1. Introduction and models of nonlinear scattering by local defects

1.1. Introduction

The intriguing optical properties of guided waves in χ ⁽³⁾ materials has been the subject of extensive research in the past decade, from both the theoretical and the experimental points of view [1]. In particular, a great deal of attention has been devoted to nonlinear “photonics” in spatially inhomogeneous waveguides, where the linear refractive index is modulated along directions which are transverse to the light propagation [2]. Considering a planar geometry in the x-z plane (and assuming beam confinement in the third dimension y), x denoting the transverse direction and z being the principal propagation direction, the well known nonlinear Schrodienger equation (NLSE) governing wave propagation in a transversally inhomogeneous Kerr medium reads:

where E(x, z) is the electric field, n(x) is the transverse refractive index profile, k ₀ is the free space wave number and n ₂ is the normalized Kerr coefficient.

Periodic modulations of the refractive index which extend over a wide transverse domain Δx [i.e., spanning over many optical wavelengths $\Delta x\gg \frac{2\pi }{k_{0}n}$ , see Fig. 1(a)] define waveguide arrays. When the coupling between adjacent array sites is intermediate, these structures can support a rich variety of nonlinear excitations exhibiting interesting interplays between nonlinearity, periodicity and discreteness. Examples of such nonlinear excitations include discrete solitons [3, 4], gap solitons [5, 6], surface solitons [7, 8] and Bragg soliton mirrors [9]. In all of the above, nonlinear localizations form within gaps of the corresponding linear system [either in inter-band gaps, in the semi-infinite gap above Band-1 or in the semi-infinite gap of the uniform unmodulated region, see the lower panel of Fig. 1(a)].

Local refractive index modulations, applied in small transverse regions ( $\Delta x\simeq \frac{2\pi }{k_{0}n}$ ) of an otherwise uniform nonlinear planar waveguide [Figs. 1(b),(c)], define scattering centers (defects) for side-coupled excitations and might support normal modes for direct excitations. In the linear-wave regime, scattered radiation modes and confined normal modes are decoupled. Introduction of nonlinearity in the wave equation of the scattering problem can lead to interesting new scenarios [10–23]. In addition to exotic new properties of the NLSE with respect to scattering states [10–19], nonlinearity may induce energy transfer from radiation modes to normal modes during the scattering process of side-coupled excitations, in which this energy can trap [20, 21]. Conversely, nonlinearity may induce energy transfer between different directly excited normal modes and radiation modes in the process of ground-state selection [22, 23].

3-layer Clad-Core-Clad AlGaAs thin films offer both large Kerr coefficients in near infrared femtosecond pulse excitations (with relatively low propagation losses) and flexibility in the incorporation of effective refractive index modulations n(x) (via variations of the top-most Clad layer, as illustrated in the upper panels in Fig. 1). Thus, these materials provide an excellent

Table 1. Physical parameters used in the simulations

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practical model system to study nonlinear guided wave optics in spatially inhomogeneous waveguides. Such media have been used in experimental studies of discrete [4], gap [6], surface [8] and vector-lattice [24] solitons, in nonlinear scattering and trapping experiments [20, 21], in the first demonstration of nonlinear ground state selection [22], as well as in recent studies of phase-related properties of nonlinear excitations using embedded reflective surfaces [25]. Therefore, we naturally adhere to the physical parameters of AlGaAs planar waveguides typically used in experiments [26], which are summarized in Table 1 above. In particular, our studies are restricted to power densities which are known in practice to be below those which cause permanent damage in such samples.

 

Fig. 1. Possible refractive index profiles (top) and their corresponding linear propagation spectra (bottom). An electric field of the form E(x, z) ~ A(x, z)e^iβz is assumed, where β is the propagation constant. (a) Periodic modulations (top) which extend over a wide transverse region Δx define waveguide arrays. These structures are characterized by Band-Gap transmission spectra (bottom). k_x denotes the beam’s transverse Bloch wave number. Localizations induced by the Kerr nonlinearity constitute new bound states, which form inside gaps of the linear system, either in the semi-infinite gap (above Band-1) upon direct excitation of the array (blue), or in a band gap (between bands) upon side excitation of the array (red). (b),(c) Local modulations (top) which are applied at a localized region Δx define scattering centers for side excitations (red), and may have linear normal modes for direct excitations (blue). Such structures are characterized by plane-wave transmission spectra (bottom) which plot the transmission coefficient T as a function of the propagation constant. Shaded green areas denotes gaps, the β values of which correspond to waves which cannot propagate in the structure. Dashed vertical lines denote normal modes of the defect.

