1. Introduction

Recently, there has been rapid progress in the performance of optical microcavities [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] and they are opening various new fields in optics [11, 12, 13, 14, 15, 16, 17, 18]. Among various forms of microcavities, it has been well established that slight structural modification of photonic-band-gap (PBG) waveguides can create ultrahigh-Q and ultrasmall optical cavities, which have theoretical Q of over 10⁸ and experimental Q of over 10⁶ with the cavity mode volume comparable to (λ/n)³ [3, 4, 5, 6, 7, 10]. Figure 1(a) shows an example of this type of cavities, which we reported previously [4, 6]. The base structure is a line-defect waveguide in a two-dimensional (2D) photonic crystal slab [19]. This type of photonic-crystal waveguides generally posses mode gaps (that is, PBGs in waveguide modes) as shown in Fig. 1(c). If the mode gap is locally modulated, it leads to 3D light confinement, which is the basic mechanism of the cavity formation [3, 20, 21]. In Ref. 3 (photonic-crystal double-heterostructure cavities), the lattice constant of the crystal is altered to realize this modulation. In the design of Fig. 1(a), the local modulation is done by shifting the position of the colored holes away from the waveguide by several nanometers [4, 21]. This tiny structural modulation locally lowers the mode gap edge, which creates cavity modes in the modulated region. We have reported that the cavity in Fig. 1(a) has theoretical Q of 1.5×10⁸ with V _eff=1.5(λ/n)³ [22].

This type of cavities (hereafter we refer them as modulated mode-gap cavities) have some unique characteristics in comparison to other type of microcavities. They are the only ultrahigh-Q cavities whose cavity volume is comparable to (λ/n)³. In addition, very slight modulation can realize ultra-strong light confinement. This latter feature motivated us to control the cavity formation not by the structural modulation, but by the dynamic refractive-index modulation. We have recently reported that we can dynamically change the confinement strength in a width-modulated mode-gap cavity by lowering the confinement potential via ultrafast nonlinearity [23]. In the present article, we investigate the possibility of on-demand cavity formation by dynamic refractive index tuning of a straight (unmodulated) mode-gap waveguide. Recently, cavity formation by post-tuning of a mode-gap waveguide was proposed [24], in which the refractive index of a rectangular area of a straight mode-gap waveguide consisting of a photo-sensitive material (chalcogenide with n=2.7) is assumed to be locally changed by a few % by selective light exposure. They found that the maximum Q of 1×10⁶ is theoretically achievable at Δn=0.04. However, such large index variation is not available in much faster optical nonlinear processes, and therefore the application may be limited. In addition, calculated Q is much smaller than Q of silicon-based structure-modulated mode-gap cavities [6, 7, 10,25] (note that the index of chalcogenide is smaller than that of conventional high index semiconductors, such as silicon). In the present study, we intend to realize ultrahigh-Q cavity formation by much smaller index variation which is achievable by fast optical nonlinear processes, such as Kerr nonlinearity or carrier-plasma nonlinearity, and discuss the possible application to dynamic control of optical signals. In the first part of this paper, we present our numerical studies of cavity formation by local index tuning of photonic-crystal line-defect waveguides. We found that ultrahigh-Q (>10⁸) cavities can be formed by extremely small index variation (<0.1%). In the second part of this paper, we dynamically change the refractive index of a mode-gap waveguide when an optical pulse is traveling in the waveguide. We will show that traveling photons can be pinned by dynamic cavity formation, which we call as “pinning photon” phenomenon.

 

Fig. 1. (a) Width-modulated line-defect mode-gap cavity in a 2D triangular-lattice air-hole photonic crystal slab. The red, green, and blue holes are shifted away from the line defect center by 9, 6, and 3 nm, respectively. (b) Index-modulated line-defect mode-gap cavity in the same photonic crystal slab. The refractive index of the red shadowed region is modulated as expressed by Eq. (1). (c) Theoretical band dispersion of the base W1 mode-gap waveguide (a row of missing holes in the Γ-K direction in a 2D triangular-lattice air-hole photonic crystal slab).

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2. Cavity formation by local index tuning of mode-gap waveguides

Our model structure is shown in Fig. 1(b), which is based on a so-called W1 line-defect waveguide [19] (consisting of a row of missing holes in the Γ-K direction, whose dispersion is shown in Fig. 1(c)) in a 2D triangular air-hole dielectric photonic crystal slab. The refractive index of the dielectric material n₀ is 3.46. The lattice constant a, the thickness of the slab t, and the hole radius r are 420 nm, 210 nm, and 108 nm, respectively. The width of the line defect (the distance between the centers of the nearest-neighbor holes at both sides) is $0.98\frac{\sqrt{3a}}{2}$. These parameters are typical numbers used in conventional photonic crystals used for 1.5 µm wavelength range [6]. In this study, we change the refractive index of the shadowed region so as to form a cavity. Since the index variation would be realized by optical nonlinearity induced by light beam illumination from the top, the index profile of the tuned region is assumed to be a two-dimensional Gaussian,

 

Fig. 2. Optical field distribution (Hz, magnetic field perpendicular to the 2D plane) of index-modulated mode-gap cavities calculated by 3D FDTD method. Size parameters for calculations are described in the text. A red circle represents r=r ₀ which describes the index modulation as in Eq. (1). r ₀ is assumed to be 3a in all calculations except for Δn/n=0.06% where r ₀=5a. The Q values are obtained with the 1244 grid condition for (a, b, c), and with the 1960 grid condition for (d, e).

