Quantitative analysis of subdiffractive light propagation in photonic crystals

Transcription

1 Optics Communications 69 (7) Quantitative analysis of subdiffractive light propagation in photonic crystals Yurii Loiko a,d, *, Carles Serrat a, Ramon Herrero b, Kestutis Staliunas a,c a Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Colom, 8 Terrassa, Spain b Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Comte Urgell 87, E-836 Barcelona, Spain c Instituciò Catalana de Reserca i Estudis Avancats (ICREA), 8, Barcelona, Spain d Institute of Molecular and Atomic Physics, National Academy of Sciences of Belarus, Nezaleznasty Ave. 7, 7 Minsk, Belarus Received 7 May 6; received in revised form 7 July 6; accepted 7 July 6 Abstract We present a qualitative study of the subdiffractive light beam propagation in photonic crystals, based on the theoretical analysis of full Mawell equation system beyond the slowly varying envelope and paraial approimations. We calculate the zero diffraction curve in the parameter space, we calculate the weak asymptotical broadening (spreading in transverse direction), and we also analyze the dependence of subdiffractive propagation on the polarization orientation (both for TM and TE polarization). Ó 6 Elsevier B.V. All rights reserved. PACS:.7.Qs;.5.F;.55.Tv;.79. e. Introduction Since the initial proposal of the concept of photonic crystals (PC) in 987 [] a number of studies revealed that these materials with a periodic modulation of the refraction inde on a spatial scale of the order of the wavelength of light are powerful tools to control and modify the propagation of electromagnetic fields. PCs offer a possibility of engineering the dispersion properties of light to yield photonic band gaps in the transmission and reflection spectra, thus they can be designed to act as light conductors or insulators []. PCs also modify (in particular reduce strongly) the phase and group velocities of light [3]. Recently it becomes apparent that the PCs can also modify the diffraction of light, in that the diffraction can become negative if the refractive inde is modulated in direction perpendicular to the propagation of the light (-dimensional PCs) []. * Corresponding author. Tel.: ; fa: address: Yurii.Loiko@upc.edu (Y. Loiko). Negative diffraction was also predicted for acoustic [5], and matter waves [6,7], in periodic materials. If the diffraction is positive at one edge of the propagation band, and is negative at the other edge, an inflection point inside of photonic band can be epected, characterized by the vanishing diffraction. The vanishing of the diffraction has been shown for the arrays of waveguides [8], and in the resonators with periodic modulation of refractive inde in one transverse direction [9]. The vanishing of diffraction means, that at some particular point of parameter space the curvature of the spatial dispersion curve o k k becomes zero (here k ok k is the longitudinal- and? k? is the transverse component of the light wavevector). The nondiffractive propagation of the monochromatic light beams in two-dimensional (D) photonic crystals has been shown analytically [], numerically [], and eperimentally [] up to now. Most of the previous studies are, however, mainly numerical, based on: () the numerical calculation of the isofrequency surfaces: with the idea that the zero diffraction can be epected on the straight segments of the isofrequency curves; or () the numerical finite-difference 3-8/$ - see front matter Ó 6 Elsevier B.V. All rights reserved. doi:.6/j.optcom

2 Y. Loiko et al. / Optics Communications 69 (7) time-domain (FDTD) calculations of the beam propagation through (usually relatively short) piece of PC, where much weaker broadening of the beam, comparing to the broadening in the homogeneous materials, is usually demonstrated. What is missing is the detailed analytical and numerical study, where the nondiffractive propagation conditions were calculated, and the zero-diffraction areas (curves) in the parameter space were determined. This knowledge is useful for the designing the optical devices based on the nondiffractive PCs. Also the knowledge on the asymptotic (long distance) propagation of the beam around the zero-diffraction point (ZDP) is missing. Nowadays the FDTD calculations, which are well described in Ref. [3], became standard tools for numerical studies of light propagation in different media. But they are usually much time consuming, and allow the calculation of relatively short propagation. An analytical study of the above properties of nondiffractive propagation has been performed in [], however in a paraial (parabolic) approimation only, which is valid for the study of light propagation in materials with the refraction inde modulated on larger spatial scale (comparing with the wavelength of the radiation). The results of [] can be thus only qualitatively applied to the real PCs, i.e. the materials with the refractive inde