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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 23 — Nov. 8, 2010
  • pp: 24245–24257
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Beam splitting at the output of photonic crystal waveguides with discrete surface point defects

Qi Wang, Lanlan Zhang, and Qi Li  »View Author Affiliations


Optics Express, Vol. 18, Issue 23, pp. 24245-24257 (2010)
http://dx.doi.org/10.1364/OE.18.024245


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Abstract

With the method of adding two point defects on modulated surface, novel photonic crystal (PC) waveguide-based beam splitters were presented. The modulated surface layer supports surface states, and introduced discrete point defects can serve as discrete light emitters. The finite-difference time-domain (FDTD) simulations show that the number of beams is sensitive to the distance of two point defects. By adjusting the positions of the point defects, 1-to-N beam splitters can be realized. These simple, easy-to-fabricate and controllable structures have important potential applications in integrated optical circuits.

© 2010 OSA

1. Introduction

In the past several years photonic crystals have attracted much interest due to their unique properties. Among many types of applications for photonic crystal, beam splitter is one of the most indispensable components. Two kinds of open-type [1

1. L. Y. Jiang, W. Jia, H. P. Li, X. Y. Li, C. X. Cong, and Z. X. Shen, “Inverse design for directional emitter and power splitter based on photonic crystal waveguide with surface corrugations,” J. Opt. Soc. Am. B 26(11), 2157–2160 (2009). [CrossRef]

] beam splitter utilizing photonic crystal (PC) have been proposed. One is self-collimation PC with modified surface [2

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

]. It is believed that the wave-front reorganization plays an important role in this design. The other one is PC waveguide with corrugated surface to the exit surface [1

1. L. Y. Jiang, W. Jia, H. P. Li, X. Y. Li, C. X. Cong, and Z. X. Shen, “Inverse design for directional emitter and power splitter based on photonic crystal waveguide with surface corrugations,” J. Opt. Soc. Am. B 26(11), 2157–2160 (2009). [CrossRef]

, 3

3. W. Jia, L. Y. Jiang, K. Chen, and X. Y. Li, “Design of photonic crystal power beam splitters via corrugated and gratinglike surfaces,” Opt. Commun. 283(20), 4078–4084 (2010). [CrossRef]

], which is designed by inverse strategy. Besides the above structures, beam splitter utilizing PC waveguide with asymmetric open cavities has also been proposed [4

4. M. Q. Xin, L. Zhang, C. Eng Png, J. H. Teng, and J. Aaron Danner, “Asymmetric open cavities for beam steering and switching from line-defect photonic crystals,” J. Opt. Soc. Am. B 27(6), 1153–1157 (2010). [CrossRef]

].

Surface waves [5

5. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B Condens. Matter 44(19), 10961–10964 (1991). [CrossRef] [PubMed]

7

7. B. Wang, W. Dai, A. Fang, L. Zhang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Surface waves in photonic crystal slabs,” Phys. Rev. B 74(19), 195104 (2006). [CrossRef]

] are electromagnetic waves localized at the interface between a PC and another medium, similar to surface plasmons [8

8. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988)

] on metallic surfaces. In 2004, Moreno et al. [9

9. 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(12), 121402 (2004). [CrossRef]

] proposed that the light emerging from a PC structure can be compressed into a narrow beam. Since then the beaming effect of PC waveguide with period corrugated surface is studied experimentally and theoretically [10

10. R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76(23), 235417 (2007). [CrossRef]

12

12. H. Caglayan, I. Bulu, and E. Ozbay, “Off-axis directional beaming via photonic crystal surface modes,” Appl. Phys. Lett. 92(9), 092114 (2008). [CrossRef]

]. The major motivation for the discovery beaming effect from PC waveguide with modified surface is largely provided by the physics of extraordinary optical transmission through subwavelength metal aperture [13

13. 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(5582), 820–822 (2002). [CrossRef] [PubMed]

16

16. D. Z. Lin, C. K. Chang, Y. C. Chen, D. L. Yang, M. W. Lin, J. T. Yeh, J. M. Liu, C. H. Kuan, C. S. Yeh, and C. K. Lee, “Beaming light from a subwavelength metal slit surrounded by dielectric surface gratings,” Opt. Express 14(8), 3503–3511 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3503. [CrossRef] [PubMed]

]. The reason is that: a metallic surface and the surface of a modulated PC have in common the fact that both surfaces can support surface propagating electromagnetic (EM) waves. All the above beaming effects that occurred in PCs result from periodic corrugation at the output surface. Recently, Y. L. Zhang et al. [17

17. Y. L. Zhang, D. Y. Zhao, C. H. Zhou, and X. Y. Jiang, “Directional light emission through a metallic nanostructure,” J. Appl. Phys. 105(11), 113124 (2009). [CrossRef]

] shown that steering a beam split is possible by designing discrete bumps on metallic nanostructure. This method is simple and easily handled experimentally.

