Design and Modeling of WDM Integrated Devices Based on Photonic Crystals

presented and analyzed using CMT and FDTD methods. It will be shown that when the phase-shift of the electromagnetic waves traveling between two adjacent PC coupled cavities is approximately equal to ( 1/2) , k   the best performance for the intersection can be achieved. In addition it will be shown that simultaneous crossing of ultra-short pulses is Optical communications systems are very important for all types of telecommunications and networks. They consists of a transmitter that encodes a message into an optical signal, a channel that carries the signal to its destination, and a receiver that reproduces the message from the received optical signal.This book presents up to date results on communication systems, along with the explanations of their relevance, from leading researchers in this field. Its chapters cover general concepts of optical and wireless optical communication systems, optical amplifiers and networks, optical multiplexing and demultiplexing for optical communication systems, and network traffic engineering. Recently, wavelength conversion and other enhanced signal processing functions are also considered in depth for optical communications systems. The researcher has also concentrated on wavelength conversion, switching, demultiplexing in the time domain and other enhanced functions for optical communications systems. This book is targeted at research, development and design engineers from the teams in manufacturing industry; academia and telecommunications service operators/ providers.

possible. In Section 3, a three-port high efficient CDF with a coupled cavity-based wavelength-selective reflector is introduced and analyzed. According to the theoretical theory using CMT in time, the performance of the proposed CDF will be investigated and the conditions which lead to 100% drop efficiency will be extracted. The performance of the designed filter will also be calculated using the 2D-FDTD method. The simulation results show that the designed CDF has a line-width of 0.78nm at the center wavelength 1550 , nm and also a multi-channel CDF with channel spacing around 10nm (1 ) nm with inter-channel crosstalk below 30 dB  (1 5 ) dB  is possible. These characteristics make the proposed CDF suitable for use in WDM optical communication systems.

Photonic crystal hybrid waveguides: Design and modeling
As mentioned before, PCs have gained great interest due to the availability of high density integrated optical circuitry (Joannopoulos et al., 2008;Yanik et al., 2004;Niemi et al., 2006;Koshiba, 2001;Fasihi & Mohammadnejad, 2009a, 2009bMekis et al., 1999;Loncar et al., 2000;Martinez et al., 2003;Shin et al., 2004;Yanik et al., 2003). PC waveguides provide an efficient means of guiding light by allowing the realization of sharp optical bends (Mekis et al., 1996). By combining the PC waveguides with PC cavities, realization of compact optical filter designs is possible . If the cavities are brought close together, they will couple and light will propagate through evanescent wave coupling from one cavity to its neighbors. This new type of waveguide is called the coupledcavity waveguide (CCW) or alternatively, a coupled-resonator optical waveguide (CROW) (Yariv et al., 1999) and has many interesting properties. With such waveguides, by appropriately positioning the coupled optical cavities, sharp bends are also possible. In this section, the hybrid waveguides based on combining of the CCWs and the conventional line defect waveguides proposed and modeled by using CMT and FDTD methods. PC hybrid waveguides are a key element in the construction of future integrated optical circuits. Orthogonal hybrid waveguide intersections are very good candidate for wideband and low cross-talk intersections, which are a crucial element in PC-based integrated circuits (Fasihi & Mohammadnejad, 2009a). In some applications, such as ultra-short pulse transmission and time delay lines, the transmission spectrum is designed to be quasi-flat. By increasing the confinement of the coupled cavities in hybrid waveguides, the continuous transmission band will be converted to a series of discrete bands with Lorentzian spectrum, which are useful for implementation of some optical devices, such as filters (Ding et al., 2009). In this section, both analytical and numerical approaches are used to design and modeling of the hybrid waveguides with quasi-flat and Lorentzian transmission spectrum.

