1. Introduction

Circling light through total internal reflection in a dielectric sphere or disk, called whispering gallery mode (WGM) [1,2], has the effect of amplifying light including laser emission [3–7], attracting attention as a device for quantum bits [8,9], nonlinear optical elements [10], highly sensitive sensors and even an optical receiver [11–14]. In a WGM, light circles around the inner circumference of a spherical dielectric while being totally reflected, resulting in optical amplification [2,3]. At this time, some of the light seeps out, so that it functions as a highly sensitive sensor or antenna [11–14]. Recently, we reported a phenomenon of light orbiting the outer circumference of a spherical dielectric on a thin SiN film [15]. When PS spheres on a SiN film were observed under a confocal laser Raman microscopy (CLRM), anomalously enhanced Raman spectra were observed at the periphery of the spheres, which varied periodically with the wavelength of the light and were more prominent for SiN films with a thickness of 50 − 75 nm [15]. We speculate that the ring-shaped light is caused by the light orbiting around the outer surface of the spherical dielectric materials, unlike the conventional WGM. This orbiting light is greatly affected by the substrate on which the spheres are placed because it orbits outside of the surface, unlike conventional WGMs, but is undetectable on ordinary glass substrates [15].

The finite difference time-domain (FDTD) method [16] is often used to analyse microscale optical devices [17–21]. The FDTD method allows the space in which optical elements exist to be finely divided and the spatio-temporal optical properties to be analysed [16]. This method has been used to analyse the optical properties of microdisks, microcylinders, and spheres [17–21]. Here, we performed an optical simulation analysis of the optical properties of 3 µm PS spheres on SiN films using the FDTD method in a 3D model. As a result, we found that the circumferential light was generated up to 650 nm from the outer circumference of the sphere. Furthermore, similar results have been obtained with CLRM. These results suggest that the light reaches 6 times further from the sphere surface than normal evanescent light. Moreover, optical simulation analysis using a 3D model showed that similar orbiting light is generated on the outer circumference of the spheres. In addition, some of the light circling the sphere was captured by the film surface and propagated parallel to the surface. Normal surface waves, known as surface plasmons, are generated only at metal-dielectric interfaces [22–25]. In contrast, they are not generated on dielectric surfaces because they do not satisfy the solution of Maxwell's electromagnetic equations [25]. However, we found that a dielectric film thinner than the wavelength has a directional surface propagating light, which formed by the thin film being placed at the centre of the light. Therefore, the surface propagating light of the thin dielectric film has different characteristics from conventional surface plasmon. In this paper, we analyse these new types of orbiting light and surface light propagation on dielectric thin films.

2. Results

2.1 Raman spectra of the periphery of dielectric spheres on SiN film

PS spheres with a diameter of 3 µm were deposited on a SiN thin film with a thickness of 50 nm and the Raman spectra were measured using CLRM [Figs. 1(a)–1(e)]. The SiN film was formed in the centre of the silicon frame of the observation window, and the spheres were attached to the top of the SiN film [Figs. 1(a) and 1(d)]. The centre of the 3 µm diameter PS sphere was irradiated from the outside with a 532 nm laser at five different positions and at each position, the Raman spectrum was measured [Fig. 1(e) and Fig. S1 in Supplement 1]. At a distance of 3 µm from the centre of the sphere (1.5 µm from the outer surface), there were no peaks in the Raman spectrum, which increased gradually towards longer wavelengths [Fig. 1(e) top]. This result suggests that the laser beam was directly irradiating the SiN film and that its wavelength shifted to longer than 532 nm due to the nonlinear optical characteristics of the SiN materials [26–28]. At 0.65 µm from the sphere surface, four weak peaks appeared in the Raman shift [Fig. 1(e), 2.15 µm]. At 0.4 µm from the sphere surface, four new peaks appeared between the four peaks described above and their intensities increased [Fig. 1(e), 1.9 µm]. At the sphere surface, there was a strong spectral signal with gentle oscillations [Fig. 1(e), 1.5 µm]. On the other hand, at the centre of the sphere, a PS Raman spectrum with a peak at 998 cm⁻¹ was detected [Fig. 1(e), 0 µm].

