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Numerical investigation of output beam quality in efficient broadband optical parametric chirped pulse amplification
Optics Communications 383 (2017) 197–207
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Numerical investigation of output beam quality in efficient broadband optical parametric chirped pulse amplification Xiao-Di Liu a,b, Lu Xu a, Xiao-Yan Liang a,c,n a State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Science, Shanghai 201800, People's Republic of China b University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China c School of Physical Science and Technology, Shanghai Tech University, Shanghai 200031, People's Republic of China
art ic l e i nf o
a b s t r a c t
Article history: Received 15 June 2016 Received in revised form 27 August 2016 Accepted 31 August 2016
We theoretically analyzed output beam quality of broad bandwidth non-collinear optical parametric chirped pulse amplification (NOPCPA) in LiB3O5 (LBO) centered at 800 nm. With a three-dimensional numerical model, the influence of the pump intensity, pump and signal spatial modulations, and the walk-off effect on the OPCPA output beam quality are presented, together with conversion efficiency and the gain spectrum. The pump modulation is a dominant factor that affects the output beam quality. Comparatively, the influence of signal modulation is insignificant. For a low-energy system with small beam sizes, walk-off effect has to be considered. Pump modulation and walk-off effect lead to asymmetric output beam profile with increased modulation. A special pump modulation type is found to optimize output beam quality and efficiency. For a high-energy system with large beam sizes, the walkoff effect can be neglected, certain back conversion is beneficial to reduce the output modulation. A trade-off must be made between the output beam quality and the conversion efficiency, especially when the pump modulation is large since. A relatively high conversion efficiency and a low output modulation are both achievable by controlling the pump modulation and intensity. & 2016 Elsevier B.V. All rights reserved.
Keywords: Nonlinear optics Parametric processes Laser amplifiers
1. Introduction Optical parametric chirped pulse amplification (OPCPA), proposed by Dubietis in 1990s [1], is considered as a high promising technology for achieving ultrafast and ultrahigh-intensity laser pulses. It has the following advantages: high gain, wavelength tunability, low B-integral accumulation, and low thermal effects. OPCPA with non-collinear geometry was favored for a broad gain bandwidth [2,3] and convenient pulse separation. Many approaches were adopted to improve the NOPCPA output. A peak power of several hundred terawatts even up to petawatts was currently available [4–13]. Few-optical-cycle even near to Fourierlimited light pulses were demonstrated [14–17]. The advances of ultra-short, high energy pulses will potentially allow further progress in probing novel physical and optical phenomena, such as ultrafast X-ray radiation, laser-matter interaction [18], particle acceleration [19], inertial confinement fusion, etc. n Corresponding author at: State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Science, Shanghai 201800, People's Republic of China. E-mail address: [email protected] (X.-Y. Liang).
http://dx.doi.org/10.1016/j.optcom.2016.08.082 0030-4018/& 2016 Elsevier B.V. All rights reserved.
For a non-collinear parametric amplifier, the conversion efficiency is crucial, and a variety of methods has been used to improve it. Using the conformal theory proposed by Begishev [20,21], the pump energy could be fully utilized in all spatial and temporal points to maximize the conversion efficiency. 67% conversion of pump energy into parametric waves was achieved by optimal profiling of interactive waves [20]. Moses used the temporal conformal theory to boost OPCPA conversion efficiency and the amplification bandwidth [22]. Flat-top beams, as the simplest conformal profile, contribute to a great improvement in the OPCPA conversion efficiency and stability [2]. A high conversion efficiency of 29% is demonstrated by Waxer using a high-order super Gaussian pump [23]. In addition to the conversion efficiency, the output beam quality is a critical factor for the application in strong-field physics. It also affects the stability and security of the whole system. Flattop spatial and temporal pulses can give optimum performance and enhance the output beam quality. However, spatial or temporal noises are inevitable in real laser systems. The OPCPA performance with modulated pulses should be discussed especially for high energy laser systems where pump pulses are far from the ideal one [9,24]. In this article, we focus on the impact of the
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spatial modulation on the overall conversion efficiency and the output modulation. Temporally, we use a Gaussian profile of the signal, and a super-Gaussian profile of the pump, which reduces gain narrowing and maintains signal bandwidth [13,25]. Spatially, we introduce some modulation to the super-Gaussian profile. To evaluate the influence of pump or signal spatial modulation separately, a super-Gaussian signal is assumed to show the impact of pump modulation. Similarly, a super-Gaussian pump with a modulated signal is set to distinguish the impact of signal. The spatial walk-off effect is also an important factor that affects the output beam quality [24]. With a three dimensional spatial and temporal model, the influence of pump and signal modulations, and the walk-off effect on output beam quality and conversion efficiency are discussed. The results show that the pump modulation has a dominant influence on the output beam quality. Besides, the walk-off effect is considerable for small beam diameters, however, it is negligible for the large beam diameters. We discuss two cases with different scales of beam sizes. In the first case, the interactive beam diameters are considerably small, compared to the walk-off length; thus, walk-off effect cannot be neglected. The pump modulation and walk-off simultaneously affect the output, which includes asymmetric intensity distribution and increased output modulation. A particular pump modulation type is found to compensate the negative influence of walk-off. In another case, however, the beam sizes are so large that walk-off length is comparatively small; hence, walk-off is neglectable owing to its little influence. Output beam quality depends mainly on the parametric amplification process. A quite low output modulation can be achieved at a specific average pump intensity, where back conversion occurs and inevitably reduces the conversion efficiency. The trade-off between the conversion efficiency and a low output modulation highlights the importance of reduction of pump modulation and selection of a proper average pump intensity, by which a relatively high conversion efficiency and a low modulation can be achieved. The results can be helpful for the design of high-energy OPCPA systems. The paper is organized as follows. The numerical simulation and initial conditions are described in detail in Section 2. In Section 3, the output features of small beam sizes are analyzed. In Section 3.1, different modulation types are set to evaluate the influence of pump modulation and the walk-off effect. In Section 3.2, the influence of input signal modulation is discussed. Section 4 concerns the output features of the large-beam-size OPCPA system. In Section 4.1, the influence of pump modulation is discussed. Section 4.2 describes the influence of signal modulation. A pump intensity range is given in Section 4.3 to ensure a relatively high conversion efficiency and low modulation, which is helpful for the design of high-energy OPCPA system. The conclusion is presented in Section 5.
2. Simulation details The theoretical investigation is based on coupled wave equations governing the parametric amplification process. According to Ref. [26], the group velocity mismatch for short pulses is considered in the frequency domain. Since the pulse durations of pump and signal in our model are of the order of nanoseconds, the group velocity mismatch between pump and signal over the crystal length is negligible compared to the pulse width, as demonstrated by Ross et al. [3]. Then considering the full spatial and temporal dependence of the three parametric waves, with type I phase matching and non-collinear geometry configuration, the following set of coupled wave equations is derived and numerically evaluated [27,28].
⎛ ∂2 ∂As ∂A ∂2 ⎞ 1 ⎜⎜ ⎟⎟As + tan ρs s − + 2 ∂z ∂y 2jns ks cos ρs ⎝ ∂x ∂y2 ⎠ ws =−j deff A p Ai*exp( − iΔkz ), ns c cos ρs
(1a)
⎛ ∂2 ∂Ai ∂A ∂2 ⎞ 1 ⎜⎜ ⎟⎟Ai + tan ρi i − + ∂z ∂y 2jni ki cos ρi ⎝ ∂x2 ∂y2 ⎠ wi =−j deff A p As*exp( − iΔkz ), ni c cos ρi
(1b)
∂A p ∂z
+ tan ρp
=−j
∂A p ∂y
wp npc cos ρp
−
⎛ 2 2 ⎞ 1 ⎜ ∂ + ∂ ⎟A p 2jnpk p cos ρp ⎜⎝ ∂x2 ∂y2 ⎟⎠
deff As Ai exp(iΔkz ).
(1c)
where subscripts s, i, and p refer to the signal, idler, and pump, respectively; A is the complex electric field amplitude; ω is the angular frequency; n is the refractive index and k is the wave number of A; ρp is the birefringence walk-off angle of pump, whereas ρs and ρi are the nonlinear angles of signal and idler wave vectors; j is the imaginary unit; c is the speed of light in vacuum; deff accounts for the effective nonlinear coefficient, Δk is the wave-vector mismatch which is derived from energy conversion and momentum conservation. There is no input idler intensity, the idler is produced during the parametric process, and it self-adjusts to ensure the maximum initial signal gain. Because of the nonlinear geometry and the broadband signal, the idler has a relative large spatial divergence angle. In this paper, the modulation of amplified signal is primarily considered. The above differential equations are numerically solved by a spit-step technique in the space and time domain. The linear process, which includes spatial walk-off, wave propagation and diffraction effect, as well as nonlinear parametric process are both considered in the procedure [29,30]. LBO (LiB3O5) is a popular nonlinear material, exhibiting a high nonlinear coefficient, a high damage threshold, and a broad gain bandwidth in the visible and near infrared. Besides, LBO with large aperture up to tens of micrometers is currently available. It is a good choice not only in the front end as a broadband amplifier but also in the final end for high-energy amplification. In order to investigate the influence of the initial pump and signal on the output signal modulation of a feasible broadband OPCPA laser system, LBO (LiB3O5) with the length of 12 mm is chosen as the nonlinear crystal [11]. Since the nonlinear process for different nonlinear crystals are similar, the numerical model and discussion in this paper can be extended to other nonlinear crystals such as BBO and DKDP. Type I phase matching (θ ¼90°, φ ¼13.85°) is adopted. A non-collinear angle between the signal and the pump of 1.26° is chosen to enlarge the amplification bandwidth. The pump walk-off angle is 0.48°. The second-harmonic of Nd: glass, at 526.5 nm, is used as the pump. The input signal is from a Ti: sapphire chirped pulse amplifier centered at 800 nm with a full spectral width of 80 nm. The pump and signal diameters of the small-beam-size OPCPA are 5 mm and 4 mm, respectively. The large-beam-size pump and signal diameters are 55 mm and 54 mm, respectively. The initial signal intensity is set to 80 MW/ cm2. The pump intensity varies from 0.53 GW/cm2 to observe the trend. The temporal profiles of pump and signal are shown in Fig. 1. The input signal is linearly chirped in time with a pulse duration of 1.6 ns (FWHM). The pump is super-Gaussian shaped with a pulse duration of 2.89 ns (FWHM). The super-Gaussian
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distribution within the spot size, i.e. a flat-top beam has a modulation of 1. A Cartesian coordinate is chosen as shown in Fig. 2(a). In that system, the direction of ks is z, the ordinary and extraordinary waves are polarized along x and y. Pump and idler walk off in the negative y direction. To better investigate the influence of input pump and signal modulation on the OPCPA output, a typical visualized modulation type is used in our simulation as can be seen in Fig. 2. The modulation is introduced along x- or y-axis to a super-Gaussian profile. Fig. 2(b) is the ideal super-Gaussian with a slope of 0 and modulation of 1; For Fig. 2(c), the modulation slope is along the x-axis, and the pump intensity in the walk-off direction is the same. For Fig. 2(d) and (e), the modulation slopes are along the y-axis, and the pump intensities in the walk-off direction differ. The different modulation types are set to evaluate the walk-off effect in NOPCPA, which will be discussed in Section 3 in detail. Fig. 1. Input temporal profiles of pump and signal.
