1. Introduction

Techniques that enable a dynamic control of coupling between different photonic modes in nano- or microstructures can be used for a sophisticated manipulation of photons. Related to this idea, various interesting optical phenomena and manipulation techniques have been proposed and demonstrated, such as periodic modulation of a coupled system to induce non-reciprocal phenomena [1–4] and synthetic dimensions [5–7], non-adiabatic control of a coupled system to manipulate Rabi oscillations [8] and to realize time reversal of light [9], and adiabatic control of a coupled system to enable manipulation of singularities [10], manipulation of geometric phases [11] and on-demand photon transfer [12–14]. In particular, we demonstrated that electrically controlled on-demand photon transfer can be implemented by using a two-dimensional (2D) silicon (Si) photonic crystal containing three high-quality-factor (high-Q) nanocavities coupled through photonic waveguides [14]. Figure 1(a) illustrates the considered system: cavities A and B are used to store photons, and the resonance frequency of cavity C is electrically controlled using an integrated p-i-n junction. In this system, a photon in cavity A can be transferred to cavity B at a time determined by the external electrical signal. This scheme is useful for the realization of devices that can buffer optical information without converting it into electrical signals. However, the performance achieved in [14] is still insufficient for actual application, since the photon lifetime in the hold state was about 1.1 ns–1.4 ns, the transfer time was about 0.8 ns–0.9 ns, and the transfer efficiency was about 70%. Therefore, methods that can significantly improve the performance need to be developed. To improve the performance, it is necessary to simultaneously increase both the coupling coefficient between neighboring nanocavities and the photon lifetime in the nanocavities. However, because a long lifetime and strong coupling are incompatible in standard photonic crystal designs, it is not trivial to achieve a simultaneous increase in both parameters.

 

Fig. 1. (a) Schematic diagram of the coupled cavity system used in this work, which consists of the three cavities A, B, and C. Cavities A and B are used for the storage of photons, and cavity C is used to control the photon transfer from A to B. For the connection between cavities A and C and that between cavities C and B, two coupling waveguides are used. Q_in and Q_v are the cavity quality factors reflecting the losses due to coupling to the waveguide and the power radiated into free space, respectively. The effective coupling coefficient μ between neighboring cavities is determined by the ratio of the cavity’s angular resonance frequency ω to 2Q_in. The ω of cavity C can be varied by injecting carriers through the integrated p-i-n junction. (b) Mode diagram of the system shown in (a). The resonance frequencies of cavities A, B, and C (dashed lines) and the frequencies of the three eigenmodes of the coupled cavity system (black solid lines) are plotted as a function of the resonance frequency of cavity C. The eigenmodes show anti-crossing behavior with gap sizes on the order of the coupling coefficient [∼o(μ)]. (c)–(f) Simulation results of the photon transfer in the system shown in (a) for different values of |dλ_c/dt| and μ, where λ_c is the resonance wavelength of cavity C. Each figure consists of two panels: The upper panels show the temporal evolution of the cavity resonance wavelengths. The lower panels show the relative numbers of photons in each cavity.

Download Full Size | PDF

In this paper, we present a type of photonic-crystal design that can overcome the above-mentioned incompatibility. We obtained an improved photonic-crystal structure by using a more symmetric coupling configuration and a unique automatic structure-tuning method. In addition, we developed a fabrication process that reduces the contamination associated with the introduction of the p-i-n junction, which usually leads to a reduction of the cavity photon lifetime. We demonstrate that these methods allow us to improve the coupling coefficient by a factor of about three and the cavity photon lifetime by a factor of about two. The developed methods enable not only the improvement of on-demand adiabatic photon transfer, but also performance improvements of various other considered devices that are based on the dynamic control of coupling between photonic modes.

2. Operating principle of the device

Figures 1(a) and (b) show a schematic diagram of the coupled three-cavity system and the principle of adiabatic optical transfer [12], respectively. The angular resonance frequencies (resonance wavelengths) of the three cavities A, B, and C are denoted with ${\mathrm{\omega }_\textrm{A}}$ (${\mathrm{\lambda }_\textrm{A}}$), ${\mathrm{\omega }_\textrm{B}}$ $({{\mathrm{\lambda }_\textrm{B}}} )$, and ${\mathrm{\omega }_\textrm{C}}$ (${\mathrm{\lambda }_\textrm{C}}$), respectively, and there are two coupling waveguides, which connect cavities A and C on the left side and cavities B and C on the right side. Cavities A and B are used to store photons, while cavity C is used to control the transfer of the photons from A to B. By separating the roles of storage and control in this way, it is possible to suppress the effects of absorption due to carrier injection into the i-layer of the p-i-n junction, which is necessary to control the system. We define Q_in as the quality factor determined by the inverse of the loss due to coupling to the waveguide (the ideal value of Q_in is calculated by assuming that the waveguide length is semi-infinite) and Q_v as the quality factor determined by the inverse of the radiation loss (the fraction of power radiated into free space). If the propagation time of the light in the coupling waveguide is sufficiently short compared to the time constant determined by Q_in, and the resonance frequencies of the cavities (${\mathrm{\omega }_\textrm{A}},{\mathrm{\omega }_\textrm{B}},{\mathrm{\omega }_\textrm{C}} \cong \mathrm{\bar{\omega }}$) are sufficiently far away from the frequencies of the Fabry-Perot modes in the coupling waveguide, the amount of light in the coupling waveguide is considerably smaller than that in the cavities. In this case, the system can be well approximated by a simple model in which the cavities are directly coupled to each other with an effective coupling coefficient $\mathrm{\mu} \cong \frac{{\mathrm{\bar{\omega }}}}{{2{Q_{\textrm{in}}}}}$, and the temporal evolution of the mode amplitude of each cavity, denoted by ${a_\textrm{A}},{a_\textrm{B}},$ and ${a_\textrm{C}}$, can be expressed by the following equation [8]:

Here, it is noted that the use of such coupling waveguides allows coupling between cavities separated by tens of the optical wavelength, and this facilitates both the introduction of an electrical control mechanism, which requires space due to the electrical contacts, and an easy observation of the individual cavities.