modulated on the scale of the wavelength. The purpose of the present article is to fill this gap. We perform, in this way the study of light propagation in PCs, based on the solution of the full set of Mawell equations beyond the slowly varying envelope (SVEA) and paraial approimations. We derive analytical relations for the PC parameters of beam subdiffractive propagation, and we check these relations by FDTD calculations of narrow beam propagation in PCs. We also derive the equations governing the (asymptotic) long range propagation, and check it by FDTD calculations. This allows to evaluate the minimal width of the nondiffractively propagating beam, or equivalently the minimal spatial modulation scale of nondiffractively propagating optical pattern. The calculations are performed for both TM and TE polarized light (with respect of the plane of two-dimensional PCs under study here). The article is organized as follows. In Section we give Mawell-equation description for TM and TE polarized light beam propagated in D plane. In Section 3 harmonics epansion is performed for the beam of TM polarization. Conditions for subdiffractive propagation of TM polarized beam are calculated in Section. Asymptotic (weak) spreading of TM polarized beam is analyzed in Section 5. Results on subdiffractive propagation of TE polarized beam are presented in Section 6. Finally in Section 7 we summarize our results.. Model Photonic crystals are usually fabricated by different mechanical, chemical and lithographical techniques producing by material echanges the refractive inde gratings with steep profile []. On the other hand, spatially periodic gratings with smooth harmonic profile can be inscribed in some materials by holographic techniques without material echanges. In particular, purely harmonic spatial modulation of the refraction inde can be provided in photorefractive crystals by the interference of the holding (crossing) light beams with the same wavelength. The two kinds of PC fabrication techniques making different internal architecture of the inde within the unit cell provide very similar dispersive and diffractive properties of PCs (types of the dispersion curves). It is, because the spatial periods, geometry (symmetry) and the overall modulation depth (or the filling factor) of refractive inde are the most significant factors that influence dispersion properties of PCs. The profile of refractive inde modulation has minor effects. Therefore, by studying the wave propagation in PCs with periodic harmonic modulation of refractive inde one can acquire general dispersion and diffraction properties of PCs fabricated by different techniques. Moreover, theoretical investigations on PCs with harmonic modulation of spatial profile of refractive inde are more easily provided, as shown below, which can give physical insight into the problems of dispersion and diffraction of waves propagated inside PCs. Therefore, in this paper we consider PCs with harmonic spatial profile of refractive inde modulation. We consider two-dimensional (D) PC of nonmagnetic isotropic media, which electric susceptibility modulated periodically and harmonically in two directions. The electric and magnetic displacements for either TM or TE polarized light can be written as follows: D r ¼ E r e e þ deðcosð q!! r Þþcosð! q! r ÞÞ ¼ E r e ðe þ de cosðq Þ cosðq z zþþ; B q ¼ l H q ; where e is the average electric susceptibility, de is the amplitude of susceptibility modulation, e and l are the electric and magnetic constants, respectively; q! and q! are the reciprocal PC lattice vectors with j q! j¼jq! j¼q (the rhombic symmetry are to be considered throughout the paper); q = qcosa =p/k and q z = qsina =p/k z ;+a( a) is the angle between the optical ais and q! ð q! Þ vector; z is the beam propagation direction; K and K z are the periods of susceptibility modulation in and z direction, respectively; E r (D r ) are the components of electric field (displacement) vector, and H q (B q ) are the components of magnetic field (magnetic induction) vector; for TM polarized beam [r = y,q ={,z}] and for TE polarized beam [r ={,z},q = y]. Here we study propagation of both TM and TE polarized light beams. For TM polarized beam, D Mawell equations can be written in the form ðþ

