1. Introduction

Photonic crystals (PC) are multidimensional periodic dielectric structures possessing specific topology with high-index-contrast materials. The periodicity of the PC structure is on the order of the wavelength of light. In two-dimensional PCs there may exist a forbidden range of frequencies within the plane of periodicity called the photonic bandgap (PBG). Electromagnetic waves can experience these bandgaps for all polarizations or only for the TE or TM polarization. The wavevector k is imaginary inside these frequency intervals; hence, the field is evanescent and cannot propagate within the PBG [1]. Introducing spatial perturbations as point or line defects into a PC enables compact cavities or waveguides, which are potential building blocks of more advanced optical circuitry.

There have been various methods proposed to modulate the parameters of basic PC structures to obtain dynamic or static changes in the optical characteristics. One common and straightforward way of creating the defects is by means of lithography. Local lithographic tuning to change the geometry and dimensions of an underlying PC structure can modify its properties [2]. Such an approach, however, does not provide dynamic control over guiding the light flow (i.e., the properties of the PC are fixed by fabrication). Using semiconductor materials and controlling the temperature and the impurity concentration, PCs can in this fashion be made tunable [3]. The temperature-dependent refractive index of an infiltrated liquid crystal and the thermo-optic effect are other possible ways of making PCs tunable [4-6]. Recently, Ref. [7] has shown a rewritable scheme based upon infiltration of liquids into the selected pores. In this Letter, we present a method to implement a reconfigurable PC waveguide (PCW) based upon the multimode nature of the structure. To the best of our knowledge this is the first time that multimode structures are proposed for the implementation of reconfigurable photonic devices. The nature of the waveguide mode under the irregularities occurred during the infiltration process is also addressed. The amount of power transferred to the output of the waveguide is calculated for different cases and it is shown that high-index liquid infiltration is necessary to have high transmission efficiency with sharp waveguide bends.

PCs are frequently designed using high-index-contrast materials to lead to a complete PBG. Usually, PCWs are obtained by perturbing the periodicity of the crystal in a systematic fashion. For example, in a hexagonal lattice of air holes in a dielectric background, if one row of holes is removed from the structure then a PCW may be permanently created. This procedure provides strong perturbation (one may assume holes are filled with the high-index-contrast material, for example, with the same type of background semiconductor material). As a result, PCW modes are strongly confined in the waveguide region.

For the purpose of reconfiguration with liquid materials, however, we should retain the holes as structural features in the waveguide region. One should note that we only focus on reconfigurable modification so that perturbing (increasing/decreasing) the radius of holes in the waveguide region is not pursued in this study. Since the refractive indices of most available liquids at optical frequencies are low (usually ∼1.5), PCWs obtained by one row of holes filled with liquids create propagating modes, which are may not be strongly confined in the waveguide region (modes which are close to band-edge frequencies are called shallow modes). Besides, during the writing process (infiltration) it is likely that one hole or several may be inadvertently left unfilled. In such a situation there can be discontinuities along the waveguide region, and the guided mode may be prone to large scattering losses.

Considering all these issues motivated us to propose multimode PCWs obtained by perturbing three rows of holes loaded with liquids for the purpose of reconfigurable devices. We will show that strong confinement can be obtained and intentionally unfilled holes do not create severe back-reflection problems for the transmission of the input light.

In fact, the idea of merging fluidics with optics is a promising route to tune photonic circuits, and this research area has been dubbed as optofluidics [8-14]. The modification of the index of the holes by infiltration of liquids provides a large refractive index variation compared with several other means, such as thermo- or electro-optics. By selective infiltration of the voids with transparent materials such as liquids, one can form an arbitrary rewritable pattern onto the PC structures. For example, an arbitrary curved waveguide can be written. We present the study of the guiding and bending properties of such designed PCWs. The analysis is carried out both in the time and frequency domain by the finite-difference time-domain (FDTD) method and the planewave-expansion (PWE) method, respectively, with the appropriate boundary conditions [15, 16].

 

Fig. 1. (a) The schematic of the photonic-crystal waveguide obtained by the infiltration of three rows of holes along ΓK direction. The dielectric background and holes in the waveguide region are assumed to be n = 3.46 and n = 1.5, respectively. The radii of holes are r = 0.3a. (b) The schematic of the photonic crystal waveguide bend (arbitrary written path) obtained by the infiltration of three rows of holes.

