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Designing finite-height photonic crystal waveguides: confinement of light and dispersion relations
15 July 2001
Optics Communications 194 (2001) 341±351
www.elsevier.com/locate/optcom
Designing ®nite-height photonic crystal waveguides: con®nement of light and dispersion relations T. Sùndergaard *, A. Bjarklev, J. Arentoft, M. Kristensen, J. Erland, J. Broeng, S.E. Barkou Libori COM, Technical University of Denmark, Building 345, DK-2800 Lyngby, Denmark Received 26 March 2001; accepted 10 May 2001
Abstract Guidelines are obtained for characteristic design parameters of ®nite-height photonic crystal waveguides using diagrams of photonic bandgaps for in®nite-height photonic crystals. This is achieved by requiring photonic crystal designs with bandgaps well below a fundamental upper frequency limit for leakage-free guidance of light related to the properties of the media above/below the ®nite-height photonic crystal waveguide. The approach has the advantage that it can be applied to a large number of diagrams for in®nite-height crystals that are already available in the literature, and furthermore the approach is not computer intensive compared to more rigorous numerical approaches to threedimensional structures. We consider optical waveguide designs based on introducing a line defect in photonic crystals with air holes arranged on a triangular lattice in a silicon slab. For the media above/below the slab we consider the choices of silica and air. Dispersion relations are calculated for various waveguide designs. The analysis reveals a complex distribution of bands related to guided modes and provides information on how the guidance properties are modi®ed as the waveguide width is changed. Furthermore, the analysis reveals the existence of bandgaps that are almost omni-directional. For a speci®c choice of photonic crystal waveguide placed on a silica substrate these bandgaps that have previously been overlooked gives the only possibility of leakage-free bandgap guidance of TM-polarized light. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Integrated optics; Guided wave optics; Photonic crystals; Photonic bandgap
1. Introduction Photonic bandgap materials represent a promising material choice for future integrated optical components because they have added new possibilities for controlling the ¯ow of light [1±9]. Concepts and ideas that have been known for a
*
Corresponding author. Fax: +45-45-93-65-81. E-mail address: [email protected] (T. Sùndergaard).
long time in solid state physics for semiconductor materials with a periodic potential for electronic wavefunctions have been transferred to optics. It is well known that as defects (impurities) are introduced in electronic bandgap materials electronic wavefunctions with an energy within the electronic bandgap may become allowed. In this case the electronic wavefunction is localized (or trapped) in the region near the impurity. The idea of introducing defects in bandgap materials to localize and control the properties of wavefunctions may also
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 3 1 6 - 5
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be applied to photonic bandgap materials making these materials ever more useful. In particular, cavities can be created by introducing point defects [10±16]. This method of trapping the photonic wavefunction may lead to high Q cavities and novel lasers. It is also possible to introduce line defects, thereby trapping the photonic wavefunction along the region of a line. In this way novel waveguides may be created [17±24] with properties that are considerably dierent from properties of more conventional optical waveguides. In particular, two-dimensional calculations have indicated that almost complete (>98%) transmission is possible around sharp corners [17] and through branching points [18]. Previous theoretical investigations of photonic crystals (see for example Refs. [4,25±28]) and photonic crystal waveguides [17±24] have to a large extent been focused on the idealized case of two-dimensional photonic crystals of in®nite height, i.e. structures being periodic in two dimensions and invariant in the third. Naturally, any realistic realization of a photonic crystal or photonic crystal waveguide cannot be of in®nite height, and in fact a small height is desirable for the design of small-scale integrated optics. Some theoretical papers have appeared taking into account in a rigorous way the ®nite height in realistic photonic crystals [12,29±32] and photonic crystal waveguides [33±36]. The investigations given in the papers [12,29±36] are based on fullvectorial numerical calculations of electromagnetic waves for three-dimensional structures. Their numerical calculations are very computer intensive. The approach that will be used in this paper to address ®nite-height photonic crystal waveguides is based on comparing dispersion relations for two-dimensional structures with dispersion relations for the media above and below the photonic crystal waveguide. Compared to the three-dimensional calculations given in Refs. [12,29±36] our approach is not computer intensive, and it furthermore has the advantage that it can be applied to a large number of diagrams of bandgaps and dispersion relations that are already available in the literature. Thereby design guidelines are straightforwardly obtained for a large number of photonic crystal structures. In these approximate
considerations we neglect some features of the scattering taking place at the boundaries between the photonic crystal waveguide and the media above and below the waveguide. The scattering here results in coupling between TE and TM-polarized electromagnetic modes, and the resulting modes are sometimes referred to instead as TE-like and TM-like [12]. Imperfections at these interfaces may also result in light losses due to scattering. However, for ®nite-height waveguides without imperfections leakage-free guidance of light is possible (see e.g. Refs. [34±36]). In the spirit of the results given in Ref. [37] we will consider waveguide designs based on introducing a line defect in photonic crystals with air holes arranged on a triangular lattice in a silicon slab. For the media above/below the slab we will consider the technologically relevant choices of silica and air. Two limiting waveguide designs of this type, known as type A and type B waveguides [22], are shown in Fig. 1. In this paper we will consider these waveguides from the point of view that a realistic waveguide is a ®nite-height waveguide and it is from this viewpoint that we decide on which frequency ranges and structure parameters (hole diameter D, lattice constant K) it is relevant to focus. We will also consider how the guidance properties of the waveguides are modi®ed as the width of the waveguides W is changed. For the type A waveguide the width W is de®ned as the separation distance between holes on each side of the waveguiding region. For type B waveguides, the width is de®ned to be zero for the width where the line defect disappears, i.e. where the structure becomes a perfect two-dimensional photonic crystal. Compared to Benisty's analysis [22], we prefer to consider waveguides with a relatively narrow width since a narrow width is required for making the waveguides single moded. Finite-height photonic crystal waveguides have previously been considered experimentally in for example Refs. [38±43]. If results for in®nite-height structures are applied directly in the design of ®nite-height waveguides without care the result may be a waveguide con®guration with extremely high losses as was reported in Ref. [39]. In Section 2 we will establish how the optical wavelength k, lattice constant K and hole diameter
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Fig. 1. Schematic of type A and type B photonic crystal waveguides. A line defect has been introduced in a photonic crystal with air holes arranged on a triangular lattice. The lattice constant K, hole diameter D and waveguide width W of the waveguides are introduced. er and eb represent dielectric constants of the air holes and the background material, respectively.
