## Abstract

Self-frequency conversion (SFC), where both laser oscillation and nonlinear frequency conversion occurs in the same laser crystal, has been used to efficiently extend the operational wavelength of lasers. Downsizing of the cavity mode volume (*V*) and increasing the quality factor (*Q*) could lead to a more efficient conversion process, mediated by enhanced *n*-th order nonlinearities that generally scale as (*Q/V*)^{n}. Here, we demonstrate nanocavity-based SFC by utilizing photonic crystal nanocavity quantum dot lasers. The high *Q* and small *V* supported in semiconductor-based nanocavities facilitate efficient SFC to generate visible light, even with only a few photons present in the laser cavity. The combined broadband quantum dot gain and small device footprint enables the monolithic integration of 26 different-color nanolasers (spanning 493-627 nm) within a micro-scale region. These nanolasers provide a new platform for studying few-photon nonlinear optics, and for realizing full-color lasers on a single semiconductor chip.

©2013 Optical Society of America

## 1. Introduction

Nonlinear optical effects are enhanced in micro/nano photonic structures with strong light confinement, such as photonic nanowires [1,2], whispering gallery mode resonators [3,4], photonic crystal (PhC) waveguides [5–7] and nanocavities [8–11]. Compared to conventional nonlinear optics in bulk crystals, such photonic structures have the potential to achieve strong nonlinear effects at lower powers and with greatly reduced device sizes, together with integratability into optical circuits. These advantages have been recently demonstrated in several works on nonlinear effects such as second/third order harmonic generation [2,4,6–10,12], by using external laser sources to introduce photons to the structures. These nonlinear optical wave mixers, whilst being useful for generating coherent light at wavelength regions inaccessible by conventional semiconductor lasers, are held back by the requirement of external laser sources and accompanying cumbersome optical setups, which tend to make the system voluminous, unwieldy and hence costly. Needless to say, this obstacle also holds true in conventional bulky light sources based on nonlinear frequency conversion.

In this context, semiconductor nanolasers with a function of self-frequency conversion (SFC) [13,14], in which both lasing action and nonlinear frequency conversion take place within the same laser material, can provide an ideal solution. Such SFC nanolasers will directly generate coherent light of various wavelengths within the nano-scale devices, potentially by current injection. Strong nonlinear effects could be available even within tiny resonators by adopting cavity designs with high quality (*Q*) factors and small mode volumes (*V*) [12,15–18]. So far, several monolithic light sources using SFC [19,20] have been studied, however, none of them have made use of the wavelength-scale optical confinement effect and have been closely investigated in terms of the effect of *Q*/*V* on their performances.

In this work, we demonstrate the hitherto-unexplored, nanocavity-based SFC lasers by means of monolithically-fabricated PhC nanocavity quantum dot (QD) lasers [21,22]. The nanocavities support continuous-wave (CW) lasing at the near infrared (NIR), which is up-converted to coherent visible (VIS) light by efficient intra-cavity second harmonic generation (SHG), facilitated by high *Q*/*V* and the large second order nonlinear susceptibility, *χ*^{(}^{2}^{)} ( = 170 pm/V at 1064 nm) [23], of GaAs-based nanocavities. The efficient conversion processes enable us to observe SHG using only a few intracavity NIR photons in average and show a further improvement by strengthening the optical confinement effect. Moreover, we establish a micro-scale integration of 26 different-color lasers, taking advantage of high integratability of our device and a broadband QD gain.

## 2. Device design, fabrication and experimental setup

A schematic of one of the PhC nanocavity lasers under investigation is shown in Fig. 1(a). The structure is based on an air bridge two dimensional PhC membrane consisting of a periodic air hole lattice. The lattice constant, *a*, is varied from 244 to 340 nm in the following experiments, and the total area of the single cavity occupies only 40 μm^{2} on average. From these tiny lasers, both coherent VIS and NIR light can be generated without any additional assembly of optical elements.

All the cavities are fabricated monolithically into the same semiconductor wafer by a simple combination of standard semiconductor fabrication technologies. The wafer was grown on a (100)-oriented semi-insulating GaAs substrate by molecular beam epitaxy and consists of a 200-nm-thick GaAs slab incorporating 6 layers of self-assembled InAs QDs, deposited on top of a 1000-nm-thick Al_{0.7}Ga_{0.3}As sacrificial layer that was later removed.

