## Abstract

We report numerical results of second-harmonic generation in a type II potassium dihydrogen phosphate crystal with a time predelay for picosecond and/or femtosecond Yb-doped solid-state lasers, and clarify the dependence of the self compression in the second-harmonic laser pulse on the initial frequency chirp, fundamental duration and intensity, and phase-mismatching angle. We also show numerically the generation possibility of a self-compressed second-harmonic laser pulse near 20 fs.

©2007 Optical Society of America

## 1. Introduction

Nd-doped solid-state lasers such as Nd:YAG, Nd:YLF and Nd:glass operating at 1.053 µm or 1.06 µm have been developed over the years. Their second harmonic generation (SHG) is used in researches ranging from micro-machining to inertial confinement fusion. Theories and simulations of SHG at these wavelengths are widely available. Meanwhile, Ytterbium (Yb) doped solid-state laser materials have received much attention because they have broad absorption and emission bandwidths, the low quantum defect and the simple electronic structures [1], which are suitable for direct diode pumping and enable efficient, broadband and high repetition rate operations. Recently, an Yb-doped laser system with high intensities has been demonstrated by using optical parametric amplification [2]. The output wavelength of the Yb-doped solid-state laser can also be converted to the visible region through SHG in a nonlinear optical crystal. In particular, a self-compressed second-harmonic (SH) pulse is possibly produced in a type II potassium dihydrogen phosphate (KDP) crystal by properly delaying the extraordinary polarized fundamental pulse at 1 ps and 1 µm [3, 4, 5, 6, 7, 8]. It is necessary for the self compression that the group velocity of the SH pulse should be close to the arithmetic average of the ordinary and extraordinary fundamental pulses under strong energy exchange. Since the Yb-doped solid-state laser at output wavelengths around 1 µm meets approximately the requirement for the group velocity, the self-compressed SH laser pulse can be expected as those demonstrated in Nd:glass lasers. However, there is no report on numerical simulations of SHG with time predelay for picosecond and/or femtosecond Yb-doped solid-state lasers around 1 µm.

The SHG process for the Yb-doped solid-state laser has some special issues. As the pulse duration is possible as short as 50 fs, not only the group-velocity mismatch (GVM) but also the group-velocity dispersion (GVD) as well as the third-order dispersion (TOD) should be involved in numerical simulations. The output of the Yb-doped solid-state laser may be frequency-chirped to provide a fundamental laser pulse longer than that of the transform limit. In addition, the third-order nonlinear optical effects involving the self- and cross-phase modulations (SPM and XPM), and self-focusing due to the high fundamental intensity are not negligible. Therefore, it is necessary to carry out numerical simulations for SHG with the time predelay at pulse durations of 50 fs-1.1 ps and peak intensity of 5-600 GW/cm^{2} including the initial frequency chirp.

In this paper, we present our simulation results of SHG in the type II KDP crystal with the time predelay for picosecond and/or femtosecond Yb-doped solid-state lasers, and clarify the dependence of the self compression on the initially frequency chirp, fundamental duration and intensity, and phase-mismatching angle. We report the optimized results for crystal thickness and corresponding SHG properties to generate a self-compressed SH laser pulse in the femtosecond regime near 20 fs.

## 2. Theoretical model

The numerical simulation is carried out with nonlinear wave equation. The electric fields *E*
_{m} of laser pulses in the *x-y-z* coordinate system are expressed as

${E}_{m}(x,y,z,t)=\frac{1}{2}{A}_{m}(x,y,t)\mathrm{exp}\left[j\left({k}_{m}z-{\omega}_{m}t\right)\right]+c.c.,$

where *m* denote 1o, 1e, and 2e, describing the ordinary and extraordinary polarizations of the fundamental pulses (1o, 1e) as well as the extraordinary SH pulse (2e), respectively, ω_{m} the carrier frequency, *k*
_{m} the wave vector and *j* the imaginary unit. By applying the slowly varying envelope approximation in the propagation, the coupled amplitude equations for *A*
_{m} are then approximately given by [8, 9, 10],

$$=j\frac{{\omega}_{0}}{{n}_{1o}c}{d}_{\mathrm{eff}}{A}_{2e}{A}_{1e}^{*}\mathrm{exp}\left(-j\Delta kz\right)+j{k}_{1o}\left({\gamma}_{1o1o}{I}_{1o}+{\gamma}_{1e1o}{I}_{1e}+{\gamma}_{2e1o}{I}_{2e}\right){A}_{1o},$$

$$=j\frac{{\omega}_{0}}{{n}_{1e}c}{d}_{\mathrm{eff}}{A}_{2e}{A}_{1o}^{*}\mathrm{exp}\left(-j\Delta kz\right)+j{k}_{1e}\left({\gamma}_{1o1e}{I}_{1o}+{\gamma}_{1e1e}{I}_{1e}+{\gamma}_{2e1e}{I}_{2e}\right){A}_{1e},$$

