JP2016154163A - Photonic crystal resonator and method of designing the same - Google Patents
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本発明は、２次元スラブ型のフォトニック結晶より構成されたＨ０型のフォトニック結晶共振器およびその設計方法に関する。 The present invention relates to a H0 type photonic crystal resonator composed of a two-dimensional slab type photonic crystal and a design method thereof.
近年、通信およびインターネットの大容量化に対応するため、光通信技術は欠くことのできないものとなっている。増大する通信容量に対応するために、単一チャンネルにおける通信速度の向上に加え、波長・偏波などの多重化による並列通信処理の拡大が進められている。通信にかかるエネルギーコストの低減、通信装置のフットプリントの縮小およびコストダウンのため、ルータなどの個別部品・装置のオンボード化、オンチップ化が進められている。 In recent years, optical communication technology has become indispensable in order to cope with an increase in capacity of communication and the Internet. In order to cope with the increasing communication capacity, in addition to improving the communication speed in a single channel, expansion of parallel communication processing by multiplexing of wavelength and polarization is being promoted. In order to reduce the energy cost for communication, reduce the footprint of communication devices, and reduce costs, on-board and on-chip use of individual components and devices such as routers has been promoted.
現行技術では、光信号を電気信号に変換した上で電子回路により信号処理を行い、再度光信号に変換して送出を行っている。しかしながら、これ以上の電子回路の集積度の拡大、微細化、および消費電力の削減には限界が指摘されている。次世代の更なる高性能化と集積化・消費エネルギーの低減を進めるためには、従来技術の延長ではなく、根本からのブレークスルーが必要という考え方が広がりつつある。光を光信号のまま処理する全光処理、またチップ内・チップ間・ボード間の大容量信号伝送を電気から光に置き換える光インタコネクト技術は、ブレークスルーのための有力候補として注目され、要素技術の開発も進められている。 In the current technology, an optical signal is converted into an electric signal, signal processing is performed by an electronic circuit, and the optical signal is converted again and transmitted. However, limits have been pointed out for further expansion, miniaturization, and reduction of power consumption of electronic circuits. The idea that fundamental breakthroughs are needed rather than an extension of the prior art is necessary to advance further performance, integration and reduction of energy consumption in the next generation. All-optical processing that processes light as an optical signal, and optical interconnect technology that replaces large-capacity signal transmission within a chip, between chips, and between boards from electricity to light, are attracting attention as potential candidates for breakthrough. Technology development is also underway.
全光処理および光インタコネクトの基盤技術として近年、シリコンフォトニクスおよびＩｎＰによる光（電子）集積回路の開発および市場導入が進められている。数μｍ程度の断面幅をもつ光配線や、この光配線をベースとした光素子を集積した光集積回路、ないし光電融合集積回路が現行の商品に採用されている。 In recent years, development and market introduction of optical (electronic) integrated circuits using silicon photonics and InP have been promoted as basic technologies for all-optical processing and optical interconnection. An optical wiring having a cross-sectional width of about several μm, an optical integrated circuit in which optical elements based on this optical wiring are integrated, or a photoelectric integrated circuit are used in current products.
更なる微細化および低消費電力化を実現する手法として、サブμｍの断面サイズを有するＳｉ細線光導波路と共に注目されているのが、フォトニック結晶（Photonic Crystal：ＰＣ）である。ＰＣは、高屈折率の媒体中に低屈折率の円柱などの要素を周期的に配置することで光のバンド構造を生成し、光の伝搬の制御を実現する人工物質である。特に光の絶縁体となるフォトニックバンドギャップ（Photonic Bandgap：ＰＢＧ）を利用すれば、光を波長スケールの微小体積に閉じ込めるナノ共振器や結晶穴１個分の幅に閉じ込めるナノ導波路を実現できる。 As a technique for realizing further miniaturization and lower power consumption, a photonic crystal (PC) is attracting attention along with a Si fine-line optical waveguide having a sub-μm cross-sectional size. PC is an artificial material that generates a light band structure by periodically arranging elements such as a low-refractive index cylinder in a high-refractive index medium, and realizes control of light propagation. In particular, if a photonic bandgap (PBG), which is an insulator of light, is used, a nanoresonator that confines light in a minute volume on the wavelength scale and a nanowaveguide that confines in the width of one crystal hole can be realized. .
フォトニック結晶周期は、真空中の光波長の１／ｎ（ｎ：媒質の屈折率）程度なので、ＰＣにおいては現行の光集積回路より１桁程度のサイズ縮小および１桁以上の消費電力低減が期待される。最近、ＰＣレーザの電流注入室温ＣＷ発振が実現され、また１００ビットのオンチップ集積光メモリが実現されるなど、ＰＣ光部品の実用化は現実に近づきつつある。 Since the photonic crystal period is about 1 / n of the wavelength of light in vacuum (n: refractive index of the medium), the size of the PC is reduced by about one digit and the power consumption is reduced by one digit or more than the current optical integrated circuit. Be expected. Recently, PC laser current injection room temperature CW oscillation has been realized, and a 100-bit on-chip integrated optical memory has been realized.
ＰＣによるナノ共振器は、結晶穴数個を取り除いて形成する欠陥部分に周囲のＰＣが有するＰＢＧにより光閉じ込めモードを形成したもので、モード体積Ｖは概ね１（λc／ｎ）3以下になるものを指す（λc：共振モード波長）。例えば、Ｌ型共振器（非特許文献１，２）およびＨ型共振器（非特許文献３）などがある。最近の共振器設計の進歩により、閉じ込めモードのＱ値を１万以上にすることが可能になり、フォトニック結晶以外では達成困難な極めて大きなＱ／Ｖを達成できるのが特長である。 The nanoresonator made of PC is one in which an optical confinement mode is formed by PBG of the surrounding PC at a defect portion formed by removing several crystal holes, and the mode volume V is approximately 1 (λ c / n) 3 or less. (Λ c : resonance mode wavelength). For example, there are an L-type resonator (Non-Patent Documents 1 and 2) and an H-type resonator (Non-Patent Document 3). Recent advances in resonator design make it possible to increase the Q value of the confinement mode to 10,000 or more, and it is possible to achieve an extremely large Q / V that is difficult to achieve except by a photonic crystal.
特に、共振器電磁力学（ｃａｖｉｔｙ−ＱＥＤ）においてパーセル効果などの大きな増強が得られることが学術的に大きな注目を集めてきた。また、ナノ共振器を用いると実用的にもレーザや光メモリ・光スイッチなどの消費パワーの大幅な低減が図れるため、光素子単体のみならず光集積回路および光電融合集積回路全体として極めて低パワーでの動作が可能になることが期待される。 In particular, it has attracted a great deal of academic attention that large enhancements such as the parcel effect can be obtained in resonator electrodynamics (cavity-QED). In addition, the use of nano-resonators can significantly reduce the power consumption of lasers, optical memories, optical switches, etc. in practical use, so that not only optical devices but also optical integrated circuits and optoelectronic integrated circuits as a whole have extremely low power. Is expected to be possible.
なお、光回路の素子としての光共振器の性能を向上させることは、Ｑ値をより高く、モード体積Ｖをより小さくすることと同義である。Ｑ値が高ければ高いほど光を長時間、かつ低損失に閉じ込めることができ、モード体積Ｖが小さければ小さいほど光の単位体積当たりのエネルギーを大きくでき、また共振器自体も小さくすることができる。ただし、モード体積Ｖが小さい共振器は、多くの場合光の外部への放射ロスが大きくなりＱ値が小さくなる。このように、Ｑ値とモード体積Ｖはトレードオフの関係となっている。このため、光共振器の性能をＱ値やモード体積Ｖを単体で評価するのではなく、Ｑ／Ｖで評価することが多い。 Note that improving the performance of an optical resonator as an element of an optical circuit is synonymous with increasing the Q value and decreasing the mode volume V. The higher the Q value, the longer the light can be confined in a low loss, and the smaller the mode volume V, the larger the energy per unit volume of light, and the smaller the resonator itself. . However, in a resonator having a small mode volume V, in many cases, radiation loss to the outside of the light increases and the Q value decreases. Thus, the Q value and the mode volume V are in a trade-off relationship. For this reason, the performance of the optical resonator is often evaluated by Q / V instead of evaluating the Q value and the mode volume V alone.
ところで、ＰＣによるナノ共振器には、結晶穴数個を取り除いて形成する欠陥部分を利用する構成の他に、無点欠陥型（Zero Cell型、Ｈ０型）と呼ばれる構成がある（非特許文献４参照）。この共振器は、最も代表的なナノ共振器であるＬ型共振器およびＨ型共振器と異なり、結晶中から１つも結晶穴を取り除いていない。 By the way, the nanoresonator by PC has a configuration called a non-spot defect type (Zero Cell type, H0 type) in addition to a configuration using a defect portion formed by removing several crystal holes (non-patent document). 4). Unlike the L-type and H-type resonators, which are the most typical nanoresonators, this resonator does not remove any crystal holes from the crystal.
