Escherlike quasiperiodic heterostructures

Escherlik e quasiperiodic heterostru ctur es Alberto G. Barriuso, 1 Juan J. Monz ´ on, 1 Luis L. S ´ a nchez-Soto , 1 and Anton io F . Costa 2 1 Departamento d e ´ Optica, F acultad de F ´ ısica, Univer sidad Co mplutense, 280 40 Mad rid, Spain 2 Departamento de Matem ´ aticas Fundamentales, F acultad de Ciencias, Universidad Nac ional de Edu caci ´ on a Distan cia, Senda del Re y 9, 28040 Madrid, Spain (Dated: No vemb er 3, 2018) W e propose quasiperiodic heterostructures associated with the tessellations of the unit disk by regular hyper - bolic triangles. W e present explicit construction rules and explore some of the properties exhibited by these geometric-based systems. P A CS numbers: 61.44.Br , 68.65.Cd, 71.55.Jv , 78.67.Pt Quasiperiod ic (QP) s ystems ha ve been receiving a lot of at- tention ov er the last years [1]. The interest was originally mo- ti vated by the theoretical predictions that they should manifest peculiar electron and pho non critical states [2, 3 ], associa ted with high ly frag mented fractal energy spectra [4, 5, 6]. On the o ther han d, the practical fabrication o f Fibo nacci [7] and Thue-Mo rse [8] superlattices h as trigge red a n umber of exper- imental ach iev ements that ha ve provided new insigh ts into the capabilities of QP structures [9]. I n par ticular , possible optical applications have deserved majo r atten tion and som e intrigu- ing p roper ties h ave been demon strated [10, 11, 12, 13]. Un- derlying all these theoretical and exp erimental efforts a cr u- cial fundam ental question remains conc erning whether QP de- vices would achieve b etter p erform ance th an usual per iodic ones for some specific application s [14]. The QP sy stems consider ed th us far rely f or their explicit construction in substitution al rules amon g the eleme nts o f a basic alp habet. In the co mmon case of a two-letter alpha- bet { A, B } , the algor ithm takes th e form A 7→ σ A ( A, B ) , B 7→ σ B ( A, B ) , where σ A and σ B can be any strin g of the letters. The sequenc es ge nerated after n applications of the algorithm are of significance in fields as diverse a s cryptog ra- phy , tim e-series analy sis, and cellular au tomata [15]. In ad- dition, th ey have in teresting alg ebraic p roperties, which are usually characterized by the nature o f the ir Fourier or multi- fractal spectra [16]. W e wish to appr oach the problem f rom an altern ativ e g e- ometrical perspective. T o this end, we first observe that in many prob lems of physical inter est [ 17] the letters of the al- phabet can be iden tified with one-dim ensional linear lo ssless systems (i.e., with two inp ut and two outp ut ch annels). Un- der these general conditio ns, it turns out th at the associated transfer m atrix be longs to the group SU(1,1 ), which is also the basic symmetry gro up o f the hyperbolic geometr y [1 8]. In consequen ce, th e unit disk appears as the natural arena to dis- cuss the ir p erform ance [19, 20, 21]. Since in the Eu clidean plane, QP beh avior is intimately linked with tessellations, one is un failingly led to consider th e role of h yperb olic tessella- tions in the unit disk , much in the spirit o f Escher’ s master- piece woodcut Cir c le Limit I II [2 2]. The answer we propose is promising : th e tessellations by different regular polygo ns provide new sequenc es with properties that may op en avenues of research in this field. A QP system can thus be seen a s a word generated by stacking different letter s of the basic alphab et. T o be spe- cific, we fo cus our attentio n on th e optical resp onse. Let us consider o ne of these letters (which in practice is made o f se veral p lane-para llel laye rs), which we