一维无序系统中的长程关联作用机理及无序耦合微球谐振腔链的光学特性研究
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摘要
为了研究格点位能之间和格点间跳跃积分之间的长程关联对一维无序系统中局域化性质的影响,本文首先利用Fourier过滤法产生了具有长程幂律关联的随机数序列,这种序列的谱密度具有S(k)∝k-α的形式,同时它描述了分形布朗运动的轨迹,适合用来模拟自然界中普遍存在的各种长程关联,例如DNA分子的核苷酸序列之间的相互关联、薄膜生长时出现的界面粗糙度之间的高度-高度关联等。然后在格点位能之间引入了这种长程幂律关联,分析和对比了一维长程关联对角无序系统中不同局域区域内的局域化性质,并在格点间跳跃积分之间也引入了这种长程幂律关联,探讨了一维长程关联非对角无序系统中局域态与扩展态之间的相互转变。
无序耦合微球谐振腔链作为一种更接近于实际应用的一维无序系统,也具有一维Anderson局域化性质,且由于其组成单元—微球谐振腔—具有极高的品质因子和极小的模式体积而使其在慢光、全光缓存、光存储和加强光物相互作用等方面具有极高的应用价值。但是,由于制备条件、方法等的不够完善而出现的微球尺寸大小不一、微球间距离不一及粗糙的微球表面等无序因素,极大的影响了光在链中的传输效率并使光损耗显著增加。为了探讨微球谐振腔的尺寸无序对耦合微球谐振腔链中光的传输性质的影响以及对光损耗进行统计分析,本文从Mie散射理论出发,利用传输矩阵方法在最近邻近似下对光在链中传输时的透射系数、反射系数和损耗进行了定义,并从数值上分别对链长远远大于局域长度的长链和链长小于局域长度的短链两种情况下链中光的传输性质进行了统计分析。
对一维长程关联对角无序系统中的局域化性质进行研究后发现,由于长程幂律关联和对角无序的共同作用,弱无序弱关联局域区域与强无序强关联局域区域的局域化性质有着显著的区别,并且都有别于一维标准Anderson模型中的局域化性质。在弱无序弱关联情况下,电子波函数具有指数衰减形式,局域长度与系统尺寸N之间存在幂律函数关系ξ∝Nμ,μ随关联指数的增大而增大；Lyapunov指数呈正态分布,而其方差var(γ)∝N-v,v也随关联指数的增大而增大。在强无序强关联情况下,电子波函数是具有蝴蝶形状分布的局域态,Lyapunov指数不再呈正态分布并且局域长度与var(γ)都是常数而不随N的变化而变化。另外,我们还发现,单参数标度理论在这两种情况下都不再有效并且长程关联表现出了二重性：它既能抑制系统的局域化也能增强系统的局域化。
在一维长程关联非对角无序系统中,尽管系统中存在着手性对称性,但格点间跳跃积分之间的长程关联与格点位能之间的长程关联都会使系统中的电子态由局域态向扩展态转变,产生这种转变的临界关联强度为αc=2.0.另一方面,在平均跳跃积分t0和能量本征值E都固定的情况下,当α>αc且无序度W较小时,增大W,系统中将出现退局域化-局域化转变,出现这种转变的临界无序度为Wc=2t0-|E|.此外,在t0和W都固定的情况下,当α>αc且W较小时,改变E,系统中也将出现退局域化-局域化转变,出现这种转变的临界能量本征值也即迁移率边界为Ec=±|2t0|-W|.
