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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 7184–7194
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Crescent shaped dielectric periodic structure for light manipulation

H. Kurt, M. Turduev, and I. H. Giden  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 7184-7194 (2012)
http://dx.doi.org/10.1364/OE.20.007184


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Abstract

We present optical properties of crescent-shaped dielectric nano-rods that comprise a square lattice periodic structure named as crescent-shaped photonic crystals (CPC). The circular symmetry of individual cells of periodic dielectric structures is broken by replacing each unit cell with a reduced symmetry crescent shaped structure. The created configuration is assumed to be formed by the intersection of circular dielectric and air rods. The degree of freedom to manipulate the light propagation arises due to the rotational sensitivity of the CPC. The interesting dispersion property of designed CPC occurs due to the anisotropic nature of the iso-frequency contours that yield tilted self-collimated wave guiding. Furthermore, this feature allows focusing, routing, splitting and deflecting light beams along certain routes which are independent of the lattice symmetry directions of regular PCs. The propagation direction of light can be tuned by means of the opening angle of the crescent shape. Finally, the property of being all-dielectric structure ensures the absence of optical absorption losses that are reminiscent of employed metallic nano-particles.

© 2012 OSA

1. Introduction

The research in the field of photonic crystals (PC) was emerged in 1987 [1

1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of the Light (Princeton, NJ: Princeton Univ. Press, 1995).

]. Since that time, highly symmetric periodic dielectric structures with a large refractive index contrast have been heavily investigated with an ultimate aim of achieving complete photonic band gap (PBG) that may appear in the dispersion diagram [1

1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of the Light (Princeton, NJ: Princeton Univ. Press, 1995).

,2

2. P. R. Villeneuve and M. Piche, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B Condens. Matter 46(8), 4969–4972 (1992). [CrossRef] [PubMed]

]. The PC acts as a mirror reflecting the entire incident light wave whose wavelength falls inside the PBG region. The band gap features of pure periodic 3D and 2D PCs were soon demonstrated [3

3. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990). [CrossRef] [PubMed]

,4

4. E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B 10(2), 283–295 (1993). [CrossRef]

]. Perturbing the periodicity of the PC may host artificially created optical modes that are surrounded by the upper and lower boundaries of PBG. Waveguides, sharp corners and cavities have become the ingredients of photonics research [5

5. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62(12), 8212–8222 (2000). [CrossRef]

7

7. M. Lončar, J. Vučković, and A. Scherer, “Methods for controlling positions of guided modes of photonic-crystal waveguides,” J. Opt. Soc. Am. B 18(9), 1362–1368 (2001). [CrossRef]

]. Meanwhile, it has been realized that the unperturbed structure also possesses rich spectral characteristics such as self-collimation, negative refraction, super-prism and super-lens [8

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

11

11. D. Chigrin, S. Enoch, C. Sotomayor Torres, and G. Tayeb, “Self-guiding in two-dimensional photonic crystals,” Opt. Express 11(10), 1203–1211 (2003). [CrossRef] [PubMed]

]. All these listed peculiar dispersion properties may not need structural defects. The two cases (unperturbed periodicity vs. broken periodicity) comprise various device applications frequently demanded for photonic devices.

Considering all of the previously investigated common PC configurations, we can conclude that PCs are highly symmetric structures and do possess fixed structural patterns. The two mostly explored and utilized PCs are square- and hexagonal- (also known as triangular) lattice photonic structures. The ingredient element of PC is usually circularly shaped unit cell although there are other types of unit cell shapes such as rectangular, elliptical or annular ones [12

12. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81(12), 2574–2577 (1998). [CrossRef]

16

16. B. Rezaei, T. Fathollahi Khalkhali, A. Soltani Vala, and M. Kalafi, “Absolute band gap properties in two-dimensional photonic crystals composed of air rings in anisotropic tellurium background,” Opt. Commun. 282(14), 2861–2869 (2009). [CrossRef]

]. The ultimate aim of these studies is to achieve complete PBG for all polarizations (TE and TM) and design polarization insensitive optical devices [17

17. Y. Zhang, L. Kong, Z. Feng, and Z. Zheng, “PBG structures of novel two-dimensional annular photonic crystals with triangular lattice,” Optoelectron. Lett. 6(4), 281–283 (2010). [CrossRef]

19

19. H. Wu, L. Y. Jiang, W. Jia, and X. Y. Li, “Imaging properties of an annular photonic crystal slab for both TM-polarization and TE-polarization,” J. Opt. 13(9), 095103 (2011). [CrossRef]

]. In a rather different perspective, the re-oriented unit cells of the PCs may give rise to the implementation of graded index (GRIN) mediums [20

20. H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express 15(3), 1240–1253 (2007). [CrossRef] [PubMed]

,21

21. E. Centeno, D. Cassagne, and J. P. Albert, “Mirage and superbending effect in two-dimensional graded photonic crystals,” Phys. Rev. B 73(23), 235119 (2006). [CrossRef]

]. The implementations of GRIN via periodic structures provide great flexibilities in terms of designing different index gradient and photonic integrated circuit components such as couplers, lenses and super-bending device [22

22. H. Kurt and D. S. Citrin, “A novel optical coupler design with graded-index photonic crystals,” IEEE Photon. Technol. Lett. 19(19), 1532–1534 (2007). [CrossRef]

27

27. B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18(19), 20321–20333 (2010). [CrossRef] [PubMed]

].

