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Abstract
We propose a new and dynamically focusable subwavelength focusing system, consisting of a negatively refracting photonic crystal and a single silicon lens. The system can adjust the focal length in the range of 370 nm to 4610 nm. We set directional light with a wavelength of 1612nm to incident on the whole system. Numerical simulation through RSOFT shows the smallest focus spot with the half-width of 0.116 λ at a distance of 0.44 λ and calculates focusing efficiency of 45%. In addition, a good result is found that the half-width of the spot is less than 1/2 wavelength at a distance of 2.85 λ from the surface of the structure.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It is of great significance to focus the incident beam into the spot with sub-wavelength resolution. With the development of integrated circuits, especially the large-scale ones, applications in the fields of optical information storage [1], nanolithography [2], bioimaging [3], and optical microscopy [4] all require focusing the spot to sub-wavelength. Therefore, the study of far-field sub-wavelength focusing and far-field sub-wavelength technology is of great significance. How to break through the constraints of diffraction limit and realize smaller focusing spot and higher resolution imaging becomes a research frontier today.
As early as 2000, Pendry [5] proposed the theory of “perfect lens” which is made of a negative-refractive-index material [6], such as metamaterial, photonic crystal, etc. Evanescent waves can be amplified and further make high-resolution imaging possible. However, most of negative-refractive-index materials have strong dispersion, in which negative refraction can really occur, but it gets quite narrow. Compared with other negative-refractive-index materials, the photonic crystal has the characteristics of the photonic band gap, photon localization, etc., which can obtain a negative refraction with a wide frequency band, a low loss and the control of the flow of light. Thus, a large number of scientists have been attracted to study photonic crystals. Notomi found that photonic crystals can adjust the effective refractive index by controlling the band structure, making it become a negative refractive material with interesting optical properties [7]. Luo et al. first discovered that two-dimensional photonic crystals can achieve full-angle negative refraction, and studied the propagation characteristics of evanescent waves in two-dimensional photonic crystals. Their research demonstrated the feasibility of using two-dimensional photonic crystals to achieve sub-wavelength focusing [8,9]. Cubukcu et al. proved that sub-wavelength resolution imaging can be achieved in the microwave band by using a two-dimensional photonic crystal superlens [10].
Unlike relying on negative-refractive-index materials to achieve sub-wavelength focusing, sub-wavelength spots can also be obtained in the near field by using the self-focusing effect of multimode fibers [11]. Another option is to use a nanopore structure, or a slit structure in a metal film so that the incident light may excite a surface wave, thereby enabling a subwavelength spot. For example, Shi et al. used metal slot arrays of different widths to modulate beam focusing, and achieved sub-wavelength focusing with a half-width of 0.42λ, and light transmission with the efficiency of 66.28% [12]. Fu showed that sub-wavelength focusing can also be achieved through the sub-wavelength micro-zone phase plate [13]. Min et al. demonstrated that nonlinear media can also be used for sub-wavelength focusing [14]. Gordon chose to achieve sub-wavelength focusing by filling the metal gap with a medium of different materials [15]. Jia et al. proposed a slit with a concave or convex profile on both sides. By manipulating the phase of the transmitted beam effectively, the focal depth can be precisely adjusted, and the spot size with the smallest focus can close to one wavelength with the maximum efficiency 15% [16]. Based on the angular spectrum theory of plane wave, Hong et al. generated an approximation of an azimuthally polarized Bessel beam with a focal spot FWHM of 0.37λ [17]. According to the theory of super-vibration, light wave from the exit face of multiple micro-nano fibers can be coherently superimposed in the diffraction region and then focused on a spot with a half-width of 0.43 λ [18]. Fu designed a tapered microtube structure and obtained a focused light with a half-width of 0.65 λ at a distance of 2.2 λ from the surface of the structure [19].
In 2009, Lee et al. found that a medium microsphere lens whose size is comparable to the working wavelength had near-field amplification and focusing phenomena [20]. In fact, using dielectric microspheres to achieve sub-wavelength focusing has become one of the simple and effective methods to break through the diffraction limit [21,22]. Guo et al. measured the near-field focusing of dielectric microspheres with FWHM value of 1/3 λ [23]. In recent years, on account of its flat surface, the compact imaging system can be designed flexibly and integrated, thus metasurfaces [24] has attracted wide attention, and metalenses with excellent focusing effect can be produced and the focusing efficiency is 65% at the wavelength of 1550 nm. Therefore, high refractive index materials are needed to replace metals.
