Beam propagation in finite size photonic crystals and metamaterials
a r X i v :0801.3161v 1 [c o n d -m a t .m t r l -s c i ] 21 J a n 2008Beam propagation in ?nite size photonic crystals and metamaterials
B.Guizal,1 D.Felbacq,2and R.Sma?a li 3
1
D′e partement d’optique,Institut FEMTO-ST,UMR 6174
Universit′e de Franche-Comt′e
16,Route de Gray 25030Besan?c on Cedex France 2Groupe d’Etude des Semiconducteurs
Unit′e Mixte de Recherche du Centre National de la Recherche Scienti?que 5650
Universit′e Montpellier II
34095Montpellier Cedex 5,France 3Laboratoire des technologies de la micro′e lectronique -17rue des Martyrs,38054Grenoble,Cedex 9France.
(Dated:February 2,2008)
The recent interest in the imaging possibilities of photonic crystals (superlensing,superprism,
optical mirages etc...)call for a detailed analysis of beam propagation inside a ?nite periodic
structure.In this paper,an answer to the question ”where does the beam emerge?”is given.
Contrarily to common knowledge,it is not always true that the shift of a beam is given by the
normal to the dispersion curve.This phenomenon is explained in terms of evanescent waves and a
renormalized diagram that gives the correct direction is given.
PACS numbers:42.70Qs,42.25.Fx INTRODUCTION AND SETTING OF THE PROBLEM Some beautiful experiments and numerical works have shown that it was possible to obtain quite unusual behaviors of light propagation inside meta-materials and photonic crystals (PhCs)[1,2,3,4,5,6,7,8].In particular,the near ?eld properties of meta-materials are intensively studied in view of the possibility of designing superlenses [9,10]or cloaking devices [11].In these structures,the evanescent waves play an important role:the point of this work is to to quantify the importance of the evanescent waves and to give a theoretical insight into the propagation of a beam [12]inside a ?nite-size PhC.In principle,the group velocity [13,14]allows to determine where the beam emerges from the PhC (see ?gure 1)by computing the shift ?.We show that,in ?nite-size structures,the shift is not always correctly predicted by the normal to the isofrequency curves.This fact is due to two reasons.First,in ?nite-size structures,there are evanescent waves near the boundaries which can contribute to the propagation of the beam [18].Therefore the ?eld inside the structure comprises not only Bloch modes but evanescent waves as well [22].The latter can have a strong in?uence on the behavior of the beam.Second,due to multiple scattering,the emerging ?eld is a sum of beams that can strongly interfere.If the beams are well separated spatially,on can clearly distinguish where the ?rst beam emerges from the structure.If the beams overlap strongly,the resulting ?eld can be strongly deformed,and it becomes di?cult to de?ne an ”emerging point”.Throughout this work,we use time-harmonic ?elds,with a time-dependence of exp(?i ?t ).The ?elds are assumed to be z -independent (this is the direction of invariance of the photonic crystal).The vectorial di?raction problem can then be reduced to the study of the two usual cases of polarization:s -polarization (electric ?eld parallel to z )or p -polarization (magnetic ?eld parallel to z ).The wavenumber in the medium surrounding the photonic crystal is denoted by k 0.The incident ?eld is a limited beam whose z component is given by u i (x,y )=
k 0?k 0A (k x )e i (k x x +k y 0y )dk x (1)where:k y 0=
4(k x ?k m )2),where k m =k 0sin θ,and θis the mean angle of incidence of the beam and w its waist.The point where the incident beam enters the photonic crystal is de?ned as the barycenter of the beam.Its abscissa is given by:X i =R x |u i (x,0)|2dx
SHIFT OF THE FIRST TRANSMITED BEAM
The crystal is described as a stack of gratings (?gure 1)and we assume that in the spectral domain de?ned by the above beam,the ratio between the wavelength and the period d of the gratings is such that there is only one re?ected and one transmitted order (the condition k 0<π/d is su?cient).Given this hypothesis,the re?ected and transmitted ?elds can be expressed as:u r (x,y )= A (k x )r N (k x )e i (k x x ?k y 0y )dk x (2)
u t (x,y )= A (k x )t N (k x )e i (k x x +k y 0y )dk x (3)
where r N and t N are the re?ection and transmission coe?cients.For a given k x ,there exists a unique real 2×2matrix T N [16,17]with determinant 1,satisfying the following relation:
T N 1+r N ik y 0(1?r N )
=t N 1ik y 0 (4)It is the dressed transfer matrix of the total structure [18].Let us denote by γand γ?1the eigenvalues of T N and by v =(φ11,φ21),w =(φ12,φ22)the associated eigenvectors (Tv =γv ,T N w =γ?1w ).The re?ection and transmission coe?cients can be written in the following form:r N = γ2?1 f γ2?g ?1f
(5)where,denoting q (x,y )=(ik y 0y ?x )/(ik y 0y +x ),the functions f and g are de?ned by g (k,θ)=q (v ),f (k,θ)=q (w )and v is chosen such that |g |≤1in the conduction bands.