 

Fig. 1. Raman shift of PS microspheres on SiN film. (a) Schematic of the setup used to measure the Raman shift of PS spheres with a diameter of 3 µm on a SiN film of 50 nm thickness supported by a Si frame. (b) OM image of the PS spheres on a 0.4 × 0.4 mm square SiN film. (c) Higher magnification OM image (×1000) of the centre region of the SiN film (broken red rectangle in (b). (d) Enlarged image of the sphere for Raman measurement (red arrow in (c)). (e) Normalized intensity plots of Raman shift at various distances from the sphere centre. (f) Top views of Raman images of a PS sphere at the PS peak (998 cm⁻¹), Raman peak at 2.15 µm distance from the sphere centre (2988 cm⁻¹) and at 1.9 µm distance (3266 cm⁻¹), Raman peak on the sphere surface at 1.5 µm distance (3394 cm⁻¹). (g) Merged Raman peak image at 2988 cm⁻¹ (red) and 3266 cm⁻¹ (green). The structure in blue indicates a peak PS image at 998 cm⁻¹. (h) Enlarged image of the left side of the sphere in (g). (i) Merged peak image when the laser is irradiated on the surface of the sphere (3394 cm⁻¹). The peak in the anomalously enhanced spectrum lies slightly outside the sphere surface. (j) Enlarged image of the left side of the sphere in (i). (k) Line plot of each peak intensity in the sphere centre. The red, green, and blue lines indicate the Raman shift intensities at 2988 cm⁻¹, 3266 cm⁻¹ and 3394 cm⁻¹, respectively. (l) Line plot converted to distance from the sphere centre (the position indicated by the bracket in (k). Scale bars, 200 µm in (b), 10 µm in (c), 3 µm in (d), 2 µm in (i) and 500 nm in (j).

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Next, we compared the Raman shifts of each peak image [Figs. 1(f)−(j)]. The mapped image of the Raman peak at 998 cm⁻¹ of PS showed a sphere-like shape [Fig. 1(f), 998 cm⁻¹]. In the peak image at 2988 cm⁻¹, 0.65 µm away from the sphere surface, the signal extended well beyond the periphery of the sphere [Fig. 1(f), 2988 cm⁻¹]. On the other hand, in the Raman peak image (3266 cm⁻¹) 0.4 µm away from the surface, the area outside the sphere was slightly reduced [Fig. 1(f), 3266 cm⁻¹]. In the merged image of the 2988 cm⁻¹ and 3266 cm⁻¹ peaks, the red region at 2988 cm⁻¹ was widely distributed around the outer edge of the sphere, and the green 3266 cm⁻¹ region partially overlapped the red region inside [Figs. 1(g) and 1(h)]. When a laser beam irradiated the sphere surface, an anomalously enhanced Raman spectrum was produced near the surface [Figs. 1(i) and 1(j)].

The line plots in the centre of each Raman peak image showed the spread of the orbiting light to the outer periphery of the spheres [Figs. 1(k) and 1(l)]. The broadening of the peak at 2988 cm⁻¹ extended significantly to the outer periphery of the sphere, reaching 0.65 µm [Figs. 1(k) and 1(l), red line]. On the other hand, at the Raman shifts of 3266 cm⁻¹ and 3394 cm⁻¹, the light coverage shifted to the sphere surface [Figs. 1(k) and 1(l), green and blue lines]. This indicates that when the number of Raman shift peaks changed from 4 to 8, it reached 6 times further than evanescent light from a normal WGM. Therefore, under the conditions where such peaks occur, it is considered that the light circles in a manner different from conventional WGM.