2. Structure and simulation results

2.1 Photonic crystal structure and surface mode

The beam splitters we proposed and simulated are based on a two-dimensional PC. The PC structure used here is the same as that used by E. Moreno [9

9. 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(12), 121402 (2004). [CrossRef]

]. The PC structure (shown in Fig. 1
Fig. 1 Schematic diagram of the photonic crystal waveguide structures, output surface cylinders with rs = 0.09a (denoted by red dots).
) consists of a square array of dielectric cylinders with a lattice constant a, dielectric constant ε = 11.56(e.g., GaAs at a wavelength of 1.5μm) and radius r = 0.18a embedded in air. For this set of parameters the system presents, the PC has a TE-polarization (electric field parallels to the axis of cylinders) band gap in the frequency range of ω = 0.30 × 2πc/a to ω = 0.44 × 2πc/a.Where ω is the angular frequency and c is the speed of light in free space. After one row of cylinders is removed, a PC waveguide can be formed, as shown in the Fig. 1. For source frequencies within the band-gap range, this waveguide supports one propagating mode.

Surface mode is a kind of surface state that decays both in air and the PC [5

5. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B Condens. Matter 44(19), 10961–10964 (1991). [CrossRef] [PubMed]

, 18

18. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystal: Molding the flow of light (Princeton University Press, Princeton, 1995).

]. In Ref. 5 the authors have found that three types of modes can be supported by an uncorrugated PC: extended both in the air and the PC, extended in the air and decaying in the PC, decaying in the air but extended in the PC. Therefore, the uncorrugated PC does not support surface modes which decay both in the air and PC. In order to create surface modes, one possible way would be to modify the parameter of surface layer, rendering it different from the air on one side and the PC bulk on the other side. In this paper, the radius of the surface layer cylinders are reduce to rs = 0.5r = 0.09a (denoted by red dots), such a surface structure supports surface mode. One nearly zero group-velocity surface mode can be found for k0 = π/a and ω0 = 0.408 × 2πc/a [9

9. 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(12), 121402 (2004). [CrossRef]

].

Figure 2(a)-(b)
Fig. 2 Simulated spatial distributions of |Ey|2 for the structure illustrated in Fig. 1. The frequency and width of the incident beam are ω = 0.408 (2πc/a) and a, respectively. (a) Unchanged surface (no surface modes allowed), (b) surface cylinders with rs = 0.09a.
shows the |Ey|2 distributions when a guided mode with frequency ω0 = 0.408(2πc/a), launched at waveguide input port, is diffracted by photonic crystal structure (shown in Fig. 1) without and with modified surface, respectively. Figure 2(a) indicates that since no surface modes are supported by structure, the radiated electric field shows a broad angular distribution because the actual aperture’s width (approximately 1.8a) of the waveguide is smaller than the wavelength (the wavelength corresponding to ω0 is λ = 2.45a) of the incident beam. From Fig. 2(b), it can be seen that when surface cylinders are modified, the presence of a surface mode is clear, however, since this surface mode is a nonradiative mode, hence the radiated power still spreads in all direction.

2.2 Design of beam splitting structures

In this section, the beam splitting capability of the structure shown in Fig. 3(b)
Fig. 3 Schematic drawing of the photonic crystal waveguide structures under study. (a) Structure with Ns modified surface cylinders rs = 0.09a (denoted by red dots) but without point defect. A part of surface cylinders (denoted by blue dots) are not modified, (b) structure with Ns modified surface cylinders rs = 0.09a (denoted by red dots) and two point defect at position (xs , 24.5), (-xs , 24.5), L = 2xs .
is studied. In order to obtain the beam splitting phenomenon, the structure is built in two steps: we first modify the radius of surface cylinders (shown in Fig. 3(a)) to excite surface mode, then introduce two point defects (see in Fig. 3(b)) to make surface modes discretely radiate into free space. The dimension of the structure shown in Fig. 3(a) and 3(b) are all 61 layers in the lateral direction (X direction) and 20 layers in the propagation direction (Z direction). The first and the last layers of PC structure are at z = 5.5a and z = 24.5a, respectively.

