基于量子絕熱捷徑技術(shù)的光波導(dǎo)分束器設(shè)計(jì)

浦珺慧,施解龍,吳仁華,陳璽

(上海大學(xué)理學(xué)院,上海　200444)

基于量子絕熱捷徑技術(shù)的光波導(dǎo)分束器設(shè)計(jì)

浦珺慧,施解龍,吳仁華,陳璽

(上海大學(xué)理學(xué)院,上海200444)

用于加快量子絕熱“慢”過程的量子絕熱捷徑技術(shù),已廣泛應(yīng)用于原子、分子和光物理.基于耦合波導(dǎo)的量子光學(xué)類比,利用量子絕熱捷徑技術(shù)設(shè)計(jì)光學(xué)波導(dǎo)的耦合系數(shù)與傳播常數(shù),實(shí)現(xiàn)快速的光波導(dǎo)分束器件.通過數(shù)值模擬,并與共振耦合和絕熱耦合波導(dǎo)進(jìn)行比較.結(jié)果表明,量子絕熱捷徑技術(shù)所設(shè)計(jì)的光學(xué)波導(dǎo)分束器具有長(zhǎng)度短、輸出穩(wěn)定性高的優(yōu)勢(shì).

集成光學(xué)；耦合波導(dǎo)；量子絕熱捷徑；光分束器

隨著集成光學(xué)的發(fā)展,耦合波導(dǎo)在光通信中有著重要和廣泛的應(yīng)用,例如設(shè)計(jì)光定向耦合器、光開關(guān)、光調(diào)制器和光分束器等.特別地,光分束器作為重要的光通信器件,因其能將光能量、偏振等區(qū)分后分配到多個(gè)波導(dǎo)中從而實(shí)現(xiàn)一路到多路或多路到一路的光傳輸,而受到了極大的關(guān)注[1-2].理想的光分束器應(yīng)具有低損耗、體積小、高帶寬等優(yōu)點(diǎn).常見的能量分配型光分束器的設(shè)計(jì)結(jié)構(gòu)有光纖拉錐型、Y分叉型和MMI型[3-5].

目前,共振耦合是實(shí)現(xiàn)波導(dǎo)耦合的一種簡(jiǎn)易的方式.采用該方式,波導(dǎo)間距不變,能量輸出呈周期性變化,只有當(dāng)波導(dǎo)長(zhǎng)度滿足特定值時(shí),才能獲取完美的能量輸出.共振耦合波導(dǎo)對(duì)于波導(dǎo)參數(shù)和入射光波長(zhǎng)等的變化較為敏感.于是,人們提出了絕熱耦合波導(dǎo)耦合器.當(dāng)滿足絕熱條件時(shí),波導(dǎo)中的能量輸出穩(wěn)定,且對(duì)波導(dǎo)參數(shù)等控制要求不高,但是該耦合器的缺點(diǎn)是需要較長(zhǎng)的波導(dǎo)距離來滿足絕熱條件.因此,人們通過對(duì)不同耦合模方程進(jìn)行最優(yōu)化設(shè)計(jì),試圖尋找到距離短、輸出能量穩(wěn)定的器件[6].但是最優(yōu)化設(shè)計(jì)的數(shù)值計(jì)算過程往往需要迭代,不但費(fèi)時(shí)而且不能得到解析表達(dá)式.

近年來,上海大學(xué)陳璽等[7-8]提出了量子絕熱捷徑技術(shù),該技術(shù)通過加快絕熱“慢”過程,實(shí)現(xiàn)了量子態(tài)的快速操控或制備.由于絕熱過程普遍存在,量子絕熱捷徑技術(shù)被推廣至導(dǎo)波光學(xué)和非線性光學(xué)并受到了廣泛關(guān)注[9].基于量子光學(xué)與波導(dǎo)光學(xué)的類比[10-11],絕熱捷徑技術(shù)已被用于設(shè)計(jì)快速且穩(wěn)定的波導(dǎo)定向耦合器[12-13]、模式轉(zhuǎn)換器等[14-15].本工作主要利用量子絕熱捷徑技術(shù),特別是Berry[16]提出的量子無摩擦動(dòng)力學(xué)法,來設(shè)計(jì)波導(dǎo)的寬度和耦合系數(shù),從而實(shí)現(xiàn)三波導(dǎo)中快速且穩(wěn)定的1×2光分束器.

