Transfer of optical signals around bends in two-dimensional linear photonic networks
TTransfer of optical signals around bends intwo-dimensional linear photonic networks
G M Nikolopoulos
Institute of Electronic Structure & Laser, FORTH, P.O.Box 1385, GR-70013Heraklion, GreeceE-mail: [email protected]
Abstract.
The ability to navigate light signals in two-dimensional networks of waveguidearrays is a prerequisite for the development of all-optical integrated circuits forinformation processing and networking. In this article, we present a theoreticalanalysis of bending losses in linear photonic lattices with engineered couplings, anddiscuss possible ways for their minimization. In contrast to previous work in thefield, the lattices under consideration operate in the linear regime, in the sense thatdiscrete solitons cannot exist. The present results suggest that the functionalityof linear waveguide networks can be extended to operations that go beyond therecently demonstrated point-to-point transfer of signals, such as blocking, routing,logic functions, etc. a r X i v : . [ qu a n t - ph ] N ov ransfer of optical signals around bends in 2D linear photonic networks
1. Introduction
Photonic lattices (PLs) are currently at the focus of extensive research for two mainreasons. Firstly, for their flexibility in simulating various phenomena, especially thoserelated to tight-binding Hamiltonians [1, 2, 3, 4], and secondly for their potentials asbuilding blocks of quantum circuits for all-optical information processing and networking[5, 6, 7], and their role in related studies on quantum random walks [8] and bosonsampling [9].PLs can be fabricated in a doped silica multilayer structure on a silicon substrate[6, 7], as well as by means of femtosecond laser-writing techniques in the bulk of glasses[5, 3]. Both of these techniques allow one to exploit Kerr nonlinearity in order to achievecertain tasks. In addition, fabrication of waveguides in LiNbO and KTP by means ofetching techniques [10] allows for integrated sources of non-classical light, paving thusthe way toward integrated quantum chips, where the generation of entangled photons,their transmission, and their processing take place on the same chip.The faithful transfer of signals is a necessary precondition for further developmentsin these directions, and the problem has attracted considerable interest in recentyears [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. The typical evolution of aninitially well-localized wavepacket in an ideal finite PL (i.e., in the absence of disorderand dissipation), is characterized by spreading, reflections from the boundaries, andinterference phenomena that give rise to diffraction effects [12]. At relatively highintensities [23], such distortion effects can be avoided by using discrete solitons asinformation carriers [1, 2, 3, 5]. Alternatively, in the linear regime (i.e., for inputpowers below the threshold for the existence of discrete solitons [23]), one may resortto the segmentation of appropriate lattice sites [11], or to the engineering of judiciouscouplings between adjacent sites [12, 13, 14, 15, 16, 17, 18]. The latter scenario has beenalso studied thoroughly in the context of quantum networks, and various solutions havebeen proposed [19, 20, 21, 22].In any case, however, the reliable transfer of signals between two distant nodes ofa network is not sufficient for large-scale information processing and networking. Tothis end, one has to be able to perform reliably and efficiently more complex signalmanipulations such as routing, splitting, switching, etc. Such communication tasks canbe performed only in higher-dimensional geometric arrangements, where the presenceof bends at different angles are inevitable [5, 24, 25, 18, 22].The transfer of signals around bent nonlinear (discrete-soliton) PLs has beeninvestigated to some extent in the literature [5, 25, 24]. On the contrary, the onlyanalogous study for the case of linear networks with engineered couplings has beenlimited to a generic theoretical model and small bend angles [22], so that the effectsof bending can be treated as a small perturbation to the unbent chain. The regimeof bend angles for which this approximation is justified could not be assessed withinthe generic model of Ref. [22], since it depends strongly on the details of the physicalsystem under consideration. One of the purposes of the present work is to address ransfer of optical signals around bends in 2D linear photonic networks ◦ . For sharper bends, the detrimental effectsof the bending become very pronounced rapidly, especially in the case of asymmetricwaveguides, and can be suppressed by introducing a defect at the corner site, whilekeeping all the other parameters in the lattice constant. Our simulations suggest that inthis way, for the particular coupling configuration under consideration, faithful transferof signals for bend angles up to 60 ◦ (120 ◦ with respect to the unbent chain), is possible.Qualitative as well as quantitative aspects of the defect required for suppression ofbending effects at different angles are also discussed, and our main results are comparedto related results for nonlinear networks, that rely on solitonic information carriers. Inthe last Sec. 4, we summarize our findings and discuss certain issues that remain open.
