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Scattering of evanescent wave by multiple parallel infinite cylinders near a surface
Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
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Scattering of evanescent wave by multiple parallel infinite cylinders near a surface Siu-Chun Lee n Applied Sciences Laboratory, Inc., Baldwin Park, CA 91706, USA
a r t i c l e in f o
abstract
Article history: Received 28 March 2014 Received in revised form 10 June 2014 Accepted 11 June 2014 Available online 19 June 2014
This paper presents an exact analytical solution for the scattering of evanescent wave by an arbitrary collection of parallel infinite cylinders located near the surface of an optically denser substrate. The evanescent wave is generated by total internal reflection of the source wave propagating within the substrate at greater than the critical angle. The source wave is arbitrarily polarized and propagates in the plane perpendicular to the axes of the cylinders. The theoretical formulation utilizes Hertz potentials, which accounts for the near-field scattering between the cylinders and Fresnel reflection of angular distribution of scattered waves from the surface. Analytical formulas are derived for the electric and magnetic fields and Poynting vector of the scattered radiation in the near field, as well as their asymptotic forms in the far-field. Numerical results are presented to illustrate the scattering of evanescent wave by several configurations of cylinders located near the surface. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Evanescent wave Surface wave interaction Optical tunneling Infinite cylinder Far-field scattering
1. Introduction There are significant interests in the scattering of evanescent wave by particles at or near a surface due to its relevance to near-field microscopy. These applications include scanning optical microscopy, scanning electron microscopy, scanning tunneling microscopy, etc. The particles are either located on, or suspended in an aqueous solution above, the surface of the substrate that has a higher refractive index. A light source illuminates the interface from within the substrate at an angle greater than critical angle. Total internal reflection occurs as a result, which gives rise to a transmitted wave that decays exponentially with distance from the surface. This decaying wave is called an evanescent wave. Optical tunneling
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http://dx.doi.org/10.1016/j.jqsrt.2014.06.006 0022-4073/& 2014 Elsevier Ltd. All rights reserved.
occurs when a particle is placed near the surface. The size, composition, and position of the particle can be determined by measuring the intensity and polarization of the scattered wave. Optical microscopy applications using evanescent wave have been designated various names in the literature. Called total internal reflection microscopy (TIRM), Prieve et al. [1,2] utilized the technique to study the equilibrium and dynamic behavior of charged particles near flat surfaces. The vertical motion of the particle above the interface is determined by measuring the intensity of the scattered light. In studies on the near-wall dynamics of colloidal systems, the penetration depth of the evanescent wave is tuned by changing the angle of incidence of the source beam. This technique is called Evanescent wave dynamic light scattering (EWDLS), which enables probing of the colloidal system at different length scales [3,4]. Aslan et al. [5] utilized the polarization of the scattered wave to characterize the particles and called this approach
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
elliptically polarized surface-wave scattering (EPSWS). Analyses of evanescent scattering generally utilize numerical and semi-analytical methods. Numerical analyses based on the discrete source and T-matrix methods have been used to treat the case of spherical particle [6–8]. Many literature citations can be found in the references of the above papers, which are not repeated here. Characterization of cylindrical obstacles on a surface is an important fundamental problem that has many applications. The objective of this paper is to present an analytical solution based on Hertz potentials for evanescent scattering by multiple infinite parallel cylinders located near the surface of a substrate. The present problem shares some commonalities with that of scattering by cylinders buried in a half space [9–11], because both cases involve scattering near a boundary. The commonalities allow the adaption of the theoretical treatment on near-field scattering between cylinders and Fresnel reflection of scattered waves from the half space interface. However, the problem of cylinders buried in a half space differs from the present study in several major aspects. The differences include: (1) the source wave was transmitted from a lower refractive index medium into a medium of higher refractive index; (2) the cylinders were located in the higher refractive index medium; and (3) backscattering transmitted out of the half space containing the cylinders was of interest. The conditions of the present problem are exactly opposite to (1)–(3), and the quantity of interest is the forward scattering in the same half space where the cylinders are located. In the present problem the source wave is arbitrarily polarized and propagates within the substrate in the plane perpendicular to the axes of the cylinders. The source wave propagates at greater than the critical angle, thus giving rise to an evanescent wave across the interface and incident on the cylinders. For generality each cylinder is radially stratified and distinct, and no restriction is placed on their size and location. In the following sections the theoretical formulation is first presented, followed by numerical examples to illustrate scattering of evanescent wave by different configurations of cylinders.
and magnetic (H) fields in the vicinity of a cylinder can be written as 0 - 1 0 -þ 1 0 -s 1 0 -r þ 1 0 -s 1 E E E E N N B E ψj C B ψj C B ψ ;kj C B ψ ;kj C B ψ j C @ - A ¼ @ - þ A þ ∑ @ -s A þ ∑ @ -r þ A þ @ -s A kaj k¼1 Hψ j H ψ ;kj Hψ j Hψ j H ψ ;kj ð1Þ where the first 3 terms on the right hand side correspond to the respective incident waves, and the last term refers to the scattered wave from the cylinder. The subscript ψ ð ¼ u; vÞ refers to the polarization of the source wave, which can be magnetic (u) or electric (v) mode. These fields satisfy Maxwell's relations given in terms of the Hertz potentials ðu; vÞ as [12] ! i ! ! E ¼ ∇ ð e z vÞ þ ∇ ∇ ð e z uÞ k
ð2Þ
~ ! i m ! ! H ¼ ∇ ð e z uÞ þ ∇ ∇ ð e z vÞ μ μko
ð3Þ
~ is refractive index, μ is the permeability, where m ~ o Þ is the propagation constant of the medium, k ð ¼ mk and ko is that for free space. The formulation for each of the terms in Eq. (1) is described in the following sections. 2.1. Source wave in substrate The primary incident wave on the cylinders arises from the source wave transmitted across the media 1–2 interface. The Hertz potential ψ ð ¼ u; vÞ of the source wave in medium 1 can be written in terms of the complex amplitudes αiψ as !i !
ψ i ¼ αiψ expð i k U R P Þ where
! RP
ð4Þ
is the radial vector to point P, and
!i
! ! k ¼ k1 ð cos θ1 e x sin θ1 e y Þ
ð5Þ
is the propagation vector in the substrate. The source wave for each polarization mode in medium 1 is formulated by
2. Theory Fig. 1 shows a schematic diagram of the present problem. An arbitrary configuration of infinite cylinders, each of which is distinct and radially stratified, is aligned parallel to the Z-axis near the medium 1–2 interface. Media 1 and 2 are both non-absorbing dielectrics with ~ 1 and m ~ 2 , respectively, and real refractive indexes m ~ 1 4m ~ 2 . In typical optical microscopy systems medium m 1 can be a glass substrate and medium 2 is either air or an aqueous solution. The source wave propagates in medium 1 in the XY plane at an angle θ1 inclined from the X axis. The pertinent scattering phenomena involving the cylinders are depicted in Fig. 2. The primary incident wave on the cylinders arises from the transmitted source wave. The secondary and tertiary incident waves include the near-field scattered waves from other cylinders and reflected waves from the substrate's surface due to scattered waves from all cylinders. The total electric (E)
253
Fig. 1. Schematic diagram of the present problem.
254
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
Fig. 2. Scattering interactions for cylinders located near a surface.
