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II KDP crystals
Optik 145 (2017) 465–472
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Research on wavefront properties of high power frequency tripling lasers based on type II/II KDP crystals Sensen Li a,b,∗ , Zhiwei Lu b , Yulei Wang b , Xiusheng Yan a , Kai Wang a , Luoxian Zhou b , Yirui Wang b , Lei Ding c a b c
Science and Technology on Electro-optical Information Security Control Laboratory, Tianjin 300308, China National Key Laboratory of Science and Technology on Tunable Laser, Harbin Institute of Technology, Harbin 150001, China Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China
a r t i c l e
i n f o
Article history: Received 22 April 2017 Accepted 3 August 2017 Keywords: High power lasers Frequency tripling KDP crystals Wavefront
a b s t r a c t The relationship between the output frequency tripling (3) wavefront and the input fundamental laser (1) wavefront is researched in theory and experiment in the frequency conversion process based on type II/II KDP (KH2 PO4 ) crystals. Theoretical results show the output 3 wavefront distortion is three times of the input 1 wavefront distortion on the phase-matching condition. The output 3 and 1 wavefront are measured by Hartman wavefront sensor in the high power frequency tripling laser system. Experimental results show that the output 3 wavefront PV value is 2.97 times of the 1 wavefront value and the output 3 wavefront RMS value is 3.57 times of the 1 wavefront value. Therefore, the theoretical relationship between the output 3 wavefront and the input 1 wavefront in the frequency conversion system is proved correct within the range of permitted experimental errors. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction In the high power laser system [1–8], the non-linear effect of the rapidly-growing large-diameter KDP (KH2PO4) crystal is widely used for the third-harmonic generation (351 nm) [9–12]. When the polarization of the fundamental frequency (1) is stable, the frequency convention efficiency is insensitive to the detuning angle of the doubling crystal in the type II/type II polarization mismatch scheme [13–17], which can greatly reduce the adjustment difficulty and improve the stability of the output 3 laser [18–20]. In the high power solid-state laser system, the uniformity of the wavefront distribution, as an important criterion of the beam quality [21], is closely related to the far-field focal spot of the laser, which affects the alignment accuracy of the laser beam [22]. In the frequency conversion process, the phase matching angle detuning in the beam aperture caused by wavefront distortion will reduce the frequency conversion efficiency [23,24]. Moreover, the spatial intensity distribution is also influenced by the wavefront distribution in the transmission process. To date, the related work on high power frequency tripling laser characteristics are mainly focused on the output pulse waveform [17], near-field [19] and far-field [25] spatial intensity distribution. Meanwhile, the theoretical calculation on coupling wave equations and wavefront distortion transmission of 3 generated by the KDP crystals with type I/type II
∗ Corresponding author at: Science and Technology on Electro-optical Information Security Control Laboratory, Tianjin 300308, China. E-mail address: [email protected] (S. Li). http://dx.doi.org/10.1016/j.ijleo.2017.08.036 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
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Fig. 1. Principle of frequency tripling lasers generated by type II/type II polarization mismatch scheme.
angle-detuned scheme has already been studied [26,27]. However, there is little research on the wavefront distribution characteristics of the output 3 laser, especially the experimental measurement of the 3 laser wavefront. In this paper, the wavefront change properties from the input 1 laser to the output 3 laser in the frequency conversion process based on KDP crystals in type II/type II polarization mismatch scheme is calculated. The relationship of wavefront distribution between 3 laser and 1 laser is theoretically analyzed and is experimentally carried out in the high power frequency tripling laser system. The wavefront distribution of the 3 laser and 1 laser is measured by Shack-Hartman wavefront sensor and the wavefront variation in the frequency conversion process is obtained.
