## Abstract

We show that two rows of photonic lattices on each side of a narrow diffraction-inhibited beam are sufficient for confinement, enabling the launching of multi-beams with an interval of one lattice constant for independent propagation. A few integrated photonic circuit building blocks including arbitrary angle bends, power splitter and intersection are designed to realize flexible controls of the diffraction-inhibited beams. In addition, under wide beam illumination, incident beam power is well separated by the lattices rows, facilitating the simultaneous excitation of multiple diffraction-inhibited beams. These novel effects and building blocks offer exceptional opportunities for multichannel photonic routing.

© 2011 OSA

## 1. Introduction

Localized light beams tend to broaden in space through diffraction, which is a natural process of electromagnetic fields. Methods to overcome the often-undesirable consequences of this effect have long been pursued. Recently, it has been reported that the diffraction inhibition including the occurrence of tunnel inhibition can occur in arrays of evanescently coupled optical waveguides [1–7]. A reasonable explanation is that the band diagram flattens when the frequency and amplitude of the longitudinal modulation are chosen properly, preventing light from spreading into the arrays [1,3,8].

Concerning photonic crystals (PCs) [9,10], i.e., the structures with periodically modulated refractive index, a similar effect known as self-collimation [11–16] has been reported as well. The self-collimation effect exploits the flat region of the equal-frequency contours (EFCs) that characterize all allowed wave vectors at a given frequency. Since the group velocity ${v}_{g}$ is always normal to the EFC [17], flat EFC means that the Bloch waves can have different *k* but same ${v}_{g}$ direction. In this case, light can only propagate in the PC along the direction perpendicular to the flat EFC. However, this non-diffractive propagation is only valid for a rather wide beam with its distribution of wave vectors restricted to the local flat portion of the EFC [14,16]. As a consequence, this diffraction suppression occurs solely at lower orders. Recently, the occurrence of diffraction inhibition under the condition that the EFC is flat across the entire first Brillouin zone has been reported in two-dimensional (2D) photonic crystals [18], mimicking the tunneling inhibition phenomenon in waveguide arrays.

In this paper, we extend the work [18] and report the design of several fundamental building blocks including arbitrary angle bends, power splitter and intersection for flexible controls of the diffraction-inhibited beams. An analysis of the performance using finite-difference time-domain (FDTD) method shows that these novel structures are capable of achieving high transmission and high compactness simultaneously.

## 2. Transmission properties of diffraction-inhibited beams

The structure under consideration is a 2D PC that consists of a rectangular lattice of air holes introduced into a dielectric background with$n=3.4$(see inset of Fig. 1(a)
). The lattice constant is *a* in the propagation direction *x* and is *b* ($b=2a$) in the *y* direction. The radius of air holes is $r=0.38a$. The EFCs calculated by a plane-wave expansion method [19] open to form almost straight vertical lines across the entire first Brillouin zone around $f=0.26$
$c/a$ [18]. It has been demonstrated analytically and numerically that, in contrary to the usual self-collimated beams which are very broad and occupy many lattices sites, a diffraction-inhibited beam can be localized between two neighboring rows of the defect-free photonic crystal [18].

As a continuation of [18], we further calculate the transmission properties with different number of lattices rows ${N}_{r}$ on each side of the beam to evaluate the efficiency of confinement. A $0.25b$ wide continuous Gaussian beam with frequency $f=0.26\cdot (c/a)$ is launched into the channel from the left. The position of light source is set as the origin, and the transmitted power at $x=10\cdot a$ with ${N}_{r}=5$ is taken as input power${P}_{i}$. When ${N}_{r}$ varies from 1 to 5, a series of transmission coefficients are calculated at the position of ${x}_{i}$ with ${x}_{i}=5\cdot i\cdot a(i=2,3,\mathrm{...12})$ and$\left|y\right|\le 0.5b$. The results are shown in Fig. 1(a). It can be observed that, when${N}_{r}=1$, transmission coefficient *T*is about 92% with a propagation distance of $50a$, so it is possible to confine light roughly with only two lattices rows. When${N}_{rows}\ge 2$, transmission *T* reaches a steady state, as can be seen from the four lines $({N}_{r}=2,3,4,5)$ with good overlap in this figure. This enables the simultaneous transmission of several closely located beams. As a particular example we illustrate in Fig. 1(b) the evolution of three beams launched with an interval of one channel. Light is trapped in each excited waveguide over a long distance with negligible crosstalk.

