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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 3 — Feb. 5, 2007
  • pp: 1234–1239
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Directional emitter and beam splitter based on self-collimation effect

W. Y. Liang, J. W. Dong, and H. Z. Wang  »View Author Affiliations


Optics Express, Vol. 15, Issue 3, pp. 1234-1239 (2007)
http://dx.doi.org/10.1364/OE.15.001234


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Abstract

We present a self-collimation-based directional emitter and compact beam splitters in a two-dimensional photonic crystal by using the surface modification method. The simulation results show that highly directional emission with a small angular divergence is achieved over a relative bandwidth of about 10.2%. Furthermore, by only modifying the monolayer structure of the output surface, the compact beam splitters, including the Y-shaped, one-to-three, and one-to-five structures are realized. Such beam splitters have remarkable properties such as symmetrical energy distribution and high transmission.

© 2007 Optical Society of America

1. Introduction

Since last decade, photonic crystals (PCs) have attracted a great deal of attention due to their unique properties and the promise in photonic integrated circuits [1

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58,2059–2062 (1987). [CrossRef] [PubMed]

7

7. J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in Planar Photonic Crystals,” IEEE J. Sel. Top. Quantum Electron. 8,1246–1257 (2002). [CrossRef]

]. Particularly, great efforts have been put into the development of defect-based devices [3

3. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425,944-–947 (2003). [CrossRef] [PubMed]

, 4

4. A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77,3787–3790 (1996). [CrossRef] [PubMed]

]. Two prominent examples are directional emission and beam splitters.

Recently enlightened by Refs. [8

8. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297,820 (2002). [CrossRef] [PubMed]

, 9

9. L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional Emission from a single subwavelength Aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90,167401 (2003). [CrossRef] [PubMed]

], two kinds of PC waveguide structures are designed for directional emission. The first one is a PC waveguide structure with surface corrugation for the excitation of surface modes at the PC output surface to realize directional emission, which has been theoretically studied [10

10. E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, Efficient coupling of light into and out of a photonic crystal waveguide via surface modes,” Photon. Nano. Fund. Appl. 2,97–102 (2004). [CrossRef]

12

12. S. K. Morrison and Y. S. Kivshar, “Engineering of directional emission from photonic-crystal waveguides,” Appl. Phys. Lett. 86,081110 (2005). [CrossRef]

] and experimentally validated [13

13. P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92,113903 (2004). [CrossRef] [PubMed]

, 14

14. I. Bulu, H. Caglayan, and E. Ozbay, “Beaming of light and enhanced transmission via surface modes of photonic crystals,” Opt. Lett. 30,3078–3080 (2005). [CrossRef] [PubMed]

]. The other one is a PC waveguide structure with some point defects at the termination [15

15. C. C. Chen, T. Pertsch, R. Iliew, F. Lederer, and A. Tünnermann, “Directional emission from photonic crystal waveguides,” Opt. Express 14,2423–2428 (2006). [CrossRef] [PubMed]

]. Besides directional emission, the research on beam splitters about the PC waveguide structures has also gotten important progress. Among them, coupled-cavity PC waveguides [16

16. M. Bayindir, B. Temelkuran, and E. Ozbay, “Photonic-crystal-based beam splitters,” Appl. Phys. Lett. 77,3902–3904 (2000). [CrossRef]

] and line-defect PC waveguides [17–19

17. T. Sóndergaard and K. H. Dridi, “Energy flow in photonic crystal waveguides,” Phys. Rev. B 61,15688–15696 (2000). [CrossRef]

] are two representative structures.

However, directional emission and beam splitters with a self-collimation PC only modified on surface structures have not been reported. These structures are not only significant to the applications due to their wide-bandwidth operation, but also are favorable to fabrication.

In this paper, we propose the directional emitter based on self-collimation effect over a large relative bandwidth by the surface modification method. Additionally, we demonstrate that the compact beam splitters with symmetrical energy distribution and high transmittance can be realized by simply modifying the monolayer structure of the output surface.