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In this paper we present a comprehensive numerical study of the trapping dynamics in nonlinear wave scattering by local single-site [Fig. 1(b)] and multi-site [Fig. 1(c)] guiding defects, of initially transform limited Gaussian beams of the form:

where I(x, z)=|E(x, z)|² is the optical power density, φ ₀(x) is the phase profile associated with an input beam which is tilted towards the defect at an angle θ (φ ₀=k ₀ n_clx sin θ), x_c is the input beam center and w is its width. Equation (1) is solved numerically with initial conditions (2) using the Beam Propagation Method. We demonstrate that diverse nonlinear trapping scenarios are possible, including the stable localization (trapping) in a single normal mode, periodic oscillations between different normal modes within a single defect site, and periodic oscillations between nearby sites of a multi-site defect, with the overall power within the defect remaining trapped. These various types of dynamics may be optically controlled by tuning the input beam’s power alone, with all other parameters remaining constant. Finally, the trapping dynamics are considered in the context of a nonlinear Fabry-Perot etalon as the defect’s refractive index is varied.

The paper is organized as follows. In Sec. 1 we discuss two alternative viewpoints by which nonlinear trapping (NLT) in local guiding optical defects may be considered: the wave optics approach (Sec. 1.2) and the ray optics approach (Sec. 1.3). In Sec. 2 NLT dynamics in a single-site multi-mode defect are studied, and in Sec. 3 scattering and trapping in the presence of multi-site defects are considered. In Sec. 4 the trapping dynamics are discussed in the context of a nonlinear Fabry-Perot etalon as function of a defect’s refractive index gradient. Section 5 concludes with a summary.

1.2. Wave optics approach

In the wave optics approach, linear dynamics of waves scattered by local guiding defects are equivalent to quantum mechanical particle scattering by potential wells. A wave packet of the form (2) has a central transverse wave number k_x and a spatial frequency width which is proportional to w ^-1. In Fig. 2(a) such a wave packet is shown incident from the left (a uniform cladding region of refractive index n_cl) towards a single-site defect (of index n_d). A structure such as this resembles a resonant cavity: in the linear regime, side coupled power which passes through the defect is transient, and leaves it as transmitted and reflected waves after a few cavity oscillations. Therefore, as z→∞ no power is found inside the center, and R+T=1 (R and T being the reflection and transmission coefficients, respectively). All linear scattering problems may be represented in terms of plane-wave transmission spectra T(β), examples of which are shown in the bottom panels of Figs. 1(b),(c) and 2(a)–(c). Note that a multi-site defect [Fig. 1(c)] has a transmission spectrum with more complex internal structure, due to multiple internal reflections which occur at interfaces between different defect sites. With a principal wave number of the incident beam $k=k_0{n}_{\mathrm{cl}}=\sqrt{{\beta }^2+k_x^2}$ , a side-coupled excitation traversing the waveguide with a tilt angle θ has β=k cos θ and k_x=k sin θ. Thus, a larger incidence angle has β values which are deeper inside the scattering regime (β<k ₀ n_cl). Normal modes of the defect (associated with discrete values of β) all exist in the interval k ₀ n_cl<β<k ₀ n_d, as indicated by the dashed vertical lines in the transmission spectra of Figs. 1(b),(c) and 2(a)–(c). Note that higher modes have β values which are lower than the fundamental mode’s β, as they are less confined inside the defect. The right-hand side of the transmission spectrum (β>k ₀ n_d) constitutes the semi-infinite gap. In addition, unlike monochromatic plane waves, wave packets of finite extent have finite-width spatial frequency content, which are represented by the purple vertical bands in the T(β) curves of Figs. 2(a)–(c).

