1. Introduction

Photonic crystals were initially introduced as synthetic crystals that can be used to prohibit the propagation of electromagnetic waves ranging from microwave to optical frequencies based on the dimensions, geometry and materials of the constituent objects. They posses almost all of the properties their predecessor atomic crystals had, a valance band, a conduction band, and a bandgap in some cases. Those properties were further engineered in the same fashion to provide functionalities including point and line defects, which were the fundamental ingredient in a majority of applications presented over the past decade. Applications such as optical waveguides[1, 2], cavity resonators[3], optical spectrometers[4], channel drop add filters[5], optical switches [6] and coupled waveguide applications[7], where the majority if not all of the attention was drawn towards bandgap based applications relying on the confinement properties of photonic crystals, not so much interest was shown for non bandgap based applications except for the early work by Kosaka, et al. [8]. This was mainly due to the complex spatial and spectral dispersion properties that photonic crystal structures had outside the bandgap of the structure if any existed. Within those regions multiple eignemodes might exist and sometimes overlap showing different degrees of degeneracy at certain wavevectors or frequencies, and hence working within those regions was not favorable to the majority of researchers in the Photonic crystal community. It was not until a three-dimensional illustration of such dispersion properties was presented to show the interesting but yet unique shapes those three-dimensional illustrations can take. Those illustrations were called dispersion surfaces.[9]

Dispersion surfaces provide the spatial variation of the spectral properties of a certain band or egienmode within a photonic crystal structure. To quantify the information content within a certain dispersion surface, a cross sectional plot of this spatial variation is taken at a constant frequency point. Since such properties were obtained at the same frequency, the plot created was referred to as an equi frequency contour. Equifrequency (EFC) contours provide the necessary information required to predict the response of a photonic crystal structure for a certain incident excitation at a constant frequency. Such response can be further used to characterize the spatial response of the photonic crystal structure and hence determine the optical properties of the photonic crystal structure described by the propagation angle(s) and further predict the effective refractive index of the photonic crystal at that frequency. Equi frequency contours of different photonic crystal geometries are different and for different polarizations, and hence they can be used to uniquely identify various geometrical orientations. Interesting applications and devices emerged utilizing different EFC shapes including non-channel waveguides and routing [10], variable beam splitters [11], self collimated optical emitters to enhance the emission from a light emitting diode.

Applications relying on the dispersive properties of photonic crystals have recently attracted the attention of the wide majority within the photonic crystal community. The reason for such is the flexibility of implementing similar functionalities offered by the confinement properties if operating in the stop or bandgap but yet without the limitation of high fabrication tolerances [12], alignment required or the need for high index contrast between the materials used to construct a PhC structure. Such requirements have greatly hindered the physical implementation of photonic bandgap based devices outside research laboratories and to only small device areas. While over 600 research papers have been published covering various PBG based applications, very small success have been attained towards their commercial integration. It is highly believed that engineering the dispersion properties of photonic crystals will truly open new frontiers towards their true introduction to the commercial market without the need for tight fabrication tolerances or high index materials to open a bandgap for confinement based applications. It is also believed that current replication methods can be utilized towards implementing photonic crystal dispersion based devices over a large area and hence overcoming slow fabrication throughput of fabricating a single operative device.

Recent experimental studies have shown that the dispersion properties in photonic crystal structures can be engineered to implement various optical devices; however, the electromagnetic response of the engineered structures was passive.

In this paper we present a possible technique to modulate the spatial response of such devices as to introduce a mechanism to change the dispersion surface and EFC shapes of photonic crystals, and hence expand the horizon of dispersion-based applications in photonic crystals to another dimension where they can be dynamically controlled. While almost all of the previous interest in examining the dispersion properties of photonic crystals in high index materials, we exploit the possibility of implementing similar functionalities using low index materials specifically capillary glass tubes, which will aid us later to modulate the optical properties of the photonic crystal structure. To introduce this paper, we begin by extracting the dispersion properties of low index capillary glass PhC structure, in section 2 and in section 3 we further expand the analysis to determine the optical properties of the capillary structure. Consequently, the remainder of the paper is organized as follows: In section 4 we present a technique to modulate the dispersion properties of a capillary glass PhC structure using microfluidics, in section 5 we present an application utilizing the analysis presented in section 4, in section 6 we present experimental results for the application analyzed in section 5 and in section 6 we present concluding remarks.