上述したＨ０型のフォトニック共振器について説明する。これは、図５に示すように、２次元スラブ型のフォトニック結晶５０１から構成する。フォトニック結晶５０１は、板状の基部５０２と、基部５０２に設けられた柱状の複数の格子要素５０３とから構成されている。格子要素５０３は、三角格子状に配列されている。ここでは、中空構造から格子要素５０３を構成している。Ｈ０型では、周囲より間隔を開けて配置した格子要素５０３ａよりなる光閉じ込め部５０４を形成し、光閉じ込め部５０４に光を閉じ込める共振器としている。２つの格子要素５０３ａは、フォトニック結晶５０１のΓ−Ｋ結晶方位に沿って、互いに遠ざける方向にシフトしている。 The above-described H0 type photonic resonator will be described. This is composed of a two-dimensional slab type photonic crystal 501 as shown in FIG. The photonic crystal 501 includes a plate-like base portion 502 and a plurality of columnar lattice elements 503 provided on the base portion 502. The lattice elements 503 are arranged in a triangular lattice shape. Here, the lattice element 503 is composed of a hollow structure. In the H0 type, a light confinement portion 504 including a lattice element 503a arranged at a distance from the periphery is formed, and a resonator that confines light in the light confinement portion 504 is formed. The two lattice elements 503a are shifted away from each other along the Γ-K crystal orientation of the photonic crystal 501.
上述したＨ０型の共振器は、２次元ナノ共振器として最小の０．２（λ／ｎ）3程度のモード体積を実現できる共振器として広く知られている。但し、当初提案された設計において、Ｑ値は１０万に留まっていた。これに対し、シフト変調の対象となる穴を増やすことで、モード体積を大きくすることなくＱ値を３０万まで高める設計が後に報告された（非特許文献５参照）。しかしながら、この構成においても、例えばＬ３型共振器（非特許文献１，２）と比べてＱ値は高いとは言えなかった。非特許文献２においては、Ｌ３型共振器でのＱ値は１００万以上である。また、共振器中心に量子ドットなどの活性層を配置する十分なスペースがないため、学術応用上もあまり進まなかった。 The H0 type resonator described above is widely known as a resonator capable of realizing a minimum mode volume of about 0.2 (λ / n) 3 as a two-dimensional nanoresonator. However, in the originally proposed design, the Q value remained at 100,000. On the other hand, a design that increases the Q value to 300,000 without increasing the mode volume by increasing the number of holes to be subjected to shift modulation was later reported (see Non-Patent Document 5). However, even in this configuration, it cannot be said that the Q value is higher than that of, for example, L3 type resonators (Non-Patent Documents 1 and 2). In Non-Patent Document 2, the Q value in the L3 type resonator is 1 million or more. Also, since there is not enough space to place an active layer such as a quantum dot at the center of the resonator, it has not made much progress in academic applications.
ごく最近、Ｈ０共振器においてＱ値を１桁程度増大できる設計例の１つが、非特許文献６により報告された。この技術では、Ｑ値を高速に計算できる計算手法を用いて７組１４個の穴のシフト量を広範な範囲に変調し、機械的に膨大なシフト量の組み合わせの各点において試行錯誤的に計算によりＱ値を取得し、遺伝的アルゴリズムにより効率的に構造を絞り込んだ上で、Ｑ値を最高にするシフト量を決定している。この設計例によれば、電磁界解析において標準的な手法の１つであるＦＤＴＤ（Finite-difference time-domain method）により求めたＱ値が、８３０万になると報告されている。これは前述したＬ型共振器のうち、欠陥数が５（Ｌ５型）以下では達成されていない非常に高いＱ値として注目される。 Very recently, Non-Patent Document 6 reported one design example that can increase the Q factor by an order of magnitude in the H0 resonator. In this technique, the shift amount of 7 sets and 14 holes is modulated over a wide range by using a calculation method capable of calculating the Q value at high speed, and trial and error at each point of a mechanically enormous shift amount combination. The Q value is obtained by calculation, the structure is narrowed down efficiently by a genetic algorithm, and the shift amount that maximizes the Q value is determined. According to this design example, it has been reported that the Q value obtained by FDTD (Finite-difference time-domain method), which is one of the standard methods in electromagnetic field analysis, is 8.30 million. This is noticed as a very high Q value that is not achieved when the number of defects is 5 (L5 type) or less among the L-type resonators described above.
非特許文献６において提案された設計は、当初のＨ０型の設計に比べモード体積Ｖを約３倍に増大させるものの、モード体積ＶはなおＬ３型共振器と同等以下であり、Ｑ／ＶがＬ３型よりも大きくなるという点は注目されるべきである。 Although the design proposed in Non-Patent Document 6 increases the mode volume V by about three times compared to the original H0 type design, the mode volume V is still equal to or less than that of the L3 type resonator, and Q / V is It should be noted that it is larger than the L3 type.
しかしながら、上述した技術では、次に示すように、２次元スラブ型のフォトニック結晶より構成されたＨ０型の共振器におけるより最適な共振構造が得られていないという問題があった。非特許文献６により示された１つの設計例は、Ｈ０型の大きな可能性を示すものであったが、その後、現在までに実用上・また学術上重要と思われる多くの点が明らかにされていない。まず、上述した７組の穴のシフトが既存の設計とどのように違うことにより高いＱ値がもたらされたのか、７組の穴のシフトの何が高Ｑ値化のポイントなのかが明らかでなかった。また、７組の穴のシフト量が、当該設計例におけるＰＣパラメータ（薄膜厚さ／格子定数＝０．５、結晶穴半径／格子定数＝０．２５）と異なる条件ではどう変わるかについても明らかではなかった。 However, the above-described technique has a problem that a more optimal resonance structure cannot be obtained in a H0 type resonator formed of a two-dimensional slab type photonic crystal, as described below. One design example shown by Non-Patent Document 6 showed the great potential of the H0 type, but since then, many points that have been considered practically and academically important have been revealed. Not. First of all, it is clear how the shift of the 7 sets of holes described above is different from the existing design, which resulted in a high Q value, and what is the key to achieving a high Q value in the shift of the 7 sets of holes It was not. It is also clear how the shift amount of the seven holes changes under different conditions from the PC parameters (thin film thickness / lattice constant = 0.5, crystal hole radius / lattice constant = 0.25) in the design example. It wasn't.
実用上は、穴のシフト量の間に相関関係があり、例えば非特許文献２の共振器設計に示されているように穴のシフト量を規則的に行うことができるのが望ましい。異なる穴のシフト量に規則性があれば、ＦＤＴＤ法のような計算負荷が高く多くの試行錯誤計算を行うことが困難な計算手法に置いても、また計算に頼ることなく実験においてシフト量を試行錯誤的に変調しながら最適化を進めＱ値を高めることも可能になる。 In practice, there is a correlation between the shift amounts of the holes, and it is desirable that the shift amounts of the holes can be regularly performed as shown in the resonator design of Non-Patent Document 2, for example. If there is regularity in the shift amount of different holes, the shift amount can be set in the experiment without relying on the calculation even if it is placed in a calculation method such as the FDTD method where the calculation load is high and it is difficult to perform many trial and error calculations. It is also possible to increase the Q value by optimizing while modulating by trial and error.
非特許文献６における著者の姿勢は、構造パラメータが変わる場合には、論文に示された手法により遺伝的アルゴリズムに任せて機械的にシフト量を最適化すれば良いので、穴通しのシフト量の相関性などは特に考慮しなくても良い、と受け取られる。 The author's attitude in Non-Patent Document 6 is that when the structural parameter changes, the shift amount of the through hole can be optimized by mechanically optimizing the shift amount by the genetic algorithm according to the method shown in the paper. It is perceived that there is no particular need to consider correlation.
しかし、非特許文献６による機械的最適化手法にはいくつかの課題があった。まず高速に膨大なシフト量の組み合わせに対しＱ値を計算するためにＧＭＥ（guided-mode expansion）法と呼ばれる計算手法が必須になる。この手法は、非特許文献６に報告されている範囲ではＦＤＴＤ法のＱ値とほぼ合致するＱ値を報告しているが、ＦＤＴＤ法並みの精度があると認知されている訳でなく、またＦＤＴＤ法のように詳細な電磁界特性を同時に取得することはできなかった。電磁界特性の解析や正確なＱ値の計算には別途ＦＤＴＤ法による計算が必要であった。 However, the mechanical optimization method according to Non-Patent Document 6 has several problems. First, a calculation method called a GME (guided-mode expansion) method is indispensable in order to calculate a Q value for a combination of a large amount of shifts at high speed. This method reports a Q value that almost matches the Q value of the FDTD method in the range reported in Non-Patent Document 6, but it is not recognized as accurate as the FDTD method, and It was impossible to acquire detailed electromagnetic field characteristics at the same time as in the FDTD method. Separate analysis by the FDTD method was required for analysis of electromagnetic field characteristics and accurate Q value calculation.
ＧＭＥ法を使わずに最初からＦＤＴＤ法で機械的最適化手法が実行できれば良いのだが、ＦＤＴＤ法は計算負荷が高く、長い計算時間を要するため、多くの試行錯誤計算を行うには適さない問題があった。また機械的最適化手法を実験に適用することは困難であった。非特許文献６においては、計算で最適化したシフト量と全く同一のシフト量を実験で設定していた。この場合、もし実験と計算の間にずれが存在する場合にＱ値が低下する恐れがあり、この補正は困難であった。 Although it is sufficient that the mechanical optimization method can be executed by the FDTD method from the beginning without using the GME method, the FDTD method has a high calculation load and requires a long calculation time, and is therefore not suitable for performing many trial and error calculations. was there. It is also difficult to apply the mechanical optimization method to the experiment. In Non-Patent Document 6, a shift amount exactly the same as the shift amount optimized by calculation is set by experiment. In this case, if there is a difference between the experiment and the calculation, the Q value may decrease, and this correction is difficult.
