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Theoretical and experimental study on a large energy potassium titanyl phosphate terahertz parametric source
Optics and Laser Technology 121 (2020) 105817
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Theoretical and experimental study on a large energy potassium titanyl phosphate terahertz parametric source
T
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Chenyang Jiaa, Xingyu Zhanga, , Zhenhua Conga, Zhaojun Liua, Xiaohan Chena, Zengguang Qina, Jie Zanga, Feilong Gaoa, Peng Wanga, Yue Jiaoa, Jinjin Xua, Weitao Wangb, Shaojun Zhangc a School of Information Science and Engineering, and Shandong Provincial Key Laboratory of Laser Technology and Application, Shandong University, Qingdao, Shandong Province 266237, China b Laser Institute of Shandong Academy of Sciences, Jinan, Shandong Province 250100, China c State Key Laboratory of Crystal Materials, Shandong University, Jinan, Shandong Province 250100, China
H I GH L IG H T S
to 17 μJ pulse energy at 5.7 THz is obtained. • Up of the KTP Stokes parametric amplifier are investigated. • Characteristics • Theoretical modelling of the Stokes parametric amplifier is carried out.
A B S T R A C T
The combination of a Stokes parametric oscillator and a Stokes parametric amplifier is used to generate high-energy terahertz pulses in KTiOPO4 (KTP) crystal at 5.70 THz. The Stokes parametric oscillator generates Stokes pulses first. And then they are amplified in the Stokes parametric amplifier and at the same time, high energy terahertz pulses are generated via stimulated polariton scattering. When the incident Stokes pulse energy is 36.6 mJ and the pump energy is 580.0 mJ, the amplified Stokes energy is 192.1 mJ and the obtained maximum terahertz pulse energy is 17.0 μJ. Compared with the reported terahertz pulse energy obtained by the KTiOPO4 terahertz parametric oscillators or generators, the terahertz pulse energy is greatly enhanced. In addition, considering the large angle non-collinear phase matching, pump beam depletion, terahertz wave surface-emitted structure, Gaussian pulse distribution, we make a theoretical simulation for the Stokes parametric amplifier through the coupled-wave equations. The theoretical results are in agreement with the experimental results on the whole.
1. Introduction Terahertz wave means the electromagnetic wave between 0.1 THz and 10 THz. It has wide applications in biomedicine, material science, safety monitoring, communication and national defense [1,2]. Terahertz parametric source is one of the important terahertz radiation sources. It has many advantages, such as narrow line-width, continuous tunability, room temperature operation, high peak power, good coherence and so on. However, the generated terahertz pulse energy is very limited. The physical basis of the terahertz parametric source is stimulated polariton scattering (SPS), that is, a pump photon is converted into a Stokes photon and a terahertz photon under the energy conservation and momentum conservation. Because of the large refractive index of the crystal in the terahertz spectral range, only non-collinear phasematching can be obtained [3–14].
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Terahertz parametric sources include terahertz parametric generator [3,4], terahertz parametric oscillator [5–20], seed injection terahertz parametric generator [21] and the combination of Stokes parametric oscillator and Stokes parametric amplifier [22]. In the fourth, the Stokes parametric oscillator generates a Stokes pulse first. It is used as the incident Stokes pulse for the Stokes parametric amplifier. The terahertz pulse can be generated by SPS when the Stokes parametric amplifier amplifies the incident Stokes pulse. By expanding the beam of the incident Stokes pulse, good spatial overlap of the pump and Stokes beams can be realized. The delay device on the path of the pump beam can ensure the good temporal overlap between the pump pulse and the Stokes pulse. This method is an important way to obtain large energy tunable terahertz pulses. The traditional nonlinear crystal for terahertz parametric sources is MgO:LiNbO3 [3–11,21–23]. It is found that KTiOPO4 (KTP), KTiOAsO4 (KTA), RbTiOPO4 (RTP) are also suitable for terahertz parametric
Corresponding author. E-mail address: [email protected] (X. Zhang).
https://doi.org/10.1016/j.optlastec.2019.105817 Received 14 April 2019; Received in revised form 18 August 2019; Accepted 5 September 2019 0030-3992/ © 2019 Elsevier Ltd. All rights reserved.
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sources in recent years [4,12–20,24–26]. KTP crystal has the advantages of mature growth technology, good optical quality and high damage threshold. The terahertz band (mainly 3.3–6.5 THz) produced by KTP crystal is different from the terahertz band (mainly 0.9–3.1 THz) produced by MgO:LiNbO3 crystal. In the previous researches, both terahertz parametric generator and terahertz parametric oscillator using KTP crystal are investigated [4,12–16,24–26]. But the obtained terahertz pulse energy is only about 5.5 μJ [14]. Therefore, it is significant to explore the potential of KTP crystal in obtaining large energy terahertz pulses by using the experimental scheme that combines Stokes parametric oscillator and Stokes parametric amplifier. In this paper, we explore the potential of KTP crystal in obtaining large energy terahertz pulses by using the experimental scheme that combines the Stokes parametric oscillator and the Stokes parametric amplifier. The terahertz wave surface-emitted structure is adopted [9–14]. When the pump pulse energy is 580.0 mJ, the incident Stokes pulse energy is 36.6 mJ, and the delay between the pumping pulse and the incident Stokes pulse is 1.7 ns, the amplified Stokes pulse energy is 192.1 mJ, and the obtained maximum terahertz pulse energy is 17.0 μJ. Compared with the reported terahertz pulse energy obtained by the KTiOPO4 terahertz parametric oscillators or generators, the terahertz pulse energy is greatly enhanced. In the theoretical part, considering the large angle non-collinear phase matching, pump wave depletion, terahertz wave surface-emitted structure, Gaussian pulse distribution, and other actual factors, we calculate the input and output characteristics of Stokes parametric amplifier and the loss characteristics of the pump wave by solving the coupled wave equations numerically. The theoretical results are in fair agreement with the experimental ones.
Output Energy of Stokes Pulse (mJ)
Fig. 2. The KTP crystal used in the Stokes parametric amplifier.
