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Dual negative refraction in a two dimension square photonic crystal
Optics Communications 350 (2015) 213–216
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Dual negative refraction in a two dimension square photonic crystal J. Derbali a,b,n, F. AbdelMalek a a b
National Institute of Applied Sciences and Technology, BP 676, 1080 Tunis Cedex, Tunisia Quantum Physics and Photonics Group, Faculty of Sciences of Tunis, Tunisia
art ic l e i nf o
a b s t r a c t
Article history: Received 14 November 2014 Received in revised form 27 February 2015 Accepted 6 April 2015 Available online 8 April 2015
Dual refraction effect based on the overlapping bands in a two dimensional (2D) photonic crystal (PhC) is demonstrated. The PhC consists of alumina rods with a dielectric constant ε ¼ 8.9, arranged in a square lattice in air. To disperse light which has special excitation frequency and a specific incident angle, by this PhC we optimize his structural parameters such as the radius of dielectric rods). It is shown that two focusing phenomena are formed in the PhC image plan; the degeneracy of modes can be applied to realize optical interference and wave front division. The simulation results are obtained by employing the PWM for analyzing bands structure and the finite-difference time-domain (FDTD) to predict the evolution of the electric fields. & 2015 Elsevier B.V. All rights reserved.
Keywords: Photonic crystal Left-hand Super-focusing Super-dispersion
1. Introduction Photonic crystals (PhCs) are periodic dielectric structures in one or two or three directions. PhCs offer the possibility to control and manipulate the propagation of the light at frequencies scale [1–9]. Among other properties, the negative refraction is one of the most promising futures for building very compact integrated circuits [10–12]. Materials with negative refraction or left hand photonic crystals have simultaneously negative permittivity and negative permeability. Negative refraction in photonic crystal has attracted many applications in light manipulation such as focusing imaging in which the photonic crystal is employed as super-lens or superprism [13], switch [14,15], router [16] and optical logic gates [17]. Several research works are based to the photonic band structure which describes the states of the photon, and especially the photonic band gap (PBG) properties. These properties are essential to design novel devices efficient for light guiding. For the proposed photonic crystal (PhC), the higher frequency bands provides dual modes also called dual negative refraction because for the same polarization state photonic bands with different wave vectors might locate in the same frequency region. So, to find the best result we trend to play on the PhC parameters (the filling ratio) and the source parameters. In this work, our idea is based on studying the overlapping between bands which generates the degeneracy of modes. We are n Corresponding author at: National Institute of Applied Sciences and Technology, BP 676, 1080 Tunis Cedex, Tunisia. E-mail address: [email protected] (J. Derbali).
http://dx.doi.org/10.1016/j.optcom.2015.04.021 0030-4018/& 2015 Elsevier B.V. All rights reserved.
interested on analyzing the overlapping to find many energy bands. The energy band overlapping generate two phenomena namely the positive refraction and the negative refraction because these bands have different wave vectors at the same frequency, which may lead to two refractions at the same frequency with different phase and group velocities, so focus our attention on the negative refraction concept because photonic crystal, used as lefthand material, amplifies the evanescent waves so it gives the possibility to fabricate new devices with diverse and efficient functionalities compared to their counterparts that are based on usual materials. In this work we trend to build a new slot array, super-lens and super-prism.
2. Results and discussions The motivation of this study is to design some structures based on left-hand photonic crystals which are anisotropic medium with different refractive indices in the x, y and z directions, therefore the propagation of light is not similar along these directions. It means that light propagation depends on the polarization direction of waves. These proposed structures play the same role as optical devices generating new phenomena and effects such as the dual self-collimation, the super-focusing and the super-dispersion. The schematic of the structure is described in Fig. 1. The structure under investigation is formed by an arrangement of cylinder rods of alumina in air. Certain conditions, by launching a single beam light into the left-hand photonic crystal, give possibility to light to disperse into two negative refracted waves with
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J. Derbali, F. AbdelMalek / Optics Communications 350 (2015) 213–216
Fig. 1. Schematic of the 2D square PhC with cylinders dielectric rods in air. Fig. 3. Dual self-collimation effect for TE polarization in 2D square photonic crystal.