 

Fig. 2. (a)–(c): Wave optics approach. Shaded purple regions indicate spatial frequency content of a wave packet; (a) Top - A side-coupled excitation scattering by a local defect: at z=0 the beam is launched towards the defect with a transverse wave number k_x=ksin θ. Upon its arrival at the interface (z=z_s) the beam is scattered by the local defect. Asymptotically (as z→∞) there remain transmitted (T) and reflected (R) wave packets. Bottom - T(β) spectrum of the single-site, two-mode defect, in which scattering states exist in β<k ₀ n_cl and normal modes in k ₀ n_cl<β<k ₀ n_d. (b) A beam which is coupled to Mode No. 2 has a double-hump intensity profile (top), and is localized around β=β ₂ in the T(β) spectrum (bottom). (c) A beam which is coupled to Mode No. 1 has a single-hump intensity profile (top), and is localized around β=β ₁ in the T(β) spectrum (bottom); (d)–(e): Ray optics approach; (d) In the linear regime, a ray incident from the left at an angle α refracts at an angle β and refracts again at the same angle γ=α. (e) In the nonlinear regime, the larger refractive index at the defect interface induces an increase in the angle β, such that γ>α. If the angle β becomes larger than the critical angle β_c, this ray will trap inside the defect.

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Introduction of a focusing Kerr nonlinearity modifies the above simple picture. For example, nonlinearity enables the creation of localized solitons inside gaps [3–9, 24–25], and also an exchange of energy between directly-excited normal defect modes and radiation modes [20–23]. Returning to our description of the scattering problem, the effect of a moderate nonlinearity can be viewed as follows: By virtue of Eq. (1), in the presence of nonlinearity there is an effective wave number k̃, satisfying k̃ ²⋍k ₀ ²(n_cl ²+2n ₂ n_clI). This increases the values of both k_x and β in the T(β) spectrum [Figs. 2(a)–(c)]. If the excitation features sufficiently small angles of incidence and sufficiently high power densities, it becomes possible to shift some of the β components of a scattering wave packet into normal modes, in which they nonlinearly trap. Higher modes are easier to excite in terms of the required shift in k̃, as their propagation constants are closer to the radiation modes’ edge [β=k ₀ n_cl, see Figs. 2(a)–(c)]. We stress that NLT is a dynamical processs which strongly depends on the beam’s power density at a defect’s interface [z⋍z_s in Fig. 2(a)]. NLT can efficiently occur only for an excitation beam that becomes sufficiently self-focused at the defect interface, thus possessing the necessary strength of nonlinearity and wave number increase.

1.3. Ray optics approach

Another way in which the NLT effect may be described is through simple arguments of ray optics. Consider the simplest case of a single ray propagating in a near-grazing angle α=90° - θ with respect to the surface normal towards a single-site defect [Figs. 2(d),(e)]. For simplicity we assume that in any reflection from a dielectric interface, from either side of the defect, the angle of reflection is equal to the angle of incidence. As the ray encounters the defect’s interface, most of the power is reflected due to the large angle α. Thus, we further assume that in the current geometry only a small fraction of the power is not reflected, such that nonlinearity inside the defect is negligible. In the linear case [Fig. 2(d)], the beam incident at an angle α refracts at an angle β<α according to Snell’s law, and is transmitted through the defect at the same angle γ=α due to symmetry. However, in the presence of the Kerr nonlinearity [Fig. 2(e)], this symmetry is broken. Specifically, nonlinearity induces an increase of the uniform (cladding) region’s refractive index by Δn_nl=n ₂ I [see the illustration in Fig. 2(e)], where I is the power density in the vicinity of the interface, such that now by virtue of the Snell’s law γ>α. In order to have NLT of this ray, a significant fraction of the input beam must enter the defect at an angle β>β_c, where β_c is the critical angle of total internal reflection inside the defect:

Therefore, a condition for NLT of a particular ray may be formulated, by virtue of Eq. (3) and the Snell’s law:

or $\alpha >\arcsin \left(\frac{n_{\mathrm{cl}}}{n_{\mathrm{cl}}+\Delta n_{\mathrm{nl}}}\right)$ . In the presence of significant nonlinearity the right-hand term has a real solution. Equation (4) clearly indicates that as the nonlinearity grows stronger, NLT can be achieved for rays of smaller angles of incidence. Recalling that a real wave packet contains a distribution of k-components, with an increasing nonlinearity (due to either a larger n ₂ coefficient or a higher power density at the defect’s interface) a larger range of k-vector “rays” may couple to normal modes of the defect, leading to a larger fraction of trapped power content. Conceptually, each ray of an input wave packet may be considered separately according to Eq. (4). This simplified ray optics approximation is most appropriate for wide beam excitations, in which the wave fronts are nearly flat.