The three black solid curves in Fig. 1(b) show the frequencies of the three eigenmodes of this system [determined by the matrix (Hamiltonian) in Eq. (1)] as a function of ${\mathrm{\omega }_\textrm{C}}$. Here, we focus on the contribution of the three cavities to the middle eigenmode, which is illustrated in the schematics of the mode amplitudes of the cavities in the insets. The composition ratio changes depending on the value of ${\mathrm{\omega }_\textrm{C}}$: When ${\mathrm{\omega }_\textrm{C}}$ is sufficiently small compared to ${\mathrm{\omega }_\textrm{A}}$ and ${\mathrm{\omega }_\textrm{B}}$ [left-hand side of Fig. 1(b)], the middle eigenmode consists almost entirely of cavity A. When ${\mathrm{\omega }_\textrm{C}}$ is sufficiently large compared to ${\mathrm{\omega }_\textrm{A}}$ and ${\mathrm{\omega }_\textrm{B}}$, it is almost entirely composed of cavity B. Therefore, if we first excite this eigenmode under the condition ${\mathrm{\omega }_\textrm{C}} \ll \; {\mathrm{\omega }_\textrm{A}}$ and then slowly increase ${\mathrm{\omega }_\textrm{C}}$, photons can be adiabatically transferred from cavity A to B. The adiabatic condition is fulfilled if the photons in each eigenmode remain in their corresponding instantaneous eigenmode during the increase in ω_C. The transfer time corresponds to the time that ${\mathrm{\omega }_\textrm{C}}$ requires to cross the frequency-gap between the eigenmodes, which is order of $\mathrm{\mu}$. Therefore, we define the transfer time as $\mathrm{\mu}/\left|{\frac{{\textrm{d}{\mathrm{\omega }_\textrm{C}}}}{{\textrm{d}t}}} \right|$. In practice, we need to use a relatively short transfer time to obtain a reasonable device operating speed and to reduce the radiation and absorption losses during the transfer. Hence, $\left|{\frac{{d{\mathrm{\omega }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ should be as large as possible, but on the other hand, too large values are incompatible with an adiabatic transfer because the frequency uncertainty is proportional to $\left|{\frac{{\textrm{d}{\mathrm{\omega }_\textrm{C}}}}{{\textrm{d}t}}} \right|/\mathrm{\mu}$ (i.e., the inverse of the transfer time). If the frequency uncertainty is larger than the gap between the eigenmodes $(\textrm{i}.\textrm{e}.,\; \left|{\frac{{\textrm{d}{\mathrm{\omega }_\textrm{C}}}}{{\textrm{d}t}}} \right|/\mathrm{\mu} >\mathrm{\mu} ,$ or $\left|{\frac{{\textrm{d}{\mathrm{\omega }_\textrm{C}}}}{{\textrm{d}t}}} \right|{ >\mathrm{\mu} ^2}$), the photons can transit to other eigenmodes, and the fraction of photons that remain in the middle eigenmode is reduced. If the frequency uncertainty is sufficiently small compared to the gap between the eigenmodes$,$ the photons remain in the middle eigenmode and are adiabatically transferred to cavity B. In [15], Landau and Zener analyzed the details of the adiabatic condition for atomic systems, and when we apply this condition to our system [13], the fraction of light that is adiabatically transferred, ${P_{\textrm{adiabatic}}}$, is given by

From this equation, we can see that, if we increase $\left|{\frac{{\textrm{d}{\mathrm{\omega }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ to shorten the transfer time, $\mathrm{\mu}$ needs to be increased by a factor equal to the square root of the change in $\left|{\frac{{\textrm{d}{\mathrm{\omega }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ in order to maintain the same value of ${P_{\textrm{adiabatic}}}$ . If the condition $\mathrm{\mu} \propto {\left|{\frac{{\textrm{d}{\mathrm{\omega }_\textrm{C}}}}{{\textrm{d}t}}} \right|^{0.5}}$ is satisfied, the transfer time ($\mathrm{\mu} {\left|{\frac{{\textrm{d}{\mathrm{\omega }_\textrm{C}}}}{{\textrm{d}t}}} \right|^{ - 1}}$) is proportional to ${\mathrm{\mu}^{ - 1}}$, and the radiation loss during the transfer time is proportional to $Q_\textrm{v}^{ - 1}{\mathrm{\mu}^{ - 1}}$.

Figures 1(c)–(f) show numerical results of the photon transfer governed by Eq. (1) including the effect of radiation losses. The resonance-wavelength change of cavity C was assumed to be due to carriers injected through the p-i-n structure, and the associated optical absorption loss [16] was also included in our simulations. The details of the simulations are explained in Appendix A. Figure 1(c) shows the results obtained by using the typical parameters achieved in [14] ($\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ = 0.2 nm/ns for a voltage of 1.8 V, $\mathrm{\mu}$ = 11 Grad/s, Q_v =1.5 × 10⁶). This calculation condition is sufficiently close to the adiabatic condition. The upper panel of Fig. 1(c) shows the resonance wavelengths as a function of time, and the lower panel shows the number of photons in each cavity. It can be seen that the carriers injected into cavity C do not significantly affect the maximum possible storage time (determined by the photon lifetime) after the transfer, which is important for optical buffering. The carriers have a small impact, because the fraction of light in cavity C is small when it is largely detuned from cavities A and B. Figure 1(d) shows the results obtained when $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ is increased by a factor of 4. Almost half of the photons remain in cavity A, because the adiabatic condition is not maintained. Figure 1(e) shows the results obtained when $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ is increased by a factor of 4 and $\mathrm{\mu}$ is increased by a factor of 2. It can be seen that the adiabatic condition is maintained, the transfer time is reduced by a factor of 2 compared to Fig. 1(c), and the loss during the transfer is also reduced. Figure 1(f) shows the results obtained when Q_v is doubled in addition to changes made in Fig. 1(e). It can be seen that the loss during the transfer is further reduced and the maximum possible storage time of the photons also increases. This is what we want to achieve by improving the structure and the fabrication process.

3. Improved photonic-crystal design

3.1 General considerations

As shown in the previous section, the ability to increase $\left|{\frac{{\textrm{d}{\mathrm{\omega }_\textrm{C}}}}{{\textrm{d}t}}} \right|$, $\mathrm{\mu}$, and Q_v is important to improve the performance of the photon-transfer in a coupled nanocavity system. Among these three parameters, only $\left|{\frac{{\textrm{d}{\mathrm{\omega }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ is determined by the p-i-n junction, and thus only this parameter can be increased by reducing the i-layer width and/or increasing the applied voltage. In this work, such electrical design considerations are discussed in Appendix B, and in the following we focus on the photonic design, which determines $\mathrm{\mu}$ and Q_v. To increase $\mathrm{\mu}$, it is generally necessary to place the cavity and the coupling waveguide closer together. However, as the distance between them decreases, the electromagnetic field distribution of the cavity is disturbed, which increases the radiation loss (probably due to the combination of the resulting asymmetry of the cavity mode field [17,18] and the more abrupt change in the envelope of the cavity mode field [19]), and thus Q_v decreases. Therefore, it is inherently difficult to increase $\mathrm{\mu}$ while keeping Q_v high.

