## Abstract

The objective of the present investigation is to demonstrate the possibility of designing compact ultra-narrow band-pass filters based on the phenomenon of non-proximity resonant tunneling in multi-core photonic band gap fibers (PBGFs). The proposed PBGF consists of three identical air-cores separated by two defected air-holes which act as highly-selective resonators. With a fine adjustment of the design parameters associated with the resonant-air-holes, phase matching at two distinct wavelengths can be achieved, thus enabling very narrow-band resonant directional coupling between the input and the two output cores. The validation of the proposed design is ensured with an accurate PBGF analysis based on finite element modal and beam propagation algorithms. Typical characteristics of the proposed device over a single polarization are: reasonable short coupling length of 2.7 mm, dual bandpass transmission response at wavelengths of 1.339 and 1.357 µm, with corresponding full width at half maximum bandwidths of 1.2 nm and 1.1 nm respectively, and a relatively high transmission of 95% at the exact resonance wavelengths. The proposed ultra-narrow band-pass filter can be employed in various applications such as all-fiber bandpass/bandstop filtering and resonant sensors.

©2006 Optical Society of America

## 1. Introduction

In the last decade, photonic crystal fibers (PCFs) [1], also known as microstructured optical fibers (MOFs) or holey fibers have witnessed an unpredictable attention due to the fact that they can provide unprecedented degrees of freedom in engineering their modal characteristics. Although PCFs are usually formed by a central defect region surrounded by multiple air-holes arranged in a regular triangular lattice, recent advancements in manufacturing technology of PCFs such as multiple-capillary drawing method [2] can readily realize multi-core PCFs [3], [4].

An important element in all-optical fiber communication systems and all-optical fiber measurements is apparently a wavelength-selective fiber device such as a fiber filter. A number of fabrication techniques have been used so far for realizing such devices [5]–[7]. The operational principle of conventional fiber filters typically involves in transferring energy over a coupling length between two distinct fiber cores coupled by proximity interaction. However in this case modes in closely separated individual cores are phase-matched at all wavelengths, thus making it difficult to engineer band pass filtering characteristics. It was recently shown that efficient band pass filters can be realized based on the resonant tunneling phenomenon [8] in multi-core PCFs [9].

One of the major trends in the development of all-fiber devices is the increasing number of functionalities in a single fiber. The ultimate goal is to be able to fabricate in a single draw a complete all-fiber component provisioned on a preform level. Some of the advantages of all-fiber devices are: simplified packaging, absence of sub-component splicing losses, environmental stability due to the absence of free space optics. While the benefits of integrated all-fiber devices are significant enough to encourage development of increasingly complex components, the major roadblock to their realization is an unavoidable complexity of the required transverse refractive index profile. These challenges can be in a certain degree met by using a novel class of microstructured optical fiber coupler that was recently introduced in Ref. [10], which operates by resonant rather than proximity coupling, where energy transfer is realized via transverse lightguides integrated into the fiber’s cross-section. Such a design allows unlimited spatial separation between interacting fibers which in turn, eliminates inter-core crosstalk via proximity coupling. Controllable energy transfer between fiber cores is then achieved on highly directional transmission through transverse lightguides or resonators. The main advantage of this coupling mechanism is its inherent scalability as additional fiber cores could be integrated into the existing fiber cross-section simply by placing them far enough from the existing circuitry to avoid proximity crosstalk, and then making the necessary inter-core connections with transverse light “wires” in a direct analogy to the “on chip electronics integration”.

Based on the above mentioned benefits that the resonant coupling mechanism can introduce, we devote the present paper to describe a novel design approach for realizing resonant bandpass filters in multi-core photonic band gap fibers (MC-PBGFs) based on the highly-selective resonant tunneling mechanism. The MC-PBGF consists of three identical air-cores separated by two defected air-holes (resonators). By adjusting the sizes of the resonant air-holes, phase-matching at two distinct wavelengths can be achieved between the input and output cores, enabling highly-selective narrowband resonant directional coupling. Although other mature technologies such as fiber Bragg gating (FBG) with circulator have been successfully used to realize narrow-band filters, perhaps one of the appealing properties of the technology of multi-core PBGFs is the exhibition of temperature and strain insensitivity in comparison to FBGs. Through an efficient modal [11] and beam propagation analysis [12], based on the finite element method (FEM), we theoretically investigate the possibility of synthesizing efficient ultra narrow-band splitters, suitable for filtering applications.

