-
Nonlinear Cherenkov radiations modulated by mode dispersion in a
Ti in-diffused LiNbO3 planar waveguide YAN CHEN,1 ZHILIN YE,1
YAODONG WU,1 YUNFEI NIU,1 RUI NI,1 XIAOPENG HU,1,* YONG ZHANG,2 AND
SHINING ZHU1 1National Laboratory of Solid State Microstructures
and School of Physics, Nanjing University, Nanjing 210093, China
2College of Engineering and Applied Sciences, Nanjing University,
Nanjing 210093, China *[email protected]
Abstract: We report nonlinear Cherenkov radiations (NCRs) in a
Ti in-diffused LiNbO3 planar waveguide. The radiations were
modulated exploiting different polarizations and orders of the
guided modes, the fundamental wavelengths and the working
temperatures. Some characteristics related to NCRs, such as
radiation angles and relative intensities were investigated in
detail. The experimental results matched well with theoretical
calculations. © 2018 Optical Society of America under the terms of
the OSA Open Access Publishing Agreement OCIS codes: (230.7390)
Waveguides, planar; (130.3730) Lithium niobate; (190.2620) Harmonic
generation and mixing; (350.5610) Radiation.
References and links 1. P. A. Cherenkov, “Visible emission of
clean liquids by action of radiation,” Dokl. Akad. Nauk SSSR 2,
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(1934). 2. S. Saltiel, Y. Sheng, N. Voloch-Bloch, D. Neshev, W.
Krolikowski, A. Arie, K. Koynov, and Y. Kivshar,
“Cerenkovtype second-harmonic generation in two-dimensional
nonlinear photonic structures,” IEEE J. Quantum Electron. 45(11),
1465–1472 (2009).
3. V. Roppo, K. Kalinowski, Y. Sheng, W. Krolikowski, C.
Cojocaru, and J. Trull, “Unified approach to Cerenkov second
harmonic generation,” Opt. Express 21(22), 25715–25726 (2013).
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“Nonlinear Čerenkov Radiation in Nonlinear Photonic Crystal
Waveguides,” Phys. Rev. Lett. 100(16), 163904 (2008).
5. C. D. Chen, Y. Zhang, G. Zhao, X. P. Hu, P. Xu, and S. N.
Zhu, “Experimental realization of Cerenkov up-conversions in a 2D
nonlinear photonic crystal,” J. Phys. D Appl. Phys. 45(40), 405101
(2012).
6. C. D. Chen, X. P. Hu, Y. L. Xu, P. Xu, G. Zhao, and S. N.
Zhu, “Čerenkov difference frequency generation in a two-dimensional
nonlinear photonic crystal,” Appl. Phys. Lett. 101(7), 071113
(2012).
7. H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear
Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev.
Lett. 108(22), 223901 (2012).
8. R. Ni, L. Du, Y. Wu, X. P. Hu, J. Zou, Y. Sheng, A. Arie, Y.
Zhang, and S. N. Zhu, “Nonlinear Cherenkov difference-frequency
generation exploiting birefringence of KTP,” Appl. Phys. Lett.
108(3), 031104 (2016).
9. M. Bazzana and C. Sada, “Optical waveguides in lithium
niobate: Recent developments and applications,” Appl. Phys. Rev.
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10. J. Lin, Y. Xu, Z. Fang, M. Wang, J. Song, N. Wang, L. Qiao,
W. Fang, and Y. Cheng, “Fabrication of high-Q lithium niobate
microresonators using femtosecond laser micromachining,” Sci. Rep.
5(1), 8072 (2015).
