## Abstract

A two-dimensional photonic crystal channel-drop filter is proposed. This device has two high group velocity waveguides that are selectively coupled by a single, low group velocity intermediate waveguide section. It exhibits computed quality factors as high as 1300, and directional dropping efficiencies as high as 90%.

©2005 Optical Society of America

## 1. Introduction

Channel-drop filters are key components for a wide range of wavelength division multiplexing (WDM) systems: in the context of photonic integration, these components are required to provide adequate means to extract wavelengths from guiding sections, with appropriate spectral and spatial resolutions. In the later property, it is expected that the channel-drop function can be achieved with the desired directionality. Compactness is also a highly desirable feature in the prospect of large-scale photonic integration.

The general scheme, which is applied to achieve channel-drop function, is based on the employment of resonators. A compact ring resonator between two optical waveguides has provided an ideal basic structure which can meet these requirements in principle [1, 2]. However, it is recognized that the performance of this type of resonator is affected by surface roughness.

Other types of resonant structures which would not be as sensitive to surface roughness have been proposed, especially based on localized defect state micro-cavities built in a Photonic Crystal (PC) [3, 4]. Wavelength selective trapping (from a waveguide) and emission (in frees-pace) of photons by a single defect in a photonic band gap structure has been demonstrated by Asano *et al*. [5]; more recently, the same group has demonstrated selective dropping between two waveguides [6]. However, as argued by Fan *et al*. [3], a single mode micro-cavity cannot, intrinsically, result in a full dropping of the incoming wavelength with the desired directionality: the maximum dropping ratio attainable is 50%, without any preferred directionality. These authors have shown that the localized state resonant structure should indeed meet very well defined conditions to achieve the desired transfer characteristics: first, it should have two degenerate modes that possess appropriate symmetries with respect to its symmetry plane functions; second, the coupling rates of these modes to the add and drop ports should be properly balanced. These conditions (which are “naturally” met in a micro-ring resonator, provided that its size exceeds significantly the operational wavelength scale) could be achieved, in principle, by using two mutually coupled single-mode localized state micro-cavity resonators. It turns however that these requirements are still beyond the current achievable technological capabilities.

Another approach has been proposed in the literature [7, 8, 9], which aims at meeting “naturally” the above exposed requirements, via the use of slow Bloch modes.

In this paper, we discuss in detail the physical principles of this approach. In section 2, we analyze the physical concepts used to achieve highly directive and wavelength selective channel drop filtering and based on the resonant coupling between “fast” and “slow” waveguided Bloch modes. In section 3, these principles are quantitatively illustrated on two dimensional (2D) PC structures, using 2D Finite Difference Time Domain (FDTD).

## 2. Theoretical concepts and analysis

The general scheme of the directional channel-drop filter based on the use of a slow Bloch mode in a resonator is shown in Fig. 1. A schematic view of the dispersion characteristics of the input/drop waveguides and of the resonator is also shown in Fig. 2.

The input and drop waveguides are identical and accommodate “fast” propagating waveguided Bloch modes (FBM), with high group velocities, as shown in Fig. 2.

The resonator consists of an intermediate waveguide section, which accommodates “slow” Bloch modes (SBM) (with a low group velocity). Indeed, as previously demonstrated by several authors [10, 11], these extremes are present in the dispersion characteristics of 2D PC “infinite length” waveguides (the concept of “infinite length” is analyzed in section 2.2.1). These extremes take place at the anti-crossing of two different and contra-propagating waveguided modes, as a result of the diffractive coupling induced by the periodic corrugation of the PC, thus giving rise to a so-called “mini-stop-band” (see Fig. 2). The extremes are formed in the dispersion characteristics at the mini-stop-band edges, which results in low group velocity Bloch modes. It can be noticed that the resulting modes are eigenstates of the “infinite length” waveguide.