The composition of the present investigation will be as follows: in Section 2 we introduce the device concept and we give exact design guidelines for achieving the narrow bandpass filtering characteristics. Then in Section 3 we validate our design’s performance by showing various numerical simulations based on FEM numerical algorithms. In Section 4 we briefly address the possibility of realizing a polarization-independent splitter operating at a single wavelength, based on the prescribed MC-PBGF technology. A final conclusion will follow in Section 5 with some suggestions for future investigations.

## 2. Schematic representation and design guidelines for engineering MC-PBGF splitters

The cross-section of the structure under investigation is shown in Fig. 1. The hollow cores are formed in a silica-based MOF with a cladding refractive index *n*=1.45, by removing two rows of tubes and smoothing the resulting core edges. The pitch constant is chosen to be Λ=2 µm, while the air-hole diameters in the cladding of the fiber is *d*/Λ=0.9, with a total of six hole layers in the cladding. Fundamental band gap where the core guided modes are found, extends between 1.29 µm < λ < 1.40 µm [13]. The formation of a splitter, operating at two distinct wavelengths, can be achieved by placing three hollow cores of *N*=8 periods apart from each other, counting from the center of each core (along the *x* axis) as shown in Fig. 1. Two dissimilar transverse resonators with *d*
_{1}/Λ and *d*
_{2}/Λ are then introduced by reducing (high index defects) the diameters of the air-holes in the middle of the line joining the cores. By an accurate modal analysis performed using an accurate FEM solver [11], in Fig. 2(a) we evaluate the effective indexes of the *x*-polarized (horizontally polarized) fundamental (blue solid curve), as well as the x-polarized excited odd resonant modes (red dashed curves), as a function of the operating wavelength and for several incremental values of the resonator’s normalized diameter *d*_{r}
/Λ, ranged from 0.6 to 0.8. The reason for showing the odd excited modes in the graph is the fact that only these modes can be coupled to the fundamental *x*-polarized mode, while the even excited modes can not be coupled at all. Note that the computation of the effective refractive index of the fundamental mode was performed assuming the core to be isolated; while the resonant excited modes have been calculated assuming these resonators isolated from the cores. This approximation was confirmed to give accurate results when comparing with the results associated with the coupled system’s performance (cores plus resonators). We can clearly see that the effective index of the fundamental mode is being crossed at certain wavelengths by the excited resonant modes. The physical interpretation of this crossing is that the excited modes at wavelengths of λ_{1}, λ_{2}, λ_{3}, λ_{n}, corresponding to different normalized resonator’s diameters *d*_{r}
/Λ, can be effectively transferred via resonant coupling through the dissimilar resonators, from the central input core-A into the output cores-B and C. This simply means that at a given normalized resonator’s diameter *d*_{r}
/Λ, there exists a wavelength where resonant tunneling can be achieved between the input (core-A) and output (core B or C) through the resonator. To identify the evolution of the resonance wavelengths as the resonator’s diameter changes, in Fig. 2(b) we plot the resonance wavelength as a function of the resonator’s normalized diameter *d*_{r}
/Λ. From the results in Fig. 2(b) we can observe that as soon as the resonator’s diameter decreases the resonance wavelength increases. Due to the fact that the resonance wavelength must lie within the photonic band gap (PBG) region (shown by the grey boundaries), the possible values of the resonator’s diameters are in the range: 0.62<*d*_{r}
/Λ<0.8, for the *x*-polarized state. The same calculations are repeated in Figs. 3(a) and (b) for the *y*-polarized (vertically polarized) fundamental mode and for the y-polarized excited even resonant modes. In the second case of y-polarization, where the odd-type excited *y*-polarized modes can not be coupled to the fundamental *y*-polarized resonant mode, we can observe a significant difference in the behavior of the evolution curve of the resonance wavelengths. By comparing the results in Figs. 2(b) and 3(b) we can see a great difference between the two polarizations. From Fig. 3(b) it is evident that the range of allowable values of the normalized resonator’s diameters for the *y*-polarized mode which will result in resonance wavelengths within the PBG of the structure is significantly larger: 0<*d*_{r}
/Λ<0.86. In addition a very interesting phenomenon occurs. This phenomenon is the insensitivity of the resonance wavelength for the following range of the normalized resonator’s diameters 0.1<dr/Λ<0.4, since at this range the resonance wavelength appears almost constant at a value of λ_{res}=1.372 µm. The physical explanation for this drastic difference among the two polarization states can be given in terms of the feature of each polarization. This means that while the *y*-polarized resonant state has an even-type profile, the *x*-polarized resonant state significantly differs because its profile is anti-symmetric (odd), as will be demonstrated qualitatively later on. Therefore as a conclusion of this section we have described the basic principle of operation for this type of multi-core PBGF splitter and we have identified the impact of each polarization state to the resonance wavelength of the coupled system consisting of the input core and the resonator. In addition we have identified the allowable range that the resonators can give a resonance wavelength within the PBG capabilities of the structure.