temporal shape of the pump is chosen to enhance the conversion efficiency and achieve full bandwidth amplification [13,25]. Near-field spatial modulation is defined by Eq. (2) [31]. It is used to evaluate beam quality.
M=
Imax − Ign Iavg − Ign
(2)
where Imax and Iavg are the maximum intensity and the average intensity within the beam, respectively, and the beam area is defined by the beam spatial FWHM. Ign is background noise. In our case, Ign ¼0. Modulation can reflect uniformity of the intensity
3. NOPCPA with small beam sizes For a small-beam-size NOPCPA system, the pump and signal diameters are 5 mm and 4 mm, respectively. The walk-off lengths of the pump and idler are approximately 0.17 mm and 0.79 mm, respectively. Walk-off causes pump, signal and idler separation; hence, not only the value of Mp but also the modulation type can influence the parametric amplification and the output properties. By carrying out the numerical simulation of Eq. (1), the influence of the pump, signal and walk-off on the OPCPA output is given in this section. For brevity, some denotations are used: average pump intensity, injected pump and signal modulation, and output signal modulation are denoted as Ip, Mp, Ms_in and Ms_out, respectively, in this article. The modulation direction orthogonal (as Fig. 2(c)) and parallel (as Fig. 2(d) and (e)) to the walk-off direction are denoted as Mp⊥ and Mp//. The influence of pump modulation is evaluated in Section 3.1, where different modulation types, Mp⊥ and Mp// þ and Mp// as Fig. 2(c)–(e) are set. Signal modulation is discussed in Section 3.2. 3.1. Influence of pump modulation on output beam quality
Fig. 2. (a) Cartesian coordinate of non-collinear geography. The propagation of signal pulse is chosen as the z axis. Walk-off occurs in the yoz-plane, the walk-off direction is along the negative direction of y axis. (b) The super-Gaussian profile; the modulation is 1. (c) The x-axis slope modulation, denoted as Mp⊥; (d) and (e) The y-axis slope modulation with positive and negative directions, denoted as Mp// þ and Mp// , respectively.
3.1.1. Pump with x-axis slope modulation An x-axis pump modulation type similar to Fig. 2(c) is set. In this case, although pump walk-off occurs, the pump intensity in the walk-off direction is identical. In addition, the pump diameter is larger than that of the signal, so pump walk-off generates little pump intensity variation among the signal beam size. Accordingly, we can identify the influence of the pump modulation while pump walk-off on the OPCPA output is diminished. The conversion efficiency (η) with different Ip and Mp⊥ is calculated and shown in Fig. 3. The curves for different Mp⊥ share a similar tendency. For certain Mp⊥, η increases and then decreases with the increase in Ip, and the maximum conversion efficiency (ηmax) can be reached at a certain pump intensity (Ieff). ηmax and Ieff for different Mp⊥ are shown in Fig. 4(a). With the increase in Mp⊥, ηmax decreases. Specifically, ηmax for Mp⊥ ¼ 1.0, 1.3 and 1.5 are 39.36%, 39.15% and 39.01%, and Ieff are 1.38, 1.30 and 1.23 GW/cm2, respectively. The reduction is mainly because of uneven amplification and back conversion cause by the maldistributed pump intensity. The short dash curves in Fig. 3 are Ms_out with different Ip and Mp⊥. First, an obvious feature is that Ms_out is strongly positively related to Mp⊥. For fixed Ip, the larger Mp⊥ is, the larger Ms_out is obtained. Moreover, Ms_out changes considerably with Ip. For fixed Mp⊥, with the increase in Ip, Ms_out tends to decrease and then increase. Minimum Ms_out (min-Ms_out) can be reached at a certain pump intensity, Imod, as shown in Fig. 4(b). Specifically, Imod for
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Table 1 Conversion efficiency (η) and output modulation (Ms_out) for different average pump intensity (Ip) and input pump modulation (Mp�). Notation
Ip (GW/cm2)
Mp�
η
Ms_out
A/D B/E C/F
1.06 1.32 1.60
1.0/1.3 1.0/1.3 1.0/1.3
38.22%/38.39% 39.34%/39.14% 38.91%/38.35%
1.11/1.35 1.12/1.39 1.18/1.46
increasing slowly until it reaches ηmax, but the raised edge contributes to increased Ms_out. For large Ip, i.e. Ip ¼1.6 GW/cm2 at points C and F, back conversion in the center can be much severer. Although amplification in the edge enlarges, the overall effect is η reduction and further increase in Ms_out, as can be seen in Figs. 3 and 5(d) and (h). Fig. 3. Conversion efficiency (η) and the output modulation (Ms_out) of different average pump intensity (Ip) and modulation (Mp⊥). A, B and C correspond to Ip ¼1.06, 1.32 and 1.6 GW/cm2, Mp ¼ 1.0; D, E, and F correspond to Ip ¼ 1.06, 1.32 and 1.6 GW/cm2, Mp⊥ ¼1.3.
Mp ¼1.0, 1.3 and 1.5 are 1.15 and 0.97 and 0.88 GW/cm2, respectively. However, Ms_out can hardly be less than the input pump modulation. This indicates that the output beam quality is closely relevant to pump quality. With a highly modulated pump, low Ms_out is difficult to maintain. η and Ms_out are closely correlated to Ip, Mp ¼1 and Mp⊥ ¼1.3 are taken as an example. Three points corresponding to Ip ¼ 1.06, 1.32 and 1.6 GW/cm2 are labeled on the output modulation curve for Mp ¼1.0 (Mp⊥ ¼ 1.3) as A, B and C (D, E and F), respectively. The input and output parameters of those points are listed in Table 1. The input pump and output signal profiles are shown in Fig. 5. When Ip increases to 1.06 GW/cm2, i.e. points A and D, gain increases as well, and the signal is amplified and has a spatial profile similar to the pump, as can be seen in Fig. 5(b) and (f). For a superGaussian pump, the central is relatively uniform, and the edges becomes steeper, as shown in Fig. 5(b); Ms_out reduces with the increase in Ip. For Mp⊥ ¼ 1.3, the signal beam edge is amplified, especially in the high-pump-intensity area. Additionally, owning to the saturation caused by the local high pump intensity, the output signal intensity in the center area is better distributed, and thus Ms_out for Mp⊥ ¼1.3 is reduced as well. Then Ip increases to 1.32 GW/cm2, i.e. point B and E, saturation and back conversion become severe especially in the beam center where pump, signal and idler always overlap. However, in the right edge, the idler walks out, and back conversion cannot occur, which results in a raise around the right edge, Fig. 5(c) and (g). The raised edge in the right bottom is more prominent for Mp⊥ ¼1.3 which is because of increasingly large pump intensity along the x-axis. η keeps
3.1.2. Pump with y-axis slope modulation Compared to the x-axis pump modulation type, the y-axis modulation type has a different intensity along the walk-off direction. As shown in Fig. 2(d), the pump intensity is gradually increased because of walk-off. It is recorded as Mp// þ . The pump modulation similar to Fig. 2(e) is recorded as Mp// . Walk-off leads to decreasing the intensity gradually. Mp// þ (Mp// ) of 1.3 is used as an example. η and Ms_out are shown in Fig. 6. First, we compare η for Mp// þ , Mp// to that of Mp⊥. Although the specific value changes slightly, i.e. η for Mp// þ is generally smaller, and that for Mp// is generally larger, the variation tendency still holds. ηmax can be achieved at Ieff. ηmax for Mp// þ and Mp// are 38.56% and 39.78%, and Ieff are 1.32 and 1.3 GW/cm2, respectively. Second, Ms_out for Mp// þ share a similar tendency with Mp⊥, but Mp// differs, as it corresponds to a much smaller Ms_out and a larger Imod. Specially, Min-Ms_out for Mp// þ and Mp// are 1.41 and 1.12, Imod are 0.9 and 1.5 GW/cm2, respectively. To be specific, three points corresponding to Ip ¼1.06, 1.32 and 1.6 GW/cm2 are labeled on the output modulation curve of Mp// þ (Mp// ) as D þ , E þ and F þ (D , E and F ), respectively. The input and output parameters are listed in Table 2. The input pump and output signal profiles are shown in Fig. 7. As Ip increases to 1.06 GW/cm2, the gain increases, the slope in central area decreases, and the beam edges become steep, as shown in Fig. 7 (b) and (f). As a result, η becomes larger and Ms_out smaller. Then with the increase in Ip, saturation and back conversion occur and become increasingly large, η cannot keep increasing and the maximum point is achieved later, at approximately 1.3 GW/cm2, as shown in Fig. 6. The pump and idler walk-off simultaneously influence the output beam quality, which is quite different for Mp// þ and Mp// . For Mp// þ , the weak part of the pump gradually walks out during the parametric process, and the pump intensity becomes
Fig. 4. (a) Maximum conversion efficiency (ηmax) and Ieff. (b) Minimum output modulation (min-Ms_out) for different Mp⊥.
