陳敏 萬婷 王征 羅朝明 劉靖

(湖南理工學(xué)院信息與通信工程學(xué)院,復(fù)雜工業(yè)物流系統(tǒng)智能控制與優(yōu)化湖南省重點(diǎn)實(shí)驗(yàn)室,岳陽 414006)(2016年4月22日收到;2016年10月16日收到修改稿)

寬絕對(duì)禁帶的一維磁性光子晶體結(jié)構(gòu)?

陳敏 萬婷 王征 羅朝明?劉靖

(湖南理工學(xué)院信息與通信工程學(xué)院,復(fù)雜工業(yè)物流系統(tǒng)智能控制與優(yōu)化湖南省重點(diǎn)實(shí)驗(yàn)室,岳陽 414006)(2016年4月22日收到;2016年10月16日收到修改稿)

提出了一種具有寬絕對(duì)禁帶的一維磁性光子晶體結(jié)構(gòu),該結(jié)構(gòu)由相同的折射率和物理厚度以及不同的波阻抗的兩種磁性材料交替組合而成.通過傳輸矩陣法分析可得,相比于非磁性光子晶體,該光子晶體的禁帶對(duì)入射角和偏振都不敏感,從而具有更寬的絕對(duì)禁帶.合適地調(diào)節(jié)兩種磁性材料的參數(shù),增加兩者波阻抗的差值,該光子晶體的絕對(duì)禁帶寬度也相應(yīng)地增加;調(diào)節(jié)兩種磁性材料的物理厚度,其絕對(duì)禁帶中心也會(huì)隨之調(diào)整;最后,將兩個(gè)滿足上述條件的一維磁性光子晶體組成異質(zhì)結(jié)構(gòu),其第一禁帶寬度與禁帶中心之間的比值可達(dá)到1.41以上.

絕對(duì)禁帶,光子晶體,磁性材料

1引 言

近年來,光子晶體的許多優(yōu)良特性和潛在應(yīng)用不斷被發(fā)現(xiàn),其研究已經(jīng)引起越來越多的研究者的關(guān)注[1,2].光子晶體又稱“光學(xué)半導(dǎo)體”,它具有的最為顯著的性質(zhì)是光子禁帶[3,4].光子晶體能夠阻止頻率落在光子禁帶中的電磁波的傳播,但是通常情況下光子禁帶對(duì)入射角和偏振態(tài)都較為敏感.為了實(shí)現(xiàn)某一頻帶的電磁波在任意偏振和角度入射下都能被阻止,可以通過合理設(shè)計(jì)光子晶體結(jié)構(gòu)使得其具有絕對(duì)禁帶[5?8].這種絕對(duì)禁帶在全方位高反射鏡[9,10]、低損耗波導(dǎo)[11]、光開關(guān)[12]和空間濾波器等[13?15]方面都具有許多潛在的應(yīng)用.

非磁性光子晶體的絕對(duì)禁帶通常是比較窄的,有必要對(duì)其禁帶進(jìn)行擴(kuò)展.目前擴(kuò)展光子晶體禁帶的方法主要有三種,第一種是通過調(diào)整光子晶體的材料和旋轉(zhuǎn)對(duì)稱性來擴(kuò)展禁帶[8,16];第二種是將幾個(gè)光子晶體組成異質(zhì)結(jié)構(gòu)來擴(kuò)展禁帶[17?20].但是前面兩種方法對(duì)非磁性光子晶體的禁帶擴(kuò)展都是在某一種偏振下進(jìn)行的,同時(shí)該禁帶還對(duì)入射角較為敏感,從而絕對(duì)禁帶擴(kuò)展比較小.隨著超常介質(zhì)材料的出現(xiàn),人們提出了第三種擴(kuò)展禁帶的方法,通過將超常介質(zhì)材料引入到光子晶體中來擴(kuò)展禁帶[21?23].由于超常介質(zhì)材料具有對(duì)入射角和偏振的不敏感性,從而能夠擴(kuò)寬光子的絕對(duì)禁帶,但這種超常介質(zhì)材料在自然界中是不存在的且其制備相當(dāng)困難.為了解決這一不足,Ouyang等[24]將磁性材料引入光子晶體中,使得其絕對(duì)禁帶得到了一定的擴(kuò)展.

