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Linear and interface defects in composite linear photonic lattice
Optics Communications 394 (2017) 6–13
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Linear and interface defects in composite linear photonic lattice a,⁎
MARK
Marija Stojanović Krasić , Ana Mančić , Slavica Kuzmanović , Snežana Đorić Veljković , Milutin Stepiće b
c
d
a
Faculty of Technology, University of Niš, Bulevar Oslobođenja 124, 16000 Leskovac, Serbia Faculty of Natural Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia Faculty of Natural Sciences and Mathematics, University of Priština, Lole Ribara 29, 38220 Kosovska Mitrovica, Serbia d Faculty of Civil Engineering and Architecture, University of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia e Vinča Institute of Nuclear Sciences, University of Belgrade, P.O.B. 522, Belgrade, Serbia b c
A R T I C L E I N F O
A BS T RAC T
Keywords: Photonic lattices Light localization Defect modes Interface modes
We numerically analysed various localized modes formed by light beam propagation through one-dimensional composite lattices consisting of two structurally different linear lattices and a linear defect (LD) in one of them. The localized modes are found in the area between the interface and the LD, near the interface and around the LD. It has been confirmed that a LD narrower than the other waveguides (WGs) in the array is better potential barrier and captures the light better than a LD that is wider than the other WGs in the array. Also, it has been shown that a LD narrower than the other WGs in the lattice captures the light more efficiently than any saturable nonlinear defect (ND) of the same width as other elements of the lattice. On the other hand, it is obtained that the influence of a LD wider than the other WGs in the array on light propagation can be mimicked by insertion of an adequate ND whose width coincides with that of the other WGs. Depending on the defect size, its position and input beam parameters, controllable beam trapping, reflection and refraction are observed.
1. Introduction Photonic crystals are periodic structures which, due to their ability to control light flow, have many practical applications such as optical switching and filtering. Their periodicity enables the existence of Bloch oscillations [1], discrete diffraction [2] and different localized structures which are not possible in homogeneous media [3]. The presence of nonlinearity in periodic systems may counteract diffraction and enable the formation of discrete solitons [4–7] or gap solitons, [8–10]. Localized structures known as lattice solitons can be found in other systems, such as DNA chains [11], electrical lattices [12], Bose – Einstein condensates [13], antiferromagnets [14], etc. The geometry of photonic systems strongly affects formation of different types of localized modes which can be found within the lattices. Examples include modulated lattices [15,16] and flat-band lattices [17–19]. One-dimensional (1D) photonic crystals or photonic lattices (PLs) enable control of the light propagation by changing the system parameters, such as the refractive index and the period of the lattice [20–22]. As a result of the system periodicity, it is possible to define zonal structure with permitted and forbidden zones (gaps) for the light propagation [23–25]. Lattice defects, which inevitably occur during either the fabrication process or usage, increase the complexity of the
⁎
zonal structure by creating defect levels in the gaps. Defects break the translation symmetry of the periodic systems and enable the occurrence of different types of stable localized defect modes [26–29] that could not exist otherwise. By increasing the number of allowed modes that can be spread through the system, defects lead to the rise of various photonic circuits and enable additional opportunities for the control of light propagation [30,31]. Defects may, for example, suppress waveguiding [32], stop light [33], shape solitons [34], trap or deflect the incident beam [35,36] and thus be used for all-optical switching and routing [37,38]. It is known that the interface of two PLs acts as a structural defect that may significantly influence the propagation of light beams in the vicinity of lattices’ compound and enables the existence of various strongly localized interface defect modes [39–45]. Recently, we have in detail investigated beam propagation through the composite PL consisting of two structurally different linear 1D lattices [46]. Linear lattices with one or more defects have also been studied in the context of condensed matter physics [47] and biology [48]. Additionally, a composite system having embedded a ND (i. e. one nonlinear WG) into one of its linear sublattices has also been examined [49]. Here, we numerically explore beam propagation in the vicinity of the interface of two dissimilar linear lattices, one of which is loaded
Corresponding author. E-mail address: [email protected] (M. Stojanović Krasić).
http://dx.doi.org/10.1016/j.optcom.2017.02.021 Received 21 September 2016; Received in revised form 26 January 2017; Accepted 7 February 2017 Available online 06 March 2017 0030-4018/ © 2017 Published by Elsevier B.V.
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review of the results obtained in the numerical experiment and their discussion. In Section 4, conclusions related to the results presented in this paper are summarized.
