Abstract

We have proposed a three-port high efficient channel-drop filter (CDF) with a coupled cavity-based wavelength-selective reflector, which can be used in wavelength division multiplexing (WDM) optical communication systems. The coupling mode theory (CMT) is employed to drive the necessary conditions for achieving 100% drop efficiency. The finite-difference time-domain (FDTD) simulation results of proposed CDF which is implemented in two dimensional photonic crystals (2D-PC), show that the analysis is valid. In the designed CDF, the drop efficiency larger than 0.95% and the spectral line-width 0.78nm at the center wavelength 1550nm have been achieved.

©2009 Optical Society of America

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, ai, i=1, 2 , are described by [21,22]

 figure: Fig. 1.

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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da1dt=jωResa1a11τ1a11τ′1+ejθ12τ1S+1+2τ′1S′+1
da2dt=jωResa2a21τ2a21τ′2+2τ2S+2+ejθ22τ′2S′+2
S1=S+1+2τ1a1
S′1=S′+1+2τ′1a1
S2=S+2+2τ2a2
S′2=S′+2+2τ′2a2

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 ai 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

S′+1=S2e
S+2=S′1e

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

S′+1=2τ2(a2a1e2jsinφ)
S+2=2τ2(a1a2e2jsinφ).

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

S+1=ja2ejθ1sinφ2/τ[1(γjsinφ)2]

where γ= [(ωω 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

T=S′2S+12=4sin2φ(1+sin2φγ2)2+(2γsinφ)2.

Using Eq. (12), it can be shown that the resonant frequencies are ω Res1,2Res, 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 nrod = 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 rd of the modified HW2 with Lorentzian spectrum.

 figure: Fig. 2.

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

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 figure: Fig. 3.

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 rd = 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.

 figure: Fig. 4.

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:

ωiT=1/2=ωRes+1τtanφ±(1τtanφ)2±12τsinφ,i=1,2,3,4.

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

FWHM=ω2ω1=ω4ω32τ=ωRes2Q.

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 τ1 = τ′ 2 = τ 2, the CMT equations that describe the temporal change of the normalized mode amplitudes of the cavities, ai, i = 1, 2 , are described by

da1dt=jωResa1a11τ1a11τ2+ejθ12τ1S+1+2τ2S″+ejθ12τ1S′+1
da2dt=jωResa2a22τ2+2τ2S′+2+ejθ22τ2S+2

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

S′1=S+1ejθ12τ1a1
S1=S′+1ejθ12τ1a1
S2=S+2ejθ22τ2a2
S′′1=S′′+1+2τ1a1
S′2=S′+2+2τ2a2.

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

S′+2=S′′1e
S′′+1=S′2e.

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

S′′+1=2τ2(a2a1e2jsinφ).
S′+2=2τ2(a1a2e2jsinφ).

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

S+1=a2(1γ2)jsinφ2τ22τ1ejθ1

where γ= [(ωω 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 ejωt time dependency, then at steady state the transmission spectrum can be determined as

D=S2S+12=(sin2φ(4τ2/τ1)((γsinφ)(1+2τ2/τ1))2+(1+(2τ2/τ1)sin2φγ2)2).

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

PT=4(τ2/τ1)(1+2(τ2/τ1))2,

and also the corresponding frequencies are

ωRes1,2=ωRes+1τ2tanφ±1τ2tanφ(112sin2φ(1+(2τ2/τ1)2)1/2).

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

R=S1S+12=14,T=S′1S+12=14

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.

 figure: Fig. 5.

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

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 figure: Fig. 6.

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, ai, i=1, 2, are described by

da1dt=jωResa1a11τ1a11τ2+ejθ12τ1S+1+2τ2S′′+1
da2dt=jωResa2a21τ2+2τ2S′+2.

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

S+1=a2(γ1/γj2τ2/τ1sinφ)jsinφ2τ22/τ1ejθ1

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

R=S1S+11=jsinφ2τ2/τ1(γ1/γj2τ2/τ1sinφ)2=γ2(2τ2/τ1)2sin2φ(γ21)2+γ2(2τ2/τ1)2sin2φ.

The frequencies of the transmission peaks obtained from

ωRes1,2=ωRes+1τ2(1tanφ±1sinφ)

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

 figure: Fig. 7.

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

da1dt=jωResaa1a11τ0a11τ1a11τ2+ejθ12τ1S+1+2τ2S″+1+ejθ12τ1S′+1
da2dt=jωResaa2a22τ2a21τ0+2τ2S′+2+ejθ22τ2S+2
db1dt=jωResbb1b11τ0b12τ3b11τ4+ejθ12τ3S+3+2τ4S″+1
db2dt=jωResbb2b21τ4b21τ0+2τ4R+2

Here, ω Resa and ω Resb 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

S′3=S+32τ3ejθ3b1
S3=S′+32τ3ejθ3b1
S′1=S+12τ1ejθ1a1
S1=S′+12τ1ejθ1a1
S2=S+2+2τ2ejθ2a1
S″1=S″+1+2τ2a1
S′2=S′+2+2τ2a2
R1=R+1+2τ4b1
R2=R+2+2τ4b2
S+3=S′1ejβd
S′+1=S3ejβd.

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

S′+2=S″1e
S″+1=S′2e
R+2=R1e
R+1=R2e.
 figure: Fig. 8.

