## Abstract

We investigated the characteristics of an ultra-high-*Q* photonic nanocavity (*Q* = ~230,000 and modal volume = ~1.2 cubic wavelengths) at various input light powers. The cavity characteristics were red-shifted as the input power increased. This nonlinearity could be explained by coupled-mode theory, taking into account two-photon absorption, the associated free-carrier absorption, plasma effect, thermo-optic effect, and a Kerr effect. Nonlinear cavity characteristics were observed at an extremely low input light power of 10 μW. We confirmed that these low-power nonlinear optical effects could be attributed to the ultra-high *Q* factor of the nanocavity.

© 2006 Optical Society of America

## 1. Introduction

High-*Q* and small-volume photonic cavities in two-dimensional (2D) photonic-crystal (PC) slabs have been reported in the past few years [1–4]. The future development of integrated photonic systems will require high-*Q* photonic nanocavities, which are small enough to integrate and which can be applied in extremely low–power nonlinear photonic devices because of their strong light-matter interactions. Advanced technologies are now available for the fabrication of such cavities in silicon (Si).

Nonlinear optical phenomena in Si nanocavities in 2D PCs [5,6], and optical bistable switches [7], have recently been reported. These findings support the concept of low-power nonlinear optical effects in 2D PC Si nanocavities. A nanocavity with a higher *Q* factor will achieve more efficient nonlinear effects. In a previous study [3], we developed an Si nanocavity in a 2D PC with a *Q* factor of between one and two orders of magnitude larger than those described elsewhere [5–7]. Here we demonstrate extremely low-power nonlinear effects in an ultra-high *Q* nanocavity, and investigate the nonlinear processes that are caused by the highly enhanced electric fields within this type of cavity.

## 2. Experiment

We investigated an Si-based photonic double-heterostructure nanocavity [3] with a modal volume of ~1.2 (λ/*n*)^{3} according to the three-dimensional (3D) finite-difference time-domain (FDTD) method (where λ is the wavelength of light in air and *n* is the refractive index of the cavity material). A scanning electron microscope (SEM) image of the fabricated sample and a schematic diagram of the measuring apparatus are shown in Fig. 1. The lattice constant of the central PC region (PC II) was set at *a*
_{2} = 420 nm, and that of the surrounding regions (PC I) at *a*
_{1} = 410 nm. The lattice constants were varied only on the longitudinal (parallel to the waveguide) direction, but not on the orthogonal direction in order to satisfy “lattice matching”. The length of PC II region was 2*a*
_{2}. The air-hole radius and the slab thickness were 0.29*a*
_{2} and 0.6*a*
_{2} for both PC I and PC II regions. An input line-defect waveguide, which was made on purpose slightly wider than the cavity one in order to guide light and couple it to the cavity, was formed six rows away from the cavity. The experimental methods have been described previously [8]. The *Q* factor of the cavity (*Q*
_{total}) was determined from the intrinsic or vertical *Q*(*Q*
_{v}), and the coupling to the waveguide or in-plane *Q*(*Q*
_{in}), following 1/*Q*
_{total} = 1/*Q*
_{v}+1/*Q*
_{in}.

Continuous-wave light with transverse-electric polarization for the slab was directed into the input waveguide and was coupled into the cavity, where it was modulated by a mechanical chopper at a frequency of ~1 kHz with a 50% duty cycle (or pulse width of ~ 0.5 ms) in order to utilize the lock-in detection technique. The power of the light radiated from the cavity was measured with varying wavelength. It should be noted that the incident light was periodically cut off by the chopper during a sufficiently long time (~ 0.5 ms) so that no hysteresis effect subsisted.

## 3. Measured results

Figure 2 shows the radiation efficiency as a function of the wavelength of the input light for various input light powers. Here, the radiation efficiency is defined as the ratio of the radiation power to the input power. The input power is defined as the power coupled to the waveguide, and was estimated from the transmission light power emitted from the waveguide exit facet^{1}. Note that we could only determine the relative radiation efficiency, as it was difficult experimentally to measure the absolute value. When the input power was 4 μW, the radiation waveform was almost symmetrical. Assuming a linear response, the estimated *Q* factor, based on the full-width to half maximum (FWHM) value, was ~230,000. Theoretically [9], if the cavity response is within a linear regime, then the radiation efficiency is determined by the ratio of *Q*
_{in} to *Q*
_{v}. When 0.5< *Q*
_{v}/*Q*
_{in} <2 we can obtain an efficiency between ~ 45% and 50%. Assuming Qv ~ 600,000 [3], *Q*
_{in} ~ 370,000 and thus *Q*
_{v}/*Q*
_{in} ~ 1.6, from which the radiation efficiency is estimated to be almost 50%.

