1. Introduction
Photonic nanojets (PNJs) are high-intensity light waves with a subwavelength beam waist generated at the shadow side of illuminated dielectric particles, which have promising applications in various fields such as micromaching, super-resolution imaging, etc. [
1,
2,
3,
4]. In 2015, Minin et al. theoretically discovered a new type of PNJ with curved optical fields [
5]. They called them photonic hooks (PHs). PHs were later experimentally proved at the terahertz frequency [
6], and then at the optical frequency [
7]. In 2021, Shang et al. [
8] and independently Minin et al. [
9] developed a contrast-enhanced microparticle-assisted microscopy, in which the incident light was converted into a PH and induced a near-field asymmetric illumination condition. Asymmetric illumination is a common technique in computational microscopic imaging, which can enhance the contrast of objects [
10,
11]. Minin’s group also reported a PH-based terahertz microscopy technique [
12].
When light waves pass through a dielectric particle whose shape or refractive index distribution is asymmetrical to the propagation direction of light, the difference in phase velocities and the interference of light waves lead to a curved high-intensity focus, i.e., a PH [
13,
14,
15]. Illuminating geometrically asymmetric dielectric particles, such as dielectric trapezoids [
5,
6,
13] and glass cubes [
16] embedded dielectric cylinders, with plane waves is a common way to generate PHs. Particles with an anisotropic refractive index distribution, such as Janus particles [
14,
17], are also suitable for the generation of PHs. In addition, PHs can be generated by partial or nonuniform illumination of symmetric and homogeneous particles [
7,
18], or by using dielectric particles partially covered with opaque films [
19]. For example, Tang et al. used 1–35 μm diameter patchy microcylinders to generate PHs. Half of the cylinder surface is covered with silver films, and PHs with a bending angle of ∼28.4
can be generated under plane wave illumination [
19]. In 2021, Liu et al. reported the generation of PHs under point-source illumination [
20]. They found that instead of using plane waves, the point-source illumination can also be used to generate PHs from Janus microcylinders, and that the properties of PHs can be effectively tuned by adjusting the parameters of the light source.
In this work, we show that the generation of PHs can be achieved using patchy particles under point-source illumination. Numerical simulations based on the finite-difference time-domain (FDTD) method were performed to investigate the characteristics of the PHs. By changing the background refractive index (n), particle diameters, the position and coverage ratio of the Ag patches, the characteristics of the PHs, such as the curvature, maximum intensity, length, etc. can be effectively tuned. We also found that the PHs from small and large microcylinders can have opposite bending directions due to the combined effect of the asymmetric energy flow inside the microcylinder and the near-field optical interaction between the light waves and the Ag patches.
3. Results and Discussion
First, we compared the optical fields generated by the 5 μm and 1 μm diameter patchy cylinder. The background refractive index was 1.33. As shown in
Figure 2a, a bended optical field can be generated by the 5 μm diameter particle. The obtained PH had a bending angle of
= 13.4
, a hook height increment of H = 0.15 μm and a subtense of L = 2.6 μm. It is interesting to see that the structure of the intensity distribution of the light field bends toward the opposite direction when the particle size decreased to 1 μm. As shown in
Figure 2b, the curved optical field had a bending angle of
= −17.1
, a hook height increment of H = 0.02 μm and a subtense of L = 0.24 μm. However, although the spatial intensity distribution generated by the 1 μm diameter patchy microcylinder had a hook-like structure, it was different from the classical PHs described in the previous work [
13,
15]. The classical PHs have a longer propagation distance of several wavelength [
3]. However, the length of the high-intensity energy flux in the 1 μm diameter particle was smaller than the incident wavelength and localized to the near-field region. The 3D trajectory of the I
points shown in
Figure 2c,d also confirmed the change of the bending directions of PHs. The I
points were obtained by slicing the optical fields along the Y-axis and then plotted the coordinates of the maximum intensity point in each slice.
