## Abstract

In the paper, a novel three-port channel drop filter in two dimensional photonic crystals (2D PCs) with a wavelength-selective reflection micro-cavity is proposed. In the structure, two micro-cavities are used. One is used for a resonant tunneling-based channel drop filter. The other is used to realize wavelength-selective reflection feedback in the bus wave-guide, which consists of a point defect micro-cavity side-coupled to a line defect waveguide based on photonic crystals. Using coupled mode theory in time, the conditions to achieve 100% drop efficiency are derived thoroughly. The simulation results by using the finite-difference time-domain (FDTD) method imply that the design is feasible.

©2006 Optical Society of America

## 1. Introduction

Since the discovery of photonic crystals more than a decade ago [1,2], the research on them has made rapid and steady progress [3–23]. With artificial periodic structures, photonic crystals can generate band gaps that forbid light propagation at certain frequency ranges [4]. Utilizing photonic band-gap effect, a lot of active and passive optical devices based on photonic crystals are designed, which may eventually pave the way for photonic integrated circuits (PICs) and dense wavelength division multiplexing (DWDM) optical communication systems.

Channel drop filter is one of the most important and essential components of PICs and DWDM systems. At present, channel spacing becomes more and more dense to make full use of the spectral bandwidth resources in DWDM system. A large number of spaced wavelength channels require a filter with a very small scale. Based on photonic crystals, channel drop filters can be made smaller in sizes than ever [5,6], and several research groups have proposed some designs based on 2D PCs to realize the channel drop functions [7–21]. Among various photonic crystal filters, Channel drop filters utilizing resonant coupling between micro-cavity modes produced by point defects and waveguide modes created by line defects have recently become the focus of much attention owing to their essential requirement in DWDM systems [10–20]. In order to achieve complete channel drop transfer at resonance (i.e., 100% channel drop efficiency), the operating principles of the four-port channel drop system found by S. Fan et al. [10–14] are that the resonators must support two degenerate modes between the two parallel waveguides, which decides a rigid and complex resonator design. The three-port channel drop filters also have been proposed, and it is easier to design multi-channel drop filters based on the three-port system than those based on the four-port system. S.Noda’s group has proposed the surface-emitting three-port channel drop filter [15,16], which consists of a cavity side-coupled to a waveguide. The resonant cavity traps photons at resonant frequency from the waveguide through evanescent coupling, and emits some of them in the vertical direction. Recently, such a channel drop filter based on the in-plane hetero photonic crystals have been proposed by them, and the drop efficiency is improved greatly by means of the reflection feedback, which is realized by the hetero-structure interface due to the presence of mode-gap in the waveguide [17]. Another three-port filter with reflection feedback has also been proposed by Hanjo Lim’s group [18]. In the structure, the bus waveguide is closed at one end for 100% reflection feedback, and close to 100% drop efficiency can be theoretically achieved. As the defect in the design is mentioned by them, however, the light over the entire wavelength range, except at the resonant wavelengths will be reflected back to the input port, which will lead to severe noise to other components if the designed structure is integrated. In fact, channel drop filters have been engineered by several researchers by means of reflection feedback to enhance the drop efficiency [15,17,19,21]. If the reflection feedback has no wavelength-selective characteristic, it is easy to introduce noise to the system when the large-scale photonic integration is performed.

In this paper, a three-port filter with a wavelength-selective reflection cavity is presented, and the conditions to achieve 100% drop efficiency in the system are derived thoroughly using coupled-mode theory in time. According to these theoretical analysis, a three-port filter is designed based on 2D PCs with high dielectric rods in air, and the numerical results by using the FDTD method with perfectly matched layers (PML) absorbing boundary conditions demonstrate complete channel drop tunneling at resonance via the three-port system.

## 2. Theoretical modeling

Figure 1 shows a structure of a cavity side-coupled to a waveguide, where both the micro-cavity and the waveguide support only one mode in the frequency range of interest. The cavity possesses a mirror reflection symmetry with respect to the reference plane, which is at the center of the resonator. The amplitudes of the incoming (outgoing) waves in the waveguide are denoted by *S*
_{+1}(*S*
_{-1}) and *S*
_{+2} (*S*
_{-2}), respectively. The time evolution of the cavity amplitude denoted by *a* can be described as [11,22]

where *ω*
_{0} is the resonant frequency of the cavity, *Q*
_{o} is the quality factor due to intrinsic loss, *Q*
_{e}is the quality factor that is related to the rate of decay into the waveguide, andθ is the phase of the coupling coefficient from the incoming wave to the resonator mode. The outgoing waves can be written as