the same frequency and polarization state, however with different phase and group velocities. By varying cylinder radius, which imposes a variation of energy bands, and the angle of incident light, we optimize the range where the beam penetrates and propagates down the photonic crystal and therefore inhibits the reflection. 2.1. TE polarization As first step, we calculate the energy bands diagram by using the plane wave method (PWM). Fig. 2 describes the bands overlapping for TE polarization. As identified with the black circle in Fig. 2, the third and fourth bands of the PhC satisfy the condition of dual negative refraction in a single band and should be the ideal candidates to induce dual refraction effects. To verify this property of dual self-collimation refraction, we use the results reported in [18] for TE polarization. Gajic et al. have demonstrated the appearance of convex equal frequency contour (EFC) around the center of Brillion zone with an inward gradient. By inspecting the EFC figures one may notice that the form of the curves are square, this is an evidence of the self-collimation effect that results in two parallel group velocities perpendicular to the EFC. After this design step and in order to validate the results of band diagrams, we consider a photonic crystal composed of 6 layers of rods in the ГM direction. We excite the structure with a continuous sino-Gaussian plane wave with frequency
Fig. 2. The band structure for TE modes in a 2D square photonic crystal composed of alumina rods in air.
f¼ a/λ ¼ 0.732, where a is the square lattice constant. The beam light hits the input surface of the structure with an incident angle θi¼ 10°, as it is shown in Fig. 1. The evolution of the electric field is performed by using the FDTD [19–21] with perfectly matched layer (PML) [22] used as the absorbing boundary conditions which absorb outgoing waves from the computation domain. The time domain snapshot of beam signal is shown in Fig. 3. This figure shows the snapshot of Hz propagating along the ГM direction, one may observe two separated parallel outgoing waves in the output surface this demonstrates the dual negative refraction effects. This effect is a dual self collimation indicated by two parallel group velocities which are perpendiculars to the EFC vg = ∇k ω . In this case, the scalar product vg . k < 0 is negative this justifies the dual negative refraction observed between the third and forth bands. This system can be used to design novel array of slots. The proposed structure is more se lective and ensures the dispersion of the polychromatic light. 2.2. TM polarization In this part we focus on studying the propagation of light along the ГX and ГM directions, we calculate the band energy as a function of the wave vector; the result is reported in Fig. 4. This figure shows multiple overlapping between bands. We select the second and the third bands as it is indicated by a blue circle where the overlapping occurs at the frequency f ¼0.545. We use this frequency to excite the electromagnetic wave propagating along the two directions ГX and ГM.
Fig. 4. TM band structure for a 2D square photonic crystal.
J. Derbali, F. AbdelMalek / Optics Communications 350 (2015) 213–216
215
Fig. 5. Electric field patterns of double focusing imaging for the PhC slab.
TE3
TE4
n<0
First Focus
Source
Second Focus
Fig. 7. The FDTD simulations of the electric field (Ez) distribution for two incident angles (a) θi ¼10° and (b) θi ¼15°. Left-hand material
Fig. 6. Ray tracing of the point source at the frequency of 0.454 placed in front of the PhC slab.
2.2.1. Study along the
ГX direction
In this case, we use a continuous source centered at the frequency f ¼a/λ ¼0.545 to investigate the propagation along ГX direction. We set the source in front of the PhC, which is composed of 10 layers of rods of a radius r ¼0.22a and a dielectric constant ε ¼8.9 in air. By inspecting Fig. 5 we observe two focusing images (two points) located near the output surface, this phenomenon is a novel double focusing imaging. It is described in Fig. 6. This result has been reported by Gajic et al. in [18] showing that the EFC curves exhibit a circular shape. The focusing image is formed as a result of the group velocity and EFCs crossing. This proposed system presents a new super-lens with high resolution and having more number of images. It ensures the stigmatism effect therefore Gauss’s conditions are no longer needed.
2.2.2. Analysis in the
frequency f¼ a/λ ¼0.545, the EFC of TM2 band extends to a concave one in air. This photonic crystal offers the possibility to build a new super-prism which is more selective compared to the usual prism and hence using the proposed structure Gauss’s conditions can be ignored.
3. Conclusion Based on the overlapping between bands, the dual negative refraction effect is demonstrated in 2D square photonic crystal with optimized parameters. These effects are manifested by a double self-collimation and a dual dispersion of the incident wave outgoing from the PhC slab. They can be predicted by analyzing the band structure and the equal frequency contours of the PhC. The difference between these two effects can be explained and shown by the calculate of the Equal Frequency counter (EFC) which gives the effect type which we must find because it provides the group velocities vectors direction. The negative refraction behavior is simulated using the FDTD method which agrees well with the theoretical analyses. We believe this double-negative refraction effect might be important and useful to the designs of novel generation of smart PhC devices.
