1. Introduction

Photonic crystals (PCs) have gained great interest due to the availability of high density integrated optical circuitry [1–5]. This is mainly due to their unique ability to control the propagation of light. The rapidly growing use of optical wavelength division multiplexing (WDM) systems, calls for ultra-compact and narrowband channel-drop filters (CDF). In a CDF, a single channel with a narrow line-width 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. Micro-cavities 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 wavelength-selective [6]. These devices attract strong interest due to their substantial demand in WDM optical communication systems. So far, different designs of CDFs in two-dimensional (2D) PCs have been proposed [7]. 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 [8,9]. 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 channel-drop 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 [10,11]. The operation principle of four-port CDF designs with and without mirror-terminated waveguides, have matured over the years [12]. 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 [7], [13,14]. In a typical three-port CDF, the power transmission efficiency is inherently less than 50% (which corresponds the transmission in the resonant 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 [15]. Lim 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 [7]. A similar design has also been proposed by Kuo et al. based on using high Q-value micro-cavities with asymmetric super-cell design [16]. 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 wavelength-selective reflection micro-cavity has been proposed by Ren et al. [17]. 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 our work a highly efficient CDF with a coupled cavity-based wavelength-selective reflection feedback has been suggested. The coupled mode theory (CMT) is employed to analyze the behavior of this system. It is shown that the suggested design leads to a CDF with an improved FWHM, and a lower inter-channel crosstalk. This design is also simulated by finite-difference time-domain (FDTD) method and the simulation results show the validity of the proposed design.

2. Theoretical model for two-PC coupled cavities system: CMT method

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 [18]. The defect could also be made by changing the index or the radius of several rods. The PC based coupled cavity waveguide can be formed by placing a series of high-Q optical PC 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 [19]. By combining the CCWs and the conventional line defect waveguides a new waveguide can be created, which is referred to as hybrid waveguide [6]. Usually, for various applications such as ultra-short pulse transmission, there is a need to have a large bandwidth (BW) and a quasi-flat transmission spectrum within the transmission band. If the confinement of the coupled cavities is increased, the continuous transmission band will be converted to a series of discrete bands, which are useful for implementation of some optical devices, such as filters [20]. Here, we consider a hybrid waveguide which contains two coupled cavities, (see Fig. 1(a)), in a 2D-PC of square lattice composed of dielectric rods in air. Now, we have chosen to name this hybrid waveguide HW2 and extend this naming to other hybrid waveguides [6]. The CMT equations that describe the temporal change of the normalized mode amplitudes of the coupled cavities, a_i, i=1, 2 , are described by [21,22]

 

Fig. 1. (a) Structure of a HW2 in a PC of square lattice. (b) Schematic diagram for HW2 based on CMT.

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where ω _Res is the cavities resonant frequency (it is assumed that the resonant frequencies of the cavities are equal), and 1/τ _i and 1/τ′_i i= 1,2 denote the decay rates of the cavity a_i into its left and right waveguides, respectively and θ ₁ (θ ₂) is the phase of the coupling coefficient between the left (right) waveguide and the cavity a₁ (a₂). Here, we ignore the internal loss of energy in PC cavities. As shown in Fig. 1(b), S _+i and S′ _+i represent the incoming electromagnetic (EM) waves into the ith PC cavity from its left and right sides, respectively, while S _−i and S′ _−i represent the corresponding outgoing EM waves. Obviously, the EM waves traveling between the two-PC coupled cavities satisfy

where φ represents the phase-shift of the EM waves traveling between the cavities. Based on Eqs. (3)–(8), when τ′ ₁ = τ ₂ we have

By substituting the two above equations in Eqs. (1) and (2), assuming that all cavities decay rates are equal to τ, we find

where γ= [(ω−ω _{Re s})τ sin φ-cos φ]. When the EM wave is launched only from the left side into the bus waveguide (S′ ₊₂ = 0), the transmission spectrum can be determined as