assume to be san d- wiched b etween two semi-in finite identical ambient ( a ) an d substrate ( s ). W e suppo se mono chroma tic plane wav es in- cident, in ge neral, from b oth the ambient and th e substrate. As a result of m ultiple reflectio ns in all the in terfaces, the to- tal electric field can be decomposed in terms of forward- and backward-traveling plane wav es, denoted by E (+) and E ( − ) , respectively . If we take these compo nents as a vecto r E =  E (+) E ( − )  , (1) then the a mplitudes at both the a mbient and the substrate s ides are related by the transfer matrix M E a = M E s . (2) It can be shown that M is of the fo rm M =  1 /T R ∗ /T ∗ R/T 1 /T ∗  , (3) where the complex num bers R an d T are, respectively , the overall reflection and transmission coefficients for a wav e in- cident f rom th e amb ient. The condition det M = +1 is equi v- alent to | R | 2 + | T | 2 = 1 , and then the set of transfe r matrices reduces to the group SU(1,1 ). Obviously , the matrix of a word obtained by putting togeth er letters of the alp habet is the prod- uct of the m atrices representing each one of them, tak en in the approp riate order . In many instances we are inter ested in the transfo rmation proper ties of field quo tients rather than th e field s th emselves. Therefo re, it seems n atural to consider the complex numbers z = E ( − ) E (+) , (4) for b oth ambient and substrate. The action of the matrix gi ven in Eq . ( 2) c an b e then seen as a fu nction z a = f ( z s ) that can be appr opriately called the transfer function. From a geomet- rical viewpoint, this functio n defines a tr ansforma tion of the 2 complex plane C , map ping the point z s into the point z a ac- cording to z a = β ∗ + α ∗ z s α + β z s , (5) where α = 1 /T and β = R ∗ /T ∗ . W hen no light is in cident from the substrate, z s = 0 and then z a = R . Equation (5) is a bilinear (or M ¨ o bius) transformation . One can check that the unit disk, the external re gion a nd the un it circle remain in v ari- ant un der (5). This un it disk is then a model fo r hyp erbolic geometry in which a line is repr esented a s an arc of a circle that meets the b ounda ry of th e disk at righ t ang les to it (and diameters are also pe rmitted). In this mo del, we ha ve three different kind s of lines: intersecting, parallel ( they inte rsect at infinity , which is precisely the b oundar y of th e disk) and ultraparallel (they are neither intersecting nor parallel). T o class ify the possible action s it proves convenient to work out th e fixed poin ts of the tra nsfer f unction ; that is, the field configur ations such that z a = z s ≡ z f in Eq . (5), whose solutions are z f = 1 2 β n − 2 i Im( α ) ± p [T r( M )] 2 − 4 o . (6) When [T r( M )] 2 < 4 the action is elliptic an d it h as only one fixed point inside the unit disk. Since in th e Eu clidean geom- etry a ro tation is characterized for having only one inv ariant point, this action can be appr opriately called a hy perbo lic ro- tation. When [T r( M )] 2 > 4 the action is h yperbo lic and it has two fixed po ints, b oth on the bound ary of the u nit d isk. The geodesic line joining these two fixed points r emains inv ariant and thus, by analogy with the Eu clidean case, this action is called a hyper bolic tr anslation. Finally , when [T r( M )] 2 = 4 the action is pa rabolic and it has on ly o ne (d ouble) fixed point on the bo undar y of the unit disk. As it is well known, SU(1, 1) is isomorp hic to the group of real unimodu lar matrices SL(2, R ), which allows us to trans- late the ge ometrical structur e d efined in the unit d isk to th e complex upper semiplane, recovering in this way an alter na- ti ve model of the hy perbolic geometry tha t is u