また、非特許文献６による機械的最適化手法においては、全ての可能なシフト量の組み合わせを、しらみ潰し的に試行錯誤計算するのではなく、遺伝的アルゴリズムによりシフト量の範囲を絞り込む最適化を行う。しかし使われている遺伝的アルゴリズムが常に正しい最適化を行うかどうかは十分実証されておらず、同手法で最適化されたシフト量が真にベストなシフト量であるという保証は無い。 In addition, in the mechanical optimization method according to Non-Patent Document 6, all combinations of possible shift amounts are not subjected to trial and error calculation in an exhaustive manner, but optimization that narrows down the range of shift amounts using a genetic algorithm. Do. However, it has not been sufficiently demonstrated whether the genetic algorithm used always performs correct optimization, and there is no guarantee that the shift amount optimized by this method is the truly best shift amount.
本発明は、以上のような問題点を解消するためになされたものであり、２次元スラブ型のフォトニック結晶より構成されたＨ０型の共振器におけるより最適な共振構造が得られるようにすることを目的とする。 The present invention has been made to solve the above-described problems, and can provide a more optimal resonance structure in a H0 type resonator formed of a two-dimensional slab type photonic crystal. For the purpose.
本発明に係るフォトニック結晶共振器は、基部および基部に対象とする光の波長以下の間隔で三角格子状に周期的に設けられて基部とは異なる屈折率の柱状の複数の格子要素を備えるフォトニック結晶本体と、フォトニック結晶本体に設けられて、フォトニック結晶のΓ−Ｋ結晶方位方向の直線上に連続する８つの第１格子要素，第２格子要素，第３格子要素，第４格子要素，第５格子要素，第６格子要素，第７格子要素，第８格子要素からなる第１列、および第１格子要素，第２格子要素，第３格子要素，第４格子要素，第５格子要素，第６格子要素，第７格子要素，第８格子要素の列の中央部でΓ−Ｋ結晶方位方向に垂直な方向に隣り合う２つの第９格子要素および第１０格子要素からなる第２列より構成された光閉じ込め部とを備え、第４格子要素および第５格子要素は、第１列の中心から対称に外側に向かって第１シフト量シフトし、第３格子要素および第６格子要素は、第１列の中心から対称に外側に向かって第２シフト量シフトし、第２格子要素および第７格子要素は、第１列の中心から対称に外側に向かって第３シフト量シフトし、第１格子要素および第８格子要素は、第１列の中心から対称に外側に向かって第４シフト量シフトし、第９格子要素および第１０格子要素は、第２列の中心から対称に内側に向かって第５シフト量シフトし、第１シフト量，第２シフト量，第３シフト量，第４シフト量，第５シフト量は、シフトによって隣り合う格子要素が接触しない範囲とされ、第１シフト量，第２シフト量，第３シフト量，第４シフト量，第５シフト量は、同一の一時結合関数により決定され、第２シフト量は、第１シフト量に第１係数を乗じた量とされ、第３シフト量は、第１シフト量に第２係数を乗じた量とされ、第４シフト量は、第１シフト量に第３係数を乗じた量とされ、第１係数は、四捨五入すると０．９になる数であり、第２係数は、四捨五入すると０．８になる数であり、第３係数は、０．５から０．６５の範囲の数である。 A photonic crystal resonator according to the present invention includes a base and a plurality of columnar lattice elements having a refractive index different from that of the base, which are periodically provided in a triangular lattice shape at intervals equal to or less than the wavelength of light of interest. A photonic crystal main body, and eight first lattice elements, second lattice elements, third lattice elements, fourth, which are provided on the photonic crystal main body and are continuous on a straight line in the Γ-K crystal orientation direction of the photonic crystal. A first row of grid elements, fifth grid elements, sixth grid elements, seventh grid elements, eighth grid elements, and first grid elements, second grid elements, third grid elements, fourth grid elements, It consists of two 9th and 10th lattice elements that are adjacent to each other in the direction perpendicular to the Γ-K crystal orientation direction at the center of the row of the 5th, 6th, 7th, and 8th lattice elements. And an optical confinement section composed of a second row, The 4th and 5th lattice elements shift the first shift amount symmetrically outward from the center of the first row, and the 3rd and 6th lattice elements symmetrically outward from the center of the first row. The second and seventh lattice elements are shifted symmetrically outward from the center of the first row by a third shift amount, and the first and eighth lattice elements are A fourth shift amount is shifted symmetrically outward from the center of the first row, and the ninth and tenth lattice elements are shifted symmetrically inward from the center of the second row by a fifth shift amount. The 1 shift amount, the 2nd shift amount, the 3rd shift amount, the 4th shift amount, and the 5th shift amount are within the range where the adjacent lattice elements do not contact with each other by the shift, and the 1st shift amount, the 2nd shift amount, the 3rd shift amount. The shift amount, the fourth shift amount, and the fifth shift amount are the same. The second shift amount is determined by the combination function, and the second shift amount is an amount obtained by multiplying the first shift amount by the first coefficient. The third shift amount is an amount obtained by multiplying the first shift amount by the second coefficient. The shift amount is an amount obtained by multiplying the first shift amount by the third coefficient. The first coefficient is a number that becomes 0.9 when rounded off, and the second coefficient is a number that becomes 0.8 when rounded off. The third coefficient is a number in the range of 0.5 to 0.65.
上記フォトニック結晶共振器において、第５シフト量を０とした場合にＱ値が最大になる第１シフト量を用いて決定された第２シフト量，第３シフト量，第４シフト量により、第１格子要素，第２格子要素，第３格子要素，第４格子要素，第５格子要素，第６格子要素，第７格子要素，第８格子要素が配置された状態で、Ｑ値が最大となる状態に第５シフト量が決定されている。 In the photonic crystal resonator, the second shift amount, the third shift amount, and the fourth shift amount determined using the first shift amount that maximizes the Q value when the fifth shift amount is set to 0, The Q value is maximum with the first lattice element, the second lattice element, the third lattice element, the fourth lattice element, the fifth lattice element, the sixth lattice element, the seventh lattice element, and the eighth lattice element arranged. The fifth shift amount is determined in such a state.
また、本発明に係るフォトニック結晶共振器の設計方法は、基部および基部に対象とする光の波長以下の間隔で三角格子状に周期的に設けられて基部とは異なる屈折率の柱状の複数の格子要素を備えるフォトニック結晶本体と、フォトニック結晶本体に設けられて、フォトニック結晶のΓ−Ｋ結晶方位方向の直線上に連続する８つの第１格子要素，第２格子要素，第３格子要素，第４格子要素，第５格子要素，第６格子要素，第７格子要素，第８格子要素からなる第１列、および第１格子要素，第２格子要素，第３格子要素，第４格子要素，第５格子要素，第６格子要素，第７格子要素，第８格子要素の列の中央部でΓ−Ｋ結晶方位方向に垂直な方向に隣り合う２つの第９格子要素および第１０格子要素からなる第２列より構成された光閉じ込め部とを備え、第４格子要素および第５格子要素は、第１列の中心から対称に外側に向かって第１シフト量シフトし、第３格子要素および第６格子要素は、第１列の中心から対称に外側に向かって第２シフト量シフトし、第２格子要素および第７格子要素は、第１列の中心から対称に外側に向かって第３シフト量シフトし、第１格子要素および第８格子要素は、第１列の中心から対称に外側に向かって第４シフト量シフトし、第９格子要素および第１０格子要素は、第２列の中心から対称に内側に向かって第５シフト量シフトし、第１シフト量，第２シフト量，第３シフト量，第４シフト量，第５シフト量は、シフトによって隣り合う格子要素が接触しない範囲とされ、第１シフト量，第２シフト量，第３シフト量，第４シフト量，第５シフト量は、同一の一時結合関数により決定され、第２シフト量は、第１シフト量に第１係数を乗じた量とされ、第３シフト量は、第１シフト量に第２係数を乗じた量とされ、第４シフト量は、第１シフト量に第３係数を乗じた量とされ、第１係数は、四捨五入すると０．９になる数であり、第２係数は、四捨五入すると０．８になる数であり、０．５から０．６５の範囲の数であるフォトニック結晶共振器の設計方法であって、第５シフト量を０としてＱ値が最大になる第１シフト量を決定する第１ステップと、第１ステップで決定された第１シフト量により、第１係数，第２係数，第３係数を用いて第２シフト量，第３シフト量，第４シフト量を決定する第２ステップと、第１ステップで決定された第１シフト量、第２ステップで決定された第２シフト量，第３シフト量，第４シフト量により、第１格子要素，第２格子要素，第３格子要素，第４格子要素，第５格子要素，第６格子要素，第７格子要素，第８格子要素を配置した状態で、Ｑ値が最大となる第５シフト量を決定する第３ステップとを備える。 Further, the design method of the photonic crystal resonator according to the present invention includes a plurality of columnar columns having a refractive index different from that of the base, which are periodically provided in a triangular lattice pattern at intervals equal to or less than the wavelength of light of interest on the base. A photonic crystal body having a plurality of lattice elements, and eight first lattice elements, second lattice elements, and third elements that are provided on the photonic crystal body and are continuous on a straight line in the Γ-K crystal orientation direction of the photonic crystal. A first row of grid elements, fourth grid elements, fifth grid elements, sixth grid elements, seventh grid elements, eighth grid elements, and first grid elements, second grid elements, third grid elements, Two ninth lattice elements adjacent to each other in the direction perpendicular to the Γ-K crystal orientation direction at the center of the four lattice elements, the fifth lattice element, the sixth lattice element, the seventh lattice element, and the eighth lattice element. Light confinement composed of the second row of 10 lattice elements A fourth lattice element and a fifth lattice element that are shifted from the center of the first row symmetrically outward by a first shift amount, and the third lattice element and the sixth lattice element are in the first row. The second lattice element and the seventh lattice element are shifted symmetrically from the center of the first row and shifted by the third shift amount symmetrically outward from the center of the first lattice element. And the eighth lattice element are shifted by the fourth shift amount symmetrically outward from the center of the first row, and the ninth lattice element and the tenth lattice element are symmetrically directed inward from the center of the second row. The first shift amount, the second shift amount, the third shift amount, the fourth shift amount, and the fifth shift amount are set to a range in which adjacent lattice elements do not contact with each other by the shift, and the first shift amount, 2nd shift amount, 3rd shift amount, 4th shift amount, 5th shift The amount is determined by the same temporary combination function, the second shift amount is obtained by multiplying the first shift amount by the first coefficient, and the third shift amount is obtained by multiplying the first shift amount by the second coefficient. The fourth shift amount is an amount obtained by multiplying the first shift amount by the third coefficient, the first coefficient is a number that becomes 0.9 when rounded off, and the second coefficient is 0. 8 is a method of designing a photonic crystal resonator that is a number in the range of 0.5 to 0.65, in which the first shift amount at which the Q value is maximized by setting the fifth shift amount to 0. The first shift amount determined in the first step and the first shift amount determined in the first step determine the second shift amount, the third shift amount, and the fourth shift amount using the first coefficient, the second coefficient, and the third coefficient. The second step, the first shift amount determined in the first step, and the second step The first lattice element, the second lattice element, the third lattice element, the fourth lattice element, the fifth lattice element, the sixth lattice element, the seventh lattice element, according to the 2 shift amount, the third shift amount, and the fourth shift amount, And a third step of determining a fifth shift amount that maximizes the Q value in a state where the eighth lattice element is arranged.
