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Abruptly autofocusing and autodefocusing optical beams with Abruptly autofocusing and autodefocusing optical beams with
arbitrary caustics arbitrary caustics
Ioannis D. Chremmos
Zhigang Chen
Demetrios N. Christodoulides
University of Central Florida
Nikolaos K. Efremidis
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Recommended Citation Recommended Citation
Chremmos, Ioannis D.; Chen, Zhigang; Christodoulides, Demetrios N.; and Efremidis, Nikolaos K., "Abruptly
autofocusing and autodefocusing optical beams with arbitrary caustics" (2012).
Faculty Bibliography
2010s
. 2421.
https://stars.library.ucf.edu/facultybib2010/2421
PHYSICAL REVIEW A 85, 023828 (2012)
Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics
Ioannis D. Chremmos,
1,*
Zhigang Chen,
2
Demetrios N. Christodoulides,
3
and Nikolaos K. Efremidis
1
1
Department of Applied Mathematics, University of Crete, Heraklion 71409, Crete, Greece
2
Department of Physics and Astronomy, San Francisco State University, San Francisco, California 94132 USA
3
CREOL/College of Optics, University of Central Florida, Orlando, Florida 32816 USA
(Received 10 November 2011; published 22 February 2012)
We propose a simple yet efficient method for generating abruptly autofocusing optical beams with arbitrary
caustics. In addition, we introduce a family of abruptly autodefocusing beams whose maximum intensity suddenly
decreases by orders of magnitude right after the target. The method relies on appropriately modulating the phase
of a circularly symmetric optical wavefront, such as that of a Gaussian, and subsequently on Fourier-transforming
it by means of a lens. If two such beams are superimposed in a Bessel-like standing wave pattern, then a complete
mirror-symmetric, with respect to the focal plane, caustic surface of revolution is formed that can be used as an
optical bottle. We also show how the same method can be used to produce accelerating 1D or 2D optical beams
with arbitrary convex caustics.
DOI:
10.1103/PhysRevA.85.023828 PACS number(s): 42.30.Kq, 42.25.Fx
I. INTRODUCTION
Recently, a family of optical waves was introduced with
abruptly autofocusing (AAF) properties [
1]. The new waves
[also termed circular Airy beams (CABs)] have a circularly
symmetric initial amplitude that oscillates outward of a dark
disk, like an exponentially truncated Airy function. By virtue
of the two outstanding features of finite-power Airy beams
[2,3], namely their self-acceleration and their resistance to
diffraction, this eventually results in light beams that can prop-
agate over several Rayleigh lengths with minimum shape dis-
tortion and almost constant maximum intensity, until an abrupt
focusing takes place right before a target, where the intensity is
suddenly enhanced by orders of magnitude. These theoretical
predictions were subsequently verified by experimental obser-
vations [
4,5]. These observations also demonstrated that AAF
beams could outperform standard Gaussian beams, especially
in settings where high-intensity contrasts must be delivered
under conditions involving long focal-distance-to-aperture
ratios (f numbers) [4]. In addition, this “silent” or low intensity
mode, at which AAF waves approach their target, makes them
ideal candidates for medical laser applications where collateral
tissue damage is supposed to be kept at a minimum. Other
possible applications include laser waveguide writing in bulk
glasses and particle trapping and guiding [5].
The focusing mechanism of AAF beams is fundamentally
different from that of Gaussian beams. In the latter case, the
wave’s constituent rays form a sharpened pencil that converges
at a single point, the focus. As the beam’s cross-sectional
area gradually decreases, the maximum intensity over the
transverse plane increases in a Lorentzian fashion, centered
at the focus. In the case of AAF beams, however, the rays
responsible for focusing are emitted from the exterior of a dark
disk on the input plane and stay tangent to a convex caustic
surface of revolution (SOR) that contracts toward the beam
axis [1]. By virtue of its Airy transverse amplitude profile, this
caustic is intrinsically diffraction-resisting, therefore keeping
its maximum intensity almost constant during propagation and
*
jochremm@central.ntua.gr
its interior almost void of optical energy. Focusing occurs as a
result of an on-axis collapse of this SOR and is as abrupt as the
transition from the dark to the lit side of an optical caustic. At
the point of “collapse,” the rays emitted from a certain circle
on the input plane interfere constructively and the gradient of
the field amplitude is maximized.
