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Based on the wavefront modulation of photonic crystal (PhC), zero phase delay of propagating electromagnetic wave (EMW) can be realized with a relaxed incident condition in the PhC. The phase velocity is modulated perpendicular to the group velocity with wavefronts extending along the direction of energy flow, which lead to the phenomenon of zero phase delay with a finite spatial period. This effect can be realized simultaneously in both positive and negative refracted waves. The phase difference between the incident and transmitted waves are measured within a wide incident angle region to demonstrate zero phase delay can be realized easily instead of zero–n or zero–averaged–n materials. Further investigations prove that the phenomena of zero phase delay induced by this way can also be realized easily in various PhC configurations and can be accurately manipulated by changing the incident angle or the flexible design of PhC configuration.
©2013 Optical Society of America
1. Introduction
Recently, great efforts have been made to construct materials with zero or near-zero-n with quasi-uniform phase and infinite wavelength because of their unusual physical properties and their potential use in many technological applications [1–3]. Materials with epsilon near zero (ENZ) may be obtained at higher frequencies, i.e., infrared and visible, by employing engineered metal-based metamaterials [4–7]. However, these materials usually suffer from strong resonance loss and hence the greatly deteriorated transmission efficiency. Matched zero-index media [8] with both ε and μ near zero at the frequency of interest have been synthesized to realize impedance match, which require very complex processing technology. Alternative approaches include the microwave waveguides below cutoff [9]. Instead of strong resonance in ENZ metamaterials, the physical phenomena of PhC [10, 11] are based on the special dispersion relations of photonic bands with weak loss. For the optical and electromagnetic properties can be engineered through the geometry design of its unit cells, the tunable diffraction gives rise to distinct optical phenomena. A type of PhC can be obtained by Stacking alternating layers of ordinary negative- and positive-index materials with zero–averaged–index gap different from that of a Bragg gap [12], which will arise naturally when the volume averaged effective refractive index equals zero. S. Kocaman [13] and colleagues have constructed a one-dimensional periodic superlattice from alternating strips of positive index homogeneous dielectric media and negative index PhCs with zero phase accumulation of a wave travelling through the whole superlattice. However, all these configurations require high fabrication precision and zero phase delay only can be achieved at a certain frequency.
In the previous work [14], we have proposed a mechanism to realize zero phase delay based on the feature of wavefront modulation of PhCs, and verified it experimentally in X-band with the triangular PhC composed of dielectric cylinders closely packed in air. In this paper, a systematical investigation on the propagation properties of zero phase delay based on wavefront modulation is carried out in a PhC slab consisting of hexagonal Si rods in air to demonstrate, by proper design, this effect not only can be realized with a relax incident condition in PhCs, but also can be induced simultaneously in both positive and negative refracted waves with different refractive angles, which reveal the relationship between EFC distribution in PhC and phase shift of propagating EMW. Moreover, it is proved that this method to realize zero phase delay can be applied in various PhCs with different lattice structures and their propagation properties can be modulated flexibly by changing the PhC parameters or the incident angles. These results may be extended to the fabrications of other artificially engineered materials and give a guideline to the design of novel optical devices with interesting functions.
2. Theory of plane wave transmission
In physics of wave propagation, the group velocity vgr is often thought as the velocity of energy flow conveyed along the wave. Wavefront is the locus of points with the same phase and wave vector k points to the normal direction of wavefront. In traditional material with an ordinary refractive index, i.e. right–handed material, EMW is transmitted away from the source with vph ⋅ vgr >0 or equivalently k⋅s >0 and wavefronts going away from the source, and accumulate a positive relative phase. The symbols vph, vgr, k and s indicate the phase velocity, group velocity, wave vector and pointing vector. In a negative index material, EMW travels toward the source with k ⋅ vgr < 0 and wavefronts moving close to the source, and accumulate a negative relative phase. No matter what kind of propagation mode, EMWs propagating in media usually undergo the phase shift with the energy of wave carried away from the source. A plane wave traveling in arbitrary direction can be described as
where A is wave amplitude, k is wave vector, r is position vector, and φ0 is initial phase, ωt is time phase factor. In spatial domain, phase shift of the plane wave is determined by the spatial phase factor of k ⋅ r. Since vgr is parallel to s in PhCs large enough [15], vgr points to the same direction with r. When the condition of k ⋅ r = 0 or equivalently k ⋅ vgr = 0 is satisfied [14], the phase difference between arbitrary locations along the direction of wave propagation will be equal to zero with the most significant phenomenon of the modulated parallel wavefronts extending along the propagation direction with zero phase delay in spatial domain and travelling along the normal direction in time domain.3. PhC structure and theory analysis
Based on the theory of wave propagation, a triangular lattice PhC consisting of hexagonal Si rods in air background with the refractive index of n = 3.4 is used to investigate the propagation properties of zero phase delay with the satisfied condition of k ⋅ vgr = 0. The magnified diagram of the fourth band by the normalized frequency is shown in Fig. 1 with the insets of the lowest four bands structure in the top left corner and the PhC schematic in the bottom right. The fourth band is denoted by the red solid line with a undulance in the ΓM direction, which implies a complicated optical phenomenon.