As the input power increased to 10, 28, and 54 μW, the radiation waveforms became asymmetrical — that is, the cavity characteristics were red-shifted. On the shorter wavelength side of the peaks, the radiation efficiencies changed more slowly with wavelength than at 4 μW; while on the longer wavelength side, the radiation efficiencies changed more sharply than at 4 μW. Moreover, the FWHMs of the waveforms became greater. However, in these cases, the *Q* factors could not be estimated because the cavity showed nonlinear responses. We suggest that these phenomena originated from a nonlinear optical response of the material comprising the cavity (Si), which implies strong enhancement of the electric fields within the cavity due to its small optical wavelength volumes and ultra-high-*Q* factor (~230,000).

## 4. Analysis and discussion

We examined the processes that might generate a nonlinear response in 2D PC nanocavities, in order to produce an analytical model of the mechanism. As second-order nonlinear effects cannot occur due to the crystal symmetry of Si, third-order nonlinear effects — that is, two-photon absorption (TPA) [5–7, 10] and a Kerr effect [10] — are expected to be dominant. The absorption of light energy in the cavity through TPA by the cavity material leads to the generation of free-carriers. The increase in the density of free-carriers may change the optical properties of the cavity material and the cavity itself. In addition, as the energy of the absorbed light is eventually converted into thermal energy, a thermo-optic effect should be taken into account. We suggest the following analytical model based on these factors: first, strongly enhanced electric fields within the cavity give rise to TPA. Second, the free-carriers generated by TPA decrease the refractive index of the cavity material by a plasma effect, and optical absorption by the free-carriers also occurs. Third, the energy of the light absorbed by TPA and free-carrier absorption (FCA) is converted into thermal energy, which increases the refractive index of the cavity material. Note that the pulse width used in the experiment (~ 0.5 ms) is long enough for the cavity to reach a thermal equilibrium [11]. Fourth and finally, another third-order nonlinear effect, a Kerr effect, increases the refractive index of the cavity material^{2}.

In order to include the abovementioned factors in our analysis, we applied coupled-mode theory (CMT) [2, 13] to a two-port system consisting of a waveguide and a cavity. Here, an amplitude with a squared magnitude equal to the energy within the cavity is denoted as *a*. At steady state,

where *S*
_{1} is an amplitude with a squared magnitude equal to the incoming power in the waveguide, and ω is its angular frequency. The resonant angular frequency of the cavity is shifted by nonlinear effects to ω′_{0}:

Here, λ_{0} is the original resonant wavelength of the cavity. ∆λ_{free}, ∆λ_{thermal}, and ∆λ_{Kerr} are the shifts in the resonant wavelength, caused by a plasma effect of TPA-generated free-carriers, a thermo-optic effect, and a Kerr effect, respectively. *c* is the velocity of light in the vacuum. 1/τ_{total} is the rate of energy decay from the cavity:

where 1/τ_{v} and 1/τ_{in} are the rates of decay into free space (vertical direction) and into the waveguide (in-plane direction), respectively, and are given by 1/τ_{v(in)} = ω′_{0}/*Q*
_{v(in)} . In this case, *Q*
_{in} = 370,000 and *Q*
_{v} = 600,000. 1/τ_{TPA} and 1/τ_{FCA} are the rates of the optical absorption within the cavity due to TPA and FCA. AAfree, ∆λ_{thermal}, ∆λ_{kerr}, 1/τ_{TPA}, and 1/τ_{FCA} are all functions of |*a*|^{2}, or the energy in the cavity, as derived below.