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Fig. 5. (a) The input super-Gaussian pump profile, Mp ¼ 1.0. (b), (c) and (d) Output beam profiles with pump (a) at points A, B and C, respectively. (e) The input pump profile, Mp⊥ ¼ 1.3. (f)–(h) Output beam profiles with pump (e) at point D, E and F, respectively.
intensity region averts serious back conversion. The overall effect is the enhanced conversion efficiency with a better beam quality and enlarged Imod.
Fig. 6. Conversion efficiency (η) and output modulation (Ms_out) of different average pump intensity (Ip) and modulation type (Mp⊥, Mp// þ , Mp// ). D þ , E þ and F þ correspond to Ip ¼1.06, 1.32 and 1.6 GW/cm2, Mp// þ ¼ 1.3; D , E and F correspond to Ip ¼ 1.06, 1.32 and 1.6 GW/cm2, Mp// ¼ 1.3.
Table 2 Conversion efficiency (η) and output modulation (Ms_out) for different average pump intensities (Ip) and input pump modulation types (Mp// þ and Mp// ). Notation
Ip (GW/cm2)
Mp ¼ 1.3
η
Ms_out
D þ /D E þ /E F þ /F
1.06 1.32 1.60
Mp// þ /Mp// Mp// þ /Mp// Mp// þ /Mp//
37.72%/39.06% 38.56%/39.76% 37.90%/38.86%
1.42/1.26 1.48/1.21 1.59/1.19
increasingly larger. However, the situation is inverse for Mp// . The walk out of the idler frees the signal right edge from back conversion, where a considerably high gain is possible. When Mp// þ is injected, the area around the signal right edge gains strongly because of an increasingly high pump intensity and absence of idler, resulting in larger Ms_out. Apart from this region, the increasingly high pump causes saturation and back conversion, preventing the increase in η. If Mp// is injected, the gain in the right edge of the signal is much smaller compared to Mp// þ , but continuous gain can compensate for the local weak pump intensity, as shown in Fig. 7 (g) and (h). Moreover, the walk-off of the ultra-high pump
3.1.3. Compensation for walk-off effect A super-Gaussian pump is favored in OPCPA, but the results shown in Figs. 3 and 5 indicate that the pump is super-Gaussian and the walk-off of the idler leads to the raised edge and increased Ms_out. In addition, Imod differs from Ieff (Imod o Ieff), which indicates that trade-off exists among the efficiency and the output beam quality. However, the output feature of Mp// suggests that Ms_out can be reduced together with larger Imod and increased η. We choose Mp// ¼ 1.1 and compare the output with a superGaussian pump, as shown in Fig. 8. Min-Ms_out is 1.06, lower than that of the super-Gaussian. Besides, Ieff and Imod coincide; they are both 1.32 GW/cm2, indicating that ηmax and min-Ms_out can be achieved simultaneously. The output profile is shown the insert in Fig. 8. The nearly flat-top spatial profile confirms the beneficial compensation effect of Mp// . To conclude, the pump beam profile can be designed to compensate the walk-off effect, optimize output beam quality and efficiency. 3.2. Influence of signal modulation on output beam quality The input signal quality, Ms_in, is another important factor of parametric amplification. In order to evaluate the influence of Ms_in, a super-Gaussian pump is assumed, i.e. Mp ¼1.0. The input signal is the modulated one with orthogonal slope as shown in Fig. 2(c). The solid curves in Fig. 9 depict the conversion efficiency (η) with different Ip and Ms_in. The variation tendency for different Ms_in are similar, ηmax can be achieved at a similar Ieff, 1.38 GW/ cm2. The dashed curves with dots show the tendency of Ms_out. Min-Ms_out for Ms_in ¼1.0, 1.3 and 1.5 are 1.107, 1.109 and 1.119, respectively. Imod for different Ms_in is identical, 1.15 GW/cm2. Unlike Fig. 4, Ms_out in Fig. 9 for different Ms_in are similar and keep small within a large pump range. For example, Ms_out for Ms_in ¼ 1.5 varies from 1.119 to 1.14 as Ip increases from Imod ¼ 1.15 GW/cm2 to Ieff ¼1.38 GW/cm2. It is a favorable feature that good output beam quality can be ensured at high conversion efficiency. Besides, the low Ms_out also indicates that output beam quality can be improved by OPCPA.
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Fig. 7. (a) The input pump profile, Mp// þ ¼ 1.3. (b), (c) and (d) The output beam profiles with pump (a), at point D þ , E þ and F þ , respectively. (e) The input pump profile, Mp// ¼ 1.3. (f), (g) and (h) Output beam profiles with pump (e) at point D , E and F , respectively.
Table 3 Conversion efficiency (η) and the output modulation (Ms_out) for different average pump intensity (Ip) and input signal modulation (Ms_in).
Fig. 8. Conversion efficiency (η) and output modulation (Ms_out) for different average pump intensity (Ip). The output beam profile at Ip ¼1.32 GW/cm2 is shown as the insert.
Ip (GW/cm2)
Ms_in
η
Ms_out
1.06 1.32 1.60
1.0/1.3/1.5 1.0/1.3/1.5 1.0/1.3/1.5
38.22%/38.08%/37.87% 39.34%/39.22%/39.03% 38.91%/38.81%/38.66%
1.11/1.11 1.12 1.12/1.12 1.13 1.18/1.18 1.19
1.3 and 1.5 are chosen. The input and output parameters are listed in Table 3. The little difference in Ms_out indicates that the influence of the signal modulation on Ms_out is insignificant. However, in Section 3.1 where a modulated pump is injected, Ms_out is strongly positively correlated with Mp. Except the value of Mp, the pump modulation type is also influential. The comparison of the results presented in Sections 3.1 and 3.2 confirms the dominant influence of the pump.
4. NOPCPA with large beam sizes For a high-energy OPCPA system, the pump and signal beam sizes are set to 55 mm and 54 mm, respectively in this section. The beam diameters are much larger than the walk-off length, and therefore walk-off is negligible [13,32,33]. Thus, the following analysis is based on the results of the orthogonal slope modulation, i.e. Mp ¼Mp⊥, and we focus on the parametric process to discuss the variation of the output beam quality. 4.1. Influence of pump modulation on output beam quality
Fig. 9. Conversion efficiency (η) and the output modulation (Ms_out) for different average pump intensity (Ip) and signal modulation (Ms_in).
Comparing Ms_out for different Ms_in, unlike Fig. 3, the variation among Ms_out is small though Ms_in differs greatly. To be specific, three different Ip as 1.06, 1.32 and 1.6 GW/cm2 for Ms_in ¼1.0,
η with different Ip and Mp are shown in Fig. 10 using the solid curves. ηmax can be reached at the certain pump intensity (Ieff), as plotted in Fig. 11(a). Specifically, ηmax of Mp ¼1.0, 1.3 and 1.5 are 47.23%, 46.26%, and 45.47%, and Ieff are 1.31, 1.17, and 1.02 GW/cm2, respectively. In this section, ηmax is larger because of a larger fill factor than that in Section 3.1. With the increase in Mp, ηmax and Ieff both decreases slightly, which is similar to the small-beam-size case. Dot curves in Fig. 10 show the variation of Ms_out. Firstly, Ms_out is strongly positively correlates with Mp. For a fixed Ip, the larger
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for the increased spatial modulation, resulting in a nearly flattop output profile with a low modulation, Fig. 12(d). Since amplification and back conversion depends mainly on Ip rather than Mp, Imod are similar for different Mp, which can be seen in Figs. 10 and 11. If Ip increases, Ep (x, y′) and Ep (x, y) are both considerably large, and back conversion occurs not only at (x, y′) but also at (x, y). The compensation effect reduces, and thus Ms_out increases once again. The amplified spectrum of the three points, A, B and C, are given in Fig. 13. Full wavelength amplification is achieved at all three points. The green curve in Fig. 13(b) shows the amplified spectral intensity at A, which is similar to the input to a large extent. However, the output spectrum at B and C show an apparent hole around the center wavelength. This is because of wavelength-dependent gain saturation and back conversion, as approved by Witte [14]. Back conversion of C is much severer, resulting in a deeper hole in the amplified spectrum. Fig. 10. Conversion efficiency (η) and the output modulation (Ms_out) for different average pump intensity (Ip) and modulation (Mp). A, B and C correspond to Ip ¼0.73, 1.17 and 2.18 GW/cm2, Mp ¼ 1.5.