Using Eq. (12), it can be shown that the resonant frequencies are ω _Res1,2 =ω_Res, ω _Res +2/(τ ₂ tanφ) and the transmission spectrum is Lorentzian if the phase-shift between the adjacent cavities is close to 0 or π. However, for the structure shown in Fig. 1(a), one can see that, using the FDTD simulation, for different radii of the coupled cavities, the phase-shift between the adjacent cavities isn’t close to zero and 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 WDM filters. To do this, we place two extra rods in both ends of the CCW. Generally, there are two types of PC lattice structures, air-hole-type and rod-type. 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 [23]. 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 [24]. We consider the structure of modified HW2 shown in Fig. 2, and investigate the effect of increasing confinement on transmission property with locating extra rods. In this structure the rods have refractive index n_rod = 3.4 and radius r = 0.20a, 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 to 0.08a. 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 are for TM polarization. The field amplitude is monitored at suitable location at the right side of the HW2. Figure 3 shows the relationship between φ and r_d of the modified HW2 with Lorentzian spectrum.

 

Fig. 2. The modified HW2 with Lorentzian transmission spectrum in a PC of square lattice.

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Fig. 3. The phase-shift between two coupled cavities versus the cavities radius of the modified HW2 in a PC of square lattice.

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To comparing the results of CMT and FDTD methods, we consider a HW2 in the same condition as mentioned previous. The radius of the coupled cavities is set to r_d = 0.04a. We utilize the FDTD simulation result to compute ω _Res and Q. The solid curve in Fig .4 shows the FDTD simulation result of transmission spectrum of the above mentioned HW2. According to this figure, the parameters ω _Res, Q and φ are equal to 0.36765 (a/λ), 472.4 and 0.07198 Rad, respectively. The dashed curve in Fig. 4 shows the calculated transmission spectrum of the HW2, based on CMT method. It is observed that the transmission spectrum calculated by CMT is in good agreement with that simulated by FDTD.

 

Fig. 4. The simulation results of transmission spectrum of the modified HW2 obtained by the FDTD and CMT method.

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From Eq. (12), the frequencies which are required to calculate the FWHM of the transmission spectrum, can be obtained as follow:

Figure 3 shows that in the modified HW2, φ is close to zero for different radii, hence the transmission spectrum will be Lorentzian. Furthermore, in this case, one can obtain

Hence, it can be seen that the FWHM in a system composed of two identical coupled cavities, when φ is close to zero, is one half its value in a similar single cavity system.

3. Theoretical model for three-port CDF based on two coupled cavities without reflector

The basic implementation of the three-port CDF, which is based on two coupled cavities, is shown in Fig. 5. In this structure, the cavities and the waveguides support only one mode in the frequency range of interest, furthermore the reference plane for the EM waves in the bus waveguide is picked at the center plane of the cavities. In this case, assuming that τ′₁ = τ′ ₂ = τ ₂, the CMT equations that describe the temporal change of the normalized mode amplitudes of the cavities, a_i, i = 1, 2 , are described by

where, ω _{Re s} is the resonant frequency (it is assumed that the resonant frequencies of the cavities are equal), 1/τ ₁ and1/τ ₂ denote the decay rates into the bus and the drop waveguides, and θ ₁ (θ ₂) is the phase of the coupling coefficient between the bus waveguide (drop waveguide) and the cavity a₁ (a₂). Here, we ignore the internal loss of energy in PC cavities. As shown in Fig. 5, the amplitudes of the incoming (outgoing) EM waves in the bus waveguide are denoted by S ₊₁ (S ₋₁) and S′ ₊₁ (S′ ₋₁), and in the drop waveguide by S ₊₂ (S ₋₂), respectively. In the case of EM waves traveling between the coupled cavities, the incoming EM wave into the cavity a₁ (a₂) is denoted by S′₊₁ (S′ ₊₂), and the outgoing EM wave by S″₋₁ (S′ ₋₂), respectively. From the power conservation and the time-reversal symmetry, the relationships among the incoming/outgoing EM wave amplitudes in the waveguides and the cavities mode amplitudes can be written as