utilizing Eq. (4) in (2) and (3): 0 1 !i B Eψ C B C @ !i A H ψ =m1 " i ! ! # αv δψ v ð sin θ1 ! e x þ cos θ1 e y Þ þ αiu δψ u e z ¼ ik1 ! ! ! αiu δψ u ð sin θ1 e x þ cos θ1 e y Þ þ αiv δψ v e z !i ! ð6Þ expð i k U R P Þ ~ 1 =μ1 and ðδψ u ; δψ v Þ are Kronecker delta where m1 ¼ m functions. The factor expðiωtÞ, where ω is the angular frequency, has been omitted for brevity. The incident wave is elliptically polarized if αiu a αiv and their phase difference is not a multiple of π =2, circularly polarized if αi ¼ αi and their phase difference is an odd multiple of u v π =2, and linearly polarized if their phase difference is a multiple of π . 2.2. Primary incident wave on cylinders We denote the transmitted source wave as the primary ~ 2 the trans~ 1 4m incident wave on the cylinders. As m mitted wave can be a propagating or an evanescent wave depending on the angle of incidence. Continuity of the tangential field components across the interface yields the primary incident wave at cylinder j as 0 !þ 1 E ψj B C @ !þ A H ψ j =m2 0 !þ 1 ! þ þ !þ α δ ψ u e z þ αv δψ v P M u Aexpð i k U ! R jP Þ ¼ ik2 εj @ þ ! ! αuþ δψ u P M þ αvþ δψ v e z ð7Þ !þ ! ! ~ where m2 ¼ m 2 =μ2 , εj ¼ expð i k U R j Þ and R j are the phase shift and radial position of the cylinder from the
! interface, respectively, R jP is the radial distance of point P relative to the cylinder, ðαuþ ; αvþ Þ ¼ ðαiu t u12 ; αiv t v12 Þðk1 =k2 Þ are the amplitudes, and ðt u12 ; t v12 Þ are the interface transmissivity [12]. The unit vectors are given by
!þ ! þ! P M ¼ ðk1 sin θ1 e x þkx e y Þ=k2
ð8Þ
!þ ! þ! k ¼ kx e x k1 sin θ1 e y
ð9Þ
where 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < k22 k21 sin 2 θ1 4 0; þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kx ¼ > : i k21 sin 2 θ1 k22 ;
θ1 o θc θ1 4 θc
ð10Þ
~ 2 =m ~ 1 Þ is the critical angle. The phase and θc ¼ sin 1 ðm shift εj is oscillatory when θ1 o θc and exponentially decaying when θ1 4 θc . Total internal reflection occurs when the source wave propagates at θ1 4 θc . The transmitted wave is an evanescent wave that decays rapidly with distance from the interface. To formulate the primary incident wave on the cylinders, we define θe based on Eq. (9) as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð11Þ expðiθe Þ ¼ k2 k1 sin 2 θ1 =k2 þ ik1 sin θ1 =k2 which yields
θe ¼
8 < sin 1 ðk1 sin
θ1 =k2 Þ;
: π =2 i ln ½ðk1 sin
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
θ1 o θ c
θ1 k21 sin 2 θ1 k22 Þ=k2 ; θ1 4 θc ð12Þ
The former is real that corresponds to Snell's law, whereas the latter is complex. Utilizing Eq. (11) allows the expansion of the exponential term in Eq. (7) as !þ ! expð i k U R jP Þ ¼
1
∑
n ¼ 1
ð iÞn exp½inðγ jP þ θe ÞJ n ðk2 RjP Þ ð13Þ
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
where γ jP is the azimuth angle of point P relative to cylinder j and J n is the integer Bessel function. The primary incident wave on cylinder j follows upon transforming Eq. (7) into cylindrical coordinates centered about the cylinder: 0 !þ 1 E ψj 1 B C @ !þ A ¼ k2 εj ∑ ð iÞn exp½inðγ jP þ θe Þ n ¼ 1 H ψ j =m2 0 1 ! ! in þ þ J αþ δ ! e J n' αv δψ v e jγ þ iJ n αu δψ u e z k2 RjP n v ψ v jR A ð14Þ @ in ! ! þ þ J αþ δ ! e þ J n' αu δψ u e jγ þ iJ n αv δψ v e z k2 RjP n u ψ u jR
255
these scatter-reflected waves is crucial to the proper treatment of the interface in evanescent wave scattering. The theoretical treatment of scattering at the interface follows that given in [9–11]. The angular distribution of backscattered waves is formulated by utilizing the expansion [9] Z 1 ð iÞn 1 exp½ in sin ðky =k2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðinγ jP ÞH n ðk2 RjP Þ ¼ 2 2 π 1 k2 ky ! ! ð18Þ expð i k s U R jP Þ dky where
where J n' is the derivative with respect to the argument.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! 2 2! k s ¼ k2 ky e x þky e y
2.3. Scattered waves and near-field secondary incident waves
is the propagation vector of the backscattered wave. Utilizing Eq. (18) in (15) yields
ð19Þ
0
The scattered wave from cylinder j is expressed similar to Eq. (14) as a function of unknown coefficients ðbjn ; ajn Þ and Hankel function of the second kind H n as 0 !s 1 E ψj 1 B C @ !s A ¼ k2 ∑ ð iÞn expðinγ jP Þ n ¼ 1 H ψ j =m2 0 1 ! ! in H a δ e H n' ajn δψ v ! e jγ þiH n bjn δψ u e z k2 RjP n jn ψ v jR A ð15Þ @ in ! ! H b δ ! e þH n' bjn δψ u e jγ þ iH n ajn δψ v e z k2 RjP n jn ψ u jR where H n' denotes the derivative with respect to the argument. Dependent scattering occurs in the near field because the cylinders are located in the near-field zone of each other due to the small spacing between them. It involves scattering of non-plane waves because the nearfield scattered waves from a cylinder have not recovered into a plane wave when incident on other cylinders. The near-field secondary incident wave on cylinder j due to scattered waves from cylinder k is formulated from Eq. (15) by applying the addition theorem of Bessel functions [13]: 0
1 !s E ψ ;kj 1 1 B C ∑ ð iÞn expðinγ jP ÞGjn @ !s A ¼ k2 ∑ ks n ¼ 1 s ¼ 1 H ψ ;kj =m2 0 ! ! ! 1 in J a δ e jR J n' aks δψ v e jγ þiJ n bks δψ u e z k2 RjP n ks ψ v A U@ ! ! ! in J b δ e jR þ J n' bks δψ u e jγ þiJ n aks δψ v e z k2 RjP n ks ψ u
ð16Þ where ¼ ð iÞs n exp½iðs nÞγ kj H s n ðk2 Rjk Þ Gjn ks
!s 1 Z 1 E ψj ! ! B C expð i k s U R jP Þ @ !s A ¼ i 1 H ψ j =m2 2 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2 2! ! 6 X ψ j e z Y ψ j ðky e x þ k2 ky e y Þ=k2 7 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 dky ! ! 2 2! X ψj ðky e x þ k2 ky e y Þ=k2 þ Y ψj e z
ð20Þ where ! X ψj Y ψj
h i ! 1 exp in sin ky =k2 bjn δψ u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ∑ ð 1Þn ajn δψ v 2 2 π n ¼ 1 k2 ky k2
1
ð21Þ are the amplitudes of the backscattered fields. The fields given by Eq. (20) are denoted as scatterincident waves because they arise from the backscattered wave incident at the half space interface. The scatterincident waves give rise to reflected and transmitted waves upon incident at the interface. We express the latter in similar forms as Eq. (20) in terms of unknown amplitudes ðX rψþj ; Y rψþj Þ and ðX tψj ; Y tψj Þ as 0
!r þ 1 Z 1 E ψj !þ ! ! ! B C expð i k s U R P þ i k s U R j Þ @ !r þ A ¼ i 1 H ψ j =m2 2 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2 2! rþ! rþ 6 X ψ j e z þY ψ j ð ky e x þ k2 ky e y Þ=k2 7 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 dky ! ! 2 2! X rψþj ðky e x k2 ky e y Þ=k2 þ Y rψþj e z
ð22Þ !t 1 Z 1 E ψj ! ! ! ! C B expð i k t U R P þ i k s U R j Þ A ¼ i @ !t 1 H ψ j =m1 0
ð17Þ
in which γ kj is the azimuth angle of cylinder j relative to k, and Rjk is the radial distance between cylinders j and k. 2.4. Scattering at the half space interface Scattered waves from the cylinders propagate into all directions. Those scattered in the backward direction ðπ =2 r γ jP r 3π =2Þ would be reflected from the half space interface and become incident waves on the cylinders. The total field in the forward direction then consists of both scattered and reflected waves. A rigorous formulation of
2
3 ! ! ! X tψj e z Y tψj ðky e x þ γ 1 e y Þ=k1 5 dky 4 ! ! ! X tψj ðky e x þ γ 1 e y Þ=k1 þ Y tψj e z
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 where γ 1 ¼ k1 ky , and ffi ! þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2 2! k s ¼ k2 ky e x þ ky e y ! ! ! k t ¼ γ 1 e x þ ky e y
ð23Þ
ð24Þ ð25Þ
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S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
are the propagation vectors of the respective fields. Over the range 1 rky r 1, the x-component of the respective propagation constant varies as 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 > < k2 ky ; k2 4 ky !þ ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð26Þ ks Uex¼ > i k2 k2 ; k o k : y 2 y 2 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 > < k1 ky ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ1 ¼ > : i k2y k21 ;
k1 4 ky k1 o ky
ð34Þ ð27Þ
The boundary conditions require the continuity of the tangential components of the field vectors across the half space interface: ðEψ j ; H ψ j Þsy;z þ ðEψ j ; H ψ j Þry;zþ ¼ ðEψ j ; H ψ j Þty;z
r vv þ ðky Þ ¼
Y sjn
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 k2 ky m2 k2 k1 ky qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 2 2 2 m1 k1 k2 ky þ m2 k2 k1 ky m1 k 1
where ψ ;jn
Rψ ;ks ¼
ð28Þ
Applying Eqs. (20), (22) and (23) in (28) yields the interface reflectivity as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 rþ m2 k1 k2 ky m1 k2 k1 ky X jn q ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð29Þ ðk Þ ¼ ¼ r uu þ y 2 2 2 2 X sjn m k k2 ky þm1 k2 k1 ky 2 1 Y rjnþ
cylinder. We obtain after some manipulations 0 !r þ 1 E ψ ;kj 1 1 B C ∑ ð iÞn expðinγ jP Þ @ !r þ A ¼ k2 ∑ n ¼ 1 s ¼ 1 H ψ ;kj =m2 0 1 v;jn u;jn in J Rv;jn a δ ! e J n' R a δ ! e þ iJ n R b δ ! e k2 RjP n v;ks ks ψ v jR v;ks ks ψ v jγ u;ks ks ψ u z @ A U u;jn u;jn v;jn ! ! ! in k2 RjP J n Ru;ks bks δψ u e jR þ J n' Ru;ks bks δψ u e jγ þiJ n Rv;ks aks δψ v e z
ð30Þ
The scatter-reflected wave is obtained by substituting Eqs. (21), (29) and (30) in (22): 0
!r þ 1 Z 1 E ψj 1 B C @ !r þ A ¼ ik2 ∑ n ¼ 1 1 H ψ j =m2
2