Mp is, the larger Ms_out is obtained. For a fixed Mp, with the increase of Ip, Ms_out tends to decrease and then increase; min-Ms_out can be achieved at Imod, as shown in Fig. 11(b). For example, minMs_out for Mp ¼ 1.3, 1.5 are 1.06, 1.1, respectively. Imod are both 2.18 GW/cm2. The important feature is that Ms_out can be much smaller than Mp if Ip is close to Imod, which indicates that for a high-energy OPCPA system with a large beam size, output beam quality can be optimized using proper Ip. We use Mp ¼1.5 as an example. The profile for Mp ¼1.5 is shown in Fig. 12(a). Specifically, three points corresponding to Ip ¼ 0.73, 1.17 and 2.18 GW/cm2, Mp ¼1.5 are labeled as A, B and C in Fig. 10. The input and output parameters at A, B and C are listed in Table 4, and their output profiles are shown in Fig. 12(b)–(d), respectively. Spatially, two points, (x, y) and (x, y′), are denoted within the pump beam size to give an explanation. The local pump intensity, input signal intensity and output signal intensity of the two points are denoted as Ep, Es_in and Es_out. From Fig. 12(a), it can be seen that Ep (x, y′)4 Ep (x, y). The input flat-top spatial profile cannot maintain, and instead, a slantwise profile that resembles the pump is achieved, Fig. 12(b). With the increase in Ip, Ep (x, y′) becomes intensely larger, the signal gain at (x, y′) reaches saturation, and back conversion occurs. While for (x, y), saturation does not occur, and the signal gain still holds. In this way, the difference between Es_out (x, y′) and Es_out (x, y) reduces, resulting in decreased Ms_out. The beam profile with a relatively smaller slope in Fig. 12(c) also confirms this. The compensation effect becomes more obvious when IP is larger than Ieff, back conversion is more severe, which can be confirmed by the decreased η shown in Fig. 10. Back conversion contributes to obvious compensation
4.2. Influence of signal on OPCPA output beam quality The solid and dash curves in Fig. 14 show the conversion efficiency (η) and Ms_out with different Ip and Ms_in. For different Ms_in, ηmax and min-Ms_out can be achieved at 1.31 GW/cm2, as shown in Figs. 14 and 15. The identical Imod and Ieff shows that the trade-off between high conversion and good output beam quality is no need. Moreover, even Ms_in is large, the min-Ms_out is close to 1. For example, min-Ms_out for Ms_in ¼1.0, 1.3 and 1.5 are 1, 1.01 and 1.02, respectively. It indicates that the signal beam quality can be improved by OPCPA. The variation tendency of Ms_out is because of parametric amplification and back conversion. Three points corresponding to Ip ¼0.73, 1.17 and 2.18 GW/cm2, Mp ¼1.5, are labeled as A′, B′ and C′. The input and output parameters at A′, B′ and C′ are listed in Table 5. The beam profiles are shown in Fig. 16. A super-Gaussian pump is used. The gain among the spot size differs because of the modulated signal. For points where the initial signal intensity is high, the signal undergoes rapid amplification first, but the rapid gain cannot maintain because of saturation and back conversion. For points where the initial signal intensity is low, back conversion cannot occur immediately. Different parametric gains among the beam size compensate for the signal spatial modulation, resulting in decreased Ms_in, and the reduced slope in Fig. 16(b) certifies this. When Ip increases to Imod, the output beam profile is nearly superGaussian, and resembles the pump profile, Fig. 16(c). This fine output indicates that the signal modulation can be compensated using proper Ip. Since the parametric gain and back conversion are closely related to Ip rather than Ms_in, Imod are similar for different Ms_in. If Ip increases from Imod to a larger one, the signal undergoes rapid amplification and backflow is severe in the initially intense area. Strong back conversion causes the initially high-intensity
Fig. 11. (a) Maximum conversion efficiency (ηmax) and Ieff. (b) The minimum output modulation (min-Ms_out) for different Mp.
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Fig. 12. (a) The input pump beam profile. (b), (c) and (d) Output signal beam profiles at points A, B and C, respectively. Table 4 Conversion efficiency (η) and the output modulation (Ms_out) for different average pump intensities (Ip) and input pump modulations (Mp). Notation
Ip (GW/cm2)
Mp
η
Ms_out
A B C
0.73 1.17 2.18
1.5 1.5 1.5
42.17% 45.62% 34.62%
1.58 1.39 1.1
area decay; meanwhile, the initially weak area undergoes a high gain and becomes strong. As a result, the compensation effect diminishes, and Ms_out increases again with the increase i Ip, which is the case in Fig. 16(d). The amplified spectra of A′, B′ and C′ are given in Fig. 17. The amplified spectral profile of A′ resembles the input spectrum, and no obvious back conversion is observed. However, a hollow is seen in the output spectrum of B′ and C′, especially of C′. Moderate back conversion occurs at B′ and improves the output beam quality. The
Fig. 14. Conversion efficiency (η) and the output modulation (Ms_out) for different average pump intensity (Ip) and initial signal modulation (Ms_in).
Fig. 13. (a) Input spectrum. (b), (c) and (d) Output spectrum at points A, B and C, respectively.
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Fig. 15. (a) Maximum conversion efficiency (ηmax) and Ieff. (b) The minimum output modulation (min-Ms_out) for different Ms_in. Table 5 Conversion efficiency (η) and output modulation (Ms_out) for different average pump intensity (Ip) and input signal modulation (Ms_in). Notation
Ip (GW/cm2)
Ms_in
η
Ms_out
A′ B′ C′
0.73 1.31 2.33
1.5 1.5 1.5
40.76% 46.63% 35.89%
1.13 1.02 1.18
deep hollow in Fig. 17(d) indicates significant back conversion at C′ which is consistent with the discussion above. Since OPCPA has the feature of wave-length dependent gain saturation, the hollow appears near the center wavelengths in the amplified spectrum. 4.3. comparison of pump and signal modulation We set η 4 40% and Ms_out o1.3 as the standards for high efficiency and low Ms_out, respectively, as the blue dot curves in Figs. 10 and 14. The common range is crucial since it corresponds to both high η and low Ms_out. For a super-Gaussian pump and signal, low output modulation always keeps, and the common range is just the range for η 4 40%, which is 0.71–2.09 GW/cm2. In this case, if the pump intensity corresponding to a high η is selected, low Ms_out is automatically ensured. However, for a modulated pump, the common range narrows greatly with the increase in Mp, for example, the range for Mp ¼1.5 is 1.46–1.75 GW/ cm2. It indicates that high η and low Ms_out are harder to maintain for a larger Mp. For a modulated signal, the common range narrows slightly as Ms_in increases. For example, the range for Ms_in ¼1.5 is 0.73–2.09 GW/cm2, similar to that for Ms_in ¼1.0. The results confirm the dominant influence of the pump on the OPCPA output. The control of the pump modulation is important in optimizing OPCPA. For a Nd: glass laser system with the energy up to order of kilojoules, which is a commonly used pump source in high-energy OPCPA systems, the spatial modulation can hardly be reduced to less than 1.3, while 1.5 can be achieved currently. For Mp ¼1.5, the
pump range that satisfies η 440% and Ms_out o 1.3 is 1.46– 1.75 GW/cm2. If Ip is chosen within this range, the output with a relatively high conversion efficiency and a good beam quality is achievable.
5. Conclusion A numerical study is carried out to investigate the output beam quality of high-efficiency broadband NOPCPA systems. The modulation type used in this paper is the linear slope along the x- or y-axis. This particular modulation is for intuitive and analytical convenience. Nevertheless, the results provide insight into pump and signal modulation effect in NOPCPA. Ms_out and η are affected by the input beam quality such as Mp and Ms_in. Generally, larger Mp or Ms_in results in larger Ms_out and lower η. Within the two factors, the pump modulation has a major impact, as proved in Sections 3 and 4. In addition, the pump intensity influences the output property. Typically, ηmax and min-Ms_out can be achieved at Ieff and Imod, respectively. For the non-collinear phase matching, the walk-off effect is considerable for small-beam-size pump and signal. The pump modulation and walk-off are related in the parametric process and affect the output simultaneously. Both the magnitude and type of Mp are influential. Mp⊥ is the modulation type that mitigates the pump walk-off; however, the idler walks out automatically and unavoidably during the parametric process. The walk-off of idler leads to beam asymmetry and a raised edge, resulting in deterioration of the output beam quality. In Sections 3.1.2 and 3.1.3, Mp// is used to show the effect of pump walk-off. For the pump with Mp// þ , the output tendency resembles Mp⊥, but η is further reduced and Ms_out increased. However, Ms_out with Mp// shows a different tendency, and much smaller Ms_out can be achieved at larger Ip. This feature can compensate for increased Ms_out result from walk-off, as confirmed by the better output property of Mp// ¼ 1.1 than that of a super-Gaussian pump. It is suggested that the negative influence of walk-off can be compensated by choosing proper pump profile. The results can be
Fig. 16. (a) Input signal beam profile. (b), (c) and (d) Output signal beam profiles at points A′, B′ and C′, respectively.
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Fig. 17. (a) Input spectrum. (b), (c) and (d) Output spectrum at points A′, B′ and C′, respectively.
useful for designing NOPCPA systems with enhanced efficiency and beam quality. The walk-off effect can be negligible when the beam sizes are considerably large. ηmax decreases with the increase in Mp or Ms_in, which indicates the negative influence of the modulation. When the pump is modulated, Imod is larger than Ieff. The difference between Imod and Ieff suggests that trade-off between minMs_out and ηmax is unavoidable. However, when signal is modulated but pump is super-Gaussian, Ieff and Imod are close, for example, Ieff and Imod for Ms_in ¼1.5 are both 1.31 GW/cm2; thus, high η and low Ms_out are reachable simultaneously. The result suggests that OPCPA output depends crucially on the pump, a high-energy pump with good spatial quality will be beneficial to a high-energy, low-modulation output. In an ultrahigh-energy OPCPA system, the output modulation from the Nd: glass based kilojoule pump source can hardly be less than 1.3. Our simulation shows that for a high-energy OPCPA system, the output can be optimized by controlling Mp and Ip. A considerably high conversion and low Ms_out can be obtained by choosing the pump energy properly. In the case of Mp ¼ 1.5, Ms_out can be less than 1.3 and η can be larger than 40% with Ip in the range of 1.46–1.75 GW/cm2.
