Q is low (~4000), because the photon lifetime in the cavity matches the pulse duration, a large amount of the pulse couples easily into the cavity, but will only be held for a brief amount of time. Conversely, for the red line where Q is high (~22,000) the photon lifetime is longer but spectral mismatch will prevent much of the light from initially coupling into the cavity at all. The blue line shows the dynamic case where initially the effective coupling of the low Q condition (~4000) is used, and then a 4 ps transition beginning at 13 ps increases Q to the higher value (~22,000) to achieve a longer photon lifetime. This result shows successful introduction of a greater component of the signal pulse for a longer period, thus combining the positive aspects of the two static cases.
Fig. 2. Coupled mode theory calculation of cavity energy vs. time calculated for various Q conditions. Black line: low Q (~4000). Red line: high Q (~22,000). Blue line: dynamic change from low to high Q over 4 ps.
Based on this concept and the dynamic Q control technique explained in the previous section, we performed an experiment to hold more light in the cavity for a longer period. Here, we used the vertical emission from the cavity as an indicator of the cavity status. The cavity energy leaks out according to the following equation:
Because Qv is constant, the rate at which energy leaves the cavity via vertical emission will always be proportional to the cavity energy at that time. Therefore the time-integral of the vertical emission, which is our experimental observable, is proportional to the time-integral of the cavity energy over time
This means that an increase of the time-integrated total vertical emission power indicates that a larger portion of the signal pulse is successfully held for a longer period. In addition, the spectrum of the total vertical emission reflects the average total Q factor of the cavity (Q sys) that the signal light experiences.
In the experiment we set the initial value of θ~0 and by the irradiation of a control pulse, changed θ to π. The energy of the signal pulse was set relatively low at ~2 pJ at the input edge of the waveguide to avoid nonlinear effects in the cavity by the signal pulse itself [18]. The total vertical emission power was measured as a function of the time difference between signal and control pulse arrivals Δt (Fig. 3(a)). If this time difference between the pulses was sufficiently large, it corresponded to one of the static Q cases. Therefore, the Δt range can be divided into three sections: (1) simultaneous arrival of both pulses (Δt~0, dynamic Q), (2) signal preceding control (Δt<0, static low Q), and (3) control preceding signal (Δt>0, static high Q). Dynamic shift to the θ=π state was achieved by varying the control pulse energy in case (3). The observed vertical emission exhibits a local minimum because before the signal pulse arrives, the cavity is effectively de-coupled from the waveguide, preventing light from entering the cavity. Once dynamic transition to the desired state was established, direct measurement of the control pulse energy showed it to be 35 pJ.
In case (1), the vertical emission power was maximum, 3 times larger than in (2) and about 9 times larger than for case (3).
The spectra of the vertical emission were also measured for the three representative Δt values (Fig. 3(b–d)). In case (1), the spectrum (Fig. 3(b)) showed a narrow linewidth of which the measured Q factor (Q sys) was 22,000. In case (2), the spectrum of the vertical emission (Fig. 3(c)) showed a low Q sys of 3800. In case (3), the vertical emission power was very low compared to the other cases so the corresponding spectrum (Fig. 3(d)) was difficult to distinguish. For case (1), the observed spectrum is almost entirely determined by the final high Q sys value as the signal light experiences this state during the majority of its interaction with the cavity as seen from Fig. 2 (blue line). Therefore, the observed Q sys (22,000) represents the photon lifetime (τ), which is calculated to be ~18 ps. For case (2), the observed Q sys (3800) represents the Q factor preset for the experiment, where the photon lifetime is calculated to be ~3 ps.
Fig. 3. Results of improved single-step dynamic Q control a) vertical emission vs. time difference between control and signal pulses b) spectrum of vertical emission for simultaneous case (low to high Q) c) spectrum of vertical emission for signal early case (low Q) d) spectrum of vertical emission for control early case.
These results can be explained as follows: (1) The light energy that enters cavity in case (1) and (2) is considered to be almost same but the vertical emission is larger for case (1) since the photon lifetime is made much longer by the dynamic change to the high Q state. (2) The photon lifetime is considered to be almost same for case (1) and (3) but the light energy that captured in the cavity is larger for case (1) due to the initial low Q factor. It can be roughly estimated from the spectral linewidths that the photon lifetime for case (1) is 6 times longer than that for case (2), and from the time-integrated radiation that the light energy captured for case (1) is at least 9 times larger than that for case (3). Therefore, we can say that the dynamic control of the Q factor makes it possible to couple a larger amount of light into the cavity and to hold it for longer period of time, which are impossible to attain simultaneously with static Q factors.
Now we discuss these cases more quantitatively. According to the above results, Q sys can be changed from 3800 to 22,000 by the irradiation of the control pulse. This indicates that Q in increases from 4100 to 36,700 assuming the constant Qv of 55,000. The difference between the dynamic range of Q in expected from the design (from 1500 to complete cutoff) and that experimentally observed can be attributed to imperfections of the fabricated device and to the absorption of light by the induced free carriers. The original Q in strongly depends on the air hole size, meaning that small variations during fabrication can lead to noticeable differences to the lower bound of Q in. The free carriers that produce the desired change in the dielectric constant of the waveguide also introduce an imaginary component. Thus as light travels from cavity to hetero-interface and back to fulfill the interference condition responsible for high Q, there is some loss due to free carrier absorption. Various methods of limiting the losses from both these factors are currently under investigation.