对光在无序耦合微球谐振腔链中的传输性质进行探讨后发现,对于链长远远大于局域长度的长链,尽管该链表现出了一维Anderson强局域化特性,但光在链中传输时的辐射损耗的统计性质与光在其他一维耗散光学系统中传输时的辐射损耗的统计性质有着显著的差别,该链中光的损耗A的统计分布具有f(A)∝A-2 exp[一(a一A)2/(b2A2)]的数学形式,并且与参数a、b相关的损耗的平均值与方差均是局域长度与耗散长度的比率的函数。对于链长远远小于局域长度的短链,尺寸无序对带边界区域光学性质的影响远远大于对带中心区域光学性质的影响,表明带边界附近的频率更适合于用来研究无序耦合微球谐振腔链的慢光效应。
To study the effect of long-range correlations (LRC) of the on-site energies or of the hopping integrals on the localization properties of one-dimensional (1D) systems, we firstly generated a sequence of random numbers with LRC by using the Fourier filtering method. This kind of sequence has a power-law spectral density of the form S(k)∞k-αand describes the trace of a fractional Brownian motion. It can be used to simulate stochastic processes in nature such as the correlations in the nucleotide sequence of DNA molecules and height-height correlations in the interface roughness appearing during growth. We then introduced this kind of LRC into the on-site energies and compared the localization properties of different localized regimes in a 1D system with long-range correlated diagonal disorder. We also introduced this kind of LRC into the hopping integrals and studied the phase transitions between the localized states and extended states in a ID system with long-range correlated off-diagonal disorder.
As a 1D system, which has practical applications, the disordered chain of coupled microspherical resonators also demonstrates Anderson localization. Since its component—microspherical resonator—is characterized by an ultra-high quality factor and extremely small mode volume, it has been proposed for applications in slow light, all-optical buffer, optical storage and intensification of light-matter interaction. In practice, however, the structural fabrication imperfections such as the random variations in the size of the microspherical resonators and in the distance between the nearest-neighbor microspherical resonators, and the surface roughness, collectively termed as "disorder", significantly influence the transport efficiency and dramatically enhance the radiative losses when light is propagating in such a chain. To study the effect of size disorder on the transport properties and to investigate statistical properties of radiative losses in a chain of coupled microspherical resonators, we defined the transmission coefficient, reflection coefficient and radiative losses by using Mie scattering theory, transfer matrix method and the nearest-neighbor approximation. We also numerically studied transport properties of two types of structures:asymptotically long chain with system size is much larger than the localization length and relatively short chain with system size is shorter larger than the localization length.
In 1D system with long-range correlated diagonal disorder, we found that, under the combined effect of LRC and disorder, the localization properties of localized regime with weak disorder and weak correlations (regime I) are significantly different from that of localized regime with strong disorder and strong correlations (regime II) and the localization properties of these two regimes are different from that of 1D standard Anderson model. In regime I, the wave functions are exponentially localized, the localization lengthξhas a power-law relation with the system size N,ξ∞Nμwithμincreases with the increasing correlation exponentα; the distribution of Lyapunov exponentγis normal and its variance var(γ) scales as var(γ)∞N-v with v also increases with the increasingα. In regime II, the localized mode exhibits a butterflylike form, which totally differs from exponentially localized modes excited in regime I; the distribution of Lyapunov exponent is not normal anymore and both of the localization length and var(γ)are constants, which don't change with the changing system size. Moreover, we found that the single parameter scaling theory is not valid anymore in both of these two regimes and LRC plays dual role in the system:it may either suppress or enhance localization.
In 1D system with long-range correlated off-diagonal disorder, we found that, even though such a system has the property of chiral symmetry, like the introducing of LRC in on-site energies induces localization-delocalization transition (LDT), the introducing of LRC in hopping integrals also induces LDT and the critical correlation strength is alsoαc= 2.0. On the other hand, under the condition of fixed average hopping integral t0 and fixed eigenenergy E, whenα>αc and disorder strength W is small, the increasing of W will induce delocalization-localization transition (DLT) and the critical disorder strength is Wc= 2t0-|E|. In addition, under the condition of fixed t0 and fixed W, whenα>αand W is small, the changing of E will also induce DLT and the critical eigenenergy is Ec=±|2t0-W|.