In the above discussion we have neglected the nonlinearity within the defect itself. However, as nonlinearity is introduced inside the defect as well, this simple model does not get much more complicated. Indeed, nonlinearity within the defect would result in a larger effective refractive index n_d+Δn_nl, thus making the angle β in Fig. 2(e) smaller, but at the same time the critical angle β_c [Eq. (3)] would become smaller by the same proportion, such that the trapping condition [Eq. (4)] remains unaltered.

2. Trapping dynamics in single-site multi-mode defects

In our numerical studies, we first consider solutions of Eq. (1) with initial excitations of the form (2) for scattering by a single-site multi-mode defect. The transverse refractive index profile of such a defect is given by:

where we choose n_d=n_cl+4×10^-3 and Δx=6 µm. Setting the other parameters as prescribed in Table 1, this defines a two-mode defect, as shown in Figs. 1(b) and 2(a)–(c). Refractive index gradients are introduced through local steps in the simulation’s transverse grid.

With the above parameters we have found it possible to nonlinearly trap power in each of the defect’s modes individually, with variation of only the beam’s angle of incidence and power density. Figure 3 shows propagation maps, along 0<z<15 mm and -0.2<x<0.2 mm, with θ=0.5° [Figs. 3(a),(b)], θ=1.1° [Figs. 3(c),(d)], and θ=1.75° [Figs. 3(e),(f)]. At low power settings [Figs. 3(a), 3(c) and 3(e)] the beam diffracts linearly in the cladding with negligible nonlinearly. In these linear cases there is no power remaining inside the defect beyond the transient region of transmission and reflection. At high powers the beam partially localizes in the defect [Figs. 3(b), 3(d) and 3(f), correspondingly] and can trap either in mode No. 2 [Fig. 3(b)] or in mode No. 1 [Fig. 3(d)]. This suggests a unique method for selective excitation of different modes in a spatially confined structure (such as, for example, a multi-mode optical fiber excited from the side) [20]. We have also found that with specific input beam parameters it is possible to induce periodic breathing of the trapped power between the modes, as shown in Fig. 3(f). Physically, we interpret this breathing as resulting from oscillations between the states illustrated in Figs. 2(b) and 2(c).

 

Fig. 3. Normalized beam intensity propagation maps I(x, z) (0<z<15 mm,-0.2<x<0.2 mm) in the presence of a single-site defect, whose index gradient is Δn=4×10^-3 (relative index step of 0.12%) and its width is Δx=6 µm. This defect supports 2 normal modes. The input Gaussian beam is of width w=20 µm and is centered at x_c=-40 µm. The insets in panels (b),(d) and (f) show close ups of the intensity distribution within the defect. The excitation angles are (a),(b) θ=0.5°; (c),(d) θ=1.1°; (e),(f) θ=1.75°.

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Fig. 4. Propagation of (a) the relative fraction of power inside the defect, and (b) position of the power distribution’s center of mass, corresponding to the simulations of Fig. 3.