When wave is launched only in the left waveguide (*S*
_{+2} =0) and *S*
_{+1} has a *e*
^{jωt} time dependence, by solving Eqs. (1)–(3), the back reflection from the input port *R* and the transmission through the waveguide *T* can be expressed as

From the above equations, close to 100% reflection (i.e. |*R*|^{2} ≈0 , |t|^{2} ≈1) at ω = ω_{0} is realized if the condition *Q*
_{o} ≫ *Q*
_{e} is satisfied. The full-width at half-maximum (FWHM) of the reflection spectrum can be expressed as σ = 2ω_{0} (1/*Q*
_{o} +1/2*Q*
_{e}). With the condition *Q*
_{o} ≫ *Q*
_{e}, the FWHM will be rewritten as σ ≈ ω_{0} /*Q*
_{e}, So at a given resonant frequency, the line-width is determined by the value of *Q*
_{e}, which is dependent on the distance between the cavity and the waveguide to a great degree. In order to obtain the narrow FWHM, the distance between the cavity and waveguide should be chosen reasonably so that *Q*
_{e} is as high as possible.

Subsequently, the wavelength-selective reflection cavity *a* is applied to the three-port channel drop filter (shown in Fig. 2), where the reflection cavity is put on the side of the bus waveguide, and the channel drop waveguide perpendicular to it is put on the other side to avoid the direct coupling between the reflection cavity and the channel drop cavity. The two cavities possess mirror reflection symmetry with respect to their own reference planes that are their center planes, respectively, and the waveguides with them support single mode in the designed spectrum range. The amplitudes of the incoming waves into the system are denoted by *S*
_{+i}or *S*′_{+i} and *S*
_{-i} or *S*′_{-i} are the amplitudes for the outgoing waves (*i* = 1,2,3). The time evolution of the amplitudes of the cavity *a* and *b*, and the incoming and outgoing waves can be described as

where *ω*
_{0a} and *ω*
_{0b}, are the resonant frequencies of the cavity *a* and *b*, respectively, *Q*
_{oa} and *Q*
_{ob} are the quality factors due to intrinsic loss of the cavity *a* and *b*, respectively, *Q*
_{3} and *Q*
_{1} are the quality factors of the cavity *a* and *b* that are connected with the rates of decay into the bus waveguide, respectively, *Q*
_{2} is the quality factor of the cavity *b* that is related to the rate of decay into the drop waveguide, θ_{1} , θ_{2} are the phases of the coupling coefficients between the cavity *b* and the bus waveguide or drop waveguide, respectively, θ_{3} is the phase of coupling coefficient between the cavity *a* and the bus waveguide, β is the propagation constant of the bus waveguide, and *d* is the distance between the two reference planes.

When wave is launched only at the entrance to the bus waveguide (*S*
_{+2} =0, *S*
_{+3} = 0) and *S*
_{+1}has a *e*
^{Jωt} time dependence, by solving Eqs. (6) to (11), *S*′_{+1} can be written as

Supposing

and then substituting Eq. (15) into Eq. (7), the amplitude of the cavity *b* can be obtained at steady state

Replacing Eq. (18) in Eqs. (12)-(14), the filter response of the system is gotten

where *T*
_{1} is the transmission through the drop waveguide, *T*
_{2} is the transmission through the
bus waveguide, and *R*′ is the back reflection from the input port.

The above results by using coupled mode theory are discussed under some methods of confinement to derive the conditions to achieve 100% drop efficiency in the system. If *Q*
_{oa} ≫ *Q*
_{3}, *Q*
_{ob} ≫ *Q*
_{1} and *ω*
_{0a} = *ω*
_{0b} = *ω*
_{0}, substituting Eq. (16) into Eq. (19), the drop efficiency η can be expressed by

In the case, the filter response is dependent on Eq. (22). Note that the term *ω*
^{4} is highest in the numerator of Eq. (22), and the highest order term is *ω*
^{6} in the denominator, so the filter response is similar to a Lorentzian line shape.

At *ω* = *ω*
_{0a} = *ω*
_{0b} = *ω*
_{0}, the drop efficiency *η*
_{0} from Eq. (22) is expressed by

Assuming *Q*
_{1} /*Q*
_{2} = *k*, where *k* is a plus value, η_{0} can be rewritten as

It is verified that the drop efficiency attains 100% from Eq. (34) at *k* = 2 and ϕ = (2*n*+ 1)π if *Q*
_{oa} ≫ *Q*
_{3} and *Q*
_{ob} ≫ *Q*
_{1}, where *n* is the integer. At the resonant frequency, Fig. 3(a) shows the drop efficiency as a function of ϕ when *k* is equal to 1, 2 and 3, respectively, and Fig. 3(b) shows the curve of drop efficiency as a function of *k* at ϕ = (2*n* + 1)π . It is clear that the drop efficiency is less sensitive to the phase error near *π* and has the maximum efficiency for a rather wide range of *k* at *k* = 2, which possibly brings flexibility with respect to the design of the filter. At *k* = 2 and ϕ=2*n*π, the drop efficiency is zero, and the light is totally reflected back to the input port.