In the present work, we propose a novel type of PC structure named as crescent-shaped photonic crystals (CPC). To the best of our knowledge, this structure has not been studied as a periodic dielectric structure, yet. In this study, the designed CPC enables us to arbitrarily route light beams by exploiting the engineered dispersion diagrams. There is no need to infiltrate any type of anisotropic material and the approach does not possess asymmetric PC patterns. In the CPC structure, the geometrical adjustments are implemented at the level of unit cells not that of structural lattice arrangements. This brings extensive parametric tunabilities in realization of ultra-compact photonic integrated devices. Moreover, although CPCs are formed by isotropic materials, designed structure exhibits anisotropic optical properties similar to optical birefringence. The other unique feature of the CPC structure is due to the fact that the operating frequency of the structure can be easily shifted to any spectral region due to the scalability of the Maxwell’s equations and availability of different lossless dielectric materials.

There have been various mechanisms that may induce optical anisotropy for light propagation in PCs. The anisotropy introduced into the periodic medium can be either in terms of selecting specific materials (dielectric parameter) or structural configuration (unit cell’s shape or type) [12

12. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81(12), 2574–2577 (1998). [CrossRef]

,14

14. X. Zhu, Y. Zhang, D. Chandra, S. C. Cheng, J. M. Kikkawa, and S. Yang, “Two-dimensional photonic crystals with anisotropic unit cells imprinted from poly (dimethylsiloxane) membranes under elastic deformation,” Appl. Phys. Lett. 93(16), 161911 (2008). [CrossRef]

,15

15. H. F. Ho, Y. F. Chau, H. Y. Yeh, and F. L. Wu, “Complete bandgap arising from the effects of hollow, veins, and intersecting veins in a square lattice of square dielectric rods photonic crystal,” Appl. Phys. Lett. 98(26), 263115 (2011). [CrossRef]

,17

17. Y. Zhang, L. Kong, Z. Feng, and Z. Zheng, “PBG structures of novel two-dimensional annular photonic crystals with triangular lattice,” Optoelectron. Lett. 6(4), 281–283 (2010). [CrossRef]

,28

28. I. Khromova and L. Melnikov, “Anisotropic photonic crystals: generalized plane wave method and dispersion symmetry properties,” Opt. Commun. 281(21), 5458–5466 (2008). [CrossRef]

35

35. F. Guan, Z. Lin, and J. Zi, “Opening up complete photonic bandgaps by tuning the orientation of birefringent dielectric spheres in three-dimensional photonic crystals,” J. Phys. Condens. Matter 17(33), L343– L349 (2005). [CrossRef]

]. The optical properties of the former approaches can be dynamically tuned by an external applied electric field. The infiltrations of liquid crystals in 2D PCs involving anisotropic media were studied for tuning their photonic band structures [31

31. S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61(4), R2389– R2392 (2000). [CrossRef]

,32

32. C. S. Kee, K. Kim, and H. Lim, “Tuning of anisotropic optical properties of two-dimensional dielectric photonic crystals,” Physica B 338(1-4), 153–158 (2003). [CrossRef]

]. The lower symmetry periodic structures have been investigated for different applications. The rectangular lattice PC was used in the study of angular super-prism effect in Ref. 36

36. A. I. Cabuz, E. Centeno, and D. Cassagne, “Superprism effect in bidimensional rectangular photonic crystals,” Appl. Phys. Lett. 84(12), 2031–2033 (2004). [CrossRef]

and broad angle self-collimation characteristic was explored in Ref. 37

37. Y. Xu, X. J. Chen, S. Lan, Q. Guo, W. Hu, and L. J. Wu, “The all-angle self-collimating phenomenon in photonic crystals with rectangular symmetry,” J. Opt. A, Pure Appl. Opt. 10(8), 085201 (2008). [CrossRef]

. PCs made of parallelogram lattice structure were investigated for light focusing device that utilize self-collimation phenomenon [38

38. Y. Ogawa, Y. Omura, and Y. Iida, “Study on Self-collimated light-focusing device using the 2-D Photonic Crystal with a Parallelogram Lattice,” J. Lightwave Technol. 23(12), 4374–4381 (2005). [CrossRef]

]. The self-collimated waveguide bends with different angles have also been implemented [39

39. D. Gao, Z. Zhou, and D. S. Citrin, “Self-collimated waveguide bends and partial bandgap reflection of photonic crystals with parallelogram lattice,” J. Opt. Soc. Am. A 25(3), 791–795 (2008). [CrossRef] [PubMed]

]. Special attention should be given while joining the rotated blocks of parallelogram lattice PC because the junction planes with complex geometries may be induced [38

38. Y. Ogawa, Y. Omura, and Y. Iida, “Study on Self-collimated light-focusing device using the 2-D Photonic Crystal with a Parallelogram Lattice,” J. Lightwave Technol. 23(12), 4374–4381 (2005). [CrossRef]

]. Similarly, the interface at the bend region should be carefully handled for the self-collimated waveguide bends [39

39. D. Gao, Z. Zhou, and D. S. Citrin, “Self-collimated waveguide bends and partial bandgap reflection of photonic crystals with parallelogram lattice,” J. Opt. Soc. Am. A 25(3), 791–795 (2008). [CrossRef] [PubMed]

]. As a result, these approaches may provide limited capabilities for beam deflecting and routing applications. However, the proposed structure in the present work enables us to easily integrate different blocks made of square-lattice crescent PC.

