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Parametric study of three-dimensional near-field Mie scattering by dielectric particles
Optics Communications 216 (2003) 1–10 www.elsevier.com/locate/optcom
Parametric study of three-dimensional near-field Mie scattering by dielectric particles Djenan Ganic, Xiaosong Gan *, Min Gu Centre for Micro-Photonics, School of Biophysical Sciences and Electrical Engineering, Swinburne University of Technology, P.O. Box 218, Hawthorn, Vic. 3122, Australia Received 8 August 2001; received in revised form 25 September 2002; accepted 25 November 2002
Abstract In this paper we present a model for studying the three-dimensional field distribution of near-field Mie scattering. The effect of the interface where near-field is generated is included in the model. We use this model to calculate the scattered electromagnetic field distribution in various planes around a dielectric particle of different sizes under S and P polarized illumination. Integrated scattered intensity into particleÕs upper half-space is investigated and it is found that it exhibits morphology-dependent resonance effect when particleÕs radius is approaching and exceeding illumination wavelength. The model is also applied to studying the dependence of the morphology-dependent resonance on various parameters associated with near-field Mie scattering by dielectric particles. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction An evanescent wave occurs when an electromagnetic wave is incident from a high refractiveindex medium into a low refractive-index medium under total internal reflection. Such an evanescent wave can be converted into a propagating wave by interaction with a small particle situated in the low refractive-index medium, provided that the distance between the particle and the interface separating the two media, d, is within a few illumination wavelengths in length. Such a scattering process is called near-field Mie scattering [1]. Un-
*
Corresponding author. Fax: +61-3-9214-5435. E-mail address: [email protected] (X. Gan).
like the plane-wave Mie scattering [2,3], near-field Mie scattering leads to an asymmetric distribution [1,4,5] and a complicated polarization nature [6,7]. The scattered field of an evanescent wave with a Mie particle has been studied [1]. It is found that the scattering cross-section exhibits a morphologydependent resonance (MDR) nature when the size of a dielectric particle is comparable to the illumination wavelength. Further, the MDR structure depends strongly on the penetration depth of the evanescent wave [4], but its peak positions and half widths do not [5]. In many applications including particle-trapped near-field scanning optical microscopy (NSOM) [8–12] and optical trapping nanometry [13], a three-dimensional (3-D) distribution of the scattered evanescent wave in the far-field region is
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(02)02252-6
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D. Ganic et al. / Optics Communications 216 (2003) 1–10
needed to understand image resolution, image contrast, detection sensitivity and accuracy. Therefore it is necessary to know the 3-D scattered evanescent wave at any point around a scattering particle. The first calculation of the scattered field [1] does not include the effect of the interface and involves an approximation, which is valid only for two principal planes at a large distance from the scattering particle. Although the effect of the interface at which an evanescent wave is generated is discussed [5–7], only the cross-sections of evanescent waves by small particles is calculated [4–7]. Recently, Doicu et al. [14,15] have investigated the differential scattering cross-section and integral response of evanescent wave scattering by small particles and a sensor tip of up to 100 nm in radius. These numerical calculations include the interface effects in the context of the discrete sources method and the T-matrix method, and the far-field is determined by extrapolation of the scattered field as r ! 1. However, these calculations have not discussed scattering properties of larger particles whose scattering characteristics are markedly different from scattering properties of small particles due to MDR effects [4,16–21]. The aim of this paper is to present a detailed parametric study of the 3-D distribution of the scattered evanescent wave around a scattering particle without using any approximation. In Section 2, our model for near-field scattering, which includes the effect of an interface, is described. A mathematical expression for calculation of the 3-D electric field scattered by a Mie particle with an evanescent wave is given. Section 3 presents the detailed parametric calculations of the 3-D intensity distribution around a dielectric particle. The phenomenon of MDR is investigated in Section 4. In particular, the dependence of MDR on the illumination polarization states is revealed. The main conclusions of this research are drawn in Section 5.
2. Three-dimensional expression for near-field Mie scattering A schematic diagram of the near-field Mie scattering model investigated in this paper is
shown in Fig. 1. The incident angle of the electromagnetic wave is represented by a. The refractive index of the scattering particle is denoted by n1 , while it is immersed into a medium of refractive index n0 . The refractive index of the substrate is n. A particle is situated at a distance d from the interface at which an evanescent wave is generated [22]. The scattered electric field Etotal at an arbitrary point P in the surrounding medium can be expressed as a sum of the field that would be scattered into point P if the interface between two media was not present, and the contribution that is reflected into point P by the interface. Mathematically it can be expressed as Etotal ðrÞ ¼ Eupper ðrÞ þ rf Ebottom ðrÞ;
ð1Þ
where Eupper is scattered field into point P (upper space) and Ebottom is scattered field into point P 0 (bottom space) without considering the interface, while rf is the Fresnel amplitude reflection coefficient for a given incident polarization state. P 0 is the mirror-reflection point of point P. The amount of the scattered field at point P 0 that is reflected to point P is determined by the Fresnel amplitude reflection coefficients under given conditions. Eupper ðrÞ and Ebottom ðrÞ can be determined without considering the effect of the interface. As shown in Fig. 1, when the incident angle a of the illumination electric wave E0 is larger than the critical angle, the scattered electric field by the Mie particle can be expressed, for the polarization state of E0 perpendicular to the plane of incidence (i.e., S polarization), as [1] h i X ic ð1Þ bE ðl; mÞr hl ðk 0 rÞYllm ð^ ESC ðrÞ ¼ rÞ 02 n x lm ð1Þ 0 þ bM ðl; mÞhl ðk rÞYllm ð^ rÞ ; ð2Þ where l ¼ 1 to 1 and m ¼ l to +l. c is the speed of light in vacuum and x is the angular frequency of the incident light. The functions bE ðl; mÞ and bM ðl; mÞ are the expansion coefficients relating the ð1Þ illumination evanescent field, hl ðk 0 rÞ is the spherical Hankel function of the first kind, and Yllm are the vector spherical harmonics. When expressed in a spherical coordinate, Eq. (2) can be reduced to
D. Ganic et al. / Optics Communications 216 (2003) 1–10
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Fig. 1. Illustration of the scattering model which includes the effect of an interface where an evanescent wave is generated. n; n0 and n1 denote refractive indices of the substrate, surrounding media and the particle, respectively. d is the distance from the center of a particle to the interface.