2. Structural analysis

A 2D PC with square lattice of dielectric cylinders in air is considered, in which the dielectric constant of dielectric cylinders εr=9 and radius of dielectric cylinders R=0.5a (a is the lattice constant). Only H-polarization (magnetic field along the cylinder axis) is studied in this paper. With this set of parameters, the dispersion and the EFS diagram of this 2D PC are calculated by plane wave expansion method [26

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

], and the results are shown in Figs. 1(a) and 1(b), respectively. The dispersion diagram clearly shows that only one band with negative group velocity lies in a wide frequency range. Moreover, from Fig. 1(b) we find that the EFSs are near square shaped around the Γ point in the frequency range 0.28–0.31(2πc/a), which means that self-collimation phenomena may occur, especially for ω0=0.30(2πc/a). This is the frequency range in which we are interest, denoted by the grey area in Fig. 1(a). We will take the frequency ω0 as an example to demonstrate both the directional emitter and beam splitters in the following discussions.

Fig. 1. (a). The H-polarization dispersion diagram of the square PC consisting of dielectric cylinders in air, with dielectric constant εr=9 and radius R=0.5a. The inset is the first Brillouin zone of the square PC. The grey area denotes the self-collimation frequency range. (b) The corresponding EFSs at several frequencies ω=0.27–0.32(2πc/a) (from outer to inner) in the K-space of the second photonic band for H-polarization.

Here, we discuss their physical mechanism. When the electromagnetic (EM) wave is incident upon the input surface, owing to the multiple scattering of cylinders of the PC, the wave-front reorganizes [27

27. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in Media with a Negative Refractive Index,” Phys. Rev. Lett. 90,107402 (2003). [CrossRef] [PubMed]

] and the EM wave undergoes self-collimation within the PC. At the output surface, the wave-front once again reorganizes. Finally, the directional emission phenomenon and various beam splitting effects reported in this paper appear. Simulation results show that surface modifications play an important role in the wave-front reorganization at the interfaces. By properly modifying the input/output surface structure, various useful devices can be designed. Here we use approximation method to optimize the parameters (the radii or refractive index of the cylinders) on the input/output surface of PC.

3. Directional emitter

We consider a 2D finite PC structure of 11a×37a with the parameters described in Fig. 1 for the directional emitter. Its surface is along the ΓX direction. The first and last layers of the PC are at Z=0 and Z=10, as shown in Fig. 2, respectively. A 2D point source (i.e. line source) with frequency ω0 is placed 3a before the input surface center of the PC structure. The finite-difference time-domain (FDTD) [28

28. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Boston, 2000), Chap. 3.

] method is employed to calculate the propagation of the EM wave. Firstly, we show the result of the PC with unchanged surface in the Fig. 2(a). The right part of Fig. 2(a) shows the spatial distribution of the Poynting vector of the EM wave. When the diverging EM wave emitted from the point source is incident upon the PC, it is collimated and then travels across the PC along a highly regular guiding channel. And in the opposite side, the EM wave once again diverges in the form of oscillating radiation patterns. We find that part of the energy spreads along the input surface and gradually loses, which is undesirable in our following discussion.

Fig. 2. The 2D PC structure with the size of 11a×37a and the spatial distribution of the Poynting vector. The PC area is denoted by the two black vertical lines. The point source is at (-3a, 0). (a) Unchanged surface; (b) The PC structure is modified as follows. For the input surface, keeping the center seven cylinders unchanged, the radius ri and the refractive index ni of the rest cylinders are changed to be ri=0.29a and ni=4.5, respectively; for the output surface, six cylinders at (10a, 3a), (10a, 4a) (10a, 5a) and (10a, -3a), (10a, -4a), (10a, -5a) are removed.

In order to achieve directional emission along the output surface normal, we modify both surface structures to modulate the wave-front of the EM wave to obtain desired effect. The PC structure is shown in Fig. 2(b). Firstly, modifying the parameters of the input surface will prevent the spread of power along the input surface and enhance the transmittance. For a square lattice with the modified parameters at each side of the input surface, the surface modes are suppressed to couple with the continuum of air modes. So the EM wave is forbidden to propagate along the input surface (i.e. the X-axis), whereas permitted along the Z-axis for some center cylinders unchanged. Further calculations show that one single line surface modification of the PC is enough to achieve this effect. Moreover, by taking these modifications, only the EM wave within a certain incident angle range can enter the PC, and then three self-collimation guiding channels are formed by the complex scattering effect within the PC, with their central positions approximately at X=0, ±4a, respectively. Secondly, by symmetrically removing several cylinders of the output surface, the wave-front of the EM wave coming out of the PC is modulated and thus produces a narrowed directional emitting beam along the output surface normal with a small divergence angle about 21.4°.