#### 2.1. Wavelength selection and selectivity

For a non limited intermediate waveguide section, the wavelength selection is achieved at the intercept of the two dispersion characteristics, where the coupling between the SBM of the intermediate waveguide and the FBM of input/drop waveguides can be achieved preferentially, due to the good matching between the propagation vectors. The dispersion characteristics show an anti-crossing bandgap (ACB), resulting from this coupling, and whose width is defined by the coupling rate between these modes. Let us point out that this coupling makes sense as far as the SBM is given birth in a time far lower than the coupling time between the FBM and the SBM. In other words, the ACB has to be much narrower than the “mini-stop-band” (see Fig. 2).

The selectivity is directly controlled by the lifetime *τ* of the mode in the intermediate waveguide, that is the time during which the mode can be confined within the waveguide. τ is also related to the quality factor *Q* of the intermediate waveguide, through the equation:

where *λ* is the operation wavelength, w the related pulsation, *c* the free-space light velocity and *δλ* the line-width of the resonance. In the absence of any other optical losses, the only factor controlling the lifetime of the mode of the intermediate waveguide is the coupling rate 1/*τ _{c}* with the FBM of the input/drop waveguides. Using Eq. (1), it comes immediately that:

with

$\tau =\frac{{\tau}_{c}}{2}$

in the case of a symmetrical structure where the coupling rates of the intermediate waveguide with the input waveguide and the drop waveguide are equally balanced.

*δλ* can be viewed as the spectral width or spectral window where the dropping process may occur. This spectral window coincides with that formed by the ACB at the anti-crossing, as described above.

#### 2.2. Directionality

The kinetics of SBM around an extreme of the dispersion characteristic has been analyzed in [12]. It was shown that a phenomenological description of this kinetics can be inferred from the dispersion characteristics. In particular, it was demonstrated that over a given time duration τ, a SBM extend over a lateral distance *L _{m}*:

where *α* is the second derivative of the dispersion characteristic at the relevant extreme (see Annex 1).

This means that, if the lifetime of the SBM is limited to *τ* (as a result of any loss mechanisms), its maximum extension will be limited to *L _{m}*. Having these considerations in mind, we analyse below the impact of the length

*L*of the intermediate waveguide on the conditions required for directional dropping.

### 2.2.1. Impact of the length of the intermediate resonator

In practice, the length *L* of the intermediate waveguide section is limited, and should be possibly made as small as possible for the sake of compactness.

Therefore, two situations can be envisaged, depending upon whether *L* is larger or longer than the longitudinal extension of the resonator’s SBM *L _{m}*, during its life time

*τ=τ*/2:

_{c}• *L* is smaller than *L _{m}*: due to the reflectivity at the boundaries, Fabry-Perot interferences take place in the intermediate section, leading to Fabry-Perot cavity modes.

• *L* is larger than *L _{m}*: during its lifetime, the slow Bloch mode will not have time to explore the boundaries of the intermediate section, which then can be viewed as “infinitely long”; in other words, it is impossible to observe any coherent Fabry-Perot cavity modes (or in an alternative way to express this concept: many successive Fabry-Perot cavity modes lie within the line-width

*δλ*of the resonance). This regime lends itself to directional dropping, but at the expense of compactness.

We therefore choose to concentrate on short (*L*<*L _{m}*) resonators.

The wave vector of the SBM is now discretized and its components fulfill the condition for constructive and coherent Fabry-Perot interferences:

where *p* is an integer and *L _{eff}* is the effective length of the resonator, which depends on the considered mode and is generally different from the physical length due to the phase shift occurring when the photons are reflected on the edges of the resonator. The successive Fabry-Perot cavity resonance wavelengths corresponding to the successive cavity

*k*-vectors are derived from the dispersion characteristic.