2. Experimental setup The schematic diagram of the experimental setup for the combination of Stokes parametric oscillator and Stokes parametric amplifier is shown in Fig. 1. There were two KTP crystals. The rectangular one with dimensions of 33 (x) mm × 6 (y) mm × 6 (z) mm was used for the Stokes parametric oscillator. The isosceles trapezoidal one was used for the Stokes parametric amplifier. The shape of the isosceles trapezoidal crystal on the xy plane is shown in Fig. 2. Its thickness along the z axis was 16 mm. The incident direction of the pump beam was approximately along the x axis of the crystal, and the polarization direction of the pump beam was parallel to the z axis of the crystal. All the input and output facets of the two crystals for the pump and the Stokes beams were antireflection coated at the wavelength range from 1060.0 to Ȝ/2
M1
T1
Ȝ/2
M7
Experimental Values Theoretical Values
195
180
165
150 0
1
1
M2
Rotating Stage
y
M8
Ȝ/2
M5
2
z Ȝ/2
Ȝ/2
x
x'
z
Ȝ/2 BW
Time Delayer M4
4
Fig. 3. The variation of the amplified Stokes energy with the time delay.
M9
M3
3
Time Delay (ns)
Pump Source BW
2
y
T2
x
y' Detector
Aperture
T3
M10
TPX Lense Attenuator
M6 BW
Rotating Stage
LPF Golay
Fig. 1. The schematic diagram of the experimental setup for the combination of Stokes parametric oscillator and Stokes parametric amplifier. 2
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Fig. 4. The waveforms of the pump and Stokes pulses for different time delays. (a) 0 ns; (b) 1.7 ns; (c) 2.7 ns; (d) 4.0 ns.
Output Energy of Stokes Pulse (mJ)
240
Experimental Values Theoretical Values
180
120
60
0 0
10
20
30
40
Incident Energy of Stokes Pulse (mJ) Fig. 5. The variation of the amplified Stokes energy with the incident Stokes energy.
Fig. 6. The variation of the terahertz wave energy with the incident Stokes energy.
1100.0 nm. The pump laser was a multi-longitudinal mode Q-switched Nd:YAG laser. The laser beam had a top-hat profile. The pulse width, the beam diameter and the repetition frequency were 7.5 ns, 6.0 mm and 1 Hz, respectively. The pump laser polarization direction was parallel to the z axis of the crystal. The laser beam from the Nd:YAG laser was divided into two parts by a beam splitter. The first part served as the pump beam of the Stokes parametric
oscillator. The Stokes wave oscillating cavity mirrors M7 and M8 were respectively coated with high reflectivity and partial transmission (T = 60%) at the Stokes wavelength. The cavity length was about 130 mm. The Stokes parametric oscillator was mounted on a rotating stage. There were two energy regulators, each of which consisted of two half wave plates and one Brewster plate. The first energy regulator was used to regulate the energy of the pump pulse and to ensure that the polarization direction of the pump beam was parallel to the z axis of the 3
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connected to a digital oscilloscope (Tektronix DPO 4104B, 1 GHz, 8GS/ s) after it passed through a TPX lens and a terahertz attenuator. The TPX lens was used to focus the terahertz beam. The terahertz attenuator was used to attenuate the terahertz wave pulse energy to adapt for the measuring range of the Golay detector. A terahertz low-pass filter (LFP 14.3-47, TYDEX) was used to block the scattered pump and Stokes beams with a transmittance of about 29% at 5.7 THz. 3. Experimental results The variations of the output Stokes pulse energy with the time delay in the Stokes parametric amplifier were measured first. The pump and incident Stokes pulse energies were fixed at 580.0 mJ and 36.6 mJ, respectively. The results are shown in Fig. 3. The dots are the experimental results while the solid line is the theoretical results (we will discuss the results further in part 4). With increasing time delay, the output Stokes pulse energy increased first, then reached the maximum, and then decreased. The same is true for the terahertz wave energy. The optimal time delay was 1.7 ns. Fig. 4 shows the waveforms of the pump and Stokes pulses for different time delays. Good temporal overlap between the pump and Stokes pulses was obtained with the time delay of 1.7 ns. The corresponding the output Stokes pulse energy was 192.1 mJ, the terahertz wave energy was 17.0 μJ. With the time delay fixed at 1.7 ns and the pump pulse energy at 580.0 mJ, the variations of the amplified Stokes pulse energy and the terahertz wave energy with the incident Stokes pulse energy are shown in Figs. 5 and 6. With the time delay fixed at 1.7 ns and the incident Stokes pulse energy at 36.6 mJ, Figs. 7 and 8 show the variations of the amplified Stokes pulse energy and the terahertz pulse energy with the pump pulse energy. The waveforms of pump pulses and residual pump pulses for different time delays are shown in Fig. 9. The residual pump pulses with different time delays were depleted in different degrees. We will discuss the results with the theoretical results together in Part 4.
Fig. 7. The variation of the amplified Stokes energy with the pump energy.
4. Theoretical simulation Fig. 8. The variation of the terahertz wave energy with the pump energy.
4.1. Coupled-wave equations for the Stokes parametric amplifier In the process of stimulated polariton scattering, the pump, Stokes and terahertz beams interact in the overlapped region according to the energy conservation ωp = ωs + ωT and the momentum conservation kp = ks + kT, where ki = ωi ni / c, (i = p , s, T ) , n is refractive index, ω is angular frequency, ki is wave vector. i = p , s, T represents the pump, Stokes and terahertz waves, respectively. Because of the large refractive index of the KTP nonlinear crystal in the terahertz spectral range, only non-collinear phase-matching for the three waves can be realized, which is shown in Fig. 10. The pump wave propagates along the x axis of the KTP crystal. The angle between the Stokes and pump waves is so small inside the crystal (about 1.8°) that the Stokes wave can be treated as parallel to the pump wave in the theoretical simulation. The terahertz wave propagates in y′ direction, the internal angle β between the terahertz and pump waves is 56.1°. The polarization directions of the three waves are parallel to the z axis of the KTP crystal. The evolutions of the pump, Stokes and terahertz waves in the process of SPS can be written as [27–29]:
crystal. The second one was used to regulate the energy of the Stokes pulse to be amplified. Telescope T1 with a ratio of 2:1 was used to reduce the pump beam size for the Stokes parametric oscillator. The pump beam diameter in the Stokes parametric oscillator was 3.0 mm. Telescope T2 with a ratio of 1:3 was used to enlarge the size of the Stokes beam to be amplified. The second part was used as the pump beam of the Stokes parametric amplifier. The time delayer on the way of the pump beam was used to adjust the temporal overlapping between the incident Stokes pulse and the pump pulse of the Stokes parametric amplifier. The energy regulator in this portion had the same function as those in the other portion. Telescope T3 and the aperture were used to adjust the pump beam size. The diameter of pump beam in the Stokes parametric amplifier was 8.0 mm. The angles between the pump and Stokes beams outside the crystals in the Stokes parametric oscillator and the Stokes parametric amplifier were fixed at 4.4°, and the corresponding angles inside the crystal were 1.8°. The wavelength of the pump wave was 1064.2 nm and the central wavelength of the Stokes wave was 1086.2 nm. The terahertz wavelength was 52.5 µm, its corresponding frequency was 5.70 THz. The maximum energy of the incident Stokes pulse is 36.6 mJ with a pulse width of 6.0 ns. The energies of the pump and Stokes pulses were measured by an energy sensor (J-50MB-YAG, Coherent Inc.) connected to an energy meter (EPM2000, Coherent Inc.). The energy of the terahertz wave pulse was measured by a calibrated Golay detector (GC-1D, TYDEX)
∂Ap ∂x =
ωp2 ⎞ ωp2 ⎛ i ⎧ ⎛ 2 −⎜kp − 2 εp ⎟ Ap − 2 4d33 + 2kp ⎨ ⎝ c ⎠ c ⎜ ⎝ ⎩ −
4
ωp2 c2
∑ dQj2 εQj∗ |As |2 Ap j
⎫ ⎬ ⎭
∑ dQi εQj∗ ⎞⎟ As AT j
⎠
(1a)
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Fig. 9. The waveforms of the pump pulses and residual pump pulses for different time delays. (a) 0 ns; (b) 1.7 ns; (c) 2.7 ns; (d) 4.0 ns.