The effects of symmetry reduction in PC were heavily explored with a goal of obtaining larger band gaps. We should emphasize that lower symmetry structures with complex configurations such as crescent-shape have not been investigated for the dispersion contours engineering and light manipulation applications. Instead of altering lattice type or introducing material anisotropy into periodic medium, we preferred to modify the circular shape of dielectric cylinders. The engineering of the iso-frequency contours (IFCs) can be performed at a level of unit cell and composite structures can be realized in such a way that the interfaces are free from complex geometries. It is possible to use other complex shape unit elements such as modified version of the crescent shape, U or V shapes instead of crescent one. However, it is expected that the degree of rotation of IFCs and focusing power may become different in each case. That aspect of the interpretation needs additional work which is kept for a future study.

The paper is organized as follows: In Section 2, we explain the geometrical details of the proposed CPC model and its dispersion characteristics. In Section 3, we analyze of the designed CPC structure using finite-difference-time-domain (FDTD) method and confirm that the expected results calculated by plane wave expansion (PWE) method agree well with the simulation results in the time-domain. The discussions of the findings and future directions are mentioned in Section 4. The conclusions will be listed in Section 5.

2. 2D crescent-shaped photonic crystals and dispersion analysis

In this work, we purposely break the circular (rotational, four-fold) symmetry of the unit cell by replacing it with a crescent shaped structure. The expectation is to enhance light manipulation capability inside the photonic structure without depending on artificially introduced structural defects. The geometrical shape of the individual cell provides the construction of complex photonic structures that may yield distinct spectral features as we show in the present work. It is versatile to tune the focal point locations and deflection angle of a beam via rotationally manipulating the structure. We show that the photon manipulation (propagation direction and focusing point) is greatly tailored due to the anisotropic nature of the IFCs. Introducing certain amount of rotational degree to each individual cells yields shifting of focal points along both x- and y-axes. It is worth noting that while rotational symmetry is lifted, we keep the translational symmetry intact. The beam flows along the direction which is dictated by the IFCs according to the following relation [40

40. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69(5), 742–756 (1979). [CrossRef]

],
vg(x,y)=kω(k=(kx,ky))=ωkxx^+ωkyy^,
(1)
where kxand kyrepresent wave-vector components along x and y directions, respectively.

PWE method is performed in order to extract the dispersion characteristics of the CPC structure [41

41. S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef] [PubMed]

]. In our case, the photonic band structure calculations are traced along the Brillouin zone edges starting at the Г point as can be seen in Fig. 1(c).

In the spatial beam routing applications, we may not need any type of structural defects. In such a case, the shapes of the IFCs become crucial. Traditional PCs composed of cylindrical rods or holes provide symmetric IFCs with respect to x- and y-directions. On the other hand, lifting the symmetry of the predefined structure by radially shifting the location of inner air-rod brings anisotropic shape to the IFCs. Hence, the light propagation direction can be arranged by solely controlling crescent open-angleθ. The first band of the IFCs is isotropic due to validity of the effective medium theory [42

42. D. E. Aspnes, “Local-Field Effects and Effective-Medium Theory: A microscopic perspective,” Am. J. Phys. 50(8), 704–709 (1982). [CrossRef]

]. The anisotropy occurs with respect toθfor the second and higher bands. For these higher order bands, there are three basic spectral characteristics. They are self-collimation, super-prism and focusing properties. The capability of the adjusting self-collimation direction (in this case it occurs not only along x- or y-directions but also along a certain angle) and the focal point of the light beam are the additional benefits of low-symmetry unit cell implementation. As a result, there is no need to alter the structure orientation or the incidence angle to adjust the focusing location.

3. FDTD analysis of the Crescent shape PC

The computational analysis of this section is based on time domain methods by employing two-dimensional FDTD [43

43. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House Publishers, 2005).

]. In order to eliminate the back reflections coming from the ends of the finite computational window, the boundaries are surrounded by the perfectly matched layers [44

44. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]

]. We launched a source with a Gaussian distribution in the time-domain. For the numerical studies, transverse magnetic (TM) guided mode is used and the concerned non-zero electric and magnetic field components are Ez, Hx, and Hy. Then, the operational frequencies are chosen according to the different regimes of anisotropic IFCs as demonstrated in Figs. 2(a)-2(c). When the crescent open angle varies from 0° to 90°, the variation of IFCs is presented in Media 1.

The steady state electric field (Ez) intensity distributions of the CPC structures for the TM mode are shown in Figs. 3(a)
Fig. 3 The steady state e-field intensity distribution of CPC structure is shown. (a) θ=300 (b) θ=+300and (c) θ=00.(d) The schematic view of the locations of focal points and the output angleαvariations for differentθvalues.
-3(c). The αparameter denotes the angle between the optical axis and focal point. The location of the focus is represented by F. We noticed that the αvalue can be changed by alteringθ. To exemplify, the crescent open angle is 300 in Fig. 3(a) and focal point occurs at above the optical axis. On the other hand, θ is 300in Fig. 3(b) and then α becomes negative. The crescent open-angle θparameter can be set as an input control parameter that is scanned between 900to900. When the crescent open-angle θ is at 0°, the focal point location in y-direction is not changed and centered at the optical axis. An oscillation occurs in the structure and a strong focusing is observed at the end face of CPC (point F1). The position of focal point is close to end face of CPC, as shown in Fig. 3(c). Due to interference of the side lobes, there occurs another secondary focal point which is represented by F2. Three important remarks can be inferred from Fig. 3. First, the closeness of the focal point to CPC’s end face is an indication of strong curvature (focusing power) due to special form of IFC. Second, the degree of anisotropy determines the amount of focal shift along y-direction, i.e. the values ofα. Finally, the output angleαdepends on the input angle θ in a rather different manner. The functional dependency between the two parameters can be summarized in three sections as follows: first case isθ=00α=00, the second case is 00<θ900f(θ)=α, and finally the third case is 900θ<00f(θ)=+α.This dependency is summarized in Fig. 3(d). The maximum shift of focal point occurs when θ=200. If one desires to obtain focal point residing on the optical axis, then θ=00,±900.

