Frequency tuning characteristics of a THzwave by varying phasematching angle and pump wavelength in a noncollinear phasematching THzwave parametric oscillator (TPO) are analyzed. A novel scheme to realize the tuning of a THzwave by moving the cavity mirror forwards and backwards is proposed in a noncollinear phasematching TPO. The parametric gain coefficients of the THzwave in a LiNbO
_{3}
crystal are explored under different working temperatures. The relationship between the poling period of periodically poled LiNbO
_{3}
(PPLN) and the THzwave frequency under the condition of a quasiphasematching configuration is deduced. Such analyses have an impact on the experiments of the TPO.
I. INTRODUCTION
Applications of the THzwave in spectrum analysis
[1
,
2]
, biology and medicine
[3
,
4]
, communications
[5]
, security technologies
[6]
and quality control
[7]
have raised much interest in terahertz photonics. Unfortunately, lack of practical terahertz sources restricts the applications of the THzwave. Until there are more practical sources available, the full potential of terahertz radiation will remain, to a great extent, unrealized. Due to the interest in exploiting this region there are many schemes proposed on source technologies over the last fifteen years or so.
[8

11]
Among many electronic and optical methods for the THzwave generation, the THzwave parametric oscillator (TPO) exhibits many advantages, such as compactness, narrow linewidth, coherent, wide tunable range, highpower output and room temperature operation
[9]
. For efficient generation of the THzwave, MgO:LiNbO
_{3}
is one of the most suitable crystals due to its large nonlinear coefficient and its wide transparency range.
[12]
In the TPO both noncollinear phasematching configuration and quasiphasematching configuration can perform well. The tuned THzwave can be realized by varying phasematching angle, pump wavelength and operation temperature.
In this letter, the frequency tuning characteristics of the THzwave by changing the pump wavelength, the phasematching angle, the operation temperature and the poling period of the PPLN crystal are investigated. Parametric gain coefficients of the THzwave under different working temperatures are analyzed.
II. PHASEMATCHING SCHEMES
The phasematching in the TPO is necessary to avoid destructive interference of the Stokes wave and the THzwave which are produced by the stimulated Raman scattering. For a LiNbO
_{3}
crystal the refractive index in the terahertz range is around 5, as compared to 2 in the near infrared, so birefringence phasematching is not applicable in this case. One method that has been used by several groups is noncollinear phasematching based on the bulk LiNbO
_{3}
crystal as the nonlinear gain medium, in which the pump wave, the Stokes wave and the THzwave are all nonparallel with each other, as is shown in
Fig. 1
(a). For the THzwave parametric process, two requirements have to be fulfilled: the energy conservation condition
and the phasematching condition
Here,
ω
_{p}
,
ω
_{s}
,
ω
_{T}
are the angular frequencies while
are the wavevectors of the pump, the Stokes and the THz wave, respectively. The phasematching condition can be rewritten as
where
θ
is the angle between the pump wave and the Stokes wave.
Usually, collinear phasematching is the preferred configuration for a nonlinear frequency conversion process because it provides the longest interaction length. In recent years PPLN has been widely investigated for generating the THz radiation, which ensures two or even three mixing waves collinearly propagate, as is shown in
Fig. 1
(be). In quasiphasematching configuration, the phasematching condition
has to be fulfilled, where
is the grating vector of an alternating secondorder nonlinearity induced by periodic poling of crystal. In the
Fig. 1
(b) the forwards parametric terahertz process is achieved by
being antiparallel to the pump
, the Stokes
and the THzwave
wavevectors. The backward parametric terahertz process is achieved by
travelling backward with respect to the pump
and the Stokes
, as is shown in
Fig. 1
(c). Most of the THz energy generated in the forwards and the backwards process is absorbed by the crystal due to the large absorption coefficients in the terahertz range. In
Fig. 1
(d), the grating vector
is arranged perpendicular to the pump wave propagation direction, thereby allowing parallel propagation of the pump and the Stokes waves while still retaining the rapid exiting of the THzwave through the side facet of the crystal. The generated THzwave is extracted from the nonlinear crystal by an array of high resistivity Siprisms avoiding total internal reflection. In
Fig. 1
(e), the pump and the Stokes wave are collinear, while the THzwave propagates perpendicular to the side facet of the crystal. The THzwave is coupled out without any coupler, so the loss is low and the beam quality is high.
Phasematching schemes. (a) Noncollinear phasematching. (b) Quasiphasematching, grating vector parallel to the pump wave propagation and the THzwave propagation direction along with pump wave propagation. (c) Quasiphasematching, grating vector parallel to the pump wave propagation and the THzwave travelling backwards with respect to the pump wave propagation. (d) Quasiphasematching scheme with grating vector perpendicular to the pump wave propagation. (e) Slantstripe periodic poling for quasiphasematching, the THzwave propagation direction perpendicular to the pump wave propagation.