本文提出了一種寬絕對(duì)禁帶的一維磁性光子晶體結(jié)構(gòu),組成該結(jié)構(gòu)的兩種磁性材料具有相同的折射率和物理厚度以及不同的波阻抗.該結(jié)構(gòu)禁帶對(duì)光的入射角和偏振態(tài)都具有不敏感性.通過改變?cè)摻Y(jié)構(gòu)的波阻抗、材料的物理厚度以及構(gòu)成的異質(zhì)結(jié)構(gòu)來研究其對(duì)光子禁帶的影響.

2結(jié)構(gòu)模型與理論

圖1(a)是由兩種磁性材料構(gòu)成的一維磁性光子晶體結(jié)構(gòu)(AB)N示意圖,組成光子晶體的A,B兩種材料的折射率和磁導(dǎo)率分別為:nA,μA,nB,μB,對(duì)應(yīng)的物理厚度分別為dA,dB,N表示周期數(shù).設(shè)平面電磁波由空氣入射到光子晶體中,θ為其入射角.為了準(zhǔn)確地描述這種結(jié)構(gòu)的傳輸特性,需要確定任意波矢分量的反射和透射,該結(jié)構(gòu)可由下面的2×2矩陣來表示[25,26]:

這里的Tm?1,m為第m?1層到第m層的轉(zhuǎn)換矩陣,

其中rm?1,m和tm?1,m則分別表示從第m?1層到第m層的反射系數(shù)和透射系數(shù);Pm為第m層的傳輸矩陣

上式中的dm和kmz分別表示第m層介質(zhì)的厚度和沿z方向的波矢分量.在本文的討論中,rAB,tAB分別表示圖1(b)所示的介質(zhì)A到介質(zhì)B界面的反射系數(shù)和透射系數(shù);rBA,tBA則分別表示圖1(c)所示的介質(zhì)B到介質(zhì)A界面的反射系數(shù)和透射系數(shù).具體表示為[25,26]

上述公式中θA和θB分別是入射電磁波在介質(zhì)A和介質(zhì)B中的傳輸角,表示介質(zhì)A對(duì)應(yīng)的波阻抗,ηB則表示介質(zhì)B對(duì)應(yīng)的波阻抗.整個(gè)光子晶體的透射系數(shù)和反射系數(shù)可由下面的公式來描述[25]:

圖1 (網(wǎng)刊彩色)光子晶體結(jié)構(gòu)示意圖及兩個(gè)不同界面的傳輸圖Fig.1.(color online)Schematic of photonic crystal structure and two transmission diagrams of di ff erent interfaces.

其中字母p,s分別對(duì)應(yīng)平行和垂直偏振情形.整個(gè)介質(zhì)的反射率和透射率用下式來表示[25]:上式中kmz,θm和k0z,θ0分別表示出射介質(zhì)m和入射介質(zhì)中沿z方向的波矢分量和傳輸角.

根據(jù)(1)式,圖1(a)所給出的一維光子晶體結(jié)構(gòu)可用下面的矩陣來表示:

M=T0A(PATABPBTBA)N?1PATABPBTB0.(10)為了獲得絕對(duì)禁帶,必須考慮其偏振特性,只有當(dāng)禁帶對(duì)偏振不敏感時(shí)才能獲得更寬的絕對(duì)禁帶.由(1)—(9)式可知,物理厚度相同的各層對(duì)應(yīng)相同的傳輸矩陣Pm,因而Pm是與偏振無關(guān)的;而轉(zhuǎn)換矩陣Tm,m?1由于rm?1,m和tm?1,m的偏振敏感性而偏振相關(guān).在圖1結(jié)構(gòu)中,很明顯影響該結(jié)構(gòu)偏振特性的是介面A到界面B的反射系數(shù)rAB和透射系數(shù)tAB,以及界面B到界面A的反射系數(shù)rBA和透射系數(shù)tBA.結(jié)合(4)—(7)式分析可知,只有當(dāng)θA和θB相等時(shí)(即組成光子晶體結(jié)構(gòu)的兩種材料的折射率相等時(shí)),不同偏振對(duì)應(yīng)的透射系數(shù)和反射系數(shù)才相等,從而使得TE和TM是偏振不敏感的,以達(dá)到擴(kuò)展光子晶體絕對(duì)禁帶的目的.因此,我們可得出滿足下面條件的磁性光子晶體結(jié)構(gòu)能擴(kuò)展其絕對(duì)禁帶寬度,即組成光子晶體結(jié)構(gòu)的兩種材料具有相同的折射率和物理厚度以及不同的波阻抗.

3數(shù)值模擬與分析

首先,我們比較研究一維磁性和非磁性光子晶體兩種結(jié)構(gòu)的禁帶特性.為了研究的方便,我們參照文獻(xiàn)[24,27,28]忽略材料損耗來選取相關(guān)參數(shù),理論探索一維磁性光子晶體的禁帶特性.根據(jù)第二部分得出的擴(kuò)展禁帶的條件(相同的折射率和物理厚度以及不同的波阻抗),我們選取的兩種磁性材料A和B的結(jié)構(gòu)參數(shù)如下:折射率nA=nB=n=3.4;磁導(dǎo)率μA=2.5,μB=1.5;每層介質(zhì)的物理厚度為dA=dB=λ0/(4n),其中λ0為真空中禁帶的中心波長(zhǎng);周期數(shù)N=6.通過采用傳輸矩陣法[13?15],得到了在不同偏振情況下以不同角度入射的一維磁性光子晶體(AB)6的傳輸譜,如圖2(a)—(d)所示.從該圖中可以看出,在禁帶處TE和TM偏振的傳輸曲線幾乎重合,從而可認(rèn)為該禁帶是偏振不敏感的;而且禁帶出現(xiàn)的位置幾乎不隨角度的改變而移動(dòng),這些都有利于絕對(duì)禁帶的擴(kuò)展,該結(jié)構(gòu)絕對(duì)禁帶的歸一化頻率帶寬(禁帶寬度與禁帶中心之間的比值)達(dá)到了0.41.為了與之進(jìn)行對(duì)比,我們選取了具有相同波阻抗比值的兩種非磁性材料C和D(即ηA/ηB= ηC/ηD=5/3),其結(jié)構(gòu)參數(shù)為nC=1.5,nD=2.5,dC= λ0/(4nC),dD= λ0/(4nD),該非磁性光子晶體 (CD)6的傳輸譜如圖2(a′)—(d′)所示.從該圖中可以看出,非磁性光子晶體的TE和TM偏振的傳輸曲線隨著入射角的增加差別越來越大,從而該禁帶是偏振相關(guān)的;并且隨著角度的增加TE偏振的禁帶左右邊緣都向右移動(dòng),但TM偏振的禁帶左邊緣向右移動(dòng)而右邊緣先向右移后又向左偏移;其絕對(duì)禁帶的歸一化頻率帶寬只有0.14,相比于前面的磁性光子晶體而言,其絕對(duì)禁帶寬度明顯窄得多.

圖2 (網(wǎng)刊彩色)一維磁性光子晶體和非磁性光子晶體的傳輸譜Fig.2.(color online)Transmission spectrum of one dimensional magnetic photonic crystals and non-magnetic photonic crystals.

