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轴向阶跃变化周期圆形介质结构中每一单元材质分布如图 1所示。其中左部分代表传统RHM,右部分为负折射率材料,这里选取DNM,其相对介电常数和相对磁导率可以用Drude模型表示[17]。每个DNM和RHM单元长度均为L;DNM和RHM折射率为nl和nr。
在笛卡尔右手坐标系下,FAiB在初始原点O处的电场强度大小为[1, 13, 15]:
$ E_{1}\left(x_{0}, y_{0}\right)=f_{\mathrm{A}}\left(\frac{x_{0}}{w_{1}}\right) f_{\mathrm{A}}\left(\frac{y_{0}}{w_{2}}\right) \exp \left(\frac{a x_{0}}{w_{1}}+\frac{a y_{0}}{w_{2}}\right) $
(1) 式中,x0和y0表示入射光波初始位置的横、纵向尺寸;0 < a < 1代表指数截断因子;w1,w2分别为x和y方向截面光斑尺寸,一般它们和光斑束腰尺寸w0相等;fA()代表艾里函数。当光波在正、负折射率交替变化的轴向阶跃周期介质传输时,圆形的透光率可用圆域函数表示为:
$ \operatorname{circ}\left(\frac{\sqrt{x^{2}+y^{2}}}{r}\right)=\left\{\begin{array}{l}{1, \left(\sqrt{x^{2}+y^{2}}<r\right)} \\ {0, (\text { other })}\end{array}\right. $
(2) 式中,r是孔径。利用有限复高斯函数展开法可将(2)式改为[18]:
$ T(x, y)=\sum\limits_{h=1}^{N} A_{h}\left\{\exp \left[-\frac{B_{h}}{r}\left(x^{2}+y^{2}\right)\right]\right\} $
(3) 式中,N为展开系数,一般取N=10即可满足计算精度要求; Ah和Bh是展开系数, 可以通过计算机优化的方法得到,具体数值见参考文献[19]。当FAiB沿光轴z传输并依次经过各材质单元时,其在圆形周期介质内任意傍轴位置电场强度可由广义惠更斯-菲涅耳积分公式表示为[20]:
$ \begin{array}{c}{E_{2}(x, y, z)=\left(-\frac{\mathrm{i}}{\lambda B}\right) \exp (\mathrm{i} k z) \iint\limits_{S_{1}} E_{1}\left(x_{0}, y_{0}, 0\right) \times} \\ {T\left(x_{0}, y_{0}\right) \exp \left\{\frac{\mathrm{i} k}{2 B}\left[A\left({x_0}^{2}+{y_0}^{2}\right)+\right.\right.} \\ {D\left(x^{2}+y^{2}\right)-2\left(x_{0} x+y_{0} y\right) ] \} \mathrm{d} x_{0} \mathrm{d} y_{0}}\end{array} $
(4) 式中,λ=1.55μm是光波波长,k=2π/λ是其对应波数, S1是入射横截面, T(x0, y0)是在入射横截面上圆孔的近似表达式; A,B,C,D代表光波通过一系列介质单元后,总光学传输矩阵T的各元素,可用下列公式表示[21]:
$ \begin{array}{c}{\boldsymbol{T}=\left[ \begin{array}{cc}{A} & {B} \\ {C} & {D}\end{array}\right]=} \\ {\left[\boldsymbol{M}\left(n_{\rm l}, n_{\mathrm{r}}\right) \;\boldsymbol{M}(L)\; \boldsymbol{M}\left(n_{\mathrm{r}}, n_{\rm l}\right) \;\boldsymbol{M}(L) \;\boldsymbol{M}\left(1, n_{\mathrm{r}}\right)\right]^{n}}\end{array} $
(5) 式中,n表示材料周期数,M(L)代表光波在均匀单一介质传输L距离的传输矩阵, M(nr, nl)为光波从RHM介质进入DNM时,其分界面传输矩阵,类似地,从DNM进入RHM材料分界面的传输矩阵为M(nl, nr)。将(5)式和(3)式代入(4)式,便可得到FAiB在空间域内传输场强的具体形式为:
$ \begin{array}{c}{E(j, z)=\frac{\mathrm{i} k}{2 B} \exp (-\mathrm{i} k z) \exp \left(D j^{2}\right) \times} \\ {\sum\limits_{h=1}^{10} A_{h}\left[\frac{1}{Q} \exp \left(\frac{q^{2}}{4 Q}\right) \exp \left(\frac{1}{96 {w_0}^{6} Q^{3}}+\right.\right.} \\ {\frac{q}{8 {w_0}^{3} Q^{2}} ) f_{\mathrm{A}}\left(\frac{1}{16 {w_0}^{4} Q^{2}}+\frac{q}{2 w_{0} Q}\right) ]}\end{array} $
(6) 式中,j=(x2+y2)0.5,而:
$ Q=\frac{B_{h}}{r^{2}}+\frac{\mathrm{i} k A}{2 B} $
(7) $ q=\frac{a}{w_{0}}+\frac{\mathrm{i} k}{B} j $
(8) 相应FAiB光强分布可由电场强度和它复共轭的乘积求得[22]。
