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Nonlinear Talbot Effect
Yong Zhang,
1,2,3
Jianming Wen,
1,2
S. N. Zhu,
1
and Min Xiao
1,2,3,
*
1
National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
2
Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA
3
School of Modern Engineering and Applied Science, Nanjing University, Nanjing 210093, China
(Received 20 January 2010; published 7 May 2010)
We propose and experimentally demonstrate the nonlinear Talbot effect from nonlinear photonic
crystals. The nonlinear Talbot effect results from self-imaging of the generated periodic intensity pattern
at the output surface of the crystal. To illustrate the effect, we experimentally observed second-harmonic
Talbot self-imaging from 1D and 2D periodically poled LiTaO3crystals. Both integer and fractional
nonlinear Talbot effects were investigated. The observation not only conceptually extends the conven-
tional Talbot effect, but also opens the door for a variety of new applications in imaging technologies.
DOI: 10.1103/PhysRevLett.104.183901 PACS numbers: 42.30.Kq, 42.65.Ky, 42.70.Mp
The Talbot effect [1,2], a near-field diffraction phenome-
non in which self-imaging of a grating or other periodic
structure replicates at certain imaging planes, holds a
variety of applications in imaging processing and synthe-
sis, photolithography, optical testing, optical metrology,
spectrometry, optical computing [3], as well as in electron
optics and microscopy [4]. Recent progress has been made
in areas such as atomic waves [5,6], nonclassical light [7],
waveguide arrays [8], and x-ray phase imaging [9].
However, all the above achievements are limited in study-
ing properties of input fundamental signals. In this Letter,
we demonstrate, for the first time, the nonlinear Talbot
effect, i.e., the formation of second-harmonic (SH) self-
imaging instead of the fundamental one from
periodically poled LiTaO3(PPLT) crystals. This demon-
stration not only maintains all characteristics of the con-
ventional Talbot effect, but offers a new way to image
objects with periodic structures with higher spatial resolu-
tion. The conceptual extension achieved here thus opens a
door for broader scopes of applications in imaging
techniques.
The conventional Talbot effect is well understood by the
Fresnel-Kirchhoff diffraction theory, as first explained an-
alytically by Lord Rayleigh in 1881 [2], attributing its
origin to the interference of diffracted beams. The sim-
plicity and beauty of such Talbot self-imaging have since
then attracted many researchers and resulted in numerous
interesting and original applications that represent com-
petitive solutions to various scientific and technological
problems. The optical self-imaging phenomenon usually
requires a highly spatially coherent illumination. The self-
imaging disappears when the lateral dimensions of the
light source are increased. As noted a long time ago,
when the source is made spatially periodic and it is placed
at the proper distance in front of a periodic structure, a
fringe pattern is formed in the space behind the structure.
The first example of this type was performed by Lau [10],
who used two amplitude gratings of the same spatial period
illuminated incoherently. However, up to today, all the
research on the self-imaging has been limited in studying
the properties of the input beams and using real gratings
for imaging. Bypassing these limitations will not only
enrich the conventional self-imaging research, but also
offer new methods for imaging technologies. Here, we
present the first experimental demonstration beyond the
conventional Talbot effect, in which the observed self-
images are not produced by the input fundamental beam
but by the SH field generated in PPLT nonlinear crystals. In
our experiment, the grating is the periodic intensity pat-
terns appearing on the output surface of the crystal due to
the periodic domain structures, i.e., the modulated second-
order nonlinear susceptibility. This difference thus distin-
guishes our results from the conventional self-imaging
research.
