Performance scaling of high-power picosecond ... - OSA Publishing

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J. Opt. Soc. Am. B / Vol. 30, No. 11 / November 2013

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Performance scaling of high-power picosecond cryogenically cooled rod-type Yb:YAG multipass amplification Xing Fu,1,2,* Kyung-Han Hong,1,4 Li-Jin Chen,1 and Franz X. Kärtner1,3 1

Department of Electrical Engineering and Computer Science and Research Laboratory of Electronics, Massachusetts Institute of Technology (MIT), Cambridge, Massachusetts 02139, USA 2 Center for Photonics and Electronics, State Key Laboratory of Tribology, Department of Precision Instrument, Tsinghua University, Beijing 100084, China 3 Center for Free-Electron Laser Science, DESY and Department of Physics, Hamburg University, Hamburg, Germany 4 e-mail: [email protected] *Corresponding author: [email protected] Received June 3, 2013; revised August 11, 2013; accepted August 31, 2013; posted September 6, 2013 (Doc. ID 191580); published October 7, 2013 The performance scaling of high-power picosecond cryogenically cooled rod-type Yb:YAG multipass amplification is studied numerically, taking into account diffraction, dispersion, self-focusing, self-phase modulation, gain guiding, and thermal lensing effects. It is shown that the beam size narrows as the beam energy rises, rapidly reaching the damage threshold of optics, mainly due to gain guiding and self-focusing effect. Simulation results predict that for a 1 kHz cryogenically cooled Yb:YAG bulk amplifier, it is difficult to obtain energies beyond 40 mJ with 100 ps seed pulse in a single stage, which is consistent with experimental results, while it is possible to produce more than 100 mJ with 300 ps pulses. © 2013 Optical Society of America OCIS codes: (140.3280) Laser amplifiers; (140.3615) Lasers, ytterbium; (320.7090) Ultrafast lasers. http://dx.doi.org/10.1364/JOSAB.30.002798

1. INTRODUCTION Ultrabroadband high-power optical parametric chirped-pulse amplifiers (OPCPA) [1] are considered a next-generation driving source for strong-field physics because of the wavelength coverage including the mid-infrared (mid-IR) wavelength range. The high kinetic energy of ionized electrons earned in the long-wavelength strong field, in comparison with the 800 nm wavelength of Ti:sapphire lasers, leads to a dramatic increase in photon energy emitted during the recombination process in high-order harmonic generation (HHG). This opens up new areas in attosecond science, coherent extreme ultraviolet and soft x-ray imaging, and high-order harmonic spectroscopy [2,3]. The energy and average power scaling of ultrabroadband OPCPA systems enables increased soft x-ray photon flux from HHG, which is important for many applications. For this reason, OPCPA systems at high repetition rates in the kHz range are of great interest even though some early studies have been performed at 10–20 Hz [4–6]. It is crucial to have high-energy (multi-mJ) mid-IR OPCPA systems for phase-matched HHG with high photon energies [7,8]. Energy scaling of kHz OPCPAs has been limited by the picosecond pump laser technology. The picosecond duration is beneficial as pump duration because of the high peak power necessary to achieve ultrabroadband parametric gain in a short crystal length, compared to nanosecond pulses. However, power scaling of amplifiers is much more difficult for picosecond pulses than for femtosecond pulses because of the lack of compact stretchers. Femtosecond pulses can 0740-3224/13/112798-12$15.00/0

easily be chirped to have a long time duration (typically >300 ps), so that the peak power is well below the damage threshold of optics in terms of both the surface and bulk damage. Already for ns pulse amplification, the self-focusing and surface damage threshold constitute a high risk. For power scaling of ps pulses, where stretching is limited due to the narrow spectral bandwidth, the peak power grows quickly with energy or average power. As will be shown later, when energy and average power increases, the beam size continuously shrinks during propagation through the laser gain medium due to gain guiding, thermal lensing, and self-focusing effect. Therefore, the amplified picosecond pulse is suffering from a rapidly rising peak intensity, which must be suppressed below a certain level to prevent optical damage. Yb:YAG crystals have been identified as an excellent ultrafast gain medium with a low quantum defect and direct diodepumping capability. However, its high saturation fluence has been the main bottleneck of scaling energy and average power from ultrafast Yb:YAG amplifiers due to the thermal and B-integral problems. The thin-disk gain geometry is one of the conventional approaches that can solve the thermal problem as well as the self-focusing issue. A thin-disk Yb:YAG chirped-pulse amplifier (CPA) regenerative amplifier was reported by Metzger et al., producing 25 mJ, ∼1 ps pulsed output at 3 kHz repetition rate [9]. It was operated in the regime of “deterministic chaos” due to the nature of thin disk configuration, such as a low single-pass gain and a large number of round trips. On the other hand, Yb:YAG bulk amplifiers at © 2013 Optical Society of America

