Abstract

Three-dimensional focusing of evanescent waves by specially configured surface arrays of nanoparticles emulating near-field optical holograms of dipole sources (located close to the surface) is suggested and analyzed. The idea is to place chains of nanoparticles along bright fringes of calculated (holographic) interference patterns so that the local nanoparticle density along these chains would be proportional to the local intensity contrast in the interference patterns. Three different configurations are considered: a holographic scheme with totally internally reflected reference and reconstructing waves, a modified scheme with the reconstructing wave being represented by a suitable surface plasmon polariton (SPP) plane wave, and a SPP holographic scheme with reference and reconstructing waves both being (phase-conjugated) laterally-confined (Gaussian) SPP beams. Our numerical approach is based on the Green’s function technique with the point-dipole approximation for radiation scattering by nanoparticles. We demonstrate that a nanoparticle array configured in accordance with the intensity interference pattern formed by a dipole field and a reference wave allows one to efficiently focus the (phase-conjugated) reconstructing wave (via its scattering by the nanoparticle array) at the site of the dipole. Influence of the polarization and wavelength of the reconstructing wave on the resulting intensity distribution is also considered. Fabrication of suitable nanoparticle arrays is discussed along with their potential applications.

©2008 Optical Society of America

1. Introduction

Modern optical technology has opened new possibilities for light manipulation (e.g., localization and directional propagation) in systems whose sizes are about the light wavelength or even smaller. For these systems, one of the most interesting and important processes to be considered is the contribution of evanescent waves in the total (incident and scattered) optical field. Evanescent waves generated by interaction of radiation with nanostructures exhibit high spatial frequencies and, thereby, can be exploited for achieving subwavelength localization of the electromagnetic energy, a phenomenon that can be in turn used to strongly affect a nano-object in question (e.g., to excite individual quantum dots, to modify a nanoparticle, to generate strong Raman scattering by a bio-molecule, etc.). The possibility to concentrate electromagnetic fields at different locations within a micro-system by advantageously exploiting the evanescent wave superposition is thereby the important aspect of modern micro- and nano-optics. For example, suitable evanescent-wave superposition can be used for creation of optical traps [1, 2] or focusing of electromagnetic radiation far beyond the diffraction limit [3]. One of the possible methods to properly shape evanescent waves, including their focusing, is by extending the holography approach into the domain of near-field optics [4, 5, 6]. It has been indeed shown [7] that a near-field hologram can be recorded with a reference wave, which is totally internally reflected by a surface with some object placed onto it, whose image can be reconstructed with the subwavelength resolution using near-field optical microscopy techniques. Detailed analysis of the reconstructed dipole field obtained in a static holographic scheme [4, 5] has shown that this method emulates the phase conjugation of optical near field, including both propagating and evanescent field components, and, as a consequence, allows one to concentrate radiation in small volumes, even smaller than that is given by the diffraction limit. Effectively, it was demonstrated that the near-field holography process can be used for three-dimensional (3D) focusing of evanescent waves beyond the diffraction limit. Note that, in order to record a near-field hologram, it was suggested to use a photosensitive layer with subwavelength thickness placed on a dielectric substrate. The hologram would then be obtained as a surface profile of the photosensitive layer developed after being illuminated by the (interfering) totally-reflected (incident) and scattered (by the object) waves. From the practical point of view, such a procedure is rather cumbersome since the usage of nm-thin photosensitive layers (e.g., its deposition and development) requires dealing with many fabrication difficulties and realization challenges [7]. Modern nanofabrication techniques allow one to carve surface profiles with nanometer precision, for example by focused ion-beam milling, but at the expense of using very complicated technological processes.

 

Fig. 1. Schematic side view of the holographic technique (a) with TIR reference wave and (b) with SPP Gaussian beam as the reference wave.

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In this paper, for the purposes of 3D focusing of evanescent waves beyond the diffraction limit, we propose the usage of specially configured surface arrays of nanoparticles (that can be straightforwardly fabricated by, for example, using electron-beam lithography) emulating near-field optical holograms of dipole sources and investigate numerically the feasibility of such an approach. Suitable nanoparticle arrays are suggested to construct by placing chains of nanoparticles along bright fringes of calculated (holographic) interference patterns so that the local nanoparticle density along these chains would be proportional to the local intensity contrast in the interference patterns. We consider different holographic configurations, including the holography based on surface plasmon polaritons (SPPs), which are electromagnetic waves of the optical range arising at a metal-dielectric interface [8]. The SPP-based near-field holography configuration is suggested and analyzed for the first time to our knowledge. We conduct extensive numerical simulations by making use of the Green’s tensor formalism and dipole approximation [9] and discuss the results obtained.

2. Model

In this section two models describing a dipole image in the a static holographic process are developed. First, the scheme on the based of the total internal reflection (TIR) configuration is considered, and, second, the holography scheme involving SPP Gaussian beams is discussed.

2.1. TIR configuration

Similar previous works [4, 5] we consider the system consisting of a point dipole situated at r d=(0,0, zd) in air above a flat surface of a glass substrate with dielectric constant ε (Fig. 1). The dipole is locally irradiated by the monochromatic field E 0(r d) with wavelength λ. A hologram of the dipole on the height zobλ above the substrate surface (Fig. 1) is obtained due to the interference between the induced dipole field E d(r) and a reference wave E r(r) being a wave (nonhomogeneous) transmitted in air for a plane wave incident from below on the substrate-air interface under the TIR condition. Note the TIR configuration eliminates the transmission of a homogeneous wave, which would affect the dipole radiation, and it facilitates the subsequent detection process of the holographic dipole image. In general one should take into account the coupling between the dipole and the substrate surface. This coupling is very small for relatively large distances between the dipole and substrate surface, that is for z dλ. Whereas in the near-field zone above the surface, when zdλ, the coupling may not be negligible resulting in an increase of the dipole magnitude [10, 11]. In order to include in our consideration the coupling effects we suppose that the dipole is the result of polarization of a spherical nanoparticle (its volume is Vd and the dielectric permittivity εd) by external optical field with wavelength being much larger than the particle’s size. In this case the dipole moment p is given by [12]:

