平板型光子晶體

1、南京理工高校周興平譯,如有不明之處敬請查看原文（附于翻譯之后）高 Q 值二維光子晶體微腔光子腔具有約束光線的特性,這個特性可以應(yīng)用到物理和工程的很多領(lǐng)域中,包括相關(guān)電 -光相互作用 1 ,超小濾光片 2,3 ,低閾值激光器 4 ,光子芯片 5 ,非線性光學(xué) 6 和處理量子信息 7 ；這些應(yīng)用的關(guān)鍵是實現(xiàn)腔的品質(zhì)因數(shù) Q,更小的模式體積；比例 Q/V 打算了各種腔的相互作用強(qiáng)度,和一個超小型腔使能夠大范疇互相作用和更寬波長范疇的單模運算；可是一個高 Q 光波長的腔范疇是很難制作,因為輻射缺失與腔的體積成反比；除了一些最近的理論爭論,在制造高值微腔方面沒有權(quán)威的理論和試驗 8-10 ；在這里我們使

2、用基于硅的二維光子晶體平板微腔,Q=45,000 和 V=；Q/V 值是以前爭論的 10-100 倍；這項爭論使我們意識到光線可以被更猛烈的約束4,11-14 ；集成與其他光子器件是特別簡潔的,已經(jīng)可以證明的光譜范疇了；腔的值打算于相對于原能量的每個周期的能量損耗；由于腔的材料對光沒有 吸取,Q 值打算于腔內(nèi)外界面之間的能量缺失；全部內(nèi)部反射（TIR）和/或 布拉格 散射常用于對光的約束；對于一個體積大于光波長的腔,已經(jīng)可以得到一個很高的值了 14,15 ；在這種情形下,被約束在腔中的光線符合光學(xué)理論,每一束在交界面處被反射的光都符合全內(nèi)反射或布拉格散射；腔越小光線對光學(xué)理論的偏移越嚴(yán) 重,因

3、此 Q 值會變??；被約束在微腔中的光線是由特別多的平面波組成的,由于光的局部化這些平面波是由很多波矢k 組成；設(shè)計出符合全內(nèi)反射或者布拉格散射的平面波特別困難,高值光子晶體諧振腔的產(chǎn)生很好的解決這個問題；解決這個問題的一個很好的方法就是在全部方向上運用布拉格散射效應(yīng)；二維 或者三維的折射率周期性變化的結(jié)構(gòu)可以產(chǎn)生這樣的效應(yīng),變化基于光波波長的數(shù)量級；這些被稱為光子晶體,類似于固體晶體5,16 ；對于一個三維光子晶體,布拉格散射可以約束全部方向特定頻率范疇的光線,稱為光子帶隙；一個小擾動或缺陷 引入三維光子晶體就會形成光子晶體微諧振腔,并具有極高的 Q/值；可是,三維 光子晶體；仍不能對光進(jìn)行很

4、好的約束；將二維光子晶體圍繞在腔的四周是一個可行的方法；一個二維晶體平板,如圖一所示,其厚度比擬與光波波長,在水平和垂直方向?qū)舛加刑貏e強(qiáng)的約束,這種 結(jié)構(gòu)式特別有期望的；光子帶隙的作用是把光的方向限制在平面內(nèi),垂直方向通過 平板與空氣的全反射來對光進(jìn)行約束；明顯,在垂直方向滿意全內(nèi)反射是制造高/V 腔的關(guān)鍵；為了進(jìn)一步爭論對二維光子晶體平板垂直方向的光線約束,我們第一考慮一簡 單模式（圖 2）,腔有厚度為,長度為的介質(zhì)材料構(gòu)成,腔的兩側(cè)是全反射鏡,約束軸方向上的光；簡潔起見,假設(shè)結(jié)構(gòu)在y 方向是勻稱的；在軸方向,光線被約束于空氣層的全反射,如上所爭論的一樣；圖2b 表示的是腔內(nèi)波長為2.5