The airbridge cavities are monolithically fabricated by a single-step electron beam lithography process, followed by an inductively coupled plasma reactive ion etching and a wet etching by a hydrofluoric acid solution. A scanning electron microscope image of a fabricated nanocavity is shown in Fig. 1(b), exhibiting successful patterning of the device. The three missing air holes (forming the defect cavity) are patterned along the crystal orientation of [110].

The cavity design is based on the so-called modified L3 type [24], a plan view of which is overlaid with the calculated electric field distribution in Fig. 1(c). This fundamental cavity mode confines light within a very small *V* of 0.032 μm^{3} (for *a* = 314 nm) and possesses a high theoretical cavity *Q* of more than 150,000. This mode volume is thousands times smaller than that of previously-reported SFC lasers based on vertical cavity surface emitting lasers [20]. The other, higher order, cavity modes have sufficiently-low cavity *Q*s such that only single mode lasing was observed in the entire operational wavelength range. The air hole radius is fixed to 0.29*a* for all cavities. In order to achieve higher *Q*, the first and third air holes are shifted outwards by 0.2*a*.

Control over the lasing wavelength, *λ*, can be done simply by changing the PhC lattice constant, *a*. We observed that *λ* obeys a linear relation to *a*: *λ* = 295 + 2.8*a* nm, for *a* ranging from 244 to 340 nm.

Gain in the PhC nanocavity lasers is provided by 6 layers of InAs QDs grown in the GaAs slab. To obtain broadband gain, the bottom two QD layers were grown at a higher temperature, so that their ground state emission was shortened to 1040 nm (130 nm shorter than that of the upper 4 layers). Figure 1(d) shows a photoluminescence (PL) spectrum from the QD wafer at 10 K. The emission covers a wide range of the NIR spectrum, spanning 950 nm to 1260 nm, when including the contribution from the QD’s excited states. The peak wavelengths of the QD ground state emission are assigned by investigating the excitation power dependence of the PL spectrum.

The samples were mounted in a liquid helium flow cryostat kept at 10 K and characterized by a confocal micro-PL setup. The sample position was controlled by a combination of a motorized stage and a piezoelectric positioner. Excitation was provided by a CW diode laser oscillating at 808 nm, focused through a x100 objective lens. The excitation power is defined as that measured after the objective lens, and the estimated spot diameter on the sample surface is about 3 μm. The PL signal was collected by the same objective lens and analyzed by spectrometers equipped with nitrogen-cooled multi-channel detectors for VIS and NIR. VIS near field images were taken one by one using a color charge coupled device camera, together with a stack of shortpass filters, which allows transmission of the VIS (450-640 nm), while strongly rejects the NIR (660-1320 nm). In this measurement, the excitation powers are set far above the thresholds of the nanolasers: 1.5 mW for most of the samples and 3 mW for samples with emission wavelength λ > 620 nm and < 510 nm. All the images are taken with a fixed accumulation time of 1 sec.

## 3. Optical characterization of SFC nanolasers

#### 3.1 Light-in versus light-out characteristic

As an initial optical characterization, the excitation power dependence of a sample with a PhC lattice constant of 314 nm was investigated by the optical pumping. The light-in versus light-out (L-L) plot for the NIR cavity mode emission at 1174 nm is shown in the upper panel of Fig. 2(a). The nanocavity mode shows a laser oscillation with a lasing threshold of 12 μW, determined from the inflection point of the L-L curve. Figure 2 (b) shows a transition of the cavity linewidth, showing a significant linewidth narrowing. From the linewidth at the lower power edge of the transition region (corresponding pumping power of 4 μW), a cold cavity *Q* factor of 15,000 is deduced. The inset of the upper panel in Fig. 2(a) shows a NIR lasing spectrum of the sample under a pumping power of 590 μW, exhibiting single mode lasing with high spectral purity. From the same cavity, at the same time, a sharp, single emission peak at 587 nm is observed, as shown in the lower panel of Fig. 2(a), along with the respective L-L curve. For pumping powers larger than 40 μW, which corresponds to the linear region of the NIR laser, the VIS emission shows a quadratic power dependence. Figure 2(c) shows a plot of the VIS light intensity, *I _{VIS}*, measured at the detector as a function of that of the NIR laser mode intensity,

*I*. The plot again displays a clear quadratic relation between the intensities of the two emission lines,

_{NIR}*I*∝

_{VIS}*I*. In addition, the peak wavelength of the VIS emission is observed to be half of that of the NIR peak throughout the measurements. Therefore, the VIS emission is attributed to the intra-cavity SHG of the NIR light in the nanolaser. This is, to the best of our knowledge, the smallest solid state yellow laser yet achieved: the color is still inaccessible by conventional semiconductor laser diodes. Remarkably, we observed the SFC signal even near the NIR lasing threshold, where only a few photons exist in the cavity on average, as demonstrated in the following section. This observation encourages the application of these nanocavities to the study of few-photon nonlinear optics: an exciting subject in various research fields such as quantum information technology [25,26].

