Generation of Photonic Nanojet Using Gold Film Dielectric Microdisk Structure

Due to their narrow beam waist size, high intensity, and long propagation distance, photonic nanojets (PNJs) can be used in various fields such as nanoparticle sensing, optical subwavelength detection, and optical data storage. In this paper, we report a strategy to realize an SPP-PNJ by exciting a surface plasmon polariton (SPP) on a gold-film dielectric microdisk. In detail, an SPP is excited by the grating–coupling method, then it irradiates the dielectric microdisk to form an SPP-PNJ. The characteristics of the SPP-PNJ, including maximum intensity, full width at half maximum (FWHM), and propagation distance, are studied by using finite difference time domain (FDTD) numerical solutions. The results demonstrate that the proposed structure can produce a high-quality SPP-PNJ, the maximum quality factor of which is 62.20, and the propagation distance of the SPP-PNJ is 3.08 λ. Furthermore, the properties of the SPP-PNJ can be modified flexibly by changing the thickness and refractive index of the dielectric microdisk.

The key properties of the PNJ, including maximum intensity, the propagating distance of the PNJ, and full width at half maximum (FWHM), depend on the geometric parameters (size, shape, etc.), refractive indices of the micro-particles, and the output wavelength of the incident light. Traditionally, the main method of generating PNJs is irradiating dielectric microparticles such as microspheres, microcylinders, and micro-cuboids. However, microspheres, microcylinders, and micro-cuboids have shortcomings such as inconvenience in experimental operations. To overcome these drawbacks, McCloskey et al. proposed a new scheme to generate SPP-PNJs using silicon nitride dielectric microdisks in 2012 [35]. Due to the limited field portion of SPPs in metals, they are often difficult to transmit over long distances, which seriously limits their application value, especially in planar photonic devices. For example, D. Ju et al. first reported nanojet effects excited by a surface plasmon polariton on the shadow-side surfaces of dielectric microdisks positioned on gold films in 2013. Additionally, in 2020, I. Minin first reported the experimental demonstration of plasmonic nanojet production by a micro-cuboid deposited on a gold film.
In this paper, we propose a structure to realize an SPP-PNJ by exciting a surface plasmon polariton (SPP) on a gold-film dielectric microdisk. In detail, by using the finite difference time domain (FDTD) method, we attempted to adjust the characteristics of an SPP-PNJ by adjusting the refractive index and thickness of the microdisk. The influences of the refractive index and thickness of the dielectric microdisk on the maximum intensity, full width at half maximum, propagation distance, and quality factor of the SPP-PNJ were analyzed in detail. The results show that the propagation distance of the SPP-PNJ is 3.08 λ. The characteristics of the SPP-PNJ are flexibly tunable by changing the thickness and refractive index of the dielectric microdisk. In addition, an SPP-PNJ with a quality factor of 62.20 can be achieved. Compared with the dielectric microsphere, microcylinder, and micro-cuboid, the dielectric microdisk structure can excite the long propagation distance of the SPP-PNJ, and the characteristics of the SPP-PNJ can be flexibly tunable. Therefore, the proposed dielectric microdisk can provide a new idea for further expanding the generation path of SPP-PNJs and conveniently tuning the characteristics of SPP-PNJs.

