A method of photonic band gap extension using mixing of periodic structures with two or more consecutively placed photonic crystals with different lattice constants is proposed. For the design of the structures with maximal photonic band gap extension the gap map imposition method is utilised. Optimal structures have been established and the gap map of photonic band gaps has been calculated at normal incidence of light for both small and large optical contrast and at oblique incidence of light for small optical contrast.
© 2005 Optical Society of America
Photonic Crystals (PCs) are materials with a periodic modulation of the dielectric constant. This modulation gives rise to what is known as a Photonic Band Gap (PBG) within the crystal . The well known Bragg mirrors widely used in modern technology is one example of one-dimensional photonic crystals (1D PCs) . Under certain conditions in a 1D PC the complete PBG, i.e. the total reflection at any polarisation and angle of incident of light, can be obtained [3–5]. Due to these particular properties the aforementioned 1D PCs are equivalent to 3D PCs and can be used in waveguide structures where total reflection from the surrounding waveguide walls is required. These 1D PCs are comprised of alternating layers with refractive indices n 1 and n 2 and widths d 1 and d 2 .
PBGs for 1D PCs can be obtained from the dispersive equation . The set of PBGs, calculated for different filling factors f(f=d 1/(d 1+d 2)), is used for drawing the PBG regions  on so-called maps of photonic gap. These gap maps represent more clearly the optical properties of PCs of different dimensions for different polarisation of light (including the regions of the complete PBGs) and the effects of changing the photonic properties due to changes of the optical properties of one of the components [7–10].
The existing technology allows fabrication of photonic crystals with a large number of periods (m ≥100). For 1D PCs, most of which are fabricated using a thin-film coating technology, any substantial increase in the layer thickness is undesirable due to the possible strain arising in the layer itself or at the interface between the layer and the substrate. This strain may result in cracks and peeling. During fabrication of PCs with different dimensions some fluctuation of geometrical parameters of the components may arise. These fluctuations are i) deviation from the required sizes, ii) roughness at the interface of different components. As a result of these deviations the transmission properties of the fabricated photonic structures with a large number of lattice periods (in particular 2D and 3D PCs) must be substantially reduced. Due to this fact the investigators have been forced to reduce the number of periods, m to the reasonable value during fabrication of the PCs. Therefore, the influence of the reduction in m value on the optical properties of the photonic crystals must be taken into account. For this purpose the calculations of the reflection/transmission spectra using the transfer matrix method (TMM)  are often applied. In this method any number of periods, m, can be chosen starting from m=1 and this is the definite advantage of this method of performing the calculations.
The calculations of photonic gap map for different values of m for 1D PCs based on silicon-air structures TMM were presented in our earlier work . In this approach the set of reflection spectra for each f value ranging from 0 to 1 was first calculated. After this, values of wavelength, λ (or wavenumber, v) satisfying the required criterion of PBG formation, RPBG were selected and the map of PBGs was plotted. We note that the PBG map we obtained compares favourably with the one calculated using a dispersive relation, if the value of m is relatively large. For smaller values of m the deviation between the ideal (obtained from dispersive relation) and designed PBG map is greater. For example, Fig. 1(b) shows clearly the difference between the first PBG regions for m=10 and m=40. It is well known that as m is reduced, the conditions for wide PBG formation deteriorate and may even result in the absence of the necessary values of RPBG in the calculated reflection spectra for certain values of m. In some cases it is sufficient to reduce the required parameter R PBG from 1 to 0.999 in order to find the necessary PBG. The criterion RPBG= 0.999 has been chosen in this study for methodological convenience.
Therefore, the gap map analysis allows an effective comparison of different PC structures depending on lattice constant, optical contrast ∆n=n 1/n 2, polarisation and angle of incidence of incoming light, number of periods and selection of the optimal structure for the required practical application. At the same time the width of the PBG is also one of the important parameters of PCs. It is desirable to have a band gap as wide as possible during the design of PCs for applications such as coating mirrors for a wide spectral range , for the fabrication of multi-frequency filters and in structures with tunable-defect mode . In general, an extension or widening of the PBG can be achieved by increasing the optical contrast ∆n. Recently a number of methods for extension of the High Reflection Range (HRR) for dielectric multilayer stack structures consisting of binary layers has been suggested. This is based on a combination of experimental observation and theoretical considerations . The theoretical framework for these methods was developed as a result of studies of the localisation of light in 1D crystals. In order to bring about this extension, disorder was introduced to the higher refractive index layers as a thickness variation along the whole structure. This approach was further developed in Ref. , where it was shown theoretically that PBG extension in the disordered 1D PCs can be even more substantial, up to three times wider, if three neighbouring PBGs can be merged into one PBG. In Ref.  the authors designed and fabricated a disordered 1D PC based on Si-air components and obtained merging of three neighbouring PBGs into one. This confirmed both the calculated spectrum and the theoretical predictions.