2. Model equations We analyse the composite PL consisting of two structurally different linear WG arrays, i. e. two PLs A and B, with LD in one of them; the interface between two PLs represents the geometric (structural) defect (GD), (see Fig. 1). The light propagation in this composite PL is modelled by the paraxial time-independent Helmholtz equation [49]: Fig. 1. Schematic representation of the system. Blue color denotes the GD position, while red color shows the LD position.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
i
∂E 1 ∂ 2E + +k 0 n 0 n (x ) E = 0, ∂z 2k 0 n 0 ∂x 2
(1)
where z is the propagation coordinate, E(x, z) is the component of the light electric field in the z-direction, k0 =2π/λ is the wave number, n0 is the refractive index of the substrate, and λ is the wavelength of light. The lattice is prepared along the transverse x direction and there are 49 WGs in each lattice. The properties of the lattice system are modelled by functional dependence of the refractive index on system parameters in the form:
with a LD. It is studied which parameters of the lattice inhomogeneities (as the width and the position of the LD) and which parameters of the input beam (such as its width, initial position and phase tilt) can enable the existence of stable localized modes. Moreover, the system parameters for which different dynamical regimes, including transmission, reflection and trapping of light occur have been investigated. We have also been searching for new dynamical regimes that can appear due to the presence of the LD. The paper has the following outline. The mathematical model of the wave propagation through the system and all the details regarding the system characteristics are formulated in Section 2. Section 3 contains a
k −1
n (x ) =
mA
∑ Gj (wgA,
sA, x )+Gk (wgk , sA, x )+
j =1 mA+ mB
+
∑ j = mA+1
∑
Gj (wgA, sA,
x)
j = k +1
Gj (wgB, sB , x ), (2)
Fig. 2. 2D plot of the averaged beam intensity profiles. Incident beam enters at the 1st WG in the lattice A, LD located at the 1st WG in the lattice B has width: (a) 2 µm; (b) 6 µm. Incident beam hits the 1st WG in the lattice B, LD placed at the 1st WG in the lattice B has width: (c) 2 µm; (d) 6 µm. (e) Incident beam enters at the 1st WG in the lattice B, 6 µm wide LD is embedded at the 1st WG in the lattice A. Vertical blue dashed lines delineate position of the interface area, whilst vertical red line indicates LD. Green arrow denotes the incident site. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
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Fig. 3. 2D plot of the averaged beam intensity profiles, incident beam enters at the 5th WG in the lattice B, LD located at the 9th WG in the lattice B has width: (a) 2 µm; (b) 6 µm. Incident beam hits the 5th WG in the lattice A, LD located at the 9th WG in the lattice A has width: (c) 2 µm; (d) 6 µm. Vertical blue dashed lines denote the interface area, while vertical red line shows LD. Green arrow shows the incident site. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
3. Results and discussion
where k is the position of the LD which is arbitrary placed within the lattice A, mA (mB) is the number of WGs in lattice A (B). Parameters wgA and wgB mark the width within the first (A) and the second (B) lattice, respectively. The parameter wgk represents the width of the LD, while parameters sA and sB represents the spacing between WGs in the lattices A and B, respectively. The distance between lattices A and B, denoted by parameter w, represents the width of the GD. Functions Gj (wgA, sA, x) and Gj (wgB, sB, x) represent Gaussians corresponding to the WGs of the lattice A and B respectively, whereas function Gk (wgk, sA, x) corresponds to the LD. Lattice potential depth is 0.011 [50]. We used dimensionless variables ξ= k0x, η= k0z and Eq. (1) obtained the following dimensionless form:
i
∂E 1 ∂ 2E + +n 0 n (ξ ) E = 0. ∂η 2n 0 ∂ξ 2
In the system with two dissimilar PLs with a LD in one of them and the interface between them, qualitatively different dynamical behaviour is observed in comparison to the cases studied previously with either a single interface [46], one nonlinear defect embedded in the uniform lattice [51] or both types of lattice inhomogeneities in the composite system [49]. Here, all dynamical regimes that can be obtained by varying the system parameters are shown giving an insight in the influence of the interface and the LDs of different widths on the beam propagation. The widths of the WGs within the lattices A and B are 5 µm and 4 µm, respectively, whereas the distance between neighbouring WGs is 4 µm in both lattices. The GD is chosen to be 3.3 µm wide [42]. The analysis started by examining the dynamics of the light beam launched in the GD area (Fig. 2). In the first case, a 2 µm wide LD is placed at the first WG in the lattice B and the light is launched into the first WG in the lattice A. Reflection of energy from the interface is observed, as shown in Fig. 2a. The LD acts as a strong potential barrier. If the width of the LD is increased to 6 µm, the energy stays reflected but not as well as in the former case (see Fig. 2b). This difference can be explained by the fact that a wider LD corresponds to a lower potential barrier. Also, it can be seen that a narrow LD (2 µm wide) is more efficient in trapping the light beam launched into it compared to the wider LD, see Fig. 2c and d. When the light is launched into the 6 µm
(3)
The split-step Fourier method [49] is used for the numerical simulation of Gaussian light beam propagation along the lattice. The light beam is TE polarized with the wavelength λ=514.5 nm. The changeable parameters of the system are the width of the LD and its position, the input beam position with respect to the defects and its transverse tilt (i. e. inclination to the propagating coordinate - z axis). The invariant parameter is the input light beam intensity. In the following, we use either a narrow Gaussian light beam with the FWHM=4 µm or a broad Gaussian beam with FWHM=48 µm.