_{NIR}^{2}#### 3.2 Estimation of intra-cavity photon number and frequency conversion efficiency

We estimate the average cavity photon number at the lasing threshold of the fundamental NIR mode by simulating the laser output characteristics using a rate equation model [27]. The model is described for cavity photon number (*N _{ph}*) and carrier number (

*N*), and formulated as follows,

*τ*= 1 nsec,

_{sp}*τ*= 100 nsec, and

_{nr}*τ*= 9.3 psec are the spontaneous emission decay time, the nonradiative life time of carriers and the cavity photon leakage time (

_{cav}*Q*= 15,000), respectively.

*β*is the spontaneous emission coupling factor into the cavity mode.

*N*= 270 is the transparency carrier number, which is set to the average QD number in the cavity area (~0.15 μm

_{tr}^{2}). L-L properties of lasers with different

*β*are simulated by numerically solving the above coupled equations. The result is shown in the left panel of Fig. 3. The best fit to the experimental data was obtained for

*β*= 0.11. This high

*β*is a characteristic of high

*Q/V*nanolasers [21]. With this numerically derived curve, we estimate that the average cavity photon number at the lasing threshold, ${N}_{ph}^{TH}$, is to be 3.4.

It is worth noting that ${N}_{ph}^{TH}$ has a strong dependence on the value of *β* in the rate equation model: once we adopt *β* of ~0.1, ${N}_{ph}^{TH}$ is found to be around a few photons even when we change the values of other parameters. This fact is also confirmed in the other rate equation models [28].

Next, we estimate the external conversion efficiency, *η*, defined by *η* = ${{P}_{VIS}^{P}/\left({P}_{NIR}^{P}\right)}^{2}$, where ${P}_{NIR}^{P}$ and ${P}_{VIS}^{P}$ are the estimated output power of the NIR lasing mode, and the detected power of the VIS emission measured after the spectrometer, respectively. *P* is the excitation power in the experiment. In order to estimate ${P}_{NIR}^{P}$, we firstly obtain the average cavity photon number of the NIR mode, ${N}_{ph}^{P}$ by using the relationship ${I}_{NIR}^{P}/{I}_{NIR}^{TH}={N}_{NIR}^{P}/{N}_{NIR}^{TH}$, where ${I}_{NIR}^{P}$ and ${I}_{NIR}^{TH}$ are the measured intensities of the NIR lasing mode at the pumping power *P* and at the lasing threshold, respectively. With this relation, a ${N}_{ph}^{P}$ value of 285 is obtained for *P* = 370 μW. Then, the output power of the NIR lasing mode, ${P}_{NIR}^{P}$, is evaluated to be 5.1 μW by using the relation, ${P}_{NIR}^{P}=\hslash {\omega}_{NIR}\kappa {N}_{ph}^{P}$, where *κ* is the cavity photon leakage rate (∝1/*Q*). At the same pumping power, the power of the VIS emission, ${P}_{VIS}^{P}$, is experimentally measured to be 26 fW. Therefore, the external conversion efficiency *η* = ${{P}_{VIS}^{P}/\left({P}_{NIR}^{P}\right)}^{2}$ is determined to be 0.1%/W: a high value in spite of the thin nonlinear material. It is worth noting that the estimated external conversion efficiency, *η*, is obtained from a comparison of the ideally-estimated NIR laser output power without any consideration on the detection process, with the experimentally-obtained VIS emission power, which is imperfectly collected by the objective lens and is damped significantly by passing through numerous optical elements in the setup. Especially, signal transmission efficiency after the collection objective lens is only ~10%. Therefore, if we take into account those losses, the estimated *η* should be at least one order of magnitude larger than the value presented above.