Structure of Gold-Film Dielectric Microdisk
Here, we consider that the FDTD method based on vector electromagnetic wave theory can accurately illustrate the propagation of a light wave in dielectric media [36]. Therefore, the structure of the gold-film dielectric microdisk was simulated by using the FDTD calculation. In order to ensure the accuracy of the calculation, a non-uniform mesh with a minimum step of 5 nm was applied. Perfectly matched layers (PML) were arranged around the boundaries. Figure 1 shows a schematic diagram of an SPP-PNJ formed by a gold-film dielectric microdisk structure. It consists of a dielectric microdisk, gold film, grating, and a dielectric substrate. An SPP was excited by the grating-coupling method, which then irradiated the dielectric microdisk with a radius R, thickness h, and refractive index n along the negative x direction. Here, the dielectric microdisk was deposited onto a gold film. The lower layer of the gold film was a dielectric substrate with a semi-infinite thickness. The whole structure was symmetric with respect to the y = 0 plane, and the upper surface of the gold film was located in the z = 0 plane. Assuming that the whole structure was enclosed by air, we set the wavelength of the incident wave as 800 nm, and the refractive indices of the background and dielectric substrate were set to 1.0 and 1.5, respectively. The material of the gold film and grating was set as Au (Gold) according to Johnson and Christy. According to the refractive indices of the grating and background, we set the period of the grating as 583.94 nm, which could couple the incident wave to the gold film to excite the 800 nm SPP. The radius of the dielectric microdisk was set to 3 µm. to the refractive indices of the grating and background, we set the period of the grating as 583.94 nm, which could couple the incident wave to the gold film to excite the 800 nm SPP. The radius of the dielectric microdisk was set to 3 μm. To evaluate the light beam quality of the SPP−PNJ more comprehensively, the quality factor Q is used to measure the SPP−PNJ, which is written as [37]: where Imax is the maximum intensity of the SPP−PNJ, which represents the maximum optical intensity reached along the light propagation direction; L is the propagation distance of the SPP−PNJ, which represents the distance of the x−axis corresponding to the optical intensity between the maximum intensity of the SPP−PNJ and 1/e of the maximum intensity; F is the FWHM of the SPP−PNJ, which represents the width of the y−axis corresponding to the SPP−PNJ's attenuation from the maximum intensity to the half maximum intensity. According to Equation (1), Q depends on Imax, L, and F. Obviously, a higher Q can be obtained by a higher Imax or L and a lower F. An SPP−PNJ with a higher Q is expected in many application fields. Here, the refractive index of the dielectric microdisk is 1.5. The thickness and radius of the dielectric microdisk are 1.0 and 3.0 μm, respectively. The illuminating wave is a z−axis polarized plane wave with a wavelength λ = 800 nm, propagating along the y−axis with an initial amplitude of 1. The plane wave is irradiated onto the grating for coupling, resulting in plasma resonance, and the generated SPP propagates along the x−axis, producing an SPP−PNJ with different properties. Figure 2a depicts an SPP−PNJ in the x−y longitudinal cut plane. Figure 2b shows a three−dimensional surface plot of the SPP−PNJ. From this figure, we see that the gold−film dielectric microdisk structure can generate the SPP−PNJ. The generated SPP−PNJ is a narrow−intensity electromagnetic beam emerging from the shadow−side surface of the gold−film dielectric microdisk structure propagating along the x−axis. To evaluate the light beam quality of the SPP-PNJ more comprehensively, the quality factor Q is used to measure the SPP-PNJ, which is written as [37]:

Simulations
where I max is the maximum intensity of the SPP-PNJ, which represents the maximum optical intensity reached along the light propagation direction; L is the propagation distance of the SPP-PNJ, which represents the distance of the x-axis corresponding to the optical intensity between the maximum intensity of the SPP-PNJ and 1/e of the maximum intensity; F is the FWHM of the SPP-PNJ, which represents the width of the y-axis corresponding to the SPP-PNJ's attenuation from the maximum intensity to the half maximum intensity. According to Equation (1), Q depends on I max , L, and F. Obviously, a higher Q can be obtained by a higher I max or L and a lower F. An SPP-PNJ with a higher Q is expected in many application fields. Figure 2 visualizes the FDTD-computed optical intensity formed by the gold-film dielectric microdisk. The optical intensity of the SPP-PNJ is shown by the color bar scale. Here, the refractive index of the dielectric microdisk is 1.5. The thickness and radius of the dielectric microdisk are 1.0 and 3.0 µm, respectively. The illuminating wave is a z-axis polarized plane wave with a wavelength λ = 800 nm, propagating along the y-axis with an initial amplitude of 1. The plane wave is irradiated onto the grating for coupling, resulting in plasma resonance, and the generated SPP propagates along the x-axis, producing an SPP-PNJ with different properties. Figure 2a depicts an SPP-PNJ in the x-y longitudinal cut plane. Figure 2b shows a three-dimensional surface plot of the SPP-PNJ. From this figure, we see that the gold-film dielectric microdisk structure can generate the SPP-PNJ. The generated SPP-PNJ is a narrow-intensity electromagnetic beam emerging from the shadow-side surface of the gold-film dielectric microdisk structure propagating along the x-axis.