D may be chosen in order to obtain leakage-free guidance of light in ®nite-height photonic crystal waveguides for various media above and below the waveguide. Based on this analysis we will choose two dierent hole diameters D. The case of small holes (D=K 0:7) being appropriate if silica is the medium above and/or below is analysed in Section 3, and the case of large holes (D=K 0:9) being appropriate for air above and below is analysed in Section 4. The conclusion is given in Section 5. 2. Con®nement of light In this section we will obtain guidelines for the hole diameter D and lattice constant K of the type A and type B photonic crystal waveguides by comparing two-dimensional calculations with dispersion relations for the media above and below the ®nite-height photonic crystal waveguides. In order to introduce the basic idea we start by considering Fig. 2 showing the allowed normalized frequencies K=k for TM-polarized light (magnetic ®eld in the xy-plane) as a function of the component of the wave vector k in the x-direction. All ®gures presented in this paper were calculated using plane-wave-expansion theory and a variational principle [44]. It is appropriate to consider the x-direction since the type A and type B waveguides are periodic structures along the x-axis. The coordinate system is de®ned in Fig. 1. Fig. 2 was
Fig. 2. Omni-directional and almost omni-directional in-plane photonic bandgaps for TM-polarized light. The air line represents the dispersion relation for light in free space, i.e. all combinations of frequencies and wave vectors (K=k, kK=2p) above this line are allowed in free space. The silica line represents the corresponding dispersion line for light in silica.
calculated for an in®nite-height photonic crystal with no defect introduced, which is equivalent to the type B waveguide structure with W 0. The structure consists of air holes (dielectric constant er 1) arranged on a triangular lattice in silicon (dielectric constant eb 12) with diameter D 0:9K. For the media above and below the ®niteheight slab of the type A and type B waveguides we will consider air and silica. If the type A or type B waveguide slabs are suspended in air we have to
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be concerned with the dispersion relation for air. This dispersion relation is shown in Fig. 2 as the air line, and all combinations of frequencies K=k and wave vectors kK=2p above this line are allowed in free space. Since this line reaches the edge of the Brillouin zone (kK=2p 0:5) with a frequency K=k 0:5 this frequency sets a fundamental upper frequency limit (air limit) for leakage-free guidance of light. The existence of such a frequency limit for linear waveguides may also be obtained from diagrams of the light cone given in recent papers devoted to three-dimensional calculations of photonic crystal waveguides [33±36]. Equivalent frequency limits can also be obtained from three-dimensional light cone considerations for con®nement of light to a photonic crystal slab (no line defect or waveguide introduced) [12,29±32]. However, the upper frequency limits obtained for con®nement of light to a slab (light con®ned to propagation in two dimensions), and the limits obtained for con®nement of light to a line defect (light con®ned to propagation in one dimension) are not the same limits. It is, however, also possible to obtain guidelines for design of ®nite-height photonic crystal waveguides without making full-vectorial three-dimensional calculations by comparing less computer-intensive twodimensional calculations with dispersion relations for the media above and below the ®nite-height photonic crystal waveguide slab [37], and this is the approach taken in the present paper. If the slab of the type A or type B waveguide is placed on a silica substrate we have to consider the silica line instead (we choose the refractive index n 1:45 for silica), and we see that in this case the fundamental upper frequency limit for leakagefree guidance of light is reduced to K=k 0:5= n 0:345 (silica limit). In Fig. 2 an omni-directional bandgap is seen for normalized frequencies in the range K=k 0:4±0.45, and an almost omni-directional bandgap is seen around the frequency K=k 0:28. If the waveguide slab considered is suspended in air we can make use of both types of bandgap since both these bandgaps are found below the air line. However, if the waveguide slab is placed on silica we see that we can no longer obtain leakagefree guidance of light using the omni-directional
bandgap (it is above the silica line), and in this case the almost omni-directional bandgap (it is below the silica line) gives the only possibility of bandgap guidance of TM-polarized light. The existence and usefulness of almost omni-directional bandgaps for guidance of light in photonic crystal waveguides has previously been overlooked. This type of bandgaps may open up for bandgap guidance of light for polarizations and frequency ranges that would not even be considered from an analysis restricted to only omni-directional bandgaps. In the literature a number of diagrams can be found showing the relation between characteristic parameters of in®nite-height two-dimensional photonic crystals and the omni-directional photonic bandgaps of the crystal (see for example Refs. [4,25±28]). We will now show how design parameters for ®nite-height photonic crystal waveguides can be extracted from such diagrams by introducing the fundamental upper frequency limits that was obtained from the air-line and silica-line considerations presented in Fig. 2. The approach is explained by giving an example for the speci®c structure of interest in this paper, but the approach is general and can be applied to many other existing diagrams of bandgaps [4,25± 28] giving simple and straightforward design guidelines for a large number of photonic crystal structures. The two-dimensional photonic crystal of interest in this paper for construction of the type A and type B waveguides is a crystal with air holes arranged on a triangular lattice in silicon. The omnidirectional bandgaps for in-plane propagation in this photonic crystal as a function of the hole diameter D relative to the crystal lattice constant K are shown in Fig. 3. The TE bandgaps shown with solid lines are the frequency ranges in which inplane propagation of light with the electric ®eld oriented in the plane of the crystal is not allowed. Similarly the TM bandgap (dashed lines) is the range of frequencies in which in-plane propagation of light with the magnetic ®eld in the plane of the crystal is not allowed. For in-plane propagation in in®nite-height structures one can consider TE and TM polarized light. In ®nite-height structures, however, the polarizations will instead be TE-like and TM-like (see for example Refs. [12,31]).
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Fig. 3. In-plane photonic bandgaps for TE and TM polarization as a function of the diameter of the air holes of the crystal. Two limiting frequencies are introduced (air limit and silica limit) above which leakage-free guidance of light is no longer possible for ®nite-height waveguides surrounded by either air or silica.