5、的電場的場分布, 是光在腔的諧振波長；在垂直方向（ z 方向）是由全內(nèi)反射對光進(jìn)約束的；將腔內(nèi)的電場通過傅里葉變換分解為一系列具有不同波矢的平面波,這樣就可以評估其約束強(qiáng)度,如參考文獻(xiàn) 10 中所述；（其中是在空氣的光波長）當(dāng)每個平面波波矢的正切分量在到 變化時,光波可以從腔中溢出到空氣層中,這是 由于守恒定律對于 | |（或 Snell s 定律的廣義）在腔和空氣界面之間產(chǎn)生作用,；這就減弱了垂直方向上的約束強(qiáng)度；留意到 | |在空氣層中 x-z 傳播方向可以取從 0 到 的值；而| |在腔取不同的值由光的位置打算,如前面所表達(dá)的一樣；當(dāng) | |在腔大于,該| |在界面出沒有守恒定律, 光在

6、腔內(nèi)被猛烈限制, 這導(dǎo)致垂直方向的猛烈限制；圖 2c 表示圖 2b 電場的 FT 光譜,其中的插圖為漏區(qū)域（| |小于）多數(shù)的重量里面存在漏區(qū)域,這說明大腔內(nèi)存在很大的輻射損耗；我們現(xiàn)在考慮缺失機(jī)理的更多細(xì)節(jié)；腔內(nèi)的電場空間分布可以表示為一個波長為 的正弦波,和一個基于腔結(jié)構(gòu)包絡(luò)函數(shù)F（x）的乘積；基波給出的一個三角形FT,其峰值為 k=,而包絡(luò)函數(shù)就形成了光譜；如圖 2b,包絡(luò)函數(shù)是 F（x）=1 （x=-L/2 到 L/2）,和 F（x）=0（x 為其他）,和相應(yīng)的 FT 光譜是一個正弦函數(shù),其寬度約為 2（圖 2c）；雖然從基波引起的頻譜峰值是在漏區(qū)域外,但是在包絡(luò)函數(shù)邊緣（ x=-L/

7、2,L/2 ）猛烈的變化形成了泄露區(qū)域的主要部分,并導(dǎo)致了大量的輻射缺失；略小的腔,邊緣效應(yīng)越嚴(yán)峻,其值越低；這就給出了一個抑制輻射損耗的重要的提示：在腔的邊緣的包絡(luò)函數(shù)的空間變化應(yīng)當(dāng)不是猛烈而是平緩,這樣的話傅里葉頻譜就不會進(jìn)入泄露區(qū)域；基于這種思想,我們使用高斯函數(shù)F（x）,如圖 2d 所示；運算所得的FT 頻譜如圖 2e所示；這里的情形發(fā)生了很大變化：,相比圖2c 泄露區(qū)域的部分很?。贿@說明可以在不轉(zhuǎn)變模式體積的情形下調(diào)整包絡(luò)函數(shù)可以大幅增加值；因此就設(shè)計出了利用二維光子晶體平板的高值微腔（圖 1b,c）；它的基本結(jié)構(gòu)是由三角形空氣柱硅晶體,其晶格常數(shù)為（.）；平板厚度和空氣柱半徑分別為

8、 .（ .）, .（ .）；我們第一去除腔中三個在同一排的空氣孔（圖1b）；這種結(jié)構(gòu),光就由于布拉格散射而被約束在平面內(nèi)；對于 z 方向,光是被空氣層限制；圖中所示為腔中的平板中心的電場；我們使用 3D 有限差分時域為運算方法；和圖中所爭論的模型不一樣,在垂直方向的約束只需考察和方向的傅里葉變換,這是由于光是被限制在腔的二維周期介質(zhì)中的；一樣的緣由, TIR 狀態(tài)（或| |守恒定律）需要分解為二維情形來爭論；考慮 2D 面內(nèi)的傳播, TIR 狀態(tài)被打破由于平面波的全部 | |在一個直徑為 的圓內(nèi)；圖 1：基于二維光子晶體平板的光子微腔a）腔的基本結(jié)構(gòu)原理圖,柱的三角晶格,晶格常數(shù)為 a=0.4

9、2,平板的厚度和空氣棒的半徑分別為 0.6a0.25 和0.29a0.12 b 腔為去除在同一排的三個空氣孔所形成；c通過移動邊上的兩個空氣柱來獲得超高的 Q/V 值；圖 2：對腔的損耗削減的分析a）厚度為長度為腔的簡化模型；為了對光進(jìn)行約束,腔的兩側(cè)是方向的全反射鏡,軸方向就是同過空氣的全內(nèi)反射來對 光進(jìn)行約束的； b,c）腔內(nèi)很短（ 2.5）長度的電場分布,與空間傅里葉變換；漏 區(qū)域顯示于藍(lán)色區(qū)； d,e）電場分布平緩的包絡(luò)函數(shù)（高斯函數(shù)）與其空間傅里葉 頻譜；圖 3b 表示圖 3a 相應(yīng)的 FT 光譜,泄露區(qū)域在灰色圓圈內(nèi),傅里葉頻譜有很大一部分在泄露區(qū)域中； 像之前已經(jīng)爭論過的一樣,