The EM waves traveling between the two-PC coupled cavities satisfy

Based on Eqs. (20)-(23), we have

By substituting the two above equations in Eqs. (15) and (16), we find

where γ= [(ω−ω _Res)τ ₂sinφ-cosφ]. When EM wave is launched only from the left side into the bus waveguide(S ₊₂, S′₊₁ = 0), and has a e^jωt time dependency, then at steady state the transmission spectrum can be determined as

From Eq. (27), the peaks of CDF transmission spectrum, given that φ ≈ 0 , is obtained by

and also the corresponding frequencies are

Under this phase condition, it can be shown that the maximum transmission of 50% is obtained when τ ₂/τ ₁ =1/ 2 . Furthermore, in this case one can see that

where T and R are the transmission through the bus waveguide and the back-reflection at the input port, respectively. Figure 6 shows the curve of three-port CDF transmission peaks as a function of τ ₂ /τ ₁.

 

Fig. 5. The basic structure of a three-port CDF which is based on two coupled cavities.

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Fig. 6. The curve of the three-port CDF transmission peaks versus τ ₂ /τ ₁.

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4. Theoretical model for a coupled cavity-based wavelength-selective reflector

As discussed previously, by using the different reflection feedback designs, the 100% drop efficiency can be theoretically achieved; but in the CDFs with unselective reflectors, the input wave over the entire wavelength range, except at the resonant wavelength, will be reflected back to the input port that may cause serious problems, especially in the integrated structures. The wavelength-selective reflector design is a way to prevent the back-reflection power to the input port, in unwanted frequencies. Here, we consider a two coupled cavity-based design of wavelength-selective reflector which is shown in Fig. 7. In this case, the CMT equations that describe the temporal change of the normalized mode amplitudes of the cavities, a_i, i=1, 2, are described by

By substituting the Eqs. (24) and (25) in the above equations, we find

where γ defined in the previous section. When EM wave is launched only from the left side into the bus waveguide (S′₊₁ = 0), the transmission spectrum can be expressed by

The frequencies of the transmission peaks obtained from

and one can see that, when φ ≈ 0, for all τ ₂/τ ₁ values, the reflected power is always 100%.

 

Fig. 7. The basic structure of a wavelength-selective reflector which is based on two coupled cavities.

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5. Theoretical model for high efficient CDF with coupled cavity-based wavelength-selective reflection feedback

The narrow band CDFs are the key element in WDM systems, and we will show that a three-port system which is based on two coupled cavities in both drop and reflector sections 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. 8, 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. 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 τ ₀, are expressed by

Here, ω _Res−a and ω _Res−b are the resonant frequencies of the coupled cavities in the drop and the reflector sections, respectively, 1/τ ₁ and 1/τ ₃ denote the decay rates of cavities a1 and b1 into the bus waveguide, respectively, 1/τ ₂ is the decay rate of cavities a₂ into the drop waveguide and also is the decay rates of the cavity a₂ into the cavity a1 and vice versa, and 1/τ ₄ is the decay rates of the cavity b₂ into the cavity b₁ and vice versa. As shown in Fig. 8, the amplitudes of the EM waves incoming the drop (reflector) section from the bus waveguide, are denoted by S ₊₁ (S ₊₃) and S′ ₊₁ (S′ ₊₃). Also, the amplitudes of the EM waves outgoing the drop (reflector) section to the bus waveguide, are denoted by S ₋₁ (S ₋₃) and S′ ₋₁ (S′ ₋₃). In the case of EM waves traveling between the coupled cavities, in the drop section, the EM wave incoming the cavity a₁ (a₂) is denoted by S″ ₊₁ (S′ ₊₂), and the EM waves outgoing the cavity a₁ (a₂) is denoted by S″ ₋₁ (S′ ₋₂), respectively, and in the reflector section, the EM wave incoming the cavity b₁ (b₂) is denoted by R″ ₊₁ (R′ ₊₂), and the EM wave outgoing the cavity b₁ (b₂) is denoted by R″ ₋₁ (R′ ₋₂), respectively. The relationships among the denoted EM waves amplitudes and the cavities mode amplitudes are