Building on the concept of a collapsing caustic SOR, the
AAF wave family was recently broadened to include general
power-law caustics that evolve from a sublinearly chirped
input amplitude [
6]. In this latter work, we showed that a
νth-power caustic requires the input amplitude to oscillate
with a chirp of the order β = 2 − ν
−1
, which generalizes the
case of CABs, whose parabolic trajectory is a result of the
3/2 chirp of the Airy function itself. Although they are not
non-diffracting as ideal CABs, these pre-engineered beams
were shown to exhibit attractive features, such as enhanced
focusing abruptness, larger intensity contrasts, and suppressed
post-focal intensity maxima.
To benefit from the attractive properties of AAF waves,
it is crucial to generate them efficiently. This is generally
a nontrivial task, since these waves evolve from initial
amplitudes that are not easy to implement directly, such as
concentric Airy rings. A possible alternative is to generate
the Fourier transform (FT) of the initial condition first and
then inverse-Fourier transform it by means of a lens. A similar
approach was adopted in the first demonstration of CABs [
4],
where the initial wavefront was produced by encoding both
amplitude and phase information on a phase-only filter. The
FT technique was also employed in Ref. [7], where a hologram
of the FT was produced. In addition in this latter work, the FT
of CABs was treated analytically and found to behave like a
Bessel function whose argument is enhanced by a cubic phase
term, i.e., a quadratic chirp. By tuning the strength of the chirp
relative to the lens’ focal distance, it was possible to generate
AAF beams with two foci, the one being defocusing while the
other one focusing, thereby defining the ends of an elegant
paraboloid optical bottle [7].
From the above it is clear that producing AAF beams can
be equally difficult, either in the real or in the Fourier space,
since, in both cases, a complicated initial condition, varying
both in phase and in amplitude, must be produced. However,
023828-1
1050-2947/2012/85(2)/023828(8) ©2012 American Physical Society
CHREMMOS, CHEN, CHRISTODOULIDES, AND EFREMIDIS PHYSICAL REVIEW A 85, 023828 (2012)
when the requirement for the exact generation of a specific
AAF wave can be relaxed, the implementation procedure
can be significantly simplified. Indeed, having explained the
AAF mechanism through both ray and wave optics [
1,6], one
realizes that the critical phenomenon is the formation of the
caustic and depends primarily on the phase modulation of
the input wavefront and secondarily on the envelope of its
amplitude. This fact has already facilitated the generation of
arbitrary convex 1D or 2D beams, by applying the appropriate
1D or 2D phase mask on a plane wave [
9]oronaGaussian
beam [
10]. This is perhaps the most obvious way to directly
produce accelerating 1D or 2D caustics. However, in some
applications, it could be more advantageous to apply the same
concept in the Fourier space. A major reason is that the FT lens
provides an additional degree of freedom for easily targeting
and sizing the generated beam [7]. Indeed, as noted in the
latter paper, the optical bottles produced with the FT method
can be arbitrarily scaled without losing their symmetry or
having to modify the input condition, by simply changing
the lens’ focal distance. This is a true advantage of the FT
approach compared to the direct (real-space) approach, which
requires that all beam characteristics must be prescribed on the
phase mask, while even a simple scaling of the beam requires
redesign of the phase modulation. In addition, the lens offers
a physical 2f separation between object and image planes,
which could be useful in applications when the phase mask
cannot be positioned very close to the target.
In this work, we present a simple yet general method for
generating in the Fourier space AAF or abruptly autodefocus-
ing (AADF) optical beams with pre-engineered caustic SOR.
From a certain viewpoint, this generalizes our previous work
on CABs and their parabolic caustics [
7] to AAF waves with
arbitrary convex caustics. We hereby show that such beams
can be produced efficiently by simple means of a circularly
symmetric phase mask and a FT lens, acting successively on
an optical wavefront with no particular amplitude information.
