## Abstract

We have measured the quality (*Q*) factors and resonant wavelengths for 80 photonic crystal nanocavities with the same heterostructure. In this statistical evaluation, the *Q* factors varied according to a normal distribution centered at 3 million and ranging between 2.3 million and 3.9 million. The resonant wavelengths also fluctuated but with a standard deviation of only 0.33 nm. Such a high average *Q* factor and highly controlled resonant wavelength will be important for the development of advanced applications of photonic crystal nanocavities. Comparing the experimental values with calculated values suggests that factors other than structural variations of air holes, which decrease the *Q* factor, are indeed present in the fabricated nanocavities.

©2011 Optical Society of America

## 1. Introduction

Optical nanocavities in two-dimensional (2D) photonic crystal (PC) slabs have the unique properties of high quality (*Q*) factors and small modal volumes [1–3]. They are currently attracting particular attention as potential components in various advanced applications such as ultrasmall wavelength-selective filters [4–6], optical pulse memories [7–9], highly sensitive environmental sensors [10,11], novel emitters [12–15], and quantum information processing [16,17]. In order to realize such applications, it is important not only to increase the *Q* factors but also to precisely control the resonant wavelengths (*λ*) of the nanocavities.

We have previously presented an important design rule for increasing the theoretical *Q* factors (*Q*
_{ideal}) of nanocavities [1] and we have proposed photonic heterostructure nanocavities with *Q*
_{ideal} of more than ten million [2,3]. However, to date the highest experimentally measured *Q* factor (*Q*
_{exp}) is approximately 2.5 million [2,18]; the values of *Q*
_{exp} for different nanocavities with the same structure varied between 2.0 million and 2.5 million. Furthermore, *λ* fluctuated by several nanometers between different cavities. This discrepancy between *Q*
_{ideal} and *Q*
_{exp} and the partially-defined *λ* can mainly be attributed to nanometer-scale random variations in the radii and positions of the air holes that form the 2D PC [19]. We have recently performed a numerical investigation of the influence of these structural variations on a heterostructure nanocavity by imposing random fluctuation patterns in which the air holes are shifted and the radii are varied. Our calculations revealed that *Q*
_{ideal} is reduced from 1.5 × 10^{7} to approximately one million even when the standard deviation of the structural variations is as small as 1 nm [20]. Furthermore, the value of *λ* randomly fluctuated on the subnanometer scale. Knowledge of the magnitude of these fluctuations is important for future applications of nanocavities, but no quantitative experimental investigation has yet been carried out. Therefore, no statistical comparison of measured and predicted values of *Q* and *λ* has been performed.

In this paper, we report on experimental evaluations of the fluctuations of *Q* and *λ* for the type of high-*Q* heterostructure nanocavity that we previously studied theoretically [20]. We measured *Q*
_{exp} and *λ* for 80 fabricated nanocavities with the same heterostructure, all of which were integrated on the same silicon (Si) chip. The values of *Q*
_{exp} varied between 2.3 million and 3.9 million according to a normal distribution; the average value was 3.0 million. The value of *λ* also fluctuated but with a standard deviation of only 0.33 nm. By comparing with calculated values of *Q* and *λ*, we conclude that the standard deviation of the air hole positions and radii in the fabricated nanocavities is less than 0.58 nm.

## 2. Sample structure

Figure 1(a)
shows the nanocavity studied in this work. The PC consists of a triangular lattice of circular air holes with radii of 110 nm, formed in a 220-nm-thick Si slab. The nanocavity is formed by a line defect of 17 missing air holes and by two successive 5 nm shifts of the lattice constant in the *x*-direction at the center of the defect [2]. The electric field distribution *E _{y}* for the high-

*Q*nanocavity mode, calculated using the three-dimensional (3D) finite difference time domain (FDTD) method, is superimposed on the structure; the values of

*Q*

_{ideal}and

*λ*were calculated to be 1.4 × 10

^{7}and 1579.21 nm, respectively. We note that the radius of the air holes and the thickness of the slab were smaller than those considered in our previous reports in order to reduce the deterioration in

*Q*

_{exp}and fluctuation in

*λ*caused by structural variations of the air holes. Reducing the radius of the air holes should ensure that the effective change of the refractive index relative to the nanocavity mode due to structural variations is smaller. Moreover, the use of a thinner slab should reduce the structural variations that arise from the fabrication process.

