## Abstract

A compact beam splitter consisting of three branches of periodic dielectric waveguides (PDW) is designed and analyzed theoretically. Both the symmetrical and asymmetrical configurations of the beam splitter are studied. The band structure for the guided modes is calculated by using finite-difference time-domain (FDTD) method with Bloch-type boundary conditions applying in an appropriate supercell. The field patterns for the whole structure and the transmissions for the output ports are calculated using the multiple scattering method. By utilizing the co-directional coupling mechanism, the light injected into the input branch can be efficiently transferred into the two output branches if the phase matching conditions are satisfied. The coupling length is short and the broad-band requirement can be achieved. Bending loss is small and high transmission (above 95 %) can be preserved for arbitrarily bent PDW if the bend radius of each bend exceeds five wavelengths. This feature indicates that the periodic dielectric waveguide beam splitter (PDWBS) is a high efficiency device for power redistribution while avoiding the lattice orientation restriction of the photonic crystal waveguides (PCW).

© 2007 Optical Society of America

## 1. Introduction

Photonic crystals (PC) are periodic dielectric structures made for manipulating light at about wavelength scale [1, 2]. The dispersion relations of the propagating modes in PC are given by the photonic band structure, which contains passbands and bandgaps. Many functional devices using photonic bandgap (PBG) effect have been proposed and are expected to play important roles in high-density optical integrated circuits. The devices based on PBG structures have the advantage of very small feature sizes comparing with their conventional counterparts. For instance, a photonic crystal waveguide (PCW) can be designed to guide light with high transmission in the sub-micron scale, which is almost an impossible task for a conventional dielectric waveguide (CDW)[3]. However, PCW also has several disadvantages. First of all, a sharp bend implies large interior reflection, which may reduce the transmitting power. In addition, the waveguide track of a PCW must follow the specific lattice orientation of the background PC structure, thus it cannot be bent arbitrarily. Furthermore, a PCW must be embedded in a wide enough background PC region (at least several lattice constants) in order to avoid the propagation loss, thus the PCW component occupies much space in the transverse dimension. These facts may cause some inconvenience toward fabricating real optical integrated circuits.

In order to control the flow of light completely, power division devices are also needed. The most commonly used power division devices are beam splitters (BS). In the past few years, a considerable number of photonic crystal beam splitters (PCBS) relying on different mechanisms have been proposed. The T-shape [4], Y-junction [5], and cross-type [6] PCW beam splitters (PCWBS) use one or more local defects to achieve impedance matching and power division. On the other hand, dispersion-based BS use the self-collimation effect of PC passband together with the evanescent wave tunnelling or partial reflection by line defects to realize the power division [7, 8, 9]. More sophisticated BS devices and interferometers consisting of PBG mirrors, partial reflection line defects, and dielectric or photonic crystal waveguides have also been proposed [10]. Although these PCBS devices can be made compact, they still have the same kind of disadvantages mentioned above.

Recently, PC power splitters and PC switches utilizing directional coupling mechanism were considered [11, 12]. In [11], PCWBS relying on multi-mode interference were considered. A 1 × 2 waveguide splitter can be designed by appropriately choosing the length of the multi-mode section of the waveguide so that the two symmetrically distributed beam images appear correctly at the entrances of the two output singlemode waveguides. In [12], the authors first showed that the switching length and bandwidth have a trade-off relationship in the conventional directional coupler switches. They then redesigned the dispersion curves of the modes via modifying the structure of the mode-coupling section, and a switch with compact size and wide bandwidth characteristics was successfully realized. Since the operation of these directional coupling devices are based on the phase matching conditions between coupling modes, they can be systematically designed by analyzing the dispersion curves of the modes.

In this paper, we continue our previous work [13] and adopt the advantage of the directional coupling mechanism to propose a new kind of BS. We have already shown in [13] that the guided modes can be transmitted through an arbitrarily bending periodic dielectric waveguide (PDW). We will further show in this paper that a wideband, small coupling length periodic dielectric waveguide beam splitter (PDWBS) can be easily designed. Like conventional directional coupler [14], light injected into the input waveguide can be switched efficiently into the two output waveguides with equal power if the phase matching condition is satisfied. In addition to the advantage of short coupling length and high transmission under arbitrary bending, PDWBS has the advantage that power division can also be achieved in its asymmetrical configurations.