Simulations
In order to analyze the influence of the gold-film dielectric microdisk on the characteristics of SPP-PNJs, we study the influence of the dielectric microdisk's refractive index and thickness. In order to analyze the influence of the gold−film dielectric microdisk on the characteristics of SPP−PNJs, we study the influence of the dielectric microdisk's refractive index and thickness. Figure 3 shows the optical intensity field and the maximum intensity Imax evolution formed by different dielectric microdisk refractive indices n = 1.3, n = 1.4, n = 1.5, n = 1.6, and n = 1.7, respectively. The optical intensity of the SPP−PNJ is shown by the color bar scale. Here, the thickness and radius of the dielectric microdisk are h = 1.0 μm and R = 3.0 μm, respectively. The effective wavelength of the SPP is λ = 800.0 nm. It can be seen in Figure 3a−e that the peak of the optical intensity field shifts toward the shadow−side surface of the dielectric microdisk by decreasing the refractive index of the dielectric microdisk n. In addition, we can see in Figure 3f that the maximum intensity Imax increases when the value of n increases from 1.3 to 1.6. However, when the value of n increases from 1.6 to 1.7, Imax decreases with n. Hence, when n reaches an appropriate value, the optical intensity field peak emerges from the shadow−side surface of the dielectric microdisk and an SPP−PNJ is formed. The maximum intensity Imax of an SPP−PNJ formed by a gold−film dielectric microdisk can be tuned by designing the refractive index n of the dielectric microdisk. Figure 4 shows the optical intensity distributions of the SPP−PNJ along the x−axis with different dielectric microdisk refractive indices n; the corresponding SPP−PNJ propagation distances L are shown in Figure 5. We can see in Figure 4 that the position of the maximum intensity along the y−axis is similar, at approximately −2 μm. In Figures 4 and 5, it can be seen that L decreases when the value of n increases from 1.3 to 1.6, while L increases when the value of n increases from 1.6 to 1.7. The value of the maximum propagation distance Lmax,n is 2.46 μm (3.08 λ) when the value of n is 1.3. The value of the minimum propagation distance Lmin,n is 0.32 μm (0.40 λ) when the value of n is 1.6. Therefore, we can design the value of n to obtain the appropriate propagation L of the SPP−PNJ according to different applications.  Figure 3 shows the optical intensity field and the maximum intensity I max evolution formed by different dielectric microdisk refractive indices n = 1.3, n = 1.4, n = 1.5, n = 1.6, and n = 1.7, respectively. The optical intensity of the SPP-PNJ is shown by the color bar scale. Here, the thickness and radius of the dielectric microdisk are h = 1.0 µm and R = 3.0 µm, respectively. The effective wavelength of the SPP is λ = 800.0 nm. It can be seen in Figure 3a-e that the peak of the optical intensity field shifts toward the shadow-side surface of the dielectric microdisk by decreasing the refractive index of the dielectric microdisk n. In addition, we can see in Figure 3f that the maximum intensity I max increases when the value of n increases from 1.3 to 1.6. However, when the value of n increases from 1.6 to 1.7, I max decreases with n. Hence, when n reaches an appropriate value, the optical intensity field peak emerges from the shadow-side surface of the dielectric microdisk and an SPP-PNJ is formed. The maximum intensity I max of an SPP-PNJ formed by a gold-film dielectric microdisk can be tuned by designing the refractive index n of the dielectric microdisk. Figure 4 shows the optical intensity distributions of the SPP-PNJ along the x-axis with different dielectric microdisk refractive indices n; the corresponding SPP-PNJ propagation distances L are shown in Figure 5. We can see in Figure 4 that the position of the maximum intensity along the y-axis is similar, at approximately −2 µm. In Figures 4 and 5, it can be seen that L decreases when the value of n increases from 1.3 to 1.6, while L increases when the value of n increases from 1.6 to 1.7. The value of the maximum propagation distance L max,n is 2.46 µm (3.08 λ) when the value of n is 1.3. The value of the minimum propagation distance L min,n is 0.32 µm (0.40 λ) when the value of n is 1.6. Therefore, we can design the value of n to obtain the appropriate propagation L of the SPP-PNJ according to different applications. Figure 6 shows the FWHM F of the SPP-PNJ with different refractive indices n of the dielectric microdisk. We can clearly see that F decreases when the value of n increases from 1.3 to 1.6, and F increases when the value of n increases from 1.6 to 1.7. Notably, when n is in the range of 1.5 to 1.7, F is less than 0.4 µm (0.5 λ), which plays an important role in super-resolution imaging. The value of F is 0.49 (0.61 λ) and 0.41 (0.51 λ) when the value of n is 1.3 and 1.4, respectively, which can be used in nanoparticle sensing and optical sub-wavelength detection. Notably, we can see that the intensity dip at y = 0 when n is in the range of 1.5 to 1.7. The reason for the significant decrease in intensity at y = 0 is that the strongest point of the focused beam formed by the interference between the field scattered by the medium's exit surface and the incident field is not at the center, but rather forms the two strongest regions symmetrically about the central axis.       