2. Physical system and modelling
Various 2D configurations of laser-written buried photonic lattices in glasses, have beendemonstrated and studied experimentally [1, 2, 3, 5, 25, 13]. Typically, the wavenumberalong the propagation direction for each waveguide can be controlled by adjusting thecorresponding size and/or the refractive-index change. Moreover, the coupling betweentwo neighboring waveguides drops exponentially with their separation, and the preciseform of the exponential law is determined by the details of the experimental set-up.Knowing the precise form of this exponential law one can engineer various configurationsof waveguides that perform certain tasks, such as the non-dispersive transfer of signalsbetween two waveguides of a PL, for input light with specific properties (i.e., wavelength,polarization) [13, 15]. In most cases, experimental observations have been shown to be inexcellent agreement with the predictions of coupled mode theory, and other theoreticalmodels that rely on the Helmhotlz equation for scalar fields.In the present work we are interested in the transfer of signals between the two ransfer of optical signals around bends in 2D linear photonic networks Θ In Out xyz C N
Figure 1. (Color online) The system under consideration: a bent quantum chain of N rectangular waveguides. Light enters the structure at z = 0 in the first waveguide,and as it propagates along z it couples to other waveguides. Our task is the faithful(ideally perfect) transfer of the input signal to the N th waveguide at the exit ( z = L )of the structure. The lower scheme shows a cross section of a rectangular waveguideof area ∆ x × ∆ y , and refractive index n g . outermost waveguides of a bent array of linear waveguides. The 2D arrangement underconsideration is depicted in Fig. 1, and pertains to N identical rectangular waveguides,of cross section σ = ∆ x × ∆ y and length L . Light of specific wavelength λ , is injected inthe first waveguide at z = 0, and couples to the neighbouring waveguides as it propagatesalong z . Our task is to achieve faithful (ideally perfect) transfer of the signal from thefirst to the N th waveguide at the exit of the structure i.e., at z = L , for a given valueof the permanent bend angle θ ≥ θ we engineer the distances between neighbouringwaveguides aiming ideally at a configuration of the coupling constants of the form G m,l = (cid:40) π L (cid:112) ( N − M ) M , | m − l | = 10 , otherwise , (1)for m, l ∈ [1 , N ] and M ≡ min { m, l } . This is a centrosymmetric configuration withrespect to the central waveguide(s) of the PL [19, 20, 16, 17, 18]. In practise, the ransfer of optical signals around bends in 2D linear photonic networks Figure 2. (Color online) Eigenmodes of two rectangular waveguides and the geometryof their overlap. The two identical waveguides (see rectangles) are shown togetherwith the electric field E ( j ) x ( x, y ). Parameters: ∆ x = 6 µ m, ∆ y = 2 µ m, n s = 1 . δn = 10 − , λ = 800nm, r = 30 µ m, θ = π/ couplings beyond nearest neighbors are never zero, but they must be negligible relativeto the nearest-neighbor couplings. For an unbent PL ( θ = 0), such engineering has beendemonstrated recently by two different groups in the framework of PLs in the bulk offused silica [13]. It requires knowledge mainly on the spatial dependence of the coupling,as well as on the refractive index profile for each waveguide and the form of the excitedeigenmodes at a given operation wavelength λ . In the following we will investigatewhether the coupling configuration (1) can be implemented in the same manner, inbent PLs (i.e., for θ > In various physical realizations the waveguides have an asymmetric elliptic profile, whichis reflected in the observed modal field distribution at a given wavelength and fieldpolarization, as well as in the dependence of the coupling on the waveguide separation[3, 5, 15, 26]. Although, as discussed in Sec. 4, the design of symmetric waveguidesis possible with current technology, throughout the present work we present resultsfor waveguides with asymmetric shape, since in this case the presence of bends turnsout to be more pronounced than for symmetric waveguides. Analogous simulations forsymmetric waveguides have also been performed, and we will refer to related findingswherever necessary, pointing out the main differences from the case of asymmetricwaveguides.We consider rectangular waveguides with ∆ x ≥ ∆ y , corresponding to the major andminor axes of the elliptic profiles typically observed experimentally. For the analysis ofthe waveguides we follow the Marcatili’s approach, which is widely used in photonics and ransfer of optical signals around bends in 2D linear photonic networks n g and n s denote the refractive indices for thecore and the cladding (substrate) respectively. Typically, their difference is very small( ∼ − ) and thus the refractive index distribution for the j th rectangular waveguide iswell approximated by n j ( x, y ) ≈ n j ; x ( x ) + n j ; y ( y ) + O ( n g − n s ) (2)with n j ; x ( x ) = (cid:40) n g / , | x | ≤ ∆ x/ n s − n g / , | x | > ∆ x/ n j ; y ( y ) = (cid:40) n g / , | y | ≤ ∆ y/ n s − n g / , | y | > ∆ y/ , (4)where we have assumed that the waveguide is centred at ( x, y ) = (0 , n g = n s √ − δn (5)where δn is the modification of the refractive index. The refractive index of theshaded (corner) areas in the lower scheme of Fig. 1, is approximated by (cid:112) n s − n g ≈ n s (1 − δn − O ( δn )). Clearly, for δn ∼ , one has (cid:112) n s − n g ≈ n s .The dimensions of the waveguides and the associated refractive-index modulationsadopted throughout this work are within the range of values one typically findsin experiments pertaining to laser-written waveguides in glasses. There are somequantitative differences, however, since the adopted rectangular profile is not expected tocapture precisely all the features of the modes observed in experiments (e.g., precise formof modes, penetration depth, etc). One of the key features in most of the experiments isthat only one (the lowest) eigenmode is excited at the operation wavelength. Hence, thewaveguide parameters we consider here are such that the lowest mode of the rectangularwaveguide is excited (i.e., E ( x )1 , or E ( y )1 , ), which means that the electric field exhibits onlyone peak along both x- and y-axis directions. The main field components for modes E [ x ( y )]1 , in the j th waveguide are E ( j ) x ( y ) and H ( j ) y ( x ) , with the electric field polarized alongthe x ( y ) direction, respectively.The focus of the present work is on the effects of bends and to this end we ignorevarious types of possible imperfections so that all the estimated losses in our simulationscan be attributed only to bending effects. Furthermore it is sufficient