2.2 Optical simulation analysis of 3D model by FDTD

Many groups have reported analyses of optical simulations of WGM using the FDTD method [17–21]. Here, we analysed the light propagation processes of a PS sphere with a diameter of 3 µm on a 50 nm thick SiN film by 3D optical simulation using the FDTD method. The laser light source of 532 nm wavelength was set to focus the light from above onto the lateral surface of the PS sphere [Fig. 2(a), see Visualization 1]. At 10 fs after the light was incident, electric-field intensity (E²) of light progressed to the surface of the sphere while gradually focusing from above [Fig. 2(a), 10 fs]. After 30 fs, a portion of the incident light contacted the side of the sphere and penetrated into the sphere [Fig. 2(a), 30 fs]. After 50 fs, the light penetrating inside the sphere reached the contact surface between the sphere and the SiN film, part of the light moved on the SiN film, and the rest of the light circled around the sphere [Fig. 2(a), 50 fs]. At this time, the light incident on the sphere spreads slightly from the central longitudinal axis of the sphere (see Visualization 1). This phenomenon is due to the curved surface of the sphere, where the incident light is irradiated. The surface of the sphere with which the incident laser light comes into contact is curved, and the light incident on the center of the vertical axis travels along the central axis plane. However, light incident at a position slightly off the central axis will travel in a direction slightly off the central axis because of the curve of the sphere's surface. Therefore, the light traveling to the contact surface of the SiN film spreads over a relatively wide width.

 

Fig. 2. Optical simulation results of PS sphere on SiN film. (a) Time course of electric-field intensity (E²) of light using the FDTD method when a 3 µm PS sphere on a 50 nm thick SiN film is applied to a laser beam from the upper left side of the film. The polarization angle of light source is 0°. The red and blue arrows in the light source indicate magnetic-field and electric-field directions, respectively. When the laser light reaches the left side of the sphere, it penetrates into the sphere (10 − 30 fs). The incident laser light then begins to circle the sphere (50 − 1000 fs). In this image, green indicates light outside the sphere above the SiN film, magenta, light inside the sphere and blue, light below the SiN film. (b) Side view of sphere on SiN film 1000 fs after laser irradiation. The laser light circles inside and outside the sphere. This orbiting light is projected onto the SiN film with directionality. (c) Light intensity image of the cross section through the sphere centre. (d) Enlarged light intensity image of the upper left of the sphere. (e) Light intensity image converted from the image in (d) to the angle and distance from the sphere centre. White arrows indicate areas with high light intensity on the surface of a sphere. (f) Line plot of the white arrow of right position in (e). Scale bars, 2 µm in (a)−(c), 500 nm in (d), and 200 nm in (e).

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After 200 fs, the intensity of the light traveling inside the sphere increased and part of the light seeped out. In addition, the light traveling on the SiN film surface was also enhanced [Fig. 2(a), 200 fs]. After 1000 fs (1 ps), the intensity of the light circling inside the sphere increased, and the circling light spread outward from the sphere [Fig. 2(a), 1000 fs]. Viewing the light state of the sphere after 1 ps from the side, we could see it clinging to the outer circumference of the sphere while greatly protruding [Figs. 2(b) − 2(d)]. Furthermore, the light also seeped out to the underside of the SiN film while circling, and part of the light was separated as a surface light of the SiN film [Fig. 2(b), blue structure]. In the map of the light state in relation to the distance and angle from the centre of the sphere, the circling light travelled almost parallel inside and outside the sphere [Fig. 2(e)]. The light on the surface of the sphere attenuated significantly at 200 nm from the surface layer and then decreased gradually, reaching 500 nm [Fig. 2(f)]. In addition, there was a faint light intensity at 500 − 750 nm from the outer surface layer [Fig. 2(f)]. These characteristics are in close correspondence with the Raman shift measurements [Fig. 1(l)].

2.3 Surface light propagation on SiN films

The light circling around the sphere seeped out significantly downward at the contact surface with the SiN film, and part of it travelled straight as a surface light propagation of the SiN film [Fig. 2]. We analysed the characteristics of this surface propagating light over longer distances by FDTD simulations [Fig. 3]. Figures 3(a) and 3(b) show the 3D structure of surface propagating light straight through the SiN film when its thickness was varied from 20 to 200 nm and light was applied onto a 3 µm diameter PS sphere. When the SiN film was 20 to 30 nm thick, the intensity of the surface propagating light in the SiN film was low and decreased as the travel distance increased [Figs. 3(a) and 3(b), 20 and 30 nm]. At thicknesses between 50 to 80 nm, the intensity of the surface propagating light became higher, and the light travelled in a straight line for a longer distance in a converged state [Figs. 3(a) and 3(b), 50 and 80 nm]. Furthermore, at a thickness of 100 nm or more, the surface propagating light travelled while meandering up and down [Figs. 3(a) and 3(b), 100 and 200 nm]. Figures 3(c) and 3(d) are magnified maps and intensity plots of Z axis for SiN surface waves 6 − 8 µm away from the centre position of the sphere.