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

R+1=2τ4(b2b1e2jsinφ)
R+2=2τ4(b1b2e2jsinφ).

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

S+3=ejθ3[γ2τ4τ0(2τ4τ3+τ4τ0)sin2φ1jγsinφ(2τ4τ3+τ4τ0)]j2τ42τ3sinφ[γj(τ4τ0)sinφ]b1

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

S3S+3=r=2τ4τ3sinφ[γj(τ4τ0)sinφ][γ2τ4τ0(2τ4τ3+τ4τ0)sin2φ1]j+γsinφ(2τ4τ3+τ4τ0)
S′+1=re(S+12τ1ejθ1a1)

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

ωRes1,2=ωResb+1τ4(1tanφ±1sinφ).

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

D=S2S+1=(2/τ1τ2)ej(θ1θ2)[(1rcosρ)+j(rsinρ)](jτ4/τ0sinφγ)1j[(ωωResa)+2rτ1sinρ1τ2tanφγατ2sinφ]+[1τ0+2τ12rτ1cosρ+(τ2/τ0+1)ατ2]

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

Dφ0=(2/τ1τ2)ej(θ1θ2)[(1rcosρ)+j(rsinρ)]j[(ωωResa)+2rτ1sinρ]+[1τ0+2τ1+1τ22rτ1cosρ].

Thus, given that ω Resa = ω Resb = ω Res and τ 0 >> τ 1,τ 2 the drop efficiency at resonant frequencies can be expressed as

ηω=ωRes1,2=D2ω=ωRes1,2=8k(1cosρ)8k2(1cosρ)+4k(1cosρ)+1

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 Resaω Resa, ρ = (2n+1)π and k=1/4 from Eq. (62), it can be shown that

ηmaxωRes1,2=64τ12(ωResbωResa)2+64=4Q12(ωResbωResa1)2+4.

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 Resb/ω Resa), is shown in Fig. 10 for modified HW1, HW2, and HW3 with rd = 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.

 figure: Fig. 9.

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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 figure: Fig. 10.

Fig. 10. Dependence of the maximum of drop efficiency on the frequency detuning factor (ω Resb/ω Resa)

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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.

 figure: Fig. 11.

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].

 figure: Fig. 12.

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

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 figure: Fig. 13.

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 rd = 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 Res1λ 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.

 figure: Fig. 14.

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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 figure: Fig. 15.

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.

References and links

1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystal: Molding the Flow of high (Princeton, Princeton Univ. Press, 2008).

2. M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightwave. Technol. 22, 2316–2322 (2004). [CrossRef]  

3. M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler,” IEEE J. Lightwave. Technol. 19, 1970–1975 (2001). [CrossRef]  

4. A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two-dimensional photonic crystal lasers,” Appl. Phys. A: Mat. Scie. Proc. 69,111–114 (1999). [CrossRef]  

5. M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28, 2506–2508 (2003). [CrossRef]   [PubMed]  

6. Z. Zhang and M. Qiu, “Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs,” Opt. Express. 13, 2596–2604 (2005). [CrossRef]   [PubMed]  

7. S. Kim, I. park, H. Lim, and C. Kee, “Highly efficient photonic crystal-based multi-channel drop filters of three-port system with reflection feedback,” Opt. Express. 12, 5518–5525 (2004). [CrossRef]   [PubMed]  

8. S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature. 407, 608–610 (2000). [CrossRef]   [PubMed]  

9. B. Song, T. Asano, Y. Akahane, and S. Noda, “Role of interfaces in hetero photonic crystals for manipulation of photons,” Phys. Rev. B 71,195101–19105 (2005). [CrossRef]  

10. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express. 3, 4–11 (1998). [CrossRef]   [PubMed]  

11. B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59–63 (2004). [CrossRef]  

12. Z. Zhang and M. Qiu, “Coupled-mode analysis of a resonant channel drop filter using waveguides with mirror boundaries,” J. Opt. Soc. Am. B 23, 104–113 (2004). [CrossRef]  

13. M. Y. Tekeste and J. M. Yarrison-Rice, “High efficiency photonic crystal based wavelength demultiplexer,” Opt. Express. 14, 7931–7942 (2006). [CrossRef]   [PubMed]  

14. M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express. 12, 1551–1561 (2004). [CrossRef]   [PubMed]  

15. C. Jin, S. Fan, S. Han, and D. Zhang, “Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration,” J. Quantum Electron. 39,160–165 (2003). [CrossRef]  

16. C. W. Kuo, C. F. Chang, M. H. Chen, S. Y. Chen, and Y. D. Wu, “A new approach of planar multi-channel wavelength division multiplexing system using asymmetric super-cell photonic crystal structures,” Opt. Express. 15, 198–206 (2006). [CrossRef]  

17. H. Ren, C. Jiang, W. Hu, M. Gao, and J. Wang, “Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity,” Opt. Express. 14, 2446–2458 (2006). [CrossRef]   [PubMed]  

18. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996). [CrossRef]  

19. A. Yariv, Y. Xu, R. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999). [CrossRef]  

20. W. Ding, L. Chen, and S. Liu, “Localization properties and the effects on multi-mode switching in discrete mode CCWs,” Opt. Commun. 248, 479–484 (2004). [CrossRef]  

21. H. A. Haus, Waves and Field in Optoelectronics (Prentice-Hall, 1984).

22. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1333 (1999). [CrossRef]  