to consider themode E ( x )1 , in the following analysis, since the calculations for the E ( y )1 , mode are thesame. We will return to this point in the concluding remarks of the present work.The components of the electric ( E ( j ) ) and magnetic ( H ( j ) ) fields that prevail in the j th waveguide are [27, 28, 29] E ( j ) x = E ( j ) x ( x, y ) exp[ i ( ωt + β j z )] (6) H ( j ) y = H ( j ) y ( x, y ) exp[ i ( ωt + β j z )] , (7) ransfer of optical signals around bends in 2D linear photonic networks H ( j ) x = 0. From Maxwell’s equations one has: ∂ H ( j ) y ∂x + ∂ H ( j ) y ∂y + [ k n j ( x, y ) − β j ] H ( j ) y = 0 , (8) E ( j ) x ≈ ωµ β j H ( j ) y , (9)where ω and k are the frequency and the wavenumber of the input light, whereas β j isthe wavenumber along the propagation direction of the waveguide (z-axis). Equation(8) can be solved numerically, or analytically for the model under consideration (seeappendix). In Fig. 2 we show the electric field E x ( x, y ) as obtained in our simulations,for a particular set of parameters. Clearly, the asymmetry of the waveguides is alsoreflected in the eigenmodes, and as will be seen later on, it also affects the couplingbetween adjacent waveguides. It has to be emphasized here, however, that the depictedmodal profile is for the sake of illustration only, and pertains to the particular parametersgiven in the caption. The quantitative aspects of the modal profile may change e.g., bychanging the dimensions of the waveguide, the wavelength of the light, etc.Finally, the normalization we have adopted throughout this work implies that thepower carried by the eigenmode of the j th waveguide along the propagation directionis [20, 28, 29, 30] P j = 12 (cid:90) (cid:90) (cid:60) (cid:2) ( E ( j ) × H ( j ) (cid:63) ) · ˆ z (cid:3) dxdy = 1 (Watt) . (10) As shown in Fig. 2, when two waveguides are brought close together, their opticalmodes overlap. For sufficiently small overlaps, the electromagnetic field distribution foreither of the neighbouring waveguide does not differ substantially from the one for anisolated waveguide, and the propagation characteristics of the coupled waveguides canbe analyzed by means of the coupled-mode theory, the details of which can be foundalmost in every textbook on photonics and optoelectronics (e.g., see [20, 28, 29, 30]).For the sake of completeness, here we sketch the main steps of the approach.The total electric field in a configuration of N evanescently coupled identicalwaveguides is well approximated by the superposition E ( x, y, z ) = N (cid:88) j =1 A j ( z ) E ( j ) x ( x, y ) exp[ i ( ωt − β j z )] ˆ x , (11)where E ( j ) x are determined by Eq. (9). Substituting this expression into the wave-equation (cid:20) ∇ + ω c n ( x, y ) (cid:21) E ( x, y, z ) = 0 , (12)where n ( x, y ) the refractive-index distribution of the entire structure of the coupledwaveguides, and following standard well-known steps one obtains a closed set of ransfer of optical signals around bends in 2D linear photonic networks A j ( z ) d A dz = J · A (13)where A ≡ ( A , A , . . . , A N ) T and J is an N × N matrix with all the diagonal elementsequal to zero, and the off-diagonal elements given by J m,l = J m,l exp[ i ( β m − β l ) z ] , (14)with the coupling between the m th and the l th waveguide given by J m,l = ωε (cid:90) (cid:90) dxdy E ( m ) ∗ x ( x, y )∆ n l ( x, y ) E ( l ) x ( x, y ) , (15)where ∆ n l ≡ n ( x, y ) − n l ( x, y ), with n l ( x, y ) the refractive-index profile for the l thwaveguide alone.Before we focus on the behaviour of the couplings for the particular setup underconsideration, it is worth mentioning that in addition to the coupling between adjacentwaveguides, in the framework of coupled-mode theory one also obtains corrections tothe propagation wavevector β l of the l th waveguide, due to the presence of the adjacentwaveguides, as well as the so-called “butt-coupling” coefficients. Such terms are typicallymuch smaller than J m,l and thus their effects are neglected here [28]. Equation (15), shows that the coupling between two waveguides originates from theoverlap between the corresponding eigenmodes. Hence, one expects the asymmetry ofthe eigenmodes to be also reflected in the couplings. Consider the directional coupler ofFig. 2. The corresponding coupling between the two waveguides is plotted in Fig. 3(a)as a function of the separation r (measured for the centers of the waveguides), at a fixedangle θ . For any value of θ the dependence of the coupling on r is well approximatedby an exponential of the form J m,l ( r, θ ) = µ ( θ ) exp[ − ξ ( θ ) r ] , (16)and it is anisotropic since the details of the exponential drop with increasing r dependon θ . Indeed, as shown in Fig. 3(b), for fixed r the coupling varies by a factor of 3, aswe change the angle θ from 0 to π/
2. The relative position of the curves for different θ in Fig. 3(a), depends strongly on the specific parameters under consideration (e.g.,wavelength of light, refractive-index modulation, height and width of the rectangularshape). In the case of symmetric waveguides, for example, the spread of the curvesfor various θ is considerably smaller [31]. Varying the refractive-index profile of thewaveguides, one can change the precise form of the modal distribution [see Fig. 2], andthus the values of the parameters µ, ξ in Eq. (16) for a given θ .In any case, the crucial point is that the present model captures the dependence ofthe coupling on both r and θ (see Eq. 16), which is also what one has in practise. The ransfer of optical signals around bends in 2D linear photonic networks C oup li ng ( c m - ) θ = 0 θ = π/6 θ = π/4 θ = π/3 θ = π/2 π/4 π/2 Angle0246 C oup li ng ( c m - ) r = 8 µ m r = 10 µ m r = 12 µ m (a)(b) Figure 3. (Color online) Geometry of the coupling between two rectangularwaveguides (see Fig. 2). (a) The coupling as a function of the separation r at differentangles θ . (b) The coupling as a function of the angle θ at different separations r . Otherparameters as in Fig. 2. precise mathematical form of this dependence on θ is not crucial for what follows andactually, it is never used explicitly. The key point is that such a type of anisotropy allowsfor the engineering of couplings at different coupling angles θ , by adjusting the distance r (e.g., see [26]). For the implementation of a centrosymmetric coupling configuration[such as the one in Eq. (1)] in a bent chain with θ >
0, the separations r j,j +1 betweenwaveguides with indices below the index of the corner site C , have to be different fromthe separations for waveguides with indices above C . In other words, a centrosymmetriccoupling configuration does not imply centrosymmetric distribution of the separations r j,j +1 in the case of an anisotropic spatial dependence of the coupling.
