Light tunneling inhibition in array of couplers with combined longitudinal modulation of refractive index
LLight tunneling inhibition in array of couplers with combined lon-gitudinal modulation of refractive index
Yaroslav V. Kartashov and Victor A. Vysloukh ICFO-Institut de Ciencies Fotoniques, and Universitat Politecnica de Catalunya, Medi-terranean Technology Park, 08860 Castelldefels (Barcelona), Spain Departamento de Fisica y Matematicas, Universidad de las Americas – Puebla, Santa Catarina Martir, 72820, Puebla, Mexico
We consider light tunneling inhibition in periodic array of optical couplers due to the spe-cially designed longitudinal and transverse modulation of the refractive index. We show that local out-of-phase longitudinal modulation of refractive index in channels of directional cou-plers in combination with the global refractive index modulation between adjacent couplers allow simultaneous suppression of both local and global energy tunneling inside each coupler and between adjacent couplers. This enables localization of light in single waveguide despite the remarkable difference of corresponding local and global energy tunneling rates.
OCIS codes: 190.0190, 190.6135
Waveguide arrays offer exceptional opportunities for control of light propagation [1,2]. Additional degrees of freedom appear if the refractive index vary also in the direction of light propagation. Such structures are capable to support discrete diffraction-managed soli-tons [3-5] and allow flexible control of beam propagation direction [6,7]. Dynamic localiza-tion of light in photonic structures with longitudinal modulation of guiding parameters un-doubtedly is among the most interesting optical phenomena. Such localization was predicted and observed in waveguide arrays [8-13] and optical couplers [14-17]. Different tools for the control of light tunneling were developed, such as periodic bending [8-10,14,15] or out-of-phase modulation of refractive index of adjacent guides [11-13,17]. All previous efforts were focused on photonic structures with a single characteristic energy exchange scale between neighboring guides. However the periodic array of optical couplers serves as an illuminating example of photonic structure where rapid local energy exchange between guides in each coupler coexists with slow global energy tunneling into adjacent couplers. The presence of two characteristic energy tunneling scales in this system makes the problem of light localiza- ξ -axis of periodic array of couplers is de-scribed by the Schrödinger equation for the normalized complex field amplitude q : q qi p η ξξ η ∂ ∂= − −∂ ∂ R q R w kw w (1) where η and are the transverse and longitudinal coordinates normalized to the character-istic transverse scale and diffraction length, respectively, while the parameter describes refractive index modulation depth. The global refractive index distribution is described by the function , where refractive index profile of k -th coupler is given by ξ p kk R +∞=−∞ = ∑ (2) s s l l s ls s l l s l [1 sin( ) sin( )] ( / 2 )[1 sin( ) sin( )] ( / 2 ), k R GG w kw μ ξ μ ξ ημ ξ μ ξ η = + Ω ± Ω + + +− Ω ± Ω − + where , is the separation between channels inside each coupler, is the channel width, stands for the distance between centers of couplers, while signs ± correspond to even/odd values of ( ) exp( / ) G η η η = − l w s w w η k . This profile corresponds to the local out-of-phase har-monic modulation of the refractive index inside two channels of each coupler with a spatial frequency and modulation depth , and simultaneous global out-of-phase modulation of refractive index between neighboring couplers with different spatial frequency and modu-lation depth [see Fig.1(a) for representative example of such an array of couplers]. Fur-ther we set , w , , and . Notice that longitudinal refractive index modulation changes not only propagation constants of guided modes but it also modi- s Ω l μ w η = s = s μ = l Ω l w p = ies coupling between waveguides due to modification of guided mode profiles and overlap integrals that affect the rate of diffraction in the structure. Figure 1(b) shows usual diffraction spreading for the case of excitation of single chan-nel in central coupler in the absence of longitudinal modulation. We use linear guided mode of a single isolated guide as an initial condition for numerical integration of Eq. (1) (other input beam shapes yield qualitatively similar results). One can observe fast local oscillations inside individual couplers (spatial frequency of intensity beatings is given by , where for our parameters ) and slow global light spreading across the array due to light tunneling between adjacent couplers. Figure 1(c) illustrates an attempt of light tunnel-ing inhibition using out-of-phase refractive index modulation only inside individual couplers ( , ). Notice that the modulation frequency and depth selected corre-spond to the optimal light tunneling inhibition in an isolated coupler. While such modula-tion drastically suppresses energy exchange inside couplers (note that suppression is re-markable, but it can not be complete) it can not suppress light tunneling to neighboring