23. T. Fujisawa and M. Koshiba, “Finite-Element Modeling of Nonlinear Mach-Zehnder Interferometers Based on Photonic-Crystal Waveguides for All-optical Signal Processing,” IEEE J. Lightwave. Technol. 24, 617–623 (2006). [CrossRef]  

24. C. C. Chen, C. Y. Chen, W. K. Wang, F. H. Huang, C. K. Lin, W. Y. Chiu, and Y. J. Chan, “Photonic crystal directional couplers formed by InAlGaAs nano-rods,” Opt. Express. 13, 38–43 (2005). [CrossRef]   [PubMed]  

References

  • View by:

  1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystal: Molding the Flow of high (Princeton, Princeton Univ. Press, 2008).
  2. M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightwave. Technol. 22, 2316–2322 (2004).
    [Crossref]
  3. M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler,” IEEE J. Lightwave. Technol. 19, 1970–1975 (2001).
    [Crossref]
  4. A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two-dimensional photonic crystal lasers,” Appl. Phys. A: Mat. Scie. Proc. 69,111–114 (1999).
    [Crossref]
  5. M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28, 2506–2508 (2003).
    [Crossref] [PubMed]
  6. Z. Zhang and M. Qiu, “Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs,” Opt. Express. 13, 2596–2604 (2005).
    [Crossref] [PubMed]
  7. S. Kim, I. park, H. Lim, and C. Kee, “Highly efficient photonic crystal-based multi-channel drop filters of three-port system with reflection feedback,” Opt. Express. 12, 5518–5525 (2004).
    [Crossref] [PubMed]
  8. S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature. 407, 608–610 (2000).
    [Crossref] [PubMed]
  9. B. Song, T. Asano, Y. Akahane, and S. Noda, “Role of interfaces in hetero photonic crystals for manipulation of photons,” Phys. Rev. B 71,195101–19105 (2005).
    [Crossref]
  10. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express. 3, 4–11 (1998).
    [Crossref] [PubMed]
  11. B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59–63 (2004).
    [Crossref]
  12. Z. Zhang and M. Qiu, “Coupled-mode analysis of a resonant channel drop filter using waveguides with mirror boundaries,” J. Opt. Soc. Am. B 23, 104–113 (2004).
    [Crossref]
  13. M. Y. Tekeste and J. M. Yarrison-Rice, “High efficiency photonic crystal based wavelength demultiplexer,” Opt. Express. 14, 7931–7942 (2006).
    [Crossref] [PubMed]
  14. M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express. 12, 1551–1561 (2004).
    [Crossref] [PubMed]
  15. C. Jin, S. Fan, S. Han, and D. Zhang, “Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration,” J. Quantum Electron. 39,160–165 (2003).
    [Crossref]
  16. C. W. Kuo, C. F. Chang, M. H. Chen, S. Y. Chen, and Y. D. Wu, “A new approach of planar multi-channel wavelength division multiplexing system using asymmetric super-cell photonic crystal structures,” Opt. Express. 15, 198–206 (2006).
    [Crossref]
  17. H. Ren, C. Jiang, W. Hu, M. Gao, and J. Wang, “Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity,” Opt. Express. 14, 2446–2458 (2006).
    [Crossref] [PubMed]
  18. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
    [Crossref]
  19. A. Yariv, Y. Xu, R. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
    [Crossref]
  20. W. Ding, L. Chen, and S. Liu, “Localization properties and the effects on multi-mode switching in discrete mode CCWs,” Opt. Commun. 248, 479–484 (2004).
    [Crossref]
  21. H. A. Haus, Waves and Field in Optoelectronics (Prentice-Hall, 1984).
  22. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1333 (1999).
    [Crossref]
  23. T. Fujisawa and M. Koshiba, “Finite-Element Modeling of Nonlinear Mach-Zehnder Interferometers Based on Photonic-Crystal Waveguides for All-optical Signal Processing,” IEEE J. Lightwave. Technol. 24, 617–623 (2006).
    [Crossref]
  24. C. C. Chen, C. Y. Chen, W. K. Wang, F. H. Huang, C. K. Lin, W. Y. Chiu, and Y. J. Chan, “Photonic crystal directional couplers formed by InAlGaAs nano-rods,” Opt. Express. 13, 38–43 (2005).
    [Crossref] [PubMed]

2006 (4)

M. Y. Tekeste and J. M. Yarrison-Rice, “High efficiency photonic crystal based wavelength demultiplexer,” Opt. Express. 14, 7931–7942 (2006).
[Crossref] [PubMed]

C. W. Kuo, C. F. Chang, M. H. Chen, S. Y. Chen, and Y. D. Wu, “A new approach of planar multi-channel wavelength division multiplexing system using asymmetric super-cell photonic crystal structures,” Opt. Express. 15, 198–206 (2006).
[Crossref]

H. Ren, C. Jiang, W. Hu, M. Gao, and J. Wang, “Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity,” Opt. Express. 14, 2446–2458 (2006).
[Crossref] [PubMed]

T. Fujisawa and M. Koshiba, “Finite-Element Modeling of Nonlinear Mach-Zehnder Interferometers Based on Photonic-Crystal Waveguides for All-optical Signal Processing,” IEEE J. Lightwave. Technol. 24, 617–623 (2006).
[Crossref]

