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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 11 — May. 30, 2005
  • pp: 4134–4140
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Scattering of surface electromagnetic waves by Sn nanoparticles

Valeriy A. Sterligov and Matthias Kretschmann  »View Author Affiliations


Optics Express, Vol. 13, Issue 11, pp. 4134-4140 (2005)
http://dx.doi.org/10.1364/OPEX.13.004134


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Abstract

We show numerically that the size of the nanoparticles (NPs) that scatter surface plasmon-polaritons (SPPs) is directly related to the angular position of the maximum in a scattered light distribution. Thus, the existence of one or two experimentally observed maxima in the angular distribution of the scattered light for different NP materials can be explained by a bimodal NP size distribution. We also invoke the polarization properties of the scattered light to estimate the contribution of multiple scattering processes to the observed light distribution. SPP excitation can be detected by a minimum in the reflectivity, or a maximum in the scattered light distribution. We show that this maximum exists for a wider range of NP sizes (or surface roughness) than the minimum in the reflectivity. This observation is interesting for the development of SPP based optical sensors.

© 2005 Optical Society of America

1. Introduction

Optical properties of nanoparticles (NPs) have been subject of numerous studies and publications, e.g., [1

1. U. Kreibig and M. Volmer, Optical Properties of Metal Clusters, (Springer-Verlag, Berlin1995).

4

4. T. Kume, S. Hayashi, and K. Yamamoto, “Light emission from surface plasmon polaritons mediated by metallic fine particles,” Phys. Rev. B 55, 4774 (1997). [CrossRef]

]. Despite these studies, many questions still remain unanswered, for example how the different phases of NP materials influence the properties of surface electromagnetic waves and their scattering processes. Previously obtained experimental data on SPP scattering by 8 nm Ga NPs [2

2. V.A. Sterligov, P. Cheyssac, R. Kofman, S.I. Lysenko, P.M. Lytvyn, B. Vohnsen, S.I. Bozhevolnyi, and A.A. Maradudin, “Near/far-field investigations of the interaction between surface waves and nanoparticles,” Phys. Stat. Sol.(b) 229, 1283 (2002). [CrossRef]

, 3

3. P. Cheyssac, V.A. Sterligov, S.I. Lysenko, and R. Kofman, “Scattering of surface plasmon-polaritons and light by metallic nanoparticles,” Opt. Commun. 175, 383 (2000). [CrossRef]

] shows the presence of one maximum in the angular distribution of the scattered light, while that for 6 nm Ag NPs [4

4. T. Kume, S. Hayashi, and K. Yamamoto, “Light emission from surface plasmon polaritons mediated by metallic fine particles,” Phys. Rev. B 55, 4774 (1997). [CrossRef]

] shows two maxima. However, an explanation for this behavior was not presented yet.

Another question concerns the estimation of the role of multiple scattering of SPPs and light. This is important because the distance between NPs is usually comparable to their size. To address these issues, we invoke the polarization properties of the scattered light, and analyze it for various scattering geometries.

One more important question concerns the possible applications of SPP scattering for an elaboration of sensors that detect changes of the optical properties of contacting media by measurements of a change of the parameters of SPPs excitation. Commonly in such sensors one measures the angular position of the specular reflectivity minimum. In the present paper we suggest an alternative method that is based on measurements of the angular dependence of the total integrated scatter (TIS). A comparison of this method with the commonly used specular reflectivity measurement promises a much better applicability.

2 Theoretical calculations

In order to analyze the angular distribution of the scattered light numerically, we employ an approach that is based on the reduced Rayleigh equation and is described in detail in [5

5. A.V. Shchegrov, I.V. Novikov, and A.A. Maradudin, “Scattering of Surface Plasmon Polaritons by a Circularly Symmetric Surface Defect,” Phys. Rev. Lett. 78, 4269 (1997). [CrossRef]

] and the references therein. The theoretical model that is considered below assumes SPPs of the form of a plane wave scattered from a single circularly symmetric surface defect on an otherwise planar metal surface. For the latter we assume a dielectric constant ε=-15.88 appropriate for silver at a vacuum wavelength of 632.8 nm. This choice of a real valued dielectric constant implies that we neglect ohmic losses. However, such a simplification is possible since the mean free path of the SPPs in question is approximately 22 µm and much larger than the diameter of the surface defects we study numerically (<0.5 µm). We obtain the differential cross section σ vac(θ, φ), measured in units of length, by dividing the power of light scattered into the vacuum region from the surface in the (θ, φ) direction by the incident power of the SPP per unit length (see Eq. (9) in Ref. [5

5. A.V. Shchegrov, I.V. Novikov, and A.A. Maradudin, “Scattering of Surface Plasmon Polaritons by a Circularly Symmetric Surface Defect,” Phys. Rev. Lett. 78, 4269 (1997). [CrossRef]

]). Here θ is the polar angle, while φ is the azimuthal angle of scattering. For our calculations we assumed that the circularly symmetric surface defect is of Gaussian form, with the profile function ζ(x)=A exp(-x 2/R 2). Shown in Fig. 1(a–b) are the contour plots of the differential cross section into the vacuum region for two surface defects of different sizes. Obviously, for a Gaussian protuberance with a height A=5 nm and a width of R=25 nm, we see that the volume waves get excited in the backward direction, while for a larger surface defect (defined by A=50 nm and R=250 nm), they get excited in the forward direction.