The required phase is the sum of a linear and a nonlinear
term. The linear term is responsible for creating an annular
focusing pattern on the image plane, while the nonlinear
term is responsible for transforming this pattern into an Airy
pattern, thus determining the shape of the caustic produced
before or after that plane. Special attention is paid to the
case of power-law caustics, for which analytical results are
readily obtained and reported. In particular, we find that a
νth-power caustic requires an nth-power phase with the orders
being related as n = (2ν − 1)/(ν − 1). Counterintuitively, for
agivenν, the stronger the nonlinear phase is, the weaker the
acceleration of the caustic, hence, the longer the distance from
the image plane to the target. The linear phase and the lens’
focal plane determine the size of the produced AAF beam in
terms of its width at the image plane and also in terms of the
distance from that plane to the target.
In Sec.
II, we describe the proposed method analytically,
and subsequently, in Sec. III, we support it with numerical
calculations that demonstrate the generation of AAF and
AADF beams and also that of optical bottles with sinusoidal
shape. Although our focus is on AAF waves, we complete this
paper by showing that the same method can actually be used
to produce accelerating 1D or 2D beams with arbitrary convex
trajectories.
II. ENGINEERING AAF AND AADF WAVES IN THE
FOURIER SPACE
To begin, consider the amplitude of a phase-modulated
circularly symmetric wavefront
u
0
(r) = A(r)exp
[
i(r)
]
= A(r)exp
[
iar + iq(r)
]
, (1)
where r is the polar distance normalized by an arbitrary
length, say x
0
,A(r) is a real envelope, and (r) is the phase
consisting of a linear term with slope a>0 and a nonlinear
(e.g., power-law) term q(r). Such an initial condition can be
easily realized by reflecting a plane wave or a collimated
Gaussian beam on the face of a spatial light modulator (SLM)
programmed with a phase (r). Subsequently, let us examine
the evolution of this wave through the single-lens FT system
of Fig.
1 under the validity of the paraxial approximation
2u
z
= i∇
2
t
u, where subscript t stands for transverse and z is
the propagation distance normalized by 2πx
2
0
/λ, λ being the
optical wavelength. Propagating the waves before and after the
lens according to the Fresnel diffraction integral [
6], and taking
into account the quadratic phase exp(−ir
2
/2f ) imprinted on
the wave transmitted through the lens, it can be shown that the
optical field on the image plane (z = 2f, where f is the focal
length) reads
u(r,z = 2f ) =−(i/f )U
0
(r/f ), (2)
where
U
0
(k) =
∞
0
u
0
(ρ)J
0
(kρ)ρdρ (3)
is the Hankel transform of the object wavefunction u
0
(ρ).
Obviously, Eq. (
2) expresses the FT property of the lens.
Substituting Eq. (3) into Eq. (2) and using the familiar integral
representation of Bessel function
J
0
(
x
)
=
1
2π
2π
0
exp(−ix cos ϕ)dϕ, (4)
r
r
2
0
r
r
2
r
r
0
f
r
1
r
0
1
2
z
f
r
c
FIG. 1. (Color online) Schematic of a ray traveling through a
single-lens FT system. The lens is positioned at z = f. The red
(convex) curve indicates the caustic formed beyond z = 2f.
023828-2
ABRUPTLY AUTOFOCUSING AND AUTODEFOCUSING ... PHYSICAL REVIEW A 85, 023828 (2012)
we obtain
u(r,z = 2f ) =−
i
2πf
∞
0
2π
0
A(ρ)
× exp
iaρ + iq(ρ) − i
ρr
f
cos ϕ
ρdρdϕ,
(5)
where u
0
was substituted from Eq. (
1). Even in its simplified
form of Eq. (3), this integral cannot be evaluated analytically
for a general function q(ρ). Hence, it is reasonable to resort to a
stationary-phase (SP) computation, which is justified by the os-
cillatory nature of the integrand. Assuming that q
′
(ρ) > 0for
all ρ>0, where the prime denotes the derivative with respect
to the argument, it is readily seen that there is only one station-
ary point (ρ
s
,ϕ
s
) = (r
0
,0), where r
0
is the solution of equation
a + q
′
(r
0
) = r/f. (6)
After some algebra, the result of integration in the neigh-
borhood of (ρ
s
,ϕ
s
)is
u
SP
(r,z = 2f ) =
r
0
fq
′′
(r
0
)r
A(r
0
)
× exp
iar
0
+ iq(r
0
) − i
r
0
r
f
. (7)
The last two equations lead to two important conclusions:
First, Eq. (
6) implies r>af,which means in essence that
only points on the image plane lying outside that disk have
appreciable amplitude. Second, since r
0
is a function of the
observation point r,Eq.(
7) shows that the wave amplitude
on the image plane is also nonlinearly phase-modulated.