3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2! u;P v;P ! R b δ e z Rv;jn ajn δψ v ðky ! k k e y Þ=k2 ψ u jn 2 y u;jn ex 6 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 5dky 2 2! v;P ! ! Ru;P b δ ðk e k k e Þ=k þ R a δ e 2 2 y y u;jn jn ψ u y x v;jn jn ψ v z ð31Þ
Z
exp½ iðs þ nÞ sin 1 ðky =k2 Þ ψ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rψ 2 2 π 1 k2 ky qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 exp i k2 ky ðxj þ xk Þ iky ðyj yk Þ dky ð 1Þs
1
ð35Þ
ψ ;jn
We denote Rψ ;ks as the integrated reflection coefficient between cylinders k and j, which accounts for scatterreflected waves over all directions. 2.6. Solution for the unknown coefficients The coefficients ðbjn ; ajn Þ are the only unknowns in the theoretical formulation. They are solved by imposing the continuity of the tangential components of the fields across all interfaces of each cylinder. The present study considers radially stratified cylinders, for which the internal fields in each interior layer have been detailed in [14]. !i The boundary conditions involve incident ð Ψ ψ ;q Þ and !s scattered ð Ψ ψ ;q Þ waves !i
!s
!i
!s
ðΨ ψ j;q 1 þ Ψ ψ j;q 1 Þjγ ;z ¼ ðΨ ψ j;q þ Ψ ψ j;q Þjγ ;z
ð36Þ
between layers q 1 and q, where Ψ ¼ ðE; HÞ, and !þ
!s
N
N
!s
!r þ
ðΨ ψ j þ Ψ ψ j þ ∑ Ψ ψ ;kj þ ∑ Ψ ψ ;kj Þj kaj
k¼1
where
!i
γ ;z
!s
¼ ðΨ ψ ;j;Q j þ Ψ ψ ;j;Q j Þj
γ ;z
ð37Þ n
1
ð 1Þ exp½ in sin ðky =k2 Þ ψ ;P qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rψ ;jn ¼ 2 2 π
k2 ky qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ψþ 2 2 exp i k2 ky ðxj þ xP Þ iky yjP r ψ
at the surface. Solving the above yields two independent sets of equations ð32Þ
N
∑
1
þ Ru;jn b gbks ∑ fδjk δns þ ½ð1 δjk ÞGjn ks u;ks jn o;I
k ¼ 1 s ¼ 1
¼ αuþ εj expðinθe Þbjn
o;I
signifies the reflection of scattered waves from cylinder j to point P. N
∑
ð38Þ
1
þ Rv;jn ao;II gaks ∑ fδjk δns þ ½ð1 δjk ÞGjn ks v;ks jn
2.5. Incident waves on cylinders due to reflection of scattered waves
k ¼ 1 s ¼ 1 ¼ vþ j expðin e Þao;II jn
The scatter-reflected wave from the interface serves as another source of incident wave on the cylinders. The incident wave on cylinder k due to the reflection of scattered waves from cylinder j is formulated by first applying the translation ! ! ! R P ¼ R kþ R ð33Þ kP '
where bjn and ao;II jn are the coefficients for the transverse
to Eq. (32). The generating function of Bessel function is next applied, followed by the transformation from Cartesian into cylindrical coordinates centered about the
α ε
θ
ð39Þ
o;I
magnetic (TM) and transverse electric (TE) mode, respectively, for an isolated cylinder in an unbounded medium 2. The influence of the various pertinent scattering interactions is evident in the functional form of Eqs. (38) and (39). The term on the right hand side is due to the source wave incident at the cylinder. Near-field scattering of nonplane waves between closely spaced cylinders is signified by Gjn that contains the Hankel function H n ðk2 Rjk Þ. Its ks
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
influence diminishes with increasing separation between ψ ;jn the cylinders. The effect of the interface is given by Rψ ;ks that captures the interaction between a pair of cylinders over all directions. If the half space interface disappears, Eqs. (38) and (39) reduce to those for multiple cylinders in an unbound medium [13,14].
0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! T ψ v;j δψ v e γ þ T ψ u;j δψ u e z 2 @ A ¼ k2 expð ik2 RP Þ ! ! iπ k2 RP T ψ u;j δψ u e γ þT ψ v;j δψ v e z ð43Þ where T uu;j
! ¼ exp½ik2 Rj cos ðγ P γ j Þ
T vv;j
2.7. Scattered waves in the near- and far-fields The scattered wave consists of scattering from the cylinders and reflection of scattered waves from the surface. The former is given by Eq. (15), which is expressed in cylindrical coordinates centered about each cylinder. The latter is given by Eq. (31), which is prescribed relative to the Cartesian coordinate system at the surface. To obtain the angular distribution of the scattered radiation, it is necessary to transform both equations into the cylindrical coordinate system at the interface. Applying straightforward coordinate transformation to Eq. (15) yields the scattered waves from cylinder j relative to the half space interface as 1 !s E ψj B C @ !s A H ψ j =m2 2 3 ! ! ! ðEsψ j;R pj þ Esψ j;γ qj Þ e R þðEsψ j;γ pj Esψ j;R qj Þ e γ þ Esψ j;z e z 5 ¼ k2 4 s ! ! ! ðHψ j;R pj þ Hsψ j;γ qj Þ e R þ ðHsψ j;γ pj H sψ j;R qj Þ e γ þ Hsψ j;z e z
257
1
bjn
n ¼ 1
ajn
∑
! expðinγ P Þ ð44Þ
Eq. (41) involves the integrals given in Eq. (31), which can be evaluated by the method of stationary phase as described in the appendix. Using the saddle point value ky ¼ k2 sin γ P gives 0 !r þ 1 E ψj B C @ !r þ A H ψ j =m2 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! Rvvj δψ v e γ þRuuj δψ u e z 2 @ A ¼ k2 expð ik2 RP Þ ! ! iπ k2 RP Ruuj δψ u e γ þ Rvvj δψ v e z
0
ð45Þ where ! Ruuj Rvvj
ψj
1
uþ r^ u bjn
n ¼ 1
vþ r^ v ajn
∑
exp½inðπ γ P Þ
ð40Þ where pj ¼ cos ðγ P γ jP Þ, qj ¼ sin ðγ P γ jP Þ, and the component fields are defined by Eq. (15). Similarly, Eq. (31) is transformed into cylindrical coordinates with respect to the interface as 0 !r þ 1 0 rþ 1 E ψj Eψ j;x Erψþj;y Erψþj;z B C A @ !r þ A ¼ k2 @ r þ H ψ j;x H rψþj;y H rψþj;z H =m
¼ exp½ ik2 Rj cos ðγ P þ γ j Þ
!
ð46Þ
uþ r^ u
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 k1 k2 sin 2 γ P qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 2 m2 k1 cos γ P þ m1 k1 k2 sin 2 γ P
ð47Þ
vþ r^ v
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 k1 k2 sin 2 γ P qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 2 m1 k1 cos γ P þ m2 k1 k2 sin 2 γ P
ð48Þ
m2 k1 cos γ P m1
m1 k1 cos γ P m2
2
0
! ! 1 cos γ P e R sin γ P e γ B ! ! C C B @ sin γ P e R þ cos γ P e γ A ! ez
ð41Þ
in which the component fields are defined by Eq. (31). The time-averaged Poynting vector of the total scattered radiation is evaluated accordingly as " # N !s N !s þ !r þ !s !r þ co S ψ ¼ Re ∑ ð E ψ j þ E ψ j Þ ∑ ð H ψ k þ H ψ k Þn ð42Þ 8π j¼1 k¼1 Eqs. (40) and (41) are general expressions that apply to any distance from the interface. There is no apparent simplification for them except in the far-field. Most applications of evanescent scattering involve measurement of the scattered radiation at large distance from the surface. In the far field, i.e., k2 RjP -1, the following approximations can be applied: RjP RP Rj cos ðγ P γ j Þ, 1=RjP 1=RP , γ jP γ kP γ P , pj 1, and qj 0. Using the asymptotic expansion of the Hankel functions, Eq. (40) becomes 0 !s 1 E ψj B C @ !s A H ψ j =m2
Combining Eqs. (43) and (45) yields the E and H fields of the total scattered wave in the far field: 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !s 2 B Eu C ¼ k expð ik2 RP Þ @ !s A 2 i π k 2 RP H u =m2 ! ! N ez ∑ ðT uu;j þ Ruuj Þ ð49Þ ! eγ j¼1 0
1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !s ! ! N eγ 2 B Ev C expð ik2 RP Þ ∑ ðT vv;j þRvv;j Þ ! @ !s A ¼ k2 iπ k2 RP j¼1 ez H v =m2
ð50Þ for the magnetic and electric mode incident wave, respectively. The time-averaged Poynting vector for the far-field scattered radiation is as follows: !s þ co m2 k22 2 ! Sψ ¼ ðI uu δψ u þ I vv δψ v Þ e R 8π π k2 RP
ð51Þ
where 2 N I uu ðγ P Þ ¼ ∑ ðT uu;j þ Ruu;j Þ j ¼ 1
ð52Þ
258
2 N I vv ðγ P Þ ¼ ∑ ðT vv;j þ Rvv;j Þ j ¼ 1
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
ð53Þ
are the scattering amplitude functions for the respective modes. Along the horizon, i.e. γ P ¼ 7 π =2, Eqs. (47) and uþ vþ (48) yield r^ u ¼ r^ v ¼ 1. We then obtain T uu;j ¼ Ruu;j and T vv;j ¼ Rvv;j from Eqs. (49) and (50), and the Poynting vector vanishes as well. This indicates that vanishing of far field scattered radiation along the surface is caused by destructive interference between the scattered and reflected waves. 3. Numerical examples An effort was made to compare numerical results based on the present solution with those given in [15], which utilized Green function and numerical solution of the boundary integral equations, for scattering by two cylinders near a half space interface. An exact comparison cannot be made because no scale is shown in Fig. 2 in [15]. A qualitative comparison indicates a similar angular variation for the far-field scattered radiation for the case of ~ 1 ¼ 1:414, whereas considerable differences exist for the m ~ 1 ¼ 1:5; 1:732Þ. The greatest discrepancy other cases ðm occurs in the far-field scattered radiation along the horizon, i.e.γ P ¼ 7 π =2, for which [15] gave finite values compared to zero from the present solution. Vanishing of the far-field scattered radiation along the surface has been shown in [7] and discussed by Small et al. [16]. The solution method of [15] may need appropriate modification in order to reconcile the discrepancy with the condition along the surface. For the purpose of illustration we examine evanescent scattering by both single and multiple parallel gold cylindrical wires located in air (medium 2) on the surface of a substrate. The wires are 0.1 μm in radius, and the clearance between adjacent wires and that between the bottom wire and the substrate's surface are both 1 Å. Different ~ 1 ¼ 1:33; 1:5; μ1 ¼ 1) are substrate refractive indexes (m utilized for the numerical examples. The amplitudes of the TM and TE mode source wave in medium 1 are both unity. The free space wavelength is λo ¼ 0:6328 μm at which ~ Au 0:166 3:15i. m