3 3 Y. Loiko et al. / Optics Communications 69 (7) 8 36 o ot B B z D y o oz C B A ¼ o o A@ o o oz o H H z E y C A; ðþ D Mawell equations for TE polarized beam propagated in (,z) plane are o D E oz o B C B o CB D z A o A@ E z A: ð3þ ot B y o oz o o In the study [], only TM polarized light beam propagation in photonic crystal has been considered under SVEA and paraial approimations. It should be noted that SVEA and paraial approimations restrict the applicability of results to the case K,z k. Here, in contrast, we focus on the regimes K,z k. Further in this paper, beam diffraction will be studied analytically by the harmonic epansion [5] of Eqs. () (3) and numerically by using FDTD method [3]. H y which corresponds to the appearance of plateau on dispersion curves. The plateau formation is essentially a result of intersection of three dispersion curves, therefore, it can be described by considering only three harmonics with indees (,) and (, ±) in epansion () Uðk z ; k Þ¼B f k ðb þ þ B Þ¼: The intersection of dispersion curve with k z ais at point k z() is given by the epression ð6þ ¼ðk k zðþ Þ f k B ; ð7þ where B ¼ B þ j k¼ ¼ B j k¼ ¼ðk ððk zðþ q z Þ þ q ÞÞ; shows how much the dispersion curves are deformed due the susceptibility modulation. At the second order of smallness of f epression (7) becomes ð8þ 3. Harmonic epansion In analytical study we decompose the electric and magnetic fields in a set of plane waves [5] A i ¼ X A ðm;nþ i epðik z;m z þ ik ;n Þ epð i tþ; ðþ m;n ðk z;m ; k ;n Þ¼ðk z þ mq z ; k þ nq Þ; where A i ={H,B,H z,b z,e y,d y } and A i ={E,D,E z, D z,h y,b y } for TM and TE polarized beam, respectively; m and n are integer numbers, and (k,k z ) denote the wavevector of the lower harmonics of waves propagating inside the photonic crystal. By substituting epansion () into the Mawell equation () and ecluding the magnetic field components one obtains coupled set of equations solely for the electric field harmonics! ¼ E ðm;nþ B mn þ fk X i¼;j¼ E ðm i;n jþ ; ð5þ p where k ¼ ffiffi e =c is the wavenumber of light in homogeneous medium with susceptibility e, f = de/(e) is the amplitude of susceptibility modulation (the relation de < e leads to the following restriction f <.5); B mn ¼ðk ððk zþ mq z Þ þðk þ nq Þ ÞÞ. Under SVEA (k z q z ) and paraial ðk k z Þ approimations, when B mn =( k z mq z + (k + nq ) ) Eqs. (5) simplify to the case considered in Ref. []. For homogeneous propagation (f = ) the dispersion curves are the circles described by the equations B mn =. Dispersion relation between k z and k in PC can be obtained from Eq. (5) as solvability condition. Numerical solutions of Eq. (5) accounting for 9 most relevant modes, are depicted in Fig.. Subdiffractive beam propagation takes place under condition d ¼ o k z =ðok Þ j k ¼ ¼, k zðþ ¼ k þ f k ððq z þ q Þ k q z Þ : Power series of k z (k ) dependence (6) result to k z ¼ k zðþ þ X ðn!þ d n k n n¼;;3;... ; ð9þ ðþ where d n are the diffraction coefficients of n order calculated below. Even orders of k appear in epression () due to the symmetry of considered problem.. Subdiffractive TM beam propagation In general the first (dominating) diffraction term, as calculated from (6), is the usual diffraction ¼ o k z ðok Þ k¼ ¼ þ f k q z B k zðþ þ B q k zðþ B þ f k ð q z A; ðþ k Þ zðþ which at the order of f reduces to! ¼ k þ f k ðq zððq z þ q Þ k q z Þ k q Þ k ððq z þ q Þ k q z Þ 3 : ðþ The zero diffraction point is given by the epression ¼ and it allows to calculate ZDP analytically: B 3 þ f k ðb þ q Þ¼: ð3þ Solution of (7) and (3) determines the surface in threedimensional space of PC parameters (q z,q,f), at which subdiffractive propagation of TM polarized beam is possible. B

4 Y. Loiko et al. / Optics Communications 69 (7) a Type I c Lz=.885. k /k z Lz=.985 k z /k Lz=.. b Type II d Lz=.5. k /k z Lz=.9 Lz=.7 k z/k k /k k /k Fig.. Two configurations (a) and (b) of transverse dispersion curves at zero diffraction point for TM polarized beam. Photonic crystal parameters are: f =.695 (e =.5, de =.65), K /k =.6; K z /k =.985 (a) and K z /k =.9 (b). (c) and (d) magnify dispersion curves at (solid lines) and close to (dashed lines) zero diffraction point. In (c) and (d) numbers near the curves indicate values of parameter L z = K z /k. Circles in (a) and (b) define domain magnified in (c) and (d), respectively. Polynom (3) is a cubic one with respect to B and has one real and two comple conjugate roots at any sets of PC parameters (q,f). Real root of polynom (3) is B ¼ f =3 k f =3 Bsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 3 þ 8 q k 8 q k A C A ; vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v! u3 u t6@ t8 3 3 f þ 8 q 8 q k k ðþ which in the leading order of smallness of f is B ¼ f =3 k. Analysis of () yields that B 6 at any set of PC parameters (q,f). Therefore, from epression (7) one can always find corresponding longitudinal component k z() of light wavevector in PC for waves with k =. Finally from the epression for B, see formula (8), one can determine two values of q z at which ¼ : sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q z k ¼ f k B q B : ð5þ k k Epressions () and (5) describe analytically the form of the subdiffractive surface. Intersection of this subdiffractive surface with plane f =.695 is depicted in Fig.. In (q z,q ) plane subdiffractive surface is a curve. Inside (outside) the domain restricted by subdiffractive curve in Fig. diffraction is anomalous (normal), as it can be seen in Fig. (c) and (d). At the parameters values of (q z,q ) that belong to the curve, propagation of TM polarized beam becomes subdiffractive. At fied value of q subdiffractive