2. Theoretical model The principle of frequency tripling lasers generated by type II/type II polarization mismatch scheme is shown in Fig. 1 [19]. The frequency conversion part includes two KDP crystals. The front KDP is a “doubler”, in which some fraction of the fundamental radiation laser (1) is converted to the second harmonic, followed by a “tripler”, in which unconverted fundamental radiation is mixed with the second harmonic to produce the third harmonic (3). The angle between the polarization of injecton 1 laser and the o axis of the first KDP crystal is 35.3◦ , which results in the maximum efficiency of 3 output efficiency [27]. The relationship of wavefront between output 3 laser and injecting 1 laser can be calculated according to the coupling wave equations and the perturbation theory. Based on the type II/type II polarization mismatch scheme, Assuming Ej (j = l, 2, 3) to be the complex amplitude of 1, 2 and 3, respectively, and the mathematical expression is as following,
Ej (x, y, z, t) =
1 A (x, y, z, t) exp[−i(ωj t − kj z)] + c.c, j = 1, 2,3 2 j
(1)
The harmonic coupling wave equations of the first KDP crystal are shown as below:
⎧ ∂A1o 1 iω 1 ∂A1o ∗ ⎪ ⎪ ⎪ ∂z + v1 ∂t = − 2 1 A1o + 2n1 c A2e A1e exp(−ikz) ⎪ ⎪ ⎨ ∂A1e
1 ∂A1e 1 = − 2 A1e + v2 ∂ t 2 1 1 ∂A2e + = − 3 A2e + v3 ∂ t 2 ∂z
⎪ ∂z ⎪ ⎪ ⎪ ⎪ ⎩ ∂A2e
+
iω A2e A∗1o exp(−ikz) 2n2 c
(2)
iω A1o A1e exp(ikz) 2n3 c
where subscripts 1,2,3 represent 1 o, 1 e and 2 e, respectively; v is the group velocity and is the absorption coefficient; A is the complex amplitude and n is the refractive index; ω is the fundamental frequency; k is the wavevector mismatch among three waves; c is the speed of light in the vacuum; is the effective nonlinear susceptibilities; = − sin cos 2ϕ, where is the angle between the vector k and z axis and ϕ is the azimuth.
S. Li et al. / Optik 145 (2017) 465–472
467
The harmonic tripling wave equations of the second KDP crystal can be expressed as:
∂ i 1 iω1 eff ∂A A3 A∗2 exp(−ikz) ∇ 2 A1 − 1 1 − ˛1 A1 + A1 = − 2 2k1 ⊥ ne (ω, )c ∂z ∂re +
iω1 ne (ω, )c
11
2
|A1 |2 + 12 |A2 |2 + 13 |A3 |2 A1
iω2 eff ∂ i 1 A3 A∗1 exp(ikz) ∇ 2⊥ A2 − ˛2 A2 + A2 = − 2 2n 2k ∂z o (2ω)c 2 +
iω2 no (2ω)c
22
(3)
|A2 |2 + 23 |A3 |2 + 21 |A1 |2 A2
2
iω3 eff i 1 ∂ ∂A A1 A2 exp(ikz) ∇ 2 A3 − 3 3 − ˛3 A3 + A3 = − 2 2k3 ⊥ 2ne (3ω, )c ∂z ∂re +
iω3 ne (3ω, )c
33
2
|A3 |2 + 31 |A1 |2 + 32 |A2 |2 A3
where k = k1 + k2 − k3 . Assuming that the mathematical expression of fundamental frequency laser (1) field consists of a plane wave and a perturbation factor [26], we get: A1 (r) = A10 [1 + (r)]
(4)
where (r) 1, so the expression above can be further written in the form of amplitude and phase disturbances as: A1 = A10 (1 + R ) exp(i I )
(5)
where R , I are amplitude and phase disturbances, respectively. Frequency tripling laser (3) field can be described as: A3 = 1/2 A10 (1 + R ) exp(iϕ3ω ) = A30 (1 + R ) exp(i I )
(6)
where ϕ3 represents the phase distribution of the frequency tripling laser. The frequency tripling efficiency is related to the 1 intensity I1 , the frequency harmonic angle 2 and the mixed harmonic angle 3 . Considering only the first derivative effect, the efficiency can be expressed as below: = 0 +
∂ ∂ ∂ I1 + 2 + 3 ∂I1 ∂2 ∂3
(7)
Noting that the mismatch angels can be written in the form as following: 2 =
1 ∂ I 1 ∂ I , 3 = k0 ∂x2 k0 ∂x3
(8)
where x2 , x3 are the sensitive axis of two KDP crystals, respectively, and x2 ⊥x3 ; k0 is the wavevector of the 1 laser. Thus the efficiency can be rewritten as: = 0 + = 0 +
∂ 1 ∂ ∂ I 1 ∂ ∂ I I1 (1 + R )2 + + k k ∂I1 0 ∂2 ∂x2 0 ∂3 ∂x3
1 ∂ I1 (1 + R )2 + k0 ∂I1
∂ ∂ eˆ 2 + eˆ 3 ∂2 ∂3
(9)
∇ I
Using the two expressions below, =
1 ∂( I1 ) ∂I1
1
= 2 k0
∂ ∂ eˆ 2 + eˆ 3 ∂2 ∂3
(10)
then we get: 1/2
1/2 = 0
[1 + ( − 1) R + ∇ I ]
(11)
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where ( − 1) R and ∇ I stand for amplitude and phase disturbances, respectively. By ignoring the second order, it can be obtained as following: A30 (1 + R ) = 1/2 A10 (1 + R ) 1/2
= 0 A10 (1 + R ) [1 + ( − 1) R + ∇ I ]
(12)
= A30 [1 + R + ( − 1) R + ∇ I ] = A30 [1 + R + ∇ I ] Thus the third harmonic amplitude disturbance is given by:
R = R + ∇ I
(13)