P=[Îk02ε0α̂Ĝs(rd,rd)]1α̂E0(rd),

where Î is the unit dyadic tensor, k 0 is the wave number in vacuum, ε 0 is the vacuum permittivity, Ĝ s(r,r′) is the surface part of the Green’s tensor of the substrate-air interface system (see below), when points r and r′ are located in the air. The polarizability α̂ used in Eq. (1) is the dipolar polarizability of a nanoparticle located in free space in the long-wavelength electrostatic approximation

α̂=3ÎVdε0εd1εd+2.

Using the Green’s tensor Ĝ(r,r′) of the substrate-air interface system the dipole field E d at a point r ob on the plane being parallel to the substrate surface on the height zob is given by

Ed(rob)=k02ε0[Ĝ0(rob,rd)+Ĝs(rob,rd)]p.

Owing to the fact that the both points r ob and r d are located in the air above the interface the Green’s tensor Ĝ(r,r′) in Eq. (3) splited into two separate contributions Ĝ0(r,r′) and Ĝs(r,r′). The first contribution is the Green’s tensor of free space determining the direct electric dipole field, and the second contribution is the scattering part of the Green’s tensor accounting for the secondary electric field that is related to reflection from the interface. The tensor Ĝ0(r,r′) has the relatively simple analytical representation in the direct space [13, 14], the tensor Ĝs(r,r′) involves numerical integrations of Sommerfeld integrals [12, 15, 16]. Note that for a dielectric-metal interface the tensor Ĝs(r,r′) also includes the excitation of SPPs. The reference electric field is given by

Er(rob)=Etexp(iktrob),

where kt=k0(εsinθ,0,iεsin2θ1), θ is the incident angle being larger than the critical one for the TIR. The amplitudes E t are linearly connected to the incident amplitude and can be found by using the transfer matrix of Fresnel coefficients (those connecting expressions for different linear polarization are clearly represented, for example, in Ref. [17]).

In order to record the hologram we suggest to use chains of identical spherical metal nanoparticles placed on the substrate surface by a special manner. Since the reconstruction of the dipole field is the result of interaction a reconstructing wave with the recorded hologram, the nanoparticles are placed along bright fringes of a interference pattern formed by the dipole field and the reference wave. Because of the intensity field distribution along the bright fringes is inhomogeneous, the densities of nanoparticle distribution along the corresponding lines should be changed proportional to the local field intensity.

Finally, illuminating the nanoparticle array by the reconstructing wave being a plane wave E c(r) incidenting from below on the substrate surface at the angle θ, one obtains the holographic image of the dipole source above the surface. Note the reconstructing plane wave should create the transmitted field E ct (r ob)=E t exp(i kt r ob) propagating in opposite direction with respect to the reference wave (4), that is kt'=k0(εsinθ,0,iεsin2θ1),, we will call just this wave as the reconstructing wave. From the calculation point of view the holography is obtained by the following manner. First, the dipole moments of the nanoparticles are found by solving the equation [12]:

pi=α̂Ect(ri)+k02ε0α̂Ĝs(ri,ri)pi+k02ε0jiNα̂[Ĝ0(ri,rj)+Ĝs(ri,rj)]pj,i=1,,N

where r i=(xi,yi, zi) and r j=(xj,yj, z j) are the radius-vectors of the centers of particles with number i and j respectively, α̂ is the free space polarizability tensor of the particles determined by Eq. (2) with corresponding volume Vp and dielectric constant εp, N is the total number of the particles in the array. Second, once the dipole moments of the particles are found from Eqs. (5), the total electric field everywhere outside the particles can be determined by

E(r)=Ect(r)+k02ε0i=1N[Ĝ0(r,ri)+Ĝs(r,ri)]pi.

2.2. SPP configuration

Another system, which is considered in the paper, is schematically shown in Fig. 1(b). The dipole field interferes with the reference surface wave being an SPP Gaussian beam excited on the surface of a metal substrate and propagating along x-axis direction. As in the previous configuration the hologram is recorded by nanoparticle array arranged on the substrate surface with correspondence of interference pattern of the intensity distribution formed by the dipole field and the reference SPP Gaussian beam. The dipole field on the calculation plane is again given by Eq. (3), the electric field of the reference beam propagating along the x-axis can be represented in the form [18]:

Er(rob)=exp(ikSPPxob+ikzzobyob2W2f(xob))f(xob){ẑ+ŷ(i2yobkz(kSPPW)2f(xob))
+x̂[kzkSPP(1+1(kSPPW)2f(xob)2(kSPPyob)2((kSPPW)2f(xob))2)]},

where r ob=(x ob,yob, zob), W is the beam waist, x̂, ŷ, ẑ are the coordinate unit vectors, kSPP=k0ε/(1+ε) is the SPP wave number, kz=k02kSPP,2f(xob)=1+i2kSPPxob/(kSPPW)2.. The reconstructing field E ct (r ob) is the phase conjugated SPP Gaussian beam with respect to the reference beam (7).