We also estimated *η* for the 9 different SFC nanolasers on the same wafer by following the same procedure above. The 9 different nanolasers have different measured cavity *Q*s, presumably due to unequal effects of fabrication errors on each cavity. Right panel of Fig. 3 summarizes the obtained values of *η* from these ten samples in total, plotted as a function of the respective experimental *Q* factors. We observe a quadratic increase of *η* with respect to *Q*, agreeing with theoretical prediction presented in the Appendix and with previous studies using passive resonantors [10,29]. This result demonstrates that, even in active SFC nanolasers, *η* can be rapidly improved by enhancing the optical confinement.

## 4. Micro-scale integration of multiple different-color SFC nanolasers

The demonstrated VIS nanolaser can be combined with the broad QD gain, so as to realize monolithic integration of many lasers with different color emission within a micro-scale semiconductor chip. We investigated PhC nanocavitiy QD lasers with 26 different values of *a*, ranging from 244 nm to 340 nm. These nanolasers are monolithically patterned within a tiny footprint of 10 × 385 μm^{2}, as shown in Fig. 4(a). A series of emission spectra for both the VIS and NIR range are shown in Fig. 4(b). In the NIR, the QD gain supports single mode lasing for a wide NIR range spanning 986 to 1254 nm, which is, to the best of our knowledge, the broadest lasing band using a single semiconductor wafer. These coherent NIR emissions are up-converted to the VIS by the intra-nanocavity SHG and result in multi-color emission seamlessly spanning 493.3 nm to 627.0 nm. As a whole, the integrated array of SFC lasers cover extremely broad wavelength range in VIS and NIR. This result highlights the advantage of active SFC nanolasers, which do not require many pumping lasers with differentwavelengths in order to produce various colors of light, in stark contrast to devices based on passive SHG. Figure 4(c) shows a series of near field images of the VIS emission from the nanocavities. Also in these images, we confirm the colorful emission from the SFC nanolasers: blue-green, green, yellow, orange and red.

The left panel of Fig. 5 shows a microscope image of a PhC cavity with a lattice constant of 314 nm. The image is taken under 650 nm light illumination and plotted in a grey scale. White lines have been added to show the edge of the PhC lattice and the defect cavity. After recording this image, a color image of the PhC nanolaser was also taken, as shown in the right panel. Using the same white lines as in the left panel, we can estimate the near field pattern of the visible light within the cavity structure. The two main bright lobes are confirmed to be located just above and below of the defect formed by three missing air holes. This main two bright lobes, separated by a dark region at the defect cavity, bear a striking resemblance to those observed in SHG processes using gallium phosphide and silicon PhC nanocavities: the former study achieved SHG through inducing a nonlinear polarization perpendicular to the PhC membrane [9], and the latter has a large contribution of surface SHG [10]. In both cases, the field maximum of the NIR mode does not coincide with that for the observed VIS field. We also note that these VIS near field patterns may also be explained by a nonlinear polarization parallel to the PhC slab. Elucidation of the exact nanoscopic origin of the nonlinear process in the present study will require further investigation, such as characterization of the optical modes for the SHG light by experiments and numerical simulations.

## 5. Conclusion and discussion

We have demonstrated SFC nanolasers by means of PhC nanocavity QD lasers. This novel platform is shown to be useful for the study of few-photon nonlinear optics, as well as for nanolasers in a broad VIS band, which open the way for realizing monolithic full-color lasers on a single chip. These devices based on a common III-V semiconductor can in principle be operated by current injection [22] at room temperature [21], which are highly desirable properties, and highlight the advantages of SFC nanolasers. To achieve these, further improvement of the laser in terms of the quantum dot quality and other device parameters will be required. We have experimentally shown that increase of *Q* factor is highly effective to improve *η* of SFC nanolasers, which provides a foundation for seeking sophisticated designs of such devices.