Unlike most of the previous studies, the minimum focal spot size (FWHM) in our work is about 0.4 λ. We put forward a new focusing system: a movable silicon lens is placed under the surface of a two-dimensional photonic crystal with an equivalent refractive index of -1, and the position of focus is adjusted by moving the lens. The system can adjust the focal length from 370 nm to 4610 nm and achieve better far-field subwavelength focusing.
2. Design and simulation
In this paper, one of the components of the focusing system is designed in a regular hexagonal lattice photonic crystal with Si (n = 3.42) rods [25] in air, the lattice constant a = 482 nm and air hole radius r = 0.365a. As shown in Fig. 1(a), with the wave vector K increasing from the inside to the outside, the corresponding frequency decreased continuously. According to the plane wave expansion method (PWEM), the group velocity and phase velocity of the electromagnetic wave in the frequency range of the first photonic band of photonic crystals are opposite, and the point multiplication of Poynting vector and wave vector K of photonic crystals is negative, so its equivalent refractive index is negative. Research shows that photonic crystals have sub-wavelength focusing ability when the equivalent refractive index approaches -1. According to the result of measuring, in the frequency range of 0.295∼0.302, the corresponding value of the effective refractive index is 0.95∼1.08. When the angular frequency ${w_0}$ = 0.299, the value of the wave vector K is 3.894. and wavelength λ = 3.344a (1612nm), while c is the speed of light in vacuum. According to Fig. 1(a), the equivalent refractive index close to -1 can be obtained. At the same time, when ${w_0}$ = 0.598$\pi $c/a, the relationship between the incident angle of the beam and the angle of refraction of the beam is shown in Fig. 1(b), the incident angle and the refraction angle are almost equal and are distributed on the same side of the normal line.
Fig. 1. (a)Several EFS contours in the first TE-polarized photonic band of the PC; (b) Relationship between incident angle and refraction angle at the same frequency.
Rsoft optical path simulation software and time domain finite difference method (FDTD) is used to simulate the focusing process. The width of PML (Perfect-matched-layer) is 0.5µm, the grid size is 0.008µm, and the width of the monitor is 24µm. The simulation results of Ma et al. [26], showed that cutting processing is indeed possible to improve the image quality, and the normalized peak value of the image is the largest when the cutting ratio of the air hole is equal to 20%. However, after a deeper investigation, we found that the peak value of G (indicating the addition of an equal-period grating on the contact surface of the photonic crystal) is higher than that of C (indicating 20% cut on the upper surface) as shown in Fig. 2(a), and the half-width is smaller as shown in Fig. 2(b). Therefore, we choose the structure shown in Fig. 2(c); the lower surface of photonic crystals is cut by 20% (d = 0.8r), at the same time, on the inclined plane; the parameters of gratings are set to ${w_1}$=0.71a, ${w_2}$ = 0.78a, ${h_1}$ = 0.513a and ${h_2}$ = 0.963a. With a silicon lens, a micro-system capable of far-field sub-wavelength focusing is constructed.
Fig. 2. (a)The peak value of the image when gratings is added to the upper surface of PCs and 20% cut on the upper surface; (b) Half-width of the image when gratings is added to the upper surface of PCs and 20% cut on the upper surface; (c) Variables schematic diagram on the focusing system.
Since the refractive index of the silicon lens is high and the focal length is small, the incident beam will probably be directly focused in the lens if the radius of curvature of the silicon lens is too small, which surely can’t meet the application requirements. By changing the radius of curvature XR (X represents the ratio) of the lens under the condition that the width (2w) of the lens remains unchanged, the beam is focused on the exit face side of the lens. As shown in Fig. 3(a), after the directional light incident a silicon lens with a curvature radius of 1.12R, the spot is focused on the exit surface of the silicon lens. Figure 3(b) shows the half-width detected by the monitor with a size of 0.118 λ, the focusing efficiency is close to 95%. The article studied the trend of the focal spot size and the distance from the exit surface of the silicon lens when the ratio of the curvature radius is changed from 1 to 8. As shown in Fig. 3(c), the results of changing the radius of curvature in a small range are very unstable. With the increase of the ratio multiplied by the curvature radius, the focusing position of the incident light is further and further away from the exit surface, and the size of the focusing spot also increases. The data shows that the incident light is almost focused on the exit surface in the range of 1.08 to 1.24. And within this range, the focal spot size is also relatively small.