2005 (3)

C. C. Chen, C. Y. Chen, W. K. Wang, F. H. Huang, C. K. Lin, W. Y. Chiu, and Y. J. Chan, “Photonic crystal directional couplers formed by InAlGaAs nano-rods,” Opt. Express. 13, 38–43 (2005).
[Crossref] [PubMed]

Z. Zhang and M. Qiu, “Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs,” Opt. Express. 13, 2596–2604 (2005).
[Crossref] [PubMed]

B. Song, T. Asano, Y. Akahane, and S. Noda, “Role of interfaces in hetero photonic crystals for manipulation of photons,” Phys. Rev. B 71,195101–19105 (2005).
[Crossref]

2004 (6)

B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59–63 (2004).
[Crossref]

Z. Zhang and M. Qiu, “Coupled-mode analysis of a resonant channel drop filter using waveguides with mirror boundaries,” J. Opt. Soc. Am. B 23, 104–113 (2004).
[Crossref]

S. Kim, I. park, H. Lim, and C. Kee, “Highly efficient photonic crystal-based multi-channel drop filters of three-port system with reflection feedback,” Opt. Express. 12, 5518–5525 (2004).
[Crossref] [PubMed]

M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightwave. Technol. 22, 2316–2322 (2004).
[Crossref]

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express. 12, 1551–1561 (2004).
[Crossref] [PubMed]

W. Ding, L. Chen, and S. Liu, “Localization properties and the effects on multi-mode switching in discrete mode CCWs,” Opt. Commun. 248, 479–484 (2004).
[Crossref]

2003 (2)

M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28, 2506–2508 (2003).
[Crossref] [PubMed]

C. Jin, S. Fan, S. Han, and D. Zhang, “Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration,” J. Quantum Electron. 39,160–165 (2003).
[Crossref]

2001 (1)

M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler,” IEEE J. Lightwave. Technol. 19, 1970–1975 (2001).
[Crossref]

2000 (1)

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature. 407, 608–610 (2000).
[Crossref] [PubMed]

1999 (3)

A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two-dimensional photonic crystal lasers,” Appl. Phys. A: Mat. Scie. Proc. 69,111–114 (1999).
[Crossref]

A. Yariv, Y. Xu, R. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
[Crossref]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1333 (1999).
[Crossref]

1998 (1)

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express. 3, 4–11 (1998).
[Crossref] [PubMed]

1996 (1)

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[Crossref]

Akahane, Y.

B. Song, T. Asano, Y. Akahane, and S. Noda, “Role of interfaces in hetero photonic crystals for manipulation of photons,” Phys. Rev. B 71,195101–19105 (2005).
[Crossref]

Altug, H.

M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightwave. Technol. 22, 2316–2322 (2004).
[Crossref]

Asano, T.

B. Song, T. Asano, Y. Akahane, and S. Noda, “Role of interfaces in hetero photonic crystals for manipulation of photons,” Phys. Rev. B 71,195101–19105 (2005).
[Crossref]

Chan, Y. J.

C. C. Chen, C. Y. Chen, W. K. Wang, F. H. Huang, C. K. Lin, W. Y. Chiu, and Y. J. Chan, “Photonic crystal directional couplers formed by InAlGaAs nano-rods,” Opt. Express. 13, 38–43 (2005).
[Crossref] [PubMed]

Chang, C. F.

C. W. Kuo, C. F. Chang, M. H. Chen, S. Y. Chen, and Y. D. Wu, “A new approach of planar multi-channel wavelength division multiplexing system using asymmetric super-cell photonic crystal structures,” Opt. Express. 15, 198–206 (2006).
[Crossref]

Chen, C. C.

C. C. Chen, C. Y. Chen, W. K. Wang, F. H. Huang, C. K. Lin, W. Y. Chiu, and Y. J. Chan, “Photonic crystal directional couplers formed by InAlGaAs nano-rods,” Opt. Express. 13, 38–43 (2005).
[Crossref] [PubMed]

Chen, C. Y.

C. C. Chen, C. Y. Chen, W. K. Wang, F. H. Huang, C. K. Lin, W. Y. Chiu, and Y. J. Chan, “Photonic crystal directional couplers formed by InAlGaAs nano-rods,” Opt. Express. 13, 38–43 (2005).
[Crossref] [PubMed]

Chen, L.

W. Ding, L. Chen, and S. Liu, “Localization properties and the effects on multi-mode switching in discrete mode CCWs,” Opt. Commun. 248, 479–484 (2004).
[Crossref]

Chen, M. H.

C. W. Kuo, C. F. Chang, M. H. Chen, S. Y. Chen, and Y. D. Wu, “A new approach of planar multi-channel wavelength division multiplexing system using asymmetric super-cell photonic crystal structures,” Opt. Express. 15, 198–206 (2006).
[Crossref]

Chen, S. Y.

C. W. Kuo, C. F. Chang, M. H. Chen, S. Y. Chen, and Y. D. Wu, “A new approach of planar multi-channel wavelength division multiplexing system using asymmetric super-cell photonic crystal structures,” Opt. Express. 15, 198–206 (2006).
[Crossref]

Chiu, W. Y.

