1. Introduction

Depolarization is one of the fundamental phenomena in optical scattering of nano-objects. It describes the polarization changes between the incident and scattered fields, and has been found important in a variety of fields, like molecular spectroscopy [1,2], optical devices [3,4], metasurfaces [5–7], to name a few. In fact, the functions of many optical materials are based on the depolarization effects of their constituent nano-units. For example, the birefringence effect of calcite is due to the anisotropy of its constituent cells [8,9]; the ability of liquid crystals to rotate the polarization of light comes from the orientation anisotropy of the inner molecules [10–12]; meta-surface can modulate the polarization state of the wave front freely because of the spatially dependent anisotropy of the meta-atom array. Therefore, it is crucial to understand the polarization properties of individual nanostructures [13,14].

In this work, we focus on plasmonic nanoparticles, one of the most important type of optical nanostructures, which are small, simple to synthesize, and exhibit strong resonant scattering capability [15–17]. To date, many works have been reported on the depolarization effects of anisotropic plasmonic nanoparticles, such as silver nanoparticles [18], gold nanorods [19,20], silver nanorods [21], silver triangles [22], silver cubes [22] and stellate gold nanoparticles [23]. However, the depolarization effects of single isotropic metallic nanoparticles, namely nanospheres (NSs), have rarely been studied, despite that their absorption and scattering theory has been established [24]. Meanwhile, in polarized light microscopy, one of the major characterization tool for microscopic materials, it is commonly believed that the optical contrast originates from the local anisotropy of the material [25]. However, as mentioned above, isotropic nanoparticles may also generate image contrast with a polarized light microscope. It is therefore necessary to quantitatively verify the depolarization effect of a single NS in experiment.

We hereby experimentally studied the depolarization effect in light scattering of single Au NSs and how the depolarization effect is related to the experimental conditions, i.e. the detailed excitation-observation polarization states, the numerical aperture of the image system and the sizes of nanoparticles.

The organization of this paper is as follows. First, the oblique-incidence dark-field microscope [26,27] used in this work is described. Then, the scattering images of single Au NSs with different excitation-observation polarization states are presented, and how the depolarization effect depends on the numerical aperture of objective lens is analyzed. Finally, we experimentally demonstrate that the optical contrast of a single gold NS can be detected in a reflective polarized light microscope with a large numerical aperture.

2. Experimental methods

Figure 1(a) depicts the schematic diagram of the experimental setup. It consists of two parts, namely the illumination part and the imaging part.

 

Fig. 1. Depolarization effect of single Au nanospheres. (a) Schematic of the oblique-incidence dark field microscope. P1: polarizer 1 for controlling the illumination light to be s or p polarization. P2: polarizer 2 for detecting polarization components along the x or y direction. In the dashed frame is the three dimensional coordinate system. (b) SEM image of monodispersed Au nanospheres (diameter=80 ± 6 nm) on a cover glass. The inset is an enlarged image of one nanosphere. (c) Measured (black) and calculated (blue) scattering spectra of a 80 nm Au nanosphere, as well as the calculated (red) extinction spectra. All the curves are normalized. (d) and (e) Polarized dark field images of single Au nanospheres with P2 along the x-axis and y-axis, respectively. In both cases P1 was along the x-axis. The exposure times are labeled at the upper-left corner of each panel. (f and g) Simulated scattering images of the E_x and E_y component of a dipole excited along the x-axis, respectively.

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In the illumination part, a supper-continuum laser (YSL, SC-PRO) was used as the light source. The laser was first coupled into a multimode fiber, and then collimated by a broadband reflective collimator. After that, the beam was linearly polarized with a Glan-Tomson polarizer (P1), and finally focused on the sample by an achromatic lens with a focal length of 3.5 cm. Since the focus is weak, the polarization state of the laser beam was maintained. In measurements, the power density illuminated on the sample was approximately 0.4 W/mm². Here, the collimator, P1 and the achromatic lens were fixed on the same board, and the incident angle can be adjusted with the help of an adjustable bracket and a three-dimensional translation stage. In this work, the incident angle was set to approximately 70 degrees to form an oblique-incidence dark-field microscope.