## 2. Review of the previous work on PDW

In our previous work [13], we have studied the transmission characteristics of finite PDW formed by dielectric cylinders. Unlike photonic crystal waveguides, PDW use index guiding mechanism to confine light. The band structure is calculated using finite-difference time-domain (FDTD) method together with supercell method. A propagating mode in the super-cell satisfies Bloch-type boundary condition along the longitudinal (propagating) direction, and perfectly matched layers (PML) are applied to the transverse regions outside the supercell. The dielectric constant of the cylinders is *ε* = 11.56 (GaAs), and the radius of the cylinders is *r* = 0.2*a*, *a* is the lattice constant. Only TM modes (with **E** field parallel to the cylinders) are considered. The first band is below the light line, which indicates that the first-band modes are indeed guided modes. Transmission for finite S-shaped bent PDW are obtained using multiple scattering method [15]. Wideband (0.2 < *a*/λ < 0.25) high transmission (> 90%) can be achieved at any bend angle (from 0*°* to 90*°*) for any bend radius larger than 11.5*a* (about three wavelengths). This feature indicates that PDW has the advantage that it can be bent to any arbitrary shape while still preserves its high transmission property, avoiding the lattice orientation restriction of PCW devices. In this work, we adopt this advantage and use the directional coupling effect to design a PDWBS.

## 3. Dispersion relations of the modes in the coupling section

In this paper, the dielectric constant *ε* = 11.56 and the radius *r* = 0.2*a* of the cylinders are chosen as the same as that used in [13]. The simulation methods are also the same. Two arrangements of rods shown in Fig. 1(a1) and (b1) are considered, which represent the mode-coupling section of two different designs of PDWBS. Both structures are multimode PDWs formed by three rows of rods in air. For the case of width *L _{y}* = 1.3

*a*, the corresponding band structures (dispersion relations) are shown in Fig. 1(a2), (b2). The number of low-frequency guiding bands is equal to the number of rows. In addition, for a definite frequency, different modes correspond to different

*k*vectors and different field patterns (See the insets of Fig. 1(a2), (b2)).

When a mode of the single-row PDW (the input waveguide) of appropriate frequency is injected into the coupling section of the PDWBS (the three rows region), mode exchange phenomenon happens. The injected mode can be expressed in this region as a superposition wave of two modes with different *k*, say, *k*
_{1} and *k*
_{2}. Assuming that *k*
_{1} > *k*
_{2}, then *k*
_{1} represents the first band mode and *k*
_{2} the second band mode. The energy of the *k*
_{1} mode is concentrated in the central row, whereas the energy of the *k*
_{2} mode is mostly distributed in the two side-rows. Along the propagation direction, the two modes interfere with each other via establishing the phase difference, thus the field pattern changes its manifestation from one to another in a propagation distance (coupling length)

Since the *k*
_{1} mode is almost the same as the original single-row mode, the length of the coupling section of the PDWBS for splitting one beam to two can be approximately set as the coupling length *L _{c}*. This is the central idea for designing the PDWBS based on the directional-coupling mechanism. Here we have utilized only the simplest concept of mode interference. More thorough treatment of coupled waves can be found in [14].