C. C. Chen, C. Y. Chen, W. K. Wang, F. H. Huang, C. K. Lin, W. Y. Chiu, and Y. J. Chan, “Photonic crystal directional couplers formed by InAlGaAs nano-rods,” Opt. Express. 13, 38–43 (2005).
[Crossref] [PubMed]

Chutinan, A.

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature. 407, 608–610 (2000).
[Crossref] [PubMed]

Ding, W.

W. Ding, L. Chen, and S. Liu, “Localization properties and the effects on multi-mode switching in discrete mode CCWs,” Opt. Commun. 248, 479–484 (2004).
[Crossref]

Dodabalapur, A.

A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two-dimensional photonic crystal lasers,” Appl. Phys. A: Mat. Scie. Proc. 69,111–114 (1999).
[Crossref]

Fan, S.

M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightwave. Technol. 22, 2316–2322 (2004).
[Crossref]

M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28, 2506–2508 (2003).
[Crossref] [PubMed]

C. Jin, S. Fan, S. Han, and D. Zhang, “Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration,” J. Quantum Electron. 39,160–165 (2003).
[Crossref]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1333 (1999).
[Crossref]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express. 3, 4–11 (1998).
[Crossref] [PubMed]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[Crossref]

Fujisawa, T.

T. Fujisawa and M. Koshiba, “Finite-Element Modeling of Nonlinear Mach-Zehnder Interferometers Based on Photonic-Crystal Waveguides for All-optical Signal Processing,” IEEE J. Lightwave. Technol. 24, 617–623 (2006).
[Crossref]

Gao, M.

H. Ren, C. Jiang, W. Hu, M. Gao, and J. Wang, “Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity,” Opt. Express. 14, 2446–2458 (2006).
[Crossref] [PubMed]

Han, S.

C. Jin, S. Fan, S. Han, and D. Zhang, “Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration,” J. Quantum Electron. 39,160–165 (2003).
[Crossref]

Haus, H. A.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1333 (1999).
[Crossref]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express. 3, 4–11 (1998).
[Crossref] [PubMed]

H. A. Haus, Waves and Field in Optoelectronics (Prentice-Hall, 1984).

Hu, W.

H. Ren, C. Jiang, W. Hu, M. Gao, and J. Wang, “Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity,” Opt. Express. 14, 2446–2458 (2006).
[Crossref] [PubMed]

Huang, F. H.

C. C. Chen, C. Y. Chen, W. K. Wang, F. H. Huang, C. K. Lin, W. Y. Chiu, and Y. J. Chan, “Photonic crystal directional couplers formed by InAlGaAs nano-rods,” Opt. Express. 13, 38–43 (2005).
[Crossref] [PubMed]

Imada, M.

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature. 407, 608–610 (2000).
[Crossref] [PubMed]

Jiang, C.

H. Ren, C. Jiang, W. Hu, M. Gao, and J. Wang, “Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity,” Opt. Express. 14, 2446–2458 (2006).
[Crossref] [PubMed]

Jin, C.

C. Jin, S. Fan, S. Han, and D. Zhang, “Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration,” J. Quantum Electron. 39,160–165 (2003).
[Crossref]

Joannopoulos, J. D.

M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28, 2506–2508 (2003).
[Crossref] [PubMed]

A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two-dimensional photonic crystal lasers,” Appl. Phys. A: Mat. Scie. Proc. 69,111–114 (1999).
[Crossref]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1333 (1999).
[Crossref]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express. 3, 4–11 (1998).
[Crossref] [PubMed]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystal: Molding the Flow of high (Princeton, Princeton Univ. Press, 2008).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystal: Molding the Flow of high (Princeton, Princeton Univ. Press, 2008).

Kee, C.

S. Kim, I. park, H. Lim, and C. Kee, “Highly efficient photonic crystal-based multi-channel drop filters of three-port system with reflection feedback,” Opt. Express. 12, 5518–5525 (2004).
[Crossref] [PubMed]

Khan, M. J.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1333 (1999).
[Crossref]

Kim, J. E.

B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59–63 (2004).
[Crossref]

Kim, S.

S. Kim, I. park, H. Lim, and C. Kee, “Highly efficient photonic crystal-based multi-channel drop filters of three-port system with reflection feedback,” Opt. Express. 12, 5518–5525 (2004).
[Crossref] [PubMed]

Koshiba, M.

T. Fujisawa and M. Koshiba, “Finite-Element Modeling of Nonlinear Mach-Zehnder Interferometers Based on Photonic-Crystal Waveguides for All-optical Signal Processing,” IEEE J. Lightwave. Technol. 24, 617–623 (2006).
[Crossref]

M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler,” IEEE J. Lightwave. Technol. 19, 1970–1975 (2001).
[Crossref]

Kuo, C. W.

C. W. Kuo, C. F. Chang, M. H. Chen, S. Y. Chen, and Y. D. Wu, “A new approach of planar multi-channel wavelength division multiplexing system using asymmetric super-cell photonic crystal structures,” Opt. Express. 15, 198–206 (2006).
[Crossref]

Kuramochi, E.

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express. 12, 1551–1561 (2004).
[Crossref] [PubMed]

Lee, R.

Lim, H.

S. Kim, I. park, H. Lim, and C. Kee, “Highly efficient photonic crystal-based multi-channel drop filters of three-port system with reflection feedback,” Opt. Express. 12, 5518–5525 (2004).
[Crossref] [PubMed]

Lin, C. K.