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Vol. 30, No. 11 / November 2013 / J. Opt. Soc. Am. B

cryogenic temperature are available for scaling both energy and average power, combining the good thermo-optical properties of the Yb:YAG crystal and the advantage of high single-pass gain. Cryogenic cooling of the Yb:YAG gain medium [10–12] dramatically reduces the saturation fluence (∼1.5 J∕cm2 at 77 K). Recently, Hong et al. demonstrated a 40 mJ, ps cryogenic Yb:YAG bulk CPA laser system operating at 2 kHz repetition rate [13]. Using this laser as the pump source, a kHz, mJ-level, carrier-envelope phase (CEP)-stable ultrabroadband OPCPA system was demonstrated at 2 μm wavelength [14]. In this paper, we numerically investigate the high power picosecond amplification based on bulk-type cryogenically cooled Yb:YAG multipass amplifiers [13], studying the limit and potential of performance scaling. The amplification process is characterized by deriving the spatiotemporal model and numerically solving the nonlinear Schrödinger wave equation (NLSE) together with the Frantz–Nodvik gain equation and the heat conduction equation. Beam propagation models have been extensively used in the past to study laser dynamics and beam evolution [15–17] in high-power and high-energy laser systems. Raybaut et al. developed a simulation code to identify the dependence of the spectral shape of gain and seed beam on the pulsed output of a Yb3 -doped regenerative amplifier, albeit without including nonlinear effects [18]. Yan et al. developed the NLSE including both gain-guiding and thermally induced refractive index guiding effects to analyze the transverse mode formation in a CW high power resonator, in which the dispersion and self-focusing effects were not included, either [19]. Chin et al. demonstrated the evolution process of the femtosecond laser pulse in an optical media based on NLSE, where the interplay of linear and nonlinear effects was discussed but the thermal lensing effect was not brought into consideration [20]. In our simulation, the pulse evolution is studied in detail including diffraction, dispersion, self-focusing, self-phase modulation, gain guiding, and thermal lensing effects that occur in the process of rod-type multipass amplification of chirped picosecond pulses. The paper is organized as follows. In Section 2, the generalized NLSE model is derived. In Section 3, parameters and boundary conditions in the modeling are defined based on the experimental setup of cryogenically cooled rod-type Yb:YAG multipass amplifiers. In Section 4, the numerical results of the model are discussed in the temporal and spatial domain, focusing on the beam narrowing mechanisms mainly due to gain guiding and selffocusing, and an attempt is presented to suppress these effects for energy scaling.

2. NLSE MODELING We derive a propagation equation for chirped laser pulses based on the standard 3 1-dimensional NLSE. The optical field can be written as Ex; y; z; t Ax; y; z; t expjβ0 z − jω0 t c:c;

(1)

where A is the complex envelope of the optical field, β0 is the propagation constant at the central angular frequency ω0 , and c.c stands for complex conjugate.

