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Süli, Endre

English

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A Riemann-Hilbert problem in a Riemann surface

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Chremmos, Ioannis

Chen, Zhigang

ACMAC’s PrePrint Repository
Abruptly autofocusing waves
Nikolaos K. Efremidis and Demetrios N. Christodoulides and Ioannis
Chremmos and Zhigang Chen
Original Citation:
Efremidis, Nikolaos K. and Christodoulides, Demetrios N. and Chremmos, Ioannis and Chen,
Zhigang
(2011)
Abruptly autofocusing waves.
The Conference on Lasers and Electro-Optics (CLEO/QELS) (595153).
This version is available at: http://preprints.acmac.uoc.gr/33/
Available in ACMAC’s PrePrint Repository: February 2012
ACMAC’s PrePrint Repository aim is to enable open access to the scholarly output of ACMAC.
http://preprints.acmac.uoc.gr/Abruptly autofocusing waves
Nikolaos K. Efremidis1;, Demetrios N. Christodoulides2, Ioannis Chremmos3, and Zhigang Chen4
1Department of Applied Mathematics, University of Crete, 71409 Heraklion, Crete, Greece
2CREOL/College of Optics, University of Central Florida, Orlando, Florida 32816
3 School of Electrical and Computer Engineering, National Technical University of Athens Zografos 15773, Athens, Greece
4Department of Physics and Astronomy, San Francisco State University, San Francisco, California 94132
Email: nefrem@tem.uoc.gr
Abstract: We introduce a new classes of waves that tend to autofocus in an abrupt fashion. These
waves can be generated through the use of radially symmetric Airy waves.
c  2009 Optical Society of America
OCIS codes: (050.1940) Diffraction; (260.2030) Dispersion; (350.5500) Propagation
The focusing characteristics of optical beams have always been an issue of great practical importance. In the case
of a Gaussian wavefront the peak intensity follows a Lorentzian distribution around the focus. However, for many
applications it is crucial that a beam abruptly focuses its energy right before a target while maintaining a low intensity
proﬁle until that very moment. Ideally this should be a linear property of the wave itself and not the outcome of any
self-focusing effects. In medical laser treatments this feature may be highly desirable since the wave should only affect
the intended area while leaving any preceding tissue intact [1]. Such behavior can also be useful in suddenly “igniting”
a particular nonlinear process, such as multi-photon absorption, stimulated Raman, optical ﬁlaments in gases etc [2, 3].
To realize this fascinating prospect we introduce new families of radially symmetric (cylindrical and spherical)
waves with this desired characteristic: their maximum intensity remains almost constant during propagation, while
close to a particular focal point they suddenly autofocus and as a result their peak intensity can increase by orders of
magnitude [4].
In the 1D case the paraxial equation is known to support the following accelerating Airy beam [5, 6]:
g(x;z) = Ai
 
