沈建琪, 邱俊程

(上海理工大學(xué) 理學(xué)院,上海 200093)

微球內(nèi)部電磁場(chǎng)的Mie理論數(shù)值計(jì)算

沈建琪, 邱俊程

(上海理工大學(xué) 理學(xué)院,上海 200093)

采用Mie散射理論對(duì)平行光束與微球顆粒相互作用的內(nèi)部場(chǎng)進(jìn)行嚴(yán)格的描述并進(jìn)行數(shù)值計(jì)算.對(duì)吸收性微球顆粒內(nèi)部場(chǎng)進(jìn)行數(shù)值計(jì)算時(shí),如果粒徑及折射率虛部較大,其徑向函數(shù)會(huì)出現(xiàn)數(shù)據(jù)溢出,從而導(dǎo)致計(jì)算失敗,得出不合理的計(jì)算結(jié)果.通過(guò)引進(jìn)徑向函數(shù)的變形,提出了一種克服數(shù)據(jù)溢出的算法,將徑向函數(shù)的數(shù)值控制在合適的范圍以?xún)?nèi).通過(guò)數(shù)值計(jì)算得到了合理的結(jié)果,證明該算法克服了數(shù)據(jù)溢出,可以穩(wěn)定地計(jì)算各種平行光照射下的微球顆粒內(nèi)部場(chǎng).

Mie理論; 內(nèi)部場(chǎng); 數(shù)值計(jì)算

球形顆粒與平面電磁波(平行光)的相互作用可以通過(guò)Maxwell方程組得到嚴(yán)格解,即Lorenz Mie理論[1-2].隨著微光學(xué)技術(shù)的不斷發(fā)展,人們對(duì)微球內(nèi)部電磁場(chǎng)及其分布的研究越來(lái)越重視.微球內(nèi)部電磁場(chǎng)分布在微激光的研制、醫(yī)學(xué)應(yīng)用等領(lǐng)域具有重要價(jià)值[3-5].對(duì)微球內(nèi)部電磁場(chǎng)的數(shù)值計(jì)算是研究微球內(nèi)部電磁場(chǎng)分布及其諧振機(jī)制、熱效應(yīng)等特性的基礎(chǔ),涉及到球坐標(biāo)系中的特殊函數(shù)計(jì)算,包括對(duì)Bessel函數(shù)和締合勒讓德函數(shù)的計(jì)算.非耗散性微球的折射率為實(shí)數(shù),內(nèi)部電磁場(chǎng)的數(shù)值計(jì)算相對(duì)容易.但耗散性微球的折射率為復(fù)數(shù),對(duì)應(yīng)的數(shù)值計(jì)算比較困難,需要克服數(shù)據(jù)溢出的問(wèn)題.現(xiàn)有文獻(xiàn)主要針對(duì)粒徑較小和非耗散性(或弱耗散)微球內(nèi)部場(chǎng)的研究,對(duì)于粒徑較大的情況主要通過(guò)幾何光學(xué)近似計(jì)算[6-9].

本文對(duì)微球內(nèi)部電磁場(chǎng)的數(shù)值計(jì)算方法進(jìn)行研究,重點(diǎn)考慮耗散性微球折射率參數(shù)對(duì)特殊函數(shù)計(jì)算的影響,提出了一種克服數(shù)據(jù)溢出的算法,并通過(guò)數(shù)值計(jì)算進(jìn)行驗(yàn)證.

1 內(nèi)部場(chǎng)表達(dá)式

如圖1所示,根據(jù)Mie散射理論,時(shí)諧項(xiàng)取exp (-iωt),且入射光是電矢量沿x方向振動(dòng)、沿z軸方向傳播的線偏振光時(shí),微球內(nèi)部電磁場(chǎng)場(chǎng)強(qiáng)在球坐標(biāo)系中的3個(gè)分量[2]分別為

(1)

圖1 偏振平行光與微球相互作用示意圖

(2)

它們滿足如下遞推關(guān)系:

(3)

其初始值為π0=0,π1=1.

Mie系數(shù)cn和dn的表達(dá)式為

(4)

第一類(lèi)Riccati-Bessel函數(shù)和第一類(lèi)Riccati-Hankel函數(shù)以及它們的導(dǎo)數(shù)滿足相同的遞推關(guān)系.

