Multicolor Čerenkov conical beams generation by cascaded-χ(2)processesin radially poled nonlinear photonic crystalsH. X. Li, S.Y. Mu, P. Xu, M. L. Zhong, C. D. Chen et al. Citation: Appl. Phys.Lett. 100, 101101 (2012); doi: 10.1063/1.3692593 View online:http://dx.doi.org/10.1063/1.3692593 View Table of Contents:http://apl.aip.org/resource/1/APPLAB/v100/i10 Published by theAmerican Institute of Physics. Related ArticlesEffect of spatialcoherence on determining the topological charge of a vortex beamAppl. Phys. Lett. 101, 261104 (2012) Broadband and tunablemid-infrared laser source based on a transversal array of chirpedfilaments Appl. Phys. Lett. 101, 261103 (2012) Energy transferbetween few-cycle laser filaments in air Appl. Phys. Lett. 101,251111 (2012) Visual and dynamical measurement of Rayleigh-Benardconvection by using fiber-based digital holographicinterferometryJ. Appl. Phys. 112, 113113 (2012) Efficient copper vapor laserusing metal (Cu, Ag) chlorides in thermal insulation andperformance with new prismresonator configurations Rev. Sci.Instrum. 83, 123101 (2012) Additional information on Appl. Phys.Lett.Journal Homepage: http://apl.aip.org/ Journal Information:http://apl.aip.org/about/about_the_journal Top downloads:http://apl.aip.org/features/most_downloaded Information forAuthors: http://apl.aip.org/authors
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Multicolor Čerenkov conical beams generation by cascaded-v(2)processesin radially poled nonlinear photonic crystals
H. X. Li,1,2 S. Y. Mu,1 P. Xu,1,a) M. L. Zhong,1 C. D. Chen,1 X.P. Hu,1 W. N. Cui,2
and S. N. Zhu1,a)1National Laboratory of Solid StateMicrostructures and School of Physics, Nanjing University,Nanjing210093, People’s Republic of China2Department of Applied Physics,Nanjing University of Science and Technology, Nanjing210094,People’s Republic of China
(Received 11 January 2012; accepted 20 February 2012; publishedonline 5 March 2012)
We observe multiple simultaneous cascaded-v(2) Čerenkov conicalradiations in radially polednonlinear photonic crystals. By usingtwo incident fundamental waves x1 and x2, a variety ofcascadednonlinear up-conversion processes occur which result inhigh-frequency Čerenkov radiations
at 2xi þ xjði; j ¼ 1; 2Þ exhibiting as multicolor conical beams.Two types of phase-matchinggeometries with different emissionangles are demonstrated for each kind of cascaded-v(2)Čerenkovradiation. The external angle of the Čerenkov radiationexhibits strong dependence on
the fundamental wavelengths. The experimental results agree wellwith the theoretical calculations.VC 2012 American Institute ofPhysics. [http://dx.doi.org/10.1063/1.3692593]
Conical optical beams have attracted great interest in
recent years due to their wide potential applications inlight
manipulation,1–3 nonlinear conversion efficiency enhance-
ment,4 domain structure diagnosis,5–7 in vivo mouseimagingreconstruction,8 and entangled photon pairsgeneration.9–11
Nonlinear optical processes,5,6 especially nonlinear
Čerenkov conical radiations,12–14 are superior ingenerating
conical beams, where the beam front is shaped by phase-
matching conditions. Similar to Čerenkov radiation of
charged particles, the nonlinear Čerenkov radiation is also
observable at a conical wave front defined by the cone angle
hc ¼ arccosðv0=vÞ, where v0 and v are the phase velocities oftheradiation wave and the nonlinear polarization under
v > v0.15 The Čerenkov-type parametric processes arehighlyenhanced due to the existence of the domain walls and tothe
modulation of v(2) in nonlinear photonic crystals. It hadevenbeen observed in an extreme case where a single boundary
between two inversely oriented ferroelectric domains was
illuminated.16 The radiation source for the nonlinearČeren-
kov emission is a spatially extended collection of dipoles
rather than a point particle in traditional electricČerenkov
radiation. Hence, any intense wave propagating in an optical
medium may induce the Čerenkov-type nonlinear polariza-
tion. The Čerenkov second harmonic,12–16 sum-frequency,13
and difference-frequency generations in waveguides17 had
been investigated. Most of previous studies focus on single
v(2) Čerenkov radiation except for the recent reports ontheČerenkov third harmonic generation (CTHG).18–21 Herein,
we investigate the more complex cascaded-v(2) nonlinearprocesseswhich result in high-frequency Čerenkov radia-
tions. By employing two collinear incident fundamental
beams, v(2) up-conversion processes such as second har-monicgeneration and sum-frequency generation may cas-
cade, leading to simultaneous multi-frequency Čerenkov
conical radiations. For each cascaded Čerenkov radiation,
two types of phase-matching geometries are demonstrated.