需要指出的是,通常情況下的磁性材料都是有損耗的,下面研究損耗對(duì)本文中磁性光子晶體禁帶特性的影響.眾所周知,損耗參數(shù)可用折射率的虛部來描述[29,30],在這里將磁性材料A和B的折射率調(diào)整為:nA=nB=3.4+0.1i,其他參數(shù)與圖2(a)—(d)保持一致,相應(yīng)的含損耗磁性材料的一維光子晶體(AB)6的傳輸譜如圖3(a)—(d)所示.由圖中可看出其透過率相對(duì)于圖2(a)—(d)中不考慮損耗時(shí)的情形略有降低,但絕對(duì)禁帶寬度變化很小,幾乎不變.因此在本文的討論中,為了研究的方便,都忽略了磁性材料的損耗參數(shù)對(duì)禁帶特性的影響,即假定磁性材料是沒有損耗的.

根據(jù)一維磁性光子晶體絕對(duì)禁帶擴(kuò)展的條件,將進(jìn)一步研究?jī)煞N材料的波阻抗對(duì)其絕對(duì)禁帶寬度的影響.假定折射率和物理厚度都與前面的分析一致,即nA=nB=n=3.4,dA=dB=λ0/(4n).在保持兩種材料介電常數(shù)不變的情況下調(diào)節(jié)磁導(dǎo)率使得波阻抗之間的差值依次增大,圖4(a)—(c)分別是兩種磁性材料比值為ηA/ηB=2,4,6的光子晶體結(jié)構(gòu)(AB)6的禁帶.從圖中可以得出,ηA/ηB=2,4,6對(duì)應(yīng)的絕對(duì)禁帶的歸一化頻率范圍分別為0.78—1.25,0.6—1.44,0.50—1.53,其歸一化的頻率帶寬分別為0.47,0.84,1.03.由此可以總結(jié)出,隨著兩種材料波阻抗差值的增加,該光子晶體結(jié)構(gòu)的絕對(duì)禁帶的頻率帶寬得到了顯著的擴(kuò)寬.

然后,將研究探討每層材料的物理厚度對(duì)該光子晶體結(jié)構(gòu)傳輸特性的影響.我們選取了圖4(b)中除每層物理厚度外相同的其他結(jié)構(gòu)參數(shù),即折射率nA=nB=n=3.4,波阻抗比值為ηA/ηB=4,磁導(dǎo)率μA=6.0,μB=1.5;同時(shí)將每層物理厚度分別調(diào)整為dA=dB=λ0/(8n)和dA=dB=λ0/(16n),其對(duì)應(yīng)的不同角度下的傳輸譜如圖5(a)—(d) 和圖5(a′)—(d′)所示. 對(duì)比兩者可以發(fā)現(xiàn),當(dāng)材料的物理厚度變小時(shí),禁帶中心依次向右移動(dòng),其歸一化頻率帶寬分別為0.78和0.80.由此我們可以總結(jié)得出,隨著材料每層的物理厚度d的縮小,絕對(duì)禁帶的禁帶中心依次右移,而對(duì)禁帶的歸一化頻率帶寬的影響很小.

圖3 (網(wǎng)刊彩色)含損耗磁性材料一維光子晶體的傳輸譜Fig.3.(color online)Transmission spectrum of one dimensional magnetic photonic crystals containing the lossy magnetic materials.

圖4 (網(wǎng)刊彩色)不同波阻抗比的一維磁性光子晶體禁帶結(jié)構(gòu)Fig.4.(color online)Bandgap structures of one-dimensional magnetic photonic crystals with di ff erent wave impedances.

圖5 (網(wǎng)刊彩色)不同厚度的一維磁性光子晶體傳輸譜Fig.5.(color online)Transmission spectra of one-dimensional magnetic photonic crystals with di ff erent thicknesses.

圖6 (網(wǎng)刊彩色)三種不同結(jié)構(gòu)的光子晶體的禁帶結(jié)構(gòu)Fig.6.(color online)Bandgap structures of three kinds of photonic crystals with di ff erent structures.