 

Fig. 3. Light propagating in SiN film generated by sphere. (a) Changes in the surface propagating light through the SiN film when the thickness is varied from 20 to 200 nm. 3D structure of light power from FDTD optical simulation when a PS sphere with a diameter of 3 µm is irradiated with a 532 nm laser beam. A highly directional surface propagating light was observed in a SiN film with a thickness of 50 to 80 nm. (b) Optical power image of the cross section through the centre of the sphere in (a). (c) Enlarged and normalised light power image of SiN film in the red framed area in (b). In a 20 nm thick SiN film, light travels while spreading widely above and below the film. As the SiN film becomes thicker, the vertical width of the light narrows and the light intensity increases. When the SiN film thickness is 80 nm or more, the optical power near the film decreases. (d) Line plot of Z-axis indicated by the red arrow in (c). (e) Changes in the light intensity and wavelength of the SiN film surface propagating light with respect to the film thickness. Twice the distance between peaks of the light intensity represents the wavelength of the light. In a 20 nm thick SiN film, the wavelength of light is 530 nm, which is almost the same as the irradiated laser light of 532 nm (red line). This wavelength gradually decreases with increasing SiN film thickness, dropping to 410 nm at a thickness of 200 nm. On the other hand, the light power peak is seen at the thickness of 60 nm (black line). (f) The generation mechanism of SiN surface propagating light. The light circling the sphere is partially projected onto the SiN film as a surface propagating light at the contact surface with the SiN film. This surface propagating light in the SiN film is stabilized so that the SiN film stays at the centre of the light due to the braking action of the SiN film. The braking action of the SiN film is caused by the refractive index of the SiN film (n = 2) being higher than the surrounding air (n = 1). Scale bars, 2 µm in (a) and (b), and 500 nm in (c).

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In the 20 nm SiN film, the SiN surface propagating light broadly spread above and below the film. This light width at Z axis gradually narrowed as the SiN film became thicker up to 100 nm. Furthermore, at a thickness of 200 nm, the light intensity exhibited periodic variations. On the other hand, the intensity of surface propagating light peaked at 60 nm [Fig. 3(e)]. The SiN surface light peak-to-peak distance of 20 nm thick was 530 nm, which is almost the same as the wavelength of the incident laser light (532 nm). This peak-to-peak distance decreased with increasing SiN film thickness, dropping to 420 nm at a thickness of 120 nm [Fig. 3(e)]. This result indicates that as the thickness of the SiN film increases, the ratio of the dielectric portion of the light propagation increases and the speed of light decreases.

This decrease in the light velocity due to the dielectric material provides an answer to the mechanism by which the SiN film captures the centre of the light and propagate the light stably [Fig. 3(f)]. The light that circles the surface of the sphere reaches the underside of the SiN film and some of the light is thrown out in the direction along the thin film. At this time, the light is shifted below the thin film and the SiN film is positioned above the light. In this state, because the refractive index of the SiN film (n = 2) is higher than that of air (n = 1), the speed of the light decreases on the upper side of the light, and the light travels upward while bending. When the light is displaced to the upper side of the thin film, the light bends slightly downward because of the SiN film underneath the light. The light gradually adjusts its position and finally stabilizes with the SiN film positioned in the centre of the light [Fig. 3(f)].

In Fig. S2 in Supplement 1, the light propagation state was examined when the deflection angle of the light source was rotated by 90° and the direction of the electric field was parallel to the contact surface of the sphere. With a light source polarized by 90°, the intensity of light circling inside the sphere is reduced. Furthermore, the light propagating through the SiN film is also reduced. On the other hand, light is emitted upwardly from the contact surface between the sphere and the SiN film at an angle of approximately 30°.

