OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 11 — May. 30, 2005
  • pp: 4160–4174
« Show journal navigation

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

Alejandro Martínez, Hernán Míguez, José Sánchez-Dehesa, and Javier Martí  »View Author Affiliations


Optics Express, Vol. 13, Issue 11, pp. 4160-4174 (2005)
http://dx.doi.org/10.1364/OPEX.13.004160


View Full Text Article

Acrobat PDF (568 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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 (S⃗·k>0), 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 (k⃗) 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

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

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

]. Veselago deduced that in this kind of medium the electric field E⃗, the magnetic field H⃗ and the wave vector k⃗ 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 S⃗ are antiparallel (S⃗·k⃗<0) 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

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

]. 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

2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

]. Following the work of Pendry and coworkers [3

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). [CrossRef] [PubMed]

,4

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). [CrossRef]

], 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

2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

]. Although there was a debate (see [5

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). [CrossRef]

], 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

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). [CrossRef] [PubMed]

].

Other materials that have shown to refract negatively the electromagnetic radiation are photonic crystals (PhCs) [7

7. J. D. Joannopoulos, P. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997). [CrossRef]

]. 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 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

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). [CrossRef]

]. 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

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). [CrossRef]

] 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

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). [CrossRef]

]. 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

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). [CrossRef]

]. Other authors have realized theoretical studies about the conditions under which negative refraction occurs in PhCs [9

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). [CrossRef]

10

10. S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2003). [CrossRef]

]. Recently, the experimental demonstration of negative refraction in a two-dimensional (2D) PhC working at microwave frequencies has also been reported [11

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). [CrossRef]

12

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). [CrossRef] [PubMed]

]. 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

13. B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Amer. A 17, 1012–1020 (2000). [CrossRef]

15

15. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Negative refraction by photonic crystals,” Nature (London) 423, 604–605 (2003). [CrossRef]

]. 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

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). [CrossRef]

]. In fact, in this case the wave is not refracted negatively in the wedge’s experiment [2

2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

], 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

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). [CrossRef]

]. 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

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. [CrossRef] [PubMed]

] 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 n eff that can be obtained from the radius of the corresponding EFS at each frequency can be associated [9

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). [CrossRef]

]. Figures 1(b) and 1(c) show n eff for the bands 1 and 2 respectively calculated as explained in Ref. 10 for ΓM and ΓK directions. The sign of n eff is positive if the radius of the EFS grows with increasing frequency and negative in case it shrinks [9

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). [CrossRef]

]. 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 n eff=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).

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 n eff of (b) the first and (c) the second photonic bands for the ΓM and ΓK directions of propagation.

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

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

21. R.W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001). [CrossRef]

]. 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 left-side 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 n eff). 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

21. R.W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001). [CrossRef]

]. Similar diagrams (not shown here) were obtained by choosing other adjacent field monitors.

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.
Fig. 4. Detected electric field at the field monitors 50 and 51 (spaced √3a/20 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).
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 √3a/20 along the ΓM direction).

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

23. K. Sakoda, Optical properties of photonic crystals (Springer, Berlin, 2001).

]:

Ez(r,k)=m=n=Em,nexp[j(k+mG1+nG2)]
(1)

where the sum applies to all the vectors on the 2D reciprocal space (G⃗=mG1+nG2, m and n being integers), E m,n is the electric field amplitude of the component with wave vector k⃗+mG1+nG2, and k⃗ is the wave vector in the first BZ or fundamental wave vector. In a triangular lattice the vectors G1 and G2 can be chosen as G1=2πa(x̂+13ŷ) and G2=2πa(x̂+13ŷ), where x̂ and ŷ are the unit vectors in the ΓK (transverse) and the ΓM (longitudinal) directions, respectively (see Fig. 6). It should be noticed that although k⃗ is the fundamental wave vector, this fact does not necessarily imply that the plane wave with wave vector k⃗ carries most of the Bloch wave energy. In other words, E 0,0>E m,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 [ki=0.315(2π/a)ŷ 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

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

,9

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). [CrossRef]

10

10. S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2003). [CrossRef]

]. The boundary conditions at the interface impose that [8

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). [CrossRef]

]

kPhC=ki+2πla
(2)

with k i and k PhC 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 k i=0 so k PhC∥=2πl/a.