3. Numerical results

For numerical simulations in the framework of the TIR configuration we consider the following set of parameters. A gold spherical nanoparticle (radius 10 nm), modeling the dipole source [Fig. 1(a)], is placed above glass substrate (ε=2.25) at the height zd=800 nm and locally eradiated by monochromatic field with wavelength 800 nm and polarization along y-axis. A reference wave, interacting with the dipole field above the substrate, is obtained due to the TIR of the plane electromagnetic wave with wavelength 800 nm and TE-polarization [Fig. 1(a)], the incident angle θ is equal to 60°. The interference (hologram) between the dipole field and the reference wave is calculated on the plane with zob=50 nm [Fig. 1(a)]. For recording the hologram we use nanoparticles with radius rp=50 nm situated on the substrate. The field interference for the described configuration and the corresponding nanoparticle array are shown in Fig. 2. Here the dipole field that reaches the substrate is basically consisted of the far-field components therefore the intensity distribution in Fig. 2 has a view of interference fringes. For the formation of the nanoparticle structure we approximate every crest of the intensity distribution shown in Fig. 2(a) by a parabolic curve and then situate the nanoparticles along these parabolas with the density being proportional to the local field intensity. The result of this procedure is shown in Fig. 2(b).

 

Fig. 2. Interference on the plane with zob=50 nm [Fig. 1(a)] between the field of dipole situated at zd=800 nm above the substrate and the reference evanescent wave being the result of the TIR for an optical plane wave with wavelength 800 nm (the TIR configuration). (a) Linear color representation. (b) Contour-representation of the interference, the black small circles indicate the positions of the nanoparticles recording the hologram.

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First, in order to demonstrate the reconstruction of the dipole field we consider intensity distribution of the reconstructed field (6) at different cross-section planes with respect to the surface of the substrate for different polarizations of the reconstructingwave (Fig. 3). It is found that the total reconstructed field indeed localizes in relatively small region above the substrate at the position of the dipole independently on the polarization of the reconstructing wave. Such behavior of the field can be considered as three-dimensional focusing of the reconstructing wave. It is interesting to remark that the field localization above the substrate with nanoparticle structure formed in the TE-polarization of the reference wave (Fig. 2) is realized also in the case of the TM-polarized reconstructing wave (Fig. 3). This possibility is connected with the fact that the interference fringes between a TM-polarized reference wave and the field created by the dipole oriented along z-axis is similar the interference pattern shown in Fig. 2. The main difference between these two cases is determined by the polarization state of the reconstructed field. This polarization is corresponded to the polarization of the reconstructing wave (Fig. 4). In the case of the TE-polarization the field at the dipole region is basically directed along y-axis [Fig. 4(a)] whereas in the TM-polarization the main field component is the z-component [Fig. 4(b)]. Note that the holography image of the dipole source was obtained only with nanoparticle structure and evanescent reconstructing waves.

In order to show the role of structural configuration in the holographic focusing process we calculated the reconstructed field for different nanoparticle structures which are corresponded partially to the interference patterns. The dependence of the field intensity distribution upon the nanoparticle structure is shown in Fig. 5. From the figure one can see that the field confinement is better for the case when the particle structure is more corresponding to the optical holograms of a dipole source, the value of the intensity also increases at the point of the dipole source.

 

Fig. 3. Normalized intensity distribution of the total electric field being the result of the interaction between the reconstructing evanescent wave and nanoparticle structure recording the hologram of the dipole in the TIR configuration (Fig. 2). (a), (c), (e) TE-polarization of the reconstructing wave, (b), (d), (f) TM-polarization of the reconstructing wave. (a) and (b) cross section z=zd=800 nm, (c) and (d) cross section y=0, (e) and (f) cross sections x=0.

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Fig. 4. Normalized intensity of the different electric field components after the reconstruction process along cross section x=0, z=zd=800 nm. (a) TE-polarization of the reconstructing wave, (b) TM-polarization of the reconstructing wave.

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Fig. 5. Dependence of the reconstructed field upon nanoparticle structures. (a)–(e) nanoparticle structure and field intensity (arb. un.) after the reconstruction process along cross section z=zd=800 nm in the TIR configuration. TE-polarization of the reconstructing wave. The red small circles indicate the positions of the nanoparticles.

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Fig. 6. Normalized intensity distribution of the total electric field being the result of the interaction between the reconstructing SPP plane wave and nanoparticle structure recording the hologram of the dipole in the TIR configuration (Fig. 2). (a) cross section y=0, (b) cross sections x=0, (c) along the z-direction for x=0, y=0.

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As it was shown the TM-polarized evanescent wave can be used for the reconstruction of the dipole field, we considered some new physical system in which the nanoparticle structure, formed in the framework of the TIR configuration and TE-polarization, is situated on gold flat surface (the permittivity ε) that sustains propagation of SPP waves. The SPP plane waves are TM-polarized evanescent waves and therefore it would be interesting to apply them for the surface holography reconstruction. Here we consider that the reconstructing wave is an SPP plane wave with the electric field E ct=exp(-ikSPPx-akSPPz)(-iax̂, 0, ̂z) in the region z>0 (a=1/ε), propagating in negative direction of the x-axis and having the real part of SPP wave number k SPP being equal to the x-component of the wave vector of the reference wave (4) in the TIR configuration. The latter means that the SPP wavelength λ SPP should be the same as the in-plane wavelength of the reconstructing evanescent wave in the TIR configuration. In our case λ SPP≈615 nm that corresponds light wavelength λ≈642 nm. The result of the reconstruction on the base of the SPP plane wave is shown in Fig. 6(a),(b) one can see good localization of the reconstructed field above the surface with shifting the position of the field maximum above the substrate as comparison with the case of the TIR-configuration (Fig. 6(c)). Note that if we selected a SPP wave with a larger wavelength (λ SPP=684 nm or light wavelength λ=705 nm), the field localization would have disappeared [the curve corresponding to λ=705 nm in Fig. 6(c)]. However if we change the light wavelength of the reconstructingwave in the TIR configuration, so that the in-plane wavelength of the evanescent wave was equal to also 684 nm, the position of the intensity maximum along the z-axis only slightly shifts [the curve corresponding to λ=887 nm in Fig. 6(c)]. The differences in the behavior are connected with different properties of SPP waves and transmitted light waves under the TIR, in particular, with different relation between the field amplitude components of SPP waves and the transmitted (evanescent) TIR waves [19]. However the both considered configurations under certain conditions allow us to efficiently focus the reconstructing evanescent wave due to its scattering by the nanoparticles.