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Fig. 4. 2D plot of the averaged beam intensity profiles. Incident beam enters at the 1st WG in the lattice B, LD located at the 3rd WG in the lattice B has width: (a) 2 µm; (b) 6 µm. Incident beam is launched at the 2nd WG in the lattice B, LD located at the 3rd WG in the lattice B has width: (c) 2 µm; (d) 6 µm. (e) 2 µm wide LD is inserted at the fifth WG in the lattice B, incident beam enters at the 3rd WG in the lattice B. (f) 6 µm wide LD is inserted at the fourth WG in the lattice B, incident beam hits the 2nd WG in the lattice B. Vertical dashed blue lines denote the interface area, whilst vertical red line indicates LD. Green arrow marks the incident site. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
the light beam is launched in the middle, an asymmetric mode starts to appear (see Fig. 3b). This asymmetry occurs because of the difference of the potentials on both sides of the light beam propagation area (interface area from one side and wider LD from the other side). The asymmetry also occurs when the LD is located within the lattice A, whether it is 2 µm or 6 µm wide (see Fig. 3c and d). The potential of the left side of the interface cannot be balanced with any LD's potential and it causes asymmetrical beam propagation regardless of the width of the LD. Other regimes of the light propagation that can be observed within the area between the defects are different types of breathing modes. These modes can be formed when the distance between the defects is less than five WGs. If the LD is located at the 3rd WG in the lattice B, two-component breathing modes start to appear (see Fig. 4a-d). Depending on the width of the LD and the input beam position, there are some shape variations of these modes. When the light is launched at the interface, there is a dominant breathing mode component on the input position, while in the WG between the defects there is a less pronounced component, (see Fig. 4a). For a wider LD (6 µm), the dominant channel vanishes and a two-component breathing mode appear (see Fig. 4b). A similar situation to the one shown in Fig. 4a occurs when the light is launched in the WG between the defects for the 2 µm wide LD (Fig. 4c): the dominant channel is the one where light is launched while the less pronounced component is located at the interface. For the wider LD (6 µm), the effect of capturing at the input beam position is not as strong (see Fig. 4d) as it was with the 2 µm wide LD. The reason for this behaviour is that the 6 µm wide LD is not as efficient potential
wide LD there is a small part of energy localized at neighbouring WGs around the LD. Also the trapped mode located on the LD is wider than it was in the former case, see Fig. 2d. A wider LD enables more space for light propagation, which causes a decrease in the height of the average beam intensity profile. In the Fig. 2e, where the 6 µm wide LD is placed at the first WG in the lattice A and the light is launched into the first WG in the lattice B, the trapping of the light at the input beam position is shown. In all the figures below, it can be seen that the potential of the lattice A is lower than the potential of the lattice B, which is due to the wider WGs in the lattice A. In the presence of the LDs, these potentials are changed additionally. Due to their different structure, the changes that are produced in the presence of a 6 µm wide LD are not the same in each lattice. For the reason of its own potential structure, which is narrower, higher and different from the rest of the lattice, the 2 µm wide LD does not give any qualitative difference regarding the light propagation whether it is located at the first WG in the lattice A or at the first WG in the lattice B. For a distance between the interface and LD larger than one WG, two types of propagation regimes are observed. If the light is launched in the area between the defects, it is being reflected from their inner sides, (see Figs. 3a and b). If the 2 µm wide LD is located within the lattice B and the light is launched precisely in the middle between the defects, it is possible to form an oscillating mode with a symmetrical shape. The shape of the oscillating mode is symmetrical with respect to the input position. It can be concluded that the potentials of the right side of the interface and the 2 µm wide LD are approximately the same and produce symmetrical effects. When the LD is 6 µm wide and when
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Fig. 5. 2D plot of the averaged beam intensity profiles. Incident beam enters at the 1st WG in the lattice A, LD located at the 3rd WG in the lattice A has width: (a) 2 µm; (b) 6 µm. (c) Incident beam enters at the second WG in the lattice A, 2 µm wide LD is located at the 3rd WG in the lattice A. (d) Incident beam hits the third WG in the lattice A, 2 µm wide LD is located at the fifth WG in the lattice A. Vertical dashed blue lines delineate the interface area, whilst vertical red line points to LD. Green arrow denotes the incident site. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
between the defects, if the beam is launched inside this area) and reflects from the inner side of the interface, (Fig. 6a). This LD is capable of strong light capturing (see Fig. 6b). It has to be mentioned that capturing of light is highly related to the profile of the input beam, ratio of the width of the waveguides and the width of the separation between the waveguides. If the interface is absent, the input beam, launched at the adjacent WG to the LD, will continue to travel through the lattice because there is no barrier to reflect from, (see Fig. 6c). If the light is launched further away from the defect, a part of the energy diffracted toward the LD gets reflected after reaching it, as shown in Fig. 6d. Different dynamical behaviour is obtained in the presence of a wider (i.e. 6 µm) LD. The WG adjacent to the LD, either on the inner or the external side of it (the inner side is faced to the interface and the external side is faced to the end of the lattice B), strongly captures the light beam launched in it, as it is presented in Fig. 7a and b. The wider defect produces potential that leads to the noticeable increase of the potentials at the surrounding WGs. This tendency for the light trapping at the WG just next to the LD position is also evident when the light is launched at the second neighbours of the LD (see Fig. 7c). At the position of the wider LD, it is also possible to obtain the entrapment effect as it is shown in Fig. 7d. We wanted to see what would happen if we replace a LD with a saturable ND of any nonlinearity (at the same position like in Figs. 6 and 7). Results for the nonlinear case are obtained by inserting nonlinear term in the form nnl(x)=−0.5n02rEpv|E|2(Id+|E|2)−1, (r is the electrooptic coefficient, Epv is photovoltaic field, Id=G/s is dark
barrier as a narrow one is, so a small amount of energy can pass through this wider LD. In Fig. 4e and f, when the distance between the defects is three and two WGs, respectively, one can notice threecomponent breathing modes with a dominant central channel for both LDs. The asymmetrical potential of the interface defect, caused by the structural difference of sublattices, leads to the dissimilar effects on the left and the right side of the interface. In Fig. 5, the LD is located within the lattice A. In comparison to the Fig. 4a and b, novel dynamical behaviour is found (see Fig. 5a and b). When the light is launched into the first WG in the lattice A, the energy is being exchanged between the interface and the second WG of the lattice A. A complex two component breathing mode is formed. One component of the breathing mode is a slowly breathing component, located just next to the interface, within the lattice A. Another component of the breathing mode, located at the interface, consists of two rapidly oscillating subcomponents. If the light beam is launched at the second WG within the lattice A, a typical twocomponent breathing mode is created (see Fig. 5c). In comparison to the three-component breathing mode given in Fig. 4e, where the LD was embedded into the lattice B, if the LD is inserted into the lattice A, oscillating modes are created (see Fig. 5d). For larger distances between the defects, oscillating modes are the only structures that exist within the area between the defects. The effect of a 2 µm wide LD on the beam propagation is presented in Fig. 6. Numerical simulations show that this defect acts as a strong barrier if the light is launched into the waveguide just next to it. Being reflected, the light travels through the lattice (i. e. through the area 10
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Fig. 6. 2D plot of the averaged beam intensity profiles. 2 µm wide LD is located at the 10th WG in the lattice B, incident beam is launched in the lattice B in the: (a) 9th WG; (b) 10th WG; (c) 11th WG; (d) 12th WG. Vertical dashed blue lines denote the interface area, while vertical red line indicates LD. Green arrow marks the incident site. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
Also, the influence of the transverse kick on broad Gaussian beam propagation has been considered because the effects are more apparent than in the case of a narrow input beam. Due to the presence of the kick, the energy is being transferred in its direction thus leading to a decay of localized structures and breathing modes. By varying the value of the transverse kick, it is possible to obtain either reflection and weak transmission of the beam or only transmission. For example, if α=π/18 it is possible to observe very strong reflection of light from the 2 µm wide LD located at the first WG either in lattice A or in lattice B (Fig. 9a). Reflection vanishes for higher values of the transverse kick (α=π/9) and transmission starts to appear, as shown in Fig. 9b. The narrow LD is placed at the first WG in the lattice A, where its potential is combined with the potential of the interface area, giving the necessary condition for dominant reflection, see Fig. 9a. If the 2 µm wide LD is placed further away from the interface area, reflection is a little bit weaker and part of the energy is transmitted, see Fig. 9c. For the 6 µm wide LD, regardless of its position and the value of the transverse kick, it is not possible to obtain pure reflection. Either mixed regimes of transmission and reflection or only transmission are effects that occur, depending on the value of the transverse kick.
