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Chapter III. Scattering and Extinction of Evanescent waves by small particles III Scattering and Evanescent of observed for size parameters of order unity Small or larger. This critical size parameter Extinction Waves by depends on the real part n of the index of Particles The theory laid out in chapter II.2 showed, refraction. It decreases, if n increases. In that, due to the strong transverse intensity order to obtain strong MDR’s, it is gradient of evanescent waves, high-order necessary to have a low absorption, i.e. a multipoles low imaginary part of are strongly enhanced in the index of scattering and extinction of evanescent refraction. The MDR’s of large dielectric waves, compared to plane waves. In many spheres are also called whispering gallery cases this leads to strongly enhanced modes (WGM’s). resonances in the corresponding spectra. If the radius of the particles increases, of the number of MDR’s and their amplitudes homogeneous spheres and for coated increase, whereas their half widths and the spheres are discussed in this chapter. In distances between them decrease in a given addition, the range of refractive indices, spectral particle sizes, and thicknesses of the applications coating layer is investigated, for which resonances can be important, e.g. in order well-defined multipolar resonances are to allow Rabi-splittings, other nonlinear obtained in the visible spectral range. A effects or altered spontaneous emission plane generally characteristics to be observed at low pump produced by total internal reflection from power. E.g. for a silica sphere with radius an interface between two media. We 65 µm and refractive index n = 1.4518 at a initially want to assume, that the distance wavelength of 900 nm the half-width of the particles from the interface is large amounts to 270 kHz, corresponding to a enough to justify the neglect of multiple quality factor Q = 1.4.109 )1. This value is scattering effects. The latter will be still limited by residual absorption and discussed later in this chapter. does not represent the intrinsic Q value. Specific examples evanescent for wave the is case range. very In quantum sharp and optical strong An example for the significant changes III.1 Isolated Particles in Free Space Homogeneous dielectric spheres For dielectric spheres, the only possible resonances are the so-called morphologydependent resonances (MDR’s), which are due to constructive interference and are in the scattering spectra due to evanescentwave excitation is shown in Fig. III.1, which displays the scattering cross sections of a spherical diamond particle of diameter 2a = 600 nm. Similar results would be obtained for particles of lower refractive 40 Chapter III. Scattering and Extinction of Evanescent waves by small particles 2 TE 6 TM TM 5 TE7 6 TE TE3 5 TM 4 TE 4 TM 3 TM 2 The quantum numbers shown in Fig. III.1 correspond to the multipolar order of the 2 Scattering Cross Section [ µm ] 1 Fig. III.1 and is not explicitly given there. plan e wa ve resonance and, hence, describe the angular 0 50 dependence of the scattered field. From the TM 7 structure of the evanescent waves it TM 6 p-p ol TM 5 TM 4 0 100 TM 3 follows, that TM modes can be excited more efficiently by p-polarized (TM) evanescent waves and TE modes by s- TE 7 s-pol TE6 polarized (TE) evanescent waves. 50 TE5 0 0.5 0.6 TE 4 0.7 In total, Fig. III.1 demonstrates the TE3 0.8 0 .9 W avelength [ µm] Fig. III.1: Scattering spectra of a spherical diamond particle of diameter 2a=600 nm for an incident plane wave (a) and for p- and spolarized evanescent waves (b, c), respectively. For the evanescent waves an angle of incidence of 60° of the totally reflected wave and a constant index of refraction of the prism np = 1.5 is assumed. Note the different scales in b, c. extremely efficient excitation of MDR’s by evanescent waves compared to the case of plane-wave excitation2. For comparison, the different scales of the ordinate axes should be taken into account. The difference is easily understood by looking at Fig. III.2, which shows the instantaneous intensity distribution of the total field for index (e.g. polystyrene spheres), if a somewhat larger diameter was considered. Obviously, evanescent-wave excitations do not modify the peak positions and the half widths, but merely the amplitudes of the resonances. Due to the oscillatory nature of the Ricatti-Bessel functions in the denominator of the wavelength-dependent Mie-coefficients multipole resonances of the same orders n exist in the UV regime but are not considered here. For this reason, the modes are characterized by a second quantum number marking the radial dependence. This radial quantum number is equal to 1 for the resonances visible in the TM14 resonance at λ = 386 nm of a polystyrene sphere, r = 700 nm, excited by a p-polarised evanescent wave. The figure suggests a mechanical analogue for evanescent-wave excitation of WGM’s consisting of a chain (representing the evanescent wave) driving a gear (representing the WGM). Fig. III.3 displays the time-averaged intensity of the scattered field alone in the same case on linear and logarithmic intensity scales. It may be easily recognised, that the intensity is confined to a narrow ring in the plane of incidence close to the inner surface of the sphere. If, contrary to the procedure laid 41 Chapter