Differentiating the phase of Eq. (7) with respect to r and using
Eq. (6), one obtains −r
0
/f < 0; i.e., the phase modulation is
of converging nature. Therefore, the wave on the image plane
satisfies the preconditions for evolving into an AAF wave.
If the phase of Eq. (7) is properly designed, then an inward
bending caustic SOR with initial width 2af will be formed and
eventually focus abruptly somewhere in the half-space z>2f.
Further understanding of this process can be gained through
a ray optics interpretation of the propagation dynamics.
Referring to Fig.
1, let us follow the path of the ray starting
from point (r
0
,0) on the input plane. According to Eq. (1),
this ray travels at an angle θ
0
with the z axis and reaches the
surface of the lens (which is assumed to be infinitesimally thin)
at (r
1
,f
−
) where r
1
= r
0
+ fs
0
and
s
0
= tan θ
0
=
′
(r
0
)(8)
is the corresponding slope. Passing through the lens, the
ray deflects inward and emerges from point (r
1
,f
+
), with
a modified slope s
1
= tan θ
1
= s
0
− r
1
/f . The transmitted ray
crosses the image plane at r
2
= r
1
+ fs
1
with a slope tan θ
2
=
s
2
= s
1
. Combining the above we obtain the equations
r
2
= fs
0
,s
2
=−r
0
/f , (9)
connecting the exit position and slope of a ray (r
2
,s
2
)tothe
input values (r
0
,s
0
). Equations (
9) are equivalent to the conclu-
sions reached previously, noting that the point here named r
2
is
the observation point r of Eqs. (
6) and (7). As (r
0
,s
0
(r
0
)) vary
continuously along the input plane, the transmitted rays form
a caustic that is expressed with coordinates (r
c
,ξ
c
), where r
c
is
the radial distance, ξ
c
= z − 2f is the distance from the image
plane, and (r
c
,ξ
c
) is interpreted as the point at which ray (r
2
,s
2
)
touches the caustic. From Fig.
1, the following equations are
also obvious:
s
2
= r
′
c
(ξ
c
),r
c
= r
2
+ ξ
c
s
2
. (10)
Differentiating the second of Eqs. (
10) with respect to ξ
c
and using Eqs. (
9), it can be shown that the caustic is expressed
in terms of the input ray characteristics as
(
r
c
,ξ
c
)
= (fs
0
(r
0
) − fr
0
s
′
0
(r
0
),f
2
s
′
0
(r
0
)), (11)
where r
0
serves as a parameter and s
0
(r
0
) is given by Eq. (
8).
Equations (
8) and (11) provide the means for a direct design
approach, namely to determine the caustic resulting from a
given input phase modulation. Alternatively, one could work
inversely and find the input ray parameters associated with a
desired caustic (r
c
(ξ
c
),ξ
c
). Again, from Eqs. (9) and (10)we
obtain directly
(r
0
,s
0
) =
−fr
′
c
(ξ
c
),
1
f
[r
c
(ξ
c
) − ξ
c
r
′
c
(ξ
c
)]
, (12)
where the parameter now is ξ
c
. From Eq. (
12) it is evident that,
for r
′′
c
(ξ
c
) < 0, i.e., for a convex caustic, we have r
′
0
(ξ
c
) > 0,
which ensures that the rays touching the caustic at different
points do not overlap on the input plane. This allows us to invert
function r
0
(ξ
c
) and determine the phase (r
0
) associated with
ray characteristics (r
0
,s
0
). Integrating Eq. (8) by introducing
the new variable ξ
c
one gets
(r
0
) =
ξ
c
(r
0
)
0
s
0
(ξ)r
′
0
(ξ)dξ
=
ξ
c
(r
0
)
0
[ξr
′
c
(ξ) − r
c
(ξ)]r
′′
c
(ξ)dξ, (13)
where ξ
c
(r
0
) is the inverse of function r
0
(ξ
c
), and functions
s
0
(ξ),r
0
(ξ) were obtained from Eqs. (
12) by substituting ξ for
ξ
c
. Using Eq. (
13), one can determine the phase (r
0
) that must
be programmed into the SLM to produce the desired caustic
SOR r
c
(ξ
c
).