2. Dispersion properties of low index PhC structure

For the analysis carried throughout this paper we study a crystal shown in Fig. 1 which consists of a hexagonal lattice of air filled capillary glass tubes (n_c=1.52) embedded within a low index material(n_back=1.66). The tubes have an inner radius of (r ₁=a/3) and an outer radius of(r ₂=a/2), where a is the lattice constant. The height of the capillary-hole structure is much greater than the lattice constant and hence the entire structure can be assumed to be two-dimensional. Due to the low index contrast between the host material and the capillary tubes, such structure has an extremely narrow bandgap (5.5%).

 

Fig. 1. Photonic Crystal Capillary Hole Structure

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However the size of the bandgap is not of great importance for the analysis and applications presented here. We are more interested in the spatial interaction of the eigenmodes existing within the photonic crystal structure with the crystalline structure in various directions. Such interaction can be better understood by analyzing the complex dispersion information contained in the equi-frequency contours for the different eigenmode(s). In order to extract such information, a dispersion surface plot is necessary. Unlike the dispersion diagram which provide a projection of the dispersion relations of the various eigenmodes for a certain polarization along a certain direction and hence it is a two dimensional plot, the dispersion surface contains a complete dispersion information necessary to describe the spatial variation of different eigenmodes in different directions for periodic structures having two dimensional periodicity and so it is a three dimensional plot between the in-plane wavevectors describing planar propagation within the photonic crystal structure and the normalized frequency in the third dimension. For structures with three-dimensional periodicity a dispersion volume can be extracted. An EFC can be obtained by taking a slice of the dispersion surface at a constant frequency. The intersection of the constant frequency slice with the various eigenmodes describes the interaction of the spatial modes with the PhC lattice at that specific frequency. The resulting EFC might be a simple contour or a multiple complex one depending on the geometry of the lattice, the refractive index contrast between the objects forming the lattice and the operating frequency range. For e.g. the higher the frequency of the eigenmode the more dispersive the EFC shape will get due to the fact that at such range, the operating wavelength is much smaller than the lattice constant, and hence each capillary hole will act as a scattering center. Dispersion diagrams, surfaces and EFC information can be obtained by casting Maxwell’s equations as an eigenvalue problem which can be solved using various available computational electromagnetic solvers utilizing either iterative plane wave expansion method or the finite difference time domain method. In our case we used EMPLabTM software package provided by EM Photonics which was suitable for this type of analysis. The solution can be represented as a dispersion surface, as shown in Fig. 2. Taking cross sections of the dispersion surface shown in Fig. 2 at constant frequencies, one obtains equifrequency contours. In the following section we present detailed discussion of the results shown in Fig.2

 

Fig. 2. Dispersion surface and EFC contours of the capillary glass structure calculated for a TE polarized wave at normalized frequency=0.7733

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2.1 Dispersion Properties of the Capillary hole structure

An example of an EFC extracted for the capillary-hole structure at normalized frequency of a/λ=0.7733 is shown in Fig. 2. For a TE polarized wave with the magnetic field parallel to the capillary hold axis. Note that the contour shown combines a slice taken from the 2nd band and a slice taken from the 3rd band at the same normalized frequency of a/λ=0.7733. Hence the overall photonic crystal structure response to such EFC can be thought of as a superposition of the response due to the 2nd band and the one due to the 3rd one. Such degeneracy, in some situations may greatly affect the operation of the PhC dispersion based device. This behavior is illustrated in Fig. 3 where an electromagnetic wave propagating through a homogeneous media with a circular EFC have an incident wavevector along k_o direction. Such wave will split and propagate along the directions defined by both the EFC contours of both the 2nd and the 3rd bands. Before we further discuss the dispersive properties of such periodic structures, we first present the details of the computational method used to generate and extract dispersion surface and contour data as the one shown in Fig. 2 and 3.

