Beam steering via peak power decay in nonlinear waveguide arrays

We report the experimental observation and theoretical analysis of a novel beam-steering effect in periodic waveguide arrays that arises from the interplay between discrete diffraction, Kerr nonlinearity and any mechanism that effectively weakens the nonlinear part of the beam. In this regime the propagation direction shows increased sensitivity to the input angle and for a certain angular range around normal incidence a nonlinear beam may be guided to a direction opposite to that initially inserted. For continuous wave beams the role of this mechanism is played by absorption of any kind, such as three photon absorption, two photon absorption or even linear absorption. For pulsed beams we show that the same dynamics can arise due to strong normal temporal dispersion, while absorption is not necessary and can be a further enhancing or alternative factor. This observation falls under a more general dissipation-assisted beam velocity control mechanism in nonlinear optical lattices, which is also theoretically predicted by the effective particle approach.


Introduction
Nonlinear optical waveguide lattices have been intensely studied in the past few years, revealing a wealth of properties that are not encountered in homogeneous media [1]. It is now well established that this is mainly due to their structural periodicity that may significantly alter their diffraction properties, a fact that is reflected in both the linear and nonlinear regime. For example, tailored diffraction properties of optical wave fronts [2] are possible in the linear domain whereas discrete optical solitons can form under nonlinear conditions [3], just to mention a few striking phenomena. It is the unique characteristics of these configurations that render them an effective medium for controlling the propagation of light.
So far, several schemes toward this direction have theoretically been proposed and experimentally realized. In the linear regime, the idea of applying a linearly varying potential in the transverse direction so as to produce photonic Bloch oscillations [4,5] has been utilized in thermo-optic waveguide arrays [6], where the beam velocity is controlled by the transverse temperature gradient. In another implementation the varying potential is achieved by a linearly growing effective index of the individual guides [7]. Quite recently discrete beam acceleration, a different concept based on the propagation of non-diffractive, self-bending Airy beams in bulk homogeneous media [8], has been theoretically demonstrated in uniform optical lattices [9]. In the nonlinear regime, power-controlled beam steering may be achieved with linearly chirped or unchirped [10,11] solitons. The beam output position may be also controlled by an either externally or internally induced velocity [12,13], the latter stemming from an input beam shift relative to the waveguides. Moreover, array inhomogeneity [14], engineered defects or a superimposed linear guided mode [15], as well as interaction with a continuous wave (CW) background [16] may induce controllable soliton steering.
In all these cases absorption was considered absent or negligible in both experiments and theoretical analysis for the configurations examined, as it was always assumed it can only degrade or even inhibit beam steering. If another competitive mechanism, such as optical gain is present whenever absorption is countable, dissipative soliton formation is possible [17] and beam velocity control is maintained [18]. Nevertheless, even in the absence of a power restoring mechanism, non-negligible absorption may lead to an interesting dynamical behavior of a focused beam [19].
Moreover, in the cases mentioned above, the interest has been focused on the spatial behavior of light, rather than its temporal evolution, by either considering CW beams or measuring time-integrated pulses at the system output. Only recently it was shown that a pulsed beam propagating in a discrete system may lead to interesting patterns, such as discrete X-waves [20], which cannot be otherwise observed if the temporal dimension is either ignored or absent.
Within this framework, in this work we observe experimentally and study numerically the spatiotemporal beam dynamics in periodic lattices and show how the temporal evolution may have a direct implication in the spatial propagation direction of light. Although the dynamics may be complex, we propose a simple concept of how to control the beam's direction in a predictable non-chaotic manner. To clarify the underlying mechanism we begin with the study of a CW system with three photon absorption (3PA) and theoretically show how absorption acts as a longitudinally dynamical potential variation. For the sake of generality we also explain why the same dynamics are feasible in similar systems that exhibit any relative kind of absorption, such as two photon absorption (2PA) or even linear absorption. Next, we extend the conclusions to pulsed systems and show how this potential variation is now induced by the spatiotemporal dynamics, rather than an absorptive mechanism, the presence of which is enhancing but not necessary. Our calculations and predictions are in good agreement with experiments preformed in an AlGaAs waveguide array system.