In the imaging part, an inverted optical microscope (Olympus IX73) was used to image the Au NSs. The scattered light of Au NSs was collected by a long working distance objective (100×, NA=0.8), and then was inspected by the second Glan-Thomson polarizer (P2). After that, the signal was refocused by the tube lens and formed scattering images, which were recorded by an EMCCD (Andor, iXon897).

Au NSs with different diameters (80 nm and 40 nm) were used in this work. The samples were prepared by spin coating dilute solution of Au NSs (Nanoseedz) onto a clean cover glass. The SEM image shows that Au NSs (diameter=80 ± 6 nm) on the cover glass are well monodispersed [Fig. 1(b)]. The enlarged image [Fig. 1(b) inset] confirms that the Au nanoparticles are spherical.

3. Results and discussion

3.1 Depolarization of a horizontally excited Au nanosphere

We first measured the depolarization effect of horizontally excited Au NSs. Figure 1(c) shows the measured scattering spectrum of an Au NS with a diameter of 80 nm (black). A single polarization-independent resonant peak was found at approx. 550 nm, which is consistent with the scattering efficiency curve (blue) by Mie theory, proving that the observed object is a single nanoparticle. This was also confirmed with SEM images of the same sample area. The normalized theoretical extinction efficiency curve (red) is also shown in Fig. 1(c). The resonant peaks in both extinction and scattering efficiency curves are mainly due to the electric dipole amplitude [28].

To investigate the polarization effect, Au NSs were first illuminated by s-polarized light (i.e. the polarization axis of P1 was along the x-axis). When the polarization axis of P2 was also along the x-axis, the scattering image of Au NSs was centro-symmetric spots [Fig. 1(d)]; when the polarization axis of P2 was along the y-axis, the scattering image became a complex pattern which consists of four equal-intensity lobes separated by dark cross lines along the x and y axes [Fig. 1(e)]. Different exposure times for different polarization conditions were used to compensate the intensity difference in the measurement.

The results demonstrate that even a single totally symmetric spherical gold nanoparticle can exhibit sophisticated depolarization effect. Since the Au NSs are subwavelength in size, their scattering can be treated as dipole radiation [28,29]. Then, the depolarization effect can be understood by considering the NS as a dipole excited along the x-axis which also generates E_y component in addition to the dominant E_x component. We calculated the point spread function with both E_x and E_y components of the dipole [Figs. 1(f) and 1(g)], and the results agree with the experimental data.

To quantify the depolarization process, we define the depolarization ratio [2] as ρ=I_y/I_x. Here I_y denotes the integrated intensity of the E_y component (the depolarized signal) and I_x denotes the integrated intensity of the E_x component (i.e., the total excitation fields). According to the experimental data, I_y is approximately two orders of magnitude weaker than I_x (i.e., ρ is on the order of 0.01). For instance, the depolarization ratio of the two NSs in Figs. 1(e) is about 1/160.

3.2 Depolarization of a vertically excited Au nanosphere

The excitation direction of Au NSs is not limited in the horizontal directions (in x-y plane) and can also be excited vertically (in z-direction). This will induce more complicated behaviors.

To demonstrate this effect, we excited Au NSs with p-polarized light (the polarization axis of P1 was nearly along the z-axis) and measured the patterns of E_x and E_y components in the image plane. Different from the case of horizontal excitation, the results are two lobes with a dark line along the x or y axis (depends on the polarization direction of P2), as shown in Figs. 2(a) and 2(b). When P2 was removed (non-polarized case), a donut shape spot was observed [Fig. 2(c)].

 

Fig. 2. Scattering images of different excitation-observation polarization configurations. The first, second, and third rows respectively represent the case where P1 is along the z, x axis and removed. The first, second, and third columns respectively represent the cases where P2 is along the x, y direction and removed. In order to ensure the signal-to-noise ratio, different exposure times were used for different configurations. The exposure times are labeled at the upper-left corner of each panel.