To design the PDWBS, we have to find the appropriate coupling length. We name the two structures in Fig. 1(a1) and (b1) as structure A and structure B, respectively. We vary the width *L _{y}* and recalculate the dispersion relations of the modes, and then derive the corresponding coupling lengths according to Eq.(1). For

*L*= 1.0

_{y}*a*, 1.2

*a*, and 1.4

*a*, the results are shown in Fig. 2(a) and (b). As one can see, when we reduce the width of the coupling section (the three rows region), the coupling length is also reduced. For a given width

*L*the coupling length for structure A is a monotonically decreasing function of frequency. Besides, there is a cut-off frequency. For example, when

_{y}*L*= 1.2

_{y}*a*, the cut-off frequency is

*a*/λ = 0.23. This frequency is the top frequency of the first band. In fact, it is impossible to couple two modes, one in the 1st-band and one in the 2nd-band, above this frequency. This explains why there is the cut-off frequency. On the other hand, the coupling length for structure B is almost a constant in a wide range of frequency from 0.2 to 0.24, insensitive to the frequency variation. This feature indicates that structure B is the better choice to be used as the coupling section of a wideband PDWBS. In the following simulations we choose structure B of length 8

*a*as the coupling section of the PDWBS, and the width

*L*is assumed to be 1.2

_{y}*a*.

## 4. System description and numerical results

We now turn to the system description before doing numerical simulations. Fig. 3(a) illustrates the structure of the beam splitter, which consists of an input waveguide, a coupling section, two S-shaped arms, and two straight output waveguides. The input and output waveguides are single row PDWs of 20 and 10 rods, respectively. The coupling section is of the type B structure of length *L _{c}* = 8

*a*and width

*L*= 1.2

_{y}*a*. An S-shaped arm (excluding the output straight PDW) contains 61 rods, forming the two oppositely curved bends and one straight PDW in between. Hereafter we denote the two arms as Arm

*and Arm*

_{a}*. For any one of the two arms, say, Arm*

_{b}*, the two bends (arcs) inside are identical, and can be characterized by the bending angle*

_{a}*θ*. The bending angles

_{a}*θ*for the two arms take arbitrary values from 0° to 90°, and the bend radius

_{a,b}*R*is chosen to be 19.1

*a*. In a bend, one lattice spacing a contributes 3° to the bending angle. Thus for a 90° bent S-shaped arm the straight PDW between the two bends disappears. In order to calculate the transmissions

*T*, we insert three planes of width 6

_{a,b}*a*in the input and output PDWs at the positions indicated by Fig. 3(b),(c) and evaluate the energy fluxes

*P*, and

_{i}*P*. The transmissions through the two output PDWs that connected with Arm

_{a,b}*are then defined as*

_{a,b}*T*=

_{a,b}*P*/

_{a,b}*P*.

_{i}We now study the influences of the bending angles *θ _{a,b}* on the transmission spectra

*T*(ω). We first study the symmetric case of

_{a,b}*θ*=

_{a}*θ*, here we define

_{b}*θ*=

_{sp}*θ*as the splitting angle. It is obvious that in this case we must have

_{a}*T*=

_{a}*T*. In order to reduce the interference effect between the two arms, we only consider the situations that

_{b}*θ*> 9°. The transmission spectrum

_{sp}*T*(ω) for different values of

_{a,b}*θ*is shown in Fig. 4. The simulation result reveals that on average for one arm the transmission is higher than 0.425, that is, the averaged total transmission is higher than 0.85, if 9° <

_{sp}*θ*< 90° are considered. The total transmission is high enough so this symmetric structure can indeed be used as a beam splitter. Furthermore, high splitting ability (> 0.46) can be achieved around

_{sp}*a*/λ = 0.219. It seems that power transfer can be accomplished more efficiently in the central part of the frequency range we considered.

Now we explore the applicability of the asymmetric configuration of the three-branch PDW structure as a beam splitter. As mentioned before, the number of rods in the two arms are the same. We first fix the bending angle *θ _{a}* and study the influence of the bending angle

*θ*on the transmission spectra

_{b}*T*(ω). Two simplest situations are considered. In the first situation we set

_{a,b}*θ*= 0° and vary

_{a}*θ*from 9° to 90°, and the calculated

_{b}*T*(ω) and

_{a}*T*(ω) are shown in Fig. 5(a1) and (b1), respectively. Similarly, in the second situation we set

_{b}*θ*= 90°, and

_{a}*T*(ω) are shown in Fig. 5(a2) and (b2). From the results of Fig. 5(a1) and (b1), we find that