3. Light transport through bent photonic lattices
Throughout our simulations we worked on a three-dimensional grid in a sufficiently largebox so that reflections from the boundaries are absent. The total electric field at a givenpoint ( x, y, z ) was estimated according to Eq. (11), through the solution of Eqs. (8), (9)and (13). Working with different parameters, we have reached similar conclusions andfor the sake of concreteness, in this section we present results pertaining to a bent PLconsisting of N = 9 nearly identical waveguides of length L = 10 cm. The waveguidesare written in the bulk of a glass with n s = 1 .
444 and the associated refractive indexchange is δn = 10 − , whereas their cross-section is σ = 6 × µ m . For a given θ ≥ r j,j ± are engineered so that the nearest-neighbour couplings J j,j ± are well approximated by Eq. (1) for L = 10 cm and N = 9.Light of wavelength λ = 800 nm, and sufficiently low power so that nonlinear effectsare negligible, is injected in the first waveguide at z = 0, and couples to the otherwaveguides as it propagates along z . ransfer of optical signals around bends in 2D linear photonic networks H a L i j H b L i j H c L i j H d L i j Figure 4. (Color online) The relative strength of the couplings between differentidentical waveguides J i,j [max { J i,j } ] − , for a bent chain of N = 9 waveguides with r j,j ± such that J j,j ± is given by Eq. (1), and bend angles: (a) θ = 0, (b) θ = 16 π/ θ = 19 π/
32, (d) θ = 20 π/
32. Other parameters as in Fig. 2.
From a theoretical point of view, the propagation of light in the PL is determined byEqs. (13), and in particular by the coupling matrix J . When the distance betweennon-neighbouring waveguides is sufficiently large, couplings beyond nearest neighboursare negligible and thus the coupling matrix J has basically tridiagonal form. Recentexperiments on the realization of the coupling configuration (1) in unbent PLs ( θ = 0)have shown that the assumptions underlying the coupled-mode theory, as well as theassumption of nearest-neighbour couplings of the form (1) can be fulfilled experimentallyfor a moderate number of waveguides, and thus faithful transfer between the two endsof the unbent chain has been observed [13]. For an unbent chain of a given length, these ransfer of optical signals around bends in 2D linear photonic networks Figure 5. (Color online) Intensity distribution at the output ( L = 10 cm) of a bentchain with N = 9 identical rectangular waveguides (also shown on each plot). Bendangles: (a) θ = 16 π/
32, (b) θ = 18 π/
32, (c) θ = 19 π/
32, (d) θ = 20 π/
32. The intensityis measured in units Wcm − and the total power in the sample at any z is normalizedto the input power. Other parameters as in Fig. 2. Note the different scale in the y axis. assumptions are expected to break down only for a large number of waveguides (e.g.,see related discussion in the work of Bellec et al . [13]).By contrast, the realization of the coupling configuration (1) for bent chains with θ > C ) come closer as one increases θ (seeFig. 1), couplings beyond nearest neighbours are expected to increase. There shouldexist, therefore, a critical angle θ c above which the couplings beyond nearest neighboursbecome comparable to the nearest-neighbour ones, and thus their effects cannot beneglected. Our first task here is to estimate the critical angle for the particular set-upunder consideration. Subsequently, for angles θ > θ c our task is to investigate whetherit is possible to improve the transfer of the signal between the two outermost waveguidesof the PL, without additional extensive engineering [32].For the reasons explained above, in our formalism the matrix J includes thecouplings for all possible pairs of waveguides. The relative strengths of the couplings inthe matrix J (with β m = β l ∀ m, l ) are plotted in Fig. 4 for a chain of N = 9 waveguides,and for increasing bend angles, with the corner site C = 5. For the unbent chain ( θ = 0) ransfer of optical signals around bends in 2D linear photonic networks θ < π/
2, the couplings beyond nearest neighbours are at least one orderof magnitude smaller than the nearest-neighbour couplings, and thus they can be safelyignored. For θ = π/ C − C +1 has been doubled, whereas couplings between higher-order neighbours emerge. For θ = 19 π/
32 and θ = 20 π/
32 the couplings between the waveguides C − C + 1 arecomparable to the nearest-neighbour couplings, and the couplings between higher-orderneighbours also increase. These observations suggest that strong deviations from thecoupling configuration of the unbent chain are expected for bend angles above θ c = 90 ◦ ,whereas the deviations for angles up to 90 ◦ are not expected to be so pronounced [33].In the case of symmetric waveguides (i.e., for ∆ x = ∆ y = 6 µ m) our simulationsshow that the couplings beyond nearest neighbours are less sensitive to bends. Forexample, for θ = 20 π/
32 we find that the coupling J C − ,C +1 is at least five timessmaller than J C ± ,C , whereas couplings between higher-order neighbours can be safelyignored. This is because the confinement of the lowest eigenmodes in all directions turnsout to be stronger than in the case of the asymmetric waveguides with ∆ x = 6 µ m and∆ y = 2 µ m. As mentioned before, one expects ideally complete transfer of the light between thetwo outermost waveguides, when the matrix elements J m,l are well approximated byEq. (1). Our simulations show that this happens for θ ≤ π/ θ = π/
2. As weincrease θ further, couplings beyond nearest neighbours distort the transfer betweenthe two outermost waveguides. For the sake of illustration, in Figs. 5(b-d) we presentthe intensity distributions at the output of a bent chain for θ = 18 π/ , π/ , and20 π/
32, respectively. One can see that the output intensity is not restricted only to the9th waveguide, but there are also non-negligible fractions in other waveguides, includingthe 1st and the 7th one. A clearer quantitative picture can be obtained by looking athow the input power is distributed among the waveguides at the output. As shown inFig. 6(a), for θ = 19 π/ , and 20 π/