Around the extreme, the Free Spectral Range (FSR) between two successive cavity SBM is approximately [13] equal to *α*(*δk*)^{2}/2, where a is the curvature at the extreme and *δk=π/L _{eff}*, is the

*k*-vector difference between two successive cavity modes; therefore:

The key parameter controlling the FSR is the curvature at the extreme, which can be made very low in photonic crystal structures. As a matter of fact, the strong coupling between optical modes induced by the strong periodic corrugation, in addition to opening up large photonic bandgaps, results in photonic band edge extremes with low curvature. The low FSR eases the condition that two or more successive cavity SBM, with the appropriate even and odd symmetry for directional transfer, can be accommodated in the spectral bandwidth of the device (which corresponds to the spectral window opened up by the ACB). Also, for a given curvature, the FSR can be significantly reduced by increasing *L _{eff}*.

### 2.2.2. Directional dropping conditions for short resonators

We remind that full forward transfer of the signal can be achieved if the intermediate resonator accommodates two degenerate modes, having the same symmetry in the direction parallel to the waveguides, and opposite symmetries in the orthogonal direction [3].

These conditions may be fulfilled, both in terms of degeneracy and symmetry, in a Fabry-Perot Cavity operating close to an extreme of the dispersion characteristic, as can be seen in Fig. 3. Indeed, for appropriate *L _{eff}*, two cavity SBM may be degenerate and have opposite symmetries with respect to the medium axis orthogonal to the resonator, since the difference between their wave-vectors is (2

*q*+1)

*π/L*.

_{eff}Note that quasi (and not strict)-degeneracy is requested, which means that the two modes have to lie within the spectral bandwidth of the device (which corresponds to the spectral selective window opened up by the ACB). It is clear that quasi-degeneracy is eased for flat low curvature extremes, which are naturally present in the dispersion characteristics of photonic crystal structures, as explained in section 2.2.1. The directionality requires also a good balance between the coupling coefficients of the FBM with the two degenerate cavity modes: these coupling coefficients depend on the transversal profile of the degenerate modes, which are different in principle, but tends to be identical closer to the extreme, that is for lower curvature and larger resonator length. Note, finally, that the extreme of the dispersion characteristic which is chosen for directional transfer should not coincide with a high symmetry point of the 2D PC, where the two degenerate modes would necessarily exhibit the same symmetry.

## 3. Numerical study of illustrating devices

The theoretical analysis presented in the section 2 was meant to be phenomenological and to provide the physical concepts and understanding which govern the process of a directional, selective channel dropping, using a SBM confined inside an intermediate resonator.

We will now proceed further with the presentation of numerical results obtained on selected devices. We have combined band diagram calculations, based upon Bloch mode expansion in a plane-wave basis [14], and 2D FDTD simulations to design a directional channel-drop filter.

#### 3.1. Device description and computation parameters

A view of the structure is shown in Fig. 4 : it contains input and drop waveguides, which are created by one missing row of holes in the 2D triangular PCs and called W1. The PC ports are connected to the “outside world” by ridge waveguides. In the following, light is assumed to be injected in Port #1 of the input waveguide and is expected to be dropped in Port #4 of the drop waveguide. Direct forwarding and back-dropping can be detected in Ports #2 and #3 respectively.

The analysis is restricted to TE waves (using the same convention as in [14]), without loss of generality. The effective refractive index approximation was applied : the effective refractive index of the slab (without the PC) was assumed to be 3.05, corresponding to a 300 nm thick slab of silicon in silica for an operation wavelength around 1.55 *µ*m. The hole filling factor is 0.4, except along the waveguide and resonator lateral edges as explained later.

#### 3.2. Dispersion Characteristics

Figure 5 shows the band diagram for the channel-drop structure “along” the GK direction, for an “infinite length” structure. In order to adjust the frequency of the coupling between the resonator and the fast waveguides, which are identical, the primitive cell includes the input and drop waveguides and the intermediate waveguide section, as can be seen in the inset of Fig. 5. The usual normalization is applied to the coordinates of the band diagram. Λ/λ stands for the normalized frequency and the normalized wave vector is *k*Λ/2*π*, where *k* is the component of the propagation wave vector along the Γ*K* direction. L is the lattice constant of the 2D PC.