F G
D
H E
Fig. 10. The non-collinear phase matching of SPS in KTP crystal.
C I
∂As ∂x
B
A
ω2 ⎛ ω2 i ⎧ ⎛ 2 = −⎜ks − 2s εs ⎟⎞ As + 2s 4d33 + 2ks ⎨ ⎝ c ⎠ c ⎜ ⎝ ⎩ +
ωp2 c2
∑ dQj2 εQj∗ |Ap |2 As j
K
∑ dQi εQj∗ ⎞⎟ Ap AT∗ j
62°
⎠
⎫ ⎬ ⎭
⎧ ⎛ 2 ω2 ⎛ ω2 ∂AT i = −⎜kT − 2T εT ⎟⎞ AT + 2T 4d33 + 2kT cos β ⎨ ⎝ c ⎜ c ∂x ⎠ ⎝ ⎩
M
⎫
⎠
⎭
z
y
(1b)
⎞
x'
x y'
Fig. 11. The terahertz wave surface-emitted structure.
∑ dQi εQj∗ ⎟ Ap As∗ ⎬ j
and Raman-active jth TO mode [31,32]. εQ can be written as [33]: (1c)
where Am (m = p , s, T ) is the complex amplitude, d33 is the secondharmonic nonlinear coefficient [30], εm (m = p , s, T ) is the dielectric constant, dQj εQj is the combined ionic responses of the infrared-active
εQ = εT − ε∞ =
∑ j
5
2 Sj ωjTO 2 ωjTO − ω2 − jΓjTO ω
(2)
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gs(2) = ωs K
(5d)
gT(2) = ωT K
(5e)
4.2. Numerical calculation Eqs. (4a)–(4c) are the coupled-wave equations describing the Stokes parametric amplifier. The corresponding parameters for the simulation can be obtained by using Eqs. (5a)–(5e). The terahertz wave surfaceemitted structure is shown in Fig. 11. AE and DF denote the width of the pump and Stokes beams, while BC is the width of the terahertz beam in the gain region. The pump and Stokes beams enter the crystal from AE plane, and are totally reflected on BC surface, and then go out from DF plane. One part of the generated terahertz wave is emitted from BC surface. The other part is reflected back.
Fig. 12. The interaction region of the three waves.
TT =
Table 1 Parameters for simulation [35–43].
αT d33 np ns nT λp λs λT β
∑j dQj Re (εQj )
Parameter
Value
The terahertz wave absorption coefficient Second-harmonic nonlinear coefficient Refractive index of the pump light Refractive index of the Stokes light Refractive index of the terahertz wave Wavelength of the pump light Wavelength of the Stokes light Wavelength of the terahertz wave The angle between the pump and terahertz wave The combined ionic responses of all the infrared-active and Raman-active TO modes
64.9 cm−1 15.4 × 10−10 cm/V 2.388 2.387 4.474 1.0642 × 10−4 cm 1.0862 × 10−4 cm 5.254 × 10-3 cm 0.979 rad 3.79 × 10-8 cm/V
p, s, T
1/2
64π 3ε02 c 4np Rj ⎤ =⎡ ⎢ Sj hωjTO ω 4 ns (XT + 1) ⎥ s ⎣ ⎦
∂Ip
(3)
= −gp(2) (Ip Is IT )1/2
(4a)
∂Is = +gs(2) (Ip Is IT )1/2 ∂x
(4b)
∂IT = −αT IT + gT(2) (Ip Is IT )1/2 ∂y′
(4c)
∂x
ωT2 Im(εT ) kT c 2 1/2
1 ⎞ K = ⎜⎛ ⎟ ⎝ np ns nT ⎠
gp(2) = ωp K
(5a) 1/2
⎛ 1 ⎞ 3 ⎝ 2c ε0 ⎠
⎜
⎟
⎡ ⎢4d33 + ⎣
∂I
Ip (0, y′ , t ) = Ipump (t )
(7a)
Is (0, y′ , t ) = IStokes (t )
(7b)
IT (0, y′ , t ) = IT (x , 0, t ) = 0
(7c)
After the pump and Stokes beams are totally reflected, the interaction region of the three waves is the trapezoidal region of BCDF. The reflected pump, Stokes, and terahertz waves will be the initial signals for the interaction in this region. The generated terahertz wave in this region will be absorbed by the nonlinear crystal because its propagating direction is close to − y′, while the residual pump and Stokes waves will be affected by the interaction in this region. Above analysis is for the xy plane of z = 0. Considering that both the pump and Stokes waves are of circular beam shapes, the three waves will vary with z axis. However, we can calculate the intensity values of the three waves in the planes for other z values in the same way. The output pulse energies of the three waves will be the integrals over the beam sizes and the pulse durations.