III. Scattering and Extinction of Evanescent waves by small particles wave excitation of WGM’s demonstrated in Fig. III.1. Correspondingly, the field distribution shown in Fig. III.4 contains significant contributions from other multipoles. Another remark should be made here concerning the angular momentum of the electromagnetic field: In the case of evanescent-wave excitation of the MDR of a homogeneous particle (Fig. III.2, Fig. III.3) the ring-shaped pattern of intensity Fig. III.2: Total instantaneous intensity for the case of the TM14-resonance (λ=386nm) of a polystyrene sphere (radius r=700 nm) excited by a p-polarized evanescent wave which propagates from top to bottom of the image. Multiple scattering at the interface, where the evanescent wave is generated by total internal reflection, is neglected here. packets is rotating in the plane of incidence out in chapter II, the axis of quantization is chosen perpendicular to this ring, then, for evanescent-wave excitation, the WGM contains predominantly the m = n component of the vectorial multipole. For comparison, Fig. III.4 shows the same quantity for the case of plane-wave excitation. Here, the plane wave propagates from left to right. The intensity Fig. III.3: Same model as in Fig. III.2, but here the time-averaged intensity of the scattered field is shown. Left: Intensity shown in the same plane as in Fig. III.2. Right: Intensity perpendicular to it. Top: linear scale, Bottom: logarithmic scale distribution is now very different from the one discussed above, and, in particular, is not confined to a narrow ring. Whereas in Fig. III.3 the WGM strongly dominates the intensity, the sphere acts mainly as a ball lens, when excited by a plane wave, and, correspondingly, the WGM is hardly recognisable in Fig. III.4. This is in Fig. III.4: Same as Fig. III.3, but for plane-wave excitation. Left: Time-averaged intensity, Right: Instantaneous intensity agreement with the inefficiency of plane42 Chapter III. Scattering and Extinction of Evanescent waves by small particles corresponding to a nonvanishing angular excitation, where two counterpropagating waves are generated, which travel close to the inner surface of the sphere and produce a nonrotating intensity pattern with vanishing angular momentum and much smaller intensity. TE 2/ TM 1 2 TE 3 TM 2 TM 3 TE TM 2 TM 3 0.4 Scattering Cross Section [ µm ] This is contrary to the case of plane-wave TE 2 0.6 momentum of the electromagnetic field. a 1 0.2 0.0 TM 2 2 b TM 3 1 TM 3 0 TE3 4 c 2 Homogeneous semiconductor spheres TE3 TE 2 0 As a further example, Fig. III.5 depicts the 0.4 sections of a compact crystalline silicon 0.6 sphere with 2a = 200 nm excited by a plane respectively. Compared to diamond (n=2.4), silicon has a much larger refractive index (n=3.56) in the visible range. In addition, because the excitation energy is above the bandgap TM 1 /TE 2 TM 3 TE2 whereas diamond is transparent in this range. The scattering spectra are 0.8 0.9 TM 3 TE3 TM 2 a TE1 2 0.2 TM 2 0.0 TM 3 TM 2 b 2 TE3 TM 3 TM 2 1 0 TE3 6 c 4 for Si, there is considerable absorption, 0.7 0.4 Extinction Cross Section [ µm ] polarized evanescent waves (Fig. III.5b,c), 0.6 W avelength [ µm] optical scattering and extinction cross wave (Fig. III.5a) and by p- and s- 0.5 TE4 2 TE 2 TE 3 TE 2 0 0.4 0.5 0.6 0.7 0.8 0.9 W avelength [ µm] qualitatively similar to those of the diamond particle. Contributions from higher-order multipoles to the extinction cross section are, however, stronger than for the scattering cross section, as the Fig. III.5: Spectra of the scattering and extinction cross sections for a homogeneous silicon particle in vacuum with 2a=200 nm for plane-wave excitation (a), and p-polarized (b), s-polarized (c) evanescent-wave excitation. Note the different scale in Fig. III.5b,c. For the calculations, indices of refraction were taken from ref. 3. squared absolute values of the Mie coefficients, which appear in (II.60) of wave excitation, the absorption (extinction chapter II.2.2.2 decrease faster with order n minus scattering) is marginal in the case of than the real parts (II.59). E.g. Fig. III.5b the TM2-mode, whereas the absorption shows, that for p-polarized evanescent- cross section case is of the same order of 43 Chapter III. Scattering and Extinction of Evanescent waves by small particles magnitude as the scattering cross section polarised evanescent-wave excitation, re- for the TM3-mode. Comparison of the TE2 spectively (compare eqs. II.59, II.60 and and TE3 modes for the case of excitation Fig. II.3 of chapter II.2.2.2). The compari- by a s-polarised evanescent wave yields son between the cross sections for plane- qualitatively the same results (Fig. III.5c). wave and evanescent-wave excitation de- It should be added, that the optical pends on the definition of the normalizati- absorption cross section generally exhibits on factor of the cross sections, but the a stronger contrast between off-resonant comparison between both polarizations in and the case of the evanescent wave excitation on-resonant conditions than the scattering cross section. does not. For larger silver particles, like those of Homogeneous metallic spheres Fig. III.7 (2a=200nm), the extinction cross Very small particles exhibit the well- section is strongly enhanced for shorter known dipolar surface plasmon resonance wavelengths in the case of evanescent- at