## 1. Introduction

By using an index gradient, a well-designed waveguide array [1], 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.,

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 [14], or even three-dimensional PCs [15]. Lately, wide-angle SC phenomena have been reported in PCs with square lattices by the composition of elliptical air holes [16]. 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° [17]. Second, the electromagnetic energy can be achieved for high degree of localization in the case of nondiffraction [5]. 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 *r* = 0.3a. Assume that the silicon rod is lossless and nondispersive near the telecommunication frequency and exhibit a refractive index of 3.5. Then, EFCs can be calculated by plane-wave expansion method. To simplify the model throughout this chapter, only one single frequency *f* = 0.2 *c*/*a* is considered for the transverse electric (TE) modes. This frequency is located in the first photonic band, and the corresponding isofrequency contour plot is shown in **Figure 1**.

### 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 *ΓX* direction. Therefore, for the purpose of improving the collimation ability, a higher EFC straightness must be obtained. As described in **Figure 1**, the straightness linearity of the EFCs is determined by β.

In order to study the correlation between length-breadth ratio “β” and EFC straightness, the least-squares method [17] 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, *i* is the number of sampling points, then, by utilizing the minimum condition, i.e., ∂ε/∂A, ∂ε/∂B, we obtain

and

From Eqs. (2) and (3), A and B can be calculated as follows:

Finally, we can quantify the straightness quality of the EFCs through the straightness factor

Where

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 _{0} and D (we assume D ≫ λ, D ≫ W_{0}) 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) = W_{0} 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 L_{0} = 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 ≤ L_{0}, for example, relative to a typical Gaussian beam with an initial waist *W*_{0} = 10λ, the beam broadening is less than 1% as the beam propagates at a distance scale of 100λ. We consider this quasi-collimated beam to be sufficient for most practical applications with self-collimation phenomenon.

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 *L* = *0.1743, L* = *0.014, L* = *0.01*, and *L* = *8.09 × 10*^{−4}, respectively. For *f = 0.2c/a*, which is the normalized frequency, the condition L = L_{0} corresponds to β = 2.3, with the corresponding EFC shown in **Figure 1(c)**. In order to provide usable operating frequency quasi-SC effect on additional information, L in the other four frequencies as a function of β is also shown in **Figure 2**, from which they can be seen with *f* = 0.2c/a very similar. Furthermore, bandwidth of quasi-SC effect becomes broader and broader by increasing β, because more and more EFCs have their straightness factor L smaller than L_{0}.

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 = *2.3*) at a single frequency *f* = 0.2c/a.

In order to study the whole-angle quasi-collimation phenomenon of the PC at the frequency *f* = 0.2c/a, with β = 2.3, *a* = 0.4 μm, and *r* = 0.3a, a Gaussian beam with the beam width 2.3 μm (a little bit larger than one wavelength 2 μm) is incident to the PC at any incident angle. **Figure 3** shows the calculation model, which can be applied to calculate the electromagnetic field by a perfectly matched layer (PML) finite-difference-time-domain (FDTD) method [10, 11] with a grid size 20 nm. The PC’s right, 50 × 71 rods array (as shown in the blue bars in **Figure 3**) are manually set to be absorbed (a small imaginary part is added to the refractive index of rod, i.e., the index becomes 3.5 + 0.02i) which set to avoid the size-dependent Fabry-Perot effect. A PML boundary condition is acted by absorption rods array for the PC. Thus, the structure may be available to simulate an infinite-length PC. As shown in **Figure 4** and the subsequent figures, the field distribution is presented. However, only the field which is located in the nonabsorption region is shown. When a Gaussian beam propagates with a different incident angle in air (top panels) and PC (lower panels), the field distribution in this case is depicted in **Figure 4(a)**–**(c)**. We can see that the beam does not diverge in the PC which shown in **Figure 4**. On the contrary, the electron beam has great divergence in the air.

### 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 *b* = 2.3a is equal to the length of the rectangular lattice of the quasi-SC PC. The silicon rods radius in the ARL is *r*_{arl}, and the distance between the ARL and the quasi-SC PC is *d*_{arl}. The parameters *d*_{arl} and *r*_{arl} should be well-designed to improve the coupling efficiency.

The coupling efficiency *r*_{tot}, which can be written as:

where β *= k*_{0} ⋅ (*d*_{arl}−*r*−*r*_{arl}) is the phase shift of the light beam as it crosses the ARL (*k*_{0} denotes the wave vector in the air), *r*_{12} is the reflection coefficient of the ARL and *r*_{23} is the reflection coefficient of the semi-infinite quasi-SC PC in the air when the light beam is striking the PC surface from air. By applying the multiple scattering method [18], both *r*_{12} and *r*_{23} can be calculated.

