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crystals
Article
Simultaneous Generation of Two Orthogonally
Polarized Terahertz Waves by Stimulated Polariton
Scattering with a Periodically Poled LiNbO3Crystal
Zhongyang Li * ID , Silei Wang, Mengtao Wang, Bin Yuan and Pibin Bing ID
College of Electric Power, North China University of Water Resources and Electric Power, Zhengzhou 450045,
China; 201610521173@stu.ncwu.edu.cn (S.W.); 201610521175@stu.ncwu.edu.cn (M.W.);
x201710518296@stu.ncwu.edu.cn (B.Y.); bing463233@163.com (P.B.)
*Correspondence: thzwave@163.com; Tel.: +86-158-3827-6960
Received: 7 April 2018; Accepted: 20 July 2018; Published: 24 July 2018
Abstract:
We present a theoretical investigation of the simultaneous generation of two orthogonally
polarized terahertz (THz) waves by stimulated polariton scattering (SPS) with a periodically poled
LiNbO
3
(PPLN) crystal. The two orthogonally polarized THz waves are generated from SPS with
A
1
and Esymmetric transverse optical (TO) modes in a LiNbO
3
crystal, respectively. The parallel
polarized THz wave is generated from A
1
symmetric TO modes with type-0 phase-matching of
e=e+e
, and the perpendicular polarized THz wave is generated from Esymmetric TO modes with
type-I phase-matching of e=o+o. The two types of phase-matching of
e=e+e
and e=o+ocan
be almost satisfied simultaneously by accurately selecting the poling period of the PPLN crystal.
We calculate the photon flux density of the two orthogonally polarized THz waves by solving the
coupled wave equations. The calculation results indicate that the two orthogonally polarized THz
waves can be efficiently generated, and the relative intensities between the two orthogonally polarized
THz waves can be modulated.
Keywords: terahertz wave; stimulated polariton scattering; periodically poled LiNbO3
1. Introduction
Stimulated polariton scattering (SPS) has proven to be an efficient scheme to generate terahertz
(THz) waves [
1
–
8
]. A polariton is a coupled quantum between the pump laser and the infrared- and
Raman-active transverse optical (TO) modes in a crystal, and it behaves like phonons near the resonant
frequency associated with the TO mode and exhibits photon-like behavior for lower non-resonant
frequencies [
1
]. SPS consists of second-order and third-order nonlinear frequency conversion processes
where a pump photon stimulates a Stokes photon at the difference frequency between the pump photon
and the polariton. At the same time, a THz wave is generated by the parametric process due to the
nonlinearity arising from both electronic and vibrational contributions of the crystal. The TO phonon
resonances can contribute substantially to the magnitude of the second-and third-order nonlinearities,
which are beneficial to the THz generation via SPS.
MgO:LiNbO
3
has been the most widely used crystal for THz wave generation via SPS [
1
–
5
].
MgO:LiNbO
3
has strong second-order nonlinear response, as well as TO phonon resonances for
efficient SPS [
9
]. MgO:LiNbO
3
has five A
1
symmetric infrared- and Raman-active TO modes polarized
parallel to the c-axis with frequencies of 248 cm−1, 274 cm−1, 307 cm−1, 628 cm−1, and 692 cm−1[10].
MgO:LiNbO
3
has eight Esymmetric infrared- and Raman-active TO modes polarized perpendicular to
the c-axis with frequencies of 152 cm
−1
, 236 cm
−1
, 265 cm
−1
, 322 cm
−1
, 363 cm
−1
, 431 cm
−1
, 586 cm
−1
,
and 670 cm
−1
[
10
]. A
1
symmetric TO modes have been the most widely used for THz wave generation
Crystals 2018,8, 304; doi:10.3390/cryst8080304 www.mdpi.com/journal/crystals
Crystals 2018,8, 304 2 of 9
via SPS [
1
–
5
]. However, Esymmetric TO modes can be also employed to generate THz waves via SPS.
In 1969, Yarborough reported the observation of tunable SPS from A
1
and Esymmetric TO modes
with a pump wave in a LiNbO
3
crystal [
11
]. If the SPS from A
1
and Esymmetric TO modes can be
simultaneously excited, then two orthogonally polarized THz waves can be simultaneously generated.
Orthogonally polarized THz waves are useful for imaging [
12
]. Yu et al. [
12
] showed that the addition
or subtraction of two images, which were taken with a perpendicularly polarized THz wave and
parallel polarized THz wave, was effective to enhance the contrast of terahertz images.