around 365 nm due to collective electron wave excitation. This effect is particularly oscillation. As a first example, we consider distinct for the case of excitation by a p- the extinction cross sections of a very polarized evanescent wave. The ratio of the small silver particle (Fig. III.6). Because in excitation-dependent cross sections in Fig. this case, the Mie coefficients an of the III.7 now depends on the wavelength, in electric multipoles are much larger than the the spectra. Furthermore, because the particle is so small, only the dipolar coefficient a1 contributes to the spectra to any significant extent. The shape of the extinction spectrum for evanescent waves is therefore unchanged in this case compared to plane-wave excitation, and the polarisation-dependent cross sections for extinction as well as for scattering are for all wavelengths very nearly in the ratio N −0 1Τ1 / 1 / N 0−1Π 1 ( Π 1 = 1 ) for p-pola- rised evanescent-wave, plane-wave, and s- 1.5 x10 -4 1.0 x10 -4 2 multipoles, the TE modes do not appear in Exti n ctio n Cross Se ction [ µm ] Mie coefficients bn of the magnetic p -p ol plan e wave 5.0 x10 -5 s-p ol 0 .0 0.3 0.4 0 .5 0 .6 W avele ngth [ µm ] Fig. III.6: Wavelength dependence of the cross section for extinction of plane waves and evanescent waves by a spherical particle with 2a=10 nm. The solid lines are calculated using (eqs. II.59 and II.60) for evanescent-wave excitation and the formulae of standard Mie theory for plane-wave excitation, respectively. The squares indicate numerical results obtained by means of the multiple multipole (MMP) method. Multiple scattering at the prism surface is neglected here. 44 Chapter III. Scattering and Extinction of Evanescent waves by small particles 2 Extinction Cross Section [ µm ] excitation 0 .4 contains stronger contributions from higher multipoles due 0 .3 to the large transverse field gradient of the p-pol 0 .2 evanescent wave. The difference in the 0 .1 cross sections for s-polarized and p- plane wave s-pol 0 .0 itself 0 .4 0 .6 0 .8 1 .0 W avele ngth [ µm] Fig. III.7: Wavelength dependence of the cross sections for extinction of plane waves and evanescent waves by a spherical particle with 2a=200nm. As in Fig. III.6a the solid lines are the analytical results from (II.59 and II.60) and the squares indicate the numerical results obtained by means of the MMP method. polarized evanescent-wave excitation, respectively, for a spherical particle is caused by the different internal structure of the exciting field in both cases. In the case of s-polarization, the electric field of the wave oscillates perpendicular to the plane of incidence, whereas in the p-polarized contrast to the spectra in Fig. III.6. In order to understand the spectra shown in Fig. III.7, Fig. III.8 displays the decomposition case, the electric field rotates in the plane of incidence due to the complex phase of the totally reflected wave. For this reason, of the extinction spectra into the different Multipole deco mposition multipolar contributions for p-polarized 0.4 plane waves (Fig. III.8b), the numbers 0.3 indicating the multipolar order n of the TMn-resonances. 0.2 1 3 4 maxima in the extinction spectra are 0.15 2 σex t [µm ] 0.0 multipoles in the case of evanescent-wave 2 0.1 Fig. III.8 clearly reveals, that the caused by a strong enhancement of higher a 2 σex t [µm ] evanescent waves (Fig. III.8a) and for b 0.10 0.05 1 2 excitation. The electric quadrupole and particularly the electric octupole give rise to increased extinction in the range of shorter wavelengths and a double-peak structure, if the excitation is a p-polarized evanescent wave. The enhancement of the higher multipoles is not caused by an excitation-dependent susceptibility of the particles, but merely by the fact, that the 3 0.00 0.3 0.4 0.5 0 .6 0.7 0.8 0.9 1.0 W avelength [µm] Fig. III.8: Decomposition of the extinction cross section into individual multipole contributions for a silver particle with 2a=200 nm. Excitation ppolarized evanescent wave (a), plane wave (b). The numbers indicate the order of the multipole, whose contribution to the total extinction cross section is shown. Results for s-polarized evanescent waves are not shown here, but are similar to those for p-polarization. 45 Chapter III. Scattering and Extinction of Evanescent waves by small particles TM 4 TM 5 * ε'' 0 -5 Extinctio n Efficie ncy D i ele ctric function of silve r 5 R e so na nc e Po sition s H om o g en eo u s S ph ere : n=1 n=2 n=3 Co a te d S ph ere : n=1 n=2 n=3 ε' -10 -15 TM 3 TM 2 TM 1 10 0 50 0 0.4 -20 0.5 0.6 0.7 0.8 0.9 Wa velen gt h [ µm] 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 W a velen gth [ µm] Fig. III.9: Resonance positions of Mie coefficients an, bn with n<=3 for silver spheres of diameter 2a=22 nm and for silver-coated spheres with core diameter 2a=20nm and shell thickness d=1 nm. The surrounding medium is assumed to be vacuum. for a spherical particle in front of a metallic mirror a polarization-dependent Fig. III.10: Extinction spectra of silica particles coated with a silver shell of thickness d=10 nm for p-polarized evanescent-wave excitation and core radius a=20nm to 150 nm in 10-mm steps (arrow marks increasing core size). To account for the variation in the cross section for the different particle sizes the cross sections have been normalized to the geometric particle cross sections to obtain