Guided modes in a two-dimensional metallic photonic crystal waveguide

Guided modes in a two-dimensional metallic photonic crystal waveguide are studied. The guided modes in the photonic crystal waveguide are related to those in a conventional metallic waveguide. There exists a cutoff frequency and Ž . consequently a mode gap at low frequencies starting from zero frequency in the photonic crystal metallic waveguide. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 42.70.Qs; 78.20.Bh Ž . Photonic band gap PBG crystals, which are periodic arrangements of dielectric or metallic materials, can suppress electromagnetic wave propagation Ž . w x within a frequency band i.e., a stop band 1–4 . If a line defect is introduced in a photonic crystal, the electromagnetic wave whose frequency is in the gap can be guided along the the line defect without appreciable losses. Thus, such a line defect is called a photonic crystal waveguide, or a PBG waveguide. w x Recently, both theoretical simulations 5–7 and exw x perimental studies 8 have shown that PBG waveguides can efficiently transmit electromagnetic waves, even for 908 bends with zero radius of curvature. Therefore, PBG waveguides are of interest as an alternative to conventional waveguides. ) Corresponding author. Fax: q46-8-108327. Ž . E-mail address: [email protected] M. Qiu . Many studies have been carried out concerning Ž . guided modes in two-dimensional 2D dielectric Ž w x. PBG crystals see e.g. 9–11 . However, very few studies can be found in the literature concerning Ž . guided modes in metallic photonic band gap MPBG w x waveguides. Danglot et al. 12 have presented a modal analysis of a T-stub guiding structure patterned on a two-dimensional metallic photonic crystal. Nevertheless, it has been suggested that periodic metallic structures have important applications w x 13,14 . It is thus of scientific and technical interest to study the properties of MPBG waveguides. In the present Letter, we study guided modes in a two-dimensional metallic photonic band gap crystal. The relationship between a MPBG waveguide and a conventional metallic waveguide is also studied. It is shown that the guided modes in a MPBG waveguide correspond to those in a conventional metallic waveguide, and the difference between them is mainly 0375-9601r00r$ see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. Ž . PII: S0375-9601 00 00049-9 ( ) M. Qiu, S. HerPhysics Letters A 266 2000 425–429 426 due to the loss of the translational symmetry in a MPBG waveguide. We also find that there exists a cutoff frequency in a MPBG waveguide, which causes a guided mode gap. Furthermore, the cutoff frequency corresponds to the cutoff frequency in a conventional metallic waveguide. For simplicity, in the present Letter we only consider a square array of metallic rods in air. The radius of the metallic rods is Rs0.2 a, where a is the lattice constant of each unit cell. We use copper as the inclusion material and ss5.80=10Srm is w x used for the copper conductivity 17 . Only the Epolarization is considered here since metallic rods are almost transparent for the H-polarization. It has been shown that there are two band gaps for such a w x metallic photonic crystal 15 . The low-frequency band gap starts from zero frequency to 0.529 Ž . = 2p cra , and the high-frequency band gap is Ž . between the frequencies 0.733 2p cra and 0.862 Ž . = 2p cra . A waveguide is introduced here by removing one or several rows of metallic rods in the photonic crystal. The eigenfrequencies and patterns of the guided modes are calculated with a finite-difference timeŽ w x. domain method details can be found in Ref. 15 , which is applicable for both dielectric and metallic inclusions. We choose a rectangular supercell, which is much larger than a lattice unit cell, as the computational domain. The waveguide is located at the center of the computational domain. The length of the supercell corresponds to the periodicity of the metallic materials in the direction of the wave guide Ž . Ž . x direction , while the width in y direction of the supercell is usually chosen to be more than 10 lattice constant. Due to the periodicity in the x direction of the waveguide, we use the periodic condition for the numerical boundary treatment in this direction. If one uses the periodic condition also in the y direction, pseudo-guided modes, which are eigenmodes for the supercell but not localized modes inside the waveguide, may be introduced into the supercell. To avoid this, we surround the computation domain with Ž . w x perfectly matched layers PML 16 in the y direction. The FDTD algorithm starts with an initial field distribution. As the FDTD time evolution proceeds, only the true guided modes remain in the computation domain, and the pseudo guided modes will w x eventually vanish. We refer to Ref. 15 for details. Fig. 1 shows the eigenfrequencies of the guided modes in the MPBG waveguide obtained by removŽ . ing one row of metallic rods in the 10 direction of the photonic crystal, as shown by the inset at the right-bottom side of the figure. The filled circles are for the even modes and the open circles are for the odd modes. For comparison, the