1. IntroductionSince the first proposal in 1987 of the full photonic bandgap concept [1,2], photoniccrystals (PhC) have attracted much interest because of their ability to confine light insmall volumes, making them a strong candidate as a platform for the realization ofcompact integrated optoelectronic circuits [3]. One of the most important elements insuch a scenario is the one responsible for routing light to and from different elementsin the circuit, namely, the photonic crystal channel waveguide [4–7], typically formedconceptually by removing lines of lattice elements. Even if it relies on a method ofguiding that is substantially different from the classical one of total internal reflec-tion, the strong lateral confinement of PhC channel waveguides also implies the pos-sibility of significant reflection at the discontinuities. The optical properties of suchwaveguides in various systems have been modeled [8,9], as well as verified experimen-tally [10–13].
To take full advantage of the positive qualities of PhC channel waveguides, onebasic challenge becomes the ability to connect PhC waveguides using the dielectricwaveguides traditionally employed in conventional integrated optoelectronics, mini-mizing the mismatch induced by the different types of waveguiding mechanisms. Pre-vious theoretical work has shown optimized geometries for this type of coupling thatemploys tapered couplers [14–16] and lenslike elements, providing coupling efficien-cies as large as 90%. However the devices under investigation incorporated intercon-nection between a dielectric waveguide and a line defect channel waveguide in asquare lattice of dielectric rods in air, a situation that is not typical in practice at opti-cal frequencies. In real planar waveguide-based devices operating at optical frequen-cies, short dielectric rods alone cannot provide the required vertical confinement. For
Vol. 6, No. 2 / February 2007 / JOURNAL OF OPTICAL NETWORKING 91
this reason, and to take advantage of their compatibility with existing planar integra-tion technologies, the configuration with PhC channel waveguides formed by a line ofdefects in a lattice of air holes created in a host dielectric is preferred in the presentwork, as elsewhere [17–25].
In the present study, the behavior of direct and abrupt junctions between ridge andPhC channel waveguides has been investigated. The transmission of the cascade ofthe two types of waveguides is evaluated in different configurations for the interfacesand has been obtained by changing the position and width of the ridge guide relativeto the PhC channel waveguide. Knowledge of the maximum transmission configura-tions and their sensitivity to dimensional and positioning variations is especially use-ful in relation to the fabrication of real device structures and to the associated errorsor deviations that occur during the fabrication process. In contrast to previous work[18,20,22], the study includes experimental verification of the simulated structures:their behavior is first computed by means of a 2D approximation and subsequentlycompared to characterized devices fabricated on low-index contrast material. More-over, targeting a measure of generality, the investigation is carried out over a largebandwidth, without focusing on specific propagation regimes of the channel wave-guide (e.g., low group velocity [25]) and concentrating on the above-the-light-linepropagation characteristic of waveguides with weak vertical confinement. This situa-tion applies to the structures investigated experimentally.
The following section will introduce the structure under investigation by providinga description of the method used to carry out the analysis. Results of simulations, fab-rication, and characterization (together with the respective techniques used) will bepresented in Sections 3–5, respectively. Conclusions will be presented in Section 6.
2. Analysis: Formulation of the ProblemAnalysis was carried out on device structures designed to operate at wavelengthsaround a central value of 850 nm. 2D plan-view representation of the structure underinvestigation is given in Fig. 1(a). The host dielectric material has a refractive indexof neff=3.4, calculated as the effective refractive index of the slab waveguide of theGaAs/AlGaAs material subsequently used for fabrication, and the photonic crystalchannel waveguide �W1� is defined as a single line defect channel formed by removinga row of holes from a triangular lattice of air holes along the nearest-neighbor direc-tion. The PhC lattice period a is 215 nm, and the surface air filling factor is 0.35, pro-viding a computed bandgap for TE light of 260 nm centered at �850 nm. The W1waveguide has a length of 18 periods (a distance equivalent to 15 optical wavelengths,long enough for the computed field to stabilize). The confining PhC regions are com-posed of eight columns of holes on either side, providing very adequate confinementwhen the channel waveguide is operated at wavelengths lying within the PhC band-gap. The ridge waveguide is defined by rectangular air areas on either side of thedielectric waveguide, each of 500 nm width and sufficient to prevent any coupling intothe lateral dielectric regions.