According to the literature [10], the TPA coefficient of bulk Si, measured using propagating light, is *β*, and the coefficient of optical absorption due to TPA is, *βcε*
_{0}
*nE*
^{2} /2 where *E* is the amplitude of the electric field of the propagating light, ϕ_{0} is the dielectric constant in the vacuum, and *n* is the refractive index of Si. As TPA is proportional to the squared magnitude of the electric fields, the optical absorption coefficient in the cavity α_{TPA} is expressed as

where *E*
_{eff} is the effective electric field for the CMT calculation, derived by considering the local TPA caused by local electric fields (Appendix 1). This gives

where *V*
_{TPA} is the effective cavity volume for TPA (Appendix 1). From these equations, we derive the following:

We know from the definition of the absorption coefficients that 1/τ_{TPA} = (*c*/*n*)α_{TPA}. The density of the free-carriers generated by TPA at steady state is expressed as

where *V*
_{cavity} is the cavity volume for free-carriers (Appendix 2) and τ_{recon} is the recombination time of free-carriers. It should be noted that the small, intrinsic density of free-carriers in the Si used to form the cavity is not included in this equation. The dielectric constant in the presence of free-carriers, ε′, is expressed within the Drude model as

where ε is the original dielectric constant of Si, τ_{relax} is the relaxation time of carriers and ω_{p} is the plasma frequency of free-carriers. Here, ω_{p} is related to the density of free-carriers by

where *e* is the elementary electric charge and *m*
^{*} is the effective mass of carriers. Eq. (8) should be applied to both electrons and holes since TPA generates equal numbers of each. The optical absorption coefficient due to FCA is given by α_{FCA} = ωε_{2} /*cn*, so 1/τ_{FCA} = (*c*/*n*)α_{FCA}. On the other hand, the refractive index change by free-carriers *n*′ is given by

where we neglect ε_{2} since it is more than two orders smaller than ε_{1} and *n*
_{0} is the original refractive index. The shift of the resonant wavelength ∆λ_{free} is given by ∆λ_{free} = (∆*n*
_{free} /*n*
_{0})λ_{0}. The temperature rise ∆*T* in the cavity due to TPA and FCA is expressed as

where *R* is the constant of proportionality of the temperature rise to the absorbed power — that is, the thermal resistance determined by the mechanical structure of the sample. Considering the temperature dependence of the refractive index of Si, the change in the refractive index due to temperature rise ∆*n*
_{thermal} is expressed as ∆*n*
_{thermal} = (*∂n*/*∂T*)∆*T* and the shift in the resonant wavelength ∆*λ*
_{thermal} is given by ∆*λ*
_{thermal} = (∆*n*
_{thermal} /*n*
_{0})λ_{0}. Finally, we must consider a Kerr effect. The change in the refractive index due to the Kerr effect ∆*n*
_{Kerr} is expressed as

where *n*
_{2} is the Kerr coefficient of bulk Si for propagating light [10]. Here, *V*
_{Kerr} is set as equal to *V*
_{TPA}, because a Kerr effect is a third-order nonlinear effect, similar to TPA. The shift of the resonant wavelength by the Kerr effect ∆λ_{Kerr} is given by ∆λ_{Kerr} = (∆*n*
_{Kerr} /*n*
_{0})λ_{0}.

Using the physical parameters shown in Table 1, we performed a nonlinear CMT calculation for the input light powers used in the experiment and derived the radiation powers from the cavity, which are expressed by |*a*|^{2}/τ_{v}. The results, shown in Fig. 3, are in good agreement with the experimental results presented in Fig. 2. We therefore propose that the nonlinear cavity model derived here effectively simulates the actual physical phenomena that take place within a 2D PC nanocavity.

Figure 4 shows the calculated radiation characteristics of a cavity with a *Q* factor of 25,000 (*Q*
_{v} = *Q*
_{in} = 50,000) for comparison. This *Q* factor is an order of magnitude lower than that used in either the experiment or the calculations described above (Figs. 2 and 3), but is of the same order of magnitude as that of cavities described in previous reports [6, 7]. In this case, the cavity characteristics seemed to be almost linear for all the input light powers shown in Figs. 2 and 3. Therefore, we concluded that the experimentally observed nonlinearity in Fig. 2, for such low input powers, resulted from the ultra-high *Q* factor of the nanocavity (~230,000). In order to understand the cavity characteristics in greater detail, we further investigated the individual contribution of each factor in the calculation model described above.

First we investigated optical absorption (TPA and FCA) within the cavity. The rates of absorbed power to input power are shown in Fig. 5 as functions of input power. Each rate was calculated using the resonant wavelengths shifted by the nonlinear effects (thermo-optic, plasma, and Kerr effects) because the nonlinear optical absorption is considered to be significant near the resonant wavelengths. As seen in Fig. 5, the optical absorption, by which the optical energy is eventually converted into thermal energy, is dominated by TPA [17].