Furthermore, the simulations for other frequencies within the vicinity of ω0 [i.e. ω=0.28, 0.29, 0.31(2πc/a)] are also carried out. The simulation results show that directional emitter can also be achieved at these frequencies with different divergence angles. The divergence angles are 34.3°, 27.4°, and 31.3° for 0.28, 0.29, and 0.31(2πc/a), respectively. By comparing these divergence angles, it is found that the divergence angle for ω0 is the smallest among the four picked frequencies, which results from the best self-collimation effect, as shown in Fig. 1(b).

4. Beam splitters

As for beam splitters, we consider a PC structure with larger size of 11a×61a. We only need to modify the output surface structure to obtain desired effect. It is shown that while the radii of the center five cylinders of the output surface are simultaneously modified to a proper value, a Y-shaped beam splitter with excellent performance can be achieved. The PC structure is shown in Fig. 3(a). A Gaussian H-polarization parallel light source propagating along Z-axis is placed at (-3a, 0). Its frequency and width are ω0 and 12a, respectively. We show some details of the Y-shaped beam splitter. In order to characterize the quality of the Y-shaped beam splitter, we measure the power PN at the far field position (49a, 0), normalized to the input power, incident upon a detector with cross-sectional length of 3a. We decrease the radius ro from 0.5a to 0 (taking 0.1a as the changing step between 0.30a and 0.10a), and the FDTD simulations are carried out. With the decreasing ro, the wave-front of the output parallel light is modulated in varying degrees and Y-shaped beam splitter with different quality are obtained. Furthermore, different values of PN at (49a, 0) are obtained, and the relation between PN and ro is plotted in Fig. 3(b). It is clearly shown that as ro decreases from 0.5a to 0, the power along the surface normal X=0, i.e. PN, drops rapidly to a minimum value around ro=0.18a and then ascends slowly. PN keeps stable and small within the range of radius ro (0.15a, 0.21a). This is the range where we can realize a Y-shaped beam splitter with high quality. As an example, Fig. 3(a) shows the spatial distribution of Poynting vector for the case ro =0.18a and we find that the incident parallel light is split into two light beams, namely, a Y-shaped beam splitter with symmetrical energy distribution is achieved. The measured transmittance of the Y-shaped beam splitter is 91.34%.

Fig. 3. (a). The 2D PC structure and the spatial distribution of the Poynting vector for the Y-shaped beam splitter. The radius of the center five cylinders of the output surface is changed to be ro=0.18a. The short black line at (49a, 0) denotes the position of the power detector to measure PN. (b) The relation between PN and ro.

Besides being designed to form a Y-shaped beam splitter, the 2D PC can also be designed to split one parallel light beam into three or even five beams. The PC structures and the distributions of the Poynting vector of the EM wave are shown in Figs. 4(a) and 4(b) for the one-to-three and one-to-five beam splitters, respectively. We find that they are both symmetrical about the output surface normal, i.e. the line X=0. For the one-to-three beam splitter, the modifications of the output surface modulate the wave-front of the output EM wave and three split light beams are produced. The power ratio of the middle to the lateral beams can be adjusted by changing the radius rc of the center cylinder of the output surface, which is 1.2:1 for the set of parameters shown in Fig. 4(a). For the one-to-five beam splitter, Fig. 4(b) clearly shows that five split light beams are formed via wave-front reorganization. The power ratio of the three light beams denoted by 1, 2, and 3 in Fig. 4(b) is 2.5:1.6:1. The measured transmittances of the one-to-three and -five beam splitters are 95.288% and 93.782%, respectively.

Fig. 4. The 2D PC structure and the spatial distribution of the Poynting vector. (a) One-to-three beam splitter. The radius ro of four cylinders at (10a, 3a), (10a, 4a) and (10a, -3a), (10a, -4a), as well as the radius rc of the center cylinder of the output surface at (10a, 0), are changed to be ro= rc=0. Two short black lines denote the positions of two detectors. (b) One-to-five beam splitter. Letting rc=0, and keeping every five cylinders next to (10a, 0) at both sides unchanged, the radius of the rest output surface cylinders is symmetrically changed to be ro=0.25a. 1, 2 and 3 denote the split beams and the three short black lines denote the positions of the detectors.