The optical fields generated by patchy microcylinders under Gaussian beam illumination are also simulated in this work (
Figure S1, Supplementary Material). As for the 5 μm diameter patchy microcylinder, the Gaussian beam-generated PH has a length of 1.6 μm, which is shorter than the PHs generated under point-source illumination (L = 2.6 μm). A larger bending angle of
= 22.5
and a higher intensity of I
= 3.2 were observed in the corresponding optical field. The optical fields of 1 μm diameter patchy microcylinder under Gaussian beam or point-source illumination showed similar characteristics in terms of the bending angle
and subtense L, but the Gaussian beam illumination led to a higher intensity (I
= 2.5).
As reported in previous work [
13], the formation mechanism of PHs can be analyzed using the time-averaged Poynting vector. In this work, the Poynting vector of the optical fields of the pristine and patchy cylinders under point-source illumination is simulated with the FDTD method, and the corresponding field-lines of the Poynting vector distribution are shown as black lines in
Figure 2a,b. As shown in
Figure 2a, because of the presence of metal films, part of the incident light is reflected backward to the free space by the Ag films, which breaks the symmetry of illumination and leads to an unbalanced flow of energy inside the microcylinders. This asymmetric flow of energy is then focused into a curved light beam after leaving the patchy cylinder. As shown in
Figure 2b, compared with the 5 μm diameter microcylinder, the starting point of the curved light field formed by the 1 μm diameter microcylinder was much closer to the Ag films. To investigate the reason for the change of the bending direction, we recorded the evolution of the E
2 field component for the duration of the simulation (
Videos S1 and S2). As shown in
Videos S1, when light propagates through a 1 μm diameter patchy microcylinder, it is reflected and diffracted by the Ag film at the right edge of the microcylinder [
3]. As a result, the direction of its propagation is significantly changed. On the contrary, the starting point of the PH formed by the 5 μm diameter patchy microcylinders was far away from the Ag patch, so the curved shape of the generated light fields was mainly attributed to the phase velocity difference of the light waves (
Video S2). We think that the different bending directions between the light fields obtained from the 1 μm and 5 μm diameter patchy microcylinders could be attributed to the near-field optical interaction between the light waves and the Ag patches.
Next, we investigated the influence of the diameter of microcylinders on the bending angle of light fields at three different refractive index contrasts (RIC) between the microcylinder and the background. The diameter of the microcylinder changed between 1 μm and 5 μm, and the refractive index of the microcylinder was 1.90. The background medium was set to air (n = 1.00), water (n = 1.33) and oil (n = 1.52) to obtain a refractive index contrast of 1.90, 1.43 and 1.25, respectively. As shown in the blue line in
Figure 3, the optical field generated by the 1 μm diameter patchy microcylinder had an intensity distribution with a bending angle
of around −50
. The bending angle gradually increased to 32
when the particle diameter increased from 1 μm to 5 μm. The curvature of the light fields formed in water (red line) and oil (black line) showed a similar trend; that is, the
increased with the particle diameter and the bending direction turned from downward to upward when the particle diameter became large. We found that the PHs formed in air had the highest curvature compared with its counterparts in water and oil. This could be attributed to the higher refractive index contrast and consequently a larger phase velocity difference between the microcylinder and the background medium. When increasing the particle diameter from 1 μm to 5 μm, the bending angle
of the PHs in water (red line) and oil (black line) increased from −17
to 13
and from −9
to 9
, respectively.
It can be seen from
Figure 3 that the bending angles were 0
at some diameters within the range of 1 μm and 5 μm, which means that the optical fields in this case became straight and were similar to a conventional photonic jet. We call this certain particle diameter the critical diameter (CD). In this study, patchy microcylinders with a diameter smaller than the CD generated optical fields with a negative bending angle, and patchy microcylinders with a diameter larger than the CD led to the formation of PHs with a positive bending angle.