Funding National Natural Science Foundation of China (Grant Nos. 61521093 and 61378030).
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Numerical investigation of output beam quality in efficient broadband optical parametric chirped pulse amplification Xiao-Di Liu a,b, Lu Xu a, Xiao-Yan Liang a,c,n a State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Science, Shanghai 201800, People's Republic of China b University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China c School of Physical Science and Technology, Shanghai Tech University, Shanghai 200031, People's Republic of China
art ic l e i nf o
a b s t r a c t
Article history: Received 15 June 2016 Received in revised form 27 August 2016 Accepted 31 August 2016
We theoretically analyzed output beam quality of broad bandwidth non-collinear optical parametric chirped pulse amplification (NOPCPA) in LiB3O5 (LBO) centered at 800 nm. With a three-dimensional numerical model, the influence of the pump intensity, pump and signal spatial modulations, and the walk-off effect on the OPCPA output beam quality are presented, together with conversion efficiency and the gain spectrum. The pump modulation is a dominant factor that affects the output beam quality. Comparatively, the influence of signal modulation is insignificant. For a low-energy system with small beam sizes, walk-off effect has to be considered. Pump modulation and walk-off effect lead to asymmetric output beam profile with increased modulation. A special pump modulation type is found to optimize output beam quality and efficiency. For a high-energy system with large beam sizes, the walkoff effect can be neglected, certain back conversion is beneficial to reduce the output modulation. A trade-off must be made between the output beam quality and the conversion efficiency, especially when the pump modulation is large since. A relatively high conversion efficiency and a low output modulation are both achievable by controlling the pump modulation and intensity. & 2016 Elsevier B.V. All rights reserved.
Keywords: Nonlinear optics Parametric processes Laser amplifiers
1. Introduction Optical parametric chirped pulse amplification (OPCPA), proposed by Dubietis in 1990s [1], is considered as a high promising technology for achieving ultrafast and ultrahigh-intensity laser pulses. It has the following advantages: high gain, wavelength tunability, low B-integral accumulation, and low thermal effects. OPCPA with non-collinear geometry was favored for a broad gain bandwidth [2,3] and convenient pulse separation. Many approaches were adopted to improve the NOPCPA output. A peak power of several hundred terawatts even up to petawatts was currently available [4–13]. Few-optical-cycle even near to Fourierlimited light pulses were demonstrated [14–17]. The advances of ultra-short, high energy pulses will potentially allow further progress in probing novel physical and optical phenomena, such as ultrafast X-ray radiation, laser-matter interaction [18], particle acceleration [19], inertial confinement fusion, etc. n Corresponding author at: State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Science, Shanghai 201800, People's Republic of China. E-mail address: [email protected] (X.-Y. Liang).
http://dx.doi.org/10.1016/j.optcom.2016.08.082 0030-4018/& 2016 Elsevier B.V. All rights reserved.
For a non-collinear parametric amplifier, the conversion efficiency is crucial, and a variety of methods has been used to improve it. Using the conformal theory proposed by Begishev [20,21], the pump energy could be fully utilized in all spatial and temporal points to maximize the conversion efficiency. 67% conversion of pump energy into parametric waves was achieved by optimal profiling of interactive waves [20]. Moses used the temporal conformal theory to boost OPCPA conversion efficiency and the amplification bandwidth [22]. Flat-top beams, as the simplest conformal profile, contribute to a great improvement in the OPCPA conversion efficiency and stability [2]. A high conversion efficiency of 29% is demonstrated by Waxer using a high-order super Gaussian pump [23]. In addition to the conversion efficiency, the output beam quality is a critical factor for the application in strong-field physics. It also affects the stability and security of the whole system. Flattop spatial and temporal pulses can give optimum performance and enhance the output beam quality. However, spatial or temporal noises are inevitable in real laser systems. The OPCPA performance with modulated pulses should be discussed especially for high energy laser systems where pump pulses are far from the ideal one [9,24]. In this article, we focus on the impact of the
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spatial modulation on the overall conversion efficiency and the output modulation. Temporally, we use a Gaussian profile of the signal, and a super-Gaussian profile of the pump, which reduces gain narrowing and maintains signal bandwidth [13,25]. Spatially, we introduce some modulation to the super-Gaussian profile. To evaluate the influence of pump or signal spatial modulation separately, a super-Gaussian signal is assumed to show the impact of pump modulation. Similarly, a super-Gaussian pump with a modulated signal is set to distinguish the impact of signal. The spatial walk-off effect is also an important factor that affects the output beam quality [24]. With a three dimensional spatial and temporal model, the influence of pump and signal modulations, and the walk-off effect on output beam quality and conversion efficiency are discussed. The results show that the pump modulation has a dominant influence on the output beam quality. Besides, the walk-off effect is considerable for small beam diameters, however, it is negligible for the large beam diameters. We discuss two cases with different scales of beam sizes. In the first case, the interactive beam diameters are considerably small, compared to the walk-off length; thus, walk-off effect cannot be neglected. The pump modulation and walk-off simultaneously affect the output, which includes asymmetric intensity distribution and increased output modulation. A particular pump modulation type is found to compensate the negative influence of walk-off. In another case, however, the beam sizes are so large that walk-off length is comparatively small; hence, walk-off is neglectable owing to its little influence. Output beam quality depends mainly on the parametric amplification process. A quite low output modulation can be achieved at a specific average pump intensity, where back conversion occurs and inevitably reduces the conversion efficiency. The trade-off between the conversion efficiency and a low output modulation highlights the importance of reduction of pump modulation and selection of a proper average pump intensity, by which a relatively high conversion efficiency and a low modulation can be achieved. The results can be helpful for the design of high-energy OPCPA systems. The paper is organized as follows. The numerical simulation and initial conditions are described in detail in Section 2. In Section 3, the output features of small beam sizes are analyzed. In Section 3.1, different modulation types are set to evaluate the influence of pump modulation and the walk-off effect. In Section 3.2, the influence of input signal modulation is discussed. Section 4 concerns the output features of the large-beam-size OPCPA system. In Section 4.1, the influence of pump modulation is discussed. Section 4.2 describes the influence of signal modulation. A pump intensity range is given in Section 4.3 to ensure a relatively high conversion efficiency and low modulation, which is helpful for the design of high-energy OPCPA system. The conclusion is presented in Section 5.
2. Simulation details The theoretical investigation is based on coupled wave equations governing the parametric amplification process. According to Ref. [26], the group velocity mismatch for short pulses is considered in the frequency domain. Since the pulse durations of pump and signal in our model are of the order of nanoseconds, the group velocity mismatch between pump and signal over the crystal length is negligible compared to the pulse width, as demonstrated by Ross et al. [3]. Then considering the full spatial and temporal dependence of the three parametric waves, with type I phase matching and non-collinear geometry configuration, the following set of coupled wave equations is derived and numerically evaluated [27,28].
⎛ ∂2 ∂As ∂A ∂2 ⎞ 1 ⎜⎜ ⎟⎟As + tan ρs s − + 2 ∂z ∂y 2jns ks cos ρs ⎝ ∂x ∂y2 ⎠ ws =−j deff A p Ai*exp( − iΔkz ), ns c cos ρs
(1a)
⎛ ∂2 ∂Ai ∂A ∂2 ⎞ 1 ⎜⎜ ⎟⎟Ai + tan ρi i − + ∂z ∂y 2jni ki cos ρi ⎝ ∂x2 ∂y2 ⎠ wi =−j deff A p As*exp( − iΔkz ), ni c cos ρi
(1b)
∂A p ∂z
+ tan ρp
=−j
∂A p ∂y
wp npc cos ρp
−
⎛ 2 2 ⎞ 1 ⎜ ∂ + ∂ ⎟A p 2jnpk p cos ρp ⎜⎝ ∂x2 ∂y2 ⎟⎠
deff As Ai exp(iΔkz ).
(1c)
where subscripts s, i, and p refer to the signal, idler, and pump, respectively; A is the complex electric field amplitude; ω is the angular frequency; n is the refractive index and k is the wave number of A; ρp is the birefringence walk-off angle of pump, whereas ρs and ρi are the nonlinear angles of signal and idler wave vectors; j is the imaginary unit; c is the speed of light in vacuum; deff accounts for the effective nonlinear coefficient, Δk is the wave-vector mismatch which is derived from energy conversion and momentum conservation. There is no input idler intensity, the idler is produced during the parametric process, and it self-adjusts to ensure the maximum initial signal gain. Because of the nonlinear geometry and the broadband signal, the idler has a relative large spatial divergence angle. In this paper, the modulation of amplified signal is primarily considered. The above differential equations are numerically solved by a spit-step technique in the space and time domain. The linear process, which includes spatial walk-off, wave propagation and diffraction effect, as well as nonlinear parametric process are both considered in the procedure [29,30]. LBO (LiB3O5) is a popular nonlinear material, exhibiting a high nonlinear coefficient, a high damage threshold, and a broad gain bandwidth in the visible and near infrared. Besides, LBO with large aperture up to tens of micrometers is currently available. It is a good choice not only in the front end as a broadband amplifier but also in the final end for high-energy amplification. In order to investigate the influence of the initial pump and signal on the output signal modulation of a feasible broadband OPCPA laser system, LBO (LiB3O5) with the length of 12 mm is chosen as the nonlinear crystal [11]. Since the nonlinear process for different nonlinear crystals are similar, the numerical model and discussion in this paper can be extended to other nonlinear crystals such as BBO and DKDP. Type I phase matching (θ ¼90°, φ ¼13.85°) is adopted. A non-collinear angle between the signal and the pump of 1.26° is chosen to enlarge the amplification bandwidth. The pump walk-off angle is 0.48°. The second-harmonic of Nd: glass, at 526.5 nm, is used as the pump. The input signal is from a Ti: sapphire chirped pulse amplifier centered at 800 nm with a full spectral width of 80 nm. The pump and signal diameters of the small-beam-size OPCPA are 5 mm and 4 mm, respectively. The large-beam-size pump and signal diameters are 55 mm and 54 mm, respectively. The initial signal intensity is set to 80 MW/ cm2. The pump intensity varies from 0.53 GW/cm2 to observe the trend. The temporal profiles of pump and signal are shown in Fig. 1. The input signal is linearly chirped in time with a pulse duration of 1.6 ns (FWHM). The pump is super-Gaussian shaped with a pulse duration of 2.89 ns (FWHM). The super-Gaussian
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distribution within the spot size, i.e. a flat-top beam has a modulation of 1. A Cartesian coordinate is chosen as shown in Fig. 2(a). In that system, the direction of ks is z, the ordinary and extraordinary waves are polarized along x and y. Pump and idler walk off in the negative y direction. To better investigate the influence of input pump and signal modulation on the OPCPA output, a typical visualized modulation type is used in our simulation as can be seen in Fig. 2. The modulation is introduced along x- or y-axis to a super-Gaussian profile. Fig. 2(b) is the ideal super-Gaussian with a slope of 0 and modulation of 1; For Fig. 2(c), the modulation slope is along the x-axis, and the pump intensity in the walk-off direction is the same. For Fig. 2(d) and (e), the modulation slopes are along the y-axis, and the pump intensities in the walk-off direction differ. The different modulation types are set to evaluate the walk-off effect in NOPCPA, which will be discussed in Section 3 in detail. Fig. 1. Input temporal profiles of pump and signal.