Figure 3(a) is used as reference structure for investigating the beam splitting phenomenon. In Fig. 3(a), at either side of the waveguide exit, the radius of Ns surface cylinders (denoted by red dots) are reduced to rs = 0.05r = 0.09a, thereby creating the conditions for a surface mode which localized at and propagating along PC-air interface.

Figure 3(b) represents the designed structure to obtain beam splitting phenomenon. This designed structure is produced by altering the surface layer geometry in two steps. Firstly, at either side of the waveguide exit, by reducing the radius of Ns surface cylinders to the value rs = 0.05r = 0.09a (same to Fig. 3(a)), and thereby creating the conditions for nonradiative surface modes to exist at PC-air interface. Because a part of surface cylinders (denoted by blue dots) are not modified, these surface modes cannot spread on the interface of these cylinders. Therefore, these blue cylinders act as reflecting mirrors in x direction. Surface modes can go through the central surface but are terminated by the reflecting mirror. The structure can be considered as a conventional Fabry-Perot resonator whose resonant wavelengths are given byλn=2π/kn, with kn=(NπΔϕ)/L, where n is the mode number, L is the cavity length, k is wave vector and Δϕ is the phase shift associated with reflection from the boundary. In this paper, Δϕ is close toπ [19

19. S. S. Xiao and M. Qiu, “Optical microcavities based on surface modes in two-dimensional photonic cyetals and silicon-on-insulator photonic crystals,” J. Opt. Soc. Am. B 24(5), 1225–1229 (2007). [CrossRef]

], we focus on thekn=π/a. Secondly, in order to obtain beam splitting, the nonradiative surface mode should couple to radiative modes discretely. This can be achieved by introducing two discrete point defects to the PC output surface.

When a TE-polarization source is placed at the PC waveguide (shown in Fig. 3(b)) input port z = 5a, it excites wave that propagate along the waveguide, at the waveguide exit, a part of energy emits to free space directly. Meanwhile, since surface modes are excited, a part of energy travels along the PC-air interface until it is scattered to free space by the combined effect of the point defects and the reflecting mirrors. The two diffraction beams and the beam exited from waveguide interfere with each other, then the beam splitting phenomenon appears in the output space. As a simple model, the waveguide exit acts as a source [20

20. Z. F. Li, K. Aydin, and E. Ozbay, “Wide band width directional beaming via waveguide ports in photonic crystals,” J. Phys. D Appl. Phys. 41(15), 155115 (2008). [CrossRef]

], in output space, the complicated radiation pattern of the transmitted light can be regarded as interference result of three discrete parts. One is the directly transmitted waves from the waveguide exit (source 2), and the others are the scattered waves by two point defects (source 1 and 3). The radiation pattern can be represented by the following formula:
Ey(x,y)=i=13Ey,i(x,y)
(1)
δij=kair(rirj)+Δϕiji,j=1,2,3
(2)
where Ey,i(x,y) is the electric field at the observation point which is emitted from the source i. Ey,i(x,y) and Ey,j(x,y) with a phase difference ofδij. δijis the total phase difference arising from the combined differences of initial phase and optical path length in air to point of observation for the sources i and j. ri is the distance between source i and observation point, kair is the wave vector in air, andΔϕijis the initial phase difference between the sources i and j. The distance between point defects (source 1 and 3) and waveguide output exit (source 2) is L/2, the wave vector of surface mode iskλff=kn=π/a, then the initial phase difference between source 2 and source 1 (or source 3):
Δϕ12=Δϕ32kλffL/2,andΔϕ13=0
(3)
According to Eq. (1)-(3), it is apparent that the interference fringe evolves gradually with periodΔϕ12=Δϕ32=π, i. e the period is L = 2π/kλff. In our design, the wave vector of surface modekλff = π/a (λsm = 2a), so the number of beams increases when the increment ΔLof the two point defects distance is 2a.

2.3 Simulation results and analysis

In the simulation, FDTD [21

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

] method was employed to numerically calculate the field distribution of the different structures under investigation. The perfect matched layers boundary conditions [22

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

] were used to surround the computational domain. In all simulations, the Gaussian light source with frequency ω = 0.408(2πc/a) and width a is placed at z = 5a, the spatial grid Δx and Δz in the computational domain are all set to a/40. All numerical results presented are only for TE-polarization.