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Using the slowly varying envelope approximation, the NLSE can be written as ∂Ar; z; t j D ∂2 ∇2⊥ j 2 2 − jδjAj2 ∂z 2k0 n0 2 ∂t jk0 ΔnT r; z ΔnG r; z Ar; z; t; (2) where k0 ω0 ∕c denotes the free-space wavenumber corresponding to the central angular frequency ω0 while c is the speed of light in a vacuum, ∇2⊥ ∂2 ∕∂x2 ∂2 ∕∂y2 is the transverse Laplacian, D2 is the group velocity dispersion coefficient of the gain medium, and δ k0 n2 ∕n0 is the nonlinear coefficient while n0 and n2 denote the linear refractive index at the central frequency and the nonlinear refractive index coefficient of the gain medium, respectively. On the right-hand side of Eq. (2), the first term represents beam diffraction and the second term accounts for dispersion. The third term presents the intensity dependence of the refractive index varying in time and space due to the nonlinear refractive index. This term leads to self-phase modulation (SPM) in the time domain and the self-focusing in the spatial domain. In this paper, we call this term the self-focusing effect for simplicity, partly because we mainly focus on the spatial domain. The last term on the right-hand side of Eq. (2) illustrates the effect caused by the change of refractive index, where ΔnT x; y; z is the variation of refractive index due to the thermal effect, and ΔnG x; y; z is the variation of refractive index due to the nonuniform gain distribution. A. Calculation of Gain Guiding Effect According to [21], the gain-induced variation of refractive index ΔnG x; y; z can be expressed as ΔnG x; y; z −

G0 x; y; z; ω j − Δω∕ψ ; 2k0 1 Δω∕ψ2

(3)

where ψ is half of the FWHM of the gain spectrum, Δω is the angular frequency detuning between the central frequency of seed laser and the gain profile, and G0 x; y; z; ω is the small-signal power gain distribution, as written by G0 x; y; z; ω expE pump x; y; z; ω∕E sat ω;

(4)

where Epump is the pump intensity and E sat is the saturation fluence, which can be illustrated as E sat ω

hp ω0 ; 2πσ e ω

(5)

where hp is Planck’s constant and σ e is the emission cross section of the gain medium. Assuming a good matching between the central frequency of seed laser and the gain profile, we have Δω 0, and obtain ΔnG x; y; z −j

G0 x; y; z; ω : 2k0

(6)

Therefore, in the calculation code, the ΔnG x; y; z term is treated with the Frantz–Nodvik equation to take the gain saturation into account, as listed subsequently.

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After a propagation length of Δz, the laser intensity Ix; y; z Δz; w can be calculated as Ix; y; z Δz; ω I sat ω lnf1 G0 x; y; z; ω × fexpIx; y; z; ω∕I sat ω − 1gg: (7) The residual gain Gr at location z after being extracted by the seed laser can thus be expressed as Gr x; y; z; ω G0 x; y; z; ω expfIx; y; z Δz; ω − Ix; y; z; ω∕I sat ωg: B. Thermal Lensing The thermally induced variation ΔnT x; y; z can be expressed as

of

(8)

refractive

index

ΔnT x; y; z ΔnT-temp x; y; z ΔnT-stress x; y; z ΔnT-end x; y; z;

(9)

where the three terms on the right-hand side of Eq. (9) represent the change of refractive index caused by thermal gradients, thermal stress, and end effect, respectively. Usually the second term ΔnT-stress is much less than the first term, and thus it is not included in our calculation. In addition, the third term ΔnT-end is neglected as well, because in this paper we discuss the crystal with end caps, which greatly suppresses the end effect. Therefore, the calculation of ΔnT x; y; z can be simplified as ΔnT x; y; z ΔnT-temp x; y; z

dn Tx; y; z − T 0 ; (10) dT

where dn∕dT is the thermo-optic coefficient of the gain medium, T 0 is the coolant temperature, and Tx; y; z is the temperature distribution within the gain medium, which can be obtained by solving the heat conduction equation kh · ∇2 Tx; y; z Qx; y; z 0;

(11)

where kh is the thermal conductivity of the gain medium and Qx; y; z is the heat energy deposited per unit volume. It should be noted that Qx; y; z is dependent on the spatial position within the laser gain medium, the calculation of which is stated as follows by assuming that the pump mode has a Gaussian profile Qx; y; z

2ηh P in αT 2x2 y2 exp − exp−αT z; w2p0 z πw2p0 z

(12)

where ηh is the fraction of absorbed pump power deposited, P in is the pump power injected on the slab end surface, αT is the effective absorption coefficient of the gain medium based on the pump spectrum, and wp0 z is the pump mode radius at position z within the crystal.