(5)

(6)

2 算法描述

根據(jù)式(1),微球內(nèi)部電磁場(chǎng)是無(wú)窮多個(gè)分波的疊加,其高階項(xiàng)較弱,因此,在數(shù)值計(jì)算中可選取一個(gè)截止項(xiàng),對(duì)應(yīng)的截止階數(shù)ns由下式?jīng)Q定[10-11]:

(7)

微球內(nèi)部電磁場(chǎng)的計(jì)算包含了散射角函數(shù)πn和τn的計(jì)算、Mie系數(shù)cn和dn的計(jì)算以及徑向函數(shù)ψn(ρ)和ψ′n(ρ)的計(jì)算.散射角函數(shù)πn和τn的數(shù)值計(jì)算可由式(3)給出的遞推關(guān)系從低階函數(shù)π0和π1依次迭代得到,其數(shù)值在計(jì)算機(jī)可執(zhí)行范圍內(nèi)不存在溢出問(wèn)題.

Mie系數(shù)cn和dn的計(jì)算和徑向函數(shù)ψn(ρ)和ψ′n(ρ)的計(jì)算需要考慮復(fù)數(shù)折射率導(dǎo)致的溢出問(wèn)題.與時(shí)諧項(xiàng)exp(-iωt)相對(duì)應(yīng),折射率的虛部為正數(shù),m=mre+imim(即mim≥0).因此,無(wú)因次徑向參數(shù)ρ=mkr也是一個(gè)復(fù)數(shù),且靠近微球邊界(即r→α)時(shí),ρ的虛部會(huì)比較大,ρim→mimα.從式(6)給出的遞推公式及其初始值可知,ψn(ρ)和ψ′n(ρ)與exp(ρim)成正比.因此,如果mimα很大,ψn(ρ)和ψ′n(ρ)可能溢出,導(dǎo)致計(jì)算失敗.基于同樣的原因,Mie系數(shù)cn和dn的數(shù)值計(jì)算存在同樣的問(wèn)題.因此,在計(jì)算內(nèi)部場(chǎng)分布時(shí),為了克服數(shù)值的溢出,需要進(jìn)行相關(guān)的處理.將式(1)的內(nèi)部場(chǎng)表達(dá)式改寫(xiě)為

(8)

對(duì)Mie系數(shù)cn和dn的表達(dá)式(式(4))作進(jìn)一步推導(dǎo),可得

(9)

(10)

(11)

(12)

式(10)中函數(shù)Nn(α,β)的計(jì)算可采用如下遞推關(guān)系:

(14)

(15)

3 數(shù)值計(jì)算驗(yàn)證

微球內(nèi)任意一點(diǎn)的電場(chǎng)能量密度

(16)

式中,ε1是微球的介電常數(shù).

當(dāng)入射光為自然光時(shí),內(nèi)部電場(chǎng)各分量為

(17)

本文僅考慮微球內(nèi)部場(chǎng)的分布,不考慮場(chǎng)的絕對(duì)大小.為簡(jiǎn)便起見(jiàn),在以下的數(shù)值計(jì)算中取ε1/4=1.

圖2給出了采用式(8)計(jì)算得到的粒徑無(wú)因次參數(shù)α=200時(shí),各種不同折射率的微球內(nèi)部場(chǎng)曲面分布圖.其中,水滴的折射率為m=1.33,Cu,Ag和Al微球在入射光波長(zhǎng)為632.8 nm時(shí)的復(fù)數(shù)折射率分別為m=0.307 03+3.434 5 i,m=0.156 67+3.804 5 i和m=1.266 7+7.281 1 i.