Next the relationship between the phase disturbance of the 3 laser and the distortion of the 1 laser is theoretically calculated. ϕ1 , the 1 phase disturbance, is determined by I . According to the coupling wave equations, the phase distributions after passing the two crystals are expressed by:
ϕ1ω = ϕ1ω + k2 L2 + k2 ϕ2ω
L2
2 (z)dz
(14)
0
1 = 2ϕ1ω + k2 L2 2
where L2 is the length of frequency doubling KDP crystal; k2 is the phase mismaching degree and 2 (z)is the coefficient associated with L2 . In the sum frequency process, the 3 phase distribution can be described as: ϕ3ω =
ϕ1ω
+ ϕ2ω
1 + k3 L3 + k2 2
L2
2 (z)dz 0
1 = 3ϕ1ω + (3k2 L2 + k3 L3 ) + k2 2
(15)
L2
2 (z)dz 0
where L3 is the length of sum frequency KDP crystal. Let F(z) =
1 L2
z
2 (z )dz
(16)
0
then the phase disturbance of 3 can be written as
I = R + 3 I + ∇ I = R + (3 + ∇ ) I
(17)
where = k2 L2 F (L2 ) =
1 2k0
∂ 2 ∂I
k2 L2 F (L2 )
∂ 2 ∂k2 1 ∂k3 + F(L2 ) eˆ 2 + eˆ 3 2k0 ∂3 ∂2 ∂2
(18)
The results above are written in the following matrix form:
R
=
I
∇
3 + ∇
R
(19)
I
Noting that is approximately equal to 1 in case of the maximum of the frequency conversion efficiency, which means |∂ /∂I1 | → 0. The order of is less than 10−6 because k0 = 2n/ ≈ 9 × 106 and |∂ /∂2 | → 0, |∂ /∂3 | → 0. is the fixed detuning angle of the frequency doubling KDP crystal, k2 L2 is on the order of 1, and the value of ranges from 0 to 1. The order of is also less than 10−6 . Since the values of and are relatively small, the Equation(19) can be simplified as:
R
I
=
0
3
R
(20)
I
After the frequency conversion process, the 3 phase distribution can be expressed as: ϕ3ω (x, y) = 3ϕ1ω (x, y) + k 1ω
(21)
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Fig. 2. The schematic of the laser system and the experiment setup for wavefront measurement.
where is a coefficient related to the length of frequency conversion crystal and the conversion efficiency; 1 is the amplitude disturbance of the 1 laser. In case of k = 0, the phase disturbance amplitude of the output 3 laser is three times larger than that of the input 1 laser. In addition, in the high efficient frequency tripling process, the values of k and the 1 amplitude disturbance are small (nearly to 0). Thus, the phase relationship between output 3 laser and the input 1 laser of the KDP crystals with type II/type II polarization mismatch scheme is established as following, ϕ3ω (x, y) = 3ϕ1ω (x, y)
(22)
3. Results and discussion The experimental setup for wavefront measurement is shown in Fig. 2. The 1 laser system has three main sections with MOPA structure: the front-end system, the preamplifier system, the main amplifier system. The front-end provides a 10 nanojoule-scale seed pulse at 1 Hz. The latter amplifiers grow progressively larger in size and the preamplifier can amplify to millijoule-scale energies at 1 Hz. Then the laser pulse passes through the main amplifier with four-stage rod amplifier which can amplify 100 J level of energy at 1053 nm with 3 ns pulse duration and 60 mm diameter beam size [18,28]. The intensity of the output intensity of the 1 laser is designed at 1.18 GW/cm2 . The laser system operats at a repetition rate of approximately 2 shots per hour. The 1 laser (1053 nm) generated by a 1 laser system is injected into the frequency conversion module to produce the 3 laser (351 nm) with 50 J level energy with the frequency conversion efficiency upwards of 50%. The frequency converter module mainly consists of two 100 × 100 × 14 mm3 KDP crystals, using Type II/type II polarization-mismatch scheme. In this frequency converter, the beam undergoes second-harmonic generation in the first KDP crystal as a doubler to yield the 527-nm light and third-harmonic generation in the second KDP crystal as a tripler to yield the 351-nm light. The 3 laser is mixed with the 1 and 2 laser, and it is separated from them by two spectral mirrors with special coating layer. Then the 3 laser enters the transport spatial filter, filtering the high spatial frequency. The KDP crystals are in a better clean-room and airtight environment with two 3-deg oblique windows on each side. There are three electric motors controlling the two crystals on three directions. In the laser system, critical beam information is gathered in two diagnostic systems: the 1 diagnostic system before