3. NUMERICAL SIMULATION The numerical simulation is carried out based on a cryogenically cooled rod-type Yb:YAG multipass amplifier (see Fig. 1),

Fig. 1. Layout of Yb:YAG multipass amplifier. Paths (1) and (2) show two-pass and four-pass amplification outputs, respectively. λ∕4, quarter-wave plate; FR, Faraday rotator; TFP, thin-film polarizer; C 1 –C 2 , lenses; LD, fiber-coupled laser diode; DM, dichroic mirror.

the configuration and parameters of which are similar to the one we previously reported [13]. The amplifier has two 20 mm long Yb:YAG crystals with a doping concentration of 2 at. %. Each of the crystals is pumped from a single end by a fiber-coupled diode module at 940 nm with an available pump power of 350 W. Both Yb:YAG crystals have 3 mm thick undoped YAG end caps on the sides where the pump beams enter, while the pump size at the crystal end can be adjusted. In addition, the gain media are cooled with liquid nitrogen in an evacuated Dewar. The amplifier can be operated in either two- or four-pass configuration by polarization beam splitters. The seed pulse that is injected into the amplifier stage is stretched with the second-order dispersion coefficients as β2 −336 ps2 , having the pulse duration of about 114 ps and a spectral width of 0.19 nm, operating at 1 kHz repetition rate. The wavelength-dependent emission cross section σ e and saturation fluence I sat of Yb:YAG at 77 K are shown in Figs. 2(a) and 2(b). It should be noted that the peak emission cross section is 1.31 × 10−19 cm2 at 1029 nm, corresponding to a saturation fluence of 1.48 J∕cm2 . Thus, in the simulation, the seed beam is assumed to have a center wavelength at 1029 nm for a perfect match in the frequency domain, as shown in Fig. 3. As for the cooling condition, the upper surface of the gain medium is in close contact with the heat sink, which transfers the heat to the liquid nitrogen. Thus, the boundary conditions for cooling can be described as ∂T ∂x ∂T ∂y ∂T ∂y ∂T ∂z

0

at x

hw T − T kh LN

0 0

w ; 2 t at y ; 2

t at y − ; 2 L at z ; 2

(13)

where hw is the heat transfer coefficient on the cooling face (y t∕2), assuming that hw 10 kW∕m2 ⋅K is a typical value for the powerful conductive cooling to account for the heat exchange process by the liquid nitrogen, while other faces are assumed to be heat insulated. T LN 77 K is the temperature of liquid nitrogen and T ∞ 288 K is the ambient room temperature.

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Table 1. Parameters Used in the Simulation of Rod-type Yb:YAG Amplifier Cryogenically Cooled at 77 K Parameter n0 n2 dn∕dT hp ηh σa

Value

Unit

1.82 [23] 7.3 × 10−20 [24] 0.9 × 10−6 [25] 6.62 × 10−34 [23] 91.3% [23] 1.7 × 10−20 [26]

/ m2 ∕W K−1 2 m · kg∕s / cm2

Fig. 4. Three-dimensional temperature distribution in the Yb:YAG crystal.

For the 2 at. % doped Yb:YAG, the value of the effective absorption coefficient at 940 nm is approximately assumed as half of the peak absorption coefficient, given by Fig. 2. (a) Emission cross section and (b) saturation fluence of Yb: YAG at 77 K.

In addition, the temperature dependence of the thermal conductivity of YAG is taken into account as [22] kh T

a1 a − 4; lna2 Ta3 T

(14)

where a1 , a2 , a3 , and a4 are constants with values a1 1.9 × 106 W∕cm · K, a2 5.33 K−1 , a3 7.14, and a4 331 W∕cm.

1 αT ηd σ a ξ; 2 1 × 1.38 × 1020 cm−3 × 1.7 × 10−20 cm2 × 2; 2 2.35 cm−1 ;

(15)

where ηd is the dopant ion density of Yb3 , ξ is the doping concentration, and σ a is the peak absorption coefficient of Yb:YAG at 940 nm (see Table 1). With the parameters listed in Table 1, Qx; y; z and Tx; y; z are calculated with the pump power of 110 W at each end, using the commercial software Femlab. The pump radius at the slab end surface is assumed to be 1.8 mm. It should be noted that all the beam diameters mentioned in this paper denote the width at which the beam intensity has fallen to 1∕e2 (13.5%) of its peak value. The simulated temperature distribution of the single Yb: YAG is illustrated in Fig. 4, with the maximum temperature of 120 K occurring at the center of the pump surface. The undoped YAG part is not displayed in Fig. 4 since it does not absorb the pump power. With the calculated three-dimensional temperature distribution, the thermal-induced variation of refractive index ΔnT x; y; z can thus be obtained according to Eq. (10). The optical path difference (OPD) along the crystal width and thickness are displayed in Fig. 5 individually.