seful in some applications. The notion of perio dicity is in timately connected with tes- sellations, i.e., tilings by identical re plicas of a unit c ell (o r fundam ental d omain) that fill the plane w ith no overlaps and no gaps. Of special interest is th e case when the primiti ve cell is a r egular polygo n with a fin ite area [ 23]. In th e Euclidean plane, the associated regular tessellation is generically noted { p, q } , wh ere p is the numbe r of poly gon edges and q is th e number of po lygon s that mee t at a vertex. W e recall that ge- ometrical con straints limit the po ssible regular tilings { p, q } to those verify ing ( p − 2)( q − 2) = 4 . This inc ludes the classical tilings { 4 , 4 } (tiling b y squar es) and { 6 , 3 } ( tiling by hexagon s), plus a third one, the tiling { 3 , 6 } by triang les (which is dual to the { 6 , 3 } ) . FIG. 1: ( color online). A realisti c implementation of the generators A and B (together with A − 1 ) of the tessellations by hyperbolic triangles in (7). The external layers (in blue) are made of cryolite (Na 3 AlF 6 ), while t he central medium (in bro wn) is zinc selenide (ZnSe). The wa veleng th in vacuu m is λ = 610 nm and normal incidence (from left t o right) has been assumed. The corresponding thicknesses are express ed in nanometers. On th e co ntrary , in the hyp erbolic disk regular tilings exist provided ( p − 2)( q − 2 ) > 4 , which now leads to an infinite number of possibilities. An essential ingred ient is th e way to o btain fu ndamen tal polyg ons. These po lygon s are d irectly connected to th e discrete sub group s of isometries (or co ngru- ent mapp ings). Such group s ar e called Fuchsian gro ups [24] and p lay for the hy perbolic geo metry a role similar to that o f crystallograp hic g roups for the Euclidean geometr y [25]. A tessellation of the hype rbolic plan e by regular poly- gons has a sym metry gro up that is gener ated by reflections in geodesics, which are inversions across circles in the unit disk. These geodesics co rrespon d to edges or axes of symmetry of the polygo ns. Th erefor e, to construct a tessellation of th e unit disk one just has to built one tile and to dup licate it b y using reflections in the edges. In this Letter , we co nsider o nly th e simple example of a tes- sellation by trian gles with vertices in th e unit cir cle, alth ough the treatmen t can be extended to other p olygo ns. The key idea is to consider the Fu chsian gro up g enerated by an ellip- tic tran sformatio n whose fixed point is the middle of an edge of the triangle an d a parabo lic o ne with its fixed point in the opposite vertex. Pr oceeding in this way we get A =  1 + i/ √ 3 1 / √ 3 1 / √ 3 1 − i / √ 3  , (7) B =  2 i/ √ 3 − 1 / √ 3 − 1 / √ 3 − 2 i/ √ 3  . The fixed point of A is − i ; while for B th e fixed point in the disk is i (2 − √ 3) . In Fig. 1 we show a possible w ay in which these matrices c an be implemented in terms of two comm only employed materials in optics. Note that, in p hysical term s, the inverses m ust be c onstructed as indep endent systems, al- though in our case only A − 1 must b e consid ered, since the action of B − 1 coincides with that of B . In Fig. 2 we h av e sho wn the tessellation ob tained by trans- forming the fundamental tr iangle with the F uchsian group 3 FIG. 2: (color online). Tiling of the unit disk with t he matrices (7). The marked points are the barycenters of the t riangles in the tessel- lation and all of them are the transformed of the origin by a matrix that hav e as reflection coefficient the complex number that links the origin with the center of the triangle. generated by the powers of { A , B } ( and th e in verses). This tri- angle is equ ilateral w ith vertices at