Fig. 1. Device under investigation. Whole device (a) and related parameters studied atthe interface (b). W represents the width of the ridge waveguide, SA represents the axialshift, and SL represents the lateral shift with respect to the W1 axis. The junction isshown at its reference position. Generic configurations are obtained by a rigid transla-tion of the ridge (i.e., of the air rectangles) and are identified by the couple �S ,S �.
L A
Vol. 6, No. 2 / February 2007 / JOURNAL OF OPTICAL NETWORKING 92
To characterize the behavior of the interface between an input ridge waveguide anda W1 PhC waveguide, the transmission of the whole system was probed over a rangeof different widths and relative positions of the feeder ridge. Three parameters associ-ated with the ridge waveguide were specified for analysis, as indicated in Fig. 1(b): thewidth of the ridge waveguide �W� and the coordinates of the position of the edge of theridge waveguide with respect to the W1 PhC channel guide axis.
To keep the situation simple and to analyze the aforementioned parameters over awide range, no further variations of the geometry of the structure were made [22].
The position was defined by SA and SL, the axial (along the z direction) and lateral(along the x direction) shifts from a normal position. Targeting a generalized descrip-tion of the system, the three parameters may be expressed in terms of the crystalperiodicity: W and SL represent the width and the lateral shift values, normalizedwith respect to the horizontal period, which is defined as Xperiod=a�3�=372 nm�, whereas SA represents the axial shift normalized with respect to the lat-tice period a �=215 nm�. At the normal initial position [as depicted in Fig. 1(b)], theedges of the input ridge waveguide touch the centers of the first two holes in the PhCchannel waveguide, and the two waveguides share the same axial symmetry axis. Ageneric configuration is obtained by a rigid translation of the ridge (i.e., of the two airboxes) and, in what follows, it will be identified by the compact notation �SL ,SA�,which provides a concise definition of the relative positions.
In the following, the analysis is first presented by describing the results of 2D simu-lations of the structure: because of the planar nature of the problem, this type ofinvestigation is useful in that it gives credible guidelines for what to expect from realdevices, with a reduced computational effort. Fabrication techniques and the charac-terization of fabricated devices are presented subsequently, and comparison is madewith simulation results, allowing the verification of the best coupling conditions andthe identification of discrepancies with a completely planar approximate description ofthe structure.
3. Simulation ResultsA commercial implementation of the 2D finite-difference time-domain algorithm wasemployed: the computational domain was discretized using a 15 nm grid, which pro-vided a good trade-off between the degree of approximation and calculation time. Per-fectly matched layers were employed as boundary conditions (width 0.5 �m, reflectiv-ity 10−8). The ridge was excited with its fundamental mode at 850 nm (computedseparately), and the response of the whole structure was probed with (computational)power monitors placed at the output end of the W1 channel waveguide withoutanother interface to a ridge. By having only one interface in the analysis, the genera-tion of spurious Fabry–Perot cavity effects was avoided, thus keeping the observationof the effect of the single interface more direct and with reduced requirements onspectral dynamics.
The investigation of the effects of varying the width of the ridge waveguide, speci-fied by the parameter W in Fig. 1, was carried out with normalized width values from0.4 up to 1.5, in steps of 0.1, at two extreme positions. These positions were the nor-mal one (0,0) and the one in which the ridge waveguide is inserted into the W1 chan-nel by 0.45 normalized units [position (0,0.45)]. These two positions provide hightransmission in the study of the axial displacements (see Fig. 4).