以上説明したことにより、本発明によれば、２次元スラブ型のフォトニック結晶より構成されたＨ０型の共振器における、より最適な共振構造が実現できるという優れた効果が得られる。 As described above, according to the present invention, it is possible to obtain an excellent effect that a more optimal resonance structure can be realized in the H0 type resonator formed of the two-dimensional slab type photonic crystal.
以下、本発明の実施の形態について図を参照して説明する。図１は、本発明の実施の形態におけるフォトニック結晶共振器の一部構成を示す平面図である。 Hereinafter, embodiments of the present invention will be described with reference to the drawings. FIG. 1 is a plan view showing a partial configuration of a photonic crystal resonator according to an embodiment of the present invention.
このフォトニック結晶共振器は、基部１０２および基部１０２に対象とする光の波長以下の間隔で三角格子状に周期的に設けられて基部１０２とは異なる屈折率の柱状の複数の格子要素１０３を備えるフォトニック結晶本体１０１から構成されている。フォトニック結晶本体１０１は、いわゆる２次元スラブ型のフォトニック結晶である。格子要素１０３は、例えば円柱状の中空構造である。 The photonic crystal resonator includes a base 102 and a plurality of columnar lattice elements 103 having a refractive index different from that of the base 102 and periodically provided in a triangular lattice shape at intervals equal to or less than the wavelength of light of interest. The photonic crystal body 101 is provided. The photonic crystal body 101 is a so-called two-dimensional slab type photonic crystal. The lattice element 103 has a cylindrical hollow structure, for example.
また、実施の形態におけるフォトニック結晶共振器は、フォトニック結晶本体１０１に設けられた、光閉じ込め部１０４を備える。光閉じ込め部１０４は、第１格子要素１３１，第２格子要素１３２，第３格子要素１３３，第４格子要素１３４，第５格子要素１３５，第６格子要素１３６，第７格子要素１３７，第８格子要素１３８，第９格子要素１３９ａ，および第１０格子要素１３９ｂから構成されている。 In addition, the photonic crystal resonator in the embodiment includes an optical confinement unit 104 provided in the photonic crystal body 101. The optical confinement unit 104 includes a first grating element 131, a second grating element 132, a third grating element 133, a fourth grating element 134, a fifth grating element 135, a sixth grating element 136, a seventh grating element 137, and an eighth grating element. It comprises a lattice element 138, a ninth lattice element 139a, and a tenth lattice element 139b.
第１格子要素１３１，第２格子要素１３２，第３格子要素１３３，第４格子要素１３４，第５格子要素１３５，第６格子要素１３６，第７格子要素１３７，第８格子要素１３８は、フォトニック結晶のΓ−Ｋ結晶方位方向の直線上に連続して配置されて第１列を構成している。また、第９格子要素１３９ａおよび第１０格子要素１３９ｂは、第１格子要素１３１，第２格子要素１３２，第３格子要素１３３，第４格子要素１３４，第５格子要素１３５，第６格子要素１３６，第７格子要素１３７，第８格子要素１３８の列の中央部で、Γ−Ｋ結晶方位方向に垂直な方向に隣り合い、第２列を構成している。 The first lattice element 131, the second lattice element 132, the third lattice element 133, the fourth lattice element 134, the fifth lattice element 135, the sixth lattice element 136, the seventh lattice element 137, and the eighth lattice element 138 The first row is arranged continuously on a straight line in the Γ-K crystal orientation direction of the nick crystal. The ninth lattice element 139a and the tenth lattice element 139b include the first lattice element 131, the second lattice element 132, the third lattice element 133, the fourth lattice element 134, the fifth lattice element 135, and the sixth lattice element 136. , The seventh lattice element 137 and the eighth lattice element 138 are adjacent to each other in the direction perpendicular to the Γ-K crystal orientation direction to form a second row.
なお、実施の形態１では、図１に示すように、図１の紙面左右方向をΓ−Ｋ結晶方位としているが、この方向に対して±６０°の方向も、Γ−Ｋ結晶方位である。 In the first embodiment, as shown in FIG. 1, the left-right direction of FIG. 1 is the Γ-K crystal orientation, but the direction of ± 60 ° relative to this direction is also the Γ-K crystal orientation. .
また、第４格子要素１３４および第５格子要素１３５は、第１列の中心から対称に外側に向かって第１シフト量シフトし、第３格子要素１３３および第６格子要素１３６は、第１列の中心から対称に外側に向かって第２シフト量シフトし、第２格子要素１３２および第７格子要素１３７は、第１列の中心から対称に外側に向かって第３シフト量シフトし、第１格子要素１３１および第８格子要素１３８は、第１列の中心から対称に外側に向かって第４シフト量シフトしている。一方、第９格子要素１３９ａおよび第１０格子要素１３９ｂは、第２列の中心から対称に内側に向かって第５シフト量シフトしている。 Further, the fourth lattice element 134 and the fifth lattice element 135 are shifted from the center of the first row symmetrically by the first shift amount, and the third lattice element 133 and the sixth lattice element 136 are shifted in the first row. The second lattice element 132 and the seventh lattice element 137 are symmetrically shifted outward from the center of the first row, and shifted by a third shift amount. The lattice element 131 and the eighth lattice element 138 are shifted by the fourth shift amount symmetrically outward from the center of the first row. On the other hand, the ninth lattice element 139a and the tenth lattice element 139b are shifted from the center of the second row by the fifth shift amount symmetrically toward the inside.
また、第１シフト量，第２シフト量，第３シフト量，第４シフト量，第５シフト量は、シフトによって隣り合う格子要素が接触しない範囲とされ、第１シフト量，第２シフト量，第３シフト量，第４シフト量，第５シフト量は、同一の一時結合関数により決定されている。第２シフト量は、第１シフト量に第１係数を乗じた量とされ、第３シフト量は、第１シフト量に第２係数を乗じた量とされ、第４シフト量は、第１シフト量に第３係数を乗じた量とされている。なお、第１係数は、四捨五入すると０．９になる数であり、第２係数は、四捨五入すると０．８になる数であり、第３係数は、０．５から０．６５の範囲の数である。なお、第１および第２係数において、小数点以下第２位を四捨五入する。 Further, the first shift amount, the second shift amount, the third shift amount, the fourth shift amount, and the fifth shift amount are within a range in which adjacent lattice elements do not contact with each other by the shift, and the first shift amount and the second shift amount. , The third shift amount, the fourth shift amount, and the fifth shift amount are determined by the same temporary combination function. The second shift amount is an amount obtained by multiplying the first shift amount by the first coefficient, the third shift amount is an amount obtained by multiplying the first shift amount by the second coefficient, and the fourth shift amount is the first shift amount. The amount obtained by multiplying the shift amount by the third coefficient. The first coefficient is a number that becomes 0.9 when rounded off, the second coefficient is a number that becomes 0.8 when rounded off, and the third coefficient is a number in the range of 0.5 to 0.65. It is. In the first and second coefficients, the second decimal place is rounded off.
非特許文献６の設計例において、厚さｔが２２０ｎｍのＳｉ薄膜フォトニック結晶（屈折率３．４６）において、格子定数ａが４００ｎｍ、穴半径ｒがａ／４、（第１シフト量ｓ１、第２シフト量ｓ２、第３シフト量ｓ３、第４シフト量ｓ４、第５シフト量ｓ５）＝（０．３８５ａ，０．３４２ａ，０．３０１ａ，０．１６２ａ、０．０１７ａ）の設定が与えられ、更に、第１格子要素１３１，第８格子要素１３８，第９格子要素１３９ａ，および第１０格子要素１３９ｂの外側の穴にも大きなシフトが与えられている。 In the design example of Non-Patent Document 6, in a Si thin film photonic crystal (refractive index 3.46) having a thickness t of 220 nm, the lattice constant a is 400 nm, the hole radius r is a / 4, (first shift amount s1, The second shift amount s2, the third shift amount s3, the fourth shift amount s4, the fifth shift amount s5) = (0.385a, 0.342a, 0.301a, 0.162a, 0.017a) is given. Furthermore, a large shift is given to the outer holes of the first lattice element 131, the eighth lattice element 138, the ninth lattice element 139a, and the tenth lattice element 139b.