ESC ðrÞ ¼
X
(
c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bE ðl; mÞ lðl þ 1Þ lm "
1 o oYlm ð1Þ 0 h ðk rÞ sin h r sin h l oh oh # ð1Þ hl ðk 0 rÞ o2 Ylm þ 0 r1 n sin h ou2 " ð1Þ ð 1ÞbM ðl; mÞhl ðk 0 rÞ oYlm pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ou isinh lðl þ 1Þ n0 x
# cbE ðl; mÞ 1 oYlm o ð1Þ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrhl ðk rÞÞ h1
n02 x lðl þ 1Þ r oh or " ð1Þ bM ðl; mÞhl ðk 0 rÞ oYlm pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ oh i lðl þ 1Þ cbE ðl; mÞ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lðl þ 1Þ r sin h # ) oYlm o ð1Þ 0 ðrh ðk rÞÞ u1 : ou or l
n02 x
ð3Þ
Here r1 ; h1 and u1 are the unit vectors in the spherical coordinate. h and u are the variables of the scalar spherical harmonics Ylm . For the polarization state E0 parallel to the plane of incidence (i.e., P polarization), the scattered electric field is also given by Eq. (3) with expansion coefficients bE ðl; mÞ and bM ðl; mÞ substituted by b~E ðl; mÞ and b~M ðl; mÞ, respectively [1]. Eq. (3) gives the electric field resulting from scattering of an evanescent wave by a Mie particle without using any approximation and without considering interface effects. In the previous calculation [1], an approximated form of Eq. (3), which is valid only in two principal planes, was used. In this paper, Eq. (3) is used together with our scattering model that includes the interface between the surrounding medium and the substrate, in the following numerical calculation for the 3-D distribution of the scattered field around a particle near the interface.
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The scattered field is dependent on the effective refractive index nef ¼ n1 =n0 and the size parameter q ¼ k 0 a, where a is the radius of a scattering Mie particle and k 0 is the propagation constant of the electromagnetic wave in the medium surrounding the particle. The strength and the distribution of the scattered field is also affected by the decay constant b of the evanescent wave, defined as [1] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 sin2 a b ¼ k0
1: ð4Þ n02 A larger decay constant of an evanescent wave, correspond to the faster decay of the evanescent wave in the depth direction (i.e., in the X-direction). A scattering particle is immersed into an evanescent wave completely or partially, depend-
ing on the particle size and the decay constant, which greatly affects the scattered field distribution around the particle. 3. Effect of the interface on the 3-D scattered intensity distribution around dielectric particles According to our mathematical model described by Eq. (3), a 3-D intensity distribution of the scattered field can be numerically calculated at any distance from the scattering particle. To understand the effect of the interface, let us first consider that a particle is situated far away from the interface. Fig. 2 shows an example of the scattered intensity distribution by a dielectric particle of a radius of 2 lm, immersed in air. To
Fig. 2. Three-dimensional distribution of the scattered intensity around a 2 lm dielectric particle situated far away from the interface (a) in the XZ-plane, (b) in the plane containing the X-axis at 45° anti-clockwise from the XZ-plane, (c) in the XY-plane and (d) in the YZ-plane. The solid and dotted curves correspond to the S and P polarization states of the illumination wave, respectively. n1 ¼ 1:6; n0 ¼ 1:0; n ¼ 1:51; k ¼ 632:8 nm and a ¼ 45°.
D. Ganic et al. / Optics Communications 216 (2003) 1–10
demonstrate the 3-D distribution of the scattered field, the scattered intensity in the XZ-plane, in the plane containing the X-axis at 45° anticlockwise from the XZ-plane, in the XY-plane and in the YZ-plane are displayed in Figs. 2(a)–(d), respectively. The scattered intensity distribution is asymmetric in the XZ-plane and in the plane at 45° from the XZ-plane because the illumination field is asymmetric in these planes and the evanescent wave propagates in the Z-direction. Fig. 2 shows that the evanescent wave is most intensely scattered into certain regions around the particle. This effect is due to the fact that the interaction of the evanescent wave with the particle is confined to the bottom of the particle because particle size is so large compared with the decay depth of the evanescent wave (in this case the decay depth is 186.5 nm). This phenomenon can be qualitatively explained using SnellÕs law and the Fresnel amplitude coefficients for reflection and transmission [2] (Fig. 3). Three rays representing the evanescent wave propagating along the Z-direction are chosen for demonstration. The length
5
of the vectors in Fig. 3 represents the relative strength of the evanescent intensity. Ray 1, the highest intensity ray of the three selected rays, interacts at the very bottom of the particle (point A) with a large incident angle. Consequently, most of the ray intensity is reflected rather than refracted because of the high incident angle with respect to the normal of the particleÕs surface. The refracted ray, after traversing through the particle, interacts with the particle-medium boundary at point B and the large amount of its intensity emerges into the medium with a small portion reflected. This reflected portion of intensity traverses through the particle again, interacts with the boundary (point C) and emerges into the medium while a negligible amount is reflected again. Rays 2 and 3, whose intensity is weaker and determined by the decay constant of the evanescent wave experience a similar process. As a result, the relative intensity distribution of the three rays after three refraction processes indicates that the scattered intensity profile is confined to certain regions around the particle and that the highest intensity region is
Fig. 3. A qualitative interpretation of the confined intensity regions in the scattering of an evanescent wave by a dielectric particle of radius 2 lm. The relative intensity of rays is denoted by the arrow length.