The results are shown in Fig. 2: the transmission for both cases exceeds 90% andexhibits a fairly flat response in the region up to a normalized ridge width of 1.Beyond that point, both responses fall off as the mismatch between the two types ofwaveguides is increased.
In the case where the ridge waveguide is axially offset by 0.45 units, the falloff intransmission is stronger than for the case where the ridge is placed at the normalposition.
Figure 3(a) demonstrates the results of lateral shifts (specified by the parameter SLin Fig. 1) of the ridge waveguide from the normal position (0,0) up to (0.6,0), withsteps of 0.05 normalized units. The different curves correspond to different ridgewidths, and the results confirm the response expected intuitively: a decrease in trans-mission with the increased offset of the axis of the two guides. For all the ridgewidths, the transmitted power drops off by 20% at a position with a lateral shift of
Vol. 6, No. 2 / February 2007 / JOURNAL OF OPTICAL NETWORKING 93
Fig. 3. (a) Transmission versus normalized ridge lateral displacement �SL� for variousridge normalized widths. (b) Intensity map of the field corresponding to the case of theridge waveguide position parameters SL=0.4, SA=0, and W=0.7, showing the field re-flection and diffraction at the interface of the two waveguides.
Fig. 2. Transmission versus normalized ridge waveguide width �W� when the ridge isplaced on the normal position �0,0� (solid curve) and when it is inserted in the PhCchannel waveguide [position �0,0.45�, dashed curve].
Vol. 6, No. 2 / February 2007 / JOURNAL OF OPTICAL NETWORKING 94
0.15 normalized units (absolute distance 56 nm) and has dropped by half for the rangeof normalized lateral shifts between 0.3 and 0.4. For greater shifts, the falloff contin-ues, with larger impact for narrower ridges, since more of the ridge section isobstructed by the side of the PhC waveguide channel, which scatters the incidentpower [Fig. 3(b)].
As a final case, the effects of axial shifts (specified by the parameter SA) of the ridgewaveguide were considered. For each value of width (from 0.6 to 1.5), the ridgewaveguides were shifted along the z axis from position �0,−1� to �0,1�.
Figure 4(a) shows the results for ridges with widths in the range from 0.6 to 1.0.Considering axial ridge positions greater than or equal to zero, the transmission of allthe ridges is high, of the order of 94%–98% and is characterized by ripples following ageneral trend of two minima (close to positions 0.25 and 0.7) and three maxima (closeto axial shift values SA equal to 0, 0.5, and 1).
When the ridge is positioned outside the PhC channel waveguide [Fig. 4(a), fornegative values of the axial ridge displacement], a minimum is observed at position�0,−0.3�, followed by a region of high transmission, on the order of 96%, which decaysrapidly for axial displacements of the ridge waveguide smaller than −0.8 normalizedunits. This decay is mainly due to the effect of spurious coupling into the horizontalchannel formed by the space between the ridge waveguide edges and the edge holes ofthe PhC region [Fig. 4(b)].
Increasing the ridge waveguide width causes a shift of the curve toward larger val-ues of SA, i.e., the transmission varies as though the ridge waveguide were shifted fur-ther from the PhC channel. The optical field leaving a narrower ridge waveguide end
Fig. 4. (a) Transmission versus normalized ridge axial displacement �SA� for normal-ized ridge widths 0.6–1. (b) Intensity map of the field corresponding to the case of theridge waveguide position parameters SL=0.0, SA=−1.0, and W=1.3, showing some ofthe field guided horizontally in the homogeneous region between the ridge and crystalregions.
Vol. 6, No. 2 / February 2007 / JOURNAL OF OPTICAL NETWORKING 95
requires a longer path to expand and reach a transverse distribution similar to that ofthe field leaving a wider ridge that is more closely located.