The bandgaps shown in Fig. 3 are for an in®nite-height photonic crystal being invariant in the z-direction. Con®nement of light to the type A and type B waveguides introduced in Section 1 using a photonic bandgap requires naturally that the hole diameter D and lattice constant K are chosen in such a way that a bandgap is available for the crystal at the wavelength of interest k. Requiring leakage-free guidance of light the waveguide must also be operated at frequencies below the air limit, silica limit or other equivalent limits depending on the choice of media above and below the crystal slab. The silica limit K=k 0:345 and the air limit K=k 0:5 are shown in Fig. 3. By introducing these limits in diagrams of bandgaps for in®niteheight photonic crystals such as Fig. 3, we are now able to obtain guidelines for the design of ®niteheight photonic crystal waveguides. Clearly, for waveguide slabs placed on a silica substrate only the TE bandgap is below the silica limit, and therefore leakage-free bandgap guidance of light will only be possible for one polarization. For the case of a slab suspended in air it is, however, possible to choose the diameter of the air holes so large that a bandgap for both polarizations exist below the air limit, and in this case leakage-free
345
guidance of light may be possible for two polarizations of the light. The bandgaps tend to move up in frequency as D=K increases, and if D=K is chosen too large the bandgaps will no longer be below the silica limit. On the other hand, if D=K is chosen too small there is no bandgap. For a ®niteheight photonic crystal waveguide placed on a silica substrate a good choice of D=K between these extremes corresponds to obtaining the largest possible photonic bandgap below the silica limit, and in the present case the ratio D=K 0:7 is a reasonable choice. An operating frequency near the centre of the bandgap is in this case K=k 0:27. For the optical wavelength 1550 nm this results in the lattice constant K 419 nm and the hole diameter D 293 nm. For waveguide slabs suspended in air we may attempt to use larger air holes given by for example D=K 0:9, and in this case we may consider making use of both polarizations of the light for waveguiding using the photonic bandgap eect. From Fig. 3 we may conclude that for ®nite-height waveguides placed on silica bandgap guidance for TM polarization is not possible using an omnidirectional bandgap. However, again from Fig. 2 bandgap guidance is in fact possible for TM polarization in this case using the almost omnidirectional bandgap discovered here below the silica limit. 3. Dispersion relations ± small holes As was clear from the previous section, if the ®nite-height type A or type B photonic crystal waveguide is placed on a silica substrate then a reasonable choice of diameter D of the air holes in the waveguide structure is 0.7K. In this section we consider the bandgap and dispersion properties of type A and type B waveguides with this size of air holes. For this choice of air-hole diameter complete bandgaps only exist for one polarization of the light, and consequently we will restrict our analysis to this polarization. The dispersion properties for the type A waveguide is shown for three waveguide widths W in Fig. 4. Similar to Fig. 2 this ®gure shows the allowed combinations of the normalized frequency
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T. Sùndergaard et al. / Optics Communications 194 (2001) 341±351 0.35 0.3
Λ /λ
0.25 0.2
0.15
Type A, TE pol.
0.1 W=0.0Λ W=0.5Λ W=1.0Λ
0.05 0 0
0.1
0.2
0.3
0.4
0.5
kΛ/2π Fig. 4. Dispersion relations (TE polarization) for three waveguide widths for the type A waveguide (air-hole diameter D 0:7K.
K=k and the component of the Bloch wave vector k in the direction of the waveguide. The gray regions correspond to the allowed modes of the photonic crystal material surrounding the waveguide. Due to the introduction of a waveguide (or a line defect) in the crystal there are discrete defect bands related to the guided modes of the waveguide. Even for the waveguide width W 0 there is a line-defect and thereby defect bands (solid lines) for the type A waveguide. A complete omni-directional photonic bandgap is not required for obtaining leakage-free guidance of light in straight waveguides, and also the defect bands for the various waveguide widths appearing in Fig. 4 for frequencies K=k < 0:2 correspond to guided modes in a frequency region where also propagating modes in the photonic crystal surrounding the waveguide are allowed. The electromagnetic modes related to the discrete defect bands in the frequency range from K=k 0:22 to K=k 0:33 are, however, con®ned to the waveguiding region due to an omni-directional photonic bandgap. The relevance of the omni-directional bandgap may appear when straight waveguides are combined to form bends and splitters such as Y-junctions and T-junctions. Only in the case of a complete omnidirectional bandgap is light prohibited from being scattered into the photonic crystal surrounding the waveguide con®guration near bends and the
branching points of splitters. However, for a ®niteheight waveguide con®guration the light reaching a bend or branching point may of course be scattered out into the media above and below the ®nite-height waveguide con®guration. For the type A waveguide we observe from Fig. 4 that for the width W 0 there are two defect bands (solid line) in the omni-directional bandgap, and the waveguide may be considered single moded for the frequency ranges covered by these bands. As the width changes to W 0:5K, several defect bands exist and cover quite dierent frequency ranges, i.e. for K=k 0:28 the waveguide structure will support a guided mode for W 0:0K but not for W 0:5K. For the width W 1:0K the waveguide supports guided modes for almost all frequencies within the omni-directional bandgap of the surrounding photonic crystal. In this case, however, the waveguide for some frequencies support two defect bands and the waveguide is for such frequencies multi-moded. There are, however, also frequency ranges for W 1:0K where the waveguide is single moded. The dispersion properties for the type B waveguide con®guration are shown in Fig. 5. In this case the waveguide width W 0 corresponds to a photonic crystal where no line defect has been introduced. For the waveguide width W 0:5K there is a large frequency interval from K=k 0:24
Fig. 5. Dispersion relations (TE polarization) for two waveguide widths for the type B waveguide (air-hole diameter D 0:7K.
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to K=k 0:3 where single-moded guidance of light is allowed. Single-moded guidance of light appears to be possible over a large frequency interval only for relatively narrow waveguides. As the width of the waveguide increases the number of defect bands related to guided modes also increases rapidly. If the width of the waveguide is increased to W 1:1K the waveguide becomes multi-moded in practically the same frequency interval from K=k 0:25 to K=k 0:3. For this waveguide width, however, there are also narrow frequency intervals where only one defect band exists and also in this case single-moded waveguidance is possible. Note that near the edge of the Brillouin zone (kK=2p 0:5) the defect bands are degenerated for the type B waveguide whereas this was not the case for the type A waveguide. The structure of the bands changes considerably as the waveguide is changed from a type A waveguide to a type B waveguide. Therefore it is possible to a high degree to control the dispersion properties of photonic crystal waveguides. But design of dispersion properties of such waveguides is also a complicated task and small deviations in the fabricated device relative to the original design may considerably change the dispersion properties and shift the frequency ranges where single-mode guidance is supported.
347
Fig. 6. Dispersion relations (TE polarization) for two waveguide widths for the type A waveguide (air-hole diameter D 0:9K.