32 we find that 85% and 60% of the input powerrespectively, has been transferred to the target waveguide at the output, whereas asignificant fraction of the input light can be found at the exit of all the other waveguides,but mainly of the first one. Hence, as shown in Fig. 6(c) (see open circles), the relativelosses are about 15% and 40%, respectively.The present scheme that relies on engineered couplings in linear PLs seems to bea bit more robust against bending losses, than schemes that rely on solitonic signalsand nonlinear PLs. For instance, the authors of Ref. [24] have estimated that for ϕ = π − θ = 90 ◦ solitons suffer about 5% bending losses, whereas for ϕ = 70 ◦ the lossesexceed 38%. As shown in Fig. 6(c), in the present scheme bending losses do not exceed5% for angles θ (cid:46) ◦ , whereas for θ ≈ ◦ (corresponding to ϕ ≈ ◦ ) bending lossesare about 40%. For the reasons discussed above, the losses for the same set-up can be ransfer of optical signals around bends in 2D linear photonic networks θ ≈ ◦ , which suggests that linear PLs with symmetric waveguidesconsiderably outperform nonlinear PLs with solitonic informations carriers.We turn now to discuss a method for minimizing the bending losses, by introducinga defect at the corner site. In the framework of solitonic signals it has been shown that bending losses can bereduced by introducing a defect at the corner site of the bend [24, 25]. More recently,it was shown that the same method also works in the framework of linear Hamiltonianswith engineered couplings [22]. The generic model of Ref. [22], however, did not allowfor an in depth investigation of various quantitative aspects of the defect. The presentmodel allows us to gain further insight into the method, and shed light on crucialquestions pertaining to the size and the refractive-index change of the defect.We assume that all of the waveguides, but the corner one, are identical, and letus denote by β the corresponding propagation wavenumber. The wavenumber for thecorner site will be denoted by β C and let ∆ ≡ β C − β be the detuning of the cornersite relative to the other waveguides of the chain. In our simulations, for a given anglethis detuning has been optimised, while keeping all other parameters of the PL fixed,so that the transfer from the first to the last waveguide is maximized (i.e. losses areminimized).Figures 7(a,b) show the intensity distribution at the output of a bent PL fortwo different angles after optimization of the wavenumber for the corner waveguide.Comparing these two figures to the corresponding figures without optimization [see Fig.5(c,d)], one sees a clear improvement of the transfer from the first to the last waveguide.Still, there are fractions of the input signal that are not found at the exit of the targetwaveguide, but certainly they are considerably smaller than in Figs. 5(c,d). Indeed, asshown in Fig. 6(b), for bend angles up to 20 π/
32, more than about 90% of the inputpower has been transferred to the target waveguide at the output, whereas the relativelosses are at least twice smaller than without optimization and they hardly exceed 10%[see filled squares in Fig. 6(c)]. We see therefore that by introducing a corner defectone can minimize bending losses for fixed θ in the present linear array with engineeredcouplings, but the same approach seems to work more efficiently for nonlinear arraysand solitons. According to Ref. [24] bending losses after the inclusion of defect arerestricted to less than approximately 1% for angles θ = 90 ◦ and 110 ◦ , whereas in ourcase we find losses approximately 1.4% for θ = 90 ◦ and 5% for θ = 110 ◦ . However,when symmetric waveguides are considered in our scheme, the corresponding bendinglosses after optimization do not exceed 1% for angles up to θ = 113 ◦ ; a performancethat is comparable to (if not better than) the performance of the bent nonlinear PLsconsidered in Ref. [24].The quantitative aspects of the defect are intimately connected to the details of ransfer of optical signals around bends in 2D linear photonic networks Π (cid:144)
32 18 Π (cid:144)
32 19 Π (cid:144)
32 20 Π (cid:144) H a L j F r ac ti ono fI npu t P o w e r H b L j F r ac ti ono fI npu t P o w e r ç No Optimization à Optimization ç ç ç ç ç ç ç à à à à à à H c L Π
32 14 Π
32 16 Π
32 18 Π
32 20 Π R e l a ti v e L o ss e s Figure 6. (Color online) Losses in the transfer of signals along a bent chain of N = 9 identical rectangular waveguides. (a) Fraction of the input power that hasbeen transferred to the j th waveguide at the output of the sample (i.e., at L = 10cm ), for different bend angles. (b) As in (a) with optimized corner site. (c) Therelative losses at the exit of the sample, and for different bend angles with and withoutoptimized corner. Other parameters as in Fig. 5. ransfer of optical signals around bends in 2D linear photonic networks Figure 7. (Color online) As in Fig. 5 after optimization of the corner site C = 5, for(a) θ = 19 π/
32 and (b) θ = 20 π/ the set-up under consideration. The optimal values of the detunings that minimizebending losses at various θ in our model are given in table 1. Clearly, in all cases ∆ isnegative and increases (in absolute value) as we increase the bend angle. Typically, thewavenumber of a waveguide can be controlled by changing the size of the waveguide, orby adjusting the associated refractive-index modification. An estimation of the changesrequired to achieve some of the detunings discussed here are shown in the last columnsof table 1. A close inspection of these values shows that in order to achieve the estimatedoptimal detunings the accuracy required in the writing of the waveguides is at least 10 − in refractive-index changes and at least 10 − cm in the cross-section of the waveguides.Interestingly enough, the present estimations for the required refractive-index changesare comparable to related estimations for solitonic schemes [24, 25].Before closing this section it is worth discussing briefly the reason for the successof the corner defect in minimizing bending losses. In the case of solitonic signals andnonlinear PLs it has been conjectured that the detuning of the corner site relative tothe rest of the lattice virtually removes the corner site from the lattice [25]. Thus,the remaining (identical) waveguides effectively constitute a smoother link, which isreflected in the improvement of the transport. This explanation does not apply to oursetup as we work in the linear regime and the coupling configuration under consideration ransfer of optical signals around bends in 2D linear photonic networks Table 1.