Figure 5 shows the dependence of the transmission on the axial shift, SA, for largerridge waveguide widths (for the range 1.0–1.5, in steps of 0.1). As the width becomeslarger, the transmission response becomes flatter, as well as more attenuated (onaverage). This behavior is the result of the larger difference between the two wave-guide widths causing increased reflection and scattering at the interface. Theresponse remains flat even when the ridge waveguide is placed some distance awayfrom the PhC channel, as the field profile experiences only a limited expansion in thehomogeneous region between the two waveguides.
In summary, the simulation results revealed that, at the central wavelength of850 nm (normalized frequency a /�=0.25), the best coupling between the ridge wave-guide and the PhC channel waveguide can be achieved using thinner ridge widths,with width values close to the distance between the rims of the neighboring holesdelimiting the waveguide channel. An axial offset close to multiples of half of the crys-tal periodicity and with no lateral offset is also required. Figure 6 shows the computedspectral response of the optimal structures (W=0.7, with SL=0 and SA=0; W=0.7,with SL=0 and SA=0.45) demonstrating a flat response, higher than 96%, over almosthalf of the bandgap region. As the figure also shows, small deviations from the opti-mal device configuration (e.g., W=0.7, SL=0.1, SA=0), especially in the case of a lat-eral shift, not only reduce the transmission but also reduce the flatness of the spectral
Fig. 5. Transmission versus normalized ridge axial displacement �SA� for normalizedridge widths W=1–1.5.
Fig. 6. Transmission spectra for the optimum device (dashed curve) and devices closeto the optimum one with respect to the axial shift (solid curve) and lateral shift (dottedcurve).
Vol. 6, No. 2 / February 2007 / JOURNAL OF OPTICAL NETWORKING 96
response. To obtain high transmission over large bandwidths, control of the position-ing of the ridge with an accuracy on the order of 0.1 normalized units (an absolutevalue of 40 nm) is shown to be required.
4. FabricationFabrication was carried out on an AlGaAs/GaAs epitaxial structure grown by metal–organic vapor epitaxy and composed of a 150 nm Al0.60Ga0.40As top cladding layer, a320 nm Al0.19Ga0.81As guiding core, and a 1.8 �m Al0.60Ga0.40As as a lower claddinglayer on a GaAs substrate. Electron-beam lithography was used to write the patternswith proximity correction tools, at an acceleration voltage of 50 kV, in a poly(methylmethacrylate) resist bilayer, which was 200 nm thick in total. The patterns were thentransferred by reactive ion etching into an intermediate SiO2 mask layer (200 nmthick) and finally into the AlGaAs/GaAs material to a depth of approximately 1 �m,using SiCl4/O2 at a flow rate of 15 SCCM/0.5 SCCM (SCCM denotes cubic centimetersper minute at standard temperature and pressure), a pressure of 0.93 Pa �7 mTorr�,and a rf power level of 250 W for 20 min.
Although a single interface between the two different waveguide types was consid-ered in the simulation, an output interface was also needed in the experimental char-acterization of the device. For this reason, in all the devices fabricated, the two inter-faces were identical except for the case of the lateral displacement of the ridge, wherethe output ridge was always placed at the normal (0,0) position. The PhC waveguideswere accessed by deeply etched 2 �m wide ridge waveguides, followed by taperedridge sections with a half-angle of 0.4° to match adiabatically with the PhC channelwaveguide width. Some simple 2 �m ridge waveguides were also included to act asreference structures. Figure 7 shows micrographs of the resulting devices.
As can be seen on the left-hand side of the figure, the ridge is deeply etched (to3.23 �m below the surface), going through the lower cladding with a smooth and ver-tical sidewall profile. The ridge waveguide width at the interface with the PhC chan-nel is 270 nm, close to the designed value of 261 nm �=0.7�Xperiod�, due to proximityeffects associated with the electron-beam writing process. The same effect appears inthe photonic crystal region defining the W1 PhC channel guide, where the hole diam-eter is 120 nm instead of the designed 134 nm.
