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

Optics Express, Vol. 13, Issue 11, pp. 4160-4174 (2005)

http://dx.doi.org/10.1364/OPEX.13.004160

Acrobat PDF (568 KB)

### 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 (*S*⃗·*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 (*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

## 1. Introduction

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

*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 (

*S*⃗·

*k*⃗<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]

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]

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]

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]

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]

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]

## 2. Negative refraction at the interface between a 2D PhC wedge and air

*ε*=10.3 and radius

*r*=0.4

*a*, 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]

*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

**62**, 10696–10705 (2000). [CrossRef]

*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

**62**, 10696–10705 (2000). [CrossRef]

*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).

19. J. P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Comput. Phys. **114**, 185–200 (1994). [CrossRef]

*fa*/

*c*=0.3153, so

*n*

_{eff}=-1 for propagation along ΓM. As it is shown in Fig. 1(c),

*n*

_{eff}is slightly different for ΓM (

*n*

_{eff}=-1) and ΓK (

*n*

_{eff}=-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 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}=60° (Fig. 2(a)) and

*ϕ*

_{i2}=30° (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}=-60° (Fig. 2(a)) and

*ϕ*

_{r2}=-30° (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.

## 3. Wave propagation inside a photonic crystal in a region with a negative effective index of refraction

*a*/20 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 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

*v*

_{g}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 (

*v*

_{g}=0.345c at

*fa*/

*c*=0.15 and

*v*

_{g}=0.276

*c*at

*fa*/

*c*=0.3153) were in excellent agreement with the group velocities (

*v*

_{g}=0.362

*c*at

*fa*/

*c*=0.15 and

*v*

_{g}=0.294

*c*at

*fa*/

*c*=0.3153) obtained from the band diagram (infinite PhC).

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

*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]

## 4. Plane-wave decomposition of a Bloch mode that propagates inside a 2D PhC

*G*⃗=

*mG*⃗

_{1}+

*nG*⃗

_{2},

*m*and

*n*being integers),

*E*

_{m,n}is the electric field amplitude of the component with wave vector

*k*⃗+

*mG*⃗

_{1}+

*nG*⃗

_{2}, and

*k*⃗ is the wave vector in the first BZ or fundamental wave vector. In a triangular lattice the vectors

*G*⃗

_{1}and

*G*⃗

_{2}can be chosen as

*x*̂ 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

*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,

*E*

_{0,0}>

*E*

_{m,n}for

*m*≠0,

*n*≠0, is not mandatory.

*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 [

*k*⃗

*=0.315(2*

_{i}*π*/

*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]

**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]

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]

*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∥}=2

*πl*/

*a*.

*k*⃗

_{0,0}=-0.315(2

*π*/

*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

**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]

*k*⃗=

*k*⃗

_{0,0}=-0.315(2

*π*/

*a*)

*ŷ*. In the following study we will only take into account the seven wave vectors of first-order, that is,

*k*⃗

_{m,n}=

*k*⃗

_{0,0}+

*mG*⃗

_{1}+

*nG*⃗

_{2}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.

*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 2

*π*/

*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.

*a*/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(2

*π*/

*a*), 0.835(2

*π*/

*a*) and 1.46(2

*π*/

*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

*k*⃗

_{1,0}=2

*π*/

*ax*̂-0.262(2

*π*/

*a*)

*ŷ*(and its symmetric counterpart,

*k*⃗

_{0,1}=2

*π*/

*ax*̂-0.262(2

*π*/

*a*)

*ŷ*,

*k*⃗

_{1,1}=0.84(

*π*/

*a*)ŷ, and

*k*⃗

_{-1,-1}=1.47(2

*π*/

*a*)

*ŷ*, respectively. Owing to the symmetry of the PhC, we can assume that both wave vectors,

*k*⃗

_{1,0}and

*k*⃗

_{0,1}, 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,

*k*⃗

_{0,0}. One reason to explain this is that, owing to an insufficient sampling accuracy, the

*k*⃗

_{0,0}component may be masked by the component

*k*⃗

_{1,0}with a longitudinal component 0.26(2

*π*/

*a*). Then we repeated the procedure but with a spacing of √3

*a*/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: √3

*a*/20 (solid curve) and √3

*a*/5 (dashed curve). From Fig. 7(c) we can see that, although when the spacing between monitors is √3

