Abstract—A novel two-dimensional photonic crystal slab wave-guide based on an antiresonant reflecting optical wavewave-guide (ARROW) structure is proposed and designed. Lightwaves prop-agating in this waveguide are confined by antiresonance reflection vertically and the photonic band gap laterally. In order to ob-tain the characteristics of the ARROW-based photonic crystal waveguides, the three-dimensional finite-difference time-domain simulations are performed. With a lateral adiabatic taper, a coupling efficiency of 80.3% from a single-mode fiber to the ARROW-based photonic crystal waveguide of a single-line defect is obtained. In addition, propagation losses less than 10 dB/mm and bend losses of 0.23 and 0.39 dB/bend for the designed 60 and 120 bends are achieved at an operating wavelength of1 55 m.
Index Terms—Antiresonant reflecting optical waveguide (ARROW), bending waveguide, finite-difference time-domain (FDTD) method, photonic crystals (PCs), propagation loss.
I. INTRODUCTION
T
WO-DIMENSIONAL (2-D) photonic crystal (PC) waveguides have the strong potential for constructing ultra-compact photonic integrated circuits due to their abil-ities of controlling lightwaves. In general, the waveguides are formed by introducing line defects into the lattices of air holes in dielectric slabs. Lightwaves are confined in-plane by the photonic band gap, in which propagation of photons is forbidden. Utilizing this attractive advantage, waveguiding in straight channels [1]–[7] and even sharp bends [8]–[11] with low losses can be realized. In the vertical direction, there have been two types of guiding mechanisms demonstrated in the previous literatures. First, the 2-D PC slab waveguidesManuscript received July 30, 2008; revised November 27, 2008. Current ver-sion published July 01, 2009. This work was supported by the National Science Council of the Republic of China under Contract NSC92-2215-E-009-047.
Y.-L. Yang is with the Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, Hsinchu 30010, Taiwan (e-mail: [email protected]).
S.-H. Hsu is with the Research Center for Applied Sciences, Academia Sinica, Taipei 11529, Taiwan (e-mail: [email protected]).
M.-F. Lu is with the Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, Hsinchu 30010, Taiwan, and also with the Department of Electronics Engineering, Minghsin University of Sci-ence and Technology, Hsinfeng, Hsinchu 304, Taiwan (e-mail: [email protected]. tw).
Y.-T. Huang is with the Department of Electronics Engineering and Insti-tute of Electronics, and the Department of Biological Science and Technology, National Chiao Tung University, Hsinchu 30010, Taiwan (e-mail: huangyt@cc. nctu.edu.tw).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JLT.2009.2014692
of high-index-contrast type are implemented by means of suspending the thin core membrane in air [1] or patterning structures in silicon-on-insulator (SOI) wafers [2], [3] for providing strong confinement. However, the low propagation loss have been achieved only in the narrow bandwidth in this configuration. Second, the low-index-contrast PC slab waveg-uides are usually investigated in InP-based [4] or GaAs-based [5] heterostructures. Since the guided modes of PC waveguides exist above the light lines of cladding layers, severe vertical radiation losses exhibit in the single-line-defect waveguides [12]. To obtain high transmission, the multimode propagation is unavoidable.
In practical applications, it is desirable to connect PC waveg-uides with fibers. However, owing to the large impedance mis-match between them, finding an efficient and direct method for coupling lightwaves into and out off PC circuits is not a trifle. To solve this difficulty, many couplers have been proposed such as gratings, mirrors, and fiber tapers [13]–[15], etc. For the butt coupling, however, it is still important to reduce the coupling losses in the vertical direction.
Compared with these two guiding mechanisms in the vertical direction, a 2-D PC slab waveguide with a relatively thick core layer is proposed and designed to improve the butt-coupling ef-ficiency with a fiber in this report. In the vertical direction, we adopt the Fabry–Pérot cavities as reflectors [16], [17] under the core layer of a PC waveguide instead of one cladding layer of a lower refractive index. Lightwaves propagating in the core layer meet total internal reflection at the air-core interface, and very high reflection from the set of interfaces of the cavities. This structure so-called an antiresonant reflecting optical waveguide (ARROW) can be designed to support low loss propagation in a core layer of a relatively large size according to flexible design rules. Laterally, the periodic arranged air holes are implemented in the ARROW structure to provide the photonic-band-gap con-finement. In our investigation, characteristics of the butt-cou-pling loss between the ARROW-based PC waveguide and a single-mode fiber, the propagation loss of the straight waveguide and bend losses at the 60 and 120 sharp bends are studied. All these losses are calculated by the finite-difference time-domain (FDTD) method performed in the commercial software (Full-WAVE by RSoft).