発明者らはまず、各格子要素のシフト量にこの設定例通りの一次相関が成り立つと仮定し、格子定数ａおよび厚さｔなどの結晶パラメータを変えてみた。この結果、１００万程度のＱ値が得られ、非特許文献６における設定例のシフト量比の設定は、任意の結晶パラメータで一定の高Ｑ値を与えることが確認された。しかしながら、非特許文献６に報告されている８３０万からすれば数分の一程度のＱ値であり、少なくとも最適なシフト量を常に与えるものでないことが確認された。 The inventors first assumed that the first-order correlation according to this setting example holds for the shift amount of each lattice element, and changed crystal parameters such as the lattice constant a and the thickness t. As a result, a Q value of about 1 million was obtained, and it was confirmed that the setting of the shift amount ratio in the setting example in Non-Patent Document 6 gives a constant high Q value with an arbitrary crystal parameter. However, from the 8.3 million reported in Non-Patent Document 6, the Q value is a fraction of a fraction, and it has been confirmed that at least the optimum shift amount is not always given.
次に、様々な結晶パラメータにおい非特許文献６における設定例より更に高Ｑ値化を行う検討を行った。この結果、まず第１格子要素１３１〜第８格子要素１３８，第９格子要素１３９ａ，第１０格子要素１３９ｂの外側の格子要素のシフトについては、これら光閉じ込め部１０４を構成する各格子要素のシフト量の設定次第ではＱ値向上に無効であり、高Ｑ値化に本質的な格子要素でないことを見出した。このため、光閉じ込め部１０４を構成する各格子要素以外のシフト量は０として除外することにした。このように、本発明では、構造調整の対象を第１格子要素１３１〜第８格子要素１３８，第９格子要素１３９ａ，第１０格子要素１３９ｂに絞り込んだ。 Next, investigations were made to further increase the Q value in the various crystal parameters compared to the setting example in Non-Patent Document 6. As a result, first, regarding the shift of the lattice elements outside the first lattice element 131 to the eighth lattice element 138, the ninth lattice element 139a, and the tenth lattice element 139b, the shift of each lattice element constituting the optical confinement unit 104 is performed. It has been found that it is ineffective for improving the Q value depending on the setting of the quantity and is not an essential lattice element for increasing the Q value. For this reason, the shift amount other than each lattice element constituting the optical confinement unit 104 is excluded as 0. Thus, in the present invention, the structural adjustment targets are narrowed down to the first lattice element 131 to the eighth lattice element 138, the ninth lattice element 139a, and the tenth lattice element 139b.
次に、第９格子要素１３９ａ，第１０格子要素１３９ｂの第５シフト量ｓ５を０に設定し、Ｑ値を最大にする第１格子要素１３１〜第８格子要素１３８の第１シフト量ｓ１，第２シフト量ｓ２，第３シフト量ｓ３，第４シフト量ｓ４を求めたところ、次の一次相関関係を見出した。 Next, the fifth shift amount s5 of the ninth lattice element 139a and the tenth lattice element 139b is set to 0, and the first shift amount s1,1 of the first lattice element 131 to the eighth lattice element 138 that maximizes the Q value. When the second shift amount s2, the third shift amount s3, and the fourth shift amount s4 were determined, the following primary correlation was found.
「第２シフト量ｓ２＝第１係数×第１シフト量ｓ１」，「第３シフト量ｓ３＝第２係数×第１シフト量ｓ１」，「第４シフト量ｓ４＝第３係数×第１シフト量ｓ１」。なお、第１係数は、小数点以下第２位を四捨五入すると０．９になる数であり、第２係数は、小数点以下第２位を四捨五入すると０．８になる数であり、第３係数は、０．５から０．６５の範囲の数である。 “Second shift amount s2 = first coefficient × first shift amount s1”, “third shift amount s3 = second coefficient × first shift amount s1”, “fourth shift amount s4 = third coefficient × first shift” Quantity s1 ". The first coefficient is a number that becomes 0.9 when the second decimal place is rounded off, the second coefficient is a number that becomes 0.8 when the second decimal place is rounded off, and the third coefficient is , A number in the range of 0.5 to 0.65.
例えばａ＝４００ｎｍ、ｒ＝ａ／４、ｔ＝２００〜２２５ｎｍにおいて、（第１シフト量ｓ１、第２シフト量ｓ２、第３シフト量ｓ３、第４シフト量ｓ４）＝（０．３８８ａ，０．３５０ａ，０．３１０ａ，０．２３３ａ）とすると、Ｑ値は１２０万程度に最適化された。この値は、非特許文献６より前に報告されたＨ０共振器のＱ値と比べれば十分高いものではあるが、最近の高Ｑ値化されたＬ３共振器（非特許文献２，６，７）と比べれば優位性があるとは言えない。 For example, at a = 400 nm, r = a / 4, and t = 200 to 225 nm, (first shift amount s1, second shift amount s2, third shift amount s3, fourth shift amount s4) = (0.388a, 0 .350a, 0.310a, 0.233a), the Q value was optimized to about 1,200,000. This value is sufficiently higher than the Q value of the H0 resonator reported before Non-Patent Document 6, but recently, a high-Q L3 resonator (Non-Patent Documents 2, 6, and 7). ) Is not superior.
次に、第９格子要素１３９ａ，第１０格子要素１３９ｂを共振器中心（光閉じ込め部１０４中心）に向けてシフトさせた（第５シフト量ｓ５を変化させた）。ｔ＝２００ｎｍの場合のＱ値の変化を図４に示す。第５シフト量ｓ５の増加に伴いＱ値は急減に増大し、ｓ５＝０．０２５ａにおいてＱ値は最高値８７０万に達した。更にｓ５を増加させるとＱ値は減少に転じ、ｓ５＝０．０５ａにおいてＱ値増強は得られなくなった。また、また、ｔ＝２２５ｎｍにおいてもｓ５＝０．０２５ａにおいてＱ値は最高値９３０万に達した。 Next, the ninth grating element 139a and the tenth grating element 139b were shifted toward the resonator center (center of the optical confinement portion 104) (the fifth shift amount s5 was changed). FIG. 4 shows the change in the Q value when t = 200 nm. As the fifth shift amount s5 increases, the Q value increases rapidly, and the maximum Q value reaches 8.7 million at s5 = 0.025a. When s5 was further increased, the Q value started to decrease, and the enhancement of the Q value was not obtained at s5 = 0.05a. Further, even at t = 225 nm, the Q value reached the maximum value of 9.3 million at s5 = 0.025a.
第５シフト量ｓ５第５シフト量ｓ５＝０に対するＱ値増倍比をＡと定義すると、Ａは７倍程度（８７０万／１２０万、もしくは９３０万／１２０万）であり、４００ｎｍ×０．０２５＝１０ｎｍ程度のわずかなシフトで著しいＱ値増倍が惹起されることを意味する。非特許文献６の報告例にも第５シフト量が与えられているが、Ｑ値の第５シフト量に関する明確な記述はなく、また第９格子要素および第１０格子要素の外側の穴にもシフトが与えられているため、第９格子要素および第１０格子要素の役割は明確にされていなかった。本発明により初めて第５シフト量の役割と効果が定量的に示された。 If the Q value multiplication ratio with respect to the fifth shift amount s5 and the fifth shift amount s5 = 0 is defined as A, A is about 7 times (8.7 million / 1.2 million or 9.3 million / 1.2 million), and 400 nm × 0. It means that a remarkable Q factor multiplication is induced by a slight shift of about 025 = 10 nm. The fifth shift amount is also given in the report example of Non-Patent Document 6, but there is no clear description regarding the fifth shift amount of the Q value, and the holes outside the ninth lattice element and the tenth lattice element are not described. Since the shift is given, the roles of the ninth lattice element and the tenth lattice element have not been clarified. The role and effect of the fifth shift amount are quantitatively shown for the first time by the present invention.
非特許文献６において実施された遺伝的アルゴリズムによる機械的最適化において構造探索の対象となった７組の穴の広範なシフト量の範囲には本発明の与える構造が含まれる。故に同手法において本発明が見い出される可能性があったが、実際に非特許文献６が報告したのは本発明とは異なる共振器構造であり、かつ本発明により更に高い共振器Ｑ値が得られている。このことは非特許文献６で行われた機械的最適化は一定の有効性はあるが現時点では完全ではなく、本発明が与える設計が非特許文献６による報告より「正しい」設計であると言える。本発明は経験的な規則により共振器構造を定めるもので、後述するように任意のａ、ｔ，ｒにおいて本発明の適用によりＱ値が高くなるようほぼ最適化される。なお、第９格子要素１３９ａ，第１０格子要素１３９ｂのシフトについては、非特許文献６において内側にシフトされているものの、この効果については文献中で一切触れられておらず、また非特許文献７および非特許文献８では外側にシフトされ、特に後者では共振器波長を制御するためのシフトでＱ値に対する影響は無いとされていた。非特許文献６およびその手法の単なる再利用では見出せなかった、任意のａ、ｔ，ｒに有効な、経験的な共振器設計規則を初めて見出したことが本発明の重要な意義である。 The structure provided by the present invention is included in a wide range of shift amounts of the seven sets of holes subjected to the structure search in the mechanical optimization performed by the genetic algorithm performed in Non-Patent Document 6. Therefore, there was a possibility that the present invention could be found in this method, but the non-patent document 6 actually reported a resonator structure different from the present invention, and a higher resonator Q value was obtained by the present invention. It has been. This is because the mechanical optimization performed in Non-Patent Document 6 has certain effectiveness, but is not perfect at present, and the design provided by the present invention can be said to be a “correct” design from the report by Non-Patent Document 6. . In the present invention, the resonator structure is determined based on empirical rules. As will be described later, the present invention is almost optimized to increase the Q value at any a, t, and r. The shift of the ninth lattice element 139a and the tenth lattice element 139b is shifted inward in Non-Patent Document 6, but this effect is not mentioned in the document at all, and Non-Patent Document 7 In Non-Patent Document 8, it is shifted outward. In particular, the latter is a shift for controlling the resonator wavelength and has no influence on the Q value. It is an important significance of the present invention that an empirical resonator design rule effective for any a, t, and r that cannot be found by simple reuse of Non-Patent Document 6 and its method is found for the first time.