In Fig. 4
Fig. 4 The dependency of α to θparameter is sketched. There are three operational frequencies used for each region. The different colors designate the three regions.
, the different operational frequency regions are displayed by different colors. The center frequencies of the input pulse for each region are set to ω1=0.416(2πc/a),ω2=0.390(2πc/a),and ω3=0.412(2πc/a), respectively. We can see that variance of αwith respect toθresembles a sinusoidal pattern. The maximum lateral shift of focal point occurs at θ=±20for a selected operating frequency. It can be seen from the figure that α initially increases quickly and then starts to decrease slowly as we increase θ. When we consider the employed discretization process in FDTD small discrepancy occurs while reading the locations of focus points. With a finer spatial resolution, odd-symmetric version of the (θα) graph can be obtained.

4. Discussions and selected applications

In the current work, we propose a novel type of photonic structure, called “CPC”, and by means of this structure, we are able to design miniaturized optical medium that control both the propagation direction and focusing behavior of the electromagnetic fields. The great capability of CPC to adjust the focusing and deflection of light beams is due to lowering the symmetry of the proposed structure. The two fundamental light manipulation schemes were investigated: focusing and self-collimation effects. In addition to these features, one can implement beam splitters, routers and deflectors as well. The design methodologies are briefly depicted in Figs. 6(a)
Fig. 6 The representation of the construction methods of CPCs for various application areas: (a) the design of beam-splitting and (b) beam-deflectors and routers.
and 6(b). This can be achieved by advisedly combining differently-positioned CPCs with various crescent open-angles. For instance, suppose that the upper half of the structure lying above the optical axis has negative value forθand the lower part has a positive value for θ, as shown in Fig. 6(a). Then, the composite structure can act as a beam splitter. Half of the beam can be directed upwards and the other part is molded in the reverse direction. On the other hand, if the θ value is adjusted as gradually varying along the propagation direction (sweeping from 0° to 90°), then beam routers can be implemented, which is schematically demonstrated in Fig. 6(b). The details of these proposals are kept the outside of the current study. However, we present an example that shows a two-step tilted light collimation process. To achieve this, a composite version of the structure can be obtained by cascading two pieces of CPC as shown in Fig. 7
Fig. 7 A composite CPC set up and steady-state electric field distribution. The cascade structure is obtained by combining two-block of CPC, one is negative θand the second part has positiveθ.The blue and red colors correspond to minimum and maximum values of e-field’s amplitude. Black arrow shows the location of source and the dashed-white one demonstrates the path of the propagation
. While the first part has negative θ, the second part may have positiveθ. The consequence of this combination yields self-collimated beam propagation having both positive and negative tilt angles. The source is placed at the left-side of the structure (the position is indicated by an arrow). The central part of the light beam follows the path that is highlighted with white arrows. When light travels inside the first part of the composite CPC, it bends upward. The second part of the structure routes the light wave downward. Due to the equal values of θ for both sections, the incidence and reflectance angles of the beam at the interface are equal to each other. One of the observations that can be deduced from Fig. 7 is that e-field concentrates strongly at the sharp edges of each crescent shaped cells.

The idea of splitting input power equally into two branches can be achieved by the help of the lower symmetry of CPC. The numerical investigation of Fig. 6(a) was performed and the result is shown in Fig. 8(a)
Fig. 8 Beam splitter configuration. The upper and lower parts of the CPC have opposite angleθ=±200. (a) The steady-state intensity distribution of electric field throughout the structure (Media 2). (b) The transverse intensity profile at the end of the structure.
. The source is placed in the middle of the structure at the left side. The normalized operating frequency is selected to be ω=0.421(2πc/a).The light is divided into two self-collimated branches as can be observed from the plot. The amount of spatial separation between the two lobes at the end of the structure can be adjusted by means of CPC’s length. The transverse e-field profile is represented in Fig. 8(b). Almost identical peaks show the successful splitting of light beam by using the designed CPC. By adjusting the location of input source, light splitting with variable intensity ratio can be achieved as well. In addition to that, splitting angle can be controlled by altering the opening angle of the crescent shape cells. The media file in Fig. 8(a) designates the propagation of the input light throughout the splitting structure.