ГM direction
The snapshots presented in Fig. 7 shows the super-dispersion effect when incident angle is 15°. Since the waves with large incident angles are reflected back to the input space or diffused in the outgoing space, the waves with limited incident angles can penetrate the PhC slab and be focused in the outgoing space. Hence, the power intensities of two focusing images are weak due to the partial energy loss. Also, the symmetrical distortions and variations of intensity distribution in the output space results from the optical interferences of the coherent outgoing rays. At the
References [1] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett. 58 (1987) 2059. [2] S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett. 58 (1987) 2486. [3] C.J. Wu, Y.C. Hsieh, H.T. Hsu, Tunable photonic band gap in a doped semiconductor photonic crystal in near infrared region, Prog. Electromagn. Res. 114 (2011) 271–27283. [4] H. Butt, Q. Dai, T.D. Wilkinson, G.A.J. Amaratunga, Photonic crystals & metamaterial filters based on 2D arrays of silicon nanopillars, Prog. Electromagn.
216
J. Derbali, F. AbdelMalek / Optics Communications 350 (2015) 213–216
Res. 113 (2011) 179–194. [5] H.T. Hsu, T.W. Chang, T.J. Yang, B.H. Chu, C.J. Wu, Analysis of wave properties in photonic crystal narrowband filters with left-handed defect, J. Electromagn. Waves Appl. 24 (16) (2010) 2285–2298. [6] H. Li, X. Yang, Larger absolute band gaps in two-dimensional photonic crystals fabricated by a three-order-effect method, Prog. Electromagn. Res. 108 (2010) 385–400. [7] H.C. Hung, C.J. Wu, T.J. Yang, S.J. Chang, Analysis of tunable multiple filltering property in a photonic crystal containing strongly extrinsic semiconductor, J. Electromagn. Waves Appl. 25 (14–15) (2011) 2089–2099. [8] H. Lu, X.M. Liu, R. Zhou, D. Mao, Y. Gong, Tunable and robust reflection-free waveguides based on a gyromagnetic photonic crystal, J. Electromagn. Waves Appl. 25 (2011) 1752–1761 11–12. [9] X. Dai, Y. Xiang, S. Wen, Broad omnidirectional, reflector in the one-dimensional ternary photonic crystals containing superconductor, Prog. Electromagn. Res. 120 (2011) 17–34. [10] W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, S. Beckx, Basic structures for photonic integrated circuits in silicon-on-insulator, Opt. Express 12 (8) (2004) 1583–1591. [11] Sharee J. McNab, Nikolaj Moll, Yurii A. Vlasov, Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides, Opt. Express 11 (22) (2003). [12] C. Monat, P. Domachuk, B.J. Eggleton, Integrated optofluidics: a new river of
light, Nat. Photonics (2007) 106–114. [13] J.B. Pendry, Negative refraction makes perfect lens, Phys. Rev. Lett. 85 (2000) 3966. [14] J. Li, J. He, Z. Hong, Terahertz wave switch based on silicon photonic crystals, Appl. Opt. 46 (2007) 5034–505037. [15] Z. Li, Y. Zhang, B. Li, Terahertz photonic crystal switch in silicon based on selfimaging principle, Opt. Express 14 (2006) 3887–3892. [16] V.R. Almeida, C.A. Barrios, R.R. Panepucci, M. Lipson, All-optical control of light on a silicon chip, Nature 431 (2004) 1081–1084. [17] Y. Zhang, B. Li, Optical switches and logic gates based on self-collimated beams in two-dimensional photonic crystals, Opt. Express 15 (2007) 9287–929292. [18] R. Gajić, R. Meisels, F. Kuchar, K. Hingerl, Refraction and rightness in photonic crystals, Opt. Express 13 (2005) 8596. [19] S. Haxha, W. Belhadj, F. AbdelMalek, H. Bouchriha, Analysis of wavelength demultiplexer based on photonic crystals, Electron. Lett. 152 (4) (2005) 193–198. [20] F. AbdelMalek, W. Belhadj, S. Haxha, H. Bouchriha, Realization of high coupling efficiency by employing concave lense based on 2D photonic crystals with negative refractive index, J. Lightwave Technol. 25 (10) (2007) 3168–3174. [21] S. Haxha, F. AbdelMalek, A Novel Design of Photonic Crystal Lens Based on Negative Refractive Index Progress in Electromagnetics Research Symposium (PIERS), China-Invited Paper, Published, March, 2008. [22] C.J.P.A. Berenger, Perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (2) (1994) 185–200.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Dual negative refraction in a two dimension square photonic crystal J. Derbali a,b,n, F. AbdelMalek a a b
National Institute of Applied Sciences and Technology, BP 676, 1080 Tunis Cedex, Tunisia Quantum Physics and Photonics Group, Faculty of Sciences of Tunis, Tunisia
art ic l e i nf o
a b s t r a c t
Article history: Received 14 November 2014 Received in revised form 27 February 2015 Accepted 6 April 2015 Available online 8 April 2015
Dual refraction effect based on the overlapping bands in a two dimensional (2D) photonic crystal (PhC) is demonstrated. The PhC consists of alumina rods with a dielectric constant ε ¼ 8.9, arranged in a square lattice in air. To disperse light which has special excitation frequency and a specific incident angle, by this PhC we optimize his structural parameters such as the radius of dielectric rods). It is shown that two focusing phenomena are formed in the PhC image plan; the degeneracy of modes can be applied to realize optical interference and wave front division. The simulation results are obtained by employing the PWM for analyzing bands structure and the finite-difference time-domain (FDTD) to predict the evolution of the electric fields. & 2015 Elsevier B.V. All rights reserved.