 

Fig. 7. Interference on the plane with zob=50 nm (Fig. 1) between the field of dipole situated at zd=800 nm above the substrate and the reference SPP Gaussian beam (the waist W=2000 nm and light wavelength 642 nm) and corresponding nanoparticle array recording the hologram in the framework of the SPP configuration. (a) Dipole orientation is along the x-axis. (b) Dipole orientation is opposite to the z-axis. The black circles indicate the positions of the dipoles, the red small circles indicate the positions of the nanoparticles recording the hologram.

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Let us now consider the holography process only in the framework of the SPP configuration [Fig. 1(b)]. In the numerical calculations we use the following parameters: λ=642 nm, zd=800 nm, the waist of the SPP Gaussian beam W=2000 nm, zob=50 nm, nanoparticle’s radius in the structure is equal 50 nm. Throughout this paper, we use the gold permittivity data from [20]. Figure 7 demonstrates both the interference between the SPP Gaussian beam and dipole field on the plane zob=50 and the corresponding nanoparticle structure for two orientations of the dipole relatively the surface of the substrate. One can see that the intensity distribution is dependent upon the dipole orientation. The intensity distribution of the reconstructed fields is represented in Fig. 8. From the figure one can see that for the reconstructing beam corresponding the same wavelength as the reference beam [Fig. 8(a),(b),(d),(e)] the total reconstructed field indeed efficiently concentrates approximately at the position of the dipole independently on the dipole orientation. However field focusing for the case when the dipole is directed perpendicular to the metal surface is better [compare Fig. 8(a),(d) and Fig. 8(b),(e)]. This difference could be connected with fact that the dipole electric field in the near-field zone (evanescent field) is directed along the dipole and therefore gives perceptible contribution in the holography in the case when the dipole is directed perpendicular to the metal surface. In any case we can say that the SPP-configuration allow us to efficiently focus the reconstructing SPP wave at a position above the surface due to its scattering by the nanoparticles. If we change the wavelength for the reconstructing beam the field localization in the region of the dipole disappears [Fig. 8(c),(f)]. It means that the obtained nanoparticle structures have some filtering properties. Analysis of the field polarization in the region of holography images shows that the main field component coincides with z-component of the reconstructing beam independently on the dipole orientation in the hologram recording process, because this component is dominant one in the reconstructing SPP beams. Concluding this section, it is important to note that, in general, locations of interference fringes are determined by relative phases of the incident (evanescent TIR or SPP) wave and the dipole field, leading to different appearances of interference patterns (cf. Figs. 3, 6 and 8).

 

Fig. 8. Cross sections [y=0 for (a), (b), (c) and x=0 for (d), (e), (f)] of the normalized intensity distribution of the total electric field being a result of the interaction between the reconstructing SPP Gaussian beam and nanoparticle structure in the SPP configuration (Fig. 7). (a) and (d) correspond the case in Fig. 7(a) (λ=642 nm), (b) and (e) correspond the case in Fig 7(b) (λ=642 nm), (c) and (f) the case when the reconstructing SPP Gaussian beam is corresponded to the light wavelength λ=705 nm, the nanoparticle structure from Fig. 7(b).

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4. Conclusions

We have proposed and analyzed the possibility of 3D focusing of evanescent waves beyond the diffraction limit by making use of specially configured surface arrays of nanoparticles emulating near-field optical holograms of dipole sources. Different holographic schemes were suggested. We have considered the TIR-configuration, in which the hologram resulted from the interference between a dipole field and a TIR reference wave, propagating along the surface of dielectric substrate, so that nanoparticle chains were placed on the substrate along lines corresponding to bright fringes of the calculated (holographic) interference pattern. Local nanoparticle density in these chains was set to be proportional to the local intensity contrast in the interference pattern. For the reconstruction, we have considered different reconstructing waves: a reconstructing optical wave being phase-conjugated with respect to the reference wave; an SPP plane wave with the wavelength being equal to the in-plane wavelength of the reconstructing optical wave in the TIR configuration. We have further extended, for the first time to our knowledge, this concept of near-field holography in the domain of plasmonics, with both the reference and reconstructing waves being represented by SPP Gaussian beams which were phase-conjugated with respect to each other. It has been demonstrated that the properly configured nanoparticle structures can be used for the efficient 3D focusing of evanescent waves (in the both considered configurations) at the site of the dipole used for calculations of the corresponding interference pattern. Furthermore, it has been shown that the polarization state of the focused field can be controlled by selecting the polarization of the reconstructing wave. However, the most efficient focusing was observed for configurations in which the main components of dipole, reference and reconstructing waves were of the same polarization: for the SPP configuration, for example, the dipole should be polarized perpendicular to the metal surface. In comparison with the previously considered holographic approach [7] based on shallow surface holograms, the main advantage is that one can make use of relatively straightforward design and fabrication approaches when employing identical nanoparticles that emulate a near-field optical hologram of an object (i.e., to be reconstructed) field, e.g. produced by dipole sources. The locations of these nanoparticles can be easily determined by calculating the corresponding (holographic) interference pattern, whose contrast sets the local density of nanoparticle chains to be placed along lines corresponding to bright fringes of the interference pattern. The nanoparticle array can be fabricated, for example, using standard techniques of electronbeam lithography and metal deposition followed by lift-off. Note that the scattering strength of nanoparticles, especially those made of metal, can be much stronger than that of shallow dielectric gratings, resulting in more efficient focusing of the incident (reconstructing) field. We believe that the suggested approach for efficient 3D focusing of evanescent waves can be advantageously exploited in all applications based on field enhancement effects, such as bio-and molecular sensing and identification.