Fig. 3. (a) Light path of silicon lens with curvature radius 1.12R; (b) The half width of the spot; (c) Blue indicates dependence of the half width and black indicates the dependence of the focus to the exit face when the radius XR changes from R to 8R.
On the one hand, the silicon lens achieves good sub-wavelength focusing near the exit surface, but its application range is greatly limited. On the other hand, we succeeded in making the beam focus more far by changing the radius of curvature of the silicon lens, unfortunately, the focal spot size also increased. To solve the problem, the silicon lens needs other structures to achieve far-field sub-wavelength focusing, while the fact is that the diffraction limit cannot be broken in the far field region due to the evanescent wave carrying the object information that is exponentially attenuated during the propagation process. Therefore, if the evanescent wave can be amplified in the middle of the propagation, it will effectively slow down the decay rate of the evanescent wave. It is possible to break through the diffraction limit of the far field. As is known to all, negative refractive metamaterials have the characteristics of enhancing evanescent waves. In this paper, the far-field subwavelength focusing is achieved by combining photonic crystals and a single silicon lens with a negative refractive effect.
To verify the far-field sub-wavelength focusing effect of the focusing system, we did the following design. A silicon lens with a radius of curvature of 1.12R is placed under the photonic crystal structure, allowing the directional light to enter the silicon lens first. After focusing on the surface of the silicon lens, the light continued to be focused a second time through the photonic crystal. Light path after a certain time is shown in Fig. 4(a) and the half-width of the focal spot is shown in Fig. 4(b). Based on MEMS (Micro Electro Mechanical System) movements [27], the silicon microlens is designed so that the microstructure can be dynamically moved within 5 µm. The change in the focus position during the movement of the silicon lens as shown in Fig. 4(c). The blue arrow indicates the direction in which the silicon lens moves. The blue dot indicates the middle position of the silicon lens's exit surface, and the red dot indicates the position of the focal spot. The data results are shown in Fig. 4(d), and the focus position is linearly distributed from the nearest 370 nm (0.23 λ) to the farthest 4610 nm (2.85 λ). And in the whole process, the half width of the focal spot is less than 1/2 λ. In addition, when the placement position of the silicon lens is closer to the angle of the left side of the photonic crystal, the focus position is also closer to the surface of the photonic crystal structure, and the half width of the focal spot is also smaller. The data shows that the minimum of the spot half-width is 0.116 λ and the highest focusing efficiency of 45% can be obtained When incident from L = 1.6µm.
Fig. 4. (a) The light path of the focusing system is incident at the position of L = 1.5µm; (b) the half-width of the focal spot. (c) When moving the silicon lens, the focus position is distributed; (d) The focal length and half width of the moving silicon lens change in the range of 4µm.
3. Conclusion
In conclusion, this paper studies the imaging characteristics of a subwavelength focusing system consisting of a single silicon lens and a photonic crystal with a negative refraction effect. This focusing system can not only focus on the focal spot half-width of 0.116 λ, which is much smaller than the previous research result of 0.4 λ. In addition, considering the practical application, we designed sub-wavelength focusing with dynamic focusing. By adjusting the placement of the silicon lens, the desired output position and half width can be selected. Moving the silicon lens in the range of 4µm, the focal length of the system can be increased from 370 nm to 4610 nm. At the same time, the half width also increases from a minimum of 0.116 λ to maximum of 0.46 λ. Compared with a single silicon lens, the imaging system consisting of a photonic crystal and a silicon lens can focus not only on the near-field region with a small half-width, but also on the beam in the far-field region while breaking through the diffraction with a half width less than 1/2 λ, and the focusing efficiency is 45%. Because of the difference between the focus direction and the incident direction of the light source, the focal spot is less affected by the leaked light. Based on the above research results, this paper proposes a method to achieve sub-wavelength focusing in far field, which will contribute to nanolithography, optical information storage, confocal microscopy and other optical fields.
Funding
National Natural Science Foundation of China (61177043); Science and Technology Commission of Shanghai Municipality (17590750300).
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