Analysis of wave propagation in a two-dimensional photonic crystal with negative index of refraction: plane wave decomposition of the Bloch modes

Analysis of wave propagation in a twodimensional photonic crystal with negative index of refraction: plane wave decomposition of the Bloch modes Alejandro Martínez, Hernán Míguez, José Sánchez-Dehesa, and Javier Martí Valencia Nanophotonics Technology Center, Universidad Politécnica de Valencia Camino de Vera s/n, 46022, Valencia (Spain) almara[email protected].es Abstract: This work presents a comprehensive analysis of electromagnetic wave propagation inside a two-dimensional photonic crystal in a spectral region in which the crystal behaves as an effective medium to which a negative effective index of refraction can be associated. It is obtained that the main plane wave component of the Bloch mode that propagates inside the photonic crystal has its wave vector 'k  out of the first Brillouin zone and it is parallel to the Poynting vector ( 0'· >kS   ), so light propagation in these composites is different from that reported for left-handed materials despite the fact that negative refraction can take place at the interface between air and both kinds of composites. However, wave coupling at the interfaces is well explained using the reduced wave vector (k  ) in the first Brillouin zone, which is opposed to the energy flow, and agrees well with previous works dealing with negative refraction in photonic crystals. ©2005 Optical Society of America OCIS codes: (160.4670) Optical materials; (260.2110) Electromagnetic theory. References and Links 1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509-514 (1968). 2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001). 3. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. 76, 4773-4776 (1996). 4. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from Conductors and Enhanced Nonlinear Phenomena,” IEEE Trans. Microwave Tech. 47, 2075-2084 (1999). 5. N. García and M. Nieto-Vesperinas, “Is there an experimental verification of a negative index of refraction yet?,” Opt. Lett. 27, 885-887 (2002). 6. C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Koltenbah, and M. Tanielian, “Experimental Verification and Simulation of Negative Index of Refraction Using Snell’s Law,” Phys. Rev. Lett. 90, 107401 (2003); A. A. Houck, J.B. Brock, and I.L. Chuang, “Experimental Observations of a Left-Handed Material That Obeys Snell’s Law,” Phys. Rev. Lett. 90, 137401 (2003). 7. J. D. Joannopoulos, P. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143-149 (1997). 8. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096-10099 (1998). 9. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000). 10. S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2003). 11. A. Martínez, H. Míguez, A. Griol, and J. Martí, “Experimental and theoretical study of the self-focusing of light by a photonic crystal lens,” Phys. Rev. B 69, 165119 (2004). (C) 2005 OSA 30 May 2005 / Vol. 13, No. 11 / OPTICS EXPRESS 4160 #7145 - $15.00 US Received 13 April 2005; revised 18 May 2005; accepted 19 May 2005 12. P.V. Parimi, W.T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative Refraction and Left- Handed Electromagnetism in Microwave Photonic Crystals,” Phys. Rev. Lett. 92, 127401 (2004). 13. B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Amer. A 17, 1012-1020 (2000). 14. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002). 15. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Negative refraction by photonic crystals,” Nature (London) 423, 604-605 (2003). 16. H.-T. Chien, H.-T. Tang, C.-H. Kuo, C.-C. Chen, and Z. Ye, “Directed diffraction without negative refraction,” Phys. Rev. B 70, 113101 (2004). 17. 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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. 18. A. Taflove, Computational Electrodynamics—The Finite Difference Time-Domain Method (Artech House, Boston, 1995). 19. J. P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Comput. Phys. 114, 185-200 (1994). 20. P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 025602 (2003). 21. R.W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001). 22. M. Qiu, L. Thylén, M. Swillo, and B. Jaskorzynska, “Wave propagation through a photonic crystal in a negative phase refractive-index region,” IEEE J. Sel. Top. Quantum Electron. 9, 106-110 (2003). 23. K. Sakoda, Optical properties of photonic crystals (Springer, Berlin, 2001). 24. A. Yariv, P. Yeh, Optical Waves in Crystals : Propagation and Control of Laser Radiation (New York, Wiley, 1984). 