Fig. 3a and b shows the scattering amplitudes ðI uu ; I vv Þ for the case of a single wire. The effect of incidence angle ~ 1 ¼ 1:5 ðθ1 ¼ 451; 601Þ is shown in Fig. 3a for m (θc ¼ 41:811). Because the amplitude of the evanescent incident wave is greater for θ1 ¼ 451 than for θ1 ¼ 601, the scattering amplitudes for θ1 ¼ 451 are generally higher than those for θ1 ¼ 601. The scattering amplitudes are asymmetric that peaks around the incident direction. Fig. 3b shows the effect of different substrate refractive ~ 1 ¼ 1:33; 1:5) for θ1 ¼ 601. The angular distriindexes (m butions show similar behavior as those in Fig. 3a. Because the amplitude of the evanescent incident wave is higher ~ 1 ¼ 1:33 (θc ¼ 48:751) than for m ~ 1 ¼ 1:5, the scatterfor m ~ 1 ¼ 1:33 are higher than those for ing amplitudes for m ~ 1 ¼ 1:5. Fig. 4a and b shows the results for a triangular m arrangement of three wires. Fig. 4a shows the effect of ~ 1 ¼ 1:5, and the results for different incidence angle for m substrate refractive indexes at the same incident angle are given in Fig. 4b. Interference of the scattered waves from the different wires gives rise to two distinct lobes for the magnetic mode ðI uu Þ that peak in roughly in both the forward and retro-reflected directions. Fig. 5a and b shows the results for a cluster of 7 wires arranged at the vertices of a hexagon. The scattering amplitudes for two incidence angles are shown in Fig. 5a, and those for two substrate refractive index at the same incident angle are shown in b. The presence of more wires resulted in complicated interference of the scattered and scatter-reflected waves, thus giving rise to more scattering lobes than in the case of 3 wires. To demonstrate the generality of the present solution, we present results for evanescent scattering by multiple dissimilar cylinders. The same hexagonal arrangement of cylinders as above is considered, except that the wires circumscribing the center gold wire are replaced by solid silica cylinders in one case and by hollow silica cylinders with an inner radius of 0:05 μm in the second case. The ~ SiO2 ¼ 1:45 i4:5 10 6 . The refractive index of silica is m far-field scattering amplitudes for the different cylinder clusters are shown in Fig. 6a and b. As silica is nearly nonabsorbing, the scattering amplitudes for the clusters with circumscribing silica rods are much higher than those for all gold wires that are strongly absorbing. Higher
~ 1 ¼ 1:33; 1:5). Fig. 3. Far-field scattering amplitude for a single gold wire: (a) two incident angles (θ1 ¼ 451; 601); (b) two substrate refractive index (m
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
scattering amplitudes are observed for the cluster containing solid silica rods, because they scatter more strongly than the hollow silica rods. The numerical results show scattering peaks in both the forward and retro-reflected directions. The former can
259
be attributed to the forward scattering characteristics generally occurs with particle scattering, whereas the latter is due to the images. The angular variation is dictated by the interference of the scattered and scatterreflected waves of all the cylinders. For a cluster of gold
~ 1 ¼ 1:33; 1:5). Fig. 4. Far-field scattering amplitude for 3 gold wires: (a) two incident angles (θ1 ¼ 451; 601); (b) two substrate refractive index (m
Fig. 5. Far-field scattering amplitude for a cluster of 7 gold wires: (a) two incident angles (θ1 ¼ 451; 601); (b) two substrate refractive index ~ 1 ¼ 1:33; 1:5). (m
~ 1 ¼ 1:5 and θ1 ¼ 601: (a) magnetic mode I uu ; (b) electric mode Ivv . Fig. 6. Far-field scattering amplitude for clusters of 7 dissimilar cylinders for m
260
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
wires, Fig. 5a and b shows that I vv exhibits more scattering lobes than I uu . The scattering pattern changed considerably when the circumscribing wires are replaced by either solid or hollow silica rods which are nearly non-absorbing. More scattering lobes then appear in I uu than I vv , as shown in Fig. 6a and b. These results indicate that the scattering pattern is strongly influenced by the size and material of the constituent cylinders. A generalization cannot be made on whether I uu or I vv exhibits more scattering lobes.
In the asymptotic limit of k2 RP -1, the modulus of exp½gðΓ Þ changes rapidly along the steepest paths along which ImðgÞ is constant. The stationary phase point is determined by g 0 ðΓ o Þ ¼ 0, which gives Γ o ¼ γ P from Eq. (iv), and ky ¼ k2 sin γ P follows from Eq. (ii). Expanding gðΓ Þ into Taylor series and keeping up to second order terms, Eq. (iii) yields Z 1 ∂ky lim I f ðΓ o Þ exp½gðΓ o Þ exp½g″ðΓ o ÞðΓ Γ o Þ2 =2 dΓ ∂ Γ k2 RP -1 1 Γo
ðvÞ
4. Summary An exact analytical solution of Maxwell's equations for evanescent wave scattering by multiple parallel infinite cylinders has been presented in this paper. The incident wave is arbitrarily polarized and propagates in the plane perpendicular to the axes of the cylinders. The present formulation utilized Hertz potentials to treat the evanescent incident wave, near-field scattering between closely spaced cylinders, and scattering and reflection at the surface of the substrate. Analytic formulas are derived for the E and H fields of the scattered waves, which consist of both forward scattered waves and boundary-reflected waves arising from the backscattered waves. The scattered waves in the far-field are shown to vanish along the horizon due to destructive interference between the forward-scattered and scatter-reflected waves. Numerical results for several configurations of cylinders were shown to illustrate the influence of substrate refractive index, incidence angle, and size and material of the cylinders on evanescent wave scattering. Appendix The method of stationary phase (MSP) is utilized to evaluate the far field scatter-reflected wave given by Eq. (31). Detailed discussion of MSP can be found in many standard texts. The brief description herein focuses on its application to the present problem. The Cartesian components of Eq. (31) all involve an integral in the form of Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 I¼ f ðky Þexpð i k2 ky xP iky yP Þ dky ðiÞ 1
where f ðky Þ is a complex function. By defining qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 expðiΓ Þ ¼ ð k2 ky þ iky Þ=k2 Eq. (i) is transformed into Z 1 ∂ky I¼ f ðky Þexp½g ðΓ Þ dΓ ∂Γ 1
ðiiÞ
ðiiiÞ
where gðΓ Þ ¼ ik2 RP cos ðΓ γ P Þ
ðivÞ
where g″ðΓ o Þ is the second order derivative. Eq. (45) readily follows after some manipulations. References [1] Prieve DC, Luo F, Lanni F. Brownian motion of a hydrosol particle in colloidal force field. Faraday Discuss Chem Soc 1987;83:297–307. [2] Prieve DC, Walz JY. Scattering of an evanescent surface wave by a microscopic dielectric sphere. Appl Opt 1993;32(9):1629–41. [3] Siegel R. Light scattering near and from interfaces using evanescent wave and ellipsometric light scattering. Curr Opin Colloid Interface Sci 2009;14(6):426–37. [4] Lisicki M, Cichocki B, Dhont KG, Lang PR. One-particle correlation function in evanescent wave dynamic light. J Chem Phys 2012;136: 204704. [5] Aslan MM, Menguc MP, Videen G. Characterization of metallic nanoparticles via surface wave scattering: B. Physical concept and numerical experiments. J Quant Spectrosc Radiat Transf 2005;93: 207–17. [6] Short MR, Tortel H, Litman A, Geffrin JM, Vaillon R, Abdeddaim R, et al. Evanescent wave scattering by particles on a surface: comparison between the discrete dipole approximation with surface interaction and the finite element method. ELS14, Lille, France; June 2013. [7] Doicu A, Eremin Y, Wriedt T. Scattering of evanescent waves by a particle on or near a plane surface. Comput Phys Commun 2001;134:1–10. [8] Eremina E, Wriedt T, Helden L. Analysis of evanescent waves scattering by a single particle in Total Internal Reflection Microscopy. In: PIER Symposium, Cambridge; 2006. [9] Lee SC, Grzesik JA. Light scattering by closely spaced parallel cylinders embedded in a semi-infinite dielectric medium. J Opt Soc Am A 1998;15:163–73. [10] Lee SC. Scattering by multiple parallel radially stratified infinite cylinders buried in a lossy half space. J Opt Soc Am A 2013;30(7): 1320–7. [11] Lee SC. Scattering by a radially stratified infinite cylinder buried in a lossy half space. J Opt Soc Am A 2013;30(4):565–72. [12] Born M, Wolf E. Principle of optics. Oxford: Pergamon Press; 1980. [13] Lee SC. Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders. J Appl Phys 1990;68(10):4952–7. [14] Lee SC. Scattering by closely-spaced radially stratified parallel cylinders. J Quant Spectrosc Radiat Transf 1992;48:119–30. [15] Belai OV, Frumin LL, Perminov SV, Shapiro DA. Scattering of evanescent wave by two cylinders near a flat boundary. Europhys Lett 2012;97(1):10007–18. [16] Small A, Fung J, Manoharan VN. Generalization of the optical theorem for light scattering from a particle at a planar interface. J Opt Soc Am A 2013;30(12):2519–25.
