August 15, 2012 / Vol. 37, No. 16 / OPTICS LETTERS
Reshaping the trajectory and spectrum
of nonlinear Airy beams
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Yi Hu,1,2,* Zhe Sun,2 Domenico Bongiovanni,1 Daohong Song,2 Cibo Lou,2 Jingjun Xu,2
Zhigang Chen,2,3 and Roberto Morandotti1
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Université du Quebec, Institute National de la Recherche Scientifique, Varennes, Quebec J3X 1S2, Canada
The Key Laboratory of Weak-Light Nonlinear Photonics, Ministry of Education and TEDA Applied Physics School,
Nankai University, Tianjin 300457, China
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Department of Physics & Astronomy, San Francisco State University, San Francisco, CA 94132, USA
*Corresponding author: [email protected]
Received April 26, 2012; revised June 16, 2012; accepted June 17, 2012;
posted June 18, 2012 (Doc. ID 167490); published 0 MONTH 0000
We demonstrate theoretically and experimentally that a finite Airy beam changes its trajectory while maintaining its
acceleration in nonlinear photorefractive media. During this process, the spatial spectrum reshapes dramatically,
leading to negative (or positive) spectral defects on the initial spectral distribution under a self-focusing (or
defocusing) nonlinearity. © 2012 Optical Society of America
OCIS codes: 190.4420, 050.1940, 350.5500.
Since the first prediction and experimental demonstration of optical Airy beams [1,2], many efforts have been
made to maintain or control their self-accelerating trajectories in both linear [3–7] and nonlinear [8–12] media. Unfortunately, nonlinearities tend to destroy the phase of
Airy beams and thus to break down the acceleration,
especially under the action of a self-focusing nonlinearity
[10,11]. Formation of accelerating self-trapped optical
beams has been proposed employing different selffocusing nonlinearities [12], and quite recently, nonlinear
accelerating Airy beams were explored experimentally
with Kerr and quadratic nonlinear media [13,4].
In this Letter we propose a scheme to control both the
trajectories and the spatial spectra of nonlinear Airy
beams. By altering the initial conditions through laterally
shifting the cubic phase mask coded on a spatial light modulator, we demonstrate experimentally and theoretically
that a finite Airy beam can change its trajectory while
maintaining its acceleration in a nonlinear photorefractive medium. Furthermore, the spatial spectrum reshapes
dramatically during nonlinear propagation leading to positive (or negative) spectral defects under a self-focusing
(-defocusing) nonlinearity. Such peculiar nonlinear phenomena may find applications in beam/pulse shaping as
well as in laser filamentation and plasma guidance.
Let us start our analysis with the one-dimensional (1D),
normalized, nonlinear equation under the paraxial and
slow varying amplitude approximations, applicable for
beam propagation under a saturable nonlinearity [10,12]:
∂Ψ ∕ ∂ξ 0.5i∂2 Ψ ∕ ∂s2 − iγΨ ∕ 1 jΨj2 ;
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(1)
where s and ξ indicate normalized transverse and longitudinal coordinates, respectively equal to x ∕ x0 and
z ∕ k0 n0 x20 , in which x0 is an arbitrary length scale, k0
is the vacuum wave number and n0 is the linear refractive
index of the photorefractive crystal. In Eq. (1), Ψ is the
complex amplitude of the optical field, and γ −k20 n0 4x20 r 33 E 0 is the normalized nonlinear coefficient
with r 33 being the electro-optics coefficient for the extraordinarily polarized beams and E 0 being the bias field.
0146-9592/12/160001-01$15.00/0
γ > 0 (<0) represents a self-focusing (-defocusing)
nonlinearity, respectively.
In the linear regime [i.e., γ 0, without the nonlinear
term in Eq. (1)], the evolution of an Airy beam can be
altered at ease through shifting the cubic phase mask
in the Fourier space [4]
ϕs;ξ 56
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p
I 0 f s;ξexp−αω2m i2α2 ωm × Aiτ − ωm ξ − ξ∕ 22 iαξ 2ωm expiωm s; (2a)
f s; ξ expαs isξ ∕ 2 −iω2m ∕ 2 iα2 ∕ 2 − 2αωm ξ
× exp−α ∕ 2 − iωm ∕ 2ξ2 − iξ3 ∕ 12;
(2b)
where I 0 is the peak intensity, α is the truncation factor,
and ωm is the shift of the cubic phase mask. From Eq. (2),
it can be seen that the Airy beam accelerates following
the parabolic trajectory s ωm ξ ξ ∕ 22 , and the peak
intensity (I 0 ) appears at ξ −2ωm . In Fig. 1(a1–a3), three
examples of numerically generated Airy beams are
shown corresponding to launching the input ϕs; ξ 0 at ωm 0, −1 and −6. As expected, the peak intensity
appears at ξ 0, 2, and 12, respectively. When the nonlinearity (jγj 2.72 is used in the whole paper) is turned
on, Airy beams (I 0 ≈ 4.08 and α 0.08) with different ωm
behave drastically different. Under a self-focusing nonlinearity (γ > 0), the evolutions of the Airy beams at
ωm 0, −1, and −6 are shown in Fig. 1(b1–b3). In the
case of ωm 0, most of the Airy beam turns into an
“off-shooting” soliton [Fig. 1(b1)]. However, at ωm −6, the Airy profile and accelerating properties are preserved [Fig. 1(b3)], while overall the transverse beam
width shrinks when compared to that associated with linear propagation depicted in Fig. 1(a3). If the absolute ωm
is not high enough, e.g., ωm −1, only part of the Airy
beam profile survives, whereas the other part undergoes
“off-shooting” along tangential directions [Fig. 1(b2)].