PPLT crystals have been extensively used as a work-
horse for laser frequency conversion [11], optical switch-
ing [12], wave-front engineering [13], and quantum
information processing [14]. Numerous interesting phe-
nomena have been discovered in both 1D and 2D nonlinear
photonic crystals, such as solitons [15], entangled photons
[16], conical second-harmonic generation (SHG) [17,18],
and nonlinear C
ˇerenkov radiation [19]. Benefitting from
their modified nonlinear properties, here we report a novel
SH Talbot self-imaging demonstrated with the use of such
crystals. Different from the conventional Talbot effect
[1,3], the observed self-imaging is a consequence of the
ð2Þnonlinear optical process, which we call the nonlinear
Talbot effect. As mentioned before, the Talbot effect is
attributed to the interference of diffracted beams from
periodic structures, a prerequisite condition to realize
such an effect [1–3]. In PPLT crystals, this condition is
fulfilled by the periodic patterns of the SH intensity differ-
ence distributed on the output surface. The SH intensity
generated from the domain walls is different from that
inside domains because the nonlinear coefficients near
the nonideal domain walls will be changed due to the
PRL 104, 183901 (2010) PHYSICAL REVIEW LETTERS week ending
7 MAY 2010
0031-9007=10=104(18)=183901(4) 183901-1 Ó2010 The American Physical Society
crystal lattice distortion after the poling process [20]. In the
experiment, we have directly observed periodically distrib-
uted SH intensity patterns in recorded self-images.
Although the observed phenomenon well resembles the
conventional Talbot effect, several interesting features dis-
tinguish this demonstration from the previous observa-
tions. Besides no real grating used in the experiment,
spatial resolution improvement by a factor of 2, due to
frequency doubling, is powerful to high-resolution imag-
ing, compared with the simple input-pump imaging. In
principle, if the material allows the Nth-order harmonic
generation, spatial resolving power can be enhanced by N
times using the nonlinear Talbot effect reported here.
Moreover, this experiment conceptually extends the con-
ventional Talbot effect and thus paves a way for new
applications.
The 1D and 2D PPLT slices were fabricated through an
electric-field poling technique at room temperature [21]. In
the experiments, we used a femtosecond mode-locked
Ti:sapphire laser operating at a wavelength of 800 nm as
the fundamental input field. The pulse width is 100 fs
with a repetition rate of 82 MHz. As shown in Fig. 1, the
fundamental beam from the Ti:sapphire laser was first
reshaped by a telescope device, composed of two focusing
lenses, to achieve a near-parallel beam with a spot size of
100 m, which propagates along the zaxis of the PPLT
sample and whose polarization is parallel to the xaxis of
the crystal. Although LiTaO3crystal has a space group of
3mðC3vÞ, only the d21 component contributes to the SHG
process in our experimental configuration [22]. After the
sample, a bandpass filter was used to filter out the near-
infrared fundamental field. The generated SH intensity
pattern was magnified by an objective lens and projected
onto a CCD camera. The SH patterns at different imaging
planes were recorded by moving the objective lens along
the SH propagation direction, which was controlled by a
precision translation stage. We emphasize that the non-
linear Talbot effect is a lensless imaging process. The
objective used here is to magnify the self-images for easy
observations.
Figs. 2and 3, respectively, show the integer SH self-
images from 1D and 2D PPLT crystals. In the experiment,
we have sequentially recorded such integer Talbot self-
images at about several Talbot lengths. For simplicity, we
arranged d21 as the only nonzero nonlinear coefficient in
the tensor to contribute to the collinear SHG process. To
verify this, we measured the polarization of the generated
SH wave and found that its polarization was indeed along
the yaxis of the crystal. As described above, d21 at the
domain walls is different from that inside the domains.
Such periodic domain structures result in the generated SH
intensity patterns displaying the same periodicity at the
sample output surface. An imaginary grating is thus
formed for self-imaging. From the Fresnel-Kirchhoff dif-
fraction theory, the diffracted field amplitude Að~
r1Þis
defined in terms of the aperture function of the object
tð~
rÞand the coherent amplitude of the source Sð~
rsÞ. Here
~
r1,~
r, and ~
rsare located at the observation, object, and
source planes, respectively. The diffracted amplitude Að~
r1Þ
is given by
Að~
r1Þ¼exp½2iðd1þd2Þ=
id1d2Zd~
rsSð~
rsÞZd~
rtð~
rÞ
expij~
r~
rsj2
d1expij~
r1~
rj2
d2;(1)
where d1is the propagation distance between the object
and the source, and d2is the distance from the object to the
observation plane. In our experiment, the SH source may
be treated as a plane wave or a Gaussian beam. The
aperture function comes from the spatially periodic domain
structures, i.e., the periodically engineered ð2Þ. One no-
table difference from conventional self-imaging is that in
Eq. (1) is the SH wavelength, which is half of the wave-
length pof the fundamental input beam. This difference
leads to a factor of 2 in imaging resolution improvement,
compared with the traditional direct input-output
measurement.