Analysis of the PC cavity modes
When a local defect is created in a PC, e.g. by removing a single rod, a cavity is formed where light is confined in one or more bound states. Depending on the quality of the confinement, these states, or modes, exist only in a narrow frequency range. In general, a defect can have any shape or size; it can be made by changing the refractive index of a rod, modifying its radius, or removing a rod altogether . The defect could also be made by changing the index or the radius of several rods. Here, we choose to modify the radius of a single rod. For ease of computation, we use a 2D-PC consisting of a hexagonal lattice of dielectric rods in air. The rods have lattice constant , a radius 0.20 , ra  and refractive index 3.4. rod n  This structure prohibits propagation of TM light (in-plane magnetic field) in the frequency range 0.280 to 0.452 (2 / ). ca  To determine the PBG regions of the PC structure, the MIT Photonic-Bands package (http://abinitio.mit.edu/mpb) is used. In order to couple energy into the cavity, it is necessary to transfer energy through the walls of the PC. Incident light can transfer energy to the resonant mode by the evanescent field across the array of rods. The setup is shown in Fig. 1. To compute the resonant frequencies, we consider a finite-sized 13 21  PC in which a single rod has been removed. We send a wide-spectrum plane wave pulse with TM polarization at the incident angle of around 15  respect to the z-axis. On the other side of the PC, the field amplitude is monitored at a short length, marked as ''Monitor''. This configuration facilitates identifying of the resonance peaks of transmission spectrum, especially in some degenerate states. In this configuration the excitation has a Gaussian profile centered at 0.37 (2 / ) ca    and a width of 0.6 (2 / ) ca     which extends beyond the edges of the PBG. The resonant frequencies of the cavities are plotted as a function of the cavity radius in Fig. 2. This figure shows that as the radius of the cavity is reduced to 0.15 , a due to the increasing perturbation, a resonant cavity mode appears at the bottom of the PBG . As the cavity radius is further reduced, since the cavity involves removing dielectric material in the PC, based on perturbation theorem, a higher frequency resonant mode is obtained (Joannopoulos et al., 2008), and eventually reaches 0.3953 (2 / ) ca    when the rod is completely removed. The corresponding electric-field ( y E ) distributions of the resonant mode is shown in Fig. 3 These states named as a dipole to reflect the two lobes in their field distributions. If the radius is further increased then a sequence of higher-order modes (with more nodal planes) are pulled down into the PBG: six dual-triply-degenerate states ( Fig. 3-(b), (c)), a higherorder monopole with an extra node in the radial direction ( Fig. 3-(e)), a hexapole state and a second-order hexapole state ( Fig. 3-(f), (g)), three triply-degenerate second order dipole states ( Fig. 3-(h)), and three triply-degenerate octapole states ( Fig. 3-(i)).
(a-1) (a-2) (a-3) The electric-field ( y E ) distributions of the resonant modes for the labled points of.

Hybrid waveguides
The PC based CCWs are formed by placing a series of high-Q optical cavities close together. In this case due to weak coupling of the cavities, light will be transferred from one cavity to its neighbors and a waveguide can be created (Yariv et al., 1999). By combining of the CCWs and the conventional line defect waveguides a new waveguide can be created, which is referred to as hybrid waveguide. Fig. 6 shows the structure of a hybrid waveguide which is implemented in a PC of square lattice. Generally, there are two types of PC lattice structures, air-hole-type and rod-type. Most theoretical studies conducted so far have investigated arrays of dielectric rods in air. The advantage of this model system is that waveguides created by removing a single line of rods are single moded. Getting light to travel around sharp bends with high transmission is then relatively straightforward, and many rod-type PC structures have been proposed. Such waveguides have been fabricated and photonic band gap guidance has been confirmed. Unfortunately, the rod in air approach does not provide sufficient vertical confinement and is difficult to implement in the optical regime. So, in practice, the PC slabs of dielectric rods in which the refractive index of the background material is higher than air, are used. Despite easier fabrication of PC waveguide based on air-hole-type structures than rod-type, there are limitations on frequency bandwidth of the single mode region and the group velocity (Fujisawa & Koshiba, 2006). Moreover in PC waveguides based on rod-type structure the large bandwidth and the large group velocity can be achieved, and recently such waveguides have been used for fabrication of photonic devices (Chen et al. 2005).

