## Abstract

We propose a general design methodology for photonic crystal (PhC) diplexers, which is carried out along a filtering T-junction. The diplexer operation is investigated while carefully analyzing the dispersion relations of the three different waveguide channels. All simulations are carried out using the multiple multipole method (MMP), which offers perfect excitation and matching conditions for all waveguide ports involved. The resulting diplexer is highly compact (it covers an area of 13×9 lattice constants) and simple when compared to other PhC diplexer designs.

© 2003 Optical Society of America

## 1. Introduction

The existence of optical band gaps [1–3] in photonic crystal (PhC) structures is convenient for the design of interesting devices. By introducing some defects in a PhC defect modes will appear [2, 4, 5] whose modal fields are localized within a small volume around the defects. Hence, it gives us the ability to control the field profile and direction of the light propagation inside of such defect crystal structures. It is well known that simple line defects in a PhC structure form very effective optical waveguides [6, 7], that provide extremely narrow waveguide bending [8, 9]. Different types of defects were adopted to waveguides [10], wavelength filters, diplexers, and multiplexers [11]. The most important part in the PhC’s defect mode analysis is the computation of the underlying dispersion relation. A robust algorithm for the automatic band diagram calculations for two-dimensional PhCs has already been developed in [12] using an accurate eigenvalue solver based on the multiple multipole (MMP) [13] code contained in MaX-1 [14, 15]. This kind of analysis for perfect PhCs is easily extendable to defect modes [16] when relying on the supercell approach [2, 4].

In the context of PhC waveguide discontinuities the MMP based analysis is able to overcome many difficulties [16] of other simulation schemes, e.g., the finite difference time domain (FDTD) analysis [9], time-domain beam propagation simulations [17], and frequency domain scattering matrix analysis [18]. According to these references, one can say that these other techniques are confronted with two major difficulties: 1. High memory requirements, due to the very long distances between excitation/termination and the discontinuity. 2. The need of complicated boundary conditions in order to tackle the residual reflections at the waveguide ports of the PhC device. Referring to the eigensolutions of our supercell analysis we have been able to introduce perfect matching conditions for PhC waveguide terminations [16]. This allows us to significantly reduce the size of the simulation domain, i.e., to lower the resulting memory requirements and computation time. Design examples for such compact PhC devices (i.e. various PhC T-junctions) have already been reported [16].

In the following we propose a design procedure for highly compact diplexer topologies (i.e., a filtering T-junction as can be seen in Fig. 3, input port on top and output ports to the left and right), where efficient power splitting and wavelength selectivity are achieved within a planar PhC’s volume of only 13×9 lattice constants.

## 2. Design of the PhC diplexer

The PhC diplexer proposed here consists of a filtering T-junction as shown in Fig. 2. Regarding its demultiplexing mode of operation the optical signal is fed through the input port into a W1 (one line of vacancies) defect waveguide of the T-junction. This W1 waveguide can support light propagation in a wide wavelength range of the photonic band gap. Wavelength selective power splitting is enabled, by introducing corresponding dispersive elements (like e.g. substitutional defects) into the two output waveguides of the T-junction. In other words, when disposing of two output waveguides each with its own dispersion relation allows *k*-matching to the input waveguide’s wave vector at two different wavelengths (labeling the two corresponding wavelenth channels of the diplexer). Efficient filtering requires some additional optimization regarding to the coupling efficiency and the set of waveguide modes involved. Within our proposed design procedure we carefully investigate the resulting dispersion relations, wherefrom the distinct operational wavelength ranges of the two output waveguides are deduced. For the diplexer design we acquire the following three steps.

The first step in the design of our PhC diplexer consists in the retrieval of the band structure of the underlying perfect PhC. For that reason we rely on a fully automatic band diagram calculation software developed on the basis of MMP [12]. A resulting band diagram is depicted in Fig. 1. Without loss of generality we have restricted our analysis to a simple structure such as the PhC with dielectric rods arranged in a square lattice and embedded in air. The lattice data are as follows: the radius of each dielectric rod is *r*=0.18·*a* (with *a*=1 µm being the lattice constant), and the rod’s dielectric constant is ε=11.56.

