Fractional second-harmonic Talbot effect.
ABSTRACT We demonstrate the fractional second-harmonic (SH) Talbot effect in a hexagonally poled LiTaO3 crystal. We carefully record the SH Talbot images at 1/2, 1/3, and 1/4 Talbot lengths, which are well matched with the simulated images by using the modified Rayleigh-Sommerfeld diffraction formula. A simplified model with a hexagonal array is adopted in the simulations. Also, we use a modified reciprocal vector theory to analytically explain the evolution of the SH array at fractional Talbot lengths. Our results show that the images are sensitive to the duty circle and the background of the array.
Fractional second-harmonic Talbot effect
Zhenhua Chen,1Dongmei Liu,1Yong Zhang,1,4Jianming Wen,1,2S. N. Zhu,1and Min Xiao1,3,5
1National Laboratory of Solid State Microstructures, School of Physics, School of Engineering
and Applied Science, Nanjing University, Nanjing 210093, China
2Institute for Quantum Information Science, University of Calgary, Calgary, Alberta T2N 1N4, Canada
3Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA
Received September 30, 2011; revised December 1, 2011; accepted December 15, 2011;
posted December 15, 2011 (Doc. ID 155181); published February 14, 2012
We demonstrate the fractional second-harmonic (SH) Talbot effect in a hexagonally poled LiTaO3crystal. We care-
fully record the SH Talbot images at 1∕2, 1∕3, and 1∕4 Talbot lengths, which are well matched with the simulated
images by using the modified Rayleigh–Sommerfeld diffraction formula. A simplified model with a hexagonal array
is adopted in the simulations. Also, we use a modified reciprocal vector theory to analytically explain the evolution
of the SH array at fractional Talbot lengths. Our results show that the images are sensitive to the duty circle and the
background of the array. © 2012 Optical Society of America
OCIS codes:110.6760, 190.2620, 140.3515.
The Talbot effect, a near-field diffraction phenomenon of
periodic object, was first observed by H. F. Talbot in 1836
when he illuminated a diffraction grating with a white
light source. . Up to today, the beauty and simplicity
of Talbot self-imaging effect still attracts many research-
ers. Recent progresses on fundamental Talbot effect are
made with, for example, single photons and entangled
photon pairs [2,3], waveguide arrays , metamaterials
, and electromagnetically induced grating . The ef-
fect is more than just an optical curiosity for physicists,
and has led to a variety of applications, such as photo-
lithography, optical testing, optical metrology, and array
illuminator . The extension to X-ray and terahertz
(THz) waves through Talbot interferometers [7,8] is par-
ticularly useful because of the lack of efficient optics for
these wavelengths. An electron Talbot interferometer
has also been demonstrated .
Recently, we reported a novel nonlinear Talbot effect
in periodically poled LiTaO3(PPLT) crystals, where the
self-imaging is formed by the generated second-harmonic
(SH) waves instead of the fundamental input beam .
The prerequisite condition to realize the effect is to have
a periodic SH pattern, which is fulfilled by the periodic
distribution of χ?2?in the PPLT crystals. The observations
also offer an optical way to image ferroelectric domain
structures in nonlinear crystals without damaging them.
Although a theoretical description was developed to in-
terpret the observed effect , the focus was mainly on
self-images at integer Talbot lengths . Therefore,
further studies are necessary for the self-imaging at frac-
tional Talbot lengths. We notice that few theoretical
methodologies [12–16] have been proposed for conven-
tional (linear) fractional Talbot effect. Among these
methods, the reciprocal vector theory (RVT) [12,13] gives
simple analytical solutions for images at any fractional
Talbot distances, while the Rayleigh–Sommerfeld (R-S)
diffraction formula  yields accurate numerical simu-
lations. In this Letter, we choose both methods to inter-
pret the recorded images at the fractional Talbot lengths.
We find that in order to obtain good agreements between
the experiment and theory [12,13,17], the spot size and
background of the hexagonal array have to be taken into
account in the theory as well as numerical simulations.
That is, the geometrical points and complete darkness
of the background assumed in original theories would re-
sult in discrepancies with our experimental observations.
Our work is further motivated by one potential applica-
tion to detect tiny defects or imperfections in fabricated
nonlinear crystals without destroying them.