## 3. Numerical results and device performance

After having explained the basic operational principle of the device under consideration, we proceed by investigating the spectral as well as its propagation characteristics. The operational principle of this band pass filter can be alternatively understood in terms of the supermodes of the “three-core” directional splitter (that is the system composed of two air-cores and one resonator). If the individual cores of the splitter are single-moded, the coupler structure supports three supermodes, two symmetric and one anti-symmetric (for *y*-polarization), and two anti-symmetric and one symmetric (for *x*-polarization), with corresponding fields defined as ϕ_{1}, ϕ_{2}, ϕ_{3} and qualitatively shown in Fig. 4. Note that the shape of the supermodes considered here are the envelopes of the supermodes, calculated by the modal analysis of the complete system. This was done only for qualitative purposes and has no particular meaning.

Let *n*
_{eff,1}, *n*
_{eff,2}, and *n*
_{eff,3} represent the effective refractive indices of the supermodes corresponding to the fields ϕ_{1}, ϕ_{2}, ϕ_{3}, respectively, for each of the polarization states. Then assuming that initially all the energy is in the input core-A, this will correspond to the excitation of a supermode combination of the following type:

After propagation over a distance-*z*, this excitation pattern will evolve into:

where β_{i}=2π*n*
_{eff,i}/λ_{0} (*i*=1, 2, 3) and λ_{0} is operating wavelength. If we design the PBGF so that the effective refractive indexes of its supermodes satisfy the condition:

or equivalently,

complete power transfer from core A to core B or C can be achieved at the exact resonant wavelength λ_{0} by choosing mode propagation length *z=L*_{c}
with