One of the interesting properties of the CPCs is that although the material of the structure is itself isotropic, the formed structure may exhibit anisotropic characteristics due to its asymmetric shape of IFCs. For normalized frequencies above 0.40, the anisotropy ratio ar, defined asar=ng(ΓΧ)/ng(ΓΜ),is higher than 1.50 andng(ΓΧ)ng(ΓΧ1) [26

26. M. Lu, B. K. Juluri, S.-C. S. Lin, B. Kiraly, T. Gao, and T. J. Huang, “Beam Aperture Modification and Beam Deflection Using Gradient-Index Photonic Crystals,” J. Appl. Phys. 108(10), 103505 (2010). [CrossRef]

]. This implies that CPCs can display different optical properties for different propagation directions of the same polarized light wave and can be approximated as an anisotropic media. Usually, the anisotropic feature of materials belongs to certain type of crystals that the nonlinear optics applications heavily use them [45

45. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley & Sons; Press, 1983).

]. The proposed structure may offer an alternative way to create similar optical effect (form birefringence) that can be realized by structural manipulation of pure transparent periodic dielectric materials. The response of the structure should be investigated for both polarizations. The scope of the present work is intended not to cover this property of the CPC.

Converting the normalized values such as lattice constant, structure dimensions etc., in terms of measurable physical quantities gives us following results. When we tune frequency at 1550 nm (for the normalized frequencyω1=0.416(2πc/a)), then the lattice constant a and the radius are 644.8nmand 193.4nm,respectively. The structure dimension becomes as(W1=6.448μm)×(L1=7.737μm). The focal point maximum shifting distance in the y-direction is equal to1.289μm.The focal lengths forθ = 30, 0, and −30 are1.2675μm,0μm and1.2675μm,respectively. Even though we outline the findings of square lattice dielectric crescent shapes in air background similar behavior can be obtained by utilizing the complementary structure (i.e., air crescent shapes in dielectric background) patterned either by triangular or square lattice type.

5. Conclusions

In summary, the reduced symmetry of photonic crystals by introducing asymmetric unit cells in terms of crescent shape instead of circular ones improves the light manipulation capability via the appearance of anisotropic iso-frequency contours in the dispersion diagram. The optical characteristics of the structure were numerically investigated by means of finite-difference time-domain and plane wave expansion methods. The crescent-shaped photonic crystal demonstrates a high degree of control over the light propagation behavior in terms of focusing and self collimation of light beams. The routes of light beam can be tuned by altering the opening angle of the crescent shape. Engineering the placement of each crescent-shape cells may offer a platform for implementing various photonic functions including beam splitters and combiners, deflectors and routers without deploying any defects inside the periodic dielectric structure.

Acknowledgments

The authors are grateful for the partial financial support from the National Science Council of Turkey, TUBITAK under Grant Number: 110T306. H.K. acknowledges partial support from the Turkish Academy of Sciences Distinguished Young Scientist Award (TUBA-GEBIP).

References and links

1.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of the Light (Princeton, NJ: Princeton Univ. Press, 1995).

2.

P. R. Villeneuve and M. Piche, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B Condens. Matter 46(8), 4969–4972 (1992). [CrossRef] [PubMed]

3.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990). [CrossRef] [PubMed]

4.

E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B 10(2), 283–295 (1993). [CrossRef]

5.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62(12), 8212–8222 (2000). [CrossRef]

6.

H. Kurt, I. H. Giden, and K. Ustun, “Highly efficient and broadband light transmission in 90° nanophotonic wire waveguide bends,” J. Opt. Soc. Am. B 28(3), 495–501 (2011). [CrossRef]

7.

M. Lončar, J. Vučković, and A. Scherer, “Methods for controlling positions of guided modes of photonic-crystal waveguides,” J. Opt. Soc. Am. B 18(9), 1362–1368 (2001). [CrossRef]

8.

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

9.

S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67(23), 235107 (2003). [CrossRef]

10.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. 74(10), 1370–1372 (1999). [CrossRef]

11.

D. Chigrin, S. Enoch, C. Sotomayor Torres, and G. Tayeb, “Self-guiding in two-dimensional photonic crystals,” Opt. Express 11(10), 1203–1211 (2003). [CrossRef] [PubMed]

12.

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81(12), 2574–2577 (1998). [CrossRef]

13.

H. Kurt and D. S. Citrin, “Annular photonic crystals,” Opt. Express 13(25), 10316–10326 (2005). [CrossRef] [PubMed]

14.

X. Zhu, Y. Zhang, D. Chandra, S. C. Cheng, J. M. Kikkawa, and S. Yang, “Two-dimensional photonic crystals with anisotropic unit cells imprinted from poly (dimethylsiloxane) membranes under elastic deformation,” Appl. Phys. Lett. 93(16), 161911 (2008). [CrossRef]

15.

H. F. Ho, Y. F. Chau, H. Y. Yeh, and F. L. Wu, “Complete bandgap arising from the effects of hollow, veins, and intersecting veins in a square lattice of square dielectric rods photonic crystal,” Appl. Phys. Lett. 98(26), 263115 (2011). [CrossRef]

16.

B. Rezaei, T. Fathollahi Khalkhali, A. Soltani Vala, and M. Kalafi, “Absolute band gap properties in two-dimensional photonic crystals composed of air rings in anisotropic tellurium background,” Opt. Commun. 282(14), 2861–2869 (2009). [CrossRef]

17.

Y. Zhang, L. Kong, Z. Feng, and Z. Zheng, “PBG structures of novel two-dimensional annular photonic crystals with triangular lattice,” Optoelectron. Lett. 6(4), 281–283 (2010). [CrossRef]

18.