5. Characterization ResultsThe fabricated devices were characterized using an end-fire measurement technique,with a tunable titanium-sapphire laser (Ti:Al2O3 pumped by an argon ion laser) asthe source, covering the range 770–875 nm with a spectral resolution of 0.2 nm. Toextrapolate the behavior in the junction region of the PhC channel guide to the ridgewaveguide system, the response of the devices was normalized to the response of the2 �m ridge by directly dividing the values at the same wavelengths. Most of the simu-lations were carried out at a single wavelength, but the transmission remains practi-cally flat in the interval used for these studies. To obtain a proper comparison betweenthe simulated and real responses, the tuning range of the laser source was dividedinto four intervals (774.4–798.3 nm, 798.5–822.2 nm, 822.5–846.2 nm, and846.5–870.3 nm), with the middle point of each range being given the average value of
Fig. 7. Scanning electron micrographs of the fabricated device structures. The 2 �mridge waveguide (left), ridge guide to the W1 channel guide interface (center), PhC re-gion (right).
Vol. 6, No. 2 / February 2007 / JOURNAL OF OPTICAL NETWORKING 97
the transmission in the interval. Consequently, due to the data averaging in eachinterval, information on Fabry–Perot type fast ripple and noise was advantageouslysuppressed.
The experimental investigation of the effect of the width of the ridge waveguide, atthe normal position (i.e., SA=SL=0), on the behavior at the interface with the PhCwas performed by fabricating ridge waveguides with a range of widths. The parameterW was varied from 0.6 up to 1.2 normalized units, in steps of 0.2 but with a width of0.7 also included. The ridge waveguide sections at both sides of the PhC channelwaveguide were positioned symmetrically. Figure 8(a) shows the normalized spectrafor W=0.7 and 1.2. All the other responses lie between these two extremes, withanalogous ripple behavior (standard deviation around the mean value of 2.8%). TheseFabry–Perot-like oscillations are due to spurious resonances between defects or dis-continuities within the device.
The responses are characterized by high transmission values in the range of85% –95%. Although the responses are rather flat, increasing the width �W� of thefeeder ridge waveguide leads to ripples of increased amplitude, as resonance effectsdue to the more mismatched junctions build up.
Figure 8(b) shows the transmission results, as a function of ridge guide width forthe four measured spectral intervals (as already explained in Section 2), comparedwith the simulated curve. Although, in the experimental response, the overall trans-mission is reduced by between 5% and 10%, the experimental curves follow the trend
Fig. 8. (a) Normalized transmission spectra for ridge widths W=0.7 (best case) andW=1.2 (worst case) placed at the normal position �SL=SA=0�. (b) Experimental trans-mission versus normalized ridge waveguide width, with the ridge being placed at thenormal position �SA=SL=0�, for the four spectral intervals within the tuning range. Thestarred curve shows the simulated response (as in Fig. 2).
Vol. 6, No. 2 / February 2007 / JOURNAL OF OPTICAL NETWORKING 98
of the simulated ones. The closest response to the simulations is observed in the spec-tral range of 774.4–798.3 nm. For all the experimental curves, optimum transmissionis observed when the ridge width W is equal to 0.7 normalized units.
The experimental behavior of laterally shifted ridges was probed by fabricatingridges of width 0.7 normalized units at 0 axial offset, with progressive misalignment.The lateral offset SL was varied from 0 to 0.5 normalized units and applied to theinput interface only. With two misaligned interfaces, more diffractive scattering wouldoccur together with more pronounced Fabry–Perot oscillations of the detected signal,thereby making it more difficult to extrapolate the actual transmission response. Theoutput interface of the ridge waveguide was positioned at the normal position �SL=SA=0�, with a width equal to 0.7 units.