A characteristic case is that of a power-law phase q(r),
which leads to a power-law caustic also. Setting q(r) = br
n
and eliminating r
0
from Eqs. (
11), the equation of the caustic
r
c
(ξ
c
) reads
r
c
= f [a − d(ξ
c
/f
2
)
ν
], (14)
where d = ν
−1
[n(n − 1)b]
1−ν
,ν = (n − 1)/(n − 2), and ξ
c
>
0. For n>2, we have ν>1, and hence a convex caustic
SOR with a waist that starts from a maximum 2af at ξ
c
= 0
to vanish on axis at ξ
c
= f
2
(a/d )
1/ν
= L. As was shown in
Ref. [
6], the point ξ
c
= L, at which the caustic collapses, is an
inflection point for the wave amplitude, i.e., a point where the
amplitude gradient along the beam axis has a local maximum.
Being very close to the focus, this point also determines
approximately the distance between the image plane and the
target. Therefore, the range of the beam and its maximum waist
size can be adjusted through the lens’ focal length f , which is
one of the advantages of the FT approach, as mentioned in the
introduction. Equation (
14) also shows that, for a given power
023828-3
CHREMMOS, CHEN, CHRISTODOULIDES, AND EFREMIDIS PHYSICAL REVIEW A 85, 023828 (2012)
ν, a larger b, i.e., a stronger nonlinear phase modulation, results
in a smaller d, i.e., a weaker accelerating caustic. This is a
rather counterintuitive property stemming from the FT relation
between object and image waves. Also counterintuitively, the
order of the caustic ν is a decreasing function of the order n
of the nonlinear phase term; as a result, the required phase
for higher-power caustics (ν = 3,4,5,...) is of subcubic order
(n = 5/2,7/3,9/4,...).
Here we would like to comment on the role of the linear
phase component in Eq. (
1). As indicated by Eqs. (6) and (14),
this term is responsible for the formation of the dark disk on
the image plane. Indeed, in the absence of q(r), the problem
reduces to Fourier-transforming a wave with a linear radial
phase, a situation encountered when, for example, working
with Bessel beams. In this case, no caustic is formed, but rather
all rays leave the object plane in parallel and are focused by
the lens on a circle on the image plane where a thin bright
annulus appears (a circle for idealized Bessel beams). The
inclusion of the nonlinear phase component subtly disturbs
this perfect ray focusing, in such a way that a smooth convex
caustic is formed and the AAF phenomenon is generated. The
single bright annulus on the image plane then transforms to
the pattern of concentric Airy rings. In conclusion, the linear
term is needed to obtain a circular focusing pattern, which
the disturbance of the nonlinear phase transforms to a CAB-
like pattern that evolves into an AAF wave. A closed-form
approximation of the beam amplitude close and exactly on
the caustic SOR can be obtained by a SP computation of the
Fresnel integral of Eq. (7). As happens in other families of
AAF waves [6], the field near the caustic is contributed by
two close stationary points on the input plane (z = 0), which
collapse into a single second-order stationary point when the
field is observed exactly on the caustic. The result is an Airy
amplitude profile. The same method can be used to find the
field at the focus, which is now contributed by a continuum of
points lying on a circle on the input plane. In the Appendix,
an outline is given of how analytical expressions for the field
in different regions can be obtained.
Returning to Eq. (
14), it is interesting to note that AAF
beams with parabolic caustics (ν = 2) require the input
wavefront to be modulated with a cubic phase (n = 3). This
is not a surprise if one takes into account our recent analytical
results on the FT of CABs [7]. In this work, the FT (expressed
as a Hankel transform) of the CAB Ai(R − r), was found to
behave as B(k)J
0
(kR + k
3
/3), where B(k) is a complicated
super-Gaussian envelope. From the asymptotic behavior of
Bessel function, it follows that, for large k, the FT behaves
proportionally to cos(kR + k
3
/3), i.e., as a real envelope
modulated with a cubic phase. Hence, from the viewpoint of
the present work, the parabolic body of a CAB is a by-product
of the cubic phase modulation of its spectrum, which can be
considered as the analogue of the same FT property of 1D Airy
beams [
3].