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为了研究圆形周期介质孔径大小对FAiB光波演变特性的影响,图 2是FAiB分别经过RHM, DNM出射横截表面光强分布图和侧面传输光强分布图随不同数值孔径的变化规律。其中,在侧面传输光强分布图中,箭头为RHM和DNM界面分界线,其它参量选取为w0=0.1mm,a=0.2,光波瑞利尺寸为ZR=πw02/λ=(k /w02)/2=0.02m,介质长度L=10ZR,RHM介质折射率nr=1.5,光波频率ν=1.94×1014Hz, 电、磁等离子角频率选取为ωpe=ωpm= 2πν×$\sqrt{2.5}$,对应DNM介质折射率为nl=-1.5,周期数n=2。从RHM出射表面的光强分布(见图 2a~图 2d)可以清楚地看出,受孔径形状的调制,FAiB外形轮廓呈圆形状,并且随着孔径r的增加,出射光波主、旁瓣已经逐步显现出来,当r=10mm时,从图 2d可以看出,FAiB已基本实现近似无衍射,主、旁瓣轮廓已可以区分,而当r=0.1mm,即孔径大小和光斑束腰尺寸可比拟时,从图 2a可知,光波外形已和高斯光束无异,并且当光波在RHM继续传输时,FAiB已无法自可愈,因此,当r逐渐减小时,衍射效应对FAiB影响越来越严重,随着r减小,光波在RHM传输时,外形轮廓逐渐从艾里光束过渡到高斯光束。当FAiB途经RHM和DNM交替排列周期结构之后,出射表面光强分布如图 2e~图 2g所示,从中可以看出,DNM可以作为理想凸镜层,实现光波的完美还原,并且随着r的增加,FAiB的旁瓣数量逐渐增多,当r=0.5mm时,从图 2e可见,在x,y方向,FAiB各有一旁瓣,而当r=10mm时,不仅在x,y方向有众多旁瓣,而且在第三象限也有旁瓣出现。从侧面光强分布(见图 2h~图 2j)可知,FAiB在RHM传输时发生的自弯曲等现象可由临近DNM补偿,使得出射光束回到初始轮廓,如图 2j所示。而当r大小和光斑束腰尺寸可比拟时,从图 2h可知,此时衍射已十分严重,FAiB在RHM传输时的自弯曲现象已无法实现,此时,即使有DNM作为补偿介质,出射光波轮廓也和FAiB大相径庭。
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由2.1节可知,对于轴向阶跃变化圆形周期介质,当r=0.5mm时,光波传输特性和外形轮廓已能满足通信对FAiB性能的要求,从光波侧面传输图(见图 2h~图 2j)可知,当绝对值函数abs(nl)=nr,并且DNM介质长度和RHM介质长度相等时,在接收横截面上,出射光波光强就回到了初始入射光强轮廓,PENDRY曾利用平面光波电磁理论阐述了在传输媒质中适当地引入负折射率材料,可以实现输出光波的完美还原,即负折射率材料犹如一个完美透镜层,补偿了光波在传统介质正传输时产生的衍射效应,从而显著地提高成像质量[23],这种补偿作用对近似无衍射FAiB也不例外,只是在这类奇异光束中,传输媒质主要起到限制光波自弯曲的作用。但在工艺制作中,RHM和DHM介质折射率很难满足互为相反数,因此,本小节中研究不同nl对FAiB传输特性的影响,其结果如图 3所示。
其中,电、磁等离子角频率分别选取为ωpe=ωpm=2πν×$\sqrt{3.0}$和ωpe=ωpm=2πν×$\sqrt{2.0}$,对应DNM折射率为nl为-2.0和-1.0,r=0.5mm固定不变,其它参量和图 2一致。从图 3a可知,当负介质折射率nl=-2.0时,FAiB在RHM中传输形成的自弯曲仍会被相邻DNM补偿,使得光波沿逆抛物线轨迹传输,但由于abs(nl)>1.5,因此光波自弯曲在DNM中不能完全被补偿,随着介质周期数增加,出射光波便不能回到初始光强分布轮廓。此时,需要探讨一个问题:每个DNM单元长度多大时,可使FAiB侧面轮廓在每一周期层传输后,其光强分布图沿RHM-DNM分界线轴对称分布?当光波依次经过RHM和DNM时,其光学传输矩阵为:
$ \begin{array}{cc}{\boldsymbol{T}=\left[ \begin{array}{cc}{A} & {B} \\ {C} & {D}\end{array}\right]=\left[ \begin{array}{cc}{1} & {0} \\ {0} & {n_{\rm l}}\end{array}\right] \times \left[ \begin{array}{cc}{1} & {L} \\ {0} & {1}\end{array}\right] \times} \\ {\left[ \begin{array}{cc}{1} & {0} \\ 0& {n_{\mathrm{r}} / n_{\rm l}}\end{array}\right] \times \left[ \begin{array}{ll}{1} & {L} \\ {0} & {1}\end{array}\right] \times \left[ \begin{array}{cc}{1} & {0} \\ {0} & {1 / n_{\mathrm{r}}}\end{array}\right]}\end{array} $
(9) 将L=10ZR, nr=1.5, nl=-2.0代入上式,可得矩阵元B的定量大小为:
$ B=-2 L / 3+4 {\rm{ \mathsf{ π} }} / 93 $
(10) 因此,光波在DNM中实现完美还原所需长度为L=13.4ZR,图 3b中给出了这种准周期结构内光波传输的仿真结果。对比图 3b和图 3a可知,FAiB在每层DNM出射表面都又回到了入射光波轮廓。与图 3a形成鲜明对比的是,当nl=-1.0时,光波实现完美还原所需的DNM长度小于10ZR,因此,当DNM单元长度固定不变,从图 3c可以看出,光波在DNM中传输时,除了补偿其自弯曲外,在剩余材质中再次出现自弯曲,导致光波在下周期出射表面一直弯曲到很远的地方,这时不仅输出中心光强微弱,且输出横截面中心主瓣远远偏离中心原点,因而光束质量很差;同样地,将L=10ZR, nr=1.5, nl=-1.0代入(9)式,则光波在这类结构模型中传输时,光学矩阵元B的定量大小为:
$ B=-L+4 {\rm{ \mathsf{ π} }} / 93 $