irradiance, G is dark generation rate and s is photoionization crosssection) in the Eq. (1). For the sake of simplicity, nonlinearity strength in the form Γ=γ/γ0, with γ0=0.0001≈0.5n03Epvr is introduced [47]. In Fig. 8, ND with nonlinearity strength Γ=1.3 (the value of nonlinearity that produces maximum trapping result, see Fig. 8a) and ND with nonlinearity Γ=10 and low trapping efficiency (see Fig. 8b) are shown. By comparing Fig. 6b and Fig. 8a it can be concluded that the narrower LD captures the light more efficiently than the saturable ND of any nonlinearity. With the increasing width of the LD, indications of diffraction start to appear until the width exceeds the width of the WGs in the lattices. Furthermore, increase of the width of the LD, again leads to the light trapping, Fig. 7d. By comparing the Fig. 7d and Fig. 8a it can be concluded that the wider LD produces a very similar trapping effect like ND does at optimum nonlinearity. In Fig. 8c, the output amplitude profiles of the modes localized at the defects (2 and 6 µm wide LDs and 4 µm wide ND with Γ=1.3) are shown. It can be seen that the trapping effect obtained for wider LDs resembles, in terms of the height of the amplitude profile and its width, the one obtained when the LD was replaced by a 4 µm wide ND, with nonlinearity Γ=1.3 [48]. A broad Gaussian beam excitation has been analysed as well. In this study, a broad Gaussian beam with FWHM=48 µm has been used. The propagation in the vicinity of the interface of two dissimilar PLs with ND placed in one of them [49] has been examined earlier, where partial trapping at the interface and the breathing cavity modes emerged. In our system, the conclusions obtained for the broad Gaussian head-on beam, are qualitatively the same with the one obtained in Ref. [49].
4. Conclusions We have numerically explored how LDs of different widths affect light beam propagation through a PL system consisting of two dissimilar uniform linear lattices with a LD placed in one of them. It is shown that, for the chosen set of system parameters, the narrower
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Fig. 7. 2D plot of the averaged beam intensity profiles. 6 µm wide LD is located at the 10th WG in the lattice B, incident beam enters in the lattice B at the: (a) 9th WG; (b) 11th WG; (c) 13th WG; (d) 10th WG. Vertical dashed blue lines mark the interface area, while vertical red line depicts LD. Green arrow shows the incident site. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
Fig. 8. 2D plot of the averaged beam intensity profiles. Incident beam enters at the 10th WG in the lattice B, 4 µm wide ND is located at the 10th WG in the lattice B with nonlinearity: (a) Γ=1.3; (b) Γ=10. Green arrow denotes the incident site. Vertical dashed blue lines mark the interface area, while vertical red line depicts ND. (c) Amplitude profiles of output light beams captured at the 2 µm wide LD (green curve), at the 6 µm wide LD (red curve) and at the 4 µm wide ND with nonlinearity Γ=1.3. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
other) in the area between them. Here, different kinds of oscillating, breathing and complex breathing regimes are observed. It is manageable to control reflection and transmission in the system by varying the input beam position and its transverse kick. We have observed stable interface and defect modes, whose reflection and transmission allow new possibilities for controlled light switching, filtering and routing.
LD is more efficient in light entrapment and it is also a better potential barrier than the wider defect. It is also shown that the LD narrower than other WGs in the array has the ability to capture light more efficiently than a saturable ND of any nonlinearity. We have examined the influence of both the interface and LD on the light beam propagation (when these lattice inhomogeneities are close to each
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Fig. 9. 2D plot of the averaged beam intensity profiles. 48 µm wide incident beam enters at the 15th WG in the lattice A. 2 µm wide LD is located at the 1st WG in the lattice A, transverse kick is: (a) (α=π/18); (b) (α=π/9). (c) 2 µm wide LD is placed at the 4th WG in the lattice A, transverse kick is: α=π/18. Vertical dashed blue lines mark the interface area, while vertical red line depicts LD. Vertical green arrow shows the incident site. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
Acknowledgment
[24] [25] [26] [27] [28]
The authors acknowledge support from the Serbian Ministry of Education, Science and Technological Development (Project No. III 45 010).