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 k0,0=-0.315(2π/a)ŷ (see Fig. 6), as at this point the group velocity, and therefore, the energy flow, points upwards so causality is not violated [9

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). [CrossRef]

10

10. S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2003). [CrossRef]

]. 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 k⃗=k0,0=-0.315(2π/a)ŷ. In the following study we will only take into account the seven wave vectors of first-order, that is, km,n=k0,0+mG1+nG2 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.

Fig. 6. Wave vector diagram of the 2D PhC under study (Fig. 1). The red circular contours correspond to the EFSs at frequency fa/c=0.3153. The fundamental vectors of the reciprocal lattice are G1 and G2. First and second BZs are highlighted with different gray tones.

The field amplitude E m,n of each plane wave is obtained by use of the space sampling of the electric field propagated inside the PhC: a sampling of the electric field along the ΓM (ΓK) directions permits to obtain the wave vectors along the y (x) direction and an estimation of the field amplitude of the plane wave with such a wave vector component. First, we perform a transverse sampling with 51 electric field monitors spaced a/20 (in the transverse direction, ΓK, the period is a), the first monitor placed on the mirror symmetry axis of the propagating signal. As observed in Figs. 2(a) and 2(b), the field has an even symmetry when propagates along ΓM at a frequency corresponding to the second band so we can double the number of samples when applying the Fourier transform. Then we choose a time step in which the steady state has been reached (as before, we consider a time step for which the wave can be assumed fully monochromatic) and calculate the space Fourier transform of the field monitored at each point in order to obtain information in the wave vector space. It should be mentioned that this procedure is just an estimation of the transverse electric-field composition as we are sampling only along a line (one-dimensional sampling) and not over the whole 2D space. The results for the transverse sampling are shown in Fig. 7(a). It should be noted that similar results were obtained for other time steps after the reaching of the steady state and also after displacing the field monitor half a period in the ΓM direction. As our FDTD calculations only allow us to obtain a measure of the amplitude but not the phase of the field, we cannot obtain information about the sign of the wave vectors. From Fig. 7(a) we can state that the main plane wave components have a zero wave vector in the x direction, as the 85.45 % of the power is carried by a zero transverse wave vector. In contrast, the power carried by plane waves with transverse wave vectors of absolute value 2π/a is only about the 12.6 % of the total power. These components are responsible for the transverse periodic modulation of the wave that is observed in Figs. 2(a) and 2(b). It should be mentioned that we estimate the power of each component as the sum of the squares of each field component.

The same analysis is done with the field monitors along the propagation (longitudinal) direction. In this case we place 201 field monitors spaced √3a/40 and also take the Fourier transform of the sampled electric field at a given time step assuming that the wave has reached its steady state and no reflected waves are present. Similar results were also obtained at other time steps and also by displacing the monitors half a period rightwards. Figure 7(b) shows the obtained distribution of field amplitudes at each wave vector (in the propagation direction). We can appreciate clearly three main contributions with absolute (at previously stated, the sign of the wave vector cannot be estimated from our simulations) wave vectors 0.26(2π/a), 0.835(2π/a) and 1.46(2π/a) that carry about the 13 %, the 85 % and the 2 % of the total power, respectively. Comparing these results with the wave vector distribution shown in Fig. 6, we can relate the obtained contributions with the wave vectors k1,0=2π/ax̂-0.262(2π/a)ŷ (and its symmetric counterpart, k0,1=2π/ax̂-0.262(2π/a)ŷ, k1,1=0.84(π/a)ŷ, and k-1,-1=1.47(2π/a)ŷ, respectively. Owing to the symmetry of the PhC, we can assume that both wave vectors, k1,0 and k0,1, are equally excited and, therefore, they carry the same power. Surprisingly, we do not find any wave vector with an absolute value close to 0.313(2π/a) that could be associated to fundamental component inside the first BZ, k0,0. One reason to explain this is that, owing to an insufficient sampling accuracy, the k0,0 component may be masked by the component k1,0 with a longitudinal component 0.26(2π/a). Then we repeated the procedure but with a spacing of √3a/5 between adjacent field monitors in order to be able to distinguish these two components. Figure 7(c) shows the distribution of field amplitudes at fa/c=0.3153 but for different samplings: √3a/20 (solid curve) and √3a/5 (dashed curve). From Fig. 7(c) we can see that, although when the spacing between monitors is √3a/20 a unique peak appears around 0.25(2π/a), when the spacing is √3a/5 two different peaks are clearly discernible: one at 0.26(2π/a) that corresponds to k1,0=2π/ax̂-0.262(2π/a)ŷ as stated before, and another one at 0.315(2π/a) that corresponds to the fundamental wave vector k0,0=-0.315(2π/a)ŷ. However, we find that the amplitude of the field component with wave vector k0,0 is much lower than the amplitude of the other main components, k1,0 and k1,1. We pay special attention on the fact that for both the transversal and the longitudinal sampling we obtain that the total power carried out by components with transverse wave vector equal to 2π/a, that is, k1,0 and k0,1, is about the 13 % of total power of the Bloch wave. The agreement between the EFS plot shown in Fig. 6 (in which the EFS radius is calculated from the band diagram) and the Fourier decomposition is excellent.