where αT is the terahertz wave absorption coefficient,
αT =
∂I
ΔIp = ∂x ΔLp , ΔIs = ∂xs ΔLs , ΔIT = ∂yT′ ΔLT . The incident pump and Stokes beams are suggested to be Gaussian temporal distributions, and the pump beam has a top-hat spatial profile and the incident Stokes beam has a Gaussian spatial profile. The boundary conditions should be:
whereε0 is the vacuum permittivity, Rj is the equivalent Raman cross section. XT is the Boson factor. Since the absorption and third-order nonlinear coefficients of the pump and Stokes waves are very small in the process of SPS, they can be neglected. Based on the relationship between intensity and complex amplitude I = 2nε0 c |A|2 , the coupled-wave equations can be written as [27,34]:
∂Ip
(6)
where TT is the intensity transmissivity of the terahertz wave on the interface between the nonlinear crystal and air, nT1 and θT1 are the refractive index and incident angle of the terahertz wave in the crystal respectively, nT2 and θT2 are the refractive index and exit angle of the terahertz wave in the air respectively. For the terahertz wave surface-emitted structure, only in the BC region can the terahertz wave be emitted efficiently. In BK region, a little terahertz wave can also be emitted because the generated terahertz wave is heavily absorbed in the journey of IK. Before the pump and Stokes beams are totally reflected, the interaction region of the three waves is the trapezoidal region of ABCE. The pump and Stokes beams propagate along x direction and the terahertz wave propagates along y′ direction. As shown in Fig. 12, we use (x m , yn′) to express the positions in this region. During a time intervalΔt , the pump and Stokes waves propagate from (x m − 1, yn′) to (x m , yn′) , and the terahertz wave propagates from (x m , yn′− 1) to (x m , yn′) . At this time interval, the interaction lengths of the pump, Stokes, and terahertz waves c are ΔLp, s, T = n Δt , the increments of the three waves are
where Sj is the oscillator strength of the jth TO mode, ωjTO and ΓjTO are its angular frequency and damping coefficient, respectively, ε∞ is the high-frequency dielectric constant. And dQj can be written as:
dQj
4nT 1 nT 2 cos θT 1 cos θT 2 (nT 1 cos θT 1 + nT 2 cos θT 2)2
⎤
∑ dQj Re(εQj) ⎥ j
⎦
(5b)
4.3. Simulation results
(5c)
In the theoretical processing, the pump pulse width is set as 7.5 ns, the Stokes pulse width is set as 6.0 ns. The pump and Stokes beams are 6
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Fig. 13. The numerical results of the waveforms of the pump pulses and residual pump pulses for different time delays. (a) 0 ns; (b) 1.7 ns; (c) 2.7 ns; (d) 4.0 ns.
volume of the three waves is a little smaller than that used in the theoretical simulation. Intuitively, it seems that the generated terahertz pulse energy should be much larger than 17 μJ. If we simply considered the energy conservation law and the fact that the Stokes pulse was amplified from 36.6 mJ to 192.1 mJ, the generated terahertz pulse energy would be around 3.2 mJ. Actually, when the terahertz wave reached the output surface, a part of it (about 40%) was reflected back into the crystal, then unable to exit. More seriously, the terahertz wave absorption loss was very large (the absorption coefficient of the KTP crystal was 64.9 cm−1). The terahertz wave was heavily absorbed during the journey to the output surface. Only in the KC region (see Fig. 11) could the terahertz wave be obtained. In other regions such as MK and CD, no terahertz wave was obtained. From this point of view, one approach to enhance the terahertz output energy is to reduce the absorption by decreasing the crystal temperature. Another approach may be the use of the pumping beam with an elliptic cross section. The beam size in zdirection is larger while that in y-direction is smaller.
in the circle shape and their diameters is 8.0 mm. The path length of the pump and Stokes beams in KTP crystal (AB + BF) is 3.2 cm. The length of AB in KTP crystal is 1.5 cm. The output plane of the pump and Stokes waves is DF plane, while the output plane of the terahertz wave is KC plane. The parameters for the simulation are shown in Table 1. The numerical results of the waveforms of the pump pulses and residual pump pulses for different time delays are shown in Fig. 13 when the incident Stokes energy is 36.6 mJ and the input pump energy is 580.0 mJ. The variations of the Stokes output energy with the time delay are shown in Fig. 3. With the time delay fixed at 1.7 ns and the pump pulse energy at 580.0 mJ, the variations of the Stokes output energy and the terahertz wave energy with the incident Stokes pulse energy are shown in Figs. 5 and 6. With the time delay fixed at 1.7 ns and the incident Stokes pulse energy at 36.6 mJ, the variations of the amplified Stokes pulse energy and the terahertz pulse energy with the pump pulse energy are shown in Figs. 7 and 8. It can be seen from Figs. 3, 5–9 and 13 that the theoretical and experimental results are basically consistent with some differences. The main causes for the small differences between the theoretical results and the experimental results may be as follows. First, there is no guarantee that the parameters used in the theoretical calculation are accurate. Second, the pump beam used in the experiment is not an absolute top-hat spatial profile, and the incident Stokes beam is not a perfect Gaussian spatial profile. Third, the pump and Stokes pulses are assumed to be Gaussian temporal distributions in the calculation. Actually, the pulses are not exact Gaussian shapes (see Figs. 4 and 8). Fourth, the perfect overlapping between the pump and Stokes beams is assumed in the calculation. In fact, due to the small phase-matching angle between the pump and Stokes beams, the actual interaction
5. Conclusion The combination of Stokes parametric oscillator and Stokes parametric amplifier is an important means to generate high-energy tunable terahertz pulses. We have applied it to KTP crystal to get high energy terahertz nanosecond pulses at 5.7 THz for the first time. When the incident Stokes pulse energy is 36.6 mJ and the pump energy is 580.0 mJ, the obtained maximum terahertz pulse energy is 17.0 μJ. In the theoretical part, considering the factors such as the large angle noncollinear phase matching, pump beam depletion, terahertz wave 7
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surface-emitted structure, and Gaussian pulse distribution, we have made a detailed theoretical simulation for the Stokes parametric amplifier through the coupled-wave equations. The theoretical and experimental results are basically consistent with some differences. The main causes for the small differences between the theoretical results and the experimental results are analyzed.
[19]
[20]
[21]
Acknowledgements
[22]
This research is supported by the National Natural Science Foundation of China (61775122, 61475087, 61605103), the Key Research and Development Program of Shandong Province (2017CXGC0809, 2017GGX10103), and the Natural Science Foundation of Shandong Province (ZR2014FM024).