[29] [30]
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Linear and interface defects in composite linear photonic lattice a,⁎
MARK
Marija Stojanović Krasić , Ana Mančić , Slavica Kuzmanović , Snežana Đorić Veljković , Milutin Stepiće b
c
d
a
Faculty of Technology, University of Niš, Bulevar Oslobođenja 124, 16000 Leskovac, Serbia Faculty of Natural Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia Faculty of Natural Sciences and Mathematics, University of Priština, Lole Ribara 29, 38220 Kosovska Mitrovica, Serbia d Faculty of Civil Engineering and Architecture, University of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia e Vinča Institute of Nuclear Sciences, University of Belgrade, P.O.B. 522, Belgrade, Serbia b c
A R T I C L E I N F O
A BS T RAC T
Keywords: Photonic lattices Light localization Defect modes Interface modes
We numerically analysed various localized modes formed by light beam propagation through one-dimensional composite lattices consisting of two structurally different linear lattices and a linear defect (LD) in one of them. The localized modes are found in the area between the interface and the LD, near the interface and around the LD. It has been confirmed that a LD narrower than the other waveguides (WGs) in the array is better potential barrier and captures the light better than a LD that is wider than the other WGs in the array. Also, it has been shown that a LD narrower than the other WGs in the lattice captures the light more efficiently than any saturable nonlinear defect (ND) of the same width as other elements of the lattice. On the other hand, it is obtained that the influence of a LD wider than the other WGs in the array on light propagation can be mimicked by insertion of an adequate ND whose width coincides with that of the other WGs. Depending on the defect size, its position and input beam parameters, controllable beam trapping, reflection and refraction are observed.
1. Introduction Photonic crystals are periodic structures which, due to their ability to control light flow, have many practical applications such as optical switching and filtering. Their periodicity enables the existence of Bloch oscillations [1], discrete diffraction [2] and different localized structures which are not possible in homogeneous media [3]. The presence of nonlinearity in periodic systems may counteract diffraction and enable the formation of discrete solitons [4–7] or gap solitons, [8–10]. Localized structures known as lattice solitons can be found in other systems, such as DNA chains [11], electrical lattices [12], Bose – Einstein condensates [13], antiferromagnets [14], etc. The geometry of photonic systems strongly affects formation of different types of localized modes which can be found within the lattices. Examples include modulated lattices [15,16] and flat-band lattices [17–19]. One-dimensional (1D) photonic crystals or photonic lattices (PLs) enable control of the light propagation by changing the system parameters, such as the refractive index and the period of the lattice [20–22]. As a result of the system periodicity, it is possible to define zonal structure with permitted and forbidden zones (gaps) for the light propagation [23–25]. Lattice defects, which inevitably occur during either the fabrication process or usage, increase the complexity of the
⁎
zonal structure by creating defect levels in the gaps. Defects break the translation symmetry of the periodic systems and enable the occurrence of different types of stable localized defect modes [26–29] that could not exist otherwise. By increasing the number of allowed modes that can be spread through the system, defects lead to the rise of various photonic circuits and enable additional opportunities for the control of light propagation [30,31]. Defects may, for example, suppress waveguiding [32], stop light [33], shape solitons [34], trap or deflect the incident beam [35,36] and thus be used for all-optical switching and routing [37,38]. It is known that the interface of two PLs acts as a structural defect that may significantly influence the propagation of light beams in the vicinity of lattices’ compound and enables the existence of various strongly localized interface defect modes [39–45]. Recently, we have in detail investigated beam propagation through the composite PL consisting of two structurally different linear 1D lattices [46]. Linear lattices with one or more defects have also been studied in the context of condensed matter physics [47] and biology [48]. Additionally, a composite system having embedded a ND (i. e. one nonlinear WG) into one of its linear sublattices has also been examined [49]. Here, we numerically explore beam propagation in the vicinity of the interface of two dissimilar linear lattices, one of which is loaded
Corresponding author. E-mail address: [email protected] (M. Stojanović Krasić).
http://dx.doi.org/10.1016/j.optcom.2017.02.021 Received 21 September 2016; Received in revised form 26 January 2017; Accepted 7 February 2017 Available online 06 March 2017 0030-4018/ © 2017 Published by Elsevier B.V.
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review of the results obtained in the numerical experiment and their discussion. In Section 4, conclusions related to the results presented in this paper are summarized.