Within the PhC diplexer design proposed here W1 defect waveguides are used in the sense of a generic waveguide type. As a second step of the design procedure it is therefore necessary to compute the dispersion relation of the W1 defect waveguide according to its propagation direction. This is done while introducing the supercell approach [1, 2] into our automatic MMP eigenvalue solver [12]. The results are presented in Fig. 2. The W1 defect waveguide supports many additional localized defect modes (above and bellow the band gap) [16], and one propagating mode covering almost the entire band gap (red curve in Fig. 2). The propagating solution is the most important one for our investigation whereas the various localized modes are – due to symmetry transformations – associated with the formation of the supercell’s 1^{st} Brillouin zone [16]. The latter can be viewed as evanescent supermode solutions of multiple parallel W1 waveguides, which are inflicted by the supercell’s implicit periodicity. Having analyzed the W1 waveguide within its entire wavelength range, the three supporting waveguide ports of the filtering T-junction are now fully characterized.

The third step is dedicated to the most critical part of the proposed diplexer design: the introduction of the wavelength filtering elements. Therefore one needs to find two additional PhC waveguide types with distinct bands within the operational range of the supporting W1 defect waveguide. Such filtering elements are easily deducible from the W1 defect waveguide just by adding some substitutional defects (dielectric rods with different radius and different permittivity) into the defect waveguide [10], or by introducing structures like coupled cavities [19]. For the diplexer design it is necessary to properly engineer the shape (position and frequency range) of the dispersion characteristics for each of the two filtering elements. While performing an exploratory analysis we found that changing the radius of the substitutional defect allows a considerable vertical (frequency) shifting of the waveguide’s dispersion curve. An additional degree of freedom is offered by the coupled-cavity waveguide where the total frequency range of the corresponding dispersion curve can be adjusted by altering the cavity volume (and the coupling strength between the cavities). For the left and right channel of the filtering T-junction we can now specify two appropriate PhC waveguide types. The choice has been led by the following design considerations: (i) In order to circumvent potential reflections at the diplexer ports, both the left and the right channel should be adapted to the W1 input waveguide as well as to the two supporting W1 output waveguides. (ii) Proper matching at the interface between two different waveguide types only occurs if both waveguides have equal group velocities and wave vector matching is maintained (as indicated by the intersections in the dispersion characteristics shown in Fig. 2).

All necessary information for the PhC diplexer design is presented in Fig. 2: The generic W1 defect waveguide shows one propagating defect mode covering the entire band gap (red dispersion curve). The left waveguide channel is formed by a coupled cavity waveguide (dimensions are given in Fig. 2) having modes with negative group velocities (blue curves). This waveguide can support light propagation within the upper part (towards higher frequencies) of the entire band gap (blue range). In addition the coupled cavity waveguide supports group velocity matching for a large frequency range within the W1 input waveguide’s dispersion characteristics. The right waveguide channel is made up of substitutional line defects (dielectric rods with a large radius as given in Fig. 2 and Fig. 4 respectively), providing light propagation at a lower frequency range within the entire band gap (yellow range). The upper (blue) range and the lower (yellow) range indicate the regime where only the modes of one corresponding waveguide channel exist. Proper k-matching occurs at each intersection of two dispersion curves, namely between those of the W1 defect waveguide and the left (or right) waveguide channel, whereas group velocity matching is achieved when both dispersion curves are intersecting with the same absolute value of the slope. The latter is usually more difficult to achieve and is therefore often subject to further optimization: In order to meet the diplexer’s specifications one has to tune the two output waveguide channels to produce intersections (between corresponding dispersion curves) at the desired frequencies while maintaining both k-matching and group velocity matching.

Within our analysis we found that the length of filtering waveguide channels can be restricted to only 3 or 4 unit cells along the propagation direction leading to a very compact and simple PhC diplexer topology as e.g. depicted in Fig. 3. It additionally turned out that a slight lateral displacement of the substitutional defects has little effect on the dispersion characteristics, but can provide an additional degree of freedom when maximizing the power coupling between the input and the corresponding output channels while minimizing cross talk between both output channels. Hence, we slightly shifted the substitutional defects of the right waveguide channel in order to align its input to the boundary of the W1 input waveguide. In the case of the right output channel we were very close to the cutoff frequency of the left W1 waveguide channel, thus, significant evanescent terms are present with very slow decay towards the left port. In order to maintain a good isolation (of the left port) the left filtering channel has been extended accordingly.