2.2 Computational method

The dispersion surface shown in Fig. 2 was computed using two-dimensional iterative Plane Wave Method (PWM), and solving for the eigenmodes of Maxwell equations not only along the high symmetry directions in the first Brillouin zone but for all k-vectors in the Irreducible Brillouin zone. For a monochromatic plane wave propagating with a phase velocity V _p=ω/|k|k̂ where ω is the radian velocity of the incident wave and |k| is the magnitude of the wavevector. If such wave is incident on a periodic structure forming the photonic crystal with an angle θ_i, the periodic structure will respond differently depending on the incident wavelength. If the wavelength of the incident wave is longer than the lattice constant of the periodic structure, the response of the structure can be approximated by a homogenized structure having an effective index equal to the average of the indices of the materials forming the holes and the background of the capillary hole structure [13]. In the other hand if the wavelength is on the order of or smaller than the lattice constant of the photonic crystal structure, the amplitude of the monochromatic plane wave will be highly modulated by the existence of the periodic dielectric structure, as a result a modulated wave packet will result in order to convey the information content of the EFC contour to the incident monochromatic plane wave. And since the propagation of information or energy in a wave always occurs as a change in the wave amplitude, it is this modulation that represents the dispersive information content of the PhC at that specific frequency. More generally, some modulation of the frequency and/or amplitude of the incident plane wave is required in order to convey this information, hence the actual speed of the dispersive information content is ∂ω/∂k which is known as the group velocity and it is usually used to denote the velocity of propagation of the energy of the excited mode inside the dispersive media. The energy propagation direction inside the PhC coincide with the direction of the group velocity vector V _g which is normal to the dispersion surface and the EFC, and hence the group velocity vector is usually written as

which means that the group velocity, V _g, or the direction of light propagation inside the periodic structure coincides with the direction of the steepest ascent of the dispersion surface, and is perpendicular to the EFC, as shown in Fig. 3. In the following section we extract the optical and spatial properties of the capillary glass structure utilizing the Equi-frequency contour analysis presented above.

 

Fig. 3. Multiple EFC contours extracted for the capillary hole structure at normalized frequency on 0.7733 will cause an incident plane wave to split and propagate along different directions. A plane wave propagating though a homogenous medium with a circular EFC (blue) incident with a wavevector k_o will propagate along k_{p 2} due to the EFC from the 2^nd band and along k_{p 3} due to the EFC from the 3^rd band.

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3. Optical properties of capillary hole structure

Going back to the EFC extracted for the capillary-hole structure shown in Fig. 1. The spatial variation of the EFC can be used to calculate the energy propagation direction through the PhC structure and the effective refractive index of the structure at that frequency. Once extracted, this information can be further used to engineer applications relying on the dispersive properties of photonic crystals instead of the commonly used confinement properties. Applications presented up to date include superprism, self-collimation, LED, and beam splitter. To study the spectral variation of various incident waves we plot the propagation angle for various incident angles for the frequency ranges covering the 2^nd band as shown in Fig. 4a(). Similar results can be computed for the 3^rd band. Hence a monochromatic wave with a single incident angle will propagate between n-various angles where n being the number of intersecting contours corresponding to n-spatial modes at that frequency. The electromagnetic energy does not necessarily propagate equally (split) between n-various directions. The response of the photonic crystal structure at that frequency can be thought of as being multimodal response (multiple angles) as opposed to a single modal one in the case of a single contour intersection or homogenous media. For each propagation angle we can approximate the PhC behavior using Snell’s law to deduce the effective refractive index at that frequency and for various incidence angles as shown in Fig. 4(b). For the analysis presented here, and without loss of generality we will consider single contour intersection case extracted at a normalized frequency equal to a/λ=0.55. For which the effective index for air filled capillary-hole structure was found to be $\mid n_{{\mathit{eff}}_{\mathit{2}\mathit{nd}}}\mid =1.35$ corresponding to the mode obtained from the 2^nd band calculated for a/λ=0.55 and for a monochromatic plane wave incident with an angle θ_in=11°. The effective refractive index extracted above can be also calculated using

From which we can extract the effective refractive index for a monochromatic wave incident with an angle of θ_in=11° and with a normalized frequency of a/λ=0.55 to be n_eff=1.3325.

 

Fig. 4(a). Propagation angle Fig.