Figure 6 shows the FWHM F of the SPP−PNJ with different refractive indices n of the dielectric microdisk. We can clearly see that F decreases when the value of n increases from 1.3 to 1.6, and F increases when the value of n increases from 1.6 to 1.7. Notably, when n is in the range of 1.5 to 1.7, F is less than 0.4 μm (0.5 λ), which plays an important role in super-resolution imaging. The value of F is 0.49 (0.61 λ) and 0.41 (0.51 λ) when the value of n is 1.3 and 1.4, respectively, which can be used in nanoparticle sensing and optical sub-wavelength detection. Notably, we can see that the intensity dip at y = 0 when n is in the range of 1.5 to 1.7. The reason for the significant decrease in intensity at y = 0 is that the strongest point of the focused beam formed by the interference between the field scattered by the medium's exit surface and the incident field is not at the center, but rather forms the two strongest regions symmetrically about the central axis. From the above analyses, we find that the gold−film dielectric microdisk can form an SPP−PNJ with a smaller diffraction limit, which is similar to the SPP−PNJs generated by irradiating dielectric microspheres or microcylinders with plane waves. The quality factor Q of an SPP−PNJ formed by a gold−film microdisk can be calculated according to Equation (1). The characteristic parameters of SPP−PNJ, including Imax, F, L, and Q with different dielectric microdisk refractive indices, are listed in Table 1. It clearly shows that the largest quality factor Qmax is 49.47 and the smallest quality factor Qmin is 20.83, which can be achieved by setting the refractive index of the dielectric microdisk to 1.5 and 1.6, respectively. Therefore, the appropriate SPP−PNJ for each application can be obtained by designing the refractive index of the dielectric microdisk.   Figure 6 shows the FWHM F of the SPP−PNJ with different refractive indices n of the dielectric microdisk. We can clearly see that F decreases when the value of n increases from 1.3 to 1.6, and F increases when the value of n increases from 1.6 to 1.7. Notably, when n is in the range of 1.5 to 1.7, F is less than 0.4 μm (0.5 λ), which plays an important role in super-resolution imaging. The value of F is 0.49 (0.61 λ) and 0.41 (0.51 λ) when the value of n is 1.3 and 1.4, respectively, which can be used in nanoparticle sensing and optical sub-wavelength detection. Notably, we can see that the intensity dip at y = 0 when n is in the range of 1.5 to 1.7. The reason for the significant decrease in intensity at y = 0 is that the strongest point of the focused beam formed by the interference between the field scattered by the medium's exit surface and the incident field is not at the center, but rather forms the two strongest regions symmetrically about the central axis. From the above analyses, we find that the gold−film dielectric microdisk can form an SPP−PNJ with a smaller diffraction limit, which is similar to the SPP−PNJs generated by irradiating dielectric microspheres or microcylinders with plane waves. The quality factor Q of an SPP−PNJ formed by a gold−film microdisk can be calculated according to Equation (1). The characteristic parameters of SPP−PNJ, including Imax, F, L, and Q with different dielectric microdisk refractive indices, are listed in Table 1. It clearly shows that the largest quality factor Qmax is 49.47 and the smallest quality factor Qmin is 20.83, which can be achieved by setting the refractive index of the dielectric microdisk to 1.5 and 1.6, respectively. Therefore, the appropriate SPP−PNJ for each application can be obtained by designing the refractive index of the dielectric microdisk. From the above analyses, we find that the gold-film dielectric microdisk can form an SPP-PNJ with a smaller diffraction limit, which is similar to the SPP-PNJs generated by irradiating dielectric microspheres or microcylinders with plane waves. The quality factor Q of an SPP-PNJ formed by a gold-film microdisk can be calculated according to Equation (1). The characteristic parameters of SPP-PNJ, including I max , F, L, and Q with different dielectric microdisk refractive indices, are listed in Table 1. It clearly shows that the largest quality factor Q max is 49.47 and the smallest quality factor Q min is 20.83, which can be achieved by setting the refractive index of the dielectric microdisk to 1.5 and 1.6, respectively. Therefore, the appropriate SPP-PNJ for each application can be obtained by designing the refractive index of the dielectric microdisk.