Acknowledgements

The authors acknowledge financial support from the Danish Research Council for Technology and Production (project No. 274-07-0258). A. B. Evlyukhin is grateful the Russian Foundation for Basic Research, Grant No. 06-02-16443, for the support.

References and links

1. V. I. Balykin, V. V. Klimov, and V. S. Letokhov, “Atom nanooptics based on photon dots and photon holes,” JETP Lett. 78, 8–12 (2003). [CrossRef]  

2. P. J. Reece, V. Garcés-Chávez, and K. Dholakia, “Near-field optical micromanipulation with cavity enhanced evanescent waves,” Appl. Phys. Lett. 88, 221116 (2006). [CrossRef]  

3. R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317, 927–929 (2007). [CrossRef]   [PubMed]  

4. S. I. Bozhevolnyi and B. Vohnsen, “Phase conjugation of optical near field by a surface hologram,” Opt. Commun. 135, 19–23 (1997). [CrossRef]  

5. B. Vohnsen and S. I. Bozhevolnyi, “Holographic approach to phase conjugation of optical near fields,” J. Opt. Soc. Am. A 14, 1491–1499 (1997). [CrossRef]  

6. L. C. Thomson, G. Whyte, M. Mazilu, and J. Courtial, “Simulated holographic three-dimensional intensity shaping of evanescent-wave fields,” J. Opt. Soc. Am. B 25, 849–853 (2008). [CrossRef]  

7. S. I. Bozhevolnyi and B. Vohnsen, “Near-field optical holography,” Phys. Rev. Lett. 77, 3351–3354 (1996). [CrossRef]   [PubMed]  

8. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). [CrossRef]   [PubMed]  

9. A. B. Evlyukhin and S. I. Bozhevolnyi, “Point-dipole approximation for surface plasmon polariton scattering: Implications and limitations,” Phys. Rev. B 71, 134303 (2005). [CrossRef]  

10. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. Total radiated power,” J. Opt. Soc. Am. 67, 1607–1615 (1977). [CrossRef]  

11. O. Keller, M. Xiao, and S. Bozhevolnyi, “Configurational resonances in optical near-field microscopy: a rigorous point-dipole approach,” Surf. Sci. 280, 217–230 (1993). [CrossRef]  

12. L. Novotny, B. Hecht, and D. W. Pohl, “Interference of locally excited surface plasmons,” J. Appl. Phys. 81, 1798–1806 (1997). [CrossRef]  

13. O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995). [CrossRef]   [PubMed]  

14. O. Keller, “Attached and radiated electromagnetic fields of an electric point dipole,” J. Opt. Soc. Am. B 16, 835–847 (1999). [CrossRef]  

15. T. Søndergaard and S. I. Bozhevolnyi, “Surface plasmon polariton scattering by a small particle placed near a metal surface: An analytical study,” Phys. Rev. B 69, 045422 (2004). [CrossRef]  

16. M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Greens tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000). [CrossRef]  

17. D. Van Labeke and D. Barchiesi, “Probes for scanning tunneling optical microscopy: a theoretical comparison,” J. Opt. Soc. Am. A 10, 2193–2201 (1993). [CrossRef]  

18. T. Søndergaard and S. I. Bozhevolnyi, “Theoretical analysis of finite-size surface plasmon polariton band-gap structures,” Phys. Rev. B 71, 125429 (2005). [CrossRef]  

19. A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. 5, S16–S50 (2003). [CrossRef]  

20. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4579 (1972). [CrossRef]  

References

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  1. V. I. Balykin, V. V. Klimov, and V. S. Letokhov, “Atom nanooptics based on photon dots and photon holes,” JETP Lett. 78, 8–12 (2003).
    [Crossref]
  2. P. J. Reece, V. Garcés-Chávez, and K. Dholakia, “Near-field optical micromanipulation with cavity enhanced evanescent waves,” Appl. Phys. Lett. 88, 221116 (2006).
    [Crossref]
  3. R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317, 927–929 (2007).
    [Crossref] [PubMed]
  4. S. I. Bozhevolnyi and B. Vohnsen, “Phase conjugation of optical near field by a surface hologram,” Opt. Commun. 135, 19–23 (1997).
    [Crossref]
  5. B. Vohnsen and S. I. Bozhevolnyi, “Holographic approach to phase conjugation of optical near fields,” J. Opt. Soc. Am. A 14, 1491–1499 (1997).
    [Crossref]
  6. L. C. Thomson, G. Whyte, M. Mazilu, and J. Courtial, “Simulated holographic three-dimensional intensity shaping of evanescent-wave fields,” J. Opt. Soc. Am. B 25, 849–853 (2008).
    [Crossref]
  7. S. I. Bozhevolnyi and B. Vohnsen, “Near-field optical holography,” Phys. Rev. Lett. 77, 3351–3354 (1996).
    [Crossref] [PubMed]
  8. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
    [Crossref] [PubMed]
  9. A. B. Evlyukhin and S. I. Bozhevolnyi, “Point-dipole approximation for surface plasmon polariton scattering: Implications and limitations,” Phys. Rev. B 71, 134303 (2005).
    [Crossref]
  10. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. Total radiated power,” J. Opt. Soc. Am. 67, 1607–1615 (1977).
    [Crossref]
  11. O. Keller, M. Xiao, and S. Bozhevolnyi, “Configurational resonances in optical near-field microscopy: a rigorous point-dipole approach,” Surf. Sci. 280, 217–230 (1993).
    [Crossref]
  12. L. Novotny, B. Hecht, and D. W. Pohl, “Interference of locally excited surface plasmons,” J. Appl. Phys. 81, 1798–1806 (1997).
    [Crossref]
  13. O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
    [Crossref] [PubMed]
  14. O. Keller, “Attached and radiated electromagnetic fields of an electric point dipole,” J. Opt. Soc. Am. B 16, 835–847 (1999).
    [Crossref]
  15. T. Søndergaard and S. I. Bozhevolnyi, “Surface plasmon polariton scattering by a small particle placed near a metal surface: An analytical study,” Phys. Rev. B 69, 045422 (2004).
    [Crossref]
  16. M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Greens tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
    [Crossref]
  17. D. Van Labeke and D. Barchiesi, “Probes for scanning tunneling optical microscopy: a theoretical comparison,” J. Opt. Soc. Am. A 10, 2193–2201 (1993).
    [Crossref]
  18. T. Søndergaard and S. I. Bozhevolnyi, “Theoretical analysis of finite-size surface plasmon polariton band-gap structures,” Phys. Rev. B 71, 125429 (2005).
    [Crossref]
  19. A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. 5, S16–S50 (2003).
    [Crossref]
  20. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4579 (1972).
    [Crossref]