Figure 9(a) shows the normalized experimental transmission responses versus dis-placement over the four spectral intervals, together with a simulated response. Asshown in Fig. 9(b), the spectra of these devices show evident Fabry–Perot features: byincreasing the lateral displacement, a reduction of the average transmission isobserved (the transmittivity of the interface decreases as the scattering increases),accompanied by an increase in the fringe visibility (as the reflectivity of the interfaceincreases). By considering the four spectral intervals, the oscillations are averagedout, and the wavelength dependence shown by Fig. 9(a) is consequently reduced.
The more rapid decline with increasing displacement shown by the simulation isnot observed in the experimental results. Although the simulated and measured
Fig. 9. (a) Experimental transmission versus normalized ridge lateral shift �SL� withno axial displacement �SA=0� and ridge width W=0.7, for the four spectral intervals inthe available tuning range. The simulation curve shows the simulated response [as inFig. 3(a)]. (b) Normalized experimental transmission versus wavelength correspondingto the data shown in (a): increasing the lateral shift from 0 to 0.4, the average trans-mission decreases while the fringe visibility increases.
Vol. 6, No. 2 / February 2007 / JOURNAL OF OPTICAL NETWORKING 99
response curves are close for the first two values of lateral shift (SL=0 and SL=0.1),substantial coupling to the W1 channel is observed experimentally even for the largestdisplacement. Such counterintuitive behavior may be due to the actual, 3D profiles ofthe fabricated structure, which cannot be captured by 2D simulations. Because of thelow out-of-plane confinement in the slab, part of the mode in the lower cladding expe-riences a less abrupt discontinuity around the interface between the ridge waveguideand the W1 channel waveguide. Furthermore smoothened edges, together with theactual cross-sectional shape of the holes (usually conical in the lower cladding) mayinduce a tapering and/or coupling effect in the experimental devices [26].
As for the axial shift, a ridge of 0.7 normalized units width (without lateral shift),together with a range of SA values from −1 (for the ridge being outside the crystal) to+1, in steps of 0.2, were examined, with an emphasis on the region approximately atSA=0.45, which seemed to be optimal in the simulations. The ridge sections were posi-tioned identically at the input and output interfaces.
Figure 10 shows a comparison between the experimental response for the four spec-tral intervals used and the simulated response for the ridge waveguide width used inthe fabricated devices �W=0.7�. The transmission of the experimental devices is glo-bally lower (�5% on average) than the simulated response, with lower minima, espe-cially for the longer wavelength regions. The oscillatory response observed in thesimulation results is quite closely repeated by the measured devices, with good agree-ment for the positions of the local minima and maxima. The highest transmission fea-tures are still observed in the region of axial displacement close to SA=0.45.
Just as for the lateral displacement case, strong reductions in transmission are notreproduced experimentally. The dip appearing in the simulations for the ridge locatedat the extreme outside position �SA=−1� is not present in the experimental results,and the transmission remains at high average levels, even when the ridge ends beforethe W1 waveguide. But the wavelength dependence becomes less flat, as shown by theaverage values in the four parts of the spectrum, which are more widely spreadaround their mean value.
As a final comment, some discrepancy between the experimental data and thesimulated data have been found and is to be expected in practice, because of theinherent differences between the simulated structures and the actual structures fab-ricated. The GaAs/AlGaAs epitaxial waveguide structure provides only weak verticalconfinement, and therefore simulation using a 2D computation space, together withan effective index approach, may not provide an adequate model. However, the overallagreement between experiment and simulation suggests that a 2D approximation cansuccessfully be used quickly to find credible design guidelines for what is otherwise acomputationally intensive 3D problem [27].
Furthermore, the experimental data are degraded by the presence of losses in theW1 channel waveguide, by losses at the two interfaces (instead of one) to the inputwaveguide, and in the transitions between the narrow ridge guide sections and the
Fig. 10. Experimental transmission versus normalized ridge axial shift �SA� with nolateral displacement �SL=0� and ridge width W=0.7, for the four spectral intervals inthe tuning range. The dotted curve shows the simulated response [as in Fig. 4(a)].