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Fig. 4. Dependence of the scattered intensity distribution in the XZ plane on the radius of a particle, when the particle is situated far away from the interface: (a) a ¼ 0:05 lm, (b) a ¼ 0:1 lm, (c) a ¼ 0:5 lm and (d) a ¼ 1 lm. The plots in the left and right columns show the intensity distributions scattered by an evanescent wave and a plane wave, respectively. The solid and dotted curves correspond to the S and P polarization states of the illumination wave, respectively. n1 ¼ 1:6; n0 ¼ 1:0; n ¼ 1:51; k ¼ 632:8 nm and a ¼ 45°.
D. Ganic et al. / Optics Communications 216 (2003) 1–10
located in the region below 0° for this particular (see Fig. 2(a)). The location of these intensity regions depends on the refractive indices of the dielectric particle and its immersion medium, the particle size and the decay constant of the nearfield wave. For particles far away from the interface the dependence of the asymmetric scattered intensity distribution on the particle size, is depicted in Fig. 4. For small particle sizes (Figs. 4(a) and (b)), the scattered intensity profiles are similar to those under plane wave illumination [2]. This similarity to plane wave Mie scattering is because when the particle is small, it is completely immersed into the evanescent field and the difference of the evanescent intensity between the top and the bottom of the particle is negligible. When the particle size becomes large, the difference of the evanescent wave intensity between the top and the bottom of the particle becomes pronounced, resulting in an asymmetric scattered intensity distribution as shown in the plane of incidence (i.e., in the XZplane). Fig. 4(c) can also be qualitatively explained using the same method as for Fig. 2(a). In this case, most of the evanescent intensity on the particle is incident with a small incident angle. As the Fresnel reflection coefficient on the particle surface
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is small for small incident angles [2], the final result is that the refracted rays lead to an intensity distribution in the region from 0° to 60°. For the 1 lm particle (Fig. 4(d)), the scattered intensity distribution for the S polarization illumination is mostly confined to a region below the Z-axis but that for the P polarization illumination is mostly above the Z-axis. According to the particle size, the major contribution to the scattered intensity distribution is from the rays with an incident angle between 70° and 80° near the bottom of the particle. The Fresnel reflection coefficients at these incident angles for S polarization illumination are larger than that for P polarization illumination (for example, the former is four times larger for the incidence at 75°). As a result, the majority of the intensity is reflected in the case of S polarization, while a majority of the intensity is refracted and transmitted through the particle in the case of P polarization (see Fig. 4(d)). When particle is brought close to the interface scattered intensity distribution in the XZ-plane is drastically changed. Fig. 5 shows scattered intensity distribution for small and large dielectric particles situated on the boundary between the surrounding medium and the substrate. It can be
Fig. 5. Dependence of the scattered intensity distribution in the XZ plane on the radius of a particle, when particle is situated on the interface: (a) a ¼ 0:05 lm, (b) a ¼ 0:1 lm, (c) a ¼ 0:5 lm and (d) a ¼ 1 lm. The solid and dotted curves correspond to the S and P polarization states of the illumination wave, respectively. n1 ¼ 1:6; n0 ¼ 1:0; n ¼ 1:51; k ¼ 632:8 nm and a ¼ 45°.
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D. Ganic et al. / Optics Communications 216 (2003) 1–10
seen that not only has the scattered intensity distribution drastically changed for both incident polarizations, but also that the scattering profile for P polarization is much stronger at smaller scattering angles (0–30° from either þz or z direction) when compared with S polarization. For larger particles, scattering profile for S polarization exhibits similar behaviour. It should be pointed out that for a particle of 100 nm in radius the scattered field distribution calculated from our model is the same as that given by the T-matrix model [14].
4. Effect of the interface on the morphologydependent resonance (MDR) Based on Fig. 5, the intensity integrated over the upper half-space of the particle can be calculated. This parameter gives a measure for the maximum signal strength in particle-trapped NSOM and therefore can be used to study the MDR caused by near-field Mie scattering. Fig. 6(a) shows the intensity integrated over the upper half-space of a particle with a refractive index of 1.6 immersed in air and situated on the boundary between the surrounding medium and the substrate for S and P polarization illumination. MDR becomes evident as the particle radius increases; the larger the particle size the sharper the resonance peaks. This is understandable because a scattering particle can be considered to be a micro-cavity. For a large particle, the evanescent wave interacts significantly with the particle near its bottom. Therefore, the beam refracted into the particle is incident on the particle boundary with a refraction angle close to the critical angle inside the particle, which results in a high reflectance or a large coefficient of finesse of such a cavity. Consequently, sharper resonance peaks emerge in the scattered field. According to Fig. 6(a), the difference of the radius between two resonance peaks is 71.2 2.0 nm, which agrees well with the result of 72.2 nm estimated by pffiffiffiffiffiffiffiffiffiffiffiffiffi k arctanð n21 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi Da ¼ ð5Þ 2p n21 1
Fig. 6. Dependence of the half-space scattered intensity on the particle radius for the S and P polarization illumination, when the particle is situated on the interface. n0 ¼ 1:0; n ¼ 1:51; k ¼ 632:8 nm and a ¼ 45°: (a) n1 ¼ 1:6, (b) n1 ¼ 1:3 and (c) n1 ¼ 1:1.