10、我們認(rèn)為這是由于腔邊緣的突變；在這我們試圖對光進(jìn)行更好的約束；方法就是在腔邊緣轉(zhuǎn)變布拉格散射；這樣散射 是由腔邊緣的空氣柱的散射的疊加所形成的；當(dāng)我們移動的一些棒靠近腔邊緣,布 拉格散射將要轉(zhuǎn)變；由于被移動過的空氣柱散射的光的相位發(fā)生了變化,結(jié)果產(chǎn)生 相位不匹配而減弱了磁場的布拉格散射；為了補(bǔ)償反射的減小,光更多進(jìn)入鏡面中 并被完全反射；這意味著腔邊緣上的電場分布被減弱,對于空氣柱適當(dāng)?shù)囊苿訒?得場分布接近由高斯函數(shù)所表示的抱負(fù)的約束；依據(jù)這種分析,我們移動腔邊緣的 兩個空氣柱（圖 1c）；圖 3c 和 d 分別表示二維傅里葉變換的電場分布,空氣柱與 原先相比移動了 .；與圖相比, 圖中在泄

11、露區(qū)域中的傅里葉變化更少；因此,可以通過這種方法使得/V 的值大幅增加；通過上面的分析,我們設(shè)計出不同位移的例子；諧振頻譜的測量使用可調(diào)諧連 續(xù)激光作為光源；腔是由線性缺陷波導(dǎo)所鼓勵的,波導(dǎo)是腔旁邊的去除一排空氣孔 所形成的（圖 4b）,可以觀看到從腔中所泄露出來的光的強(qiáng)度；結(jié)構(gòu)和試驗方法的細(xì)節(jié)在別的地方說明 17 ；腔的本征 Q 因素取決于去除腔和波導(dǎo)之間的相互作用后的輻射頻譜；有效地模式體積取決于電場空間分布 x . 4 ；從這結(jié)果發(fā)覺 V 值小 ,在（6-7）圖 4a,b 分別表示移動了不同距離空氣孔的腔的諧振頻譜以及相應(yīng)的電子顯微 鏡掃描圖；諧振峰值的帶寬隨著空氣孔的移動發(fā)生猛烈的變化

12、；當(dāng)空氣孔的移動為0.15a 時頻譜帶寬顯現(xiàn)最小值0.045nm,值 ,（考慮到與波導(dǎo)的相互作用；圖為以空氣孔移動為自變量的/V 的函數(shù)；在空氣孔移動距離到0.15a之前, /V 以 10 的速度增加；這樣就得到了6.4x或者 120,000/的 Q/V；這比之前報道的腔值高了一到兩個數(shù)量級,如環(huán)形微諧振腔、微磁盤和光子晶體諧振腔4,11-14 ；通過對腔邊緣空氣孔恰當(dāng)?shù)恼{(diào)整可以獲得更高的Q/V；插圖 4a 表示寬的波長范疇的光譜測量,這說明在1,500 到 1,600nm 的范疇不存在其他共振峰值；這結(jié)果說明單模適用于很寬范疇的波長,這對于不同的應(yīng)用很有幫忙；圖 3：高 Q/V 腔的結(jié)構(gòu)；

13、a） 圖 1b 顯示腔的基本模式的電場（Ey）分布的基本模式； b）a 的 FT 光譜；灰色圓內(nèi)部是相應(yīng)的泄露區(qū)域 c,d）分別是對應(yīng)圖 1c 腔的電場空間分布和 氣孔移動了 0.15a；2D FT 光譜；相對于圖 1b 中的原始位置,邊緣上的空圖 4：試驗結(jié)果； a,b 不同位移的腔的頻譜及其顯微鏡掃面圖；插圖 a 為腔的諧振頻譜（ 0.15a 位移）； c）以位移為自變量的所估算的函數(shù)；已經(jīng)的得到了最大值是 6.4x 或 120,000/ 的；我們描述了很重要的設(shè)計規(guī)章因這種設(shè)計可以對光進(jìn)行很好的約束來獲得高 Q值并且保持很小的模式體積；通過在2D 平板光子晶體引入腔的空氣棒兩個邊緣的位移