Experimental results
The system under investigation is a 5 mm long waveguide array, fabricated in an AlGaAs planar waveguide by way of conventional photolithography techniques. The waveguides are 4 µm wide, and the array period is D = 8.5 µm. The areas between waveguides exhibit a low refractive index (RI) of n 0 = 3.3244, while the lateral effective RI step is n = 0.0011. This relatively small value that stems from the shallow etching depth, results in a strong coupling constant of 950 m −1 between waveguides. The laser beam that enters the sample operates at 1530 nm and the self-focusing Kerr coefficient isn 2 = 1.5 × 10 −13 cm 2 W −1 . Although 2PA is negligible, 3PA is present, yielding a coefficient α 3 = 10 −4 m −1 W −2 . The dispersion in the AlGaAs array is approximately β 2 = 1.35 ps 2 m −1 . The experimental setup is shown in figure 1.
Our laser source is an optical parametric oscillator (Spectra-Physics OPAL), pumped by a 810 nm Ti:sapphire (Tsunami) laser, producing 170 fs pulses at a repetition rate of 80 MHz at a wavelength of 1530 nm. The input power to the sample is adjusted by a variable filter. A cylindrical lens at the back focal plane of the 40× input objective was used to shape the input beam. Shifts of the beam's axis before the input objective was used to induce a linear phase chirp to the beam at the input facet of the array, thus controlling the initial propagation angle in the array. Light is coupled into the waveguide array, propagates in it and is collected from the output facet, to be imaged using a CCD camera.
In order to scan the beam output position in terms of the input transverse velocity, the beam is tilted within a range of at least −13.5 and 13.5 mrad, which for the specific system covers the (−π/2, π/2) range of the transverse Bloch momentum Q = k x D. In this configuration, a Gaussian beam of 12 µm full-width half-maximum (FWHM) input spatial extent is used, which covers approximately 2-3 waveguides. The scan starts at low power levels and is repeated with  higher power beams. This procedure is carried out twice, with respect to the input position of the beam's peak. The first time it overlaps with the midpoint of the central waveguide and in the second case it is positioned between the central and the adjacent waveguide. The experimental output beam position versus the input beam tilt for both scans is presented in figures 2 and 3, for low as well as high power pulsed beams. In these figures, the beam output for every scanned input angle is taken at the sample exit (z = 5 mm) and integrated in time. The horizontal axis is always normalized to the array period D, unless stated otherwise.
At low input power levels, as light propagates in the sample, it couples to adjacent waveguides and undergoes discrete diffraction. When the beam deviates from normal incidence, the wave front is directed according to the curvature of the diffraction diagram, essentially following up the input tilt as theoretically expected. Hence, during linear propagation the output position versus input beam tilt pattern follows the familiar cosine-like behavior, which reflects the first band curvature of the periodic system [2] (see e.g. figures 2(a) and 3(a)). However, at high input power levels where the beam propagates in a nonlinear fashion, the output pattern can be significantly altered, depending on the initial beam peak position. For a certain angular range around normal incidence the beam is guided to a direction opposite to that initially inserted when the beam is centered on the central waveguide (figure 2), behavior that does not take place when the beam is positioned between waveguides ( figure 3). Typically, a narrow beam at such high power levels would be expected to form a discrete soliton in the input waveguide for a certain angular range around normal incidence [13], fact that raises the question of how this counter-intuitive behavior takes place.

Continuous wave (CW) systems: the effective particle approach
To gain insight into the complex beam dynamics, let us first begin with a pertinent CW system. In this case beam propagation is described by a modified nonlinear Schrödinger (NLS) equation, which includes the transverse variation of the RI. In its general form with the appropriate absorption terms added, the equation reads where k 0 = 2π/λ 0 is the free-space wavenumber, n(x) the RI modulation and n 0 the lowest value of the RI.n 2 is the nonlinear Kerr coefficient and β 2 the group velocity dispersion (GVD) term, while α 1 , α 2 and α 3 are the linear absorption, 2PA and 3PA coefficients, respectively.
To stress the generality of the dynamics, let us normalize equation (1) with a simple coordinate transformation, i.e. ζ = z/Z 0 , χ = x/X 0 and U = u/U 0 . Under this normalization equation (1) is written as with Equation (2) with zero rhs corresponds to the integrable NLS equation with the soliton . η is the amplitude or the inverse width of the soliton solution, χ 0 is the position of its center (called sometimes the center of mass due to the effective-particle analogy of the solitons), v is the transverse velocity and φ is the nonlinear phase shift. If we consider a relatively shallow lattice with low absorption, for which equation (2) is allowed to be treated via a perturbative approach, then the soliton can be treated as an effective particle [21] of variable mass m = |U | 2 dχ = 2η and momentum p = mv at a position χ 0 moving in an effective potential due to the lattice according to the equations dm dζ Let us for simplicity consider a sinusoidal RI modulation, varying as n(χ) = A sin(K χ ), for which the effective potential is simply calculated as In the absence of absorption (σ 1 = σ 2 = σ 3 = 0) the effective particle motion described by equations (4) and (5) is integrable and stationary stable and unstable solitons are formed in the minima and the maxima of the effective potential, respectively. Moreover, two distinct dynamical behaviors are described, one corresponding to trapped solitons oscillating around a potential minimum and the other to traveling solitons moving transversely across the potential, depending on the initial energy of the effective particle. A soliton initially located at the minimum of the potential can be untrapped only if its initial velocity exceeds a critical value v cr = 2 2AπK /m sinh(πK /m) From equations (6) and (7) it is obvious that in the same periodic medium solitons of different mass move in effective potentials of different strength and consequently different critical velocities. In fact, it is apparent from equation (7) that solitons of higher mass require higher velocities to escape their periodic orbit, which is trapped within the potential minimum they are launched, in other words they experience a stronger effective potential.
In the presence of absorption (σ 1 , σ 2 , σ 3 or their combination) the soliton mass reduces according to equation (4) until it essentially settles down after a transient propagation distance where the nonlinear effects are relaxed due to mass reduction. Consequently, via this procedure absorption is translated into a dynamical modulation (weakening) of the effective potential V eff . If the soliton is initially launched at a potential minimum with v < v cr it starts to oscillate within this minimum, until its mass has been reduced enough so that it may escape. Depending on the propagation distance at which this happens, the output angle may follow a complex pattern, since this can occur at any random moment of the oscillatory motion. If, however, absorption is such that the escape can occur within the first half-period of oscillation, then an interesting and predictable pattern emerges, as depicted in the example of figure 4(a) for a system with 3PA (σ 1 = σ 2 = 0, σ 3 = 0).
For small initial velocities the soliton escapes the periodic motion within the first halfcycle of oscillation at a direction opposite to that initially launched. For larger initial velocities, the soliton directly escapes the input minimum (within the first quarter cycle of oscillation) . The x-axis is shifted π/2K units to the left and normalized to the period 2π/K of the RI modulation, so that the output position ticks mark the potential minima.
following a trajectory more or less along the initial angle, giving birth to this wiggle pattern. On the other hand, a soliton launched at a potential maximum is unstable and even the slightest initial velocity is sufficient to repel it from its input position, thus approximately following the input direction, as figure 4(b) implies. Similar patterns arise when considering σ 2 = 0 or σ 1 = 0 and this behavior falls under a more general case for dissipative optical lattices, which has been analyzed in detail in our previous work [18].

CW systems: simulations
Let us now consider a system with stronger lattice modulation and 3PA, similar to the scenario realized in our experiment. Although the experiment was carried out with 170 fs pulses, let us in this section assume a hypothetical experiment with a CW beam, where only discrete diffraction, Kerr nonlinearity and 3PA are present (no thermal or other effects are considered).
To simulate the results numerically, the coefficients of equation (1) as well as the input beam parameters are set to the values used in our experiment. The temporal term has been removed and α 1 = α 2 = 0 has been considered. To make the role of absorption clearer, in these CW simulations 3PA has been set to α 3 = 7 × 10 −4 m −1 W −2 , i.e. seven times higher than the real value. The output beam position versus the input beam tilt of the hypothetical experiment and the respective simulations is presented in figure 5 for low as well as high power beams.
It is worth mentioning that, in contrast to the scenario analyzed with the effective particle approach, here not only the lattice exhibits strong RI modulation and strong 3PA, but also the beam is spatially Gaussian and not sech-like, yet, the two systems retain similar properties.
The key to understanding this resemblance lies in the stability of the initial state that is excited in each case. In case the beam peak is placed on the central waveguide this position corresponds to a potential minimum of the former theoretical analysis. When the input power is increased, the initially narrow Gaussian beam undergoes self-focusing and is 'locked' within the same waveguide, forming a discrete soliton. Under an initial tilt, if no absorption mechanism is present, the self-trapped beam starts to oscillate within the waveguide. The presence of 3PA intervenes in the oscillatory motion, exactly as explained with the effective particle approach and is depicted in figure 6 for the system in hand, where it is demonstrated how the distance, at which the beam escapes, approaches z = 0 as absorption increases. Finally, for further tilt, the oscillation amplitude becomes greater (while the period shortens) until the beam escapes, approximately following the input angle.
Since the optical power diminishes, the beam eventually propagates in a linear manner, thus discretely diffracting toward the new direction. In this procedure, side radiation acts as a further power damping factor for the initially localized state (see figure 6(a) for example), thus facilitating beam delocalization.
Interestingly, when 3PA is very intense, regardless of the input tilt the beam peak is very quickly damped, and the output beam trace resembles that of linear propagation since the massdependent effective potential drastically reduces. On the other hand, when 3PA is rather small, then the intense nonlinearity combined with discrete diffraction may lead to complex patterns. Both scenarios are presented in figure 7.
In case the input beam peak is placed between two waveguides, position corresponding to a potential maximum, the excited nonlinear state is unstable [22]. A slight transverse velocity, inaccurate displacement or even material imperfection will push the beam to the nearest waveguide, while undergoing an uneven power splitting among the two adjacent sites.
Inevitably, the small part of the beam that remains on one waveguide weakens the major beam part that jumps on the other and starts oscillating, while being further deteriorated by side radiation. Hence, even at small angles the beam is less prone to trapping and consequently lower absorption levels are capable of making such a phenomenon possible. This is why for the 3PA level of the given sample such a pattern is not observed, as already demonstrated in figure 5(d).

Extension to CW systems with linear absorption or two photon absorption
From the preceding analysis it is apparent that in the CW case the key to this extraordinary effect is the presence of an absorptive mechanism that acts effectively within the first half-cycle of beam oscillation in its initial confinement. To demonstrate how this can also be accomplished by any kind of relative absorption we set α 3 = 0 in equation (1) and assume the linear absorption and 2PA coefficients to be α 1 = 370 m −1 and α 2 = 0.5 m −1 W −1 , respectively. Successively switching them on, the simulations for a hypothetical system of the exact same parameters, which instead of 3PA exhibits either linear absorption or 2PA, are presented in figure 8. As expected, the simulations reveal a performance similar to that of figures 5(c) and (d). The main difference lies in the propagating distance along which each kind of absorption acts. Since 3PA acts proportionally to ∼|u| 4 , it switches off sooner than the respective system with 2PA (∼|u| 2 ) or with linear absorption, where optical power constantly deteriorates. Consequently, in terms of power consumption, a system with 3PA would be considered more advantageous. As already mentioned, similar patterns are also predicted by the effective particle approach when considering σ 2 = 0 or σ 1 = 0 and hence this counter-intuitive effect of the dissipative mechanisms is simply explained.