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2. Results and discussion

To start, assume that an arbitrary waveguide profile, as shown in Fig. 1(a), is drawn onto a PC. For purely computational purposes, the length of the PCW is taken to be 60a where a is the lattice constant. The mechanism employed to create a waveguide along the TK symmetry direction comes from selective infiltration of a liquid. The structure is a two-dimensional triangular-lattice PC with air holes (r = 0.3a) in a dielectric background (n = 3.46). The bandgap of the PC is from a/λ = 0.207 to 0.273 for the TE (electric field is in the plane) polarization. For the TM polarization, there is no bandgap. Hence, only the TE polarization is considered in this study. The dispersion diagram of the waveguide is obtained by the PWE method using the supercell approach. The refractive index of the infiltrated liquid is taken to be 1.50. The multimode nature of the waveguide can be seen from the figure and the even and odd modes create pairs in the dispersion diagram. By inspecting the dispersion diagram in Fig. 2(a), we can take the normalized frequency a/λ = 0.26 as the central frequency of a modulated Gaussian input source. To show the waveguide-transmission behavior in the presence and absence of the liquid we employ the FDTD method. The steady-state field map presented in Fig. 3(a) shows the input field propagation through a PCW written with a liquid. It can be seen from the plot that the light is confined in the waveguide region and power is transmitted by means of the infiltration process. On the other hand, if the holes are not filled with the liquids, the same PC behaves as a pure mirror to the input source.

We next consider the effect of bends in the PCW, which are potentially useful to route the light through an arbitrary angle [17]. An example of a PCW bend is shown in Fig. 1(b). The same type of the source as before is launched at the entrance of this PCW bend. We can conclude by examining the field profile in Fig. 3(b) that the beam is guided and routed through the PCW bend without experiencing major loss. Again, we can expect that field cannot propagate through this waveguide bend if the infiltration is not performed (all the incident light will be reflected back).

 

Fig. 2. (a) The dispersion diagram of the waveguide used in the study. The plane-wave expansion method with the supercell approach is used in order to calculate the dispersion plot. (b) The dispersion diagram of the waveguide in case the infiltrated holes have higher-refractive indices n = 2.0 .

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Fig. 3. (a) Steady-state magnetic-field distribution along the waveguide for a frequency a/λ = 0.26a corresponding to a propagating mode within the bandgap. The colorbar indicates the amplitude variation between the minimum and maximum values. (b) Steady-state magnetic field distribution along the waveguide bend for a frequency a/λ = 0.26a . The dashed-vertical lines show the locations of sources.

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In order to quantify the power transmitted through the reconfigurable waveguide we calculated the Poynting vector integrated across the entrance and exit of the PCW. The power at the exit of the PCW is normalized with respect to the input power that is obtained in the absence of the PC. The transmission coefficient of Fig. 3(a) is found to be around 20 %. This relatively poor transmission is mainly due to the input and output coupling losses (Fresnel reflections at the input and output section of the waveguide lower the amount of the total transmitted power). There is, however, also another loss mechanism in the multimode reconfigurable waveguide. It is known that fundamental mode can be converted to higher-order modes due to corrugations along the sides of the guide. As a result, some portion of the higher order modes can leak in the transverse direction, thus representing loss. Besides, if the propagating mode is confined shallowly, some part of the power will leak out of the PCW.

The reflections at the entrance and the exit of the waveguide can be lowered both by reducing the refractive index of the dielectric background and/or by the infiltration of the higher-refractive-index material into the holes. The loss due to higher-order-mode conversion may also be altered by the modification of the row of holes along the side of the PCW. We will show later that infiltration with a high-index liquid enhances the power transmission dramatically through the reconfigurable PCWs.

Multimode PCWs possess small gaps between the fundamental and higher-order modes which can be coupled by inhomogeneities in the PCW, such as due to corrugation along the PCW sides. One should select the waveguide frequency with care. If the frequency is near these small gaps, it is likely that higher-order modes can be directly generated and the electromagnetic field will have a broader transverse profile.

 

Fig. 4. (a) Photonic-crystal waveguide with some of the holes unfilled intentionally. In total 10 holes left unfilled with liquids. (b) The schematic representation of photonic crystal waveguide bend with 90° angle.

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Fig. 5. Steady-state field map of the structure as the schematic is indicated in Fig. 4(a). (b) Steady-state magnetic- field distribution along the waveguide [schematic is shown in Fig. 1(a)] for a frequency a/λ = 0.26a corresponding to a propagating mode within the bandgap. The infiltrated holes in this case have higher-refractive indices n = 2.0. The dashed-vertical lines show the locations of sources.

