姜其暢，蘇艷麗，聶合賢，馬紫微，李永宏

(運(yùn)城學(xué)院 物理與電子工程系，運(yùn)城 044000)

厄米-高斯光束在飽和非線性介質(zhì)中的傳輸特性

姜其暢，蘇艷麗，聶合賢，馬紫微，李永宏

(運(yùn)城學(xué)院 物理與電子工程系，運(yùn)城 044000)

為了研究厄米-高斯光束在光折變飽和非線性介質(zhì)中的傳輸特性，采用有限差分方法數(shù)值求解了光波演化方程,理論分析了厄米-高斯光束的傳輸特性。結(jié)果表明，1維1階、2階和3階厄米-高斯光束在光折變非線性介質(zhì)中傳輸時(shí)，在合適的非線性條件下，均可以形成呼吸模式的孤子；隨著非線性的加大，厄米-高斯光束的光場分量之間的相互分離趨勢(shì)將逐漸變?nèi)酰瑫r(shí)，每個(gè)光場分量的振幅起伏效應(yīng)會(huì)更加明顯；改變厄米-高斯光束的入射位置、入射角度對(duì)其傳輸特性沒有影響；2維厄米-高斯光束的傳輸特性和1維情況是類似的。厄米-高斯光束的這些特性在光開關(guān)領(lǐng)域有一定的應(yīng)用前景。

非線性光學(xué)；厄米-高斯光束；光折變效應(yīng)；空間孤子

引 言

1998年，CASPERSON和TOVAR給出了近軸近似直角坐標(biāo)下亥姆霍茲方程的一類特解，即所謂厄米-正弦類-高斯光束，它是具有廣泛代表性意義的一類光束[1-5]。厄米-高斯光束被認(rèn)為是厄米-正弦類-高斯光束的特例。目前，人們對(duì)厄米-高斯光束的研究重要集中在兩個(gè)方面，一方面是研究厄米-高斯光束對(duì)各種介質(zhì)圓柱、介質(zhì)球的散射問題[6-8];另一方面是研究厄米-高斯光束在大氣湍流[9-11]和非線性介質(zhì)[12-15]中的傳輸問題。近幾年，人們報(bào)道了厄米-高斯光束在強(qiáng)非局域非線性介質(zhì)中的傳輸特性,發(fā)現(xiàn)厄米-高斯光束可以在一定條件下形成厄米-高斯孤子，由于厄米多項(xiàng)式的調(diào)制作用，厄米-高斯孤子相比于傳統(tǒng)的基模高斯孤子[16-17]表現(xiàn)出更豐富的傳輸特性，但是關(guān)于厄米-高斯光束在光折變飽和非線性介質(zhì)中的傳輸特性還少有相關(guān)報(bào)道。本文中借助光波演化的非線性薛定諤方程，數(shù)值研究了厄米-高斯光束在光折變飽和非線性介質(zhì)中的傳輸特性。

1 基本理論

考慮厄米-高斯光束在光折變晶體中沿z軸傳輸，其偏振沿x方向；假定光束只在x方向衍射，晶體光軸和外加電場均沿x方向。為了便于數(shù)值分析，取歸一化的坐標(biāo)參量s=x/x0，ξ=z/(kx02)。其中，x0是任意空間寬度，k=k0ne=(2π/λ0)ne，λ0是自由空間波長，ne是未受擾動(dòng)的非尋常光折射率。光波傳輸?shù)臍w一化方程可以表示為:

式中，U(s,ξ)是歸一化光波包絡(luò)，非線性系數(shù)β=(k0x0)2(ne4r33/2)E0，r33是電光系數(shù)，E0是外加電場強(qiáng)度。入射面處的厄米-高斯光束可以表示為:

式中，Hn是厄米多項(xiàng)式，n表示厄米多項(xiàng)式的階數(shù)。如果x0=w0=50μm(w0表示高斯光束的光斑大小)，λ0=0.5μm，相應(yīng)的高斯光束的瑞利距離ZR=πw02/λ0=15.7mm。下面基于(1)式和(2)式數(shù)值分析各階厄米-高斯光束的傳輸特性。

2 數(shù)值結(jié)果

當(dāng)厄米多項(xiàng)式H0=1時(shí)，(2)式即退化為基模高斯光束的表達(dá)式。如果外加電場為零，則非線性參量β=0，可得到圖1a所示結(jié)果，圖中橫縱坐標(biāo)均是歸一化的無量綱坐標(biāo)。ξ=1對(duì)應(yīng)實(shí)際傳輸距離大約是70mm，可以看到超過瑞利距離(約ξ=0.2)，由于光束的衍射效應(yīng)，高斯光束的光斑能量逐漸彌散。為了抑制光束的這種衍射效應(yīng)，增加外加電場強(qiáng)度，當(dāng)非線性系數(shù)β=5.5，基本可以抑制光束的衍射，形成呼吸模式的高斯孤子，為了更清楚看到其呼吸模式，可以將傳輸距離增加到ξ=6，如圖1b和圖1c所示。