2.4 Propagation of light between spheres through thin films

Next, an optical simulation was performed for the case where a sphere exists at the end of the light propagation through the thin film [Fig. 4]. Light incident from above on the left sphere travels along the surface of the SiN film and reaches the contact surface of the right sphere (Fig. S3 in Supplement 1). Most of the light from the SiN film is then captured by the right-side sphere and begins to travel around the sphere [Fig. 4, Fig. S3 in Supplement 1]. Therefore, the light on the SiN film does not travel beyond the right sphere [Figs. 4(b) − 4(d)]. This simulation result indicates that the surface propagating light of the SiN film can be easily manipulated by the object on the SiN film.

 

Fig. 4. Surface light propagation in SiN film between two spheres. (a) FDTD simulation results of light propagating through two spheres on SiN film. 3D structure of optical power after 1 ps when 3 µm diameter spheres are placed 9 µm apart on a 50 nm thick SiN film and the sphere on the left is irradiated with laser light. The red and blue arrows in the light source indicate magnetic-field and electric-field directions, respectively. The light irradiating the sphere on the left from directly above propagates through the SiN film and reaches the sphere on the right. (b) Light power image of the cross section through the centre of the sphere in (a). (c) Optical power image of 20 nm under the SiN film in (a). (d) Line plot of optical power in the centre of c. Light propagating through the SiN film is captured by a sphere on the right but does not propagate thereafter. Scale bars, 2 µm in (a) and (c).

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2.5 Direct observation of light propagating on SiN films

Finally, we performed a direct surface light propagation of SiN film formed by a PS sphere. The propagation of light in the SiN film was sharply directed, and if other spheres existed in the direction of the travel, the light was captured in the spheres. This light then circled the spheres and scattered in various directions. Therefore, if the scattered light from the spheres can be observed, it is possible to reveal the existence of light traveling along the SiN film. Here, the sphere was irradiated by a laser beam from above, and a small digital microscope was placed below the sphere to observe the state of light in the SiN film [Fig. 5(a), Fig. S4 in Supplement 1].

 

Fig. 5. Direct observation of light propagating in SiN film by digital OM. (a) Schematic of digital OM set on the stage of confocal Raman microscope. A small digital microscope was set on the XY stage of the Raman microscope, and a SiN film with 3 µm spheres attached was set directly above via a cover glass. (b) Optical image of PS spheres on SiN film by Raman microscope. (c) Enlarged image of the sphere in the centre of (b) and observation position by laser irradiation. (d) Observed images by digital OM at each laser irradiated position. For the digital OM image, the image before irradiation was subtracted from the image after laser irradiation. The direction of propagation of the surface light in the SiN film changes depending on the irradiation position of the laser beam on the sphere surface. The surface propagating light in the SiN film travels from the laser irradiation position toward the centre of the sphere. No surface traveling light wave is observed in the case of laser irradiation to the centre of the sphere (No. 4). The middle figure shows surface propagating light at each laser position merged with the optical image of the Raman microscope. The yellow framed area in a CLMS OM image of centre panel indicates the light propagation range at each laser position. The surface propagating light in the SiN film disappears in the Si frame. (e) Raman shift at each laser irradiation position. Spectra with gradual oscillations are observed on the sphere surface. The centre part of the figure shows the Raman image (3400 cm⁻¹) of the gradual oscillation peak and the observed position of Raman shift. Scale bars, 2 µm in (c), (d) laser position and (e) centre, 100 µm in (d) Digital OM, 50 µm in the centre of (d) CLMS OM image.

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PS spheres with a diameter of 3 µm were dispersed on a SiN film of the thickness of 50 nm [Fig. 5(b)]. A laser beam was applied at various positions on the sphere shown at the centre of Figs. 5(b) and 5(c), and the Raman spectra at each position were simultaneously observed from below with a small optical microscope [Figs. 5(d) and 5(e), Fig. S4 in Supplement 1]. When the laser beam was applied slightly to the left of the top surface of the sphere, the incident light was projected through the centre of the sphere toward the lower right and became a surface propagating light on the SiN film [Fig. 5(d), No.1]. At this time, the spheres in the direction of a propagating light on the film surface were observed to shine by the optical microscope because they scattered the surface light [Fig. 5(d), No.1 OM]. On the other hand, when the laser was applied slightly to the right from the upper surface, the propagating light in the SiN film traveling to the lower left through the centre of the spheres [Fig. 5(d), No.2]. Similarly, when the laser beam was applied on the outer circumference of the sphere, the surface propagating light was formed in the direction from the irradiated position through the centre of the sphere [Fig. 5(d), No.3 and 5 − 7]. In contrast, when the laser beam was applied at the centre of the sphere, no surface propagating light were observed in the SiN film [Fig. 5(d), No. 4]. These results indicate that when the laser beam is applied on the outer circumference of the sphere, the light circles around the spheres and forms highly directional SiN surface waves. This direction can be easily formed at an arbitrary position by changing the irradiation position on the sphere surface.