band structure for Ž . the perfect photonic crystal without the waveguide is also shown in the same figure by the gray areas and the thick lines. One can see from this figure that there are two even-guided modes and one odd-guided Ž mode in the high-frequency band gap. Here the parity of the guided modes is distinguished by the symmetry of the electric field with respect to the central plane of the waveguide, i.e., the x axis; see w x . e.g. Ref. 9 . In the low-frequency band gap, there is only one even guided mode. One can notice that there exists a cutoff frequency in the MPBG waveguide, i.e., there exists a mode gap between zero and Fig. 1. Eigen frequencies of the guided modes in a metallic Ž photonic crystal waveguide a square array of copper rods in air; . Ž . the radius of the rods is Rs0.2 a . The waveguide is in the 10 direction of the photonic crystal, and is obtained by removing one row of metallic rods. The filled circles are for the even modes and the open circles are for the odd modes. For comparison, the solid lines and the dot lines show the even and odd modes, respectively, for the guided modes in a conventional metallic waveguide with width ds1.8a. The gray areas and the thick lines show the band Ž . structure for the perfect photonic crystal without the waveguide . ( ) M. Qiu, S. HerPhysics Letters A 266 2000 425–429 427 the cutoff frequency. For E-polarization modes, the electric field E of the guided modes are almost zero z near the boundaries of the MPBG waveguide since the guided modes can not propagate in the otherwise perfect photonic crystal. These properties including the cutoff frequency and zero electric field at the boundaries, etc., are similar to those for a conventional metallic waveguide. Now we consider a 2D metallic waveguide with width b, and assume that the fields have harmonic time dependence e i v . For the E-polarization case, it is easy to obtain the following solution of Maxwell’s equations for the spatial part of the electric field, Ž . Ž . E x, y , with the boundary conditions E x, y s0 z z at ys"br2, i k x x E x , y ssin mp yrbq1r2 e , Ž . Ž . z ms1,2,3, . . . , 1 Ž . where k is the wave vector in the guiding direction. x In order to relate the metallic waveguide to a periodic structure and then compare it with a MPBG Ž waveguide, we impose an artificial periodicity with . period d in the x direction. Then the solution becomes i 2p n dqk x Ž . E x , y ssin mp yrbq1r2 e , Ž . Ž . z ns0,"1,"2, . . . , 2 Ž . where k is the irreducible wave vector in the Brillouin zone. Thus, one obtains the following eigenfrequencies for the guided modes in such a 2D metallic waveguide, 2 2 ( vs 2p crd mdr2b q nqkdr2p . 3 Ž . Ž . Ž . Ž . The cutoff frequency is given by v sp crb correc sponds to the case when ms1, ns0, and ks0. Ž . From Eq. 2 one sees that the even modes correspond to ms"1,"2,"3, . . . and the odd modes correspond to ms"2,"4,"6, . . . . The eigenfrequencies of the guided modes in the metallic waveguide with a width bs1.8a and an artificial period dsa are shown in Fig. 1. The solid lines are for the even modes and the dot lines are for the odd modes. w x These lines are labeled by m,n according to Eq. Ž . 3 . From Fig. 1 one can notice that the guided modes in a MPBG waveguide can be related Ž . Ž roughly to those in a metallic waveguide with an . effective width . For the example of Fig. 1, the effective width is bs1.8a. The cutoff frequency is Ž . 0.278 2p cra for the MPBG waveguide, while it is Ž . 0.273 2p cra for the metallic waveguide. They are also in a good agreement. However, some odd modes, Ž w x . e.g., the 2,0 modes , which exist in the corresponding metallic waveguide, can not be found in the MPBG waveguide. We think the reason is that these Ž modes are too close to the band structure propagat. Ž ing modes of the perfect photonic crystal without . the waveguide . Eigen frequencies of the guided modes in a MPBG Ž . waveguide in the 11 direction of the crystal are shown in Fig. 2. The waveguide is obtained by removing one row of metallic rods. All these modes are similar to those in a metallic waveguide with a ' waveguide width bs 2 a and an artificial period ' ds 2 a. However, no odd mode can exist in this MPBG waveguide. From the inset of Fig. 2, one sees that the boundary of the MPBG waveguide has a ' zig-zag shape, while the effective width bs 2 a is very small. Therefore, there are few guided modes Ž . with one or more nodes i.e., m)1 inside the Ž waveguide note that ms2 for the two odd modes . in the metallic waveguide . Ž . Ž . Fig. 3 a and b show the eigenfrequencies of guided modes for a wider MPBG waveguide in the Fig. 2. Eigen frequencies of guided modes for the MPBG waveŽ . guide in the 11 direction of the photonic crystal. The waveguide is obtained by removing one row of metallic rods. All the notations and labels are the same as in Fig. 1. ( ) M. Qiu, S. HerPhysics Letters A 266 2000 425–429 428 Fig. 3. Eigen frequencies of guided modes for the MPBG waveŽ . Ž . Ž . Ž . guide in a the 10 direction and b the 11 direction of the photonic crystal. The waveguide is obtained by removing two rows of metallic rods. All the notations and labels are the same as