- Email: [email protected]
Modeling of a surface plasmon polariton interferometer
Optics Communications 240 (2004) 345–350 www.elsevier.com/locate/optcom
Modeling of a surface plasmon polariton interferometer Victor Coello a
a,*
, Thomas Søndergaard b, Sergey I. Bozhevolnyi
b
CICESE Unidad Monterrey, Pedro de Alba S/N Posgrado FCFM-UANL, C.P. 66450 San Nicolas des los Garza Nuevo Leon, Mexico b Micro Managed Photons A/S, Department of Physics and Nanotechnology, Aalborg University, Pontoppidanstræde 103, DK-9220 Aalborg Øst, Denmark Received 9 February 2004; received in revised form 27 May 2004; accepted 21 June 2004
Abstract We model the operation of a micro-optical interferometer for surface plasmon polaritons (SPPs) that comprises an SPP beam-splitter formed by equivalent scatterers lined up and equally spaced. The numerical calculations are carried out by using a relatively simple vectorial dipolar model for multiple SPP scattering [Phys. Rev. B 67 (2003) 165405]. The SPP beam-splitter is simulated elucidating the influence of system parameters, such as the angle of SPP beam incidence, scattering particle size, and inter-particle distance, on the splitting efficiency and phase difference between the transmitted and reflected beams. It is found that the splitting efficiency is very sensitive to the size of scatterers and angle of incidence. Comparing our simulations with experimental data available in the literature, we conclude that this approach can be used, with certain limitations, for modelling of SPP components assembled of individual scatterers, e.g., beamsplitters and interferometers, and suggest further improvements of the model used. 2004 Elsevier B.V. All rights reserved. PACS: 73.20. Mf; 78.67.n Keywords: Surface plasmon polaritons; Nanostructures
1. Introduction Surface plasmon polaritons (SPPs) i.e., collective oscillations of surface electron charge density, represent (quasi) two-dimensional waves [1]. Associated with SPPs there exist electromagnetic fields *
Corresponding author. Tel.: +52-81-8478-0507; fax: +5281-8478-0508. E-mail address: [email protected] (V. Coello).
propagating along the (metal-dielectric) interface and exponentially decaying perpendicular to it. Due to their electromagnetic nature, SPPs propagating along the surface can diffract and reflect from surface features and interfere. These properties are clearly exhibited in the course of elastic (in the plane) SPP scattering. Usually, elastic scattering of SPPs and related phenomena [1] have been generated because of randomly (and inadvertently) situated surface defects (e.g. because of surface
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.06.042
346
V. Coello et al. / Optics Communications 240 (2004) 345–350
roughness). However, in the last years, several studies of two-dimensional optics of SPPs based on artificially fabricated micro-components were reported. Thus, first examples of SPP micro-lens, micro-mirrors [2–4] and SPP band gap structures [5,6] have been demonstrated. Additionally, SPP propagation along thin metal stripes was realized showing a potential tool for optical addressing purposes at nanometer scales [7]. In general, this new direction of SPP investigations has revealed several features such as wavelength dispersion and stability (with respect to geometric parameters) of the micro-components that have to be elucidated. One could gain more understanding in this context investigating several configurations (e.g., varying the size and number of individual scatterers) of a particular micro-component. This task could be well complemented by means of numerical simulations. The problem is not simple, as even a circularly symmetric surface defect requires elaborated numerical calculations [8] and a complete theory that could deal with the SPP scattering by surface features has not yet been developed. A scalar multiple-scattering approach was used for simulations of SPP optical micro-components [4] and photonic band gap structures formed by sets of individual scatterers [9]. These scatterers were considered in a two-dimensional geometry as isotropic point-like scatterers characterized by their effective polarizabilities (related directly to the total scattering cross sections). Despite the apparent success, the model has some limitations, one of them being that the effective polarizability of an individual scatterer is a phenomenological quantity which is difficult to relate to scattererÕs parameters (e.g., size, dielectric susceptibility, etc). Such an approach has been extended into a vector dipolar multiple-scattering theory and used to calculate SPP scattering produced by band-gap structures [10]. This approach entails point-like dipolar scatterers interacting via SPPs so that the multiple-scattering problem in question can be explicitly formulated, making it very attractive for modelling of SPP components assembled of individual scatterers. The approach should be accurate for large inter-particle distances and seems to be suitable (to some extent) even for modelling of SPP band-gap structures, in which these distances are
smaller than the light wavelength [10]. However, there exist several issues that have to be elucidated such as the model accuracy and limitations with respect to both the size and separation of scatterers. A recently demonstrated SPP beam-splitter formed by equivalent scatterers lined up and equally spaced [11] represents an appropriate scattering configuration for testing the model, because its proper modelling requires accurate simulations of both amplitude and phase of scattered SPP waves. In this communication, we report the results of modelling of an SPP beam-splitter formed by equivalent scatterers and compare our simulations with experimental data available in the literature [11], commenting on both the applicability and limitations of the model.