If *Q*
_{ob} ≫ *Q*
_{1} and *Q*
_{ob} ≫ *Q*
_{2}, we assume

and the drop efficiency η_{0} at *ω* = *ω*
_{0a} = *ω*
_{0b} = *ω*
_{0} from Eq. (19) can be expressed as,

At *Q*
_{1} /*Q*
_{2} = 2, η_{0} can be rewritten as

Fig. 3(c) shows the dependence of drop efficiency on the various value of *l*, where *l* is the reflection ratio of the cavity *a* at the resonant frequency. It is proved that the enhanced drop efficiency is less sensitivity to the reflection ratio of the reflection cavity *a* when the phase ϕ is around π, and the ultimate drop efficiency is always higher than the reflection ratio *l* at ϕ = (2*n* + 1)>π.

If *Q*
_{oa}≫*Q*
_{3}, assuming

under the conditions of ϕ = (2*n* + 1)π and *Q*
_{1} /*Q*
_{2} = 2, at ω = ω_{0a} = ω_{0b} = ω_{0}, from Eq. (19), the drop efficiency η0 can be rewritten as,

The value of *Q*
_{1}/*Q*
_{ob}is also an important parameter affecting the drop efficiency in the three-port system. As shown in Fig. 3(d), the drop efficiency as a function of *p* is plotted when the phase term, quality factor ratio *k* and 100% reflection feedback are well satisfied. It can be seen that a small value of *p* is needed to improve the drop efficiency. In a practical PC slab structure, the *Q*
_{o} factor of the point-defect cavity due to loss is mainly determined by the effective vertical quality factor *Q*
_{v}, which is related to radiation loss from the cavity to the vertical direction [23]. So *Q*
_{1}/*Q*
_{v} should be small in such a structure to enhance the drop efficiency.

The above analysis is on the basis of the two cavities with the same resonant frequency. In the practical fabrication process, it is inevitable that the detuning between the resonant frequencies of the two cavities appears. Its influence on the drop efficiency is discussed.

At *Q*
_{oa} ≫*Q*
_{3} and *ω* = *ω*
_{0a} , when the resonant frequency of the channel drop cavity co satisfies the term as follows,

from Eq. (19), the drop efficiency η_{0} can be expressed as,

where *k* is equal to *Q*
_{1}/*Q*
_{2}. Fig. 3(e) shows the curve of drop efficiency as a function of phase ϕ for *k* = 1, 2 and 3. At *k* = 2, the curve has a maximum value in the rather wide phase range around π, where close to 100% drop efficiency can be attained in the phase range between 0.8 π and 1.2π if the frequency term Eq. (40) is satisfied. The continuous phase range may be available to design the multi-channel drop filters conveniently while the phase term ϕ = (2*n* + 1)π is limited by the discrete number of lattice constant. If *k*<2, it is also possible to achieve 100% drop efficiency, where the phase ϕ is not equal to(2*n* + 1)π.

Under the phase term ϕ = (2*n* + 1)π and quality factor ratio *k* = 2, if *Q*
_{oa} ≫ *Q*
_{3} and *Q*
_{ob} ≫ *Q*
_{1}, supposing

at *ω* = *ω*
_{0a}, from Eq. (19), the drop efficiency η_{0} can be rewritten as,

From Eq. (44), it is clear that the drop efficiency is also affected by the value of *Q*
_{1} in the case. Fig. 3(f) plots the drop efficiency as a function of the frequency detuning coefficient *m* for *Q*
_{1} =10^{3},10^{4} and 10^{5}. As *Q*
_{1} is increased, the filter signal become narrow, and the drop
efficiency sharply reduces with the detuning between the two resonant frequencies varied. In order to achieve the high drop efficiency, the detuning needs to be reduced sufficiently in the high Q three-port system, and this implies that the precise fabrication technology is necessary.

## 3. Design and numerical calculations

Figure 4(a) shows a three-port filter based on 2D PCs with high dielectric rods in air. The photonic crystals are made up of a square lattice of high-index dielectric rods with a refractive index of 3.4 and a radius of 0.20*a*, where a is the lattice constant of the square array, and it possesses a band-gap only for the transverse magnetic (TM) mode that has its electric field parallel to the rods. The smaller rod that defines the defect of the channel drop cavity has a radius of 0.042*a*, and the two rods at the interface between the cavity and drop wave-guide or bus wave-guide have a radius of 0.211*a*. They all have the same dielectric constant as the background rods. The structure has been simulated using the FDTD method with PML absorbing boundary conditions to calculate the transmission performance.