1. Introduction In 1968, Soviet physicist Veselago analyzed theoretically the electromagnetic properties of media in which the real part of the magnetic permeability μ and the electric permittivity ε were both negative [1]. Veselago deduced that in this kind of medium the electric field E  , the magnetic field H  and the wave vector k  would form a left-handed set of vectors (this is the reason why these media are usually known as left-handed materials, LHMs), which means that the wave vector and the Poynting vector S  are antiparallel ( 0<⋅kS   ) and the phase velocity is opposite to the energy flow. This particular property would give rise to unexpected phenomena such as negative refraction at the interface between air and an LHM, focusing of electromagnetic radiation by a negative-index flat plate, reversed Doppler effect and reversed Cerenkov radiation [1]. Although Veselago also suggested some ways about how to construct a real LHM, it has been more than 30 years after that LHMs have been demonstrated experimentally [2]. Following the work of Pendry and coworkers [3,4], a composite structure consisting of square metallic split-ring resonators and wire strips was demonstrated to refract negatively (that is, to the same side of the normal to the interface) the electromagnetic waves impinging from free space to the LHM [2]. Although there was a debate (see [5], for instance) about the right interpretation of the experimental results reported in Ref. 2, more recent works have demonstrated that the composite described above follows the Snell’s law provided that a negative effective index is associated to the LHM [6]. Other materials that have shown to refract negatively the electromagnetic radiation are photonic crystals (PhCs) [7]. One of the advantages of negative refraction in PhCs compared to the LHMs described above is that PhCs can be properly scaled in space to work at any frequency range, for example, infrared or visible wavelengths, if the materials are properly chosen. In a PhC the permitivitty ε is always locally positive whereas the permeability is μ = 1. Electromagnetic propagation inside a PhC takes place in the form of Bloch waves governed by the dispersion diagram that relates the frequency and the wave vector for each electromagnetic mode. One of the most interesting properties of PhCs is the existence of certain frequency intervals, commonly known as photonic band gaps, in which wave propagation is forbidden regardless of the wave vector. PhCs have also very interesting (C) 2005 OSA 30 May 2005 / Vol. 13, No. 11 / OPTICS EXPRESS 4161 #7145 - $15.00 US Received 13 April 2005; revised 18 May 2005; accepted 19 May 2005 properties as photonic conductors, one of them being their ability to deflect the electromagnetic radiation in the “wrong” way as it occurs in a LHM [8]. At this point, let us stress that the propagation of electromagnetic waves inside a PhC is a consequence of the multiple diffraction through the strongly-modulated periodic lattice of dielectric scatterers, which can give rise to extraordinary physical phenomena such as that studied in this paper, i. e., the so-called negative refraction. Notomi [9] showed that PhCs can behave as dielectric materials with an effective index of refraction in certain spectral regions where the equifrequency surfaces (EFSs) become rounded, despite of the fact that the underlying physical phenomenon is not really refraction. If the EFS shrinks with increasing frequency then the group velocity points inwards and a phenomenon of negative refraction can be expected at the interface between the PhC and air [9]. This is the case analyzed throughout this paper, in which the term “refractive” will be used to describe the studied phenomenon since it has been widely used in the literature. Negative refraction in a PhC was first observed experimentally at optical frequencies by Kosaka et al [8]. Other authors have realized theoretical studies about the conditions under which negative refraction occurs in PhCs [9-10]. Recently, the experimental demonstration of negative refraction in a two-dimensional (2D) PhC working at microwave frequencies has also been reported [11-12]. It should be noticed that it has been reported that negative refraction in PhCs can also occur at frequencies in the first photonic band [13-15]. In this case, an effective index of refraction cannot be associated to the PhC since the corresponding EFSs do not become rounded and the observed negative deflection of the beam inside the PhC can be explained by considering that the existence