the extinction efficiency. Ppolarized evanescent-wave excitation. The line marked with a star arises from all multipoles appearing in the spectrum (compare Fig. III.9). Again εM =1. cross section is also expected even for the case of plane-wave excitation. In order to demonstrate, that the numerical calculations yield the same results as the analytical formulae, the symbols representing the numerical results have been added in Fig. III.6 and Fig. III.7. The numerical method will be used below to take multiple coated spheres can be calculated by formula (II.84 and II.85). The imaginary part of the index of refraction can be neglected for silver-coated particles, because it is sufficiently small for silver and does not vary drastically with the wavelength. The index of refraction for silver, as well as the resonance positions scattering at the interface into account. for a compact silver sphere with diameter 2a=22 nm and for a coated sphere with a Coated spheres Whereas for homogeneous dielectric spheres either a large sphere radius or a sufficiently large refractive index is required to obtain morphology-dependent resonances, comparatively small particles can exhibit resonances in the visible spectral region, if they are coated by a metal. The resonance position for the silica core of diameter 2a= 20 nm and a silver shell of thickness d=1nm are shown in Fig. III.9 for a broad spectral range. Fig. III.10 displays the dependence of the extinction spectra of silica particles coated with a silver shell of thickness d = 10 nm for the case of p-polarised evanescent-wave excitation and core radii 46 Chapter III. Scattering and Extinction of Evanescent waves by small particles 1.0 1.0 n=1 0.9 n=2 0.8 0.8 n=3 0.7 0.7 d = 1 nm 0.6 d = 2 nm 0.6 0.5 0.4 0.5 0 20 40 60 80 100 120 140 0 20 Partic le Radius a [nm ] 40 60 80 100 120 140 0.4 Partic le Radius a [nm ] 1.0 1.0 n=2 d = 5 nm 0.9 d = 10 nm 0.9 0.8 n=1 0.7 0.7 n=2 n=4 0.6 0.8 n=1 n=3 n=3 n =5 0.6 0.5 0.5 W avel ength [µ m ] Wavelength [ µm] W avelength [µm ] Wavelength [µ m] n=1 n=2 0.9 n=5 0.4 0 20 40 60 80 100 120 140 Partic le Radius a [nm ] 0 20 40 60 80 100 120 140 0.4 Partic le Radius a [nm ] Fig. III.11: Resonance positions of the electric multipoles TMn for silver-coated silica particles as a function of core radius and thickness of the shell. The straight lines are guidelines for the eye. a = 20 to 150 nm in 10 nm steps. For orders broaden and shift to the red. The clarity the spectra are shifted vertically by resonance at λ ≈ 340 nm corresponding to a individual ε1(Ag) ≈ 0 remains at approximately the multipolar contributions are identified. The same position for all core sizes. It should weak line marked with a star arises from be mentioned, that the calculated spectra all multipoles appearing in the spectrum shown in Fig. III.10 do not take into and corresponds to the resonances close to account size-dependent corrections of the ε1(Ag) = 0, compare Fig. III.9. For the dielectric function, which are known4 to be smallest core radius of a = 20 nm only the necessary for small metallic structures. dipole contribution is observed. With These corrections essentially lead to a increasing core size higher multipoles broadening of the calculated resonances. constant amount. The appear and the splitting between the Fig. III.11 shows the dependence of the resonance positions increases. At the same observable resonance positions on core time resonances due to lower multipole size for thicknesses d = 1,2,5,10 nm. The 47 Chapter III. Scattering and Extinction of Evanescent waves by small particles lines through the calculated positions are discussed. The following discussion is also only guidelines to the eye, but to a good relevant for apertureless near-field optical approximation a linear relationship is microscopy (chapter VI). observed. In an attempt to verify the calculations Homogeneous silver spheres: for metal-coated spherical particles shown As an example, the cross sections for a above homogeneously silver-coated latex spherical silver particle in contact with a spheres have been prepared in our group in glass prism were caclulated via the MMP the meantime5. Smooth silver coatings of technique and are shown in Fig. III.12: a thickness d=11.5 nm were obtained. redshift of the spectra due to the presence Confocal wavelength-resolved scattering of the interface is universallyly recognised. microscopy of individual particles of this kind did not reproduce, however, the discussed above, and a particle. This is ascribed to the strong influence of the unavoidable Extinction spectra was obtained from partricle to 2 considerable variation of the scattering Cross Section [µm ] resonances grain 0,2 plane wave Scattering Effects of Multiple Scattering at p-pol 0,2 plane wave 0,0 p-pol In evanescent-wave scattering experiments the particle is almost always in contact with the interface, where the evanescentwave is generated by total internal Absorption 2 the Surface of a Substrate Cross section [µm ] 2 Cross Section [µm ] scattering spectra (compare here also III.2 p-pol 0,0 structure of the silver shell on the chapter V). 