4. Dispersion relations ± large holes For the ®nite-height waveguide with air above and below it was possible to consider using air holes with a large diameter D 0:9K. In this section we investigate the dispersion properties of the type A and type B waveguides with this air-hole diameter. In this case there are omni-directional in-plane bandgaps for both polarizations of the light, and we will therefore investigate dispersion relations for both TE and TM polarization. The dispersion relations for TE polarization are shown in Fig. 6 for a type A waveguide and in Fig. 7 for a type B waveguide. Clearly, the TE bandgap has moved up in frequency due to the larger hole diameter D (as expected from Fig. 3). The bandgap is now considerably larger, and therefore for the
Fig. 7. Dispersion relations (TE polarization) for two waveguide widths for the type B waveguide (air-hole diameter D 0:9K.
same waveguide width the waveguide will generally support more defect bands. From Fig. 6 it is clear that for the frequency ranges covered by the defect bands related to the width W 0 the type A waveguide with this width is single moded. Already for the waveguide width W 0:6K practically all frequencies within the gap are supported by a guided mode. As was also the case for waveguides with smaller air holes there are now frequency ranges where the waveguide is
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single moded, and others where it is multi-moded. For W 0:6K there is for example a large frequency range from K=k 0:36 to 0.44 where the waveguide is single moded. The dispersion relations shown in Fig. 7 for a type B waveguide with large air holes should be compared to the similar calculation for a waveguide with small air holes in Fig. 5. Consider the case of a narrow waveguide with W 0:2K in Fig. 7. In Fig. 7 the defect bands (solid line) takes up only a small part of the bandgap relatively to corresponding defect bands (W 0:5K) for a narrow waveguide in Fig. 4. The frequency ranges covered in both cases by the defect bands are, however, comparable in size. As was also the case in Fig. 5, the defect bands are degenerated at the edge of the Brillouin zone kK=2p 0:5. For TM polarized light we found in Section 2 the existence of almost omni-directional bandgaps for a photonic crystal without defects introduced. As a line defect is introduced defect bands related to guided modes may also appear in this type of bandgap. This is illustrated in Fig. 8, where we show the defect bands for a type A waveguide with width W 0:5K. For this choice of polarization and waveguide we show in Fig. 9 the ¯ow of energy for a guided mode corresponding to kK= 2p 0:15 and k=K 0:28. Here, the direction and length of the arrows illustrates the Poynting vector energy ¯ow [45] for the guided mode in the
Fig. 8. Dispersion relations (TM polarization) for the width W 0:5K for the type A waveguide (air-hole diameter D 0:9K).
Fig. 9. Energy ¯ow of a mode localized by an almost omnidirectional bandgap (TM polarization) in a type A waveguide. The frequency of the mode is K=k 0:28 and the Bloch wave vector is kK=2p 0:15. The arrows show the magnitude and direction of the Poynting vector. The amplitude of the electric ®eld squared is also shown. Here, dark shading corresponds to high intensity and light shading to low intensity.
waveguide. Fig. 9 also shows the amplitude of the electric ®eld squared for the mode. The dark shading represents high intensity and light shading low intensity. The guided modes illustrated by Figs. 8 and 9 may also be considered as bandgap guided modes, and indeed guidance of light at frequencies around K=k 0:28 could be considered as a possibility for making waveguides with the advantage that this may also work if the ®nite-height waveguide is placed on top of silica as pointed out in Section 2. An interesting question when using almost omni-directional bandgaps is how much light will be scattered out into the photonic crystal near bends and branching points. Due to the fact that the bandgap is almost completely omni-directional this scattering loss might turn out to be acceptably low, and this loss mechanism may not be high compared to scattering into the media above and below the ®nite-height waveguide. Almost omni-directional bandgaps should be kept in mind for future designs of ®nite-height photonic crystal waveguides. Fig. 10 shows for the type A waveguide and TM-polarization the dispersion relations for the omni-directional bandgap region relevant for
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zone (kK=2p 0:5). Even for the small width W 0:1K a guided mode is supported for nearly all frequencies within the omni-directional bandgap. Also for TM-polarization is it clear that the behaviour of the defect bands is highly complex and depends considerably on type of waveguide (A or B) and waveguide width. 5. Conclusion
Fig. 10. Dispersion relations (TM polarization) for two waveguide widths for the type A waveguide (air-hole diameter D 0:9K.
waveguides suspended in air. A striking feature of the defect band dispersion relations in this case is that these bands may be almost completely ¯at over a large wave vector range. Flat defect bands at the operating frequency results in low energy velocity and high dispersion [18]. For the width W 0:5K the waveguide is already multi-moded over most of the omni-directional bandgap. Fig. 11 shows a similar calculation for the type B waveguide. Also for TM polarization the defect bands are degenerated at the edge of the Brillouin 0.5
Type B, TM pol.
Λ /λ
0.45
0.4
0.35 W=0.1Λ W=0.3Λ 0.3 0
0.1
0.2
0.3
0.4
0.5
kΛ/2π Fig. 11. Dispersion relations (TM polarization) for two waveguide widths for the type B waveguide (air-hole diameter D 0:9K.
A method for obtaining design guidelines for ®nite-height photonic crystal waveguides has been described. The method is based on comparing calculations for in®nite-height structures with dispersion relations for the media above and below the ®nite-height waveguides. The approach can be applied to a large number of diagrams already available in the literature giving straightforward design guidelines for a large number of ®niteheight photonic crystals. By introducing fundamental upper frequency limits in a diagram of bandgaps versus the hole diameter for holes arranged on a triangular lattice in silicon it was possible to evaluate the proper choice of hole diameters depending on the media above and below the ®nite-height waveguides. For the photonic crystal with air holes arranged on a triangular lattice in silicon the existence of an omni-directional bandgap for both polarizations of the light requires air holes with a large diameter. In this case, however, leakage-free guidance of light by taking advantage of an omni-directional bandgap is not possible with a ®nite-height version of the investigated photonic crystal waveguides if placed on for example a silica substrate (the bandgaps have moved to frequencies higher than the fundamental limit related to silica). In this case the ®nite-height waveguide must be suspended in air to have a medium above and below the waveguide with a suciently low refractive index. Waveguides placed on silica supporting leakage-free guidance of light using an omni-directional bandgap can, however, be designed for one polarization of the light, but not for both polarizations, by using a smaller air-hole diameter. Taking advantage of an almost omni-directional bandgap for TM polarization it is also
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possible to obtain bandgap guidance of light using both polarizations if the ®nite-height waveguide is placed on silica. Almost omni-directional bandgaps should be kept in mind for future designs of photonic crystal waveguides. Dispersion relations for type A and type B waveguides has been given for the relevant polarizations of the light for both the case of small and large air-hole diameters in the photonic crystal. The variety of dispersion relations given in the paper show that the structure of the bands describing the guidance properties changes considerably as the waveguide is changed from a type A to a type B waveguide con®guration. The width of the waveguide is also an important parameter determining if the waveguide will be single moded, multi-moded or not guide light at all. It is indeed possible to a high degree to control the dispersion properties of photonic crystal waveguides by appropriate design. However, designing dispersion properties of such waveguides is also a complicated task, and small deviations in the fabricated device relative to the original design may lead to modi®ed dispersion properties and shift the frequency ranges where the waveguide supports single-mode guidance. Acknowledgements This work was supported by the EU-project PICCO (photonic integrated circuits using photonic crystal optics).
