Negative refraction in Photonic Crystals: thickness dependence and Pendellösung phenomenon

Negative refraction in Photonic Crystals: thickness dependence and Pendellösung phenomenon

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  Negative refraction in Photonic Crystals: thickness dependence and Pendellösung phenomenon Vito   Mocella    Istituto per la Microelettronica e Microsistemi – Consiglio Nazionale delle  Ricerche (IMM-CNR) - Sezione di Napoli, Via P. Castellino 111, 80131 Napoli,  Italy; Abstract: We show that the refracted wave at the exit surface of a Photonic Crystal (PhC) slab is periodically modulated, in positive or in negative direction, changing the slab thickness. In spite of an always increasing literature, the effect of the thickness in negative refraction on PhC’s does not seem to be appropriately considered. However such an effect is not surprising if interpreted with the help of Dynamical Diffraction Theory (DDT), which is generally applied in the x-ray diffraction. The thickness dependence is a direct result of the so-called Pendellösung phenomenon. That explains the periodic exchange, inside the crystal, of the energy among direct beam (or positively refracted) and diffracted beam (or negatively refracted). The Pendellösung phenomenon is an outstanding example of the application of the DDT as a powerful and simple tool for the analysis of s electromagnetic interaction in PhC’s. © 2005 Optical Society of America OCIS codes : (050.1960) Diffraction theory; (260.2110) Electromagnetic theory; (290.4210) Multiple scattering; (999.9999) Photonic crystal. ___________________________________________________________________________ References and links 1.   J. B. Pendry and D. R. Smith. "Reversing light with negative refraction,” Physics Today 57 , 37-43 (2004). 2.   D. R. Smith, J. B. Pendry, M.C.K. Wiltshire, “Metamaterials and Negative refractive index,” Science 305 , 788-792 (2004). 3.   R. A. Shelby, D. R. Smith and S. Schultz. "Experimental verification of a negative index of refraction," Science 292  , 77-79 (2001) 4.   S. 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E 70 , 026608 (2004). 1.   Introduction Negative refraction was the subject of a large amount of papers in the last years [1-7], also in view of very interesting new phenomena like the super-lens effect [8-16]. A negative refractive index arises in meta-materials where inhomogeneities are much smaller than the wavelength of the incoming radiation and it has been demonstrated at microwave wavelengths. On the other hand, it has been shown that using photonic crystals (PhC’s) it is possible to obtain a negative refraction behaviour also at optical wavelengths [17-27], i.e., a refracted beam going to the “wrong direction” compared to the direction expected on the base of the classical refraction laws. Most of the theoretical analysis of negative refraction in PhC’s makes use of the equi-frequency surface (EFS), (ex. Refs. [18, 19, 23, 25, 26, 27]) which are contour diagrams in the (C) 2005 OSA7 March 2005 / Vol. 13, No. 5 / OPTICS EXPRESS 1362 #6107 - $15.00 USReceived 20 December 2004; revised 15 February 2005; accepted 15 February 2005  wavevector space for a constant frequency. The normal to such curves gives the Poynting vector direction, indicating the energy flow in PhC’s. In such an analysis, using the conservation of the parallel wavevector component across the boundary, conditions are determined under which negative refraction behaviour appears in a PhC even without a negative refraction index. We follow here the classical approach of Dynamical Diffraction Theory (DDT) [28, 29] making also use of EFS (dispersion surfaces, adopting the DDT terminology). We will show however that, in general, it is not correct to select among the different possible wavevectors in the crystal. In general the possible different wavevectors in the crystal coexist, and it is really their difference which explains the modulation: this is the base of the phenomenon that we analyze in this paper. In the following we will show that, at the exit surface of the crystal, the intensity will be at a maximum in the positive (forward diffracted), or in the negative (diffracted) direction depending on the difference of the two wavevectors inside the crystal: this is a manifestation of the Pendellösung effect. Pendellösung is a fundamental result of DDT, as worked out by Ewald at beginning of the 20 th  century [30-32]. This