3.2 Choosing a design with higher symmetry

In our previous work on on-demand photon transfer between nanocavities [14], a so-called side-coupling configuration was adopted, where the coupling waveguide is placed parallel to the cavity axis with a certain offset as shown in Fig. 2(a). In this figure, the circles indicate the air holes of the photonic crystal (the actual sample uses an air-bridge Si photonic-crystal with a slab thickness of 220 nm). At the center of the figure, we can see a missing row of air holes, which basically defines a short waveguide. Our high-Q cavity is defined by an additional modulation of the lattice constant a to form a heterostructure (the areas with a = 413 nm and 416 nm). Due to the heterostructure, a cavity mode forms on the low frequency side of the propagation band of the waveguide. The distance between the centers of the air holes on both sides of the short waveguide is defined as W₀. The width of the coupling waveguide is larger, 1.12W₀, which leads to a shift of its propagation band to the lower frequency side and allows the photons stored in the cavity to propagate in the coupling waveguide. In the case of no structural optimization, $\mathrm{\mu}$ is approximately 3.8 Grad/s and Q_v ≈ 1 × 10⁷. To obtain an optimized structure, the spatial relationship between the coupling waveguide and the cavity was set appropriately and the positions of the air holes within the rectangular area indicated by the red dashed line in Fig. 2(a) were adjusted by using an optimization method based on machine learning [20]. In this method, several sample structures are generated by randomly shifting the air-hole positions, and their $\mathrm{\mu}$ and Q_v values are calculated using three-dimensional (3D) finite-difference time-domain (FDTD) simulations (3D simulations are needed to evaluate the radiation losses, which are used to determine the Q_v values). A deep neural network is then trained on this data set consisting of air-hole displacements, $\mathrm{\mu}$ and Q_v. The trained neural network is used to approximately evaluate the objective function (which we set to the product μ×Q_v) for a given air-hole pattern and its derivative with respect to the air-hole positions. New sample structures are generated by shifting the air-hole positions based on the gradient method [20]. By repeating these steps, we obtained a design with a $\mathrm{\mu}$ of 10 Grad/s and a high Q_v exceeding 4 × 10⁷ ($\mathrm{\mu}$ and Q_v were evaluated as explained in Appendix C).

 

Fig. 2. Details of different 2D photonic-crystal designs (we consider the use of air-bridge Si photonic crystals with a slab thickness of 220 nm). The circles indicate the air holes of the photonic crystal, of which radii are 110 nm. (a) The design of cavity A and the coupling waveguide in the case of a side-coupling configuration, which was used in our previous work. (b) The design of cavity A and the coupling waveguide in the case of a coaxial-coupling configuration, which is used in this work. W₀ is the distance between the centers of the air holes on both sides of the line defect at the center. The high-Q cavity is formed by modulating the lattice constant in the x-direction: a is changed from 410 nm to 416 nm. The coupling waveguide has a larger width to ensure the propagation of the photons stored in cavity A. The distribution of the y-component of the cavity-mode electric field (E_y) is shown by the color plot behind the photonic-crystal structure. The positions of the air holes in the red rectangle are tuned in such a way to decrease Q_in (increase μ) while maintaining a sufficiently large Q_v. The horizontally flipped structure is used for cavity B, and the structure obtained by mirroring the right side of the figure about the central y-axis is used for cavity C.

Download Full Size | PDF

Note that the side-coupling configuration in Fig. 2(a) has no mirror symmetry about the cavity axis. In contrast, in this work we adopt the coaxial-coupling configuration shown in Fig. 2(b) to preserve mirror symmetry about the cavity axis. Since a design with a higher symmetry has lower radiation losses, the coaxial-coupling configuration is considered more suitable for obtaining a large $\mathrm{\mu}$ while maintaining a high Q_v. When the boundary between the coupling waveguide (1.12W₀) and the cavity (1.0W₀) is set to the position indicated by thick black dashed line in Fig. 2(b) and no optimization of the air-hole positions is performed, a Q_v exceeding 4.8 × 10⁷ is obtained and $\mathrm{\mu}$ is around 3.6 Grad/s.

3.3 Improvement of the optimization method

Regarding the optimization of the air-hole positions, we previously used a rather simple neural-network-based scheme and succeeded in obtaining a structure with a relatively large product $\mathrm{\mu} \times $Q_v. However, we realized that this scheme makes it difficult to control the $\mathrm{\mu}$ and Q_v values of the newly generated structures independently, because the gradient with respect to a scalar objective function ($\mathrm{\mu} \times $Q_v) was used: the result was often a structure with a large $\mathrm{\mu}$ but a very small Q_v. We also tried to control the direction of the sample evolution in the parameter space by setting up a more complex objective function, but it was difficult to achieve the desired independent control.

In this work, we use the Covariance Matrix Adaptation Evolution Strategy (CMAES) [21–23] in combination with two additional methods [explained in Figs. 3(a) and (b)] to control the direction of the sample evolution. In CMAES, a multivariate normal distribution $\mathrm{\vec{{\cal N}}}$ is used to generate the next generation of sample structures, and the covariance matrix $\boldsymbol{C}$ and the mean vector $\vec{m}$ of $\mathrm{\vec{{\cal N}}}({\vec{m},\boldsymbol{C}} )\cong \vec{m} + \mathrm{\vec{{\cal N}}}({0,\boldsymbol{C}} )$ are set using a certain number of structures selected from the previous generation of sample structures based on a given criterion. To explain this scheme mathematically, we first define a structural parameter vector $\vec{s}$ consisting of the set of displacements of all air holes in the tuning region for a given structure, and then consider to choose M structures from the previous iteration cycle. This results in M structural parameter vectors, i.e., if we use the index i to identify each structure, we obtain ${\vec{s}_{\textrm{selected},i}},\; i = 0, \ldots M - 1$. Then, $\vec{m}$ and $\boldsymbol{C}$ are updated by using $\frac{1}{M}\mathop \sum \limits_i {\vec{s}_{\textrm{selected},i}}$ and $\frac{1}{M}\mathop \sum \limits_i ({{{\vec{s}}_{\textrm{selected},i}} - \vec{m}} ){({{{\vec{s}}_{\textrm{selected},i}} - \vec{m}} )^\textrm{T}}$, respectively [22]. Finally, new structures are generated by randomly sampling from $\vec{m} + \mathrm{\vec{{\cal N}}}({0,\boldsymbol{C}} )$. Therefore, the new structures are random while reflecting the statistical characteristics of the selected structures. Such a preservation of the statistical characteristic is even useful in exploring new regions in the parameter space by non-deterministic generation [24]. Furthermore, since the gradient method is not used, it is not necessary to set up a differentiable scalar objective function.