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Scattering of evanescent wave by multiple parallel infinite cylinders near a surface Siu-Chun Lee n Applied Sciences Laboratory, Inc., Baldwin Park, CA 91706, USA
a r t i c l e in f o
abstract
Article history: Received 28 March 2014 Received in revised form 10 June 2014 Accepted 11 June 2014 Available online 19 June 2014
This paper presents an exact analytical solution for the scattering of evanescent wave by an arbitrary collection of parallel infinite cylinders located near the surface of an optically denser substrate. The evanescent wave is generated by total internal reflection of the source wave propagating within the substrate at greater than the critical angle. The source wave is arbitrarily polarized and propagates in the plane perpendicular to the axes of the cylinders. The theoretical formulation utilizes Hertz potentials, which accounts for the near-field scattering between the cylinders and Fresnel reflection of angular distribution of scattered waves from the surface. Analytical formulas are derived for the electric and magnetic fields and Poynting vector of the scattered radiation in the near field, as well as their asymptotic forms in the far-field. Numerical results are presented to illustrate the scattering of evanescent wave by several configurations of cylinders located near the surface. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Evanescent wave Surface wave interaction Optical tunneling Infinite cylinder Far-field scattering
1. Introduction There are significant interests in the scattering of evanescent wave by particles at or near a surface due to its relevance to near-field microscopy. These applications include scanning optical microscopy, scanning electron microscopy, scanning tunneling microscopy, etc. The particles are either located on, or suspended in an aqueous solution above, the surface of the substrate that has a higher refractive index. A light source illuminates the interface from within the substrate at an angle greater than critical angle. Total internal reflection occurs as a result, which gives rise to a transmitted wave that decays exponentially with distance from the surface. This decaying wave is called an evanescent wave. Optical tunneling
n
Tel.: þ1 626 960 8800; fax: þ1 626 960 8810. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.jqsrt.2014.06.006 0022-4073/& 2014 Elsevier Ltd. All rights reserved.
occurs when a particle is placed near the surface. The size, composition, and position of the particle can be determined by measuring the intensity and polarization of the scattered wave. Optical microscopy applications using evanescent wave have been designated various names in the literature. Called total internal reflection microscopy (TIRM), Prieve et al. [1,2] utilized the technique to study the equilibrium and dynamic behavior of charged particles near flat surfaces. The vertical motion of the particle above the interface is determined by measuring the intensity of the scattered light. In studies on the near-wall dynamics of colloidal systems, the penetration depth of the evanescent wave is tuned by changing the angle of incidence of the source beam. This technique is called Evanescent wave dynamic light scattering (EWDLS), which enables probing of the colloidal system at different length scales [3,4]. Aslan et al. [5] utilized the polarization of the scattered wave to characterize the particles and called this approach
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
elliptically polarized surface-wave scattering (EPSWS). Analyses of evanescent scattering generally utilize numerical and semi-analytical methods. Numerical analyses based on the discrete source and T-matrix methods have been used to treat the case of spherical particle [6–8]. Many literature citations can be found in the references of the above papers, which are not repeated here. Characterization of cylindrical obstacles on a surface is an important fundamental problem that has many applications. The objective of this paper is to present an analytical solution based on Hertz potentials for evanescent scattering by multiple infinite parallel cylinders located near the surface of a substrate. The present problem shares some commonalities with that of scattering by cylinders buried in a half space [9–11], because both cases involve scattering near a boundary. The commonalities allow the adaption of the theoretical treatment on near-field scattering between cylinders and Fresnel reflection of scattered waves from the half space interface. However, the problem of cylinders buried in a half space differs from the present study in several major aspects. The differences include: (1) the source wave was transmitted from a lower refractive index medium into a medium of higher refractive index; (2) the cylinders were located in the higher refractive index medium; and (3) backscattering transmitted out of the half space containing the cylinders was of interest. The conditions of the present problem are exactly opposite to (1)–(3), and the quantity of interest is the forward scattering in the same half space where the cylinders are located. In the present problem the source wave is arbitrarily polarized and propagates within the substrate in the plane perpendicular to the axes of the cylinders. The source wave propagates at greater than the critical angle, thus giving rise to an evanescent wave across the interface and incident on the cylinders. For generality each cylinder is radially stratified and distinct, and no restriction is placed on their size and location. In the following sections the theoretical formulation is first presented, followed by numerical examples to illustrate scattering of evanescent wave by different configurations of cylinders.
and magnetic (H) fields in the vicinity of a cylinder can be written as 0 - 1 0 -þ 1 0 -s 1 0 -r þ 1 0 -s 1 E E E E N N B E ψj C B ψj C B ψ ;kj C B ψ ;kj C B ψ j C @ - A ¼ @ - þ A þ ∑ @ -s A þ ∑ @ -r þ A þ @ -s A kaj k¼1 Hψ j H ψ ;kj Hψ j Hψ j H ψ ;kj ð1Þ where the first 3 terms on the right hand side correspond to the respective incident waves, and the last term refers to the scattered wave from the cylinder. The subscript ψ ð ¼ u; vÞ refers to the polarization of the source wave, which can be magnetic (u) or electric (v) mode. These fields satisfy Maxwell's relations given in terms of the Hertz potentials ðu; vÞ as [12] ! i ! ! E ¼ ∇ ð e z vÞ þ ∇ ∇ ð e z uÞ k
ð2Þ
~ ! i m ! ! H ¼ ∇ ð e z uÞ þ ∇ ∇ ð e z vÞ μ μko
ð3Þ
~ is refractive index, μ is the permeability, where m ~ o Þ is the propagation constant of the medium, k ð ¼ mk and ko is that for free space. The formulation for each of the terms in Eq. (1) is described in the following sections. 2.1. Source wave in substrate The primary incident wave on the cylinders arises from the source wave transmitted across the media 1–2 interface. The Hertz potential ψ ð ¼ u; vÞ of the source wave in medium 1 can be written in terms of the complex amplitudes αiψ as !i !
ψ i ¼ αiψ expð i k U R P Þ where
! RP
ð4Þ
is the radial vector to point P, and
!i
! ! k ¼ k1 ð cos θ1 e x sin θ1 e y Þ
ð5Þ
is the propagation vector in the substrate. The source wave for each polarization mode in medium 1 is formulated by
2. Theory Fig. 1 shows a schematic diagram of the present problem. An arbitrary configuration of infinite cylinders, each of which is distinct and radially stratified, is aligned parallel to the Z-axis near the medium 1–2 interface. Media 1 and 2 are both non-absorbing dielectrics with ~ 1 and m ~ 2 , respectively, and real refractive indexes m ~ 1 4m ~ 2 . In typical optical microscopy systems medium m 1 can be a glass substrate and medium 2 is either air or an aqueous solution. The source wave propagates in medium 1 in the XY plane at an angle θ1 inclined from the X axis. The pertinent scattering phenomena involving the cylinders are depicted in Fig. 2. The primary incident wave on the cylinders arises from the transmitted source wave. The secondary and tertiary incident waves include the near-field scattered waves from other cylinders and reflected waves from the substrate's surface due to scattered waves from all cylinders. The total electric (E)
253
Fig. 1. Schematic diagram of the present problem.
254
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
Fig. 2. Scattering interactions for cylinders located near a surface.