Fig. 1 Diagram of the fourth band of the triangular PhC with hexagonal Si rods in air for TM polarization.
The EFC plot of the fourth band and the corresponding equal frequency surface of the fourth band are shown in Figs. 2(a) and 2(b) with different colors of red and blue to distinguish the high and low frequencies. Supposing a source plane wave with the relative frequency of ω = 0.36 is incident from air with the incident angle of θinc = 30° upon the interface between air and the PhC slab with the surfaces cut along the ΓK direction, the corresponding wave vector diagram is illustrated in Fig. 2(a) with the highlighted black bold EFC of 0.36 like six slim leaves gathering around the center point Γ symmetrically, where the blue circle represents the air EFC at the same frequency, the black arrow denotes the incident wave vector ki in air and the dashed line notes the conservation of the parallel component of wave vector. In this case, the k-conservation line intersects the EFC of ω = 0.36 with four intersections. Since the group velocity vgr would not give an energy flow away from the source, the waves at two circled points do not exist and only two refracted waves at points A and B can be excited with different transmission properties in this PhC. Furthermore, it is noted that these refracted wave vectors almost overlap the corresponding EFCs around the centre in the radial directions. By the definition of group velocity vgr = ∇kω, the group velocity vector is oriented perpendicular to the EFC surface in the frequency–increasing direction. The red and green arrows denote the group velocity directions of two refracted waves A and B, respectively, which are almost perpendicular to kra and krb . In spatial domain, the schematic diagram of phase shift for two refracted waves is shown in Fig. 2(c), where the phases are invariable in the PhC slab and the phase shifts between the incident wave and the transmitted waves are equal to zero, just like one plane wave split into two plane waves without experiencing the spatial separation of PhC slab.
Fig. 2 (a) EFCs plot of the fourth band with the wave vector diagram at ω = 0.36 with θinc = 30°; (b) the corresponding equal frequency surface; (c) schematic diagram of EMW propagating through the PhC slab with two refracted waves.
Due to the six-fold rotational symmetry of this PhC, two refracted waves are excited in this PhC slab. In order to investigate the phase shift of EMW propagating in this PhC, the trend chart of the included angle between k and υgr at ω = 0.36 is shown in Fig. 3 to illustrate the modulation effect of the incident angle changing from 5° to 50°. The red solid curve denotes the trend of the positive refraction and the blue dashed curve denotes the negative one. Obviously, apart from the beginning, both of them are close to the line of 90° and increase slowly with the increase of incident angle. For the positive refracted wave A, the included angle between k and υgr is greater than 90° with k ⋅ vgr <0 in the region of incident angle θinc <13.3° and θinc >41° and negative spatial phase shift can be achieved in the process of EMW propagating in the PhC. The red curve passes through the line of 90° twice, one is at θinc = 13.3° with the refractive angle of θref = 14.3° and another happens in a wide region of about 29.5°<θinc<41° with a stationary refractive angle of θref = 11.5°. In this region, the included angle θ is very close to 90° with a slow change, which should be attributed to its radial slim shape of EFC. In this region, the spatial phase of EMW propagating in this PhC is almost invariable with parallel wavefronts spreading perpendicular to the direction of energy flow. In the scope of 13.3°<θinc<29.5°, the included angle between k and υgr is slightly less than 90° with k ⋅ vgr >0, which lead to the spatial phase of propagating EMW increase slowly along the spread direction.