本発明の有効性を確認するため、様々な格子定数ａおよび厚さｔの組み合わせにおいて本発明を適用し最適なシフト量を見出した。具体的には。まず第５シフト量ｓ５＝０においてＱ値が最高になるように第１シフト量ｓ１を定め、次に、第５シフト量ｓ５を変えてＱ値を最高化する。結果を図２に示す。増倍係数Ａは全ての構造パラメータにおいて７から１０程度とほぼ一桁のＱ値増大をあり、第９格子要素１３９ａ，第１０格子要素１３９ｂのシフトによるＱ値の著しい増大がＨ０共振器に共通する、また同共振器を特徴づける性質であることが示された。図４と同様の第５シフト量ｓ５の変化に対するＱ値の定性的変化が全ての構造パラメータにおいて確認された。 In order to confirm the effectiveness of the present invention, the present invention was applied to various combinations of the lattice constant a and the thickness t, and the optimum shift amount was found. In particular. First, the first shift amount s1 is determined so that the Q value becomes maximum when the fifth shift amount s5 = 0, and then the Q value is maximized by changing the fifth shift amount s5. The results are shown in FIG. The multiplication factor A has a Q value increase of about one order of magnitude between 7 and 10 in all structural parameters, and a significant increase in the Q value due to the shift of the ninth lattice element 139a and the tenth lattice element 139b is common to the HO resonator. It has been shown that this is a property that characterizes the resonator. The qualitative change of the Q value with respect to the change of the fifth shift amount s5 similar to FIG. 4 was confirmed in all the structural parameters.
非特許文献６の設定例において第５シフト量ｓ５は０．０１６ａに設定されているが、このシフトがＱ値にどのように影響を与えるかについては一切記述が無かった。発明者らは、Ｑ値はｔが厚くなるほど、また規格化周波数ａ／λcが小さくなるほど高くなる傾向があり、ｔ＝３００ｎｍにおいて１７００万程度のＱ値が得られることを初めて見出した。 In the setting example of Non-Patent Document 6, the fifth shift amount s5 is set to 0.016a. However, there is no description as to how this shift affects the Q value. The inventors found for the first time that the Q value tends to increase as t becomes thicker and the normalized frequency a / λ c become smaller, and a Q value of about 17 million can be obtained at t = 300 nm.
本発明に係るフォトニック結晶共振器においてＱ値を最高になるように最適化した際のモード体積Ｖは、ｒ＝０．２５ａのＳｉの場合に概ね０．７（λc／ｎ）3 程度となり、標準的なＬ３共振器と同等か少し小さい値になり、最近報告されているＱ値の高いＬ型共振器（非特許文献２，６，７）よりは明らかに小さくなる。Ｌ型共振器との比較においては上述したの大きな増倍係数Ａの効果で同じ結晶パラメータａ，ｔ，ｒにおいて常に、本発明に係るフォトニック結晶共振器が、Ｌ１−Ｌ３共振器をＱ値およびＱ／Ｖで大きく上回る。本発明により、少数個の点欠陥からなるあらゆるＬ型およびＨ型共振器より優れたナノ共振器モードを有するＨ０型のフォトニック結晶共振器が実現できる。 The mode volume V when the Q value is optimized in the photonic crystal resonator according to the present invention is about 0.7 (λ c / n) 3 in the case of Si with r = 0.25a. Thus, the value is equal to or slightly smaller than that of the standard L3 resonator, and is clearly smaller than the L-type resonator having a high Q value reported recently (Non-Patent Documents 2, 6, and 7). In comparison with the L-type resonator, the photonic crystal resonator according to the present invention always has the Q value of the L1-L3 resonator at the same crystal parameters a, t, r due to the effect of the large multiplication factor A described above. And greatly surpassed by Q / V. According to the present invention, a H0 type photonic crystal resonator having a nanocavity mode superior to all L type and H type resonators composed of a small number of point defects can be realized.
図３の（ａ）に、本発明の実施の形態に係るフォトニック結晶共振器の１例である、「ａ＝４００ｎｍ、ｒ＝ａ／４、ｔ＝２００ｎｍ」における垂直方向の共振器モード磁場Ｈｙの分布を、結晶格子要素配置に重ねた図を示す。 FIG. 3A shows an example of the photonic crystal resonator according to the embodiment of the present invention, which is an example of the resonator mode magnetic field in the vertical direction at “a = 400 nm, r = a / 4, t = 200 nm”. The figure which overlaid the distribution of Hy on the crystal lattice element arrangement | positioning is shown.
このフォトニック共振器の基本共振モードは、Ｌ型共振器と同様、一列抜き線欠陥の偶対称伝搬モード（非特許文献９）を起源としている。この基本共振モードは、２次元薄膜ＴＥモードの特徴を有し、薄膜に垂直なｙ方向の磁場成分Ｈｙと、薄膜面内方向の電場成分Ｅｘ、Ｅｚを有する。また電磁界モード形状は、Ｌ型共振器のものに近く、対称性もＬ型（Ｌ２，Ｌ４）に準拠している。一方で、中心部の格子要素の無い（欠陥）領域にＬ型共振器が２つ以上のモードの腹(antinode)を有するのに対し、本発明に係るフォトニック結晶共振器は、１つのみの腹を有することを特徴としており、よりＶが小さいことと合わせ、共振器中心に量子ドットなどの活性層を配置すれば、共振モードによる相互作用が最も効率良く得られることが期待される。 The fundamental resonance mode of this photonic resonator originates from the even-symmetric propagation mode (Non-Patent Document 9) of a single-line-drawn line defect, as in the L-type resonator. This basic resonance mode has the characteristics of a two-dimensional thin film TE mode, and has a magnetic field component Hy in the y direction perpendicular to the thin film and electric field components Ex and Ez in the in-plane direction of the thin film. The electromagnetic field mode shape is close to that of the L-type resonator, and the symmetry conforms to the L-type (L2, L4). On the other hand, an L-type resonator has two or more antinodes in a (defect) region without a lattice element in the center, whereas only one photonic crystal resonator according to the present invention is present. If the active layer such as a quantum dot is arranged at the center of the resonator in combination with the smaller V, it is expected that the interaction by the resonance mode can be obtained most efficiently.
また、本発明に係るフォトニック結晶共振器は、点光源に近い近視野放射特性を有している。従来知られるＨ０共振器の設計では、第４格子要素１３４，第５格子要素１３５の外側へのシフト第１シフト量ｓ１が、より小さかったため活性層の配置が難しかった。これに対し、本発明においては第１シフト量ｓ１が０．４ａ程度と大きく、中心の格子要素の無い領域のサイズが既に量子ドット配置の実績のあるＨ１共振器に近いため、活性層を配置できる可能性が高くなっている。 The photonic crystal resonator according to the present invention has near-field radiation characteristics close to a point light source. In the conventional design of the H0 resonator, the first shift amount s1 shifted outward from the fourth grating element 134 and the fifth grating element 135 is smaller, so that it is difficult to dispose the active layer. On the other hand, in the present invention, the first shift amount s1 is as large as about 0.4a, and the size of the region without the central lattice element is close to the H1 resonator that has already been proven in the quantum dot arrangement. The possibility of being able to be increased.
図３の（ｂ）に、共振モードの電磁界分布をフーリエ変換し波数空間における分布を示す。一般にＱ値の低い共振器は、図中に白点線の円で示したライトコーン内の波数成分の電磁界強度が高いため、光が共振器外に放射されＱ値が制限される。薄膜型２次元フォトニック結晶は、ｙ方向の光閉じ込めを、高屈折率な材料（例えばＳｉ）からなる薄膜と低屈折率のクラッド（空気など）との屈折率差により実現しているが、ライトコーンは、薄膜とクラッドとの境界において全反射の臨界角より大きい入射角に相当する波数成分の領域である。 FIG. 3B shows the distribution in the wave number space by Fourier transforming the electromagnetic field distribution in the resonance mode. In general, a resonator having a low Q value has a high electromagnetic field strength of a wave number component in a light cone indicated by a white dotted line circle in the figure, so that light is emitted outside the resonator and the Q value is limited. The thin-film type two-dimensional photonic crystal realizes light confinement in the y direction by a refractive index difference between a thin film made of a high refractive index material (for example, Si) and a low refractive index cladding (such as air). The light cone is a region of a wave number component corresponding to an incident angle larger than the critical angle of total reflection at the boundary between the thin film and the clad.