0,4 0,2 0,0 plane wave 0,4 0,6 W avelength [nm] reflection. Modifications of the scattering spectra due to backscattering from the interface must then be taken into account. These effects will now be discussed for a Fig. III.12. Comparison of the cross sections of a silver sphere with diameter 2a=200 nm in free space (solid lines) and on the glass prism (squares) for excitation with a plane (lower two curves) and a p-polarized evanescent wave (upper two curves). number of examples. Applications to total internal reflection microscopy6 are 48 Chapter III. Scattering and Extinction of Evanescent waves by small particles Moreover, the scattering cross section is 1.5 enhanced in the wavelength range λ > 0.4 absorption is strongly reduced in ppolarisation for λ < 0.4 µm. Homogeneous dielectric spheres In this case, again considerable modifications of the scattering spectra occur. As an example, in Fig. III.13 the 2 S ca tte rin g Cros s S ec tion [µm ] µm in both polarisations, whereas the 1.0 TM TE 5 TM 6 6 TE 5 TM 4 TE4 TM 3 a 0.5 0.0 TM 6 TM 7 TM 5 5 b TM 4 TM 3 TM 2 0 10 T E7 TE 6 TE5 c TE 4 TE 3 5 cross sections for a spherical diamond particle in contact with a glass prism are shown: Compared to the isolated particle, the resonance wavelengths are again slightly shifted to longer wavelengths. The resonances are broadened considerably, and, correspondingly, their amplitudes are reduced substantially. 0 0.5 0.6 0.7 0.8 0.9 W avelength [µm] Fig. III.13: Scattering spectra for the spherical diamond particle of Fig. III.1 (diameter 2a = 600 nm), when it is in contact with the prism surface. Multiple scattering is included in the calculation. Refractive index of the prism nP = 1.5. Evanescent waves as in Fig. III.1. In the plane-wave case a plane wave is assumed to be incident normally on the prism surface from the prism side. An equivalent model consisting of a spherical scattering object, which is surface, which means that the impedances embedded in a medium with a higher index are parallel. Simplifying the surface as an of refraction approximately describes the equivalent electric circuit consisting of two spectra. If the index of refraction of the serial or parallel capacitances, respectively, environment increases, the resonances shift a larger effective dielectric constant for the to longer wavelengths and in this way this TM modes is obtained. Comparison of Fig. simple model can describe the observed III.1 and Fig. III.13 makes it clear, that red shift.. The red shift is larger for TM large resonance enhancements are possible, modes than for TE modes. This may be when the particle is held at some distance explained in a heuristic fashion by the fact, from the prism surface. On the other hand, that for TE modes the electric field is even when the sphere is in contact with the parallel to the surface, which corresponds surface, the spectra for evanescent-wave to a serial arrangement of the impedances excitation differ strongly from those for of both halfspaces, whereas for TM modes plane-wave excitation, and resonances with there is a field component normal to the large peak amplitudes are observed for s49 Chapter III. Scattering and Extinction of Evanescent waves by small particles as well as p-polarised evanescent waves. changes of the scattered field due to The broadening and the reduction of the multiple scattering processes, if the particle peak heights may be explained by the is in contact with the interface. disturbance of the total internal reflection Multiple scattering effects like those due to multiple scattering processes. The shown in Fig. III.14 are obviously highly latter result in a smaller transverse field relevant for Total Internal Reflection gradient, equivalent to reduction of the Microscopy (TIRM)6. In this experimental multipolar enhancement factors, and lead technique light scattering by particles, to a strong energy flux into the prism, which move in an evanescent wave, is used equivalent to a reduction of the quality to study the motion of the particle factor. This energy flux is described by the perpendicular to the interface with high Poynting vector. Fig. III.14 shows the sensitivity to small changes in the distance time-averaged Poynting vector for the TM6 to the interface, and to characterise in this resonance of the particles of Fig. III.1 and way particle-interface potentials. Fig. III.13 and demonstrates the large It is thereby generally assumed, that the Fig. III.14: Time-averaged Poynting vector of the scattered field for a diamond particle of radius a=300nm and for p-polarized evanescent-wave excitation at the TM6 resonance at 456 nm in free space (a), and at 458 nm, when the particle is in contact with the prism surface (b). The latter is indicated by the straight line, and the prism is to the right of the line. The evanescent wave propagates from the top of the figure to the bottom and decays from the right to the left. 50 Chapter III. Scattering and Extinction of Evanescent waves by small particles scattered power is proportional to that of size. Scattering cross sections for a p- the exciting evanescent wave at the center polarized evanescent wave at two different of the sphere, and, hence, decreases expo- wavelengths, λ = 458 nm, corresponding to nentially with the distance d to the inter- the resonance ( TM 16 ) of the isolated face. Under this assumption, the particle- sphere, and λ = 488 nm, corresponding to wall interaction potential can be calculated from the probability distribution of z and the use of the Boltzmann probability distribution: w(d) ~ exp(φ(d)/kT). This was first applied by Prieve et al.6 in order to study colloidal forces