11. A. Yariv, Quantum Electronics (Wiley, 1975), Chap. 16. 12.
A. N. Kaul and K. Thyagarajan, “Inverse WKB method for refractive
index profile estimation of monomode
graded index planar optical waveguides,” Opt. Commun. 48(5),
313–316 (1984). 13. P. Ganguly, D. C. Sen, S. Datt, J. C. Biswas,
and S. K. Lahiri, “Simulation of refractive index profiles for
titanium indiffused lithium niobate channel waveguides,” Fiber
Integr. Opt. 15(2), 135–147 (1996). 14. P. Ganguly, J. C. Biswas,
and S. K. Lahiri, “Analysis of titanium concentration and
refractive index profiles of
Ti: LiNbO3 channel waveguide,” J. Opt. 39(4), 175–180 (2010).
15. G. J. Edwards and M. Lawrence, “A temperature-dependent
dispersion equation for congruently grown lithium
niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984). 16. A.
Gedeon, “Comparison between rigorous theory and WKB-analysis of
modes in graded-index waveguides,”
Opt. Commun. 12(3), 329–332 (1974). 17. H. Tamada, “Coupled-mode
analysis of second harmonic generation in the form of Cerenkov
radiation from a
planar optical waveguide,” IEEE J. Quantum Electron. 27(3),
502–508 (1991).
Vol. 26, No. 2 | 22 Jan 2018 | OPTICS EXPRESS 2006
#315233 Journal © 2018
https://doi.org/10.1364/OE.26.002006 Received 7 Dec 2017;
revised 16 Jan 2018; accepted 16 Jan 2018; published 18 Jan
2018
https://doi.org/10.1364/OA_License_v1https://crossmark.crossref.org/dialog/?doi=10.1364/OE.26.002006&domain=pdf&date_stamp=2018-01-18
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1. Introduction When the velocity of a charged particle exceeds
that of light in medium, coherent light would be emitted at a
specific angle, which is called Cherenkov radiation in conventional
particle physics [1]. Similar effect was applied to nonlinear
optics as the nonlinear Cherenkov radiation (NCR), which requires
that the phase velocity of the nonlinear polarization wave (NPW)
exceeds that of the harmonic wave. NCR corresponds to an
automatically accomplished phase-matching diagram, which means the
Cherenkov phase matching angle will change accordingly to the
varying of the parameters such as the input wavelength or the
working temperature. NCR can be observed in both bulk ferroelectric
crystals where the radiation is enhanced by the domain walls, and
in nonlinear waveguides where the value of the momentum uncertainty
in the thin film is more than the transverse momentum mismatch due
to uncertainty principle [2, 3]. In both material platforms,
several approaches can be exploited to modulate the behavior of
NCRs. Forward and backward reciprocal vectors parallel to the
direction of the NPW can deaccelerate or accelerate the phase
velocity of the NPWs in periodically-poled nonlinear waveguides [4]
or two-dimensional nonlinear photonic crystals [5, 6], thus
changing of the radiation angles as well as the nonlinear
conversion efficiencies. Besides, when the interacting waves are
set to be at different polarization states, i.e., using the
birefringence of ferroelectric crystals, the phase velocity of the
NPW can be modulated as well. By this means, Cherenkov type second
harmonic generation (SHG), was experimentally demonstrated in an
anomalous dispersive ultra-thin MgO doped LiNbO3 without phase
velocity threshold [7]. Following the same idea, Cherenkov type
different frequency generation (DFG), which cannot automatically be
accomplished in normal dispersion materials, was realized in a KTP
crystal [8]. In addition to utilizing the birefringence properties,
the mode dispersion in waveguides can be engineered by choosing
proper waveguide parameters, thus provide more flexibility in
modulating NCRs. In this paper, we will report our results on NCR
modulation in a Ti in-diffused LiNbO3 planar waveguide by mode
dispersion.
To fabricate optical waveguides in LiNbO3, several methods, such
as proton exchange, metal in-diffusion, femtosecond laser writing,
ion implantation, optical grade dicing, and smart cut could be used
[9, 10]. Here we chose Ti in-diffused LiNbO3 waveguide to
investigate NCRs, because both the ordinary and extraordinary
refractive indices are increased after Ti in-diffusion, thus both
TE and TM modes are supported in these waveguides. Moreover, the
fabrication procedure is relatively simple.