beam propagation occurs with the same value of longitudinal wavevector k z independently to which of two branches (upper or lower) of subdiffractive curve the value of q belongs. From Fig. it follows that these two branches merge at certain value of q TM. Above this value, subdiffractive TM beam propagation in considered PC is impossible. The subdiffractive TM beam propagation is impossible when q z becomes comple (i.e. when the epression under the second square root in (5) becomes negative). (Since B < for any set of (q,f) the epression under the first square root is always positive.) The subdiffractive propagation threshold, as obtained from (5) and () by ecluding B, is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ ¼ k B þ f =3 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ð f =Þ ðq TM f 5=3 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ C q p A: ð6þ 3 ð f =Þ Fig. represents the typical slice of subdiffractive surface in the plane of constant modulation amplitude f. The maimum value of ratio q /k, above which subdiffractive TM beam propagation pffiffi is impossible, corresponds to K TM ¼ kk =ðq TM e Þ. Dependance of q TM on susceptibility modulation amplitude f is depicted in Fig. 3. Subdiffractive TM beam propagation is possible in the domain of parameter q below the curve in Fig. 3. As one can see, the maimum modulation wavenumber q TM (minimum modulation length K TM ) increases (decreases) with increasing the modulation amplitude f. The other feature of subdiffractive surface is that at fied value of PC parameters (q,f) there are two values of longitudinal modulation period, i.e. two values of q z, at which subdiffractive TM beam propagation is possible, see Fig.. For both these values of q z the value of

5 3 Y. Loiko et al. / Optics Communications 69 (7) 8 36 q TM 8 q /k z Qz q /k..8. F Fig.. Zero diffraction curves for subdiffractive TM beam propagation at f =.695 as given by (3) and (7). Circles represent numerical data as obtained by solution of Mawell equation () by FDTD method. Right part of figure represents the same data in the parameter space as used in Ref. [], namely, Q z ¼ q z k =q and F ¼ fk =q. q k TM.... Fig. 3. Maimum value of PC modulation wavenumber in transverse direction q TM depending on f for subdiffractive propagation of TM polarized beam as obtained form (6). B is the same. Parameter B corresponds to the lift of degeneracy due to the effect of periodic susceptibility modulation. Two values of q z, at which subdiffractive propagation of TM beam is possible at fied magnitude of (q,f), correspond to two different geometries with q z < k depicted in Fig. (a) and q z > k depicted in Fig. (b). The former case, i.e. q z < k, has been reported previously in []. Here we focus on subdiffractive propagation in the latter case of q z > k. The domain of subdiffractive TM beam propagation is restricted from below in q z direction, i.e. we found that at fied amplitude of susceptibility modulation, subdiffractive TM beam propagation is impossible in PC when modulation length in z direction (direction of injected beam) becomes larger than certain value K z. The restriction from above on the values of q means that at q > q TM different dispersion branches (dispersion circles) are not intersected with each other and therefore they are decoupled. As a result changes of PC parameters do not influence strongly the dispersion properties of PC and dispersion curves are reduced to ordinary dispersion circles, for which subdiffractive propagation is impossible. The analytical results were checked by numerical integration of Mawell equation () by FDTD method. FDTD simulation is nowadays already a standard technique for solving the partial differential equations by integrating in time, and by approimating the spatial derivatives by finite differences f [3]. In FDTD simulation we considered propagation of light with wavelength k = 8 nm. The discretization was 3 points per wavelength if K z, > k or 3 points per smallest period of modulation in a case when K z or K < k. Calculations were performed on grid which size is 3k in transverse and k in longitudinal direction. Computational domain was terminated by absorptive boundaries with uniaial perfectly matched layers (UPML) [3]. Numerical results (circles in Fig. ) have been obtained by managing the smallest value of beam spreading in transverse direction (beam width) after propagation distance of light wavelengths. It is seen that numerical results are consistent with the analytical predictions for large q z and q. For small values of q z and q, FDTD solution of the problem demands huge computer resources, and, therefore, it is not presented here. The analytic results presented in this study were obtained by decomposition in three harmonics. It should be mentioned that in some cases, for instance, at both small and large values of (q z,q ), i.e. large and small values of space modulation periods (K z,k ), respectively, one should account for more than three harmonics in epansion (). Such analysis is out of the scope of this paper. 