Optimal detunings of the corner site that minimize bending lossesat different angles in a linear chain with N = 9 asymmetric waveguides of length L = 10 cm. The detuning is defined as ∆ = β C − β where for the parameters underconsideration, β (cid:39) . × cm − . The strength of the detuning relative to thecoupling G C,C ± (see Eq. (1)) is given in the third column. The fourth and the fifthcolumn give the control required on the refractive-index modulation and the size ofthe corner site relative to δn = 10 − and σ = 12 µ m respectively, in order to achievethe optimal detunings of the second column.Angle ( × π/
32) ∆ (cm − ) | ∆ | /G C,C ± Relative change of refractive index Relative change of size
18 -0.1955 0.275 0.46 0.2519 -0.4317 0.608 1.60 0.8520 -1.0733 1.512 5.15 2.50 Defined as 100 × ( δn − δn C ) /δn . Defined as 100 × ( σ − σ C ) /σ . [see Eq. (1)] is rather sensitive to the details of the lattice (i.e., number of waveguides,length, etc). First of all, as shown in table 1, the optimal detunings are smaller or atmost comparable to the couplings of the corner site to its neighbours. Furthermore, asshown in Fig. 8 the corner site may acquire at least 10% of the input power as the lightpropagates from z = 0 to L , which is not a negligible amount. These two observationstogether suggest that there is no solid ground for omission of the corner site relative tothe others, and thus the derivation of an effective chain cannot be justified.The coupling configuration under consideration is a member of a large class ofstate-transfer Hamiltonians, whose operation relies on the commensurate eigenenergies[19, 20]. As explained in Ref. [22], for such a type of Hamiltonians a corner defectminimizes the bending losses by rearranging the spectrum that has been distorted bythe bend. To confirm this once more, in Fig. 9 we plot the separation between successiveeigenvalues of the matrix J in Eq. (13), for different angles before and after optimizationof the corner site. For the unbent chain ( θ = 0) the eigenvalues are commensurate (i.e.,equidistant), and that is why the coupling configuration of Eq. (1), ensures ideallyperfect transfer of light between the two outermost waveguides. As we bend the lattice,however, the commensurate nature of the eigenvalues is distorted, and the distortionis getting larger for sharper bends [see Fig. 9(a)]. As depicted in Fig. 9(b), theinclusion of a defect at the corner site of the bend tends to restore the relative positionof the eigenvalues (i.e., the deviations from the case of the unbent chain are gettingsmaller). The remaining deviations at the borders are not of great importance since thecontribution of eigenvectors with small/large indices to the evolution of the system isnegligible [22]. ransfer of optical signals around bends in 2D linear photonic networks Length (cm) F r ac ti on o f I npu t P o w e r θ = 0θ = 18π/32θ = 19π/32θ = 20π/32 Figure 8. (Color online) Fraction of the input power that is found at the cornerwaveguide at different lengths. The detuning of the corner waveguide has beenoptimized to minimize bending losses. Other parameters as in Fig. 5. ø æ Π (cid:144) à Π (cid:144) ò Π (cid:144) ø ø ø ø ø ø ø ø æ æ æ æ æ æ æ æà à à à à à à àò ò ò ò ò ò ò ò H a L j D i ff e r e n ce H c m - L ø ø ø ø ø ø ø ø æ æ æ æ æ æ æ æà à à à à à à àò ò ò ò ò ò ò ò H b L j D i ff e r e n ce H c m - L Figure 9. (Color online) Spectrum of matrix J for various bend angles. (a) Differencebetween successive eigenvalues before optimization of the corner site. (b) As in (a),after optimization of the corner site. Other parameters as in Fig. 2 and table 1. ransfer of optical signals around bends in 2D linear photonic networks
4. Concluding remarks
We have analyzed the effects of bends on the transport of photonic signals between thetwo outermost waveguides of a 2D photonic lattice with engineered nearest-neighbourcouplings that operates in the linear regime. In contrast to previous studies, in thepresent scheme the suppression of dispersion effects and the faithful transport of lightdoes not rely on Kerr nonlinearities, but rather on the engineering of judicious couplingsbetween nearest neighbours. It has been shown that our scheme works reliably for bendsat least up to θ c = 90 ◦ (with respect to the unbent chain). Sharper bends (with θ > θ c )have been shown to distort the transport, with the distortion being more pronouncedfor asymmetric waveguides. In this case, one has to find ways for minimizing bendingeffects, and in this direction it has been shown that the inclusion of a defect at thecorner site can be a rather useful approach. Although our findings suggest that thepresent scheme outperforms its nonlinear (soliton-based) counterparts, further analysisis required for definite conclusions in this respect.Laser-written buried waveguides in glasses typically have elliptic shape, due