Fig. 1. The contour plots of the differential cross section (in µm) for the light scattered into the vacuum region from a Gaussian surface defect described by (a): A=5 nm and R=25 nm, and (b): A=50 nm and R=250 nm. The direction of SPPs propagation is from left to right; (c): θ max, as a function of the surface defect size s.

We denote the polar angle of maximum scattering into the vacuum region as θ max and plot the latter in the plane of incidence as a function of the surface defect size. We define the size s as a scalar factor such that the height of the surface defect is given as A=s*50 nm, while the width of the surface defect is given as R=s*250 nm. From Fig. 1(c) one can see that the maximum of the differential cross section lies in the backward direction for surface defects that are smaller than s=0.6, while for larger surface defects, the maximum moves towards the forward direction.

3. Experimental setup and analysis

The samples used by us were glass BK7 prisms, with a 70 nm Au film on a hypotenuse face, evaporated in UHV. This film supports the propagation of surface waves. On top of it, Sn NPs of ø8, ø16 and ø32 nm average size were evaporated and condensed. Finally, the obtained structure was protected by 3 nm film of SiOx. For comparison, some area without NPs was incorporated onto the sample.

The normalized half-sphere distribution of scattered light, Angle Resolved Scattering (ARS) [6

6. J.C. Stover, “Optical Scattering: Measurement and Analysis,” (SPIE Optical Engineering Press, Bellingham, Washington1995).

], was obtained by normalizing the scattered intensity I to the incident intensity Io and the solid angle dΩ of the photo detector: ARS(θ, φ)=I(θ, φ)/(Io dΩ). It was studied with a setup described in Refs. [7

7. V.A. Sterligov and P. Cheyssac, “Appareil et procédé de caractérisation optique d’un objet,” French Patent FR2832795, G01B-011/30 (2001).

, 8

8. C. Métayer, V.A. Sterligov, A. Meunier, G. Bossis, J. Persello, and S.V. Svechnikov, “Field induced structures and phase separation in electrorheological and magnetorheological colloidal suspensions,” J. Phys.: Cond. Matt. 16, S3975 (2004). [CrossRef]

]. In brief, the sample is placed in the first focus of an elliptical mirror that covers the whole half sphere above the sample and reflects all scattered light in the direction of a CCD camera which is placed in the second focus of the same mirror and aligned in a way that enables one to write the half sphere distribution of scattered light intensity. The polarizer in the incident beam and the analyzer in front of the CCD camera enables to control the polarization state of the analyzed light. We use the XY notation to indicate its polarization state: X denotes the polarization of the incident beam (X=S or P), while Y (Y=N, S or P) indicates the absence of an analyzer (N) or its corresponding position (S or P). Within the experimental setup it is possible to vary the angle of incidence of the beam i to record the I(θ, φ, i) dependence.

Fig. 2. Angular dependence of Rp and TIS for a zone without NPs and zones with different NP sizes.

The dependencies of the specular reflectivity for P polarization (Rp) and the TIS versus i for different sample zones with NP size d are presented in Fig. 2. For a zone without NPs, the minimum of Rp (Rmin) that corresponds to the angle of optimal SPP excitation (imin) is deeper and narrower than that for the zones with NPs. We observe that increasing d increased the half width of the minimum and decreased the Rmin value. However, for d>8 nm, the minimum of the Rp(i, d) dependence disappears. A weak interference structure of Rp(i, d) near 45° is due to interference effects with the beam that is reflected from the leg face of the prism.

In Fig. 2 the data of TIS(i, d) for different zones is presented for comparison. These dependencies are characterized by the presence of a well-pronounced maximum. As d increases, the angular position of this maximum i max shifts to smaller i, and its half-width increases. A comparison of TIS(i, d) with the Rp(i, d) dependencies reveals the fact that the range of d values for which the maximum of TIS(i, d) exists is much larger than the corresponding range for Rp(i, d). The extended range of d values for which a well-visible maximum of TIS(i, d) exists is an additional reason for choosing this investigative tool for our further studies of SPP excitation.