Next, we have carried out 2D FDTD simulations of the present device to evaluate the performance more precisely. The structure, the value of Qv and the range of Q in were set to reproduce the parameters of the experiment. The 4 ps signal pulse was assumed to be transform-limited, and free carrier absorption was included. The results are shown in Fig. 4. It is found from the calculation that the portion of the total signal pulse energy that enters the cavity at the peak time for cases (1) and (2) is 43%, with an additional 33% in the waveguide between the cavity and the hetero-interface producing the interference conditions responsible for the dynamic Q in. In case (1) the high Q condition causes all 76% of this signal pulse energy to remain held, coupling back and forth between the cavity and this section of the waveguide. Therefore, case (1) could hold as much as three quarters of the signal pulse energy with a photon lifetime of ~18ps, which is 6 times longer than that for the static low Q case (2). To extend this hold time further, the upper bound of the dynamic Q range could be increased by using cavities with inherently higher Qv and reducing free carrier losses. This would also slightly increase the amount of the signal pulse captured.
Fig. 4. 2D FDTD simulation of cavity energy behavior over time for different Q conditions using experimental parameters. Black line: low Q case (Q sys~4000). Red line: high Q case (Q sys~22,000). Blue line: dynamic shift of Q from low to high by a 4 ps control pulse.
It is also found that the design of this device results in less light being held in the static, high Q (22,000) condition in case (3), than could be held in a simpler system with the same high quality factor as shown in Fig. 2. This is because dynamic Q control depends on the interference of light coupling from the cavity back to the waveguide. The distance between cavity and hetero-interface contributes a time delay to one of the optical paths, meaning the high Q condition takes longer to establish. The cavity energy response under this condition (red curve in Fig. 4) shows a low peak of energy entering then quickly leaving the cavity until the interference condition stabilizes with a much smaller quantity of the pulse energy still contained in the cavity. Therefore, even though the quantity of case (3) vertical emission observed in Fig. 3(a) is small, the bulk of it comes from this initial quantity of light that soon leaves the cavity and not from the portion of light that experiences the 18 ps lifetime, which might explain the relatively broad spectrum just visible above the noise floor in Fig. 3(d). This means that the amount of light actually held in the cavity for 18 ps by case (3) is much less than 1/9 times the light held in case (1). The influence of the delay in the interference should be also taken into account to optimize the portion of energy captured in the dynamic case. To significantly increasing the captured energy the time would require the high Q condition to be established more quickly by using shorter control pulses and reducing the travel time from cavity to hetero-interface.
4. Decreasing Q by an additional control pulse
Having established a system for introducing and holding a greater portion of the signal pulse within the cavity, the next step is to develop a method to eject it again on demand. This requires that the Q factor be dynamically returned to its minimum value. Since the system’s Q factor is determined by Eq. (1), Q can be lowered again by moving θ forward to 2π. This can be accomplished by a second control pulse arriving some time T after the first increased Q. By suddenly lowering Q in, this second control pulse would cause the cavity energy E cav to rapidly drop as the light in the cavity would once again be ejected preferentially down the waveguide. In this device design, the signal pulse exits from the same port used to inject it, making it difficult to observe directly. As in the first experiment, the time-integrated vertical emission from the cavity can be used to appraise the new operation. As mentioned before, the time-integral of the vertical emission, which is our experimental observable, is proportional to the time-integral of the cavity energy over time. Thus suddenly discharging the light from the cavity via the waveguide should be observed by a decrease of the vertical emission. For experimental identification, the response of the vertical emission to both the timing and magnitude of the second change of θ was first analyzed using the FDTD method.
First we examine the system where timed Q increase has already occurred to hold light in the cavity and a short time later (T=5 ps) a second phase shift of variable magnitude is induced. At this timing, most of the signal pulse energy is still in the system, making the vertical emission’s response to the second phase change clearly observable. Figure 5 shows the cavity energy and flux of energy exiting the waveguide over time for varying quantities of phase shift from the high Q condition. The first case shows only the increase of Q, with the initial shift of θ from 0 to π, as described in the previous section. The cavity energy E cav decays slowly (a) and none of it escapes through the waveguide (b). The total observed vertical emission, being proportional to the time integral of (a), will result in the maximum observed vertical output. If a small second change is introduced, it increases the decay rate of E cav slightly (c) as the lowered Q in allows some light to leak back out into the waveguide (d). The observed vertical emission would thus be reduced. At the minimum Q condition (shift θ from π to 2π), E cav falls sharply (e) and a pulse is clearly seen exiting the waveguide (f), minimizing the total vertical emission. Increasing the phase difference beyond 2π yields interesting results: Because the change of θ is not instantaneous, when θ moves from π to 3π, effectively shifting the system to the next pulse capture condition, there is an opportunity for some of the light to escape the cavity as the phase difference moves past 2π. A portion of the light does not escape quickly enough and thus remains in the cavity under the new high Q state (g) but the rest exits via the waveguide (h). Further increase to the phase difference causes the cycle to repeat, the light remaining in the cavity has a faster decay rate (i) as it leaks out the waveguide (j).