Modeling of hybrid waveguides by CMT method
Here, we consider the CCWs that are formed by periodically introducing defects along one direction in 2D-PCs. Generally, the coupling between two PC cavities depends on the leakage rate of energy amplitude into the adjacent cavity (1 / ),  which defines the quality factor of the cavity, and the phase-shift between two adjacent defects () .  In a straight CCW which contains N PC cavities (3 ) , N  the transmission spectrum is given as (Sheng et al., 2005) The parameter min T is the minimum in transmission band of a PC molecule. The peaks of the equation (3), which are equal to unity, appear at 0    and 1 0 1( 2t a n) Q         . Hence, using (4) and the simulated transmission spectrum of one PC molecule, 0  and Q can be extracted. It must be noted that the analytical results of equation (1) can be extended to CCWs of any dimensions (Sheng et al., 2005). Now, we consider the hybrid waveguides that contain N identical cavities in 2D-PCs and generalize CMT analytical method to obtain the transmission spectrum. According to (1), it can be seen that for a given N, the transmission spectrum curve has 2N -1 number of extremums and the minimum in transmission band ( min T ), is independent of 0  and , Q and we can obtain  as a function of the radius of the coupled cavities ( d r ), as follows (Fasihi & Mohammadnejad, 2009a):


The relationship between min T and d r can be calculated by repeating a numerical simulation, such as FDTD method, for different values of d r .
Here we consider a hybrid waveguide which contains three coupled cavities, 3 N  (see Fig.  6), in the 2D-PC of square lattice composed of dielectric rods in air. Now, we have chosen to name this hybrid waveguide HW3 and extend this naming to other hybrid waveguides. The rods have refractive index 3.4 rod n  and radius 0.20 , ra  where a is the lattice constant. By normalizing every parameter with respect to the lattice constant a, we can scale the waveguide structure to any length scale simply by scaling a. The radius of the coupled cavities are varied from 0.27a to 0.345 .
a The grid size parameter in the FDTD simulation is set to 0.046a and the excitations are electromagnetic pulses with Gaussian envelope, which are applied to the input port from the left side. All the FDTD simulations below are for TM polarization. The field amplitude is monitored at suitable location at the right side of the HW3. Table 1 shows the relationship between min T and d r for the HW3 which are obtained from the FDTD simulations.


The relationship between min T and  for the HW3 can be calculated from (1).  Fig. 7 shows this relationship over one-half period of (1). Therefore, the relationship between  and d r of the HW3 can be demonstrated in Fig. 8.  In order to compare the results of CMT and FDTD methods, we consider a HW2 under the same conditions as mentioned previously and utilize the FDTD simulation results to compute 0  and .
Q The radius of the coupled cavities are set to 0.32 . d ra  The transmission spectrum of HW2 computed by the FDTD is shown in Fig. 9. According to this figure, the parameters 0 ,  Q and  are equal to 0.3428 (2 / ), ca  130.3 and 0.4066 ,  respectively. Hence, the CMT transmission spectrum can be calculated from (3) (see Fig. 9). It is observed that the transmission spectrum calculated by CMT is in good agreement with that simulated by FDTD. As another example, we take a HW3 under the same condition as mentioned previously, with 0.32 d ra  which corresponds to 0.1509 .

  
The transmission spectra of the above HW3 simulated by FDTD and CMT are shown in Fig. 10 for comparison. Although there is a difference in the minimum transmission spectrum between www.intechopen.com the first and second peaks, it is observed that the spectrum calculated by analytical method is nearly in good agreement with that simulated by the numerical simulation. Now, we consider a hybrid waveguide which contains three coupled cavities, in the 2D-PC of hexagonal lattice composed of dielectric rods in air. All conditions are the same as the previous structure and the radius of the coupled cavities is varied from 0 to 0.08 .
a The relationship between  and d r of the HW3 in PC of hexagonal lattice is shown in Fig. 11. Tables 2 and 3 show the transmission regions and 3dB  bandwidths (BW) of the proposed hybrid waveguide for different values of the coupled cavities radii in PC of square and hexagonal lattices, respectively.