Figure 5(e) shows the Raman spectra at each irradiation position of the sphere. When the laser was applied at the periphery of the sphere, an anomalously enhanced Raman spectrum was produced [Fig. 5(e), No. 1 − 3, 5 − 7]. On the other hand, near the centre of the sphere, the Raman spectrum was that of polystyrene [Fig. 5(e), No. 4]. This result suggests that the orbiting light from laser irradiation around the sphere is linked to the generation of propagating light in the SiN film. On the other hand, the intensity of the surface propagating light of the SiN film by digital OM does not change much for No. 3, which has the highest intensity of the anomalous Raman spectrum, and No. 7, which has the lowest intensity of the anomalous Raman spectrum. Therefore, the intensity of the anomalous Raman spectra has little relationship with the intensity of the surface propagating light along the SiN film.

3. Discussion

For PS spheres on a SiN film, extremely strong Raman spectra with slow oscillations were observed when the outer circumference of the sphere was irradiated with laser light [15]. This was thought to be caused by the interference of the light on the periphery when it circled around the surface of the sphere and further by its interaction with the SiN film [15]. Here, we analysed the mechanism of the generation of light circling the sphere surface using CLRM and FDTD optical simulation.

In the Raman shift of the sphere periphery, circumferential light with different spectral peaks was observed depending on the distance from the periphery [Fig. 1]. The spectrum occurring at the outermost side showed four peaks, with the peak intensity reaching at 250 nm to 500 nm from the sphere surface [Figs. 1(e) and 1(h)]. This is more than six times the range of evanescent light reached by normal WGM shorter than 100 nm [12,14], confirming that this circumferential light differs from that of normal WGM. Below 250 nm from the sphere surface, four more peaks were detected between the peaks mentioned above for a total of eight peaks [Fig. 1(e), green line]. Furthermore, an extremely strong spectrum with gradual periodic changes was observed near the sphere surface [Figs. 1(i) and 1(j)]. The circumferential light can be attributed to the difference in optical path length and interaction with the SiN thin film and the optical nonlinearity of the SiN film. SiN materials have nonlinear optical properties and are used as nonlinear optical devices and waveguides [26–29]. Here we used a 50 nm thick SiN film, much thinner than the wavelength of the light. The interaction of the nonlinear optical properties of such a SiN film with the circumferential light of the PS sphere is considered to form wavelength transitions and periodic spectral amplitudes.

The sharp Raman peaks away from the edge of the sphere are probably due to the resonance of the light circling the sphere [4 − 6]. The phenomenon in which the position of the peak changes with the laser irradiation position is probably due to the change in the resonance position of the spectrum caused by the difference in the radius of the light orbiting the sphere. On the other hand, the blunt strong peak is due to the interference effect of the light orbiting the sphere [15]. Therefore, it is expected that the different radii of circumference of the light for the sharp peak due to resonance and the blunt peak due to interference determine the position of their occurrence on the Raman spectrum. However, new optical experiments and simulation analysis are needed to demonstrate whether this is actually the mechanism. This will be the subject of our next studies.