II. DESIGN OFARROW-BASEDPC WAVEGUIDES
As shown in Fig. 1(a), the ARROW structure is a multilayer planar waveguide consisting of a thick core layer of refractive index , and two cladding layers of a high refractive index
Fig. 1. Schematic pictures of (a) an ARROW structure and (b) an ARROW-based PC waveguide.
and a low refractive index between the core layer and sub-strate of index . Lightwaves in the core layer are confined by means of total internal reflection at the air-core interface and ultrahigh reflection from the cladding layers. To attain a low in-sertion loss with a single-mode fiber, the thickness of the core layer is chosen as m. The optimized and thinnest thick-nesses of these two claddings and satisfy the antiresonance condition as [16]
(1) (2) where is the free-space operating wavelength and equal to m in our design. Many materials are transparent in the objective wavelength range and can be exploited in this struc-ture. Here, we adopt the following parameters for the design: . The derived thicknesses of the first cladding and the second cladding are 0.13 and 3.50 m, respectively. First, the modal characteristics of the planar ARROW structure are calculated by the transfer matrix method to obtain the effective index and the propagation loss for each transverse electric (TE) and transverse magnetic (TM) mode. For the mode, the propagation loss is lower than that of higher order modes at least one order of magnitude, as listed in Table I. Reflections of TM modes from interfaces of claddings are always lower because of the existing of the Brewster angles. Therefore, the TM and higher order TE modes can be effec-tively filtered out due to the loss discrimination to maintain the quasi-single mode propagation. The electric field profiles for the planar ARROW structure are shown in Fig. 2. The quasi-guided mode behaves like a fundamental guided mode in a con-ventional slab waveguide. The overlap integrals between the electric fields of the higher order modes and the mode are all lower than 0.03%. This result indicates that the amounts of mode converting into higher order modes are extremely small. In addition, the excitation coefficient of the mode from the launched Gaussian profile is about 95.9%. Therefore,
Fig. 2. Electric field profiles of theTE to TE modes for the planar ARROW structure.
TABLE I
EFFECTIVEINDICES AND THEPROPAGATIONLOSSES OFTEANDTM MODES
the losses from higher order modes can be neglected, and our discussion is focused on the mode.
The PC pattern is embedded in the core layer of the ARROW structure. The periodicity of air holes is arranged with a trian-gular lattice structure, and a PC waveguide is formed by re-moving one row of air holes along the -direction ( -K direc-tion), as shown in Fig. 1(b). Air hole radius is equal to , where is one lattice constant. To probe the confinement of the PC waveguide in this guiding layer of a low refractive index , the distribution of defect modes in the band dia-gram has to be confirmed. We convert the 3-D structure to an effective 2-D PC plane. The effective index of the mode in the ARROW structure is used as the background index in the 2-D calculations, and the refractive index of air holes is still 1.0. By this approximation, the projected band structure is calculated with the 2-D plane-wave expansion (PWE) method and the su-percell technique in the commercial software (BandSOLVE by RSoft), as shown in Fig. 3(a). It is found that two defect modes are induced in the range of the band gap, . One is an even mode, and the other is an odd mode. The former’s field is symmetric to the bisecting plane of the PC wave-guide, and the latter’s is antisymmetric to that. The even mode existing from frequency 0.378 to 0.45 with a negative slope, stemming from the guided mode being folded at the edge of the Brillouin zone, is dominated by the index confinement. In the flat dispersion range from 0.370 to 0.378 , it can be ex-plained with the mechanism of the band gap confinement. In ad-dition, the 2-D FDTD method with the effective index is used to compute related transmission spectra under the boundary condi-tion of the perfectly matched layer (PML). The lattice constant
is specified as m and the spatial resolution is set as in the PC plane ( -plane). A pulse with a Gaussian profile of TE mode is inputted in the front of the PC waveguide to mainly excite the even mode in the waveguide over a wide range of fre-quencies. The input power is set as unity. The time monitor is located at the end of the waveguide to detect the power trans-mission through it. By Fourier transforming the fields recorded
Fig. 3. (a) Projected TE band structure for the 2-D PC waveguide of the single-line defect. The light gray area is the region of PC bulk bands. (b) Related trans-mission spectra of the straight waveguides with different propagation lengths and the waveguide incorporated with a conventional 60 bend. The total prop-agation length of the bending waveguide is20a. These waveguides are all of nine rows of PCs on each side of guiding channels.
on the monitor in every time step, transmission spectra are cal-culated and shown in Fig. 3(b). It can be seen that the trans-mission spectra match well with the band structure except some blue shifts. In the frequency range of index guiding, transmis-sion efficiencies can reach high values above 92%, and the prop-agation losses of the band-gap guided mode near the band edge are extremely low.