where ${n}_{\mathit{\text{eff}}\mathit{,}}^{x\mathit{,}y}$
_{1} and ${n}_{\mathit{\text{eff}}}^{x\mathit{,}y}$
_{,3} denote the effective indexes of the supermodes, for *x*-polarization or *y*-polarization respectively. Assuming the following geometrical parameters of the PBGF for the *y*-polarized state: Λ=2 µm, *d*/Λ=0.9, *d*
_{1}/Λ=0.7 and *d*
_{2}/Λ=0.6, from the results in Fig. 3(a) we can clearly see that the excited resonant modes cross the effective index curve of the fundamental mode, at two distinct wavelengths of λ_{1}, *y*=1.339 µm and λ_{2, y}=1.357 µm. In case of *x*-polarization, by choosing the resonators sizes to be *d*
_{1}/Λ=0.74 and *d*
_{2}/Λ=0.70, the resonant wavelengths will be different: λ_{1,x}=1.331 µm and λ_{2,x}=1.356 µm. Using the relation in Eq. (5) combined with an accurate modal solver based on the FEM computational algorithm, in Fig. 5 we plot the calculated coupling lengths as a function of the normalized resonator’s diameter *d*_{r}
/Λ for (a) *x*-polarization and (b) *y*-polarization, respectively. From the results in Fig. 5 we can at first observe a significant difference of the coupling lengths evolution for the two different polarization states. Particularly from Fig. 5(a) we can see that the coupling length changes linearly as the resonator’s diameter increases, while a different rate of change occurs in Fig. 5(b), where we can clearly see the asymptotic behavior of the coupling length as the resonator’s diameter approaches lower values. The partial coupling lengths in this case are: ${L}_{c}^{x}$
(λ_{1}=1.331 µm)=7.3 mm, ${L}_{c}^{x}$
(λ_{2}=1.356 µm)=6.4 mm, ${L}_{c}^{y}$
(λ_{1}=1.339 µm)=2.9 mm, and ${L}_{c}^{y}$
(λ_{2}=1.357 µm)=2.6 mm. In Fig. 6 we plot the obtained spectral characteristics of this novel type of PBGF splitter with total fiber length of 2.7 mm, for the y-polarized state, by using an accurate analysis based on the BPM algorithm [12]. From these results we can observe a dual band-pass transmission response centered at the prescribed wavelengths of λ_{1, y}=1.339 µm and λ_{2, y}=1.357 µm. The highly selectivity in the filter’s response we could obtain in this case, indicates the potential capability of the non-proximity resonator’s states to synthesize highly selective resonant coupling characteristics. The full width at half maximum (FWHM) bandwidths of this filter are 1.2 nm and 1.1 nm for the y-polarized state at wavelengths of λ1, y=1.339 µm and λ2, y=1.357 µm, respectively, while for the *x*-polarized state the FWHM bandwidths were found to be a bit smaller. In both cases a transmission of about 95 % at the resonant wavelengths of λ_{1} and λ_{2} could be achieved. The difference in the values of the FWHM bandwidths for the two different polarization states, is associated with the larger coupling length of the *x*-polarization in comparison to the *y*-polarization (see Fig. 5), a fact which in general will result in a weaker coupling between the input core-A and the output core-B or C for the *x*-polarization, thus resulting in a lower FWHM.

To visualize the power splitting mechanism in our proposed PBGF splitter, in Fig. 7 we plot the coupling characteristics of the field distribution at different propagation distances, obtained by using a BPM [12]. Specifically Fig. 7 shows the snapshots of the electric field distribution, that is y-polarized mode (*E*_{y}
), for (a) λ_{1, y}=1.339 µm, (b) λ_{2, y}=1.357 µm, calculated at a distance of *z*=0 mm, (c) λ_{1, y}=1.339 µm (d) λ_{2, y}=1.357 µm, at a distance of *z*=1.0 mm, (e) λ_{1, y}=1.339 µm, and (f) λ*2, *_{y}
=1.357 µm, at a distance of *z*=1.5 mm, (g) λ_{1, y}=1.339 µm, and (h) λ_{2, y}=1.357 µm, at a distance of *z*=2.0 mm, and finally for (i) λ_{1, y}=1.339 µm, (j) λ_{2, y}=1.357 µm, calculated at the coupling length of *z*=*L*_{c}
=2.7 mm. We can clearly observe that at the coupling length of *L*_{c}
=2.7 mm the two different wavelengths were splitted in the output cores B and C within a power decrement of 5 % from the targeted level of 100 %, associated with the slightly difference between the partial coupling lengths corresponding to the two different operating wavelengths.