Design of hybrid waveguides with quasi-flat transmission spectrum
As discussed earlier, usually, the transmission spectrum is designed to be quasi-flat within the transmission region for various applications, such as ultra-short pulse transmission and also time delay lines. By comparing the results of Fig. 8 and Table 2, and also the results of Fig. 11 and Table 3, it can be seen that the optimum values of bandwidth is obtained when the phase-shift between two adjacent cavities are close to /2( ). rad  In this case, the transmission spectrum of the HW3 is quasi-flat. Fig. 12 shows the FDTD simulation results of the HW3 transmission spectrum (in PC of square lattice) for different radii, in which the phase-shift between the adjacent cavities is nearly /2( ). rad  Fig. 12. The transmission behavior of the HW3 in a PC of square lattice when

Design of hybrid waveguides with Lorentzian transmission spectrum
Using (1), it can be shown that the transmission spectrum is Lorentzian if the phase-shift between the adjacent cavities is close to 0 or .  Fig. 8 shows that for different radii of the coupled cavities, the phase-shift between the adjacent cavities isn't close to zero, hence we have a continuous transmission spectrum. By increasing the confinement of the coupled cavities, the continuous transmission spectrum tends to reduce into a series of discrete modes, which are useful for some optical devices, such as the WDM filters. To do this, we place two extra rods in both ends of the CCW. We consider the structure shown in Fig. 13, and investigate the effect of increasing confinement on transmission property with locating extra rods with radius of 0.2 .
a All conditions are the same as the previous structure studied at section 3.2 and the radius of the coupled cavities is varied from 0 to 0.08 .
a Fig. 14 shows the relationship between  and d r of the modified HW3 with Lorentzian spectrum. The www.intechopen.com FDTD simulation results of the modified HW3 transmission spectrum for different radii, in which the phase-shift between the adjacent cavities is nearly zero, is shown in Fig. 15. Fig. 13. The modified HW3 in a PC of square lattice with Lorentzian transmission spectrum.

Orthogonal hybrid waveguides: An approach to low cross-talk and wideband intersection design
In the implementation of PC-based integrated circuits, such as those used in WDM systems, it is necessary to have intersections in which crossing of ultra-short lightwave signals are possible. In another study, we show that the orthogonal hybrid waveguide intersections are very good candidate for wideband and low cross-talk intersections, which are a key element in PC-based integrated circuits (Fasihi & Mohammadnejad, 2009a). In 1998 Johnson et al. proposed a scheme to eliminate cross-talk for a waveguide intersection based on a 2D-PC of square lattice by using a single defect with doubly degenerate modes (Johnson et al., 1998). They also presented general criteria for designing such waveguide intersections based on symmetry consideration. Lan and Ishikawa presented another mechanism where the defect coupling is highly dependent on the field patterns in the defects and the alignment of the defects (i.e., the coupling angle) (Lan & Ishikawa, 2002). They asserted that their design leads to a 10 nm wide region at the central wavelength of 1310 nm with cross-talk as low as -10 to -45 dB, while in Ref. (Johnson et al., 1998) the width of the transmission band with comparable cross-talk is only 7.8 nm. In the above mentioned design, the central wavelength value of the low cross-talk transmission band is related to the air-holes radii of PC structure and therefore, adjusting the wavelength domain of transmission band is a challenge. Furthermore, Liu et al. proposed another waveguide intersection for lightwaves with no cross-talk and excellent transmission which was based on non-identical PC coupled resonator optical waveguide (CROW), without transmission band overlap (Liu et al., 2005). Li et al. proposed a different approach that utilizes a vanishing overlap of the propagation modes in the waveguides created by line defects which support dipole-like defect modes (Li et al., 2007). They claimed that in their design, over a BW of 30 nm with the central wavelength at 1300 nm, transmission efficiency above 90% with cross-talk below -30 dB can be obtained. It is obvious that in that proposal -and also in , simultaneous propagation of lightwaves with equal frequencies through the intersection is impossible and due to using of taper structure to solve the mode mismatch problem, total length of the intersection is increased. In our solution an approach to design of low cross-talk and wideband PC waveguide intersections based on two orthogonal hybrid waveguides in a crossbar configuration, is proposed. Without losing generality, once again we consider a 2D square lattice of infinitely long dielectric rods in the air. Fig. 16 shows the structures of an orthogonal hybrid waveguide intersection in which the rods have refractive index n rod = 3.4 and radius r = 0.20a. www.intechopen.com   Table 4. Values of -3dB BW and cross-talk in orthogonal HW3 intersection for various radii of the coupled cavities To evaluate the performance of the proposed device, the FDTD method is used for simulation, under the same conditions as mentioned previously. The excitations are electromagnetic pulses with Gaussian envelope, which are launched to the input port from the left side. The field amplitudes are monitored at suitable locations around the intersection in horizontal and perpendicular waveguides. exists around 0.0415a and 0.0228a regions in which the transmission is over 50%. Also, it must be noted that that the transmission properties of the proposed intersection are the same as transmission properties of the corresponding hybrid waveguide. Furthermore, by varying the radius of the coupled cavities of the hybrid waveguides, a wide frequency domain of transmission band will be obtained which proves the flexibility of the proposed design. Table  II shows 3  dB BW and the cross-talk of the proposed intersection for different values of the coupled cavities radii. By comparing the results of Fig. 8 and Table 4, it can be seen that the optimum values of BW and cross-talk are obtained when (1 / 2 ) . k    In this case, the transmission spectra of the intersection is quasi-flat (see Fig. 12).

