A LOW-PASS FILTER OF WIDE STOPBAND WITH A NOVEL MULTILAYER FRACTAL PHOTONIC BANDGAP STRUCTURE

: A novel multilayer fractal photonic bandgap (PBG) is presented as the substrate for a microstrip line and resulting conﬁgura-tion builds a low-pass ﬁlter (LPF) of wide stop-band. Experimental re-sults in comparison with the corresponding two ﬁlters with monoplayer PBG show that the proposed ﬁlter drastically enhances the width of the stop-band. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 40: 431–432, 2004; Published online in Wiley InterScience (www. interscience.wiley.com). DOI 10.1002/mop.11401

In this paper, it has been found that, to more correctly and logically verify the numerical dispersions of the conventional FDTD and ADI-FDTD methods, the test structure must be excited with the plane wave, rather than the commonly used point source. This is because the numerical dispersion relations of all kinds of FDTDrelated methods are derived from the assumption of the plane wave, and the cylindrical wave created by the point source has a different numerical dispersion relation from the one based on the plane wave. Moreover, it has been demonstrated that the propagation behaviors of the cylindrical and plane waves are different. For example, the phase velocity calculated from the point-source excitation highly depends upon the positions of the observation points, whereas for plane-wave excitation the phase velocity computed from the observation points that are arbitrarily located in the computational space is the same. Furthermore, numerical investigation indicates that using a point source to validate the dispersion relation based on the plane wave results in errors, unless the observation points are located infinitely far away from the point source (which is impossible in reality). Moreover, from the analysis of this paper, one can see that with the capability of current computers it is almost impossible to validate the numerical dispersion relations of 3D FDTD-related methods, even the validation is rough. However, an accurate and reasonable validation of 3D cases can be easily done if plane-wave excitation is used. Consequently, the results of this paper correct the common mistake (namely, using the point source as excitation) that was illogically used for the numerical validation of the numerical dispersion relations of the FDTD-related methods.

INTRODUCTION
Filtering of undesired frequencies in microstrip designs can be implemented with shunt stubs or stepped-impedance lines, but these techniques typically provide narrow band and a spurious passband in stopband, and occupy valuable circuit-layout area. Photonic bandgap (PBG) structures have been considered as an alternate to solve this problem in microwave and millimeter-wave circuit applications recently [1][2][3][4][5][6][7][8]. (PBG structures are periodic structures which exhibit a bandgap within a certain band of electromagnetic propagation is prohibited [9,10].) But most PBG structures show a periodic frequency stopband which is termed harmonic in the higher-order stopband. In order to achieve wide stopband characteristics, several efforts have been proposed using serial or parallel PBG structures to meet this goal [11,12]. But this requires large size and has limited microstrip-circuit application in compact sizes. In this paper, a low-pass filter (LPF) of wide stop-band with a novel multilayer and fractal PBG structure is introduced. Moreover, the comparisons of a LPF with the conventional PBG structure, a LPF with one-order Sierpinski carpet PBG, structure and the proposed LPF are shown. The experimental results of the three LPF comparisons are presented. At the same time, the results show that the transmission characteristic of the proposed LPF is better than the other two.

DESIGN OF THE PROPOSED LPF
The proposed LPF with multilayer fractal PBG structures is described in Figure 1. It is a microstrip structure in which the ground plate has been replaced by a two-level fractal PBG plate arrange-  Figure 2. The intermediate fractal PBG structure is made of eight three-order Sierpinski gaskets with a period of p 1 ϭ 38.2 mm and the bottom fractal PBG structure is made of seven one-order Sierpinski carpets (in fact, the proposed Sierpinski carpet is not a totally fractal shape, strictly speaking [13]) with a period of p 2 ϭ 22.6 mm.
The bottom PBG is surrounded by a metallic region connected to the ground, while the intermediate PBG is grounded by holes. The Teflon-based substrate has a dielectric constant of 2.22 with a thickness of h 1 ϭ h 2 ϭ 0.254 mm. A 50⍀ microstrip transmission line on the top plate has a width of 1.77 mm, corresponding to a gap center frequency of 5 GHz. The photograph of the structure is shown in Figure 2.

EXPERIMENTAL RESULTS
Three LPFs with different PBG structures are designed: one LPF has a monolayer PBG structure, with typical square holes etched in the ground plane (LPF1), and one LPF has a monolayer PBG structure with the proposed one-order Sierpinski carpets etched in the ground plane (LPF2), and the proposed one (LPF3), is described in Figure 1. The experimental results of the transmission characteristic of the three LPF are shown in Figure 3. In the 3-20-GHz-wide stopband, shown in Figure 3, the performance of the proposed PBG structure is better than the performance of the other two, and the transmission coefficients of the proposed one are below Ϫ20 dB while the other two have a more than Ϫ10-dB transmission peak at about 7 GHz. The measurements were performed with a network analyzer (HP 8722D) over the frequency range 50 -40 GHz.

CONCLUSION
An LPF with novel multilayer fractal PBG structures has been proposed, realized, and tested. In this specific design, the multilayer PBGs include two fractal structures with different patterns. Experimental results show that the proposed filter drastically enhances the width of the stop-band, in comparison with the corresponding two filters with monolayer PBG.