(11) 这样,光波在这类介质中实现完美还原所需长度为L=6.7ZR,图 3d中描绘了其传输图样,与理论分析结果一致。进一步由应用光学基本原理可知,在傍轴近似下,实现光波完美还原的介质折射率nl和DNM单元长度L之间具有良好的线性关系,其定量拟合关系如图 4所示。
圆形周期介质内艾里光束的传输特性
Characteristics of Airy beam propagating in circular periodic media
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摘要: 为了研究无衍射光波在特异材质内的传输特性,实现更优良的光波通信,将传统右手材料和双负折射率材料相结合,提出了一种轴向阶跃变化周期圆形介质结构。基于广义惠更斯-菲涅耳光学积分公式,结合光学传输矩阵,分析了艾里光束在这种传输媒质中出射表面光强分布特性和侧面传输光强分布图;分析了负折射率参量对这类光波演变的影响及其补偿机理;分析了实现输出光波完美还原时,负折射率大小同介质单元长度的定量关系。结果表明,当介质孔径逐渐减小时,有限艾里光束衍射效应越来越严重,并且出射光强外形轮廓逐渐从艾里光束过渡到高斯光束;当双负折射率材料的折射率nl的绝对值大于右手材料的折射率nr时,出射表面实现光波完美还原的双负折射率材料单元层越长,反之则越短。该研究对分析周期或准周期轴向阶跃变化的圆形平板介质光波通信是有帮助的。
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关键词:
- 激光光学 /
- 光强演变 /
- 有限艾里光束 /
- 广义惠更斯-菲涅耳光学公式 /
- 传输矩阵
Abstract: In order to study the propagation characteristics of non-diffracting light waves in special materials and achieve better optical communication, one periodic circular dielectric structure with axial step change was proposed by combining traditional right-handed materials with bi-negative refractive index materials. Based on the generalized Huygens-Fresnel optical integral formula, the distribution characteristics of the emitted surface light intensity and the profile of the side light intensity of the Airy beam in this transmission medium were analyzed by using optical transmission matrix. The influence of negative refractive index parameters on the evolution of such light waves and its compensation mechanism were analyzed. The quantitative relationship between the negative refractive index and the length of the dielectric unit was analyzed when the output light wave was perfectly restored. The results show that, when the pore size of the medium decreases gradually, the diffraction effect of finite Airy beam is getting worse and worse. The profile of the emitted light intensity gradually transits from the Airy beam to the Gaussian beam. When the absolute value of nl is greater than nr, the bi-negative refractive index material layer is longer when the perfect light wave reduction is achieved on the exit surface. Conversely, the shorter. The study is helpful for analyzing optical wave communication in circular flat dielectrics with periodic or quasi-periodic axial step changes. -
Figure 2. Intensity distribution of FAiB with different aperture sizes
a—RHM cross section,r=0.1mm b—RHM cross section,r=0.5mm c—RHM cross section,r=1mm d—RHM cross section,r=10mm e—DNM cross section, r=0.5mm f—DNM cross section, r=1mm g—DNM cross section,r=10mm h—side transmission view, r=0.1mm i—side transmission view, r=0.5mm j—side transmission view, r=10mm
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