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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 8 — Apr. 18, 2005
  • pp: 2814–2827
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Modulating dispersion properties of low index photonic crystal structures using microfluidics

Ahmed Sharkawy, David Pustai, Shouyuan Shi, Dennis W. Prather, Sterling McBride, and Peter Zanzucchi  »View Author Affiliations


Optics Express, Vol. 13, Issue 8, pp. 2814-2827 (2005)
http://dx.doi.org/10.1364/OPEX.13.002814


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Abstract

We present a technique for manipulating the dispersive properties of low index periodic structures using microfluidic materials that fill the lattice with various fluids of different refractive indices. In order to quantify the modulation of the optical properties of the periodic structure we use Equi-frequency contours (EFC) data to calculate the frequency dependant refractive index and the refractive angle. We further introduce various types of defects by selectively filling specific lattice sites and measuring the relative change in the index of refraction. Finally we design and optically characterize an adaptive low index photonic crystal based lens with tunable optical properties using various microfluidics. We also present experimental results for a silicon based PhC lens used an optical coupling element.

© 2005 Optical Society of America

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

1. T. Sondergaard, A. Bjarklev, J. Arentoft, M. Kristensen, J. Erland, J. Broeng, and S. E. B. Libori, “Designing finite-height photonic crystal waveguides: confinement of light and dispersion relations,” Opt. Commun. 194. 341–351, (2001). [CrossRef]

, 2

2. S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The road from Theory to Practice. Norwell, MA: Kluwer Academic Publishers, (2002).

], cavity resonators[3

3. S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Microcavities in Photonic Crystals: Mode Symmetry, Tunability, and Coupling Efficiency,” Phys. Rev. B 54. 7837–7842, (1996). [CrossRef]

], optical spectrometers[4

4. A. Sharkawy, S. Shi, and D. W. Prather, “Multichannel Wavelength Division Multiplexing Using Photonic Crystals,” Appl. Opt. 40. 2247–2252, (2001). [CrossRef]

], channel drop add filters[5

5. H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Channel Drop Filters in Photonic Crystals,” Opt. Express 3. 4–11, (1998). [CrossRef] [PubMed]

], optical switches [6

6. A. Sharkawy, S. Shouyuan, D. W. Prather, and R. A. Soref, “Electro-optical switching using coupled photonic crystal waveguides,” Opt. Express 10. 1048–1059, (2002). [PubMed]

] and coupled waveguide applications[7

7. D. M. Pustai, A. Sharkawy, S. Y. Shi, G. Jin, J. Murakowski, and D. W. Prather, “Characterization and analysis of photonic crystal coupled waveguides,” Journal of Microlithography Microfabrication and Microsystems 2. 292–299, (2003). [CrossRef]

], 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

8. H. Kosaka, T. Kawashima, Akihisa Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism Phenomena in Photnic Crystals,” Phys. Rev. B 58. R10096–R10099, (1998). [CrossRef]

]. 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

9. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” App. Phys. Lett. 74. 1212–1214, (1999). [CrossRef]

]

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

10. J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Topics Quantum Electron.8, (2002). [CrossRef]

], variable beam splitters [11

11. S. Y. Shi, A. Sharkawy, C. H. Chen, D. M. Pustai, and D. W. Prather, “Dispersion-based beam splitter in photonic crystals,” Opt. Lett. 29. 617–619, (2004). [CrossRef] [PubMed]

], 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

12. D. W. Prather, S. Shi, D. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. J. Scheider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. 29. 50–52, (2004). [CrossRef] [PubMed]

], 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.

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 (nc=1.52) embedded within a low index material(nback=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

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

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 ko 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| 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 θi, 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

13. P. Halevi, “Photonic Crystal optics and Homogenization of 2D periodic Composites,” Phys. Rev. Lett. 82. 719–722, (1999). [CrossRef]

]. 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

Vg=kω(k)=ωkxx̂+ωkyŷ
(1)

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 ko will propagate along kp 2 due to the EFC from the 2nd band and along kp 3 due to the EFC from the 3rd band.

3. Optical properties of capillary hole structure

kx2+ky2=(koneff)2
(2)

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

Fig. 4(a). Propagation angle Fig.
Fig. 4(b). Effective refractive index

4. Modulating dispersion properties of capillary hole structure using microfluidics

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

5. Tunable microfluidic PhC dispersion based lens

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.
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.

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.

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

table-icon
View This Table

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

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 2nd 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

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.

Acknowledgments

This work was funded under DARPA Bio-Optic Synthetic Systems under contract No. 4900000131.

References and links

1.

T. Sondergaard, A. Bjarklev, J. Arentoft, M. Kristensen, J. Erland, J. Broeng, and S. E. B. Libori, “Designing finite-height photonic crystal waveguides: confinement of light and dispersion relations,” Opt. Commun. 194. 341–351, (2001). [CrossRef]

2.

S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The road from Theory to Practice. Norwell, MA: Kluwer Academic Publishers, (2002).

3.

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Microcavities in Photonic Crystals: Mode Symmetry, Tunability, and Coupling Efficiency,” Phys. Rev. B 54. 7837–7842, (1996). [CrossRef]

4.

A. Sharkawy, S. Shi, and D. W. Prather, “Multichannel Wavelength Division Multiplexing Using Photonic Crystals,” Appl. Opt. 40. 2247–2252, (2001). [CrossRef]

5.

H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Channel Drop Filters in Photonic Crystals,” Opt. Express 3. 4–11, (1998). [CrossRef] [PubMed]

6.

A. Sharkawy, S. Shouyuan, D. W. Prather, and R. A. Soref, “Electro-optical switching using coupled photonic crystal waveguides,” Opt. Express 10. 1048–1059, (2002). [PubMed]

7.