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Optics Communications 194 (2001) 341±351
www.elsevier.com/locate/optcom
Designing ®nite-height photonic crystal waveguides: con®nement of light and dispersion relations T. Sùndergaard *, A. Bjarklev, J. Arentoft, M. Kristensen, J. Erland, J. Broeng, S.E. Barkou Libori COM, Technical University of Denmark, Building 345, DK-2800 Lyngby, Denmark Received 26 March 2001; accepted 10 May 2001
Abstract Guidelines are obtained for characteristic design parameters of ®nite-height photonic crystal waveguides using diagrams of photonic bandgaps for in®nite-height photonic crystals. This is achieved by requiring photonic crystal designs with bandgaps well below a fundamental upper frequency limit for leakage-free guidance of light related to the properties of the media above/below the ®nite-height photonic crystal waveguide. The approach has the advantage that it can be applied to a large number of diagrams for in®nite-height crystals that are already available in the literature, and furthermore the approach is not computer intensive compared to more rigorous numerical approaches to threedimensional structures. We consider optical waveguide designs based on introducing a line defect in photonic crystals with air holes arranged on a triangular lattice in a silicon slab. For the media above/below the slab we consider the choices of silica and air. Dispersion relations are calculated for various waveguide designs. The analysis reveals a complex distribution of bands related to guided modes and provides information on how the guidance properties are modi®ed as the waveguide width is changed. Furthermore, the analysis reveals the existence of bandgaps that are almost omni-directional. For a speci®c choice of photonic crystal waveguide placed on a silica substrate these bandgaps that have previously been overlooked gives the only possibility of leakage-free bandgap guidance of TM-polarized light. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Integrated optics; Guided wave optics; Photonic crystals; Photonic bandgap
1. Introduction Photonic bandgap materials represent a promising material choice for future integrated optical components because they have added new possibilities for controlling the ¯ow of light [1±9]. Concepts and ideas that have been known for a
*
Corresponding author. Fax: +45-45-93-65-81. E-mail address: [email protected] (T. Sùndergaard).
long time in solid state physics for semiconductor materials with a periodic potential for electronic wavefunctions have been transferred to optics. It is well known that as defects (impurities) are introduced in electronic bandgap materials electronic wavefunctions with an energy within the electronic bandgap may become allowed. In this case the electronic wavefunction is localized (or trapped) in the region near the impurity. The idea of introducing defects in bandgap materials to localize and control the properties of wavefunctions may also
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 3 1 6 - 5
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be applied to photonic bandgap materials making these materials ever more useful. In particular, cavities can be created by introducing point defects [10±16]. This method of trapping the photonic wavefunction may lead to high Q cavities and novel lasers. It is also possible to introduce line defects, thereby trapping the photonic wavefunction along the region of a line. In this way novel waveguides may be created [17±24] with properties that are considerably dierent from properties of more conventional optical waveguides. In particular, two-dimensional calculations have indicated that almost complete (>98%) transmission is possible around sharp corners [17] and through branching points [18]. Previous theoretical investigations of photonic crystals (see for example Refs. [4,25±28]) and photonic crystal waveguides [17±24] have to a large extent been focused on the idealized case of two-dimensional photonic crystals of in®nite height, i.e. structures being periodic in two dimensions and invariant in the third. Naturally, any realistic realization of a photonic crystal or photonic crystal waveguide cannot be of in®nite height, and in fact a small height is desirable for the design of small-scale integrated optics. Some theoretical papers have appeared taking into account in a rigorous way the ®nite height in realistic photonic crystals [12,29±32] and photonic crystal waveguides [33±36]. The investigations given in the papers [12,29±36] are based on fullvectorial numerical calculations of electromagnetic waves for three-dimensional structures. Their numerical calculations are very computer intensive. The approach that will be used in this paper to address ®nite-height photonic crystal waveguides is based on comparing dispersion relations for two-dimensional structures with dispersion relations for the media above and below the photonic crystal waveguide. Compared to the three-dimensional calculations given in Refs. [12,29±36] our approach is not computer intensive, and it furthermore has the advantage that it can be applied to a large number of diagrams of bandgaps and dispersion relations that are already available in the literature. Thereby design guidelines are straightforwardly obtained for a large number of photonic crystal structures. In these approximate
considerations we neglect some features of the scattering taking place at the boundaries between the photonic crystal waveguide and the media above and below the waveguide. The scattering here results in coupling between TE and TM-polarized electromagnetic modes, and the resulting modes are sometimes referred to instead as TE-like and TM-like [12]. Imperfections at these interfaces may also result in light losses due to scattering. However, for ®nite-height waveguides without imperfections leakage-free guidance of light is possible (see e.g. Refs. [34±36]). In the spirit of the results given in Ref. [37] we will consider waveguide designs based on introducing a line defect in photonic crystals with air holes arranged on a triangular lattice in a silicon slab. For the media above/below the slab we will consider the technologically relevant choices of silica and air. Two limiting waveguide designs of this type, known as type A and type B waveguides [22], are shown in Fig. 1. In this paper we will consider these waveguides from the point of view that a realistic waveguide is a ®nite-height waveguide and it is from this viewpoint that we decide on which frequency ranges and structure parameters (hole diameter D, lattice constant K) it is relevant to focus. We will also consider how the guidance properties of the waveguides are modi®ed as the width of the waveguides W is changed. For the type A waveguide the width W is de®ned as the separation distance between holes on each side of the waveguiding region. For type B waveguides, the width is de®ned to be zero for the width where the line defect disappears, i.e. where the structure becomes a perfect two-dimensional photonic crystal. Compared to Benisty's analysis [22], we prefer to consider waveguides with a relatively narrow width since a narrow width is required for making the waveguides single moded. Finite-height photonic crystal waveguides have previously been considered experimentally in for example Refs. [38±43]. If results for in®nite-height structures are applied directly in the design of ®nite-height waveguides without care the result may be a waveguide con®guration with extremely high losses as was reported in Ref. [39]. In Section 2 we will establish how the optical wavelength k, lattice constant K and hole diameter
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343
Fig. 1. Schematic of type A and type B photonic crystal waveguides. A line defect has been introduced in a photonic crystal with air holes arranged on a triangular lattice. The lattice constant K, hole diameter D and waveguide width W of the waveguides are introduced. er and eb represent dielectric constants of the air holes and the background material, respectively.