2. Model for elastic SPP scattering Our modelling is based on the assumption that the elastic SPP scattering is dominant with respect to the inelastic (out of the plane) SPP scattering. This fact is justified since it has been demonstrated that for an SPP being resonantly excited at relatively smooth surfaces, the near field intensity maps are indeed directly related to the total SPP field, i.e., field of the excited and elastically scattered SPPs [4]. Therefore it is possible, at least to some extent, to avoid the complicated mathematical treatment involved in the problem of SPP scattering by surface inhomogeneities [8]. This assumption led to the construction of an approximate GreenÕs tensor describing SPP–SPP scattering by a dipolar point-like scatterer located at a metal/dielectric interface [10]. The validity of the model has been established for relatively large inter-particle distances, whereas for smaller distances it is more accurate to use a total GreenÕs tensor and include multipolar contributions in the scattered field ([10] and references therein). The self-consistent polarization of each scatterer established in the process of multiple scattering is obtained by solving the following equation: Pi ¼ ai E0 ðri Þ þ
k 20 X ai Gðri ; rn Þ Pn ; e0 n6¼i
ð1Þ
V. Coello et al. / Optics Communications 240 (2004) 345–350
where Pi is the polarization of the particle i, ai is the polarizability tensor for particle i with the multiple scattering between the particle and the metal surface taken into account (surface dressing effect), E0 is an incoming electric field, k0 is the free space wave number, e0 is the vacuum permittivity and G(ri,rn) is the GreenÕs tensor for the reference structure (total field propagator). The GreenÕs tensor G is the sum of a direct contribution Gd, in this case the free space GreenÕs tensor, and an indirect contribution GS that describes both reflection from the metal/dielectric interface and excitation of SPPs. The incoming E0 describes a Gaussian SPP field impinging on the arrangement of scatterers. For a spherical particle made of the same metal as the substrate, the polarizability tensor is given by 1 e1 e1 1 1 1 ^x^x þ ^y ^y þ ^z^z a I a0 ; eþ1 eþ2 8 8 4 ð2Þ where I is the unit dyadic tensor, e is the metal dielectric constant, xˆ, yˆ, zˆ are unit vectors in a cartesian coordinate system with ^z being perpendicular to the air–metal interface, and a0 ¼ e0 I4pa3 e1 is eþ2 the free space polarizability tensor in the longwave electrostatic approximation with a being the sphere radius. The polarizations (Eq. (1)) and the total field, EðrÞ ¼ E0 ðrÞ þ
k 20 X Gðr; rn Þ Pn ; e0 n
ð3Þ
can be calculated using the appropriate GreenÕs tensor for the reference structure G(r,r 0 ). Finally, based on the initial assumptions, it was proposed to use a three-dimensional dyadic GreenÕs tensor approximation which accounts only for the SPP elastic scattering channel [10]. The complete analysis of the validity domain of such an approximation is beyond the scope of this report and can be found elsewhere [10].