the points − i , exp( iπ / 6) and exp( i 5 π / 6) (which are the fixed po ints of A , AB and BA , respectively). Mo reover , all the other triangles are equal, with an area π . In the figure we hav e plotted also the barycenters of each triangle tog ether with the resulting tree (that is c alled the dual gra ph of the tessellation), which turn s out to be a Farey tree [26]. In fact, each line con necting two of these bary cen- ters rep resent the a ction of a word ( with alp habet { A , B } and the in verses). T o give a n e xplicit construction ru le for the possible words, we pro ceed as fo llows. First, we arb itrarily assign the num- ber 0 to th e u pper side of the f undame ntal triangle, while the other two sides ar e clockwise n umber ed as 1 an d 2. It is easy to c onvince oneself that th is assignm ent fixes o nce fo r all the number ing f or the sides of the oth er triangles in th e tessella- tion. Howe ver , the se triangles can be distinguished by the ir orientation (as seen from the cor respond ing barycen ter): th e clockwise oriented are filled in red , while the cou nterclock - T ABLE I: E xplicit rules to obtain the barycenter z n +1 from the z n . W e ha ve indicated th e correspo nding transformations, which depend on the color jumps and the sides crossed by going from z n to z n +1 . red → yello w yello w → red Side Trans A n +1 B n +1 T rans A n +1 B n +1 0 B n B n A n B n B n B n B n A − 1 n B n B n 1 A n A n A n B n A − 1 n A − 1 n A n A − 1 n B n A n 2 A − 1 n A − 1 n A − 1 n B n A n A n A − 1 n A n B n A − 1 n FIG. 3: (color online). Normalized structure factor for the word (made of 60 lett ers) connecting t he origin with the point z 8 in the zig-zag path sho wn in Fig. 2. wise a re filled in y ellow . In sho rt, we have determined a fun- damental coloring of the tessellation [27]. T o ob tain on e bary center z n +1 from the previous one z n , one looks first at the cor respond ing color jump. Next, the ma- trix that take z n into z n +1 depend s on the numb ering of the side (0, 1 , or 2 ) one must cr oss, and ap pear in the ap propr iate column “Trans” in T ab le I . The next g eneration is obtain ed much in the sam e way , except for the fact th at A n and B n must be replaced b y A n +1 and B n +1 , respectively , as in dicated in the T able. In obtainin g recursiv ely an y word, the origin is de- noted as z 0 and the matrices A 0 and B 0 coincide with A a nd B . W ith this rule, o ne can co nstruct any word proceed ing step by step. For example, th e word that tr ansform z 0 into z 5 in the zig-zag path sketched in Fig. 2 results z 0 → z 1 : B , z 1 → z 2 : BA − 1 B , z 2 → z 3 : BA − 1 B , z 3 → z 4 : BA − 1 A − 1 BAAB , z 4 → z 5 : BA − 1 A − 1 BABAAB . (8) Obviously , th e total word is ob tained by composing these par - tial words. In fact, one can show tha t given a n a rbitrary sequenc e o f nonzer o integers { k 1 , k 2 , . . . , k r } , the word r epresented by the transfer matrix M ( s, k 1 , . . . , k r ) = B s 1 A k 1 BA k 2 . . . BA k r B s 2 , (9) where { s 1 , s 2 } ⊂ { 0 , 1 } , tran sforms the origin in a bary center of the tessellation. Giv en th e geometric regularity o f the construction sketched in th is Letter , the sequences obtained must play a key r ole in the theory an d practice o f QP systems. Of course, to put for- ward the relevant ph ysical featur es of these sequence s, th ere are a numb er of quantities one can look at. Per haps one of the most app ropriate o nes to assess the perfo rmance o f these systems is the structure factor [15]. For a word M (compo sed of L letters) we d efine a num erical sequen ce f n by assignin g 1, e 2 π i/ 3 , an d e − 2 π i/ 3 to th e letters B , A , and A − 1 , r espec- ti vely . Next, we