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908 Vol. 9, No. 8 / August 2022 / Optica Research Article
Observation of a triangular-lattice pattern in
nonlinear wave mixing with optical vortices
B. Pinheiro da Silva,1,2,* G. H. dos Santos,3A. G. de Oliveira,3
N. Rubiano da Silva,3W. T. Buono,4R. M. Gomes,5W. C. Soares,6A. J. Jesus-Silva,7
E. J. S. Fonseca,7P. H. Souto Ribeiro,3AND A. Z. Khoury1
1Instituto de Física, Universidade Federal Fluminense, 24210-346 Niterói, RJ, Brazil
2School of Science and Engineering, University of Dundee, Dundee, DD1 4HN, Scotland, UK
3Departamento de Física, Universidade Federal de Santa Catarina, CEP 88040-900, Florianóplis, SC, Brazil
4School of Physics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa
5Instituto de Física, Universidade Federal de Goiás, CEP 74690-900, Goiânia, GO, Brazil
6Núcleo de Ciências Exatas—NCEx, Universidade Federal de Alagoas, CEP 57309-005, Arapiraca, AL, Brazil
7Instituto de Física, Universidade Federal de Alagoas, CEP 57072-970, Maceió, AL, Brazil
*Corresponding author: braianps@gmail.com
Received 1 April 2022; revised 7 June 2022; accepted 25 June 2022; published 4 August 2022
Preparation, control, and measurement of optical vortices are increasingly important, as they play essential roles in
both fundamental science and optical technology applications. Spatial light modulation is the main approach behind
the control strategies, although there are limitations concerning the controllable wavelength. It is therefore crucial to
develop approaches that expand the spectral range of light modulation. Here, we demonstrate the modulation of light
by light in nonlinear optical interactions to demonstrate the identification of the topological charge of optical vortices.
A triangular-lattice pattern is observed in light beams resulting from the spatial cross modulation between an optical
vortex and a triangular shaped beam undergoing parametric interaction. Both up- and downconversion processes are
investigated, and the far-field image of the converted beam exhibits a triangular lattice. The number of sites and the lat-
tice orientation are determined by the topological charge of the vortex beam. In the downconversion process, the lattice
orientation can also be affected by phase conjugation. The observed cross modulation works for a large variety of spa-
tial field structures. Our results show that modulation of light by light can be used at wavelengths for which solid-state
devices are not yet available. © 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
https://doi.org/10.1364/OPTICA.459812
1. INTRODUCTION
The cross talk between spatial structures in nonlinear wave mixing
is widely relevant in both classical and quantum regimes. The
nonlinear optical process of parametric downconversion (PDC)
has been extensively employed to generate quantum states of light
structured in the transverse spatial degrees of freedom [1]. In the
classical regime, the same process can be operated in the stimulated
emission mode (StimPDC) [2,3], providing a convenient platform
for the design of quantum optical schemes [4–6], and for the study
of the interplay between the spatial structures of the interacting
light fields in the parametric process [7–11]. In the same way,
parametric upconversion plays an important role in a wide variety
of applications in quantum and classical optical schemes, as, for
instance, frequency conversion of squeezed light fields [12,13]
and imaging with visible and invisible light [14,15]. The spatial
structure of light beams, including the so-called optical vortex
[16], gives rise to interesting effects in upconversion [17–21].
Therefore, frequency conversion of structured light paves the way
for an increasing number of applications [22,23].
In the present work, we investigate the fields generated in
the process of parametric upconversion and stimulated down-
conversion. It is known that the nonlinear evolution of optical
vortex beams undergoing parametric up- and downconversion
is subjected to selection rules, which determine orbital angular
momentum (OAM) conservation as a ubiquitous condition
[8,17], and the appearance of radial modes as a possible side effect
depending on the relative chirality of the interacting beams [24–
28]. Both conditions naturally appear from the straightforward
calculation of the spatial overlap between the interacting modes.
However, a more appealing physical picture is to consider the
propagation properties of the outgoing field as a result of the spa-
tial cross modulation due to the nonlinear interaction between
incoming beams, which is equivalent to diffraction through an
aperture.