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Fig. 4(b). Effective refractive index

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4. Modulating dispersion properties of capillary hole structure using microfluidics

Next we explore the possibility of modulating the dispersive properties of a large area PhC by introducing defects. It is well known that defects can be introduced to a PhC structure by either reducing or increasing the refractive index of a local lattice site. Commonly, the radius of the capillary tubes was varied (increased or decreased) to introduce such defects. Such technique provides a static mean for modulating the dispersion properties of the PhC structure. Here, we will use microfluidics with refractive indices, n ₁=1.46, n ₂=1.55 and n ₃=1.66 to selectively or globally fill certain lattice sites and hence introduce a periodic defect pattern and examine the dispersive and optical properties of the resulting structure. Microfluidics has been previously used to modulate the location of the bandgap in photonic crystal structures [14]Such technique allow for dynamic modulation of the effective refractive index and hence of the dispersion properties of the entire PhC structure. Various scenarios can emerge following this approach [15]; typically for the two dimensional capillary hole structure analysis considered here, a capillary can be full or empty, providing different levels of tunability. In this work we will assume that the entire capillary is full without loss of generality. Fig. 5(a) shows a periodic defect pattern introduced by filling every other lattice site with the various microfluidics. In doing so we have engineered a new periodic structure which is related to the original periodic structure by having the same lattice constant and air hole diameters and introducing a new defect pattern of filling every other lattice site. The EFC for this newly created periodic structure using various microfluidics is shown in Fig. 5(b) from which we can see that using a microfluid with a matching background index n=1.55 to fill every other capillary-hole indeed modulates the overall dispersive response of the periodic dielectric structure which will reflect as a change in the effective refractive index calculated from Fig. 5(c) to be n=1.3578 also it is clear from Fig. 5(b) that the EFC radius will also be changed from the original structure. Using a microfluid with an index higher than the background index will further modulate the effective refractive index and EFC radius as shown in Fig. 5(b). Next consider the case of introducing the defects every two lattice sites using the same microfluidcs used previously (n=1.46, n=1.55, and n=1.66), the results are shown in Fig. 6. Results for the case of introducing defects every three-lattice site are shown in Fig. 7. In each of the cases shown in Fig. 5, 6 and 7, a new periodic structure was created with a new effective refractive index, which was otherwise unachievable from the original material without introducing geometrical changes to the periodic structure by selectively filling certain regions of the capillary hole structure. Based on the EFC information shown in Figs 5, 6, and 7 we can see that increasing the order of the defect by spacing the defects further apart can be used to finely tune the effective refractive index, the EFC shape and radius. If we consider the extreme case of flooding the entire structure with a fluid having index n=1.55 this will have the effect of converting the PhC structure to a homogenous structure with refractive index n_eff=1.533 for which the dispersion relations take circular shape, while flooding the entire structure with a microfluidic material with index n=1.66 changes to PhC material to a homogenous material with effective refractive index equal to n_eff=1.6459 and such mechanism offers an alternative for coarsely tuning the effective refractive index of the periodic structure.

 

Fig. 5. Modulating the dispersive properties of the capillary glass structure using a one dimensional defect pattern, (a) Capillary glass structure filled every other lattice site with various microfluidics (b) Equi-frequency contours for the modulated periodic structure in (a) using fluids with n=1.46, n-1.52 and n=1.66 (c) Spectral variation of the effective refractive index of the structure in (a) under different fluids. Incident angle is 5 degrees (d) Spectral variation of the angular dispersion through the structure in (a) with different fluids using a fixed incident angle of 5 degrees

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Fig. 6. Modulating the dispersive properties of the capillary glass structure using a two dimensional defect pattern, (a) Capillary glass structure filled every two lattice site with various microfluidics (b) Equifrequency contours for the modulated periodic structure in (a) using fluids with n=1.46, n-1.52 and n=1.66 (c) Spectral variation of the effective refractive index of the structure in (a) under different fluids. Incident angle is 9 degrees.

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Fig. 7. Equi-frequency contours for the capillary glass structure for a three lattice sites defect pattern (filling every three lattice sites).

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This will indeed open a new paradigm for large area effective refractive index modulation using microfluidics based PhC devices. Such behavior can be used to tune the PhC dispersion response for applications such as superprism and was previously used to tune the resonance frequency of a periodic structure operating in the bandgap of the structure. Next we explore the possibility of designing a tunable microfluidic PhC dispersion based lens in the capillary glass structure analyzed in this section.