D may be chosen in order to obtain leakage-free guidance of light in ®nite-height photonic crystal waveguides for various media above and below the waveguide. Based on this analysis we will choose two dierent hole diameters D. The case of small holes (D=K 0:7) being appropriate if silica is the medium above and/or below is analysed in Section 3, and the case of large holes (D=K 0:9) being appropriate for air above and below is analysed in Section 4. The conclusion is given in Section 5. 2. Con®nement of light In this section we will obtain guidelines for the hole diameter D and lattice constant K of the type A and type B photonic crystal waveguides by comparing two-dimensional calculations with dispersion relations for the media above and below the ®nite-height photonic crystal waveguides. In order to introduce the basic idea we start by considering Fig. 2 showing the allowed normalized frequencies K=k for TM-polarized light (magnetic ®eld in the xy-plane) as a function of the component of the wave vector k in the x-direction. All ®gures presented in this paper were calculated using plane-wave-expansion theory and a variational principle [44]. It is appropriate to consider the x-direction since the type A and type B waveguides are periodic structures along the x-axis. The coordinate system is de®ned in Fig. 1. Fig. 2 was
Fig. 2. Omni-directional and almost omni-directional in-plane photonic bandgaps for TM-polarized light. The air line represents the dispersion relation for light in free space, i.e. all combinations of frequencies and wave vectors (K=k, kK=2p) above this line are allowed in free space. The silica line represents the corresponding dispersion line for light in silica.
calculated for an in®nite-height photonic crystal with no defect introduced, which is equivalent to the type B waveguide structure with W 0. The structure consists of air holes (dielectric constant er 1) arranged on a triangular lattice in silicon (dielectric constant eb 12) with diameter D 0:9K. For the media above and below the ®niteheight slab of the type A and type B waveguides we will consider air and silica. If the type A or type B waveguide slabs are suspended in air we have to
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be concerned with the dispersion relation for air. This dispersion relation is shown in Fig. 2 as the air line, and all combinations of frequencies K=k and wave vectors kK=2p above this line are allowed in free space. Since this line reaches the edge of the Brillouin zone (kK=2p 0:5) with a frequency K=k 0:5 this frequency sets a fundamental upper frequency limit (air limit) for leakage-free guidance of light. The existence of such a frequency limit for linear waveguides may also be obtained from diagrams of the light cone given in recent papers devoted to three-dimensional calculations of photonic crystal waveguides [33±36]. Equivalent frequency limits can also be obtained from three-dimensional light cone considerations for con®nement of light to a photonic crystal slab (no line defect or waveguide introduced) [12,29±32]. However, the upper frequency limits obtained for con®nement of light to a slab (light con®ned to propagation in two dimensions), and the limits obtained for con®nement of light to a line defect (light con®ned to propagation in one dimension) are not the same limits. It is, however, also possible to obtain guidelines for design of ®nite-height photonic crystal waveguides without making full-vectorial three-dimensional calculations by comparing less computer-intensive twodimensional calculations with dispersion relations for the media above and below the ®nite-height photonic crystal waveguide slab [37], and this is the approach taken in the present paper. If the slab of the type A or type B waveguide is placed on a silica substrate we have to consider the silica line instead (we choose the refractive index n 1:45 for silica), and we see that in this case the fundamental upper frequency limit for leakagefree guidance of light is reduced to K=k 0:5= n 0:345 (silica limit). In Fig. 2 an omni-directional bandgap is seen for normalized frequencies in the range K=k 0:4±0.45, and an almost omni-directional bandgap is seen around the frequency K=k 0:28. If the waveguide slab considered is suspended in air we can make use of both types of bandgap since both these bandgaps are found below the air line. However, if the waveguide slab is placed on silica we see that we can no longer obtain leakagefree guidance of light using the omni-directional
bandgap (it is above the silica line), and in this case the almost omni-directional bandgap (it is below the silica line) gives the only possibility of bandgap guidance of TM-polarized light. The existence and usefulness of almost omni-directional bandgaps for guidance of light in photonic crystal waveguides has previously been overlooked. This type of bandgaps may open up for bandgap guidance of light for polarizations and frequency ranges that would not even be considered from an analysis restricted to only omni-directional bandgaps. In the literature a number of diagrams can be found showing the relation between characteristic parameters of in®nite-height two-dimensional photonic crystals and the omni-directional photonic bandgaps of the crystal (see for example Refs. [4,25±28]). We will now show how design parameters for ®nite-height photonic crystal waveguides can be extracted from such diagrams by introducing the fundamental upper frequency limits that was obtained from the air-line and silica-line considerations presented in Fig. 2. The approach is explained by giving an example for the speci®c structure of interest in this paper, but the approach is general and can be applied to many other existing diagrams of bandgaps [4,25± 28] giving simple and straightforward design guidelines for a large number of photonic crystal structures. The two-dimensional photonic crystal of interest in this paper for construction of the type A and type B waveguides is a crystal with air holes arranged on a triangular lattice in silicon. The omnidirectional bandgaps for in-plane propagation in this photonic crystal as a function of the hole diameter D relative to the crystal lattice constant K are shown in Fig. 3. The TE bandgaps shown with solid lines are the frequency ranges in which inplane propagation of light with the electric ®eld oriented in the plane of the crystal is not allowed. Similarly the TM bandgap (dashed lines) is the range of frequencies in which in-plane propagation of light with the magnetic ®eld in the plane of the crystal is not allowed. For in-plane propagation in in®nite-height structures one can consider TE and TM polarized light. In ®nite-height structures, however, the polarizations will instead be TE-like and TM-like (see for example Refs. [12,31]).
T. Sùndergaard et al. / Optics Communications 194 (2001) 341±351
Fig. 3. In-plane photonic bandgaps for TE and TM polarization as a function of the diameter of the air holes of the crystal. Two limiting frequencies are introduced (air limit and silica limit) above which leakage-free guidance of light is no longer possible for ®nite-height waveguides surrounded by either air or silica.