the KDP crystals and the 3 diagnostic system efore the 3 output side. Each of these diagnostic systems includes a wavefront sensors, a near-field scientific-grade CCD camera, a fast photodiode to provide temporal measurements and a calorimeter for energy information. In the diagnostics system, The 1 and 3 output wavefronts are directly measured by Shack–Hartman wavefront sensors (Thorlabs WFS150-5C) at the same time [29,30]. The laser beam passes through a 4f system with a 20:1 reduction ratio, and the image plane is relayed on the wavefront sensor with 39 × 31 lenslets with each lenslet focal length of 5.2 mm and pitch of 150 m. The main laser is sampled into the wavefront sensor through some bandpass filters and attenuators, which can reduce the interference of stray light and make the laser pulse attenuated to the affordable energy of the wavefront sensor. Before measuring the laser wavefront, the output laser nearfield and temporal pulse have been active shaped to be uniformable. After spatial beam shaping by using a liquid-crystal spaitial light modulator (SLM), the flat-top output laser is achieved. The 1 and 3 laser fluence contrast (RMS value) reaches 9% and 15.4%, relatively [see Fig. 3(a) and (b)]. As shown in Fig. 3(c) and (d), the laser system can provide the output 1 and 3 pulse with a flat-top super Gaussian distribution with duration of 3 ns [19,31–34]. The uniform nearfield and temporal pulse are necessary condition for a good wavefront output.
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Fig. 3. The output nearfield and the temporal pulse of the 1 and 3 laser. The 1 near field (a), the 3 near field (c), the 1 temporal pulse (c) and the 3 temporal pulse (c).
For estimating the output wavefront quality before and after the KDP crystals, the peak-to-velly (PV) and RMS value are introduced here. Generally, the PV value can be used to describe the whole wavefront distribution and the RMS value can be used to describe the wavefront details. The PV value is mathematically given by: PV = ϕmax (x, y) − ϕmin (x, y)
(23)
where ϕmax (x,y), ϕmin (x,y) are the maximum and minimum values of the wavefront respectively. The units of PV value are usually expressed by the laser wavelength , or m. The PV value is a good reflection of the overall wavefront fluctuation, which is the structure information of spatial low-frequency slow wavefront distribution. The RMS value is defined as the root mean square of the error distribution of the laser wavefront, which is the reflection of the fluctuation of the wavefront distribution, and is expressed by:
N 1 2 RMS = ϕi (x, y) − ϕavg (x, y) N
(24)
i=1
where ϕi (x,y) is the wavefront value of a certain point; ϕavg (x,y) is the average value of the beam wavefront and N is the total number of discrete points distributed in the light spot. The experimental results of wavefront distribution are described in Fig. 4. In case of 1 laser, the PV value is 0.38 m and the RMS value is 0.07 m. While for the 3 output laser, the PV value is 1.13 m and the RMS value is 0.25 m. Results show that the PV value of 3 output is 2.97 times of the 1 laser, and the RMS value is 3.57 times. This output wavefront data are achieved in experiment and they account for the local beam distortions (such as hot spots or an intensity roll-off across the beam profile), pulse-shape and pulse-width issues (that might be inherently present in the seed or might occur in going from 1 to 3), leading to slight disagreements with the ideal data. In the high-power solid-state laser system, for PV value, the difference between theoretical and experimental results is less than 1%, and for RMS, less than 16%. In the whole beam wavefront profile, the wavefront error of 3 is about 3 times that of 1 according to the experimental results, which is nearly the same to the theoretical results when considering experimental errors. 4. Conclusions The relationship between the 3 laser wavefront and the 1 laser wavefront is discussed both theoretically and experimentally in the frequency conversion process based on KDP crystals in the type II/type II polarization mismatch scheme.
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Fig. 4. The output laser wavefront. (a) 1 laser; (b) 3 laser.