temporal shape of the pump is chosen to enhance the conversion efficiency and achieve full bandwidth amplification [13,25]. Near-field spatial modulation is defined by Eq. (2) [31]. It is used to evaluate beam quality.
M=
Imax − Ign Iavg − Ign
(2)
where Imax and Iavg are the maximum intensity and the average intensity within the beam, respectively, and the beam area is defined by the beam spatial FWHM. Ign is background noise. In our case, Ign ¼0. Modulation can reflect uniformity of the intensity
3. NOPCPA with small beam sizes For a small-beam-size NOPCPA system, the pump and signal diameters are 5 mm and 4 mm, respectively. The walk-off lengths of the pump and idler are approximately 0.17 mm and 0.79 mm, respectively. Walk-off causes pump, signal and idler separation; hence, not only the value of Mp but also the modulation type can influence the parametric amplification and the output properties. By carrying out the numerical simulation of Eq. (1), the influence of the pump, signal and walk-off on the OPCPA output is given in this section. For brevity, some denotations are used: average pump intensity, injected pump and signal modulation, and output signal modulation are denoted as Ip, Mp, Ms_in and Ms_out, respectively, in this article. The modulation direction orthogonal (as Fig. 2(c)) and parallel (as Fig. 2(d) and (e)) to the walk-off direction are denoted as Mp⊥ and Mp//. The influence of pump modulation is evaluated in Section 3.1, where different modulation types, Mp⊥ and Mp// þ and Mp// as Fig. 2(c)–(e) are set. Signal modulation is discussed in Section 3.2. 3.1. Influence of pump modulation on output beam quality
Fig. 2. (a) Cartesian coordinate of non-collinear geography. The propagation of signal pulse is chosen as the z axis. Walk-off occurs in the yoz-plane, the walk-off direction is along the negative direction of y axis. (b) The super-Gaussian profile; the modulation is 1. (c) The x-axis slope modulation, denoted as Mp⊥; (d) and (e) The y-axis slope modulation with positive and negative directions, denoted as Mp// þ and Mp// , respectively.
3.1.1. Pump with x-axis slope modulation An x-axis pump modulation type similar to Fig. 2(c) is set. In this case, although pump walk-off occurs, the pump intensity in the walk-off direction is identical. In addition, the pump diameter is larger than that of the signal, so pump walk-off generates little pump intensity variation among the signal beam size. Accordingly, we can identify the influence of the pump modulation while pump walk-off on the OPCPA output is diminished. The conversion efficiency (η) with different Ip and Mp⊥ is calculated and shown in Fig. 3. The curves for different Mp⊥ share a similar tendency. For certain Mp⊥, η increases and then decreases with the increase in Ip, and the maximum conversion efficiency (ηmax) can be reached at a certain pump intensity (Ieff). ηmax and Ieff for different Mp⊥ are shown in Fig. 4(a). With the increase in Mp⊥, ηmax decreases. Specifically, ηmax for Mp⊥ ¼ 1.0, 1.3 and 1.5 are 39.36%, 39.15% and 39.01%, and Ieff are 1.38, 1.30 and 1.23 GW/cm2, respectively. The reduction is mainly because of uneven amplification and back conversion cause by the maldistributed pump intensity. The short dash curves in Fig. 3 are Ms_out with different Ip and Mp⊥. First, an obvious feature is that Ms_out is strongly positively related to Mp⊥. For fixed Ip, the larger Mp⊥ is, the larger Ms_out is obtained. Moreover, Ms_out changes considerably with Ip. For fixed Mp⊥, with the increase in Ip, Ms_out tends to decrease and then increase. Minimum Ms_out (min-Ms_out) can be reached at a certain pump intensity, Imod, as shown in Fig. 4(b). Specifically, Imod for
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Table 1 Conversion efficiency (η) and output modulation (Ms_out) for different average pump intensity (Ip) and input pump modulation (Mp�). Notation
Ip (GW/cm2)
Mp�
η
Ms_out
A/D B/E C/F
1.06 1.32 1.60
1.0/1.3 1.0/1.3 1.0/1.3
38.22%/38.39% 39.34%/39.14% 38.91%/38.35%
1.11/1.35 1.12/1.39 1.18/1.46
increasing slowly until it reaches ηmax, but the raised edge contributes to increased Ms_out. For large Ip, i.e. Ip ¼1.6 GW/cm2 at points C and F, back conversion in the center can be much severer. Although amplification in the edge enlarges, the overall effect is η reduction and further increase in Ms_out, as can be seen in Figs. 3 and 5(d) and (h). Fig. 3. Conversion efficiency (η) and the output modulation (Ms_out) of different average pump intensity (Ip) and modulation (Mp⊥). A, B and C correspond to Ip ¼1.06, 1.32 and 1.6 GW/cm2, Mp ¼ 1.0; D, E, and F correspond to Ip ¼ 1.06, 1.32 and 1.6 GW/cm2, Mp⊥ ¼1.3.
Mp ¼1.0, 1.3 and 1.5 are 1.15 and 0.97 and 0.88 GW/cm2, respectively. However, Ms_out can hardly be less than the input pump modulation. This indicates that the output beam quality is closely relevant to pump quality. With a highly modulated pump, low Ms_out is difficult to maintain. η and Ms_out are closely correlated to Ip, Mp ¼1 and Mp⊥ ¼1.3 are taken as an example. Three points corresponding to Ip ¼ 1.06, 1.32 and 1.6 GW/cm2 are labeled on the output modulation curve for Mp ¼1.0 (Mp⊥ ¼ 1.3) as A, B and C (D, E and F), respectively. The input and output parameters of those points are listed in Table 1. The input pump and output signal profiles are shown in Fig. 5. When Ip increases to 1.06 GW/cm2, i.e. points A and D, gain increases as well, and the signal is amplified and has a spatial profile similar to the pump, as can be seen in Fig. 5(b) and (f). For a superGaussian pump, the central is relatively uniform, and the edges becomes steeper, as shown in Fig. 5(b); Ms_out reduces with the increase in Ip. For Mp⊥ ¼ 1.3, the signal beam edge is amplified, especially in the high-pump-intensity area. Additionally, owning to the saturation caused by the local high pump intensity, the output signal intensity in the center area is better distributed, and thus Ms_out for Mp⊥ ¼1.3 is reduced as well. Then Ip increases to 1.32 GW/cm2, i.e. point B and E, saturation and back conversion become severe especially in the beam center where pump, signal and idler always overlap. However, in the right edge, the idler walks out, and back conversion cannot occur, which results in a raise around the right edge, Fig. 5(c) and (g). The raised edge in the right bottom is more prominent for Mp⊥ ¼1.3 which is because of increasingly large pump intensity along the x-axis. η keeps
3.1.2. Pump with y-axis slope modulation Compared to the x-axis pump modulation type, the y-axis modulation type has a different intensity along the walk-off direction. As shown in Fig. 2(d), the pump intensity is gradually increased because of walk-off. It is recorded as Mp// þ . The pump modulation similar to Fig. 2(e) is recorded as Mp// . Walk-off leads to decreasing the intensity gradually. Mp// þ (Mp// ) of 1.3 is used as an example. η and Ms_out are shown in Fig. 6. First, we compare η for Mp// þ , Mp// to that of Mp⊥. Although the specific value changes slightly, i.e. η for Mp// þ is generally smaller, and that for Mp// is generally larger, the variation tendency still holds. ηmax can be achieved at Ieff. ηmax for Mp// þ and Mp// are 38.56% and 39.78%, and Ieff are 1.32 and 1.3 GW/cm2, respectively. Second, Ms_out for Mp// þ share a similar tendency with Mp⊥, but Mp// differs, as it corresponds to a much smaller Ms_out and a larger Imod. Specially, Min-Ms_out for Mp// þ and Mp// are 1.41 and 1.12, Imod are 0.9 and 1.5 GW/cm2, respectively. To be specific, three points corresponding to Ip ¼1.06, 1.32 and 1.6 GW/cm2 are labeled on the output modulation curve of Mp// þ (Mp// ) as D þ , E þ and F þ (D , E and F ), respectively. The input and output parameters are listed in Table 2. The input pump and output signal profiles are shown in Fig. 7. As Ip increases to 1.06 GW/cm2, the gain increases, the slope in central area decreases, and the beam edges become steep, as shown in Fig. 7 (b) and (f). As a result, η becomes larger and Ms_out smaller. Then with the increase in Ip, saturation and back conversion occur and become increasingly large, η cannot keep increasing and the maximum point is achieved later, at approximately 1.3 GW/cm2, as shown in Fig. 6. The pump and idler walk-off simultaneously influence the output beam quality, which is quite different for Mp// þ and Mp// . For Mp// þ , the weak part of the pump gradually walks out during the parametric process, and the pump intensity becomes
Fig. 4. (a) Maximum conversion efficiency (ηmax) and Ieff. (b) Minimum output modulation (min-Ms_out) for different Mp⊥.