The bandgaps shown in Fig. 3 are for an in®nite-height photonic crystal being invariant in the z-direction. Con®nement of light to the type A and type B waveguides introduced in Section 1 using a photonic bandgap requires naturally that the hole diameter D and lattice constant K are chosen in such a way that a bandgap is available for the crystal at the wavelength of interest k. Requiring leakage-free guidance of light the waveguide must also be operated at frequencies below the air limit, silica limit or other equivalent limits depending on the choice of media above and below the crystal slab. The silica limit K=k 0:345 and the air limit K=k 0:5 are shown in Fig. 3. By introducing these limits in diagrams of bandgaps for in®niteheight photonic crystals such as Fig. 3, we are now able to obtain guidelines for the design of ®niteheight photonic crystal waveguides. Clearly, for waveguide slabs placed on a silica substrate only the TE bandgap is below the silica limit, and therefore leakage-free bandgap guidance of light will only be possible for one polarization. For the case of a slab suspended in air it is, however, possible to choose the diameter of the air holes so large that a bandgap for both polarizations exist below the air limit, and in this case leakage-free
345
guidance of light may be possible for two polarizations of the light. The bandgaps tend to move up in frequency as D=K increases, and if D=K is chosen too large the bandgaps will no longer be below the silica limit. On the other hand, if D=K is chosen too small there is no bandgap. For a ®niteheight photonic crystal waveguide placed on a silica substrate a good choice of D=K between these extremes corresponds to obtaining the largest possible photonic bandgap below the silica limit, and in the present case the ratio D=K 0:7 is a reasonable choice. An operating frequency near the centre of the bandgap is in this case K=k 0:27. For the optical wavelength 1550 nm this results in the lattice constant K 419 nm and the hole diameter D 293 nm. For waveguide slabs suspended in air we may attempt to use larger air holes given by for example D=K 0:9, and in this case we may consider making use of both polarizations of the light for waveguiding using the photonic bandgap eect. From Fig. 3 we may conclude that for ®nite-height waveguides placed on silica bandgap guidance for TM polarization is not possible using an omnidirectional bandgap. However, again from Fig. 2 bandgap guidance is in fact possible for TM polarization in this case using the almost omnidirectional bandgap discovered here below the silica limit. 3. Dispersion relations ± small holes As was clear from the previous section, if the ®nite-height type A or type B photonic crystal waveguide is placed on a silica substrate then a reasonable choice of diameter D of the air holes in the waveguide structure is 0.7K. In this section we consider the bandgap and dispersion properties of type A and type B waveguides with this size of air holes. For this choice of air-hole diameter complete bandgaps only exist for one polarization of the light, and consequently we will restrict our analysis to this polarization. The dispersion properties for the type A waveguide is shown for three waveguide widths W in Fig. 4. Similar to Fig. 2 this ®gure shows the allowed combinations of the normalized frequency
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T. Sùndergaard et al. / Optics Communications 194 (2001) 341±351 0.35 0.3
Λ /λ
0.25 0.2
0.15
Type A, TE pol.
0.1 W=0.0Λ W=0.5Λ W=1.0Λ
0.05 0 0
0.1
0.2
0.3
0.4
0.5
kΛ/2π Fig. 4. Dispersion relations (TE polarization) for three waveguide widths for the type A waveguide (air-hole diameter D 0:7K.
K=k and the component of the Bloch wave vector k in the direction of the waveguide. The gray regions correspond to the allowed modes of the photonic crystal material surrounding the waveguide. Due to the introduction of a waveguide (or a line defect) in the crystal there are discrete defect bands related to the guided modes of the waveguide. Even for the waveguide width W 0 there is a line-defect and thereby defect bands (solid lines) for the type A waveguide. A complete omni-directional photonic bandgap is not required for obtaining leakage-free guidance of light in straight waveguides, and also the defect bands for the various waveguide widths appearing in Fig. 4 for frequencies K=k < 0:2 correspond to guided modes in a frequency region where also propagating modes in the photonic crystal surrounding the waveguide are allowed. The electromagnetic modes related to the discrete defect bands in the frequency range from K=k 0:22 to K=k 0:33 are, however, con®ned to the waveguiding region due to an omni-directional photonic bandgap. The relevance of the omni-directional bandgap may appear when straight waveguides are combined to form bends and splitters such as Y-junctions and T-junctions. Only in the case of a complete omnidirectional bandgap is light prohibited from being scattered into the photonic crystal surrounding the waveguide con®guration near bends and the
branching points of splitters. However, for a ®niteheight waveguide con®guration the light reaching a bend or branching point may of course be scattered out into the media above and below the ®nite-height waveguide con®guration. For the type A waveguide we observe from Fig. 4 that for the width W 0 there are two defect bands (solid line) in the omni-directional bandgap, and the waveguide may be considered single moded for the frequency ranges covered by these bands. As the width changes to W 0:5K, several defect bands exist and cover quite dierent frequency ranges, i.e. for K=k 0:28 the waveguide structure will support a guided mode for W 0:0K but not for W 0:5K. For the width W 1:0K the waveguide supports guided modes for almost all frequencies within the omni-directional bandgap of the surrounding photonic crystal. In this case, however, the waveguide for some frequencies support two defect bands and the waveguide is for such frequencies multi-moded. There are, however, also frequency ranges for W 1:0K where the waveguide is single moded. The dispersion properties for the type B waveguide con®guration are shown in Fig. 5. In this case the waveguide width W 0 corresponds to a photonic crystal where no line defect has been introduced. For the waveguide width W 0:5K there is a large frequency interval from K=k 0:24
Fig. 5. Dispersion relations (TE polarization) for two waveguide widths for the type B waveguide (air-hole diameter D 0:7K.
T. Sùndergaard et al. / Optics Communications 194 (2001) 341±351
to K=k 0:3 where single-moded guidance of light is allowed. Single-moded guidance of light appears to be possible over a large frequency interval only for relatively narrow waveguides. As the width of the waveguide increases the number of defect bands related to guided modes also increases rapidly. If the width of the waveguide is increased to W 1:1K the waveguide becomes multi-moded in practically the same frequency interval from K=k 0:25 to K=k 0:3. For this waveguide width, however, there are also narrow frequency intervals where only one defect band exists and also in this case single-moded waveguidance is possible. Note that near the edge of the Brillouin zone (kK=2p 0:5) the defect bands are degenerated for the type B waveguide whereas this was not the case for the type A waveguide. The structure of the bands changes considerably as the waveguide is changed from a type A waveguide to a type B waveguide. Therefore it is possible to a high degree to control the dispersion properties of photonic crystal waveguides. But design of dispersion properties of such waveguides is also a complicated task and small deviations in the fabricated device relative to the original design may considerably change the dispersion properties and shift the frequency ranges where single-mode guidance is supported.
347
Fig. 6. Dispersion relations (TE polarization) for two waveguide widths for the type A waveguide (air-hole diameter D 0:9K.