Detailed simulation results indicate that when ωm 0,
even a weak self-focusing nonlinearity would affect the
trajectory of the Airy beam [13,14]. However, the scheme
© 2012 Optical Society of America
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described in Fig. 1(b3) seems to be an efficient way to
maintain the acceleration property of the nonlinear Airy
beam. On the other hand, Airy beams perform much better under a self-defocusing nonlinearity (γ < 0), as the acceleration persists for all cases and even under the
condition ωm 0, although in these cases the transverse
beam width broadens slightly [Fig. 1(c1–c3)].
To have a better understanding of the nonlinear Airy
beam at ωm −6, let us look into the corresponding
spectra. Although the beam dynamics look similar in
the linear and nonlinear cases except for the lateral shifts
(towards the positive and negative s direction under a
self-focusing and -defocusing nonlinearity, respectively),
their spectral distributions have dramatic differences, as
shown in Fig. 2. The spectrum of an exponentially truncated Airy beam in the linear regime has a Gaussian
shape without any change along the propagation direction [Fig. 2(a)]. However, in the presence of a selffocusing nonlinearity, a distance-dependent self-shifting
spectral gap appears from ξ ≈ 7 to ξ ≈ 17 (regions where
the Airy beam intensity is high), as shown in Fig. 2(b).
After this spatial-spectral gap emerges, most of the power
spectra concentrate in the vicinity of the gap with surrounding ripples. However, before ξ ≈ 7 and after ξ ≈ 17,
the spectrum keeps the Gaussian distribution since the
nonlinearity has no, or only a weak, effect on the Airy
beam in low-intensity regions. Likewise, under a selfdefocusing nonlinearity, the spectrum of the Airy beam
exhibits a positive distance-dependent self-shifting defect [Fig. 2(c)]. This nonlinear spectrum reshaping phenomenon is robust even in presence of high-level noise
(results not shown here), providing an experimentally
practical yet effective way for spectrum selection, and
the idea could be readily extended to the temporal
spectral domain.
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Fig. 1. (Color online) Linear (row 1) and nonlinear (rows 2
and 3) propagation of an Airy beam under different conditions.
The panels from left to right correspond to different shifting
ωm 0, −1, and −6 of the cubic phase mask, and from top
to bottom to beam propagation under (a) linear, (b) nonlinear
self-focusing, and (c) nonlinear self-defocusing conditions.
Fig. 2. (Color online) Spatial power spectra of Airy beams at
ωm −6 during beam propagation under (a) linear, (b) nonlinear
self-focusing, and (c) nonlinear self-defocusing conditions.
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Based on recent work in [12–14], one may postulate
that the persistence of the acceleration of finite-energy
Airy beams might originate from the existence of ideal
infinite-energy nonlinear self-accelerating modes. To find
those nonlinear modes, let us define ς s − hξ2 ∕ 2 (h is
the rate of acceleration), in such a way that Eq. (1)
becomes
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∂Ψ ∕ ∂ξ − hξ∂Ψ ∕ ∂ς 0.5i∂2 Ψ ∕ ∂ς2 − iγΨ ∕ 1 jΨj2 : (3)
By inserting Ψ uς expih2 ξ3 ∕ 6 ihςξ into Eq. (3)
and letting u be a real function, we get
d2 u ∕ dς2 − 2hςu − 2γu ∕ 1 u2 0:
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(4)
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By solving Eq. (4) numerically (here we use h 1 ∕ 2), 1 133
solutions with different peak intensities are found. Typical
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results are shown in Fig. 3, where self-focusing and defo135
cusing nonlinearities lead to slightly shrinking and broad136
ening of the main Airy lobe with respect to the linear
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infinite Airy beam, in agreement with the beam pro138
pagation simulation presented in Fig. 1. The spectra of
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Figs. 3(a) and 3(d) are plotted in Figs. 3(b) and 3(e), where
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the formation of negative and positive spectral defects is
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clearly evident. Moreover, the spectrum in Fig. 3(b) has
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more noticeable ripples outside the defect compared to
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the one in Fig. 3(e). After Fourier transforming Ψ uς
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Fig. 3. (Color online) Nonlinear self-accelerating solutions
under saturable nonlinearities. (a) and (d): Self-accelerating
modes under self-focusing and -defocusing nonlinearities, respectively. (The red dashed line corresponds to a linear infinite
Airy beam); (b) and (e): Spectra corresponding to (a) and (d);
(c) and (f): spectral distribution during the nonlinear propagation of the self-accelerating modes in (a) and (d), respectively.