utilizing Eq. (4) in (2) and (3): 0 1 !i B Eψ C B C @ !i A H ψ =m1 " i ! ! # αv δψ v ð sin θ1 ! e x þ cos θ1 e y Þ þ αiu δψ u e z ¼ ik1 ! ! ! αiu δψ u ð sin θ1 e x þ cos θ1 e y Þ þ αiv δψ v e z !i ! ð6Þ expð i k U R P Þ ~ 1 =μ1 and ðδψ u ; δψ v Þ are Kronecker delta where m1 ¼ m functions. The factor expðiωtÞ, where ω is the angular frequency, has been omitted for brevity. The incident wave is elliptically polarized if αiu a αiv and their phase difference is not a multiple of π =2, circularly polarized if αi ¼ αi and their phase difference is an odd multiple of u v π =2, and linearly polarized if their phase difference is a multiple of π . 2.2. Primary incident wave on cylinders We denote the transmitted source wave as the primary ~ 2 the trans~ 1 4m incident wave on the cylinders. As m mitted wave can be a propagating or an evanescent wave depending on the angle of incidence. Continuity of the tangential field components across the interface yields the primary incident wave at cylinder j as 0 !þ 1 E ψj B C @ !þ A H ψ j =m2 0 !þ 1 ! þ þ !þ α δ ψ u e z þ αv δψ v P M u Aexpð i k U ! R jP Þ ¼ ik2 εj @ þ ! ! αuþ δψ u P M þ αvþ δψ v e z ð7Þ !þ ! ! ~ where m2 ¼ m 2 =μ2 , εj ¼ expð i k U R j Þ and R j are the phase shift and radial position of the cylinder from the
! interface, respectively, R jP is the radial distance of point P relative to the cylinder, ðαuþ ; αvþ Þ ¼ ðαiu t u12 ; αiv t v12 Þðk1 =k2 Þ are the amplitudes, and ðt u12 ; t v12 Þ are the interface transmissivity [12]. The unit vectors are given by
!þ ! þ! P M ¼ ðk1 sin θ1 e x þkx e y Þ=k2
ð8Þ
!þ ! þ! k ¼ kx e x k1 sin θ1 e y
ð9Þ
where 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < k22 k21 sin 2 θ1 4 0; þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kx ¼ > : i k21 sin 2 θ1 k22 ;
θ1 o θc θ1 4 θc
ð10Þ
~ 2 =m ~ 1 Þ is the critical angle. The phase and θc ¼ sin 1 ðm shift εj is oscillatory when θ1 o θc and exponentially decaying when θ1 4 θc . Total internal reflection occurs when the source wave propagates at θ1 4 θc . The transmitted wave is an evanescent wave that decays rapidly with distance from the interface. To formulate the primary incident wave on the cylinders, we define θe based on Eq. (9) as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð11Þ expðiθe Þ ¼ k2 k1 sin 2 θ1 =k2 þ ik1 sin θ1 =k2 which yields
θe ¼
8 < sin 1 ðk1 sin
θ1 =k2 Þ;
: π =2 i ln ½ðk1 sin
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
θ1 o θ c
θ1 k21 sin 2 θ1 k22 Þ=k2 ; θ1 4 θc ð12Þ
The former is real that corresponds to Snell's law, whereas the latter is complex. Utilizing Eq. (11) allows the expansion of the exponential term in Eq. (7) as !þ ! expð i k U R jP Þ ¼
1
∑
n ¼ 1
ð iÞn exp½inðγ jP þ θe ÞJ n ðk2 RjP Þ ð13Þ
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
where γ jP is the azimuth angle of point P relative to cylinder j and J n is the integer Bessel function. The primary incident wave on cylinder j follows upon transforming Eq. (7) into cylindrical coordinates centered about the cylinder: 0 !þ 1 E ψj 1 B C @ !þ A ¼ k2 εj ∑ ð iÞn exp½inðγ jP þ θe Þ n ¼ 1 H ψ j =m2 0 1 ! ! in þ þ J αþ δ ! e J n' αv δψ v e jγ þ iJ n αu δψ u e z k2 RjP n v ψ v jR A ð14Þ @ in ! ! þ þ J αþ δ ! e þ J n' αu δψ u e jγ þ iJ n αv δψ v e z k2 RjP n u ψ u jR
255
these scatter-reflected waves is crucial to the proper treatment of the interface in evanescent wave scattering. The theoretical treatment of scattering at the interface follows that given in [9–11]. The angular distribution of backscattered waves is formulated by utilizing the expansion [9] Z 1 ð iÞn 1 exp½ in sin ðky =k2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðinγ jP ÞH n ðk2 RjP Þ ¼ 2 2 π 1 k2 ky ! ! ð18Þ expð i k s U R jP Þ dky where
where J n' is the derivative with respect to the argument.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! 2 2! k s ¼ k2 ky e x þky e y
2.3. Scattered waves and near-field secondary incident waves
is the propagation vector of the backscattered wave. Utilizing Eq. (18) in (15) yields
ð19Þ
0
The scattered wave from cylinder j is expressed similar to Eq. (14) as a function of unknown coefficients ðbjn ; ajn Þ and Hankel function of the second kind H n as 0 !s 1 E ψj 1 B C @ !s A ¼ k2 ∑ ð iÞn expðinγ jP Þ n ¼ 1 H ψ j =m2 0 1 ! ! in H a δ e H n' ajn δψ v ! e jγ þiH n bjn δψ u e z k2 RjP n jn ψ v jR A ð15Þ @ in ! ! H b δ ! e þH n' bjn δψ u e jγ þ iH n ajn δψ v e z k2 RjP n jn ψ u jR where H n' denotes the derivative with respect to the argument. Dependent scattering occurs in the near field because the cylinders are located in the near-field zone of each other due to the small spacing between them. It involves scattering of non-plane waves because the nearfield scattered waves from a cylinder have not recovered into a plane wave when incident on other cylinders. The near-field secondary incident wave on cylinder j due to scattered waves from cylinder k is formulated from Eq. (15) by applying the addition theorem of Bessel functions [13]: 0
1 !s E ψ ;kj 1 1 B C ∑ ð iÞn expðinγ jP ÞGjn @ !s A ¼ k2 ∑ ks n ¼ 1 s ¼ 1 H ψ ;kj =m2 0 ! ! ! 1 in J a δ e jR J n' aks δψ v e jγ þiJ n bks δψ u e z k2 RjP n ks ψ v A U@ ! ! ! in J b δ e jR þ J n' bks δψ u e jγ þiJ n aks δψ v e z k2 RjP n ks ψ u
ð16Þ where ¼ ð iÞs n exp½iðs nÞγ kj H s n ðk2 Rjk Þ Gjn ks
!s 1 Z 1 E ψj ! ! B C expð i k s U R jP Þ @ !s A ¼ i 1 H ψ j =m2 2 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2 2! ! 6 X ψ j e z Y ψ j ðky e x þ k2 ky e y Þ=k2 7 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 dky ! ! 2 2! X ψj ðky e x þ k2 ky e y Þ=k2 þ Y ψj e z
ð20Þ where ! X ψj Y ψj
h i ! 1 exp in sin ky =k2 bjn δψ u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ∑ ð 1Þn ajn δψ v 2 2 π n ¼ 1 k2 ky k2
1
ð21Þ are the amplitudes of the backscattered fields. The fields given by Eq. (20) are denoted as scatterincident waves because they arise from the backscattered wave incident at the half space interface. The scatterincident waves give rise to reflected and transmitted waves upon incident at the interface. We express the latter in similar forms as Eq. (20) in terms of unknown amplitudes ðX rψþj ; Y rψþj Þ and ðX tψj ; Y tψj Þ as 0
!r þ 1 Z 1 E ψj !þ ! ! ! B C expð i k s U R P þ i k s U R j Þ @ !r þ A ¼ i 1 H ψ j =m2 2 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2 2! rþ! rþ 6 X ψ j e z þY ψ j ð ky e x þ k2 ky e y Þ=k2 7 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 dky ! ! 2 2! X rψþj ðky e x k2 ky e y Þ=k2 þ Y rψþj e z
ð22Þ !t 1 Z 1 E ψj ! ! ! ! C B expð i k t U R P þ i k s U R j Þ A ¼ i @ !t 1 H ψ j =m1 0
ð17Þ
in which γ kj is the azimuth angle of cylinder j relative to k, and Rjk is the radial distance between cylinders j and k. 2.4. Scattering at the half space interface Scattered waves from the cylinders propagate into all directions. Those scattered in the backward direction ðπ =2 r γ jP r 3π =2Þ would be reflected from the half space interface and become incident waves on the cylinders. The total field in the forward direction then consists of both scattered and reflected waves. A rigorous formulation of
2
3 ! ! ! X tψj e z Y tψj ðky e x þ γ 1 e y Þ=k1 5 dky 4 ! ! ! X tψj ðky e x þ γ 1 e y Þ=k1 þ Y tψj e z
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 where γ 1 ¼ k1 ky , and ffi ! þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2 2! k s ¼ k2 ky e x þ ky e y ! ! ! k t ¼ γ 1 e x þ ky e y
ð23Þ
ð24Þ ð25Þ
256
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
are the propagation vectors of the respective fields. Over the range 1 rky r 1, the x-component of the respective propagation constant varies as 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 > < k2 ky ; k2 4 ky !þ ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð26Þ ks Uex¼ > i k2 k2 ; k o k : y 2 y 2 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 > < k1 ky ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ1 ¼ > : i k2y k21 ;
k1 4 ky k1 o ky
ð34Þ ð27Þ
The boundary conditions require the continuity of the tangential components of the field vectors across the half space interface: ðEψ j ; H ψ j Þsy;z þ ðEψ j ; H ψ j Þry;zþ ¼ ðEψ j ; H ψ j Þty;z
r vv þ ðky Þ ¼
Y sjn
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 k2 ky m2 k2 k1 ky qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 2 2 2 m1 k1 k2 ky þ m2 k2 k1 ky m1 k 1
where ψ ;jn
Rψ ;ks ¼
ð28Þ
Applying Eqs. (20), (22) and (23) in (28) yields the interface reflectivity as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 rþ m2 k1 k2 ky m1 k2 k1 ky X jn q ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð29Þ ðk Þ ¼ ¼ r uu þ y 2 2 2 2 X sjn m k k2 ky þm1 k2 k1 ky 2 1 Y rjnþ
cylinder. We obtain after some manipulations 0 !r þ 1 E ψ ;kj 1 1 B C ∑ ð iÞn expðinγ jP Þ @ !r þ A ¼ k2 ∑ n ¼ 1 s ¼ 1 H ψ ;kj =m2 0 1 v;jn u;jn in J Rv;jn a δ ! e J n' R a δ ! e þ iJ n R b δ ! e k2 RjP n v;ks ks ψ v jR v;ks ks ψ v jγ u;ks ks ψ u z @ A U u;jn u;jn v;jn ! ! ! in k2 RjP J n Ru;ks bks δψ u e jR þ J n' Ru;ks bks δψ u e jγ þiJ n Rv;ks aks δψ v e z
ð30Þ
The scatter-reflected wave is obtained by substituting Eqs. (21), (29) and (30) in (22): 0
!r þ 1 Z 1 E ψj 1 B C @ !r þ A ¼ ik2 ∑ n ¼ 1 1 H ψ j =m2
2