We try to find the appropriate r_{arl} and d_{arl} values to reduce total reflection |*r*_{tot}|^{2}. First, we calculate the coupling efficiency for an initial value of *r*_{arl} with *r*_{arl}*= r*, and the resulting color map is shown in **Figure 6(a)**. In this figure, the red area corresponds to high coupling efficiency. As shown in **Figure 5**, there are two regions, which allow for high coupling efficiency over a relatively wide incidence angle. We can select the appropriate d_{arl} according to **Figure 1**. As shown in **Figure 5**, for instance, 3.28a for an incident angle of 0°–20°, or 3.44a for an incident angle of 20°–30°. Then, for the optimization of d_{arl}, r_{arl} can be further optimized by improving the coupling efficiency, as shown in **Figure 6(b)** and **(c)**. The *d*_{arl} values are fixed to be 3.28a and 3.44a, respectively. From **Figure 6(d)**, the enhanced coupling efficiency is described versus the incident angle. The coupling efficiency is greatly improved over a wide incidence angle range by using the well-designed ARL.

In order to further improve the coupling efficiency, graded multiple ARLs are provided with same lattice constant (*b* = 2.3a) as shown in **Figure 7(a)**. They locate in front of the photonic crystal, which the distance is *a*. There are 20 layers in the graded multiple ARLs, and the rods radius is varying from the left to the right slowly, i.e., for the *i*th layer, the radius of the rods is *i* = 1, 2,…, 20. By using the multiple ARLs, the coupling efficiency is improved for nearly all incident angles, as shown in **Figure 7(b)**.

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 [19]. 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 *b*/*a* = 2.4. A top silicon layer with a height *h* = 2:025*a* was used as a board layer. The buried oxide (BOX) layer forms a high refractive index-contrast structure with the air as the cladding. According to this particular structure, out-of-plane light confinement is achieved owing to total internal reflection. For the silicon and BOX, the refractive indices are *n*_{Si} = 3.5 and *n*_{SiO2} = 1.5, respectively. Using plane wave expansion (PWE), we can obtain the band diagrams for the TM-like polarization (with the E-field along the *y*-direction), as shown in **Figure 9(b)**. The colored area is a light cone, below which are guided mode to be positioned in the plate [20, 21].

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 *v*_{g} = ∇kω (k), where ω is the optical frequency corresponding the wave vector k [22]. Thus, we can consider that the self-collimation effect is derived from the flat portion of the EFC. However, the nondiffractive propagation is confined owing to the use of a relatively broad beam of light, which the wave vectors are limited to the local flat portion of the EFC. For a narrow beam including a large range of k vectors, for instance, to obtain all-angle quasi-self-collimation, we require a flat EFC across the entire first Brillouin zone (as we discussed in the above section). Then, we calculate the corresponding EFC of the second frequency band as shown in **Figure 9(c)**. It is noteworthy that, at the *f* = 0.258 c/a, the flat EFC is flat across the first Brillouin zone, which is expected to lead to a large angle self-collimation. To validate our prediction, 3D finite difference time domain (FDTD) method is used to perform the simulation. A Gaussian beam having a width of 0.25b at *f* = 0.258 c/a is emitted into the designed PC. The calculated electric field intensity is drawn in **Figure 10(a)**. The self-calibration is achieved even though the small beam circumference, which the light source contains a wide range of k vectors. The *E*_{y} field intensity is presented along the *z*-direction at *x* = 40a in **Figure 10(b)**. It is noteworthy that the EM energy is highly positioned along the array of a single nanorod. The E_{y} field intensity is only 1/10^{5} which located in the third row to each one side of this single nanorod array, for instance, a very weak electric field. It will ensure good confinement for light waves when placing of four rows of nanorods on each side of the central single nanorod array.

### 3.2. Experimental results and discussion

The parameters of the dielectric nanorod chain design is *a* = 400 nm, *r* = 160 nm, and *h* = 810 nm for which these operate at frequencies of optical communications. The PC is fabricated on a platform of silicon-on-insulator (SOI) wafers by four main processing steps. First, we require that thermal oxidation is needed to form an oxide layer, which serves as a hard mask for the purpose of transferring the pattern into the Si layer by using reactive ion etching (RIE). Second, we apply electron beam (E-beam) lithography to transfer this pattern to the high-sensitivity and high-resolution E-beam Resist ZEP-520. Third, the hard mask is etched using RIE in a CHF_{3} plasma. Finally, after using the inductively coupled plasma (ICP) etching process, which a thermally grown SiO_{2} layer is used as a hard mask, the resulting dielectric rods are about 810 nm. **Figure 11** shows the scanning electron microscope (SEM) image of the nanorods, the rods have sidewall profiles approximately 90° to the substrate; the screw thread is caused by deep etching with using the Bosch process.

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 [23]. 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 [24]. 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 [25] 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.

## 4. Conclusion

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 [28]. Such a large-angle SC phenomenon resulting from the nanorod structure may also find applications in sensing and optofluidics.