The two modes involved in the directional channel dropping process are shown in Fig. 5. These are an even FBM in the input/drop waveguides, and an odd SBM in the intermediate waveguide section. The dotted curve represents the dispersion characteristic of the even Bloch mode of the intermediate waveguide section, which is not used in the process.

It was previously shown by several authors [10, 11] that even and odd Bloch modes in a Wn PC waveguide are generated by the coupling of guided modes in an equivalent ridge waveguide through the periodic structure. In a W1 waveguide, according to reference [11], these guided modes can be denoted as e1, e2, e3 and e4. An odd Bloch mode is created by the coupling of e2 and e4, while an even Bloch mode is created by the coupling of e1 and e3. Moreover, the coupling of e2 and e4 opens a band gap (so called “mini-stop-band” in Fig. 2) whose the lower edge corresponds to the extreme of the odd Bloch mode, which is used here.

As previously explained, the most efficient directional, selective dropping occurs around the “intercept” of the dispersion characteristic of the even FBM with the odd SBM of the intermediate waveguide section used as a Fabry-Perot resonator, close to the extreme point of the dispersion characteristic. Thus, the hole filling factors, which is 0.4 in the lattice, has been adjusted to 0.6 along the input/drop waveguides and to 0.3 for the two closer rows around the resonator, in order to set this “intercept” close to the extreme of the dispersion characteristic of the SBM. In a real device with coupled waveguides, this “intercept” is forbidden and an ACB is created, as previously argued in section 2 (see the band diagram in Fig. 2).

Using this band diagram, the lattice constant L of the 2D PC was set at 420 nm for a wavelength dropping range around 1.55 *µ*m.

The determination of the curvature a at the relevant extreme of the dispersion characteristic gives *α*=50-60 m^{2}s^{-1}. Besides, the bandwith of the selectivity of the dropping structure, as derived from the spectral width of the ACB, is around 1 nm, yielding to *τ*=1·10^{-12} s, according to equation 2. Then, using equation 3, the longitudinal extension of the mode is the range 10-20 µm. As discussed in section 2.2.1, we choose to concentrate this contribution on compact structures, which implies the choice of resonator lengths *L* significantly smaller than 25 *µ*m, and to operate in the regime of “short length” resonator (that is for *L*<*L _{m}*, as opposed to the regime of “infinitely long” resonator which applies when

*L*>

*L*).

_{m}#### 3.3. FDTD simulations

### 3.3.1. Case of a short resonator with non degenerate modes

Figure 6 shows FDTD simulation of the resonance wavelengths (resulting from the SBM Fabry-Perot interferences) around 1.55 *µ*m, for a short, 3.54 *µ*m long, isolated resonator (8 missing holes). Two odd modes with different symmetries in the orthogonal direction are allowed in the resonator, but they are 5.6 nm apart. Since the spectral separation of these two modes is larger than the width of the ACB, which is around 1 nm, they would be unable to ensure an efficient and directional dropping.

### 3.3.2. Optimization of the mode degeneracy

If this spectral separation between the modes is decreased and becomes lower than the width of the ACB, a directional and selective dropping is expected. As explained in section 2.2.1, a simple way to reduce the FSR between the modes consists in increasing the length of the resonator. Figure 7 shows the FDTD simulation of the transfer spectral characteristics of the complete device, for a 4.5 µm long resonator (10 missing holes). Also the holes at the ends of the resonator have been shifted by 0.14Λ, in order to tentatively adjust finely the effective length of the resonator to promote degeneracy of the two modes (phase adjustment of the reflectance at the ends of the wave-guiding resonator section). The outgoing signals, i.e. dropped (D), back-coupled (BC), direct transmission (DT), reflected (R) are normalized with respect to the whole in-going power; the device is assumed to be free of intrinsic losses, resulting, for example, from unwanted coupling with radiated modes. At the resonant wavelength λ=1.5858 µm, D=75%, DT=10%, BC=5%, R=10%. The spectral linewidth (full width at half maximum) is about 0.7 nm, providing a Q of about 2300.