3. Numerical calculations In general, the operation of an SPP beam-splitter [11] depends on the scattering configuration (shape, size and separation of scattering particles),
347
splitting angle, light wavelength and metal used. Here, following the experimental conditions reported [11], we investigated the in-plane scattered SPP field created with a 5-lm-wide Gaussian SPP beam (k = 750 nm) of unit amplitude, impinging on an equally spaced line of spherical silver particles placed on a silver surface (dielectric constant e = 23.11 + 1.4i). First, we considered 280nm-spaced and 64-nm-radius lined-up spheres splitting the beam incident at 30 with respect to the particle line direction. This configuration can be considered as fairly similar to the experimentally realized one, if one takes into account that the scattering volume (that determines the scattering strength for small scatterers) of a 64-nm-radius sphere is the same as that of a 140-nm-diameter and 70-nm-high cylinder [11]. The splitting ratio (the ratio between the reflected and transmitted SPP intensities) simulated was surprisingly low, i.e., only 0.1 that is much smaller than the reported ratio of unity [11]. There can be many factors amounting together to yield such a difference, but we believe that the main reason is related to the presence (in the experiment) of a 30-nm-thick polymer film on the silver surface, since it was established that its presence has noticeably decreased the SPP wavelength (down to 610 nm [11]) and it is known that the scattering cross-section increases drastically with the decrease in the wavelength [8]. We have realized that the configuration parameters should be changed in order to increase the splitting ratio (Fig. 1), and investigated the influence of the angle of incidence (Fig. 2), inter-particle separation (Fig. 3) and sphere radius (Fig. 4) on the balance between the reflected and transmitted SPP intensities. It is seen that the splitting efficiency depends most strongly on the angle of incidence and the radius of (spherical) scatterers. In the first case, the efficiency is influenced by the number of particles that actually scatter the incident SPP beam and anisotropy of the SPP part of GreenÕs tensor [10], both leading to its increase with the decrease of the angle of incidence (Fig. 2). The dependence on the sphere radius is clearly due to the corresponding (cubic) dependence of polarizability (see Eq. (2) and the following description). One can also notice that the sum of the calculated intensities of reflected and transmitted
348
V. Coello et al. / Optics Communications 240 (2004) 345–350
Fig. 1. Gray-scale representation (a) of the total SPP intensity distribution in the area of 50 · 50 lm2 calculated for an SPP beamsplitter composed of 200 spherical particles with the radius of 64 nm separated by 280 nm and for the light wavelength of 750 nm. The angle of incidence is set at 16 with regard to the line of particles. The white arrow in (a) indicates the cross-section shown in (b).
1.0
Intensity (arb.units)
Intensity (arb.units)
0.8
0.6 transmitted
0.4
reflected total
0.8 transmitted reflected
0.6
total
0.4
0.2
14
16
18
20
22
24
26
28
30
Angle of incidence (degrees) Fig. 2. Dependencies of the transmitted, reflected and total intensities determined from the intensity distributions similar to that shown in Fig. 1(a) as functions of the angle of incidence on a beam-splitter with the parameters as in Fig. 1.
beams is not constant. One of the reasons is related to the fact the (self-consistent) field acting on particles, whose dielectric constant is complex with a nonzero imaginary part, can be noticeably influenced by any of considered parameters, leading to variations in the absorption. In addition, the calculation of the self-consistent polarizations (Eq. (1)) is accurate only as long as the GreenÕs tensor approximation used here is accurate, which
0.2
200
220
240
260
280
300
320
340
360
Inter-particle distance (nm) Fig. 3. Dependencies of the transmitted, reflected and total intensities as functions of the inter-particle distance. All else is as in Fig. 1.
is reasonably accurate for large inter-particle distances and small scatterers [10]. The sum of the reflected and transmitted intensities, which is proportional to the total power of reflected and transmitted beams, is most strongly dependent on the radius of particles, increasing rapidly for a > 50 nm (Fig. 4). This behaviour indicates that the particle polarizability and its scattering cross-section are becoming too large than it is allowed by the energy conservation [12]. The latter
V. Coello et al. / Optics Communications 240 (2004) 345–350 1.0 transmitted reflected
Intensity (arb.units)
0.8
total
0.6
0.4
0.2
0.0
40
45
50
55
60
65
70
Sphere radius (nm) Fig. 4. Dependencies of the transmitted, reflected and total intensities as functions of the particle radius. All else is as in Fig. 1.
120
Phase (degrees)
100 80 transmitted
60
reflected difference
40 20 0 40
45
50
55
60
65
70
Sphere radius (nm) Fig. 5. Dependencies of the phase of the transmitted and reflected beams and their difference as functions of the particle radius. All else is as in Fig. 1.