4. RESULTS AND DISCUSSION Fig. 3. Match between the seed beam and gain profile in the frequency domain.

The numerical scheme to solve Eq. (2) is based on the splitstep Fourier method, in which the Hankel transform [27] and

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Fig. 5. OPD profile along the crystal length: (a) versus width at thickness center and (b) versus thickness at width center.

its inverse transform are applied. For a problem with cylindrical symmetry, the Hankel transform technique is preferred to the two-dimensional fast Fourier transform (FFT), making the computing speed comparable to that of the one-dimensional FFT. The parameters used in the code are summarized in Table 1. A. Double-Pass Amplifier The simplified propagation model used for the double-pass amplifier is shown in Fig. 6, unfolding the beam paths between dichroic mirrors. A series of positions along the propagation length are marked with A to I, with L1 0.45 m, L2 0.9 m, and L3 0.9 m. The beam energy is amplified from 2.5 mJ at the input to 6.8 mJ after the first passage and to 21.2 mJ after the two-pass amplification. The evolution of the beam size during propagation is shown in Fig. 7 for a pump power of 110 W injected at each end with a pump radius of 1.8 mm. The normalized beam intensity as well as the gain profile after the first and second

Fig. 6. Simplified propagation model for the double-pass amplifier.

Fig. 7. Beam size along the beam path shown in Fig. 6.

passage are illustrated in Figs. 8(a) and 8(b), respectively, compared to the intensity profile of the seed beam and the small-signal gain profile. The beam radius is significantly reduced from 1.8 to 0.74 mm after the two-pass amplification, while the gain is saturated at the center. The pulse shapes of the amplified beam in the time and frequency domain are shown in Fig. 9, together with the seed beam. After the energy scaling through the Yb:YAG crystals, the pulse duration drops from 114 to 107 ps, while the spectrum width slightly expands from 0.19 to 0.22 nm due to self-phase modulation. In order to identify individual effects influencing the beam shape and size in the multipass amplification process the diffraction, gain guiding, thermal lensing, dispersion, and selffocusing terms are added consecutively to the model. The parameters of the amplified beam considering different combination of effects are summarized in Table 2, with a pump radius of 1.8 mm, while the normalized intensity profiles of the amplified beam in different conditions are described in Fig. 10. It can be seen from Table 2 and Fig. 10 that the gain guiding effect and the self-focusing effect play major roles in beam size narrowing, while the gain guiding effect is more dominant. In the simulation case with only diffraction and gain guiding effect considered, the gain guiding effect dramatically reduces the beam size by 36.6%, from 1.86 to 1.18 mm. In addition, as illustrated by the comparison between the dash dotted line and the square-marked line in Fig. 10, introduction of the self-focusing effect changes the distribution of beam intensity, making it more center-focused and producing longer wings, deviating from a Gaussian shape. Furthermore, the weak thermal lensing contributes very little to the beam narrowing. By comparing the dashed line and dash dotted line, it is seen that the diffraction effect slightly expands the beam size along the propagation length. With all effects included, the beam radius is finally decreased to 0.74 mm, with the B-integral as high as 0.845. We also investigate the dependence of the amplified beam size and energy on the seed beam size and pump beam size. Figures 11(a) and 11(b) describe the amplified beam radius and beam energy as a function of the seed beam radius, respectively, with the pump beam size of wp 1.6, 1.8, and 2.0 mm. It is shown that, as the seed beam radius increases, the amplified energy decreases due to the reduced intensity and gain, and thus the amplified beam size increases because

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Fig. 8. Beam after the first and second passage: (a) amplified beam profile and (b) gain profile.

Fig. 9. Pulse profile: (a) time domain and (b) frequency domain.