 

Fig. 3. Visual explanation of the methods used to automatically optimize the positions of the air holes with respect to μ while maintaining a sufficiently large Q_v. (a) The essence of the CMAES algorithm: new samples are generated by sampling from a multivariate normal distribution, whose mean and covariance matrix are updated by using preferable structures selected from the structures generated in the previous round. In addition, the sample selection method was improved as illustrated by the difference between the left and right figures: Instead of simply selecting samples that result in a large value of the scalar valued objective function μ×Q_v (left), only samples in the preferred region of the two-dimensional space spanned by μ and Q_v are selected in this work (right). This avoids undesirable samples with a very low Q_v and very high μ. (b) Weighting method for tuning the air-hole positions by avoiding large displacements of holes that are likely to lead to a reduction of Q_v if they are moved. The real-space distribution of the leaky components of the cavity is first calculated, and then a new air-hole pattern is generated using the multivariate normal distribution weighted by the normalized intensity of the leakage at each air hole.

Download Full Size | PDF

The right-hand side of Fig. 3(a) illustrates our improved sample selection method, which is one of the two additional methods introduced to control the direction of the sample evolution. Here, we first select samples with a high Q_v (here, the upper half of the Q_v distribution), and then we select a certain number of samples in the order of their $\mathrm{\mu}$ values. Based on these selected samples, new structures are generated according to the procedure explained above. This allows us to explore the parameter space where both Q_v and $\mathrm{\mu}$ are large. This is not a selection based on a scalar-valued objective function, but is equivalent to considering a two-dimensional space (spanned by $\mathrm{\mu}$ and Q_v) and selecting the preferred region within that space. This method is generally considered useful in optimizing multiple parameters simultaneously. Note that independent control of multiple parameters is difficult even with CMAES, if the preferred structures are selected based on a scalar objective function as shown on the left-hand side of Fig. 3(a).

The second additional method introduced to control the direction of the sample evolution is the leaky-component visualization method [25]. In this method, first the Fourier transform of the electromagnetic field distribution of the cavity needs to be calculated, and then all components except those in the light cone are removed before the inverse Fourier transform is performed to visualize the distribution of the light leakage (the radiation loss) [25]. In the resulting image, we can determine whether a given hole is in an area of low or high leakage. It is almost certain that moving the air holes in areas of low leakage will increase radiation losses and decrease Q_v. In contrast, moving the air holes in areas of high leakage may increase or decrease Q_v, depending on the adjustment. Therefore, we find that optimizing the positions of air-holes in high-leakage areas is more likely to result in structures with larger Q_v and $\mathrm{\mu}$ values compared to the case of optimizing those in low-leakage areas. To implement this finding into our automated optimization procedure, we normalized the leaky-component distribution, constructed a weight vector $\vec{L}$ by using the intensity at the center of each hole, and generated new structures by using the element-wise product of $\vec{L}$ and $\mathrm{\vec{{\cal N}}}$ [Fig. 3(b)].

3.4 Theoretical results

The result of the structure tuning using the above methods is shown in Fig. 4(a). The magenta triangle indicates the starting point [the structure shown in Fig. 2(b)], and we generated 30 samples per generation, selected 15 out of them for the optimization in the current iteration cycle, and repeated this 100 times. As can be seen from the figure, the structures evolve mainly toward larger $\mathrm{\mu}$ values while maintaining a high Q_v of at least about 2 × 10⁷. By using Q_v > 4 × 10⁷ as the final criterion, which is the theoretical Q_v of the structure used in the previous work (green circle), a structure with $\mathrm{\mu}$ = 32 Grad/s and Q_v = 4.1 × 10⁷ (magenta square) was obtained. This $\mathrm{\mu}$ is about 9 times larger than that of the initial coaxial structure and about 3.5 times larger than that of the optimized side-coupling structure with a similar Q_v. The displacements of the air holes with respect to the hole positions of the initial structure are shown in Fig. 4(b), where the circles (their sizes and relative positions with respect to each other) are plotted on the same scale as the lattice constant a, but the displacements indicated by the black arrows are plotted 100 times larger (when the length of the arrow is a, the actual displacement is a/100). During the optimization, the waveguide in the vicinity of the boundary between the cavity and the coupling waveguide (the thick black dashed line) became wider on the cavity side and narrower on the other side, which means that the change in the waveguide width near the boundary became smoother. It can also be seen that the electric field amplitude in the coupling waveguide of this structure is significantly larger than that of the initial structure [Fig. 2(b)], indicating that Q_in has decreased ($\mathrm{\mu}$ has increased). Figure 4(c) shows the distribution of the light leakage. Compared to the distribution before the tuning procedure [Fig. 3(b); right side], where the leakage at the cavity/waveguide boundary is large, the leakage after the tuning is larger on the right-hand side of the initial boundary line. It can be considered that the change in the width near the boundary results in an increase of the horizontal leakage into the waveguide and a decrease of scattering at the initial boundary line, resulting in radiation loss mainly at the right-hand side of the initial boundary line. The effects of the other small hole shifts are not yet completely understood.

 

Fig. 4. Results of the automatic structure optimization procedure aimed at obtaining large μ values while maintaining a sufficiently large Q_v. (a) The evolution of the system performance in terms of Q_v and μ, where each dot corresponds to one structure generated during the optimization procedure (a set of displacements of the air holes) and the colors of the dots represent the optimization cycle. (b) The finally selected structure. The displacements of the air holes are represented by the black arrows (note the scale for the arrows at the bottom). The higher electric field in the coupling waveguide compared to that before optimization [Fig. 2(b)] directly indicates the increase of μ. (c) The distribution of the leaky components of the cavity mode of the final structure. Compared to Fig. 3(b), the leakage is concentrated more to the right side.

Download Full Size | PDF

4. Sample fabrication

In the previous section, the coupling coefficient $\mathrm{\mu}$ was theoretically improved by a factor of 3.5 while maintaining Q_v at the same level as that of the previous structure. However, we need to consider that the quality factors of actually fabricated cavities are usually affected by absorption losses due to surface contamination and scattering losses due to air-hole fabrication fluctuations, in addition to radiation losses [26]. In our previous work on on-demand photon transfer, the theoretical Q_v exceeded 4 × 10⁷, while the experimental Q values obtained from the photon lifetimes of the fabricated cavities were around 1 × 10⁶ to 2 × 10⁶ [14]. Considering that experimental Q values exceeding 1.1 × 10⁷ have been achieved for similar cavities without the additional p-i-n junction [26], the previously obtained low experimental Q values can be attributed to increased surface contamination due to the more complex fabrication process required for the introduction of the p-i-n junction. It is therefore necessary to improve the fabrication process with respect to surface contamination. (The influence of air-hole fabrication fluctuations in the presently considered cavities is relatively small as discussed in Appendix D.)