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The index of the air region is kept constant. Note that this modulation is equivalent to the structural modulation in Fig. 1(a). We have performed numerical simulations to find resonant modes in the tuned waveguides by 3D finite-difference time-domain (FDTD) method. Q is obtained by an exponential fit of the electromagnetic energy decay. The calculation volume is 168×312×1224 grids with perfect-matching layers (8 grids) otherwise specified. The grid size is λ/60.

As a result of numerical calculations, we found that significantly small index modulation can create cavity modes in our structures. Figure 2 shows optical field strength profiles of the tuned waveguides with different Δn/n, and Figure 3 shows corresponding Q and effective mode volume V _eff as a function of Δn/n. In a wide range of the parameters, ultrahigh-Q and ultrasmall cavities can be formed by the index modulation. At relatively large Δn/n (>10^-2), we observed the cavity formation in a similar manner to Ref. 24. At smaller Δn/n, we found that Q becomes extremely high. Note that when Δn/n=0.3% and r₀=3a, the calculated Q for this case is surprisingly high (Q=4.8×10⁹), and the effective cavity mode volume is still 0.206 µm³ (V _eff=2.1(λ/n)³). It is worth noting that this Q value is even higher than Q of structure-modulated mode-gap cavities [6, 7, 10, 25], and it is the highest theoretical Q reported for any wavelength-sized cavities as far as we know. This proves that the present index modulation method is a very effective way to create ultrahigh-Q cavities.

 

Fig. 3. Summarized performance of index-modulated cavities. Q and V _eff as a function of Δn/n. Q ₁ is obtained with the 1224 grids in the horizontal direction, and Q ₂ is obtained with the 1960 grids. r ₀ is 3a except the data point at Δn/n=0.06% and 0.04% where r ₀=5a.

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As is evident from Fig. 2, when Δn/n becomes much smaller, the mode distribution becomes spread along the waveguide direction. This accompanies the gradual increase of V _eff for smaller Δn/n. This makes the calculation increasingly difficult for smaller Δn/n. In Fig. 3, when the number of grids in the horizontal direction is 1224, Q (Q ₁) abruptly deceases when Δn/n is smaller than 0.3%. But when we increase the number of grids from 1224 to 1960 with the same grid size, we observed that Q (Q ₂) is still extremely high even down to Δn/n=0.04%, which clearly shows that Q with 1224 grids at Δn/n<0.3% is limited by the insufficient number of grids in the horizontal direction (that is, leakage in the mode-gap waveguide direction). In contrast, the calculated Q at Δn/n>0.3% does not change by increasing the calculation volume, which proves that Q at Δn/n>0.3% is intrinsic Q (that is, dominated by the vertical radiation loss). We are not sure if the Q values (Q ₂) at Δn/n<0.3% with 1960 grids is intrinsic Q or still limited by the horizontal leakage, because further increase of the calculation volume requires large memory and long calculation time, but we guess that they are limited by the horizontal leakage. Concerning V _eff and the mode profile, we confirmed that there is no noticeable difference when the number of grids is changed in Figs. 2 and 3.

Although there is still some ambiguity in the intrinsic Q values at small index modulation condition, our results have proven the lower limits of the intrinsic Q _s which are already extremely high (e.g. Q at Δn/n=0.06% is 6.7×10⁸). Considering from these findings, we believe that further smaller index modulation (~10^-4) can create ultrahigh-Q with some cost in increasing mode volume. In real experiments, however, there must be lower limits for the smallest Δn/n. One of the reason is because smaller Δn/n leads to larger cavity volume as shown in Fig. 2 and 3. But the issue of the cavity size depends on the application, and note that the cavity volume is still smaller than conventional microcavities. As regards the dynamic photon capturing application described in the next section, relatively large cavity volume will be acceptable. Another issue we have to consider is that the smallest Δn would be also limited by the size homogeneity of mode-gap waveguides, because if the waveguide size is fluctuated, fine local index tuning becomes ineffective. Concerning this issue, we have recently succeeded in dynamically tuning the confinement strength of structure-modulated mode-gap cavities in silicon photonic crystals by changing the refractive index of the order of Δn/n=10^-4 via the optical nonlinear process [23]. Although the situation is different, it is suggesting that Δn/n=10^-4 can be still effective index perturbation for this system. We think that the smallest Δn/n should be determined by performing real experiments.