The caustic SOR of Eq. (14) develops in the half-space z>
2f and the generated beam is AAF with its focus occurring
on axis shortly after z = 2f + L.This is a result of the phase
modulation in Eq. (
1) being of diverging nature. If, instead,
the complex conjugate initial condition u
∗
0
(r) is assumed
(converging phase modulation), then the entire field in the half-
space behind the lens becomes −u
∗
(4f − z); i.e., it is mirrored
with respect to the focal plane and the mirror-symmetric of
Eq. (
14) caustic is formed in f<z<2f. Moreover, as a result
of the converging phase modulation itself, another caustic SOR
is formed before the lens (z<f), having the expression r
c
=
z(dz
−ν
− a). This caustic is asymptotic (varies as z
1−ν
)tothe
input plane and, after passing through the lens, it experiences
an inward slope discontinuity and transforms into a power-law
caustic. If additionally the beam parameters are chosen so that
L<f ⇔ d>af
ν
, then the transmitted power-law caustic
collapses at z = 2f − L, a point of maximum but negative
amplitude gradient, thus imparting to the transmitted beam an
AADF character. In the spirit of our previous work [
7], we
term this condition as the weak-chirp regime. On the other
hand, when L>f ⇔ d<af
ν
, the chirp is strong enough to
make the caustic collapse before the lens at z = (d/a)
1/ν
and
no focus occurs after the lens.
As shown in Ref. [
7], if the input amplitude is properly
engineered, the generated CAB can have two foci, the first
being AADF and the second AAF. As a result, an elegant,
perfectly mirror-symmetric optical bottle is formed between
the two foci, which can be used as an optical trap. Optical
bottles can also be built by the approach presented here,
however, with some additional effort, by letting the input
beams u
0
(r) and u
∗
0
(r) interfere in a standing wave pattern
of the form A(r) cos[(r)]. In that case, each of the two
components creates half of the full caustic. The parameters
should, of course, be tuned in the weak-chirp regime, so that
the bottle lies entirely behind the lens. An illustrative example
is given in Sec. III for an optical bottle with sinusoidal shape.
III. NUMERICAL EXAMPLES
To illustrate our analytical arguments, we devote this
section to numerical simulations. Note that, in all of the
following figures, the spatial coordinates are normalized. To
give a sense of the beam’s actual extent, typical values for
the length scales can be x
0
= 50 μm in the transverse and
2πx
2
0
/λ = π cm in the longitudinal direction, at a wavelength
around λ = 500 nm.
Let us first demonstrate the AAF and AADF mechanisms
through the ray optics picture. Figure
2(a) presents the results
of ray tracing for a wave with the phase modulation of Eq. (1)
and the parameters n = 3,a = 1,b= 1/3000, being Fourier-
transformed by a lens with f = 10 (also in 2πx
2
0
/λ units).
The chirp parameter has been chosen to satisfy b = 1/(3f
3
),
resulting in the formation of the parabolic caustic r
c
= 10 −
ξ
2
c
/4 (indicated with a dashed curve), which is familiar from
1D Airy beams [
3]. In Fig. 2(b), the corresponding ray pattern
is depicted for a beam with the same parameters but with
the complex conjugate input amplitude. Note how the exactly
symmetric caustic now develops in f<z<2f, and also the
caustic r
c
= z(250z
−2
− 1) developing in 0 <z<f.The two
curves meet at the lens’ plane with different slopes, as a result
of the lens’ focusing action.
Wave simulations of the two previous configurations are
showninFigs.3 and 4, respectively, as obtained by numerically
solving the paraxial equation of propagation. In both cases,
the envelope of the input beam has been assumed to be the
Gaussian A(r) = exp(−r
2
/45
2
).The results clearly verify our
expectations from the ray-optics approach. In Fig.
3(a),the
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