III. BUTT-COUPLING ANDPROPAGATIONLOSSES IN
ARROW-BASEDPC WAVEGUIDES
To obtain more accurate results, a fully 3-D calculation is necessary to compute the power radiate out of plane. The 3-D FDTD method and the boundary condition of PML are used for ARROW-based PC waveguides with grid sizes of in the -plane and m in the -direction. Our devices are cal-culated with continuous waves at the wavelength of m as an input light source. The spatial distribution of the launched field is a Gaussian profile with a width of and a height of m. Without losing generality, the launched field is put in the center of the core layer. To verify the guiding mechanism of the ARROW-based PC waveguide, it is essential to find the rela-tion between the transmission efficiency and the air hole depth. As shown in Fig. 4, the transmission reaches a saturation value as air hole depth is increased to m, which is the same as the thickness of the core layer. This is because the field of the mode for an ARROW structure is mainly confined in
Fig. 4. Transmission versus air hole depth for the ARROW-based PC wave-guide. The lattice constanta is 0:674 m. The waveguide length is 20a with nine rows of PCs on each side of the guiding channel.
the core layer, where the TE mode in the 3-D configuration is defined that the electric field is primarily along the -direction. The air hole depth should be deep enough to enclose the tail of the mode sufficiently and confine the field laterally. As the depth of air holes is reduced, the field of lightwaves would pen-etrate into the PC bulk laterally to result in increase of propaga-tion losses. The lightwaves becomes intrinsically radiative for the depth m, and the transmission still maintains around dB due to the short propagation length in this simulation. Increasing the thickness of the core layer is beneficial for butt coupling with a single-mode fiber, whereas gives a challenge for the process of the deep etching. With the advanced integrated circuit technology, etching air holes of high aspect ratios has been achieved in previous papers [4], [18], [19]. This makes the ARROW-based PC waveguides realized more possibly.
For the SOI-type PC slab waveguides, the thickness of core layer is smaller than m to keep single-mode propagation. By simulating with the 3-D FDTD method, the butt coupling efficiency between the waveguide and the single-mode fiber, which has a core diameter m with a refractive index 1.451 and a cladding diameter m with an index 1.445, is shown as low as only 3.3% without the assistance of couplers. This efficiency can be significantly improved to 30.3% for the ARROW-based PC waveguide due to the relatively large core size and the matched refractive index of the core layer. There have been many lateral couplers proposed for efficiently converting guided modes from wide waveguides to narrow waveguides. Here, we consider the simplest case of an adiabatic taper, waveguide structure varing linearly along the propagation direction from the 7- m width to the width, where is equal to m. The single-mode fiber is directly butted in contact with the taper as shown in Fig. 5. Through this mode converter, it is found that the 95.1% of power from the fiber transmits through the interface between the fiber and the taper by the 3-D FDTD simulation. The taper itself conserves distribution of the power in the waveguide during propagation and provides a 89.1% transmission. In addition, the 3-D FDTD simulation gives a transmission of 94.8% through the interface of the taper and the ARROW-based PC waveguide. By means of multiplying these three values, the overall butt-coupling
efficiency of 80.3% is obtained between the fiber and the PC waveguide.