Theoretically results show that the wavefront error of the output 3 laser is 3 times of that of the input 1 laser under the condition that the phase matching is satisfied. In the high-power solid-state laser system, for PV value, the difference between theoretical and experimental results is less than 1%, and for RMS, less than 16%. The correctness of the theoretical model of the relationship between the 3 laser and the 1 laser wavefront is verified within the range of the experimental error. The results provide a theoretical reference for wavefront control or compensation in high power solid-state laser systems. Acknowledgments This work is supported by the project of the National Natural Science Foundation of China (NSFC) under Grant No. 61622501. The support does not constitute an endorsement by the NSFC of the views expressed in this article. References [1] E.I. Moses, Ignition on the National Ignition Facility: a path towards inertial fusion energy, Nucl. Fusion 49 (2009) 104022. 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Optik journal homepage: www.elsevier.de/ijleo
Original research article
Research on wavefront properties of high power frequency tripling lasers based on type II/II KDP crystals Sensen Li a,b,∗ , Zhiwei Lu b , Yulei Wang b , Xiusheng Yan a , Kai Wang a , Luoxian Zhou b , Yirui Wang b , Lei Ding c a b c
Science and Technology on Electro-optical Information Security Control Laboratory, Tianjin 300308, China National Key Laboratory of Science and Technology on Tunable Laser, Harbin Institute of Technology, Harbin 150001, China Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China
a r t i c l e
i n f o
Article history: Received 22 April 2017 Accepted 3 August 2017 Keywords: High power lasers Frequency tripling KDP crystals Wavefront
a b s t r a c t The relationship between the output frequency tripling (3) wavefront and the input fundamental laser (1) wavefront is researched in theory and experiment in the frequency conversion process based on type II/II KDP (KH2 PO4 ) crystals. Theoretical results show the output 3 wavefront distortion is three times of the input 1 wavefront distortion on the phase-matching condition. The output 3 and 1 wavefront are measured by Hartman wavefront sensor in the high power frequency tripling laser system. Experimental results show that the output 3 wavefront PV value is 2.97 times of the 1 wavefront value and the output 3 wavefront RMS value is 3.57 times of the 1 wavefront value. Therefore, the theoretical relationship between the output 3 wavefront and the input 1 wavefront in the frequency conversion system is proved correct within the range of permitted experimental errors. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction In the high power laser system [1–8], the non-linear effect of the rapidly-growing large-diameter KDP (KH2PO4) crystal is widely used for the third-harmonic generation (351 nm) [9–12]. When the polarization of the fundamental frequency (1) is stable, the frequency convention efficiency is insensitive to the detuning angle of the doubling crystal in the type II/type II polarization mismatch scheme [13–17], which can greatly reduce the adjustment difficulty and improve the stability of the output 3 laser [18–20]. In the high power solid-state laser system, the uniformity of the wavefront distribution, as an important criterion of the beam quality [21], is closely related to the far-field focal spot of the laser, which affects the alignment accuracy of the laser beam [22]. In the frequency conversion process, the phase matching angle detuning in the beam aperture caused by wavefront distortion will reduce the frequency conversion efficiency [23,24]. Moreover, the spatial intensity distribution is also influenced by the wavefront distribution in the transmission process. To date, the related work on high power frequency tripling laser characteristics are mainly focused on the output pulse waveform [17], near-field [19] and far-field [25] spatial intensity distribution. Meanwhile, the theoretical calculation on coupling wave equations and wavefront distortion transmission of 3 generated by the KDP crystals with type I/type II
∗ Corresponding author at: Science and Technology on Electro-optical Information Security Control Laboratory, Tianjin 300308, China. E-mail address: [email protected] (S. Li). http://dx.doi.org/10.1016/j.ijleo.2017.08.036 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
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Fig. 1. Principle of frequency tripling lasers generated by type II/type II polarization mismatch scheme.
angle-detuned scheme has already been studied [26,27]. However, there is little research on the wavefront distribution characteristics of the output 3 laser, especially the experimental measurement of the 3 laser wavefront. In this paper, the wavefront change properties from the input 1 laser to the output 3 laser in the frequency conversion process based on KDP crystals in type II/type II polarization mismatch scheme is calculated. The relationship of wavefront distribution between 3 laser and 1 laser is theoretically analyzed and is experimentally carried out in the high power frequency tripling laser system. The wavefront distribution of the 3 laser and 1 laser is measured by Shack-Hartman wavefront sensor and the wavefront variation in the frequency conversion process is obtained.
