Numerical evidence of ultrarefractive optics in photonic crystals

Transcription

1 15 Marc 1999 Optics Communications Numerical evidence of ultrarefractive optics in potonic crystals S. Enoc 1, G. Tayeb, D. Maystre ) Laboratoire d Optique Electromagnetique, ESA 6079, Faculte des Sciences et Tecniques, Centre de Saint-Jerome, ˆ Marseille Cedex 20, France Received 13 July 1998; revised 22 October 1998; accepted 5 January 1999 Abstract Tis paper is devoted to a numerical demonstration of a conjecture of specialists of potonic band structures: te existence of ultrarefractive optics penomena. Near te edges of a transmission gap, te permittivity of a dielectric potonic crystal becomes close to zero. As a consequence, surprising refractive effects sould be observed on te ligt transmitted and reflected by a slice of potonic crystal. A property of beam translation very similar to te Goos Hanscen effect Žbut in transmission. is sown, as well as a strange penomenon: a beam incident on a slice of potonic crystal can be enlarged or even split into some separate transmitted beams. q 1999 Elsevier Science B.V. All rigts reserved. PACS: Qs; Bs; Fx; Ct Keywords: Potonic crystals; Ultrarefractive optics; Homogenization 1. Introduction Arguments based on dispersion diagrams of ligt inside potonic crystals ave allowed some specialists in tis field to predict te penomenon of ultrarefraction of ligt w1 4 x. Comparable considerations can be found in Ref. w5x in te context of a modulated planar waveguide. In outline, ligt velocity at edges of a transmission gap may tend to infinity and tus a potonic crystal can simulate an effective medium aving a permittivity close to zero. In tis paper, we ave used numerical tools based on rigorous electromagnetic teories w6,7x in order to demonstrate te validity of tese predictions. Some surprising applications of ultrarefractive optics will be sown. For example, field maps of transmitted field generated by a slice of potonic crystals illuminated by a ligt beam bear evidence of a penomenon similar to te Goos Hanscen effect wx 8 : te translation of te transmitted beam wit respect to te location predicted by geometrical optics. In contrast to te classical Goos Hanscen penomenon te potonic crystal allows to get tis translation penomenon for te transmitted beam, and close to normal incidence. Even toug te matematical demonstration of te Goos Hanscen effect does not old in tat context, it can be intuited tat te beam translation penomenon is linked to a rapid variation of te argument of te transmitted wave wit incidence angle. A euristic reasoning allows one to predict a surprising penomenon: te beam transmitted by a potonic crystal under ultrarefractive condition can be significantly widened wit respect to te prediction of geometrical optics. Field maps sow tat tis intuitive conjecture is rigt. Furtermore, it is sown tat under some conditions, te transmitted ligt may be split into some separate beams. 2. Te concept of effective medium ) Corresponding autor. maystre@loe.u-3mrs.fr 1 enoc@loe.u-3mrs.fr It can be noticed tat te concept of effective medium wx near Brillouin zone edges as been used by Lin et al. 9 in r99r$ - see front matter q 1999 Elsevier Science B.V. All rigts reserved. PII: S