Keywords: Photonic crystal Left-hand Super-focusing Super-dispersion
1. Introduction Photonic crystals (PhCs) are periodic dielectric structures in one or two or three directions. PhCs offer the possibility to control and manipulate the propagation of the light at frequencies scale [1–9]. Among other properties, the negative refraction is one of the most promising futures for building very compact integrated circuits [10–12]. Materials with negative refraction or left hand photonic crystals have simultaneously negative permittivity and negative permeability. Negative refraction in photonic crystal has attracted many applications in light manipulation such as focusing imaging in which the photonic crystal is employed as super-lens or superprism [13], switch [14,15], router [16] and optical logic gates [17]. Several research works are based to the photonic band structure which describes the states of the photon, and especially the photonic band gap (PBG) properties. These properties are essential to design novel devices efficient for light guiding. For the proposed photonic crystal (PhC), the higher frequency bands provides dual modes also called dual negative refraction because for the same polarization state photonic bands with different wave vectors might locate in the same frequency region. So, to find the best result we trend to play on the PhC parameters (the filling ratio) and the source parameters. In this work, our idea is based on studying the overlapping between bands which generates the degeneracy of modes. We are n Corresponding author at: National Institute of Applied Sciences and Technology, BP 676, 1080 Tunis Cedex, Tunisia. E-mail address: [email protected] (J. Derbali).
http://dx.doi.org/10.1016/j.optcom.2015.04.021 0030-4018/& 2015 Elsevier B.V. All rights reserved.
interested on analyzing the overlapping to find many energy bands. The energy band overlapping generate two phenomena namely the positive refraction and the negative refraction because these bands have different wave vectors at the same frequency, which may lead to two refractions at the same frequency with different phase and group velocities, so focus our attention on the negative refraction concept because photonic crystal, used as lefthand material, amplifies the evanescent waves so it gives the possibility to fabricate new devices with diverse and efficient functionalities compared to their counterparts that are based on usual materials. In this work we trend to build a new slot array, super-lens and super-prism.
2. Results and discussions The motivation of this study is to design some structures based on left-hand photonic crystals which are anisotropic medium with different refractive indices in the x, y and z directions, therefore the propagation of light is not similar along these directions. It means that light propagation depends on the polarization direction of waves. These proposed structures play the same role as optical devices generating new phenomena and effects such as the dual self-collimation, the super-focusing and the super-dispersion. The schematic of the structure is described in Fig. 1. The structure under investigation is formed by an arrangement of cylinder rods of alumina in air. Certain conditions, by launching a single beam light into the left-hand photonic crystal, give possibility to light to disperse into two negative refracted waves with
214
J. Derbali, F. AbdelMalek / Optics Communications 350 (2015) 213–216
Fig. 1. Schematic of the 2D square PhC with cylinders dielectric rods in air. Fig. 3. Dual self-collimation effect for TE polarization in 2D square photonic crystal.