In this work, we theoretically study the simultaneous generation of two orthogonally polarized
THz waves by SPS with a periodically poled LiNbO
3
(PPLN) crystal. The two orthogonally polarized
THz waves are generated from SPS with A
1
and Esymmetric TO modes in a MgO:LiNbO
3
crystal,
respectively. We calculate the photon flux density of the two orthogonally polarized THz waves by
solving the coupled wave equations.
2. Theoretical Model
Figure 1shows a schematic diagram of THz wave generation by the SPS processes by a PPLN
crystal with a quasi-phase-matching (QPM) condition. A pump wave and two seed waves (Seed
e
and Seed
o
) propagate along the x-axis of the PPLN crystal. The electric field of the pump wave and
Seed
e
is along the z-axis of the PPLN crystal, whereas the electric field of Seed
o
is perpendicular to the
z-axis of the PPLN crystal. The z-axis is the optical axis of the LiNbO
3
crystal. The poling period of
the PPLN crystal is
Λ
. Two orthogonally polarized THz waves (THz
e
and THz
o
) are generated by the
SPS processes. The electric field of THz
e
is along the z-axis of the PPLN crystal, whereas the electric
field of THz
o
is perpendicular to the z-axis of the PPLN crystal. The pump, Seed
e
, and THz
e
waves
satisfy the type-0 phase-matching of e=e+e, whereas the pump, Seed
o
, and THz
o
waves satisfy the
type-I phase-matching of e=o+o. The above two types of phase-matching can also be applied to the
forward SPS processes and backward SPS processes by accurately selecting the poling period
Λ
of the
PPLN crystal. The generated THz waves are deflected by parabolic mirrors, which transmit the pump
and two seed waves.
Figure 1.
Schematic diagram of terahertz (THz) wave generation by stimulated polariton scattering
(SPS) processes in a periodically poled LiNbO
3
(PPLN) crystal with a quasi-phase-matching (QPM)
condition.
Λ
is the poling period of the PPLN crystal. P
1
and P
2
are parabolic mirrors which transmit
the pump and two seed waves, and couple out the two THz waves.
3. Phase-Matching Characteristics
Due to the different eigenfrequency, oscillator strength, and bandwidth of the TO modes, the two
orthogonally polarized THz waves, THz
e
and THz
o
, have different dispersion and absorption
characteristics. Figure 2shows the dispersion and absorption characteristics of the THz
e
and THz
o
waves. n
Te
and n
To
are the refractive indices of THz
e
and THz
o
, respectively, and
αTe
and
αTo
are the
absorption coefficients of THz
e
and THz
o
, respectively. The curves of n
Te
and
αTe
are below the lowest
A
1
symmetric TO mode, 248 cm
−1
, and the curves of n
To
and
αTo
are below the lowest Esymmetric
TO mode, 152 cm
−1
. The theoretical parameters of the refractive index and absorption coefficient for
LiNbO
3
in the THz range are cited in [
9
]. From the figure, we find that n
Te
and n
To
are larger than 5.
Crystals 2018,8, 304 3 of 9
The value of the refractive index in the THz range is much larger than that in the optical range, so the
collinear phase-matching is impossible to realize. The absorption coefficients
αTe
and
αTo
are very
large, especially in the high THz frequency range.
Figure 2.
The dispersion and absorption characteristics of the two orthogonally polarized THz waves,
THz
e
and THz
o
.n
Te
and n
To
are the refractive indices of THz
e
and THz
o
, respectively, and
αTe
and
αTo
are the absorption coefficient of THzeand THzo, respectively.
In the optical SPS processes, the THz waves are generated, and the seed waves are amplified.
The amplified seed waves are Stokes waves. In order to achieve efficient conversion of the SPS
processes from the pump wave to the THz waves, a precise phase-matching condition must be satisfied.
For the forward SPS processes, the pump, Seed
e
, and THz
e
waves satisfy the type-0 phase-matching of
e=e+e, and the phase mismatch ∆
*
keis as follows:
∆
*
ke=
*
kp−
*
kse −
*
kTe +
*
kΛ. (1)
The pump, Seed
o
, and THz
o
waves satisfy the type-I phase-matching of e=o+o, and the phase
mismatch ∆
*
kois as follows:
∆
*
ko=
*
kp−
*
kso −
*
kTo +
*
kΛ(2)
where
*
kp
is the wave vector of the pump wave,
*
kse
and
*
kso
are the wave vectors of the two Seed
e
and Seed
o
waves, respectively, and
*
kTe
*
kTo
are the wave vectors of the two THz
e
and THz
o
waves,
respectively.
*
kΛ=2π/Λis the grating vector, and Λis the poling period of the PPLN crystal.