Table 2. Influence of Various Effects on the Parameters of the Amplified Beam

Radius (mm) B-integral Pulse duration (ps) Spectrum width (nm)

Seed

Diffraction

Diffraction Gain − Guiding

Diffraction Gain − Guiding Thermal-Lens Dispersion

1.80 / 114 0.190

1.86 0.082 114 0.190

1.18 0.715 110 0.183

1.06 0.764 110 0.183

of the weakened beam narrowing mechanism mainly driven by the gain guiding effect. In addition, a smaller pump size leads to a higher gain, a smaller amplified beam radius, and a higher amplified beam energy. Furthermore, the amplified beam radius wa is plotted again in Figs. 12(a)–12(c) versus seed beam radius, for the cases of wp 1.6, 1.8, and 2.0 mm, respectively, together with the allowed minimum beam size wmin that is determined by the surface damage threshold corresponding to each energy level. As shown in Fig. 11(b), the energy of the amplified beam can be

Diffraction Gain-Guiding Thermal-Lens Dispersion Self-Focusing 0.74 0.845 107 0.220

determined by the given seed beam size ws and pump beam size wp . As for each energy level, Ek , wmin Ek can be calculated as s Ek wmin ; (16) 2πρ where ρ is the averaged damage threshold of the amplified optics. The damage threshold of high-energy dielectric mirrors is typically in the range of ∼10 J∕cm2 for 10–20 ns pulses,

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(2) Reduce the crystal length to weaken the self-focusing effect. (3) Expand the seed pulse duration with a larger stretching ratio, as an attempt to weaken the self-focusing effect and gain-guiding effect at the same time.

Fig. 10. Amplified beam profile at 21.2 mJ with different beam shaping effects included.

corresponding to ∼1 J∕cm2 for 100–200 ps pulses [13,28]. The effective damage threshold is lower than ∼1 J∕cm2 for highrepetition-rate operation in the kHz range because a thermal relaxation cannot fully occur. It can be found in Fig. 12 that, for the case of wp 1.6 mm, none of the seed beam radii from 1.6 to 2.8 mm can be used because of the high risk of optics damage. For the case of wp 2.0 mm, all the listed seed beam radii satisfy the requirement of the damage threshold, with the best choice of ws 1.6 mm corresponding to a relatively high output energy of 16.8 mJ. It can be expected that the damage threshold will be reached for smaller seed beam size. For the case of wp 1.8 mm, a seed beam radius larger than 2 mm can be chosen, which corresponds to the highest output energy of no more than 20 mJ at a total pump power of 220 W. In addition, the amplified beam size gets smaller as the pump power rises, due to the increased gain guiding effect. As shown in Fig. 13, the amplified beam radius decreases with a slope efficiency of ∼0.003 mm∕W for a seed energy of 2.5 mJ and it drops even faster for a seed energy of 5 mJ due to a higher amplified pulse energy. In order to scale the energy with the beam size far below the damage threshold, three methods can be used to suppress the beam narrowing and self-focusing. (1) Use a curved mirror to compensate the gain guiding and thermal lensing.

B. Adoption of Curved Mirror A convex mirror is replacing the flat mirror in position E in Fig. 6 to expand the beam size between the first and second passage. Figure 14 illustrates the amplified intensity profile versus different mirror curvature, while Table 3 lists the corresponding beam radius, energy, B-integral, and twodimensional near-field beam profile. As shown in Fig. 14 and Table 3, the curvature of the mirror compensates the beam narrowing to some extent. With a mirror focal length of −1 m, the beam radius gets expanded to 1.78 mm, with a little drop in beam energy. However, it should be noted the radius of 1.78 mm is calculated at point I (Fig. 6). As shown in Fig. 15, along the beam paths from E to F and from H to I the beam size is rapidly expanded thanks to the curved mirror, but along the beam path from F to H the beam size is rapidly reduced due to the combination of gain guiding, self-focusing, and thermal lensing effects. As a result, the beam size at point H (crystal end surface) is as small as 1.22 mm with an amplified energy of 17.7 mJ. If the beam size is expanded even larger to prevent the potential damage by further reducing the curvature of the mirror or increasing the propagation length from E to F, the beam size at the end surface (F) of the crystal before the second passage would become so large that the output energy is below 10 mJ. C. Reduction of Crystal Length In this section, we run simulations with a shorter crystal length. Amplified beam profiles with the total crystal length of LG 46, 23, and 10 mm are compared in Fig. 16, where the values of B-integral, amplified energy, and amplified beam radius are marked for each case. Reduction of crystal length indeed suppresses the self-focusing effect, indicated by a much smaller B-integral from 0.845 to 0.130. However, since the gain guiding effect is dominant and almost unchanged in all three cases, the weakened self-focusing effect does not lead to a significant expansion of beam size. Based on the crystal with a short length (10 mm), we take a further step to use a large pump mode and seed beam size to

Fig. 11. (a) Amplified beam size and (b) amplified beam energy versus seed and pump mode size.

