Guiding on-chip optical beams without diffraction is very important in the future’s all-photonic circuits. Herein, both theoretically and experimentally, we study an all-angle quasi-self-collimation phenomenon occurring in photonic crystals composed of silicon nanorods. When the all-angle quasi-self-collimation phenomenon occurs, the optical beams can be incident onto such photonic crystals from directions covering a wide range (extremely close to all-angle) of incident angles direction and become highly localized along even a single array of rods, which finally achieve results in the narrow-beam propagation without divergence. The propagation length is expected to be 1000 times larger than the wavelength of light. Theoretically, it is shown that such all-angle quasi-self-collimation phenomenon is owing to the symmetry change of the lattice of photonic crystals. By changing the symmetry of a photonic crystal to straighten the isofrequency contours, the photonic crystal shows the all-angle quasi-self-collimation effect. Experimentally, we show the observation of all-angle quasi-self-collimation phenomenon occurring in a rod-type silicon photonic crystal fabricated on by patterning a silicon-on-insulator (SOI) wafer. The experimentally observed propagation length is more than 0.4 mm over the telecom wavelength range, even though at large angle of incidence, which is a relatively large length scale for on-chip optical interconnection.
- without diffraction
- silicon photonic crystal
- all-angle quasi-self-collimation
By using an index gradient, a well-designed waveguide array , or nonlinear optical effects [2, 3], beam spreading arising from geometrical origin can be manipulated in many ways. Photonic crystals (PCs) provide an alternative to beam steering based on photonic bandgap (PBG) formation and related dispersion phenomena such as negative refraction, superprisms, and slow light. Self-collimation (SC) as a dispersion-related phenomenon is worth studying because it has important implications for subwavelength focusing, channel-less waveguiding and diffraction suppression [4–7], etc. In a SC PC, the light beams can propagate without diffraction, since propagation directions are paralleled to the group velocity, i.e., , where ω is the optical frequency for a determined wave vector k. Thus, we can determine that the SC effect is ascribed to the flat part of the equifrequency contours (EFCs) [8, 9]. Based on the SC effect, it can be used to design novel SC-based photonic devices, such as devices for diffraction inhibition, channel-less waveguides, subwavelength focusing or imaging [4, 5, 10, 11].
In recent years, the development of fabrication techniques has enabled the SC PCs to function at optical frequencies. In experiments, it has been recognized that SC phenomenon exists in many different kinds of structures, containing pillar-type and hole-type PCs [4, 12, 13], and quasi-zero-average-index structures , or even three-dimensional PCs . Lately, wide-angle SC phenomena have been reported in PCs with square lattices by the composition of elliptical air holes . Technically speaking, contrary to the hole-type PC, there are a great potential for active components with the pillar-type structure, benefiting from the possibility of the heat dissipation capability and electrical contraction. In addition, the light interacts with the medium (to be sensed) which more strongly surrounds the rods, and in optofluidics, the rod structure allows for fluid penetration better than the structure composed of air holes in a dielectric slab, therefore, the rods are more suitable for sensing applications.
However, there are still some problems to be solved. For example, the SC effect may be limited by the angle of incidence of light, which makes SC-based super-integrated devices difficult to manufacture. In this paper, the model is clarified for a full-angle quasi-SC PC, and the lower reflections for our quasi-SC PC by utilizing the destructive interference-based method. This all-angle quasi-SC is related to a flat equifrequency contour (EFC) across the entire Brillouin zone. In contrast to the conventional self-collimation, the all-angle quasi-SC shown in this chapter exhibits two unique properties. First, the light waves can be injected from any angle, even up to 90° . Second, the electromagnetic energy can be achieved for high degree of localization in the case of nondiffraction . For hole-type PCs, light waves are confined between the pores owing to the full-angle quasi-SC phenomenon, whereas for pillar-type PC, the electromagnetic energy can be positioned highly along a narrow path just as in a single nanorod array, as presented in this paper. Moreover, we will also show our recent experimental work on the all-angle quasi-SC in PCs composed of silicon nanorods. We believe this work motivates the current research on silicon photonics.