in the proximity of a surface. In recent experiments7 the Coulomb interaction between negatively charged particles (assumed to be infinitely hard in first approximation) and a negatively charged surface was determined in this way. Gravity appears as an additional potential in this experiment. The method is very sensitive, because off-resonant excitation, were determined as a function of d. The differential scattering cross section was integrated I) over a solid angle of 4π in order to obtain the total power scattered by the particle and II) over a finite solid angle (aperture angle 53°) in the vacuum half space above the particle in order to calculate the signal obtained experimentally with a N = 0.8 microscope lens. Fig. III.15 displays the results for the case of resonant excitation of the whispering-gallery resonance TM 16 at λ = 458 nm, the forces acting on the particles can be as well as for off-resonant excitation λ = measured in the range of 10 fN. The 488 nm. As in Fig. III.14b strong scattering calculated potential, however, will be into the prism is easily recognised, when affected by any deviation from the the particle is in contact with the substrate exponential dependence due to multiple (Fig. III.15a,b). Moving the particle away scattering effects in the vicinity of the from the surface by a few hundred surface. The method can be used for the nanometers results in a drastic reduction of examination of surfaces, e.g. to distinguish this directional scttering into the prism between surfaces of different polarity. (Fig. III.15c,d). Furthermore, in the case of We have calculated the power scattered resonant excitation of a WGM, the diffe- from a diamond sphere (r=300 nm, same as rential cross section for scattering scatter- above) as a function of the distance d from ing into the vacuum half space exhibits the interface, where the evanescent wave is much stronger modulations as a funtion of generated. Similar results are expected for the scattering angle than in the case of off- polymeric particles of somewhat larger resonant excitation. This holds independent of the particle-interface distance. The 51 Chapter III. Scattering and Extinction of Evanescent waves by small particles Fig. III.15: Absolute value of time-averaged Poynting vector for a diamond particle of radius a=300nm, which is excited by a p-polarized evanescent-wave of wavelength of λ=458 nm (TM16 resonance: part (a,c)) and 488 nm (off-resonant excitation, b,d), respectively. The images a,b show the results for the particle in contact with the interface, where the evanescent wave is generated, and b,d those for the particle 550 nm above it. The boundary is indicated by the straight line, and the optically denser medium is to the right of this line. The evanescent wave propagates from the top of the figure to the bottom and decays from the right to the left. The scattered fields are shown (left and middle part) in the plane of incidence. The modification of the corresponding potentials (explanation in the text) due to multiple scattering processes are shown in the right part for the resonant (e) and the non-resonant (f) case. The inserts in (e,f) show the integrated powers on a logarithmic scale. The main graphs show the integrated powers on a linear scale after normalisation to the expected exponential dependence. The black line represents the corrections necessary in TIRM. inserts in the graphs (Fig. III.15e,f) show exponential is clearly observed. This is the integrated powers on a logarithmic related to the strong scattering into the scale. In the main graphs these results are prism, when the particle is close to contact shown on a linear scale after normalisation with the interface. The power scattered into to the expected exponential dependence. the lens (black lines), on the other hand Due to multiple scattering effects decreases by about 25% in the case of involving the substrate and particle boun- resonant excitation and by about 10% for daries, an increase of the total scattered off- resonant excitation, relative to the power expected exponential dependence. This can (red lines) over the simple 52 Chapter III. Scattering and Extinction of Evanescent waves by small particles be explained by the strong directional almost tangential to the interface. This emission into the medium with the higher applies to the scattered intensity in both index of refraction discussed above, which leads to reduced scattering into the vacuum half space. The inserts in both graphs show, that the intensities have to be determined with high accuracy over several orders of magnitude, which is experimentally quite difficult. Our calculations show, that it is important at which wavelength the TIRM experiment described above, is carried out and that sizable deviations from the exponential distance dependence indeed exist, especially at wavelengths, where whispering gallery modes (WGM’s) of the sphere are resonantly excited. These corrections lead, of course, to altered particle-wall potentials. We have investigated not only diamond particles in air, but also polystyrene spheres in water on a substrate surface, because real experiments often employ this set-up. The corresponding results at a wavelength of λ = 500 nm are presented in Fig. III.16. If the particle is in contact with the surface, a highly directional emission into the optically denser medium is again