2. Model equations We analyse the composite PL consisting of two structurally different linear WG arrays, i. e. two PLs A and B, with LD in one of them; the interface between two PLs represents the geometric (structural) defect (GD), (see Fig. 1). The light propagation in this composite PL is modelled by the paraxial time-independent Helmholtz equation [49]: Fig. 1. Schematic representation of the system. Blue color denotes the GD position, while red color shows the LD position.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
i
∂E 1 ∂ 2E + +k 0 n 0 n (x ) E = 0, ∂z 2k 0 n 0 ∂x 2
(1)
where z is the propagation coordinate, E(x, z) is the component of the light electric field in the z-direction, k0 =2π/λ is the wave number, n0 is the refractive index of the substrate, and λ is the wavelength of light. The lattice is prepared along the transverse x direction and there are 49 WGs in each lattice. The properties of the lattice system are modelled by functional dependence of the refractive index on system parameters in the form:
with a LD. It is studied which parameters of the lattice inhomogeneities (as the width and the position of the LD) and which parameters of the input beam (such as its width, initial position and phase tilt) can enable the existence of stable localized modes. Moreover, the system parameters for which different dynamical regimes, including transmission, reflection and trapping of light occur have been investigated. We have also been searching for new dynamical regimes that can appear due to the presence of the LD. The paper has the following outline. The mathematical model of the wave propagation through the system and all the details regarding the system characteristics are formulated in Section 2. Section 3 contains a
k −1
n (x ) =
mA
∑ Gj (wgA,
sA, x )+Gk (wgk , sA, x )+
j =1 mA+ mB
+
∑ j = mA+1
∑
Gj (wgA, sA,
x)
j = k +1
Gj (wgB, sB , x ), (2)
Fig. 2. 2D plot of the averaged beam intensity profiles. Incident beam enters at the 1st WG in the lattice A, LD located at the 1st WG in the lattice B has width: (a) 2 µm; (b) 6 µm. Incident beam hits the 1st WG in the lattice B, LD placed at the 1st WG in the lattice B has width: (c) 2 µm; (d) 6 µm. (e) Incident beam enters at the 1st WG in the lattice B, 6 µm wide LD is embedded at the 1st WG in the lattice A. Vertical blue dashed lines delineate position of the interface area, whilst vertical red line indicates LD. Green arrow denotes the incident site. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
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Fig. 3. 2D plot of the averaged beam intensity profiles, incident beam enters at the 5th WG in the lattice B, LD located at the 9th WG in the lattice B has width: (a) 2 µm; (b) 6 µm. Incident beam hits the 5th WG in the lattice A, LD located at the 9th WG in the lattice A has width: (c) 2 µm; (d) 6 µm. Vertical blue dashed lines denote the interface area, while vertical red line shows LD. Green arrow shows the incident site. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
3. Results and discussion
where k is the position of the LD which is arbitrary placed within the lattice A, mA (mB) is the number of WGs in lattice A (B). Parameters wgA and wgB mark the width within the first (A) and the second (B) lattice, respectively. The parameter wgk represents the width of the LD, while parameters sA and sB represents the spacing between WGs in the lattices A and B, respectively. The distance between lattices A and B, denoted by parameter w, represents the width of the GD. Functions Gj (wgA, sA, x) and Gj (wgB, sB, x) represent Gaussians corresponding to the WGs of the lattice A and B respectively, whereas function Gk (wgk, sA, x) corresponds to the LD. Lattice potential depth is 0.011 [50]. We used dimensionless variables ξ= k0x, η= k0z and Eq. (1) obtained the following dimensionless form:
i
∂E 1 ∂ 2E + +n 0 n (ξ ) E = 0. ∂η 2n 0 ∂ξ 2
In the system with two dissimilar PLs with a LD in one of them and the interface between them, qualitatively different dynamical behaviour is observed in comparison to the cases studied previously with either a single interface [46], one nonlinear defect embedded in the uniform lattice [51] or both types of lattice inhomogeneities in the composite system [49]. Here, all dynamical regimes that can be obtained by varying the system parameters are shown giving an insight in the influence of the interface and the LDs of different widths on the beam propagation. The widths of the WGs within the lattices A and B are 5 µm and 4 µm, respectively, whereas the distance between neighbouring WGs is 4 µm in both lattices. The GD is chosen to be 3.3 µm wide [42]. The analysis started by examining the dynamics of the light beam launched in the GD area (Fig. 2). In the first case, a 2 µm wide LD is placed at the first WG in the lattice B and the light is launched into the first WG in the lattice A. Reflection of energy from the interface is observed, as shown in Fig. 2a. The LD acts as a strong potential barrier. If the width of the LD is increased to 6 µm, the energy stays reflected but not as well as in the former case (see Fig. 2b). This difference can be explained by the fact that a wider LD corresponds to a lower potential barrier. Also, it can be seen that a narrow LD (2 µm wide) is more efficient in trapping the light beam launched into it compared to the wider LD, see Fig. 2c and d. When the light is launched into the 6 µm
(3)
The split-step Fourier method [49] is used for the numerical simulation of Gaussian light beam propagation along the lattice. The light beam is TE polarized with the wavelength λ=514.5 nm. The changeable parameters of the system are the width of the LD and its position, the input beam position with respect to the defects and its transverse tilt (i. e. inclination to the propagating coordinate - z axis). The invariant parameter is the input light beam intensity. In the following, we use either a narrow Gaussian light beam with the FWHM=4 µm or a broad Gaussian beam with FWHM=48 µm.