For the backward SPS processes, the pump, Seed
e
and THz
e
waves satisfy the type-0
phase-matching of e=e+e, and the phase mismatch ∆
*
keis as follows:
∆
*
ke=
*
kp−
*
kse +
*
kTe −
*
kΛ. (3)
The pump, seed
o
, and THz
o
waves satisfy the type-I phase-matching of e=o+o, and the phase
mismatch ∆
*
kois as follows:
∆
*
ko=
*
kp−
*
kso +
*
kTo −
*
kΛ. (4)
The energy conservation condition has to be fulfilled according to the following:
1
λp
−1
λse
−1
λTe
=0 (5)
Crystals 2018,8, 304 4 of 9
1
λp
−1
λso
−1
λTo
=0 (6)
where
λp
is the wavelength of pump wave,
λse
and
λso
are the wavelengths of the two Seed
e
and Seed
o
waves, respectively, and
λTe λTo
are the wavelengths of the two THz
e
and THz
o
waves, respectively.
If both the phase mismatches
∆
*
ke
and
∆
*
ko
are small enough, two perpendicular THz waves THz
e
and THzocan be generated simultaneously with a single pump wave.
For the SPS processes, we calculate the phase mismatches
∆
*
ke
and
∆
*
ko
according to Equations (1)
and (2), respectively, at a fixed pump wavelength. The wavelengths of the two seed waves and the two
THz waves are dependent on Equations (5) and (6). The sum phase mismatch
∆ks=
∆
*
ke
+
∆
*
ko
.
If the sum phase mismatch
∆ks
is small enough, the two phase mismatches
∆
*
ke
and
∆
*
ko
are small
enough to realize the two phase-matching conditions of e=e+eand e=o+o.
Figure 3shows the phase-matching characteristics for the forward SPS processes with a pump
wavelength of 1550 nm.
νTe
and
νTo
are the frequencies of the THz
e
and THz
o
waves, respectively.
The theoretical values of the refractive indices are calculated using a Sellmeier equation for LiNbO
3
in the infrared range [
13
]. From Figure 3a, we find that as
Λ
varies from 9 to 18
µ
m, there are many
points of
∆ks
with values below
π
cm
−1
, which indicates that the two SPS processes generating the
THz
e
and THz
o
waves can be efficiently realized. Most frequencies from 4.6 to 6 THz of THz
e
and
most frequencies from 0.4 to 2.8 THz of THz
o
can be efficiently generated. The minimum value of
∆ks
is 0.064 cm
−1
with
Λ
of 17.1
µ
m, corresponding to
νTe
of 4.66 THz and
νTo
of 0.56 THz. Figure 3b
shows the detailed phase-matching characteristics with
Λ
from 17.096 to 17.100
µ
m. As
Λ
varies
from 17.0981 to 17.0983
µ
m,
∆ks
with a value of 0.0369 cm
−1
is small enough to stimulate the two SPS
processes. In particular, as
Λ
is 17.0982
µ
m,
∆ke
equals
∆ko
, which indicates that the two SPS processes
can be realized to equal degrees.
Figure 3.
The phase-matching characteristics for the forward SPS processes.
νTe and νTo
are the
frequencies of the THz
e
and THz
o
waves, respectively. The sum phase mismatch
∆ks=
∆
*
ke
+
∆
*
ko
,
and
λp
= 1550 nm. (
a
) The phase-matching characteristics with
Λ
from 9 to 18
µ
m. (
b
) The detailed
phase-matching characteristics with Λfrom 17.096 to 17.100 µm.
Figure 4shows the phase-matching characteristics for the backward SPS processes with a pump
wavelength of 1550 nm. From Figure 4a, we find that as
Λ
varies from 20 to 100
µ
m, there are also
many points of
∆ks
with values below
π
cm
−1
, particularly below 1 cm
−1
. Most frequencies from
0.45 to 2.03 THz of THz
e
and most frequencies from 2.01 to 3 THz of THz
o
can be efficiently generated.
The minimum value of
∆ks
is 0.344 cm
−1
with
Λ
of 80.98
µ
m, corresponding to
νTe
of 0.52 THz and
νTo
of 2.06 THz. Figure 4b shows the detailed phase-matching characteristics with
Λ
around 80.98
µ
m.
As
Λ
varies from 80.960 to 80.995
µ
m,
∆ks
with a value of 0.344 cm
−1
is small enough to stimulate the
Crystals 2018,8, 304 5 of 9
two SPS processes. In particular, as
Λ
is 80.978
µ
m,
∆ke
equals
∆ko
, which indicates that the two SPS
processes can be realized to equal degrees.