As mentioned above, we use air-bridge Si photonic crystals, which are typically fabricated as follows: first, the top Si layer of a silicon-on-insulator (SOI) wafer is patterned by e-beam lithography and plasma etching, and then the etching of the SiO₂ layer below the top Si layer is performed using an aqueous solution of hydrogen fluoride (HF). The introduction of a p-i-n junction [14] adds the following fabrication steps: (a) implantation and activation of the dopants in the p- and n-regions before the fabrication of the photonic crystal pattern, (b) fabrication of metal electrodes (the contacts of the p- and n-regions as well as the microheaters required for temperature adjustments) after the fabrication of the photonic crystal pattern, and (c) undercutting using dry HF instead of an HF solution after (b), since the Al electrodes are affected by HF solutions [27]. Note that (d) cleaning by oxidation annealing and immersion in a dilute HF solution [26] cannot be applied to such a sample since Al was used. Regarding (a), we were already able to sufficiently suppress the contamination due to this process step in the previous study by using a SiO₂ film as a protective film during the dopant activation [14]. In this work, we used a similar protective film during the dopant implantation process to better protect the i-layer. Additionally, we deposited a SiO₂ film on the photonic crystal pattern before (b) to avoid contamination during electrode fabrication process. Regarding (b) and (d), we changed the electrode material from Al to Au/Cr [28] to enable cleaning by an acid (we used a weak piranha solution) prior to undercutting. Besides this modification of the fabrication procedure, in order to improve $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$, the width of the i-layer was reduced to 3.5 μm (that of the previous sample was 4.5 μm), and the doping density was increased to 6 × 10¹⁹ cm³ for both the p- and n-regions (that of the previous sample was 3 × 10¹⁹ cm³). It is expected that a high $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ of more than 1.2 nm/ns can be achieved in the new structure for an applied voltage of 3.5 V. Details regarding this modification are discussed in Appendix B.

Figure 5(a) shows an optical microscope image of the fabricated sample. At the center of the figure we can identify the three cavities A, B, and C (the slightly thinner green lines) and the coupling waveguides that connect cavities A and C as well as cavities C and B (the slightly thicker green lines). The long waveguide at the upper side of the cavities is the excitation waveguide used to characterize each cavity. The excitation waveguide extends to the sample edges, and the end facet is used to couple light into the waveguide. The p- and n-regions are indicated by the half-transparent orange and blue regions. Furthermore, the two microheaters and the surrounding trenches for thermal insulation are also clearly visible.

 

Fig. 5. Experimental results. (a) Optical microscope image of the fabricated sample. The thin green lines (which we surrounded by white dotted lines) are the three cavities and the thicker green lines between them are the two coupling waveguides. The areas with implanted boron and phosphorous are indicated by the half-transparent orange and blue areas labelled “P” and “N”, respectively. (b) Time-resolved emission from cavities A (red), B (blue) and C (green). The dashed lines are the results of fitting the data to an exponential decay and the quality factors are determined from the obtained photon lifetimes. (c) The peak wavelengths of the eigenstates of the system (black dots) determined while λ_c was changed by heating cavity C. The black dashed lines are the wavelengths of the eigenmodes of the coupled cavity system obtained by fitting the data, from which the coupling coefficients μ were determined.

Download Full Size | PDF

5. Optical characterization

We performed three different experiments to determine the resonance wavelengths, the Q values, and the coupling coefficients. Firstly, the resonance wavelengths of the cavities were confirmed by coupling light from a tunable laser (Santec TSL-710) into the excitation waveguide and measuring the intensity of the light emitted from each cavity using an infrared camera (InGaAs, Hamamatsu Photonics C12741-03) as a function of the laser wavelength. It was found that cavity C has the shortest resonance wavelength (1566.4 nm), and cavity A has the longest wavelength (≈ 1566.9 nm). We also confirmed that the FP modes of the coupling waveguides are more than 1 nm away from the cavity resonance wavelengths.

Secondly, using a LiNbO₃ optical modulator, we modulated the input light to obtain pulses with a temporal width of about 10 ns at about 10 MHz, and excited each cavity at its resonance wavelength to determine the photon lifetime. The emitted photons were detected by a photomultiplier tube (Hamamatsu Photonics H10330A-75, time resolution ∼200 ps), and we performed time-resolved measurements using the time-correlated single photon counting (TCSPC) method (Pico Quant Picoharp300). The results in Fig. 5(b) show that the photon lifetimes of cavities A, B, and C are 3.0 ns, 3.1 ns, and 3.6 ns, respectively. These experimental values correspond to quality factors of 3.6 × 10⁶, 3.7 × 10⁶, and 4.3 × 10⁶, respectively. In comparison, the experimental quality factors of cavities A, B, and C in [14] were 1.8 × 10⁶, 2.2 × 10⁶, and 0.86 × 10⁶, respectively. Therefore, an approximately 2-fold improvement was achieved for cavities A and B, and an approximately 5-fold improvement was achieved for cavity C. Since the theoretical Q_v values of the cavities fabricated in this work are almost the same as those in the previous work, we conclude that the reason for the improvement in Q lies in the improvement of the fabrication process. The increased Q of cavity A as well as that of cavity B can be directly attributed to lower surface contamination. For cavity C, we need to consider that the experimental Q value is significantly higher than that of the previous device despite the reduced width of the i-layer and the increased doping densities. The increase in the experimental Q value can thus also be attributed to an improved alignment accuracy between the ion implantation mask and the photonic crystal as well as the improvements of the protective mask that was used during ion implantation, in addition to lower surface contamination.