for plane-wave Mie scattering [18]. The dependence of MDR on the effective refractive index nef ¼ n1 =n0 is shown in Figs. 6(b) and (c). As
D. Ganic et al. / Optics Communications 216 (2003) 1–10
expected from Eq. (5), the spacing between two adjacent peaks increases as n1 decreases. As the effective refractive index of the particle increases, MDR effects become more pronounced with sharper resonance peaks because the coefficient of finesse of the cavity (i.e. the reflectance of the cavity) becomes larger. It can be seen from Fig. 7 that MDR peak positions are independent of the incident angle a of the illumination wave. This feature indicates that MDR is mainly caused by the wave interacting at the bottom of the particle. As the decay constant b of the evanescent wave increases with the incident angle a, the scattering of the evanescent wave with a particle is mainly confined to the bottom of the particle. As a result, the coefficients of finesse of the cavity are effectively increased, resulting in the shaper peaks (see Fig. 7 (bottom)). The calculation step in Fig. 7 is chosen to be 5 nm. It is found from our calculation that resonances are found much more precisely and more significant in strength if smaller step is used. However, smaller step calculation also requires significantly more computational time. Signal strength of the light scattered by a polystyrene particle of radius 0.25 lm, and illuminated by an evanescent wave produced by a He– Ne laser incident at an angle larger than the critical angle, is shown in Fig. 8. The particle is immersed in water and placed on the interface. Both calculated and experimental results agree that the scattered intensity decreases with incident angle a for both S and P polarization states of the incident illumination. Furthermore, it shows that the rate of decrease of the scattered intensity under S polarization is slower than that under P polarized illumination. Our model also confirms the experimental finding that the scattered intensity for S and P polarization states of incident illumination becomes equal at an incident angle, a, of approximately 58°. Provided that the refractive index of the particle does not change appreciably when the illumination wavelength varies, the dependence of MDR described in Figs. 6 and 7 changes only by a scaling factor because MDR is dependent on the size parameter q ¼ k 0 a. In other words, MDR in near-field Mie scattering can be dem-
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Fig. 7. Dependence of the half-space scattered intensity as a function of the particle radius for the S (solid) and P (dotted) polarization illumination for different incident angles a, when the particle is situated on the interface. Top: a ¼ 42°, middle: a ¼ 43°, bottom: a ¼ 45°. n1 ¼ 1:6; n0 ¼ 1:0; n ¼ 1:51 and k ¼ 632:8 nm.
onstrated from the fluorescence spectrum of a fluorescent particle excited by an evanescent wave.
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D. Ganic et al. / Optics Communications 216 (2003) 1–10
nounced and sharper. Increasing the effective refractive index causes sharper resonant peaks but reduces the separation between neighboring resonance peaks. It is also shown that the main contribution to MDR results from the evanescent wave at the bottom region of the particle because the MDR peak positions are independent of incident angle a.
Acknowledgement The authors thank the Australian Research Council for its support. Fig. 8. A comparison of the calculated half-space scattered intensity with the intensity measured by a NA ¼ 1.3 objective in particle-trapped near-field microscopy. The solid and dotted curves represent S and P incident polarization states, respectively. a is the incident angle and a polystyrene particle of radius 0.25 lm immersed in water is placed on the interface between the surrounding medium and the substrate.
5. Conclusion In this paper, we have developed a model for scattering of evanescent waves by a particle. Our scattering model includes the effect of an interface where an evanescent wave is generated. The 3-D intensity distribution around a dielectric particle and the integrated scattered intensity in the upper half-space of the particle for S and P incident polarized illumination is calculated. When particles are situated far away from the interface, the scattered intensity profile is similar to that obtained for plane-wave Mie scattering for small dielectric particles but becomes asymmetric when dielectric particles are larger. For particles close to the interface scattered intensity profiles are markedly different from those expected when the interface is not considered for both small and large dielectric particles. It has been found that MDR is evident in the scattered field of an evanescent wave and its period is the same as that for plane wave scattering. The peak position, sharpness and separation of MDR depend on the size parameter and effective refractive index of the particle. As the size parameter increases resonance peaks become more pro-
References [1] H. Chew, D. Wang, M. Kerker, Appl. Opt. 18 (1979) 2679. [2] M. Born, E. Wolf, Principles of Optics, seventh ed., Cambridge University Press, Cambridge, 1999. [3] C.F. Bohern, D.R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, 1983. [4] C. Liu, T. Kaiser, S. Lange, G. Schweiger, Opt. Commun. 117 (1995) 521. [5] R. Wannemacher, A. Pack, M. Quinten, Appl. Phys. B 68 (1999) 225. [6] R. Wannemacher, M. Quinten, A. Pack, J. Microsc. 194 (1999) 260. [7] M. Quinten, A. Pack, R. Wannemacher, Appl. Phys. B 68 (1999) 87. [8] S. Kawata, T. Sugiura, Opt. Lett. 17 (1992) 772. [9] M. Gu, P.C. Ke, Opt. Lett. 24 (1999) 74. [10] M. Gu, P.C. Ke, Appl. Phys. Lett. 75 (1999) 175. [11] P.C. Ke, M. Gu, Opt. Commun. 171 (1999) 205. [12] M. Gu, P.C. Ke, J. Appl. Phys. 88 (2000) 5415. [13] Y. Ishii, T. Yanagida, Single Mol. 1 (2000) 5. [14] A. Doicu, Y. Eremin, T. Wriedt, Comp. Phys. Commun. 134 (2001) 1. [15] A. Doicu, Y. Eremin, T. Wriedt, Opt. Commun. 190 (2001) 5. [16] P.W. Barber, R.K. Chang, in: Optical Effects Associated with Small Particles, World Scientific, Singapore, 1988, p. 5. [17] A. Ashkin, J.M. Dziedzic, Phys. Rev. Lett. 38 (1977) 1351. [18] P. Chylek, J.T. Kiehl, M.K.W. Ko, Phys. Rev. A 18 (1978) 2229. [19] R.E. Benner, P.W. Barber, J.F. Owen, R.K. Chang, Phys. Rev. Lett. 44 (1980) 475. [20] A. Ashkin, J.M. Dziedzic, Appl. Opt. 20 (1981) 1803. [21] P. Chylek, V. Ramaswamy, A. Ashkin, J.M. Dziedzic, Appl. Opt. 22 (1983) 2302. [22] D.C. Prieve, J.Y. Walz, Appl. Opt. 32 (1993) 1629.