of a pseudogap for one of the two main symmetry directions makes the waves to travel along the allowed direction, which in some cases can be confused with negative refraction [16]. In fact, in this case the wave is not refracted negatively in the wedge’s experiment [2], as reported in Ref. 10. This phenomenon will not be considered here. In this paper we intend to shed more light in the phenomenon of negative refraction in PhCs. To this end, the evolution of the phase fronts of a wave propagating inside a PhC is analyzed in a frequency region in which the PhC can be considered as an effective medium in which negative refraction takes place at the interface air-PhC [9]. Surprisingly, we find that, in contrast with conventional LHMs in which the phase fronts advance is opposite to the energy flow, in a PhC both the phase velocity (defined for the plane wave component of the Bloch mode with the largest amplitude) and the energy flow (parallel to the group velocity in an infinite PhC) point in the same direction, both away from the source. We will see that this behavior can be explained by considering the excitation of states with wave vectors out of the first Brillouin zone that carry the main part of the Bloch wave energy. Despite this fact, negative refraction occurs at the interface between the PhC and air because of the existence of another plane wave component whose wave vector is inside the first Brillouin zone and is opposed to the energy flow inside the PhC. The refracted angles can be predicted with quite good accuracy by means of the Snell’s law with a negative effective index associated to the wave vector inside the first Brillouin zone, as in the case of LHMs. 2. Negative refraction at the interface between a 2D PhC wedge and air In our study, we work with a 2D PhC made of dielectric rods with ε = 10.3 and radius r = 0.4a, a being the lattice constant, arranged in a triangular lattice. The results shown here can be easily extended to other types of 2D as well as three dimensional PhCs. We use a plane wave expansion method [17] to calculate the band diagram of this PhC for TM polarization (electric field parallel to the rods’ axis). This band diagram is shown in Fig. 1(a). Frequency is represented in normalized units fa/c, f being the absolute frequency and c the light speed in vacuum. We will pay special attention to the two first frequency bands, highlighted in Fig. 1(a) as 1 and 2. For both bands there are frequency intervals for which the EFSs in the reciprocal space are circles. In these regions the PhC behaves as an isotropic dielectric material to which an effective refractive index neff that can be obtained from the radius of the corresponding EFS at each frequency can be associated [9]. Figures 1(b) and 1(c) show neff for (C) 2005 OSA 30 May 2005 / Vol. 13, No. 11 / OPTICS EXPRESS 4162 #7145 - $15.00 US Received 13 April 2005; revised 18 May 2005; accepted 19 May 2005 the bands 1 and 2 respectively calculated as explained in Ref. 10 for ΓM and ΓK directions. The sign of neff is positive if the radius of the EFS grows with increasing frequency and negative in case it shrinks [9]. The refractionlike behavior is evident when neff is identical for both ΓM and ΓK directions. For example, at the frequency 0.15 (first band) the PhC behaves as an effective medium with neff = 2.588. The same occurs in the second band for frequencies above 0.33. Following the results of Refs. 9 and 10 it could be expected that for frequencies in the second band the phase velocity and the group velocity are antiparallel and waves that travelling through free space impinge in this PhC should be negatively refracted with an angle that can be predicted from the Snell’s law using the effective index plotted in Fig. 1(c). Γ M K Γ Wave vector Frequency (fa/c) 2 1M K Γ Frequency (fa/c) ΓΜ ΓΚ fa/c= 0.15  neff = 2.588 ΓM ΓK ΓM: fa/c = 0.3153  neff = -1 (a) neff neff Frequency (fa/c) (b) (c) Fig. 1. (a) TM-polarization photonic band structure of the 2D PhC under study: a triangular lattice (the inset shows its first BZ) of dielectric rods with ε = 10.3 and radius r = 0.4a. Effective refractive index neff of (b) the first and (c) the second photonic bands for the ΓM and ΓK directions of propagation. To demonstrate the negative refraction behavior we simulate the wedge experiment reported in Refs. 2, 6, 10 and 12 using the finite-difference time-domain (FDTD) method [18] with perfectly-matched layer boundary conditions [19]. The first air-PhC interface is chosen to be along the ΓK direction (so the interface normal is parallel to the ΓM