To study the effect of refractive index contrast on the CD of the microcylinder, we increased the refractive index contrast from RIC = 1.19 to RIC = 1.60 and found the CD at each refractive index. Different illumination wavelengths of
= 400, 550 and 700 nm were used to show the influence of the wavelength on the CD of the particle. The microcylinder here had a constant refractive index of n = 1.9. As can be seen from the red line in
Figure 4, when the illumination wavelength was
= 700 nm, the CD increased from 1.5 μm to 4.8 μm as the refractive index contrast increased from 1.19 to 1.90. We also found that a shorter illumination wavelength led to a smaller CD, as shown in the black and blue lines in
Figure 4. The CD increased from 1.2 μm (RIC = 1.19) to 4.0 μm (RIC = 1.90) when the illumination wavelength was
= 550 nm (red line in
Figure 3). For the light fields formed with 400 nm wavelength, the CD increased from 0.9 μm (RIC = 1.19) to 3.2 μm (RIC = 1.90) (blue line in
Figure 4).
The influence of the distance between point sources and patchy microcylinders on the characteristics of the light fields was also investigated in this work. Microcylinders with a diameter of 1 μm were created for simulation. The upper part of the cylinders was covered with 100 nm thick Ag films. Point sources were positioned at different distances between 0.5 and 20 μm. We defined the distance between the light source and the left edge of the particle as S. As shown in
Figure 5a–c, we compared the optical fields formed by the microcylinders when the distance was S = 0.5 μm (
Figure 5a), 7 μm (
Figure 5b) and 20 μm (
Figure 5c). We found that the I
position changed with the distance, a closer point-source distance made a further I
position. The light fields formed at S = 0.5 μm also had the smallest curvature. As shown in
Figure 5d,e, when the distance increased from 0–20 μm, the normalized I
decreased from ∼ 0.96 to ∼ 0.1 and the curvature became larger as the bending angle
changed from −2
to −25
. As the distance between the point source and the microcylinder increased, the amount of light waves that can be captured and focused by the microcylinder decreased, resulting in a decrease in the intensity of the light field.
Then, we studied the influence of the position of Ag films on the formation of curved light fields. As shown in
Figure 6, a patchy microcylinder with a diameter of 1 μm and a refractive index of 1.9 was used for this study. The cylinder was assumed to be fully immersed in water, so the background refractive index was set to be 1.33. The top surface of the cylinder was partially covered with 100 nm thick Ag films. The opening angle
of the Ag film was constant at 90
. To generate a curved light field at different positions of Ag films, the patchy microcylinder was rotated clockwise around its center, and the corresponding optical field of the patchy microcylinder under point-source illumination was simulated with the FDTD method. The distance between the point source and the left edge of the cylinder was 5 μm. The rotation angle of the microcylinder was defined as the angle
between the horizontal line and the line connecting the center of the microcylinder to the left edge of the Ag film. As shown in
Figure 6a–f, we found that the position of Ag films can significantly affect the properties of the light field. The light field generated at
= 0
bend upward with a bending angle of
= 38
(
Figure 6a). After the microcylinder was rotated 60
clockwise, the spatial distribution of the generated optical field became straight, such as a classical PNJ, and had no curved structure (
Figure 6d). When the patchy microcylinder was further rotated clockwise, the generated optical field bends downward toward the opposite direction. As shown in
Figure 6f, the bending angle reached
= −35.7
when the Ag films had a rotation angle of
= 90
.
The main characteristics of the light fields as a function of rotation angle
are shown in
Figure 6g–i. As shown in
Figure 6g, the bending angle
of the light fields decreases as the rotation angle increases.
Figure 6h shows the I
(black line) and the corresponding full width at half maximum (FWHM) (red line) of the light fields obtained at different rotation angles. We found that the focusing ability of the microcylinder became stronger as the rotation angle of the Ag film increased, which could be attributed to the fact that more incident light can be collected by the microcylinder when the Ag film has a larger rotation angle
. The black line in
Figure 6i shows that the subtense L of the light fields increased from 0.42 μm to 0.69 μm when increasing
from 0
to 45
, then L decreased to 0.13 μm at
= 90
. The height increment (H) of the light fields generated at different rotation angles
is shown as a red line in
Figure 6i. We found that the maximum height increment was obtained at
= 30
with a value of H = 0.09 μm. Then, the light beams focused by the patchy cylinder had a structure similar to a PJ when
was 60
with a value of H = 0 μm. When
was between 60
and 90
, the height increment increased to 0.02 μm as the rotation angle
increased to 90
. The data of the corresponding optical fields is shown in the
Table S1 (supplementary material).