Simultaneous crossing of lightwave signals and transmission of ultra-short pulses through the orthogonal hybrid waveguide intersections
In the implementation of PC-based integrated circuits, it is necessary to have intersections in which simultaneous crossing of lightwaves is possible. In the orthogonal hybrid waveguide intersections, lightwave signals can cross through the intersection simultaneously because each resonant state of the intersection will couple to modes in just one waveguide and be orthogonal to modes in the other waveguide. We consider the structure shown in Fig. 16  am   During simulation, two input pulses with Gaussian envelope are applied to input ports from the top and the left sides. The monitors are placed at right and bottom output ports at suitable locations. The intensities of 500-fs pulses are adjusted to unity and 0.5, while their central wavelengths are set at 1550nm and the phase difference between them is 180 .  Fig. 18 shows the transmission www.intechopen.com behavior of simultaneous crossing of lightwave signals through the orthogonal HW3 intersection. It can be seen that the input pulses are transmitted through the intersection with negligible interference effect. In a separate assessment, we again consider the structure shown in Fig. 16  and investigate the transmission property of the intersection for ultra-short pulses by using the FDTD method. Fig. 19 shows the transmission behavior of a 200-fs pulse whose central wavelength is 1550 .
nm We can see that not only the cross-talk is negligible, but also the distortion of the pulse shape is very small.