The ARS(θ, φ) dependencies for different polarization states of incident and scattered light are presented in Fig. 3. The shape of the ARS(θ, φ, PN) dependence in Fig. 3(a) is rather complicated. However, an analysis of the polarization of the scattered light provides important information about its essential components: the ARS(θ, φ, PP) distribution clearly shows the presence of two maxima, one in the forward and one in the backward direction relative to that of the direction of propagation of the SPP. For orthogonal polarization, ARS(θ, φ, PS) shows a well-pronounced minimum along the plane of incidence; out of this plane, the magnitude of ARS(θ, φ, PS) significantly increases and becomes comparable to that of ARS(θ, φ, PP). ARS(θ, φ, SP) is characterized by a significantly lower value of scattered intensity and the presence of a unique maximum in the forward direction. This dependence reflects the space distribution of light scattered without SPP excitation, because of the S polarization of the incident beam.

Fig. 3. The polarization properties of ARS(θ, φ) for ø8 nm Sn NP: (a) - PN, (b) - PP, (c) - PS, (d) - SP. The direction of SPP propagation is from left to right. The angle of incidence i=46.17°. Anisotropy of ARS(θ, φ, XY) relative to the horizontal axis is probably related to some error in the prism shape.

The possible role of Coherent Backscattering (CBS) was analyzed with a traditional experiment of such kin d [9

9. D.S. Wiersma, M.P. van Albada, and A. Lagendijk, “Coherent Backscattering of Light from Amplifying Random Media,” Phys. Rev. Lett. 75, 1739 (1995). [CrossRef] [PubMed]

]. After a careful examination of the obtained angular distribution of the scattered light, we did not find the presence of a backscattering cone, at least in the backward direction: Im/I1-1<3%, where I1 and Im are the beam intensities for single and multiple scattering, respectively.

Figure 4 and the corresponding video-recording illustrate variations of the half-sphere distribution of the SPPs scattered intensity as a function of the angle of incidence i. The range of i (43°<i<53°) was chosen, so that the maximum of SPP excitation is inside it. These values are suggested by the results presented in Fig. 1, even though, strictly speaking our calculations dealt with the reciprocal process. The experimental data for the SPP scattering shows the presence of forward and backward maxima in the half-sphere light distribution. The variations of the shape and amplitude of these maxima with the angle of incidence are different.

Fig. 4. Video-recording (1.5 Mb) of the dependence of I(θ, φ, i) with PP polarization on the angle of incidence (i=43°-53°) for Ø16 nm Sn NPs. The direction of SPP propagation is from right to left.

4. Discussion of results

A change of the NPs’ size can be regarded as a change of effective surface roughness. A comparison of the TIS and Rp data presented in Fig. 2 shows that the range of NPs sizes, i.e. the effective surface roughness for which a minimum of Rp(i) exists is much smaller than the range for which the TIS(i) maximum exists, while the angular half-widths of these dependences are comparable in the ranges of existence. This observation demonstrates a very important application possibility of the scattered light measurement for developments of different surface optical sensors that use SPPs excitation.

As was noted in the introduction, there are big differences between the ARS(θ, φ) dependencies for different metal NPs of comparable sizes. From a theoretical point of view, we believe that a small change in the dielectric function, caused, for example, by the phase change of the metal from solid to liquid, does not change the general shape of the ARS(θ, φ) dependence significantly.

Moreover, the video-recording of the I(θ, φ, i) dependence clearly shows that, as the angle of incidence varies, both shapes and heights of these two maxima change in a different way, thus additionally confirming the above explanation.

The comparable values of the ARS(θ, φ, PS) distribution - Fig. 3(c), relative to the ARS(θ, φ, PP) distribution- Fig. 3(b) enables us to draw the conclusion that the b) →a) processes (incident light → multiple SPP → SPP scattering) are a very important contribution to the system studied in the present work.

5. Conclusion

Acknowledgments

The authors are grateful to P. Cheyssac for his research contribution; they also highly appreciate fruitful discussions and collaboration with Prof. A.A. Maradudin. Partial support for this work from CNRS (French-Ukrainian collaboration project n°6987) is also appreciated.

References and links

1.

U. Kreibig and M. Volmer, Optical Properties of Metal Clusters, (Springer-Verlag, Berlin1995).

2.

V.A. Sterligov, P. Cheyssac, R. Kofman, S.I. Lysenko, P.M. Lytvyn, B. Vohnsen, S.I. Bozhevolnyi, and A.A. Maradudin, “Near/far-field investigations of the interaction between surface waves and nanoparticles,” Phys. Stat. Sol.(b) 229, 1283 (2002). [CrossRef]

3.