observed. Because of the relatively small difference of the index of refraction of water and polystyrene and the relatively small particle diameter, the light is scattered mainly in the direction of propagation of the evanescent wave, Fig. III.16: Absolute value of the time-averaged Poynting vector for a polystyrene particle of radius a = 300 nm in water, which is excited by a ppolarized evanescent wave at a wavelength of λ = 500 nm. The images (a,c) show the results for the particle in contact with the interface, where the evanescent wave is generated, and the images (b,d) those for the particle 550 nm above it. The boundary is indicated by the straight line, and the optically denser medium is to the right of this line. The evanescent wave propagates from the top of the figure to the bottom and decays from the right to the left. The scattered fields are shown in the plane of incidence (a,b) and perpendicular to it (c,d). The main graphs in part (e) show the integrated powers on a logarithmic scale. In the insert, these results are shown on a linear scale after normalisation to the expected exponential dependence. 53 that most of the scattered power does not enter the lens in this case. The total scattered power depends on the surface-to-surface distance in a way quite different from the case considered above. Here, not only the power scattered 2 media (glass, H2O). It is easy to recognize, scattering cross section [ µm ] Chapter III. Scattering and Extinction of Evanescent waves by small particles 2x10 monomer, p-pol monomer, s-pol dimer, p-pol dimer, s-pol 2 T E14 T M 14 1x10 2 T E13 T E15 TM 13 T M 12 0 380 400 420 440 wav elength [n m] into the solid angle in water, corresponding to the numerical aperture of the lens, but also the total scattered power decreases with decreasing surface-to-surface Fig. III.17: Scattering cross sections for two contacting polystyrene spheres (r=700nm) excited by an evanescent wave. The spectra were calculated using the MMP technique. Multiple scattering processes due to the presence of the substrate surface have been neglected here. distance. Besides, the power scattered into the lens decreases already for a distance of about 500 nm. This must be ascribed to the change in the field distribution with increasing distance, which is different in the present case, compared to the case considered in Fig. III.15. III.3 Interacting Dielectric Spheres in free Space interact due to the evanescent fields of these modes outside of the particles. The coupling is, however, relatively weak, and the condition for constructive interference of the WGM is approximately unchanged. Therefore, the resonant modes of two interacting spheres (‘photonic molecule’8) can still be classified according to the TE and TM modes of the individual spheres. In many cases scattering and extinction of The main effect of the interaction is then a more than one particle is of interest. If the splitting of the whispering-gallery resonan- particles are close to each other the most ces. This is demonstrated in Fig. III.17, important change in the spectra is a which displays the scattering spectra of splitting of the resonances due to the two interacting polystyrene spheres, r = interaction of the resonant modes. For 700 nm, in a narrow spectral range. The metallic particles the situation becomes spectra have been calculated using the actually quite involved the delocalised MMP technique, and multiple scattering at nature of the electronic wavefunction. This the substrate surface has been neglected. is discussed in detail in chapter V. In the Fig. III.19 displays the absolute value of present paragraph only dielectric particles the time-averaged (a,c) and instantaneous are considered. In this case the whispering- (b,d) Poynting vector for the components gallery modes of the individual spheres of the split TM14 resonance at 384 nm (a,b) 54 Chapter III. Scattering and Extinction of Evanescent waves by small particles Fig. III.18: Dimer (same configuration as in Fig. III.17 and Fig. III.19) on the substrate of a surface. (a), (b): instantaneous intensity (c): timeaveraged intensity of the scattered field. (a) linear scale (ratio of the maximum and the minimum intensity: 10), (b,c) logarithmic scale (30dB). The bright colors feature the regions of high field intensity (see colorscale at the bottom) Fig. III.19: Absolute value of the time-averaged (a,c) and instantantaneous (b,d) Poynting vector of the scattered field (total field minus field in the absence of the particles) for the split components of the TM14 mode of the same configuration as in Fig. III.17. λ=384nm (a,b) and λ=388nm (c,d), respectively. and 388 nm (c,d). The two resonances III.4 Interacting Dielectric Spheres on the Surface of a Substrate differ, because the phase difference of the two WGM’s of both spheres is different at If the particle dimer is now placed on a the two wavelengths. This phase difference substrate surface, multiple scattering pro- is determined by the wavelength of the cesses lead again to a substantial modifi- evanescent wave and the distance of the cation of the intensity distribution (Fig. two particles. It controls the coupling of III.18). As in the case of an isolated the two WGM’s at the contact point. Fig. particle on a substrate (Fig. III.14, Fig. III.19 shows, that the almost isotropic III.15) strong emission into the substrate is intensity distribution for a single particle in observed. The emission pattern, however, the plane of incidence (compare the left is much more complicated than in the case column of images in Fig. III.3 becomes of isolated spheres. This applies also to the highly anisotropic for a dimer of such emission pattern into the vacuum half particles. Light is emitted predominantly in space. two directions corresponding to the upper right and lower left corners of the images in Fig. III.19. It is immediately evident, that the field distribution will become even more complicated, if particle arrays containing more than two particles are considered. 