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Fig. 4. 2D plot of the averaged beam intensity profiles. Incident beam enters at the 1st WG in the lattice B, LD located at the 3rd WG in the lattice B has width: (a) 2 µm; (b) 6 µm. Incident beam is launched at the 2nd WG in the lattice B, LD located at the 3rd WG in the lattice B has width: (c) 2 µm; (d) 6 µm. (e) 2 µm wide LD is inserted at the fifth WG in the lattice B, incident beam enters at the 3rd WG in the lattice B. (f) 6 µm wide LD is inserted at the fourth WG in the lattice B, incident beam hits the 2nd WG in the lattice B. Vertical dashed blue lines denote the interface area, whilst vertical red line indicates LD. Green arrow marks the incident site. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
the light beam is launched in the middle, an asymmetric mode starts to appear (see Fig. 3b). This asymmetry occurs because of the difference of the potentials on both sides of the light beam propagation area (interface area from one side and wider LD from the other side). The asymmetry also occurs when the LD is located within the lattice A, whether it is 2 µm or 6 µm wide (see Fig. 3c and d). The potential of the left side of the interface cannot be balanced with any LD's potential and it causes asymmetrical beam propagation regardless of the width of the LD. Other regimes of the light propagation that can be observed within the area between the defects are different types of breathing modes. These modes can be formed when the distance between the defects is less than five WGs. If the LD is located at the 3rd WG in the lattice B, two-component breathing modes start to appear (see Fig. 4a-d). Depending on the width of the LD and the input beam position, there are some shape variations of these modes. When the light is launched at the interface, there is a dominant breathing mode component on the input position, while in the WG between the defects there is a less pronounced component, (see Fig. 4a). For a wider LD (6 µm), the dominant channel vanishes and a two-component breathing mode appear (see Fig. 4b). A similar situation to the one shown in Fig. 4a occurs when the light is launched in the WG between the defects for the 2 µm wide LD (Fig. 4c): the dominant channel is the one where light is launched while the less pronounced component is located at the interface. For the wider LD (6 µm), the effect of capturing at the input beam position is not as strong (see Fig. 4d) as it was with the 2 µm wide LD. The reason for this behaviour is that the 6 µm wide LD is not as efficient potential
wide LD there is a small part of energy localized at neighbouring WGs around the LD. Also the trapped mode located on the LD is wider than it was in the former case, see Fig. 2d. A wider LD enables more space for light propagation, which causes a decrease in the height of the average beam intensity profile. In the Fig. 2e, where the 6 µm wide LD is placed at the first WG in the lattice A and the light is launched into the first WG in the lattice B, the trapping of the light at the input beam position is shown. In all the figures below, it can be seen that the potential of the lattice A is lower than the potential of the lattice B, which is due to the wider WGs in the lattice A. In the presence of the LDs, these potentials are changed additionally. Due to their different structure, the changes that are produced in the presence of a 6 µm wide LD are not the same in each lattice. For the reason of its own potential structure, which is narrower, higher and different from the rest of the lattice, the 2 µm wide LD does not give any qualitative difference regarding the light propagation whether it is located at the first WG in the lattice A or at the first WG in the lattice B. For a distance between the interface and LD larger than one WG, two types of propagation regimes are observed. If the light is launched in the area between the defects, it is being reflected from their inner sides, (see Figs. 3a and b). If the 2 µm wide LD is located within the lattice B and the light is launched precisely in the middle between the defects, it is possible to form an oscillating mode with a symmetrical shape. The shape of the oscillating mode is symmetrical with respect to the input position. It can be concluded that the potentials of the right side of the interface and the 2 µm wide LD are approximately the same and produce symmetrical effects. When the LD is 6 µm wide and when
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Fig. 5. 2D plot of the averaged beam intensity profiles. Incident beam enters at the 1st WG in the lattice A, LD located at the 3rd WG in the lattice A has width: (a) 2 µm; (b) 6 µm. (c) Incident beam enters at the second WG in the lattice A, 2 µm wide LD is located at the 3rd WG in the lattice A. (d) Incident beam hits the third WG in the lattice A, 2 µm wide LD is located at the fifth WG in the lattice A. Vertical dashed blue lines delineate the interface area, whilst vertical red line points to LD. Green arrow denotes the incident site. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
between the defects, if the beam is launched inside this area) and reflects from the inner side of the interface, (Fig. 6a). This LD is capable of strong light capturing (see Fig. 6b). It has to be mentioned that capturing of light is highly related to the profile of the input beam, ratio of the width of the waveguides and the width of the separation between the waveguides. If the interface is absent, the input beam, launched at the adjacent WG to the LD, will continue to travel through the lattice because there is no barrier to reflect from, (see Fig. 6c). If the light is launched further away from the defect, a part of the energy diffracted toward the LD gets reflected after reaching it, as shown in Fig. 6d. Different dynamical behaviour is obtained in the presence of a wider (i.e. 6 µm) LD. The WG adjacent to the LD, either on the inner or the external side of it (the inner side is faced to the interface and the external side is faced to the end of the lattice B), strongly captures the light beam launched in it, as it is presented in Fig. 7a and b. The wider defect produces potential that leads to the noticeable increase of the potentials at the surrounding WGs. This tendency for the light trapping at the WG just next to the LD position is also evident when the light is launched at the second neighbours of the LD (see Fig. 7c). At the position of the wider LD, it is also possible to obtain the entrapment effect as it is shown in Fig. 7d. We wanted to see what would happen if we replace a LD with a saturable ND of any nonlinearity (at the same position like in Figs. 6 and 7). Results for the nonlinear case are obtained by inserting nonlinear term in the form nnl(x)=−0.5n02rEpv|E|2(Id+|E|2)−1, (r is the electrooptic coefficient, Epv is photovoltaic field, Id=G/s is dark
barrier as a narrow one is, so a small amount of energy can pass through this wider LD. In Fig. 4e and f, when the distance between the defects is three and two WGs, respectively, one can notice threecomponent breathing modes with a dominant central channel for both LDs. The asymmetrical potential of the interface defect, caused by the structural difference of sublattices, leads to the dissimilar effects on the left and the right side of the interface. In Fig. 5, the LD is located within the lattice A. In comparison to the Fig. 4a and b, novel dynamical behaviour is found (see Fig. 5a and b). When the light is launched into the first WG in the lattice A, the energy is being exchanged between the interface and the second WG of the lattice A. A complex two component breathing mode is formed. One component of the breathing mode is a slowly breathing component, located just next to the interface, within the lattice A. Another component of the breathing mode, located at the interface, consists of two rapidly oscillating subcomponents. If the light beam is launched at the second WG within the lattice A, a typical twocomponent breathing mode is created (see Fig. 5c). In comparison to the three-component breathing mode given in Fig. 4e, where the LD was embedded into the lattice B, if the LD is inserted into the lattice A, oscillating modes are created (see Fig. 5d). For larger distances between the defects, oscillating modes are the only structures that exist within the area between the defects. The effect of a 2 µm wide LD on the beam propagation is presented in Fig. 6. Numerical simulations show that this defect acts as a strong barrier if the light is launched into the waveguide just next to it. Being reflected, the light travels through the lattice (i. e. through the area 10
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Fig. 6. 2D plot of the averaged beam intensity profiles. 2 µm wide LD is located at the 10th WG in the lattice B, incident beam is launched in the lattice B in the: (a) 9th WG; (b) 10th WG; (c) 11th WG; (d) 12th WG. Vertical dashed blue lines denote the interface area, while vertical red line indicates LD. Green arrow marks the incident site. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
Also, the influence of the transverse kick on broad Gaussian beam propagation has been considered because the effects are more apparent than in the case of a narrow input beam. Due to the presence of the kick, the energy is being transferred in its direction thus leading to a decay of localized structures and breathing modes. By varying the value of the transverse kick, it is possible to obtain either reflection and weak transmission of the beam or only transmission. For example, if α=π/18 it is possible to observe very strong reflection of light from the 2 µm wide LD located at the first WG either in lattice A or in lattice B (Fig. 9a). Reflection vanishes for higher values of the transverse kick (α=π/9) and transmission starts to appear, as shown in Fig. 9b. The narrow LD is placed at the first WG in the lattice A, where its potential is combined with the potential of the interface area, giving the necessary condition for dominant reflection, see Fig. 9a. If the 2 µm wide LD is placed further away from the interface area, reflection is a little bit weaker and part of the energy is transmitted, see Fig. 9c. For the 6 µm wide LD, regardless of its position and the value of the transverse kick, it is not possible to obtain pure reflection. Either mixed regimes of transmission and reflection or only transmission are effects that occur, depending on the value of the transverse kick.