III. TUNING CHARACTERISTICS OF THE THZWAVE
Optical parametric oscillators are more versatile because of their tuning properties. In this section we analyze the tuning characteristics of the THzwave based on the noncollinear phasematching and the quasiphasematching configuration. According to the Eqs. (1) and (3), the tunable THzwave frequency vT can be realized by varying the pump wavelength
λ
_{p}
and the phasematching angle
θ
. Such tuning is shown in
Fig. 2
. The THzwave frequency
v
_{T}
is sensitive to the angle
θ
, so the rapid tuning can be reached by changing the angle
θ
. Different from the methods provided by other groups by rotating the gain medium or mirrors,
[9
,
13]
here we propose a method for realizing the tuning output of the THzwave for the first time. A symmetric resonant cavity of the Stokes wave with a diamond configuration is proposed, as is shown in
Fig. 3
. The angle
θ
between the pump wave and the Stokes wave is tuned by moving the cavity mirror M
_{3}
backwards and forwards, as a result, the tuned THzwave can be reached. The incidence angle of the pump wave
θ
_{0}
THzwave frequency vT versus the phasematching angle θ and the Stokes wavelength λ_{s} at room temperature, λ_{p}=1064 nm.
A symmetric resonant cavity of the Stokes wave with a diamond configuration. The cavity mirror M_{3} can move backwards and forwards to acquire the tuning angle θ.
is set to ensure that the THzwave with the frequency of 1.5 THz emits perpendicularly from the LiNbO
_{3}
crystal. The relationship between the initial position of the mirror M3 and the exit point of the THzwave
L
is
Where
R
is half the distance between the mirror M
_{1}
and M
_{2}
. The relationship between the movement distance Δ
L
of the mirror M
_{3}
and the phasematching angle
θ
is
According to the Eqs. (3) and (6) the tuning THzwave can be realized by moving the mirror M3 backwards and forwards. Such tuning is shown in
Fig. 4
. The tuning range of 0.83 THz can be obtained by moving the M3 forwards from 0.87 to 3.51 mm. The method is simple and practical for the tuning output of the THzwave.
In the process of the THzwave generation, the THzwave parametric gain is of vital importance. According to the Ref. (14), the analytical expressions of the exponential gain for the THzwave can be written as
The movement distance ΔL of the mirror M_{3} versus the angle θ and the THzwave frequency v_{T}, R=60 mm, θ_{0}= 63.53°.
where
φ
is the phasematching angle between the THzwave and the pump wave,
ω
_{0j}
and
S_{j}
are the eigenfrequency and the oscillator strength of the lowest
A
_{1}
symmetry phonon mode, respectively.
I
_{p}
is the pump power density,
g
_{s}
is the gain coefficient of the Stokes wave.
n
_{p}
,
n
_{s}
and
n
_{T}
are the refractive indices of the pump wave, the Stokes wave and the THzwave, respectively.
d_{E}
′ and
d_{Q}
′ are related to the secondorder and thirdorder nonlinear parametric processes, respectively. The values of parameters of Eqs. (7)(9) are presented in Ref. (14).
Fig. 5
shows the relationship between the THzwave parametric gain coefficient gT and the angle
θ
as
I
_{p}
equals to 60, 100 and 150 MW/cm
^{2}
, respectively. From the figure we find the gT increases rapidly to the peak, and then decreases slowly to the lower values. The maximum values of the tuning curves move to the high frequency band as the
I
_{p}
changes from 60 to 100 and 150 MW/cm
^{2}
.
The tuned THzwave can be achieved also by varying the pump wavelength
λ
_{p}
.
Fig. 6
shows the relationship between the THzwave frequency
v
_{T}
and the pump wavelength
The parametric gain coefficient g_{T} versus the angle θ at room temperature, λ_{p}=1064 nm, I_{p}=60, 100 and 150 MW/cm^{2}, respectively.
The THzwave frequency v_{T} versus the pump wavelength λ_{p} and the Stokes wavelength λ_{s} at room temperature, θ=0.7°.
λ
_{p}
at room temperature. The
v
_{T}
decreases rapidly and slowly with the increase of the pump wavelength
λ
_{p}
. The pump wavelength not only varies the THzwave frequency, but also affects the parametric gain of the THzwave.
Fig. 7
shows the parametric gain coefficient gT with the changing of the pump wavelength
λ
_{p}
. As the
λ
_{p}
changes from 0.5 to 4 μm, the
g
_{T}
increases rapidly to the peak, and then decreases slowly to the lower values. From the figure we find that the maximum values of the tuning curves move to the lower wavelength band as the
I
_{p}
changes from 60 to 100 and 150 MW/cm
^{2}
.