A z-cut congruent LiTaO3 crystal with a thickness of0.5 mm waselectrically poled to possess a radial pattern with
an azimuthal angle of 0.3�. It has a smallest period of2.1 lm, aduty cycle of 30%, and an outer/inner diameter of3.0/1.2 mm. Figure1(a) shows its schematic radial structure
as well as the partial microscopic image of the surface-
etched sample. It can be seen that the reversed domains are
FIG. 1. (Color) (a) Schematic of the radially poled nonlinearphotonic crys-
tal (LiTaO3) and the partial optical microscopic image of thesample with an
azimuthal angle of 0.3� and a smallest poling period of 2.1 lm.The meas-ured diffraction patterns of the central and edge parts ofthe radial structure
for green incident light are also presented. (b) Schematic ofthe experimental
setup with two fundamental waves propagating along the opticaxis of the
crystal. (c) and (d) Multiple rings of Čerenkov conicalradiations for cases
with two fundamental wavelengths (signal/idler) of 1474.1/1744.7and
1456/1771 nm, respectively.
a)Authors to whom correspondence should be addressed.Electronic
addresses: [emailprotected] and [emailprotected].
0003-6951/2012/100(10)/101101/4/$30.00 VC 2012 AmericanInstitute of Physics100, 101101-1
APPLIED PHYSICS LETTERS 100, 101101 (2012)
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http://dx.doi.org/10.1063/1.3692593http://dx.doi.org/10.1063/1.3692593http://dx.doi.org/10.1063/1.3692593http://dx.doi.org/10.1063/1.3692593http://dx.doi.org/10.1063/1.3692593
uniform over the whole sample despite of its small polingpe-
riod. Figure 1(a) also displays the diffraction pattern, i.e.,the
Fourier spectrum of the radial structure under illuminationof
green light. The pattern exhibits as rings for lightincident
onto the central region of the radial structure. Thecircular
arc lines are observed for light incident on an areadeviating
from the central part. All the diffraction patterns were
recorded far from the sample. The experimental setup was
depicted in Fig. 1(b). A passively mode-locked femtosecond
oscillator combined with an optical parametric amplifiergen-
See AlsoIdentificação de Sistemas a Eventos Discretos Maxplus lineares - AutomaçãoFashion cap Imports in New Zealand New Zealand To New Zealand To New Zealand To New Zealand To New Zealand - Import data with price, buyer, supplier, HSN codeerates the 800 nm pump (pulse width: 150 fs; repetitionrate:
5 kHz). Two fundamental beams, signal (F1) and idler (F2),
are down converted from the 800 nm pump, whose wave-
lengths can be tuned between 1.3-2.1 lm. They areperpen-dicularly polarized and propagate along the z-axis ofthesample. A lens was used to loosely focus the beam, and the
sample was placed off the focus plane. The beam waist of
the fundamental waves on the surface of the crystal isaround
180 lm. Figures 1(c) and 1(d) display the typical images ofthemultiple Čerenkov rings for the cases with fundamental
wavelengths (F1/F2) of 1474.1/1744.7 nm and 1456/
1771 nm, respectively. For the former case, the powers ofthe
fundamental beams are 400/330 mW, while the total power
intensity on the sample is 3.8 TW/cm2. The magnified partial
image of Fig. 1(c) is displayed in Fig. 2(a). The ringsexhibit
higher intensity for more inner incidence of the fundamental
beams on the surface of the radially poled nonlinearphotonic
crystals. Considering that the more inner region of thecrystal
possesses greater overall area of domains walls, hence, this
observation supports the conjecture that the domain walls of
the nonlinear photonic crystals can highly enhance the
Čerenkov radiation.
The measured values of the cone angles and the frequen-
cies of the rings corresponding to fundamental wavelengths
of 1474.1/1744.7 nm are shown in Table I. These Čerenkov
conical beams can be identified as cascaded-v(2) or single-v(2)processes, respectively. The rings 1 and 2 are assigned tothesecond harmonic of the fundamental beams. The ring 1 is
infrared and can only be identified through a digitalcamera.