Fig. 5. Simulation of system’s response to phase shifts induced by the second control pulse a short time after first control pulse. (a–j) Calculated energy flux leaving the cavity vertically or down the waveguide over time for various phases.
Having established this relationship between E cav over time and the pulse successfully exiting the cavity, the former is used to predict the device response in experiment. The simulated response of the total vertical emission to increasing magnitude of the second change of phase is shown in Fig. 6(a). Maximum emission occurs when there is no second change, is minimized when the minimum Q condition is reached, and then cycles on to a local maximum when the phase difference reaches another odd integer of π.
To confirm this behavior experimentally, a second, identical control pulse was installed in the experimental set-up with independently controllable delay and intensity. The timing and phase change of the original control pulse were set to dynamically increase Q and thus hold the signal pulse as before, with the second control pulse striking the system approximately 5 ps later. By increasing the second pulse’s energy, the magnitude of phase shift caused by refractive index change increased. The observed behavior of the vertical emission shown in Fig. 6(b) closely follows the predicted results from simulation and continued to oscillate as the second control pulse energy rose further. The oscillatory pattern of the vertical emission measurements demonstrates that the second control pulse is indeed capable of causing the cavity energy to once again couple preferentially to the waveguide. By setting the second control pulse energy to the first minimum of Fig. 6(b), we can cause the light remaining in the cavity to exit down the waveguide most effectively.
Fig. 6. (a) Predicted observable vertical emission as a function of second phase shift, using the data from Fig. 5(b) Measured results of vertical emission for increasing energies of the second control pulse.
Next we consider the system response to the quantity of delay time T between the two dynamic Q events. The simulation conditions that achieved θ=2π (Fig. 5(e) and (f)) were repeated for various T as shown in Fig. 7(a). When light is ejected from the cavity, the trailing tail of the cavity energy over time is cut off (see inset), so the total vertical emission is decreased. The predicted response of total vertical emission as a function of T is shown by the black curve. If T is very large, then the total emission approaches the level observed when only the first control pulse is used (blue line). The difference in vertical emission between these two lines for a given T is proportional to the light remaining in the cavity at the instant that Q is lowered. This dependence on T is determined by the photon lifetime of the cavity after the first control pulse.
Using the established experimental conditions for successful increase and subsequent decrease of Q, the experiment was repeated for various T, (Fig. 7(b)). When T was sufficiently long (e.g.: 60 ps), effectively all of the light had leaked out of the cavity before the second control pulse arrived, thus there was no discernable difference in vertical emission from the pulse capture case. However, as T was shortened, the total measured vertical emission decreased, approaching to the value without either control pulse. The dependence of the vertical emission on T agreed well with the curve predicted by the photon life time determined in the first experiment (18 ps). These results (Fig. 6 and 7) clearly indicate that the induced exit was successfully demonstrated: the light remaining in the cavity preferentially escapes down the waveguide when the second control pulse lowers Q in, and by controlling when this occurs, we can dictate when the held light exits the cavity. The device design no longer dictates how long the light is held in the cavity, it can be dynamically ejected at any time of our choosing within the photon lifetime.
Fig. 7. Vertical emission as a function of delay between the control pulses T (a) Simulated response of cavity energy to different times between the two changes of Q, compared to the case with only Q increase (blue line). (b) Experimental response of the vertical emission between the Q increase only case (blue line) and static Q case (red line) for various T. Dashed line indicates a photon lifetime of 18 ps.
In order to extend the photon lifetime in the cavity still further, the upper bound of the dynamic Q range can be increased by using cavities with higher vertical Q factors, such as the heterostructure-based designs [2], improving the fabrication process and by refining the control pulse systems to limit free carrier losses. Investigating alternative methods for rapidly changing the phase difference of the interference should also be considered to increase the quantity of the pulse that can be held in addition to extending the photon lifetime. In future work the goal will be to eject the held light down an alternate waveguide, permitting direct observation of the outgoing pulse and bringing us closer to practical dynamic systems for light pulse control and eventually even the dynamic control of single photons.
5. Summary
In conclusion, by developing an all-optical system that can both dynamically increase and decrease the Q factor of a PC nanocavity system within its photon lifetime, we have succeeded in capturing a 4 ps signal pulse in a cavity with a Q value of 22,000, and then discharging it at a time of our choosing up to 18 ps later. This multi-step dynamic control of the Q factor is a significant advancement for nanocavities because it overcomes the spectral mismatch problem and allows pulses to be held for periods of time much longer than their temporal duration. We believe these results are very useful for various research and application areas such as optical nonlinear optics, slowing/stopping light, and quantum electrodynamics.
References and links
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