In the above equations, θ ₁ and θ ₂ are the phases of the coupling coefficients between the bus waveguide and the cavities a₁ and b₁, respectively, θ ₃ is the phase of the coupling coefficient between the drop waveguide and cavity a₂, β 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

 

Fig. 8. The basic structure of the proposed three-port CDF with coupled cavity based wavelength selective-reflector.

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Based on Eqs. (47)–(48) and (53)–(54), when τ ₃ = τ ₄ we have

By substituting the Eqs. (55)–(56), in Eqs. (38)–(39), when EM wave is launched only from the left side into the bus waveguide (S ₊₁, S′ ₊₃ = 0), we find

where γ = [(ω−ω _Res−b)τ ₄ sinφ-cosφ]. Using Eqs. (41), (49)–(50), and (57) the reflectivity and S′+1 can be written as

where ρ = 2βd. The frequencies of the reflectivity peaks, given that τ ₀>>τ ₃,τ ₄, can be determined as

From Eqs. (36), (46) and (59) the transmission spectrum of the CDF can be expressed as

where α = γ ² +sin² φ (τ ₂/τ ₀ +1)². Assuming that φ ≈ 0, Eq. (61) can be much simplified as

Thus, given that ω _Res−a = ω _Res−b = ω _Res and τ ₀ >> τ ₁,τ ₂ the drop efficiency at resonant frequencies can be expressed as

where k = τ ₂/τ ₁. In this case, assuming ρ = 2βd = (2n + 1)π for either ω _Res1 or ω _Res2, where n is an integer, one can see that the channel drop efficiency of 100% will be obtained when k = τ ₂/τ ₁ =1/4. The dependence of the maximum of drop efficiency on k parameter is shown in Fig. 9(a). Figure 9(b) shows the dependence of the maximum of drop efficiency on ρ 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 three-port CDFs with narrow FWHM, which are the key element in WDM systems. On the other hand, at resonant frequencies and given that φ ≈ 0, from Eq. (58) the reflectivity can be simplified to r = τ ₃/(τ ₃ +τ ₀) and in order to obtain 100% reflectivity, the condition τ ₀ >> τ ₃ must be satisfied. Furthermore, concerning the sensitivity of the design and fabrication tolerance, the effect of the resonant frequency difference on the transmission spectrum is considerable. Assuming τ ₀ >> τ ₃, ω _Res−a ≤ ω _Res−a, ρ = (2n+1)π and k=1/4 from Eq. (62), it can be shown that

This implies that by increasing the value of the quality factor, the detuning between the resonant frequencies, leads to the reduction in drop efficiency, and an advanced fabrication technology will be necessary. The drop efficiency as a function of the frequency detuning factor (ω _Res−b/ω _Res−a), is shown in Fig. 10 for modified HW1, HW2, and HW3 with r_d = 0.04a. Accordingly, by using appropriate structure with suitable values for the cavities quality factor, a narrowband three-port CDF with high transmission efficiency can be achieved.

 

Fig. 9. (a) Dependence of drop efficiency at resonant frequencies on the ratio of decay rates τ ₂/τ ₁ when ρ = (2n + 1)π. (b) Dependence of the maximum of drop efficiency on ρ.

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Fig. 10. Dependence of the maximum of drop efficiency on the frequency detuning factor (ω _Res−b/ω _Res−a)

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6. Design and numerical simulations

We investigate the validity of the proposed PC coupled cavity based CDF by employing the FDTD method with PML absorbing boundary conditions. Figure 11 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 structure 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. Figure 12 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. 13.