Fig. 3 (a) The included angle between k and vgr for the refracted waves A and B with different incident angles.
Although the negative refractive angle of the refracted wave B is opposite to the refracted wave A, the blue trend basically coincides with the red one with a small shift in Fig. 3. The blue curve intersects the line of 90° at two points of θinc = 9° and 37.67°, which is less than the corresponding incident angles of θinc = 13.3° and 41° for the red curve of the positive refracted wave A. The similar propagating properties of two refracted waves are attributed to the rotation symmetrical EFC of the fourth band. Compared with the flat slope of EFC for the refracted wave A in Fig. 2(a), the slope for the refracted wave B is more steep than the wave A along the direction of incident interface, which leads to the included angle θ between k and υgr changing more significantly and faster with the variation of incident angle. The more flat radial EFC may satisfy the condition of k ⋅ vgr = 0 more easily in the wider region of incident angle. As shown in Fig. 3, the red curve is smoother than the blue one with a small variation, which is the reason why the refracted wave A can propagate in the PhC slab with zero phase delay in a wide region of incident angle. Since EFCs are determined by the PhC structure, by proper design, the effect of zero phase delay can be realized in a wide frequency region instead of the certain frequency in zero-n or near-zero-n materials. Compared with the longitudinal direction of the path–averaged zero–index nanofabricated superlattices in Ref [13], the propagation direction of the transmitted wave with zero phase delay in this PhC can be modulated by changing the incident angle.
4. FDTD results and discussion
In order to verify the above analysis results, a continuous Gaussian wave source at the relative frequency of a/λ = 0.36 with a spatial width of 10a is located in front of the PhC slab with the thickness of 40 layers, which can be considered as a plane wave source. The finite-difference time-domain (FDTD) simulation of electric field distribution in the PhC slab for transmitted refractive waves at θinc = 30° is shown in Fig. 4(a), where the incident wave is splitted into the positive and negative refracted waves with different refractive angles at the interface between air and the PhC slab, then two refracted waves pass through the PhC slab with the parallel wavefronts extending along the direction of energy flow, finally exit from the output surface with two parallel outgoing waves as the incident wave, which indicate the separate transmitted waves originate from the same source wave. The linked Media 1 shows the direction of wavefronts travelling in the PhC slab which is perpendicular to the wave propagation direction with the finite spatial periods, as predicted in the theoretical analysis. Since the horizontal component of energy flow vector of the positive refracted wave A has the same direction with the incident wave, the transmittance of the positive refracted wave A is much larger than that of the negative refracted wave B with a value of nearly 70% as shown in Fig. 4(b), and the pattern of electric field distribution for the positive refracted wave A can be seen more clearly in Fig. 4(a). According to the expression of wave vector k = 2π/λ, the wavelength is inversely proportional to the wave vector. From the wave vector diagram of Fig. 2(a), it is easy to find that the wave vectors will reduce with the decrease of incident angle, therefore, the spatial wavelength will enlarge with the decrease of incident angle. The FDTD simulation of electric field distribution at the incident angle of θinc = 13.3° is shown in Fig. 5(a) with the larger spatial period than the state at θinc = 30° and the EMW can also be transmitted in PhCs with modulated wavefronts extending along the propagation direction to realize zero phase delay. Obviously, these numerical simulations agree well with the above analysis results.
Fig. 4 (a) Electric field distribution (Media 1) and (b) transmissivity of two refracted waves in the holographic triangular PhC slab at the frequency of 0.36 with θinc = 30°.
Fig. 5 Electric field distributions with zero phase delay in various PhC configurations. (a) The triangular PhC composed of hexagonal Si rods at ω = 0.36 with θinc = 13.3°, (b) the honeycomb PhC composed of air rods with r = 0.1a in dielectric background at ω = 0.345 and θinc = 20°, (c) the triangular PhC composed of round rods with r = 0.5a at ω = 0.35 and θinc = 30°, (d) the square PhC composed of dielectric rods with r = 0.4a at ω = 0.365 and θinc = 40°.