2. Theoretical model and analysis
First, the two-dimensional (2D) rod-shaped silicon PC is considered with a rectangular lattice in the air, as shown on the inset in Figure 1(a). The breadth and length of the rectangular lattice are denoted by a and b, respectively, and the radius of the rod is
2.1. Straightness of the EFCs and collimation ability of the beams
As shown in Figure 1(a)–(d), the large angle SC effect in PC can be obtained by changing the symmetry of the PC, that is, by increasing the aspect ratio of the length to breadth “β = b/a”. As shown in Figure 1(d), the large angle SC effect is indicated by the straightness of the EFC plotted over the entire Brillouin zone. The light beams can reduce beam divergence along with the increase of β value, resulting in self-collimation propagation along the
In order to study the correlation between length-breadth ratio “β” and EFC straightness, the least-squares method  is used to quantify EFC straightness as shown in Figure 1. According to this method, a quasistraight EFC whose curve can be illustrated by a function “Y = F(X)” (Y representing ky and X representing kx). Assuming that a straight line with the equation
can be applied to fit the quasistraight EFC, where A and B are undetermined coefficients. For the purpose of obtaining A and B, is defined, where
Finally, we can quantify the straightness quality of the EFCs through the straightness factor
Where and are the maximum and the minimum convexities, respectively.
According to the straightness factor L is defined by Eq. (5), the averaged deviation angle of the propagation direction can be illustrated for a quasi-collimated light beam. Furthermore, we note that the beam width will become broader while the beam propagates through the quasi-SC PC, with determining the beam waist by where λ is the wavelength in the air, W0 and D (we assume D ≫ λ, D ≫ W0) describe the initial waist and propagation distance of the beam, respectively. The smaller L corresponds to a straighter EFC and better self-collimation behavior. Just in case L = 0, a perfectly straight EFC corresponding to a strict self-collimation without beam divergence is obtained, W(D) = W0 which the D can take any value.
For most practical applications, the condition L = 0 is too strict. A sufficiently small L is usually acceptable. We proposed L0 = 0.01 as the critical value for the straightness factor (corresponding to the pink dash line in Figure 2). In this case, the quasi-collimated beam shows almost no diffraction if L ≤ L0, for example, relative to a typical Gaussian beam with an initial waist
Using Eq. (5), the straightness factor L of the EFCs can be calculated. Figure 2 shows the relationship between the straightness factor of the EFCs and the rectangular lattice structure. As β increases, the straightness factor L decreases obviously. This result is consistent with the result in Figure 1. As shown in Figure 1(a)–(d), we can measure that the corresponding straightness factors are
As shown in Figure 2, it is possible to realize a better and better SC by increasing β. For the case of β≫1, the system will become separated from each other, so in this case, we can investigate only one of the row of rods. We find that the SC effect is still valid in this single row of rods. However, there are many differences between their single row, and also there are many differences between a single row and a PC having a rectangular lattice. For the sake of simplicity, we limit ourselves to only explore the PC shown in Figure 1(with β =
In order to study the whole-angle quasi-collimation phenomenon of the PC at the frequency
2.2. Coupling analysis
We investigate the optical coupling efficiency of the SC PC in this section. Owing to strong reflection, there are the large coupling losses (about 50%), which are shown in Figure 4. For improving the coupling efficiency, we propose a method based on destructive interference. Figure 5 depicts an antireflection layer (ARL), which is used in front of the quasi SC PC. The ARL is also constituted of the silicon rods array, which a lattice constant
The coupling efficiency is directly determined by the total reflection coefficient
We try to find the appropriate rarl and darl values to reduce total reflection |
In order to further improve the coupling efficiency, graded multiple ARLs are provided with same lattice constant (
In order to verify the improvement of the coupling efficiency, the field distribution is calculated for the Gaussian beam at four angles of incidence emitted at a PC with multiple ARLs, and the resulting field distribution is described in Figure 8(a)–(d), respectively. For multiple ARLs structure, the Gaussian beams are almost completely coupled into the PC with little reflection. Comparing Figure 8 with Figure 4, it is indicated that the multiple ARLs can suppress the reflection.
3. Experimental model, results and discussion
We present the demonstration of a full-angle quasi-SC phenomenon based on pillar-type PCs in this section. Light waves are received into these pillar-type PCs from a wide range of incidence angles and can be positioned highly along the array of rods. We observed that the propagation length was 0.4 mm in the wavelength range of 1540–1570 nm. For on-chip photonic applications, this value might be sufficient propagation distance.