*a*/20 a unique peak appears around 0.25(2

*π*/

*a*), when the spacing is √3

*a*/5 two different peaks are clearly discernible: one at 0.26(2π/

*a*) that corresponds to

*k*⃗

_{1,0}=2

*π*/

*ax*̂-0.262(2

*π*/

*a*)

*ŷ*as stated before, and another one at 0.315(2

*π*/

*a*) that corresponds to the fundamental wave vector

*k*⃗

_{0,0}=-0.315(2

*π*/

*a*)

*ŷ*. However, we find that the amplitude of the field component with wave vector

*k*⃗

_{0,0}is much lower than the amplitude of the other main components,

*k*⃗

_{1,0}and

*k*⃗

_{1,1}. 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 2

*π*/

*a*, that is,

*k*⃗

_{1,0}and

*k*⃗

_{0,1}, 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.

*k*⃗

_{1,1}=0.84(2

*π*/

*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

*k*⃗

_{1,1}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

*k*⃗

_{0,0}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.

*a*/20 between adjacent field monitors. For example, Fig. 8(a) depicts the wave vector decomposition of the electric field when monochromatic waves with frequencies corresponding to the first band (

*fa*/

*c*=0.1 and 0.15 respectively) are injected in our PhC. Only positive wave vectors are shown to better appreciate the amplitude peaks. We can see that there is a unique relevant field component for each frequency and this component has a wave vector that agrees well with the wave vector inside the first BZ obtained from the band diagram:

*k*⃗

_{0,0}=0.25(2

*π*/

*a*)

*ŷ*for

*fa*/

*c*=0.1 and

*k*⃗

_{0,0}=0.39(2

*π*/

*a*)

*ŷ*for

*fa*/

*c*=0.15. Then, in clear contrast with the results shown previously in Fig. 7, we can state that for frequencies within the first band the plane wave component that carries almost all the power of the Bloch state is that with

*k*⃗

_{0,0}, which is inside the first BZ. This result agrees with the main phase front slope highlighted in Fig. 3(b) with a dashed line. The lack of other significant components in the plane wave decomposition also agrees with the slightly-modulated field shown in Fig. 3(b). This effect can be simply explained by the fact that at low frequencies the wavelength is much larger than the PhC periodicity and the wave sees an almost homogeneous medium so the wave that propagates inside the PhC mostly behaves as a plane wave with wave vector

*k*⃗

_{0,0}. In other words, at low frequencies the system is highly refractive. Figure 8(b) shows the same plane-wave decomposition but for frequencies corresponding to the second band:

*fa*/

*c*=0.3, 0.33 and 0.36. In the three cases, we observe similar features to those previously analyzed for the frequency

*fa*/

*c*=0.3153: i) the main plane wave component is that corresponding to a wave vector

*k*⃗

_{1,1}; ii) the existence of a component with wave vector

*k*⃗

_{0,0}is noticed in Fig. 8(b) as the value of this wave vector should diminish with increasing frequency, in contrast to the

*k*⃗

_{1,0}and

*k*⃗

_{1,1}whose longitudinal components increase with frequency; and iii) the obtained wave vectors are in good agreement with those that can be obtained by replicating the fundamental wave vector

*k*⃗

_{0,0}obtained from the band diagram on the reciprocal space as shown in Fig. 6. It is also interesting to notice that the amplitude of the component with wave vector

*k*⃗

_{0,0}is very small compared to the components with wave vectors

*k*⃗

_{1,0}and

*k*⃗

_{1,1}at

*fa*/

*c*=0.36.

*fa*/

*c*=0.5 corresponding to the fifth band (with a positive index, as the first band) and in the decomposition we also found that the main plane wave components had wave vectors out of the first BZ. In this case we also obtained a time-space diagram (not shown here) as those depicted in Fig. 3 and the result was that the main phase front had also a positive slope as the rest of the cases considered in the text. We also calculated the plane wave decomposition (not shown here) for the case described in Fig. 5 [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]

*fa*/

*c*=0.8648), and the main plane wave components were those with a transverse wave vector 2π/

*a*. However, and in agreement with the previous case, we found that the amplitude of the component with wave vector

*k*⃗

_{0,0}was much smaller than other components with wave vectors out of the first BZ.