 

Fig. 11. The structure of three-port CDF with coupled cavitiy-based wavelength-selective reflection feedback, in 2D-PC of square lattice composed of dielectric rods in air. The dashed-line and the dotted-line rectangulars are the drop and the reflector sections, respectively.

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Given that the radii of the coupled cavities in the drop and reflector sections are set to 0.055a, from Fig. 13, the corresponding resonant frequencies of the CDF coupled-cavities are ω _Res1 =0.36076 and ω _Res2 =0.36573 (2πc/a). The τ ₀ parameter, which is due to the internal loss of energy, is infinite in the desired 2D-PCs [17] and the total quality factors of the cavities are 1925. So, the condition τ ₀ >> τ ₃ is satisfied and the perfect reflection can be realized. The condition τ ₂/τ ₁= 1/4 can be easily satisfied using the coupled mode theory [7].

 

Fig. 12. Dispersion curve of the bus/drop line-defect waveguides versus the wave vector component k along the defect.

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Fig. 13. Dependence of the resonant frequencies of the coupled cavities in the HW2 structure on the coupled cavities radii.

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From Fig. 13 the guided mode has wave vectors 0.2325×(2π/a) and 0.2428 ×(2π/a) at ω _Res1 and ω _Res2, respectively, and when the distance between the drop and reflector sections, d, is set to 14a, the condition ρ(ω _Res1) = 2βd = (2n + 1)π = 13π will be satisfied (in this case ρ(ω _Res2) = 2βd =13.59π, which is not desired). Figure 14(a) shows the transmission spectra of the designed CDF calculated using the 2D-FDTD method. The simulated transmission spectrum through the drop waveguide (the dashed curve) represents that the proposed CDF has the ability of dropping a wavelength channel (at frequency ω _Res1) with the dropping efficiency 0.95% and the spectral line-width 0.0014a. Assuming the lattice constant a = 0.56μm, considering that in this case the wavelength corresponds to ω _Res1 is equal to 1550nm when r_d = 0.055a, the line-width is equal to 0.78nm. Figure 14(b) shows the transmission spectrum of the drop waveguide in dB. In this case, it can be seen that if channel spacing δλ, is chosen as δλ >(λ _Res1 − λ _Res2)/2 ≈ 10nm the inter channel crosstalk is reduced to below −30 dB which shows very good ability for WDM devices in practical applications. 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 [16]. Figure 15 shows the steady filed patterns at the resonant frequencies ω _Res1 = 0.36076 and ω _Res1 = 0.36573 (2πc/a). at the bus and drop 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.

 

Fig. 14. Transmission spectra for the designed CDF calculated using the 2D-FDTD method.(a) The drop port (the dashed curve) and the bus port transmission spectrum (the solid curve).(b) The drop port transmission spectrum in dB.

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Fig. 15. The steady state wave propagation at the resonant frequencies of the designed CDF. (a) ω _Res1 =0.36076 (2πc/a). (b) ω _Res2 =0.36573 (2πc/a).

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7. Conclusion

This paper has presented a highly efficient three-port CDF with a coupled cavity based wavelength reflection feedback. A modified HW2 with extra rods in both ends of the CCW and Lorentzian transmission spectrum was proposed, which can be used as the key element in implementation of WDM filters. It was shown that the phase-shift of the EM waves traveling between the modified HW2 cavities is close to zero. According to the theoretical theory using CMT in time, the performance of the proposed CDF was investigated and the conditions which lead to 100% drop efficiency were extracted. The performance of the designed filter was 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 1550nm, and also a multi-channel CDF with channel spacing around 10nm (1nm) with inter-channel crosstalk below −30dB (-15 dB) is possible. These characteristics make the proposed CDF suitable for use in WDM optical communication systems.