1. Introduction

Highly valuable information about interactions in biological processes has been obtained by determining absolute distances and their evolution over time. Most of this information has been obtained using fluorescence resonance energy transfer (FRET) [1–4]. The desire to extend this technique to many other biological systems has been limited by three aspects of FRET: (a) photostability of the probes, (b) brightness and (c) range of interaction. (a) The photostability of the probes (or properly the lack of it) limits the length of the process to be studied and blinking creates dark windows where information is unavailable. (b) The low brightness results in poor signal to noise ratio (SNR). To improve SNR long integration times must be used, leading to poor temporal resolution to perform single molecule-single probe experiments. Although many developments helped partially to overcome this problem (robust and bright organic dyes and Quantum Dots [5, 6]), we are still far from the dreamed everlasting chromophore. (c) The range of interaction is inherently limited due to the fact that FRET occurs through dipole-dipole interactions of donor and acceptor thus decaying as 1/distance⁶. The distance at which FRET is 50% efficient (called Förster distance and usually named as Ro) is different for each pair of fluorophores. Typical values for Ro are around 5 nm, preventing the measurement of distances over 10 nm due to the negligible sensitivity in that range.

The use of scattering of metal nanoparticles as probes provides an interesting alternative to fluorescence. As the phenomenon does not rely in electronic transitions, it does not bleach nor blink. Photons scattered depends linearly on photons received, and thus the process is not rate limited by the lifetime as in fluorescent probes. These two facts result in experiments of indefinite length, with fast acquisition rates and high signal [7]. It is well known that two metallic particles couple when are brought together, and that the electromagnetic response differs from the single particle response [8–10]. Recently Reinhardt et al. [11, 12] showed that measuring the spectra of coupled metal nanoparticles, the distance between them can be inferred as it differs from the monomer spectrum. This technique can be used to fill the gap in the measurement of distances between those determined directly by optical microscopy (above 200nm) and by FRET (below 10nm).

In addition to distance, orientation information is very important to understand many biological systems. An example of this is the study of structural dynamics of molecular motors using fluorescent techniques sensitive to orientation [13–15]. In these techniques, the image of the fluorophore is collected and fitted by a theoretical function. When imaging a defocused fluorophore, it is possible to determine position and orientation. The accuracy is limited by the number of photons collected. As the noise due to counting statistics is equal to the square root of the collected photons in each detector (pixel) it is clear that the sensitivity of the technique will be limited by the integration time, the photostabilty and lifetime of the dyes. Highly sensitive and low noise detectors must be used to follow fast processes with the needed accuracy and time resolution.

We propose a novel method for measuring distance and orientation based on the fact that scattering of two metallic particles will change depending on the angle between the polarization and the line that joins them. It requires only one wavelength, provides very high orientation sensitivity and could also be used in tracking experiments.

2. Theoretical model

When two metallic particles are brought close together, the boundary conditions in the surface of one particle include the field scattered by the other. The effect of this coupling makes that the scattering field of a dimer is stronger than the sum of the scattering fields of two isolated particles. The closer the particles, the stronger the effect; and when the two particles are far away, the effect is negligible.

 

Fig. 1. Gold nanoparticles (in grey) are illuminated with a plane wave polarized an angle β measured from the vector that joins the two particles. Cases where polarization is parallel (a) and perpendicular (b) are shown.

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To calculate the effect of the coupling lets assume we will identical metallic particles of diameter D separated at a distance d (center to center) with a complex index of refraction n(w) relative to the medium. As the sizes and distances involved are much smaller than the wavelength, the retardation free approximation can be used. The polarization of the external field is in the plane that contains the two particles, being β the polarization angle measured from the vector that joins them. As it is shown in Fig. 1, the scattering is expected to be larger in the parallel (β=0) case than to the perpendicular one (β=π/2) as the scattered field of each particle reinforces the effect of the external field.

The solution to the electromagnetic problem must satisfy the relations imposed by the boundary conditions.

where E⃗_i (j) (H⃗_i (j)) is the internal electric (magnetic) field of the particle j at its surface, E⃗ _o(j) (H⃗ _o(j)) is the external applied electric (magnetic) field over the surface of the particle j and E⃗_s (l, j) (H⃗_s (l, j)) is the electric (magnetic) scattered field of particle l on the surface of particle j.

We have solved this problem using Generalized Multiparticle Mie [16–18]. In this formalism, the fields (external, scattered and internal) are expanded in a sum of infinite Vector Spherical Harmonics (VSH) centered in each particle. The VSH are usually expressed as ${\overrightarrow{N}}_{\mathit{\text{mn}}}^{\left(X\right)}$ and ${\overrightarrow{M}}_{\mathit{\text{mn}}}^{\left(X\right)}$ , being m it’s degree, n it’s order and X the indicator of the appropriate function to choose for the radial part depending on whether the field is internal or external. As an example, the scattered fields are expanded in the following equations:

Internal fields can be expanded in a similar way, replacing the ${\overrightarrow{N}}_{\mathit{\text{mn}}}^{\left(3\right)}$ and ${\overrightarrow{M}}_{\mathit{\text{mn}}}^{\left(3\right)}$ by ${\overrightarrow{N}}_{\mathit{\text{mn}}}^{\left(1\right)}$ and ${\overrightarrow{M}}_{\mathit{\text{mn}}}^{\left(1\right)}$ to fulfill the correct limits at the origin and at the infinite.

In order to write the boundary equation of a particle, the fields of the rest of the particles are re-expanded in its basis using the translation relations of the VSH. The process is repeated for each particle resulting in an infinite set of equations relating the expansion coefficients in the VSH basis of the scattered field of each particle with the external field. Schematically, it is a linear system where diagonal elements are related to the isolated Mie coefficients [19] and non diagonal elements contain information about the coupling of the element of the basis of the different spheres.