the same frequency and polarization state, however with different phase and group velocities. By varying cylinder radius, which imposes a variation of energy bands, and the angle of incident light, we optimize the range where the beam penetrates and propagates down the photonic crystal and therefore inhibits the reflection. 2.1. TE polarization As first step, we calculate the energy bands diagram by using the plane wave method (PWM). Fig. 2 describes the bands overlapping for TE polarization. As identified with the black circle in Fig. 2, the third and fourth bands of the PhC satisfy the condition of dual negative refraction in a single band and should be the ideal candidates to induce dual refraction effects. To verify this property of dual self-collimation refraction, we use the results reported in [18] for TE polarization. Gajic et al. have demonstrated the appearance of convex equal frequency contour (EFC) around the center of Brillion zone with an inward gradient. By inspecting the EFC figures one may notice that the form of the curves are square, this is an evidence of the self-collimation effect that results in two parallel group velocities perpendicular to the EFC. After this design step and in order to validate the results of band diagrams, we consider a photonic crystal composed of 6 layers of rods in the ГM direction. We excite the structure with a continuous sino-Gaussian plane wave with frequency
Fig. 2. The band structure for TE modes in a 2D square photonic crystal composed of alumina rods in air.
f¼ a/λ ¼ 0.732, where a is the square lattice constant. The beam light hits the input surface of the structure with an incident angle θi¼ 10°, as it is shown in Fig. 1. The evolution of the electric field is performed by using the FDTD [19–21] with perfectly matched layer (PML) [22] used as the absorbing boundary conditions which absorb outgoing waves from the computation domain. The time domain snapshot of beam signal is shown in Fig. 3. This figure shows the snapshot of Hz propagating along the ГM direction, one may observe two separated parallel outgoing waves in the output surface this demonstrates the dual negative refraction effects. This effect is a dual self collimation indicated by two parallel group velocities which are perpendiculars to the EFC vg = ∇k ω . In this case, the scalar product vg . k < 0 is negative this justifies the dual negative refraction observed between the third and forth bands. This system can be used to design novel array of slots. The proposed structure is more se lective and ensures the dispersion of the polychromatic light. 2.2. TM polarization In this part we focus on studying the propagation of light along the ГX and ГM directions, we calculate the band energy as a function of the wave vector; the result is reported in Fig. 4. This figure shows multiple overlapping between bands. We select the second and the third bands as it is indicated by a blue circle where the overlapping occurs at the frequency f ¼0.545. We use this frequency to excite the electromagnetic wave propagating along the two directions ГX and ГM.
Fig. 4. TM band structure for a 2D square photonic crystal.
J. Derbali, F. AbdelMalek / Optics Communications 350 (2015) 213–216
215
Fig. 5. Electric field patterns of double focusing imaging for the PhC slab.
TE3
TE4
n<0
First Focus
Source
Second Focus
Fig. 7. The FDTD simulations of the electric field (Ez) distribution for two incident angles (a) θi ¼10° and (b) θi ¼15°. Left-hand material
Fig. 6. Ray tracing of the point source at the frequency of 0.454 placed in front of the PhC slab.
2.2.1. Study along the
ГX direction
In this case, we use a continuous source centered at the frequency f ¼a/λ ¼0.545 to investigate the propagation along ГX direction. We set the source in front of the PhC, which is composed of 10 layers of rods of a radius r ¼0.22a and a dielectric constant ε ¼8.9 in air. By inspecting Fig. 5 we observe two focusing images (two points) located near the output surface, this phenomenon is a novel double focusing imaging. It is described in Fig. 6. This result has been reported by Gajic et al. in [18] showing that the EFC curves exhibit a circular shape. The focusing image is formed as a result of the group velocity and EFCs crossing. This proposed system presents a new super-lens with high resolution and having more number of images. It ensures the stigmatism effect therefore Gauss’s conditions are no longer needed.
2.2.2. Analysis in the
frequency f¼ a/λ ¼0.545, the EFC of TM2 band extends to a concave one in air. This photonic crystal offers the possibility to build a new super-prism which is more selective compared to the usual prism and hence using the proposed structure Gauss’s conditions can be ignored.
3. Conclusion Based on the overlapping between bands, the dual negative refraction effect is demonstrated in 2D square photonic crystal with optimized parameters. These effects are manifested by a double self-collimation and a dual dispersion of the incident wave outgoing from the PhC slab. They can be predicted by analyzing the band structure and the equal frequency contours of the PhC. The difference between these two effects can be explained and shown by the calculate of the Equal Frequency counter (EFC) which gives the effect type which we must find because it provides the group velocities vectors direction. The negative refraction behavior is simulated using the FDTD method which agrees well with the theoretical analyses. We believe this double-negative refraction effect might be important and useful to the designs of novel generation of smart PhC devices.