5. Tunable microfluidic PhC dispersion based lens

In this section we utilize the analysis and results presented in previous sections to design and numerically characterize a PhC based lens in the capillary-hole structure, and we further modulate the optical properties of the lens using the available microfluidics. Using the calculated effective refractive index of the air filled capillary-hole photonic crystal structure to design the lens profile operating in the 2^nd band and using normalized frequency of a/λ=0.55 as our design frequency. The lens profile was designed using the effective refractive index of the capillary-hole structure extracted from the 2^nd band for a/λ=0.55 and was found to be n=1.33. The lens parameters were selected to be as follows: Diameter=20a, focal length=1.5×Diameter which results in an F-number equivalent to f=1.5. To validate the designed lens numerically, we used a rectangular module of capillary holes arranged on a hexagonal lattice. To introduce the designed lens profile to the rectangular module we used a glass mask to simulate the effect of backfilling a region where we do not want the capillary hole structure to exist as shown in Fig 8. Once designed, the capillary-hole photonic crystal based lens was then numerically characterized using the 2D-FDTD engine built in EMPLab^™, where a monochromatic plane wave normally incident on the cleaved edge of our designed lens was used to excite the lens structure. If the frequency of the incident plane wave lied in the bandgap of the capillary hole photonic crystal structure, the incident plane wave will be reflected back toward the initiating source. However since we deliberately chose the frequency of the incident plane wave to coincide with the frequency of the EFC from the 2^nd band at normalized frequency of a/λ=0.55, the incident plane wave will propagate through the capillary-hole structure with minimal transmission loss. The propagation path of the incident plane wave will be however modulated by the existence of the lens profile previously engineered using the dispersive properties of the capillary hole structure to alter or bend the propagation path of the incident plane wave to come to a focus as shown in Fig. 8. Examining the results shown in Fig. 8 we can characterize the performance of our designed lens by running a line scan through the focal plane to extract an intensity profile of the electric field in the focal plane located at 45µm and numerically measure the transmission efficiency to be 92%. We further measured the diffraction efficiency of our designed lens in Fig. 8. The diffraction efficiency is defined as the ratio of the optical energy passing through a detector window to the optical energy passing through the focal plane. According to the diffraction theory, the detection window width B is chosen as 2.44 fλ/Diameter, which is the spacing between the first two zeros of the focal beam shown in red in Fig. 8. As a result the diffraction efficiency was found to be 88%.

To this end we utilized the dispersion information content of the EFC’s to engineer a capillary hole photonic crystal based lens structure in a low index material (glass). At this point we will utilize the analysis presented in section 4 to introduce local or global defects to modulate the effective refractive index of the capillary-hole structure. Of the various techniques presented in section 4 and without loss of generality we choose to completely fill the entire capillary-hole photonic crystal area forming the lens region. In doing so we aim to coarsely tune the optical properties of our designed lens in Fig 8. Fine tuning will be attained by selectively filling partial regions of the capillary hole photonic crystal structure as previously presented in section 4.

 

Fig. 8. A Photonic crystal based lens in the capillary glass structure analyzed in section 4. A normally incident plane wave with a normalized frequency of 0.55 is used to excited the lens structure and produced a focus at 45micon. The transmission and diffraction efficiencies were numerically measured to be 92% and 88% respectively.

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Fig. 9. A Photonic crystal based lens in the capillary glass structure filled with an index matching fluid n=1.55. A normally incident plane wave with a normalized frequency of 0.55 is used to excited the lens structure and produced a focus at 22micon. The transmission and diffraction efficiencies were numerically measured to be 92% and 85% respectively.

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If we start with an index matching fluid with a refractive index n=1.55 this is equivalent to diminishing the periodic structure and creating a homogenous structure with effective refractive index of n=1.533 as previously calculated in section 3. This modulation in effective refractive index will result in modulation in the optical properties of the PhC lens which is shown in Fig. 9 as a change in the focal length from 45µm for the unfilled structure to 22µm for a structure filled with an index matching fluid with refractive index of n=1.55. This corresponds to 50% modulation in the focal length of the dispersion based PhC lens; which will in turn allow us to implement an adaptively controlled lens by varying the microfluid used. Fine modulation can be attained by utilizing one the defect pattern filling discussed in section 4.