2008 (1)

2007 (1)

R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317, 927–929 (2007).
[Crossref] [PubMed]

2006 (1)

P. J. Reece, V. Garcés-Chávez, and K. Dholakia, “Near-field optical micromanipulation with cavity enhanced evanescent waves,” Appl. Phys. Lett. 88, 221116 (2006).
[Crossref]

2005 (2)

A. B. Evlyukhin and S. I. Bozhevolnyi, “Point-dipole approximation for surface plasmon polariton scattering: Implications and limitations,” Phys. Rev. B 71, 134303 (2005).
[Crossref]

T. Søndergaard and S. I. Bozhevolnyi, “Theoretical analysis of finite-size surface plasmon polariton band-gap structures,” Phys. Rev. B 71, 125429 (2005).
[Crossref]

2004 (1)

T. Søndergaard and S. I. Bozhevolnyi, “Surface plasmon polariton scattering by a small particle placed near a metal surface: An analytical study,” Phys. Rev. B 69, 045422 (2004).
[Crossref]

2003 (3)

A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. 5, S16–S50 (2003).
[Crossref]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

V. I. Balykin, V. V. Klimov, and V. S. Letokhov, “Atom nanooptics based on photon dots and photon holes,” JETP Lett. 78, 8–12 (2003).
[Crossref]

2000 (1)

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Greens tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[Crossref]

1999 (1)

1997 (3)

L. Novotny, B. Hecht, and D. W. Pohl, “Interference of locally excited surface plasmons,” J. Appl. Phys. 81, 1798–1806 (1997).
[Crossref]

S. I. Bozhevolnyi and B. Vohnsen, “Phase conjugation of optical near field by a surface hologram,” Opt. Commun. 135, 19–23 (1997).
[Crossref]

B. Vohnsen and S. I. Bozhevolnyi, “Holographic approach to phase conjugation of optical near fields,” J. Opt. Soc. Am. A 14, 1491–1499 (1997).
[Crossref]

1996 (1)

S. I. Bozhevolnyi and B. Vohnsen, “Near-field optical holography,” Phys. Rev. Lett. 77, 3351–3354 (1996).
[Crossref] [PubMed]

1995 (1)

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[Crossref] [PubMed]

1993 (2)

O. Keller, M. Xiao, and S. Bozhevolnyi, “Configurational resonances in optical near-field microscopy: a rigorous point-dipole approach,” Surf. Sci. 280, 217–230 (1993).
[Crossref]

D. Van Labeke and D. Barchiesi, “Probes for scanning tunneling optical microscopy: a theoretical comparison,” J. Opt. Soc. Am. A 10, 2193–2201 (1993).
[Crossref]

1977 (1)

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4579 (1972).
[Crossref]

Balykin, V. I.

V. I. Balykin, V. V. Klimov, and V. S. Letokhov, “Atom nanooptics based on photon dots and photon holes,” JETP Lett. 78, 8–12 (2003).
[Crossref]

Barchiesi, D.

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Bozhevolnyi, S.

O. Keller, M. Xiao, and S. Bozhevolnyi, “Configurational resonances in optical near-field microscopy: a rigorous point-dipole approach,” Surf. Sci. 280, 217–230 (1993).
[Crossref]

Bozhevolnyi, S. I.

A. B. Evlyukhin and S. I. Bozhevolnyi, “Point-dipole approximation for surface plasmon polariton scattering: Implications and limitations,” Phys. Rev. B 71, 134303 (2005).
[Crossref]

T. Søndergaard and S. I. Bozhevolnyi, “Theoretical analysis of finite-size surface plasmon polariton band-gap structures,” Phys. Rev. B 71, 125429 (2005).
[Crossref]

T. Søndergaard and S. I. Bozhevolnyi, “Surface plasmon polariton scattering by a small particle placed near a metal surface: An analytical study,” Phys. Rev. B 69, 045422 (2004).
[Crossref]

B. Vohnsen and S. I. Bozhevolnyi, “Holographic approach to phase conjugation of optical near fields,” J. Opt. Soc. Am. A 14, 1491–1499 (1997).
[Crossref]

S. I. Bozhevolnyi and B. Vohnsen, “Phase conjugation of optical near field by a surface hologram,” Opt. Commun. 135, 19–23 (1997).
[Crossref]

S. I. Bozhevolnyi and B. Vohnsen, “Near-field optical holography,” Phys. Rev. Lett. 77, 3351–3354 (1996).
[Crossref] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4579 (1972).
[Crossref]

Courtial, J.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[Crossref] [PubMed]

Dholakia, K.