C. C. Chen, C. Y. Chen, W. K. Wang, F. H. Huang, C. K. Lin, W. Y. Chiu, and Y. J. Chan, “Photonic crystal directional couplers formed by InAlGaAs nano-rods,” Opt. Express. 13, 38–43 (2005).
[Crossref] [PubMed]

Liu, S.

W. Ding, L. Chen, and S. Liu, “Localization properties and the effects on multi-mode switching in discrete mode CCWs,” Opt. Commun. 248, 479–484 (2004).
[Crossref]

Manolatou, C.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1333 (1999).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystal: Molding the Flow of high (Princeton, Princeton Univ. Press, 2008).

Meier, M.

A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two-dimensional photonic crystal lasers,” Appl. Phys. A: Mat. Scie. Proc. 69,111–114 (1999).
[Crossref]

Mekis, A.

A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two-dimensional photonic crystal lasers,” Appl. Phys. A: Mat. Scie. Proc. 69,111–114 (1999).
[Crossref]

Min, B. K.

B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59–63 (2004).
[Crossref]

Mitsugi, S.

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express. 12, 1551–1561 (2004).
[Crossref] [PubMed]

Noda, S.

B. Song, T. Asano, Y. Akahane, and S. Noda, “Role of interfaces in hetero photonic crystals for manipulation of photons,” Phys. Rev. B 71,195101–19105 (2005).
[Crossref]

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature. 407, 608–610 (2000).
[Crossref] [PubMed]

Notomi, M.

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express. 12, 1551–1561 (2004).
[Crossref] [PubMed]

Park, H. Y.

B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59–63 (2004).
[Crossref]

park, I.

S. Kim, I. park, H. Lim, and C. Kee, “Highly efficient photonic crystal-based multi-channel drop filters of three-port system with reflection feedback,” Opt. Express. 12, 5518–5525 (2004).
[Crossref] [PubMed]

Qiu, M.

Z. Zhang and M. Qiu, “Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs,” Opt. Express. 13, 2596–2604 (2005).
[Crossref] [PubMed]

Z. Zhang and M. Qiu, “Coupled-mode analysis of a resonant channel drop filter using waveguides with mirror boundaries,” J. Opt. Soc. Am. B 23, 104–113 (2004).
[Crossref]

Ren, H.

H. Ren, C. Jiang, W. Hu, M. Gao, and J. Wang, “Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity,” Opt. Express. 14, 2446–2458 (2006).
[Crossref] [PubMed]

Ryu, H.

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express. 12, 1551–1561 (2004).
[Crossref] [PubMed]

Scherer, A.

Shinya, A.

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express. 12, 1551–1561 (2004).
[Crossref] [PubMed]

Slusher, R. E.

A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two-dimensional photonic crystal lasers,” Appl. Phys. A: Mat. Scie. Proc. 69,111–114 (1999).
[Crossref]

Soljacic, M.

Song, B.

B. Song, T. Asano, Y. Akahane, and S. Noda, “Role of interfaces in hetero photonic crystals for manipulation of photons,” Phys. Rev. B 71,195101–19105 (2005).
[Crossref]

Tekeste, M. Y.

M. Y. Tekeste and J. M. Yarrison-Rice, “High efficiency photonic crystal based wavelength demultiplexer,” Opt. Express. 14, 7931–7942 (2006).
[Crossref] [PubMed]

Villeneuve, P. R.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1333 (1999).
[Crossref]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express. 3, 4–11 (1998).
[Crossref] [PubMed]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[Crossref]

Vuckovic, J.

M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightwave. Technol. 22, 2316–2322 (2004).
[Crossref]

Wang, J.

H. Ren, C. Jiang, W. Hu, M. Gao, and J. Wang, “Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity,” Opt. Express. 14, 2446–2458 (2006).
[Crossref] [PubMed]

Wang, W. K.

C. C. Chen, C. Y. Chen, W. K. Wang, F. H. Huang, C. K. Lin, W. Y. Chiu, and Y. J. Chan, “Photonic crystal directional couplers formed by InAlGaAs nano-rods,” Opt. Express. 13, 38–43 (2005).
[Crossref] [PubMed]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystal: Molding the Flow of high (Princeton, Princeton Univ. Press, 2008).

Wu, Y. D.

C. W. Kuo, C. F. Chang, M. H. Chen, S. Y. Chen, and Y. D. Wu, “A new approach of planar multi-channel wavelength division multiplexing system using asymmetric super-cell photonic crystal structures,” Opt. Express. 15, 198–206 (2006).
[Crossref]

Xu, Y.

Yanik, M. F.

M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightwave. Technol. 22, 2316–2322 (2004).
[Crossref]

M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28, 2506–2508 (2003).
[Crossref] [PubMed]

Yariv, A.

Yarrison-Rice, J. M.

M. Y. Tekeste and J. M. Yarrison-Rice, “High efficiency photonic crystal based wavelength demultiplexer,” Opt. Express. 14, 7931–7942 (2006).
[Crossref] [PubMed]

Zhang, D.

C. Jin, S. Fan, S. Han, and D. Zhang, “Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration,” J. Quantum Electron. 39,160–165 (2003).
[Crossref]

Zhang, Z.