J. Hou, D. Gao, H. Wu, and Z. Zhou, “Polarization insensitive self-collimation waveguide in square lattice annular photonic crystals,” Opt. Commun. 282(15), 3172–3176 (2009). [CrossRef]

19.

H. Wu, L. Y. Jiang, W. Jia, and X. Y. Li, “Imaging properties of an annular photonic crystal slab for both TM-polarization and TE-polarization,” J. Opt. 13(9), 095103 (2011). [CrossRef]

20.

H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express 15(3), 1240–1253 (2007). [CrossRef] [PubMed]

21.

E. Centeno, D. Cassagne, and J. P. Albert, “Mirage and superbending effect in two-dimensional graded photonic crystals,” Phys. Rev. B 73(23), 235119 (2006). [CrossRef]

22.

H. Kurt and D. S. Citrin, “A novel optical coupler design with graded-index photonic crystals,” IEEE Photon. Technol. Lett. 19(19), 1532–1534 (2007). [CrossRef]

23.

C. Tan, T. Niemi, C. Peng, and M. Pessa, “Focusing effect of a graded index photonic crystal lens,” Opt. Commun. 284(12), 3140–3143 (2011). [CrossRef]

24.

H. Kurt, E. Colak, O. Cakmak, H. Caglayan, and E. Ozbay, “The focusing effect of graded index photonic crystals,” Appl. Phys. Lett. 93(17), 171108 (2008). [CrossRef]

25.

B. Vasić and R. Gajić, “Self-focusing media using graded photonic crystals: Focusing, Fourier transforming and imaging, directive emission, and directional cloaking,” J. Appl. Phys. 110(5), 053103 (2011). [CrossRef]

26.

M. Lu, B. K. Juluri, S.-C. S. Lin, B. Kiraly, T. Gao, and T. J. Huang, “Beam Aperture Modification and Beam Deflection Using Gradient-Index Photonic Crystals,” J. Appl. Phys. 108(10), 103505 (2010). [CrossRef]

27.

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18(19), 20321–20333 (2010). [CrossRef] [PubMed]

28.

I. Khromova and L. Melnikov, “Anisotropic photonic crystals: generalized plane wave method and dispersion symmetry properties,” Opt. Commun. 281(21), 5458–5466 (2008). [CrossRef]

29.

H. Xie and Y. Y. Lu, “Modeling two-dimensional anisotropic photonic crystals by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 26(7), 1606–1614 (2009). [CrossRef] [PubMed]

30.

B. Rezaei and M. Kalafi, “Tunable full band gap in two-dimensional anisotropic photonic crystals infiltrated with liquid crystals,” Opt. Commun. 282(8), 1584–1588 (2009). [CrossRef]

31.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61(4), R2389– R2392 (2000). [CrossRef]

32.

C. S. Kee, K. Kim, and H. Lim, “Tuning of anisotropic optical properties of two-dimensional dielectric photonic crystals,” Physica B 338(1-4), 153–158 (2003). [CrossRef]

33.

T. Trifonov, L. F. Marsal, A. Rodríguez, J. Pallarès, and R. Alcubilla, “Effects of symmetry reduction in two dimensional square and triangular lattices,” Phys. Rev. B 69(23), 235112 (2004). [CrossRef]

34.

R. Proietti Zaccaria, P. Verma, S. Kawaguchi, S. Shoji, and S. Kawata, “Manipulating full photonic band gaps in two dimensional birefringent photonic crystals,” Opt. Express 16(19), 14812–14820 (2008). [CrossRef] [PubMed]

35.

F. Guan, Z. Lin, and J. Zi, “Opening up complete photonic bandgaps by tuning the orientation of birefringent dielectric spheres in three-dimensional photonic crystals,” J. Phys. Condens. Matter 17(33), L343– L349 (2005). [CrossRef]

36.

A. I. Cabuz, E. Centeno, and D. Cassagne, “Superprism effect in bidimensional rectangular photonic crystals,” Appl. Phys. Lett. 84(12), 2031–2033 (2004). [CrossRef]

37.

Y. Xu, X. J. Chen, S. Lan, Q. Guo, W. Hu, and L. J. Wu, “The all-angle self-collimating phenomenon in photonic crystals with rectangular symmetry,” J. Opt. A, Pure Appl. Opt. 10(8), 085201 (2008). [CrossRef]

38.

Y. Ogawa, Y. Omura, and Y. Iida, “Study on Self-collimated light-focusing device using the 2-D Photonic Crystal with a Parallelogram Lattice,” J. Lightwave Technol. 23(12), 4374–4381 (2005). [CrossRef]

39.

D. Gao, Z. Zhou, and D. S. Citrin, “Self-collimated waveguide bends and partial bandgap reflection of photonic crystals with parallelogram lattice,” J. Opt. Soc. Am. A 25(3), 791–795 (2008). [CrossRef] [PubMed]

40.

P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69(5), 742–756 (1979). [CrossRef]

41.

S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef] [PubMed]

42.

D. E. Aspnes, “Local-Field Effects and Effective-Medium Theory: A microscopic perspective,” Am. J. Phys. 50(8), 704–709 (1982). [CrossRef]

43.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House Publishers, 2005).

44.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]

45.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley & Sons; Press, 1983).

46.

G. Si, A. J. Danner, S. Lang Teo, E. J. Teo, J. Teng, and A. A. Bettiol, “Photonic crystal structures with ultrahigh aspect ratio in lithium niobate fabricated by focused ion beam milling,” J. Vac. Sci. Technol. B 29(2), 021205–021209 (2011). [CrossRef]

47.