In our first experiment, a 1D PPLT sample with a domain
period of a¼8:0mand a duty cycle of 50% [see the
SEM picture shown in Fig. 2(a)] was chosen to illustrate
the effect. Figs. 2(b) and 2(c) are the recorded self-images
on the first and third Talbot planes, respectively. For a
plane-wave illumination, the self-images repeat at multi-
ples of the SH Talbot length zT¼4a2=p, which gives
320 mand 960 mfor the first and third Talbot imaging
planes. The experimentally measured lengths were
330 m5mand 1020 m10 m, respectively.
FIG. 1 (color online). Experimental setup. The PPLT sample is
placed at the waist plane of the fundamental wave. The patterns
on different imaging planes are recorded by a CCD camera
through moving the objective.
FIG. 2 (color online). 1D SH Talbot self-imaging. (a) The
domain structure of the 1D PPLT crystal; (b) the SH self-image
at the first Talbot plane; (c) the SH self-images at the third Talbot
plane. The marked domain in (a) is a little narrower than the
others.
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The discrepancy between the experiment and theory can be
readily corrected by taking into account the Gaussian
profile of the beam, which reinterprets the SH Talbot length
as zT¼4ða2=pÞðrz=r0Þ2, where rzand r0represent the
Gaussian beam radii at the imaging plane and the waist
plane. In the experiments, the PPLT sample was placed at
the waist plane of the Gaussian beam. The corrected theo-
retical lengths are now 320 mwith rz=r0¼1:00 and
1018 mwith rz=r0¼1:03, consistent with the experi-
mental data. Periodic SH interference fringes are clearly
observable on the first Talbot plane [Fig. 2(b)] and the
period coincides with the periodicity of the domain struc-
ture. The bright fringes correspond to the domains exactly.
The intensities from positive or negative domains have no
appreciable difference because the value of the nonlinear
coefficient inside the domains does not change during the
poling process. The dark fringes correspond to the domain
walls. In the self-imaging, the SH intensity produced at the
domain walls is much weaker than that inside the domains.
From Fig. 2(b), the width of the domain walls is estimated
to be 0:5m, which agrees with the reported range from
100 nm to few microns [23]. One difference between the
SEM image [Fig. 2(a)] and the self-image [Fig. 2(b)] is that
the SEM imaging is sensitive to the imperfections in the
domain structure (the marked square) while the self-
imaging produces a uniform interference pattern. It is
because only the periodically distributed SH light is self-
imaged at the Talbot planes while the nonperiodic part is
not reproduced. This could be useful for the application of
optical lithography. We also experimentally found that the
image qualities after the third Talbot plane became worse
than those at the first and second planes. One reason is that
higher-order diffraction fields cannot be totally collected
by the CCD camera with a limited numerical aperture.