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is known to be a rather complicated issue in modelling of strong light scattering since, as a rule, some approximations have to be made to complete the simulations (at least, in the course of numerical calculations). In the case of a beam-splitter, the energy conservation can be controlled via calculations of the phases of reflected and transmitted beams that should differ by 90. The reflected beam phase was calculated as a function of the radius of particles by directly superimposing an incident (at h0) SPP beam having an adjustable phase shift with the reflected beam. Their interference allowed us to determine the reflected beam phase as a phase shift maximizing the total SPP field propagating in the reflected direction. In a similar manner, the transmitted phase has been calculated showing that the relative phase difference between the transmitted and reflected beams in a beam-splitter is close to 90 for radii smaller than 60 nm (Fig. 5). The SPP interferometer has been completed by adding a second beam impinging the line of scatterers at the incident angle of h0 (Fig. 6). The numerical simulations showed that the intensities of two output beams, which result from the interaction of the incident beams with the beam-splitter, vary as a function of the introduced phase difference u between the incident beams (Fig. 7) in a fashion that is very similar to the experimental results (cf. Figs. 6(b) and (d) with Fig. 5 from 11). Deviations from the behaviour expected in an ideal loss-less interferometer are related to the aforementioned problems with the energy conservation: the radius of 64 nm used in simulations was probably too large in this respect but just large
Fig. 6. Gray-scale representations of the total SPP intensity distributions in the area of 50 · 50 lm2 calculated for an SPP interferometer with the parameters as in Fig. 1. The SPP maps have been calculated for the relative phase differences between the incident beams of u = (a) 0, (b) p/2, (c) p, and (d) 3p/2 rad.
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V. Coello et al. / Optics Communications 240 (2004) 345–350 2.0 loweroutput beam
Intensity (arb. units)
upper output beam
1.6
1.2
0.8
0.4
0.0
0
50
100
150
200
250
300
350
Phase difference (degrees) Fig. 7. Dependencies of the lower and upper output beam intensities as functions of the phase difference between the two input beams. All else is as in Fig. 1.
enough to ensure the efficient operation of the beam-splitter and the interferometer.
4. Conclusions We have modelled the operation of SPP-based beam-splitters and an interferometer whose main element represents individual scatterers lined up and equally spaced. The numerical calculations were carried out by using a relatively simple vectorial dipolar model for multiple SPP scattering [10] that allows one to explicitly formulate the set of linear equations for the self-consistent field, facilitating greatly computer-aided design considerations. The SPP beam-splitter was simulated numerically elucidating the influence of system parameters, such as the angle of SPP beam incidence, scattering particle size, and inter-particle distance, on the splitting efficiency and phase difference between the transmitted and reflected beams. It was found that the splitting efficiency is very sensitive to the size of scatterers and angle of incidence. It
was also established that the sum of the calculated intensities of reflected and transmitted beams is not constant increasing rapidly for particle radii larger than 50 nm, a circumstance that is most probably related to the violation of energy conservation [12]. However, our simulations conducted for a relatively large radius of 64 nm did reproduce main features observed experimentally [11], indicating that this approach can be used, with certain limitations, for modelling of SPP components assembled of individual scatterers, e.g., beam-splitters and interferometers. One can try to further improve this model by, for example, developing another (but analytic as well) approximation of the GreenÕs tensor for relatively small inter-particle distances. We conduct further investigations in this area.
References [1] H. Raether, Surface Plasmons, Springer, Berlin, 1988. [2] I.I. Smolanyinov, D.L. Mazzoni, C.C. Davis, Phys. Rev. Lett. 77 (1996) 3877. [3] S.I. Bozhevolnyi, F.A. Pudonin, Phys. Rev. Lett. 78 (1997) 2823. [4] S.I. Bozhevolnyi, V. Coello, Phys. Rev. B. 58 (1998) 10899. [5] S.I. Bozhevolnyi, J.E. Erland, K. Leossson, P.M.W. Skovgaard, J.M. Hvam, Phys. Rev. Lett. 86 (2001) 3008. [6] S. Bozhevolnyi, V. Volkov, K. Leosson, Opt. Commun. 196 (2001) 41. [7] B. Lamprecht, J.R. Krenn, G. Schider, H. Ditlbacher, M. Salemo, N. Felidj, A. Leitner, F.R. Aussenegg, J.C. Weeber, Appl. Phys. Lett. 79 (2001) 51. [8] A.V. Shchegrov, I.V. Novikov, A.A. Maradudin, Phys. Rev. Lett. 78 (1997) 4269. [9] S.I. Bozhevolnyi, V. Volkov, Opt. Commun. 198 (2001) 241. [10] T. Søndergaard, S.I. Bozhevolnyi, Phys. Rev. B. 67 (2003) 165405. [11] H. Dittlbacher, J.R. Krenn, G. Shider, A. Leitner, F.R. Aussenegg, Appl. Phys. Lett. 81 (2002) 1762. [12] P. de Vries, D.V. Van Coevorden, A. Lagendijk, Rev. Mod. Phys. 70 (1998) 447.
Modeling of a surface plasmon polariton interferometer Victor Coello a
a,*
, Thomas Søndergaard b, Sergey I. Bozhevolnyi
b
CICESE Unidad Monterrey, Pedro de Alba S/N Posgrado FCFM-UANL, C.P. 66450 San Nicolas des los Garza Nuevo Leon, Mexico b Micro Managed Photons A/S, Department of Physics and Nanotechnology, Aalborg University, Pontoppidanstræde 103, DK-9220 Aalborg Øst, Denmark Received 9 February 2004; received in revised form 27 May 2004; accepted 21 June 2004
Abstract We model the operation of a micro-optical interferometer for surface plasmon polaritons (SPPs) that comprises an SPP beam-splitter formed by equivalent scatterers lined up and equally spaced. The numerical calculations are carried out by using a relatively simple vectorial dipolar model for multiple SPP scattering [Phys. Rev. B 67 (2003) 165405]. The SPP beam-splitter is simulated elucidating the influence of system parameters, such as the angle of SPP beam incidence, scattering particle size, and inter-particle distance, on the splitting efficiency and phase difference between the transmitted and reflected beams. It is found that the splitting efficiency is very sensitive to the size of scatterers and angle of incidence. Comparing our simulations with experimental data available in the literature, we conclude that this approach can be used, with certain limitations, for modelling of SPP components assembled of individual scatterers, e.g., beamsplitters and interferometers, and suggest further improvements of the model used. 2004 Elsevier B.V. All rights reserved. PACS: 73.20. Mf; 78.67.n Keywords: Surface plasmon polaritons; Nanostructures
1. Introduction Surface plasmon polaritons (SPPs) i.e., collective oscillations of surface electron charge density, represent (quasi) two-dimensional waves [1]. Associated with SPPs there exist electromagnetic fields *
Corresponding author. Tel.: +52-81-8478-0507; fax: +5281-8478-0508. E-mail address: [email protected] (V. Coello).