2 172 ( ) S. Enoc et al.roptics Communications order to realize a prism in a two-dimensional Ž 2D. potonic crystal material wic could serve as a dispersive element in a ultracompact miniature spectrometer. It as been sown in a recent paper w10x tat a metallic potonic crystal can simulate a medium aving an effective index close to zero. In tis paper, we are concerned wit dielectric potonic crystals, te properties of wic are very different from tose of metallic crystals. However, let us sow tat a simple reasoning can lead us to te same conclusions: a dielectric potonic crystal can simulate an effective medium aving a permittivity close to zero. Fig. 1 sows a slice of a 2D dielectric potonic crystal wit square symmetry and Fig. 2 gives te transmission factor Ž in energy. and te pase sift of te transmitted wave wit respect to te incident one, wic is a s-polarized plane wave in normal incidence. It is important to notice tat in te range of wavelengts lgw2.4, 3.3 x, te only reflected and transmitted waves generated by tis grating are te zero order ones. A transmission gap is obtained in te range lgw2.5, 3.2 x. Let us sow tat, at a given wavelengt and a given incidence, tis slice of potonic crystal can simulate a slice of omogeneous medium wose permittivity is close to zero. First, let us notice tat te searc for te parameters and e of te equivalent tin film defined in Fig. 3 is a well posed problem, at least if one is interested in te transmitted field only. Indeed, tis transmitted field is caracterized by te transmission coefficient t wen te crystal is illuminated by a s-polarized wave wit given incidence and wavelengt. Te equivalent tin film must give te same coefficient t wen it is stroked by te same incident wave. In order to obtain tis complex transmission coefficient, we can coose two parameters: te widt e and te relative permittivity of te dielectric medium. Since te potonic crystal is a lossless structure, we will assume tat te tin film is lossless as well, tus is real. In conclusion, te transmission coefficient t, wic is a complex number, may be obtained by coosing two real numbers: e and. Tis reasoning sows tat our searc is consistent, but does not provide a rigorous proof of te existence of suc a tin film. However, it exists a range of Fig. 2. Energy solid line and pase sift dased line of te wave transmitted by te dielectric potonic crystal of Fig. 1 in normal incidence wit s-polarized ligt versus te wavelengt l in vacuum. Te small inserted grap sows te energy on a larger range of wavelengts. wavelengts were tis existence can be predicted. Let us consider very large wavelengts Žmuc larger tan te size of te elementary cell of te potonic crystal.. It is well known tat, under tese conditions, te eterogeneous structure can be omogenized. In tis omogenization domain, te effective index of te crystal is given by simple rules w11 x. For s-polarized ligt, te effective permittivity,` of te medium is te mean value of in te elementary cell of te potonic crystal. Tus it is a real number greater tan unity Ž rigt-and side of Fig. 4.. Wen te wavelengt l is reduced, we are inside te gap and te crystal becomes opaque: te ligt is exponentially attenuated during propagation. As a consequence, te permittivity must be a negative real number. Indeed, positive permittivities allow propagation, and complex permittivities imply losses and must also be rejected since te rods are lossless. Te reader can notice tat suc a permittivity generates a pure imaginary optical index. If we conjecture tat is a continuous function of l, te curve must cross te abscissa axis, tus tere exists a wavelengt were s0 Ž point B of Fig. 4.. Te same reasoning can be made for te left-and side of Fig. 2. Indeed, for Fig. 1. Scema of a slice of a 2D dielectric potonic crystal. Tis crystal is made of seven grids of infinite extension in x wit period ds1.27, separated in y by te same distance d. Eac grid is formed by parallel dielectric infinite rods of index n s3 and diameters Ds0.95. Fig. 3. Equivalence of a slice of potonic crystal Ž left. wit a 2 dielectric omogeneous tin film rigt of permittivity sn Ž n is te corresponding optical index..

3 ( ) S. Enoc et al.roptics Communications Fig. 4. Pysical predictions of te variation of te effective permittivity of te tin film as a function of te wavelengt under te conditions of Fig. 2. smaller wavelengts te ligt can propagate again inside te crystal and tus te permittivity is positive, wic sows tat tere exists a second wavelengt for wic te permittivity vanises Ž point A of Fig. 4.. Of course, te value of for a given wavelengt l sould depend on te angle of incidence u. However, te goal of a potonic crystal is to provide ligt propagation properties wic are almost independent of te direction of propagation. 3. Conjectures about ultrarefractive optics Let us consider te dielectric tin film of Fig. 5, wit a positive relative permittivity close to zero, illuminated by a monocromatic incident beam wit incidence angle close to zero. Due to te refraction penomenon, te beam inside te dielectric can propagate wit a large refraction angle and te emerging beam below te tin film is sifted wit respect to te incident beam. It is wort noticing tat te direction of tis sift is just opposite to tat obtained for classical materials wit permittivities larger tan unity. Moreover, te amplitude of te sift can be very large if te permittivity is very close to zero. A slice of potonic Fig. 6. Widening of te emerging beam after transmission by a tin film wit small permittivity. crystal illuminated by an incident beam at a wavelengt close to te edge of a gap sould generate te same penomenon. Now, let us illuminate te dielectric tin film of Fig. 5 by a beam in normal incidence. It is well known tat suc a beam is composed by a sum of plane waves propagating in a given range of incidence angles around null incidence wx 8. Wen te widt of tis range of incidence angles tends to zero, te widt of te beam increases and tends to infinity, tus te beam becomes a plane wave. In Fig. 6, tree directions of propagation ave been represented: te normal and te two edges of te range of te incidence angle. Due to te refractive penomenon inside te material of small permittivity, te emerging beam sould be muc larger tan te beam deduced from te laws of geometrical optics. However, tis prediction must be corrected since te slice of omogeneous material acts like a Fabry Perot interferometer. In oter words, te different plane waves of te beam are reflected at air dielectric interfaces, in suc a way tat te transmission factor depends on te angle of incidence. We can conjecture from tis remark tat wen te tin film of Fig. 6 is illuminated by a Gaussian beam, te emerging beam could be very different from a Gaussian beam. If te transmission factor strongly depends on te angle of incidence, te emerging beam could be composed of some separate beams, like te separate rings generated by a Fabry Perot. 4. Numerical results 4.1. Sift of te beam transmitted by a slice of potonic crystal Fig. 5. Anomalous sift of te emerging beam after transmission by a tin film wit small permittivity. Te first problem tat arises is to determine te wavelengt for wic te permittivity of te potonic crystal is close to zero. Wit tis aim, it can be observed tat te penomenon of beam sift scematised in Fig. 5 looks very similar to te Goos Hanscen penomenon wx 8, but for te transmitted wave. Remembering tat te potonic