3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2! u;P v;P ! R b δ e z Rv;jn ajn δψ v ðky ! k k e y Þ=k2 ψ u jn 2 y u;jn ex 6 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 5dky 2 2! v;P ! ! Ru;P b δ ðk e k k e Þ=k þ R a δ e 2 2 y y u;jn jn ψ u y x v;jn jn ψ v z ð31Þ
Z
exp½ iðs þ nÞ sin 1 ðky =k2 Þ ψ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rψ 2 2 π 1 k2 ky qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 exp i k2 ky ðxj þ xk Þ iky ðyj yk Þ dky ð 1Þs
1
ð35Þ
ψ ;jn
We denote Rψ ;ks as the integrated reflection coefficient between cylinders k and j, which accounts for scatterreflected waves over all directions. 2.6. Solution for the unknown coefficients The coefficients ðbjn ; ajn Þ are the only unknowns in the theoretical formulation. They are solved by imposing the continuity of the tangential components of the fields across all interfaces of each cylinder. The present study considers radially stratified cylinders, for which the internal fields in each interior layer have been detailed in [14]. !i The boundary conditions involve incident ð Ψ ψ ;q Þ and !s scattered ð Ψ ψ ;q Þ waves !i
!s
!i
!s
ðΨ ψ j;q 1 þ Ψ ψ j;q 1 Þjγ ;z ¼ ðΨ ψ j;q þ Ψ ψ j;q Þjγ ;z
ð36Þ
between layers q 1 and q, where Ψ ¼ ðE; HÞ, and !þ
!s
N
N
!s
!r þ
ðΨ ψ j þ Ψ ψ j þ ∑ Ψ ψ ;kj þ ∑ Ψ ψ ;kj Þj kaj
k¼1
where
!i
γ ;z
!s
¼ ðΨ ψ ;j;Q j þ Ψ ψ ;j;Q j Þj
γ ;z
ð37Þ n
1
ð 1Þ exp½ in sin ðky =k2 Þ ψ ;P qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rψ ;jn ¼ 2 2 π
k2 ky qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ψþ 2 2 exp i k2 ky ðxj þ xP Þ iky yjP r ψ
at the surface. Solving the above yields two independent sets of equations ð32Þ
N
∑
1
þ Ru;jn b gbks ∑ fδjk δns þ ½ð1 δjk ÞGjn ks u;ks jn o;I
k ¼ 1 s ¼ 1
¼ αuþ εj expðinθe Þbjn
o;I
signifies the reflection of scattered waves from cylinder j to point P. N
∑
ð38Þ
1
þ Rv;jn ao;II gaks ∑ fδjk δns þ ½ð1 δjk ÞGjn ks v;ks jn
2.5. Incident waves on cylinders due to reflection of scattered waves
k ¼ 1 s ¼ 1 ¼ vþ j expðin e Þao;II jn
The scatter-reflected wave from the interface serves as another source of incident wave on the cylinders. The incident wave on cylinder k due to the reflection of scattered waves from cylinder j is formulated by first applying the translation ! ! ! R P ¼ R kþ R ð33Þ kP '
where bjn and ao;II jn are the coefficients for the transverse
to Eq. (32). The generating function of Bessel function is next applied, followed by the transformation from Cartesian into cylindrical coordinates centered about the
α ε
θ
ð39Þ
o;I
magnetic (TM) and transverse electric (TE) mode, respectively, for an isolated cylinder in an unbounded medium 2. The influence of the various pertinent scattering interactions is evident in the functional form of Eqs. (38) and (39). The term on the right hand side is due to the source wave incident at the cylinder. Near-field scattering of nonplane waves between closely spaced cylinders is signified by Gjn that contains the Hankel function H n ðk2 Rjk Þ. Its ks
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
influence diminishes with increasing separation between ψ ;jn the cylinders. The effect of the interface is given by Rψ ;ks that captures the interaction between a pair of cylinders over all directions. If the half space interface disappears, Eqs. (38) and (39) reduce to those for multiple cylinders in an unbound medium [13,14].
0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! T ψ v;j δψ v e γ þ T ψ u;j δψ u e z 2 @ A ¼ k2 expð ik2 RP Þ ! ! iπ k2 RP T ψ u;j δψ u e γ þT ψ v;j δψ v e z ð43Þ where T uu;j
! ¼ exp½ik2 Rj cos ðγ P γ j Þ
T vv;j
2.7. Scattered waves in the near- and far-fields The scattered wave consists of scattering from the cylinders and reflection of scattered waves from the surface. The former is given by Eq. (15), which is expressed in cylindrical coordinates centered about each cylinder. The latter is given by Eq. (31), which is prescribed relative to the Cartesian coordinate system at the surface. To obtain the angular distribution of the scattered radiation, it is necessary to transform both equations into the cylindrical coordinate system at the interface. Applying straightforward coordinate transformation to Eq. (15) yields the scattered waves from cylinder j relative to the half space interface as 1 !s E ψj B C @ !s A H ψ j =m2 2 3 ! ! ! ðEsψ j;R pj þ Esψ j;γ qj Þ e R þðEsψ j;γ pj Esψ j;R qj Þ e γ þ Esψ j;z e z 5 ¼ k2 4 s ! ! ! ðHψ j;R pj þ Hsψ j;γ qj Þ e R þ ðHsψ j;γ pj H sψ j;R qj Þ e γ þ Hsψ j;z e z
257
1
bjn
n ¼ 1
ajn
∑
! expðinγ P Þ ð44Þ
Eq. (41) involves the integrals given in Eq. (31), which can be evaluated by the method of stationary phase as described in the appendix. Using the saddle point value ky ¼ k2 sin γ P gives 0 !r þ 1 E ψj B C @ !r þ A H ψ j =m2 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! Rvvj δψ v e γ þRuuj δψ u e z 2 @ A ¼ k2 expð ik2 RP Þ ! ! iπ k2 RP Ruuj δψ u e γ þ Rvvj δψ v e z
0
ð45Þ where ! Ruuj Rvvj
ψj
1
uþ r^ u bjn
n ¼ 1
vþ r^ v ajn
∑
exp½inðπ γ P Þ
ð40Þ where pj ¼ cos ðγ P γ jP Þ, qj ¼ sin ðγ P γ jP Þ, and the component fields are defined by Eq. (15). Similarly, Eq. (31) is transformed into cylindrical coordinates with respect to the interface as 0 !r þ 1 0 rþ 1 E ψj Eψ j;x Erψþj;y Erψþj;z B C A @ !r þ A ¼ k2 @ r þ H ψ j;x H rψþj;y H rψþj;z H =m
¼ exp½ ik2 Rj cos ðγ P þ γ j Þ
!
ð46Þ
uþ r^ u
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 k1 k2 sin 2 γ P qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 2 m2 k1 cos γ P þ m1 k1 k2 sin 2 γ P
ð47Þ
vþ r^ v
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 k1 k2 sin 2 γ P qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 2 m1 k1 cos γ P þ m2 k1 k2 sin 2 γ P
ð48Þ
m2 k1 cos γ P m1
m1 k1 cos γ P m2
2
0
! ! 1 cos γ P e R sin γ P e γ B ! ! C C B @ sin γ P e R þ cos γ P e γ A ! ez
ð41Þ
in which the component fields are defined by Eq. (31). The time-averaged Poynting vector of the total scattered radiation is evaluated accordingly as " # N !s N !s þ !r þ !s !r þ co S ψ ¼ Re ∑ ð E ψ j þ E ψ j Þ ∑ ð H ψ k þ H ψ k Þn ð42Þ 8π j¼1 k¼1 Eqs. (40) and (41) are general expressions that apply to any distance from the interface. There is no apparent simplification for them except in the far-field. Most applications of evanescent scattering involve measurement of the scattered radiation at large distance from the surface. In the far field, i.e., k2 RjP -1, the following approximations can be applied: RjP RP Rj cos ðγ P γ j Þ, 1=RjP 1=RP , γ jP γ kP γ P , pj 1, and qj 0. Using the asymptotic expansion of the Hankel functions, Eq. (40) becomes 0 !s 1 E ψj B C @ !s A H ψ j =m2
Combining Eqs. (43) and (45) yields the E and H fields of the total scattered wave in the far field: 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !s 2 B Eu C ¼ k expð ik2 RP Þ @ !s A 2 i π k 2 RP H u =m2 ! ! N ez ∑ ðT uu;j þ Ruuj Þ ð49Þ ! eγ j¼1 0
1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !s ! ! N eγ 2 B Ev C expð ik2 RP Þ ∑ ðT vv;j þRvv;j Þ ! @ !s A ¼ k2 iπ k2 RP j¼1 ez H v =m2
ð50Þ for the magnetic and electric mode incident wave, respectively. The time-averaged Poynting vector for the far-field scattered radiation is as follows: !s þ co m2 k22 2 ! Sψ ¼ ðI uu δψ u þ I vv δψ v Þ e R 8π π k2 RP
ð51Þ
where 2 N I uu ðγ P Þ ¼ ∑ ðT uu;j þ Ruu;j Þ j ¼ 1
ð52Þ
258
2 N I vv ðγ P Þ ¼ ∑ ðT vv;j þ Rvv;j Þ j ¼ 1
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
ð53Þ
are the scattering amplitude functions for the respective modes. Along the horizon, i.e. γ P ¼ 7 π =2, Eqs. (47) and uþ vþ (48) yield r^ u ¼ r^ v ¼ 1. We then obtain T uu;j ¼ Ruu;j and T vv;j ¼ Rvv;j from Eqs. (49) and (50), and the Poynting vector vanishes as well. This indicates that vanishing of far field scattered radiation along the surface is caused by destructive interference between the scattered and reflected waves. 3. Numerical examples An effort was made to compare numerical results based on the present solution with those given in [15], which utilized Green function and numerical solution of the boundary integral equations, for scattering by two cylinders near a half space interface. An exact comparison cannot be made because no scale is shown in Fig. 2 in [15]. A qualitative comparison indicates a similar angular variation for the far-field scattered radiation for the case of ~ 1 ¼ 1:414, whereas considerable differences exist for the m ~ 1 ¼ 1:5; 1:732Þ. The greatest discrepancy other cases ðm occurs in the far-field scattered radiation along the horizon, i.e.γ P ¼ 7 π =2, for which [15] gave finite values compared to zero from the present solution. Vanishing of the far-field scattered radiation along the surface has been shown in [7] and discussed by Small et al. [16]. The solution method of [15] may need appropriate modification in order to reconcile the discrepancy with the condition along the surface. For the purpose of illustration we examine evanescent scattering by both single and multiple parallel gold cylindrical wires located in air (medium 2) on the surface of a substrate. The wires are 0.1 μm in radius, and the clearance between adjacent wires and that between the bottom wire and the substrate's surface are both 1 Å. Different ~ 1 ¼ 1:33; 1:5; μ1 ¼ 1) are substrate refractive indexes (m utilized for the numerical examples. The amplitudes of the TM and TE mode source wave in medium 1 are both unity. The free space wavelength is λo ¼ 0:6328 μm at which ~ Au 0:166 3:15i. m