5. Asymptotic analysis of subdiffractive propagation of TM beam Subdiffractive propagation of light beam (d =) is described by the higher order diffraction coefficients in (), in particular of the fourth order d. This means that asymptotically (at a large distance) the beam spreading should follows w z /, see Ref. [], i.e. logðwþ logðzþ, where w is the beam width in transverse direction. The fourth order diffraction coefficient d at PC parameters for which d = reads ¼ o k z ok k¼;d ¼ ¼ k zðþ þ q B B fk B 3 þ ðfk Þ ð q z=k zðþ Þ B : ð7þ At the leading order of smallness parameter f f =3 in consistence with [] under SVEA and paraial approimations. At fied (q,f) variables B and k z() have

6 Y. Loiko et al. / Optics Communications 69 (7) the same values at upper and lower branches of subdiffractive curve, see Fig.. In a case when q z > k (which corresponds to the upper curve in Fig. ) fourth order diffraction coefficient can be represented as ðq U z Þ a a and the same representation for a case q z < k gives ðq L z Þ a þa (in both epressions the coefficients a,, are positive). Therefore, for the upper subdiffractive curve (q z > k ) the fourth order diffraction coefficient is higher in magnitude than that for the lower subdiffractive curve (q z < k )asit is shown in Fig.. This means that in PC with identical transverse modulation period and modulation amplitude of optical susceptibility, under subdiffractive propagation condition ð ¼ Þ the transverse width of TM beam will be higher at smaller longitudinal modulation period (larger value of q z ). We have checked asymptotic behavior of TM beam spreading under subdiffractive propagation conditions by FDTD solution of Eq. () for several PC parameter values (q z,q,f) that belong to subdiffractive surface. A typical result for subdiffractive beam propagation is presented in Fig. 5, where in the upper figure the electric field component averaged in time during one period of light oscillations is represented. White line in the upper figure and dash-dotted line (in logarithmic scale) in the lower figure represent evolution of Gaussian beam in homogeneous case. One can see that near the input point the TM beam (see also solid line in the lower part of the figure) spreads faster in PC than in homogeneous case. However, at larger distance the width of subdiffractively propagated beam increases slowly in accordance with w z / law, valid for fourth order diffraction []. In Fig. 6 typical spatial spectrum of the beam propagated under subdiffractive condition is depicted. One can see that the main peak is centered around zero diffraction point (k z(),), but in addition the backreflected waves (with wavevectors ( k z(),k )) as well as the forward and backward Bloch modes with k 5 (which belong to the other plateaus of dispersion curve under subdiffraction propagation condition) contribute to the formation of subdiffractively propagated beam. To analyze more precisely their contribution more than three modes should be accounted for in the epansion (). Fig. 5. FDTD simulation of subdiffractive propagation of TM polarized beam in PCs with parameters indicated in Fig. (b). Spatial distribution in PCs of the electric field component averaged during one period of light oscillations (top part) and asymptotic behavior of transverse width of the same beam (bottom part). Numerical calculation of Eq. () has been performed by FDTD method (details see in the tet). Gaussian beam with transverse waist w =k was injected from the left of the presented domain. Lengths in the longitudinal and transverse directions of the domain presented in the top part of the figure are l z =8k and l =7k, respectively. Solid line in bottom part of the figure represents the transverse width (transverse spreading) of the beam in PCs. Dashed line represents guide line for beam which spreading in PCs is governed by fourth diffraction order w z /. White line on the top figure and dashed-dotted line in the bottom figure represents the transverse width of the same gaussian beam propagated in homogeneous material (f = ). k z / k q /k z q / k > z - q /k -q / k z q / k < z q / k Fig.. Fourth order diffraction coefficient as obtained from (7). Upper (lower) curve corresponds to the upper (lower) branch of subdiffractive curve in Fig.. - k / k Fig. 6. Spatial spectra of subdiffractive propagation of TM beam presented in Fig. 5 in region l z = 8k. Open circles denote localization domains (plateaus) of waves propagated with positive group velocity, i.e. the waves which belong to the same Bloch mode propagated in the forward direction. Open triangles denote localization domains of waves which belong to the other Bloch mode, which group velocity is opposite to that of the Bloch mode denoted by open circles. Dashed lines denote the directions in the basis of reciprocal lattice.