Highly efficient channel-drop filter with a coupled cavity-based wavelength-selective reflection feedback
The rapidly growing use of optical WDM systems, calls for ultra-compact and narrowband channel-drop filters (CDF). In a CDF, a single channel with a narrow linewidth can be selected, while other passing channels remain undisturbed. The means to control the propagation of light is mainly obtained by introducing defects in PCs. Microcavities formed by point defects and waveguides formed by line defects in PCs. In particular, the resonant CDFs implemented in PC, which are based on the interaction of waveguides with micro-cavities, can be made ultra-compact and highly wavelengthselective (Zhang & Qiu, 2006). These devices attract strong interest due to their substantial demand in WDM optical communication systems. So far, different designs of CDFs in 2D-PCs have been proposed (Kim et al., 2007). These designs can be basically classified into two categories: surface emitting designs and in-plane designs. The surface emitting designs make use of side-coupling of a cavity to a waveguide. The input signal at resonant frequency tunnels from the waveguide into the cavity and is emitted vertically into the air (Noda et al., 2000;Song et al., 2005). The in-plane designs usually may be classified into two categories: four-port CDF designs and three-port CDF designs. The four-port CDF designs usually involve the resonant tunneling through a cavity with two degenerate modes of different symmetry, which is located between the two parallel waveguides (bus and drop). Although in this design a complete channeldrop transfer at resonant frequency is possible (i.e., 100% channel-drop efficiency), but enforcing degeneracy between the two resonant modes of different symmetry requires a complicated resonator design Min et al., 2004). The operation principle of four-port CDF designs with and without mirror-terminated waveguides, have matured over the years (Zhang & Qiu, 2004). The basic concept of three-port CDF designs is based on direct resonant tunneling of input signal from bus waveguide to drop waveguide. This kind of CDF designs have simple structures and can be easily extended to design multi-channel drop filters , (Tekeste & Yarrison-Rice, 2006;Notomi et al., 2004). In a typical three-port CDF, the power transmission efficiency i s i n h e r e n t l y l e s s t h a n 5 0 % ( w h i c h c o r r e s p o n d s t h e t r a n s m i s s i o n i n t h e r e s o n a n t frequency), because a part of trapped signal in the cavity is reflected back to the bus waveguide when channel-drop tunneling process occurs. So far, different approaches have been proposed to solve this problem. Fan et al. proposed an approach to enhance the drop efficiency using controlled reflection to cancel the overall reflection in a full demultiplexer system. This structure is realized by coupling among an ultra low-quality factor cavity and micro-cavities with high-quality factor (Jin, 2003). Kim et al. proposed a three-port CDF with reflection feedback, in which nearly 100% drop efficiency can be theoretically achieved. In this design, the reflected back power to input port, except at the resonant frequencies, is close to 100% which leads to noise if the designed structure is incorporated in photonic integrated circuits . A similar design has also been proposed by Kuo et al. based on using high Q-value micro-cavities with asymmetric super-cell design (Kuo et al., 2006). This design leads to an improvement in the drop efficiency and the full-width at half-maximum (FWHM), respect to the corresponding symmetric super-cell. Another three-port CDF with a wavelengthselective reflection micro-cavity has been proposed by Ren et al. (Ren et al., 2006). In the proposed design two micro-cavities are used. One is used for a resonant tunneling-based CDF, and another is used to realize wavelength-selective reflection feedback. In this section we study a three-port system which is based on two coupled cavities in both drop and reflector sections. We show that the proposed structure can provide a practical approach to attain a high efficient CDF with narrow FWHM, with no reduction in transmission efficiency parameter. Here, we consider the structure shown in Fig. 20, where the coupled cavities of the drop and the reflector sections are located at opposite sides of the bus waveguide to prevent the direct coupling between them. Fig. 20. The basic structure of the proposed three-port CDF with coupled cavity based wavelength selective-reflector (Fasihi & Mohammadnejad, 2009b).
The time evolution of the cavities modes, given that all of the cavities decay rates which are due to internal loss of energy be equal to 0 ,  are expressed by (Fasihi & Mohammadnejad, 2009b;Haus, 1984;Manolatou et al., 1999) Here, Re sa   and Re sb   are the resonant frequencies of the coupled cavities in the drop and the reflector sections, respectively, 1 1/ and 3 1/ denote the decay rates of cavities a 1 and b 1 into the bus waveguide, respectively, 2 1/ is the decay rate of cavities a 2 into the drop waveguide and also is the decay rates of the cavity a 2 into the cavity a 1 and vice versa, and 21, the amplitudes of the electromagnetic waves (EM) incoming the drop (reflector) section from the bus waveguide, are denoted by 1 S  3 () S  and 1 ' S  3 (' ) . S  Also, the amplitudes of the EM waves outgoing the drop (reflector) section to the bus waveguide, are denoted by In the case of EM waves traveling between the coupled cavities, in the drop section, the EM wave incoming the cavity a 1 (a 2 ) is denoted by 1 " S  2 (' ) , S  and the EM waves outgoing the cavity a 1 (a 2 ) is denoted by 1 " S  2 (' ) , S  respectively, and in the reflector section, the EM wave incoming the cavity b 1 (b 2 ) is denoted by 1 " R  2 (') , R  and the EM wave outgoing the cavity b 1 (b 2 ) is denoted by In the above equations, 1  and 2  are the phases of the coupling coefficients between the bus waveguide and the cavities a 1 and b 1 , respectively, 3  is the phase of the coupling coefficient between the drop waveguide and cavity a 2 ,  is the propagation constant of the bus waveguide, and d is the distance between two reference planes. The EM waves traveling between the two coupled cavities in drop and reflector sections, satisfy Based on Eqs. (16)- (17) and (22) The dependence of the maximum of drop efficiency on k parameter is shown in Fig. 21-(a). Fig. 21- (b) shows the dependence of the maximum of drop efficiency on  parameter. The value of the cavities quality factor has an important role in the CDF performance.
On one hand, the cavities with high quality factor are necessary for implementation of threeport CDFs with narrow FWHM, which are the key element in WDM systems. On the other hand, at resonant frequencies and given that 0,    Fig. 23 shows the structure of the three-port CDF with wavelength-selective reflection feedback, in 2D-PC of square lattice composed of dielectric rods in air. All conditions are the same as the previous structures studied at section 2. The excitations are electromagnetic pulses with Gaussian envelope, which are applied to the bus waveguide from the top side. The field amplitudes are monitored at suitable locations at the bus and the drop waveguides. Fig. 24 shows the dispersion curve of the bus and the drop line-defect waveguides versus the wave vector component k along the defect. The resonant frequencies of the coupled cavities in the modified HW2 structure as a function of the coupled cavities radii are shown in Fig. 25. Given that the radii of the coupled cavities in the drop and reflector sections are set to 0.055 , a from Fig. 25  parameter, which is due to the internal loss of energy, is infinite in the desired 2D-PCs (Ren et al., 2006) and the total quality factors of the cavities are 1925. So, the condition 03    is satisfied and the perfect reflection can be realized. The condition 21 /1 / 4    can be easily satisfied using the coupled mode theory . From Fig. 25    The channel spacing can be reduced to 1nm for the 15dB  inter channel crosstalk. In a single cavity based CDF with reflector, the crosstalk with channel spacing of 20nm is between 18  to 23 dB  (Kuo et al., 2006). waveguides. For more optimal CDF design, the sizes of the rods between the cavities and the bus and drop waveguides, in both drop and reflector sections can be trimmed. In fact, by adjust tuning the resonant frequencies of the drop and reflector sections, further improve in CDF performance can be achieved and also the back reflection power into the input port, around the resonant frequencies, can be reduced. Even though, we don't use the additional trimming in the design. Because the add operation is the "time-reversed" process of the channel drop operation, the tunneling-based channels add and drop operation can be combined into a compact form as shown in Fig. 28. The wavelength-selective reflection section locates in the central of the structure, and it ensures full power transfer between the bus waveguide and channel-add/drop sections. The top takes the narrow-band signal out of the bus waveguide while the bottom one couples signal from the transmitters into the buswaveguide.