The set of equations is infinite due to the fact that the fields are expanded into an infinite set of VSH. In order to find a numerical solution, the set must be truncated at n=N. In each case we have numerically tested the convergence of the series to find an appropriate N.

3. Results

We have studied numerically the behavior of a dimer of gold nanoparticles in water illuminated by a plane wave (λ=532 nm). The complex index of refraction of gold (0.51+2.22 i) was interpolated from values available in the literature [20]. The polarization of the incident electric field is rotated in the plane that contains both nanoparticles and the scattering cross section (C_sca ) is calculated in each case. Figure 2 shows C_sca vs. β for gold dimers formed by 20 nm gold nanoparticles separated 22 nm, 26 nm and 62 nm respectively. C_sca is normalized to the scattering cross section of a single 20 nm gold nanoparticle. The results agree with the expected situation mentioned above: in the parallel case the scattering is larger than in the perpendicular. Comparing the curves it can be seen that the closer the particles, the larger the effect. The mean value of the normalized C_sca also increases when particles get closer and is 2 with 0 anisotropy as the particles can be considered isolated. The periodicity of the scattering is π due to the symmetry of the system.

 

Fig. 2. Scattering cross section at 532 nm as function of the angle between the polarization of the external field and the orientation of the dimer composed of 20 nm gold nanoparticles. Three different interparticle distances (center to center) are plot: 22 nm for the dotted line, 26 nm for the dashed line and 62 nm for the solid line.

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To account for this fact we define the anisotropy (η) as the ratio between the difference and the sum of the maximum and minimum values.

This dimensionless magnitude takes a value of 0 when the scattering does not depend on the polarization of the external field (isotropic case), and 1 when the system is completely anisotropic.

Calculating η for different distances we studied the dependence of the anisotropy with the interparticle distance. Figure 3 shows the anisotropy as a function of dimensionless distance δ defined as the distance in units of the diameter of the particle. As expected the anisotropy tends to 0 when the distance increases. From this graph it can be seen that for δ>3 the effect is negligible and the particles can be considered as isolated.

 

Fig. 3. Anisotropy of the scattering response as a function of the dimensionless distance defined as interparticle Distance/Diameter

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As the scattering single gold spherical nanoparticles does not depends of the polarization, a diluted sample containing only monomers can be used as a test for the sphericity of the particles. Non-spherical particles (i.e. ellipsoidal) will provide an intrinsic anisotropy not included above, limiting the ability to infer distance from anisotropy from previous calculation. However, eccentricity can be included to correct for it [21].

Depending on the sample preparation, it is possible that the presence of trimers, tetramers or even n-mers cannot be avoided. Nevertheless, in most cases they will be easily distinguished from dimers as the mean values of C_sca will be higher. In those cases where this is not possible (i.e. 20 nm gold particles separated 10.55 nm has same mean C_sca as a non interacting trimer) the symmetry of the n-mer will be reflected in the periodicity of the C_sca vs. β graph.

Presence of monomers can be actually desired in order to have an in situ normalization factor for scattering of the dimer. This makes the analysis much simpler and straight forward as the factors depending on illumination are cancelled out.

Non-uniformity of the particles can lead to an uncertainty in the determination of distances as two dimers with different particle sizes can yield the same polarization anisotropy. A simple example is that two 20 nm particles, 40 nm apart yield the same anisotropy (within the numerical and experimental error) as two 22 nm particles, 44 nm apart. However, in the last case mean value of C_sca will be 1.34 times higher.

In order to account for this detrimental effect we performed numerical calculations for asymmetric dimers considering that gold nanoparticles are available commercially with dispersion lower than 10%. Sweeping the particles within this range, our calculations show that if anisotropy and mean scattering cross section is determined with an error of 5%, distances of 25 nm can be obtained with an uncertainty lower than ±1 nm. It is important to mention that orientation information will not be lost nor diminished due to this cause. Orientation is obtained by finding the polarization at which the dimer has the maximum scattering and that is independent of the sizes of the particles (as long as the particles are big enough to interact at a given distance).

4. Conclusion

These results show that scattering polarization microscopy of coupled nanoparticles should provide an alternative method for determining the orientation of a dimer and the relative distance between the particles. In the gap we would like to fill with this technique (from 10 nm to 200 nm), the particles will be closer than the resolution limit of the microscope and will appear in the camera as a single spot. By taking images of the same sample with different incident polarization and measuring the intensity of the spot in each image, the theoretical model presented above can be used to fit distance from anisotropy and mean value of scattering, and orientation from polarization of maximum intensity. As this process can be done in a microscope with a camera, many dimers can be imaged together, thus parallelizing the acquisition of data.

This method contains those benefits mentioned by Reinhardt et al, might the most important be the exceptional photostability and it brightness. This allows taking images with better SNR with shorter exposure times expanding the range of work to regions of low sensitivity (i.e. δ~3 in this case). As the effect scales with the diameter of the particle, the range of work can be selected by choosing the appropriate particle. Besides these benefits, only wavelength is needed in order to perform the experiment.

Orientation information is an additional benefit that makes this method unique, providing a way to gather important data to understand many biological processes like movement of molecular motors or conformational changes of macromolecules. In single molecule experiments, orientation information can be collected fast and efficiently by rotating the polarization at f and performing a lock-in detection at 2f. Noise estimation with a 1mW laser focused in 25 µm² and 2 ms of integrantion time, and using a dimer of 20 nm gold nanoparticles separated 40 nm, results in a 2 mrad accuracy in the determination of the angle.