Figure 4(b) shows the transmission characteristics of the three-port system in Fig. 4(a). The drop efficiency is only 42% at the resonant frequency. The cavity quality factors are calculated by the FDTD method. The intrinsic *Q* factor *Q*
_{o} is larger than 10^{6}, and the system *Q* factor *Q*
_{s} is calculated to be 1020 (The total computation domain is 30*a*×30*a* along the x, y directions in the plane, where *a* is the lattice constant.). So it is evident that *Q*
_{o} ≫*Q*
_{s} is satisfied. Using the coupled mode theory [18], *Q*
_{1} /*Q*
_{2} ≈ 2 is easily obtained, where *Q*
_{1}and *Q*
_{2} represent the quality factors due to the rates of decay into the bus waveguide or drop waveguide, respectively.

Based upon the theoretical analysis in section 2, a three-port channel drop filter with a wavelength-selective reflection cavity is designed (shown in Fig. 5(a)). The parameters of the channel drop cavity are same as that in Fig. 4(a). The wavelength-selective reflection cavity consists of the point defect cavity side-coupled to the bus waveguide. The smaller rod of the wavelength-selective reflection cavity has a radius of 0.042*a*, and the rod between the cavity and the bus waveguide has a radius of 0.23*a* to guarantee the same resonant frequency of the two cavities, where they all have a refractive index of 3.4. The cavity quality factor due to loss *Q*
_{o} is calculated to be larger than 10^{6}, and the total quality factor of the cavity is 2460. So the condition *Q*
_{o} ≫ *Q*
_{e} is satisfied, and the wavelength-selective reflection function is realized in the desired 2D PCs, where *Q*
_{e} is the quality factor that is related to the rate of decay into the bus waveguide. The distance between the two cavities is set to 5*a*, and ϕ = 2*βd* = 5π is satisfied at the resonant frequency, where the guided mode has a wave vector of 0.25(2*πa*
^{-1}).

Figure 5(b) shows the transmission characteristics of the three-port system in Fig.5 (a). The transmission spectra through port A, B and the back reflection spectrum from port C are shown as thin solid, dotted, and thick solid lines, respectively. The spectrum through port A is similar to a Lorentzian line shape with a maximum close to 100% at the resonant wavelength, and the transmission spectrum through port B is close to 100% over the entire spectrum, except at resonance, where it drops to zero. The back reflection at port C is barely observable over entire frequency range. The steady field pattern is obtained by launching a continuous-wave (CW) at resonant wavelength (shown in Fig. 5(c)). FDTD simulations with PML absorbing boundary condition calculate the results. The simulation does indeed verify complete channel drop tunneling via the three-port system. For a lattice constant of 573nm, the selecting wavelength is 1552.05nm and the FWHM of the filter is around 1.51nm corresponding to a *Q* factor of 1020. It is clear that the realization of extremely high- *Q* cavities is necessary in the filter for DWDM system.

If the distance between the two cavities is 3*a* and 7*a*, respectively, the numerical results are also in very good agreement with the theory analysis perfectly due to the satisfaction of the phase condition. At *d* = *a*, the results do not match the theoretical analysis very well although the condition ϕ = π is satisfied. The reason is that the strong coupling between the two cavities via the bus waveguide occurs, which is on account of so little distance between them, and it is unsuitable to apply the theory analysis in section 2 to the case. To get a compact filter, we choose the distance as 3a or 5a in the example.

## 4. Conclusion

In this paper, a three-port channel drop filter with a wavelength-selective cavity is proposed. The wavelength-selective cavity consists of a cavity side-coupled to a waveguide. Using coupled mode theory in time, the system is analyzed in detail to achieve 100% drop efficiency at resonance. Based on the theoretical analysis, a three-port structure with a wavelength-selective cavity is engineered in 2D PCs, and close to 100% drop efficiency is numerically calculated by the FDTD method.

In the structure, the functions of two cavities are independent, and the direct coupling between the cavities is not required. The theoretic analysis imply that such a structure maybe bring flexibility with respect to the device design. The distance between the cavities is also easily decided by the phase condition, which is convenient to design a multi-channel drop filter. In the practical filter, it is necessary to consider vertical confinement of photons, such as photonic crystal slabs, in which the loss in the vertical direction needs to be taken into account. In this situation, the design of 2D PCs-based filter needs to be extended, and a further research is necessary.

## Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant No.60377023) and New Century Excellent Talents of Universities (NCET), Shanghai Optical Science and Technology Project.

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