Z. Zhang and M. Qiu, “Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs,” Opt. Express. 13, 2596–2604 (2005).
[Crossref] [PubMed]

Z. Zhang and M. Qiu, “Coupled-mode analysis of a resonant channel drop filter using waveguides with mirror boundaries,” J. Opt. Soc. Am. B 23, 104–113 (2004).
[Crossref]

Appl. Phys. A: Mat. Scie. Proc. (1)

A. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two-dimensional photonic crystal lasers,” Appl. Phys. A: Mat. Scie. Proc. 69,111–114 (1999).
[Crossref]

IEEE J. Lightwave. Technol. (3)

M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightwave. Technol. 22, 2316–2322 (2004).
[Crossref]

M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler,” IEEE J. Lightwave. Technol. 19, 1970–1975 (2001).
[Crossref]

T. Fujisawa and M. Koshiba, “Finite-Element Modeling of Nonlinear Mach-Zehnder Interferometers Based on Photonic-Crystal Waveguides for All-optical Signal Processing,” IEEE J. Lightwave. Technol. 24, 617–623 (2006).
[Crossref]

IEEE J. Quantum Electron. (1)

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1333 (1999).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Quantum Electron. (1)

C. Jin, S. Fan, S. Han, and D. Zhang, “Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration,” J. Quantum Electron. 39,160–165 (2003).
[Crossref]

Nature. (1)

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature. 407, 608–610 (2000).
[Crossref] [PubMed]

Opt. Commun. (2)

B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two dimensional triangular lattice photonic crystal slabs,” Opt. Commun. 237, 59–63 (2004).
[Crossref]

W. Ding, L. Chen, and S. Liu, “Localization properties and the effects on multi-mode switching in discrete mode CCWs,” Opt. Commun. 248, 479–484 (2004).
[Crossref]

Opt. Express. (8)

C. C. Chen, C. Y. Chen, W. K. Wang, F. H. Huang, C. K. Lin, W. Y. Chiu, and Y. J. Chan, “Photonic crystal directional couplers formed by InAlGaAs nano-rods,” Opt. Express. 13, 38–43 (2005).
[Crossref] [PubMed]

Z. Zhang and M. Qiu, “Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs,” Opt. Express. 13, 2596–2604 (2005).
[Crossref] [PubMed]

S. Kim, I. park, H. Lim, and C. Kee, “Highly efficient photonic crystal-based multi-channel drop filters of three-port system with reflection feedback,” Opt. Express. 12, 5518–5525 (2004).
[Crossref] [PubMed]

C. W. Kuo, C. F. Chang, M. H. Chen, S. Y. Chen, and Y. D. Wu, “A new approach of planar multi-channel wavelength division multiplexing system using asymmetric super-cell photonic crystal structures,” Opt. Express. 15, 198–206 (2006).
[Crossref]

H. Ren, C. Jiang, W. Hu, M. Gao, and J. Wang, “Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity,” Opt. Express. 14, 2446–2458 (2006).
[Crossref] [PubMed]

M. Y. Tekeste and J. M. Yarrison-Rice, “High efficiency photonic crystal based wavelength demultiplexer,” Opt. Express. 14, 7931–7942 (2006).
[Crossref] [PubMed]

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express. 12, 1551–1561 (2004).
[Crossref] [PubMed]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express. 3, 4–11 (1998).
[Crossref] [PubMed]

Opt. Lett. (2)

Phys. Rev. B (2)

B. Song, T. Asano, Y. Akahane, and S. Noda, “Role of interfaces in hetero photonic crystals for manipulation of photons,” Phys. Rev. B 71,195101–19105 (2005).
[Crossref]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[Crossref]

Other (2)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystal: Molding the Flow of high (Princeton, Princeton Univ. Press, 2008).

H. A. Haus, Waves and Field in Optoelectronics (Prentice-Hall, 1984).

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Figures (15)

Fig. 1.
Fig. 1. (a) Structure of a HW2 in a PC of square lattice. (b) Schematic diagram for HW2 based on CMT.
Fig. 2.
Fig. 2. The modified HW2 with Lorentzian transmission spectrum in a PC of square lattice.
Fig. 3.
Fig. 3. The phase-shift between two coupled cavities versus the cavities radius of the modified HW2 in a PC of square lattice.
Fig. 4.
Fig. 4. The simulation results of transmission spectrum of the modified HW2 obtained by the FDTD and CMT method.
Fig. 5.
Fig. 5. The basic structure of a three-port CDF which is based on two coupled cavities.
Fig. 6.
Fig. 6. The curve of the three-port CDF transmission peaks versus τ 2 /τ 1.
Fig. 7.
Fig. 7. The basic structure of a wavelength-selective reflector which is based on two coupled cavities.
Fig. 8.
Fig. 8. The basic structure of the proposed three-port CDF with coupled cavity based wavelength selective-reflector.
Fig. 9.
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.
Fig. 10.
Fig. 10. Dependence of the maximum of drop efficiency on the frequency detuning factor (ω Resb /ω Resa )
Fig. 11.
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.
Fig. 12.
Fig. 12. Dispersion curve of the bus/drop line-defect waveguides versus the wave vector component k along the defect.
Fig. 13.
Fig. 13. Dependence of the resonant frequencies of the coupled cavities in the HW2 structure on the coupled cavities radii.
Fig. 14.
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.
Fig. 15.
Fig. 15. The steady state wave propagation at the resonant frequencies of the designed CDF. (a) ω Res 1 =0.36076 (2πc/a). (b) ω Res 2 =0.36573 (2πc/a).