direction). Then the wave propagation inside the PhC is along the ΓM direction for which the modes corresponding to the first two bands have an even symmetry, which makes easier the excitation of the Bloch modes by an external plane wave. Figures 2(a) and 2(b) represent the propagation of the TM-polarized electric field through the wedge structure obtained from FDTD calculations for two different output interfaces (highlighted with the bold solid lines in Figs. 2(a) and (b)): (a) along ΓK, and (b) along ΓM. The frequency of the wave is fa/c = 0.3153, so neff = -1 for propagation along ΓM. As it is shown in Fig. 1(c), neff is slightly different for ΓM (neff = -1) and ΓK (neff = -1.15) at the frequency 0.3153. This means that the EFS is a hexagon-like circle, not really a circle. However, this fact has almost no influence on (C) 2005 OSA 30 May 2005 / Vol. 13, No. 11 / OPTICS EXPRESS 4163 #7145 - $15.00 US Received 13 April 2005; revised 18 May 2005; accepted 19 May 2005 the study carried out in this paper. The reason to choose such a frequency is that the EFS of the PhC is almost the same that for free space propagation, so the analysis to be carried out is more straightforward. The incident angles with respect to the normal (dashed lines in Figs. 2(a) and (b)) to the output interfaces are respectively φ i1 = 60º (Fig. 2(a)) and φ i2 = 30º (Fig. 2(b)). It is observed that refraction occurs to the same side of the interface normal and there is only a single ray propagating outside the PhC. The refracted angles can be roughly estimated from the phase fronts of the waves in free space and they are close to the values φ r1 = -60º (Fig. 2(a)) and φ r2 = -30º (Fig. 2(b)) that can be obtained by applying the Snell’s law with a negative effective index at the PhC side. From these results and taking into account previous works, one might conclude that inside the PhC the wave propagates backwards (in the sense of the evolution of the phase fronts) and opposed to the energy flow, as in a LHM, and that when the wave reaches the output interface the wave vector component normal to the surface changes its sign so negative refraction takes place. The next section is devoted to deeply analyze the wave propagation in the bulk of the PhC. ΓM ΓK (a) (b) PhC air φr2 φi2 φr1 φi1 Ez Fig. 2. FDTD simulation of the refraction of a monochromatic wave (fa/c = 0.3153) that propagates along the ΓM direction in a 2D PhC wedge. The output interface is along (a) ΓM and (b) ΓK directions. The electric field parallel to the rods’ axis is shown. The output PhC-air interface is highlighted with a bold solid line. The normal to the interface is highlighted with a bold dashed line. Arrows show the propagation direction of the waves inside the PhC and in air. 3. Wave propagation inside a photonic crystal in a region with a negative effective index of refraction To study wave propagation inside the PhC we consider a semi-infinite PhC like the studied above with rectangular instead of wedge shape in which a monochromatic TM-polarized wave is injected. The input interface is along the ΓK direction as in the wedge’s type calculations and the wave propagates along the ΓM direction. One lattice constant after the input interface we place a series of 101 electric field monitors (the field component parallel to the rods is measured) along the ΓM direction with a spacing of 20/3a between them and centered with respect to the symmetry axis of the incident wave in order to spatially sample the wave in the propagation direction. We make the PhC large enough in the propagation direction (more than 40 periods along the ΓM direction) in order to ensure that the reflected wave does not modify the monitored fields (the simulations end before the reflected wave reaches the monitors). In other words, we can say that our PhC is semi-infinite in the sense that only the wave that propagates away from the source is considered in the analysis. We present the results as in Ref. 20: a diagram that represents the electric field amplitude with the simulation time in the (C) 2005 OSA 30 May 2005 / Vol. 13, No. 11 / OPTICS EXPRESS 4164 #7145 - $15.00 US Received 13 April 2005; revised 18 May 2005; accepted 19 May 2005 vertical direction (in ct/a units, t being the absolute time) and the space (monitor position) in the horizontal direction. This kind of diagram allows a straightforward visualization of the phase fronts, as shown in Fig. 3 where the time-spatial evolution of an electromagnetic wave is presented for three different cases: (a) free-space propagation with fa/c = 