Exploiting this simple physical picture, we demonstrate the
occurrence of one striking effect in the diffraction phenomena of
vortex beams generated in the nonlinear optical process, namely,
the formation of a triangular lattice in far-field patterns [29–31].
2334-2536/22/080908-05 Journal © 2022 Optica Publishing Group
Research Article Vol. 9, No. 8 / August 2022 / Optica 909
We observe this outcome in both frequency up- and downcon-
version by mixing a vortex beam with a triangular shaped beam.
The triangular lattice in the converted field evinces the effect,
and the lattice orientation and number of sites are determined
by the topological charge of the incoming vortex beam. In the
downconversion process, the lattice orientation is also affected by
phase conjugation [7,32], depending on whether the vortex struc-
ture is prepared in the pump or seed beam. Our findings advance
the understanding of the role of spatial transverse structures in
light fields generated from interaction in a nonlinear medium.
Moreover, the fact that these fields have different wavelengths for
the pump and seed allows wavefront manipulation and sensing in
frequency ranges for which there are no commercial modulation
devices.
2. SPATIAL CROSS MODULATION IN NONLINEAR
WAVE MIXING
The wave mixing of two input signals inside a nonlinear crystal
generates a new field contribution, which is coherently amplified
along the interaction length, provided the phase matching condi-
tion is fulfilled. Pphase matching implies a constraint between the
wave vectors of the interacting fields [33]. In the paraxial regime,
it is useful to analyze this constraint separately in longitudinal and
transverse directions. In the case of an optically thin nonlinear
medium, the bandwidth of the longitudinal phase matching is
large, and the spatial profile of the field generated in the nonlinear
interaction is essentially determined by the product of the trans-
verse structures carried by the input beams [34]. The output field
carries the combined information of the input beams, in a situation
that is quite equivalent to usual diffraction problems, where the
field distribution immediately after an obstacle is the product
between the incident field distribution and the transmission func-
tion T(r)that characterizes the obstacle: Eout(r)=T(r)Ein (r).
Therefore, the patterns generated in nonlinear wave mixing can be
viewed as an effective diffraction problem where one input beam
plays the role of an obstacle or spatial modulator. We next analyze
the up- and downconversion processes separately, demonstrating
the striking triangular pattern formed by transmission of an optical
vortex through a triangular aperture.
Upconversion. In the upconversion configuration, two input
beams E1and E2are mixed in the nonlinear medium and generate
the output field E3, satisfying energy ω1+ω2=ω3and momen-
tum k1+k2=k3conservation. Each field component carries a
spatial structure Ej(r)(j=1,2,3) and a polarization unit vector
ˆ
ej, so that
Ej=Ej(r)ˆ
ej. (1)
The spatial structure of the upconverted beam is proportional
to the product of those carried by the incoming beams [26,27]:
E3(r)=gE1(r)E2(r), (2)
where gis the effective coupling constant. Therefore, the pattern
formed by the upconverted beam after the interaction region
corresponds to the cross modulation between the input (usually
infrared) beams. In this sense, the resulting pattern can be viewed
as the diffraction of one beam through an effective transmission
function embodied by the other. This interpretation is illustrated
in Fig. 1(left panel).
Stimulated parametric downconversion. In the stimulated
downconversion configuration (StimPDC), two input beams Ep
(pump) and Es(seed) are mixed in the nonlinear medium and
generate the output field Ei(idler), satisfying energy ωp−ωs=ωi
and momentum kp−ks=kiconservation. Each field compo-
nent has a spatial structure Ej(r)(j=p,s,i) and a polarization
unit vector ˆ
ej, as before. The spatial structure of the downcon-
verted beam is proportional to the product between the structure
carried by the pump and the conjugate of the one carried by the
seed beam [7]:
Ei(r)=gEp(r)E∗
s(r), (3)
where gis the effective coupling constant. Therefore, the pattern
formed by the downconverted beam after the interaction region
corresponds to the cross modulation between the pump and the
conjugate seed structures. In this case, the role of the effective
transmission function is played differently by the pump and seed
beams. Figure 1(right panel) illustrates the situation of having the
triangular aperture in the pump field.