図３の（ｂ）は、ライトコーン内の電磁界成分が極めて低く抑えられていることを示しており、本発明に係るフォトニック結晶共振器が１０００万程度の理論Ｑ値を有するのは、放射モードの低減によることを説明している。 FIG. 3 (b) shows that the electromagnetic field component in the light cone is extremely low, and the photonic crystal resonator according to the present invention has a theoretical Q value of about 10 million. It explains that the radiation mode is reduced.
以上に説明したように、本発明では、フォトニック結晶のΓ−Ｋ結晶方位方向の直線上に連続して配置された第１格子要素，第２格子要素，第３格子要素，第４格子要素，第５格子要素，第６格子要素，第７格子要素，第８格子要素、これらに直交して配置された第９格子要素第１０格子要素１３９ｂのシフト量を、同一の一時結合関数により決定するようにした。この結果、本発明によれば、２次元スラブ型のフォトニック結晶より構成されたＨ０型の共振器におけるより最適な共振構造が得られるようになる。 As described above, in the present invention, the first lattice element, the second lattice element, the third lattice element, and the fourth lattice element that are continuously arranged on the straight line in the Γ-K crystal orientation direction of the photonic crystal. , The fifth lattice element, the sixth lattice element, the seventh lattice element, the eighth lattice element, and the shift amount of the ninth lattice element and the tenth lattice element 139b arranged orthogonal to these are determined by the same temporary coupling function I tried to do it. As a result, according to the present invention, a more optimal resonance structure can be obtained in the H0 type resonator composed of the two-dimensional slab type photonic crystal.
本発明では、１０００万程度の理論Ｑ値を有しモード体積が０．７（λc／ｎ）3程度のＨ０型のPhotonic共振器の設計が、５組１０個の特定の穴を一次相関による極めて単純な関係に従い規則的にシフトすることで、任意の格子定数・薄膜厚さ・穴径の組み合わせにおいて概ね最適化されることを提示した。非特許文献６に記載された機械的最適化手法は、結果的に本発明と同等あるいは類似の近い共振器構造および特性を与える可能性はあるが、現時点で同手法が常にほぼ最適な共振器構造を与える保証は無かった。現に同手法による本発明の構造の確定はまだ報告されていない。また、ＧＭＥ法という計算負荷が少ない手法を利用可能であるとはいえ、機械的最適化手法は遺伝的アルゴリズムを組み合わせても数千〜数万程度の共振器構造を試行錯誤的に解析するため一定の計算機リソースと計算時間が必要であった。 In the present invention, the design of a HO type photonic resonator having a theoretical Q value of about 10 million and a mode volume of about 0.7 (λ c / n) 3 is a linear correlation between 5 sets and 10 specific holes. By regularly shifting according to the extremely simple relation of, it is proposed that the combination of arbitrary lattice constant, thin film thickness, and hole diameter can be optimized. Although the mechanical optimization method described in Non-Patent Document 6 may result in a similar resonator structure and characteristics similar to or similar to those of the present invention, this method is always an almost optimal resonator. There was no guarantee to give structure. In fact, the determination of the structure of the present invention by this method has not yet been reported. In addition, although the GME method with a low calculation load can be used, the mechanical optimization method is used to analyze several thousand to several tens of thousands of resonator structures by trial and error even when combined with a genetic algorithm. A certain amount of computer resources and calculation time were required.
一方、本発明では、４組８個の穴のシフト量に規則性が与えられ、１つの変数で制御されるため、残りの１組２個の穴のシフトと合わせ２変数の最適化に単純化される。加えて図２に示す通り、例えば、フォトニック結晶の基部をＳｉから構成し、格子要素の穴範型ｒ＝０．２５ａの場合、経験的に最適解がほぼ第１シフト量ｓ１＝０．３９０ａ、第５シフト量ｓ５＝０．０２５ａになることを明らかにしている。 On the other hand, in the present invention, since the regularity is given to the shift amount of the four sets of eight holes and the shift is controlled by one variable, it is simple to optimize the two variables in combination with the shift of the remaining one set of two holes. It becomes. In addition, as shown in FIG. 2, for example, when the base portion of the photonic crystal is made of Si and the lattice element hole type r = 0.25a, the optimum solution is empirically determined by the first shift amount s1 = 0. It is clarified that 390a and the fifth shift amount s5 = 0.025a.
本発明は経験的にシフト対象となる穴およびそのシフト量のリファレンス値を与える。これに対し穴径・薄膜厚さ・屈折率などの薄膜材料物性・等の構造パラメータに依存しリファレンスからのシフト量の若干のずれが発生する。本発明の与える共振器においては各シフト量ｓ１〜ｓ５が最適値から数ｎｍずれると大きなＱ値低下が発生するため、シフト量のずれの補正は重要である。Ｑ値を最高にするためには試行錯誤的なシフト量の微調整が必要であるが、そのための試行錯誤ステップ数は数回から数１０回の少数回で済む。よって計算負荷が高いＦＤＴＤ法によっても短時間で容易に最適化が可能である。試行錯誤のコストの更に高い実験においても、試行錯誤的にシフト量を最適化することが現実的になる。このようなアプローチの有効性については非特許文献２に述べられている。 The present invention empirically gives a hole to be shifted and a reference value for the shift amount. On the other hand, a slight shift of the shift amount from the reference occurs depending on the structural parameters such as the physical properties of the thin film material such as the hole diameter, the thin film thickness, and the refractive index. In the resonator according to the present invention, when the shift amounts s1 to s5 deviate from the optimum value by several nm, a large Q value decrease occurs. Therefore, correction of the shift amount deviation is important. In order to maximize the Q value, it is necessary to finely adjust the shift amount by trial and error, but the number of trial and error steps required for this is only a few from several to several tens. Therefore, optimization can be easily performed in a short time even by the FDTD method having a high calculation load. Even in experiments with higher trial and error costs, it becomes practical to optimize the shift amount by trial and error. The effectiveness of such an approach is described in Non-Patent Document 2.
本発明は、モード体積Ｖが１（λc／ｎ）3を下回る極微小ナノ共振器において最高の共振器Ｑ値および最大のＱ／Ｖを与える。本発明は、性能がＱ値又はＱ／Ｖに依存するあらゆる共振器デバイスにおいて極めて有用である。また、本発明は、共振器電磁力学（ｃａｖｉｔｙ−ＱＥＤ）を初めとする学術面においても極めて価値が高い。 The present invention gives the highest resonator Q value and the highest Q / V in a micro nanoresonator with a mode volume V of less than 1 (λ c / n) 3 . The present invention is very useful in any resonator device whose performance depends on Q factor or Q / V. The present invention is also extremely valuable in academic fields including resonator electromagnetics (cavity-QED).
図２に示すように、本発明においてはフォトニック結晶共振器の共振器波長λcを構造パラメータの設定により任意の光通信に有用な波長に設定し、同時に極めて高いＱ値を与えることが可能である。よって、応用上の利用価値も高い。 As shown in FIG. 2, in the present invention, it is possible to set the resonator wavelength λc of the photonic crystal resonator to a wavelength useful for any optical communication by setting a structural parameter, and at the same time, it is possible to give an extremely high Q value. is there. Therefore, application value is also high.
なお、本発明は以上に説明した実施の形態に限定されるものではなく、本発明の技術的思想内で、当分野において通常の知識を有する者により、多くの変形および組み合わせが実施可能であることは明白である。 The present invention is not limited to the embodiment described above, and many modifications and combinations can be implemented by those having ordinary knowledge in the art within the technical idea of the present invention. It is obvious.
１０１…フォトニック結晶本体１０１、１０２…基部、１０３…格子要素、１０４…光閉じ込め部、１３１…第１格子要素、１３２…第２格子要素、１３３…第３格子要素、１３４…第４格子要素、１３５…第５格子要素、１３６…第６格子要素、１３７…第７格子要素、１３８…第８格子要素、１３９ａ…第９格子要素、１３９ｂ…第１０格子要素。 DESCRIPTION OF SYMBOLS 101 ... Photonic crystal main body 101, 102 ... Base, 103 ... Lattice element, 104 ... Light confinement part, 131 ... 1st lattice element, 132 ... 2nd lattice element, 133 ... 3rd lattice element, 134 ... 4th lattice element 135, fifth lattice element, 136, sixth lattice element, 137, seventh lattice element, 138, eighth lattice element, 139a, ninth lattice element, 139b, tenth lattice element.