Next, we investigated the nonlinearity of the cavity characteristics caused by the change in the refractive index of the cavity material. The changes in the refractive index, due to a thermo-optic effect, a plasma effect, and a Kerr effect, as well as the total change, are shown in Fig. 6 as functions of the input powers at the shifted resonant wavelengths. The results show that the total refractive index change is dominated by the increase due to a thermo-optic effect. The cavity characteristics are consequently red-shifted. The waveforms of the cavity characteristics shown in Fig. 2 appear to “chase” the red-shifted resonant wavelengths from the shorter wavelength side to the longer wavelength side. A sharp drop in the radiation efficiencies occurs when the wavelength of the input light passes the red-shifted resonant wavelength.

## 5. Summary

This study investigated the characteristics of an ultra-high-*Q* photonic nanocavity (*Q* = ~230,000 and modal volume = ~1.2 (λ/*n*)^{3} ) at various input light powers. Nonlinear cavity characteristics were observed even at the low input light power of 10 μW. Analysis, using the nonlinear coupled-mode theory derived from the analytical model proposed in this paper, revealed that the observed nonlinearity was mainly attributable to the refractive index change caused by a thermo-optic effect due to two-photon absorption. Comparison with a cavity with a lower *Q* factor confirmed that the nonlinear effects produced at a low input power of 10 μW were caused by the ultra-high *Q* factor of the nanocavity used here. We believe that the extremely low-power nonlinear effects demonstrated here mean that ultra-high-*Q* photonic nanocavities have great potential for use in nonlinear optical applications.

## Appendix 1.Effective cavity volume for TPA

The *V*
_{TPA} used for the CMT calculation is derived as follows. First, we consider the local electromagnetic energy associated with the electric field, *u*, which is given by *u* = ε_{0}
*n*(**r**)^{2}|*E*(**r**)|^{2}/2, where *E*(**r**) is the amplitude of the local electric field. The local optical absorption rate due to TPA, 1/τ_{TPA}, is given by 1/τ_{TPA} = ε_{0}
*c*
^{2}
*β*(**r**)|*E*(**r**)|^{2}/2. The time derivative of *u* is expressed as

Therefore, the total energy loss due to TPA in the cavity is expressed as

We define the effective absorption rate in the cavity, 1 / τ^{eff}
_{TPA} , as

Therefore, we can define the effective electric field used for the CMT calculation, *E*
_{eff}, by

*V*
_{TPA} is defined such that the electromagnetic energy of *E*
_{eff} distributed over *V*
_{TPA} is equal to the total energy in the cavity. So,

Thus, from Eqs. (15)–(17), we derive the expression

## Appendix 2. Cavity volume used to derive the density of free-carriers

The cavity volume *V*
_{cavity}, used to derive the density of free-carriers, is defined as follows. The length of the cavity in the waveguide direction can be calculated from the diffusion length of the carriers, which is ~1.1 μm (calculated from the electron mobility used to derive the relaxation time shown in Table 1). The length of the cavity in the orthogonal direction should be defined as the distance between the air holes nearest to the center of the cavity in the orthogonal direction, because the surface recombination of the carriers occurs around the air holes. Therefore, we can define *V*
_{cavity} as follows: *V*
_{cavity} = (2 × the diffusion length) × (distance between the air holes nearest to the center of the cavity in the orthogonal direction) × (slab thickness).

## Acknowledgments

We would like to thank Mr. Y. Tanaka for his help with the computational calculations. This work was partly supported by a Grant-in-Aid (no. 15GS0209) and an IT project grant from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and by the Japan Science and Technology Agency (CREST).

## Footnotes

^{1} | We first checked the coupling efficiency of the measuring apparatus without the sample and then we considered the coupling of the light emitted from a line-defect waveguide to objective lens using the FDTD analysis. |

^{2} | Recently, optical absorption due to the surface-state effects in Si ring resonators has been reported [12]. This may be another factor for the cavity characteristics. However, in this paper we did not consider it because we do not know the surface state of the cavity and as shown below, we could explain the experimental results without considering it. This factor may need to be investigated in the future. |

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**17. **
We confirmed that the FCA exceeded the TPA when we used the different value of free-carriers relaxation time in Ref. 6 (where it is described as free-carriers cross-section) for the calculation. However, the cavity characteristics remained similar, because the contributions of both the TPA and the FCA to the cavity characteristics occurred via a thermo-optic effect. It was therefore difficult to experimentally distinguish between these factors in the current work. We think that the TPA was larger than the FCA in this case because FCA is optical absorption consequent from TPA and also because free-carriers recombination time is as short as 0.5 ns.