Vol. 6, No. 2 / February 2007 / JOURNAL OF OPTICAL NETWORKING 100
tapered input and output sections. The effect of these losses is not eliminated by thenormalization with respect to the transmission of a 2 �m waveguide, and these factorswere not taken into account in the simulated design. On the other hand, the (weak)resonance effects that occur in the Fabry–Perot cavity created between the two dis-continuities may induce a partially compensating small overestimation of the trans-mission for some wavelengths.
6. ConclusionsIn this study, the coupling between two different guiding elements, a ridge waveguideand a PhC W1 channel waveguide, has been investigated. The study has concentratedon the effects that different dimensions in the ridge waveguide, as well as positionswith respect to the W1 channel, have on the coupling and therefore on the transmis-sion through the whole cascaded device. The simulated behavior showed more than90% transmitted power for ridge widths in the range [0.5a�3, 1.1a�3], and the axialposition shifts showed oscillating features related to the lattice periodicity. Asexpected, lateral shifts produced substantial degradation of the interface transmissionefficiency.
Characterization of devices fabricated in GaAs/AlGaAs material showed that, inmost cases, the experimentally observed maximum transmission features were lowerthan the simulated ones, linked to the combined effects of the quality of the fabricateddevices and the normalization technique. On the other hand, high transmission wasobserved even in the case of heavily displaced junctions (lateral and axial displace-ments) but with good reproduction of the predicted periodiclike behavior, in the case ofaxial displacement.
Finally, the experimental verification of optimum coupling and transmission wasdemonstrated for the case of a 0.7*a�3 wide ridge waveguide (axially) inserted intothe W1-PhC channel waveguide by 0.4a, for which a predicted transmission value of98% was achieved over a relative bandwidth of 6.5% (a /� from 0.2471 to 0.2638).
AcknowledgmentsThis work was supported by the Engineering and Physical Sciences Research Council(EPSRC) through the Ultrafast Research Collaboration and by the Faculty of Engi-neering of the University of Bologna.
References1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,”
Phys. Rev. Lett. 58, 2059–2062 (1987).2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys.
Rev. Lett. 58, 2486–2489 (1987).3. T. F. Krauss and R. M. De la Rue, “Photonic crystals in the optical regime—past, present
and future,” Prog. Quantum Electron. 23, 51–96 (1999).4. R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and K. Kash,
“Novel applications of photonic band-gap materials—low-loss bends and high Q-cavities,” J.Appl. Phys. 75, 4753–4755 (1994).
5. J. Arentoft, T. Sondergaard, M. Kristensen, A. Boltasseva, M. Thorhauge, and L. Frandsen,“Low-loss silicon-on-insulator photonic crystal waveguides,” Electron. Lett. 38, 274–275(2002).
6. W. Kuang, C. Kim, A. Stapleton, W. J. Kim, and J. D. O’Brien, “Calculated out-of-planetransmission loss for photonic crystal slab waveguides,” Opt. Lett. 28, 1781–1783 (2003).
7. A. Shinya, M. Notomi, E. Kuramochi, T. Shoji, T. Watanabe, T. Tsuchizawa, K. Yamada, andH. Morita, “Functional components in SOI photonic crystal slabs,” in Photonic CrystalMaterials and Devices, A. Adibi, A. Scherer, S. Y. Lin, eds., Proc. SPIE 5000, 104–117 (2003).
8. S. G. Johnson, P. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, “Linear waveguides inphotonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
9. S. Boscolo, M. Midrio, and T. F. Krauss, “Y junctions in photonic crystal channelwaveguides: high transmission and impedance matching,” Opt. Lett. 27, 1001–1003 (2002).
10. M. Loncar, T. Doll, J. Vuckovic, and A. Scherer, “Design and fabrication of silicon photoniccrystal optical waveguides,” J. Lightwave Technol. 18, 1402–1411 (2000).