P. J. Reece, V. Garcés-Chávez, and K. Dholakia, “Near-field optical micromanipulation with cavity enhanced evanescent waves,” Appl. Phys. Lett. 88, 221116 (2006).
[Crossref]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Evlyukhin, A. B.

A. B. Evlyukhin and S. I. Bozhevolnyi, “Point-dipole approximation for surface plasmon polariton scattering: Implications and limitations,” Phys. Rev. B 71, 134303 (2005).
[Crossref]

Garcés-Chávez, V.

P. J. Reece, V. Garcés-Chávez, and K. Dholakia, “Near-field optical micromanipulation with cavity enhanced evanescent waves,” Appl. Phys. Lett. 88, 221116 (2006).
[Crossref]

Gay-Balmaz, P.

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Greens tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[Crossref]

Girard, C.

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[Crossref] [PubMed]

Hecht, B.

L. Novotny, B. Hecht, and D. W. Pohl, “Interference of locally excited surface plasmons,” J. Appl. Phys. 81, 1798–1806 (1997).
[Crossref]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4579 (1972).
[Crossref]

Keller, O.

O. Keller, “Attached and radiated electromagnetic fields of an electric point dipole,” J. Opt. Soc. Am. B 16, 835–847 (1999).
[Crossref]

O. Keller, M. Xiao, and S. Bozhevolnyi, “Configurational resonances in optical near-field microscopy: a rigorous point-dipole approach,” Surf. Sci. 280, 217–230 (1993).
[Crossref]

Klimov, V. V.

V. I. Balykin, V. V. Klimov, and V. S. Letokhov, “Atom nanooptics based on photon dots and photon holes,” JETP Lett. 78, 8–12 (2003).
[Crossref]

Kunz, R. E.

Letokhov, V. S.

V. I. Balykin, V. V. Klimov, and V. S. Letokhov, “Atom nanooptics based on photon dots and photon holes,” JETP Lett. 78, 8–12 (2003).
[Crossref]

Lukosz, W.

Martin, O. J. F.

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Greens tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[Crossref]

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[Crossref] [PubMed]

Mazilu, M.

Merlin, R.

R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317, 927–929 (2007).
[Crossref] [PubMed]

Novotny, L.

L. Novotny, B. Hecht, and D. W. Pohl, “Interference of locally excited surface plasmons,” J. Appl. Phys. 81, 1798–1806 (1997).
[Crossref]

Paulus, M.

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Greens tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[Crossref]

Pohl, D. W.

L. Novotny, B. Hecht, and D. W. Pohl, “Interference of locally excited surface plasmons,” J. Appl. Phys. 81, 1798–1806 (1997).
[Crossref]

Reece, P. J.

P. J. Reece, V. Garcés-Chávez, and K. Dholakia, “Near-field optical micromanipulation with cavity enhanced evanescent waves,” Appl. Phys. Lett. 88, 221116 (2006).
[Crossref]

Smolyaninov, I. I.

A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. 5, S16–S50 (2003).
[Crossref]

Søndergaard, T.

T. Søndergaard and S. I. Bozhevolnyi, “Theoretical analysis of finite-size surface plasmon polariton band-gap structures,” Phys. Rev. B 71, 125429 (2005).
[Crossref]

T. Søndergaard and S. I. Bozhevolnyi, “Surface plasmon polariton scattering by a small particle placed near a metal surface: An analytical study,” Phys. Rev. B 69, 045422 (2004).
[Crossref]

Thomson, L. C.

Van Labeke, D.

Vohnsen, B.

S. I. Bozhevolnyi and B. Vohnsen, “Phase conjugation of optical near field by a surface hologram,” Opt. Commun. 135, 19–23 (1997).
[Crossref]

B. Vohnsen and S. I. Bozhevolnyi, “Holographic approach to phase conjugation of optical near fields,” J. Opt. Soc. Am. A 14, 1491–1499 (1997).
[Crossref]

S. I. Bozhevolnyi and B. Vohnsen, “Near-field optical holography,” Phys. Rev. Lett. 77, 3351–3354 (1996).
[Crossref] [PubMed]

Whyte, G.

Xiao, M.

O. Keller, M. Xiao, and S. Bozhevolnyi, “Configurational resonances in optical near-field microscopy: a rigorous point-dipole approach,” Surf. Sci. 280, 217–230 (1993).
[Crossref]

Zayats, A. V.

A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. 5, S16–S50 (2003).
[Crossref]

Appl. Phys. Lett. (1)

P. J. Reece, V. Garcés-Chávez, and K. Dholakia, “Near-field optical micromanipulation with cavity enhanced evanescent waves,” Appl. Phys. Lett. 88, 221116 (2006).
[Crossref]

J. Appl. Phys. (1)

L. Novotny, B. Hecht, and D. W. Pohl, “Interference of locally excited surface plasmons,” J. Appl. Phys. 81, 1798–1806 (1997).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. 5, S16–S50 (2003).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

JETP Lett. (1)

V. I. Balykin, V. V. Klimov, and V. S. Letokhov, “Atom nanooptics based on photon dots and photon holes,” JETP Lett. 78, 8–12 (2003).
[Crossref]

Nature (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Opt. Commun. (1)

S. I. Bozhevolnyi and B. Vohnsen, “Phase conjugation of optical near field by a surface hologram,” Opt. Commun. 135, 19–23 (1997).
[Crossref]

Phys. Rev. B (4)

A. B. Evlyukhin and S. I. Bozhevolnyi, “Point-dipole approximation for surface plasmon polariton scattering: Implications and limitations,” Phys. Rev. B 71, 134303 (2005).
[Crossref]