3. EXPERIMENT
Upconversion setup. We start by describing the experiment of
sum-frequency generation. The experimental setup is sketched in
Fig. 2(a). The horizontally polarized Gaussian beam is produced
by a 100 mW, c.w. Nd:YAG laser (λ=1064 nm), which is split in
a beam splitter (BS). One spatial light modulator (SLM) divided
in two panels is used to produce a triangular-shaped beam, which
Fig. 1. Cross modulation of light fields in nonlinear wave mixing. Left panel: input fields (in red) with orthogonal polarizations, equal frequencies, and
different spatial structures incident on a polarizing beam splitter for second-harmonic generation (SHG). Right panel: input fields of different frequencies
and spatial structures (in red and purple) in stimulated parametric downconversion (StimPDC).
Research Article Vol. 9, No. 8 / August 2022 / Optica 910
Fig. 2. (a) Experimental scheme for spatial cross modulation in upcon-
version. BS, beam splitter; SLM, spatial light modulator; HWP, half-wave
plate; L1–L6, lenses; PBS, polarizing beam splitter; KTP, potassium
titanyl phosphate nonlinear crystal; F, bandpass filter; CCD, camera.
The power ratio between triangle and LG beam is one. (b) Measured
far-field intensity patterns for the LG input fields (top row) and for
the upconverted ones (middle row). The bottom row shows the theo-
retical upconverted patterns. The dashed white triangle illustrates the
orientation of the triangular beam.
is transmitted, and also a Laguerre–Gaussian (LG) mode that is
reflected by the BS. In both cases, we use the standard modula-
tion approach based on blazed phase gratings and forked masks
for the LG modes. To preserve the transverse structure along the
propagation to the nonlinear crystal, we use 4 fimaging lens sys-
tems ( f=10 cm) L1/L2 and L3/L4 in upper and lower paths,
respectively. The polarization for proper phase matching is set
by a half-wave plate oriented at 45◦in the path of the triangular
beam, resulting in vertical polarization. The beams are focused on
a potassium titanyl phosphate (KTP) crystal cut for type-II phase
matching using a 10 cm focal length lens. A bandpass filter is used
to prevent the non-converted infrared beams from reaching the
CCD camera, while the upconverted green light (λ=532 nm)
is imaged after collimation by a 10 cm focal length lens. Far-field
intensity profiles are registered with the camera.
Figure 2(b) displays the experimental results for LG input
beams having topological charges ranging from −3 to +3. The
images in the upper row show the measured LG beam intensity
profiles. In the second and bottom rows, the measured and theo-
retical far-field intensity patterns for the upconverted beam are
respectively shown.
StimPDC setup. We have also investigated the StimPDC proc-
ess. The sketch of the experimental setup is shown in Fig. 3(a).
We use a vertically polarized, 30 mW, c.w. 405 nm laser beam to
pump a beta barium borate (BBO) nonlinear crystal. The beam is
transmitted through a mechanical (not SLM) triangular aperture,
and then imaged in the crystal plane using a 30 cm focal length
lens. As the seed beam, we use laser light of 780 nm wavelength
and horizontal polarization. We use a SLM to shape the seed beam
as LG modes, and the SLM plane is imaged onto the crystal plane
using a 30 cm focal length lens. The pump beam is incident nearly
perpendicular to the BBO crystal surface, while the seed beam
Fig. 3. (a) Experimental scheme for spatial cross modulation in
stimulated downconversion. SLM, spatial light modulator; L7–L10,
lenses; BBO, beta barium borate nonlinear crystal; CCD, camera. The
power ratio between triangle and LG beam is two. (b) Measured far-field
intensity patterns for LG seed (top row) and idler (middle row) fields.
The bottom row shows the theoretical idler patterns. The dashed white
triangle illustrates the orientation of the triangular beam.
is incident at about 4 deg with respect it. The far-field intensity
distributions of both seed and idler beams are registered by CCD
cameras with the aid of 40 cm focal length lenses.