x   z2=4 + iz

exp[i
 
62z   6i
 
2x   z2
+ 6xz   z3
=12]: (1)
In Eq. (1), the decay parameter  ensures that the wave conveys ﬁnite energy (is thus realizable) and is typically small
so as the behavior of this wave approximates in many respects [5, 6] that of an ideal ( = 0) diffraction-free Airy
wavepacket. Perhaps the most intriguing feature of this solution is its lateral parabolic acceleration.
Let us start by considering the diffraction of a radially symmetric beam propagating in a linear medium. In the
paraxial approximation the normalized beam dynamics satisfy uz = (i=2)(ur=r + urr). The propagation of an arbi-
trary radially symmetric initial condition u(r;z = 0) = u0(r) can be computed in terms of the Hankel transform. In
particular, we analyze the dynamics of radially symmetric Airy beams of the form u0(r) = Ai(r0  r)exp[(r0  r)]
where r0 is the initial radius of the main ring. In Fig. 1(a) the amplitude of this cylindrical Airy beam is depicted as a
function of z. Our simulations show that the maximum amplitude that the beam reaches during propagation is almost
constant up to the point where the beam approaches the center [Figs. 1(a)-(b)]. Close to the focal point the power of
the ﬁrst Airy ring is concentrated in a small area around r = 0. While the peak intensity remains around unity up
0 5 10
0
50
100
150
z
I
m
a
x
0 10 20 30
0
50
100
150
200
r
0
I
m
a
x
0 2 4 6 8
−40
−20
0
20
40
60
k
S
p
e
c
t
r
u
m
I
m
a
x
I
m
a
x
S
p
e
c
t
r
u
m
z r0 k
(a) (b) (c) (d)
Fig. 1. Dynamics of radially symmetric Airy beams for  = 0:05, r0 = 10, and Imax(z = 0) = 1. (a) Propagation dynamics. (b) Maximum
intensity as a function of z. (c) Maximum intensity that the Airy beam reaches during propagation for different values of the initial radius r0. (d)
Hankel transform of the initial condition.0 10 20
0
1000
2000
3000
4000
z
I
m
a
x
0 20 40
0
1000
2000
3000
4000
5000
6000
7000
r
0
I
m
a
x
I
m
a
x
I
m
a
x
r0 z
(a) (b) (c)
Fig. 2. (Color online) (a) Isointensity hemisphere of the Airy wave. (b), (c) Same as in Fig. 1 for the three-dimensional Airy solution given by
Eq. (2) for  = 0:05, r0 = 15, and Imax(z = 0) = 1.
to z  6 it then very rapidly increases by more than 135 times at the focal point [Fig. 1(b)]. For longer propagation
distances the maximum intensity starts to decrease. In Fig. 1(c) the maximum intensity that the beam reaches during
propagation is shown as a function of r0. Note that large intensity contrasts are possible for a wide range of values of
r0. Much higher values of Imax are possible by further suppressing diffraction (decreasing ). Fig. 1(d) depicts the
Hankel transform of the input ﬁeld proﬁle of this beam as a function of the radial spectral component k.
Particularly engineered superpositions of Airy functions can exhibit enhanced properties as compared to radially
symmetric Airy beams. Here we introduce the transformations x0(r;r0;0) = rcos(0) + r0
p
2=2; y0(r;r0;0) =
 rsin(0) + r0
p
2=2; which are used to deﬁne the following radially symmetric superposition of two-dimensional
Airy beams u(r;r0;z) =
R 2
0 g(x0(r;r0;0);z)g(y0(r;r0;0);z)d0; where g(x;z) is given by Eq. (1). In comparison
with Fig. 1 we notice two main advantages of this latter conﬁguration: (a) The intensity contrasts achieved are larger.
(b) The maximum intensities of the subsequent oscillations are smaller.
Three-dimensional spherically symmetric abruptly autofocusing waves are also possible in the region of anoma-
lous dispersion provided that dispersion and diffraction effects are equalized. The corresponding normalized spatio-
temporal equation reads uz = (i=2)(2ur=r + urr) where r =
p
x2 + y2 + t2 and supports the following exact
Airy-like solution
u(r;z) = [g(r0   r;z)   g(r0 + r;z)]=r; (2)
where g(r;z) is given by Eq. (1) and close to the origin u(r;z)   2gr0(r0;z)   gr0r0r0(r0;z)(r2=3). Since the
energy is initially spread out on a 3D ring even higher intensity contrasts are attained [see Fig. 2].
Nonlinear processes and, in particular, multiphonon absorption and Kerr are the fundamental phenomena that ac-
company the aforementioned potential application of abrupt autofocusing. For this reason we study the autofocusing
dynamics in the presence of these phenomena. The interplay between Kerr nonlinearity can lead to instabilities, de-
pending on the strength of the nonlinearity. In addition, strong Kerr nonlinearities might lead to self-focusing that can
play a synergetic role to abrupt autofocusing.
References
1. T. Juhasz, F. Loesel, R. Kurtz, C. Horvath, J. Bille, and G. Mourou, “Corneal refractive surgery with femtosecond lasers,” IEEE J. Sel. Top.
Quantum Electron. 5, 902–910 (1999).
2. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21, 1729–1731 (1996).
3. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23,
382–384 (1998).
4. N. K. Efremidis and D. N. Chirstodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35, 4045–4047 (2010).
5. G. A. Siviloglou and D. N. Christodoulides, “Accelerating ﬁnite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
6. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901
(2007).

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