J. Feng, Y. Chen, J. Blair, H. Kurt, R. Hao, D. S. Citrin, C. J. Summers, and Z. Zhou, “Fabrication of annular photonic crystals by atomic layer deposition and sacrificial etching,” J. Vac. Sci. Technol. B 27(2), 568–572 (2009). [CrossRef]

48.

R. R. Panepucci, H. B. Kim, R. V. Almeida, and M. D. Jones, “Photonic crystals in polymers by direct electron-beam lithography presenting a photonic band gap,” J. Vac. Sci. Technol. B 22(6), 3348–3351 (2004). [CrossRef]

49.

P. Borel, A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12(9), 1996–2001 (2004). [CrossRef] [PubMed]

OCIS Codes
(130.0130) Integrated optics : Integrated optics
(130.2790) Integrated optics : Guided waves
(130.3120) Integrated optics : Integrated optics devices
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Integrated Optics

History
Original Manuscript: November 23, 2011
Revised Manuscript: February 3, 2012
Manuscript Accepted: March 11, 2012
Published: March 14, 2012

Virtual Issues
Vol. 7, Iss. 5 Virtual Journal for Biomedical Optics

Citation
H. Kurt, M. Turduev, and I. H. Giden, "Crescent shaped dielectric periodic structure for light manipulation," Opt. Express 20, 7184-7194 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7184