1　理論模型

首先考慮一個(gè)量子三能級(jí)體系[17],具體模型如圖1所示.

圖1　三能級(jí)體系的光學(xué)波導(dǎo)類比Fig.1　Optical waveguides analogy of three-level systems

若令變量?=1,則該體系的哈密頓量可以寫為

由式(1)～(4)可知,當(dāng)滿足絕熱條件|˙?|?|χ|時(shí),H0緩慢變化,則系統(tǒng)沿著瞬時(shí)本征態(tài)演化至末態(tài)輸出,即絕熱演變.然而,當(dāng)絕熱條件不滿足時(shí),系統(tǒng)則發(fā)生躍遷,轉(zhuǎn)換效率降低.為了加快系統(tǒng)的絕熱過程,本工作采用Berry[16]提出的量子無摩擦動(dòng)力學(xué)法,構(gòu)造哈密頓量H=H0+Hcd,其中Hcd的表達(dá)式為

新的哈密頓量H的薛定諤方程的解恰恰是原哈密頓量H0的絕熱近似解,但是Hcd在物理上難以實(shí)現(xiàn)[18].為此,引入變換矩陣

并對(duì)哈密頓量H作幺正變換,

同時(shí),設(shè)定

以滿足Hnew(0)=0,Hnew(zf)=0,使得波導(dǎo)的輸出和輸入端處波導(dǎo)之間的耦合效應(yīng)近乎為0.另外,進(jìn)一步設(shè)定

圖2　參數(shù)χ(z),?(z)隨距離z的變化Fig.2　Parameters χ(z),?(z)versus z

將χ(z)與?(z)代入式(8)～(9),得到光波導(dǎo)的耦合系數(shù)~?和傳播常數(shù)差~?,如圖3所示.由于選取的波導(dǎo)長(zhǎng)λ=1.55μm較小,絕熱條件不滿足,因此運(yùn)用該絕熱耦合型波導(dǎo)無法實(shí)現(xiàn)完美的1×2光分束.

圖3　3種光分束器系統(tǒng)的耦合系數(shù)和傳播常數(shù)差Fig.3　Coupling efficiency and mismatch of propagation constant for three different waveguide systems

2　數(shù)值模擬與比較

為了驗(yàn)證上述理論結(jié)果,利用光束傳播法[19]進(jìn)行數(shù)值模擬.假設(shè)二維波導(dǎo)由折射率nc=2.3的材料構(gòu)成,包層為空氣(折射率ncl=1).入射光中心波長(zhǎng)λ=1.55μm,TE偏振.要設(shè)計(jì)上述二維波導(dǎo),就需要對(duì)3根波導(dǎo)的寬度以及相互間的距離進(jìn)行設(shè)計(jì).在近似對(duì)稱波導(dǎo)中耦合系數(shù)?與波導(dǎo)間距滿足如下指數(shù)關(guān)系[20]：

式中,?0=0.661 907,γ=7.596 92,D0=0.5μm.以中間波導(dǎo)的中心為基準(zhǔn)線,將式(9)代入式(14),計(jì)算得到D(z),即左、右波導(dǎo)的中心到中間波導(dǎo)中心的距離.另外,利用傳播常數(shù)與波導(dǎo)寬度之間的近似關(guān)系[21-22]：

通過調(diào)整中間波導(dǎo)的寬度變化,可以計(jì)算得到中間波導(dǎo)與兩邊波導(dǎo)的傳播常數(shù)差~?.這里選取WM為波導(dǎo)實(shí)際寬度,左右波導(dǎo)的基準(zhǔn)寬度為W=0.5μm.根據(jù)式(7)所設(shè)計(jì)的耦合光分束器的光能量分布和功率演化如圖4所示.