Figure 4.
The phase-matching characteristics for the backward SPS processes,
λp
= 1550 nm. (
a
)
The phase-matching characteristics with
Λ
from 20 to 100
µ
m. (
b
) The detailed phase-matching
characteristics with Λfrom 80.93 to 81.02 µm.
4. THz Photon Flux Density
The coupled wave equations for the SPS processes can be found in [
9
,
14
]. The coupled wave
equations describe the field envelope variation of the pump, Stokes, and THz waves. The analytical
expression of THz parametric gain coefficient g
T
under the QPM condition in the international system
of units can be written as follows:
gT=αT
2(1+16 cos ϕ(g0
αT
)21
2
−1)(7)
g2
0=ωsωT
128π2ε0c3npnsnT
Ip(dE+∑
j
Sjω2
0jdQj
ω2
0j
−ω2
T
)2(8)
αT=2ωT
cIm(ε∞+∑
j
Sjω2
0j
ω2
0j
−ω2
T−iωTΓj
)1
2(9)
where
ω0j
,
Sj
, and
Γj
denote the eigenfrequency, the oscillator strength of the polariton modes, and the
bandwidth of the jth TO mode in the LiNbO
3
crystal, respectively. I
p
is the power density of the pump
wave, and g
0
is the low-loss parametric gain.
np
,
ns
, and
nT
are the refractive indices of the pump,
Stokes, and THz waves, respectively.
ϕ
is the angle between the wavevectors of the pump wave and
THz wave.
αT
is material absorption coefficient in THz region.
dE
and
dQ
are nonlinear coefficients
related to pure parametric (second-order) and Raman (third-order) scattering processes, respectively.
When THz frequencies are far below the lowest A
1
symmetry TO mode of 248 cm
−1
and the
lowest Esymmetry TO mode of 152 cm−1, Equation (8) can be rewritten as follows [9]:
g2
0=ωsωT
128π2ε0c3npnsnT
Ip(dE+∑
j
SjdQj)2. (10)
For SPS with type-0 phase-matching of e=e+e, the relationship between
dE
and
dQ
is given by [
9
,
15
,
16
]
the following:
dE+∑
j
SjdQj=1
4r33np4. (11)
Crystals 2018,8, 304 6 of 9
For SPS with type-I phase-matching of e=o+o, the relationship between dEand dQis as follows:
dE+∑
j
SjdQj=1
4r13np4(12)
where r33 and r13 are the linear electro-optic coefficients of LiNbO3.
With strong THz wave absorption and phase mismatch and without pump depletion, the coupled
wave equations can be solved to give the THz photon flux density
φT
with a general solution [
17
],
given by the following:
φT=φs(0)e−αTL/2 g2
T
g2
T+αT
4−j∆k
22×
sinh rg2
T+αT
42L!
2
(13)
where
∆k
is the phase mismatching and Lis the crystal length. The initial THz photon flux density
φT
is assumed to be zero, and
φs(0)
is the initial seed wave photon flux density. The initial photon
flux densities of Seed
e
and seed
o
are
φse(0)
and
φso(0)
, respectively. The THz photon flux densities of
THzeand THzoare φTe and φTo, respectively. The ratio Rof φso(0)to φse (0)is as follows:
R=φso(0)
φse(0). (14)
Figure 5shows the THz wave photon flux densities
φTe
and
φTo
for the forward SPS processes.
From Figure 5a–d , we find that when
Λ
varies from 15 to 19
µ
m,
φTe
and
φTo
increase first
and then decrease. When
Λ
is equal to 17.0982
µ
m,
φTe
and
φTo
reach their maximum values as
the phase mismatches
∆ke
and
∆ko
reach their minimum values. The maximum value of
φTe
is
5.74 ×10−6s−1cm−2
. The value of
φTe
is so small, because the THz absorption coefficient of 4.66 THz
is very large.
φTo
increases with the increase of R. The relative photon flux densities between
φTe
and
φTo
can be tuned by varying R. When Ris 0.0037, the maximum values of
φTe
and
φTo
are approximately
equal. From Figure 5e, we find that when crystal length Lvaries from 0 to 30 mm,
φTe
and
φTo
increase
rapidly and smoothly. When Ris 0.00323, the values of
φTe
and
φTo
are approximately equal, as Lis
larger than 20 mm.
Figure 6shows the THz wave photon flux densities
φTe
and
φTo
for the backward SPS processes.
From Figure 6a, we find that when
Λ
varies from 74 to 88
µ
m,
φTe
and
φTo
increase first and then
decrease. When
Λ
is equal to 80.978
µ
m,
φTe
and
φTo
reach their maximum values as the phase
mismatches
∆ke
and
∆ko
reach their minimum values. The maximum value of
φTe
is 43.62 s
−1
cm
−2
.
