楊明惠，金　琪，劉勁松，王可嘉，楊振剛

(華中科技大學(xué) 武漢光電國(guó)家實(shí)驗(yàn)室，武漢430074)

飛秒激光結(jié)合啁啾太赫茲脈沖控制CO分子取向

楊明惠，金琪，劉勁松，王可嘉*，楊振剛

(華中科技大學(xué) 武漢光電國(guó)家實(shí)驗(yàn)室，武漢430074)

摘要：為了研究啁啾太赫茲脈沖誘導(dǎo)后的CO分子取向，采用剛性轉(zhuǎn)子近似求解含時(shí)薛定諤方程的方法，進(jìn)行了理論分析和數(shù)值仿真。為了在較低太赫茲場(chǎng)強(qiáng)時(shí)獲得較好的取向效果，采用了飛秒激光結(jié)合啁啾太赫茲脈沖的方案。結(jié)果表明，分子取向程度可增強(qiáng)81%。在有限溫度條件下，飛秒激光結(jié)合啁啾太赫茲脈沖誘導(dǎo)分子取向的效率隨溫度的升高而降低;相對(duì)于少周期太赫茲脈沖，啁啾太赫茲脈沖有誘導(dǎo)分子取向的優(yōu)越性。這一結(jié)果對(duì)提高CO分子取向程度是有意義的。

關(guān)鍵詞：激光物理；分子取向；剛性轉(zhuǎn)子模型；啁啾太赫茲脈沖

*通訊聯(lián)系人。E-mail: wkjtode@sina.com

引言

分子的各向異性和空間分布在許多物理及化學(xué)過(guò)程中起著重要作用，如高次諧波產(chǎn)生[1]、化學(xué)反應(yīng)動(dòng)力學(xué)[2]、光電離及解離[3-4]、超快分子成像[5-6]等。近年來(lái)，控制分子的準(zhǔn)直及取向引起了人們的廣泛關(guān)注。分子的準(zhǔn)直指的是分子軸沿著某一固定的坐標(biāo)軸，如激光場(chǎng)的偏振方向。而分子的取向指的是在準(zhǔn)直的基礎(chǔ)上控制分子指向固定的方向，這就意味著要打破分子的反演對(duì)稱性，因而分子取向的實(shí)現(xiàn)要比準(zhǔn)直更加困難。

作者以CO氣體為例，研究了利用啁啾太赫茲脈沖來(lái)控制大轉(zhuǎn)動(dòng)周期分子氣體的取向，通過(guò)對(duì)比少周期太赫茲脈沖，理論上證明了啁啾太赫茲脈沖的優(yōu)越性。為了能夠在啁啾太赫茲脈沖場(chǎng)強(qiáng)較低的情況下得到大轉(zhuǎn)動(dòng)周期的氣體分子較高的取向，采用了飛秒激光結(jié)合啁啾太赫茲脈沖的方法來(lái)控制分子取向，即先利用飛秒激光預(yù)激發(fā)氣體分子，在適當(dāng)?shù)难訒r(shí)后分子再與啁啾太赫茲脈沖相互作用，發(fā)現(xiàn)可增強(qiáng)CO分子取向程度大約81%。最后研究了溫度對(duì)飛秒激光結(jié)合啁啾太赫茲脈沖誘導(dǎo)分子取向的影響。該方案從理論上證明了飛秒激光結(jié)合啁啾太赫茲脈沖可增強(qiáng)CO氣體的分子取向程度，可望用于利用低場(chǎng)強(qiáng)的啁啾太赫茲脈沖控制大轉(zhuǎn)動(dòng)周期分子氣體的化學(xué)反應(yīng)及光電離、解離等過(guò)程。

1理論方法

由Born-Oppenheimer近似，氣體分子與線偏振場(chǎng)相互作用，分子轉(zhuǎn)動(dòng)波函數(shù)ψ(t)的時(shí)間演變可由含時(shí)薛定諤方程求出：

(1)

作者采用剛性轉(zhuǎn)子近似[21]，即只需考慮分子轉(zhuǎn)動(dòng)自由度，忽略其它自由度的影響。因而，線偏振的太赫茲脈沖及飛秒激光脈沖與分子相互作用的哈密頓量[19]可以寫為：

(2)

(3)

接下來(lái)，用自由空間中剛性轉(zhuǎn)子的本征函數(shù)球諧函數(shù)球諧函數(shù)YJ,M(θ,φ)(φ為球坐標(biāo)系下的方位角)展開(kāi)轉(zhuǎn)動(dòng)波函數(shù)[22]:

(4)

式中，Jmax設(shè)為一個(gè)較大值，大于脈沖可激發(fā)分子至的最高轉(zhuǎn)動(dòng)態(tài)，CJ,M(t)為布居系數(shù)，EJ表示J轉(zhuǎn)動(dòng)態(tài)的本征能量，J為轉(zhuǎn)動(dòng)態(tài)能級(jí)的角量子數(shù)，M表示角動(dòng)量在激光(或太赫茲)場(chǎng)偏振方向上的投影，對(duì)于線偏振的激光場(chǎng)，M保持不變，即ΔM=0。由于仿真環(huán)境溫度為0K，此時(shí)分子初態(tài)處于轉(zhuǎn)動(dòng)基態(tài)，即M恒等于0。