圖4　1×2絕熱捷徑耦合光分束器的光傳輸模擬Fig.4　Beam propagation simulations for 1×2 shortcut coupler beam splitter

為了進(jìn)一步與原有的波導(dǎo)分束器作比較,這里給出共振耦合分束器的光能量分布和功率演化,如圖5所示.研究結(jié)果表明：在波導(dǎo)參數(shù)相同的情況下,絕熱捷徑耦合分束器的輸出能量更為穩(wěn)定,對(duì)波導(dǎo)折射率和入射光波長(zhǎng)等參數(shù)的變化不敏感.

圖6為絕熱捷徑耦合分束器和共振耦合分束器波導(dǎo)輸出功率損耗隨波長(zhǎng)的變化，可以看出利用量子絕熱捷徑所設(shè)計(jì)的分束器對(duì)波長(zhǎng)的敏感性小于共振分束器,其光譜頻帶較寬,有利于設(shè)計(jì)寬波帶的光分束器.

圖5　共振耦合分束器的光傳輸模擬Fig.5　Beam propagation simulations for resonant coupler beam splitter

圖6　波導(dǎo)輸出功率損耗隨波長(zhǎng)的變化Fig.6　Output power loss versus input wavelength

3　結(jié)束語(yǔ)

本工作研究了量子絕熱捷徑技術(shù)在耦合波導(dǎo)中的應(yīng)用,設(shè)計(jì)了1×2耦合光分束器.通過與共振分束器的比較,發(fā)現(xiàn)基于量子絕熱捷徑技術(shù)所設(shè)計(jì)的光分束器具有長(zhǎng)度短、輸出穩(wěn)定、對(duì)入射波長(zhǎng)變化不敏感等優(yōu)勢(shì).這不僅拓展了量子絕熱捷徑技術(shù)的運(yùn)用,而且對(duì)集成光波導(dǎo)的設(shè)計(jì)具有重要的指導(dǎo)意義.在接下來的工作中,還將針對(duì)入射波長(zhǎng)變化等微擾因素對(duì)波導(dǎo)進(jìn)行最優(yōu)化設(shè)計(jì),并進(jìn)一步推廣至1×3甚至1×N的光分束器.

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Beam splitter in optical waveguides designed by shortcuts to adiabaticity

PU Junhui,SHI Jielong,WU Renhua,CHEN Xi
(College of Sciences,Shanghai University,Shanghai 200444,China)

Shortcuts to adiabaticity have been proposed to accelerate“slow”adiabatic processes with the applications in atom,molecular and optical physics.Based on the quantum optical analogy of coupled waveguide,the coupling coefficient and propagating constant are designed by using shortcuts to adiabaticity,to realize optical beam splitters in short length.Compared with resonant and adiabatic couplers by numerical simulation,the designed waveguide is demonstrated its advantages on shorter length and high stability.

integrated optics；waveguide coupler；shortcuts to adiabaticity；beam splitter

O 436

A

1007-2861(2016)05-0545-07

10.3969/j.issn.1007-2861.2015.02.007

2015-03-24

國(guó)家自然科學(xué)基金資助項(xiàng)目(11474193,61176118)；上海市浦江人才計(jì)劃資助項(xiàng)目(13PJ1403000)；教育部博士點(diǎn)基金資助項(xiàng)目(2013310811003)；上海市高校特聘教授“東方學(xué)者”資助項(xiàng)目

施解龍(1960—),男,副教授,博士,研究方向?yàn)楣獠▽?dǎo)理論.E-mail:sjlong@staff.shu.edu.cn