11. A. Talneau, M. Mulot, S. Anand, and P. Lalanne, “Compound cavity measurement oftransmission and reflection of a tapered single-line photonic-crystal waveguide,” Appl. Phys.Lett. 82, 2577–2579 (2003).
12. E. A. Camargo, H. M. H. Chong, and R. M. De la Rue, “2D Photonic crystal thermo-opticswitch based on AlGaAs/GaAs epitaxial structure,” Opt. Express 12, 588–592 (2004).
Vol. 6, No. 2 / February 2007 / JOURNAL OF OPTICAL NETWORKING 101
13. M. H. Shih, W. J. Kim, W. Kuang, J. R. Cao, H. Yukawa, S. J. Choi, J. D. O’Brien, P. D.Dapkus, and W. K. Marshall, “Two-dimensional photonic crystal Mach–Zehnderinterferometers,” Appl. Phys. Lett. 84, 460–462 (2004).
14. Y. Xu, R. K. Lee, and A. Yariv, “Adiabatic coupling between conventional dielectricwaveguides and waveguides with discrete translational symmetry,” Opt. Lett. 25, 755–757(2000).
15. A. Mekis and J. D. Joannopoulos, “Tapered couplers for efficient interfacing betweendielectric and photonic crystal waveguide,” J. Lightwave Technol. 19, 861–865 (2001).
16. M. E. Potter and R. W. Ziolkowski, “Two compact structures for perpendicular coupling ofoptical signals between dielectric and photonic crystal waveguides,” Opt. Express 10,691–698 (2002).
17. T. D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact modeconversion,” Opt. Lett. 26, 1102–1104 (2001).
18. P. Sanchis, J. Marti, B. Luyssaert, P. Dumon, P. Bienstman, and R. Baets, “Analysis anddesign of efficient coupling in photonic crystal circuits,” Opt. Quantum Electron. 37,133–147 (2005).
19. P. Pottier, I. Ntakis, and R. M. De La Rue, “Photonic crystal continuous taper for low-lossdirect coupling into 2D photonic crystal channel waveguides and further devicefunctionality,” Opt. Commun. 223, 339–347 (2003).
20. N. Moll and G. L. Bona, “Comparison of three-dimensional photonic crystal slab waveguideswith two-dimensional photonic crystal waveguides: efficient butt coupling into thesephotonic crystal waveguides,” J. Appl. Phys. 93, 4986–4991 (2003).
21. P. Sanchis, P. Bienstman, B. Luyssaert, R. Baets, and J. Marti, “Analysis of butt coupling inphotonic crystals,” IEEE J. Quantum Electron. 40, 541–550 (2004).
22. E. Miyai and S. Noda, “Structural dependence of coupling between a two-dimensionalphotonic crystal waveguide and a wire waveguide,” J. Opt. Soc. Am. B 21, 67–72 (2004).
23. M. Badieirostami, B. Momeni, M. Soltani, A. Adibi, Y. Xu, and R. K. Lee, “Investigation ofphysical mechanisms in coupling photonic crystal waveguiding structures,” Opt. Express12, 4781–4789 (2004).
24. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on achip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
25. Y. A. Vlasov and S. J. McNab, “Coupling into the slow light mode in slab-type photoniccrystal waveguides,” Opt. Lett. 31, 50–52 (2006).
26. Y. Tanaka, T. Asano, Y. Akahane, B. S. Song, and S. Noda, “Theoretical investigation of atwo-dimensional photonic crystal slab with truncated cone air holes,” Appl. Phys. Lett. 82,1661–1663 (2003).
27. M. Gnan, G. Bellanca, H. M. H. Chong, P. Bassi, and R. M. De la Rue, “Modelling ofphotonic wire Bragg gratings,” Opt. Quantum Electron. 38, 133–148 (2006).