Fig. 7. Plane-wave decomposition of the space sampling of the electric field that propagates inside the PhC under study along the ΓM direction. The field correspond to a simulation time step for which the wave has reached its steady state. (a) Transverse sampling (a/20 spacing); (b) longitudinal sampling (√3a/40 spacing); (c) longitudinal sampling: √3a/20 spacing (solid curve) and √3a/5 spacing (dashed curve). The peaks corresponds to the amplitude of a certain plane wave component of the whole Bloch wave.

From the results depicted in Fig. 7 we can also state that the main plane-wave component of the Bloch wave propagation though the 2D PhC has a wave vector k1,1=0.84(2π/a)ŷ, which is well outside the first BZ. Specifically, as shown in Fig. 6, it is located in the second BZ, as we could have deduced intuitively by considering that we are exciting the second photonic band. Moreover, if we associate a phase velocity to the plane wave with wave vector k1,1 we obtain that it corresponds to the main phase front component shown before with a dashed line in Fig. 3(c). And more surprisingly, we find that the main plane wave component is not a backward wave (the components corresponding to the backwards wave vectors k0,0 and k-1,-1 carry a small amount of power) as in a conventional LHM, but a forward one, with it phase front moving in the same direction that the energy flow. In this way, PhCs behave clearly in a different way than LHMs. In fact, we obtain that in PhCs it takes more sense to define the phase velocity as that of the main plane wave component, and not that of the fundamental component (inside the first BZ), paying attention to the evolution of phase fronts inside the PhC. This results agrees with that given in Ref. 24 when analyzing wave propagation in one dimensional periodic layered media. However, the value of n eff to be used in the Snell’s law to analyze wave refraction should be defined by choosing the fundamental wave vector.

Fig. 8. Plane-wave decomposition of the longitudinal sampling (√3a/20 spacing) of the electric field that propagates inside the PhC under study along the ΓM direction. The field correspond to a simulation time step for which the wave has reached its steady state. (a) first band: fa/c=0.1 (solid curve); fa/c=0.15 (dotted curve); (b) second band: fa/c=0.3 (solid curve); fa/c=0.33 (dashed curve), fa/c=0.36 (dotted curve). Only positive wave vectors are shown.

To conclude this section, let us describe the physical phenomenon that occurs at the output interface in the wedge structure. A schematic plot of our explanation is shown in Figs. 9(a) and 9(b), for the interfaces along ΓK and ΓM respectively. When the wave reaches the interface, the component of the wave vectors parallel to the interface is conserved. This condition is established by the dashed lines in Figs. 9(a) and 9(b). These lines are parallel and the spacing between them depends on the periodicity of the interface. The wave vectors that propagate in air are given by the crossing of these lines with the EFS of air, which, as we mentioned previously, coincides with that of the PhC, although in this case, as the air is an homogeneous medium, the EFS is not replicated on the reciprocal space. First let us discuss the case the case in which the output interface is along ΓK direction [see Fig. 9(a)]. The incident angle is ϕ i1=60° as in Fig. 2(a). Only the main wave vectors that contribute to the propagating the whole Bloch wave are plotted. From these k0,0 and k1,0 are the ones whose component parallel to the interface intersects with the air EFS. Both components give rise to the same refracted ray with wave vector kr. The direction of the energy flow in air points to the same direction than kr and is given by the arrow in Fig. 9(a), in fairly agreement with the field distribution shown in Fig. 2(a).

Fig. 9. Schematic explanation of the refraction at the output PhC-air interface using a wave vector diagram. The circle corresponds to the EFS of air. The hexagon is the first BZ of the PhC. The bold solid gray lines show the interface: (a) along ΓK; (b) along ΓM. The dashed lines represent the condition of conservation of the wave vector components parallel to the interface. The arrows stand for the energy flow of the wave before and after the interface.

