Terahertz (THz) wave generation from bulk and periodically poled LiTaO
_{3}
(PPLT) with a Cherenkov phase matching scheme is numerically investigated. It is shown that by using the crystal birefringence of bulk LiTaO
_{3}
and a grating vector of PPLT, THz waves can be efficiently generated by difference frequency generation (DFG) with a Cherenkov phase matching scheme. The frequency tuning characteristics of the THz wave via varying wavelength of difference frequency waves, phase matching angle, poling period of PPLT and working temperature are theoretically analyzed. The parametric gain coefficient in the lowloss limit and the absorption coefficient of the THz wave during the DFG process in the vicinity of polariton resonances are numerically analyzed. A THz wave can be efficiently generated by utilizing the giant second order nonlinearities of LiTaO
_{3}
in the vicinity of polariton resonances.
I. INTRODUCTION
In the study of modern terahertz (THz) optoelectronics, monochromatic THz sources play an important role in high resolution THz applications, such as environmental gas monitoring and highdensity and highspeed wireless communications
[1

4]
. Difference frequency generation (DFG) with two closely spaced laser frequencies
ω
_{1}
and
ω
_{2}
in secondorder nonlinear optics crystals is one of the promising processes for efficient monochromatic THz wave output in the wide frequency tuning range and for room temperature operation
[5
,
6]
. Recently, highefficiency THz wave generation based on DFG employing Cherenkov phase matching has been intensively researched
[7

9]
. THz wave generation with Cherenkov phase matching is an attractive scheme because strong THz absorption originated from lattice vibration modes of the nonlinear optical crystal can be overcome. Unfortunately, the maximum frequencies generated by DFG with a Cherenkov phase matching scheme are limited to 7.2 THz
[10]
. Moreover, the output power of the THz wave within the highfrequency band is extremely low. Such limitations on output frequencies and power are caused by the dramatically increased absorption of the nonlinear materials in the vicinity of polariton resonances. However, at the same time, polariton resonances can induce giant second order nonlinearities
[11

13]
. The giant second order nonlinearities in the vicinity of polariton resonances can be exploited to extend the output frequencies and enhance the output power of the THz wave. In this paper, we explore THz wave generation based on DFG in the vicinity of polariton resonances with a Cherenkov phase matching scheme. A surfaceemitted configuration is employed to overcome the strong absorption problem in the vicinity of polariton resonances. The nonlinear crystals utilized in this paper are bulk and periodically poled LiTaO
_{3}
(PPLT). To the best of our knowledge, THz wave generation based on DFG utilizing bulk LiTaO
_{3}
and PPLT with a Cherenkov phase matching scheme has not been investigated. The advantage of utilizing LiTaO
_{3}
over LiNbO
_{3}
lies in the fact that LiTaO
_{3}
has a significantly reduced rate of opticallyinduced index change damage due to the photorefractive effect
[14]
. According to Ref.
[14]
, the optically induced indexchange damage increases at rates of 2.8×10
^{−3}
cm
^{2}
/μW and 4.9×10
^{−4}
cm
^{2}
/μW in LiNbO
_{3}
and LiTaO
_{3}
, respectively. The second advantage is that the poling electric field required for LiTaO
_{3}
is one order of magnitude lower than that for LiNbO
_{3}
[15]
. The coercive fields are 1.7 kV/mm and 21 kV/mm for stoichiometric LiTaO
_{3}
and congruent LiNbO
_{3}
, respectively
[15]
. In this paper, a Cherenkov phase matching scheme is realized by using the crystal birefringence of bulk LiTaO
_{3}
and a grating vector of PPLT. We numerically simulate the frequency tuning characteristics via varying pump wavelengths, phase matching angle, poling period of PPLT and working temperature. Parametric gain coefficient in the lowloss limit of the THz wave during the DFG process in the vicinity of polariton resonances is numerically analyzed.
II. THEORETICAL MODEL
The schematic drawing of surfaceemitted DFG in bulk LiTaO
_{3}
and PPLT is presented in
Fig. 1
. As shown in
Fig. 1(a)
, two input optical waves,
λ
_{1}
and
λ
_{2}
, at the frequencies of
ω
_{1}
and
ω
_{2}
in the infrared (IR) domain are propagating collinearly along the
X
axis in a bulk LiTaO
_{3}
crystal and are polarized along the
Z
and
Y
axes, respectively. The two optical waves are located close to the lateral surface (
XY
) of the crystal. A THz wave is emitted due to the oscillating nonlinear polarization at the difference frequency between the two input waves. The radiation angle
α
between directions of the optical and THz wave propagation is determined by the refractive index of input optical waves in the crystal, and the refractive index of the THz wave,
Schematic diagram of the surfaceemitted DFG with a Cherenkov phase matching scheme. (a) LiTaO_{3}, (b) PPLT.
where
K
_{THz}
=
ω
_{THz}
n
_{THz}
/
c
is the wave vector,
n
_{THz}
is the refractive index at
ω
_{THz}
frequency;
K
_{IR}
=
n
_{1}
^{e}
(
θ
)
ω
_{1}
/
c

n
_{2}
^{o}
ω
_{2}
/
c
,
n
_{1}
^{e}
(
θ
) and
n
_{2}
^{o}
are the extraordinary and ordinary refractive indices of optical waves
λ
_{1}
and
λ
_{2}
at
ω
_{1}
and
ω
_{2}
frequencies, respectively.
θ
is the angle between the directions of the optical wave propagation and the optical axis of LiTaO
_{3}
. For the THz wave generation based on DFG, the energy conservation condition has to be fulfilled
where
λ
_{THz}
is the wavelength of the THz wave. From Eq. (1), it follows that the THz wave is emitted perpendicular to the directions of the optical wave propagation if
K
_{IR}
=0. The effective nonlinear coefficient
d_{eff}
in the DFG process is given by
where
d
_{15}
and
d
_{22}
are the nonlinear coefficients, the angle
φ
is an azimuthal angle of the optical wave vector with respect to the
X
axis.
As shown in
Fig. 1(b)
, two input optical waves,
λ
_{1}
and
λ
_{2}
, at the frequencies of
ω
_{1}
and
ω
_{2}
in the infrared (IR) domain are propagating collinearly along the
X
axis in the PPLT crystal. Both
λ
_{1}
and
λ
_{2}
are polarized along the
Z
axis. The two optical waves are located close to the lateral surface (
XZ
) of the crystal. The optical axis of the LiTaO
_{3}
crystal parallels the
Z
axis. A THz wave can be generated in a direction perpendicular to the optical wave propagation if poling period Λ of the PPLT satisfies
The parametric gain coefficient
g
_{0}
in the lowloss limit during DFG processes in cgs units can be determined by the following expression
[16]
:
where
ω
_{THz}
is the absorption coefficient in the THz region,
ω
_{0j}
,
S_{j}
and Γ
_{j}
denote eigenfrequency, oscillator strength of the polariton modes and the bandwidth of the
j
th A
_{1}
symmetry phonon mode in the LiTaO
_{3}
crystal, respectively.
I
_{λ1}
is the power density of the optical wave
λ
_{1}
.
d'
_{E}
and
d'_{Q}
are nonlinear coefficients related to pure parametric (secondorder) and Raman (thirdorder) scattering processes, respectively. In this letter, the data for LiTaO
_{3}
is taken from reference
[17]
.
The theoretical values of the optical wavelengths are calculated using a wavelength and temperatureindependent Sellmeier equation for 0.5% MgOdoped stoichiometric LiTaO
_{3}
(MgO: LiTaO
_{3}
)in the IR
[18]
. The Sellmeier equation for MgO:LiTaO
_{3}
of Dolev et al.
[18]
in the IR range in a temperature range from room temperature to 200℃ can be written as
where
a
_{1e}
= 4.5615,
a
_{2e}
= 0.08488,
a
_{3e}
= 0.1927,
a
_{4e}
= 5.5832,
a
_{5e}
= 8.3067,
a
_{6e}
= 0.021696,
b
_{1e}
= 4.782×10
^{−7}
,
b
_{2e}
= 3.0913 ×10
^{−8}
,
b
_{3e}
= 2.7326×10
^{−8}
,
b
_{4e}
= 1.4837×10
^{−5}
,
b
_{5e}
= 1.3647×10
^{−7}
,
a
_{10}
= 4.5082,
a
_{20}
= 0.084888,
a
_{30}
= 0.19552,
a
_{40}
= 1.1570,
a
_{50}
= 8.2517,
a
_{60}
= 0.0237,
b
_{10}
= 2.0704×10
^{−8}
,
b
_{20}
= 1.4449 ×10
^{−8}
,
b
_{30}
= 1.5978×10
^{−8}
,
b
_{40}
= 4.7686×10
^{−6}
,
b
_{50}
= 1.1127×10
^{−5}
and
f
= (
T
−24.5)×(
T
+570.82),
T
is the crystal temperature in ℃. All the figures in this work were numerically simulated according to the Eqs (18) using MATLAB software.
III. FREQUENCY TUNING CHARACTERISTICS
 3.1. THz Wave Generation with Bulk LiTaO3
DFG processes are more versatile because of their wide tuning properties. According to Eq. (12), the frequencies of the THz wave can be tuned by varying phasematching angle
θ
, wavelength of optical waves
λ
_{1}
and
λ
_{2}
and working temperature
T
.
Figure 2
shows the tuning characteristics versus optical wavelength
λ
_{1}
. From the figure we find that as
λ
_{1}
changes from 0.5 to 3 μm, a THz wave with frequency varying from 0.68 to 0.027 THz can be attained. The frequency of the THz wave is insensitive to the wavelength
λ
_{1}
because the refractive index difference between
n
_{1}
^{e}
(
θ
) of optical wave
λ
_{1}
and
n
_{2}
^{o}
of optical wave
λ
_{2}
changes relatively little with optical wavelength in the infrared range.
THz wave frequency versus the wavelength of optical wave λ_{1}. T=23℃, α = 90°, θ = 90°.
The frequency tuning can be realized by using the dependence of crystal birefringence on angle
θ
between the directions of the optical wave propagation and optical axis.
Figure 3
shows the tuning characteristics versus angle
θ
. From the figure we find that as
θ
changes from 0° to 90°, THz wave frequency varying from 0 to 0.3 THz can be attained. The frequency of the THz wave is insensitive to angle
θ
. As
θ
changes from 0° to 90°, the effective nonlinear coefficient
d_{eff}
gradually increases. When
θ
equals 90°, the
d_{eff}
reaches the maximum value.
THz wave frequency versus the angle θ . T=23℃, α = 90°, λ_{1}=1.064 μm.
Figure 4
shows the tuning characteristics versus working temperature
T
. As working temperature
T
varies from 20℃ to 77℃, THz wave frequency changes are relatively small. The tuning range is relatively limited because the difference between
n
_{1}
^{e}
(
θ
) of optical wave
λ
_{1}
and
n
_{2}
^{o}
of optical wave
λ
_{2}
in the infrared range changes relatively little with working temperature
T
. According to Eq. (6), with the increase of working temperature
T
the absorption coefficient
α
_{THz}
intensively increases due to the increase of the linewidth Γ
_{j}
of the A
_{1}
symmetry phonon modes.
THz wave frequency versus working temperature T. λ_{1}=1.064 μm, α = 90°.
 3.2. THz Wave Generation with PPLT
According to Eq. (4), the frequencies of the THz wave can be tuned by varying poling period Λ, wavelength of optical wave
λ
_{1}
and
λ
_{2}
, and working temperature
T
.
Figure 5
shows the tuning characteristics versus optical wavelength
λ
_{1}
when the poling period Λ equals 13 μm. From the figure we find that as
λ
_{1}
changes from 0.4 to 3 μm, frequencies of the THz wave varying from 8.5 to 10.7 THz can be attained. The frequency of the THz wave can be rapidly tuned by varying the wavelength
λ
_{1}
when the wavelength
λ
_{1}
is less than 1μm. Different from THz wave generation with bulk LiTaO
_{3}
where the frequencies of the generated THz wave are under 1 THz, the frequencies of THz wave generated from PPLT approach the polariton resonances of the LiTaO
_{3}
crystal.
THz wave frequency versus the wavelength of optical wave λ_{l}. T = 25℃, Λ=13 μm.
As
λ
_{1}
equals to 1.064 μm, the frequencies of THz wave versus the poling period Λ of PPLT are depicted in
Fig. 6
. From the figure we find that as Λ varies from 5 to 60 μm, a THz wave covering a tuning of 2.3 to 27.6 THz can be realized. The tuning range can cover the entire polariton resonances of the LiTaO
_{3}
crystal. The frequency of THz wave can be rapidly tuned by varying the poling period Λ for Λ less than 20 μm.
THz wave frequency versus the poling period Λ of PPLT. T = 25℃, λ_{1}=1.064 μm
Figure 7
shows the tuning characteristics versus working temperature
T
. As working temperature
T
varies from 20℃ to 200℃, THz wave frequency changes relatively little. The rapid tuning of a THz wave cannot be realized by varying working temperature.
THz wave frequency versus working temperature T. λ_{1}=1.064 μm, Λ=15 μm.
IV. PARAMETRIC GAIN CHARACTERISTICS
LiTaO
_{3}
crystal has five infrared and Ramanactive transverse optical (TO) phonon modes at the frequencies of 200 cm
^{−1}
, 241 cm
^{−1}
, 357 cm
^{−1}
, 596 cm
^{−1}
, 657 cm
^{−1}
, which are called A
_{1}
symmetry modes. The five phonon modes which correspond to the THz wave frequencies of 6 THz, 7.23 THz, 10.71 THz, 17.88 THz and 19.71 THz, are useful for efficient THz wave generation because of the largest parametric gain in the vicinity of phonon polariton resonances, as shown in
Fig. 8
. From the figure we find that as the THz wave approaches polariton resonances, the parametric gain coefficient
g
_{0}
reaches sharply a maximum value and the absorption coefficient
α_{THz}
increases intensively to a great value. Such dramatic enhancements of parametric gain coefficients can be exploited for improving the output powers and extending the frequency bands of THz waves if a surfaceemitted configuration is employed to minimize the propagation path of the THz wave within the crystal.
Parametric gain coefficient in the lowloss limit g_{0} and absorption coefficient α_{THz} versus THz wave frequency at room temperature. λ_{1} = 1.064 μm, I_{λl} = 200 MW/cm^{2}.
Compared with other works in which THz wave generations are far from polariton resonances to avoid intensive absorption, the theoretical model proposed in this letter can exploit dramatic enhancement of parametric gain in the vicinity of polariton resonances with a surfaceemitted configuration. The theoretical model proposed in this letter is useful to other materials with crystal birefringence properties, such as LiNbO
_{3}
, KTA, in which polaritons are both infraredactive and Ramanactive. The theoretical model is also useful to semiconductor optical waveguides with modal birefringence properties. By utilizing modal birefringence of fundamental TE and TM modes in a planar waveguide, the absorption at polariton resonances can be efficiently reduced
[19]
.
V. CONCLUSION
THz wave generation at polariton resonance of LiTaO
_{3}
by DFG processes is investigated. It is shown that by using crystal birefringence of bulk LiTaO
_{3}
and grating vector of PPLT, a THz wave can be efficiently generated by DFG in the vicinity of polariton resonances with a Cherenkov phase matching scheme. In the case of bulk LiTaO
_{3}
, the frequency of the THz wave is insensitive to the wavelength
λ
_{1}
, phase matching angle and working temperature. In the case of PPLT, the tuning range of a THz wave is widened by varying the poling period Λ. Dramatic enhancement of parametric gain coefficients in the vicinity of polariton resonances can be exploited for improving the output power and extending the frequency bands of a THz wave by using a surfaceemitted configuration to minimize the propagation path of a THz wave within the crystal.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 61201101, 61205003 and 61172010).
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