2. Theoretical model The principle of frequency tripling lasers generated by type II/type II polarization mismatch scheme is shown in Fig. 1 [19]. The frequency conversion part includes two KDP crystals. The front KDP is a “doubler”, in which some fraction of the fundamental radiation laser (1) is converted to the second harmonic, followed by a “tripler”, in which unconverted fundamental radiation is mixed with the second harmonic to produce the third harmonic (3). The angle between the polarization of injecton 1 laser and the o axis of the first KDP crystal is 35.3◦ , which results in the maximum efficiency of 3 output efficiency [27]. The relationship of wavefront between output 3 laser and injecting 1 laser can be calculated according to the coupling wave equations and the perturbation theory. Based on the type II/type II polarization mismatch scheme, Assuming Ej (j = l, 2, 3) to be the complex amplitude of 1, 2 and 3, respectively, and the mathematical expression is as following,
Ej (x, y, z, t) =
1 A (x, y, z, t) exp[−i(ωj t − kj z)] + c.c, j = 1, 2,3 2 j
(1)
The harmonic coupling wave equations of the first KDP crystal are shown as below:
⎧ ∂A1o 1 iω 1 ∂A1o ∗ ⎪ ⎪ ⎪ ∂z + v1 ∂t = − 2 1 A1o + 2n1 c A2e A1e exp(−ikz) ⎪ ⎪ ⎨ ∂A1e
1 ∂A1e 1 = − 2 A1e + v2 ∂ t 2 1 1 ∂A2e + = − 3 A2e + v3 ∂ t 2 ∂z
⎪ ∂z ⎪ ⎪ ⎪ ⎪ ⎩ ∂A2e
+
iω A2e A∗1o exp(−ikz) 2n2 c
(2)
iω A1o A1e exp(ikz) 2n3 c
where subscripts 1,2,3 represent 1 o, 1 e and 2 e, respectively; v is the group velocity and is the absorption coefficient; A is the complex amplitude and n is the refractive index; ω is the fundamental frequency; k is the wavevector mismatch among three waves; c is the speed of light in the vacuum; is the effective nonlinear susceptibilities; = − sin cos 2ϕ, where is the angle between the vector k and z axis and ϕ is the azimuth.
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The harmonic tripling wave equations of the second KDP crystal can be expressed as:
∂ i 1 iω1 eff ∂A A3 A∗2 exp(−ikz) ∇ 2 A1 − 1 1 − ˛1 A1 + A1 = − 2 2k1 ⊥ ne (ω, )c ∂z ∂re +
iω1 ne (ω, )c
11
2
|A1 |2 + 12 |A2 |2 + 13 |A3 |2 A1
iω2 eff ∂ i 1 A3 A∗1 exp(ikz) ∇ 2⊥ A2 − ˛2 A2 + A2 = − 2 2n 2k ∂z o (2ω)c 2 +
iω2 no (2ω)c
22
(3)
|A2 |2 + 23 |A3 |2 + 21 |A1 |2 A2
2
iω3 eff i 1 ∂ ∂A A1 A2 exp(ikz) ∇ 2 A3 − 3 3 − ˛3 A3 + A3 = − 2 2k3 ⊥ 2ne (3ω, )c ∂z ∂re +
iω3 ne (3ω, )c
33
2
|A3 |2 + 31 |A1 |2 + 32 |A2 |2 A3
where k = k1 + k2 − k3 . Assuming that the mathematical expression of fundamental frequency laser (1) field consists of a plane wave and a perturbation factor [26], we get: A1 (r) = A10 [1 + (r)]
(4)
where (r) 1, so the expression above can be further written in the form of amplitude and phase disturbances as: A1 = A10 (1 + R ) exp(i I )
(5)
where R , I are amplitude and phase disturbances, respectively. Frequency tripling laser (3) field can be described as: A3 = 1/2 A10 (1 + R ) exp(iϕ3ω ) = A30 (1 + R ) exp(i I )
(6)
where ϕ3 represents the phase distribution of the frequency tripling laser. The frequency tripling efficiency is related to the 1 intensity I1 , the frequency harmonic angle 2 and the mixed harmonic angle 3 . Considering only the first derivative effect, the efficiency can be expressed as below: = 0 +
∂ ∂ ∂ I1 + 2 + 3 ∂I1 ∂2 ∂3
(7)
Noting that the mismatch angels can be written in the form as following: 2 =
1 ∂ I 1 ∂ I , 3 = k0 ∂x2 k0 ∂x3
(8)
where x2 , x3 are the sensitive axis of two KDP crystals, respectively, and x2 ⊥x3 ; k0 is the wavevector of the 1 laser. Thus the efficiency can be rewritten as: = 0 + = 0 +
∂ 1 ∂ ∂ I 1 ∂ ∂ I I1 (1 + R )2 + + k k ∂I1 0 ∂2 ∂x2 0 ∂3 ∂x3
1 ∂ I1 (1 + R )2 + k0 ∂I1
∂ ∂ eˆ 2 + eˆ 3 ∂2 ∂3
(9)
∇ I
Using the two expressions below, =
1 ∂( I1 ) ∂I1
1
= 2 k0
∂ ∂ eˆ 2 + eˆ 3 ∂2 ∂3
(10)
then we get: 1/2
1/2 = 0
[1 + ( − 1) R + ∇ I ]
(11)
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where ( − 1) R and ∇ I stand for amplitude and phase disturbances, respectively. By ignoring the second order, it can be obtained as following: A30 (1 + R ) = 1/2 A10 (1 + R ) 1/2
= 0 A10 (1 + R ) [1 + ( − 1) R + ∇ I ]
(12)
= A30 [1 + R + ( − 1) R + ∇ I ] = A30 [1 + R + ∇ I ] Thus the third harmonic amplitude disturbance is given by:
R = R + ∇ I
(13)
Next the relationship between the phase disturbance of the 3 laser and the distortion of the 1 laser is theoretically calculated. ϕ1 , the 1 phase disturbance, is determined by I . According to the coupling wave equations, the phase distributions after passing the two crystals are expressed by:
ϕ1ω = ϕ1ω + k2 L2 + k2 ϕ2ω
L2
2 (z)dz
(14)
0
1 = 2ϕ1ω + k2 L2 2