## 4. Realization of polarization-insensitive PBGF splitters operating at a single wavelength

In the previous sections we have devoted our efforts to design PBGF splitters operating at two distinct wavelengths and for two different polarization states. One of the main conclusions was that the selection of polarization is very important in terms of the device propagation characteristics. Recently there has been much effort in realizing polarization-independent splitters based on the PCF technology [14], [15]. In this section we will show that by using the prescribed multi-core PBGF topology with an appropriate selection of the design parameters, polarization-independent propagation characteristics can be achieved at a single operational wavelength. We refer to the results obtained in Figs. 2(b) and 3(b), and we combine these results to generate Fig. 8, for the structure shown in Fig. 9, where we can see the evolution of the resonance wavelength curves as a function of the resonator’s normalized diameter *d*_{r}
/Λ, for *x*-polarization (blue curve) and *y*-polarization (red curve). These two curves cross each other at a unique point corresponding to *d*_{r}
/Λ=0.75 with resonance wavelength of λ_{res}=1.325 µm. This unique selection of the resonator’s diameter will lead to polarizationindependent propagation characteristics, for the structure shown in Fig. 9, at the prescribed single resonance wavelength. So by choosing the resonator’s diameter as *d*_{r}
/Λ=0.75, the resulting structure will operate as an effectively 100 % coupler from core-A into core-B, with polarization-independent propagation characteristics, operating at a single wavelength. The corresponding coupling length in this case was calculated by the modal analysis to be *L*_{c}
=22.3 mm, a relatively short coupling length, acceptable for most practical applications. In order to verify the exact coupling length, in Fig. 10 we perform a simulation of the normalized power propagation along the PBGF splitter, using an accurate BPM algorithm [12]. Specifically Fig. 10(a) shows the normalized power propagation at the resonance wavelength of λ_{res}=1.325 µm, for *x*-polarization (blue curve) and y-polarization (red curve), in the input core-A as a function of the propagating distance in mm. The same simulation is shown in Fig. 10(b) for the output core-B. From these results we can see that at the coupling length of Lc=22.3 mm, the power is transferred from the input core-A to the output core-B with a transmission of more than 90 %, independent of the polarization state of the input signal. The main conclusion arising from this section is that the proposed MC-PBGF technology has indeed the potential capabilities of realizing polarization-independent devices by a judicious choice of the design parameters.

## 5. Conclusions

To summarize our work, we have proposed and numerically investigated the propagation properties of a novel band pass filter based on the resonant tunneling phenomenon in a three-core PBGF. The design strategy of realizing multi-core couplers based on the resonant tunneling effect in PBGFs, according to the best of our knowledge, is reported in the international literature for the first time. Results of a full vectorial finite element modal analysis confirmed by BPM simulations have been presented for a variety of quantities related to the fiber’s propagation characteristics. The high suppression of the side-lobes in comparison to previous reported filters based on conventional fiber technology [16]–[18] as well as the ultra-narrowband response and the reasonable short coupling length are the main advantages of the proposed PBGF architecture. Additionally we have showed that by using this MC-PBGF platform we can even achieve polarization-independent propagation characteristics at a single operational wavelength. Regarding the feasibility of our proposed fiber, maybe somehow difficult in the present stage of technology, but we strongly believe that with more advanced fabrication technologies that will be introduced in the near future, the proposed MC-PBGF splitter will be a challenging fabrication task for the experimentalists. Our three-core PBGF coupler can be employed in multifunctional all-fiber bandpass/bandstop filtering applications and resonant sensors.

A generalized wavelength splitter based on the resonant tunneling effect in multi-core PBGFs with multiple integrated resonators in its profile, is proposed for future analysis in Fig. 11. The operational principle is exactly the same as that of the splitter in Fig. 1. By an appropriate choice of the design parameters the generalized wavelength splitter in Fig. 11 can perform a four-wavelength splitting operation within reasonably short coupling length, appropriate for the realization of multi-operational all-fiber devices. We believe that the inclusion of multiple integrated components in a single fiber for multifunctional purposes is a challenging problem and currently is an active research topic in our group.

## Acknowledgments

The authors would like to kindly acknowledge partial financial support of this project from the Postdoctoral Fellowship Program of the Japan Society for the Promotion of Science (JSPS).

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