propagating along the (metal-dielectric) interface and exponentially decaying perpendicular to it. Due to their electromagnetic nature, SPPs propagating along the surface can diffract and reflect from surface features and interfere. These properties are clearly exhibited in the course of elastic (in the plane) SPP scattering. Usually, elastic scattering of SPPs and related phenomena [1] have been generated because of randomly (and inadvertently) situated surface defects (e.g. because of surface
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.06.042
346
V. Coello et al. / Optics Communications 240 (2004) 345–350
roughness). However, in the last years, several studies of two-dimensional optics of SPPs based on artificially fabricated micro-components were reported. Thus, first examples of SPP micro-lens, micro-mirrors [2–4] and SPP band gap structures [5,6] have been demonstrated. Additionally, SPP propagation along thin metal stripes was realized showing a potential tool for optical addressing purposes at nanometer scales [7]. In general, this new direction of SPP investigations has revealed several features such as wavelength dispersion and stability (with respect to geometric parameters) of the micro-components that have to be elucidated. One could gain more understanding in this context investigating several configurations (e.g., varying the size and number of individual scatterers) of a particular micro-component. This task could be well complemented by means of numerical simulations. The problem is not simple, as even a circularly symmetric surface defect requires elaborated numerical calculations [8] and a complete theory that could deal with the SPP scattering by surface features has not yet been developed. A scalar multiple-scattering approach was used for simulations of SPP optical micro-components [4] and photonic band gap structures formed by sets of individual scatterers [9]. These scatterers were considered in a two-dimensional geometry as isotropic point-like scatterers characterized by their effective polarizabilities (related directly to the total scattering cross sections). Despite the apparent success, the model has some limitations, one of them being that the effective polarizability of an individual scatterer is a phenomenological quantity which is difficult to relate to scattererÕs parameters (e.g., size, dielectric susceptibility, etc). Such an approach has been extended into a vector dipolar multiple-scattering theory and used to calculate SPP scattering produced by band-gap structures [10]. This approach entails point-like dipolar scatterers interacting via SPPs so that the multiple-scattering problem in question can be explicitly formulated, making it very attractive for modelling of SPP components assembled of individual scatterers. The approach should be accurate for large inter-particle distances and seems to be suitable (to some extent) even for modelling of SPP band-gap structures, in which these distances are
smaller than the light wavelength [10]. However, there exist several issues that have to be elucidated such as the model accuracy and limitations with respect to both the size and separation of scatterers. A recently demonstrated SPP beam-splitter formed by equivalent scatterers lined up and equally spaced [11] represents an appropriate scattering configuration for testing the model, because its proper modelling requires accurate simulations of both amplitude and phase of scattered SPP waves. In this communication, we report the results of modelling of an SPP beam-splitter formed by equivalent scatterers and compare our simulations with experimental data available in the literature [11], commenting on both the applicability and limitations of the model.
2. Model for elastic SPP scattering Our modelling is based on the assumption that the elastic SPP scattering is dominant with respect to the inelastic (out of the plane) SPP scattering. This fact is justified since it has been demonstrated that for an SPP being resonantly excited at relatively smooth surfaces, the near field intensity maps are indeed directly related to the total SPP field, i.e., field of the excited and elastically scattered SPPs [4]. Therefore it is possible, at least to some extent, to avoid the complicated mathematical treatment involved in the problem of SPP scattering by surface inhomogeneities [8]. This assumption led to the construction of an approximate GreenÕs tensor describing SPP–SPP scattering by a dipolar point-like scatterer located at a metal/dielectric interface [10]. The validity of the model has been established for relatively large inter-particle distances, whereas for smaller distances it is more accurate to use a total GreenÕs tensor and include multipolar contributions in the scattered field ([10] and references therein). The self-consistent polarization of each scatterer established in the process of multiple scattering is obtained by solving the following equation: Pi ¼ ai E0 ðri Þ þ
k 20 X ai Gðri ; rn Þ Pn ; e0 n6¼i
ð1Þ
V. Coello et al. / Optics Communications 240 (2004) 345–350
where Pi is the polarization of the particle i, ai is the polarizability tensor for particle i with the multiple scattering between the particle and the metal surface taken into account (surface dressing effect), E0 is an incoming electric field, k0 is the free space wave number, e0 is the vacuum permittivity and G(ri,rn) is the GreenÕs tensor for the reference structure (total field propagator). The GreenÕs tensor G is the sum of a direct contribution Gd, in this case the free space GreenÕs tensor, and an indirect contribution GS that describes both reflection from the metal/dielectric interface and excitation of SPPs. The incoming E0 describes a Gaussian SPP field impinging on the arrangement of scatterers. For a spherical particle made of the same metal as the substrate, the polarizability tensor is given by 1 e1 e1 1 1 1 ^x^x þ ^y ^y þ ^z^z a I a0 ; eþ1 eþ2 8 8 4 ð2Þ where I is the unit dyadic tensor, e is the metal dielectric constant, xˆ, yˆ, zˆ are unit vectors in a cartesian coordinate system with ^z being perpendicular to the air–metal interface, and a0 ¼ e0 I4pa3 e1 is eþ2 the free space polarizability tensor in the longwave electrostatic approximation with a being the sphere radius. The polarizations (Eq. (1)) and the total field, EðrÞ ¼ E0 ðrÞ þ
k 20 X Gðr; rn Þ Pn ; e0 n
ð3Þ
can be calculated using the appropriate GreenÕs tensor for the reference structure G(r,r 0 ). Finally, based on the initial assumptions, it was proposed to use a three-dimensional dyadic GreenÕs tensor approximation which accounts only for the SPP elastic scattering channel [10]. The complete analysis of the validity domain of such an approximation is beyond the scope of this report and can be found elsewhere [10].