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Fig. 5. (a) The input super-Gaussian pump profile, Mp ¼ 1.0. (b), (c) and (d) Output beam profiles with pump (a) at points A, B and C, respectively. (e) The input pump profile, Mp⊥ ¼ 1.3. (f)–(h) Output beam profiles with pump (e) at point D, E and F, respectively.
intensity region averts serious back conversion. The overall effect is the enhanced conversion efficiency with a better beam quality and enlarged Imod.
Fig. 6. Conversion efficiency (η) and output modulation (Ms_out) of different average pump intensity (Ip) and modulation type (Mp⊥, Mp// þ , Mp// ). D þ , E þ and F þ correspond to Ip ¼1.06, 1.32 and 1.6 GW/cm2, Mp// þ ¼ 1.3; D , E and F correspond to Ip ¼ 1.06, 1.32 and 1.6 GW/cm2, Mp// ¼ 1.3.
Table 2 Conversion efficiency (η) and output modulation (Ms_out) for different average pump intensities (Ip) and input pump modulation types (Mp// þ and Mp// ). Notation
Ip (GW/cm2)
Mp ¼ 1.3
η
Ms_out
D þ /D E þ /E F þ /F
1.06 1.32 1.60
Mp// þ /Mp// Mp// þ /Mp// Mp// þ /Mp//
37.72%/39.06% 38.56%/39.76% 37.90%/38.86%
1.42/1.26 1.48/1.21 1.59/1.19
increasingly larger. However, the situation is inverse for Mp// . The walk out of the idler frees the signal right edge from back conversion, where a considerably high gain is possible. When Mp// þ is injected, the area around the signal right edge gains strongly because of an increasingly high pump intensity and absence of idler, resulting in larger Ms_out. Apart from this region, the increasingly high pump causes saturation and back conversion, preventing the increase in η. If Mp// is injected, the gain in the right edge of the signal is much smaller compared to Mp// þ , but continuous gain can compensate for the local weak pump intensity, as shown in Fig. 7 (g) and (h). Moreover, the walk-off of the ultra-high pump
3.1.3. Compensation for walk-off effect A super-Gaussian pump is favored in OPCPA, but the results shown in Figs. 3 and 5 indicate that the pump is super-Gaussian and the walk-off of the idler leads to the raised edge and increased Ms_out. In addition, Imod differs from Ieff (Imod o Ieff), which indicates that trade-off exists among the efficiency and the output beam quality. However, the output feature of Mp// suggests that Ms_out can be reduced together with larger Imod and increased η. We choose Mp// ¼ 1.1 and compare the output with a superGaussian pump, as shown in Fig. 8. Min-Ms_out is 1.06, lower than that of the super-Gaussian. Besides, Ieff and Imod coincide; they are both 1.32 GW/cm2, indicating that ηmax and min-Ms_out can be achieved simultaneously. The output profile is shown the insert in Fig. 8. The nearly flat-top spatial profile confirms the beneficial compensation effect of Mp// . To conclude, the pump beam profile can be designed to compensate the walk-off effect, optimize output beam quality and efficiency. 3.2. Influence of signal modulation on output beam quality The input signal quality, Ms_in, is another important factor of parametric amplification. In order to evaluate the influence of Ms_in, a super-Gaussian pump is assumed, i.e. Mp ¼1.0. The input signal is the modulated one with orthogonal slope as shown in Fig. 2(c). The solid curves in Fig. 9 depict the conversion efficiency (η) with different Ip and Ms_in. The variation tendency for different Ms_in are similar, ηmax can be achieved at a similar Ieff, 1.38 GW/ cm2. The dashed curves with dots show the tendency of Ms_out. Min-Ms_out for Ms_in ¼1.0, 1.3 and 1.5 are 1.107, 1.109 and 1.119, respectively. Imod for different Ms_in is identical, 1.15 GW/cm2. Unlike Fig. 4, Ms_out in Fig. 9 for different Ms_in are similar and keep small within a large pump range. For example, Ms_out for Ms_in ¼ 1.5 varies from 1.119 to 1.14 as Ip increases from Imod ¼ 1.15 GW/cm2 to Ieff ¼1.38 GW/cm2. It is a favorable feature that good output beam quality can be ensured at high conversion efficiency. Besides, the low Ms_out also indicates that output beam quality can be improved by OPCPA.
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Fig. 7. (a) The input pump profile, Mp// þ ¼ 1.3. (b), (c) and (d) The output beam profiles with pump (a), at point D þ , E þ and F þ , respectively. (e) The input pump profile, Mp// ¼ 1.3. (f), (g) and (h) Output beam profiles with pump (e) at point D , E and F , respectively.
Table 3 Conversion efficiency (η) and the output modulation (Ms_out) for different average pump intensity (Ip) and input signal modulation (Ms_in).
Fig. 8. Conversion efficiency (η) and output modulation (Ms_out) for different average pump intensity (Ip). The output beam profile at Ip ¼1.32 GW/cm2 is shown as the insert.
Ip (GW/cm2)
Ms_in
η
Ms_out
1.06 1.32 1.60
1.0/1.3/1.5 1.0/1.3/1.5 1.0/1.3/1.5
38.22%/38.08%/37.87% 39.34%/39.22%/39.03% 38.91%/38.81%/38.66%
1.11/1.11 1.12 1.12/1.12 1.13 1.18/1.18 1.19
1.3 and 1.5 are chosen. The input and output parameters are listed in Table 3. The little difference in Ms_out indicates that the influence of the signal modulation on Ms_out is insignificant. However, in Section 3.1 where a modulated pump is injected, Ms_out is strongly positively correlated with Mp. Except the value of Mp, the pump modulation type is also influential. The comparison of the results presented in Sections 3.1 and 3.2 confirms the dominant influence of the pump.
4. NOPCPA with large beam sizes For a high-energy OPCPA system, the pump and signal beam sizes are set to 55 mm and 54 mm, respectively in this section. The beam diameters are much larger than the walk-off length, and therefore walk-off is negligible [13,32,33]. Thus, the following analysis is based on the results of the orthogonal slope modulation, i.e. Mp ¼Mp⊥, and we focus on the parametric process to discuss the variation of the output beam quality. 4.1. Influence of pump modulation on output beam quality
Fig. 9. Conversion efficiency (η) and the output modulation (Ms_out) for different average pump intensity (Ip) and signal modulation (Ms_in).
Comparing Ms_out for different Ms_in, unlike Fig. 3, the variation among Ms_out is small though Ms_in differs greatly. To be specific, three different Ip as 1.06, 1.32 and 1.6 GW/cm2 for Ms_in ¼1.0,
η with different Ip and Mp are shown in Fig. 10 using the solid curves. ηmax can be reached at the certain pump intensity (Ieff), as plotted in Fig. 11(a). Specifically, ηmax of Mp ¼1.0, 1.3 and 1.5 are 47.23%, 46.26%, and 45.47%, and Ieff are 1.31, 1.17, and 1.02 GW/cm2, respectively. In this section, ηmax is larger because of a larger fill factor than that in Section 3.1. With the increase in Mp, ηmax and Ieff both decreases slightly, which is similar to the small-beam-size case. Dot curves in Fig. 10 show the variation of Ms_out. Firstly, Ms_out is strongly positively correlates with Mp. For a fixed Ip, the larger
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for the increased spatial modulation, resulting in a nearly flattop output profile with a low modulation, Fig. 12(d). Since amplification and back conversion depends mainly on Ip rather than Mp, Imod are similar for different Mp, which can be seen in Figs. 10 and 11. If Ip increases, Ep (x, y′) and Ep (x, y) are both considerably large, and back conversion occurs not only at (x, y′) but also at (x, y). The compensation effect reduces, and thus Ms_out increases once again. The amplified spectrum of the three points, A, B and C, are given in Fig. 13. Full wavelength amplification is achieved at all three points. The green curve in Fig. 13(b) shows the amplified spectral intensity at A, which is similar to the input to a large extent. However, the output spectrum at B and C show an apparent hole around the center wavelength. This is because of wavelength-dependent gain saturation and back conversion, as approved by Witte [14]. Back conversion of C is much severer, resulting in a deeper hole in the amplified spectrum. Fig. 10. Conversion efficiency (η) and the output modulation (Ms_out) for different average pump intensity (Ip) and modulation (Mp). A, B and C correspond to Ip ¼0.73, 1.17 and 2.18 GW/cm2, Mp ¼ 1.5.