Next, we analysed the optical properties of PS spheres on SiN films by a 3D optical model using FDTD simulations [Fig. 2]. Light incident on the side surface of the sphere from above penetrated into the sphere and reached the junction with the SiN film [Fig. 2(a)]. This light was expected to largely seep through to the lower surface of the SiN film and form wavelength transitions due to the nonlinear optical properties of the SiN film [Fig. 2(b)]. The light then split into light circling the sphere and surface propagating light on the SiN film [Figs. 2(b) and 3]. The light split from the sphere is stabilized to pass through the light centre in the SiN film and travels straight [Fig. 3]. Furthermore, the direction of the light traveling through the SiN film can be easily changed by modifying the position of the sphere irradiated by the laser [Fig. 5]. Therefore, it is clear that the sphere functions as a projector to the SiN film [Fig. 5]. In addition, the light propagating through the thin film surface can be easily manipulated from outside the thin film because the light travels with a large overhang at the top and bottom of the thin film [Figs. 3 and 4].

The light seeping as it travels around the surface of the sphere and SiN film would be affected by coatings with different surface refractive indices. We anticipate that varying the material and thickness will allow light to seep out over a wider area. Recently, nanostructures and/or microsphere finer than the wavelength of light have been shown to exert special optical effects; e. g. hyper-hemi-microsphere [30], truncated microspheres with a patterned Fresnel zone plate (FZP) [31] and fragmented FZP regions [32]. Therefore, nanoscale grooves or structures on the sphere surface and/or SiN film could produce special optical effects. These properties could be applied to new optical devices and nonlinear optical elements. Furthermore, microspheres can be used to control the direction of light, which may serve as a new device for constructing optical quantum qubits including optical integrated circuits.

In FDTD simulations, Maxwell's electromagnetic equations are calculated spatio-temporally. Therefore, our simulation results include the influence of the magnetic field caused by the evanescent wave. Moreover, in our optical model, the calculations are performed in 3D, using a 3D model of a sphere and a thin film. Therefore, the 3D model is more indicative of actual conditions than the 1D model [33]. However, in our optical model, the spatial mesh size is 10 nm, which may include some spatial errors. Therefore, it is preferable to subdivide the mesh size to 5 nm or less, but in that case, the memory capacity required for one calculation is about 640 GB, and the calculation time is about 15 days using a high-performance PC. More precise simulation analysis will be the subject of our future work.

In conclusion, we analysed the characteristics of light traveling through dielectric spheres on SiN films by Raman microscopy and 3D model simulations. The light traveling around the spheres reaches 650 nm from the surface of the sphere, and the Raman spectral characteristics indicate that the light consists of various modes. Furthermore, when the light circling the sphere reached the contact surface with the SiN film, it seeped out significantly below the SiN film, and part travelled through the SiN film surface. The direction of the surface propagating light changed depending on the irradiation position of the sphere circumference, indicating that the sphere functioned as a projector to the SiN film. These properties are expected to lead to the development of new devices such as highly sensitive sensors, antennas, quantum optical qubits and optical integrated circuits.

4. Methods

4.1 Sample preparation

PS microspheres with a diameter of 3 µm in an aqueous buffer (Micromer, micromod Partikeltechnologie GmbH, Germany) were diluted 3-fold with ultrapure water, vortexed for 10 s and sonicated for 1 min. A microsphere suspension (5 µL) was placed on a SiN film supported by a Si frame attached on an aluminium holder, and 10 s later, the liquid of the suspension was absorbed from the droplet by a piece of filter paper. On a glass substrate, 10 µL of the microsphere suspension was placed on a slide glass (Micro slide glass S1111, Matsunami Glass Ind. Ltd., Japan), and after absorbing the suspension liquid from the droplet with filter paper, the sample was dried for 5 min at a room temperature of 23 °C.

4.2 Raman microscopy

The PS microspheres on a SiN film or slide glass were observed under a confocal Raman microscope using a × 100 objective lens (Epiplan Neofluar ×100 with a 0.9 Numerical Aperture, Carl Zeiss, Oberkochen, Germany) and a 532-nm Nd-YAG laser (alpha300R, WITec, Ulm, Germany). The Abbe resolution is 295 nm. Spectra were acquired with a Peltier-cooled charge-coupled device detector (DV401-BV, Andor, UK) with 600 gratings/mm (UHTS 300VIS, WITec, Germany). The WITec suite (version 7.0, WITec, Germany) was used for data acquisition. For 2D Raman spectra, the laser intensity was 1.5 mW and the number of pixels in the XY axis was 180 × 180. The pixel step width was 50 nm, and the measurement time for each pixel was set to 0.05 s. Raman spectral data were calculated using MATLAB R2020a (Math Works Inc., Natick, MA, USA) and plotted using Origin 2021J (Origin-Lab Co., Northampton, MA, USA).