Firstly, for comparison, the field distributions for the corresponding PC structure shown in Fig. 3(a) (with modified surface cylinders but without point defects) are calculated. Figure 4(a)-(d)
Fig. 4 Simulated spatial distributions of |Ey|2 for the structure illustrated in Fig. 3(a). (a) Ns = 2, (b) Ns = 3, (c) Ns = 4, (d) Ns = 5.
show the spatial |Ey|2 distributions for Ns = 2, 3, 4, 5. The presence of a surface mode is clear and the beam splitting effects appear but the splitting remains unclear. The numerical results presented here can be understood as follows: when a surface mode is created, at either side of the waveguide exit, a part of the energy coming from the waveguide does spread on the modified interface but is scattered by the blue rods (acting as reflecting mirror in x direction). The intensities of two scattering beams which are radiated into free space are slight. This lead to the interaction of scattering beams and beam exited from waveguide is weak; therefore, splitting phenomena remains unclear.

The spatial |Ey|2 distributions for the corresponding PC structure shown in Fig. 3(b) (with modified surface cylinders and point defects) are depicted in Fig. 5
Fig. 5 Simulated spatial distributions of |Ey|2 for the structure illustrated in Fig. 3(b), (a)L = 4a (b)L = 6a (c)L = 8a (d) L = 10a.
. Figure 5 shows that when the positions of point defects are xs = ± 2a, ± 3a, ± 4a, ± 5a, i.e. the distances between the defects are 2λsm, 3λsm, 4λsm and 5λsm, two, three, four, five symmetrical beams would emerge in the output half-space, respectively. The angular distributions of the time-averaged intensity along a half circle of radius of r = 20λ are depicted in Fig. 6
Fig. 6 Calculated angular distribution of |Ey|2 for radiation pattern plotted in Fig. 5 (along a half circle of r = 20λ). All the above results are obtained at a frequency of ω = 0.408 (2πc/a).
. From this figure, the divergence angle of each beam can be found. When xs = ± 2a, along two direction of θ = ± 33°, emission intensity is very large. When xs = ± 3a, the beams are along the directions of θ = 0, ± 47°. When xs = ± 4a the four beaming directions are θ = ± 17° and θ = ± 56°. When xs = ± 5a, the five beaming directions are θ = 0, θ = ± 29° and θ = ± 63°.

Figure 5 and Fig. 6 show some interesting phenomena. (1) When point defects are introduced, the beam splitting emerges in the output half-space. (2) The beams have symmetrical energy distribution. (3) The beam number depends on the distance between two point defects (L). (4) When the distance of the defects L is sm, the radiation field presents an n beam pattern. Different beam patterns can be realized when the discrete point defect positions are varied. Based on these results, 1-to-N beam splitter can be proposed. The property can be used in application of beam splitter.

In order to get more information about the underlying mechanism that yields the beam splitting phenomena, the Ponynting vector Sz can be scrutinized along the output PC-air interface. The simulation results are shown in Fig. 7
Fig. 7 Poynting field Sz(x, z = 24.5a) along the PC-air interface for the case of L = 4a, 6a, 8a, 10a.
. This figure shows Sz (x, z = 24.5a) for the case of Fig. 3(b). For L = 4a, the first peak value of Sz occurs when x = 0 which correspond to the waveguide port, the second peak value of Sz occurs when x ≈ ± 2a which correspond to the position near the point defect central. At other positions, the Sz is almost zero or negative. For L = 6a, 8a, 10a, the results are similar to L = 4a. These results agree with three source model discussed above.