where L2 is the length of frequency doubling KDP crystal; k2 is the phase mismaching degree and 2 (z)is the coefficient associated with L2 . In the sum frequency process, the 3 phase distribution can be described as: ϕ3ω =
ϕ1ω
+ ϕ2ω
1 + k3 L3 + k2 2
L2
2 (z)dz 0
1 = 3ϕ1ω + (3k2 L2 + k3 L3 ) + k2 2
(15)
L2
2 (z)dz 0
where L3 is the length of sum frequency KDP crystal. Let F(z) =
1 L2
z
2 (z )dz
(16)
0
then the phase disturbance of 3 can be written as
I = R + 3 I + ∇ I = R + (3 + ∇ ) I
(17)
where = k2 L2 F (L2 ) =
1 2k0
∂ 2 ∂I
k2 L2 F (L2 )
∂ 2 ∂k2 1 ∂k3 + F(L2 ) eˆ 2 + eˆ 3 2k0 ∂3 ∂2 ∂2
(18)
The results above are written in the following matrix form:
R
=
I
∇
3 + ∇
R
(19)
I
Noting that is approximately equal to 1 in case of the maximum of the frequency conversion efficiency, which means |∂ /∂I1 | → 0. The order of is less than 10−6 because k0 = 2n/ ≈ 9 × 106 and |∂ /∂2 | → 0, |∂ /∂3 | → 0. is the fixed detuning angle of the frequency doubling KDP crystal, k2 L2 is on the order of 1, and the value of ranges from 0 to 1. The order of is also less than 10−6 . Since the values of and are relatively small, the Equation(19) can be simplified as:
R
I
=
0
3
R
(20)
I
After the frequency conversion process, the 3 phase distribution can be expressed as: ϕ3ω (x, y) = 3ϕ1ω (x, y) + k 1ω
(21)
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469
Fig. 2. The schematic of the laser system and the experiment setup for wavefront measurement.
where is a coefficient related to the length of frequency conversion crystal and the conversion efficiency; 1 is the amplitude disturbance of the 1 laser. In case of k = 0, the phase disturbance amplitude of the output 3 laser is three times larger than that of the input 1 laser. In addition, in the high efficient frequency tripling process, the values of k and the 1 amplitude disturbance are small (nearly to 0). Thus, the phase relationship between output 3 laser and the input 1 laser of the KDP crystals with type II/type II polarization mismatch scheme is established as following, ϕ3ω (x, y) = 3ϕ1ω (x, y)
(22)
3. Results and discussion The experimental setup for wavefront measurement is shown in Fig. 2. The 1 laser system has three main sections with MOPA structure: the front-end system, the preamplifier system, the main amplifier system. The front-end provides a 10 nanojoule-scale seed pulse at 1 Hz. The latter amplifiers grow progressively larger in size and the preamplifier can amplify to millijoule-scale energies at 1 Hz. Then the laser pulse passes through the main amplifier with four-stage rod amplifier which can amplify 100 J level of energy at 1053 nm with 3 ns pulse duration and 60 mm diameter beam size [18,28]. The intensity of the output intensity of the 1 laser is designed at 1.18 GW/cm2 . The laser system operats at a repetition rate of approximately 2 shots per hour. The 1 laser (1053 nm) generated by a 1 laser system is injected into the frequency conversion module to produce the 3 laser (351 nm) with 50 J level energy with the frequency conversion efficiency upwards of 50%. The frequency converter module mainly consists of two 100 × 100 × 14 mm3 KDP crystals, using Type II/type II polarization-mismatch scheme. In this frequency converter, the beam undergoes second-harmonic generation in the first KDP crystal as a doubler to yield the 527-nm light and third-harmonic generation in the second KDP crystal as a tripler to yield the 351-nm light. The 3 laser is mixed with the 1 and 2 laser, and it is separated from them by two spectral mirrors with special coating layer. Then the 3 laser enters the transport spatial filter, filtering the high spatial frequency. The KDP crystals are in a better clean-room and airtight environment with two 3-deg oblique windows on each side. There are three electric motors controlling the two crystals on three directions. In the laser system, critical beam information is gathered in two diagnostic systems: the 1 diagnostic system before the KDP crystals and the 3 diagnostic system efore the 3 output side. Each of these diagnostic systems includes a wavefront sensors, a near-field scientific-grade CCD camera, a fast photodiode to provide temporal measurements and a calorimeter for energy information. In the diagnostics system, The 1 and 3 output wavefronts are directly measured by Shack–Hartman wavefront sensors (Thorlabs WFS150-5C) at the same time [29,30]. The laser beam passes through a 4f system with a 20:1 reduction ratio, and the image plane is relayed on the wavefront sensor with 39 × 31 lenslets with each lenslet focal length of 5.2 mm and pitch of 150 m. The main laser is sampled into the wavefront sensor through