After studied the optical transport properties of the disordered chain of microspherical resonators, we found that, for a long chain, even though the chain behaves in some aspects as a typical strong Anderson localized system, its radiative loss statistics is strongly different from that observed for other lossy optical systems. The distribution function of loss A in this chain is found as f(A)∞A-2 exp[-(a-A)2/(b2A2)], the mean value and variance of A, which are related to parameters a and b, are the functions of a single scaling parameter:ratio between the localization length and the loss length in ordered chains. For a short chain, on another hand, we found that the band edges are much more sensitive to disorder than the band center showing that the frequencies near the band edges are more suitable to be used to study the effect of slow light.
无序耦合微球谐振腔链作为一种更接近于实际应用的一维无序系统,也具有一维Anderson局域化性质,且由于其组成单元—微球谐振腔—具有极高的品质因子和极小的模式体积而使其在慢光、全光缓存、光存储和加强光物相互作用等方面具有极高的应用价值。但是,由于制备条件、方法等的不够完善而出现的微球尺寸大小不一、微球间距离不一及粗糙的微球表面等无序因素,极大的影响了光在链中的传输效率并使光损耗显著增加。为了探讨微球谐振腔的尺寸无序对耦合微球谐振腔链中光的传输性质的影响以及对光损耗进行统计分析,本文从Mie散射理论出发,利用传输矩阵方法在最近邻近似下对光在链中传输时的透射系数、反射系数和损耗进行了定义,并从数值上分别对链长远远大于局域长度的长链和链长小于局域长度的短链两种情况下链中光的传输性质进行了统计分析。
对一维长程关联对角无序系统中的局域化性质进行研究后发现,由于长程幂律关联和对角无序的共同作用,弱无序弱关联局域区域与强无序强关联局域区域的局域化性质有着显著的区别,并且都有别于一维标准Anderson模型中的局域化性质。在弱无序弱关联情况下,电子波函数具有指数衰减形式,局域长度与系统尺寸N之间存在幂律函数关系ξ∝Nμ,μ随关联指数的增大而增大；Lyapunov指数呈正态分布,而其方差var(γ)∝N-v,v也随关联指数的增大而增大。在强无序强关联情况下,电子波函数是具有蝴蝶形状分布的局域态,Lyapunov指数不再呈正态分布并且局域长度与var(γ)都是常数而不随N的变化而变化。另外,我们还发现,单参数标度理论在这两种情况下都不再有效并且长程关联表现出了二重性：它既能抑制系统的局域化也能增强系统的局域化。
在一维长程关联非对角无序系统中,尽管系统中存在着手性对称性,但格点间跳跃积分之间的长程关联与格点位能之间的长程关联都会使系统中的电子态由局域态向扩展态转变,产生这种转变的临界关联强度为αc=2.0.另一方面,在平均跳跃积分t0和能量本征值E都固定的情况下,当α>αc且无序度W较小时,增大W,系统中将出现退局域化-局域化转变,出现这种转变的临界无序度为Wc=2t0-|E|.此外,在t0和W都固定的情况下,当α>αc且W较小时,改变E,系统中也将出现退局域化-局域化转变,出现这种转变的临界能量本征值也即迁移率边界为Ec=±|2t0|-W|.
对光在无序耦合微球谐振腔链中的传输性质进行探讨后发现,对于链长远远大于局域长度的长链,尽管该链表现出了一维Anderson强局域化特性,但光在链中传输时的辐射损耗的统计性质与光在其他一维耗散光学系统中传输时的辐射损耗的统计性质有着显著的差别,该链中光的损耗A的统计分布具有f(A)∝A-2 exp[一(a一A)2/(b2A2)]的数学形式,并且与参数a、b相关的损耗的平均值与方差均是局域长度与耗散长度的比率的函数。对于链长远远小于局域长度的短链,尺寸无序对带边界区域光学性质的影响远远大于对带中心区域光学性质的影响,表明带边界附近的频率更适合于用来研究无序耦合微球谐振腔链的慢光效应。
To study the effect of long-range correlations (LRC) of the on-site energies or of the hopping integrals on the localization properties of one-dimensional (1D) systems, we firstly generated a sequence of random numbers with LRC by using the Fourier filtering method. This kind of sequence has a power-law spectral density of the form S(k)∞k-αand describes the trace of a fractional Brownian motion. It can be used to simulate stochastic processes in nature such as the correlations in the nucleotide sequence of DNA molecules and height-height correlations in the interface roughness appearing during growth. We then introduced this kind of LRC into the on-site energies and compared the localization properties of different localized regimes in a 1D system with long-range correlated diagonal disorder. We also introduced this kind of LRC into the hopping integrals and studied the phase transitions between the localized states and extended states in a ID system with long-range correlated off-diagonal disorder.