3. Effects of the modified surface cylinder number

When the parameter Nb is changed from 0 to 2, and L is changed from 4a to 10a, the characters of beam splitting will vary correspondingly. The simulated |Ey|2 field intensity distributions are plotted in Fig. 9(a)-(l)
Fig. 9 Spatial distributions of |Ey|2 for the structure illustrated in Fig. 8, (a) L = 4a, Nb = 0. (b) L = 4a, Nb = 1. (c) L = 4a, Nb = 2. (d) L = 6a, Nb = 0. (e) L = 6a, Nb = 1. (f) L = 6a, Nb = 2. (g) L = 8a, Nb = 0 (h) L = 8a, Nb = 1. (i) L = 8a, Nb = 2. (j) L = 10a, Nb = 0. (k) L = 10a, Nb = 1. (l) L = 10a, Nb = 2.
, and the angular distributions are plotted in Fig. 10(a)-(d)
Fig. 10 Calculated angular distribution of |Ey|2 for radiation pattern plotted in Fig. 9 (along a half circle of r = 20λ), (a) L = 4a. (b) L = 6a. (c) L = 8a. (d) L = 10a.
. Figure 9 and 10 display distinct features. 1) The number of beam depends on the distance between two point defects and has nothing to do with Nb. 2) For all the cases of L = 4a, 6a, 8a and 10a, with the Nb increased from 0 to 2, the azimuth angle of bilateral beams increases clearly, but central beams change slightly. 3) With Nb increased from 0 to 2, the power ratio of the middle to lateral beams get closer to 1, when Nb = 2, the beams have almost the same width and intensity.

To clearly understand the underlying physical mechanism of the observed phenomenon, we measure the distribution of Ponynting vector Sz along the output PC-air interface for Nb = 0, 1, 2. The results are shown in Fig. 11
Fig. 11 Poynting field Sz(x, z = 24.5a) along the PC-air interface for Nb = 0, 1, 2, (a) L = 4a. (b) L = 6a. (c) L = 8a. (d) L = 10a.
. When Nb increased from 0 to 2, the location and amplitude of the three peaks will vary correspondingly, that is to say, the position and intensity of three point sources will change correspondingly. According to the analysis in section 2, the beam splitting of the output space is caused by the interference of these three light sources. The variation of the source at position 1, 3 would cause the variation of the beam angle in the output space. The beam intensity in the output space changes when intensity ratio of these three sources changes. However, the numbers of the beam in the output space do not change because the variation of the source at position 1,3 is minute.

4. Effects of the extra point defects (A)

In this section, extra point defects are introduced to the adjacent layer of output surface. The effects of these extra point defects on radiation patterns are investigated. In Fig. 12
Fig. 12 Schematic drawing of the photonic crystal waveguide structure with two extra point defects are introduced to the adjacent layer of output surface. These two extra point defects at position (xs -1, zs -1), here zs -1 = 23.5a, xs -1 = xs + a.
, besides the same structure shown in Fig. 3(b), two extra point defects are introduced to the position( ± xs -1, zs -1), here zs -1 = 23.5a. These two extra point defects are indicated by A. When extra point defects are introduced to PC structure, some energy is stored in the extra point defects which can affect the intensity of the three discrete sources on the surface layer. We change the xs -1 from 2a to xs + a (taking a as the changing step), and FDTD simulations are carried out. The simulation results indicate that when xs -1 changes from 2a to xs + a, the intensity of some beams in the output space increase, and the intensity of other beams in the output space decrease. The reason is that the influences on intensities of every discrete source are different when the positions of the extra defects are different. When xs- 1 = xs + a, the intensities of all the beams in the output space increase.

Figure 13(a)-(h)
Fig. 13 Spatial distributions of |Ey|2 for the structure illustrated in Fig. 12, (a) L = 4a, without A. (b) L = 4a, with A. (c) L = 6a, without A. (d) L = 6a, with A. (e) L = 8a, without A. (f) L = 8a, with A. (g) L = 10a, without A. (h) L = 10a, with A.
shows the |Ey|2 distributions of the structure (shown in Fig. 12) without A and with A (xs -1 = xs + a.). The angular distributions (20λ away from the output exit of the PC waveguide) of the time-averaged intensity are plotted in Fig. 14(a)-(d)
Fig. 14 Calculated angular distribution of |Ey|2 for radiation pattern plotted in Fig. 12 (along a half circle of r = 20λ). (a) L = 4a (b) L = 6a. (c) L = 8a. (d) L = 10a.
. Compared with the structure without A, when extra point defects (A) are introduced to the adjacent layer of surface, the energy is stronger and the energy of central beams increases clearly, but bilateral beams increase slightly. The numerical results presented above can be understood as follows: when two extra point defects (A) are introduced to PC structure, some energy is stored in the two extra point defects which can be regarded as radiating sources. The coupling between these radiating sources and the three discrete sources in the output interface will lead to the change of the interference fringes in output space.