some bandpass filters and attenuators, which can reduce the interference of stray light and make the laser pulse attenuated to the affordable energy of the wavefront sensor. Before measuring the laser wavefront, the output laser nearfield and temporal pulse have been active shaped to be uniformable. After spatial beam shaping by using a liquid-crystal spaitial light modulator (SLM), the flat-top output laser is achieved. The 1 and 3 laser fluence contrast (RMS value) reaches 9% and 15.4%, relatively [see Fig. 3(a) and (b)]. As shown in Fig. 3(c) and (d), the laser system can provide the output 1 and 3 pulse with a flat-top super Gaussian distribution with duration of 3 ns [19,31–34]. The uniform nearfield and temporal pulse are necessary condition for a good wavefront output.
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Fig. 3. The output nearfield and the temporal pulse of the 1 and 3 laser. The 1 near field (a), the 3 near field (c), the 1 temporal pulse (c) and the 3 temporal pulse (c).
For estimating the output wavefront quality before and after the KDP crystals, the peak-to-velly (PV) and RMS value are introduced here. Generally, the PV value can be used to describe the whole wavefront distribution and the RMS value can be used to describe the wavefront details. The PV value is mathematically given by: PV = ϕmax (x, y) − ϕmin (x, y)
(23)
where ϕmax (x,y), ϕmin (x,y) are the maximum and minimum values of the wavefront respectively. The units of PV value are usually expressed by the laser wavelength , or m. The PV value is a good reflection of the overall wavefront fluctuation, which is the structure information of spatial low-frequency slow wavefront distribution. The RMS value is defined as the root mean square of the error distribution of the laser wavefront, which is the reflection of the fluctuation of the wavefront distribution, and is expressed by:
N 1 2 RMS = ϕi (x, y) − ϕavg (x, y) N
(24)
i=1
where ϕi (x,y) is the wavefront value of a certain point; ϕavg (x,y) is the average value of the beam wavefront and N is the total number of discrete points distributed in the light spot. The experimental results of wavefront distribution are described in Fig. 4. In case of 1 laser, the PV value is 0.38 m and the RMS value is 0.07 m. While for the 3 output laser, the PV value is 1.13 m and the RMS value is 0.25 m. Results show that the PV value of 3 output is 2.97 times of the 1 laser, and the RMS value is 3.57 times. This output wavefront data are achieved in experiment and they account for the local beam distortions (such as hot spots or an intensity roll-off across the beam profile), pulse-shape and pulse-width issues (that might be inherently present in the seed or might occur in going from 1 to 3), leading to slight disagreements with the ideal data. In the high-power solid-state laser system, for PV value, the difference between theoretical and experimental results is less than 1%, and for RMS, less than 16%. In the whole beam wavefront profile, the wavefront error of 3 is about 3 times that of 1 according to the experimental results, which is nearly the same to the theoretical results when considering experimental errors. 4. Conclusions The relationship between the 3 laser wavefront and the 1 laser wavefront is discussed both theoretically and experimentally in the frequency conversion process based on KDP crystals in the type II/type II polarization mismatch scheme.
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Fig. 4. The output laser wavefront. (a) 1 laser; (b) 3 laser.
Theoretically results show that the wavefront error of the output 3 laser is 3 times of that of the input 1 laser under the condition that the phase matching is satisfied. In the high-power solid-state laser system, for PV value, the difference between theoretical and experimental results is less than 1%, and for RMS, less than 16%. The correctness of the theoretical model of the relationship between the 3 laser and the 1 laser wavefront is verified within the range of the experimental error. The results provide a theoretical reference for wavefront control or compensation in high power solid-state laser systems. Acknowledgments This work is supported by the project of the National Natural Science Foundation of China (NSFC) under Grant No. 61622501. The support does not constitute an endorsement by the NSFC of the views expressed in this article. References [1] E.I. Moses, Ignition on the National Ignition Facility: a path towards inertial fusion energy, Nucl. Fusion 49 (2009) 104022. 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