to thebeam focus, and they exhibit “form birefringence”, as a result of which fields withdifferent polarizations experience different effective refractive indices [15]. Moreover,due to the formation of self-aligned nanogratings in the material during the irradiationprocess, one may also have “material birefringence” [34, 35].Birefringence is a detrimental effect for quantum circuits that are intended forefficient guide and manipulation of qubits that are encoded in the polarization ofphotons. In general, the shape of the waveguides can be controlled efficiently by shapingthe writing beam using standard techniques [36], and thus “form birefringence” can be,in principle, eliminated. The elimination of “material birefringence” is also feasible if onechooses the right material/substrate, and the right combination of writing parameters(i.e., wavelength, duration and energy of the laser pulses, repetition rate, objectivenumerical aperture, translation speed, etc). In this way, the overall birefringence canbe reduced by at least an order of magnitude facilitating thus the design of photonicprimitives (e.g., directional couplers), that operate efficiently for polarization-encodedqubits [34]. One has to keep in mind, however, that “material birefringence” can beuseful in the design of crucial polarization-sensitive components of quantum circuits,such as integrated wave plates [37], polarization routers [35], etc. In this context, forinstance, one can have waveguides that allow for transmission of light with specificpolarization, whereas light with the orthogonal polarization is totally reflected.For the sake of simplicity, the present analysis of bending losses has been restrictedto one of the lowest polarization modes (the calculations and conclusions for the otherpolarization are the same given that birefringence is not included in our formalism).Strictly speaking, the present results are valid for optical networks and communicationschemes in which qubits are not encoded in the polarization of the signal. In the case ofpolarization-based qubits, the present results and conclusions are expected to be validonly for photonic lattices that are polarization-independent i.e., non-birefringent. As ransfer of optical signals around bends in 2D linear photonic networks θ c beyond which bending effects cannot be ignored, depends on the details of particularset-up under consideration. The present analysis can be performed for any couplingconfiguration, and it is pertinent to on-going experiments on photonic lattices, inthe framework of which a point-to-point link that relies on the coupling configurationdiscussed here has been demonstrated [13]. When combined with the ideas of [5], thepresent findings suggest that linear waveguide arrays with engineered nearest-neighbourcouplings can be used as building blocks of fundamental optical primitives that performmore demanding communication tasks such as routing, splitting, blocking, as well aslogical functions.
5. Acknowledgements
The author acknowledges with pleasure discussions at various times with M. Bellec onexperimental issues pertaining to laser-written photonic lattices. He is also grateful toP. Lambropoulos, T. Brougham and M. Bellec, for helpful comments on this paper.
Appendix A. Calculation of eigenmodes of a rectangular waveguide
Here we discuss briefly an analytic solution of Eq. (8) using the separation of variables.This is a well known standard procedure and the details are discussed in various standardtextbooks and papers [27, 28, 29]. Setting H ( j ) x ( x, y ) = X ( x ) Y ( y ) , (A.1)Eq. (8) splits into two independent parts (one for each direction). The solutions (up tonormalization factors) are the following X ( x ) = (cid:40) cos( k x x ) , | x | ≤ ∆ x cos( k x ∆ x/ e − γ x ( x − ∆ x/ | x | > ∆ x . (A.2)and Y ( y ) = (cid:40) cos( k y y ) , | y | ≤ ∆ y cos( k y ∆ y/ e − γ y ( y − ∆ y/ , | y | > ∆ y (A.3)with γ x + k x = k ( n g − n s ) (A.4) γ y + k y = k ( n g − n s ) (A.5) β = k n g − k x − k y . (A.6)Boundary conditions on the electric field imply also that k x ∆ x = arctan (cid:18) γ x k x (cid:19) (A.7) ransfer of optical signals around bends in 2D linear photonic networks k y ∆ y = arctan (cid:18) γ y k y (cid:19) (A.8)where we have used the fact that n g ≈ n s [38]. Equations (A.4) - (A.8) form a closedset and determine all the parameters entering Eq. (A.1). References [1] D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature , 817 (2003); F. Lederer, G. I.Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. , 1(2008).[2] S. Longhi, Laser & Photonics Reviews , 243 (2009).[3] A. Szameit and S. Nolte, J. Phys. B: At. Mol. Opt. Phys. , 163001 (2010).[4] I. L. Garanovicha, S. Longhi, A. A. Sukhorukova, Y. S. Kivshar, Phys. Rep. , 1 (2012).[5] R. Keil, M. Heinrich, F. Dreisow, T. Pertsch, A. T¨unnermann, S. Nolte, D. N. Christodoulidesand A. Szameit, Sci. Rep. , 1 (2011); D. N. Christodoulides and E. D. Eugenieva, Phys. Rev.Lett. , 233901 (2001); E. D. Eugenieva, N. K. Efremides, and D. N. Christoulides, Opt. Lett. , 1978 (2001).[6] J. L. O’Brien, A. Furusawa, and J. Vuckovi´c, Nature Photonics