Finally, to experimentally verify Fig. 1(b), the resonance wavelengths of the coupled cavity system were measured by statically changing ${\mathrm{\lambda }_\textrm{C}}$ (we heated the i-region using voltage pulses with a small duty ratio of 8%) and scanning the wavelength of the tunable laser. (It was originally intended to obtain a sample with ${\mathrm{\lambda }_\textrm{A}}$ and ${\mathrm{\lambda }_\textrm{B}}$ < ${\mathrm{\lambda }_\textrm{C}}$, and this would have allowed us to achieve a crossover between C and A (or B) by increasing ${\mathrm{\lambda }_\textrm{A}}$ (or ${\mathrm{\lambda }_\textrm{B}}$) using the microheaters, but it was not possible to use this method because cavity C had the shortest wavelength.) Although voltage application results in a transient shift of ${\mathrm{\lambda }_\textrm{C}}$ to shorter wavelengths due to the plasma shift induced by the injected carriers, this had a negligible impact on the spectral peak measurements by the infrared camera because of the used duty ratio. The black points in Fig. 5(c) show the measured resonance wavelengths as a function of the average electrical power injected into the p-i-n junction, which leads to an increase of the temperature in the i-region due to Joule heating. It can be clearly seen that there are two distinct anti-crossing regions: when ${\mathrm{\lambda }_\textrm{C}}$ crosses ${\mathrm{\lambda }_\textrm{B}}$ at an injected power of about 2.2 mW, and when ${\mathrm{\lambda }_\textrm{C}}$ crosses ${\mathrm{\lambda }_\textrm{A}}$ at about 5 mW. The black dashed lines are the results of fitting the data to the ${\mathrm{\lambda }_\textrm{C}}$-dependence of the eigenvalues of the matrix in Eq. (1) in the case that the two coupling coefficients (for coupling between cavities A and C, $\mathrm{\mu}$_AC, and coupling between cavities C and B, $\mathrm{\mu}$_BC) are different. The red, blue, and green dashed lines in Fig. 5(c) are the wavelengths of cavities A, B, and C obtained by the fitting procedure. Furthermore, regarding the coupling coefficients we find $\mathrm{\mu}$_AC = 31 Grad/s and $\mathrm{\mu}$_BC = 43 Grad/s. In other words, $\mathrm{\mu}$_AC is almost equal to the theoretical value (32 Grad/s), while $\mathrm{\mu}$_BC is larger than the theoretical value. The larger $\mathrm{\mu}$_BC may be due to air-hole fabrication fluctuations or a small wavelength spacing between the modes of cavities B and C and the FP modes of the right coupling waveguide (between B and C) when cavity C was heated. Since the experimentally obtained $\mathrm{\mu}$ values of the side-coupling structure in the previous work were 9.4 Grad/s ($\mathrm{\mu}$_AC) and 12.6 Grad/s ($\mathrm{\mu}$_BC), an approximately three times higher $\mathrm{\mu}$ was experimentally obtained in the optimized coaxial-coupling structure.

6. Discussion

In our previous work, we achieved a μ of about 11 Grad/s, a $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ of about 0.3 nm/ns (using a voltage of 2.5 V), and a Q of about 1.5 × 10⁶ for cavities A and B [14]. According to Section 2, the increase in $\mathrm{\mu}$ by a factor of 3 achieved in this work means that the adiabatic condition can be maintained even if $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ is increased by a factor of 9. The two-times larger Q values of cavities A and B imply that the maximum possible photon storage time before and after the transfer is improved by a factor of 2, and this also leads to an increased transfer efficiency [Fig. 1(f)]. However, in this work, we encountered also three difficulties: (1) The resonance wavelength of cavity C was shorter than those of the other cavities, which made it difficult to cross the resonance wavelengths of cavities A and B by a fast wavelength shift due to carrier injection. (2) The fact that the experimental Q of cavity C is very high indicates that the actual Q_v is also very high, and we consider that this high Q_v made it impossible to measure the actual value of $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ for voltages larger than 2.3 V (the voltage dependence is discussed in Appendix B). (3) When carriers were injected into cavity C, the Q values of cavities A and B transiently decreased to values <0.6 × 10⁶ even when cavity C was significantly detuned from cavities A and B. Mainly due to issue (3), unfortunately, we have not yet been able to demonstrate photon transfer with a high efficiency and long storage time. Regarding issue (1), we plan to use a slightly different design for cavity C to ensure that ${\mathrm{\lambda }_\textrm{C}}$ is longer than ${\mathrm{\lambda }_\textrm{A}}$ and ${\mathrm{\lambda }_\textrm{B}}$ by about 1 nm even if the air-hole fabrication fluctuations lead to ${\mathrm{\lambda }_\textrm{A}}$ and ${\mathrm{\lambda }_\textrm{B}}$ values that are larger than the design values. Issue (2) can also be solved by adjusting the design of cavity C: If we choose a design with a relative low Q_v of about 1 × 10⁶, this will increase the emission intensity in the vertical direction, which makes it easier to determine ${\mathrm{\lambda }_\textrm{C}}$ even if the absorption loss increases due to carrier injection. Regarding issue (3), even when cavity C is not in resonance with cavities A and B, a significant fraction of the light in cavities A and B leaks into the coupling waveguide [see Fig. 4(b) and Section 3.4]. Since this light can be absorbed by the carriers injected around cavity C, we consider that this effect leads to the increased absorption loss for cavities A and B. We plan to address this issue in the future by a 3D-FDTD simulation of this effect and applying the results to our automatic structure tuning procedure.

7. Conclusion

We optimized the design and fabrication of an electrically controllable coupled nanocavity system to simultaneously achieve large coupling coefficients and long photon lifetimes, which is necessary to improve the performance of on-demand photon transfer based on an adiabatic process. The basic device layout was changed from side coupling to coaxial coupling, and an automatic structure tuning method based on CMAES was applied. To improve the coupling coefficient without reducing Q_v, we introduced two additional methods (an appropriate sample selection in 2D target parameter space and a structure modification weighted by the leaky component distribution) to control the direction of the structure evolution. We found a structure in which the theoretical coupling coefficient is increased by a factor of 3.5 compared to that of the previously demonstrated side-coupling structure, while Q_v remained at the same level as that of the previous structure. We also developed a fabrication process that can suppress the reduction in the cavity photon lifetime caused by the introduction of the p-i-n junction. We have shown that these improvements allow us to increase the coupling strength by a factor of about 3 and the photon lifetime by a factor of about 2, compared to the previous system. This is not only important for optical buffer memories based on adiabatic photon transfer, but also for performance and functionality improvements of various optical phenomena based on the dynamic control of coupling between photonic modes.