4. Dispersion relations ± large holes For the ®nite-height waveguide with air above and below it was possible to consider using air holes with a large diameter D 0:9K. In this section we investigate the dispersion properties of the type A and type B waveguides with this air-hole diameter. In this case there are omni-directional in-plane bandgaps for both polarizations of the light, and we will therefore investigate dispersion relations for both TE and TM polarization. The dispersion relations for TE polarization are shown in Fig. 6 for a type A waveguide and in Fig. 7 for a type B waveguide. Clearly, the TE bandgap has moved up in frequency due to the larger hole diameter D (as expected from Fig. 3). The bandgap is now considerably larger, and therefore for the
Fig. 7. Dispersion relations (TE polarization) for two waveguide widths for the type B waveguide (air-hole diameter D 0:9K.
same waveguide width the waveguide will generally support more defect bands. From Fig. 6 it is clear that for the frequency ranges covered by the defect bands related to the width W 0 the type A waveguide with this width is single moded. Already for the waveguide width W 0:6K practically all frequencies within the gap are supported by a guided mode. As was also the case for waveguides with smaller air holes there are now frequency ranges where the waveguide is
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single moded, and others where it is multi-moded. For W 0:6K there is for example a large frequency range from K=k 0:36 to 0.44 where the waveguide is single moded. The dispersion relations shown in Fig. 7 for a type B waveguide with large air holes should be compared to the similar calculation for a waveguide with small air holes in Fig. 5. Consider the case of a narrow waveguide with W 0:2K in Fig. 7. In Fig. 7 the defect bands (solid line) takes up only a small part of the bandgap relatively to corresponding defect bands (W 0:5K) for a narrow waveguide in Fig. 4. The frequency ranges covered in both cases by the defect bands are, however, comparable in size. As was also the case in Fig. 5, the defect bands are degenerated at the edge of the Brillouin zone kK=2p 0:5. For TM polarized light we found in Section 2 the existence of almost omni-directional bandgaps for a photonic crystal without defects introduced. As a line defect is introduced defect bands related to guided modes may also appear in this type of bandgap. This is illustrated in Fig. 8, where we show the defect bands for a type A waveguide with width W 0:5K. For this choice of polarization and waveguide we show in Fig. 9 the ¯ow of energy for a guided mode corresponding to kK= 2p 0:15 and k=K 0:28. Here, the direction and length of the arrows illustrates the Poynting vector energy ¯ow [45] for the guided mode in the
Fig. 8. Dispersion relations (TM polarization) for the width W 0:5K for the type A waveguide (air-hole diameter D 0:9K).
Fig. 9. Energy ¯ow of a mode localized by an almost omnidirectional bandgap (TM polarization) in a type A waveguide. The frequency of the mode is K=k 0:28 and the Bloch wave vector is kK=2p 0:15. The arrows show the magnitude and direction of the Poynting vector. The amplitude of the electric ®eld squared is also shown. Here, dark shading corresponds to high intensity and light shading to low intensity.
waveguide. Fig. 9 also shows the amplitude of the electric ®eld squared for the mode. The dark shading represents high intensity and light shading low intensity. The guided modes illustrated by Figs. 8 and 9 may also be considered as bandgap guided modes, and indeed guidance of light at frequencies around K=k 0:28 could be considered as a possibility for making waveguides with the advantage that this may also work if the ®nite-height waveguide is placed on top of silica as pointed out in Section 2. An interesting question when using almost omni-directional bandgaps is how much light will be scattered out into the photonic crystal near bends and branching points. Due to the fact that the bandgap is almost completely omni-directional this scattering loss might turn out to be acceptably low, and this loss mechanism may not be high compared to scattering into the media above and below the ®nite-height waveguide. Almost omni-directional bandgaps should be kept in mind for future designs of ®nite-height photonic crystal waveguides. Fig. 10 shows for the type A waveguide and TM-polarization the dispersion relations for the omni-directional bandgap region relevant for
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zone (kK=2p 0:5). Even for the small width W 0:1K a guided mode is supported for nearly all frequencies within the omni-directional bandgap. Also for TM-polarization is it clear that the behaviour of the defect bands is highly complex and depends considerably on type of waveguide (A or B) and waveguide width. 5. Conclusion
Fig. 10. Dispersion relations (TM polarization) for two waveguide widths for the type A waveguide (air-hole diameter D 0:9K.
waveguides suspended in air. A striking feature of the defect band dispersion relations in this case is that these bands may be almost completely ¯at over a large wave vector range. Flat defect bands at the operating frequency results in low energy velocity and high dispersion [18]. For the width W 0:5K the waveguide is already multi-moded over most of the omni-directional bandgap. Fig. 11 shows a similar calculation for the type B waveguide. Also for TM polarization the defect bands are degenerated at the edge of the Brillouin 0.5
Type B, TM pol.
Λ /λ
0.45
0.4
0.35 W=0.1Λ W=0.3Λ 0.3 0
0.1
0.2
0.3
0.4
0.5
kΛ/2π Fig. 11. Dispersion relations (TM polarization) for two waveguide widths for the type B waveguide (air-hole diameter D 0:9K.
A method for obtaining design guidelines for ®nite-height photonic crystal waveguides has been described. The method is based on comparing calculations for in®nite-height structures with dispersion relations for the media above and below the ®nite-height waveguides. The approach can be applied to a large number of diagrams already available in the literature giving straightforward design guidelines for a large number of ®niteheight photonic crystals. By introducing fundamental upper frequency limits in a diagram of bandgaps versus the hole diameter for holes arranged on a triangular lattice in silicon it was possible to evaluate the proper choice of hole diameters depending on the media above and below the ®nite-height waveguides. For the photonic crystal with air holes arranged on a triangular lattice in silicon the existence of an omni-directional bandgap for both polarizations of the light requires air holes with a large diameter. In this case, however, leakage-free guidance of light by taking advantage of an omni-directional bandgap is not possible with a ®nite-height version of the investigated photonic crystal waveguides if placed on for example a silica substrate (the bandgaps have moved to frequencies higher than the fundamental limit related to silica). In this case the ®nite-height waveguide must be suspended in air to have a medium above and below the waveguide with a suciently low refractive index. Waveguides placed on silica supporting leakage-free guidance of light using an omni-directional bandgap can, however, be designed for one polarization of the light, but not for both polarizations, by using a smaller air-hole diameter. Taking advantage of an almost omni-directional bandgap for TM polarization it is also
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possible to obtain bandgap guidance of light using both polarizations if the ®nite-height waveguide is placed on silica. Almost omni-directional bandgaps should be kept in mind for future designs of photonic crystal waveguides. Dispersion relations for type A and type B waveguides has been given for the relevant polarizations of the light for both the case of small and large air-hole diameters in the photonic crystal. The variety of dispersion relations given in the paper show that the structure of the bands describing the guidance properties changes considerably as the waveguide is changed from a type A to a type B waveguide con®guration. The width of the waveguide is also an important parameter determining if the waveguide will be single moded, multi-moded or not guide light at all. It is indeed possible to a high degree to control the dispersion properties of photonic crystal waveguides by appropriate design. However, designing dispersion properties of such waveguides is also a complicated task, and small deviations in the fabricated device relative to the original design may lead to modi®ed dispersion properties and shift the frequency ranges where the waveguide supports single-mode guidance. Acknowledgements This work was supported by the EU-project PICCO (photonic integrated circuits using photonic crystal optics).
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