The following expansions hold:
r N (k,θ)=g +(g ?f )
+∞ p =1γ2p |g |2p (6)t N (k,θ)=(1?|g |2)γ+∞ p =0γ2p |g |2p
(7)These series expansion show that,due to the multiple scattering inside the photonic crystal,the transmitted (and re?ected)?elds consist of a sum of beams u t (x,y )= +∞p =0u p t ,where:
u p t (x,y )= A (k x )(1?|g |2)|g |2p γ2p +1e i (k x x +k y 0y )dk x (8)
The position where a beam emerges from the photonic crystal is de?ned as its barycenter.We are interested in the direction that is followed by the beam inside the structure.This direction is given by the shift of the ?rst transmitted beam (see ?gure 1),the shift being de?ned as the di?erence between the barycenter of the incident ?eld and that of the ?rst transmitted ?eld:?=X t ?X i ,where:X t =
R x |u 0
t (x,0)|2dx Nh =? dk y
A 2(k x ) 1?|g |2 2dk x (10)
A series expansion of?can be obtained provided the phase function is analytic with respect to k x in a neighborhood of k m.Then we obtain:
?=?Nh +∞
m=0C m dk m+1x k x0(11)
where C m=
R A2(k x)(1?|g|2)2(k x?k x0)m dk x
dk x does not vary too quickly in the vicinity of k m,we obtain the
well-know crude approximation:
?～?Nh
dk y
dk x
(k m),?1 is a vector that is normal to the dispersion curve at wavelengthλ.
We retrieve the well-known fact that for a spatially large beam,the direction of propagation is given by the normal to the isofrequency Bloch diagram[15]but here it is the”renormalized”Bloch diagram that is involved. We will see in the numerical application that it can be quite di?erent from the usual Bloch diagram.
In the particular case of a one dimensional strati?ed medium,i.e.when the relative permittivity is constant in the horizontal direction,the direction of propagation of a beam is given by the normal to the usual dispersion curve. Indeed,denoting T the transfer matrix for1period,the transfer matrix for N periods is T N.This matrix coincides with the matrix T N:this is due to the absence of evanescent waves.A direct consequence is that the Bloch vectors obtained from T N and T N are the same,hence the renormalized Bloch diagram coincides with the non-renormalized one.
DISCUSSION
In the following,we present numerical computations illustrating the behavior of a beam inside?nite PhCs.All the numerical results were obtained by means of a rigourous di?raction code for gratings based on the Fourier Modal Method(see[21]).We will denote:
??n the shift of the?rst transmitted beam computed by a direct numerical computation of the?eld.
??B the shift computed through the isofrequency Bloch diagram.
??R the shift computed through the renormalized isofrequency diagram
??f the shift of the entire transmitted?eld comprising all the beams,computed numerically.
In order to quantify the role of the evanescent waves,let us remark that,inside the photonic crystal,the?eld can be expanded over three types of modes[22].These modes are the eigenvectors of the transfer matrix T of the photonic crystal,i.e.the matrix that relates the?elds below the crystal to the?elds above the crystal:
1.the propagative modes,i.e.the Bloch modes(corresponding to the eigenvalues of T of modulus1),
2.the evanescent modes(corresponding to the eigenvalues of T of modulus less than1),
3.the anti-evanescent modes(corresponding to the eigenvalues of T of modulus greater than1).
By projecting the?eld on these modes,it is possible to compute the ratio of the?eld that is carried by the evanescent, anti-evanescent and propagating waves[22].