The Characteristic Parameters of SPP-PNJs with Different Thicknesses of the Dielectric Microdisk
According to the above analysis, the maximum value of Q of an SPP-PNJ can be realized when n = 1.5. Therefore, the thickness of the dielectric microdisk is analyzed with n set to 1.5.
The optical intensity field and the maximum intensity I max of SPP-PNJs for different thicknesses of dielectric microdisks are shown in Figure 7. We can see the SPP-PNJs formed by dielectric microdisks with different thicknesses with a radius R = 3.0 µm and refractive index n = 1.5. The color bar scale demonstrates the intensity of the SPP-PNJ. Figure 7 shows that the maximum intensity I max increases when the value of h increases from 0.6 to 0.8 µm.
On the contrary, when the value of n increases from 0.8 to 1.0, I max decreases with h. We can find that the peak of the optical intensity field shifts toward the shadow-side surface of the dielectric microdisk by increasing the refractive index of the dielectric microdisk h from 0.6 to 0.8 µm.

The Characteristic Parameters of SPP−PNJs with Different Thicknesses of the Dielectric Microdisk
According to the above analysis, the maximum value of Q of an SPP−PNJ can be realized when n = 1.5. Therefore, the thickness of the dielectric microdisk is analyzed with n set to 1.5.
The optical intensity field and the maximum intensity Imax of SPP-PNJs for different thicknesses of dielectric microdisks are shown in Figure 7. We can see the SPP−PNJs formed by dielectric microdisks with different thicknesses with a radius R = 3.0 μm and refractive index n = 1.5. The color bar scale demonstrates the intensity of the SPP−PNJ. Figure 7 shows that the maximum intensity Imax increases when the value of h increases from 0.6 to 0.8 μm. On the contrary, when the value of n increases from 0.8 to 1.0, Imax decreases with h. We can find that the peak of the optical intensity field shifts toward the shadow−side surface of the dielectric microdisk by increasing the refractive index of the dielectric microdisk h from 0.6 to 0.8 μm. To analyze the characteristic parameters of the SPP-PNJ, Figure 8 shows the optical intensity distributions of the SPP-PNJ along the y-axis with different dielectric microdisk refractive indices n; the corresponding SPP-PNJ propagation distances L are shown in Figure 9. We can clearly see that the position of maximum intensity along the y-axis is −2.3 µm when h is 0.8 µm. However, the attenuation of the SPP-PNJ is faster. Furthermore, when the h is 0.9 µm, the position of maximum intensity along the y-axis is −2 µm and the attenuation of the SPP-PNJ is slower. Therefore, in Figures 8 and 9, we can see that L decreases when the value of h increases from 0.6 to 0.8 µm. When h is 0.9 µm, L increases suddenly and then decreases continuously. The value of the maximum propagation distance L max,h is 1.72 µm (2.15 λ) when the value of h is 0.9 µm. The value of the minimum propagation distance L min,h is 0.76 (0.95 λ) when the value of h is 0.7 µm. Obviously, the value of the maximum propagation distance L max,h (2.15 λ) is shorter than the value of L max,n (3.08 λ). Therefore, we can design n and h to obtain the appropriate propagation L of the SPP-PNJ according to different applications.