Figure 1(b) shows the entire structure of a measured sample. Ten nanocavities and an extended line-defect waveguide to excite the nanocavities were fabricated in parallel, separated by 5 rows of air holes. All 10 cavities had the structure shown in Fig. 1(a). The length of the waveguide is 300 μm with a separation of 20 μm between cavities. In order to ensure that any field distortion influencing the accuracy of the electron beam (EB) lithography was steady for all 10 cavities, the pattern for each cavity was drawn at the same position within the EB field by instead displacing the EB stage (Conversely, this method may produce slight changes in the height of the EB stage from cavity to cavity). We performed optical microscopy measurements to determine the values of *Q*
_{exp} and *λ*. Input light from a tunable-wavelength laser was coupled into a facet of the excitation waveguide and dropped light emitted from the nanocavities to free space was measured. The probability that the spectral resonant peaks of the 10 nanocavities overlap is small because the linewidths of the peaks (<1 pm) are much smaller than the random fluctuation of *λ*, as shown in Fig. 3
. We fabricated 8 such samples on the same Si chip by a process described previously [2,18] and thus efficiently measured 80 nanocavities with the same structure.

## 3. Experimental results

Figure 2(a)
shows the values of *Q*
_{exp} of the 80 nanocavities, which were derived from the photon lifetimes of the nanocavities in time-domain measurements [3]. The identification numbers of the nanocavities are shown on the x-axis. The value of *Q*
_{exp} is greater than 2 million for all 80 nanocavities and the distribution of values is essentially random across the Si chip. The inset shows that the variation of *Q*
_{exp} corresponds approximately to a normal distribution, which agrees with our previous calculations: the reduction of *Q*
_{exp} from *Q*
_{ideal} and the fluctuation is due to the air hole variations [20]. The highest and lowest values are 3.87 × 10^{6} and 2.30 × 10^{6}, which correspond to photon lifetimes of 3.23 ns and 1.92 ns, respectively. The average *Q*
_{exp} is 3.04 × 10^{6}, which is higher than any previously reported values for photonic crystal cavities. Based on the average *Q*
_{exp}, the *Q*
_{loss} factor is estimated to be 3.88 × 10^{6} from the relation:

*Q*

_{exp}for nanocavities with

*Q*

_{ideal}less than 10

^{5}, such as the L3 cavity [1].

Figure 2(b) shows the variation of *λ* for the 80 nanocavities, determined using spectral-domain measurements. During measurements the surrounding temperature fluctuated about by 1 degree corresponding to a *λ* shift of 0.08 nm. The inset shows that, in similar fashion to *Q*
_{exp}, *λ* randomly fluctuates according to a normal distribution; the average value is 1574.37 nm and the standard deviation is only 0.33 nm. Such a highly controlled *λ* is very useful not only for the passive devices as wavelength filters [5,6] and environmental sensors [10,11], but also for active devices as nanolasers [12]. The difference between the highest and lowest values of *λ* among the 80 nanocavities is 1.90 nm. This precision might be sufficient for coarse wavelength division devices. It should be emphasized that the yield rate of the high-*Q* nanocavities was 100% in this study. Similar statistical results in *Q* and *λ* were obtained in different chips. We believe that these developments have been brought about both by improvements in the fabrication process and by tuning the PC structural parameters as described above.

## 4. Comparison with simulation results

Finally, we estimate the standard deviation (*σ*
_{hole}) associated with remaining variations in the radii and positions of the air holes in the measured nanocavities. Although it is difficult to specify the reasons for any remaining variations, it is important to make such an estimation in order to further improve the fabrication process and to design nanocavity devices. Because *σ*
_{hole} is expected to be less than 1 nm, corresponding to structural deviations that are barely visible using conventional observation apparatus, we adopted a method to compare our experimental results with 3D FDTD simulations that take these structural variations into account. We have presented such a comparison in [18], although statistical experimental data was not included. Here, our calculations used the structural parameters shown in Fig. 1(a), and random variations in the air hole positions and radii were added using 30 different fluctuation patterns, according to a normal distribution with *σ*
_{hole} = 1 nm. Details of these calculations are given in [20].

Figures 3(a) and 3(b) show the calculated values of *Q* and *λ* for the 30 fluctuation patterns. The calculated *Q* factors in Fig. 3(a), which we denote as *Q*
_{fluc}, are significantly smaller than the *Q*
_{ideal} of 1.4 × 10^{7} and are randomly distributed between 6.0 × 10^{5} and 2.8 × 10^{6}. In Fig. 3(b), the average calculated value of *λ* is 1579.32 nm with a standard deviation of 0.48 nm. Because the lowest value of *Q*
_{exp} is 2.3 × 10^{6} and the standard deviation of the experimentally determined *λ* is 0.33 nm, we can be certain that *σ*
_{hole} in our fabricated samples is below 1 nm.