Claims (3)
前記フォトニック結晶本体に設けられて、フォトニック結晶のΓ−Ｋ結晶方位方向の直線上に連続する８つの第１格子要素，第２格子要素，第３格子要素，第４格子要素，第５格子要素，第６格子要素，第７格子要素，第８格子要素からなる第１列、および前記第１格子要素，前記第２格子要素，前記第３格子要素，前記第４格子要素，前記第５格子要素，前記第６格子要素，前記第７格子要素，前記第８格子要素の列の中央部で前記Γ−Ｋ結晶方位方向に垂直な方向に隣り合う２つの第９格子要素および第１０格子要素からなる第２列より構成された光閉じ込め部と
を備え、
前記第４格子要素および前記第５格子要素は、前記第１列の中心から対称に外側に向かって第１シフト量シフトし、
前記第３格子要素および前記第６格子要素は、前記第１列の中心から対称に外側に向かって第２シフト量シフトし、
前記第２格子要素および前記第７格子要素は、前記第１列の中心から対称に外側に向かって第３シフト量シフトし、
前記第１格子要素および前記第８格子要素は、前記第１列の中心から対称に外側に向かって第４シフト量シフトし、
前記第９格子要素および前記第１０格子要素は、前記第２列の中心から対称に内側に向かって第５シフト量シフトし、
前記第１シフト量，前記第２シフト量，前記第３シフト量，前記第４シフト量，前記第５シフト量は、シフトによって隣り合う格子要素が接触しない範囲とされ、
前記第１シフト量，前記第２シフト量，前記第３シフト量，前記第４シフト量，前記第５シフト量は、同一の一時結合関数により決定され、
前記第２シフト量は、前記第１シフト量に第１係数を乗じた量とされ、
前記第３シフト量は、前記第１シフト量に第２係数を乗じた量とされ、
前記第４シフト量は、前記第１シフト量に第３係数を乗じた量とされ、
前記第１係数は、四捨五入すると０．９になる数であり、
前記第２係数は、四捨五入すると０．８になる数であり、
前記第３係数は、０．５から０．６５の範囲の数である
ことを特徴とするフォトニック結晶共振器。 A photonic crystal body comprising a base and a plurality of columnar lattice elements having a refractive index different from that of the base, which are periodically provided in a triangular lattice shape at intervals equal to or less than the wavelength of light of interest on the base,
Eight 1st lattice elements, 2nd lattice elements, 3rd lattice elements, 4th lattice elements, 5th which are provided in the photonic crystal body and continue on a straight line in the Γ-K crystal orientation direction of the photonic crystal A first row of grid elements, sixth grid elements, seventh grid elements, eighth grid elements, and the first grid elements, second grid elements, third grid elements, fourth grid elements, 5th lattice element, 6th lattice element, 7th lattice element, 2nd 9th lattice element and 10th adjacent to the direction perpendicular to the Γ-K crystal orientation direction at the center of the row of the 8th lattice element An optical confinement section composed of a second row of lattice elements,
The fourth lattice element and the fifth lattice element are shifted by a first shift amount symmetrically outward from the center of the first row,
The third lattice element and the sixth lattice element are shifted by a second shift amount symmetrically outward from the center of the first row,
The second lattice element and the seventh lattice element are shifted from the center of the first row symmetrically outward by a third shift amount,
The first lattice element and the eighth lattice element are shifted by a fourth shift amount symmetrically outward from the center of the first row,
The ninth and tenth lattice elements are shifted inwardly by a fifth shift amount symmetrically from the center of the second row;
The first shift amount, the second shift amount, the third shift amount, the fourth shift amount, and the fifth shift amount are within a range in which adjacent lattice elements do not contact with each other by the shift,
The first shift amount, the second shift amount, the third shift amount, the fourth shift amount, and the fifth shift amount are determined by the same temporary combination function,
The second shift amount is an amount obtained by multiplying the first shift amount by a first coefficient,
The third shift amount is an amount obtained by multiplying the first shift amount by a second coefficient,
The fourth shift amount is an amount obtained by multiplying the first shift amount by a third coefficient,
The first coefficient is a number that becomes 0.9 when rounded off;
The second coefficient is a number that when rounded to 0.8.
The third coefficient is a number in the range of 0.5 to 0.65. A photonic crystal resonator.
前記第５シフト量を０とした場合にＱ値が最大になる前記第１シフト量を用いて決定された前記第２シフト量，前記第３シフト量，前記第４シフト量により、前記第１格子要素，前記第２格子要素，前記第３格子要素，前記第４格子要素，前記第５格子要素，前記第６格子要素，前記第７格子要素，前記第８格子要素が配置された状態で、Ｑ値が最大となる状態に第５シフト量が決定されている
ことを特徴とするフォトニック結晶共振器。 The photonic crystal resonator according to claim 1, wherein
Based on the second shift amount, the third shift amount, and the fourth shift amount determined using the first shift amount that maximizes the Q value when the fifth shift amount is set to 0, the first shift amount In a state in which the lattice element, the second lattice element, the third lattice element, the fourth lattice element, the fifth lattice element, the sixth lattice element, the seventh lattice element, and the eighth lattice element are arranged. The fifth shift amount is determined so that the Q value becomes maximum. A photonic crystal resonator, wherein:
前記フォトニック結晶本体に設けられて、フォトニック結晶のΓ−Ｋ結晶方位方向の直線上に連続する８つの第１格子要素，第２格子要素，第３格子要素，第４格子要素，第５格子要素，第６格子要素，第７格子要素，第８格子要素からなる第１列、および前記第１格子要素，前記第２格子要素，前記第３格子要素，前記第４格子要素，前記第５格子要素，前記第６格子要素，前記第７格子要素，前記第８格子要素の列の中央部で前記Γ−Ｋ結晶方位方向に垂直な方向に隣り合う２つの第９格子要素および第１０格子要素からなる第２列より構成された光閉じ込め部と
を備え、
前記第４格子要素および前記第５格子要素は、前記第１列の中心から対称に外側に向かって第１シフト量シフトし、
前記第３格子要素および前記第６格子要素は、前記第１列の中心から対称に外側に向かって第２シフト量シフトし、
前記第２格子要素および前記第７格子要素は、前記第１列の中心から対称に外側に向かって第３シフト量シフトし、
前記第１格子要素および前記第８格子要素は、前記第１列の中心から対称に外側に向かって第４シフト量シフトし、
前記第９格子要素および前記第１０格子要素は、前記第２列の中心から対称に内側に向かって第５シフト量シフトし、
前記第１シフト量，前記第２シフト量，前記第３シフト量，前記第４シフト量，前記第５シフト量は、シフトによって隣り合う格子要素が接触しない範囲とされ、
前記第１シフト量，前記第２シフト量，前記第３シフト量，前記第４シフト量，前記第５シフト量は、同一の一時結合関数により決定され、
前記第２シフト量は、前記第１シフト量に第１係数を乗じた量とされ、
前記第３シフト量は、前記第１シフト量に第２係数を乗じた量とされ、
前記第４シフト量は、前記第１シフト量に第３係数を乗じた量とされ、
前記第１係数は、四捨五入すると０．９になる数であり、
前記第２係数は、四捨五入すると０．８になる数であり、
前記第３係数は、０．５から０．６５の範囲の数である
フォトニック結晶共振器の設計方法であって、
前記第５シフト量を０としてＱ値が最大になる前記第１シフト量を決定する第１ステップと、
前記第１ステップで決定された前記第１シフト量により、前記第１係数，前記第２係数，前記第３係数を用いて前記第２シフト量，前記第３シフト量，前記第４シフト量を決定する第２ステップと、
前記第１ステップで決定された前記第１シフト量、前記第２ステップで決定された前記第２シフト量，前記第３シフト量，前記第４シフト量により、前記第１格子要素，前記第２格子要素，前記第３格子要素，前記第４格子要素，前記第５格子要素，前記第６格子要素，前記第７格子要素，前記第８格子要素を配置した状態で、Ｑ値が最大となる前記第５シフト量を決定する第３ステップと
を備えることを特徴とするフォトニック結晶共振器の設計方法。 A photonic crystal body comprising a base and a plurality of columnar lattice elements having a refractive index different from that of the base, which are periodically provided in a triangular lattice shape at intervals equal to or less than the wavelength of light of interest on the base,
Eight 1st lattice elements, 2nd lattice elements, 3rd lattice elements, 4th lattice elements, 5th which are provided in the photonic crystal body and continue on a straight line in the Γ-K crystal orientation direction of the photonic crystal A first row of grid elements, sixth grid elements, seventh grid elements, eighth grid elements, and the first grid elements, second grid elements, third grid elements, fourth grid elements, 5th lattice element, 6th lattice element, 7th lattice element, 2nd 9th lattice element and 10th adjacent to the direction perpendicular to the Γ-K crystal orientation direction at the center of the row of the 8th lattice element An optical confinement section composed of a second row of lattice elements,
The fourth lattice element and the fifth lattice element are shifted by a first shift amount symmetrically outward from the center of the first row,
The third lattice element and the sixth lattice element are shifted by a second shift amount symmetrically outward from the center of the first row,
The second lattice element and the seventh lattice element are shifted from the center of the first row symmetrically outward by a third shift amount,
The first lattice element and the eighth lattice element are shifted by a fourth shift amount symmetrically outward from the center of the first row,
The ninth and tenth lattice elements are shifted inwardly by a fifth shift amount symmetrically from the center of the second row;
The first shift amount, the second shift amount, the third shift amount, the fourth shift amount, and the fifth shift amount are within a range in which adjacent lattice elements do not contact with each other by the shift,
The first shift amount, the second shift amount, the third shift amount, the fourth shift amount, and the fifth shift amount are determined by the same temporary combination function,
The second shift amount is an amount obtained by multiplying the first shift amount by a first coefficient,
The third shift amount is an amount obtained by multiplying the first shift amount by a second coefficient,
The fourth shift amount is an amount obtained by multiplying the first shift amount by a third coefficient,
The first coefficient is a number that becomes 0.9 when rounded off;
The second coefficient is a number that when rounded to 0.8.
The third coefficient is a number in the range of 0.5 to 0.65.
A first step of determining the first shift amount at which the Q value is maximized by setting the fifth shift amount to 0;
Based on the first shift amount determined in the first step, the second shift amount, the third shift amount, and the fourth shift amount are calculated using the first coefficient, the second coefficient, and the third coefficient. A second step of determining;
Based on the first shift amount determined in the first step, the second shift amount, the third shift amount, and the fourth shift amount determined in the second step, the first lattice element, the second shift amount, When the lattice element, the third lattice element, the fourth lattice element, the fifth lattice element, the sixth lattice element, the seventh lattice element, and the eighth lattice element are arranged, the Q value is maximized. And a third step of determining the fifth shift amount. A method of designing a photonic crystal resonator, comprising:
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