4.3 Calculation of Raman spectra and 2D Raman images of PS spheres

The 2D Raman spectral data of the spheres were converted to a Matlab data file using the WITec suite and transferred to a personal computer (Intel Core i7, 3.2 GHz, Windows 10). 2D Raman spectral images were calculated by MATLAB R2020a with the Image Processing Toolbox and Signal Processing Toolbox. The average Raman spectrum of the sphere centre was calculated from the pixels within a circle of radius 5 pixels from the sphere centre. For the spectrum of the sphere periphery, the Raman image of the PS peak of 998 cm⁻¹ was normalized after applying a 0.5σ Gaussian filter, and Raman spectra of the sphere periphery were averaged from pixels with normalized PS peak intensities between 0.2 and 0.5.

4.4 Optical simulation by FDTD

Optical simulations of PS spheres on a 50 nm thick SiN film were performed using the FDTD method with the commercial software using Ansys Lumerical, 2022 R1.3 (Ansys Inc., Pennsylvania, USA). The diameter of the polystyrene sphere was set to 3 µm and the refractive index, to n = 1.6. The SiN thin film was varied in thickness from 20 to 200 nm using Lumerical data of Philipp [34], with its refractive index being n = 2.025 at a wavelength of 537 nm.

In the simulation of Fig. 2, the size of the SiN film was set to X = 12.2 µm, Y = 14 µm, and a PS sphere with a diameter of 3 µm was placed in the centre. The FDTD simulation area was set to X = 11 µm, Y = 10.4 µm, and Z = 6 µm, and this area was calculated with a mesh span of 10 nm in all axes. The boundary area was set to a perfectly matched layer absorbing boundary (PML). The light for the simulation was focused at 1.65 µm from the centre of the 3 µm diameter PS sphere, with a focusing width of 150 nm, a focusing angle of 48.5°, and a Gaussian distribution intensity at a wavelength of 532 nm. The simulation end time was set to 1.0 psec.

In the simulation of Figs. 3 and 4, the size of the 50 nm thick SiN film was set to X = 22 µm, Y = 14 µm, and a PS sphere with a diameter of 3 µm was placed on the left side or both sides. The area of the FDTD simulation was X = 19 µm, Y = 10.4 µm, and Z = 6 µm, and this area was calculated with a mesh with 10 nm spacing on all axes. The light source and boundary conditions are the same as in Fig. 2. Simulation results were saved as Matlab format files and analysed using MATLAB R2020a with Image processing Toolbox and Signal processing Toolbox.

4.5 Digital microscopy

A small USB-connected digital microscope (TinyScope Cam, Convergence Ltd., Wuhan, China) was attached to the XY stage of a confocal laser Raman microscope (Fig. S4(a) in Supplement 1). Image data were imported to a PC via a USB connection. A 50 nm thick SiN film with dispersed PS spheres was attached to a cover glass (Micro cover glass, 0.13 − 0.17 mm thick, 18 × 18 mm, Matsunami Glass Ind. LTD, Japan) with double-sided tape, which was placed on the digital microscope. The cover glass under the SiN film was coated with black ink to reduce the light intensity to 1/1000. Laser irradiation of the sphere by the confocal laser microscope scanned a 10 × 10 µm area at 0.2 µm steps, the observation time was 50 ms and the laser power was 0.1 mW. Video recording by a digital microscope was started 3 s before laser irradiation. The recording conditions were 15 frame per second with a pixel size of 3264 × 2478. After the recording was completed, analysis processing was performed on each frame of the video. First, to remove background, the image of the video before laser irradiation was averaged and this was subtracted from each frame after laser irradiation. Next, a 30 × 30 µm area centred on the laser irradiation position was cut out to create a pseudo colour image (Figs. S4(b) − 4(g) in Supplement 1). This image analysis was calculated using MATLAB R2020a. This allowed us to confirm the irradiation position of the laser light and the propagation state of the light in the SiN film.