5. Conclusion

In conclusion, we present novel photonic crystal waveguide-based beam splitters by method of adding two point defects on the modulated surface. When the positions of point defects are properly designed, the 1-to-N (n = 2, 3, 4, 5) beam splitter can be achieved. The main mechanism of beam splitting lies in the interference of light beams from waveguide exit and two point defects. Using the FDTD method, we obtained field patterns for several geometries of the photonic crystal structures. The simulation results show that when the distance between two point defect L is equal to an integer multiple of the wavelength of surface modes (L = sm, λsm = 2a), the radiation field presents an n beam pattern. In addition, the simulation results also shown that the beam properties (intensity, direction) can be well adjusted by varying modified surface cylinder number and introducing extra point defects to adjacent layer of output surface.

References and links

1.

L. Y. Jiang, W. Jia, H. P. Li, X. Y. Li, C. X. Cong, and Z. X. Shen, “Inverse design for directional emitter and power splitter based on photonic crystal waveguide with surface corrugations,” J. Opt. Soc. Am. B 26(11), 2157–2160 (2009). [CrossRef]

2.

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

3.

W. Jia, L. Y. Jiang, K. Chen, and X. Y. Li, “Design of photonic crystal power beam splitters via corrugated and gratinglike surfaces,” Opt. Commun. 283(20), 4078–4084 (2010). [CrossRef]

4.

M. Q. Xin, L. Zhang, C. Eng Png, J. H. Teng, and J. Aaron Danner, “Asymmetric open cavities for beam steering and switching from line-defect photonic crystals,” J. Opt. Soc. Am. B 27(6), 1153–1157 (2010). [CrossRef]

5.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B Condens. Matter 44(19), 10961–10964 (1991). [CrossRef] [PubMed]

6.

R. Mousse, Th. Koschny, and C. M. Soukoulis, “Excitation of surface waves in a photonic crystal with negative refraction: The role of surface termination,” Phys. Rev. B 74(11), 115111 (2006). [CrossRef]

7.

B. Wang, W. Dai, A. Fang, L. Zhang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Surface waves in photonic crystal slabs,” Phys. Rev. B 74(19), 195104 (2006). [CrossRef]

8.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988)

9.

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(12), 121402 (2004). [CrossRef]

10.

R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76(23), 235417 (2007). [CrossRef]

11.

S. K. Morrison and Y. S. Kivshar, “Engineering of directional emission from photonic-crystal waveguide,” Appl. Phys. Lett. 86(8), 081110 (2005). [CrossRef]

12.

H. Caglayan, I. Bulu, and E. Ozbay, “Off-axis directional beaming via photonic crystal surface modes,” Appl. Phys. Lett. 92(9), 092114 (2008). [CrossRef]

13.

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(5582), 820–822 (2002). [CrossRef] [PubMed]

14.

F. J. García-Vidal, L. Martín-Moreno, H. J. Lezec, and T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83(22), 4500–4502 (2003). [CrossRef]

15.

A. Degiron and T. W. Ebbesen, “Analysis of the transmission process through single apertures surrounded by periodic corrugations,” Opt. Express12, 3694–3700 (2004). http://www.opticsinfobase.org/oe/abstract. cfm?URI=oe-12-16-3694. [CrossRef] [PubMed]

16.

D. Z. Lin, C. K. Chang, Y. C. Chen, D. L. Yang, M. W. Lin, J. T. Yeh, J. M. Liu, C. H. Kuan, C. S. Yeh, and C. K. Lee, “Beaming light from a subwavelength metal slit surrounded by dielectric surface gratings,” Opt. Express 14(8), 3503–3511 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3503. [CrossRef] [PubMed]

17.

Y. L. Zhang, D. Y. Zhao, C. H. Zhou, and X. Y. Jiang, “Directional light emission through a metallic nanostructure,” J. Appl. Phys. 105(11), 113124 (2009). [CrossRef]

18.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystal: Molding the flow of light (Princeton University Press, Princeton, 1995).

19.

S. S. Xiao and M. Qiu, “Optical microcavities based on surface modes in two-dimensional photonic cyetals and silicon-on-insulator photonic crystals,” J. Opt. Soc. Am. B 24(5), 1225–1229 (2007). [CrossRef]

20.

Z. F. Li, K. Aydin, and E. Ozbay, “Wide band width directional beaming via waveguide ports in photonic crystals,” J. Phys. D Appl. Phys. 41(15), 155115 (2008). [CrossRef]

21.

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

22.