將(2)式、(3)式、(4)式帶入(1)式，可以得到一個(gè)關(guān)于CJ,M(t)的微分方程組。采用4階Runge-Kutta法可以求出CJ,M(t)隨時(shí)間的變化，進(jìn)而可以表示出分子的轉(zhuǎn)動(dòng)波函數(shù)ψ(t)。

分子取向程度可由cosθ的期望值表示：

(5)

式中,*表示共軛，′表示J和Jmax不同的取值。

2結(jié)果與討論

2.1　利用啁啾或少周期太赫茲脈沖誘導(dǎo)分子取向的比較

啁啾太赫茲脈沖電場(chǎng)表達(dá)式可寫為[20]：

(6)

式中，τ表示啁啾太赫茲場(chǎng)的半峰全寬(full width at half maximum,FWHM),啁啾太赫茲脈沖頻率隨時(shí)間變化，ω為中心時(shí)刻tp時(shí)的角頻率，χ為線性啁啾率，E0表示太赫茲脈沖峰值場(chǎng)強(qiáng)。

少周期太赫茲脈沖[17]由于電場(chǎng)不對(duì)稱性很強(qiáng)，因而其與分子作用后可以得到很大的取向值，作為參照，選取同樣參量下的少周期太赫茲脈沖與啁啾太赫茲脈沖進(jìn)行對(duì)比。

圖1為啁啾太赫茲脈沖與少周期太赫茲脈沖的時(shí)域及頻域的對(duì)比圖。其中τchirped=τfew-cycle=500fs，ωchirped=ωfew-cycle=2πTHz，χ=1.014×1025s-2，中心時(shí)刻tp=0ps。

Fig.1a—time-dependent electric field of chirped terahertz pulse and few-cycle pulseb—frequency distribution of chirped terahertz pulse and few-cycle pulse

由于太赫茲脈沖頻率與CO分子轉(zhuǎn)動(dòng)躍遷頻率接近，因而它們的相互作用屬于共振激發(fā)，分子隨著太赫茲場(chǎng)強(qiáng)的變化而加速或減速轉(zhuǎn)動(dòng)。由圖1b可以看出，相比而言，啁啾太赫茲脈沖在0.6THz以下及1.7THz以上的頻率分量較多，而少周期太赫茲脈沖在0.6THz~1.7THz之間的頻率分量較多。而小于0.6THz的頻率分量對(duì)應(yīng)的正好是CO分子轉(zhuǎn)動(dòng)態(tài)能級(jí)的角量子數(shù)J=4以下的躍遷過(guò)程。同樣的電場(chǎng)強(qiáng)度下，由于啁啾太赫茲脈沖的低頻分量(0THz~0.6THz)更多，因而能更大程度地激發(fā)轉(zhuǎn)動(dòng)基態(tài)及低轉(zhuǎn)動(dòng)態(tài)躍遷，而少周期太赫茲脈沖即便高頻分量(0.6THz~1.7THz)更多，但由于低頻分量較少，不能夠有效激發(fā)轉(zhuǎn)動(dòng)基態(tài)及低轉(zhuǎn)動(dòng)態(tài)躍遷至高轉(zhuǎn)動(dòng)態(tài)，實(shí)現(xiàn)轉(zhuǎn)動(dòng)態(tài)布居的有效轉(zhuǎn)移，因而不易得到高取向程度。

圖3a和圖3b分別為1×107V/cm，6.6×107V/cm電場(chǎng)下，少周期太赫茲脈沖及啁啾太赫茲脈沖與分子相互作用后的轉(zhuǎn)動(dòng)態(tài)布居圖。1×107V/cm場(chǎng)強(qiáng)下，啁啾太赫茲脈沖可以把分子激發(fā)至J=2轉(zhuǎn)動(dòng)態(tài)，而少周期太赫茲脈沖只能激發(fā)分子至J=1轉(zhuǎn)動(dòng)態(tài)，6.6×107V/cm場(chǎng)強(qiáng)下，啁啾太赫茲脈沖可以把分子激發(fā)至J=11轉(zhuǎn)動(dòng)態(tài)，而少周期太赫茲脈沖只能激發(fā)分子至J=7轉(zhuǎn)動(dòng)態(tài)。

Fig.3Rotational population of the oriented CO molecules when chirped terahertz pulse or few-cycle pulse

a—peak amplitude of 1×107V/cmb—peak amplitude of 6.6×107V/cm

2.2　飛秒激光結(jié)合啁啾太赫茲脈沖誘導(dǎo)CO分子取向

盡管單獨(dú)的啁啾太赫茲脈沖能夠取得較高的取向值，但需要非常高的場(chǎng)強(qiáng)(108V/cm量級(jí))，為了便于實(shí)現(xiàn)，需要在相對(duì)較低場(chǎng)強(qiáng)時(shí)就能夠得到較大的取向值，因而采用了飛秒激光預(yù)激發(fā)，在適當(dāng)?shù)难訒r(shí)τ2后加入啁啾太赫茲脈沖，以期增強(qiáng)分子取向程度。