14、得到特別大的；我們認(rèn)為這概念能夠應(yīng)用到各種類型的光子微腔；這樣 高 Q 微腔能夠應(yīng)用到科學(xué)和工程的不同場交叉；包括納米激光,非線性光學(xué),納米生物材料,原子物理學(xué),與量子運算；目前的結(jié)果是同樣重要由于光子晶體的制造以集成電路為基礎(chǔ),例如已經(jīng)使用三維光子晶體實現(xiàn)了微小體積的微腔,其在垂直方向的泄露得到充分的抑制；參考文獻(xiàn)看原文；附件 2：外文原文 （復(fù)印件）High-Q photonic nanocavity in a two-dimensional photonic crystal Yoshihiro Akahane 1,2 , Takashi Asano 1 , Bong-Shik Song

15、1 & Susumu Noda Photonic cavities that strongly con.ne light are finding applications in many areas of 1physics and engineering, including coherent electronphoton interactions , ultra-small filters 2,3 ,low-threshold lasers 4 , photonic chips 5 , nonlinear optics 6and quantum 7information processing

16、 . Critical for these applicationsis the realization of a cavity with both high quality factor, Q, and small modal volume, V. The ratio Q/V determines the strength of the various cavity interactions, and an ultra-small cavity enables large-scale integration and single-mode operation for a broad rang

17、e of wavelengths. However, a high-Q cavity of optical wavelength size is dificult to fabricate, as radiation loss increases in inverse proportion to cavity size. With the exception of a few recent theoretical studies 8-10 , definitive theories and experiments for creating high-Q nanocavities have no

18、t been extensively investigated. Here we use a silicon-based two-dimensional photonic-crystal slab to fabricate a nanocavity with Q=45,000 and V= ; the value of Q/V is 10100 times larger than in previous studies 4,11-14 . Underlying this development is the realization that light should be confined g

19、ently in order to be confined strongly. Integration with other photonic elements is straightforward, and a large free spectral range of 100 nm has been demonstrated. The Q of a cavity is determined by the energy loss per cycle versus the energy stored. With no absorption by the cavity material, Q is

20、 determined by the reflection loss at the interface between the interior and exterior of the cavity. Total internal re.ection TIR and/or Bragg reflection are generally used for light confinement. For a cavity with a size much larger than the wavelength of light, a very high Q has already been achiev

21、ed 14,15 . In that case, the behaviour of light confined in a large cavity obeys ray optics theory, and each ray of light reflected at the interface can be designed to fulfil TIR or Bragg reflection conditions. For much smaller cavities, deviation from ray optics becomes serious, and Q is greatly re

22、duced. Light confined in a very small cavity consists of numerous plane wave components with wavevectors k of various magnitudes k and directions owing to the localization of light. As it is difficult to design all such plane wave components to obey TIR or Bragg reflection conditions, photonic nanoc

23、avities with very high Q factors have yet to be realized. One of the best approaches to resolving the problem is the extension of the Bragg reflection effect in multiple directions. Structures having a two- or three-dimensional 2D or 3D periodic change of refractive index on the scale of the light w

24、avelength are required for such extension. These are known as photonic crystals,from an analogy to solid crystals 5,16 . For a 3D photonic crystal,Bragg reflection conditions can be fulfilled for all the propagation directions of light in a certain frequency range, known as the photonic bandgap. A s

25、mall disorder or defect introduced into the 3D photonic crystal would become an ultimate photonic nano- cavity, with ultra-large Q/V. However, 3D photonic crystals with sufficiently strong optical confinement have yet to be created 5 . A cavity surrounded by a 2D photonic crystal is considered a fea

26、sible solution. A 2D photonic-crystal slab, as shown in Fig. 1a, with a thickness of the order of the light wavelength is very promising, owing to strong optical cofinement for both in-plane and vertical directions 2,3 . The photonic-bandgap effect is used for light cofinement in the in-plane direct

27、ion, and TIR, at the interface between the slab and the air clad, in the vertical direction. Apparently, fulfilment of the TIR condition in the vertical direction is crucial in designing high-Q/V cavities. To investigate vertical confinement in 2D photonic-crystal slabs,we first consider a simplfied