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Fig. 12. Amplified beam size wa and allowed minimum beam size wmin : (a) wp 1.6 mm and (b) wp 1.8 mm; wp 2.0 mm.

slow down the beam narrowing over the propagation on one hand while increasing the number of passes through the gain medium to four times to obtain a higher total gain, trying to investigate the potential of energy scaling of the amplifier. Both the pump beam radius and seed beam radius are expanded to 2.8 mm in the four-pass amplifier. Furthermore, a 4f image relay system is applied with an extended arm, in order to avoid the beam narrowing between neighboring passes. The equivalent geometry is shown in Fig. 17, in which f can be determined as 300 mm. In an actual experimental setup, a vacuum tube needs to be installed at the focal spot

to avoid the air breakdown and a flat reflector is located at the dashed vertical line. The separation between the lenses in the telescope needs to be set slightly larger than 2f for the 1∶1 image relay in the presence of positive thermal lensing at the crystal. The intensity profiles of the amplified beam as well as the gain profile under the total pump power of 330 W are described in Fig. 18 versus different passage numbers. After the four-pass amplification, the energy is raised to 34.5 mJ with the beam size of 1.20 mm, which again approaches the damage threshold. Figure 19 compares the case with

Fig. 13. Amplified beam radius versus pump power.

Fig. 14. Normalized intensity profile versus mirror curvature.

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Table 3. Amplified Beam Parameters with Varying Curvature of the Mirror Focal Length of Curved Mirror (m) Radius (mm) Energy B-integral Near-field beam profile

Flat Mirror

−4

−2

−1

0.87 21.2 0.84

1.00 20.4 0.72

1.28 19.5 0.63

1.78 17.7 0.50

and without the 4f relay imaging system, which shows that the amplified beam size after four-pass amplification with the 4f system is only slightly larger than that without the 4f system. Since the additional image relay optics (lenses and windows in the vacuum tube) accumulate additional B-integral, the adoption of a 4f image relay system is not an effective way to reduce the beam narrowing or avoid the self-focusing.

Fig. 15. Beam size along the beam length when using a convex mirror.

Fig. 16. Amplified beam intensity profile with different crystal length.

D. Seeding with Increased Pulse Duration The pulse duration of the seed for the chirped pulse amplifier is increased to 300 ps with an expanded spectral bandwidth of 0.503 nm. Increasing stretching ratio without increasing the bandwidth is not simple because the grating stretcher and compressor can become extraordinarily bulky. Using the new seed pulse, the simulation has been run for the two-pass amplification (same configuration as that in Section 4.A), with a pump power of 110 W at each end and a pump radius of 1.8 mm. The beam energy is amplified from 2.5 to 18.5 mJ after the two-pass amplification. The parameters of the amplified beam, considering different combinations of effects, are summarized in Table 4. Comparing Table 4 with Table 2, one can find that using the seed pulse with a larger pulse duration, the beam narrowing mechanism is much weaker. The beam radius is 0.95 mm after two-pass amplification for the 300 ps seed in contrast to 0.74 mm for the seed with a 104 ps duration. The pulse shapes of both seed beam and amplified beam are shown in Figs. 20(a) and 20(b), respectively, in the time and frequency domains. Furthermore, the 300 ps seed is adopted for the four-pass amplification, as the configuration shown in Fig. 17, in which a 4f system is used, and the crystal length is 10 mm. Both the pump beam radius and seed beam radius are 2.8 mm, the same as that in Section 4.C. Amplified beam intensity profiles are illustrated in Fig. 21 for the pump power of 330, 400, and 500 W, respectively. It is worth mentioning that, for the pulse

Fig. 17. Four-pass amplifier model with 4f system.