5. Conclusion

In conclusion, we accomplish the directional emitter and compact beam splitters based on self-collimation effect by using the surface modification method. The FDTD method is employed to demonstrate these phenomena where the surface structures play an important role. The simulation results show that the directional emitter can be achieved over a relative bandwidth of about 10.2% by taking modifications to both of the input and output surfaces. Furthermore, the compact beam splitters, including the Y-shaped, one-to-three, and -five structures, can be achieved by only modifying the monolayer structure of the output surface. All these beam splitters possess symmetrical energy distribution and high transmittance. These self-collimation-based devices can be used to enhance the light coupling efficiency, narrow the beam divergence of microcavity laser over a large frequency range, or divide an optical beam into multiple signals. Moreover, they may have potential applications in integrated optical and microphotonic circuits.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (10674183), National 973 of China (2004CB719804), and Ph.D Degrees Foundation of Ministry of Education of China.

References and links

1.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58,2059–2062 (1987). [CrossRef] [PubMed]

2.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58,2486–2489 (1987). [CrossRef] [PubMed]

3.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425,944-–947 (2003). [CrossRef] [PubMed]

4.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77,3787–3790 (1996). [CrossRef] [PubMed]

5.

Y. W. Li, J. Y Pan, J. Zeng, J. W. Dong, and H. Z. Wang, “Band engineering and periodic defects doping by lattices compounding,” Opt. Express 13,8526–8531 (2005). [CrossRef] [PubMed]

6.

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

7.

J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in Planar Photonic Crystals,” IEEE J. Sel. Top. Quantum Electron. 8,1246–1257 (2002). [CrossRef]

8.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297,820 (2002). [CrossRef] [PubMed]

9.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional Emission from a single subwavelength Aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90,167401 (2003). [CrossRef] [PubMed]

10.

E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, Efficient coupling of light into and out of a photonic crystal waveguide via surface modes,” Photon. Nano. Fund. Appl. 2,97–102 (2004). [CrossRef]

11.

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B 69,121402 (2004). [CrossRef]

12.

S. K. Morrison and Y. S. Kivshar, “Engineering of directional emission from photonic-crystal waveguides,” Appl. Phys. Lett. 86,081110 (2005). [CrossRef]

13.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly directional emission from photonic crystal waveguides of subwavelength width,” Phys. Rev. Lett. 92,113903 (2004). [CrossRef] [PubMed]

14.

I. Bulu, H. Caglayan, and E. Ozbay, “Beaming of light and enhanced transmission via surface modes of photonic crystals,” Opt. Lett. 30,3078–3080 (2005). [CrossRef] [PubMed]

15.

C. C. Chen, T. Pertsch, R. Iliew, F. Lederer, and A. Tünnermann, “Directional emission from photonic crystal waveguides,” Opt. Express 14,2423–2428 (2006). [CrossRef] [PubMed]

16.

M. Bayindir, B. Temelkuran, and E. Ozbay, “Photonic-crystal-based beam splitters,” Appl. Phys. Lett. 77,3902–3904 (2000). [CrossRef]

17.

T. Sóndergaard and K. H. Dridi, “Energy flow in photonic crystal waveguides,” Phys. Rev. B 61,15688–15696 (2000). [CrossRef]

18.

P. Pottier, S. Mastroiacovo, and R. M. De La Rue, “Power and polarization beam-splitters, mirrors, and integrated interferometers based on air-hole photonic crystals and lateral large index-contrast waveguides,” Opt. Express 14,5617–5633, (2006). [CrossRef] [PubMed]

19.

C. C. Chen, H. D. Chien, and P. G. Luan, “Photonic crystal beam splitters,” Appl. Opt. 43,6187–6190 (2004). [CrossRef] [PubMed]

20.

X. F. Yu and S. H. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83,3251–3253, (2003). [CrossRef]

21.

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]

22.

D. M. Pustai, S. Y. Shi, C. H. Chen, A. Sharkawy, and D. W. Prather, “Analysis of splitters for self-collimated beams in planar photonic crystals,” Opt. Express 12,1823–1831 (2004). [CrossRef] [PubMed]

23.

P. J. Yao, B. Chen, J. Y. Zhang, X. Y. Chen, P. Wang, and H. Ming, “Y-shaped beam splitters for self-collimated beams in 2D photonic crystals,” Europhys. Lett. 70,197–203 (2005). [CrossRef]

24.

G. P. Nordin, S. Kim, J. B. Cai, and J. H. Jiang, “Hybrid integration of conventional waveguide and photonic crystal structures,” Opt. Express 10,1334–1341 (2002). [PubMed]

25.