couplers. Analogously, if only modulation between couplers is present ( μ , ), one can effectively suppress energy exchange between the couplers but not beatings inside input coupler [Fig. 1(d)]. This indicates that simple single-frequency longitudinal refractive index modulation does not allow to achieve tunneling inhibition in system with two characteristic energy tunneling scales and one has to resort to more complicated simultaneous local and global longitudinal modulation of refractive index, such as the one described by Eq. (2) with . The key issue is thus optimal selection of parameters of such a modulation. b b T π Ω = l μ ≠ b T = s μ ≠ s l , μ μ ≠ l μ = s Ω s μ = s As a criterion of optimization we used the distance-averaged energy flow trapped in the input channel of central coupler as well as distance-averaged energy flow in the entire central coupler: U U s sl ll l ( , ) ( , 0) ,( , ) ( , 0) , L w wL w ww w
U L d q d q dU L d q d q d ξ η ξ η η ηξ η ξ η η − − −− − − == ∫ ∫ ∫∫ ∫ ∫ η (3) A characteristic feature of light tunneling inhibition is the resonant behavior of the distance-averaged energy flow trapped in the input channel of central coupler as well as dis-tance-averaged energy in the entire central coupler , as shown in Fig. 2. Panel 2(a) depicts dependence on the frequency for nonzero in the absence of global modu-
1m s ( ) U Ω ( ) U Ω s μ U s Ω ation, while panel 2(b) illustrates dependence on for nonzero in the absence of local modulation. Notice that the resonant peaks in the dependence are remarkably sharper. Interestingly, the frequencies of primary resonances (i.e., resonances that occur for largest or values) do not differ considerably for and dependencies from Fig. 2. Figure 2(c) exemplify almost linear dependencies of the principal resonance fre-quency and the frequency of second resonance Ω on the depth of global modulation at . It should be stressed that the resonances in Figs. 2(a) and 2(b) correspond to suppression of coupling either inside couplers or between couplers, but overall localization that can be characterized by the combined localization criterion remains low (i.e. U is considerably smaller than 1) in both cases because only one type of modulation is present. U U l Ω U U s μ =Ω l μ s Ω s μ s ( ) Ω s ( ) Ω l μ tm ( Ω s l μ μ ss Ω r1 Ω s = tm Ω l Ω l ( ) U Ω U U U = l Ω = s μ r 2 l μ n l Ω μ tm l ) Ω s Ω μ , ≠ l Ω Figure 3 demonstrates combined localization criterion as a function of modulation fre-quencies and for simultaneous local and global longitudinal modulation of refractive index for the case when [Fig. 3(a)] and when [Fig. 3(b)]. The combined lo-calization criterion attains a maximal value when energy tunneling between channels of individual coupler and between neighboring couplers is suppressed simultaneously. Impor-tantly, the strongest overall localization with U (or the principal resonance for com-bined modulation) always appears in the vicinity of value that corresponds to light tun-neling inhibition inside individual couplers and even small detuning of from this resonant value leads to remarkable diminishing of localization. The U dependence is character-ized by the narrow bands of strong delocalization around the frequencies , where is an integer, while maxima of corresponding to overall inhibition of tunneling appear exactly in between these delocalization bands. For a fixed the comparable inhibi-tion of tunneling can be achieved for several values. Figure 3(c) illustrates the depend-ence of combined localization criterion on the local modulation depth when , , and are fixed and correspond to strongest global resonance. Typical features of U pro-file are sharp localization peak and well defined localization minimum which appear due to almost linear dependence of resonant frequencies on the modulation depth [see also Figs. 2(a)-2(c)]. The examples of complete light tunneling inhibition are shown in Figs. 1(e) and 1(f) for the case of combined longitudinal modulation with . One can see that by properly selecting corresponding modulation frequencies and one can simultaneously suppress energy exchange between channels of coupler and between neighboring couplers, so that upon propagation the light remains in the excited channel. l Ω U s μ ≠ s l μ tm tm l Ω ≈ s Ω s / n l s Ω tm s ( μ ≥ tm ) eferences with titles F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, "Discrete solitons in optics," Phys. Rep. , 1 (2008). 2.
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Ω = Ω b Ω b T s μ = l Ω l μ = b l b Ω = Ω s l μ μ = = b L T = Ω =
Figure 2. (a) Distance-averaged energy flow in the input channel versus at , . (b) Distance-averaged energy flow in the input coupler versus at , . (c) Frequencies of first and second resonances versus for . s Ω s μ = l Ω l μ l μ = s μ = s μ = l μ = Figure 3. The product of distance-averaged energy flows versus modula-tion frequencies and at (a) , , and (b) , . Red regions corresponds to strongest inhibition of tunneling when , while in blue regions one has strongest delocalization when . (c) versus at , , . tm 1m 2 m
U U U = l μ = l b Ω = Ω s Ω s μ l Ω= s μ = s b Ω = Ω s μ = m U → l μ = m U ≈ tm U l μ11