calculate the discrete Fourier tran sform of the 4 sequence f n F k = L − 1 X n =0 f n exp  − 2 π ik n L  , (10) where k = 0 , 1 , ..L − 1 . The stru cture factor ( or power spec- trum) is ju st | F k | 2 . In Fig. 3 we h av e plotted this structure factor in te rms of k for the word conne cting th e o rigin with the point z 8 in the zig-zag path of Fig. 2. The pea ks r ev eal a rich be havior: a f ull an alysis of th ese q uestions is outside the scope of this Letter and will be presented else where. As a final and r ather technic al r emark, we no te that the quo - tient of the hyper bolic disk by the Fu chsian gro up gen erated by A and B is a 2 -orbif old of genus 0, with a co nical point of order two and a cu sp. E ach word as g iv en in Eq. (9) repre- sents a hyper bolic tran sformation o f the disk, and the axis of the transformation is pr ojected onto a closed geodesic of such an or bifold. This provid es an orbifold inter pretation o f ou r QP sequences. In su mmary , we expect to h av e pre sented new sch emes to generate QP sequ ences based on hyp erbolic tessellations of the unit disk. Ap art fr om th e intrinsic beauty of th e f ormalism, our preliminar y results seem to be qu ite enco uragin g f or f uture applications of these systems. The au thors wish to express their warmest gr atitude to E. Maci ´ a an d J. M . M ontesinos f or th eir h elp and in terest in the present work. [1] E. Maci ´ a, Rep. Prog. Phys. 69 , 397 (2006). [2] S. Ostlund and R. Pandit, Phys. Re v . B 29 , 1394 (1984). [3] M. K ohmoto, B. Suthe rland, and K . Iguc hi, Phys. Rev . Lett . 58 , 2436 (1987). [4] M. K ohmoto, L. P . Kadanof f, and C. T ang, Phys. Rev . Lett. 50 , 1870 (1983). [5] A. S ¨ ut ¨ o, J. S tat. Phys. 56 , 525 (1989). [6] J. B ellissard, B . Iochum, E. Scoppola, and D. T estard, Comm. Math. Phys. 125 , 527 (1989). [7] R. Merlin, K. Bajema, R. Clarke, F . Y . Juang, and P . K. Bhat- tacharya, Phys. Re v . Lett. 55 , 1768 (1985). [8] R. Merlin, K. Bajema, J. Nagle, and K. Ploog, J. Phys. C olloq. 48 , C5 503 (1987). [9] V . R. V elasco and F . Garc´ ıa-Moliner , P rog. Surf. Sci . 74 , 343 (2003). [10] S. T amura and F . Nori, Phys. Rev . B 40 , 9790 (1989). [11] M. S. V asconcelos and E. L. Albuque rque, Phys. Rev . B 59 , 11128 (1999). [12] D. Lusk, I. Abdulhalim, and F . P lacido, Opt. Commun. 198 , 273 (2001). [13] A. G. Barriuso, J. J. Monz ´ on, L. L. S ´ anchez-Soto, and A. Fe- lipe, Opt. Express 13 , 3913 (2005). [14] E. Maci ´ a, Phys. Rev . B 63 , 205421 (2001). [15] Z. Cheng and R. Savit, J. Stat. Phys. 60 , 383 (1990). [16] V . W . Spinadel, Nonlinear Anal. 36 , 721 (1999). [17] E. Maci ´ a and R. Rodr ´ ıguez-Oliv eros, Phys. Rev . B 74 , 144202 (2006). [18] H. S. M. Coxeter , Non-Euclidean Geometry (Uni ve rsity of T oronto Press, T oronto, 1968). [19] T . Y onte, J. J. Monz ´ on, L. L. S ´ anchez-Soto, J. F . Cari ˜ nena, and C. L ´ opez-Lacasta, J. Opt. Soc. Am. A 19 , 603 (2002). [20] J. J. Monz ´ on, T . Y onte, L. L. S ´ anchez-Soto, and J. F . Cari ˜ nena, J. Opt. Soc. Am. A 19 , 985 (2002). [21] A. G. Barriuso, J. J. Monz ´ on, and L. L. S ´ anchez-Soto, Opt. Lett. 28 , 1501 (2003). [22] H. S. M. Coxeter , Math. Intell. 18 , 42 (1996). [23] H. Ziesch ang, E. V ogt, and H. D. Coldew ay , Surfaces an d planar discontinuous gr oups , vol. 835 of Lect. Not. Math. (Springer , Berlin, 1980 ). [24] L. R. Ford, Automorph ic functions (Chelsea Publising Com- pany , Ne w Y ork, 1972). [25] A. F . Beardon, The geometry o f discr ete gr oups (Springer , Berlin, 1983). [26] M. R. S chr ¨ oder , Number Theory in Science and Communication (Springer , Ne w Y ork, 2006), 4th ed. [27] B. Gr ¨ unbaum and G . C. Shepard, T i lings and P atterns (F ree- man, Ne w Y ork, 1987).