0.3153, (b) propagation inside the PhC for fa/c = 0.15 (first band); and (c) propagation inside the PhC for fa/c = 0.3153 (second band). From Fig. 3 we can state that only in the case of free-space propagation (Fig. 3(a)) the phase fronts (lines with the same color in the diagrams) can be easily identified as they correspond to a plane wave. The phase velocity can be estimated from the slope of the phase fronts and is approximately equal to c, as expected. However, for the cases of propagation inside the PhC (Figs. 3(b) and (c)) a periodic modulation of the phase fronts is observed, as the wave inside the PhC is a Bloch mode that can be expressed as a set of plane waves. It is not easy to associate a phase index from these diagrams, owing to the modulation of the field, but what is really surprising is that the behavior of the wave inside the PhC is very similar for both bands 1 (positive index) and 2 (negative index), discounting the fact that the modulation is stronger for the second band. Moreover, from Fig. 3(c) we can see that the modulated phase fronts have a positive slope, in clear contrast to the diagram shown in Ref. 20 for a LHM. If we consider the main phase front (this concept will be analyzed more in detail in the next section), highlighted with a dashed line in Figs. 3(b) and (c) and corresponding to the main plane wave component that forms the Bloch wave, it travels upwards in the direction of the energy flow. Similar results were observed for other frequencies corresponding to the first and second band and employing a different spacing between field monitors, and backward-wave propagation was not identified in any case. We also tried to estimate the group velocity vg for each case from the slope of the leading edge of the monochromatic wave as shown by the bold solid line that forms an angle α with respect to the horizontal in Fig. 3(c). The results so obtained (vg = 0.345c at fa/c = 0.15 and vg = 0.276c at fa/c = 0.3153) were in excellent agreement with the group velocities (vg = 0.362c at fa/c = 0.15 and vg = 0.294c at fa/c = 0.3153) obtained from the band diagram (infinite PhC). The detected electric field for two adjacent field monitors (numbers 50 and 51, specifically) is shown in Figs. 4(a)-(d), as in Ref. [21]. Figures 4(a) and 4(b) correspond to the frequency 0.15 whereas Figs. 4(c) and 4(d) are obtained for the frequency 0.3153. The leftside diagrams shows the time step when the edge of the incoming wave reaches the monitors whereas the right-side ones show a time interval of the field once the steady state has been approximately reached (we can assume that the wave is monochromatic). As expected from causality considerations, in both cases the wave reaches the first monitor before than the second (see Figs. 4(a) and 4(c)). We also find that the phase front reaches before the monitor 50 than the monitor 51 when the steady state has been achieved regardless of the frequency (and, therefore, the sign of neff). The results shown in Fig. 4(d) are in clear disagreement with what would be expected for a LHM, in which the phase fronts would reach first the monitor 51 [21]. Similar diagrams (not shown here) were obtained by choosing other adjacent field monitors. (C) 2005 OSA 30 May 2005 / Vol. 13, No. 11 / OPTICS EXPRESS 4165 #7145 - $15.00 US Received 13 April 2005; revised 18 May 2005; accepted 19 May 2005 ct/a Electric field meter α main phase front ct/a Electric field meter ct/a Electric field meter ct/a Electric field meter phase fronts (a) (b) (c) Electric field monitor Electric field monitor Electric field monitor ct/a ct/act/a main phase front Fig. 3. Diagrams showing the evolution of the electric field with time (vertical axis) and space (horizontal axis). 101 field monitors are employed in a 2D FDTD simulation. In all cases the wave is monochromatic and TM-polarized. (a) Propagation in air, fa/c = 0.3153; (b) Propagation inside the PhC described in Fig. 1, fa/c = 0.15, (c) Propagation inside the PhC described in Fig. 1, fa/c = 0.3153. The phase fronts are the lines of the same color and the inverse of their slope gives the phase velocity. The slope of the arriving impulse stands for the group velocity, which can be obtained from the angle α in Fig. 3(c). The dashed lines in (b) and (c) highlight the slope of the main phase front. (C) 2005 OSA 30 May 2005 / Vol. 13, No. 11 / OPTICS EXPRESS 4166 #7145 - $15.00 US Received 13 April 2005; revised 18 May 2005; accepted 19 May 2005 (a) (b) Monitor 50 Monitor 51 Field amplitude (a. u.) Field amplitude (a. u.) Field amplitude (a. u.) Field amplitude (a. u.) ct/act/a ct/a ct/a (c) (d) Fig. 4. Detected electric field at the field monitors 50 and 51 (spaced 20/3a along the ΓM direction). A TM-polarized monochromatic wave propagating along ΓM and with frequencies fa/c = 0.15 [(a) and (b)] and 0.3153 [(c) and (d)] is injected in the 2D PhC. The diagrams (a) and (c) corresponds to the arrival of the leading edge (related to the group velocity). The diagrams (b) and (d) correspond to a time step for which the steady state has been reached and the signal can be considered almost monochromatic (related to the phase velocity). Ez φi air PhC φr Field amplitude (a. u.) ct/a (a) (b) (c) Electric field monitor ct/a main phase front α Fig. 5. Study of the 2D PhC in Ref. 11: a triangular lattice of dielectric rods (r = 0.182 and ε = 11), fa/c = 0.8648. (a) FDTD simulation (electric field) of the refraction of a monochromatic wave that propagates along the ΓM direction in a 2D PhC wedge. Output interface along the ΓM direction. (b) Diagram showing the evolution of the electric field with time (vertical axis) and space (horizontal axis) inside the PhC. 101 field monitors are employed. (c) Detected electric field at the field monitors 50 and 51 (spaced 20/3a along the ΓM direction). (C) 2005 OSA 30 May 2005 / Vol. 13, No. 11 / OPTICS EXPRESS 4167 #7145 - $15.00 US Received 13 April 2005; revised 18 May 2005; accepted 19 May 2005 4. Plane-wave decomposition of a Bloch mode that propagates inside a 2D PhC A TM-polarized Bloch mode that propagates inside a 2D PhC can be written as a superposition of plane waves as follows [23]: [] ∑∑ ∞ −∞= ∞ −∞= ++= mn nmz GnGmkjEkrE )(exp),( 21,      (1) where the sum applies to all the vectors on the 2D reciprocal space ( 21 GnGmG    += , m and n being integers), Em,n is the electric field amplitude of the component with wave vector 21 GnGmk    ++ , and k  is the wave vector in the first BZ or fundamental wave vector. In a triangular lattice the vectors 1 G  and 2 G  can be chosen as ) ˆ 3 1 ˆ ( 2 1yx a G+= π  and ) ˆ 3 1 ˆ ( 2 2yx a G+−= π  , where x ˆ and y ˆ are the unit vectors in the ΓK (transverse) and the ΓM (longitudinal) directions, respectively (see Fig. 6). It should be noticed that although k  is the fundamental wave vector, this fact does not necessarily imply that the plane wave with wave vector k  carries most of the Bloch wave energy. In other words, E0,0 > Em,n for m ≠ 0, n ≠ 0, is not mandatory. Suppose we inject a TM-polarized monochromatic plane wave with frequency fa/c = 0.3153 propagating through air into the 2D PhC previously analyzed (see Fig. 1). The interface is along ΓK, as in Fig. 2(a), and the incidence is normal to the surface, so the propagation inside the PhC is along ΓM. At the chosen frequency, the EFSs of the air and the 2D PhC are almost identical, although in the PhC the EFS is replicated on the whole reciprocal space due to the 2D periodicity of the structure. When the wave enters the 2D PhC the normal component of the incident wave vector [ yakiˆ )/2(315.0 π =  in Fig. 6] should reverse its sign since we are in a region in which the EFS of the PhC is identical to that of the air but with the group velocity pointing inwards [1,9-10]. The boundary conditions at the interface impose that [8] alkk iPhC /2 //// π += (2) with ki// and kPhC// being the wave vector components parallel to the interface in air and inside the 2D PhC respectively, a the periodicity of the interface along ΓK (that in this case is the lattice constant) and l an integer. As the incidence is normal to the interface, we have ki// = 0 so alkPhC /2 // π =. In agreement with the band diagram shown in Fig. 1(a) and the refraction results shown in Fig. 2, we can conclude that the fundamental wave vector of the Bloch mode is yak ˆ )/2(315.0 0,0 π −=  (see Fig. 6), as at this point the group velocity, and therefore, the energy flow, points upwards so causality is not violated [9-10]. However, this is not the only wave vector that forms the Bloch wave. Instead, the Bloch wave has the form of Eq. (1) with fundamental wave vector yakk ˆ )/2(315.0 0,0 π −==  . In the following study we will only take into account the seven wave vectors of first-order, that is, 210,0, GnGmkk nm     ++= with |m|, |n| ≤ 1 (see Fig. 6). Since we are analyzing the second band we think that the results obtained taking into account only this set of wave vectors can be enough to give a good picture of the propagation inside the PhC. (C) 2005 OSA 30 May 2005 / Vol. 13, No. 11 / OPTICS EXPRESS 4168 #7145 - $15.00 US Received 13 April 2005; revised 18 May 2005; accepted 19 May 2005