These results are provoking interesting possibilities. First, the above results indicate that we may be able to realize extremely high-Q cavities by static gradual index modulation induced by various methods, such as photo-sensitive process, temperature variation, pressure variation and etc. Our calculation suggests that one can expect much higher Q for index-modulated mode-gap cavities than Q for structure-modulated mode-gap cavities. We believe that this is because the index modulation is more advantageous to realize gradual modulation than the structural modulation, which eventually leads to less out-of-plane radiation loss. Second, the results mean that ultrahigh-Q cavities can be formed in a reconfigurable way. We can create an ultrahigh-Q cavity at any position in the waveguide, which may be useful in various situations. Third, the above results suggest that ultrahigh-Q cavities can be dynamically formed by optical nonlinear process induced by ultra-fast light beam illumination. Since conventional high-Q cavities are static in an ultra-fast time scale, such situations have been hardly available. In fact, this dynamic cavity formation process should be useful in trapping and releasing photons in a chip. For this application, the predicted ultrahigh-Q will be important because it means a long photon capture time (Note that Q of 4.8×10⁹ means a photon lifetime of ~4 µsec at 1.5 µm wavelength). In the next section, we investigate such dynamic cavity formation process as a time-dependent phenomenon, and study the possibility of capturing traveling photons by shining a light beam.

3. Dynamic pinning of photons traveling in a waveguide

In this section, we study a situation shown in Fig. 4(a). A relatively long light pulse is traveling in a mode-gap waveguide, and at a certain timing, we start to illuminate a certain point of the waveguide by a focused optical beam, which induces Gaussian index modulation. We expect that traveling photons can be captured in the dynamically-formed cavity. The holding time of the photons should be the same as the photon lifetime of the formed cavity, which can be extraordinarily long as shown in the previous section. The base waveguide structure is the same as that in Fig. 2. For this simulation, we employ 2D FDTD method because this simulation requires large computation area. In 2D FDTD calculation, the vertical radiation loss is neglected, which is justified by the fact that the calculation time is much shorter than the photon lifetime of the formed cavity obtained from Fig. 3.

First, a long light pulse is injected from the left end of the waveguide, and it propagates toward the right end. The temporal profile of the pulse without tuning is shown in Fig. 5(b). The wavelength of the injected light (1641 nm) was determined such that the wavelength of the light pulse after the index tuning matches the resonant wavelength of the would-be-formed cavity (In this process, the dynamic wavelength shift takes place. The detail of the wavelength shift will be discussed later in Fig. 6). Figure 4(b–h) shows snapshots of field intensity profiles of traveling optical pulse in the waveguide with index tuning. As can be seen in Fig. 4(b, c), the input pulse is propagating from left to right. Here, we change the refractive index of the circular region (with r₀=3a) by Δn/n=0.8% at the position indicated by a red circle in Fig. 4(a) at t=11.025 ps (t ₀) (right after Fig. 4(c)). The assumed index variation as a function of time is shown in Fig. 5(a). The characteristic tuning time τ (defined as the time when t=t ₀-τ, Δn/n=(Δn/n)_max/(1+e)) is 0.1 ps which is achievable by Kerr or carrier-plasma effect. As clearly visible in Fig. 4(d–h), a significant portion of the light intensity is indeed captured in the tuned area after t=t ₀, and the captured intensity stays there long after the pulse passed. That is, photons in the propagating pulse are pinned to the spatial point where the index is tuned. Note that the field profile of the pinned state is exactly the same as the cavity mode profile shown in Fig. 2, which proves that the light intensity is captured as the cavity mode.

 

Fig. 4. Snap shots of optical field distribution with index tuning. An optical pulse with the wavelength of 1641 nm is injected from the right end of a line defect waveguide. At t=1.025 ps, the refractive index of the red shadowed region is tuned by 0.8%. The temporal profile of the index tuning is shown in Fig. 5(a). The structural parameters are the same as in Fig. 2 except we employ the effective refractive index of n=2.8.

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In order to know how much portion of the light intensity is pinned, we monitor the light strength at the center of the tuned area. Figure 5(b–e) shows the field strength at this point with and without tuning. Without tuning (Fig. 5(b)), the plot shows that a single light pulse is simply passing through this point. With tuning (Fig. 5(c)), the plot clearly shows that the most of the field intensity at t=11.025 ps is captured in this point after the tuning, which means that photons in this region is literally pinned by the index tuning. Next, we examined how the tuning rate affects this process, as shown in Fig. 5(d and e), which showed that faster tuning rate leads to larger pinning efficiency. This makes sense if we consider the light traveling speed of the light pulse because when the tuning rate is slow, photons may pass over the tuned region before they are pinned.