Mp is, the larger Ms_out is obtained. For a fixed Mp, with the increase of Ip, Ms_out tends to decrease and then increase; min-Ms_out can be achieved at Imod, as shown in Fig. 11(b). For example, minMs_out for Mp ¼ 1.3, 1.5 are 1.06, 1.1, respectively. Imod are both 2.18 GW/cm2. The important feature is that Ms_out can be much smaller than Mp if Ip is close to Imod, which indicates that for a high-energy OPCPA system with a large beam size, output beam quality can be optimized using proper Ip. We use Mp ¼1.5 as an example. The profile for Mp ¼1.5 is shown in Fig. 12(a). Specifically, three points corresponding to Ip ¼ 0.73, 1.17 and 2.18 GW/cm2, Mp ¼1.5 are labeled as A, B and C in Fig. 10. The input and output parameters at A, B and C are listed in Table 4, and their output profiles are shown in Fig. 12(b)–(d), respectively. Spatially, two points, (x, y) and (x, y′), are denoted within the pump beam size to give an explanation. The local pump intensity, input signal intensity and output signal intensity of the two points are denoted as Ep, Es_in and Es_out. From Fig. 12(a), it can be seen that Ep (x, y′)4 Ep (x, y). The input flat-top spatial profile cannot maintain, and instead, a slantwise profile that resembles the pump is achieved, Fig. 12(b). With the increase in Ip, Ep (x, y′) becomes intensely larger, the signal gain at (x, y′) reaches saturation, and back conversion occurs. While for (x, y), saturation does not occur, and the signal gain still holds. In this way, the difference between Es_out (x, y′) and Es_out (x, y) reduces, resulting in decreased Ms_out. The beam profile with a relatively smaller slope in Fig. 12(c) also confirms this. The compensation effect becomes more obvious when IP is larger than Ieff, back conversion is more severe, which can be confirmed by the decreased η shown in Fig. 10. Back conversion contributes to obvious compensation
4.2. Influence of signal on OPCPA output beam quality The solid and dash curves in Fig. 14 show the conversion efficiency (η) and Ms_out with different Ip and Ms_in. For different Ms_in, ηmax and min-Ms_out can be achieved at 1.31 GW/cm2, as shown in Figs. 14 and 15. The identical Imod and Ieff shows that the trade-off between high conversion and good output beam quality is no need. Moreover, even Ms_in is large, the min-Ms_out is close to 1. For example, min-Ms_out for Ms_in ¼1.0, 1.3 and 1.5 are 1, 1.01 and 1.02, respectively. It indicates that the signal beam quality can be improved by OPCPA. The variation tendency of Ms_out is because of parametric amplification and back conversion. Three points corresponding to Ip ¼0.73, 1.17 and 2.18 GW/cm2, Mp ¼1.5, are labeled as A′, B′ and C′. The input and output parameters at A′, B′ and C′ are listed in Table 5. The beam profiles are shown in Fig. 16. A super-Gaussian pump is used. The gain among the spot size differs because of the modulated signal. For points where the initial signal intensity is high, the signal undergoes rapid amplification first, but the rapid gain cannot maintain because of saturation and back conversion. For points where the initial signal intensity is low, back conversion cannot occur immediately. Different parametric gains among the beam size compensate for the signal spatial modulation, resulting in decreased Ms_in, and the reduced slope in Fig. 16(b) certifies this. When Ip increases to Imod, the output beam profile is nearly superGaussian, and resembles the pump profile, Fig. 16(c). This fine output indicates that the signal modulation can be compensated using proper Ip. Since the parametric gain and back conversion are closely related to Ip rather than Ms_in, Imod are similar for different Ms_in. If Ip increases from Imod to a larger one, the signal undergoes rapid amplification and backflow is severe in the initially intense area. Strong back conversion causes the initially high-intensity
Fig. 11. (a) Maximum conversion efficiency (ηmax) and Ieff. (b) The minimum output modulation (min-Ms_out) for different Mp.
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Fig. 12. (a) The input pump beam profile. (b), (c) and (d) Output signal beam profiles at points A, B and C, respectively. Table 4 Conversion efficiency (η) and the output modulation (Ms_out) for different average pump intensities (Ip) and input pump modulations (Mp). Notation
Ip (GW/cm2)
Mp
η
Ms_out
A B C
0.73 1.17 2.18
1.5 1.5 1.5
42.17% 45.62% 34.62%
1.58 1.39 1.1
area decay; meanwhile, the initially weak area undergoes a high gain and becomes strong. As a result, the compensation effect diminishes, and Ms_out increases again with the increase i Ip, which is the case in Fig. 16(d). The amplified spectra of A′, B′ and C′ are given in Fig. 17. The amplified spectral profile of A′ resembles the input spectrum, and no obvious back conversion is observed. However, a hollow is seen in the output spectrum of B′ and C′, especially of C′. Moderate back conversion occurs at B′ and improves the output beam quality. The
Fig. 14. Conversion efficiency (η) and the output modulation (Ms_out) for different average pump intensity (Ip) and initial signal modulation (Ms_in).
Fig. 13. (a) Input spectrum. (b), (c) and (d) Output spectrum at points A, B and C, respectively.
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Fig. 15. (a) Maximum conversion efficiency (ηmax) and Ieff. (b) The minimum output modulation (min-Ms_out) for different Ms_in. Table 5 Conversion efficiency (η) and output modulation (Ms_out) for different average pump intensity (Ip) and input signal modulation (Ms_in). Notation
Ip (GW/cm2)
Ms_in
η
Ms_out
A′ B′ C′
0.73 1.31 2.33
1.5 1.5 1.5
40.76% 46.63% 35.89%
1.13 1.02 1.18
deep hollow in Fig. 17(d) indicates significant back conversion at C′ which is consistent with the discussion above. Since OPCPA has the feature of wave-length dependent gain saturation, the hollow appears near the center wavelengths in the amplified spectrum. 4.3. comparison of pump and signal modulation We set η 4 40% and Ms_out o1.3 as the standards for high efficiency and low Ms_out, respectively, as the blue dot curves in Figs. 10 and 14. The common range is crucial since it corresponds to both high η and low Ms_out. For a super-Gaussian pump and signal, low output modulation always keeps, and the common range is just the range for η 4 40%, which is 0.71–2.09 GW/cm2. In this case, if the pump intensity corresponding to a high η is selected, low Ms_out is automatically ensured. However, for a modulated pump, the common range narrows greatly with the increase in Mp, for example, the range for Mp ¼1.5 is 1.46–1.75 GW/ cm2. It indicates that high η and low Ms_out are harder to maintain for a larger Mp. For a modulated signal, the common range narrows slightly as Ms_in increases. For example, the range for Ms_in ¼1.5 is 0.73–2.09 GW/cm2, similar to that for Ms_in ¼1.0. The results confirm the dominant influence of the pump on the OPCPA output. The control of the pump modulation is important in optimizing OPCPA. For a Nd: glass laser system with the energy up to order of kilojoules, which is a commonly used pump source in high-energy OPCPA systems, the spatial modulation can hardly be reduced to less than 1.3, while 1.5 can be achieved currently. For Mp ¼1.5, the
pump range that satisfies η 440% and Ms_out o 1.3 is 1.46– 1.75 GW/cm2. If Ip is chosen within this range, the output with a relatively high conversion efficiency and a good beam quality is achievable.
5. Conclusion A numerical study is carried out to investigate the output beam quality of high-efficiency broadband NOPCPA systems. The modulation type used in this paper is the linear slope along the x- or y-axis. This particular modulation is for intuitive and analytical convenience. Nevertheless, the results provide insight into pump and signal modulation effect in NOPCPA. Ms_out and η are affected by the input beam quality such as Mp and Ms_in. Generally, larger Mp or Ms_in results in larger Ms_out and lower η. Within the two factors, the pump modulation has a major impact, as proved in Sections 3 and 4. In addition, the pump intensity influences the output property. Typically, ηmax and min-Ms_out can be achieved at Ieff and Imod, respectively. For the non-collinear phase matching, the walk-off effect is considerable for small-beam-size pump and signal. The pump modulation and walk-off are related in the parametric process and affect the output simultaneously. Both the magnitude and type of Mp are influential. Mp⊥ is the modulation type that mitigates the pump walk-off; however, the idler walks out automatically and unavoidably during the parametric process. The walk-off of idler leads to beam asymmetry and a raised edge, resulting in deterioration of the output beam quality. In Sections 3.1.2 and 3.1.3, Mp// is used to show the effect of pump walk-off. For the pump with Mp// þ , the output tendency resembles Mp⊥, but η is further reduced and Ms_out increased. However, Ms_out with Mp// shows a different tendency, and much smaller Ms_out can be achieved at larger Ip. This feature can compensate for increased Ms_out result from walk-off, as confirmed by the better output property of Mp// ¼ 1.1 than that of a super-Gaussian pump. It is suggested that the negative influence of walk-off can be compensated by choosing proper pump profile. The results can be
Fig. 16. (a) Input signal beam profile. (b), (c) and (d) Output signal beam profiles at points A′, B′ and C′, respectively.
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Fig. 17. (a) Input spectrum. (b), (c) and (d) Output spectrum at points A′, B′ and C′, respectively.
useful for designing NOPCPA systems with enhanced efficiency and beam quality. The walk-off effect can be negligible when the beam sizes are considerably large. ηmax decreases with the increase in Mp or Ms_in, which indicates the negative influence of the modulation. When the pump is modulated, Imod is larger than Ieff. The difference between Imod and Ieff suggests that trade-off between minMs_out and ηmax is unavoidable. However, when signal is modulated but pump is super-Gaussian, Ieff and Imod are close, for example, Ieff and Imod for Ms_in ¼1.5 are both 1.31 GW/cm2; thus, high η and low Ms_out are reachable simultaneously. The result suggests that OPCPA output depends crucially on the pump, a high-energy pump with good spatial quality will be beneficial to a high-energy, low-modulation output. In an ultrahigh-energy OPCPA system, the output modulation from the Nd: glass based kilojoule pump source can hardly be less than 1.3. Our simulation shows that for a high-energy OPCPA system, the output can be optimized by controlling Mp and Ip. A considerably high conversion and low Ms_out can be obtained by choosing the pump energy properly. In the case of Mp ¼ 1.5, Ms_out can be less than 1.3 and η can be larger than 40% with Ip in the range of 1.46–1.75 GW/cm2.
Funding National Natural Science Foundation of China (Grant Nos. 61521093 and 61378030).
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