Parametric study of three-dimensional near-field Mie scattering by dielectric particles Djenan Ganic, Xiaosong Gan *, Min Gu Centre for Micro-Photonics, School of Biophysical Sciences and Electrical Engineering, Swinburne University of Technology, P.O. Box 218, Hawthorn, Vic. 3122, Australia Received 8 August 2001; received in revised form 25 September 2002; accepted 25 November 2002
Abstract In this paper we present a model for studying the three-dimensional field distribution of near-field Mie scattering. The effect of the interface where near-field is generated is included in the model. We use this model to calculate the scattered electromagnetic field distribution in various planes around a dielectric particle of different sizes under S and P polarized illumination. Integrated scattered intensity into particleÕs upper half-space is investigated and it is found that it exhibits morphology-dependent resonance effect when particleÕs radius is approaching and exceeding illumination wavelength. The model is also applied to studying the dependence of the morphology-dependent resonance on various parameters associated with near-field Mie scattering by dielectric particles. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction An evanescent wave occurs when an electromagnetic wave is incident from a high refractiveindex medium into a low refractive-index medium under total internal reflection. Such an evanescent wave can be converted into a propagating wave by interaction with a small particle situated in the low refractive-index medium, provided that the distance between the particle and the interface separating the two media, d, is within a few illumination wavelengths in length. Such a scattering process is called near-field Mie scattering [1]. Un-
*
Corresponding author. Fax: +61-3-9214-5435. E-mail address: [email protected] (X. Gan).
like the plane-wave Mie scattering [2,3], near-field Mie scattering leads to an asymmetric distribution [1,4,5] and a complicated polarization nature [6,7]. The scattered field of an evanescent wave with a Mie particle has been studied [1]. It is found that the scattering cross-section exhibits a morphologydependent resonance (MDR) nature when the size of a dielectric particle is comparable to the illumination wavelength. Further, the MDR structure depends strongly on the penetration depth of the evanescent wave [4], but its peak positions and half widths do not [5]. In many applications including particle-trapped near-field scanning optical microscopy (NSOM) [8–12] and optical trapping nanometry [13], a three-dimensional (3-D) distribution of the scattered evanescent wave in the far-field region is
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(02)02252-6
2
D. Ganic et al. / Optics Communications 216 (2003) 1–10
needed to understand image resolution, image contrast, detection sensitivity and accuracy. Therefore it is necessary to know the 3-D scattered evanescent wave at any point around a scattering particle. The first calculation of the scattered field [1] does not include the effect of the interface and involves an approximation, which is valid only for two principal planes at a large distance from the scattering particle. Although the effect of the interface at which an evanescent wave is generated is discussed [5–7], only the cross-sections of evanescent waves by small particles is calculated [4–7]. Recently, Doicu et al. [14,15] have investigated the differential scattering cross-section and integral response of evanescent wave scattering by small particles and a sensor tip of up to 100 nm in radius. These numerical calculations include the interface effects in the context of the discrete sources method and the T-matrix method, and the far-field is determined by extrapolation of the scattered field as r ! 1. However, these calculations have not discussed scattering properties of larger particles whose scattering characteristics are markedly different from scattering properties of small particles due to MDR effects [4,16–21]. The aim of this paper is to present a detailed parametric study of the 3-D distribution of the scattered evanescent wave around a scattering particle without using any approximation. In Section 2, our model for near-field scattering, which includes the effect of an interface, is described. A mathematical expression for calculation of the 3-D electric field scattered by a Mie particle with an evanescent wave is given. Section 3 presents the detailed parametric calculations of the 3-D intensity distribution around a dielectric particle. The phenomenon of MDR is investigated in Section 4. In particular, the dependence of MDR on the illumination polarization states is revealed. The main conclusions of this research are drawn in Section 5.