The external angles of rings 1 and 2 are 23.4� and 25.7�,
respectively, well obeying the Čerenkov second harmonic
relation k2cosh ¼ 2k1. The calculated values of theexternalconical angles by using Snell’s law12 are also presentedin
Table I. This type of Čerenkov radiation had previouslybeen
studied.12,13
In the following sections, we shall model and explain
the observed various cascaded-v(2) Čerenkov radiations.Suc-cessive second harmonic generation and sum-frequency gen-
eration will lead to high-frequency Čerenkov radiations at
2xi þ xjði; j ¼ 1; 2Þ, where x1 and x2 are the frequencies ofthefundamental waves. The first kind of Čerenkov radiation
is the Čerenkov third harmonic generation. The rings 3 and5
(Fig. 2(a)) which possess radiation angles of 29.3� and35.6�
along with an identical wavelength of 582 nm are third har-
monic of the idler fundamental wave (1744.7 nm). The ring
5 is so weak that it seemed to have different color with
respect to ring 3 ascribed to color distortion of the CCDcam-
era. Our theoretical calculation shows that there are two
types of phase-matching geometries (Fig. 2(b)) for the Če-
renkov third harmonic generation. Considering the small
value of v(3) for the congruent LiTaO3, the observed third
FIG. 2. (Color online) (a) Magnified
central part of the image shown in Fig.
1(d). Numbers 1-8 refer to the radiated
Čerenkov second harmonic generation,
type I and type II third harmonic genera-
tions, and type I and type II cascaded
sum-frequency generations (Table I),
respectively. (b) and (c) The schematic
diagram showing the phase-matching
geometries of the Čerenkov third har-
monic and sum-frequency generations.
TABLE I. Experimental and calculated external radiate angles ofČerenkov
ringsa.
Number
Wavelength
(nm) Type
Calculated
angle (�)Experimental
angle (�)
1 863 SH of F2 23.7 23.4
2 737 SH of F1 25.5 25.7
5 582 Type I TH of F2 35.6 35.6
3 582 Type II TH of F2 29.6 29.3
8 491 Type I TH of F1 41.9 41.9
10 491 Type II TH of F1 35.4 -
6 548 Type I SF (SH2 þ F1) 37.4 37.24 548 Type II SF (SH2 þ F1)31.9 32.27 517 Type I SF (SH1 þ F2) 39.4 39.59 517 Type II SF (SH1þ F2) 32.5 …
aF1/F2 refers to the fundamental wave with wavelength of1474.0/
1744.7 nm, and SH1 (863 nm)/SH2 (737 nm) refers to the secondharmonic
of F1/F2, respectively.
101101-2 Li et al. Appl. Phys. Lett. 100, 101101 (2012)
Downloaded 07 Jan 2013 to 180.209.15.130. Redistribution subjectto AIP license or copyright; seehttp://apl.aip.org/about/rights_and_permissions
harmonic generation (THG) should result from a two-step
cascading process.19–21 The first step involves the second
harmonic generation (SHG) and the second involves the
sum-frequency of the second harmonic and the fundamental
wave. For the first SHG process, the harmonic beam may be
collinear or noncollinear relative to the fundamental beam.
The noncollinear SHG is a Čerenkov radiation, whose longi-
tudinal component (along the z axis) of the wavevectors is
conserved. The resulting Čerenkov THG is called type I
CTHG (ring 5 for F2). The cascaded sum-frequency of the
collinear SHG and the fundamental beam results in type II
CTHG (ring 3 for F2). The collinear SHG generates at the
high peak power of the femtosecond laser although the
phase-matching condition is not strictly satisfied. Thephase-
matching conditions for these two types of Čerenkov THGs
can be expressed as
k3xcos hITH ¼ k2xcos hSH þ kx; (1)
k3xcos hIITH ¼ k2x þ kx; (2)
where kx, k2x, and k3x are wavevectors of the fundamental,secondharmonic, and third harmonic beams; and hSH, h
ITH,
and hIITH represent the emission angles of the ČerenkovSHG,type I CTHG, and type II CTHG, respectively. Then,
we have hITH ¼ arccosðn1=n3Þ for type I CTHG and hIITH¼arccosð2n2þn1
3n3Þ for type II CTHG, where n1, n2, and n3
are the refraction indices of the fundamental, second har-
monic, and third harmonic beams, respectively. It is easy to
verify that hIITH < hITH. Measured values of h
ITH and h
IITH are
35.6� and 29.3�, respectively, which are in good agreementwiththeoretical ones. The type I CTHG had been observed
recently.20 The CTHG process for the signal was observed in
our experiment too. Our theoretical calculation gives theval-
ues of the external conical angles of type I and type IICTHG
of the signal light as 35.4� and 41.9�. In experiment, onlythering 8 with a conical angle of 41.9� for type I CTHG canbeidentified. The expected ring 10 with a conical angle of
35.4� for type II CTHG cannot be well identified because itisvery weak due to phase-mismatching and that it is very
close to the much brighter ring 5 (type I CTHG of idler).The
ring 9 cannot be identified due to the same reason.