## 3. Arbitrary angle bends

A mechanism for beam bending is essential for practical applications in highly compact integrated photonic circuits. For typical self-collimated beams, some simple and effective bending concepts for ${90}^{o}$ bending have been suggested [13,20,21]. However, it is difficult to form other bending angles as the guiding track must follow the specific lattice orientation of the PC structure. In this particular case, the light beam is only allowed to propagate along one direction, so the proposed method such as by introducing defects or mirrors in PCs cannot work. However, since the rows of photonic lattices can serve as physical boundaries for light beams with very shallow penetration depths, bends can be achieved by connecting curved physical boundaries with the straight ones. Figure 2(a)
shows a bend of arbitrary angle *θ* with the central radius $R=2.5b$ using four annular air grooves. The circular arc profiles are tangent with the lattices rows so that the direction of the propagating light can be changed with little scattering or reflecting losses. The inner groove radius is *b*, and radius of each annular is an integer multiples of the lattice constant. As shown in Fig. 2(b), the transmission is over 98% for arbitrary bending angle. Without losing generality, the steady-state magnetic field distributions for ${40}^{o}$, ${90}^{o}$and ${135}^{o}$ bends are shown in Figs. 2(c)–2(e), respectively. We can see that mode mismatch between the linear and curve parts are small at the interfaces and the shape of the fields is little distorted during propagating along the bends.

## 4. Beam splitter

Besides straight guiding and arbitrary bending, another basic routing mechanism for integrated optics applications is the power splitting which allows the division of an optical beam into multiple beams or, conversely, for combination of individual beams into one beam. In this section, we present the design for a one-to-two beam splitter in a scheme analogous to that employed for bends. As shown in Fig. 3(a)
, the splitter is essentially a T-branch formed by the intersection of $\pm {90}^{o}$ waveguide bends. The inner and outer radiuses of the air grooves are *b* and $2b$, respectively. Considering the radiation loss caused by the abrupt directional change near the crossing point of the two outer air grooves, we place five extra air holes on both sides of the corner to improve the confinement of light at the junction. The distribution of the $Hz$ field component in this splitter is shown in Fig. 3(b). The total transmittance of the two outputs is 91%, and the loss is 7%, while the reflectivity is 2%. Due to the symmetry, the transmitted power is equally distributed in the two outputs.

## 5. Intersection

For systems within limited space, it is often necessary to route beams cross each other from different channels. We present an analysis of a perpendicular intersection as shown if Fig. 4(a)
. The five extra air holes on each side of the four corners play the same role as in the beam splitter. A Gaussian beam with width of $0.25b$ and frequency $f=0.26$
$c/a$ is launched into the PC in between the lattice sites along the *x* direction. As can be seen in Fig. 4(b), the light beam is able to pass through such intersection with no leakage into the *y* direction with a transmittance of 88%. The main loss is the back reflection and the scattering loss at the junction that totally account for 12% of the input energy. However, there is negligible crosstalk when two perpendicular beams pass through the intersection.

## 6. Multichannel bends and splitters under wide beam excitation

Until now, we have studied only light propagation with narrow beam excitations such that the initial widths of beams are less than one lattice period. For broad beam, excitation of diffraction-inhibition beams happens easier as it excites an even narrower range of spatial frequencies. Figure 5
shows the light evolution when input beams covers approximately three waveguides with a uniform intensity distribution described by a rectangular step function. Unlike the uniform distribution of input field, the energy is well separated by the lattices rows after the beam enters the PC. This phenomenon provides an alternative way for multichannel routing. As mentioned previously, when${N}_{r}=1$, transmission coefficients *T*is about 92% with a propagation distance of $50a$, so most of the electromagnetic energy is still confined to the channel even with one row of lattices on each side. Figure 5(a) (Media 1) shows the bending of wide beams using the structure similar to Fig. 2(d). The input beam evolves into three separate beams during the initial stage after entering the PC structure, and then the individual beams propagate independently. As the propagation length is longer for the outer channel than the inner channel, the field profile at the output site is different from the input side. Figure 5(b) presents the results for power splitter with the same excitation condition. The power splitter has three smooth junctions, which split the launched power almost equally into three waveguides. Because of the unique feature of simplicity, these bending and splitting mechanisms may be widely used in the situations where there is no strong demand for high transmission or the required transmission distance is short.