P. Cheyssac, V.A. Sterligov, S.I. Lysenko, and R. Kofman, “Scattering of surface plasmon-polaritons and light by metallic nanoparticles,” Opt. Commun. 175, 383 (2000). [CrossRef]

4.

T. Kume, S. Hayashi, and K. Yamamoto, “Light emission from surface plasmon polaritons mediated by metallic fine particles,” Phys. Rev. B 55, 4774 (1997). [CrossRef]

5.

A.V. Shchegrov, I.V. Novikov, and A.A. Maradudin, “Scattering of Surface Plasmon Polaritons by a Circularly Symmetric Surface Defect,” Phys. Rev. Lett. 78, 4269 (1997). [CrossRef]

6.

J.C. Stover, “Optical Scattering: Measurement and Analysis,” (SPIE Optical Engineering Press, Bellingham, Washington1995).

7.

V.A. Sterligov and P. Cheyssac, “Appareil et procédé de caractérisation optique d’un objet,” French Patent FR2832795, G01B-011/30 (2001).

8.

C. Métayer, V.A. Sterligov, A. Meunier, G. Bossis, J. Persello, and S.V. Svechnikov, “Field induced structures and phase separation in electrorheological and magnetorheological colloidal suspensions,” J. Phys.: Cond. Matt. 16, S3975 (2004). [CrossRef]

9.

D.S. Wiersma, M.P. van Albada, and A. Lagendijk, “Coherent Backscattering of Light from Amplifying Random Media,” Phys. Rev. Lett. 75, 1739 (1995). [CrossRef] [PubMed]

10.

A. Taubert, U.-M. Wiesler, and K. Müllen, “Dendrimer-controlled one-pot synthesis of gold nanoparticles with a bimodal size distribution and their self-assembly in the solid state,” J. Mater. Chem. 13(5), 1090 (2003). [CrossRef]

OCIS Codes
(240.0310) Optics at surfaces : Thin films
(240.6680) Optics at surfaces : Surface plasmons
(260.3910) Physical optics : Metal optics

ToC Category:
Research Papers

History
Original Manuscript: April 5, 2005
Revised Manuscript: May 16, 2005
Published: May 30, 2005

Citation
Valeriy Sterligov and Matthias Kretschmann, "Scattering of surface electromagnetic waves by Sn nanoparticles," Opt. Express 13, 4134-4140 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-4134


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References

  1. U. Kreibig, M. Volmer, Optical Properties of Metal Clusters, (Springer-Verlag, Berlin 1995).
  2. V.A. Sterligov, P. Cheyssac, R. Kofman, S.I. Lysenko, P.M. Lytvyn, B. Vohnsen, S.I. Bozhevolnyi, and A.A. Maradudin, �??Near/far-field investigations of the interaction between surface waves and nanoparticles,�?? Phys. Stat. Sol.(b) 229, 1283 (2002). [CrossRef]
  3. P. Cheyssac, V.A. Sterligov, S.I. Lysenko, R. Kofman, �??Scattering of surface plasmon-polaritons and light by metallic nanoparticles,�?? Opt. Commun. 175, 383 (2000). [CrossRef]
  4. T. Kume, S. Hayashi, and K. Yamamoto, �??Light emission from surface plasmon polaritons mediated by metallic fine particles,�?? Phys. Rev. B 55, 4774 (1997). [CrossRef]
  5. A.V. Shchegrov, I.V. Novikov, and A.A. Maradudin, �??Scattering of Surface Plasmon Polaritons by a Circularly Symmetric Surface Defect,�?? Phys. Rev. Lett. 78, 4269 (1997). [CrossRef]
  6. J.C. Stover, �??Optical Scattering: Measurement and Analysis,�?? (SPIE Optical Engineering Press, Bellingham, Washington 1995).
  7. V.A. Sterligov, P. Cheyssac, �??Appareil et procédé de caractérisation optique d'un objet,�?? French Patent FR2832795, G01B-011/30 (2001).
  8. C. Métayer, V.A. Sterligov, A. Meunier, G. Bossis, J. Persello, S.V. Svechnikov, �??Field induced structures and phase separation in electrorheological and magnetorheological colloidal suspensions,�?? J. Phys.: Cond. Matt. 16, S3975 (2004). [CrossRef]
  9. D.S. Wiersma, M.P. van Albada, and A. Lagendijk, �??Coherent Backscattering of Light from Amplifying Random Media,�?? Phys. Rev. Lett. 75, 1739 (1995). [CrossRef] [PubMed]
  10. A. Taubert, U.-M. Wiesler, and K. Müllen, �??Dendrimer-controlled one-pot synthesis of gold nanoparticles with a bimodal size distribution and their self-assembly in the solid state,�?? J. Mater. Chem. 13(5), 1090 (2003). [CrossRef]

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