To analyze the characteristic parameters of the SPP−PNJ, Figure 8 shows the optical intensity distributions of the SPP−PNJ along the y−axis with different dielectric microdisk refractive indices n; the corresponding SPP−PNJ propagation distances L are shown in Figure 9. We can clearly see that the position of maximum intensity along the y−axis is −2.3 μm when h is 0.8 μm. However, the attenuation of the SPP−PNJ is faster. Furthermore, when the h is 0.9 μm, the position of maximum intensity along the y−axis is −2 μm and the attenuation of the SPP−PNJ is slower. Therefore, in Figures 8 and 9, we can see that L decreases when the value of h increases from 0.6 to 0.8 μm. When h is 0.9 μm, L increases suddenly and then decreases continuously. The value of the maximum propagation distance Lmax,h is 1.72 μm (2.15 λ) when the value of h is 0.9 μm. The value of the minimum propagation distance Lmin,h is 0.76 (0.95 λ) when the value of h is 0.7 μm. Obviously, the value of the maximum propagation distance Lmax,h (2.15 λ) is shorter than the value of Lmax,n (3.08 λ). Therefore, we can design n and h to obtain the appropriate propagation L of the SPP−PNJ according to different applications.   Figure 10 shows the full width at half maximum F of the SPP−PNJ with different thicknesses of h of the dielectric microdisk. It can be seen that when h increases from 0.6 μm to 0.8 μm, F increases with h. However, while h increases from 0.8 to 1.0 μm, F decreases with h. It is noteworthy that F is always less than 0.4 μm (0.5 λ) when h is in the refractive indices n; the corresponding SPP−PNJ propagation distances L are shown in Figure 9. We can clearly see that the position of maximum intensity along the y−axis is −2.3 μm when h is 0.8 μm. However, the attenuation of the SPP−PNJ is faster. Furthermore when the h is 0.9 μm, the position of maximum intensity along the y−axis is −2 μm and the attenuation of the SPP−PNJ is slower. Therefore, in Figures 8 and 9, we can see that L decreases when the value of h increases from 0.6 to 0.8 μm. When h is 0.9 μm, L increases suddenly and then decreases continuously. The value of the maximum propagation distance Lmax,h is 1.72 μm (2.15 λ) when the value of h is 0.9 μm. The value of the minimum propagation distance Lmin,h is 0.76 (0.95 λ) when the value of h is 0.7 μm. Obviously, the value of the maximum propagation distance Lmax,h (2.15 λ) is shorter than the value of Lmax,n (3.08 λ). Therefore, we can design n and h to obtain the appropriate propagation L of the SPP−PNJ according to different applications.   Figure 10 shows the full width at half maximum F of the SPP−PNJ with different thicknesses of h of the dielectric microdisk. It can be seen that when h increases from 0.6 μm to 0.8 μm, F increases with h. However, while h increases from 0.8 to 1.0 μm, F decreases with h. It is noteworthy that F is always less than 0.4 μm (0.5 λ) when h is in the  Figure 10 shows the full width at half maximum F of the SPP-PNJ with different thicknesses of h of the dielectric microdisk. It can be seen that when h increases from 0.6 µm to 0.8 µm, F increases with h. However, while h increases from 0.8 to 1.0 µm, F decreases with h. It is noteworthy that F is always less than 0.4 µm (0.5 λ) when h is in the range of 0.6 to 1.0 µm. This means that the incident wave can form an SPP-PNJ with a smaller diffraction limit by designing the appropriate value of h. range of 0.6 to 1.0 μm. This means that the incident wave can form an SPP−PNJ with a smaller diffraction limit by designing the appropriate value of h.  Table 2 shows the characteristic parameters of the SPP−PNJ with different thicknesses h of the dielectric microdisk. It can be seen in Table 2 that when h = 0.8 μm and the Imax of the SPP−PNJ is 14.33 times stronger than that of the exciting light, F = 0.37 μm (0.46 λ) and L = 0.98 μm (1.23 λ). The largest obtained value of Q for the SPP−PNJ was 62.20. The smallest value of Q was 23.92, which can be realized when h = 0.7 μm. This will benefit the application of SPP−PNJs in super−resolution optical imaging. This indicates that with the increase of h, Imax and F increase first and then decrease. However, the relationship between h and the propagation distance L of the SPP−PNJ is more complicated. According to Tables 