Another suggestion for extension of the spectral region of the HRR has been made in Refs. [18,19]. The main idea in both papers is based on the stacking of multilayer components with different lattice parameters. In this work we propose to use a similar idea but with a different approach. This approach is based on using the overlapping areas of the gap map in order to determine the composite extended PBG. Our approach visually demonstrates the possibility of designing the periodic structures with any optical thickness and with the variation of both the lattice constant and the filling factor. Different methods and opportunities for PBG extension for PCs with both small and large optical contrast and at different angles of incident light are discussed.
2. Method for producing band gap extension
2.1 Structures with small optical contrast
We begin by drawing the gap map for the periodical structure with optical contrast ∆n=2.3/1.45 (n 1=2.3, n 2=1.45). We take the initial PC1 with lattice constant A1=0.21 μm, number of periods m=10 and the refractive indices of external media and substrate equal to 1 and 1.5, respectively. For each value of f ranging from 0 to 1 we calculate the reflection spectra of the periodic structure using the TMM for normal incidence of light. The calculations were performed with a wavelength step increase equal ∆λ=0.0001 and a step in A value equal to ∆A=0.01. Here f=d 1/(d 1+d 2) and d 1+d 2=A1. Then we select all the wavelength values for which R>0.999 (since R PBG=0.999 as mentioned above) and draw the gap map for the structure under investigation. Strictly speaking if one considers a finite PC there is no photonic band gap but regions of high reflectivity. However, because the method proposed in this paper is based on gap map presentation suggested in Ref.  we used the term PBG instead of regions of high reflectivity for consistency with the original name of the procedure.
The calculated PBG region (thin line), which is lowest in frequency, is presented in Fig. 1 as a closed region with the empty space surrounding this region representing the ranges of transmission, T. Note that transmission means the regions of wavelength where T can vary from 0.001 up to 1. We now calculate the gap map for PC2 with A2=0.185 μm. This value of A2 was chosen so as the PBG region of PC2 partly overlaps the region of PBG for PC1 with A1=0.21 μm (Fig. 1 (a)). The same procedure was used to calculate PBG for PC3 and PC4 with A3=0.156 and A4=0.13 μm, see Fig. 1 (a). The values A2–A4 were obtained simply by selection of the lowest PBG for the second PC overlapping with the lowest PBG for the first PC not only for one value of f but for a range of f values. This range of f values was chosen quite arbitrarily in this paper only for the purpose of demonstrating clearly the effect of PBG overlap. Also shown in this figure is the merging of four PBGs leading to one extended PBG (EPBG) in the wavelength range λ≈0.4-0.8 μm. For this the value of f for each PC must be within 0.3-0.5, but the f value must be the same for all PCs included in a stack. Due to the fact that in composite PC the lattice constant will be changed for the second and following PCs, the values of d 1 and d 2 for these PCs must be changed simultaneously to keep f as a constant value. Now we can build the gap map for a composite PC (CPC), which includes four PCs consecutively located in one structure: external medium-A4-A3-A2-A1-substrate with number of periods, m=10 for each single PC (m=40 in total). Let us now calculate the reflection spectra, R of this composite PC using TMM for different f values, select values of λ with R≥0.999 and build the gap map for CPC (see Fig. 1 (b)).
As can be seen from Fig. 1 (b), the region of extended PBG is close to the predicted one. Therefore, the spectra R calculated for the values f≈0.25-0.5 will reach the value of R>0.999 in the range from 0.4-0.5 up to 0.8-0.9 μm. Obviously, by changing the A value, we can obtain composite PCs for a different range of spectra (taking into account the optical contrast of the periodic structure). Moreover, we can either add or remove one of the CPC elements and this will result in further reduction or extension of the EPBG for the new CPC. The gap map for a PC with A=0.21 μm and m=40 is presented in Fig. 1 (b) for comparison. As can be seen from the insert in Fig. 1 (b), by increasing m from 10 to 40 this leads to a 1.5x extension in the PBG. The design of the composite PC (comb) allows additional PBG extension by a factor of 2 to 3.