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The trapping dynamics can be conveniently characterized by the relative fraction of power which is found inside the defect [I _in=∫^Δx/2 _-Δx/2 I(x)dx/I ₀] and the power distribution’s “center of mass” (CM) within the defect [x_CM=∫^Δx/2 _-Δx/2 xI(x)dx/∫^Δx/2 _-Δx/2 I(x)dx, with x=0 being the defect’s “center of inertia”]. Figure 4 shows these quantities as a function of the propagation distance z. While in low input intensities there is an exponential decay of the power within the defect, in the nonlinear cases power localizes inside the defects [see Fig. 4(a)]. In the two stationary cases of single mode trapping [Figs. 3(b) and 3(d)], the CM becomes fixed at the defect’s center (which is indeed the CM of the symmetric waveguide modes). In the nonlinear breathing case [Fig. 3(f)] the CM oscillates around the center in the transition between the two modes [green squares in Fig. 4(b)]. We note that such oscillations should in principle be accompanied by power losses to cladding modes. This is due to the partial mismatch between the defect’s normal mode profiles and the beam’s profile during the oscillations. However, since the “1+1”-propagation problem considered here (1+1 denoting one transverse dimension of the dynamics and one propagation direction) is conservative (i.e., an Hamiltonian system) [1], and an approximation of effective mode confinement is assumed in the third dimension (y) throughout the dynamics, these losses are less pronounced in our simulations. We therefore anticipate the weak power decay from within the defect in the oscillating case (green squares in Fig. 4), to be more substantial in the experiment. In the stationary trapped nonlinear cases (black triangles and red circles in Fig. 4), the power content within the defect was found to remain constant indefinitely with z.

3. Trapping dynamics in multi-site local defects

We now turn to numerical solutions of Eq. (1) with initial conditions (2) for waves scattered by multi-site defects, as illustrated in Fig. 1(c). We choose a refractive index profile of the form:

 

Fig. 5. (a) Refractive index profile of the 3-site defect (top), and the corresponding 3-layer AlGaAs sample cross section (bottom). (b)–(e) Normalized beam intensity propagation (0<z<15 mm, -0.3<x<0.3 mm) in the presence of a 3-site defect, with parameters Δn=3.3×10^-3 (0.1% index step), d=18 µm, w=4 µm and Δx=62 µm. The input Gaussian beam is of width w=40 µm, centered at x_c=-40 µm and tilted at an angle of θ=1.34°. The input power densities are (b) 10 W; (c) 1.8 kW; (d) 3.4 kW; (e) 3.5 kW.

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A cross section of the index profile and the 3-layer sample composition which defines it are illustrated in Fig. 5(a). Such a defect, of total width Δx, consists of 3 guiding sites of widths d and separations w. Specifically, we have chosen n_d=n_cl+3.3×10^-3, d=18 µm and w=4 µm. With these parameters, each guiding site supports 3 normal modes and adjacent sites are strongly coupled to each other when they are directly excited. The total defect width is Δx=62 µm, which is still sufficiently small to be regarded as a local scattering center for side excitations propagating through lengths of severeal mm’s.

With the above parameters we have found several possibilities of nonlinear trapping, which can be distinguished by variations in the input power alone. Propagation maps of linear and NLT dynamics are illustrated in Figs. 5(b)–(e). Figure 6 shows the relative fraction of power inside the entire defect region (I _in=∫^Δx/2 _-Δx/2 I(x)dx/I ₀) [Fig. 6(a)] and the power within the left, central and right sites individually [Figs. 6(b)–(e)]. While at low power excitation there is again an exponential decay of power inside the defect with z [Figs. 5(b),6(b)], at intermediate power settings a trapped power fraction emerges, which periodically oscillates in space between the defect’s sites [Figs. 5(c),6(c)], and at high power settings the trapped power can localize at the central site [Figs. 5(d),6(d)] or at the left site [Figs. 5(e),6(e)] (while still weakly oscillating between the site’s different normal modes). The physical mechanism underlying the oscillations observed in Figs. 5(c),6(c) is tunneling between the strongly-coupled adjacent defect sites [21]. We again stress that in our simulations, radiation losses in the oscillating solutions are not fully accounted for, due to the assumption of energy conservation and complete beam confinement in the y dimension. In practice, we anticipate each period of oscillation between sites to be accompanied by a more substantial shedding of power to radiation modes. The nearly complete localization at a single site of the defect [Figs. 5(d),(e) and 6(d),(e)] is achieved due to the delicate interplay between the nonlinearly strength at the defect boundaries and the nearest-neighbor coupling between adjacent sites of the defect. In particular, these results constitute a unique demonstration of all-optical spatial switching.