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August 15, 2012 / Vol. 37, No. 16 / OPTICS LETTERS
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Fig. 4. (Color online) Experimental observations of Airy
beams propagating in a biased photorefractive crystal. From
top to bottom the panels corresponds to beam propagation under (a, d) linear, (b, e) nonlinear self-focusing, and (c, f) nonlinear self-defocusing conditions. (a–c) and (d–f) correspond,
respectively, to the cases in which the peak intensity is set
at the middle (a–c) and output (d–f) of the crystal. (a2–f2) show
the averaged spectral distributions of the outputs in (a1–f1).
The downward pointing (red) arrows indicate the position of
the spectral defects in k-space.
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expih2 ξ3 ∕ 6 ihςξ, we can get Φω; ξ exp−ih2 ξ3 ∕ 6
Uω − hξ, where Φ F Ψs; ξ, U F us, and F ⋅
stands for the Fourier transform. Therefore, the power
spectrum I Φ ω; ξ jΦω; ξj2 U 2 ω − hξ can be calculated, showing that the intensity profile of the spectrum
shifts linearly towards the right with a shifting rate of
h 1 ∕ 2. This is further proved by the spectra distribution
during propagation of the nonlinear modes [see Figs. 3(c)
and 3(f)]. With regard to this aspect, the spectral defects in
Figs. 2(b) and 2(c) have similar characteristics, indicating
that the Airy beams in Figs. 1(b3) and 1(c3) tend to evolve
to nonlinear self-accelerating modes under the action of
nonlinearities, which might offer a new way (different
from that in [13]) to test the formation of nonlinear selfaccelerating modes.
The above predicted phenomena have been observed in
our experiment, carried out in a biased photorefractive
crystal with a setup similar to that used in [10], except that
here we employ a 1D cubic phase mask and a cylindrical
lens for 1D Airy beam generation. In our experiments, the
absolute value of the bias voltage is kept at 600 V across
the 5 mm-wide crystal (corresponding to jγj 2.72 as
used in our simulation). By shifting the cubic mask, we
conveniently set the peak intensity of the Airy beam in
the middle [as indicated by line II in Fig. 1(a3)] of the crystal along the propagation direction ξ [4]. Figure 4(a1)
shows the linear output profile whose main hump has
the same position of the input and locates at the right side
of the peak intensity. The dashed lines I and III in Fig. 1(a3)
mark the input and output facets of the crystal, respectively, which in turn indicate the nonlinear regime along
the propagation direction in the simulation (from ξ ≈ 7 to
17, as the propagation outside the crystal is basically linear). The dashed vertical line in Fig. 1(a3) also marks the
transversal positions of the peak intensity for the linear
beam, as does the one in Fig. 4(a1). As mentioned before,
the evolution is almost linear at ξ < 7, so the nonlinear
evolution in the experiments is similar to the simulation
shown in Figs. 1(a3–c3). Under a self-focusing nonlinearity, the main hump of the output is not self-trapped, but
keeps its acceleration [Fig. 4(b1)]. In addition, it shrinks
and shifts to the right when compared to the linear case.
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On the other hand, under a self-defocusing nonlinearity,
the main hump expands and shifts to the left when compared to the linear output [Fig. 4(c1)]. Meanwhile, negative and positive defects appear in the initial Gaussian
spectrum [presented in Fig. 4(a2)] under self-focusing
and -defocusing nonlinearities, respectively. Those defects reside at the right side of center in the Fourier space
[Figs. 4(b2) and 4(c2)]. By shifting the crystal along the
ξ-direction, it is possible to position the output facet at
the dashed line II in Fig. 1(a3). The beam evolutions
[Figs. 4(d1–f1)] have a similar behavior to those shown
in Figs. 4(a1–c1). However, in the latter case, the spectral
defects move to the center of the Fourier space due to the
actually half-shortened distance experienced in a regime
of nonlinear propagation. Our experimental results
agree well with the simulations in Figs. 1(a3–c3) and
Figs. 2(a–c).
In summary, we have demonstrated nonlinear control
of both trajectories and spatial spectra of accelerating
Airy beams. Such nonlinear control may be applicable
to nonparaxial accelerating beams studied recently
[15–17], and may prove useful for exploiting these beams
for various important applications.
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This work is supported by NSERC Discovery, and
Strategic grant, the 111 Project, NSFC, the Program
for Changjiang Scholars and Innovative Research Teams,
and by NSF and AFOSR.
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