The situation is slightly different when the interface is along ΓM [see Fig. 9(b)] since in this case k0,0 is the only wave vector having a component parallel to the interface that intersects the air EFS. The refracted ray with wave vector kr obtained from this intersection, as depicted in Fig. 9(b), defines the direction of the refracted energy flow in air, in perfect agreement with the result in Fig. 2(b). The rest of components, k0,1,k1,0 and k1,1, which carry the main part of the energy of the Bloch wave propagating through the PhC, do not produce any intersection with the air EFS. An issue to be solved in a future work is to understand the behavior of the components with wave vectors k0,1,k1,0 and k1,1 when they reach the interface. The possible choices are: (i) these components are totally reflected at the interface since they are below the air light cone; (ii) these components transfer their energy to the k0,0 component so they can cross the interface and be refracted into air.

5. Conclusion

Acknowledgments

This work has been partially funded by the Spanish Ministry of Science and Technology under grant TIC2002-01553.

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). [CrossRef]

2.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef] [PubMed]

7.

J. D. Joannopoulos, P. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997). [CrossRef]

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). [CrossRef]

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). [CrossRef]

10.

S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2003). [CrossRef]

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). [CrossRef]

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). [CrossRef] [PubMed]

13.

B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Amer. A 17, 1012–1020 (2000). [CrossRef]

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). [CrossRef]

15.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Negative refraction by photonic crystals,” Nature (London) 423, 604–605 (2003). [CrossRef]

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). [CrossRef]

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. [CrossRef] [PubMed]

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). [CrossRef]

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). [CrossRef]

21.

R.W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001). [CrossRef]

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). [CrossRef]

23.

K. Sakoda, Optical properties of photonic crystals (Springer, Berlin, 2001).

24.

A. Yariv and P. Yeh, Optical Waves in Crystals : Propagation and Control of Laser Radiation (New York, Wiley, 1984).

OCIS Codes
(160.4670) Materials : Optical materials
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Research Papers

History
Original Manuscript: April 13, 2005
Revised Manuscript: May 18, 2005
Published: May 30, 2005

Citation
Alejandro Mart�nez, Hern�n M�guez, Jos� S�nchez-Dehesa, and Javier Mart�, "Analysis of wave propagation in a two-dimensional photonic crystal with negative index of refraction: plane wave decomposition of the Bloch modes," Opt. Express 13, 4160-4174 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-4160


Sort:  Journal  |  Reset  

References

  1. V. G. Veselago, �The electrodynamics of substances with simultaneously negative values of ? and ?,� Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
  2. R. A. Shelby, D. R. Smith, and S. Schultz, �Experimental verification of a negative index of refraction,� Science 292, 77-79 (2001). [CrossRef] [PubMed]
  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). [CrossRef] [PubMed]
  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). [CrossRef]
  5. N. Garc�a and M. Nieto-Vesperinas, �Is there an experimental verification of a negative index of refraction yet?,� Opt. Lett. 27, 885-887 (2002). [CrossRef]
  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) [CrossRef] [PubMed]
  7. J. D. Joannopoulos, P. Villeneuve, and S. Fan, �Photonic crystals: putting a new twist on light,� Nature (London) 386, 143-149 (1997). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  10. S. Foteinopoulou and C. M. Soukoulis, �Negative refraction and left-handed behavior in two-dimensional photonic crystals,� Phys. Rev. B 67, 235107 (2003). [CrossRef]
  11. A. Mart�nez, H. M�guez, A. Griol, and J. Mart�, �Experimental and theoretical study of the self-focusing of light by a photonic crystal lens,� Phys. Rev. B 69, 165119 (2004). [CrossRef]
  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). [CrossRef] [PubMed]
  13. B. Gralak, S. Enoch, and G. Tayeb, �Anomalous refractive properties of photonic crystals,� J. Opt. Soc. Amer. A 17, 1012-1020 (2000). [CrossRef]
  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). [CrossRef]
  15. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, �Negative refraction by photonic crystals,� Nature (London) 423, 604-605 (2003). [CrossRef]
  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). [CrossRef]
  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), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173."> http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.<a/> [CrossRef] [PubMed]
  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). [CrossRef]
  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). [CrossRef]
  21. R.W. Ziolkowski and E. Heyman, �Wave propagation in media having negative permittivity and permeability,� Phys. Rev. E 64, 056625 (2001). [CrossRef]
  22. M. Qiu, L. Thyl�n, 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). [CrossRef]
  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).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited