## Abstract

We demonstrate a technique for a single shot mapping of nonlinear phase shift profiles in spatial solitons that are formed during short pulse propagation through one-dimensional slab AlGaAs waveguides, in the presence of a focusing Kerr nonlinearity. The technique uses a single beam and relies on the introduction of a lithographically etched reflective planar mirror surface positioned in proximity to the beam’s input position. Using this setup we demonstrate nonlinearity-induced sharp lateral phase variations for certain initial conditions, and creation of higher spatial harmonics when the beam is in close proximity to the mirror.

© 2007 Optical Society of America

## 1. Introduction, theoretical and numerical background

#### 1.1. Introduction

Solitons are self-regulating nonlinear excitations, with fascinating properties that
are intermediate between waves and particles [1]. Nonlinear optics provide the most common and elementary manifestation
of solitons [2], including spatial solitons
[3]. Among the most important quantities
that characterize a nonlinear excitation are the phase profile that is accumulated
during its propagation, and the phase shift with respect to its linear counterpart.
Traditionally, these quantities are measured using rather complex two-beam techniques
[4, 5, 6]. In this paper we demonstrate
a simple, single beam technique for measuring the phase shift profile of spatial
solitons in 2D slab waveguides. Our method relies on the introduction of an embedded
planar mirror inside the waveguide, the excitation of a soliton in close proximity to
this mirror and the interference formed by the soliton and the wave reflected from
the mirror. The paper is organized as follows: in Sec. 1 we briefly discuss the
theoretical background of solitons and their nonlinear phase evolution. In Sec. 2 the
optical experiment is described, the geometry of the sample is introduced, and our
sample is compared to a sample with shallow-etched barrier interfaces, of the type
that is traditionally used, *i.e*. in the study of the interaction of
spatial solitons with micro-structured inhomogeneities [7, 8, 9]. In Sec. 3 our experimental results are shown,
including the measurement of the nonlinear phase shift profile of a spatial soliton,
the observation of nonlinearity-induced sharp lateral phase variations, and the
creation of higher spatial harmonics when the soliton is launched near the mirror,
due to proximity effects. In Sec. 4 we summarize our main results and
conclusions.

#### 1.2. Theoretical and numerical background

While the intensity properties of spatial solitons in slab waveguides have been
studied extensively [3], it is interesting to
note that several phase-related properties of these solitons are usually assumed
without their explicit measurement. Writing the electric field associated with an
initial excitation as
*E*(*x,z*)=|*E*(*x,z*)|*e*
^{iϕ}(*x,z*), with *x*
being the lateral waveguide direction and *z* being the propagation
direction (assuming confinement in the *y* direction),
*ϕ*(*x,z*) is the phase accumulation and
$\beta (x,Z)=\frac{\partial \varphi}{\partial z}{\mid}_{z=Z}$ is the propagation constant. While a linear wave that traverses the
waveguide has a propagation constant that falls in the allowed region of the
geometrical waveguide dispersion [10], a
nonlinear soliton has a higher propagation constant, corresponding to a bound state
penetrating into the semi-infinite gap where no linear guided modes exist; indeed,
this is the key property that enables the non-dispersive nature of these nonlinear
excitations, *i.e*. a spatial localization occurring at higher optical
intensities. Assuming the slowly varying envelope approximation, the paraxial
approximation, and energy conservation for the forward traveling wave, the 2D
electromagnetic field dynamics in a slab Kerr waveguide can be described by the well
known nonlinear Schrödinger equation (NLSE) [1–3]. The solution of the
NLSE of the form
*E*(*x,z*)~*E*(*x*)*e*
^{iβz} leads to an eigenvalue equation $i\frac{\partial E}{\partial z}\sim -\beta E$, where *β* is in the gap.

To illustrate the importance of *ϕ* and
*β* in spatial soliton formation, Figs. 1 and 2 show
solutions of the NLSE obtained from beam propagation method (BPM) simulations [11].

It is well known that a low-power beam slowly diffracts when propagating along the
*z* direction (Fig. 1(a)).
The accumulated phase at the center of the beam (Fig. 1(b)) is linear with a slope of approximately unity in the
dimensionless units of Fig. 1(b). Note that
the slope ${\beta}_{0}=\frac{\Delta \varphi}{\Delta z}\approx 2\pi \u2044{\lambda}_{0}$ is approximately the plane-wave propagation constant. As the input
power is increased, a solitary wave is formed (Fig.
1(c)), and the accumulated phase is different from the case presented in
Fig. 1(b). The accumulated phase
*difference* relative to Fig.
1(b) is shown in Fig. 1(d), and is
indeed nonlinear. The positive slope corresponds to a self-focusing nonlinearity
*n*
_{2}>0, with a local increase of the propagation constant, associated with
its penetration into the semi-infinite gap. Lateral Profiles of the output phases and
propagation constants obtained from the above simulations are shown in Figs. 2(a),(b) for different input powers.
Beyond a certain threshold power, a spatial breakup occurs that is accompanied by a
breakup of the phase profile (Fig. 2(a)) and
by the recovery of the linear propagation constant (Fig. 2(b)).

In the well known special case of input beams with the hyperbolic Secant intensity
form, while the phase front profile of a low power diffracting beam has a particular
pattern (green line in Fig. 2(c)), as a
soliton forms it becomes flat and stationary at the beam’s center (blue line
in Fig. 2(c)). In this case
*β* is uniform and larger than *β*
_{0} (Fig. 2(d)).

## 2. Optical setup and sample composition

#### 2.1. Optical setup

The conventional techniques for measuring nonlinear phase variations are cumbersome,
as they require at least *two* beams: the soliton beam of interest,
and a reference low-power beam not undergoing the phase change, *i.e*.
it linearly diffracts along with the soliton [4, 5, 6]. These methods are illustrated in Figs. 3(a),(b). One realization (Fig. 3(a)) involves coupling the beams at remote locations of
the input facet with separate lenses, under similar conditions and with different
input powers. When the beams reach the output facet, they have different linear and
nonlinear phases. As a result, their overlapping on a nonlinear crystal yields a
phase related cross-correlation signal [4,
5]. In another realization the two beams
are coupled in close proximity, using the same input lens (Fig. 3(b)). The beams’ partial overlap and the
resulting coherent interference pattern recorded by an imaging camera contains the
required information regarding the soliton’s added phase [6]. In both of these techniques the two beams
must be coherent and individually stable following their splitting, also implying
that the alignment of the sample with respect to both of the input beams is
critical.

Here we show that by using an embedded mirror surface, oriented parallel to the
propagation direction (Figs. 3(c),(d)), a
single beam experiment can yield the spatially resolved phase profile, eliminating
any need for multiple beams. In particular, a low-power diffracting beam that is
coupled near a mirror (Fig. 3(c)) exhibits
interference between its central part and the reflected tails, all having the same
wavelength. Even as the power is increased and a spatial soliton is formed, there is
some weak background undergoing linear diffraction. Some extent of these diffracting
tails then interferes with the soliton (Fig.
3(d)). The output phase difference profile
Δ*ϕ(x)* is then directly measured as it results in
phase shifts of the imaging interference pattern. We stress that the interference
fringe shift is exactly equal to the nonlinear phase only in the case of
*spatial* solitons (which are dispersive in the temporal domain)
[2, 3], such as in AlGaAs waveguides. When *spatiotemporal*
compression is considered (*i.e*. as self-focusing is observed in both
space and time dimensions) [12, 13], it is generally accompanied by a shift of
the soliton wavelength [14]. Since this added
wavelength shift is unknown without additional measurements, neither of the
contributions to the phase change can be extracted from the proposed experimental
technique.

#### 2.2. Sample geometry

3-layer semiconductor Al_{x}GaAs_{1-x} waveguides have been used extensively for nonlinear optics applications [7], as well as for studies of soliton propagation
in micro-structured waveguide arrays [8, 9]. These samples consist of Clad-Core-Clad
sadwiches deposited on top of a GaAs substrate with vertical dimensions designed for
single-mode propagation [10] in the near
infrared (see Figs. 4(a),(c)). The Aluminum
doping levels *x*=0.18 in the core layer and *x*=0.24
in the clad layer give rise to a physical vertical refractive index difference of
Δ*n*=0.03 at *λ*
_{0}=1.5 *µm*, with a core index of *n*
_{0}=3.34. A 25 *µm*-wide shallow etching applied to
the top clad (Fig. 4(a)) effectively
decreases the mode area, and therefore the effective refractive index, of a beam
confined to the core [8, 9]. Therefore, shallow etchings serve as local barriers to
confined beams.

In contrast, a deeply etched interface that penetrates into the core layer (Fig. 4(c)), can inhibit coupled wave transfer from one side of the center to the other, serving as a reflective barrier. The corresponding output facet images, as a function of the input beam position, are shown in Figs. 4(b),(d). Clearly, the diffracting beam crosses almost completely the shallow etched interface, while completely reflecting from the deeply etched interface. In the latter case, the sharp beam cutoff at the barrier interface is evident, as well as the fringe patterns associated with the interference between the original and reflected beams, and the increased degree of overlap between the two beams as the input beam is coupled closer and closer to the barrier. Also note that at input locations that are distant from the center, the fringe spacing changes linearly with the input position, while at locations that are adjacent to the center the fringe spacing changes parabolically. In the former case only the weak diffracting wings are reflected by the mirror (as in Fig. 3(c)), and we therefore refer to this region as the “weak perturbation regime”. In the latter case, however, both beams are of comparable powers as part of the input beam is already at the interface, and the interference pattern changes nonlinearly with the input position. To this region of excitation we will refer below as the “strong perturbation regime”.

## 3. Experimental results

#### 3.1. Phase shift profile mapping

In order to excite spatial solitons in our 6.5 mm-long AlGaAs waveguides, which
include an embedded mirror, we use laser pulses of 100 fs duration, 1 kHz repetition
rate, and peak powers of up to 5 kW. The input beam is shaped to be elliptical with
an height of ≃1.5 *µm* (to enable efficient coupling
to the core layer, see Fig. 4(c)) and an
input width of ≃100 *µm* at the beam’s waist.
The formation of a spatial soliton as a function of the input peak power is shown in
Fig. 5(a) for a coupling location that is
far away from the mirror, *i.e*. in an homogeneous region.

As the coupling location is in the weak perturbation regime, the spatial soliton that is formed is accompanied by a positive phase shift in the fringe pattern (Fig. 5(b)). Following integration along the vertical axis (Fig. 5(c)), the comparison between the fringe positions in low power (green) and in soliton power (orange) yields the relative phase shifts of the interference patterns. Note that the fringe movement is gradual between adjacent fringes, implying a slowly-varying phase profile. By the arguments discussed in Sections 1 and 2, we can infer that in the first approximation (that is applicable as long as the mirror perturbation is weak) the phase shift is exactly equal to the soliton’s local phase increase. The extracted soliton’s lateral phase shift profile is shown in Fig. 5(d). This result is in qualitative agreement with the corresponding numerical simulation (Fig. 2(a)).

#### 3.2. Observation of sharp phase gradients

By applying tilts to the input beam relative to the sample, different excitation
conditions were explored, *i.e*. with initial conditions in which the
input phase front is not flat. An interesting phenomenon that we have observed in
such cases was the occurrence of a series of nonlinearity-induced sharp phase shifts
(larger than 2*π*) at *x* separations that are
smaller than the fringe period, even when the excitation is in the weak perturbation
regime. An example is shown in Fig. 6(a).
Sharp nonlinear phase gradients, which are extended in *x*, were
observed as local “blurring” of the interference pattern around these
regions, as shown in Fig. 6(b).

#### 3.3. Strong perturbation regime

As the beam is coupled in close proximity to the mirror (lower part of Fig. 4(d) with *p*>800
*µm*), the power-dependent characteristics of the
interference pattern are substantially different.

As shown in Fig. 7(a), in this case there is
an increase in the number of visible fringes for a high power input beam, in
comparison to a low power excitation. The discrete Fourier transforms of these scans
(Fig. 7(b)) confirms that new spatial
frequency components *k _{x}* appear in the spectrum of the interference pattern in the intermediate and
high power cases.

We speculate that these new harmonics are signatures of cross phase modulation and wave mixing [15–18] between the original and reflected components, as both of them now possess comparable high power. We also note that there are a few other possibilities, such as the resonant scattering of dispersive waves by the soliton [19] and nonlinearity-induced ground state selection [20], which can be regarded as a four-wave mixing process [21]. In any event, phase retrieval is impossible using simple imaging in the strong perturbation regime, as the physical influence of the mirror becomes substantial rather than perturbative.

## 4. Conclusion

We have introduced a unique single beam method for the single shot measurement of phase shift profiles formed in spatial solitons, following their propagation through 2D slab nonlinear waveguides. The method uses a lithographically etched reflective surface positioned near the beam’s launching position. In addition to phase shift profile mapping in spatial solitons, this setup enabled us to observe nonlinearity-induced sharp phase gradients, and to record the creation of high spatial harmonics as the beam is launched in near proximity to the mirror surface.

## Acknowledgments

We gratefully acknowledge financial support by the Israel Science Foundation (ISF), through grants 8006/03 and 944/05, and by the Natural Sciences and Engineering Research Council of Canada (NSERC). We thank Prof. Victor Fleurov and his student Gali Dekel for fruitful discussions regarding different aspects of this work.

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