irradiance, G is dark generation rate and s is photoionization crosssection) in the Eq. (1). For the sake of simplicity, nonlinearity strength in the form Γ=γ/γ0, with γ0=0.0001≈0.5n03Epvr is introduced [47]. In Fig. 8, ND with nonlinearity strength Γ=1.3 (the value of nonlinearity that produces maximum trapping result, see Fig. 8a) and ND with nonlinearity Γ=10 and low trapping efficiency (see Fig. 8b) are shown. By comparing Fig. 6b and Fig. 8a it can be concluded that the narrower LD captures the light more efficiently than the saturable ND of any nonlinearity. With the increasing width of the LD, indications of diffraction start to appear until the width exceeds the width of the WGs in the lattices. Furthermore, increase of the width of the LD, again leads to the light trapping, Fig. 7d. By comparing the Fig. 7d and Fig. 8a it can be concluded that the wider LD produces a very similar trapping effect like ND does at optimum nonlinearity. In Fig. 8c, the output amplitude profiles of the modes localized at the defects (2 and 6 µm wide LDs and 4 µm wide ND with Γ=1.3) are shown. It can be seen that the trapping effect obtained for wider LDs resembles, in terms of the height of the amplitude profile and its width, the one obtained when the LD was replaced by a 4 µm wide ND, with nonlinearity Γ=1.3 [48]. A broad Gaussian beam excitation has been analysed as well. In this study, a broad Gaussian beam with FWHM=48 µm has been used. The propagation in the vicinity of the interface of two dissimilar PLs with ND placed in one of them [49] has been examined earlier, where partial trapping at the interface and the breathing cavity modes emerged. In our system, the conclusions obtained for the broad Gaussian head-on beam, are qualitatively the same with the one obtained in Ref. [49].
4. Conclusions We have numerically explored how LDs of different widths affect light beam propagation through a PL system consisting of two dissimilar uniform linear lattices with a LD placed in one of them. It is shown that, for the chosen set of system parameters, the narrower
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Fig. 7. 2D plot of the averaged beam intensity profiles. 6 µm wide LD is located at the 10th WG in the lattice B, incident beam enters in the lattice B at the: (a) 9th WG; (b) 11th WG; (c) 13th WG; (d) 10th WG. Vertical dashed blue lines mark the interface area, while vertical red line depicts LD. Green arrow shows the incident site. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
Fig. 8. 2D plot of the averaged beam intensity profiles. Incident beam enters at the 10th WG in the lattice B, 4 µm wide ND is located at the 10th WG in the lattice B with nonlinearity: (a) Γ=1.3; (b) Γ=10. Green arrow denotes the incident site. Vertical dashed blue lines mark the interface area, while vertical red line depicts ND. (c) Amplitude profiles of output light beams captured at the 2 µm wide LD (green curve), at the 6 µm wide LD (red curve) and at the 4 µm wide ND with nonlinearity Γ=1.3. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
other) in the area between them. Here, different kinds of oscillating, breathing and complex breathing regimes are observed. It is manageable to control reflection and transmission in the system by varying the input beam position and its transverse kick. We have observed stable interface and defect modes, whose reflection and transmission allow new possibilities for controlled light switching, filtering and routing.
LD is more efficient in light entrapment and it is also a better potential barrier than the wider defect. It is also shown that the LD narrower than other WGs in the array has the ability to capture light more efficiently than a saturable ND of any nonlinearity. We have examined the influence of both the interface and LD on the light beam propagation (when these lattice inhomogeneities are close to each
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Fig. 9. 2D plot of the averaged beam intensity profiles. 48 µm wide incident beam enters at the 15th WG in the lattice A. 2 µm wide LD is located at the 1st WG in the lattice A, transverse kick is: (a) (α=π/18); (b) (α=π/9). (c) 2 µm wide LD is placed at the 4th WG in the lattice A, transverse kick is: α=π/18. Vertical dashed blue lines mark the interface area, while vertical red line depicts LD. Vertical green arrow shows the incident site. The potential of the lattice is schematically represented within all plots – black curve.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
Acknowledgment
[24] [25] [26] [27] [28]
The authors acknowledge support from the Serbian Ministry of Education, Science and Technological Development (Project No. III 45 010).
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