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References

  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of the Light (Princeton, NJ: Princeton Univ. Press, 1995).
  2. P. R. Villeneuve, M. Piche, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B Condens. Matter 46(8), 4969–4972 (1992). [CrossRef] [PubMed]
  3. K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990). [CrossRef] [PubMed]
  4. E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B 10(2), 283–295 (1993). [CrossRef]
  5. S. G. Johnson, P. R. Villeneuve, S. Fan, J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62(12), 8212–8222 (2000). [CrossRef]
  6. H. Kurt, I. H. Giden, K. Ustun, “Highly efficient and broadband light transmission in 90° nanophotonic wire waveguide bends,” J. Opt. Soc. Am. B 28(3), 495–501 (2011). [CrossRef]
  7. M. Lončar, J. Vučković, A. Scherer, “Methods for controlling positions of guided modes of photonic-crystal waveguides,” J. Opt. Soc. Am. B 18(9), 1362–1368 (2001). [CrossRef]
  8. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74(9), 1212–1214 (1999). [CrossRef]
  9. S. Foteinopoulou, C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67(23), 235107 (2003). [CrossRef]
  10. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. 74(10), 1370–1372 (1999). [CrossRef]
  11. D. Chigrin, S. Enoch, C. Sotomayor Torres, G. Tayeb, “Self-guiding in two-dimensional photonic crystals,” Opt. Express 11(10), 1203–1211 (2003). [CrossRef] [PubMed]
  12. Z. Y. Li, B. Y. Gu, G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. 81(12), 2574–2577 (1998). [CrossRef]
  13. H. Kurt, D. S. Citrin, “Annular photonic crystals,” Opt. Express 13(25), 10316–10326 (2005). [CrossRef] [PubMed]
  14. X. Zhu, Y. Zhang, D. Chandra, S. C. Cheng, J. M. Kikkawa, S. Yang, “Two-dimensional photonic crystals with anisotropic unit cells imprinted from poly (dimethylsiloxane) membranes under elastic deformation,” Appl. Phys. Lett. 93(16), 161911 (2008). [CrossRef]
  15. H. F. Ho, Y. F. Chau, H. Y. Yeh, F. L. Wu, “Complete bandgap arising from the effects of hollow, veins, and intersecting veins in a square lattice of square dielectric rods photonic crystal,” Appl. Phys. Lett. 98(26), 263115 (2011). [CrossRef]
  16. B. Rezaei, T. Fathollahi Khalkhali, A. Soltani Vala, M. Kalafi, “Absolute band gap properties in two-dimensional photonic crystals composed of air rings in anisotropic tellurium background,” Opt. Commun. 282(14), 2861–2869 (2009). [CrossRef]
  17. Y. Zhang, L. Kong, Z. Feng, Z. Zheng, “PBG structures of novel two-dimensional annular photonic crystals with triangular lattice,” Optoelectron. Lett. 6(4), 281–283 (2010). [CrossRef]
  18. J. Hou, D. Gao, H. Wu, Z. Zhou, “Polarization insensitive self-collimation waveguide in square lattice annular photonic crystals,” Opt. Commun. 282(15), 3172–3176 (2009). [CrossRef]
  19. H. Wu, L. Y. Jiang, W. Jia, X. Y. Li, “Imaging properties of an annular photonic crystal slab for both TM-polarization and TE-polarization,” J. Opt. 13(9), 095103 (2011). [CrossRef]
  20. H. Kurt, D. S. Citrin, “Graded index photonic crystals,” Opt. Express 15(3), 1240–1253 (2007). [CrossRef] [PubMed]
  21. E. Centeno, D. Cassagne, J. P. Albert, “Mirage and superbending effect in two-dimensional graded photonic crystals,” Phys. Rev. B 73(23), 235119 (2006). [CrossRef]
  22. H. Kurt, D. S. Citrin, “A novel optical coupler design with graded-index photonic crystals,” IEEE Photon. Technol. Lett. 19(19), 1532–1534 (2007). [CrossRef]
  23. C. Tan, T. Niemi, C. Peng, M. Pessa, “Focusing effect of a graded index photonic crystal lens,” Opt. Commun. 284(12), 3140–3143 (2011). [CrossRef]
  24. H. Kurt, E. Colak, O. Cakmak, H. Caglayan, E. Ozbay, “The focusing effect of graded index photonic crystals,” Appl. Phys. Lett. 93(17), 171108 (2008). [CrossRef]
  25. B. Vasić, R. Gajić, “Self-focusing media using graded photonic crystals: Focusing, Fourier transforming and imaging, directive emission, and directional cloaking,” J. Appl. Phys. 110(5), 053103 (2011). [CrossRef]
  26. M. Lu, B. K. Juluri, S.-C. S. Lin, B. Kiraly, T. Gao, T. J. Huang, “Beam Aperture Modification and Beam Deflection Using Gradient-Index Photonic Crystals,” J. Appl. Phys. 108(10), 103505 (2010). [CrossRef]
  27. B. Vasić, G. Isić, R. Gajić, K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18(19), 20321–20333 (2010). [CrossRef] [PubMed]
  28. I. Khromova, L. Melnikov, “Anisotropic photonic crystals: generalized plane wave method and dispersion symmetry properties,” Opt. Commun. 281(21), 5458–5466 (2008). [CrossRef]
  29. H. Xie, Y. Y. Lu, “Modeling two-dimensional anisotropic photonic crystals by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 26(7), 1606–1614 (2009). [CrossRef] [PubMed]
  30. B. Rezaei, M. Kalafi, “Tunable full band gap in two-dimensional anisotropic photonic crystals infiltrated with liquid crystals,” Opt. Commun. 282(8), 1584–1588 (2009). [CrossRef]
  31. S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61(4), R2389– R2392 (2000). [CrossRef]
  32. C. S. Kee, K. Kim, H. Lim, “Tuning of anisotropic optical properties of two-dimensional dielectric photonic crystals,” Physica B 338(1-4), 153–158 (2003). [CrossRef]
  33. T. Trifonov, L. F. Marsal, A. Rodríguez, J. Pallarès, R. Alcubilla, “Effects of symmetry reduction in two dimensional square and triangular lattices,” Phys. Rev. B 69(23), 235112 (2004). [CrossRef]
  34. R. Proietti Zaccaria, P. Verma, S. Kawaguchi, S. Shoji, S. Kawata, “Manipulating full photonic band gaps in two dimensional birefringent photonic crystals,” Opt. Express 16(19), 14812–14820 (2008). [CrossRef] [PubMed]
  35. F. Guan, Z. Lin, J. Zi, “Opening up complete photonic bandgaps by tuning the orientation of birefringent dielectric spheres in three-dimensional photonic crystals,” J. Phys. Condens. Matter 17(33), L343– L349 (2005). [CrossRef]
  36. A. I. Cabuz, E. Centeno, D. Cassagne, “Superprism effect in bidimensional rectangular photonic crystals,” Appl. Phys. Lett. 84(12), 2031–2033 (2004). [CrossRef]
  37. Y. Xu, X. J. Chen, S. Lan, Q. Guo, W. Hu, L. J. Wu, “The all-angle self-collimating phenomenon in photonic crystals with rectangular symmetry,” J. Opt. A, Pure Appl. Opt. 10(8), 085201 (2008). [CrossRef]
  38. Y. Ogawa, Y. Omura, Y. Iida, “Study on Self-collimated light-focusing device using the 2-D Photonic Crystal with a Parallelogram Lattice,” J. Lightwave Technol. 23(12), 4374–4381 (2005). [CrossRef]
  39. D. Gao, Z. Zhou, D. S. Citrin, “Self-collimated waveguide bends and partial bandgap reflection of photonic crystals with parallelogram lattice,” J. Opt. Soc. Am. A 25(3), 791–795 (2008). [CrossRef] [PubMed]
  40. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69(5), 742–756 (1979). [CrossRef]
  41. S. Johnson, J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef] [PubMed]
  42. D. E. Aspnes, “Local-Field Effects and Effective-Medium Theory: A microscopic perspective,” Am. J. Phys. 50(8), 704–709 (1982). [CrossRef]
  43. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House Publishers, 2005).
  44. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]
  45. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley & Sons; Press, 1983).
  46. G. Si, A. J. Danner, S. Lang Teo, E. J. Teo, J. Teng, A. A. Bettiol, “Photonic crystal structures with ultrahigh aspect ratio in lithium niobate fabricated by focused ion beam milling,” J. Vac. Sci. Technol. B 29(2), 021205–021209 (2011). [CrossRef]
  47. J. Feng, Y. Chen, J. Blair, H. Kurt, R. Hao, D. S. Citrin, C. J. Summers, Z. Zhou, “Fabrication of annular photonic crystals by atomic layer deposition and sacrificial etching,” J. Vac. Sci. Technol. B 27(2), 568–572 (2009). [CrossRef]
  48. R. R. Panepucci, H. B. Kim, R. V. Almeida, M. D. Jones, “Photonic crystals in polymers by direct electron-beam lithography presenting a photonic band gap,” J. Vac. Sci. Technol. B 22(6), 3348–3351 (2004). [CrossRef]
  49. P. Borel, A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12(9), 1996–2001 (2004). [CrossRef] [PubMed]

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