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Fig. 18. Simulated result for four-pass amplification versus different passage numbers: (a) intensity profile and (b) gain profile.

Fig. 19. Output intensity with and without the 4f system.

duration of 300 ps, the damage threshold of the amplified optics is increased to ∼1.5 J∕cm2 . As shown in Fig. 21, with the pump power of 330 W, the beam energy is amplified to 29.8 mJ, similar to the energy for the case of 104-ps seed, but with a much larger beam size (1.42 mm rather than 1.20 mm). As the pump power increases to 400 W, the output energy increases to 50.9 mJ with wa 1.31 mm, operating at a safe condition well below the damage threshold (wmin 0.75 mm. At a pump power of 500 W, the beam energy goes up to 103.7 mJ with the beam radius of wa 1.14 mm, almost reaching the damage threshold, where the corresponding wmin is 1.06 mm. In summary, our simulations disclose that in the process of scaling up the energy, the gain guiding, self-focusing,

and thermal lensing act in such a way that the beam size gets smaller and smaller, which would eventually induce optical damage at some point. The beam narrowing mechanism can be significantly relaxed by larger stretching of the seed pulse. Increasing the pulse duration increases the damage threshold and reduces the self-focusing effect, so this is more beneficial than using shorter crystals. This can be achieved by using a room-temperature Yb:YAG regenerative amplifier that provides a much broader spectral bandwidth and thereby much longer pulse duration for seeding a multipass chirped pulse amplifier. However, the beam-narrowing phenomenon is a general problem of solid-state bulk amplifiers when scaling average power and peak power simultaneously. If the multipass geometry is carefully designed, a thin-disk approach even at cryogenic temperature [29] can be a promising way of solving this problem because it can significantly mitigate the thermal lensing, the self-focusing, and the gain guiding effects.

5. CONCLUSION The performance scaling of high-power picosecond cryogenically cooled rod-type Yb:YAG multipass amplification is numerically studied, taking into account diffraction, dispersion, self-focusing, self-phase modulation, gain guiding, and thermal lensing effects. It is shown that the beam size narrows as the amplified energy increases, rapidly reaching the damage threshold of optics, mainly due to the gain guiding and self-focusing effect. In addition, the beam narrowing mechanism cannot be well compensated, but can be weakened at the cost of extraction efficiency and system complexity. Based on the simulation results, it is of great difficulty to produce energy of higher than 40 mJ from the cryogenically cooled Yb:YAG bulk amplifier at 1 kHz using 100 ps seed pulses in

Table 4. Influence of Various Effects on the Parameters of the Amplified Beam

Seed Radius (mm) B-integral Pulse duration (ps) Spectrum width (nm)

1.80 / 300 0.503

Diffraction

Diffraction Gain-Guiding

Diffraction Gain-Guiding Thermal-Lens Dispersion

1.86 0.037 300 0.503

1.19 0.278 240 0.401

1.07 0.296 240 0.401

Diffraction Gain-Guiding Thermal-Lens Dispersion Self-Focusing 0.95 0.307 237 0.404

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Fig. 20. Pulse profile: (a) time domain and (b) frequency domain.

Fig. 21. Amplified beam intensity profile under different pump power.

a single stage, given the damage threshold for picosecond pulses. However, for the 300 ps pulse with a larger stretching ratio, it is possible to reach the 100 mJ level at 1 kHz. For further energy scaling, a sequence of multipass or single-pass amplifiers will be required with different optimization parameters. It should be noted that cryogenic multipass thin-disk amplifier configuration would be a promising approach for obtaining higher energy at kHz repetition rate because the self-focusing and gain guiding effects, as well as the thermal lensing, can be minimized while it is possible to keep a reasonably high single-pass gain by cryogenic cooling.

ACKNOWLEDGMENTS X. F. acknowledges support by the China Scholarship Council Postgraduate Scholarship Program, and the authors are grateful for helpful discussions with Dr. Luis Zapata from MIT, Dr. Daniel Miller and Dr. Darren Rand from MIT Lincoln Laboratory, and Prof. Mali Gong from Tsinghua University. This work was partly supported by Center for Free Electron Laser Science, DESY.

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