*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

*k*⃗

_{0,0}and

*k*⃗

_{1,0}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

*k*⃗

*. The direction of the energy flow in air points to the same direction than*

_{r}*k*⃗

*and is given by the arrow in Fig. 9(a), in fairly agreement with the field distribution shown in Fig. 2(a).*

_{r}*k*⃗

_{0,0}is the only wave vector having a component parallel to the interface that intersects the air EFS. The refracted ray with wave vector

*k*⃗

*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,*

_{r}*k*⃗

_{0,1},

*k*⃗

_{1,0}and

*k*⃗

_{1,1}, 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

*k*⃗

_{0,1},

*k*⃗

_{1,0}and

*k*⃗

_{1,1}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

*k*⃗

_{0,0}component so they can cross the interface and be refracted into air.

## 5. Conclusion

*n*

_{eff}is strongly different from the propagation inside a left-handed material in which the phase fronts evolve in the opposite direction of the energy flow. It is concluded that the main difference comes from the fact that in bulk of the PhC, the electromagnetic energy propagates in the form of a Bloch wave. It has been shown that at the frequency in which negative refraction occurs, the main component of the Bloch wave is a plane wave whose wave vector is outside the first BZ and that propagates in the same direction that the energy flow. In this sense, PhCs present a very different behavior in comparison to conventional LHMs, although negative refraction takes place for both kinds of composites.

## Acknowledgments

## References and Links

1. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. |

2. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

3. | J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. |

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. |

5. | N. García and M. Nieto-Vesperinas, “Is there an experimental verification of a negative index of refraction yet?,” Opt. Lett. |

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) |

8. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B |

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 |

10. | S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B |

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 |

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. |

13. | B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Amer. A |

14. | C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B |

15. | E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Negative refraction by photonic crystals,” Nature (London) |

16. | H.-T. Chien, H.-T. Tang, C.-H. Kuo, C.-C. Chen, and Z. Ye, “Directed diffraction without negative refraction,” Phys. Rev. B |

17. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

18. | A. Taflove, |

19. | J. P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Comput. Phys. |

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 |

21. | R.W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E |

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. |

23. | K. Sakoda, |

24. | A. Yariv and P. Yeh, |

**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

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### References

- V. G. Veselago, �The electrodynamics of substances with simultaneously negative values of ? and ?,� Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- R. A. Shelby, D. R. Smith, and S. Schultz, �Experimental verification of a negative index of refraction,� Science 292, 77-79 (2001). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- J. D. Joannopoulos, P. Villeneuve, and S. Fan, �Photonic crystals: putting a new twist on light,� Nature (London) 386, 143-149 (1997). [CrossRef]
- 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]
- 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]
- S. Foteinopoulou and C. M. Soukoulis, �Negative refraction and left-handed behavior in two-dimensional photonic crystals,� Phys. Rev. B 67, 235107 (2003). [CrossRef]
- 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]
- 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]
- B. Gralak, S. Enoch, and G. Tayeb, �Anomalous refractive properties of photonic crystals,� J. Opt. Soc. Amer. A 17, 1012-1020 (2000). [CrossRef]
- 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]
- E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, �Negative refraction by photonic crystals,� Nature (London) 423, 604-605 (2003). [CrossRef]
- 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]
- 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]
- A. Taflove, Computational Electrodynamics�The Finite Difference Time-Domain Method (Artech House, Boston, 1995).
- J. P. Berenger, �A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,� J. Comput. Phys. 114, 185-200 (1994). [CrossRef]
- 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]
- R.W. Ziolkowski and E. Heyman, �Wave propagation in media having negative permittivity and permeability,� Phys. Rev. E 64, 056625 (2001). [CrossRef]
- 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]
- K. Sakoda, Optical properties of photonic crystals (Springer, Berlin, 2001).
- A. Yariv, P. Yeh, Optical Waves in Crystals : Propagation and Control of Laser Radiation (New York, Wiley, 1984).

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