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To summarize the polarization effects in scattering, we list all the possible combinations of the polarization states of P1 and P2 together, as shown in Fig. 2. Theoretically, the total intensity of the scattered field equals the sum of the two orthogonal polarization components. Our measured results are in accordance with the theory, i.e. the results shown in the third row/column are the sum of the results shown in the first two rows/columns. For example, the two transverse lobes in Fig. 2(a) and the two longitudinal lobes in Fig. 2(b) constitute the donut shape in Fig. 2(c).

Note that the signal intensity of the cross-polarization case [Fig. 2(e)] is much weaker than other cases and a longer exposure time was used to make the signal-to-noise ratio comparable with others. As a result, the sum of Fig. 2(b) and Fig. 2(e) still exhibits a two-lobe shape [Fig. 2(h)] instead of a complex multi-lobes pattern; the sum of Fig. 2(d) and Fig. 2(e) is still a centro-symmetric spot [Fig. 2(f)].

3.3 Theoretical model of the depolarization effect

In this section, we explain the theoretical model of the depolarization effect based on the imaging process. The input of the imaging process is the scattered light of single Au NSs excited by the linearly polarized light. As mentioned above, the Au NSs are subwavelength in size and their scattering fields can be described by dipole radiation. Here, we discuss two cases, namely horizontally excited dipole (along the x-axis) and vertically excited dipole (along the z-axis) which correspond to the cases of s-polarization and p-polarization excitation, respectively. Figure 3(a) shows how the scattered fields evolves along the optical path. The imaging process can be treated as a two-step coordinate transformation and a subsequent interference process. First, the scattered fields from an Au NS can naturally be treated as a spherical wave radiation, and then are transformed into plane waves by the objective (infinite corrected system). Since the system is axial symmetric, the fields after the objective can be described using cylindrical coordinate. The second transformation occurs when the fields meet the tube lens, which refocuses the scattered fields and converts them into spherical waves again. Finally, the scattered fields interfere at the focus plane and generate images of the Au NSs. By selecting the polarization direction of the polarizers, images with different patterns can be observed on the image plane. The light field distribution on the object plane [Fig. 3(b)], Fourier plane [Fig. 3(c)] and image plane [Fig. 3(d)] are sequentially shown below Fig. 3(a).

 

Fig. 3. Theoretical model of the depolarization effect in light scattering of single Au nanospheres. (a) Schematic of the imaging process of a dipole excited along the x (red) or z (blue) axis. The arrows indicate the polarization direction of light in the x-z plane. (b), (c) and (d) depict the distributions of the scattered light of an Au NS in the objective plane, Fourier plane and image plane, respectively. The upper panels in (b), (c), (d) are the results for the x-excited dipole, and the lower panels are the results of z-excited dipole. The insets in the corner of (d) are the simulated results of the case where the dipole is excited along the direction which is 20 degrees off the z-axis. (c) is drawn in polar coordinate and its polar diameter is sinθ. Here we assume the focal length of the objective lens is 1. (d) is drawn in rectangular coordinate.

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Using the above physical picture, the sophisticated polarization behaviors can be understood by considering the scattered field of the dipole collected by the objective lens. Figure 3(b) depicts the projection of the scattering electric field of the dipole onto the x-y plane. The dipole is located at the origin of the coordinate system. The dotted lines represent the electric field lines of the dipole and the blue arrows indicate the direction of the electric field.