4 174 ( ) S. Enoc et al.roptics Communications origin of tis penomenon lies on a rapid variation of te argument of te reflected wave wit incidence angle, we are led to te searc for a rapid variation of te pase sift of te transmission factor of te potonic crystal wit incidence. Studies of grating anomalies w12x ave sown tat rapid variation of te caracteristics of te scattered wave wit incidence angle and wit wavelengt appens togeter. One can find suc a variation in Fig. 2, at te left-and side of te gap, for example, at ls Tis rapid variation of te pase sift corresponds to a sarp peak of transmission. Fig. 7 sows te transmittance and pase sift versus te angle of incidence at tis wavelengt. It is to be noticed tat te pase variation between 08 and 88 is close to Te same penomenon arises at ls2.5575, wile for oter wavelengts te pase variation in te same range of incidence angles remains of te order of 108. Te two peaks observed in Fig. 7 are similar to te peaks generated by a Fabry Perot interferometer wen te incidence angle is varied, te wavelengt being fixed. Using te rectangular coordinate system depicted in Fig. 1, let te incident beam be limited in te x-direction. Te only component of te s-polarized incident complex electric field can tus be expressed as te integral: q` inc E x, y sh A a exp Ž ia xyib a y. da, 1 y` 2 2 wit ask sin u, b a s k ya and ks2prl. ' We consider Gaussian beams wit mean incidence u 0: 2 2 Ž 0. ž / W aya W AŽ a. s exp y, Ž 2. 2' p 4 were a0sk sin u 0 and define te angular range as te range were te amplitude AŽ a. is greater tan AŽ a. 0 r10. It can be noticed tat te parameter W appearing in Eq. Ž. 2 is directly linked to te incident beam widt. Fig. 8 sows te field map Žmodulus of te electric field. around and inside te crystal, te incident field being Fig. 8. Field map of te total field modulus wit a potonic crystal similar to tat of Fig. 2, but of finite extension in te x-direction. Tis crystal is made of 69=7 rods, as sown in te figure. Above te crystal te beam reflected by te crystal interferes wit te incident beam and generates a system of stationary waves. Below te crystal, te figure sows te transmitted beam. Straigt lines sow te locus of te maximum incident Ž black line. and transmitted Ž wite line. fields. a Gaussian beam of wavelengt ls2.5447, of angular range w1.78, x, te mean incidence angle being equal to Te calculation as been acieved using te rigorwx 6. Te ous teory of scattering by a finite number of rods beam directly reflected by te crystal interferes wit te incident beam and generates a system of stationary waves wic can be observed at te top left corner of te figure. Below te crystal, te center of te transmitted beam is sifted to te rigt by four wavelengts wit respect to te center of te incident beam, wic sows tat te permittivity of te effective medium is small. At te top rigt side of te figure, one can observe a second reflected beam Fig. 7. Transmittance and pase sift at ls Fig. 9. Te same as Fig. 8, but from anoter rigorous teory, te crystal of Fig. 8 aving an infinite extension in te orizontal direction.