Fig. 3a and b shows the scattering amplitudes ðI uu ; I vv Þ for the case of a single wire. The effect of incidence angle ~ 1 ¼ 1:5 ðθ1 ¼ 451; 601Þ is shown in Fig. 3a for m (θc ¼ 41:811). Because the amplitude of the evanescent incident wave is greater for θ1 ¼ 451 than for θ1 ¼ 601, the scattering amplitudes for θ1 ¼ 451 are generally higher than those for θ1 ¼ 601. The scattering amplitudes are asymmetric that peaks around the incident direction. Fig. 3b shows the effect of different substrate refractive ~ 1 ¼ 1:33; 1:5) for θ1 ¼ 601. The angular distriindexes (m butions show similar behavior as those in Fig. 3a. Because the amplitude of the evanescent incident wave is higher ~ 1 ¼ 1:33 (θc ¼ 48:751) than for m ~ 1 ¼ 1:5, the scatterfor m ~ 1 ¼ 1:33 are higher than those for ing amplitudes for m ~ 1 ¼ 1:5. Fig. 4a and b shows the results for a triangular m arrangement of three wires. Fig. 4a shows the effect of ~ 1 ¼ 1:5, and the results for different incidence angle for m substrate refractive indexes at the same incident angle are given in Fig. 4b. Interference of the scattered waves from the different wires gives rise to two distinct lobes for the magnetic mode ðI uu Þ that peak in roughly in both the forward and retro-reflected directions. Fig. 5a and b shows the results for a cluster of 7 wires arranged at the vertices of a hexagon. The scattering amplitudes for two incidence angles are shown in Fig. 5a, and those for two substrate refractive index at the same incident angle are shown in b. The presence of more wires resulted in complicated interference of the scattered and scatter-reflected waves, thus giving rise to more scattering lobes than in the case of 3 wires. To demonstrate the generality of the present solution, we present results for evanescent scattering by multiple dissimilar cylinders. The same hexagonal arrangement of cylinders as above is considered, except that the wires circumscribing the center gold wire are replaced by solid silica cylinders in one case and by hollow silica cylinders with an inner radius of 0:05 μm in the second case. The ~ SiO2 ¼ 1:45 i4:5 10 6 . The refractive index of silica is m far-field scattering amplitudes for the different cylinder clusters are shown in Fig. 6a and b. As silica is nearly nonabsorbing, the scattering amplitudes for the clusters with circumscribing silica rods are much higher than those for all gold wires that are strongly absorbing. Higher
~ 1 ¼ 1:33; 1:5). Fig. 3. Far-field scattering amplitude for a single gold wire: (a) two incident angles (θ1 ¼ 451; 601); (b) two substrate refractive index (m
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
scattering amplitudes are observed for the cluster containing solid silica rods, because they scatter more strongly than the hollow silica rods. The numerical results show scattering peaks in both the forward and retro-reflected directions. The former can
259
be attributed to the forward scattering characteristics generally occurs with particle scattering, whereas the latter is due to the images. The angular variation is dictated by the interference of the scattered and scatterreflected waves of all the cylinders. For a cluster of gold
~ 1 ¼ 1:33; 1:5). Fig. 4. Far-field scattering amplitude for 3 gold wires: (a) two incident angles (θ1 ¼ 451; 601); (b) two substrate refractive index (m
Fig. 5. Far-field scattering amplitude for a cluster of 7 gold wires: (a) two incident angles (θ1 ¼ 451; 601); (b) two substrate refractive index ~ 1 ¼ 1:33; 1:5). (m
~ 1 ¼ 1:5 and θ1 ¼ 601: (a) magnetic mode I uu ; (b) electric mode Ivv . Fig. 6. Far-field scattering amplitude for clusters of 7 dissimilar cylinders for m
260
S.-C. Lee / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 252–260
wires, Fig. 5a and b shows that I vv exhibits more scattering lobes than I uu . The scattering pattern changed considerably when the circumscribing wires are replaced by either solid or hollow silica rods which are nearly non-absorbing. More scattering lobes then appear in I uu than I vv , as shown in Fig. 6a and b. These results indicate that the scattering pattern is strongly influenced by the size and material of the constituent cylinders. A generalization cannot be made on whether I uu or I vv exhibits more scattering lobes.
In the asymptotic limit of k2 RP -1, the modulus of exp½gðΓ Þ changes rapidly along the steepest paths along which ImðgÞ is constant. The stationary phase point is determined by g 0 ðΓ o Þ ¼ 0, which gives Γ o ¼ γ P from Eq. (iv), and ky ¼ k2 sin γ P follows from Eq. (ii). Expanding gðΓ Þ into Taylor series and keeping up to second order terms, Eq. (iii) yields Z 1 ∂ky lim I f ðΓ o Þ exp½gðΓ o Þ exp½g″ðΓ o ÞðΓ Γ o Þ2 =2 dΓ ∂ Γ k2 RP -1 1 Γo
ðvÞ
4. Summary An exact analytical solution of Maxwell's equations for evanescent wave scattering by multiple parallel infinite cylinders has been presented in this paper. The incident wave is arbitrarily polarized and propagates in the plane perpendicular to the axes of the cylinders. The present formulation utilized Hertz potentials to treat the evanescent incident wave, near-field scattering between closely spaced cylinders, and scattering and reflection at the surface of the substrate. Analytic formulas are derived for the E and H fields of the scattered waves, which consist of both forward scattered waves and boundary-reflected waves arising from the backscattered waves. The scattered waves in the far-field are shown to vanish along the horizon due to destructive interference between the forward-scattered and scatter-reflected waves. Numerical results for several configurations of cylinders were shown to illustrate the influence of substrate refractive index, incidence angle, and size and material of the cylinders on evanescent wave scattering. Appendix The method of stationary phase (MSP) is utilized to evaluate the far field scatter-reflected wave given by Eq. (31). Detailed discussion of MSP can be found in many standard texts. The brief description herein focuses on its application to the present problem. The Cartesian components of Eq. (31) all involve an integral in the form of Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 I¼ f ðky Þexpð i k2 ky xP iky yP Þ dky ðiÞ 1
where f ðky Þ is a complex function. By defining qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 expðiΓ Þ ¼ ð k2 ky þ iky Þ=k2 Eq. (i) is transformed into Z 1 ∂ky I¼ f ðky Þexp½g ðΓ Þ dΓ ∂Γ 1
ðiiÞ
ðiiiÞ
where gðΓ Þ ¼ ik2 RP cos ðΓ γ P Þ
ðivÞ
where g″ðΓ o Þ is the second order derivative. Eq. (45) readily follows after some manipulations. References [1] Prieve DC, Luo F, Lanni F. Brownian motion of a hydrosol particle in colloidal force field. Faraday Discuss Chem Soc 1987;83:297–307. [2] Prieve DC, Walz JY. Scattering of an evanescent surface wave by a microscopic dielectric sphere. Appl Opt 1993;32(9):1629–41. [3] Siegel R. Light scattering near and from interfaces using evanescent wave and ellipsometric light scattering. Curr Opin Colloid Interface Sci 2009;14(6):426–37. [4] Lisicki M, Cichocki B, Dhont KG, Lang PR. One-particle correlation function in evanescent wave dynamic light. J Chem Phys 2012;136: 204704. [5] Aslan MM, Menguc MP, Videen G. Characterization of metallic nanoparticles via surface wave scattering: B. Physical concept and numerical experiments. J Quant Spectrosc Radiat Transf 2005;93: 207–17. [6] Short MR, Tortel H, Litman A, Geffrin JM, Vaillon R, Abdeddaim R, et al. Evanescent wave scattering by particles on a surface: comparison between the discrete dipole approximation with surface interaction and the finite element method. ELS14, Lille, France; June 2013. [7] Doicu A, Eremin Y, Wriedt T. Scattering of evanescent waves by a particle on or near a plane surface. Comput Phys Commun 2001;134:1–10. [8] Eremina E, Wriedt T, Helden L. Analysis of evanescent waves scattering by a single particle in Total Internal Reflection Microscopy. In: PIER Symposium, Cambridge; 2006. [9] Lee SC, Grzesik JA. Light scattering by closely spaced parallel cylinders embedded in a semi-infinite dielectric medium. J Opt Soc Am A 1998;15:163–73. [10] Lee SC. Scattering by multiple parallel radially stratified infinite cylinders buried in a lossy half space. J Opt Soc Am A 2013;30(7): 1320–7. [11] Lee SC. Scattering by a radially stratified infinite cylinder buried in a lossy half space. J Opt Soc Am A 2013;30(4):565–72. [12] Born M, Wolf E. Principle of optics. Oxford: Pergamon Press; 1980. [13] Lee SC. Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders. J Appl Phys 1990;68(10):4952–7. [14] Lee SC. Scattering by closely-spaced radially stratified parallel cylinders. J Quant Spectrosc Radiat Transf 1992;48:119–30. [15] Belai OV, Frumin LL, Perminov SV, Shapiro DA. Scattering of evanescent wave by two cylinders near a flat boundary. Europhys Lett 2012;97(1):10007–18. [16] Small A, Fung J, Manoharan VN. Generalization of the optical theorem for light scattering from a particle at a planar interface. J Opt Soc Am A 2013;30(12):2519–25.