In the other hand if instead of using an index matching fluid we use a fluid with refractive index n=1.66, the overall effective refractive index of the capillary hole photonic crystal lens structure will be modulated from the designed value of n=1.33 to n=1.66 for which we will expect such effect to modulate the operating optical properties of the designed lens further, which was numerically measured as a change in the focal length to 15µm. We further examined the variation in the focal length of our PhC based lens upon varying the wavelength of the incident beam and summarized the results in Table 1. Our next experiment was to test the performance of our designed lens to oblique incident plane waves and measure the transmission and diffraction efficiencies. This analysis is shown in Fig. 10.

Table 1. Modulating Focal Point of PhC Designed Lens using various Fluids at Different Wavelengths

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In the following section we redesign our PhC based lens in silicon and utilize it as an optical coupling element to PhC Si based circuits.

6. Photonic crystal lens as a coupling element

The lens designed and analyzed in section 5 can be used in integrated optics as a coupling device to couple electromagnetic light wave to narrow waveguide structures such as photonic bandgap based waveguides. Coupling to PBG based waveguides has been a true challenge since their initial inception. Several approaches have been successfully capable of efficiently coupling a wide dielectric waveguide to a submicron PBG based waveguide. In this section we present another possible solution to such challenge. An example of which is shown in Fig. 11(a), where a coupling structure was fabricated in a 260nm thick silicon-on-insulator wafer.

The photonic crystal structure forming the dispersion based lens consists of a periodic array of air holes arranged on a hexagonal lattice with a lattice constant a=455nm and a hole diameter 2r=364nm.

 

Fig. 10. A Photonic crystal based lens in the capillary glass structure filled with an index matching fluid n=1.55. An oblique plane wave at normalized frequency of 0.55 is incident with 5 degrees off axis is used to excite the lens structure and produced a focus at 25micon the transmission and diffraction efficiencies were numerically measured to be 92% and 77% respectively

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The photonic bandgap structure forming the line defect waveguide consists of a periodic array of air holes arranged on a hexagonal lattice with lattice constant a=455nm and a hole diameter 2r=273nm. Such structure was designed to operate at 1300nm wavelength. The PhC lens and the PBG waveguide structures were fabricated at the University of Delaware using direct write electron beam lithography to pattern the PhC lens and the PBG waveguide. Then the pattern, developed in polymethylmethacrylate was transferred into the silicon layer by a dry etching process with a reactive ion etching. Initial prototype steady state simulation result for the coupling structure was obtained using EMPLab^™ and is shown in Fig. 11(b). To experimentally characterize the structure shown in Fig. 11(a), a collimated 1300nm laser diode incident on a 10x magnification objective was used which was end fire coupled to the feed waveguide. We imaged the top surface of the entire sample. The imaging system consisted of a 2^nd microscope objective (40X magnification) and an IR camera. The Experimental results shown in Fig. 11(c) are shown to be in good agreement with the simulation results in Fig. 11(b). The structure shown in Fig. 11 combines two applications operating both outside the bandgap of the SOI based photonic crystal structure manifested by the existence of the lens, and inside the bandgap of the SOI based photonic crystal structure manifested by the existence of the photonic bandgap based waveguide.

7. Conclusion

In conclusion we have unveiled a possible technique for modulating the dispersion properties of low index photonic structures such as capillary hole structure using microfluidics. Different degrees of tuning are available based on the density of the defects introduced varying from simple point defects for fine-tuning to wide area defects for coarse tuning. In each case we were able to characterize the dispersion properties of the capillary hole photonic crystal structure by measuring the effective refractive index and the effective propagation angles. We further introduced a lens profile into a rectangular array of capillary holes arranged on a hexagonal lattice. We examined the optical properties of the capillary hole photonic crystal lens structure and compared the diffraction efficiency with a square slit. Next we introduced various micorfuildincs to the entire lens region and re measured its optical response. Numerical results included show that a change in the optical properties by 50% can be attained using various fluidics. We finally presented an application utilizing our designed lens in integrated optics as a coupling element to single mode photonic crystal waveguides.

 

Fig. 11. A photonic crystal based lens in silicon background can be used as a coupling element in a Photonic crystal based circuit. (a) SEM image of a fabricated lens coupling structure (b) Steady state simulation results for the structure in (a). (c) Experimental characterization of the device in (a) at 1300nm.

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