2. Three-dimensional expression for near-field Mie scattering A schematic diagram of the near-field Mie scattering model investigated in this paper is
shown in Fig. 1. The incident angle of the electromagnetic wave is represented by a. The refractive index of the scattering particle is denoted by n1 , while it is immersed into a medium of refractive index n0 . The refractive index of the substrate is n. A particle is situated at a distance d from the interface at which an evanescent wave is generated [22]. The scattered electric field Etotal at an arbitrary point P in the surrounding medium can be expressed as a sum of the field that would be scattered into point P if the interface between two media was not present, and the contribution that is reflected into point P by the interface. Mathematically it can be expressed as Etotal ðrÞ ¼ Eupper ðrÞ þ rf Ebottom ðrÞ;
ð1Þ
where Eupper is scattered field into point P (upper space) and Ebottom is scattered field into point P 0 (bottom space) without considering the interface, while rf is the Fresnel amplitude reflection coefficient for a given incident polarization state. P 0 is the mirror-reflection point of point P. The amount of the scattered field at point P 0 that is reflected to point P is determined by the Fresnel amplitude reflection coefficients under given conditions. Eupper ðrÞ and Ebottom ðrÞ can be determined without considering the effect of the interface. As shown in Fig. 1, when the incident angle a of the illumination electric wave E0 is larger than the critical angle, the scattered electric field by the Mie particle can be expressed, for the polarization state of E0 perpendicular to the plane of incidence (i.e., S polarization), as [1] h i X ic ð1Þ bE ðl; mÞr hl ðk 0 rÞYllm ð^ ESC ðrÞ ¼ rÞ 02 n x lm ð1Þ 0 þ bM ðl; mÞhl ðk rÞYllm ð^ rÞ ; ð2Þ where l ¼ 1 to 1 and m ¼ l to +l. c is the speed of light in vacuum and x is the angular frequency of the incident light. The functions bE ðl; mÞ and bM ðl; mÞ are the expansion coefficients relating the ð1Þ illumination evanescent field, hl ðk 0 rÞ is the spherical Hankel function of the first kind, and Yllm are the vector spherical harmonics. When expressed in a spherical coordinate, Eq. (2) can be reduced to
D. Ganic et al. / Optics Communications 216 (2003) 1–10
3
Fig. 1. Illustration of the scattering model which includes the effect of an interface where an evanescent wave is generated. n; n0 and n1 denote refractive indices of the substrate, surrounding media and the particle, respectively. d is the distance from the center of a particle to the interface.
ESC ðrÞ ¼
X
(
c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bE ðl; mÞ lðl þ 1Þ lm "
1 o oYlm ð1Þ 0 h ðk rÞ sin h r sin h l oh oh # ð1Þ hl ðk 0 rÞ o2 Ylm þ 0 r1 n sin h ou2 " ð1Þ ð 1ÞbM ðl; mÞhl ðk 0 rÞ oYlm pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ou isinh lðl þ 1Þ n0 x
# cbE ðl; mÞ 1 oYlm o ð1Þ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrhl ðk rÞÞ h1
n02 x lðl þ 1Þ r oh or " ð1Þ bM ðl; mÞhl ðk 0 rÞ oYlm pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ oh i lðl þ 1Þ cbE ðl; mÞ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lðl þ 1Þ r sin h # ) oYlm o ð1Þ 0 ðrh ðk rÞÞ u1 : ou or l
n02 x
ð3Þ
Here r1 ; h1 and u1 are the unit vectors in the spherical coordinate. h and u are the variables of the scalar spherical harmonics Ylm . For the polarization state E0 parallel to the plane of incidence (i.e., P polarization), the scattered electric field is also given by Eq. (3) with expansion coefficients bE ðl; mÞ and bM ðl; mÞ substituted by b~E ðl; mÞ and b~M ðl; mÞ, respectively [1]. Eq. (3) gives the electric field resulting from scattering of an evanescent wave by a Mie particle without using any approximation and without considering interface effects. In the previous calculation [1], an approximated form of Eq. (3), which is valid only in two principal planes, was used. In this paper, Eq. (3) is used together with our scattering model that includes the interface between the surrounding medium and the substrate, in the following numerical calculation for the 3-D distribution of the scattered field around a particle near the interface.
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The scattered field is dependent on the effective refractive index nef ¼ n1 =n0 and the size parameter q ¼ k 0 a, where a is the radius of a scattering Mie particle and k 0 is the propagation constant of the electromagnetic wave in the medium surrounding the particle. The strength and the distribution of the scattered field is also affected by the decay constant b of the evanescent wave, defined as [1] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 sin2 a b ¼ k0
1: ð4Þ n02 A larger decay constant of an evanescent wave, correspond to the faster decay of the evanescent wave in the depth direction (i.e., in the X-direction). A scattering particle is immersed into an evanescent wave completely or partially, depend-
ing on the particle size and the decay constant, which greatly affects the scattered field distribution around the particle. 3. Effect of the interface on the 3-D scattered intensity distribution around dielectric particles According to our mathematical model described by Eq. (3), a 3-D intensity distribution of the scattered field can be numerically calculated at any distance from the scattering particle. To understand the effect of the interface, let us first consider that a particle is situated far away from the interface. Fig. 2 shows an example of the scattered intensity distribution by a dielectric particle of a radius of 2 lm, immersed in air. To
Fig. 2. Three-dimensional distribution of the scattered intensity around a 2 lm dielectric particle situated far away from the interface (a) in the XZ-plane, (b) in the plane containing the X-axis at 45° anti-clockwise from the XZ-plane, (c) in the XY-plane and (d) in the YZ-plane. The solid and dotted curves correspond to the S and P polarization states of the illumination wave, respectively. n1 ¼ 1:6; n0 ¼ 1:0; n ¼ 1:51; k ¼ 632:8 nm and a ¼ 45°.