The maximum value of
φTe
in the backward SPS processes is larger than that in the forward SPS
processes, because the THz absorption coefficient of 0.52 THz in the backward SPS processes is smaller
than that of 4.66 THz in the forward SPS processes. The relative photon flux densities between
φTe
and
φTo
can be tuned by varying R. When Ris 2.5
×
10
7
, the maximum values of
φTe
and
φTo
are
approximately equal. From Figure 6b, we find that when crystal length Lvaries from 0 to 50 mm,
φTe
and
φTo
increase rapidly and smoothly. When Ris 5.21
×
10
7
and Lis larger than 40 mm, the values
of φTe and φTo are approximately equal.
The intensities of generated THz waves are very low, because the THz waves are heavily absorbed
by the PPLN crystal. However, the intensities of the THz waves can be enhanced by injection
intense seed waves, as shown in Equation (13). Moreover, one can use organic crystals with QPM,
because organic crystals have larger nonlinear optical coefficients and lower absorption coefficients in
the THz region [
18
]. Furthermore, the enhancement of the THz intensities can be realized by cryogenic
cooling. At liquid nitrogen temperature, the gain coefficients of the THz waves in the SPS processes
are enhanced. At the same time, the absorption coefficients of the THz waves decrease.
Crystals 2018,8, 304 7 of 9
Figure 5.
THz wave photon flux density
φT
for the forward SPS processes.
λp
= 1550 nm,
νTe
= 4.66
THz,
νTo
= 0.56 THz, I
p
= 100 MW/cm
2
, and
φse(0)
= 10
6
s
−1
cm
−2
. (
a
–
d
)
φT
versus
Λ
with R= 0.002,
0.003, 0.0037, and 0.005, respectively. L= 10 mm. (e)φTversus crystal length L.Λ= 17.0982 µm.
Figure 6.
THz wave photon flux density
φT
for the backward SPS processes.
Λ
p = 1550 nm,
νTe
= 0.52
THz,
νTo
= 2.06 THz, I
p
= 100 MW/cm
2
, and
φse(0)
= 10
6
s
−1
cm
−2
. (
a
)
φT
versus
Λ
from 74 to 88
µ
m.
L= 10 mm. (b)φTversus crystal length L.Λ= 80.978 µm.
The scheme in this work of generating two orthogonally polarized THz waves by SPS processes
has certain advantages. First of all, the two orthogonally polarized THz waves are simultaneously
Crystals 2018,8, 304 8 of 9
generated by a pump wave, which means that the two THz waves are phase-conjugate. Second,
the two orthogonally polarized THz waves are generated only by a PPLN crystal. Third, the intensities
of the two orthogonally polarized THz waves can be tuned by varying the intensities of the input
seed waves.
5. Conclusions
We present the simultaneous generation of two orthogonally polarized THz waves by forward and
backward SPS processes with a PPLN crystal. The minimum values of
∆ks
of 0.064 cm
−1
in the forward
SPS processes and 0.344 cm
−1
in the backward SPS processes indicate that the type-0 phase-matching
generating parallel polarized THz wave and the type-I phase-matching generating perpendicular
polarized THz wave can almost be satisfied simultaneously. In particular, the two SPS processes can
be excited to equal degrees by accurately selecting the poling period of the PPLN crystal. We calculate
the photon flux densities of the two orthogonally polarized THz waves by solving the coupled wave
equations. The theoretical calculations show that the photon flux densities of the two orthogonally
polarized THz waves are very small. The relative intensities between the two orthogonally polarized
THz waves can be modulated by varying the intensities of the input seed waves.
Author Contributions:
Z.L., S.W., and M.W. conceived of the original idea; B.Y. and P.B. contributed useful
and deep discussions; and Z.L. wrote the manuscript. All authors read and approved the final version of
the manuscript.
Funding:
This work was supported by the National Natural Science Foundation of China (61601183); the Natural
Science Foundation of Henan Province (162300410190); the Program for Innovative Talents (in Science and
Technology) in University of Henan Province (18HASTIT023); the Young Backbone Teachers in University of
Henan Province (2014GGJS-065); and the Program for Innovative Research Team (in Science and Technology) in
University of Henan Province (16IRTSTHN017).
Conflicts of Interest:
All contributing authors declare no conflicts of interest. The founding sponsors had no role
in the design of the study; in the collection, analyses, or interpretation of the data; in the writing of the manuscript;
or in the decision to publish the results.
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