3. Numerical calculations In general, the operation of an SPP beam-splitter [11] depends on the scattering configuration (shape, size and separation of scattering particles),
347
splitting angle, light wavelength and metal used. Here, following the experimental conditions reported [11], we investigated the in-plane scattered SPP field created with a 5-lm-wide Gaussian SPP beam (k = 750 nm) of unit amplitude, impinging on an equally spaced line of spherical silver particles placed on a silver surface (dielectric constant e = 23.11 + 1.4i). First, we considered 280nm-spaced and 64-nm-radius lined-up spheres splitting the beam incident at 30 with respect to the particle line direction. This configuration can be considered as fairly similar to the experimentally realized one, if one takes into account that the scattering volume (that determines the scattering strength for small scatterers) of a 64-nm-radius sphere is the same as that of a 140-nm-diameter and 70-nm-high cylinder [11]. The splitting ratio (the ratio between the reflected and transmitted SPP intensities) simulated was surprisingly low, i.e., only 0.1 that is much smaller than the reported ratio of unity [11]. There can be many factors amounting together to yield such a difference, but we believe that the main reason is related to the presence (in the experiment) of a 30-nm-thick polymer film on the silver surface, since it was established that its presence has noticeably decreased the SPP wavelength (down to 610 nm [11]) and it is known that the scattering cross-section increases drastically with the decrease in the wavelength [8]. We have realized that the configuration parameters should be changed in order to increase the splitting ratio (Fig. 1), and investigated the influence of the angle of incidence (Fig. 2), inter-particle separation (Fig. 3) and sphere radius (Fig. 4) on the balance between the reflected and transmitted SPP intensities. It is seen that the splitting efficiency depends most strongly on the angle of incidence and the radius of (spherical) scatterers. In the first case, the efficiency is influenced by the number of particles that actually scatter the incident SPP beam and anisotropy of the SPP part of GreenÕs tensor [10], both leading to its increase with the decrease of the angle of incidence (Fig. 2). The dependence on the sphere radius is clearly due to the corresponding (cubic) dependence of polarizability (see Eq. (2) and the following description). One can also notice that the sum of the calculated intensities of reflected and transmitted
348
V. Coello et al. / Optics Communications 240 (2004) 345–350
Fig. 1. Gray-scale representation (a) of the total SPP intensity distribution in the area of 50 · 50 lm2 calculated for an SPP beamsplitter composed of 200 spherical particles with the radius of 64 nm separated by 280 nm and for the light wavelength of 750 nm. The angle of incidence is set at 16 with regard to the line of particles. The white arrow in (a) indicates the cross-section shown in (b).
1.0
Intensity (arb.units)
Intensity (arb.units)
0.8
0.6 transmitted
0.4
reflected total
0.8 transmitted reflected
0.6
total
0.4
0.2
14
16
18
20
22
24
26
28
30
Angle of incidence (degrees) Fig. 2. Dependencies of the transmitted, reflected and total intensities determined from the intensity distributions similar to that shown in Fig. 1(a) as functions of the angle of incidence on a beam-splitter with the parameters as in Fig. 1.
beams is not constant. One of the reasons is related to the fact the (self-consistent) field acting on particles, whose dielectric constant is complex with a nonzero imaginary part, can be noticeably influenced by any of considered parameters, leading to variations in the absorption. In addition, the calculation of the self-consistent polarizations (Eq. (1)) is accurate only as long as the GreenÕs tensor approximation used here is accurate, which
0.2
200
220
240
260
280
300
320
340
360
Inter-particle distance (nm) Fig. 3. Dependencies of the transmitted, reflected and total intensities as functions of the inter-particle distance. All else is as in Fig. 1.