5 ( ) S. Enoc et al.roptics Communications Fig. 10. Field map of te incident Ž above te potonic crystal. and transmitted fields Ž below te potonic crystal. modulus wen te potonic crystal used in Fig. 9 is illuminated by a Gaussian beam in normal incidence. Te transmitted field as been multiplied by a factor 2.25 in order to get te same maximum value as te incident field. generated by te ligt transmitted inside te crystal and ten reflected. In order to ceck te validity of tis surprising result, we acieved te same calculation using a quite different rigorous teory wx 7. Tis teory is able to deal wit periodic Ž in x. structures and tus te potonic crystal sown in Fig. 8 as been continued in te x-direction in order to become infinite. Fig. 9 sows te field above and below te crystal. Te very good agreement between Figs. 8 and 9 sows tat te edges of te crystal of Fig. 8 Ž in te x-direction. ave no effect on te scattering penomenon, a fact wic could be predicted by observing in Fig. 8 tat te field at te limits Ž in x. of te crystal vanises. Furtermore, tis agreement sows te validity of bot calculations Widening and splitting of te transmitted beam In order to ceck te predictions of Fig. 6, we ave used te potonic crystal of Section 4.1 at te same Fig. 12. Split of an incident beam Ž above te potonic crystal. into four separated transmitted beams Ž below te potonic crystal.. Te transmitted field as been multiplied by a factor wavelengt but wit a Gaussian beam in normal incidence. Te angular widt of te beam as previously defined is equal to Fig. 10 sows te field map of te incident beam Ž top. and te transmitted beam Ž bottom. obtained from te rigorous teory described in Ref. wx 4. Obviously, te beam is significantly widened, even toug tis penomenon is partially idden by te fact tat te transmitted beam contains 72% of te energy of te incident beam only. Te structure of te field obtained by adjusting te maximum of te transmitted field modulus to te same value Ž unity. as te incident field modulus provides a better estimate of te widening, wic is close to 200%. Looking at Fig. 7, it appears tat te transmission factor contains tree peaks, including negative incidence angles. It can be conjectured tat te transmitted field could be split into tree beams, te first is propagating along te y-axis and te two oters are propagating symmetrically at an angle of "6.48 from te first one. In fact, te field map sows a complicated interference system between tese tree beams since tey are not separated. On te oter and, a sligt cange of te wavelengt from ls to ls2.543, wic significantly modifies te transmittance of te crystal Ž Fig. 11., leads to te transmitted field sown in Fig. 12, wit four separated transmitted beams corresponding to te four peaks of te transmittance. Te angular widt of te incident beam extends from y14.28 to It is wort noticing tat te four transmitted beams, wic correspond to te rings of a Fabry Perot, ave very close directions of propagation despite te moderate widt of te crystal. Tis is a consequence of te small value of te permittivity. 5. Conclusion Fig. 11. Te same as Fig. 7 but for ls It as been proved from electromagnetic teory tat ultrarefractive optics penomena can be generated by dielectric potonic crystals.

6 176 ( ) S. Enoc et al.roptics Communications Surprising penomena like anomalous refraction, widening or splitting of a ligt beam ave been sown. In te present work, we ave not tried to optimize te amplitude of ultrarefractive effects and tus it can be conjectured tat more pronounced penomena could be obtained if practical applications are envisaged. Acknowledgements Te work described in tis paper as been done under a contract between te Laboratoire d Optique Electromagne- tique and te Direction Generale de l Armement ŽFrenc Ministry of Defense.. References wx 1 J.P. Dowling, C.M. Bowden, J. Mod. Opt wx 2 P.M. Wisser, G. Nienuis, Opt. Commun. 136 Ž wx 3 C.M. Soukoulis Ž Ed.., Potonic Band Gap Materials, Kluwer, 1996, pp wx 4 E. Burstein, C. Weisbuc Ž Eds.., Confined Electrons and Potons: New Pysics and Applications, Plenum, 1995, pp wx 5 R. Zengerle, J. Mod. Opt. 34 Ž wx 6 D. Felbacq, G. Tayeb, D. Maystre, J. Opt. Soc. Am. A 11 Ž wx 7 D. Maystre, Pure Appl. Opt. 3 Ž wx 8 R. Petit, M. Cadilac, J. Opt. 8 Ž wx 9 S.Y. Lin, V.M. Hietala, L. Wang, E.D. Jones, Opt. Lett. 21 Ž w10x G. Guida, D. Maystre, G. Tayeb, P. Vincent, J. Opt. Soc. Am. B 15 Ž w11x D. Felbacq, G. Boucitte, Waves Random Media 7 Ž w12x D. Maystre, in: A.D. Boardman Ž Ed.., Electromagnetic Surface Modes, Wiley, 1982, pp