## 7. Conclusions

We have studied the linear propagation of light beams in two-dimensional photonic crystals with flat EFC across the entire first Brillouin zone. Under this condition, light can only propagate along the direction perpendicular to the flat EFC, and the energy is highly localized in the channel between two neighboring rows of photonic lattices due to coherent superposition of Bloch waves. Such transverse localization of light is interesting from a practical application point of view. It is shown that light beams launched with an interval of one channel width are capable of independent transmission. In addition, high transmission (over 98%) through arbitrary angle bends can be obtained by introducing annular air grooves. We also analyze of the performance of beam splitter and intersection with compact sizes and high efficiency. Furthermore, we show that multiple diffraction-inhibited beams can be launched simultaneously with wide excitation beam. The uniform input field evolves into separate beams after entering the PC structure, which can be consequently rerouted. Our study therefore suggests a great potential of using these multichannel routing structures in integrated photonics circuitry.

## References and links

**1. **S. Longhi, “Multiband diffraction and refraction control in binary arrays of periodically curved waveguides,” Opt. Lett. **31**(12), 1857–1859 (2006). [CrossRef] [PubMed]

**2. **Y. V. Kartashov, A. Szameit, V. A. Vysloukh, and L. Torner, “Light tunneling inhibition and anisotropic diffraction engineering in two-dimensional waveguide arrays,” Opt. Lett. **34**(19), 2906–2908 (2009). [CrossRef] [PubMed]

**3. **A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, V. A. Vysloukh, F. Lederer, and L. Torner, “Inhibition of light tunneling in waveguide arrays,” Phys. Rev. Lett. **102**(15), 153901 (2009). [CrossRef] [PubMed]

**4. **Y. V. Kartashov and V. A. Vysloukh, “Light tunneling inhibition in longitudinally modulated Bragg-guiding arrays,” Opt. Lett. **35**(12), 2097–2099 (2010). [CrossRef] [PubMed]

**5. **Y. V. Kartashov and V. A. Vysloukh, “Light tunneling inhibition in array of couplers with longitudinal refractive index modulation,” Opt. Lett. **35**(2), 205–207 (2010). [CrossRef] [PubMed]

**6. **V. E. Lobanov, V. A. Vysloukh, and Y. V. Kartashov, “Inhibition of light tunneling for multichannel excitations in longitudinally modulated waveguide arrays,” Phys. Rev. A **81**(2), 023803 (2010). [CrossRef]

**7. **P. Zhang, N. K. Efremidis, A. Miller, Y. Hu, and Z. G. Chen, “Observation of coherent destruction of tunneling and unusual beam dynamics due to negative coupling in three-dimensional photonic lattices,” Opt. Lett. **35**(19), 3252–3254 (2010). [CrossRef] [PubMed]

**8. **D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. **90**(5), 053902 (2003). [CrossRef] [PubMed]

**9. **S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**(23), 2486–2489 (1987). [CrossRef] [PubMed]

**10. **E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**(20), 2059–2062 (1987). [CrossRef] [PubMed]

**11. **H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. **74**(9), 1212–1214 (1999). [CrossRef]

**12. **J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. **8**(6), 1246–1257 (2002). [CrossRef]

**13. **X. F. Yu and S. H. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. **83**(16), 3251–3253 (2003). [CrossRef]

**14. **Z. F. Li, H. B. Chen, Z. T. Song, F. H. Yang, and S. L. Feng, “Finite-width waveguide and waveguide intersections for self-collimated beams in photonic crystals,” Appl. Phys. Lett. **85**(21), 4834–4836 (2004). [CrossRef]

**15. **P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. **5**(2), 93–96 (2006). [CrossRef] [PubMed]

**16. **D. W. Prather, S. Y. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. H. Chen, B. L. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D Appl. Phys. **40**(9), 2635–2651 (2007). [CrossRef]

**17. **R. S. Chu and T. Tamir, “Group velocity in space-time periodic media,” Electron. Lett. **7**(14), 410–412 (1971). [CrossRef]

**18. **L. L. Zhang, Q. W. Zhan, J. Y. Zhang, and Y. P. Cui, “Diffraction inhibition in two-dimensional photonic crystals,” Opt. Lett. **36**(5), 651–653 (2011). [CrossRef] [PubMed]

**19. **S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**(3), 173–190 (2001). [CrossRef] [PubMed]

**20. **S. G. Lee, S. S. Oh, J. E. Kim, H. Y. Park, and C. S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. **87**(18), 181106 (2005). [CrossRef]

**21. **M. Wang, M. J. Yun, W. J. Kong, and C. L. Cui, “Beam splitter and beam bends based on self-collimation effect in two-dimensional photonic crystals,” J. Mod. Opt. **56**(10), 1159–1162 (2009). [CrossRef]