7 3 Y. Loiko et al. / Optics Communications 69 (7) TE polarized beam in PC We performed the study of TE polarized beam propagated in D photonic crystal. Accounting for three harmonics (m, n) = (+, ±) and (, ) in each polarization component in decomposition () the dispersion curve is Wðk z ; k Þ¼CA D ;!! A ¼ ðk k zðþ Þ f k þ k z;þ þ k z; B þþ B þ!! C ¼ ðk k ; Þ f k þ k ;þ þ k ; B þþ B þ D ¼ k zðþ k ; þ f k k z;þ k ;þ þ k z;þk ; : B þþ B þ ð8þ The dispersion curve intersects the k z ais at point k z(), which intrinsically is determined by the following epression: ¼ðk k zðþ Þ f k B ½ q =k Š: ð9þ Eq. (9) is slightly different from the epression (7) obtained for TM polarized beam. The difference is marked by square bracket. Approimate epression (9) obtained above for TM beam in the case of TE beam is also respectively modified. The difference is negligibly small for /q /k, but it becomes significant for /q /k. We obtain dispersion relation k z ¼ k z ðk Þ and epression for subdiffractive curve analogously as in the case of TM beam: TE d TE ¼ o k z ðok Þ k ¼ k zðþ þ f k k z;þ ðbþq þf Þ k k z;þð Bþq Þ B f þ q B 3 B ¼ : k f þ q k B zðþ þ f k k z;þ ðk q Þ B ðþ Subdiffractive beam propagation with TE polarization takes place when d TE ¼. Under this condition one can obtain from epression () the 6th order polynom with respect to B, from which variable B can be determined at fied values of PC parameters (q,f). It is difficult to analyze this polynom and therefore we omit this analysis. To give more analytics we simplified 6th order polynom by considering only terms up to the second order in f. By neglecting the terms of fourth and high order of f one can obtain the following 3d order polynomial for B X a i B i ¼ i¼;3 a ¼ 6f k 6 ðq =k Þ ð ðq =k Þ Þ a 3 ¼ f k ð 5ðq =k Þ Þ a ¼ f k ð ðq =k Þ Þ a ¼ : ðþ Solution of () gives zero diffraction curve presented in Fig. 7. It follows that at fied values of PC parameters (q,f) there are two values of susceptibility modulation period, i.e. two values of q z, at which subdiffractive TE beam propagation is possible. At such two values the dispersion curves for TE beam are presented in Fig. 8. This figure shows that transverse dispersion curves for TE polarized beam have the same types as presented in Fig. (a) and (b) in the case of TM beam. We have verified for TE polarized beams eistence of these two values of q z by numerical solution of Mawell equation (3) with FDTD method. Results of this integration are presented by open circles in Fig. 7. They agree with analytical predictions in the considered domain of q value. From Fig. 7 one can see, that in the same way as it has been found for TM beam, subdiffractive propagation of TE beam is possible when length of spatial modulation in transverse direction accedes some limit value. Analysis of () gives possibility to establish the dependence on susceptibility modulation depth f of maimum value q TE for which subdiffractive propagation of TE beam is possible. This relation can by written in implicit form as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 5 Þð þ 6 9 þ 8 6 Þ ð 6 þ þ Þ 3 f ¼ : ð þ 3 þ 6 6 Þð þ Þ ¼ q TE =k : ðþ From the last epression it follows that there are at least two values of q TE that restrict domain of PC parameters in which subdiffractive TE beam propagation is possible at fied f. The smaller value of q TE corresponds to the point where two branches in (q z,q ) plane intersect near the point q z, = (left point of subdiffractive curve intersection in Fig. 7) and the larger value of q TE corresponds to the right point in Fig. 7. In Fig. 7 one can also see that in certain domain of q there are four values of q z for which subdiffractive propagation of TE beam is possible at fied f. To study this domain more than three harmonics should be accounted for in (). Studies of this domain will be provided elsewhere. q / k z q TE q / k Fig. 7. Zero diffraction curve for TE polarized beam at f =.695 as obtained by solution of Eqs. () and (9). Circles represent data obtained for TE beam by numerical solution of Eq. (3) with FDTD method.