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Full length article
Theoretical and experimental study on a large energy potassium titanyl phosphate terahertz parametric source
T
⁎
Chenyang Jiaa, Xingyu Zhanga, , Zhenhua Conga, Zhaojun Liua, Xiaohan Chena, Zengguang Qina, Jie Zanga, Feilong Gaoa, Peng Wanga, Yue Jiaoa, Jinjin Xua, Weitao Wangb, Shaojun Zhangc a School of Information Science and Engineering, and Shandong Provincial Key Laboratory of Laser Technology and Application, Shandong University, Qingdao, Shandong Province 266237, China b Laser Institute of Shandong Academy of Sciences, Jinan, Shandong Province 250100, China c State Key Laboratory of Crystal Materials, Shandong University, Jinan, Shandong Province 250100, China
H I GH L IG H T S
to 17 μJ pulse energy at 5.7 THz is obtained. • Up of the KTP Stokes parametric amplifier are investigated. • Characteristics • Theoretical modelling of the Stokes parametric amplifier is carried out.
A B S T R A C T
The combination of a Stokes parametric oscillator and a Stokes parametric amplifier is used to generate high-energy terahertz pulses in KTiOPO4 (KTP) crystal at 5.70 THz. The Stokes parametric oscillator generates Stokes pulses first. And then they are amplified in the Stokes parametric amplifier and at the same time, high energy terahertz pulses are generated via stimulated polariton scattering. When the incident Stokes pulse energy is 36.6 mJ and the pump energy is 580.0 mJ, the amplified Stokes energy is 192.1 mJ and the obtained maximum terahertz pulse energy is 17.0 μJ. Compared with the reported terahertz pulse energy obtained by the KTiOPO4 terahertz parametric oscillators or generators, the terahertz pulse energy is greatly enhanced. In addition, considering the large angle non-collinear phase matching, pump beam depletion, terahertz wave surface-emitted structure, Gaussian pulse distribution, we make a theoretical simulation for the Stokes parametric amplifier through the coupled-wave equations. The theoretical results are in agreement with the experimental results on the whole.
1. Introduction Terahertz wave means the electromagnetic wave between 0.1 THz and 10 THz. It has wide applications in biomedicine, material science, safety monitoring, communication and national defense [1,2]. Terahertz parametric source is one of the important terahertz radiation sources. It has many advantages, such as narrow line-width, continuous tunability, room temperature operation, high peak power, good coherence and so on. However, the generated terahertz pulse energy is very limited. The physical basis of the terahertz parametric source is stimulated polariton scattering (SPS), that is, a pump photon is converted into a Stokes photon and a terahertz photon under the energy conservation and momentum conservation. Because of the large refractive index of the crystal in the terahertz spectral range, only non-collinear phasematching can be obtained [3–14].
⁎
Terahertz parametric sources include terahertz parametric generator [3,4], terahertz parametric oscillator [5–20], seed injection terahertz parametric generator [21] and the combination of Stokes parametric oscillator and Stokes parametric amplifier [22]. In the fourth, the Stokes parametric oscillator generates a Stokes pulse first. It is used as the incident Stokes pulse for the Stokes parametric amplifier. The terahertz pulse can be generated by SPS when the Stokes parametric amplifier amplifies the incident Stokes pulse. By expanding the beam of the incident Stokes pulse, good spatial overlap of the pump and Stokes beams can be realized. The delay device on the path of the pump beam can ensure the good temporal overlap between the pump pulse and the Stokes pulse. This method is an important way to obtain large energy tunable terahertz pulses. The traditional nonlinear crystal for terahertz parametric sources is MgO:LiNbO3 [3–11,21–23]. It is found that KTiOPO4 (KTP), KTiOAsO4 (KTA), RbTiOPO4 (RTP) are also suitable for terahertz parametric
Corresponding author. E-mail address: [email protected] (X. Zhang).
https://doi.org/10.1016/j.optlastec.2019.105817 Received 14 April 2019; Received in revised form 18 August 2019; Accepted 5 September 2019 0030-3992/ © 2019 Elsevier Ltd. All rights reserved.
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sources in recent years [4,12–20,24–26]. KTP crystal has the advantages of mature growth technology, good optical quality and high damage threshold. The terahertz band (mainly 3.3–6.5 THz) produced by KTP crystal is different from the terahertz band (mainly 0.9–3.1 THz) produced by MgO:LiNbO3 crystal. In the previous researches, both terahertz parametric generator and terahertz parametric oscillator using KTP crystal are investigated [4,12–16,24–26]. But the obtained terahertz pulse energy is only about 5.5 μJ [14]. Therefore, it is significant to explore the potential of KTP crystal in obtaining large energy terahertz pulses by using the experimental scheme that combines Stokes parametric oscillator and Stokes parametric amplifier. In this paper, we explore the potential of KTP crystal in obtaining large energy terahertz pulses by using the experimental scheme that combines the Stokes parametric oscillator and the Stokes parametric amplifier. The terahertz wave surface-emitted structure is adopted [9–14]. When the pump pulse energy is 580.0 mJ, the incident Stokes pulse energy is 36.6 mJ, and the delay between the pumping pulse and the incident Stokes pulse is 1.7 ns, the amplified Stokes pulse energy is 192.1 mJ, and the obtained maximum terahertz pulse energy is 17.0 μJ. Compared with the reported terahertz pulse energy obtained by the KTiOPO4 terahertz parametric oscillators or generators, the terahertz pulse energy is greatly enhanced. In the theoretical part, considering the large angle non-collinear phase matching, pump wave depletion, terahertz wave surface-emitted structure, Gaussian pulse distribution, and other actual factors, we calculate the input and output characteristics of Stokes parametric amplifier and the loss characteristics of the pump wave by solving the coupled wave equations numerically. The theoretical results are in fair agreement with the experimental ones.
Output Energy of Stokes Pulse (mJ)
Fig. 2. The KTP crystal used in the Stokes parametric amplifier.