1 and 2, the relationship between Q, n, and h is shown in Figure  11. It can be seen in Figure 10 that Q is greatly different with different h and n values, respectively. However, it is found that the maximum value of Q can be achieved by setting the thickness of the dielectric microdisk h. For example, when the value of h is 0.9 μm, Q reaches a maximum value of 62.20, and when the value of n is 1.5, Q reaches a maximum value of 49.47. However, compared with the influence of different thicknesses of h, the longest propagation distance L can be realized by different refractive indices. When the value of n is 1.3, L reaches a maximum value of 2.46 μm (3.08 λ). Therefore, the most suitable SPP−PNJ can be achieved by adjusting the refractive index and thickness of the dielectric microdisk in practical applications.  Table 2 shows the characteristic parameters of the SPP-PNJ with different thicknesses h of the dielectric microdisk. It can be seen in Table 2 that when h = 0.8 µm and the I max of the SPP-PNJ is 14.33 times stronger than that of the exciting light, F = 0.37 µm (0.46 λ) and L = 0.98 µm (1.23 λ). The largest obtained value of Q for the SPP-PNJ was 62.20. The smallest value of Q was 23.92, which can be realized when h = 0.7 µm. This will benefit the application of SPP-PNJs in super-resolution optical imaging. This indicates that with the increase of h, I max and F increase first and then decrease. However, the relationship between h and the propagation distance L of the SPP-PNJ is more complicated. According to Tables 1 and 2, the relationship between Q, n, and h is shown in Figure 11. It can be seen in Figure 10 that Q is greatly different with different h and n values, respectively. However, it is found that the maximum value of Q can be achieved by setting the thickness of the dielectric microdisk h. For example, when the value of h is 0.9 µm, Q reaches a maximum value of 62.20, and when the value of n is 1.5, Q reaches a maximum value of 49.47. However, compared with the influence of different thicknesses of h, the longest propagation distance L can be realized by different refractive indices. When the value of n is 1.3, L reaches a maximum value of 2.46 µm (3.08 λ). Therefore, the most suitable SPP-PNJ can be achieved by adjusting the refractive index and thickness of the dielectric microdisk in practical applications.

Conclusions
In this paper, we study the characteristics of an SPP−PNJ generated by a gold−film dielectric microdisk structure. In detail, the effects of the dielectric microdisk's refractive index and thickness on the maximum intensity, full width at half maximum, propagation distance, and quality factor of the SPP−PNJ are analyzed. The results show that the gold−film dielectric microdisk structure can generate high−quality SPP-PNJs, which are similar to the SPP-PNJs generated by irradiating dielectric microspheres or microcylinders with plane waves. In addition, compared with dielectric microspheres or microcylinders, the gold−film dielectric microdisk structure can adjust the characteristics of the SPP−PNJ by changing the refractive index and thickness of the dielectric microdisk. Therefore, the proposed dielectric microdisk can provide new ideas for further expanding the generation path of SPP−PNJs and conveniently tuning their characteristics.

Conclusions
In this paper, we study the characteristics of an SPP-PNJ generated by a gold-film dielectric microdisk structure. In detail, the effects of the dielectric microdisk's refractive index and thickness on the maximum intensity, full width at half maximum, propagation distance, and quality factor of the SPP-PNJ are analyzed. The results show that the goldfilm dielectric microdisk structure can generate high-quality SPP-PNJs, which are similar to the SPP-PNJs generated by irradiating dielectric microspheres or microcylinders with plane waves. In addition, compared with dielectric microspheres or microcylinders, the gold-film dielectric microdisk structure can adjust the characteristics of the SPP-PNJ by changing the refractive index and thickness of the dielectric microdisk. Therefore, the proposed dielectric microdisk can provide new ideas for further expanding the generation path of SPP-PNJs and conveniently tuning their characteristics.