is reasonably accurate for large inter-particle distances and small scatterers [10]. The sum of the reflected and transmitted intensities, which is proportional to the total power of reflected and transmitted beams, is most strongly dependent on the radius of particles, increasing rapidly for a > 50 nm (Fig. 4). This behaviour indicates that the particle polarizability and its scattering cross-section are becoming too large than it is allowed by the energy conservation [12]. The latter
V. Coello et al. / Optics Communications 240 (2004) 345–350 1.0 transmitted reflected
Intensity (arb.units)
0.8
total
0.6
0.4
0.2
0.0
40
45
50
55
60
65
70
Sphere radius (nm) Fig. 4. Dependencies of the transmitted, reflected and total intensities as functions of the particle radius. All else is as in Fig. 1.
120
Phase (degrees)
100 80 transmitted
60
reflected difference
40 20 0 40
45
50
55
60
65
70
Sphere radius (nm) Fig. 5. Dependencies of the phase of the transmitted and reflected beams and their difference as functions of the particle radius. All else is as in Fig. 1.
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is known to be a rather complicated issue in modelling of strong light scattering since, as a rule, some approximations have to be made to complete the simulations (at least, in the course of numerical calculations). In the case of a beam-splitter, the energy conservation can be controlled via calculations of the phases of reflected and transmitted beams that should differ by 90. The reflected beam phase was calculated as a function of the radius of particles by directly superimposing an incident (at h0) SPP beam having an adjustable phase shift with the reflected beam. Their interference allowed us to determine the reflected beam phase as a phase shift maximizing the total SPP field propagating in the reflected direction. In a similar manner, the transmitted phase has been calculated showing that the relative phase difference between the transmitted and reflected beams in a beam-splitter is close to 90 for radii smaller than 60 nm (Fig. 5). The SPP interferometer has been completed by adding a second beam impinging the line of scatterers at the incident angle of h0 (Fig. 6). The numerical simulations showed that the intensities of two output beams, which result from the interaction of the incident beams with the beam-splitter, vary as a function of the introduced phase difference u between the incident beams (Fig. 7) in a fashion that is very similar to the experimental results (cf. Figs. 6(b) and (d) with Fig. 5 from 11). Deviations from the behaviour expected in an ideal loss-less interferometer are related to the aforementioned problems with the energy conservation: the radius of 64 nm used in simulations was probably too large in this respect but just large
Fig. 6. Gray-scale representations of the total SPP intensity distributions in the area of 50 · 50 lm2 calculated for an SPP interferometer with the parameters as in Fig. 1. The SPP maps have been calculated for the relative phase differences between the incident beams of u = (a) 0, (b) p/2, (c) p, and (d) 3p/2 rad.
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Phase difference (degrees) Fig. 7. Dependencies of the lower and upper output beam intensities as functions of the phase difference between the two input beams. All else is as in Fig. 1.
enough to ensure the efficient operation of the beam-splitter and the interferometer.
4. Conclusions We have modelled the operation of SPP-based beam-splitters and an interferometer whose main element represents individual scatterers lined up and equally spaced. The numerical calculations were carried out by using a relatively simple vectorial dipolar model for multiple SPP scattering [10] that allows one to explicitly formulate the set of linear equations for the self-consistent field, facilitating greatly computer-aided design considerations. The SPP beam-splitter was simulated numerically elucidating the influence of system parameters, such as the angle of SPP beam incidence, scattering particle size, and inter-particle distance, on the splitting efficiency and phase difference between the transmitted and reflected beams. It was found that the splitting efficiency is very sensitive to the size of scatterers and angle of incidence. It
was also established that the sum of the calculated intensities of reflected and transmitted beams is not constant increasing rapidly for particle radii larger than 50 nm, a circumstance that is most probably related to the violation of energy conservation [12]. However, our simulations conducted for a relatively large radius of 64 nm did reproduce main features observed experimentally [11], indicating that this approach can be used, with certain limitations, for modelling of SPP components assembled of individual scatterers, e.g., beam-splitters and interferometers. One can try to further improve this model by, for example, developing another (but analytic as well) approximation of the GreenÕs tensor for relatively small inter-particle distances. We conduct further investigations in this area.
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