In the experiment, a hexagonally poled LiTaO3crystal
was fabricated through an electric-field poling technique
at room temperature . The size of the sample slice is
20 mm?x? × 20 mm?y? × 0.5 mm?z?, and its domain struc-
ture with period 9 um and duty cycle ∼30% is shown in
Fig. 1. The experimental setup is shown in Fig. 2, where a
femtosecond laser operating at a wavelength of 800 nm
serves as the pump laser. It is focused by a lens with a
focal length of 200 mm and propagates along the z axis of
the crystal. The PPLT sample is placed at the focal plane
The SEM image of the hexagonally poled LiTaO3
February 15, 2012 / Vol. 37, No. 4 / OPTICS LETTERS 689
0146-9592/12/040689-03$15.00/0© 2012 Optical Society of America
of the lens. The laser spot size on the sample surface is
about 100 μm in radius. The long-pass filter (filter 1 in
Fig. 2) in front of the sample removes visible noise in the
fundamental beam. A short-pass filter (filter 2 in Fig. 2) is
placed between the crystal and the objective to filter out
the near-IR pump field. A 50× objective is used to magnify
the SH images. The SH patterns at different imaging
planes are imaged by moving the objective along the
SH propagation direction. Since the confocal length
(20 mm) is much larger than the sample thickness
(0.5 mm) in our setup, the input pump is treated as a
plane wave in both experiments and simulations.
By taking the PPLT crystal as the SH source, the dif-
fraction amplitude at a distance of z from the object
can be deduced from the R-S diffraction formula as 
where λ is the input wavelength, Ui?r
the incident light at the object plane, t?r
function, and r
ject and imaging planes, respectively. n
unit vector normal in the object plane. r
length is zt? 3a2∕λ, where a is the structure period .
We first checked the images at integer SH Talbot
planes. In our previous work , we chose the patterns
exhibiting at the output surface of the crystals as the ob-
ject. To ease the numerical simulations here, we chose
the pattern about a few micrometers from the output sur-
face as the object [Fig. 3(a)]. Figure 3(b) is the self-image
at the first SH Talbot plane. The measured SH Talbot
length is 300 μm and the calculated ztis 304 μm. The evo-
lution of the SH images from the object plane to the first
Talbot plane was recorded in the video (see Media). The
images at 1∕N Talbot planes (N ? 2, 3, 4) are carefully
measured and shown in Figs. 4(a), 4(b), and 4(c), respec-
tively. In comparison with the integer case [Fig. 3(b)],
remarkable and complicated features, such as period
change, lattice rotation, and phase shift of the array, ap-
pear in the fractional self-images.
To quantitatively understand these images, we first
built a model for t?r
as shown in Fig. 3(c). The parameters used in the model
are based on the actual object in Fig. 3(a), where the per-
iod is 9 μm and the spot size is 2.6 μm in radius. In current
experiments, the hexagonal array in our model consists
of more than 750 SH spots. Careful examination on
Fig. 3(a) indicates that, behind the periodic object, a no-
table background exists. One possible origin of this back-
ground comes from the unpoled parts of the sample. In
→? is the amplitude of
→? is the aperture
1are spatial coordinates at the ob-
→is an outward
1. After some algebra, the SH Talbot
→? to simulate the integer case 
the model, we assume a uniform background, for simpli-
city, and take Is∶Ib? 2.5∶1, where Isand Ibare the in-
tensities of the SH spot and the background, respectively.
From the simulations, we notice that although the back-
ground does not affect the period and rotation of the
array, it has important contributions to the details in
the SH self-images. The quantitative analysis is based
on the modified RVT, and the numerical simulations start
with the R-S diffraction formula by assuming uniform
phase over the object plane. The integer case is easily
calculated and understood with the model described
above. For instance, we checked the simulated image
at the first Talbot plane [see Fig. 3(d)], which is an exact
replica of the pattern in Fig. 3(c).
Next, we look at the image at the half Talbot plane, i.e.,
the SH image at z ? 152 μm. Quantitative analysis based
on the RVT [12,13] states that the image at 1∕2 Talbot
length should also be a hexagonal array but with half the
input period. However, a discrepancy appears by com-
paring Fig. 4(a) with Fig. 3(a). That is, the experiment
Fig. 2.(Color online) Experimental setup.
from the sample output surface. (b) Recorded SH self-imaging
at the first SH Talbot plane. (c) Modeled object used in our
simulations. (d) Simulated mage at the first SH Talbot plane.