In the case of excitation along the x-axis [see the top figure in Fig. 3(b)], there is no electric field along the y direction on the x-axis or y-axis, which means there is no E_y component on the x-axis or y-axis. As a result, in the image plane [the upper-right figure of Fig. 3(d)], the field distribution of |E_y|² appears as a four-lobe shape with dark lines on the x and y axes. Meanwhile, in the central region of the x-y plane, there always exist electric field along x direction, so the pattern of |E_x|² appears as a centro-symmetric spot [the upper-left figure of Fig. 3(d)]. From the aspect of depolarization effect, for a dipole excited along the x-axis, at the symmetric point (center point) in the image plane, there is no depolarized light. The depolarization effect only occurs at the surrounding area in the image plane, where the light path is not symmetric anymore [see the upper-right figure of Fig. 3(d)].

In the case of excitation along the z-axis [see the bottom figure of Fig. 3(b)], there is no E_y component on the x-axis, and there is no E_x component on the y-axis. This explains why the field distribution of |E_x|² in the image plane [the lower-left figure of Fig. 3(d)] is zero on the y-axis, showing two lobes on the left and right. Similarly, the field distribution of |E_y|² is zero on the x-axis, showing two lobes up and down [the lower-right figure of Fig. 3(d)].

The insets in the subfigures of Fig. 3(d) (in the dashed frame) show the results when the excitation direction is 20 degrees off the z-axis, which is in accordance with the actual experimental condition for the p-polarized incident light. Although the polarization is mainly along the z-axis, there is still a few component along the y-axis. Therefore, in addition to the dipole excitation along the z-axis, another component of the dipole oscillated along the y-axis is considered in our simulation, generating a circular spot of weak intensity in the center of the x-y plane. Therefore, the two lobes in the pattern of |E_y|² are not totally separated as that of |E_x|².

3.4 NA-dependent depolarization effect

In the previous section, we see that the depolarization effect is related to the scattered field collected by the objective lens. If we increase the numerical aperture (NA) of the collection optics [namely sinθ, θ is marked in Fig. 3(a)], the scattered light over a larger angle range will be collected. In order to quantitatively study how the collection angle (i.e. NA) of the system influences the depolarization effect, we placed an iris with adjustable aperture behind the objective [see Fig. 3(a)] to change the effective NA.

We recorded the images of a single Au NS excited along the x-axis with different aperture sizes, as shown in Fig. 4(a). The first and second row are the field maps of |E_x|² and |E_y|² components, respectively. The values of effective NA were derived from the sizes of the aperture.

 

Fig. 4. NA-dependent depolarization effect. (a) Dark field images of a single Au nanosphere excited along the x-axis with different effective numerical apertures. The first and second row are images of |E_x|² and |E_y|², respectively. (b) and (c) are the total signal intensity of |E_x|² and |E_y|² pattern as a function of NA², respectively. The experimental data (squares) are in line with the theoretical calculations (black lines).

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In order to quantify the influence of NA on the depolarization effect, we integrated the intensity of |E_x|² and |E_y|² in the images, and plot them as functions of NA² [Fig. 4(b) and Fig. 4(c)]. Here, NA² is used because it is proportional to the solid angle in which scattered fields are collected. It is found that the total intensity of |E_x|² is almost linear with NA², while the total intensity of |E_y|² increases rapidly with the increase of NA².

The different behaviors of |E_x|² and |E_y|² can be understood from the field distribution of |E_x|² and |E_y|² in the Fourier plane [see Fig. 3(c)]. For a dipole excited along the x-axis, |E_x|² is almost uniformly distributed in the center area, whereas the intensity of |E_y|² is zero at the center of the x-y plane and strong at the four corners. As a result, when sinθ (i.e. NA, the radius of the polar coordinate) increases from 0 to 1, |E_x|² increases almost linearly with NA², whereas |E_y|² increases nonlinearly with NA². The curves obtained by theoretical calculations (black lines) in Figs. 4(b) and 4(c) are consistent with the experimental data (squares).

This result indicates that the larger the numerical aperture is, the larger the proportion of E_y component (in other words, stronger depolarization effect) we will have. In Fig. 4(a), the E_x components are visible even when NA is as small as 0.5.

3.5 Depolarization effect of a sub-50 nm Au nanoshphere

In order to prove that the depolarization effect of sub-50 nm Au NSs can also be observed, experiments were performed for Au NSs with a diameter of 40 nm. Figure 5 shows the scattering patterns of different polarization states. Due to the smaller particle size, the scattering signals of 40 nm nanoparticles were considerably weaker compared with those of 80 nm nanoparticles. But their depolarization behaviors were similar since they are essentially the depolarization effects of dipole radiation. For particles which support high order resonance modes, such as high refractive index spheres, more complex depolarization effects may occur due to the complex scattering patterns of them.

 

Fig. 5. Polarization effects of single Au nanospheres with a smaller diameter (40 nm). (a) SEM image of monodispersed Au nanospheres with a diameter of 40 nm on a cover glass. The inset is an enlarged image of one nanosphere. (b) Scattering images of a single Au nanosphere with a diameter of 40 nm with different excitation-observation polarization configurations. The first, second, and third rows respectively represent the case where P1 is along the z, x axis and removed. The first, second, and third columns respectively represent the cases where P2 is along the x, y direction and removed. The patterns are almost the same as those of a single Au NS with a diameter of 80 nm except the signal intensity is weaker. The exposure time of each image is labeled at the upper-left corner of each panel.

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3.6 Observing a single gold nanosphere with a polarized light microscope

It has been shown that the depolarization effect of a single Au NS can be observed with a dark field microscope. Therefore, it is achievable to observe the optical contrast of a NS with a polarized light microscope. To this end, we implemented a reflective polarized light microscope [30] to observe the isotropic gold nanoparticles [Fig. 6(a)]. The white light beam produced by a xenon lamp was collimated and linearly polarized, and then passed through the central region of the objective lens (NA = 0.8) to illuminate the sample. The reflected light of the cover glass and the scattered light of the Au NSs were collected by the objective lens and finally detected by the EMCCD. Two polarizers (P1 and P2) were placed into the optical path to construct the crossed or parallel polarization states.

 

Fig. 6. Imaging single Au nanospheres with a polarized light microscope. (a) Schematic diagram of the reflective polarized light microscope. P1 and P2 are polarizers for excitation and detection, respectively. (b) and (c) Extinction images of a single Au nanosphere with a diameter of 80 nm. In (b), the polarization axes of P1 and P2 were parallel to each other; In (c), the polarization axes of P1 and P2 were perpendicular to each other. The exposure time of (b) and (c) were 0.2s and 90 s, respectively.

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Figure 6(b) shows the extinction image of an 80 nm single Au NS with the polarization axis of P2 parallel to that of P1, and a dark spot can be observed. When the polarization axis of P2 was perpendicular to that of P1, the signal of the nanoparticle is still visible [Fig. 6(c)] but with a considerably lower signal intensity (here, the exposure time has been increased to make it visible). This result is contradictive to the assumption in polarized light microscopy that the optical contrast in a polarized light microscope solely generates from the structural anisotropy of samples.

It is worth mentioning that the signal in Fig. 6(c) is very weak [more than two orders of magnitude smaller than the case in Fig. 6(b)]. Therefore, in cases where quantitative analysis is not necessary, the depolarization effect of NS can be neglected.

4. Conclusion

In this work, we investigated the depolarization effects in the light scattering of single Au nanospheres, and sophisticated polarization-dependent scattering image patterns were observed. Depending on the polarization states of excitation and observation, the scattering images of a single Au NS can be a circular spot, donut-like structure, two equal intensity lobes or four equal intensity lobes. In the case of non-polarized light, only a circular spot was observed. Theoretical analysis and numerical simulations were also performed, showing well agreements with the experimental results. Meanwhile, analysis shows that the in-plane depolarization effect is related to the NA of the objective, and this was confirmed by experiments. All the results indicate that even an ideally symmetric spherical nanoparticle can generate observable depolarization effect in the light scattering process. As a result, we need to reconsider the imaging interpretation in the widely used polarized light microscopy, which states that the image contrasts are solely caused by samples’ anisotropy.