687 (2009).[7] A. Politi et al. , Science , 646 (2008); J. C. F. Matthews, A. Politi, A. Stefanov, J. L. O’Brien,Nature Photon. , 346 (2009). A. Politi, J. C. F. Matthews, and J. L. O’Brien, Science ,1221 (2009).[8] A. Peruzzo et al. , Science , 1500 (2010); L. Sansoni, et al. , Phys. Rev. Lett. , 010502 (2012);K. Poulios et al. , 143604 (2014).[9] J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X. M. Jin, M. Barbieri, A.Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. Smith,I. A. Walmsley, Science , 798 (2013); M. Tillmann, B. Dakic, R. Heilmann, S. Nolte, A.Szameit, and P. Walther, Nat. Photon. , 540 (2013); N. Spagnolo, C. Vitelli, M. Bentivegna,D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, and G. Milani, R. Ramponi, P. Mataloni, R.Osellame, E. F. Galvao, and F. Sciarrino, Nat. Photon. , 615 (2014).[10] S. Krapick, H. Herrmann, V. Quiring, B. Brecht, H. Suche and C. Silberhorn, New J. Phys. ,033010 (2013); R. Kruse, F. Katzschmann, A. Christ, A. Schreiber, S. Wilhelm, K. Laiho, Aur´elG´abris, C. S. Hamilton, I. Jex, and C. Silberhorn, New J. Phys. , 083046 (2013).[11] S. Longhi et al. , Opt. Lett. , 473 (2008); A. Szameit et al. , Appl. Phys. Lett. , 181109 (2008).[12] S. Longhi, Phys. Rev. B , 041106 (R) (2010).[13] M. Bellec, G. M. Nikolopoulos, and S. Tzortzakis, Opt. Lett. , 4504 (2012); A. Perez-Leija, R.Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. M. Rodriguez-Lara, A. Szameit, and D.N. Christo- doulides, Phys. Rev. A , 012309 (2013).[14] Y. N. Joglekar, C. Thompson, and G. Vemuri, Phys. Rev. A , 063817 (2011).[15] S. Weimann, A. Kay, R. Keil, S. Nolte, and A. Szameit, Opt. Lett. , 123 (2014).[16] M. Christandl et al. , Phys.Rev. Lett. , 187902 (2004); G. M. Nikolopoulos, D. Petrosyan and P.Lambropoulos, Europhys. Lett. , 297 (2004); J. Phys.: Cond. Matter , 4991 (2004).[17] R. Gordon, Opt. Lett. , 2752 (2004).[18] G. M. Nikolopoulos, Phys. Rev. Lett. , 200502 (2008).[19] See the reviews S. Bose, Contemp. Phys. , 13 (2007); D. Burgarth, Eur. Phys. J. Special Topics , 147 (2007); A. Kay, Int. J. Quant. Inform. , 641 (2010).[20] G. M. Nikolopoulos and I. Jex, Quantum State Transfer and Network Engineering (Springer-Verlag,Berlin Heidelberg, 2014).[21] A. Kay, Phys. Rev. A , 032306 (2006); V. Kostak, G. M. Nikolopoulos and I. Jex, Phys. Rev.A , 042319 (2007); T. Brougham, G. M. Nikolopoulos and I. Jex, Phys. Rev. A , 052325(2009). ransfer of optical signals around bends in 2D linear photonic networks [22] G. M. Nikolopoulos, A. Hoskovec, and I. Jex, Phys. Rev. A , 62319 (2012).[23] Discrete solitons are self-trapped states that exist as a result of balance between linear couplingeffects and material Kerr nonlinearity. Their existence is only possible for input powers above acertain power threshold.[24] D. N. Christodoulides and E. D. Eugenieva, Opt. Lett. , 1876 (2001).[25] M. Heinrich, R. Keil, F. Dreisow, A. T¨unnermann, A. Szameit, and S. Nolte, Appl. Phys. B ,469 (2011).[26] A. Szameit, F. Dreisow, T. Pertsch, S. Nolte and A. T¨unnermann, Opt. Express , 1579 (2006).[27] E. A. J. Marcatili, Bell Syst. Tech. J. Fundamentals of Optical Waveguides (Academic Press, 2006).[29] K. Kawano and T. Kitoh,
Solving Maxwell’s Equations and the Schr¨odinger equation (John Wiley& Sons, 2001).[30] A. Yariv and P. Yue,
Photonics: Optical Electronics in Modern Communications (OxfordUniversity Press, New York, 2006).[31] In the present model, there will be some dependence on θ because of the approximation of therefractive index at the shaded (corner) areas in Fig. 1. For symmetric waveguides, however, thisdependence is very weak.[32] One can always engineer new coupling configurations that go beyond nearest neighbours, andensure faithful transfer of signals between the two ends of the chain. The engineering of networksthat perform certain tasks has been discussed extensively in the literature, and goes beyond thescope of the present work [20, 21][33] In order to ensure min { r l,m } > r C,C ± (or equivalently max { J l,m } < J C,C ± ), we have restrictedour simulations to θ > ◦ .[34] L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, Phys.Rev. Lett. , 200503 (2010).[35] G. Cheng, K. Mishchik, C. Mauclair, E. Audouard, and R. Stoian, Opt. Express , 9515 (2009).[36] R, Osellame, S. Taccheo, M. Marangoni, R. Ramponi, P. Laporta, D. Polli, S. De Silvestri, and G.Cerullo, J. Opt. Soc. Am. B , 1595 (2003).[37] L. A. Fernandes, J. R. Grenier, P. R. Herman, J. S. Aitchison, and P. V. S. Marques, Opt. Express , 1824 (2011).[38] For the sake of simplicity, the above equations are given for a waveguide that is centred at ( x, yx, y