 

Fig. 5. (a) Temporal profile of the index variation with Δn/n=0.8% at τ=0.1 ps and t₀=11.025 ps. (b–e) Magnetic field at the center of the tuned point (shown in Fig. 5(a)) as a function of time. (b) No tuning case. (c) Tuned by Δn/n=0.8% at τ=0.1 ps at t=11.025 ps. (d) Tuned by Δn/n=1% at τ=0.05 ps at t=11.025 ps. (e) Tuned by Δn/n=1% at τ=0.5 ps at t=11.025 ps.

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In these simulations, we assumed relatively large index variation than that in the previous section, which is simply due to the limitation of the computation time. If we want to pin photons at smaller index variation, we are forced to input much longer light pulse whose spectral width is sharper, which is difficult to perform in the present FDTD calculation. However, in real experiments, it is not difficult to input a longer optical pulse, and we believe that the photon pinning phenomenon discussed in the section should be achievable in real experiments.

 

Fig. 6. Frequency spectrum of the field intensity at the tuned point. The wavelength of the input pulse is 1641 nm. (a) Spectrum for the whole process in Fig. 4 (the time period is approximately 62 ps). (b) Spectrum after tuning (last half of the time period in (a), approximately 31 ps).

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This process is related to dynamic (adiabatic) tuning of optical micro-systems which is attracting much attention recently [23, 26, 27, 28, 29], which literally means tuning an optical small system within its photon dwell time. In fact, the pinning of photons demonstrated in Figs. 4 and 5 is realized by adiabatic wavelength conversion induced by the dynamic index tuning. That is, photons initially at the propagating states of the mode-gap waveguide are adiabatically shifted to the pinned state (cavity mode). During this process, the wavelength of the pinned photons is actually shifted, which is equivalent to adiabatic wavelength conversion discussed in [23, 27, 28]. In terms of the photon capturing, this process is also related to stopping light by electromagnetically-induced transparency (or its equivalent processes) [26, 29] where time-varying operation transforms traveling light into a frozen state.

To directly examine this issue, we performed Fourier conversion of the field data at the tuned point. The resultant spectra are shown in Fig. 6. Figure 6(a) shows a spectrum obtained from the field data for the whole process in Fig. 4. There are two distinctive peaks in the spectrum, which are assigned to the input running pulse and the pinned pulse after the tuning. To directly check this assignment, we made another spectrum obtained from the last half of the field data used in Fig. 6(a). That is, it corresponds to the spectrum after the tuning. Apparently, we observe only a single peak located at 1651.3 nm, which unambiguously demonstrates that the peak at 1651.3 nm corresponds to the pinned state observed in Fig. 4(f, g, h). This result clearly shows that the wavelength of the pulse is shifted from 1641 nm to 1651.3 nm. The final wavelength is exactly the same as the static resonant wavelength of the index tuned cavities with Δn/n=0.8%. Note that the peak at the pinned state is remarkably sharper than that for the input pulse (although the calculation time is not sufficiently long to resolve the intrinsic width of the pinned state), which manifests the bandwidth compression as is expected for the stopping light process [26] and is consistent with the high Q for this state.

4. Discussions and conclusions

Before concluding, we add some remarks which were not touched in the above sections. In all through the works presented here, we assumed red shift of the index variation (that is, n is increased). It might be worth noting that index blue shift also can function in a similar manner when we employ a mode-gap waveguide having opposite sign of the dispersion, such as in [10, 30, 31]. In addition, the photon pinning process presented in the last section does not capture the whole single pulse packet. To accomplish this, we would need to carefully design several parameters including the pulse length, the index variation profile, and the tuning rate, which is left for the future study.

In summary, we have shown that we can reconfigurably create an ultrahigh-Q wavelength-sized cavity in a mode-gap line-defect waveguide in a photonic crystal by local index tuning. The most important aspect of our findings is the fact that extraordinarily high Q can be obtained by this tuning with very small index variation. The maximum theoretical Q of this index-modulated mode-gap cavities is as high as 5×10⁹, which is higher than Q of structure-modulated mode-gap cavities, and the highest among any wavelength-sized cavities as far as we know. The required index variation (Δn/n) is ~10^-3 or even smaller, which is making possible to create a cavity dynamically by fast optical nonlinear process. In our simulation, we confirmed that high-Q cavities can be formed even at Δn/n=0.06%, and we believe that much smaller index variation would be sufficient since our calculation is limited by the computation resource. To show the applicability employing ultrafast tuning, we have performed numerical simulations which clarified that traveling photons in a straight mode-gap waveguide can be pinned by slight index variation induced by optical beam illumination. This photon pinning process can be useful in various situations for photon memories, photon manipulations, and all-optical on-chip processing.