D. M. Pustai, A. Sharkawy, S. Y. Shi, G. Jin, J. Murakowski, and D. W. Prather, “Characterization and analysis of photonic crystal coupled waveguides,” Journal of Microlithography Microfabrication and Microsystems 2. 292–299, (2003). [CrossRef]

8.

H. Kosaka, T. Kawashima, Akihisa Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism Phenomena in Photnic Crystals,” Phys. Rev. B 58. R10096–R10099, (1998). [CrossRef]

9.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” App. Phys. Lett. 74. 1212–1214, (1999). [CrossRef]

10.

J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Topics Quantum Electron.8, (2002). [CrossRef]

11.

S. Y. Shi, A. Sharkawy, C. H. Chen, D. M. Pustai, and D. W. Prather, “Dispersion-based beam splitter in photonic crystals,” Opt. Lett. 29. 617–619, (2004). [CrossRef] [PubMed]

12.

D. W. Prather, S. Shi, D. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. J. Scheider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. 29. 50–52, (2004). [CrossRef] [PubMed]

13.

P. Halevi, “Photonic Crystal optics and Homogenization of 2D periodic Composites,” Phys. Rev. Lett. 82. 719–722, (1999). [CrossRef]

14.

H. C. Nguyen, P. Domachuk, M. J. Steel, and B. J. Eggleton, “Experimental and finite difference time domain technique characterization of transverse in-line photonic crystal fiber,” IEEE Photonics Technology Letters 16. 1852–1854, (2004). [CrossRef]

15.

Y. Y. Huang, Y. Xu, and A. Yariv, “Fabrication of functional microstructured optical fibers through a selective-filling technique,” Appl. Phys. Lett. 85. 5182–5184, (2004). [CrossRef]

OCIS Codes
(130.3130) Integrated optics : Integrated optics materials
(220.0220) Optical design and fabrication : Optical design and fabrication
(250.5300) Optoelectronics : Photonic integrated circuits
(350.4600) Other areas of optics : Optical engineering

ToC Category:
Research Papers

History
Original Manuscript: February 28, 2005
Revised Manuscript: March 25, 2005
Published: April 18, 2005

Citation
Ahmed Sharkawy, David Pustai, Shouyuan Shi, Dennis Prather, Sterling McBride, and Peter Zanzucchi, "Modulating dispersion properties of low index photonic crystal structures using microfluidics," Opt. Express 13, 2814-2827 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-2814


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References

  1. T. Sondergaard, A. Bjarklev, J. Arentoft, M. Kristensen, J. Erland, J. Broeng, and S. E. B. Libori, "Designing finite-height photonic crystal waveguides: confinement of light and dispersion relations," Opt. Commun. 194. 341-351, (2001). [CrossRef]
  2. S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The road from Theory to Practice. Norwell, MA: Kluwer Academic Publishers, (2002).
  3. S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Microcavities in Photonic Crystals: Mode Symmetry, Tunability, and Coupling Efficiency," Phys. Rev. B 54. 7837-7842, (1996). [CrossRef]
  4. A. Sharkawy, S. Shi, and D. W. Prather, "Multichannel Wavelength Division Multiplexing Using Photonic Crystals," Appl. Opt. 40. 2247-2252, (2001). [CrossRef]
  5. H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Channel drop filters in photonic crystals," Opt. Express 3, 4-11 (1998) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4</a>. [CrossRef] [PubMed]
  6. A. Sharkawy, S. Shouyuan, D. W. Prather, and R. A. Soref, "Electro-optical switching using coupled photonic crystal waveguides," Opt. Express 10, 1048-1059 (2002) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1048">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1048</a>. [PubMed]
  7. D. M. Pustai, A. Sharkawy, S. Y. Shi, G. Jin, J. Murakowski, and D. W. Prather, "Characterization and analysis of photonic crystal coupled waveguides," Journal of Microlithography Microfabrication and Microsystems 2. 292-299, (2003). [CrossRef]
  8. H. Kosaka, T. Kawashima, AkihisaTomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami,"Superprism Phenomena in Photnic Crystals," Phys. Rev. B 58. R10096-R10099, (1998). [CrossRef]
  9. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," App. Phys. Lett. 74. 1212-1214, (1999). [CrossRef]
  10. J. Witzens, M. Loncar, and A. Scherer, "Self-collimation in planar photonic crystals," IEEE J. Sel. Topics Quantum Electron. 8, (2002). [CrossRef]
  11. S. Y. Shi, A. Sharkawy, C. H. Chen, D. M. Pustai, and D. W. Prather, "Dispersion-based beam splitter in photonic crystals," Opt. Lett. 29. 617-619, (2004). [CrossRef] [PubMed]
  12. D. W. Prather, S. Shi, D. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. J. Scheider, and J. Murakowski, "Dispersion-based optical routing in photonic crystals," Opt. Lett. 29. 50-52, (2004). [CrossRef] [PubMed]
  13. P. Halevi, "Photonic Crystal optics and Homogenization of 2D periodic Composites," Phys. Rev. Lett. 82. 719-722, (1999). [CrossRef]
  14. H. C. Nguyen, P. Domachuk, M. J. Steel, and B. J. Eggleton, "Experimental and finite difference time domain technique characterization of transverse in-line photonic crystal fiber," IEEE Photonics Technology Letters 16. 1852-1854, (2004). [CrossRef]
  15. Y. Y. Huang, Y. Xu, and A. Yariv, "Fabrication of functional microstructured optical fibers through a selective-filling technique," Appl. Phys. Lett. 85. 5182-5184, (2004). [CrossRef]

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