28、 model Fig. 2a, where the cavity consists of a dielectric material with thickness T and length L. Both sides of the cavity are closed by perfect mirrors, cofining light in the x direction. The structure is assumed to be uniform in the y direction for simplicity. Light is confined by TIR in the z dir

29、ection by the air clad, as discussed above. Figure 2b shows an example of the electric field pro.le inside a cavity with a very short length, 2.5 , where is the resonant wavelength of light in the cavity.The strength of the vertical z-direction cofinement by TIR can be investigated by decomposing th

30、e electric field inside a cavity into a set of plane wave components with various k-vectors by spatial Fourier transformation FT, which is a similar approach to that reported in ref. 10.When the tangential component of the k-vector | |,| | of each plane wave lies within the range 0 to where is the w

31、avelength of light in air, the wave can escape from the cavity to the air clad, because the conservation law for | | or Snell s law in the broad sense is satis fied at the interface between the cavity and air clad, which leads to weak vertical confinement. Note that | |in the air clad can take a val

32、ue from 0 to depending on the propagation direction in the x z plane, while | |in the cavity takes a variety of values owing to the localization of light, as explained before. When | | in the cavity is larger than , the | | conservation law does not hold at the interface, and light becomes strongly

33、confined inside the cavity, which leads to strong vertical confinement. Figure 2c shows the spatial FT spectra of the electric field of Fig. 2b, where the leaky region |is smaller than is indicated. Large components exist inside the leaky region, which indicates that large radiation loss occurs in t

34、he cavity. We now consider the loss mechanism in more detail. The electric field pro .le inside the cavity can be expressed as a product of a fundamental sinusoidal wave with wavelength , and an envelope function Fx that is determined by the cavity structure. The fundamental wave gives a delta funct

35、ional FT spectrum with peaks at k= , while the envelope function modifies the spectrum. In the case of Fig. 2b, the envelope function is Fx= 1 for x=-L/2 to L/2 and Fx= 0 for all other x, and the corresponding FT spectrum is a sinc function with a width of about k=Fig. 2c. Although the peak of the s

36、pectrum originating from the fundamental wave is outside the leaky region, an abrupt change in the envelope function at the edges x = -L/2, L/2 generates large components inside the leaky region, leading to large radiation loss. The smaller the cavity, the more serious the edge effect, drastically d

37、ecreasing the Q factor. This gives an important hint for suppressing radiation loss: the spatial variation of the envelope function at the cavity edges should not be abrupt but gentle, so that the FT spectrum does not have components inside the leaky region. On the basis of this idea, we have used a

38、 gaussian function for Fx, as shown schematically in Fig. 2d; the calculated FT spectrumis shown in Fig. 2e. The situation has drastically changed: there are very small components inside the leaky region, when compared with Fig. 2c. This suggests that the Q factor can be increased significantly by t

39、ailoring the envelope function while keeping the mode volume small. A physical design of a high-Q photonic nanocavity has thus been carried out using a 2D photonic-crystal slab Fig. 1b and c. The base structure is composed of Si with a triangular lattice of air with lattice constant a=0.42 . The thi

40、ckness of the slab and the radii of the air rods are 0.6a 0.25 and 0.29a 0.12 , respectively. We made the initia structure of the cavity with three missing air rods in a line17 Fig. 1b.With this structure, light can be confined by Bragg reflection for the in-plane directions. For the z direction, li

41、ght is confined by the air clad. The electric field profile （）of the fundamental mode of the cavity at the centre plane of the slab is shown in Fig. 3a.We used 3D finite-difference time-domain methods for the calculation. Unlike the model discussed in Fig. 2, x- and y-directional 2D FT spectra are n

42、ecessary for the investigation of the vertical confinement, as light is confined two-dimensionally in the cavity. For the same reason, the TIR condition or | |conservation law should be expanded two-dimensionally. Considering in-plane 2D propagation, the TIR condition is broken for plane waves havin

43、g | |inside a circle of diameter Figure 1: Photonic nanocavities using a 2D photonic-crystal slab. a, Schematic of the base cavity structure having a triangular lattice of air rods with lattice constant a=0.42 . The thickness T of the slab and the radius R of the air rods are 0.6a0.25 and 0.29a 0.12

44、 , respectively. b, Starting cavity structure with three missing air rods I a line. c, Designed cavity structure created by displacing the air rods aboth edges to obtain an ultrahigh Q/V value. Figure 2: Analysis and reduction of cavity loss. a, Simplified model of a cavity consisting of a dielectri

45、c material with thickness T and length L. For confinement of light, both sides of the cavity are closed by perfect mirrors for the x direction, and by the air clad based on TIR for the z direction. b, c, The electric field pro a cavity with a very short （2.5）length, and the spatial FT spectra. The l

46、eaky region is indicated as a blue area. d, e, The electric field pro.le with a gentle envelope function gaussian curve and its spatial FT spectrum. a.u., arbitrary units. Figure 3b: Shows the FT spectra corresponding to Fig. 3a, where the leaky region is inside the grey circle. The FT spectrum cont

47、ains large components inside the leaky region. As discussed, we considerthat this is due to the abrupt change at the cavity edges. Here we try to make confinement gentler. The strategy to obtain gentler confinement is to change the condition for Bragg reflection at the cavity edge. Such reflection i

48、s determined by a summation of partial reflections at a series of rods near the cavity edge. When we move several rods near the cavity edge, the Bragg reflection condition should be modified. Because the phases of partial reflections at the moved rods are changed, the resultant phase-mismatch weaken

49、s the magnitude of Bragg reflection. To compensate for the reduction of the reflection, light is considered to penetrate more inside the mirror and be reflected perfectly. It means that the electric field pro the cavity edge becomes gentler. With the appropriate movement of rods, the pro considered

50、to be close to the ideal confinement expressed by the gaussian function as discussed above. Using this strategy, the air rods at both edges of the cavity are shifted Fig. 1c. Figure 3c and d show the electric field pro .le and 2D FT spectrum, respectively, where the shift of the air rods is 0.15a fr

51、om their original position. As in Fig. 3d, the FT spectrum contains much smaller components inside the leaky region compared with Fig. 3b.The mode volume itself is confirmed as almost unchanged. Therefore, a significant increase of Q/V is expected to be achieved by this method. Encouraged by the abo

52、ve analysis, we fabricated samples with various displacements. The resonant spectra were measured using a tunable c.w. laser as a light source. The cavities were excited through a line defect waveguide constructed by filling a row of air holes near the cavity Fig. 4b, and the intensity of the light

53、emitted from the cavities to free space was observed. Details of construction and experimental methods are given elsewhere 17 . The intrinsic Q factor of the cavity was determined fromits radiation spectra by removing the effect of coupling between the cavity and the waveguide. The effective modal v

54、olume was calculated from the electric field pro the cavity 4 ; from this it was found that V is small and constant at （6-7）x . Figures 4a and b show resonant spectra of cavities with various air-rod shifts and the corresponding scanning electron microscope SEM pictures, respectively. The width of t

55、he resonant peak changes drastically with shift of air rods. The spectral width becomes a minimum 0.045 nm for the sample with shift 0.15a, from which a Q factor of 45,000 is derived considering the coupling effect with the waveguide. In Fig. 4c, the Q/V values are plotted as a function of shift of

56、air rods. Q/V increases by a factor of 10 upon increasing the air-rod shift up to ,0.15a.A Q/V as large as 6.4x or 120,000/ has thus been obtained. This is one to two orders of magnitude higher than values for previously reported cavities such as toroid microcavities, microdisks and photonic-crystal

57、 cavities 4,11-14 . Further increases of Q/V should be possible by .ne-tuning the arrangement of air rods at the cavity edges to obtain the perfect gaussian curve for light confinement. The inset of Fig. 4a shows the spectrum measured for a wide wavelength range, indicating that no other resonant pe

58、ak exists in the range 1,500 to 1,600 nm. The result shows that single-mode operation is possible for a broad range of wavelengths, which is very useful for various applications. Figure 3: Physical design of high-Q/V cavity. a, The electric field profile （Ey）of the fundamental mode of the cavity sho

59、wn in Fig. 1b as the starting structure. b, The FT spectra of a. The region inside the grey circle corresponds to the leaky region. c, d, The electric field pro 2D FT spectrum, respectively, for the designed cavity shown in Fig. 1c. The displacement of the air rods at the edges is set at 0.15a from

60、the starting structure shown in Fig. 1b. Figure 4: Experimental results. a, b, Resonant spectra of cavities with various shifts of air rods and their SEM pictures, respectively. PC, photonic crystal. The inset in a shows the resonant spectrum of the cavity with 0.15a displacement measured over a wid