As discussed above, the light can be confined laterally by 2-D PCs based on the photonic band gap, and vertically in the core layer of the ARROW structure by satisfying the antires-onance condition. A propagation loss of a PC slab waveguide is an essential issue, however, performing its calculation is still difficult owing to large consumption of time and computational memories [12], [20], [21]. In this paper, the 3-D FDTD sim-ulation is used for calculating the power of lightwaves trans-mission through the detecting monitors, which are placed along the waveguide of a length of . The distance between two adjacent monitors is fixed at three lattice constants. The propa-gation loss at each frequency can be obtained from the linear fit of the recorded transmission efficiencies on the monitors. The launched field is a Gaussian profile for exciting the even mode of PC in the band gap laterally and the mode of ARROW structure vertically. The operating wavelength is fixed as m to keep the optimized performance of the ARROW structure, and the lattice constant of the PC is varied to obtain the transmission spectrum, as shown in Fig. 6. In comparison with Fig. 3, the transmission spectrum of the ARROW-based PC waveguide agrees with the 2-D case and its corresponding projected TE band structure. Further, it can be seen that the losses gradually decrease from low to high frequencies, which is unobvious in transmission spectra of 2-D case. This can be simply explained that a mode of higher frequency is more re-fractive-like, and its field profile is well confined in the channel of the PC waveguide as shown in Fig. 7(a) and (b). Conse-quently, this causes less interaction between lightwaves and air holes [12], [21], resulting in less coupling to radiation modes in light cone of air. It is noticeable that the attenuation less than 10 dB/mm predicted for the ARROW-based PC waveguide is of the same order of magnitude for the PC slab waveguides of low-index-contrast type with a three-line defect [4]. In ad-dition, PC waveguides on an ARROW platform still maintain single-mode propagation both in the vertical and the lateral di-rections. Lightwaves of the mode propagating in the core layer of the ARROW structure with larger ray angles results in the less number of reflection at the air-core interface. Even op-erating in the radiation region (above the light line), therefore, the ARROW-based PC waveguides can keep acceptable losses for the photonic integrated circuits in a compact chip. For the same reason, it can be explained that losses of air-bridge-type PC slab waveguides are as high as 120 dB/mm when operating above the light line [1].
Fig. 6. Transmission spectrum for the ARROW-based PC waveguide. The ARROW-based PC waveguides display propagation losses less than 10 dB/mm in the range of the lattice constanta from 0.651 to 0:704 m, in which the lowest loss is 6.31 dB/mm(a = 0:694 m).
Fig. 7. Power distributions at the frequency of (a) 0.382 and (b) 0.447(a=) in the cross section of the ARROW-based PC waveguide. The corresponding lattice constants are 0.594 and0:694 m, respectively.
IV. BENDLOSSES INARROW-BASEDPC WAVEGUIDES
For building up compact optical circuits, not only straight waveguides but also bends are required. One of the most attrac-tive features for PC waveguides is to allow sharp bends with low losses. However, the spectral dependence of the bend losses and the propagation losses should be considered at the same time. Some designs of sharp bends focus on achieving high transmission through the corners of the bends, but the spec-tral range of the high transmission is located above the light line, suffering from severe out-of-plane losses for the PC slab waveguides of high-index-contrast type. As shown in Fig. 3(b), the conventional 60 bend exhibits high transmission in the fre-quency range where the straight waveguide has high losses. We adopt the same design structure of a 60 bend in [9] and use the similar fashion to design a 120 bend for obtaining high trans-mission in the frequency range of 0.420 to 0.450 , in which the ARROW-based PC waveguide has low propagation losses. For the designed 60 bend, air holes A and B around the corner of the bend are linked by a rectangular air defect with a width equal to the diameter of the air holes and a length of , which is the distance between the centers of the holes A and B as shown in Fig. 8(a). The linked hole defect is merged with the air hole C which is located on the site of the triangular lattice in the con-ventional 60 bend to form the reshaped defect. As shown in Fig. 8(b), the air hole D with a radius is inserted on the site of the lattice, and an identical linked hole defect except a longer length is incorporated in a distance of to the hole D in the structure of the designed 120 bend. The designed 60 and 120 bends are connected with two single-line defects with
Fig. 8. Schematic pictures of (a) designed 60 and (b) 120 bends. Air holes A, B, and C are located on the sites of the triangular lattice in the conventional 60 bend. For the designed 120 bend, the air hole defect surrounded with a solid line is additionally placed in the structure of the conventional 120 bend. is the tilt angle of the linked hole defect with respect to the z-axis and set as 60 in our design.
length of along the direction before and after bends. The bend losses in PC waveguides are mainly due to the large re-flections around the corners of the bends, in which periodicity of the lattice is discontinuous. In addition, lightwaves propagating through sharp bends would cause redistribution of the energy and excite the higher order modes, but only the modes matching the guided modes in the output single-line defect can propagate efficiently. The coupling mismatch between the modes at the bends and in the straight waveguides also leads to considerable losses.