圖2 各種不同折射率的微球內(nèi)部場(chǎng)曲面分布圖

數(shù)值計(jì)算顯示,當(dāng)微球折射率為實(shí)數(shù)或折射率虛部較小(圖2(a)和2(b))時(shí),式(1)和式(8)的數(shù)值計(jì)算結(jié)果相同.但是,當(dāng)微粒折射率虛部較大(圖2(c)和2(d))時(shí),采用式(1)計(jì)算內(nèi)部場(chǎng)遇到數(shù)據(jù)溢出問(wèn)題;而采用式(17)計(jì)算依然可以得到合理的結(jié)果.這是由徑向參數(shù)ρ=mkr虛部(即ρim)的大小決定的.在式(1)的計(jì)算中,其中間變量與e-ρim有關(guān),當(dāng)ρim≥710時(shí),可導(dǎo)致數(shù)值出現(xiàn)溢出.在所考慮的幾個(gè)算例中,水滴的ρim=0,因此,式(1)和式(8)實(shí)際上是一致的,可以得到完全一致的計(jì)算結(jié)果.對(duì)于Cu,Ag和Al微球,徑向參數(shù)虛部的值分別為ρim=686.9,ρim=760.9和ρim=145 6.因此,Cu微球的兩種計(jì)算方法結(jié)果一致,而Ag和Al微球的計(jì)算只能采用式(17).

從各種不同的吸收情況(對(duì)應(yīng)不同的復(fù)數(shù)折射率)來(lái)看,非吸收性微球內(nèi)部場(chǎng)遍及了整個(gè)微球,且在前向和后向具有較強(qiáng)的分布.對(duì)于吸收性微球,由于媒質(zhì)的吸收效應(yīng),內(nèi)部場(chǎng)主要分布在微球表面部分.

圖3給出了采用式(17)計(jì)算得到的粒徑無(wú)因次參數(shù)α=200時(shí),不同波長(zhǎng)的入射波照射下Li微球內(nèi)部場(chǎng)曲面分布圖.

數(shù)據(jù)計(jì)算結(jié)果顯示,隨著入射波波長(zhǎng)的增大,Li微球?qū)饽芰康奈赵鰪?qiáng),與之相對(duì)應(yīng),在顆粒吸收能力較弱時(shí)內(nèi)部場(chǎng)比較強(qiáng).反之,隨著吸收的增強(qiáng),內(nèi)部場(chǎng)逐漸減小,并趨向于在微球表面集中.

圖3 不同波長(zhǎng)入射波照射下Li微球內(nèi)部場(chǎng)曲面分布圖

4 結(jié) 論

在Mie理論內(nèi)部場(chǎng)數(shù)值計(jì)算中,其徑向函數(shù)(即Riccati-Bessel函數(shù)和Riccati-Hankel函數(shù))的數(shù)值范圍與微球顆粒的吸收特征(即復(fù)數(shù)折射率的虛部)有關(guān),強(qiáng)吸收情況下會(huì)導(dǎo)致數(shù)據(jù)溢出.本文對(duì)徑向函數(shù)進(jìn)行變形處理,將數(shù)值控制在合理的范圍內(nèi),克服了數(shù)據(jù)計(jì)算溢出問(wèn)題.不同吸收參數(shù)情況下的數(shù)值計(jì)算結(jié)果表明,該算法穩(wěn)定可靠,可以得到合理的計(jì)算結(jié)果.

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(編輯:石 瑛)

Numerical Calculation of the Electromagnetic Field Inside the Microsphere with Mie Theory

SHEN Jianqi, QIU Juncheng

(CollegeofScience,UniversityofShanghaiforScienceandTechnology,Shanghai200093,China)

The classical Mie theory was used to exactly describe the field inside the sphere in the interaction between the parallel beams and microsphere particles.If the particle size and the imaginary part of the refractive index are large,The data overflow by the calculation of radial function for an absorbing microsphere particle will appear,which leads to the failure of the calculation,and the unreasonable results are obtained.In order to overcome the overflow,an algorithm was proposed by introducing modified radial functions so as to keep their values in an appropriate range.Numerical calculations were conducted and reasonable results were obtained.The results demonstrate that the algorithm can overcome the data overflow successfully,and it can also be used to calculate any internal field of microsphere particles interacting with various parallel optical beams.

Mietheory;internalfield;numericalcalculation

1007-6735(2017)02-0159-06

10.13255/j.cnki.jusst.2017.02.011

2016-05-19

國(guó)家自然科學(xué)基金資助項(xiàng)目(NSFC51476104)

沈建琪(1965-),男,教授.研究方向:光散射理論與顆粒測(cè)試技術(shù)研究.E-mail:jqshenk@163.com

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