ГM direction
The snapshots presented in Fig. 7 shows the super-dispersion effect when incident angle is 15°. Since the waves with large incident angles are reflected back to the input space or diffused in the outgoing space, the waves with limited incident angles can penetrate the PhC slab and be focused in the outgoing space. Hence, the power intensities of two focusing images are weak due to the partial energy loss. Also, the symmetrical distortions and variations of intensity distribution in the output space results from the optical interferences of the coherent outgoing rays. At the
References [1] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett. 58 (1987) 2059. [2] S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett. 58 (1987) 2486. [3] C.J. Wu, Y.C. Hsieh, H.T. Hsu, Tunable photonic band gap in a doped semiconductor photonic crystal in near infrared region, Prog. Electromagn. Res. 114 (2011) 271–27283. [4] H. Butt, Q. Dai, T.D. Wilkinson, G.A.J. Amaratunga, Photonic crystals & metamaterial filters based on 2D arrays of silicon nanopillars, Prog. Electromagn.
216
J. Derbali, F. AbdelMalek / Optics Communications 350 (2015) 213–216
Res. 113 (2011) 179–194. [5] H.T. Hsu, T.W. Chang, T.J. Yang, B.H. Chu, C.J. Wu, Analysis of wave properties in photonic crystal narrowband filters with left-handed defect, J. Electromagn. Waves Appl. 24 (16) (2010) 2285–2298. [6] H. Li, X. Yang, Larger absolute band gaps in two-dimensional photonic crystals fabricated by a three-order-effect method, Prog. Electromagn. Res. 108 (2010) 385–400. [7] H.C. Hung, C.J. Wu, T.J. Yang, S.J. Chang, Analysis of tunable multiple filltering property in a photonic crystal containing strongly extrinsic semiconductor, J. Electromagn. Waves Appl. 25 (14–15) (2011) 2089–2099. [8] H. Lu, X.M. Liu, R. Zhou, D. Mao, Y. Gong, Tunable and robust reflection-free waveguides based on a gyromagnetic photonic crystal, J. Electromagn. Waves Appl. 25 (2011) 1752–1761 11–12. [9] X. Dai, Y. Xiang, S. Wen, Broad omnidirectional, reflector in the one-dimensional ternary photonic crystals containing superconductor, Prog. Electromagn. Res. 120 (2011) 17–34. [10] W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, S. Beckx, Basic structures for photonic integrated circuits in silicon-on-insulator, Opt. Express 12 (8) (2004) 1583–1591. [11] Sharee J. McNab, Nikolaj Moll, Yurii A. Vlasov, Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides, Opt. Express 11 (22) (2003). [12] C. Monat, P. Domachuk, B.J. Eggleton, Integrated optofluidics: a new river of
light, Nat. Photonics (2007) 106–114. [13] J.B. Pendry, Negative refraction makes perfect lens, Phys. Rev. Lett. 85 (2000) 3966. [14] J. Li, J. He, Z. Hong, Terahertz wave switch based on silicon photonic crystals, Appl. Opt. 46 (2007) 5034–505037. [15] Z. Li, Y. Zhang, B. Li, Terahertz photonic crystal switch in silicon based on selfimaging principle, Opt. Express 14 (2006) 3887–3892. [16] V.R. Almeida, C.A. Barrios, R.R. Panepucci, M. Lipson, All-optical control of light on a silicon chip, Nature 431 (2004) 1081–1084. [17] Y. Zhang, B. Li, Optical switches and logic gates based on self-collimated beams in two-dimensional photonic crystals, Opt. Express 15 (2007) 9287–929292. [18] R. Gajić, R. Meisels, F. Kuchar, K. Hingerl, Refraction and rightness in photonic crystals, Opt. Express 13 (2005) 8596. [19] S. Haxha, W. Belhadj, F. AbdelMalek, H. Bouchriha, Analysis of wavelength demultiplexer based on photonic crystals, Electron. Lett. 152 (4) (2005) 193–198. [20] F. AbdelMalek, W. Belhadj, S. Haxha, H. Bouchriha, Realization of high coupling efficiency by employing concave lense based on 2D photonic crystals with negative refractive index, J. Lightwave Technol. 25 (10) (2007) 3168–3174. [21] S. Haxha, F. AbdelMalek, A Novel Design of Photonic Crystal Lens Based on Negative Refractive Index Progress in Electromagnetics Research Symposium (PIERS), China-Invited Paper, Published, March, 2008. [22] C.J.P.A. Berenger, Perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (2) (1994) 185–200.