We now introduce an additional factor *Q*
_{loss_fluc} associated with the structural variation of the air holes, defined as

*Q*

_{loss_fluc}increases with the square of

*σ*

_{hole}when the fluctuation pattern is fixed:where the coefficient

*A*differs from pattern to pattern. By applying the data in Fig. 3(a) to Eqs. (2) and (3), the dependence of 1/

_{m}*Q*

_{loss_fluc}on

*σ*

_{hole}can be obtained for the 30 fluctuation patterns without carrying out simulations for individual values of

*σ*

_{hole}. As a result, the dependence of the average value of 1/

*Q*

_{loss_fluc}and its standard deviation on

*σ*

_{hole}can be obtained as follows:

*Q*

_{loss_fluc}) and S.D.(1/

*Q*

_{loss_fluc}) for the 80 measured nanocavities by substituting

*Q*

_{exp}for

*Q*

_{fluc}in Eq. (2); we obtain values of 2.58 × 10

^{−7}and 3.85 × 10

^{−8}, respectively. By substituting these values into Eqs. (4) and (5),

*σ*

_{hole}for the 80 nanocavities is evaluated to be 0.58 nm and 0.36 nm, respectively.

An estimation of *σ*
_{hole} is also possible using the standard deviation of *λ*, which is known to be proportional to *σ*
_{hole} when the fluctuation pattern is fixed. This coefficient is evaluated from Fig. 3(b) as

*σ*

_{hole}= 0.69 nm, which is larger than from the analysis based on 1/

*Q*

_{loss_fluc}.

It is noted that estimated values from Avg.(1/*Q*
_{loss_fluc}) and S.D.(1/*Q*
_{loss_fluc}) are clearly different. The corresponding *Q*
_{loss_fluc} for *σ*
_{hole} of 0.36 nm is 9.98 × 10^{6} from Eq. (4), which is different by 6.27 × 10^{6} from the value for Avg.(1/*Q*
_{loss_fluc}). Here, it should be noted that our estimations based on 1/*Q*
_{loss_fluc} only take into account *Q*
_{loss} factors due to structural variations of the air holes. If other factors in the fabricated cavities contribute, the evaluation of *σ*
_{hole} from Avg.(1/*Q*
_{loss_fluc}) might be overestimated. If one assumes that S.D.(1/*Q*
_{loss_fluc}) is unaffected by any additional factors, the smaller *σ*
_{hole} estimated from S.D.(1/*Q*
_{loss_fluc}) is likely to be closer to the actual value, which suggests that other constant *Q*
_{loss} factors of 6.27 × 10^{6} are indeed present in the fabricated nanocavities, such as absorption at the Si surface [21]. In other words, we can still increase *Q*
_{exp} factor by decreasing the constant *Q*
_{loss} factors. The larger value of *σ*
_{hole} estimated from S.D.(*λ*) is probably due either to the fluctuation of the measurement temperature or to tiny fluctuations in the height of the EB stage occurring during EB lithography, which would only increase the fluctuation of *λ*. The standard error arising from the finite number of fluctuation patterns is also a probable cause. Although unknown factors remain, we conclude that the value of *σ*
_{hole} in our nanocavities is less than 0.58 nm.

## 5. Summary

In summary, we have measured the *Q*
_{exp} factors and resonant wavelengths *λ* for 80 high-*Q* nanocavities with the same heterostructure. The values of *Q*
_{exp} follow a normal distribution between 2.3 million and 3.9 million, with an average *Q*
_{exp} of three million. The value of *λ* also fluctuates but with a standard deviation of only 0.33 nm. Such a high average *Q*
_{exp} and stable *λ* will be important for expanding the area of nanocavity applications. Furthermore, we have used a statistical comparison of our experimental results and FDTD calculations to estimate that the standard deviation of the structural variations of our air holes is less than 0.58 nm and that *Q*
_{loss} factors other than *Q*
_{loss_fluc} due to the air hole variations might be present. This method will be an important tool for further developing high-*Q* nanocavities .

## Acknowledgments

This work was partly supported by the Special Coordination Funds for Promoting Science and Technology commissioned by MEXT, by the CREST and PRESTO programs of the JST, by the Global COE Program, and by KAKENHI (No. 23104721).

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