Further increase of *η* is envisaged by improvements in various areas, such as increasing *Q*/*V* [10,12,17,18] (which is of major importance due to *η*’s quadratic dependence on it), and employing resonant cavity modes simultaneously for both VIS and NIR [12,15,16,30]. The latter improvement may enable almost 100% absolute conversion efficiency with a very low NIR power [17,18] (see also Appendix). However, in our devices, the generated harmonic light could be significantly absorbed by the host material when trying to confine it for long times, and hence we should keep the structure-dependent *Q* factor for the corresponding optical mode lower than that limited by the material absorption. Even while keeping the low *Q*, improvement of *η* can be achieved by optimizing the VIS mode in terms of its mode volume, out-coupling efficiency, and spatial mode overlap with the fundamental cavity mode. One further possibility is the use of transparent material even for the harmonic mode, including rare-earth doped crystals with optical nonlinearity, like Nd^{3+}:LiNbO_{3} [13,14] and Er^{3+}:GaN [31]. Also of interest is the possible use of nanolasers as monolithic, ultra-compact far- and mid-infrared light sources via difference frequency generation between two lasing modes. Such multi-mode lasing can be supported by the QD gain [32] that is also expected to possess a large *χ*^{(}^{2}^{)} [33]. The SFC nanolasers demonstrated here would find applications in display, spectroscopy, medical and biochemical sensing technology, where controllable narrow emission lines, together with monolithic, cost-effective fabrication, are necessary.

## Appendix

In this part, we derive a theoretical expression for the nonlinear frequency conversion efficiency, *η*. We deal with an NIR cavity mode at frequency *ω _{NIR}*, which is coupled to multiple VIS modes through the second order nonlinear interaction. Here, we assume that complex photonic bands at the VIS region could provide multiple VIS modes coupling to the second harmonic light. Moreover, the NIR mode is considered to be coherently driven by an intracavity field, representing the NIR lasing oscillation. Our model can be regarded as an extension of previous studies, which consider only a single VIS mode [34–37] and injection of laser fields from outside the system [17,18]. After taking an appropriate rotating frame, the Hamiltonian of this system can be expressed as,

*Δ*=

_{k}*ω*is the detuning of the generated second harmonic light from the

^{k}_{VIS}−2ω_{NIR}*k*-th VIS mode at the frequency of

*ω*.

^{k}_{VIS}*b*(

*a*) is the annihilation operator for the NIR (

_{k}*k*-th VIS) mode.

*E*determines the strength of the coherent drive for the NIR intracavity field.

*g*is the nonlinear coupling strength between the NIR and

_{k}*k*-th VIS mode and is expressed as [26],

*ε*is the vacuum permittivity,

_{0}*ε*and

_{NIR}*ε*are relative permittivities of the host material at the NIR and VIS mode frequencies, respectively.

_{VIS}*V*(

_{NIR}*V*) is the mode volume of the NIR (

_{VIS}*k*-th VIS) cavity mode.

*χ*

^{(2)}

*(*

_{lmn}*r*) is the second order nonlinear susceptibility tensor of the host material at a position

*r*.

*b*(

*r*) (

*a*(

^{k}*r*)) represent the normalized spatial distribution of the NIR (

*k*-th VIS) mode and the repeated index summation convention is employed within the integral. The integration is performed over the whole space under consideration. In this formula, the restriction of phase matching is rendered to the requirement of large nonlinear spatial mode overlap among the related modes [17,38]. The time evolution of a system density operator,

*ρ*, is governed by the following master equation,The term

*Lρ*constitutes the system-bath interaction, whose expression is,

*k*-th VIS) mode at a rate of

*κ*(

_{b}*κ*). We derive a following set of evolution equations for expectation values,

^{k}_{a}*b*=

*β*and

*a*=

_{k}*α*. In this approximation, the output power from the NIR cavity mode is expressed as

_{k}*P*= η

_{NIR}*ω*, and for the

_{NIR}κ_{b}|β|^{2}*k*-th VIS mode,

*P*= η2

^{k}_{VIS}*ω*. Then, for the steady state (d/dt = 0), we obtain an expression for

_{NIR}κ^{k}_{a}|α_{k}|^{2}*η*from the first equation in the set of evolution equations,

*η*holds a quadratic relation to

*Q*/

_{NIR}*V*, where

_{NIR}*Q*represents the quality factor of the NIR mode. For a simplified discussion, we further assume the VIS mode as single,

_{NIR}*Δ*= 0 and having the quality factor of

*Q*, and then find a relationship,

_{VIS}*η*=

*P*/(

_{VIS}*P*)

_{NIR}^{2}∝ (

*Q*/

_{NIR}*V*)

_{NIR}^{2}(

*Q*/

_{VIS}*V*). Therefore, the improvement of

_{VIS}*Q*has a primal importance for achieving high conversion efficiency in this system. Fully quantum mechanical treatment of the investigated model will be given in a future publication.