To obtain the transmission property of bending waveguides based on the ARROW structure, the approximation in [22] and [23] is adopted at the bends. In this approximation, the air holes on the inner and outer sides of the bend are all assumed as ho-mogeneous air claddings, so the bend region is taken as a short ridge waveguide, in which the confinement of light is support by total internal reflection in the lateral plane and by antiresonance reflection in the vertical direction. The width of the ridge wave-guide is , an average width of the bend region. The bend loss would be low when the guided mode in the straight wave-guide and the transient mode at the bend region are matched. The propagation constants of the transient mode are calculated by the transfer matrix method and the effective index method. The dispersion curves of the guided mode in the ARROW-based straight PC waveguide and the transient mode at the bend are both shown in Fig. 9(a). For simplicity, the 1-D scattering model [22] is used to estimate the reflection coefficient R through the bend, which is given as
(3) where and are wave vectors of the straight PC and the ap-proximate ridge waveguide, respectively. is the propagation length of the ridge waveguide, for the 60 bend and for the 120 bend, respectively. By this model, only the compo-nent of wave vectors along the direction is considered, and high transmission through the bends can be obtained in the frequency range where and are close. To verify the va-lidity of this model, the 3-D FDTD simulation is performed at the operating m as shown in Fig. 9(b).
Fig. 9. (a) Dispersion curves of the guided mode(k ) in the straight PC waveguide and the transient mode(k ) at the designed bends. (b) Transmission spectra of the 60 , 120 bending and straight PC waveguides based on the ARROW structure. The inset shows the bend loss spectra of the designed bends, in which low loss region( 0:5 dB/bend) is from frequency 0.425 to 0.437(a=). The corresponding lattice constant a is from 0.659 to 0:677 m.
In order to extract the bend losses, transmission spectra of a straight waveguide with the same propagation length is shown together with that of 60 and 120 bending waveguides. Ac-cording to the distribution of the dispersion curves, the highest transmission is at the frequency about 0.415 where these two curves cross, but it shifts to 0.438 for the 60 bending waveguide and 0.432 for the 120 one in the 3-D FDTD simulation. This stems from the discrepancy between the 2-D approximation and the 3-D calculation. For the 60 bend, the low bend loss range (transmission dB) is shown in the frequency range of 0.422 to 0.441 . On the other hand, the 120 bend exhibits a lower transmission in the same fre-quency range because the scattering loss is increased around the corner of the bend. Below the frequency of 0.410 , the difference between and becomes large, resulting in invalidity of the 1-D scattering model. It is noticeable that the frequency range of high transmission for the bends agrees with that for the straight waveguide. The bend losses are obtained by normalizing the transmission spectra of bends to that of the straight waveguide as shown in the inset of Fig. 9. For example, bend losses are 0.23 and 0.39 dB/bend at the frequency of 0.435 m for the 60 and 120 bends, respectively. At the same frequency, the propagation loss for the straight waveguide is around 8 dB/mm. As a result, PC waveguides based on the ARROW structure can provide low loss propaga-tion for both straight and bending waveguides in the same fre-quency range.
based PC waveguide can has flexible design rules with a large core size for improving coupling efficiency with a single-mode fiber to 80.3%. For our designed device operating at a wave-length of m, the propagation losses less than 10 dB/mm can be obtained in the range of the lattice constant from 0.651 to m with the hole radius-to-lattice constant ratio of 0.35. As the designed sharp bends incorporate into the ARROW-based PC waveguide, the normalized frequency range of high transmission for the bends agrees with that for the straight wave-guide. For the lattice constant of m, the bend losses are as low as 0.23 and 0.39 dB/bend for the 60 and 120 bends, respectively.
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Yu-Lin Yang was born in Taipei, Taiwan, in 1980.
He received the B.S. degree in physics from National Central University, Jhongli, Taiwan, in 2002 and the M.S. degree in electronics from National Chiao Tung University, Hsinchu, Taiwan, in 2004. He is currently working toward the Ph.D. degree in electronics at Na-tional Chiao Tung University.
His current research is focused on 2-D photonic crystal waveguides and optical properties of 3-D metallic photonic crystals.
Shih-Hsin Hsu (S’99–M’06) received the B.S.
de-gree in physics from the National Taiwan University, Taipei, Taiwan, in 1994 and the Ph.D. degree in electronics from the National Chiao Tung University, Hsinchu, Taiwan, in 2005.
He is a Postdoctoral Research Fellow with the Research Center for Applied Sciences, Academia Sinica, Taipei. His current research is focused on nanophotonics and optical characterizations.
grated optics, photonic crystal waveguides, bio-optoelectronics, and optoelec-tronic switching networks.
Prof. Huang received the Outstanding Research Award from the National Sci-ence Council in 1998.