As a 1D system, which has practical applications, the disordered chain of coupled microspherical resonators also demonstrates Anderson localization. Since its component—microspherical resonator—is characterized by an ultra-high quality factor and extremely small mode volume, it has been proposed for applications in slow light, all-optical buffer, optical storage and intensification of light-matter interaction. In practice, however, the structural fabrication imperfections such as the random variations in the size of the microspherical resonators and in the distance between the nearest-neighbor microspherical resonators, and the surface roughness, collectively termed as "disorder", significantly influence the transport efficiency and dramatically enhance the radiative losses when light is propagating in such a chain. To study the effect of size disorder on the transport properties and to investigate statistical properties of radiative losses in a chain of coupled microspherical resonators, we defined the transmission coefficient, reflection coefficient and radiative losses by using Mie scattering theory, transfer matrix method and the nearest-neighbor approximation. We also numerically studied transport properties of two types of structures:asymptotically long chain with system size is much larger than the localization length and relatively short chain with system size is shorter larger than the localization length.
In 1D system with long-range correlated diagonal disorder, we found that, under the combined effect of LRC and disorder, the localization properties of localized regime with weak disorder and weak correlations (regime I) are significantly different from that of localized regime with strong disorder and strong correlations (regime II) and the localization properties of these two regimes are different from that of 1D standard Anderson model. In regime I, the wave functions are exponentially localized, the localization lengthξhas a power-law relation with the system size N,ξ∞Nμwithμincreases with the increasing correlation exponentα; the distribution of Lyapunov exponentγis normal and its variance var(γ) scales as var(γ)∞N-v with v also increases with the increasingα. In regime II, the localized mode exhibits a butterflylike form, which totally differs from exponentially localized modes excited in regime I; the distribution of Lyapunov exponent is not normal anymore and both of the localization length and var(γ)are constants, which don't change with the changing system size. Moreover, we found that the single parameter scaling theory is not valid anymore in both of these two regimes and LRC plays dual role in the system:it may either suppress or enhance localization.
In 1D system with long-range correlated off-diagonal disorder, we found that, even though such a system has the property of chiral symmetry, like the introducing of LRC in on-site energies induces localization-delocalization transition (LDT), the introducing of LRC in hopping integrals also induces LDT and the critical correlation strength is alsoαc= 2.0. On the other hand, under the condition of fixed average hopping integral t0 and fixed eigenenergy E, whenα>αc and disorder strength W is small, the increasing of W will induce delocalization-localization transition (DLT) and the critical disorder strength is Wc= 2t0-|E|. In addition, under the condition of fixed t0 and fixed W, whenα>αand W is small, the changing of E will also induce DLT and the critical eigenenergy is Ec=±|2t0-W|.
After studied the optical transport properties of the disordered chain of microspherical resonators, we found that, for a long chain, even though the chain behaves in some aspects as a typical strong Anderson localized system, its radiative loss statistics is strongly different from that observed for other lossy optical systems. The distribution function of loss A in this chain is found as f(A)∞A-2 exp[-(a-A)2/(b2A2)], the mean value and variance of A, which are related to parameters a and b, are the functions of a single scaling parameter:ratio between the localization length and the loss length in ordered chains. For a short chain, on another hand, we found that the band edges are much more sensitive to disorder than the band center showing that the frequencies near the band edges are more suitable to be used to study the effect of slow light.
引文
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