_{a,b}*T*is in general larger than

_{a}*T*, and on average we have

_{b}*T*:

_{a}*T*≈ 0.63 : 0.37, implying that the internal reflection of wave in the straight arm is much smaller than in the bent arm. Moreover, the typical value of total transmission is larger than 95%, exceeding that of the symmetric structure (see Fig.4). For case of

_{b}*θ*= 90°, the results are very different, as shown in Fig. 5(a2) and (b2). In this situation, light powers are more equally distributed between the two arms. The average splitting ratio

_{a}*T*:

_{a}*T*is 0.44 : 0.56. It indicates that the asymmetric configuration of three-branch structure can indeed be used as a practical beam splitter.

_{b}We now study the influence of varying both *θ _{a}* and

*θ*on the transmissions

_{b}*T*. We choose

_{a,b}*a*/

*λ*= 0.219 and plot the calculated

*T*in Fig. 6. The horizontal and vertical axes are

_{b}*θ*and

_{a}*θ*, respectively. By symmetry consideration, the result for

_{b}*T*can be easily obtained by the reflection mapping with respect to the 45°-oriented line (the line described by the equation

_{a}*θ*=

_{a}*θ*). According to the numerical results, the ratio

_{a}*θ*:

_{a}*θ*on average is 0.48 : 0.52. In addition, for the major portion of the transmission map shown in Fig. 6,

_{b}*T*and

_{a}*T*are almost equal. This indicates that asymmetric PDWBS can work well in most situations.

_{b}To demonstrate the splitting abilities of the symmetric and asymmetric configurations of PDWBS, we plot the corresponding field patterns at frequency *a*/λ = 0.219 in Fig.7 (a) and (b). In Fig. 7(a), the splitting angle for the symmetric PDWBS is taken to be *θ _{sp}* = 60°. In Fig. 7(b), we set

*θ*= 75° and

_{a}*θ*= 39° for the asymmetric PDWBS. The field patterns confirm again the beam splitting abilities of the two kinds of structures. We believe both the symmetric and asymmetric structures of PDWBS can be used as practical devices in the optical integrating circuits.

_{b}In this paper the band structure of the PDWBS is calculated using the FDTD method, whereas the transmission data in Fig. 4–6 and the field pattern in Fig. 7 are obtained by implementing the multiple scattering method. One can also calculate the transmission data using FDTD method, however, it takes a lot of time. The multiple scattering method is the faster method in dealing with such kind of calculation when compared with the FDTD method. However, the multiple scattering method also has its weakness. For instance, the method is hard to implement directly if the cross section of the dielectric cylinder is not circular. Besides these differences, the FDTD method is as good as the multiple scattering method when be used to calculate the field distribution. We re-calculate the field distribution for the symmetric structure of Fig. 7(a) using FDTD method by launching the light from the end of single row waveguide. A snapshot of the field pattern (the real part) around the coupling section of the PDWBS is shown in Fig. 8. The result confirms that the light energy can be successfully switched to the two output rows of rods.

Recently, Zhang and Li proposed to use line-defect bending waveguides in photonic crystal for optical interconnections [16]. According to their numerical simulations, high transmission over 98.5% can be achieved if the various bending waveguides are operated at frequencies within the photonic band gap. It seemed possible to design compact beam splitter based on this kind of waveguide for power division while avoiding the lattice orientation restriction.

## 5. Conclusion

In conclusion, we have studied the transmission properties of PDWBS numerically. Effects of symmetric and asymmetric bending of the arms on the transmissions are considered. By employing a simple analysis on the mode coupling condition, the coupling length can be determined. Numerical results reveal that high transmissions can be preserved for both the symmetric and asymmetric structures. The advantage that the geometric restriction of PCW can be avoided might be helpful in making compact optical circuits.

## Acknowledgments

The authors gratefully acknowledge financial support from National Science Council (Grant No. NSC 95-2221-E-008-114-MY3) and Ministry of Economic Affairs (Grant No. 95-EC-17-A-08-S1-0006) of the Republic of China, Taiwan.

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