For Cherenkov type SHG in waveguides, it is a
guided-to-radiation nonlinear process, i.e., the fundamental wave
(FW) is in the guided mode while the second harmonic (SH) wave is
radiated into the substrate. In a z-cut, x-propagating Ti
in-diffused LiNbO3 waveguide, considering the non-zero elements of
the second order nonlinear susceptibility of LiNbO3, the y and
z-components of the second order nonlinear polarization 2P ω are
given by [11]:
2
0 22 31
2 20 31 33
2 ( 2 )
2 ( )y y y z
z y z
P d E d E E
P d E d E
ε
ε
= +
= + (1)
Ey corresponds to the electric field of the TE mode and Ez
corresponds to that of the TM mode. Equation (1) contains four
types of nonlinear processes, where the FWs take different
waveguide modes and the radiated SH waves are of different
polarizations. For simplicity, we adopt the notations which
describe the nonlinear frequency conversions involving different
polarizations in bulk crystals for the Cherenkov type SHGs: oo-o,
eo-o, ee-e, and oo-e. It should be noted that ‘e’ is not exact for
the SH wave, because there is a small angle, typically ten or more
degrees, between the radiated SH wave and the x-axis of the
crystal. The NCR’s phase-matching condition can be written as:
Vol. 26, No. 2 | 22 Jan 2018 | OPTICS EXPRESS 2007
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i ( ) ( ) (2 )cos( )j kβ ω β ω ω θ+ = (2)
Where , ,( ) 2 ( ) / ( )i j i jNeffβ ω π ω λ ω= ⋅ , (2 ) 2 (2 )
/ (2 )k nω π ω λ ω= ⋅ . ,i jNeff denotes the effective refractive
index of two guided modes at the FW, n is the refractive index of
the SH wave in the substrate, and θ is the internal Cherenkov
angle. Equation (2) is satisfied only when the wave-vector mismatch
k ( ) ( ) (2 )i j kβ ω β ω ω∆ = + − is less than zero. The cladding
layer of Ti in-diffused LiNbO3 waveguide is the air, while the
refractive index of the guided layer is around 2.2, so Cherenkov SH
waves can only radiate into the substrate, instead of into the air,
due to the phase-matching requirement.
The refractive index profile of a Ti in-diffused LiNbO3 planar
waveguide is determined by the thickness of the Ti film, the
diffusion temperature and the diffusion time. After the waveguides
have been fabricated at certain process parameters, NCRs could be
modulated by changing the working temperature, the input wavelength
and exploiting different modes involved in the nonlinear
interactions.
The waveguide used in our experiment was fabricated by Ti
in-diffusion. A sample with dimensions 20 × 10 × 0.5mm3 along X, Y,
Z axes was prepared from a congruent LiNbO3 wafer. After cleaning,
an 80-nm thick Ti layer was deposited on the –z face of the sample
by electron-beam evaporation. Then Ti in-diffusion was carried out
at 1050 °C for 9 hours in air ambient. The end facets of the
waveguide were polished for optical measurements. Subsequently, the
effective refractive index was characterized through a prism
coupler at 632.8 nm. With the inverse WKB analysis [12], the
diffusion depth was fitted to be 4.03 μm, thus we could obtain the
refractive index profiles at different wavelengths according to
[13–15]. We calculated the mode dispersion by the WKB analysis
[16], finding four guided modes (TM0, TM1, TE0 and TE1) were
supported at 1064 nm, while three modes (TM0, TM1 and TE0) were
supported at the longer wavelength of 1342nm in this waveguide.
Figure 1(a) shows the temperature dependent dispersion relations of
the FW at 1064 nm and the SH wave at 532 nm. It is evident that for
ee-e, oo-o, and eo-o processes, Eq. (2) is automatically completed
since the relation n(2ω)>Neffi,j(ω) is always satisfied. While
for the oo-e process, as shown in Fig. 1(b), NCR is phase-matched
at the temperatures above 70°C when the FW is in the TE0 mode,
while the critical phase-matching temperature goes down to 20°C
when the FW is in the TE1 mode.