2. Experimental setup The schematic diagram of the experimental setup for the combination of Stokes parametric oscillator and Stokes parametric amplifier is shown in Fig. 1. There were two KTP crystals. The rectangular one with dimensions of 33 (x) mm × 6 (y) mm × 6 (z) mm was used for the Stokes parametric oscillator. The isosceles trapezoidal one was used for the Stokes parametric amplifier. The shape of the isosceles trapezoidal crystal on the xy plane is shown in Fig. 2. Its thickness along the z axis was 16 mm. The incident direction of the pump beam was approximately along the x axis of the crystal, and the polarization direction of the pump beam was parallel to the z axis of the crystal. All the input and output facets of the two crystals for the pump and the Stokes beams were antireflection coated at the wavelength range from 1060.0 to Ȝ/2
M1
T1
Ȝ/2
M7
Experimental Values Theoretical Values
195
180
165
150 0
1
1
M2
Rotating Stage
y
M8
Ȝ/2
M5
2
z Ȝ/2
Ȝ/2
x
x'
z
Ȝ/2 BW
Time Delayer M4
4
Fig. 3. The variation of the amplified Stokes energy with the time delay.
M9
M3
3
Time Delay (ns)
Pump Source BW
2
y
T2
x
y' Detector
Aperture
T3
M10
TPX Lense Attenuator
M6 BW
Rotating Stage
LPF Golay
Fig. 1. The schematic diagram of the experimental setup for the combination of Stokes parametric oscillator and Stokes parametric amplifier. 2
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Fig. 4. The waveforms of the pump and Stokes pulses for different time delays. (a) 0 ns; (b) 1.7 ns; (c) 2.7 ns; (d) 4.0 ns.
Output Energy of Stokes Pulse (mJ)
240
Experimental Values Theoretical Values
180
120
60
0 0
10
20
30
40
Incident Energy of Stokes Pulse (mJ) Fig. 5. The variation of the amplified Stokes energy with the incident Stokes energy.
Fig. 6. The variation of the terahertz wave energy with the incident Stokes energy.
1100.0 nm. The pump laser was a multi-longitudinal mode Q-switched Nd:YAG laser. The laser beam had a top-hat profile. The pulse width, the beam diameter and the repetition frequency were 7.5 ns, 6.0 mm and 1 Hz, respectively. The pump laser polarization direction was parallel to the z axis of the crystal. The laser beam from the Nd:YAG laser was divided into two parts by a beam splitter. The first part served as the pump beam of the Stokes parametric
oscillator. The Stokes wave oscillating cavity mirrors M7 and M8 were respectively coated with high reflectivity and partial transmission (T = 60%) at the Stokes wavelength. The cavity length was about 130 mm. The Stokes parametric oscillator was mounted on a rotating stage. There were two energy regulators, each of which consisted of two half wave plates and one Brewster plate. The first energy regulator was used to regulate the energy of the pump pulse and to ensure that the polarization direction of the pump beam was parallel to the z axis of the 3
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connected to a digital oscilloscope (Tektronix DPO 4104B, 1 GHz, 8GS/ s) after it passed through a TPX lens and a terahertz attenuator. The TPX lens was used to focus the terahertz beam. The terahertz attenuator was used to attenuate the terahertz wave pulse energy to adapt for the measuring range of the Golay detector. A terahertz low-pass filter (LFP 14.3-47, TYDEX) was used to block the scattered pump and Stokes beams with a transmittance of about 29% at 5.7 THz. 3. Experimental results The variations of the output Stokes pulse energy with the time delay in the Stokes parametric amplifier were measured first. The pump and incident Stokes pulse energies were fixed at 580.0 mJ and 36.6 mJ, respectively. The results are shown in Fig. 3. The dots are the experimental results while the solid line is the theoretical results (we will discuss the results further in part 4). With increasing time delay, the output Stokes pulse energy increased first, then reached the maximum, and then decreased. The same is true for the terahertz wave energy. The optimal time delay was 1.7 ns. Fig. 4 shows the waveforms of the pump and Stokes pulses for different time delays. Good temporal overlap between the pump and Stokes pulses was obtained with the time delay of 1.7 ns. The corresponding the output Stokes pulse energy was 192.1 mJ, the terahertz wave energy was 17.0 μJ. With the time delay fixed at 1.7 ns and the pump pulse energy at 580.0 mJ, the variations of the amplified Stokes pulse energy and the terahertz wave energy with the incident Stokes pulse energy are shown in Figs. 5 and 6. With the time delay fixed at 1.7 ns and the incident Stokes pulse energy at 36.6 mJ, Figs. 7 and 8 show the variations of the amplified Stokes pulse energy and the terahertz pulse energy with the pump pulse energy. The waveforms of pump pulses and residual pump pulses for different time delays are shown in Fig. 9. The residual pump pulses with different time delays were depleted in different degrees. We will discuss the results with the theoretical results together in Part 4.
Fig. 7. The variation of the amplified Stokes energy with the pump energy.
4. Theoretical simulation Fig. 8. The variation of the terahertz wave energy with the pump energy.
4.1. Coupled-wave equations for the Stokes parametric amplifier In the process of stimulated polariton scattering, the pump, Stokes and terahertz beams interact in the overlapped region according to the energy conservation ωp = ωs + ωT and the momentum conservation kp = ks + kT, where ki = ωi ni / c, (i = p , s, T ) , n is refractive index, ω is angular frequency, ki is wave vector. i = p , s, T represents the pump, Stokes and terahertz waves, respectively. Because of the large refractive index of the KTP nonlinear crystal in the terahertz spectral range, only non-collinear phase-matching for the three waves can be realized, which is shown in Fig. 10. The pump wave propagates along the x axis of the KTP crystal. The angle between the Stokes and pump waves is so small inside the crystal (about 1.8°) that the Stokes wave can be treated as parallel to the pump wave in the theoretical simulation. The terahertz wave propagates in y′ direction, the internal angle β between the terahertz and pump waves is 56.1°. The polarization directions of the three waves are parallel to the z axis of the KTP crystal. The evolutions of the pump, Stokes and terahertz waves in the process of SPS can be written as [27–29]:
crystal. The second one was used to regulate the energy of the Stokes pulse to be amplified. Telescope T1 with a ratio of 2:1 was used to reduce the pump beam size for the Stokes parametric oscillator. The pump beam diameter in the Stokes parametric oscillator was 3.0 mm. Telescope T2 with a ratio of 1:3 was used to enlarge the size of the Stokes beam to be amplified. The second part was used as the pump beam of the Stokes parametric amplifier. The time delayer on the way of the pump beam was used to adjust the temporal overlapping between the incident Stokes pulse and the pump pulse of the Stokes parametric amplifier. The energy regulator in this portion had the same function as those in the other portion. Telescope T3 and the aperture were used to adjust the pump beam size. The diameter of pump beam in the Stokes parametric amplifier was 8.0 mm. The angles between the pump and Stokes beams outside the crystals in the Stokes parametric oscillator and the Stokes parametric amplifier were fixed at 4.4°, and the corresponding angles inside the crystal were 1.8°. The wavelength of the pump wave was 1064.2 nm and the central wavelength of the Stokes wave was 1086.2 nm. The terahertz wavelength was 52.5 µm, its corresponding frequency was 5.70 THz. The maximum energy of the incident Stokes pulse is 36.6 mJ with a pulse width of 6.0 ns. The energies of the pump and Stokes pulses were measured by an energy sensor (J-50MB-YAG, Coherent Inc.) connected to an energy meter (EPM2000, Coherent Inc.). The energy of the terahertz wave pulse was measured by a calibrated Golay detector (GC-1D, TYDEX)
∂Ap ∂x =
ωp2 ⎞ ωp2 ⎛ i ⎧ ⎛ 2 −⎜kp − 2 εp ⎟ Ap − 2 4d33 + 2kp ⎨ ⎝ c ⎠ c ⎜ ⎝ ⎩ −
4
ωp2 c2
∑ dQj2 εQj∗ |As |2 Ap j
⎫ ⎬ ⎭
∑ dQi εQj∗ ⎞⎟ As AT j
⎠
(1a)
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Fig. 9. The waveforms of the pump pulses and residual pump pulses for different time delays. (a) 0 ns; (b) 1.7 ns; (c) 2.7 ns; (d) 4.0 ns.