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demonstrate the 3-D distribution of the scattered field, the scattered intensity in the XZ-plane, in the plane containing the X-axis at 45° anticlockwise from the XZ-plane, in the XY-plane and in the YZ-plane are displayed in Figs. 2(a)–(d), respectively. The scattered intensity distribution is asymmetric in the XZ-plane and in the plane at 45° from the XZ-plane because the illumination field is asymmetric in these planes and the evanescent wave propagates in the Z-direction. Fig. 2 shows that the evanescent wave is most intensely scattered into certain regions around the particle. This effect is due to the fact that the interaction of the evanescent wave with the particle is confined to the bottom of the particle because particle size is so large compared with the decay depth of the evanescent wave (in this case the decay depth is 186.5 nm). This phenomenon can be qualitatively explained using SnellÕs law and the Fresnel amplitude coefficients for reflection and transmission [2] (Fig. 3). Three rays representing the evanescent wave propagating along the Z-direction are chosen for demonstration. The length
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of the vectors in Fig. 3 represents the relative strength of the evanescent intensity. Ray 1, the highest intensity ray of the three selected rays, interacts at the very bottom of the particle (point A) with a large incident angle. Consequently, most of the ray intensity is reflected rather than refracted because of the high incident angle with respect to the normal of the particleÕs surface. The refracted ray, after traversing through the particle, interacts with the particle-medium boundary at point B and the large amount of its intensity emerges into the medium with a small portion reflected. This reflected portion of intensity traverses through the particle again, interacts with the boundary (point C) and emerges into the medium while a negligible amount is reflected again. Rays 2 and 3, whose intensity is weaker and determined by the decay constant of the evanescent wave experience a similar process. As a result, the relative intensity distribution of the three rays after three refraction processes indicates that the scattered intensity profile is confined to certain regions around the particle and that the highest intensity region is
Fig. 3. A qualitative interpretation of the confined intensity regions in the scattering of an evanescent wave by a dielectric particle of radius 2 lm. The relative intensity of rays is denoted by the arrow length.
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Fig. 4. Dependence of the scattered intensity distribution in the XZ plane on the radius of a particle, when the particle is situated far away from the interface: (a) a ¼ 0:05 lm, (b) a ¼ 0:1 lm, (c) a ¼ 0:5 lm and (d) a ¼ 1 lm. The plots in the left and right columns show the intensity distributions scattered by an evanescent wave and a plane wave, respectively. The solid and dotted curves correspond to the S and P polarization states of the illumination wave, respectively. n1 ¼ 1:6; n0 ¼ 1:0; n ¼ 1:51; k ¼ 632:8 nm and a ¼ 45°.
D. Ganic et al. / Optics Communications 216 (2003) 1–10
located in the region below 0° for this particular (see Fig. 2(a)). The location of these intensity regions depends on the refractive indices of the dielectric particle and its immersion medium, the particle size and the decay constant of the nearfield wave. For particles far away from the interface the dependence of the asymmetric scattered intensity distribution on the particle size, is depicted in Fig. 4. For small particle sizes (Figs. 4(a) and (b)), the scattered intensity profiles are similar to those under plane wave illumination [2]. This similarity to plane wave Mie scattering is because when the particle is small, it is completely immersed into the evanescent field and the difference of the evanescent intensity between the top and the bottom of the particle is negligible. When the particle size becomes large, the difference of the evanescent wave intensity between the top and the bottom of the particle becomes pronounced, resulting in an asymmetric scattered intensity distribution as shown in the plane of incidence (i.e., in the XZplane). Fig. 4(c) can also be qualitatively explained using the same method as for Fig. 2(a). In this case, most of the evanescent intensity on the particle is incident with a small incident angle. As the Fresnel reflection coefficient on the particle surface
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is small for small incident angles [2], the final result is that the refracted rays lead to an intensity distribution in the region from 0° to 60°. For the 1 lm particle (Fig. 4(d)), the scattered intensity distribution for the S polarization illumination is mostly confined to a region below the Z-axis but that for the P polarization illumination is mostly above the Z-axis. According to the particle size, the major contribution to the scattered intensity distribution is from the rays with an incident angle between 70° and 80° near the bottom of the particle. The Fresnel reflection coefficients at these incident angles for S polarization illumination are larger than that for P polarization illumination (for example, the former is four times larger for the incidence at 75°). As a result, the majority of the intensity is reflected in the case of S polarization, while a majority of the intensity is refracted and transmitted through the particle in the case of P polarization (see Fig. 4(d)). When particle is brought close to the interface scattered intensity distribution in the XZ-plane is drastically changed. Fig. 5 shows scattered intensity distribution for small and large dielectric particles situated on the boundary between the surrounding medium and the substrate. It can be
Fig. 5. Dependence of the scattered intensity distribution in the XZ plane on the radius of a particle, when particle is situated on the interface: (a) a ¼ 0:05 lm, (b) a ¼ 0:1 lm, (c) a ¼ 0:5 lm and (d) a ¼ 1 lm. The solid and dotted curves correspond to the S and P polarization states of the illumination wave, respectively. n1 ¼ 1:6; n0 ¼ 1:0; n ¼ 1:51; k ¼ 632:8 nm and a ¼ 45°.
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seen that not only has the scattered intensity distribution drastically changed for both incident polarizations, but also that the scattering profile for P polarization is much stronger at smaller scattering angles (0–30° from either þz or z direction) when compared with S polarization. For larger particles, scattering profile for S polarization exhibits similar behaviour. It should be pointed out that for a particle of 100 nm in radius the scattered field distribution calculated from our model is the same as that given by the T-matrix model [14].