_{NIR}/V_{NIR}Next we proceed with analysis of the model under the single VIS mode assumption with *Δ* = 0, and derive analytical expressions for the output powers from the cavity modes. We again visit the set of evolution equations and treat them under the steady state condition and with the classical approximation. It is found that the field amplitudes obey the flowing relations:

*|β|*and these equations can be analytically solved for

*|β|*and

*|α|*. The second equation ensures that the VIS output is quadratic to that of NIR, which is also experimentally observed in our devices. For a weak excitation case (

*|E|*= 1), the average output power in the NIR cavity mode can be expressed as ${P}_{NIR}=\hslash {\omega}_{NIR}\left(4{\left|E\right|}^{2}/{\kappa}_{b}\right)$ and for the VIS mode, ${P}_{VIS}=\hslash {\omega}_{VIS}\left(4{g}^{2}/{\kappa}_{a}\right){\left(4{\left|E\right|}^{2}/{\kappa}_{b}\right)}^{2}$. The term, $\left(4{\left|E\right|}^{2}/{\kappa}_{b}\right)$, is equivalent to the steady state photon emission rate from the NIR mode under

*g*= 0. For strong excitation (

*|E|*>>1), those expressions turn out to be ${P}_{NIR}=\hslash {\omega}_{NIR}{\left({\kappa}_{a}\left|E\right|/4{g}^{2}\right)}^{\frac{2}{3}}$ and ${P}_{VIS}=\hslash {\omega}_{VIS}\left(4{g}^{2}/{\kappa}_{a}\right){\left({\kappa}_{a}\left|E\right|/4{g}^{2}\right)}^{\frac{4}{3}}$. In this case, the output from the system is dominated by

*P*(>>

_{VIS}*P*), meaning nearly-perfect frequency conversion. The field intensity satisfying

_{NIR}*P*=

_{VIS}*P*is equal to $\left|E\right|=\sqrt{\frac{{\kappa}_{a}{\kappa}_{b}^{3}}{8{g}^{2}}}$. Above this intensity, the output from the system rapidly becomes dominated by the VIS emission, as increasing $\left|E\right|$. Thus, improvement of

_{NIR}*Q*is primarily effective to lower the power required to achieve the almost-perfect frequency conversion, which scales as (

_{NIR}/V_{NIR}*Q*)

_{NIR}/V_{NIR}^{−2}. It is worth noting that the single VIS mode assumption used for the last part of discussions may be suitable even in our photonic crystal SFC nanolasers. According to the two-photon-loss model proposed by Collet and Levien [37], it could be possible to pick up a single mode (

*A*) from a continuum to provide a simple second nonlinear interaction Hamiltonian ($g\left({A}_{}^{\u2020}{b}_{}^{2}+A{b}_{}^{\u2020}{}_{}^{2}\right)$). This may be the case for our photonic crystals, in which the photonic modes at the VIS band, although colored, should behave almost like a continuum for the narrow-band second harmonic light.

Finally, we compare the experimentally obtained *η* with that estimated from the above theory. For a simple discussion, let us first assume that there is a single VIS cavity mode resonant with the generated SHG light. We also assume that the cavity *Q* for the VIS mode to be 10 and use experimentally obtained parameters for the NIR fundamental mode in the following discussion. In this situation, the efficiency derived from the Eq. (7) in the manuscript becomes about 0.3%/W for *g* = 0.01 μeV and 30%/W for *g* = 0.1 μeV. If a detection efficiency of 1% is assumed, the latter value becomes 0.3%/W, which is comparable to the experimentally obtained value of 0.1%/W. For the case assuming *Q* = 100 and *g* = 0.01 μeV, the efficiency is estimated to be 0.03%/W after accounting for the detection loss. The assumption for the *Q* factor for the VIS mode is made considering the material absorption limit. In the worst case, in which all the electric field of the VIS mode is distributed in the material, *Q* becomes about 10. *Q* = 100 may be obtained by extending the electric field outside of the material. We consider that the values of *g* used above discussion are realistic. A study in ref [26]. reported a simulated value of *g* ~0.2 μeV in a similar PhC nanocavity. This *g* value is easily reduced when the field overlap between the related cavity modes reduces and their mode volumes increase.

## Acknowledgments

We thank M. Holmes, E. Harbord, H. Takagi, N. Kumagai, S. Kako and S. Ishida for their technical support and for fruitful discussions. This work was supported by the Project for Developing Innovation Systems, MEXT, Japan and by JSPS through its FIRST Program.

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