Appendix A: Details of the photon transfer simulations

The photon transfer was simulated by numerically solving the coupled mode equation [Eq. (1)]. First, the values of ${\mathrm{\lambda }_\textrm{A}}$, ${\mathrm{\lambda }_\textrm{B}}$, and ${\mathrm{\lambda }_\textrm{C}}$ (and the respective values of ${\mathrm{\omega }_\textrm{A}}$, ${\mathrm{\omega }_\textrm{B}}$ and ${\mathrm{\omega }_\textrm{C}}$) were set as follows: in the case of $\mathrm{\mu}$ = 11 Grad/s, ${\mathrm{\lambda }_\textrm{B}}$ is shorter than ${\mathrm{\lambda }_\textrm{A}}$ by 12.5 pm, in the case of $\mathrm{\mu}$ = 22 Grad/s, ${\mathrm{\lambda }_\textrm{B}}$ is shorter than ${\mathrm{\lambda }_\textrm{A}}$ by 25 pm (the optimum value of ${\mathrm{\omega }_\textrm{B}} - {\mathrm{\omega }_\textrm{A}}$ is proportional to $\mathrm{\mu}$ [12]), and ${\mathrm{\lambda }_\textrm{C}}$ is initially longer than ${\mathrm{\lambda }_\textrm{A}}$ by 150 pm. Then, we solved the eigenvalue problem for the stationary case to obtain the eigenfrequencies of the eigenmodes. Finally, we calculated the time evolution of the amplitudes ${a_\textrm{A}}$, ${a_\textrm{B}}$, and ${a_\textrm{C}}$ in Eq. (1) with the initial condition ${a_\textrm{A}} = \; {a_\textrm{B}} = \; {a_\textrm{C}} = 0$ for the following process: first, cavity A is excited [with the eigenfrequency of the middle eigenmode, see Fig. 1(b)] and then, after the excitation of the middle mode, ${\mathrm{\omega }_\textrm{C}}$ is changed with the speed $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$, which is 0.2 nm/ns in Fig. 1(c) and 0.8 nm/s in Figs. 1(d)-(f). Since $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ was assumed to be induced by carriers injected through the p-i-n structure, we also included the associated optical absorption loss [16] in this simulation, in addition to the radiation losses of the cavities determined by the assumed Q_v values. The obtained time-dependent values ${|{{a_\textrm{A}}(t )} |^2}$, ${|{{a_\textrm{B}}(t )} |^2}$, and ${|{{a_\textrm{C}}(t )} |^2}$ are shown in Figs. 1(c)–(f), and the transfer efficiencies and speeds were determined from these curves.

Appendix B: Design of the p-i-n structure and the rate of cavity C's wavelength change

Figure 6 shows the structure of the p-i-n diode integrated at the center of the coupled cavity system (cavity C is located in the i-region). Carriers can be injected into the i-region by applying a forward voltage to the diode, and the refractive index of the i-region changes due to the carrier plasma effect. Therefore, the resonance wavelength of cavity C, ${\mathrm{\lambda }_\textrm{C}}$, can be dynamically changed. The actual behavior of $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ caused by a given voltage pulse can be reconstructed by performing time-resolved measurements of the emission from the cavity excited by continuous-wave light while changing the excitation wavelength [14].

 

Fig. 6. The geometrical properties of the p- and n- regions in the vicinity of cavity C in the case of the new structure (the width of the i-layer shown in this figure corresponds to 3.5 µm). The electric field distribution of the cavity mode is also plotted to clarify the size-relation between the photonic and electronic structures.

Download Full Size | PDF

In the previous report [14], we used an i-layer width of 4.5 µm and a doping density of 3 × 10¹⁹cm^-3 in both the p- and n-regions. The blue dots in Fig. 7 show the measured rate of the resonance wavelength shift of cavity C for this structure; a $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ of 0.2 nm/ns was achieved for an applied voltage of 1.8 V, and a $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ of 0.4 nm/ns was achieved for 3 V. These results are in good agreement with the numerical simulation of the carrier transport in the p-i-n structure [14]. On the other hand, in this work, we used an i-layer width of 3.5 µm and a doping density of 6 × 10¹⁹cm^-3. The measurement results for this structure are shown by the orange dots in Fig. 7. It can be seen that $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ of the new structure is about two times larger than that of the previous structure for the same applied voltage. However, it was not possible to measure the $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ values for voltages higher than 2.3 V, probably because the experimental Q of cavity C was too high (4.29 × 10⁶), which leads to a significant reduction in the cavity emission intensity when carriers are injected and the absorption loss is increased. However, the linear fitting result shown by the black dashed curve in Fig. 7 suggests that a high $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ of more than 1.2 nm/ns can be achieved for a voltage of 3.5 V. A further increase in $\left|{\frac{{\textrm{d}{\mathrm{\lambda }_\textrm{C}}}}{{\textrm{d}t}}} \right|$ should be possible by reducing the i-layer width to 3.0 µm.

 

Fig. 7. The experimentally determined |dλ_c/dt| values for different voltages; the blue dots show the data for the previous structure and the orange dots show the data for the new structure.

Download Full Size | PDF

Appendix C: Method to determine the in-plane quality factor and the radiation quality factor

For this work, we calculated the in-plane and radiation quality factors, Q_in and Q_v, using 3D FDTD simulations. Here, we used an FDTD cell size of $({\mathrm{\Delta x},\mathrm{\Delta y},\mathrm{\Delta z}} )= \left( {\frac{a}{{10}},\frac{{\sqrt 3 a}}{{16}},\frac{{0.536 \times a}}{6}} \right)$, and the distribution of the dielectric constants was calculated using subcells with a size of $({\mathrm{\delta x},\mathrm{\delta y},\mathrm{\delta z}} )= \left( {\frac{a}{{10 \times 400}},\frac{{\sqrt 3 a}}{{16 \times 400}},\frac{{0.536 \times a}}{{6 \times 50}}} \right)$. Therefore, our FDTD calculations can reflect even small changes in the distribution of the dielectric constant according to air-hole displacements on the order of a/4000. The size of the computational domain was 460 × 384 × 170 cells in x-, y-, and z-directions, respectively. We placed the cavity at the center of the computational domain and added a semi-infinite coupling waveguide [see Fig. 4(b)]. To determine Q_in and Q_v, we defined a so-called observation volume centered around the cavity (this volume had a size of 320 × 244 × 110 cells in x-, y-, and z-directions, respectively) and obtained the energy U in the observation volume and the power escaping through each of the six surface planes of the observation volume in the case of resonant excitation of the cavity. The power escaping in the x-direction, P_x, was used to calculate Q_in (=${\mathrm{\omega }_\textrm{A}}U/{P_x}$), and the power escaping in the z-direction, P_z, was used to calculate Q_v (=${\mathrm{\omega }_\textrm{A}}U/{P_z}$). Although this separation into Q_in and Q_v is not exact due to the finite size of the observation volume, this approach is considered to be sufficient, because ${P_x}$ (radiation via the coupling waveguide) is much larger (by about three orders of magnitude) than ${P_z}$ (the out-of-plane radiation), and P_y is negligible compared to ${P_z}$.

Appendix D: Robustness of the new design to air-hole fabrication fluctuations

The robustness of the device performance to deviations from the ideal air-hole geometry due to random fabrication errors is an important point, but the impact of such air-hole fabrication fluctuations is relatively small in this work:

In [26], we estimated the magnitudes of the fluctuations of the air-hole radius and position based on the observed variation of the Q value in a set of fabricated cavities with the same design. The estimated standard deviation was less than 0.3 nm. To clarify the robustness of our new design, we calculated the coupling coefficients and radiation quality factors for ten structures with random deviations from the ideal air-hole geometry using 3D FDTD simulations. Here, we randomly changed the position (x, y) and radius (r) of each air hole in the ideal structure [Fig. 4(b)] according to a normal distribution, where the standard deviations of x, y, and r were set to 0.41 nm. The results are shown in Table 1, which clarifies that the average coupling coefficient of the structures with random deviations is 28 Grad/s, which is smaller than the 32 Grad/s of the ideal structure. The standard deviation of the coupling coefficient is 3.9 Grad/s.