(Color online) (a) SH pattern at few micrometers away
1∕N SH Talbot lengths in CCD camera (a) N ? 2, (b) N ? 3,
(c) N ? 4) and (d)–(f) their corresponding numerical si-
mulations, respectively (Media 1).
(Color online) Recorded fractional self-images at
690OPTICS LETTERS / Vol. 37, No. 4 / February 15, 2012
further shows that all the bright SH spots in the
object [Fig. 3(a)] evolve to the dark holes in the image
[Fig. 4(a)], while the dark background in the object be-
comes bright SH boundary. Alternatively, a phase shift of
π occurs at the 1∕2 Talbot length. The apparent contra-
diction between the theory and experiment can be easily
resolved by taking into account the spot size of the array.
Recall that the spot size is 2.6 um in radius while the per-
iodicity at the 1∕2 Talbot length is 4.5 um. This implies
that the fields from two nearest adjacent spots in the
image will partially overlap with each other and conse-
quently, lead to the interference. After noting this correc-
tion, the simulated image [Fig. 4(d)] using the modified
R-S formula is now consistent with the experimental
pattern [Fig. 4(a)]. The fractal structures in the bright
boundary in Fig. 4(d) were not experimentally observa-
ble due to the limitation of the intensity contrast of the
At 1∕3 Talbot plane (z ? 101 μm), the RVT predicts
that: (1)the SHimage is a hexagonal array with areduced
period equal to 3 × 31∕2μm and (2) the basis vectors of
the image lattice are rotated 30° with respect to the input.
Our experiment confirms these two conclusions, see
Figs. 4(b) and 4(e). In this case, because the period
5.2 μm is comparable to the distance between two near-
est spots, the fields from the corresponding spots do not
overlap and accordingly, no interference is produced to
further modify the image. Yet, a careful examination of
Fig. 4(b) further reveals that the center SH spot in the
marked area is relatively weaker than its six neighbors.
This finding does not show in the RVT. Our simulations
indicate that the background SH waves from the object
are responsible for such a discrepancy. Without consid-
ering the background in the object (Ib? 0), the calcu-
lated pattern would simply be a composition of SH
spots with same intensities. We also numerically find that
as the intensity of the background increases, the bright-
ness of center SH spots decreases and becomes invisible
eventually. From the measurements we experimentally
estimate Is∶Ib? 2.5∶1. This unexpected background is
detrimental in realizing meshlike SH patterns. The way
to eliminate its effect is to achieve good phase-matching
condition and high-quality fabrication.
At 1∕4 Talbot length (z ? 76 μm), the period of the im-
age reduces to 2.25 μm according to the theory. In this
case, the interference due to the fields emitted from
two neighboring spots becomes a dominant effect, and
shall be carefully counted in both theories and simula-
tions. The simulated image given in Fig. 4(f) now coin-
cides well with the recorded one, Fig. 4(c). Note that the
background is also considered in the simulation and it
leads to a hexagonal array of weak SH spots (which were
not clearly observed in the experiment). One possible
reason is that the SH background in the object, assumed
to be homogeneous in our model, actually consists of
small structures beyond the resolution of the used CCD
In summary, we have investigated the fractional SH
Talbot self-imaging effect in a PPLT crystal. In particular,
our experiment indicates that the duty cycle of the array
in the object has to be taken into account for analyzing
the image details at fractional Talbot lengths, so does the
background of SH waves. By including these two factors,
the modified RVT and R-S diffraction formula agree well
with the experiment. Our result also offers a way to
shape the fractional images details by engineering phase
matching and fabrication, which would be difficult for
the conventional case. Further, by making slight changes
to the current experimental setup, this work could allow
detecting tiny defects in crystals, which will be presented
This work was supported by the National Basic
Research Program of China (contracts 2012CB921804
and 2011CBA00205), the National Natural Science Foun-
dation of China (NSFC; grants 11004097 and 11021403),
the Fundamental Research Funds for the Central Univer-
sities (contracts 1095021339 and 1117021306), and the
Priority Academic Program Development of Jiangsu
Higher Education Institutions. J. Wen was supported
by an AI-TF New Faculty Grant and an NSERC Discovery
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