2.2 Structures with large optical contrast
It is possible to calculate and to plot the joined PBGs, in the same way as outlined above, using addition of the lowest PBGs for PCs with different values of A. However, we proposed that another approach is to proceed from the key features of the gap map of structures with large optical contrast. These features are that the PBGs of higher order are separated both from each other and from the lowest PBG, by the regions of transparency comparable in area to the PBG regions. This fact creates conditions for their overlapping and for PBG combination. Applications of this idea will be published in more details elsewhere . In order to demonstrate this let us build the gap map for a periodic structure Si-Air with optical contrast ∆n=3.42/1 (n 1=3.42 and n 2=1). For the purpose of these calculations the light beam is assumed to enter the PC from an ambient with n=1 and, after travelling through the PC, to reenter the same ambient.
We first calculate and plot the gap map for PC1 with A1=3 μm and m=10. Next we calculate the gap map for PC2 with A2=A1×0.71=2.13 μm and perform merging of the two maps obtained (Fig. 2 (a)). The reason for this value for the parameter A2 is that this particular parameter provides the widest range of possible f values (f=0.45-0.49) at which four PBGs of the PC stack will overlap. As happened for the structures with small optical contrast, the PBGs of PC2 (with A2=2.13 μm) are shifted to the shorter wavelength (blue shift) in comparison to the PBGs for the original PC1. We can see from Fig. 2 (a) that for the region of f=0.45-0.49, the PBG regions of four PCs overlap with possible formation of EPBG, whilst in the region of f=0.09-0.1 there is a possibility to form another EPBG. Once we know the possible regions of EPBG formation we can plot the gap map for the composite PC. For this purpose we construct a CPC which, as in the previous case, consists of two consecutively located photonic crystals PC1 and PC2 with A1=3 and A2=2.13 μm.
We then calculate the reflection spectra of this composite PC by the TMM for different values of f , select the λ (or v) values for which R≥0.999 and plot the gap map of the CPC obtained, see Fig. 2 (b). From Fig. 2 (b) the closed regions of the PBGs can be seen, as was the case with ordinary PCs. As expected, the PBG regions are located in predicted regions of f values (for f=0.42-0.53 and f=0.08-0.11). At the same time the extension of other PBGs is also seen. The PBG regions corresponding to the lowest frequency scale regions of CPC are also extended (Fig. 2 (b)) due to the partial overlapping of PBGs of the original component of the PCs (see Fig. 2 (a)). Therefore, we can confirm that the combination of the optical properties of both original PCs has occurred and these are complementary to one another. However, the contours of the extended band gap regions shown in Fig. 2 for composite PC with high refractive index contrast are quite similar to the overlapping region obtained for 4 gaps of the individual PCs. This is not the case shown in Fig. 1 for combined PC structure with small refractive index contrast. We believe that there could be a few reasons for this effect such as i) influence of the refractive index of the substrate or ii) influence of the refractive index contrast. We plan to investigate these effects in the future.
We note that for the CPC a reduction in the m value of both original PCs (Fig. 2 (b)) is possible up to a certain limit (m=7 in this case) below which the regions of transparency will appear and no substantial extension of the EPBG is observed. The reduction of the m value, even up to 7, is however, quite important from a practical point of view, since the structure obtained consists of only two PCs with a total number of periods m=14. For these cases the size of the EPBG regions is λ=2.2-12 and 4-20 μm. We note that the change in order in the original PCs has practically no influence on the EPBG of the composite PC. An investigation of the influence of the f value (at constant A) on the EPBG in the composite PC was also carried out. We found that in this case the size of the EPBG extension is not as great as in the case of the variation in A . It is more difficult to investigate the effect on EPBG of changing both f and A parameters simultaneously, however this was not attempted in the work reported here. It was also found that for structures with large optical contrast we can also obtain a larger EPBG extension by increasing the number of single PCs with different values of A, as was observed in the case of structures with small optical contrast.