Conclusions
In summary, the resonance frequencies and the field distributions of 2D-PC have been investigated. PC hybrid waveguides with quasi-flat and Lorentzian transmission spectrum were analyzed and modeled by using FDTD and CMT methods. The theoretical results derived by CMT were in good agreement with FDTD simulation results. It was shown that when the phase-shift of the electromagnetic waves traveling between two adjacent PC coupled cavities ()  , is close to (1 / 2 ) , k   the transmission spectrum of the hybrid waveguide is quasi-flat. A modified HW3 with extra rods in both ends of the CCW and Lorentzian transmission spectrum was proposed, which can be used in implementation of WDM filters. It was shown that in this case  is close to zero. Transmission of ultra-short pulses through the hybrid waveguide was also investigated. A low cross-talk and wideband PC waveguide intersection design based on two orthogonal hybrid waveguides in a crossbar configuration was proposed. Also, it has been shown that when (1 / 2 ) , k     optimum performance results for the intersection can be achieved. In addition, it has been clearly proved that simultaneous crossing of ultra-short pulses through the intersection is possible with negligible interference. The transmission of a 200-fs pulse at 1550 nm was simulated by using the FDTD method, and the transmitted pulse showed negligible cross-talk and very little distortion. A three-port high efficient CDF with a coupled cavity-based wavelength-selective reflector, which can be used in WDM optical communication systems, was proposed. The CMT was employed to drive the necessary conditions for achieving 100% drop efficiency. The FDTD simulation results of proposed CDF which was implemented in 2D-PC, showed that the analysis was valid. The simulation results show that the designed CDF has a line-width of 0.78nm at the center wavelength 1550 , nm and also a multi-channel CDF with channel spacing around 10nm (1 ) nm with inter-channel crosstalk below 30 dB  (1 5 ) dB  is possible. These characteristics make the proposed CDF suitable for use in WDM optical communication systems.