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

OCIS Codes
(230.1360) Optical devices : Beam splitters
(240.6690) Optics at surfaces : Surface waves
(250.5300) Optoelectronics : Photonic integrated circuits
(230.5298) Optical devices : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: September 7, 2010
Revised Manuscript: October 14, 2010
Manuscript Accepted: October 24, 2010
Published: November 4, 2010

Citation
Qi Wang, Lanlan Zhang, and Qi Li, "Beam splitting at the output of photonic crystal waveguides with discrete surface point defects," Opt. Express 18, 24245-24257 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-24245


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References

  1. L. Y. Jiang, W. Jia, H. P. Li, X. Y. Li, C. X. Cong, and Z. X. Shen, “Inverse design for directional emitter and power splitter based on photonic crystal waveguide with surface corrugations,” J. Opt. Soc. Am. B 26(11), 2157–2160 (2009). [CrossRef]
  2. W. Y. Liang, J. W. Dong, and H. Z. Wang, “Directional emitter and beam splitter based on self-collimation effect,” Opt. Express 15(3), 1234–1239 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-3-1234 . [CrossRef]
  3. W. Jia, L. Y. Jiang, K. Chen, and X. Y. Li, “Design of photonic crystal power beam splitters via corrugated and gratinglike surfaces,” Opt. Commun. 283(20), 4078–4084 (2010). [CrossRef]
  4. M. Q. Xin, L. Zhang, C. Eng Png, J. H. Teng, and J. Aaron Danner, “Asymmetric open cavities for beam steering and switching from line-defect photonic crystals,” J. Opt. Soc. Am. B 27(6), 1153–1157 (2010). [CrossRef]
  5. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B Condens. Matter 44(19), 10961–10964 (1991). [CrossRef] [PubMed]
  6. R. Mousse, Th. Koschny, and C. M. Soukoulis, “Excitation of surface waves in a photonic crystal with negative refraction: The role of surface termination,” Phys. Rev. B 74(11), 115111 (2006). [CrossRef]
  7. B. Wang, W. Dai, A. Fang, L. Zhang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Surface waves in photonic crystal slabs,” Phys. Rev. B 74(19), 195104 (2006). [CrossRef]
  8. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988)
  9. 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(12), 121402 (2004). [CrossRef]
  10. R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76(23), 235417 (2007). [CrossRef]
  11. S. K. Morrison and Y. S. Kivshar, “Engineering of directional emission from photonic-crystal waveguide,” Appl. Phys. Lett. 86(8), 081110 (2005). [CrossRef]
  12. H. Caglayan, I. Bulu, and E. Ozbay, “Off-axis directional beaming via photonic crystal surface modes,” Appl. Phys. Lett. 92(9), 092114 (2008). [CrossRef]
  13. 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(5582), 820–822 (2002). [CrossRef] [PubMed]
  14. F. J. García-Vidal, L. Martín-Moreno, H. J. Lezec, and T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83(22), 4500–4502 (2003). [CrossRef]
  15. A. Degiron and T. W. Ebbesen, “Analysis of the transmission process through single apertures surrounded by periodic corrugations,” Opt. Express 12, 3694–3700 (2004). http://www.opticsinfobase.org/oe/abstract. cfm?URI=oe-12-16-3694 . [CrossRef] [PubMed]
  16. D. Z. Lin, C. K. Chang, Y. C. Chen, D. L. Yang, M. W. Lin, J. T. Yeh, J. M. Liu, C. H. Kuan, C. S. Yeh, and C. K. Lee, “Beaming light from a subwavelength metal slit surrounded by dielectric surface gratings,” Opt. Express 14(8), 3503–3511 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3503 . [CrossRef] [PubMed]
  17. Y. L. Zhang, D. Y. Zhao, C. H. Zhou, and X. Y. Jiang, “Directional light emission through a metallic nanostructure,” J. Appl. Phys. 105(11), 113124 (2009). [CrossRef]
  18. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystal: Molding the flow of light (Princeton University Press, Princeton, 1995).
  19. S. S. Xiao and M. Qiu, “Optical microcavities based on surface modes in two-dimensional photonic cyetals and silicon-on-insulator photonic crystals,” J. Opt. Soc. Am. B 24(5), 1225–1229 (2007). [CrossRef]
  20. Z. F. Li, K. Aydin, and E. Ozbay, “Wide band width directional beaming via waveguide ports in photonic crystals,” J. Phys. D Appl. Phys. 41(15), 155115 (2008). [CrossRef]
  21. A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 2000).
  22. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]

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