The simulation of the compact PhC diplexer is performed as follows: The MMP expansions parameters of the generic W1 defect waveguide’s supercell are estimated at all wavelengths involved and made available for any further computation. Thus, these expansions are packed into connections [13, 14], and used as perfect matching conditions at every port [16] of the filtering T-junction. This approach allows us to perform a highly accurate analysis at relatively short arm lengths of the filtering T-junction [16] without using complicated boundary conditions [20]. The length of the output branches is limited only by the decay of the evanescent modes produced by waveguide discontinuity itself [16]. The small size of the simulation domain makes the resulting MMP model of the overall diplexer topology easily manageable. Referring to the Fig. 2, group velocity and wave vector matching occurs at the intersection points marked with the black circles. The upper circle labels the relative frequency for the left output channel (ω·*a*/2·π·*c*=0.387) and the lower circle represents the relative frequency for the right output channel (ω·*a*/2·π·*c*=0.346). Both cases of diplexer operation are shown in the Fig. 3.

This preliminary diplexer design acts as an illustration of our design methodology. A close look at the performance of the filtering T-junction indicates that there is still room for further improvements. The main reason for the moderate performance of the right output channel as shown in Table 1 is due to crosstalk and to residual reflections stemming both from poor group velocity matching at the upper edge of the (yellow) frequency range.

## 3. Design improvement

A phenomenological argument helps to improve the design: A small increase in the radius of the right waveguide’s substitutional defect up to *r*=*0.375·a* is sufficient to obtain the desired downward shifting of the dispersion curve (black lines) where better group velocity matching is achieved. A fact that can be easily reproduced just by inspecting Fig. 4. With this new waveguide we obtain intersection with the dispersion curve of the generic W1 defect waveguide at a lower relative frequency of ω·*a*/2·π·*c*=0.336. Due to the better group velocity matching smaller reflection at the input of the right output channel is achieved.

Moreover, we are further away from cutoff frequency of the left output waveguide channel (marked by the black horizontal line in Fig. 4) and hence crosstalk will be lowered as well. The light propagation through the new diplexer design (at the relative frequency of the right channel) is shown in Fig. 5. The resulting reflectance and transmittance calculation for of the improved design are given in the Table 2.

Obviously, the new design performs much better than initial one. It shows power reflection below 10% and power transmission above 90%, for both wavelength channels. Furthermore the level of the crosstalk is very low. Compared to other references on PhC diplexers [21, 22] our design turns out to be very promising in terms of both compactness and transmission ability.

The MMP computations of band structures for perfect PhC crystals are of very high accuracy, which is confirmed in the context of metallic PhCs [23] and when comparing to other algorithms [12]. A similar conclusion can be drawn concerning the MMP treatment of classical waveguide discontinuities like e.g. waveguides and bends [9, 16]. The internal MMP check detects average relative errors below 1%. Up to now, a reliable validation of the filtering T-junction still lacks of alternative data. However, power conservation including all device ports may foster confidence in the numerical results obtained. As one can see from Table I and II total power is 99% and 98% respectively, i.e. we have accuracy in the order of 1%. It is worth mentioning that the quality of power conservation is only warranted if the supporting waveguides are sufficiently long (more than one effective wavelength [16]) in order to omit evanescent field coupling between the waveguide discontinuity and the corresponding device port.

## 4. Conclusion

We have presented a functional straightforward design procedure for PhC diplexers. The general methodology is carried out along a filtering T-junction, which has been implemented into a simple PhC with dielectric rods arranged in a square lattice. Proper diplexer operation is investigated while carefully analyzing the dispersion relation of the different PhC waveguide types involved. An improvement of the initial design is suggested and executed as well. Finally, the accuracy of our device simulations is discussed in the light of the available data. The resulting diplexer is extremely compact (it covers an area of around 13×9 lattice constants) and simple when compared to other PhC diplexer designs [21, 24]. Our general design procedure should give a good starting point especially when looking into other PhC designs based on structures with different lattice geometries and material systems.