Fig.4Time-dependant orientation degree of CO molecules induced by chirped terahertz pulse in combination with femtosecond laser pulse

圖5為分子分別與啁啾太赫茲脈沖或飛秒激光結(jié)合啁啾太赫茲脈沖相互作用后的轉(zhuǎn)動(dòng)態(tài)布居對(duì)比圖。取向程度的增強(qiáng)原因同飛秒激光結(jié)合其它類型太赫茲脈沖類似。先由飛秒激光非共振激發(fā)分子至更高的轉(zhuǎn)動(dòng)態(tài)，使分子共振躍遷頻率更接近太赫茲脈沖頻率，再由啁啾太赫茲脈沖共振激發(fā)分子至相鄰的轉(zhuǎn)動(dòng)態(tài)，實(shí)現(xiàn)了更多的轉(zhuǎn)動(dòng)態(tài)分布，與此同時(shí)通過(guò)調(diào)節(jié)飛秒激光與太赫茲之間的延時(shí)控制分子相鄰轉(zhuǎn)動(dòng)態(tài)間的相位，最終達(dá)到增強(qiáng)分子取向的效果。由圖5可以看出，單獨(dú)的啁啾太赫茲脈沖只能激發(fā)分子至轉(zhuǎn)動(dòng)態(tài)J=2，而加入飛秒激光后可以激發(fā)分子至轉(zhuǎn)動(dòng)態(tài)J=8，且奇偶轉(zhuǎn)動(dòng)態(tài)的布居為:Podd=0.4962，Peven=0.5038，兩者相互接近，符合獲得高取向值的特征，與參考文獻(xiàn)[22]中描述的一致。

Fig.5Rotational population of oriented CO molecules induced by chirped terahertz pulse or chirped terahertz pulse in combination with a femtosecond laser pulse when the pulses are over

Fig.6Relationship between maximum degrees of CO orientation and peak amplitude of femtosecond laser pulse electric fieldEfs

a—induced by chirped terahertz pulse in combination with femtosecond laser pulseb—induced by few-cycle pulse in combination with femtosecond laser pulse

2.3　有限溫度下的分子取向

在前兩節(jié)中，作者假定分子轉(zhuǎn)動(dòng)溫度為0K，然而在現(xiàn)實(shí)實(shí)驗(yàn)中，分子是處在有限溫度下的，分子轉(zhuǎn)動(dòng)態(tài)布居應(yīng)滿足Boltzmann分布。取向程度〈cosθ〉可以寫成如下形式[22]：

(7)

式中,kB為玻爾茲曼常數(shù)，gJ為分子自旋簡(jiǎn)并系數(shù)，Q為配分函數(shù)：

(8)

Fig.7Time-dependant orientation degrees for CO molecules at different rotational temperatures

3結(jié)論

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CO molecular orientation controlled by combination of

chirped THz pulse and femtosecond laser pulse

YANGMinghui,JINQi,LIUJinsong,WANGKejia,YANGZhengang

(Wuhan National Laboratory for Optoelectronics, Huazhong University of Science & Technology, Wuhan 430074， China)

Abstract：In order to study the orientation behaviors of CO molecules induced by chirped terahertz pulse, the time-dependent Schr?dinger equation was solved by means of rigid rotor approximation for theoretical analysis. In order to achieve high degree of molecular orientation in chirped terahertz pulse at a low intensity, the method by utilizing chirped terahertz pulse in combination with femtosecond laser pulse was proposed. The results show that the molecular orientation degree can be enhanced about 81%, and that the orientation efficiency decreases with the rise of temperature in finite temperature range. Compared with the few-cycle terahertz pulse, chirped terahertz pulse has more advantages in inducing molecular orientation. The result is meaningful to enhance the CO orientation degree.

Key words:laser physics；molecular orientation；rigid rotor model；chirped terahertz pulse

收稿日期：2015-01-13；收到修改稿日期：2015-03-16

作者簡(jiǎn)介：楊明惠(1990-)，男，碩士研究生，現(xiàn)主要從事太赫茲分子取向方面的研究。

基金項(xiàng)目：國(guó)家自然科學(xué)基金資助項(xiàng)目(61177095;61475054;61405063);湖北省自然科學(xué)基金資助項(xiàng)目(2012FFA074;2013BAA002);武漢市科技計(jì)劃基金資助項(xiàng)目(2014010101010009);中央高?；究蒲袠I(yè)務(wù)費(fèi)資助項(xiàng)目(2013KXYQ004;2014ZZGH021;2014QN023)

中圖分類號(hào)：TN241

文獻(xiàn)標(biāo)志碼：A

doi:10.7510/jgjs.issn.1001-3806.2015.06.001

文章編號(hào)：1001-3806(2015)06-0735-06