最后研究利用該一維磁性光子晶體組成的異質(zhì)結(jié)構(gòu)來進(jìn)一步擴(kuò)展其絕對(duì)禁帶.異質(zhì)結(jié)構(gòu)的絕對(duì)禁帶的擴(kuò)展應(yīng)滿足疊加原理,即要求組合成異質(zhì)結(jié)構(gòu)的兩個(gè)光子晶體結(jié)構(gòu)中的一個(gè)結(jié)構(gòu)所產(chǎn)生的導(dǎo)帶被包含在另一個(gè)結(jié)構(gòu)的絕對(duì)禁帶中.由此,我們選擇了圖5中的兩個(gè)磁性光子晶體,并重畫了它們的禁帶結(jié)構(gòu)圖,即每層材料的物理厚度為dA=dB= λ0/(8n)和dA′=dB′= λ0/(16n)時(shí)的光子晶體的禁帶結(jié)構(gòu)圖,具體如圖6(a)—(b)所示.從圖6(a)中可以看出,第一個(gè)光子晶體結(jié)構(gòu)(AB)6的第一和第二絕對(duì)禁帶的歸一化頻率范圍依次為 1.18—2.85和5.37—6.85;圖6(b)為第二個(gè)光子晶體結(jié)構(gòu)(A′B′)6的禁帶結(jié)構(gòu),其絕對(duì)禁帶的歸一化頻率范圍為2.37—5.68.很明顯,第一個(gè)光子晶體的第一和第二絕對(duì)禁帶之間的導(dǎo)帶被包含在第二個(gè)光子晶體的絕對(duì)禁帶中,從而有利于禁帶的擴(kuò)展.圖6(c)為這兩個(gè)光子晶體組成的異質(zhì)結(jié)構(gòu)(AB)6(A′B′)6的禁帶結(jié)構(gòu),其絕對(duì)禁帶的歸一化頻率范圍為1.18—6.85,對(duì)應(yīng)的歸一化頻率帶寬達(dá)1.41.結(jié)合圖6(a)—(c)可以總結(jié)出,滿足上述疊加條件的兩個(gè)光子晶體結(jié)構(gòu)所構(gòu)成的異質(zhì)結(jié)構(gòu)的絕對(duì)禁帶就是這兩個(gè)光子晶體絕對(duì)禁帶的疊加,從而實(shí)現(xiàn)了絕對(duì)禁帶的擴(kuò)展.

4結(jié) 論

本文提出了一種具有寬絕對(duì)禁帶的一維磁性光子晶體結(jié)構(gòu),該結(jié)構(gòu)由折射率相同和物理厚度相同但波阻抗不同的兩種磁性材料分別交替組合而成.首先,我們對(duì)比分析了磁性和非磁性一維光子晶體的禁帶特性,進(jìn)一步論證了上述絕對(duì)禁帶擴(kuò)展的條件.然后研究了波阻抗的改變對(duì)該絕對(duì)禁帶擴(kuò)展的影響,結(jié)果發(fā)現(xiàn):隨著兩種材料波阻抗差值的增大,絕對(duì)禁帶得到了很明顯的擴(kuò)展.同時(shí)也得出了在保持所有波阻抗不變的情況下,通過磁性材料的物理厚度的調(diào)整,其禁帶中心隨之改變,但絕對(duì)禁帶寬度幾乎不變.最后我們用兩個(gè)一維磁性光子晶體構(gòu)成異質(zhì)結(jié)構(gòu),研究發(fā)現(xiàn)該異質(zhì)結(jié)構(gòu)的絕對(duì)禁帶寬度是這兩個(gè)光子晶體絕對(duì)禁帶的疊加.這種磁性光子晶體具有寬的絕對(duì)禁帶,在集成光學(xué)、光纖通信以及激光系統(tǒng)等領(lǐng)域有著潛在的應(yīng)用,比如可實(shí)現(xiàn)具有偏振無關(guān)、全方向等良好性能的反射鏡、光開關(guān)、光學(xué)濾波器等.