The selective dropping is not fully directional, due to the non complete degeneracy of the modes. Unfortunately, these results could not be improved by further refining the hole position at the ends of the resonator, due to the limited spatial resolution of the FDTD. This indicates that this tuning scheme is rather critical in terms of technological constraints, similarly to the case of directional dropping filters recently proposed in the literature (see the introduction and reference [3]) based on the use of coupled cavities, where a fine tuning of the coupling strength is requested.

The degeneracy may be promoted if the device is made to operate closer to the extreme of the dispersion characteristic: this can be achieved either by reducing the curvature at the extreme of the characteristic, or by further increasing the length of the resonator. This last solution is again chosen hereafter.

Figure 8 shows the FDTD computation results concerning the complete device, with a 8.58 *µ*m long resonator (20 missing holes), without any shift of the holes at the ends of the resonator. At the resonant wavelength λ=1.5769 µm, D=91%, DT=8.5%, BC=0.25%, R=0.25%. The spectral linewidth is about 1.2 nm, providing a *Q* of about 1300. This smaller *Q* is due to the stronger coupling between the resonator modes and the input/drop waveguides modes, since their wave vector are closer to each other than in the former case.

Figure 9 shows the magnetic field distribution at resonance: besides the fact that some amount of the power keeps on travelling forward, most of the power is effectively dropped. Although relatively compact, this resonator owns, to a large extent, the periodic topology of a true periodic infinite length structure: it results that the fabrication of the device is certainly robust against technological imperfections, the detrimental effect of local deviations being minimized by the averaging over the number of periods (20 in this example).

## 4. Conclusion

An original device for directional and selective transfer of wavelengths between two waveguides has been presented in this work. It is based on a Fabry-Perot cavity resonator, which operates close to an extreme of the dispersion characteristics, where two quasi degenerate slow Bloch modes are used for the resonant transfer of photons. A directional dropping efficiency of 91% is determined using 2D FDTD simulations, with a Q factor of 1300. The device is compact and its fabrication should be robust against technological imperfections. Devices, based on a silicon membrane suspended in air, are under realization. Under these conditions, device operation below the light line is easily achievable: this makes reasonable the assumption made in this contribution of negligible intrinsic losses (negligible coupling of waveguided modes with radiated modes). The issue of unwanted optical losses occurring at the edges of the resonator has been addressed recently by [15, 16]. These authors have shown that these losses can be controlled by carefully designing the edges of the resonator.

The concept proposed in this work can also be applied to other wavelength transfer schemes where, for example, the incoming photons would be guided in a PC waveguide and emitted into free 3D space, as proposed by Noda and co-workers [5, 16], who used single localized defect state resonators : use of slow degenerate Bloch modes as analyzed in the present work, would insure a full removal and dropping of the photons at resonance, with, in addition, an improved directionality of the emission diagram, unlike solutions based on localized defect state resonators. This aspect is under development in our group.

## Acknowledgments

This work was supported by the Ministère Délégué à la Recherche in the frame of the Lamb-daconnect program. E. Drouard is grateful to the Network of Excellence ePIXnet for his postdoctoral fellowship.

Annex 1

Around the extreme (*ω*
_{0}, *k*
_{0}), the dispersion characteristics (*ω, k*) can be expanded as :

$\omega ={{\omega}_{0}+\frac{1}{2}\alpha (k-{k}_{0})}^{2}$,

where *α* is the second derivative or the curvature (parabolic approximation). The limited lifetime *τ* of the optical mode results in a spectral broadening:

$\delta \omega =\frac{1}{\tau}$,

which relates to the extension of momentum *δk* in k-space through:

$\delta \omega =\frac{1}{\tau}=\frac{\alpha}{2}{\left(\delta k\right)}^{2}$.

from where the corresponding longitudinal extension of the optical mode in real space can be derived:

${L}_{m}=\frac{1}{\delta k}={\left(\frac{\alpha \tau}{2}\right)}^{1\u20442}$.

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