phenomenon is a periodic exchange of energy between two interfering beams inside the crystal, as between two coupled pendulums, hence the name given to that effect by Ewald [31] who also gave a mechanical analogy [32]. It is not surprising that we find the same result for PhC’s. The optical properties of PhC’s [33-36] are commonly derived in analogy with the electron band theory in solid state physics; on the other hand, the analogy with the x-ray diffraction in the crystals has been evident since the publications of the first papers on PhC’s [37-38]. In the next sections we illustrate the essential characteristics of the dispersion surfaces (or EFS), analytically derived within DDT, and also applied to the negative refraction analysis [39]. Starting from dispersion surfaces, we will show that it is possible to understand the Pendellösung phenomenon and to quantify the thickness (Pendellösung distance) where the periodic exchange of energy happens. Finally a numerical study, based on a Finite Element Method (FEM) will prove the existence of this phenomenon for the PhCs and the validity of the parameters calculated by DDT. 2.   Dispersion surface and Pendellösung phenomenon for photonic crystals In vacuum, as in any homogeneous material, the dispersion relation in reciprocal space is simply a sphere of radius 2 k  πλ  = , where λ   is the wavelength in the medium (in vacuum kc ω  = ). In photonic crystals, like for the real crystals in x-ray diffraction, a periodic spatial variation of dielectric properties has to be considered. To a first approximation, when the medium can be considered as homogeneous (i.e., in the long wavelength limit) and λ  av  is the average wavelength in the medium, dispersion surfaces are spheres of radius 2 av k  πλ  =  centred in each reciprocal lattice point (Fig.1(a)). Increasing the energy, av λ  decreases and the medium cannot be more considered as homogeneous because spheres approach each others. As a consequence, around the point  L 0  where the Bragg law is satisfied, the spheres partly split resulting in a more complex dispersion surface where a Bragg gap appears (Fig. 1(b)). In the following we will limit ourselves to consider only the “two-beam case”, i.e., when only two Fourier components of the fields in the crystal are significant [28], which is usually the case in x-ray diffraction. This assumption is well satisfied for PhCs if the contrast, in the dielectric spatial distribution, is not very high. In such a case when the Bragg condition is satisfied for a considered reciprocal lattice vector, hHO =    (see the Fig. 1), the dispersion surfaces are only affected by h   and the associated component in the Fourier expansion of the field. In the two-beam approximation, the influence on the Bragg diffraction from other reciprocal lattice vectors can be neglected. Fixing the srcin in  L 0  and indicating by  B k     the wavevector which satisfies the Bragg law for the considered reciprocal lattice vector h     ( ) 2 i. e. /2  B khh ⋅=−   the modulus of which is equal (C) 2005 OSA7 March 2005 / Vol. 13, No. 5 / OPTICS EXPRESS 1363 #6107 - $15.00 USReceived 20 December 2004; revised 15 February 2005; accepted 15 February 2005  to the average wavenumber in the crystal, the dispersion surface can be conveniently described by the displacement δ    from the point  L 0 . Fig. 1. Dispersion surfaces in crystal in the long wavelength limit: the medium can be considered as homogeneous (a). Decreasing the wavelength, the spheres approach and a Bragg gap appears (b). By indicating with  X  0  and  X  h  the projection of δ    on the spheres centred in O  and  H  , following the classical DDT [28,29] approach, one gets a simplified analytical expression of the dispersion surface: ( ) 200 41 hhh k  XX   χ  − =+   (1)   This is the equation of a hyperbola. In fact, the product of the distances  X  0  and  X  h  from the asymptotes, which are the tangent planes of the spheres of radius 2 av k  πλ  =  centered in O and H, respectively, is a constant (see Fig. 2 caption for details). In (1)  χ  0   ,  χ  h  and  χ  -h  are the Fourier components of the function  χ defined by  ( ) ( ) ( ) 0 1 rr  εεχ  =+   , which is equivalent to the susceptibility definition. Fig. 2. The crystal dispersion surface, which determines the permitted wavevectors in the structure for a given frequency is a hyperbola (thick line) close to the Bragg gap. In the figure are also shown dispersion