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PACS:42.70.Qs,42.55.Tv,75.50.–yDOI:10.7498/aps.66.014204

*Project supported by the National Natural Science Foundation of China(Grant No.61205126),the Science and Technology Program of Hunan Province,China(Grant No.2016TP1021),and the Experimental Project of College Students in Hunan Province and Hunan Institute of Science and Technology,China.

?Corresponding author.E-mail:zhaomingluo@hnu.edu.cn

One-dimensional magnetic photonic crystal structures with wide absolute bandgaps?

Chen Min Wan Ting Wang Zheng Luo Zhao-Ming?Liu Jing

(Key Laboratory of Hunan Province on Intelligent Control and Optimization of Complex Industrial Logistics System,College of Information and Telecommunications Engineering,Hunan Institute of Science and Technology,Yueyang 414006,China)(Received 22 April 2016;revised manuscript received 16 October 2016)

The photonic absolute bandgaps have many potential applications in speci fi c fi elds,and some methods to enlarge the absolute bandgaps,such as adjusting the material and the rotational symmetry,constituting a heterostructure have been explored.Recently,with the occurring of metamaterial,the photonic crystal based on metamaterial has also realized the wide absolute bandgaps.However,the metamaterial is an arti fi cially structured material of which the construction is more complicated.In this paper,one-dimensional magnetic photonic crystal structure with wide absolute bandgaps is proposed,which is composed of two kinds of magnetic materials with the same refractive index and physical thickness but di ff erent wave impedances.First of all,the transmission properties of one-dimensional magnetic and non-magnetic photonic crystals with the same wave impedance ratio are studied by using transfer matrix method.It is shown that the normalized frequency bandwidth of magnetic photonic crystal,i.e.the ratio of the band of bandgap to its center,is 0.41,while the normalized frequency bandwidth of the non-magnetic photonic crystal is 0.14.From the results,we can conclude that the absolute bandgap of the above magnetic photonic crystal is wider than that of non-magnetic photonic crystal because the former bandgap is not sensitive to the incident angle nor polarization.Secondly,we adjust the wave impedance ratios of the two kinds of magnetic materials and make them respectively reach 2,4and 6,with the refractive index and the physical thickness kept unchanged.By analyzing their transmission properties,it is found that the normalized frequency bandwidths of the absolute bandgaps are respectively 0.47,0.84and 1.03,and the greater the di ff erence between the two wave impedances,the wider the normalized frequency bandwidth is.Thirdly,we investigate the in fl uence of the per-layer physical thickness of the magnetic material on the bandgap,with the other parameters remaining unchanged.It is shown that the center of the absolute bandgap shifts toward high frequency with the decrease of the per-layer physical thickness.Finally,a kind of heterostructure is constructed by the above two one-dimensional magnetic photonic crystals.The normalized frequency ranges of the fi rst and the second absolute bandgap of one magnetic photonic crystal structure are respectively 1.18–2.85and 5.37–6.85.The normalized frequency range of the absolute bandgap of the other magnetic photonic crystal is 2.37–5.68.The normalized frequency range of the absolute bandgap of the heterostructure can be enlarged to 1.18–6.85and the corresponding normalized frequency bandwidth can reach more than 1.41.The wide absolute bandgaps can be applied to integrated optics,optical fi ber communication and high-power laser systems,according to which we may design the polarization-independent and omnidirectional devices such as re fl ectors,optical switchers and optical fi lters.

absolute bandgap,photonic crystals,magnetic material

10.7498/aps.66.014204

?國家自然科學(xué)基金(批準(zhǔn)號(hào):61205126)、湖南省科技計(jì)劃項(xiàng)目(批準(zhǔn)號(hào):2016TP1021)和湖南省和湖南理工學(xué)院大學(xué)生實(shí)驗(yàn)項(xiàng)目(批準(zhǔn)號(hào):湘教通[2016]283號(hào),校[2016]21號(hào))資助的課題.

?通信作者.E-mail:zhaomingluo@hnu.edu.cn