S. Kim, G. P. Nordin, J. B. Cai, and J. H. Jiang, “Ultracompact high-efficiency polarizing beam splitter with a hybrid photonic crystal and conventional waveguide structure,” Opt. Lett. 28,2384–2386 (2003). [CrossRef] [PubMed]

26.

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

27.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in Media with a Negative Refractive Index,” Phys. Rev. Lett. 90,107402 (2003). [CrossRef] [PubMed]

28.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Boston, 2000), Chap. 3.

OCIS Codes
(230.1360) Optical devices : Beam splitters
(250.5300) Optoelectronics : Photonic integrated circuits

ToC Category:
Photonic Crystals

History
Original Manuscript: November 14, 2006
Revised Manuscript: December 22, 2006
Manuscript Accepted: January 4, 2007
Published: February 5, 2007

Citation
W. Y. Liang , J. W. Dong, and H. Z. Wang, "Directional emitter and beam splitter based on self-collimation effect," Opt. Express 15, 1234-1239 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-1234


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References

  1. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059 - 2062 (1987). [CrossRef] [PubMed]
  2. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486 - 2489 (1987). [CrossRef] [PubMed]
  3. Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944 - 947 (2003). [CrossRef] [PubMed]
  4. A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in Photonic Crystal Waveguides," Phys. Rev. Lett. 77, 3787 - 3790 (1996). [CrossRef] [PubMed]
  5. Y. W. Li, J. Y Pan, J. Zeng, J. W. Dong, and H. Z. Wang, "Band engineering and periodic defects doping by lattices compounding," Opt. Express 13, 8526-8531 (2005). [CrossRef] [PubMed]
  6. 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, 1212 - 1214 (1999). [CrossRef]
  7. J. Witzens, M. Loncar, and A. Scherer, "Self-collimation in Planar Photonic Crystals," IEEE J. Sel. Top. Quantum Electron. 8, 1246 - 1257 (2002). [CrossRef]
  8. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820 - 822 (2002). [CrossRef] [PubMed]
  9. L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional Emission from a single subwavelength Aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003). [CrossRef] [PubMed]
  10. E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, "Efficient coupling of light into and out of a photonic crystal waveguide via surface modes," Photon. Nano. Fundam. Appl. 2, 97-102 (2004). [CrossRef]
  11. E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402 (2004). [CrossRef]
  12. S. K. Morrison and Y. S. Kivshar, "Engineering of directional emission from photonic-crystal waveguides," Appl. Phys. Lett. 86, 081110 (2005). [CrossRef]
  13. P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, "Highly directional emission from photonic crystal waveguides of subwavelength width," Phys. Rev. Lett. 92, 113903 (2004). [CrossRef] [PubMed]
  14. I. Bulu, H. Caglayan, and E. Ozbay, "Beaming of light and enhanced transmission via surface modes of photonic crystals," Opt. Lett. 30, 3078-3080 (2005). [CrossRef] [PubMed]
  15. C. C. Chen, T. Pertsch, R. Iliew, F. Lederer, and A. Tünnermann, "Directional emission from photonic crystal waveguides," Opt. Express 14, 2423-2428 (2006). [CrossRef] [PubMed]
  16. M. Bayindir, B. Temelkuran, and E. Ozbay, "Photonic-crystal-based beam splitters," Appl. Phys. Lett. 77, 3902-3904 (2000). [CrossRef]
  17. T. Sóndergaard and K. H. Dridi, "Energy flow in photonic crystal waveguides," Phys. Rev. B 61, 15688-15696 (2000). [CrossRef]
  18. P. Pottier, S. Mastroiacovo, and R. M. De La Rue, "Power and polarization beam-splitters, mirrors, and integrated interferometers based on air-hole photonic crystals and lateral large index-contrast waveguides," Opt. Express 14, 5617-5633 (2006). [CrossRef] [PubMed]
  19. C. C. Chen, H. D. Chien, and P. G. Luan, "Photonic crystal beam splitters," Appl. Opt. 43, 6187-6190 (2004). [CrossRef] [PubMed]
  20. X. F. Yu and S. H. Fan, "Bends and splitters for self-collimated beams in photonic crystals," Appl. Phys. Lett. 83, 3251-3253 (2003). [CrossRef]
  21. 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]
  22. D. M. Pustai, S. Y. Shi, C. H. Chen, A. Sharkawy, and D. W. Prather, "Analysis of splitters for self-collimated beams in planar photonic crystals," Opt. Express 12, 1823-1831 (2004). [CrossRef] [PubMed]
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