Fig. 1. (a) Temperature dependent dispersion relations of the FW
at 1064nm and SH wave at 532 nm; (b) Zoom in view of the dispersion
relations when the FW is in the TE0 and TE1 guided mode, while SH
is e-polarized.
Similar calculations were carried out at 1342 nm and the results
were plotted in Fig. 2. One can see that three types of NCRs, which
are ee-e, oo-o, and eo-o, can meet the NCR phase-matching
requirements. However, for the oo-e process, NCR cannot occur at
the temperatures below 200°C where ,(2 ) ( )i jn Neffω ω< .
Vol. 26, No. 2 | 22 Jan 2018 | OPTICS EXPRESS 2008
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0 50 100 150 200
Temperature/℃
2.14
2.16
2.18
2.2
2.22
2.24
2.26
2.28
2.3
(Eff
ectiv
e) R
efra
ctiv
e In
dex
no (2ω)
n e (2ω)
NeffT E 0
(ω)
NeffT M 0
(ω)
NeffT M 1
(ω)
Fig. 2. Temperature dependent dispersion relations of the FW at
1342nm and SH wave at 671 nm.
2. Experimental setup The schematic experimental setup is shown
in Fig. 3. The fundamental source is a Q-switched Nd:YVO4 laser
operating at 1064nm/1342nm with a pulse width of 20ns/30ns and a
repetition rate of 10 kHz. The polarization of the near infrared
beam was controlled by a half wave plate and was focused into the
planar waveguide through a cylindrical lens with a 5-mm focal
length. The waveguide was put in an oven for accurate temperature
control. The generated NCRs were displayed on a screen which was
placed 85 mm away from the end facet of the sample. Normally, one
can observe two groups of SH spots symmetrically displayed on both
sides of the horizontal line of the waveguide, and it can be
explained as the following. Because the internal radiation angles
are normally less than 20°, so part of the SH waves will experience
multiple total internal reflections in the LiNbO3 substrate and
exit from the end facet of the waveguide, see the inset in Fig. 3.
The lower spots on the screen came from the SH waves directly
hitting the end facet, as well as the ones last reflected at the
upper surface of the substrate before hitting the end facet of the
waveguide. While the upper spots came from the SH waves last
reflected from the lower surface of the substrate.
Fig. 3. Schematic experimental setup. The inset on the upper
left illustrates the origination of the two SH spots on the
screen.
3. Results and discussions Firstly, the FW was chosen to be at
1064 nm with 45-degree polarization and the temperature of the
waveguide was set to 50°C. Four groups of NCRs should be observed
in total, which corresponded to the four processes utilizing
different nonlinear coefficients described in Eq. (1). As shown in
left part of Fig. 4, group , and can be clearly observed and each
group of radiations included two or more Cherenkov radiation spots,
which were attributed to Cherenkov SHG or sum frequency generation
(SFG) between different guided modes. Group and were Cherenkov type
frequency up-conversions via the oo-e and ee-e process,
Vol. 26, No. 2 | 22 Jan 2018 | OPTICS EXPRESS 2009
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where the FWs in each group were in the waveguide modes with
same polarization but different orders. As for group , the spots
therein were eo-o type Cherenkov SFGs between different order TE
and TM modes. The radiations in Group through the oo-o process were
much weaker than the radiations in group , which can be seen from
the relative output intensities in Table 1. In experiment, to
obtain a suitable contrast ratio between group and which were close
to each other, we set the polarization of the FW to be mainly at
the horizontal, thus group could be visualized as shown in the
middle part of Fig. 4.
Fig. 4. Cherenkov SHGs and SFGs at the fundamental wavelength of
1064nm. The left part illustrates the radiation patterns on the
screen. The middle part is a zoom in view of the radiations in
group , and . The right part gives the phase-matching types and the
involved guide modes for the four groups of radiations.