2.3 Omni-directional 1D photonic crystals
The omni-directional (OD) reflection region is normally determined by overlapping of regions of high reflectivity for both polarisations and at all angles (from 0 to 90°) of the incident light. Calculations of the gap maps for 1D PCs with different refractive index contrast and different angles and polarisations of the incident light have been performed using the TMM approach. A detailed description and discussion of the results obtained will be published elsewhere. Due to the importance of these results for demonstration of the viability of the proposed approach we have included one of the results we obtained in this section. One of the main conclusion arising from these results is that for structures with small refractive index contrast (∆n =2.35/1.45) and m=40 the OD region is relatively small, not more than ∆λ/λ=2.3 %. We propose here to form wider OD regions using the PBG extension method suggested in this paper. We will use the normal notation for TM polarisation when the electric vector of the incident light is parallel to the plane of incidence and TE polarisation when the electric vector of the incident light is perpendicular to the plane of incidence. Typically as the angle of incidence increases the TM region decreases. Therefore, the most important and simple check on the gap maps if the overlapping of the TM and TE regions obtained at the maximum angle of 85° (marked as TM-85 and TE-85) with PBG region for normal incidence (marked as region 0) has happened.
Let us first obtain the extended PBG for two PCs at normal incidence of light, as described above. We then calculate the map of PBG regions for TE И TM polarisation for φ ranging from 0° to 85° with step of 1°. For this, we calculate separately the gap maps for two PCs, the joining of which can provide the extended OD region. The following lattice parameters were chosen for PC1: A1=0.21 μm, m=20, the refractive index of the external medium and substrate are 1 and 1.5 respectively. Then we draw the map for PBG regions of PC2 with A2=0.857×A1=0.18 μm (m=20) and merge gap maps for PC1 and PC2. These particular parameters for A1 and A2 were selected for the purpose of demonstrating the possibilities for the extension of OD region. As was already demonstrated in Sections 2.1 and 2.2 the first PBG will be extended for normal incidence of light (Figs.3 (a) and 3 (b)). Similarly, the PBG regions for TE polarisation will also be extended. The PBG regions for TM wave decrease with increase of the angle of incidence in comparison with the regions for TE wave and normal incidence of light and as a result the overlapping of these TM-85 regions for two PCs does not occur (Fig. 3 (a)). By overlapping the gap map for both PCs for both polarisations and extreme angles of incidence (0° and 85°) we obtain a new gap map shown in Fig. 3 (a). It can be seen from this figure that the second region for TM-85 (at the bottom) overlaps with the extended PBG for the case of normal incidence of light and TE-85 region. Therefore, we can predict the appearance of OD region, in particular in the range of overlap. Based on these preliminary estimations we now design the composite PC contain a stack of PC1 and PC2 with total number of periods m=20+20=40. We now calculate the gap map for composite PC with lattice parameters A1 and A2. The procedure is the same as described above: using TMM we calculate the reflection spectra for the multilayer stack: external media-PC 1 (m=20)-PC2 (m=20)-substrate for angle of incidence φ=0° and φ=85°. Based on these calculations we can draw the gap map shown in Fig. 3 (b). Fig. 3 (b) shows that, in fact, in accordance with our predictions, both PBG regions for normal incidence of light and for TE polarisation at φ=85° join together and are extended. Moreover, as was expected, the second PBG region TM-85 is formed and at the same time falls into both the extended TE region and the PBG region for normal incidence of light. Then, the gap maps were calculated for angles <85° and merged, as described in Sections 2.1 and 2.2, which confirmed the existence of OD region for the composite PC. Thus, we can conclude that using the method of PBG extension, based on the gap map overlapping we have received the omni-directional region for structure with small refractive index contrast (∆n=2.35/1.45) and m=40 for the region of f values ranging from 0.28 up to 0.4 in vicinity of λ~0.6. The relative width of ∆λ/λ=11.8 % for f=0.35.
A novel method for the extension of Photonic Band Gaps by designing a composite periodic structure consisting of conventional PCs with different lattice constants and filling factors has been presented. The choice of the additional PC structure is done by merging the gap maps for original PCs. The gap map of composite photonic crystals with optical contrast ∆n=3.42/1 for the middle infrared range and structure with small optical contrast for the visible range of spectra at normal and oblique incidence of light has been calculated. The merging of regions of transparency with photonic band gaps resulted in substantial extension of the original PBGs due to the high optical contrast of photonic crystals. The suggested approach significantly reduces the range of searching the possible structures with extended PBGs, by directly pointing to the limited range of structures with optimal characteristics. The range of the omnidirectional reflection in composite PC was enlarged from 2% up to ~11% as a result of the PBG extension.
This work has been supported by the Science Foundation Ireland Basic Research Program (Grant 04/BR/P0698) and Russian Programs “Physics of Solid-State Nanostructures”, Laser Physics and Scientific school -758.2003.2. The authors would like to thank Ekaterina Astrova for useful discussions.
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