Table 1. Predicted values of $\mathrm{\mu}$ and Q_v for ten structures with random deviations from the ideal air-hole geometry

View Table

We consider that the influence of the air-hole fabrication fluctuations on the coupling coefficient is relatively small, because the magnitudes of the air-hole shifts used to ensure a smooth change of the waveguide width (to achieve a large coupling coefficient) are larger than 4.1 nm as can be seen in Fig. 4(b), while the assumed magnitudes of the fluctuations are only 0.41 nm (and this is still larger than the experimentally evaluated magnitudes of the fluctuations).

Regarding radiation quality factor, the average Q_v of the structures with random deviations is 10.6 million and the standard deviation is 2.7 million (the Q_v of the ideal structure is 41 million). These values are acceptable, because the value of 10.6 million is still more than two to three times larger than the experimental Q values of the fabricated structures, which means that the air-hole fabrication fluctuations are not yet the main limiting factor of the experimental Q values.

Funding

Japan Society for the Promotion of Science (KAKENHI 22H01988).

Acknowledgments

We thank Dr. Heungjoon Kim for the fruitful discussion on the fabrication process. We also thank the author of the blog site https://horomary.hatenablog.com/entry/2021/01/23/013508 for providing information concerning the implementation of the (μ/μw, λ) CMAES algorithm.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108(15), 153901 (2012). [CrossRef]  

2. N. A. Estep, D. L. Sounas, J. Soric, et al., “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled- resonator loops,” Nat. Phys. 10(12), 923–927 (2014). [CrossRef]  

3. L. D. Tzuang, K. Fang, P. Nussenzveig, et al., “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8(9), 701–705 (2014). [CrossRef]  

4. M. Nakadai, T. Asano, and S. Noda, “Dynamic switching of propagation direction in coupled photonic nanocavities,” 7th International Symposium on Photonics and Electronics Convergence (2017).

5. T. Ozawa, H. M. Price, N. Goldman, et al., “Synthetic dimensions in integrated photonics: from optical isolation to four-dimensional quantum Hall physics,” Phys. Rev. A 93(4), 043827 (2016). [CrossRef]  

6. L. Yuan, Q. Lin, M. Xiao, et al., “Synthetic dimension in photonics,” Optica 5(11), 1396–1404 (2018). [CrossRef]  

7. A. Dutt, Q. Lin, L. Yuan, et al., “A single photonic cavity with two independent physical synthetic dimensions,” Science 367(6473), 59–64 (2020). [CrossRef]  

8. Y. Sato, Y. Tanaka, J. Upham, et al., “Strong coupling between distant photonic nanocavities and its dynamic photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2012). [CrossRef]  

9. R. Konoike, T. Asano, and S. Noda, “On-chip dynamic time reversal of light in a coupled-cavity system,” APL Photonics 4(3), 030806 (2019). [CrossRef]  

10. M. A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019). [CrossRef]  

11. R. Konoike, T. Asano, Y. Tanaka, et al., “Coupled mode theory analysis of geometric phases in a coupled system of four optical cavities,” International Conference on Computational Physics (2015).

12. R. Konoike, Y. Sato, Y. Tanaka, et al., “Adiabatic transfer scheme of light between strongly coupled photonic crystal nanocavities,” Phys. Rev. B 87(16), 165138 (2013). [CrossRef]  

13. R. Konoike, H. Nakagawa, M. Nakadai, et al., “On-demand transfer of trapped photons on a chip,” Sci. Adv. 2(5), e1501690 (2016). [CrossRef]  

14. M. Nakadai, T. Asano, and S. Noda, “Electrically controlled on-demand photon transfer between high-Q photonic crystal nanocavities on a silicon chip,” Nat. Photonics 16(2), 113–118 (2022). [CrossRef]  

15. C. Zener, “Non-adiabatic crossing of energy levels,” Proc. R. Soc. London Ser. A 137(833), 696–702 (1932). [CrossRef]  

16. M. Nedeljkovic, R. Soref, and G. Z. Mashanovich, “Free-carrier electrorefraction and electroabsorption modulation predictions for silicon over the 1-14-μm infrared wavelength range,” IEEE Photonics J. 3(6), 1171–1180 (2011). [CrossRef]  

17. S. G. Johnson, S. Fan, A. Mekis, et al., “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78(22), 3388–3390 (2001). [CrossRef]  

18. H. Y. Ryu, M. Notomi, and Y. H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83(21), 4294–4296 (2003). [CrossRef]  

19. Y. Akahane, T. Asano, B. Song, et al., “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef]  

20. T. Asano and S. Noda, “Optimization of photonic crystal nanocavities based on deep learning,” Opt. Express 26(25), 32704–32716 (2018). [CrossRef]  

21. N. Hansen and A. Ostermeie, “Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation,” International Conference on Evolutionary Computation (IEEE, 1996).

22. N. Hansen, “The CMA Evolution Strategy: A Tutorial,” arXiv, arXiv:1604.00772v2 (2016). [CrossRef]  

23. K. Takahashi and T. Baba, “Optimization of a Photonic Crystal Nanocavity Using Covariance Matrix Adaptation Evolution Strategy,” IEEE Photonics J. 14(3), 1–5 (2022). [CrossRef]  

24. T. Asano and S. Noda, “Iterative optimization of photonic crystal nanocavity designs by using deep neural networks,” Nanophotonics 8(12), 2243–2256 (2019). [CrossRef]  

25. T. Nakamura, Y. Takahashi, Y. Tanaka, et al., “Improvement in the quality factors for photonic crystal nanocavities via visualization of the leaky components,” Opt. Express 24(9), 9541–9549 (2016). [CrossRef]  

26. T. Asano, Y. Ochi, Y. Takahashi, et al., “Photonic crystal nanocavity with a Q factor exceeding eleven million,” Opt. Express 25(3), 1769–1777 (2017). [CrossRef]  

27. Y. I. Lee, K. H. Park, J. Lee, et al., “Dry release for surface micromachining with HF vapor-phase etching,” J. Microelectromechanical Syst. 6(3), 226–233 (1997). [CrossRef]  

28. J. Park, G. Jung, S. Hong, et al., “Analysis of Cr/Au contact reliability in embedded poly-Si micro-heater for FET-type gas sensor,” Sens. Actuators, B 360, 131673 (2022). [CrossRef]