4. Effect of the interface on the morphologydependent resonance (MDR) Based on Fig. 5, the intensity integrated over the upper half-space of the particle can be calculated. This parameter gives a measure for the maximum signal strength in particle-trapped NSOM and therefore can be used to study the MDR caused by near-field Mie scattering. Fig. 6(a) shows the intensity integrated over the upper half-space of a particle with a refractive index of 1.6 immersed in air and situated on the boundary between the surrounding medium and the substrate for S and P polarization illumination. MDR becomes evident as the particle radius increases; the larger the particle size the sharper the resonance peaks. This is understandable because a scattering particle can be considered to be a micro-cavity. For a large particle, the evanescent wave interacts significantly with the particle near its bottom. Therefore, the beam refracted into the particle is incident on the particle boundary with a refraction angle close to the critical angle inside the particle, which results in a high reflectance or a large coefficient of finesse of such a cavity. Consequently, sharper resonance peaks emerge in the scattered field. According to Fig. 6(a), the difference of the radius between two resonance peaks is 71.2 2.0 nm, which agrees well with the result of 72.2 nm estimated by pffiffiffiffiffiffiffiffiffiffiffiffiffi k arctanð n21 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi Da ¼ ð5Þ 2p n21 1
Fig. 6. Dependence of the half-space scattered intensity on the particle radius for the S and P polarization illumination, when the particle is situated on the interface. n0 ¼ 1:0; n ¼ 1:51; k ¼ 632:8 nm and a ¼ 45°: (a) n1 ¼ 1:6, (b) n1 ¼ 1:3 and (c) n1 ¼ 1:1.
for plane-wave Mie scattering [18]. The dependence of MDR on the effective refractive index nef ¼ n1 =n0 is shown in Figs. 6(b) and (c). As
D. Ganic et al. / Optics Communications 216 (2003) 1–10
expected from Eq. (5), the spacing between two adjacent peaks increases as n1 decreases. As the effective refractive index of the particle increases, MDR effects become more pronounced with sharper resonance peaks because the coefficient of finesse of the cavity (i.e. the reflectance of the cavity) becomes larger. It can be seen from Fig. 7 that MDR peak positions are independent of the incident angle a of the illumination wave. This feature indicates that MDR is mainly caused by the wave interacting at the bottom of the particle. As the decay constant b of the evanescent wave increases with the incident angle a, the scattering of the evanescent wave with a particle is mainly confined to the bottom of the particle. As a result, the coefficients of finesse of the cavity are effectively increased, resulting in the shaper peaks (see Fig. 7 (bottom)). The calculation step in Fig. 7 is chosen to be 5 nm. It is found from our calculation that resonances are found much more precisely and more significant in strength if smaller step is used. However, smaller step calculation also requires significantly more computational time. Signal strength of the light scattered by a polystyrene particle of radius 0.25 lm, and illuminated by an evanescent wave produced by a He– Ne laser incident at an angle larger than the critical angle, is shown in Fig. 8. The particle is immersed in water and placed on the interface. Both calculated and experimental results agree that the scattered intensity decreases with incident angle a for both S and P polarization states of the incident illumination. Furthermore, it shows that the rate of decrease of the scattered intensity under S polarization is slower than that under P polarized illumination. Our model also confirms the experimental finding that the scattered intensity for S and P polarization states of incident illumination becomes equal at an incident angle, a, of approximately 58°. Provided that the refractive index of the particle does not change appreciably when the illumination wavelength varies, the dependence of MDR described in Figs. 6 and 7 changes only by a scaling factor because MDR is dependent on the size parameter q ¼ k 0 a. In other words, MDR in near-field Mie scattering can be dem-
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Fig. 7. Dependence of the half-space scattered intensity as a function of the particle radius for the S (solid) and P (dotted) polarization illumination for different incident angles a, when the particle is situated on the interface. Top: a ¼ 42°, middle: a ¼ 43°, bottom: a ¼ 45°. n1 ¼ 1:6; n0 ¼ 1:0; n ¼ 1:51 and k ¼ 632:8 nm.
onstrated from the fluorescence spectrum of a fluorescent particle excited by an evanescent wave.
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nounced and sharper. Increasing the effective refractive index causes sharper resonant peaks but reduces the separation between neighboring resonance peaks. It is also shown that the main contribution to MDR results from the evanescent wave at the bottom region of the particle because the MDR peak positions are independent of incident angle a.
Acknowledgement The authors thank the Australian Research Council for its support. Fig. 8. A comparison of the calculated half-space scattered intensity with the intensity measured by a NA ¼ 1.3 objective in particle-trapped near-field microscopy. The solid and dotted curves represent S and P incident polarization states, respectively. a is the incident angle and a polystyrene particle of radius 0.25 lm immersed in water is placed on the interface between the surrounding medium and the substrate.
5. Conclusion In this paper, we have developed a model for scattering of evanescent waves by a particle. Our scattering model includes the effect of an interface where an evanescent wave is generated. The 3-D intensity distribution around a dielectric particle and the integrated scattered intensity in the upper half-space of the particle for S and P incident polarized illumination is calculated. When particles are situated far away from the interface, the scattered intensity profile is similar to that obtained for plane-wave Mie scattering for small dielectric particles but becomes asymmetric when dielectric particles are larger. For particles close to the interface scattered intensity profiles are markedly different from those expected when the interface is not considered for both small and large dielectric particles. It has been found that MDR is evident in the scattered field of an evanescent wave and its period is the same as that for plane wave scattering. The peak position, sharpness and separation of MDR depend on the size parameter and effective refractive index of the particle. As the size parameter increases resonance peaks become more pro-
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