55 Chapter III. Scattering and Extinction of Evanescent waves by small particles Such arrays may be interesting, e.g. for rapid intensity variations as a function of optical computing purposes, or in order to excitation wavelength can hence be related create to multiple splitting of the single-particle photonic crystals with band structures in two or three dimensions. resonances. For large chains or two- Using the MMP method, we have dimensional structures, which may be modeled the scattering of evanescent considered as ‘photonic crystals’, these waves by a chain of six latex spheres splittings correspond, of course, to the (r=0.7µm) on the surface of a substrate band structure of the crystal. The k vector (n=1.5) for different wavelengths of the of the mode excited by the evanescent exciting light. Multiple scattering effects at wave is then determined by the angle of the boundary of the surface are taken into incidence of the wave inside the substrate. account in this calculation. We have A scan of the latex chain with a SNOM chosen a range of wavelengths, where tip at a constant height above the chain resonances of the single particle occur. The would scan images depending strongly on scattering spectrum of such a particle was the scan height. The images displayed in shown in Fig. III.17. The images in Fig. Fig. III.20 show highly directed radiation III.20 show the absolute value of the time- into the optically denser medium and, less averaged Poynting-vector in the plane of intense, into the vacuum. It is possible to incidence. Similar models were recently address different spheres by choosing the examined by other groups9. Within the appropriate wavelength. For example, the spheres immediate first image in the top row of Fig. III.20 intensity (λ=408nm) shows a high intensity at the distribution is obtained. Noticeable is the top sphere, and the first image in the high intensity contrast in the images, and bottom row (λ=414nm) a high intensity at the sensitive dependence of the local the second latex sphere from the bottom. intensity on the excitation wavelength. In This ‘addressing’ of individual spheres order to understand the basic features of concerns both, the radiation into the these results, it is useful to reconsider the substrate and into the vacuum domain. spectra for two such spheres in contact This may be of interest for the construction with each other (Fig. III.17). These spectra of optical switches in the field of optical show a splitting of the whispering-gallery computing. In order to address individual resonances into two modes. It is expected, spheres in a 2-dimensional lattice of latex that in the case of a chain of n spheres, the spheres, it is necessary to generate two resonances split into n components. The exciting evanescent waves, which pro- and neighborhood in a the complicated 56 Chapter III. Scattering and Extinction of Evanescent waves by small particles pagate in two different directions on the substrate surface. If the latex array becomes very large, however, the addressing of the inner spheres will become more difficult, because the inner spheres in the array become more and more equivalent. As may be recognized from Fig. III.20, the radiation is emitted mainly from the spheres at the end of the chain and only weakly from the spheres in the middle part. Whispering-gallery modes can be efficiently excited not only by evanescent waves, but also by light-emitting dipoles within the spheres. Experimentally, such active spheres can be realized by doping the spheres with fluorescent molecules, semiconductor quantum dots or other fluorescent particles10. The latter may be excited by a plane wave, whose frequency fits to an absorption of the dopant. Lasing in micro-cavities is currently studied in many groups, and in this context, stimulated emission in photonic crystals made of latex spheres is of considerable interest. 57 Chapter III. Scattering and Extinction of Evanescent waves by small particles Fig. III.20: Linear chain of six latex spheres (radius r=700 nm) in vacuum on a glass substrate (n=1.5), where an evanescent wave is generated by total internal reflection of a plane wave (white arrows) incident under an angle of 60° on the glass-vacuum interface. In the glass medium a standing wave is formed. The evanescent wave propagating in the vacuum domain is scattered by the latex spheres. The intensity (absolute value of the time-averaged Poynting vector) was calculated by the MMPmethod for 8 different wavelengths (values given on top of the individual images) and plotted using a logarithmic color-coded scale. 58 Chapter III. Scattering and Extinction of Evanescent waves by small particles References 1 L. Colet, V. 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