8 Y. Loiko et al. / Optics Communications 69 (7) a Type I c Lz=.36. k z /k Lz=.39 Lz=.396 k z/k b d Lz=..5 k z /k k /k Lz=. Type II Lz=. k /k k z /k Fig. 8. Two configurations (a) and (b) of transverse dispersion curves at zero diffraction point for TE polarized beam. Photonic crystal parameters are: f =.695 (e =.5, de =.65 ), K /k =.7; K z /k =.39 (a) and K z /k =. (b). (c) and (d) magnify dispersion curves at (solid lines) and close to (dashed lines) zero diffraction point. In (c) and (d) numbers near the curves indicate values of parameter L z = K z /k. Circles in (a) and (b) define domain magnified in (c) and (d), respectively. We have also found that asymptotic behavior of TE polarized beam under subdiffractive propagation condition is governed by fourth order diffractive coefficient d TE, in the same way as it has been found in the case of TM polarized beam, see Fig Conclusion Subdiffractive TM and TE beam propagation in photonic crystal (with harmonic modulation of electric susceptibility in two spatial dimensions) is considered both analytically (by harmonic epansion method) and numerically (by FDTD method) beyond paraial and slowly varying envelop approimations. Subdiffractive propagation of beam means that the second order diffraction coefficient d becomes equal to zero, and beam spreading is much slower than that for Gaussian beam in homogeneous medium. We show by FDTD numerical solution of Mawell equations that asymptotic behavior of beam spreading under subdiffractive propagation condition is governed by the fourth order diffraction coefficient d, which verifies analytical predictions. Analytical epressions for the second and fourth order diffraction coefficients, that govern normal and subdiffractive beam propagation, as well as the analytical epressions for the zero diffraction curve have been obtained without paraial and slowly varying envelope approimations. It has been shown that there are two values of the period /q z of medium susceptibility modulation in longitudinal (along beam propagation) direction under which subdiffractive propagation of TM and TE beam is possible for other PC parameters (q,f) being fied. One of this period is larger but the other is smaller than the light wavelength, when transverse period of susceptibility modulation is on the order of light wavelength. Restrictions on spatial periods of susceptibility modulation (i.e. on PC parameters), under which subdiffractive TM and TE beam propagation becomes impossible, have been found. Namely, it has been shown that there is a minimum spatial length of susceptibility modulation in transverse direction below which subdiffractive beam propagation becomes impossible. This occurs when different branches of dispersion curves become uncoupled from each other. This condition imposes a limit on the modulation period, and eventually on the width of input beam which could be subdiffractively propagated in PCs. Analytical epressions are more transparent for TM polarization. The TE case, however, brings no substantial difference into subdiffractive propagation. Acknowledgements Support from the CRED-program of the Generalitat de Catalunya, from the Programa Ramón y Cajal of the Spanish Ministry of Science and Technology, and from Projects FIS5-793-C3-3 and FIS5-97 is acknowledged. References [] E. Yablonovitch, Phys. Rev. Lett. 58 (987) 59; S. John, Phys. Rev. Lett. 58 (987) 86. [] See e.g. Photonic Band Gaps and Localization C.M. Soukoulis (Ed.), NATO Advanced Studies Institute, Series B: Physics, vol. 38, Plenum, New York, 993. [3] M. Scalora, R.J. Flynn, S.B. Reinhardt, R.L. Fork, M.J. Bloemer, M.D. Tocci, C.M. Bowden, H.S. Ledbetter, J.M. Bendickson, J.P. Dowling, R.P. Leavitt, Phys. Rev. E 5 (996) R78; A. Imhof, W.L. Vos, R. Sprik, A. Lagendijk, Phys. Rev. Lett. 83 (999) 9; K. Sakoda, Opt. Epress (999) 67. [] R. Morandotti, H.S. Eisenberg, Y. Silberberg, M. Sorel, J.S. Aitchison, Phys. Rev. Lett. 86 () 396; M.J. Ablowitz, Z.H. Musslimani, Phys. Rev. Lett. 87 () 5.

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