F G
D
H E
Fig. 10. The non-collinear phase matching of SPS in KTP crystal.
C I
∂As ∂x
B
A
ω2 ⎛ ω2 i ⎧ ⎛ 2 = −⎜ks − 2s εs ⎟⎞ As + 2s 4d33 + 2ks ⎨ ⎝ c ⎠ c ⎜ ⎝ ⎩ +
ωp2 c2
∑ dQj2 εQj∗ |Ap |2 As j
K
∑ dQi εQj∗ ⎞⎟ Ap AT∗ j
62°
⎠
⎫ ⎬ ⎭
⎧ ⎛ 2 ω2 ⎛ ω2 ∂AT i = −⎜kT − 2T εT ⎟⎞ AT + 2T 4d33 + 2kT cos β ⎨ ⎝ c ⎜ c ∂x ⎠ ⎝ ⎩
M
⎫
⎠
⎭
z
y
(1b)
⎞
x'
x y'
Fig. 11. The terahertz wave surface-emitted structure.
∑ dQi εQj∗ ⎟ Ap As∗ ⎬ j
and Raman-active jth TO mode [31,32]. εQ can be written as [33]: (1c)
where Am (m = p , s, T ) is the complex amplitude, d33 is the secondharmonic nonlinear coefficient [30], εm (m = p , s, T ) is the dielectric constant, dQj εQj is the combined ionic responses of the infrared-active
εQ = εT − ε∞ =
∑ j
5
2 Sj ωjTO 2 ωjTO − ω2 − jΓjTO ω
(2)
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gs(2) = ωs K
(5d)
gT(2) = ωT K
(5e)
4.2. Numerical calculation Eqs. (4a)–(4c) are the coupled-wave equations describing the Stokes parametric amplifier. The corresponding parameters for the simulation can be obtained by using Eqs. (5a)–(5e). The terahertz wave surfaceemitted structure is shown in Fig. 11. AE and DF denote the width of the pump and Stokes beams, while BC is the width of the terahertz beam in the gain region. The pump and Stokes beams enter the crystal from AE plane, and are totally reflected on BC surface, and then go out from DF plane. One part of the generated terahertz wave is emitted from BC surface. The other part is reflected back.
Fig. 12. The interaction region of the three waves.
TT =
Table 1 Parameters for simulation [35–43].
αT d33 np ns nT λp λs λT β
∑j dQj Re (εQj )
Parameter
Value
The terahertz wave absorption coefficient Second-harmonic nonlinear coefficient Refractive index of the pump light Refractive index of the Stokes light Refractive index of the terahertz wave Wavelength of the pump light Wavelength of the Stokes light Wavelength of the terahertz wave The angle between the pump and terahertz wave The combined ionic responses of all the infrared-active and Raman-active TO modes
64.9 cm−1 15.4 × 10−10 cm/V 2.388 2.387 4.474 1.0642 × 10−4 cm 1.0862 × 10−4 cm 5.254 × 10-3 cm 0.979 rad 3.79 × 10-8 cm/V
p, s, T
1/2
64π 3ε02 c 4np Rj ⎤ =⎡ ⎢ Sj hωjTO ω 4 ns (XT + 1) ⎥ s ⎣ ⎦
∂Ip
(3)
= −gp(2) (Ip Is IT )1/2
(4a)
∂Is = +gs(2) (Ip Is IT )1/2 ∂x
(4b)
∂IT = −αT IT + gT(2) (Ip Is IT )1/2 ∂y′
(4c)
∂x
ωT2 Im(εT ) kT c 2 1/2
1 ⎞ K = ⎜⎛ ⎟ ⎝ np ns nT ⎠
gp(2) = ωp K
(5a) 1/2
⎛ 1 ⎞ 3 ⎝ 2c ε0 ⎠
⎜
⎟
⎡ ⎢4d33 + ⎣
∂I
Ip (0, y′ , t ) = Ipump (t )
(7a)
Is (0, y′ , t ) = IStokes (t )
(7b)
IT (0, y′ , t ) = IT (x , 0, t ) = 0
(7c)
After the pump and Stokes beams are totally reflected, the interaction region of the three waves is the trapezoidal region of BCDF. The reflected pump, Stokes, and terahertz waves will be the initial signals for the interaction in this region. The generated terahertz wave in this region will be absorbed by the nonlinear crystal because its propagating direction is close to − y′, while the residual pump and Stokes waves will be affected by the interaction in this region. Above analysis is for the xy plane of z = 0. Considering that both the pump and Stokes waves are of circular beam shapes, the three waves will vary with z axis. However, we can calculate the intensity values of the three waves in the planes for other z values in the same way. The output pulse energies of the three waves will be the integrals over the beam sizes and the pulse durations.