Figure 5 shows the behavior of the oo-e type NCR at different
temperatures. There was only one pair of spots induced by Cherenkov
SHG of the FW in the TE1 mode at 25°C, between which was a bright
spot generated from the collinear guided-to-guided SHG process.
When the temperature was increased to 50°C, a new pair of spots
appeared, which were generated through Cherenkov SFG of the TE1 and
TE0 modes. Meanwhile, the collinear SHG spot weakened obviously
because the working temperature was far from the collinear
phase-matching temperature at about 20°C. When the temperature was
further increased to 75°C, the phase-matching condition for
Cherenkov SHG of the FW in the TE0 mode was satisfied, which was
consistent with our predictions in Fig. 1(b).
Fig. 5. oo-e type NCRs at different temperatures. The guide
modes involved are indicated beside the radiation spots.
The measured external radiation angles varying with the working
temperature of the waveguide was given in Fig. 6. With temperature
increasing, the radiation angles of the oo-e process became larger,
while for ee-e, eo-o, and oo-o processes the radiation angles
remained
Vol. 26, No. 2 | 22 Jan 2018 | OPTICS EXPRESS 2010
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almost the same because temperature has little influence on the
wave-vector mismatches for the three processes.
20 40 60 80 100 120 140 160
Temperature/℃
0
10
20
30
40
50
Ext
erna
l NC
R a
ngle
/°
ee-eTM0+TM0
eo-oTM0+TE0
oo-oTE0+TE0
oo-eTE0+TE0
oo-eTE1+TE1
Fig. 6. Experimental (the circle) and theoretical (the line)
external radiation angles varying with temperatures for different
types of NCRs.
In order to investigate the behavior of the NCRs in the Ti
in-diffused waveguide at different wavelengths, we changed the
incident light to a nanosecond pulsed 1342 nm laser. Then three
kinds of radiation spots was observed, as shown in Fig. 7, when the
input beam was 45° polarized. Group , and were obtained through
eo-o, oo-o and ee-e type NCRs respectively. The number of radiation
spots in each group was determined by the guided mode supported at
1342 nm, where were TE0, TM0 and TM1 as mentioned previously. The
external NCR angles of the ee-e, oo-o and eo-o processes, where the
FW only took the lowest order guided modes, were measured to be
26.3°, 29.7° and 39.5°, which matched well with the calculated
values 26.0°, 29.7° and 39.0°, respectively. In experiment, oo-e
type NCR could not be observed in the temperature range from 25 °C
to 200 °C, which was consistent with the theoretical predictions in
Fig. 2.
Fig. 7. Three groups of NCRs at the fundamental wavelength of
1342nm were shown in the left part. Right part is the zoom in view
of the radiation spots in group and . The guide modes involved are
indicated beside the radiation spots.
We also investigated the efficiencies of the NCRs. By applying
the nonlinear coupled-mode theory [17], the output power of the
NCRs can be written as:
2 (2)
2 2 1[ ]tanSH FW SH
dP P Sωβ θ
∝ ⋅ ⋅ ⋅ (3)
where FWP is the power of the FW coupled into the waveguide, SHβ
is the propagation constant of the SH wave, (2)d denotes the second
order nonlinear coefficient. S is the
Vol. 26, No. 2 | 22 Jan 2018 | OPTICS EXPRESS 2011
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normalized overlap integral between the FW and SH
electromagnetic field. Assuming FWP to be the same, we can
calculate the relative radiation intensities for different
processes at 50 °C. As shown in Table 1, oo-e type NCR at 1064 nm
has the most intense SH output, which has a small radiation angle
and a large overlap integral due to small phase mismatch, though
the nonlinear coefficient for this process is not the biggest. The
output of the two ee-e processes were less intense, which utilizes
the largest nonlinear coefficient. As for oo-o and eo-o processes,
the small nonlinear coefficients, small overlap integrals and large
radiation angles lead to far less SH powers than the former two
processes. Calculations were in good accordance with the
experimental observations.