 

Fig. 6. Propagation of (a) the relative power fraction inside the entire defect, and (b)–(e) the power fraction found within each site, corresponding to the simulations of Fig. 5.

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4. Nonlinear Fabry-Perot etalon

Further insight into the process of nonlinear trapping may be gained if we recall that an optical defect, or a single waveguide coupled from the side in the simplest case, serves as a Fabry-Perot etalon, in which light is multiply reflected and transmitted by the parallel surface boundaries [see Fig. 7(a)]. The transmission function of such an etalon has maxima values corresponding to constructive interference of the transmitted beams [T ₁+T ₂+… in Fig. 7(a)] and minima values corresponding to constructive interference of the reflected beams [R ₁+R ₂+… in Fig. 7(a)]. The transmission function of this simple single-site etalon is given by:

where ℜ is the reflectance coefficient of each surface and n=n_d/n_cl>1 is the ratio between the defect and cladding refractive indices.

Figure 7(b) shows transmission and reflection curves, as a function of n (i.e., with a variation of the defect refractive index while keeping the cladding index constant, as specified in Table 1 above) of a low power linear beam which is scattered by a single-site defect. These curves were obtained by numerical integration of the simulation output at z=15.4 mm, with respect to the defect boundaries. Here R+T=1, and no power remains inside the defect at the observation point for any value of n. A fit of the transmission curve in Fig. 7(b) to Eq. (7) yields ℜ=0.92. In Fig. 7(c) we show curves of the reflected, transmitted and trapped power fractions for a high-power nonlinear beam scattered by the same defect. It is readily seen that the trapped fraction of the power exhibits the same periodic nature as the transmitted and reflected fractions, and it therefore depends strongly on the same etalon parameters that dictate the transmission function. We thus conclude that the defect supporting the NLT effect may be regarded as a nonlinear Fabry-Perot etalon. We also note the strong influence of the physical parameters, in particular the refractive indices and the angle of incidence, on the transmission, reflection and trapping curves. With each specific set of input beam parameters, there are certain index ratios in which NLT is most pronounced. This generally occurs when the reflection and transmission coefficients are equal [see Fig. 7(c)]. With a multi-site etalon [Fig. 7(d)], the periodic nature of the reflection, transmission and trapping curves is still prominent. However, there are subtle differences: Here a larger fraction of the overall power can nonlinearly trap in the triple-site defect [~0.5 in Fig. 7(d) rather than ~0.3 in Fig. 7(c)], the trapping curve has a finer internal structure, and NLT occurs over a slightly larger region of n values in each period.

 

Fig. 7. (a) Illustration of a Fabry-Perot etalon, consisting of two parallel surfaces of reflectance ℜ coupled from the side. (b)–(d) Transmission (red triangle), reflection (green square), and trapping (blue circle) curves as a function of the index ratio n=n_d/n_cl, with input beam parameters w=20 µm, x_c=-70 µm and θ=0.7°; The output observation plane is at Z=15.4 mm. (b) Low-power input (I ₀=10 W) and (c) high-power input (I ₀=2.5 kW) with a single-site scatterer (Δx=18 µm); (d) high power input (I ₀=2.5 kW) with a 3-site scatterer [Δx=62 µm, d=18 µm and w=4 µm, see Fig. 5(a)].

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5. Summary

We have presented a comprehensive numerical study of the trapping dynamics in nonlinear scattering by local guiding defects that support normal modes. The dynamics of linear and nonlinear scattering from local defects have been described in terms of a wave optics approach and a complimentary ray optics approach. In scattering by a single-site multi-mode defect, we have demonstrated that power may localize at a certain normal mode or periodically oscillate between the different modes of the defect. In the presence of strongly-coupled multi-site defects, we have identified two regimes of nonlinear trapping: at intermediate nonlinear powers, periodic tunneling between adjacent sites and their normal modes has been observed, with the overall trapped power within the defect remaining constant. At highly nonlinear powers the radiation was mostly localized at a single site with almost no oscillations. As these dynamical properties can be tuned with variation of the input power only, this constitutes a unique demonstration of all-optical spatial switching. With a varying refractive index of the defect, the scattering and trapping dynamics were described in the context of a nonlinear Fabry-Perot etalon.