3.1. Experimental model
In order to increase the angular collimation range, a simple and feasible approach is to reduce the symmetry of the photonic construct by using a rectangular lattice . Here, the 2D pillar-type PC composed of a rectangular lattice fabricated by patterning silicon on insulator (SOI) wafer, as presented in Figure 9(a). By increasing a rectangular lattice of side ratio, a flat EFC across the entire Brillouin zone can be obtained. However, due to the small filling factor of the dielectric material, an excess ratio will result in leakage of the light wave in the other direction (z-direction). As a compromise between the full-angle quasi-self-collimation effect and the collimating efficiency, we choose a PC structure with
The light propagation direction in PCs depends entirely on the gradient direction of the corresponding EFC, since the propagation direction is the same as the group velocity
3.2. Experimental results and discussion
The parameters of the dielectric nanorod chain design is
To show the large angle of self-collimation effect, three different samples with tilted PC areas were designed in the experiment. We introduce light waves to irradiate the PCs by a waveguide, which a width is 3 m as shown in left panels (a), (b), and (c) of Figure 12, it is designed that the tilt angles with respect to the incident waveguide are set to be 0°, 45°, and 75°, respectively. Theoretically, note that self-collimation can even be observed for an incident angle of almost 90°. However, due to impedance mismatch, larger incident angle causes more reflection losses. In our experiments, the maximum incident angle is set at 75°, so that our camera can be used to capture a sufficient amount of light scattering due to structural defects. The TM polarized light from the tunable laser source is introduced into the lensed fiber. In order to visualize the EM energy transfer in the PC, we use an infrared camera to capture the ray trajectory of the EM energy (through the scattered light). Light traces were observed clearly in the 1540–1570 nm wavelength range. Coupled resonator waveguides (CROWs) based on weak coupling between high-Q cavities can also achieve functions similar to those of our guided-wave PC structures. Unfortunately, their operating frequencies are limited to the resonant frequencies of the whispering gallery modes of individual rods . In contrast, our SC effect is based on the dispersion relation of the entire PC, operating over a bandwidth wider than the resonant frequency of a single rod. Figure 12(d)–(f) corresponds to capturing the results at 1550 nm wavelength with incident angles of 0°, 45°, and 75°, respectively. At the point of A, this light spot is contributed by strong scattering which located on the interface of the waveguide surface and the PCs. At the point of B, this light spot which situated at the opposite end of the PC region, also provides an interface for optical scattering. There are 1000 lattice periods in the routing from A and B, corresponding to the propagation length value of 0.4 mm. For the on-chip optical interconnection, it is a very considerable length scale. For the large-angle self-collimation effect, previous experiments shown that the light propagating had at least 100 lattice periods without significant divergence . In our experiment, the propagation length can be as high as 1000 lattice periods, and ray trajectory of the light is clearer than the previous.
For a detailed description of the quantitative results, the field intensity of the nanorod array determined from the FDTD calculations  is plotted in Figure 13(a). The experimentally determined propagation loss of the self-collimated beam is measured by plotting the relative intensity along the nanorods in Figure 12(d). The experimental propagation loss is determined to be 17.6 dB/mm, which is higher than the simulation result as shown in Figure 13. It is believed that the extra propagation loss is mainly attributed to the roughness of the pillars, but we can reduce this roughness by improving the manufacturing process. Our simulation shows that the coupling efficiency can be as high as 92, 90, and 45% corresponding to the incident angles of 0°, 45°, and 75°, respectively. Furthermore, through modifying the interface [26, 27], the coupling efficiency can be enhanced by suppressing scattering at the point A. Therefore, it is believed that the millimeter-scale propagation length can be easily observed with the rod structure based on the above-mentioned improvements.
In conclusion, an all-angle quasi-self-collimation effect can be achieved by changing the rectangular-lattice symmetry of photonic crystal. The quasi-self-collimation effect is identified and quantified by a straightness factor L, which is based on the least-squares method. With the decrease of straightness factor L, the photonic crystal possesses a more powerful self-collimation effect. Besides, the efficiency of light coupling to the quasi-self-collimation photonic crystal is investigated and is greatly improved by applying a carefully designed antireflection structure. Experimentally, we have designed, fabricated, and characterized a structure based on silicon nanorods that exhibits the all-angle quasi-SC phenomenon. The millimeter-scale propagation length and broad wavelength range exhibited by our structure may be sufficient for on-chip photonic applications. Our recent study shows such all-angle quasi-self-collimation effect can also be used to design a compact silicon nanoantenna array for high-efficiency vertical light emission . Such a large-angle SC phenomenon resulting from the nanorod structure may also find applications in sensing and optofluidics.
This research was sponsored by the National Natural Science Foundation of China (Grant Nos. 61475180, and 61275112), and the Science and Technology Commission of Shanghai Municipality (Grant Nos. 14JC1407601, and 14JC1407602). The figures in this chapter were extracted from our previous papers (H. Li et al, IEEE Photon. J. 5, 2201306 (2013) and M. Li et al., IEEE Photon. J. 7, 4700108 (2015)).