Since the optical properties of PhC can be engineered by the geometry of its unit cells with a large freedom, numerical calculations demonstrate the novel effect of zero phase delay with k ⋅ vgr = 0 can also be achieved in other various PhC configurations. Figures 5(a)-5(d) give the electric field distributions of zero phase delay in different PhCs, such as the honeycomb lattice composed of air rods with r = 0.1a in dielectric background as shown in Fig. 5(b), the triangular lattice PhC in Fig. 5(c) composed by dielectric cylinders close packed in air and the square lattice PhC composed of dielectric rod with r = 0.1a in air, as shown in Fig. 5(d). Obviously, although the transmission direction and transmittance of these refracted waves in PhC slabs are different, all of them have the same propagation characteristic with parallel wavefronts extending along the respective propagation directions with different spatial wavelengths. In spatial domain, the phase difference between arbitrary spatial locations along the propagation direction is static and equal to zero, while remaining dynamic in time domain. Due to the engineered system parameters of PhC, such as period scaling, dielectric contrast and filling ratio etc., the frequency and the propagation direction of EMW propagating with zero phase delay can be rigorously modulated easily by changing the PhC parameters and incident conditions.
5. Conclusions
In conclusion, the mechanism of wavefront modulation in PhCs is a simple and effective way to realize zero phase delay of propagating EMW instead of zero-index or zero–averaged– index materials. Numerical calculation and analyses demonstrate the effect of zero phase delay can be achieved by this way in various PhC configurations with different lattice structures and rotational symmetries. With the satisfied condition of k ⋅ vgr = 0, this effect has a close relation with the design of PhC configuration and the center radial EFC with a flat slope is the critical factor to realize zero phase delay in a relax incident condition. Since the PhC structure can be engineered easily and have more design flexibility, it is convenient to manipulate the frequency and direction of EMW propagating in PhC with zero phase delay by changing the parameters of PhC configuration or the incident angle. Due to the prosperous nanofabrication technology, this method for engineered control of the phase delay in PhCs may have significant potential applications in chip–scale transmission lines, information processing and the design of new type optical devices.
Acknowledgments
This work is supported by the financial support from National Natural Science Foundation (51102148, 51032003, 11274198 and 61275014), National High Technology Research and Development Program of China (863 Program) (2012AA030403), President Foundation of University of Chinese Academy of Science B (Y35102DN00).
References and links
1. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010). [CrossRef]
2. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010). [CrossRef] [PubMed]
3. J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96(10), 101109 (2010). [CrossRef]
4. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007). [CrossRef]
5. N. Garcia, E. V. Ponizovskaya, and J. Q. Xiao, “Zero permittivity materials: Band gaps at the visible,” Appl. Phys. Lett. 80(7), 1120–1122 (2002). [CrossRef]
6. C. Rizza, A. Di Falco, and A. Ciattoni, “Gain assisted nanocomposite multilayers with near zero permittivity modulus at visible frequencies,” Appl. Phys. Lett. 99(22), 221107 (2011). [CrossRef]
7. A. Ciattoni, R. Marinelli, C. Rizza, and E. Palange, “|ε|-near-zero materials in the near-infrared,” Appl. Phys. B. 110(1), 23–26 (2013). [CrossRef]
8. M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75(7), 075119 (2007). [CrossRef]
9. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008). [CrossRef] [PubMed]
10. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987). [CrossRef] [PubMed]
11. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987). [CrossRef] [PubMed]
12. J. S. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90(8), 083901 (2003). [CrossRef] [PubMed]
13. S. Kocaman, M. S. Aras, P. Hsieh, J. F. McMillan, C. G. Biris, N. C. Panoiu, M. B. Yu, D. L. Kwong, A. Stein, and C. W. Wong, “Zero phase delay in negative-refractive-index photonic crystal superlattices,” Nat. Photonics 5(8), 499–505 (2011). [CrossRef]
14. G. Y. Dong, J. Zhou, and L. Cai, “Zero phase delay induced by wavefront modulation in photonic crystals,” Phys. Rev. B 87(12), 125107 (2013). [CrossRef]
15. S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72(16), 165112 (2005). [CrossRef]