Similar to the upconversion measurements, we used LG seed
beams having topological charges ranging from −3 to +3. The
results are shown in Fig. 3(b). The top row displays the measured
LG beam intensity profiles. Images of the measured and theoreti-
cally calculated far-field intensity patterns of the idler beam are
shown in the middle and bottom rows, respectively.
4. DISCUSSION
Figures 2(b) and 3(b) demonstrate the good agreement between
experimental and theoretical intensity patterns. For each conver-
sion process alone, we observe the formation of a triangular lattice
with topological charge |`| = N−1, where Nis the number of
high intensity lobes at the edges, and orientation is dependent on
the sign of `. These results reinforce the physical picture presented
above, since they follow what is observed when diffracting a LG
beam through a triangular aperture [29].
The opposite orientations of the triangular lattices for upcon-
version and StimPDC emphasize the phase conjugation effect
existing only in StimPDC. Equation (3) shows the dependence of
the idler field on the phase conjugated seed field E∗
s(r). In this case,
because the seed is prepared as a LG beam with topological charge
+`, the triangular lattice formed in the idler looks like it was the
diffraction of a −`beam.
The results also show that the effective spatial modulation
in nonlinear wave mixing is of both amplitude and phase. Even
though the phase modulation effect is more clearly demonstrated
for StimPDC, it also works for upconversion.
5. CONCLUSION
In conclusion, we demonstrated triangular-lattice patterns gener-
ated by nonlinear wave mixing of an optical vortex with a triangular
Research Article Vol. 9, No. 8 / August 2022 / Optica 911
aperture-shaped beam, which works as a spatial modulation device.
The cross modulation between input optical fields in the con-
version schemes is, however, more general, and could be used to
overcome the lack of devices in certain frequency ranges, whereas
its counterpart in the visible range is readily available. Wavefront
shaping in the THz, in the extreme ultraviolet, and in the x-ray
ranges, for instance, can be achieved by using a SLM to control
the visible input field. In the THz range, for instance, nonlinear
optical conversion from visible light is already used to generate
[35] and detect [36] THz fields, and a scheme to optically control
metasurfaces generating THz radiation has been recently demon-
strated [37]. In addition, wavefront sensing of telecom and x-ray
fields can be accomplished by conversion to the visible range. The
phase information thus transferred to the visible output field can
be recorded using a common CCD camera. One example of such
application is the conversion of infrared images to visibile [15]
using upconversion. In the upconversion experiment of our work,
we use the same kind of cross modulation for a different purpose, to
demonstrate detection of topological charges.
Moreover, our scheme of StimPDC allows for filtering phase
information. Recently, Rocha et al. [38] introduced a way of
filtering the random phase from speckles through nonlinear
wave mixing. The configuration presented there works only if
the conjugate phase is present in one of the patterns, a restric-
tion that is naturally lifted in StimPDC. A possible application
would be using a random phase or medium as an encryption
key in optical communication. An image (seed beam) encodes
information, which is transferred to the idler beam in StimPDC.
The decoded information would be obtained by propagating
the idler through the key. Directly filtering the random phase
in a speckle pattern is of paramount importance also to imaging
systems, and mode sorters, where the information is commonly
achieved through computationally extensive post-processing using
statistical correlation.
Our findings advance the knowledge of the role of spatially
structured light in nonlinear wave mixing. The theoretical mod-
eling and experimental control of frequency conversion processes
is crucial in applications such as quantum communication and
quantum memories and relays [39].
Funding. Coordenação de Aperfeiçoamento de Pessoal de Nível Superior;
Instituto Nacional de Ciência e Tecnologia de Informação Quântica; Fundação
de Amparo à Pesquisa do Estado de Goiás; Fundação de Amparo à Pesquisa e
Inovação do Estado de Santa Catarina; Fundação Carlos Chagas Filho de Amparo
à Pesquisa do Estado do Rio de Janeiro; Conselho Nacional de Desenvolvimento
Científico e Tecnológico.
Disclosures. The authors declare no conflicts of interest.
Data availability. The data analyzed in the presented research are included in
the main text.
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