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We claimed in the introduction that multimode PCWs may still provide satisfactory power transmission even though some of the holes in the waveguide region are left unfilled. To test this idea we perform additional FDTD simulations for the PCW shown in Fig. 4(a). A few holes (total 10 holes) are left unfilled along the both sides of the PCW and the same input source as before is launched. Although the unfilled holes produce a back reflection as presented in Fig. 5(a), the input field is able to reach the end of the PCW. In the case where waveguide is composed of only one row of holes the situation may be far worse as the PCWs integrity is more severely disturbed due to the unfilled holes.

It is important to lower the power loss as the light is traveling through a reconfigurable photonic circuit. In order to achieve this goal, we look at the case in which the infiltrated liquid has a refractive index of 2.0. The dispersion diagram of the PCW thus formed is shown in Fig. 2(b). The multimode nature of the PCW can be clearly seen. Also, the waveguide modes move further away from the air bands well within the bandgap region. That means we can obtain more strongly confined waveguide modes. The field map is shown in Fig. 5(b) and can be compared to Fig. 3(a); one notes that the power transmission is enhanced dramatically. The transmission coefficient increases up to 90 %. The high-refractive-index-liquid infiltration has two advantages. It allows the light to be confines strongly in the waveguide region and the Fresnel reflections at the input and output ends of the PCW are lowered. As a result, high transmission efficiency can be more easily obtained.

One crucial aspect of the reconfigurable waveguides is the limiting value of the bend angle or bend radius at which the light propagation manages to reach the waveguide exit with a satisfactory field amplitude. This is important in order to have dense photonic integrated devices on the same substrate. For this purpose, we increased the bend angles of the waveguides starting from the straight one to a 90° bend angle. The schematic of the waveguide bend is shown in Fig. 4(b). The transmission coefficient of such structure is expected to be lower than the straight waveguide and this is the case with an efficiency of approximately 6 %. The steady-state magnetic field map is presented in Fig. 6(a) for the case in which the infiltrated substance has refractive index of 1.5. The idea proposed to enhance the transmission efficiency via higher-refractive-index-liquid infiltration will also work for such a large-bend-angle waveguides as shown in Fig. 6(b). In this case, the transmission coefficient reaches around 40 %. The sharp corners degrade significantly the amount of power exiting the waveguide.

 

Fig. 6. The steady-state magnetic field variation of the waveguide bend with bend angle of 90°. The filled holes have refractive index of 1.5 in (a) and 2.0 in (b). The dashed-vertical lines show the locations of sources.

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Finally, a purely two-dimensional study is incapable of treating the inhomogeneities along the third dimension that may happen after the liquid infiltration. This situation can be analyzed by employing fully 3D analysis at the expense of considerable computational resources. The inhomogeneities will affect the out-of-plane scattering losses which may contribute as an additional loss mechanism for the field propagating in-plane. Even though we present the results using a two-dimensional analysis, the main findings of this work are not expected to change significantly if one carries the analysis with a three-dimensional calculation for a PC slab. Due to the finite thickness of such a structure, the refractive index of the background material will be lower than the one used here, and the bandgap occurs at slightly higher frequencies while the width of the bandgap will be somewhat smaller. The confinement in the plane along the waveguide occurs due to the PBG effect and the refractive index contrast between the PC slab and the air governs the out-of-plane confinement. If the slab is partly covered by the liquid during the infiltration process then we may speculate that the thickness of this layer and the refractive index of the substance are the important parameters. There will be additional interfaces; PC slab-liquid layer and liquid layer-air. Depending on the thickness and the refractive index of the liquid, this artificial layer may alter the out-of-plane scattering losses but to make concrete conclusion, 3D study should be employed.

3. Conclusion

Dynamically tuning the properties of photonic devices is one of the key steps to enhance widespread deployment of photonic integrated circuits. Therefore, reshaping the structure in a reconfigurable manner is of paramount importance. We demonstrate a novel approach to obtain reconfigurable PCWs. An effective waveguide medium is obtained by infiltration of liquids in air holes in a two-dimensional PC. Light can be directed along an arbitrary written path. The efficiency of the power transfer can be increased dramatically from 20 % for low-index substances to 90 % with liquids possessing refractive index of 2.0. The realization of waveguide bends with tight angles may become feasible with the presented work. This idea may find many applications in integrated optics, including optical switches, filters, and modulators operating in a reconfigurable manner. Although the studied structure is a triangular lattice of holes in a dielectric background, other complimentary type of configurations such as square lattice dielectric rods in a low refractive-index background can be also used for reconfigurable photonic devices.