We consider a photonic crystal with square symmetry,in?nite in the horizontal direction and comprising N periods in the vertical direction.The basic cell is given in?g.2:it is made of square air holes(side d/2)inside a dielectric matrix of permittivityε=9.We?rst compute the transmitted?eld for N=1:the structure is made of one single grating.The incident beam is a gaussian p-polarized beam(w=50d,λ/d=2.5,θ=40o).As it has been explained before the transmitted?eld is a sum of beams:the total?eld on the upper interface is given in?g.3and the amplitudes of the di?erent beams|u p t(x,Nh)|for p=0,1,2,3,4in?g. 4.It is clearly seen that there is a strong
overlap between the beams.The shift of the?rst beam is?n/d=1.66while the shift of the total transmitted beam is ?f/d=0.27.The Bloch diagram is given in?g.5(black curve).The value of k m is:k m×d=2π/2.5×sin(40o)～1.6. The predicted shift is?B/d=?1.36,the renormalized Bloch diagram(grey curve in?g.5)gives:?R/d=1.66.In this situation,the overlap between the multiple beams is strong and the transmitted?eld cannot be reduced to the ?rst beam.
Let us now keep the same parameters except for the number of periods:we take N=4.This time the situation changes drastically:the transmitted?eld shows clearly a negative refraction(?g.6),we have?n/d～?36,the shift of the total?eld is?f/d～?32and the Bloch diagram gives:?B/d～?11.In this situation the overlap between the beams is less important(?g.7).The renormalized Bloch diagram gives:?R/d～?27.The discrepancy between ?n and?R is due to the fact that the beam is not spectrally narrow enough.If we use a narrower beam by taking w=200d we get a better result:?n/d～?28.Let us look at the modulus of the eigenvalues of T and the ratii on the di?erent modes(?g.8).On this?gure,the modulus of the eigenvalues are given in thin solid line,the ratio over the propagative waves is shown in thick solid line and the ratio over the evanescent waves in dashed line.It is clearly seen that nearλ/d～2.5there are more evanescent waves than propagative ones:this explains why the Bloch diagram gives a false description.Let us pursue and take N=16,keeping the same parameters as for N=4,we get a transmitted?eld given in?g.9the beam is hugely deformed and the value of its barycenter is not relevant:here the behavior of the?eld is much more complicated than could be expected from the Bloch diagram.
Although the above results show that the prediction of the Bloch diagram can be quite false,it should be said that, provided the in?uence of the evanescent waves is not too strong,the Bloch diagram can give very accurate results. This is the case if we takeλ/d=6with the same parameters as above.We get:?n/d=6.55and?B/d=6.45.This time the Bloch diagram predicts fairly well the position of the?rst transmitted beam.
CONCLUSION
We have shown that the shift of the?rst transmitted beam is given by the normal to the renormalized isofrequency diagram and that it can be quite di?erent from the prediction of the Bloch diagram.We have proven that this e?ect is due to the existence of evanescent waves inside the structure.The analysis uses a dressed transfer matrix that could be extended to aperiodic or random structures[23,24].We have also pointed out the strong in?uence of the number of periods and of the spectral widths of the beams:these parameters can drastically modify the behavior of the beam. Moreover,it may happen that the total transmitted?eld be not properly described by the?rst transmitted beam.In that situation,neither the Bloch diagram nor the renormalized diagram can give reliable results.This study aims at showing that one should be prudent when using the Bloch diagram to predict the behavior of the?eld inside a true (i.e.?nite)photonic crystal.Beyond that,it shows that many new e?ects can be expected due to the presence of evanescent waves.These are not a drawback but rather represent new possibilities to imagine new fonctionnalities. Acknowledgments
This work was realized in the framework of the ANR project POEM PNANO06-0030.
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Figures captions
Figure1:2D Photonic crystal as stacks of gratings:periodic in x and?nite in y.
Figure2:Basic cell of the photonic crystal used in the numerical experiments.
Figure3:Amplitude of the transmitted?eld on the upper interface.
Figure4:Amplitude of the?rst?ve transmitted beams.
Figure5:Black line:Bloch diagram for the photonic crystal.Gray line:renormalized Bloch diagram.k x×d and k y×d belong to(?π,+π).The vertical dashed line indicates the average value of k x in the incident?eld.
Figure6:Amplitude of the transmitted?eld on the upper interface.
Figure7:Amplitude of the three?rst transmitted beams.
Figure8:Thin solid line:modulus of the eigenvalues of the transfer matrix.Thick solid line(blue online):ratio of the?eld on the propagating modes.Dashed line(red online):ratio of the?eld on the evanescent modes.Dashed dotted line(green online):ratio of the?eld on the anti-evanescent modes.This arrows indicate the wavelengths were the computations were performed.
Figure9:Amplitude of the transmitted?eld on the upper interface.
?100?50050100