Pulsed systems
Thus far it has been evident that the excitation of a pattern similar to the one observed in the experiments requires the presence of a mechanism that is able to weaken the nonlinear part of the beam, i.e. the beam peak power, in an effective way. Returning to our experimental situation, it can now be simply explained why in the case of pulses this can be directly provided by any process leading to pulse broadening, such as the combined effect of normal dispersion with Kerr nonlinearity.
The output pattern depends on the beam's peak input position as explained in the CW case. The dynamical potential variation is now realized by the combined action of normal dispersion and discrete diffraction in the presence of Kerr nonlinearity. As the pulse propagates, the nonlinear main part of the pulse behaves in the oscillatory fashion described previously, while the weak edges undergo discrete diffraction. Due to the high normal dispersion of AlGaAs and the presence of Kerr nonlinearity the pulse is also subject to severe broadening. Consequently, the pulse broadening, which leads to peak power decrease, weakens the initially strongly nonlinear main part of the pulse, intervening in the oscillatory motion in a fashion similar to that of absorption in a CW system.
To simulate the pulsed beam propagation numerically we have included the appropriate temporal term in equation (1), which for α 1 = α 2 = 0 now reads The simulated output beam position versus the input beam tilt for high input peak power of 875 W is presented in figure 9. Here, the beam output for every scanned input angle is taken at the sample exit (z = 5 mm) and integrated in time, exactly as in experimental figures 2 and 3. Figures 9(a) and (b) correspond to figures 2(c) and 3(c), respectively. In all cases, the numerical simulations are in excellent agreement with the experimental data.
To better illustrate the essence of this mechanism, in figure 10 we present the full spatiotemporal profile of the simulated pulse together with the time-integrated spatial profile. . The respective output profile for a system without 3PA is almost identical to (c).
The input pulse depicted in figure 10(a) is given an initial tilt of 4 mrad and after 5 mm of linear propagation the output is clearly shifted, following the input tilt (see figure 10(b)). For nonlinear propagation at 875 W peak power (see figure 10(c)) the same pulse is shifted to a direction opposite to the initial after almost half a cycle of oscillation. It is evident that due to pulse broadening and discrete diffraction, pulse splitting is intense.
Although it is apparent from figure 10(c) that absorption is not necessary in this procedure, its presence can serve as an enhancing factor or even an alternative mechanism to this performance (for low normal GVD systems). To this end, and to emphasize the resemblance to the conclusions of figure 7 which concerned the CW case, in figure 10(d) we demonstrate the spatiotemporal distribution of the initial pulse, after propagating in a hypothetical system with intense 3PA (20 times that of our experiment). Again, as in the CW case, with strong absorption a linear-like pattern as in figure 7(b) is possible, as the similar propagation directions in figures 10(b) and (d) indicate. Naturally, in the presence of countable absorption, the lower the GVD coefficient, the more the time-integrated output will resemble the CW beam behavior.
The fact that directed pulse propagation at will can be implemented either with the aid of normal dispersion or absorption provides an extremely useful versatility. Depending on whether pulse shaping is a desired effect, one can have the same results (in terms of beam direction) with low GVD and countable absorption, thus not significantly altering the temporal unity of the pulse.

Conclusions
In this work we have showed how the interplay between discrete diffraction, Kerr nonlinearity and a mechanism that effectively weakens the nonlinear part of the beam along the propagation can result in counter-intuitive beam steering effects. The idea is based on using nonlinearity to lock optical power in a waveguide and subsequently release it with the assistance of a peak power reducing mechanism. Although such beam dynamics can be complex, if this mechanism acts within the 1st cycle of beam oscillation, the beam direction can be engineered in a predictable non-chaotic manner. Although in CW dynamics this mechanism must be some sort of absorption, in pulsed systems this role can be played by the combined action of normal dispersion and Kerr nonlinearity. In the latter case absorption, although not necessary, acts as a further or alternative beam control factor. We experimentally demonstrated the proposed idea in an AlGaAs waveguide system with intense normal GVD and low 3PA and theoretically showed that the same dynamics are feasible in similar systems that exhibit any relative kind of absorption, such as 2PA or even linear absorption.