where αT is the terahertz wave absorption coefficient,
αT =
∂I
ΔIp = ∂x ΔLp , ΔIs = ∂xs ΔLs , ΔIT = ∂yT′ ΔLT . The incident pump and Stokes beams are suggested to be Gaussian temporal distributions, and the pump beam has a top-hat spatial profile and the incident Stokes beam has a Gaussian spatial profile. The boundary conditions should be:
whereε0 is the vacuum permittivity, Rj is the equivalent Raman cross section. XT is the Boson factor. Since the absorption and third-order nonlinear coefficients of the pump and Stokes waves are very small in the process of SPS, they can be neglected. Based on the relationship between intensity and complex amplitude I = 2nε0 c |A|2 , the coupled-wave equations can be written as [27,34]:
∂Ip
(6)
where TT is the intensity transmissivity of the terahertz wave on the interface between the nonlinear crystal and air, nT1 and θT1 are the refractive index and incident angle of the terahertz wave in the crystal respectively, nT2 and θT2 are the refractive index and exit angle of the terahertz wave in the air respectively. For the terahertz wave surface-emitted structure, only in the BC region can the terahertz wave be emitted efficiently. In BK region, a little terahertz wave can also be emitted because the generated terahertz wave is heavily absorbed in the journey of IK. Before the pump and Stokes beams are totally reflected, the interaction region of the three waves is the trapezoidal region of ABCE. The pump and Stokes beams propagate along x direction and the terahertz wave propagates along y′ direction. As shown in Fig. 12, we use (x m , yn′) to express the positions in this region. During a time intervalΔt , the pump and Stokes waves propagate from (x m − 1, yn′) to (x m , yn′) , and the terahertz wave propagates from (x m , yn′− 1) to (x m , yn′) . At this time interval, the interaction lengths of the pump, Stokes, and terahertz waves c are ΔLp, s, T = n Δt , the increments of the three waves are
where Sj is the oscillator strength of the jth TO mode, ωjTO and ΓjTO are its angular frequency and damping coefficient, respectively, ε∞ is the high-frequency dielectric constant. And dQj can be written as:
dQj
4nT 1 nT 2 cos θT 1 cos θT 2 (nT 1 cos θT 1 + nT 2 cos θT 2)2
⎤
∑ dQj Re(εQj) ⎥ j
⎦
(5b)
4.3. Simulation results
(5c)
In the theoretical processing, the pump pulse width is set as 7.5 ns, the Stokes pulse width is set as 6.0 ns. The pump and Stokes beams are 6
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Fig. 13. The numerical results of the waveforms of the pump pulses and residual pump pulses for different time delays. (a) 0 ns; (b) 1.7 ns; (c) 2.7 ns; (d) 4.0 ns.
volume of the three waves is a little smaller than that used in the theoretical simulation. Intuitively, it seems that the generated terahertz pulse energy should be much larger than 17 μJ. If we simply considered the energy conservation law and the fact that the Stokes pulse was amplified from 36.6 mJ to 192.1 mJ, the generated terahertz pulse energy would be around 3.2 mJ. Actually, when the terahertz wave reached the output surface, a part of it (about 40%) was reflected back into the crystal, then unable to exit. More seriously, the terahertz wave absorption loss was very large (the absorption coefficient of the KTP crystal was 64.9 cm−1). The terahertz wave was heavily absorbed during the journey to the output surface. Only in the KC region (see Fig. 11) could the terahertz wave be obtained. In other regions such as MK and CD, no terahertz wave was obtained. From this point of view, one approach to enhance the terahertz output energy is to reduce the absorption by decreasing the crystal temperature. Another approach may be the use of the pumping beam with an elliptic cross section. The beam size in zdirection is larger while that in y-direction is smaller.
in the circle shape and their diameters is 8.0 mm. The path length of the pump and Stokes beams in KTP crystal (AB + BF) is 3.2 cm. The length of AB in KTP crystal is 1.5 cm. The output plane of the pump and Stokes waves is DF plane, while the output plane of the terahertz wave is KC plane. The parameters for the simulation are shown in Table 1. The numerical results of the waveforms of the pump pulses and residual pump pulses for different time delays are shown in Fig. 13 when the incident Stokes energy is 36.6 mJ and the input pump energy is 580.0 mJ. The variations of the Stokes output energy with the time delay are shown in Fig. 3. With the time delay fixed at 1.7 ns and the pump pulse energy at 580.0 mJ, the variations of the Stokes output energy and the terahertz wave energy with the incident Stokes pulse energy are shown in Figs. 5 and 6. With the time delay fixed at 1.7 ns and the incident Stokes pulse energy at 36.6 mJ, the variations of the amplified Stokes pulse energy and the terahertz pulse energy with the pump pulse energy are shown in Figs. 7 and 8. It can be seen from Figs. 3, 5–9 and 13 that the theoretical and experimental results are basically consistent with some differences. The main causes for the small differences between the theoretical results and the experimental results may be as follows. First, there is no guarantee that the parameters used in the theoretical calculation are accurate. Second, the pump beam used in the experiment is not an absolute top-hat spatial profile, and the incident Stokes beam is not a perfect Gaussian spatial profile. Third, the pump and Stokes pulses are assumed to be Gaussian temporal distributions in the calculation. Actually, the pulses are not exact Gaussian shapes (see Figs. 4 and 8). Fourth, the perfect overlapping between the pump and Stokes beams is assumed in the calculation. In fact, due to the small phase-matching angle between the pump and Stokes beams, the actual interaction
5. Conclusion The combination of Stokes parametric oscillator and Stokes parametric amplifier is an important means to generate high-energy tunable terahertz pulses. We have applied it to KTP crystal to get high energy terahertz nanosecond pulses at 5.7 THz for the first time. When the incident Stokes pulse energy is 36.6 mJ and the pump energy is 580.0 mJ, the obtained maximum terahertz pulse energy is 17.0 μJ. In the theoretical part, considering the factors such as the large angle noncollinear phase matching, pump beam depletion, terahertz wave 7
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surface-emitted structure, and Gaussian pulse distribution, we have made a detailed theoretical simulation for the Stokes parametric amplifier through the coupled-wave equations. The theoretical and experimental results are basically consistent with some differences. The main causes for the small differences between the theoretical results and the experimental results are analyzed.
[19]
[20]
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Acknowledgements
[22]
This research is supported by the National Natural Science Foundation of China (61775122, 61475087, 61605103), the Key Research and Development Program of Shandong Province (2017CXGC0809, 2017GGX10103), and the Natural Science Foundation of Shandong Province (ZR2014FM024).
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