Table 1. Calculated relative output intensities of NCRs
Wavelength 1064 nm 1064 nm 1064 nm 1064 nm 1342 nm 1342 nm 1342
nm Type of NCR ee-e
TM0 + TM0
eo-o TM0 +
TE0
oo-e TE1 + TE1
oo-o TE0 + TE0
ee-e TM0 + TM0
eo-o TM0 +
TE0
oo-o TE0 + TE0
(2) ( / )d pm V 33 27d = 31 4.5d = 31 4.5d = 22 2.7d = 33 27d =
31 4.5d = 22 2.7d = 2 2( / )S V A m 0.038 0.013 0.197 0.010 0.093
0.027 0.026
1/ tanθ 3.97 2.97 28.00 3.56 5.10 3.48 4.48 Normalized
Intensity (a.u.)
1.000 0.002 4.944 0.001 4.896 0.007 0.003
4. Conclusions Ti in-diffused LiNbO3 is a promising platform for
integrated nonlinear photonics, and in this paper, a comprehensive
theoretical and experimental study on NCRs in a Ti in-diffused
LiNbO3 planar waveguide was presented. The mode dispersions were
numerically calculated to find out the phase-matching requirements
for NCRs. In experiment, we observed rich phenomena of NCRs which
were realized through Cherenkov type second harmonic generation of
the same guided mode as well as sum frequency generation between
guided modes with different polarizations and different orders. The
dependence of the radiations on the wavelength of the input FW and
the working temperature were also studied and experimentally
confirmed. The relative output intensity of the radiations were
mainly determined by the second-order nonlinear coefficient, the
overlapping integral and the radiation angle, and the numerical
results were consistent with experimental observations. The
phase-matching conditions required in this work are for Cherenkov
type frequency up-conversions. When the requirements are not
satisfied, for example, below some critical temperatures in Fig.
1(b), Cherenkov type frequency down-conversions could be realized.
Thus, Cherenkov type different frequency generations (DFG) and
spontaneous parametric down conversions (SPDC) based on mode
dispersion tailoring might be further research topics in Ti
in-diffused LiNbO3 waveguides, which would provide compact tunable
laser light sources and entangled photon sources respectively.
Funding National Key R&D Program of China (2017YFA0303700);
National Natural Science Foundation of China (NSFC) (11674171,
91636106, 11627810, 11621091); Jiangsu Science Foundation
(BK20151374); Fundamental Research Funds for the Central
Universities (020414380064); PAPD of Jiangsu Higher Education
Institutions.
Vol. 26, No. 2 | 22 Jan 2018 | OPTICS EXPRESS 2012
References and links1. IntroductionFig. 1. (a) Temperature
dependent dispersion relations of the FW at 1064nm and SH wave at
532 nm; (b) Zoom in view of the dispersion relations when the FW is
in the TE0 and TE1 guided mode, while SH is e-polarized.Fig. 2.
Temperature dependent dispersion relations of the FW at 1342nm and
SH wave at 671 nm.2. Experimental setupFig. 3. Schematic
experimental setup. The inset on the upper left illustrates the
origination of the two SH spots on the screen.3. Results and
discussionsFig. 4. Cherenkov SHGs and SFGs at the fundamental
wavelength of 1064nm. The left part illustrates the radiation
patterns on the screen. The middle part is a zoom in view of the
radiations in group , and . The right part gives the phase-matching
t...Fig. 5. oo-e type NCRs at different temperatures. The guide
modes involved are indicated beside the radiation spots.Fig. 6.
Experimental (the circle) and theoretical (the line) external
radiation angles varying with temperatures for different types of
NCRs.Fig. 7. Three groups of NCRs at the fundamental wavelength of
1342nm were shown in the left part. Right part is the zoom in view
of the radiation spots in group and . The guide modes involved are
indicated beside the radiation spots.Table 1. Calculated relative
output intensities of NCRs4. Conclusions