To illustrate the integer SH Talbot effect from a 2D
modulated nonlinear crystal, a hexagonally-poled LiTaO3
slice, was adopted in the second experiment. The period
of the domain structure of the sample is a¼9:0mand
the duty cycle is 30% as shown in the SEM image in
Fig. 3(a). The SH self-images were also successively ob-
served at several Talbot imaging planes. For the ideal
plane-wave illumination, the Talbot length is deduced to
be zT¼3a2=p. Taking into account the Gaussian profile,
the SH Talbot length is corrected to be zT¼3ða2=pÞ
ðrz=r0Þ2. Figs. 3(b) and 3(c) show the images recorded at
the first and third SH Talbot planes, respectively. Same as
for the 1D case, the SH waves generated at the domain
walls are weaker than that created inside the domains. The
domain wall in the 2D structure exhibits a ring-shaped
structure [Fig. 3(b)], while it appears as a dark line in the
1D PPLT [Fig. 2(b)]. Similar as the 1D structure, the
qualities of the self-images after the third Talbot plane
[Fig. 3(c)] become worse due to the loss of higher-order
SH diffractions. The measured first and third SH Talbot
lengths were 315 m5mand 970 m10 m,
which agree well with the calculated 304 m(rz=r0¼
1:00) and 966 m(rz=r0¼1:03), respectively.
Besides these integer SH Talbot effect, in the third
experiment we have also investigated the fractional SH
self-images occurring at the intermediate Talbot distances,
z¼ðp=qÞzT; where pand qare integers with no common
factor. It is well known that fractional Talbot effect [24–26]
has a close connection with fractional revivals, quantum
carpet [27], and Gauss sums [28]. Compared with the
integer self-imaging, fractional SH Talbot effect exhibits
more interesting and complicated interference patterns. We
have experimentally observed such patterns for both 1D
and 2D PPLT crystals. In the 1D case, because of the
limited illumination area on the sample, the quality of frac-
tional self-images is greatly reduced and becomes blurry in
contrast with the integer cases [Figs. 2(b) and 2(c)]. The
situation is dramatically changed for the 2D case, where
the SH factional Talbot images have high quality and show
number of unique properties in comparison with the integer
case. In Figs. 4(a)–4(f), we present several representative
fractional self-images with different fractional (p=q) pa-
rameters. For instance, the SH interference pattern at the
zT=7plane [Fig. 4(a)] is periodically distributed bright
spots with the same lattice structure as in the integer
case. In Figs. 4(b) and 4(c), the period of the hexagonal
structure is halved. In Figs. 4(c)–4(f),a30
rotation of the
FIG. 3 (color online). 2D SH Talbot self-imaging. (a) The
domain structure of the hexagonally poled LiTaO3crystal;
(b) the SH self-image at the first Talbot plane; (c) the SH self-
images at the third Talbot plane.
FIG. 4 (color online). The SH patterns (a)–(f ) are correspond-
ing to fractional Talbot images at different fractional Talbot
planes.
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structure is obviously observable and even a -phase shift
can be deduced from Figs. 4(d)–4(f). We found that the
fractional Talbot images are very sensitive to the precision
positions of the imaging planes. Further detailed studies of
these fractional SH Talbot effects will be presented
elsewhere.
In recent years, SH imaging technique has been devel-
oped into a powerful near-field imaging tool for visualizing
various ferroic domains [29], carrier motion [30], biomo-
lecular array [31], and collagen modulation [32].
Comparing with the conventional SH imaging, such SH
Talbot self-imaging does not need an imaging lens and no
reference SH wave is needed, which greatly simplifies the
experimental setup. For periodically-poled ferroelectric
domains, although chemical etching method is adopted
as a standard procedure to look at the domain structures
[33], one major disadvantage of this method is that the
sample surface is damaged. The newly observed nonlinear
Talbot effect provides a better (optical) way to easily check
the domains without damaging the sample surfaces. This
will be very helpful for inspecting integrated nonlinear
optical devices such as nonlinear photonic waveguides
and wavelength conversion devices, which cannot be
done by the conventional Talbot effect. More importantly,
the conceptual generalization demonstrated here is not
limited to the optical signals, and could also apply to other
research fields where similar situations may exist. This
effect can also be further considered with nonclassical light
states [7] to achieve sub-Rayleigh images. Generally
speaking, the demonstrated effect not only enriches con-
ventional imaging techniques, but also offers a new method
for imaging in broad applications.
We acknowledge partial support from the National
Science Foundation (U.S.A.) and partial support by the
111 Project B07026 (China).
*mxiao@uark.edu
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