Fig.1 Natural diffraction and soliton propagation of fundamental-mode Gaussian beama—β=0 b—β=5.5 c—β=5.5

Fig.2 Propagation of first-order Hermite-Gaussian beam

如果加大非線性即加大外加電場，兩個(gè)光場分量之間的分離距離可以明顯減小，但是相應(yīng)的每個(gè)光場分量的起伏效應(yīng)會(huì)顯著加大，如圖3所示。通過改變外加電場的大小可以靈活調(diào)節(jié)兩個(gè)光場分量之間的距離，從而可以控制出射面某一點(diǎn)處光信號(hào)的有無，這在光開關(guān)領(lǐng)域有一定的應(yīng)用前景。

同樣，當(dāng)厄米多項(xiàng)式取H2=8s2-2時(shí)，(2)式就是2階厄米-高斯光束的表達(dá)式。此時(shí)，在初始入射面，2階厄米-高斯光束呈現(xiàn)對(duì)稱的3個(gè)光場分量，取合適的外加電場強(qiáng)度如β=15，同樣可以形成呼吸模式的孤子，中間光場分量直線傳輸，兩側(cè)的光場分量彼此分離，而且分離的距離隨著傳輸距離的加大而增加；當(dāng)外加電場強(qiáng)度加大時(shí)，比如β=25，兩側(cè)光場分量之間的分離距離明顯減小，但是3個(gè)光場分量振幅的起伏效應(yīng)會(huì)更加明顯(見圖4)。

Fig.3 Propagation of the first-order Hermite-Gaussian beam at β=40

Fig.4 Propagation of the second-order Hermite-Gaussian beam under different nonlinear conditions

Fig.5 Propagation of the third-order Hermite-Gaussian beam under different conditions

光波傳輸?shù)臍w一化方程(1)式在2維情況時(shí)可以表示為[15]:

式中，sx=x/x0，sy=y/x0，分別是x方向和y方向的歸一化坐標(biāo)。其它參量的意義和(1)式相同。這里以2階厄米-高斯光束為例，2維厄米-高斯光束表示為U(sx,sy,0)=(8sx2-2)(8sy2-2)exp(-sx2-sy2)，同樣取非線性參量β=15，2維厄米-高斯光束在ξ=0，ξ=1.5和ξ=3處的橫截面強(qiáng)度分布如圖6所示?？梢钥吹?，在合適非線性條件下，2階厄米-高斯光束的每一個(gè)光場分量都以呼吸模式的孤子形式傳輸，而且隨著傳輸距離的加大，各個(gè)光場分量之間的距離會(huì)逐漸增加。

Fig.6 Propagation of 2-D second-order Hermite-Gaussian beam

3 結(jié) 論

分析了1階、2階和3階厄米-高斯光束在光折變飽和非線性介質(zhì)中的傳輸特性，在合適的外加電場強(qiáng)度條件下，厄米-高斯光束的各個(gè)光場分量都可以形成呼吸模式的孤子，各個(gè)光場分量之間的分離距離可以由外加電場強(qiáng)度靈活操控。厄米-高斯光束的這些特性在光開關(guān)領(lǐng)域有一定的應(yīng)用前景。

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PropagationcharacteristicsofHermite-Gaussianbeaminsaturablenonlinearmedia

JIANGQichang,SUYanli,NIEHexian,MAZiwei,LIYonghong

(Department of Physics and Electronic Engineering, Yuncheng University, Yuncheng 044000, China)

In order to study propagation properties of Hermite-Gaussian beams in photorefractive saturable nonlinear media, finite difference method was used to solve the evolution equation of light wave numerically and analyze the propagation properties of Hermite-Gaussian beams theoretically. The results show that, under suitable nonlinear conditions, 1-D Hermite-Gaussian beams of 1-order, 2-order and 3-order can form the solitons in respiratory mode during the propagation in photorefractive nonlinear media. With the increase of nonlinearity, the separation tendency among light field components of Hermite-Gaussian beams would become weaker. At the same time, the amplitude fluctuation effect of each light field component would be more obvious. The changes of incident position and incident angle of Hermite-Gaussian beams have no influence on its propagation characteristics. The transmission characteristics of 2-D Hermite-Gaussian beams are similar to those of 1-D. These properties of Hermite-Gaussian beams have certain application prospects in the field of optical switching.

nonlinear optics; Hermite-Gaussian beam; photorefractive effect; spatial solitons

1001-3806(2018)01-0141-04

山西省自然科學(xué)基金資助項(xiàng)目(2011011003-2)

姜其暢(1980-),男，博士，副教授，主要從事非線性光學(xué)和光場調(diào)控方面的研究。

E-mail:jiangsir009@163.com

2017-02-13；

2017-03-17

O437.5

A

10.7510/jgjs.issn.1001-3806.2018.01.028