## Acknowledgement

This work was supported by the Swiss National Science Foundation.

## References and Links

**1. **J. D. Joannopoulos, R. D. Meade, and J. N. Winn, *Photonic Crystals - Molding the Flow of Light* (Princeton University Press, New Jersey, 1995).

**2. **K. Sakoda, *Optical Properties of Photonic Crystals* (Springer-Verlag, Berlin, 2001).

**3. **K. M. Ho, C. T. Chan, and C. M. Soukolis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. **65**, 3152–3155 (1990). [CrossRef] [PubMed]

**4. **E. Yablonovich, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. **67**, 3380–3383 (1993). [CrossRef]

**5. **M. Sigalas, C. M. Soukolis, E. N. Economou, C. T. Chan, and K. M. Ho,“Photonic band gaps and defects in two dimensions: Studies of the transmission coefficient,” Phys. Rev. B **48**, 14121–14126 (1993). [CrossRef]

**6. **E. Centeno and D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. **160**, 57 (1999). [CrossRef]

**7. **H. Benisty, “Modal analysis of optical guides with two-dimensional photonic cand-gap boundaries,” J. Appl. Phys. **79**, 7483–7492 (1996). [CrossRef]

**8. **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). [CrossRef]

**9. **A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. **77**, 3787–3790 (1996). [CrossRef] [PubMed]

**10. **M. Loncar, J. Vuckovic, and A. Scherer, “Methods for controlling positions of guided modes of photonic crystal waveguides,” J. Opt. Soc. Am. B **18**, 1362–1368 (2001). [CrossRef]

**11. **E. Centeno, B. Guizal, and D. Felbacq, “Multiplexing and demultiplexing with photonic crystals,” J. Opt. A **1**, L10 (1999). [CrossRef]

**12. **J. Smajic, Ch. Hafner, and D. Erni, “Automatic calculation of band diagrams of photonic crystals using the multiple multipole method,” ACES Journal, (to be published).

**13. **Christian Hafner, *Post-modern Electromagnetics Using Intelligent MaXwell Solvers* (John Wiley & Sons, Chichester, 1999).

**14. **Christian Hafner, *MaX-1: A Visual Electromagnetics Platform* (John Wiley & Sons, Chichester, 1998).

**15. **Christian Hafner and Jasmin Smajic, The Computationa Optics Group Web Page (IFH, ETH Zurich), http://alphard.ethz.ch/.

**16. **E. Moreno, D. Erni, and Ch. Hafner, “Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method,” Phys. Rev. E66, 036618 (2002). [CrossRef]

**17. **M. Koshiba, Y. Tsui, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technol. **LT18**, 102–110 (2000). [CrossRef]

**18. **J. Yonekura, M. Ikeda, and T. Baba, “Analysis of finite 2-D photonic crystals of columns and lightwave devices using the scattering matrix method,” J. Lightwave Technol. **LT17**, 1500–1508 (1999). [CrossRef]

**19. **A. Boag and B. Z. Steinberg, “Narrow-band microcavity waveguides in photonic crystals,” J. Opt. Soc. Am. A **18**, 2799–2805 (2001). [CrossRef]

**20. **A. Mekis, S. Fan, and J. D. Yoannopoulos, “Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides,” IEEE Microwave Guided Wave Lett. **9**, 502–504 (1999). [CrossRef]

**21. **K. B. Chung and S. W. Hong, “Wavelength demultiplexers based on the superprism phenomena in photonic crystals,” Appl. Phys. Lett. **81**, 1549–1551 (2002). [CrossRef]

**22. **S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals, “Opt. Express **3**, 4–10 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4 [CrossRef] [PubMed]

**23. **E. Moreno, D. Erni, and Ch. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B65 155120 (2002). [CrossRef]

**24. **M. Koshiba, “Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers,” J. Lightwave Technol. **LT19**, 1970–1975 (2001). [CrossRef]