surfaces in the air, which are spheres. The intersections of the hyperbola asymptotes in vacuum, the Lorentz point (Lo) and the Laue point (La) respectively, are also indicated. (C) 2005 OSA7 March 2005 / Vol. 13, No. 5 / OPTICS EXPRESS 1364 #6107 - $15.00 USReceived 20 December 2004; revised 15 February 2005; accepted 15 February 2005   The dispersion surface is the locus of permitted wavevectors, for a given frequency ck  ω  = , inside the crystal. In the case of the incident wave on the crystal (in vacuum for instance), it is necessary to consider the dispersion surfaces in the two half-spaces and apply the continuity conditions across the boundary of the vacuum-crystal interface. In order to study the Pendellösung phenomenon, in the following we consider the transmission geometry (also known as Laue geometry in x-ray diffraction), where the surface normal vector n    is orthogonal to the vector h   . In Fig. 2 the wavevector i k     of the incident plane wave is shown. Its modulus is equal to the inverse of the wavelength in vacuum: 2 i kc πλω  ==  and it lies in vacuum on the dispersion surfaces which are simply spheres of radius k  , centered in O and H. The conservation of the tangential wavevector component implies that the vertices of the wavevectors inside the crystal, which have to be on the dispersion surface by definition, are aligned along the normal to the surface. This assures the continuity of the tangential component of the wavevectors. As illustrated in the Fig. 2 there are always two intersections with the dispersion surfaces ( P 1  and P 2 in the Fig.   2). Therefore, inside the crystal there are two wavevectors oriented in the O-direction, 101 kPO =     and 202 kPO =     and two in the H-direction 11 h kPH  =    and 22 h kPH  =    . All these components form the wavefield in the crystal [28,29]. There is no reason to choose only one intersection with the dispersion surface [23] because both are present in the PhC’s. The difference of such wavevectors, 12 PP   , is oriented along the normal and a modulation is expected in this direction with a period Λ, where   12 1 PP =Λ   . At the exit surface we have to apply again the conservation of the tangential wavevector component. If we limit ourselves to considering the case of a crystal slab, the exit normal coincides with that at the entrance surface. In such a case, the tip of the wavevector at the exit surface will point again on  I  O  for an in-vacuum dispersion surface centered in O, and on  I  h  for an in-vacuum dispersion surface centered on H. Applying some straightforward DDT considerations, it is possible to show that the intensity  I  O  transmitted to the exit surface along the O direction and  I  h  along the H direction, is periodically modulated throughout the thickness, and  I  0   has a maximum where  I  h  has a minimum and vice versa [28,29]. When the incident wavevector exactly fulfills the Bragg law (i.e., O I  La ≡ ) the modulation is given by the Pendellösung distance  ( ) 0 iB θθ  Λ=Λ= , which is the inverse of the minimum distance in reciprocal space between the hyperbola branches (Fig. 2): 00 1cos  Bhh λθ  χ  − +Λ=   (2)   3.   Numerical results Hereafter we study the Pendellösung phenomenon in a 2D PhC square lattice with parameters indicated in the caption of Fig. 3. FEM Simulations are performed using the commercial package FEMLAB, considering a plane monochromatic incident wave (  E   along the cylinder axis) forming an angle θ  i =60° with the normal and having a Gaussian transverse profile. This wave satisfies the Bragg law for the reciprocal lattice vector indicated in Fig. 3, thus implying that 3 a λ  = . We have considered the case of a quite low-contrast refractive index because it allows visualizing the effect in a more pictorial way. In fact, in such a case 0  12 a Λ∼ ( a  is the lattice parameter). From Eq. (2) it is clear that the period of oscillation, Λ 0 , depends on the function  χ which is directly linked to the index contrast. If contrast is high, Λ 0 is small and can also become a fraction of the period a , so that it is not possible to give a picture as in Fig. 3. In the considered example  χ  0 = 0.0633;  χ  h =  χ  -h = 0.0632. The low-contrast refractive index (C) 2005 OSA7 March 2005 / Vol. 13, No. 5 / OPTICS EXPRESS 1365 #6107 - $15.00 USReceived 20 December 2004; revised 15 February 2005; accepted 15 February 2005
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