The second kind of cascaded-v(2) Čerenkov radiation isthegeneration of new frequencies at 2x1 þ x2 or 2x2 þ x1from thecombination of two fundamental beams. For each
Čerenkov radiation frequency, there also exist two types of
phase-matching geometries (Fig. 2(c)), i.e., the sum-
frequency of the noncollinear or collinear SHG of one funda-
mental beam with another fundamental beam. The former is
called type I cascade-v(2) Čerenkov sum-frequency genera-tion(CSFG) and the latter type II CSFG. The corresponding
phase-matching conditions are
kSFcos hISF ¼ k2x2 cos hSH þ kx1 ; (3)
kSFcos hIISF ¼ k2x2 þ kx1 ; (4)
where kx1 , k2x2 , and kSF are the wavevectors of thefunda-mental wave x1, second harmonics of x2, and theirsum-freqeuncy 2x2 þ x1; hSH, hISF, and h
IISF are the external radi-
ate angles of the Čerenkov SHG, type I CSFG, and type II
CSFG, respectively. We, thus, can obtain
hISF ¼ arccos�
2n2x2 þ n1x1nSFð2x2 þ x1Þ
�; (5)
hIISF ¼ arccos�
2n2x2x2 þ n1x1nSFð2x2 þ x1Þ
�; (6)
where n1, n2, n2x2 , and nSF are the refraction indices offre-quency x1, x2, 2x2, and 2x2 þ x1, respectively. It is easytoverify that hIISF < h
ISF. The measured values of h
ISF and h
IISF
are 37.2� and 32.2�, respectively, in good agreementwithcalculated values (Table I). The rings 4 and 6 with anidenti-
cal wavelength of 548 nm are assigned to the two types of
CSFG. Note that the CSFG at 2x1 þ x2 shows similarbehavior tothose of CSFG at 2x2 þ x1. Our results showthat some differentČerenkov nonlinear radiations, such as
type II CSFG of 2x1 þ x2 and 2x2 þ x1, can have an iden-ticalexternal angle, thereby their rings overlap.
The external angles of the cascade-v(2) Čerenkov conicalbeamsdepend on the fundamental wavelengths. Figure 3 dis-
plays the measured values of the external angles of the Če-
renkov CTHG and CSFG versus the fundamental
wavelengths. It can be seen that all of the external angles
decrease monotonically with increasing wavelength of sig-
nal. The external angle of type I CSFG changes only slightly
with varying fundamental wavelengths.
We observe also the cascade-v(2) Čerenkov radiations inothertypes (one-dimensional periodically poled and two-
FIG. 3. (Color online) Plot of the experimental (signs)
and calculated values (solid lines) of the external radi-
ate angles of (a) Čerenkov third harmonic generation
and (b) Čerenkov sum-frequency generation versus fun-
damental wavelengths (F1: signal; F2: idler).
101101-3 Li et al. Appl. Phys. Lett. 100, 101101 (2012)
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dimensional hexgonally poled) of nonlinear photonic crys-
tals. They exhibit different diffraction patterns withrespect
to those of the radially poled nonlinear photonic crystals,and
there exists sixfold modulation of the azimuthal intensitydis-
tribution for them. Additionally, the difference between the
refractive indices of the ordinary and extraordinary wavesof
LiTaO3 is rather small, leading to almost overlapped rings
for them; and as a result, the light of the rings generally
occupy elliptical polarization at each azimuthal angle.12,20
In summary, we have studied the simultaneously arising
colorful Čerenkov conical beams generated by cascade-v(2)
processes in radially poled nonlinear photonic crystals. Two
types of phase-matching geometries exist for each kind of
cascade-v(2) Čerenkov radiation. These nonlinear responsesofthe nonlinear photonic crystals can be tuned over a very
wide frequency range demonstrating their advantage in
obtaining conical beams with different wavelengths and dif-
ferent emission angles. Therefore, they can find wide appli-
cations in many areas such as light manipulation and domain
structure diagnosis.
This work was supported by the State Key Program for
Basic Research of China (Nos. 2012CB921802 and
2011CBA00205) and the National Natural Science Founda-
tions of China (Nos. 11104145, 91121001, and 11021403). It
was also supported by the Priority Academic Program Devel-
opment of Jiangsu Higher Education Institutions (PAPD).
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- https://www.passeidireto.com/arquivo/138331296/identificacao-de-sistemas-a-eventos-discretos-maxplus-lineares
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Introduction: My name is Nathanael Baumbach, I am a fantastic, nice, victorious, brave, healthy, cute, glorious person who loves writing and wants to share my knowledge and understanding with you.