Equations (64)

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da1dt=jωResa1a11τ1a11τ′1+ejθ12τ1S+1+2τ′1S′+1
da2dt=jωResa2a21τ2a21τ′2+2τ2S+2+ejθ22τ′2S′+2
S1=S+1+2τ1a1
S′1=S′+1+2τ′1a1
S2=S+2+2τ2a2
S′2=S′+2+2τ′2a2
S′+1=S2e
S+2=S′1e
S′+1=2τ2(a2a1e2jsinφ)
S+2=2τ2(a1a2e2jsinφ).
S+1=ja2ejθ1sinφ2/τ [1(γjsinφ)2]
T=S′2S+12=4sin2φ(1+sin2φγ2)2+(2γsinφ)2.
ωiT=1/2=ωRes+1τtanφ±(1τtanφ)2±12τsinφ,i=1,2,3,4.
FWHM=ω2ω1=ω4ω32τ=ωRes2Q .
da1dt=jωResa1a11τ1a11τ2+ejθ12τ1S+1+2τ2S″+ejθ12τ1S′+1
da2dt=jωResa2a22τ2+2τ2S′+2+ejθ22τ2S+2
S′1=S+1ejθ12τ1a1
S1=S′+1ejθ12τ1a1
S2=S+2ejθ22τ2a2
S′′1=S′′+1+2τ1a1
S′2=S′+2+2τ2a2.
S′+2=S′′1e
S′′+1=S′2e.
S′′+1=2τ2(a2a1e2jsinφ).
S′+2=2τ2(a1a2e2jsinφ).
S+1=a2(1γ2)jsinφ2τ22τ1ejθ1
D=S2S+12=(sin2φ(4τ2/τ1)((γsinφ)(1+2τ2/τ1))2+(1+(2τ2/τ1)sin2φγ2)2).
PT=4(τ2/τ1)(1+2(τ2/τ1))2,
ωRes1,2=ωRes+1τ2tanφ±1τ2tanφ(112sin2φ(1+(2τ2/τ1)2)1/2).
R=S1S+12=14 , T=S′1S+12=14
da1dt=jωResa1a11τ1a11τ2+ejθ12τ1S+1+2τ2S′′+1
da2dt=jωResa2a21τ2+2τ2S′+2.
S+1= a2(γ1/γj2τ2/τ1sinφ)jsinφ2τ22/τ1ejθ1
R=S1S+11=jsinφ2τ2/τ1(γ1/γj2τ2/τ1sinφ)2=γ2(2τ2/τ1)2sin2φ(γ21)2+γ2(2τ2/τ1)2sin2φ.
ωRes1,2=ωRes+1τ2(1tanφ±1sinφ)
da1dt=jωResaa1a11τ0a11τ1a11τ2+ejθ12τ1S+1+2τ2S″+1+ejθ12τ1S′+1
da2dt=jωResaa2a22τ2a21τ0+2τ2S′+2+ejθ22τ2S+2
db1dt=jωResbb1b11τ0b12τ3b11τ4+ejθ12τ3S+3+2τ4S″+1
db2dt=jωResbb2b21τ4b21τ0+2τ4R+2
S′3=S+32τ3ejθ3b1
S3=S′+32τ3ejθ3b1
S′1=S+12τ1ejθ1a1
S1=S′+12τ1ejθ1a1
S2=S+2+2τ2ejθ2a1
S″1=S″+1+2τ2a1
S′2=S′+2+2τ2a2
R1=R+1+2τ4b1
R2=R+2+2τ4b2
S+3=S′1ejβd
S′+1=S3ejβd.
S′+2=S″1e
S″+1=S′2e
R+2=R1e
R+1=R2e.
R+1=2τ4(b2b1e2jsinφ)
R+2=2τ4(b1b2e2jsinφ).
S+3=ejθ3[γ2τ4τ0(2τ4τ3+τ4τ0)sin2φ1jγsinφ(2τ4τ3+τ4τ0)]j2τ42τ3sinφ[γj(τ4τ0)sinφ]b1
S3S+3=r=2τ4τ3sinφ[γj(τ4τ0)sinφ][γ2τ4τ0(2τ4τ3+τ4τ0)sin2φ1]j+γsinφ(2τ4τ3+τ4τ0)
S′+1=re(S+12τ1ejθ1a1)
ωRes1,2=ωResb+1τ4(1tanφ±1sinφ).
D=S2S+1=(2/τ1τ2)ej(θ1θ2)[(1rcosρ)+j(rsinρ)](jτ4/τ0sinφγ)1j[(ωωResa)+2rτ1sinρ1τ2tanφγατ2sinφ]+[1τ0+2τ12rτ1cosρ+(τ2/τ0+1)ατ2]
Dφ0=(2/τ1τ2)ej(θ1θ2)[(1rcosρ)+j(rsinρ)]j[(ωωResa)+2rτ1sinρ]+[1τ0+2τ1+1τ22rτ1cosρ].
ηω=ωRes1,2=D2ω=ωRes1,2=8k(1cosρ)8k2(1cosρ)+4k(1cosρ)+1
ηmaxωRes1,2=64τ12(ωResbωResa)2+64=4Q12(ωResbωResa1)2+4 .

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