T. Søndergaard and S. I. Bozhevolnyi, “Surface plasmon polariton scattering by a small particle placed near a metal surface: An analytical study,” Phys. Rev. B 69, 045422 (2004).
[Crossref]

T. Søndergaard and S. I. Bozhevolnyi, “Theoretical analysis of finite-size surface plasmon polariton band-gap structures,” Phys. Rev. B 71, 125429 (2005).
[Crossref]

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4579 (1972).
[Crossref]

Phys. Rev. E (1)

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Greens tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[Crossref]

Phys. Rev. Lett. (2)

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[Crossref] [PubMed]

S. I. Bozhevolnyi and B. Vohnsen, “Near-field optical holography,” Phys. Rev. Lett. 77, 3351–3354 (1996).
[Crossref] [PubMed]

Science (1)

R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317, 927–929 (2007).
[Crossref] [PubMed]

Surf. Sci. (1)

O. Keller, M. Xiao, and S. Bozhevolnyi, “Configurational resonances in optical near-field microscopy: a rigorous point-dipole approach,” Surf. Sci. 280, 217–230 (1993).
[Crossref]

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Figures (8)

Fig. 1.
Fig. 1. Schematic side view of the holographic technique (a) with TIR reference wave and (b) with SPP Gaussian beam as the reference wave.
Fig. 2.
Fig. 2. Interference on the plane with zob =50 nm [Fig. 1(a)] between the field of dipole situated at zd =800 nm above the substrate and the reference evanescent wave being the result of the TIR for an optical plane wave with wavelength 800 nm (the TIR configuration). (a) Linear color representation. (b) Contour-representation of the interference, the black small circles indicate the positions of the nanoparticles recording the hologram.
Fig. 3.
Fig. 3. Normalized intensity distribution of the total electric field being the result of the interaction between the reconstructing evanescent wave and nanoparticle structure recording the hologram of the dipole in the TIR configuration (Fig. 2). (a), (c), (e) TE-polarization of the reconstructing wave, (b), (d), (f) TM-polarization of the reconstructing wave. (a) and (b) cross section z=zd =800 nm, (c) and (d) cross section y=0, (e) and (f) cross sections x=0.
Fig. 4.
Fig. 4. Normalized intensity of the different electric field components after the reconstruction process along cross section x=0, z=zd =800 nm. (a) TE-polarization of the reconstructing wave, (b) TM-polarization of the reconstructing wave.
Fig. 5.
Fig. 5. Dependence of the reconstructed field upon nanoparticle structures. (a)–(e) nanoparticle structure and field intensity (arb. un.) after the reconstruction process along cross section z=zd =800 nm in the TIR configuration. TE-polarization of the reconstructing wave. The red small circles indicate the positions of the nanoparticles.
Fig. 6.
Fig. 6. Normalized intensity distribution of the total electric field being the result of the interaction between the reconstructing SPP plane wave and nanoparticle structure recording the hologram of the dipole in the TIR configuration (Fig. 2). (a) cross section y=0, (b) cross sections x=0, (c) along the z-direction for x=0, y=0.
Fig. 7.
Fig. 7. Interference on the plane with zob =50 nm (Fig. 1) between the field of dipole situated at zd =800 nm above the substrate and the reference SPP Gaussian beam (the waist W=2000 nm and light wavelength 642 nm) and corresponding nanoparticle array recording the hologram in the framework of the SPP configuration. (a) Dipole orientation is along the x-axis. (b) Dipole orientation is opposite to the z-axis. The black circles indicate the positions of the dipoles, the red small circles indicate the positions of the nanoparticles recording the hologram.
Fig. 8.
Fig. 8. Cross sections [y=0 for (a), (b), (c) and x=0 for (d), (e), (f)] of the normalized intensity distribution of the total electric field being a result of the interaction between the reconstructing SPP Gaussian beam and nanoparticle structure in the SPP configuration (Fig. 7). (a) and (d) correspond the case in Fig. 7(a) (λ=642 nm), (b) and (e) correspond the case in Fig 7(b) (λ=642 nm), (c) and (f) the case when the reconstructing SPP Gaussian beam is corresponded to the light wavelength λ=705 nm, the nanoparticle structure from Fig. 7(b).

Equations (8)

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P = [ I ̂ k 0 2 ε 0 α ̂ G ̂ s ( r d , r d ) ] 1 α ̂ E 0 ( r d ) ,
α ̂ = 3 I ̂ V d ε 0 ε d 1 ε d + 2 .
E d ( r o b ) = k 0 2 ε 0 [ G ̂ 0 ( r ob , r d ) + G ̂ s ( r ob , r d ) ] p .
E r ( r ob ) = E t exp ( i k t r ob ) ,
p i = α ̂ E ct ( r i ) + k 0 2 ε 0 α ̂ G ̂ s ( r i , r i ) p i + k 0 2 ε 0 j i N α ̂ [ G ̂ 0 ( r i , r j ) + G ̂ s ( r i , r j ) ] p j , i = 1 , , N
E ( r ) = E ct ( r ) + k 0 2 ε 0 i = 1 N [ G ̂ 0 ( r , r i ) + G ̂ s ( r , r i ) ] p i .
E r ( r ob ) = exp ( i k SPP x ob + i k z z ob y ob 2 W 2 f ( x ob ) ) f ( x ob ) { z ̂ + y ̂ ( i 2 y ob k z ( k SPP W ) 2 f ( x ob ) )
+ x ̂ [ k z k SPP ( 1 + 1 ( k SPP W ) 2 f ( x ob ) 2 ( k SPP y ob ) 2 ( ( k SPP W ) 2 f ( x ob ) ) 2 ) ] } ,

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