The tuning THzwave can be achieved by changing the working temperature of the LiNbO
_{3}
crystal. The relationship among the crystal temperature, the THzwave frequency vT and the Stokes wavelength
λ
_{p}
is shown in
Fig. 8
. Temperature dependence of the refractive index of the LiNbO
_{3}
crystal in the terahertz range is reported in Ref. (12). As the temperature varies from 40℃ to 200℃, the THzwave in the range of 1.811.84 THz can be obtained. Compared with the tuning characteristics by varying the
The parametric gain coefficient g_{T} versus the pump wavelength λ_{p} at room temperature, θ=0.7o, I_{p}=60, 100 and 150 MW/cm^{2}, respectively.
The temperature tuning characteristics of the THzwave, λ_{p}=1064 nm, θ=0.7°.
Gain coefficients of the THzwave and the Stokes wave, λ_{p}=1064 nm, I_{p}=100 MW/cm^{2}.
phasematching and the pump wavelength, the THzwave frequency is insensitive to the working temperature. The crystal temperature not only affects the phasematching condition, but also has a significant impact on the parametric gain coefficient
g
_{T}
and
g
_{s}
. The characteristics of
g
_{T}
and
g
_{s}
at different temperatures are shown in
Fig. 9
. From the figure we find that the
g
_{T}
and
g
_{s}
increase along with the decrease of the temperature. The damping coefficient of the lowest
A
_{1}
symmetry phonon mode in the LiNbO
_{3}
crystal reduces with the decrease of the temperature
[15]
, resulting in the enlargement of the parametric gain. As discussed above, the enhanced output of the THzwave can be realized by reducing the working temperature.
The tuning output of the THzwave can be realized in quasiphasematching configuration by varying the poling period of the PPLN crystal and the phasematching angle. In this section we analyze the tuning characteristics based on the model shown in
Fig. 1
(e), since the THzwave is coupled out perpendicularly to the side surface of the PPLN crystal without using any output coupler. According to the
Fig. 1
(e), the poling period Λ and the phasematching angle
β
between the THzwave propagation direction and the grating vector are
Where
λ
_{T}
is the wavelength of the THzwave. The tuning THzwave versus the phasematching angle
β
and poling period Λ at room temperature is shown in
Fig. 10
. With the increase of the THzwave frequency vT, the poling period Λ decreases rapidly and then slowly, while the
The THzwave frequency vT versus the phasematching angle β and poling period Λ at room temperature, λ_{p}=1064 nm.
Schematic diagram of the quasiphasematching in PPLN crystal as the THzwave wavevector is not perpendicular to the the side surface of the PPLN crystal.
angle
β
decreases slowly and then rapidly. At the point of 1.5 THz where the output of the THzwave is of the most intensity
[16]
, the poling period Λ equals to 36.5 μm and the angle
β
equals to 23.8°.
As the THzwave propagation direction is not perpendicular to the side surface of the PPLN crystal, the THzwave can be coupled out by employing an array of Siprisms to avoid total internal reflection
[16
,
17]
, as is shown in
Fig. 11
. The angle α is between the THzwave wavevector and the pump wave wavevector. The relationship between the angle α and the THzwave wavelength
λ
_{T}
is
According to the Eqs. (1) and (12), the tuning THzwave with different propagation directions can be realized in a PPLN crystal with a fixed poling period Λ and a fixed angle
β
by employing a tuning Stokes seed beam. Such tuning is shown in
Fig. 12
, assuming the poling period Λ of 36.5 μm and the angle
β
of 23.8° where the THzwave output is the most intense. Since the refractive indexes of the THzwave in LiNbO
_{3}
crystal and highresistivity Si are
The THzwave frequency vT versus the angle α and the Stokes wavelength λ_{s} at room temperature, Λ=36.5 μm, β =23.8o, λ_{p}=1064 nm.
approximately 5.1 and 3.4 respectively, the minimum value of the angle
α
is 47.2° to avoid total internal reflection of the THzwave. From the figure we find that by injecting the tuning Stokes seed beam in the range of 1069.71071.7 nm we can obtain the tuning THzwave from 1.5 to 2.02 THz. The analysis here provides a choice for the tuning THzwave by employing a PPLN crystal with a fixed poling period Λ and a fixed angle
β
.
IV. CONCLUSION
The THzwave tuning characteristics of the noncollinear phasematching TPO and the quasiphasematching TPO are investigated. In the condition of the noncollinear phasematching configuration, the THzwave frequency is sensitive to the variation of the phasematching angle
θ
and the pump wavelength
λ
_{p}
, while insensitive to the variation of the crystal temperature. The phasematching angle
θ
, the pump wavelength
λ
_{p}
and the crystal temperature affect the parametric gain coefficients of the THzwave. The tuning THzwave can be realized in quasiphasematching configuration by varying the poling period of the PPLN crystal and the phasematching angle. Employing the PPLN crystal with the poling period Λ of 36.5 μm and the angle
β
of 23.8°, we can obtain the tuning THzwave from 1.5 to 2.02 THz by injecting the Stokes seed beam with the wavelength
λ
_{s}
in the range of 1069.71071.7 nm.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 61201101 and 61172010).
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