$$=j\frac{{\omega}_{0}}{{n}_{2e}c}{d}_{\mathrm{eff}}{A}_{1e}{A}_{1o}\mathrm{exp}\left(j\Delta kz\right)+j{k}_{2e}\left({\gamma}_{1o2e}{I}_{1o}+{\gamma}_{1e2e}{I}_{1e}+{\gamma}_{2e2e}{I}_{2e}\right){A}_{2e},$$

where ∇^{2}
_{⊥}=∂^{2}/∂*x*
^{2}+∂^{2}/∂*y*
^{2}, *A*
_{1o} and *A*
_{1e} are the envelopes of the two orthogonally polarized fundamental pulses, *A*
_{2e} the envelope of the generated SH pulse, ω_{0} the carrier frequency of the fundamental pulse, Δ*k=k*
_{2e}(*θ*
_{0})-*k*
_{1o}-*k*
_{1e}(*θ*
_{0}) the wave-vector mismatch, *θ*
_{0} the phase-matching angle, ρ_{1e} and *ρ*
_{2e} the walk-off angles of the extraordinary fundamental and SH pulses, respectively, *δ*
_{xm} and *δ*
_{ym} (*m*=1o, 1e, 2e) the incident angles in the *x* and *y* directions, α_{m} the absorption coefficients, *v*
_{m} the group velocities, *g _{m}*=∂

^{2}

*k*/∂

*ω*

^{2}the GVD coefficients,

*β*

_{m}=∂^{3}k/∂ω^{3}the TOD coefficients,

*d*

_{eff}the nonlinear constant and γ

_{mm}’ (m’=1o, 1e, 2e) the coefficients of the third-order nonlinear susceptibility. The intensity

*I*

_{m}are given by

*I*=

_{m}*cε*

_{0}nm|A_{m}|^{2}/2, where

*c*is the vacuum speed of the light,

*n*

_{m}the index of the material and ε0 the vacuum permittivity.

The evolution of the fundamental wave with a super-Gaussian transverse-mode profile and a Gaussian pulse shape is expressed as

where *A*
_{0m} (m=1o, 1e) are the peak amplitudes of the fundamental pulses, *G* the positive integer controlling the degree of the edge sharpness of the super-Gaussian function, t0m the initial times of the peak positions, *D*
_{m} the beam diameters (FWHM), *τ*
_{m} the pulse durations (FWHM) and *b* the initial chirp rate. We define *z*=0 as the vacuum-crystal boundary. The central positions *x*
_{0m} and *y*
_{0m} are assumed to be zero.

The product of the fundamental duration *τ*
_{m} (*m*=1o, 1e) and the corresponding frequency bandwidth Δ*v _{m}* (FWHM) is given by

where Φ=[1+*bτ*
^{2}
_{m}/2ln ^{2})]^{1/2} is the chirp-rate coefficient describing the frequency chirp of the fundamental pulse. The pulse duration of the chirp-free laser beam (Φ=1) is corresponding to the transform limit, and is subsequently stretched for the initial chirp (Φ>1). The chirp rate *b* is calculated from the differentiation of the instantaneous frequency with respect to time.

The SHG optimization in this work is numerically evaluated by the product of the peak-intensity and energy conversion efficiencies, which are directly obtained through the temporal peak and spatiotemporal integration of the SH pulse. The optimized thickness of the crystal is corresponding to the maximum value of the product so that both high energy conversion and short SH duration can be achieved simultaneously. The SH duration τ_{2e} (FWHM) is calculated from the Gaussian curve fit to the SH pulse.

## 3. Numerical results

The set of nonlinear wave Eqs. (1), (2), and (3) is numerically solved by using a standard split-step beam-propagation algorithm with a fourth-order Runge-Kutta nonlinear integration [9]. A type II KDP crystal with a phase-matching angle of *θ*
_{0}=59.62° is used in the SHG process at a central fundamental wavelength of 1017 nm. The Yb-doped solid-state laser used in the simulation has a temporal duration in the range of 50 fs-1.1 ps (*τ*
_{1o}=*τ*
_{1e}). Under the transform-limit condition (Φ=1), the laser operating at a spectral bandwidth of 30.4 nm produces a chirp-free fundamental pulse as short as 50 fs. If the laser pulse is frequency-chirped with the same bandwidth, the corresponding pulse duration becomes longer than that of the transform limit. The fundamental laser beam normal to the crystal (δ_{xm}=δ_{ym}=0) is assumed to be a super-Gaussian distribution (G=8) with a diameter of 1 mm (*D*
_{1o}=*D*
_{1e}). The peak intensity of the fundamental pulse is variable in the range of 5–600 GW/cm^{2} (*I*
_{1o}=*I*
_{1e}). The walk-off angles *ρ*
_{1e} and *ρ*
_{2e} for the extraordinary fundamental and SH pulses are 20.47 and 24.59 mrad, respectively. The nonlinear coefficients γ_{mm}’ are drawn from Refs. [10, 11, 12] and other parameters involving the group velocities as well as the second- and third-order dispersion coefficients are calculated with the data of Ref. [13]. The extraordinary polarized fundamental laser pulse is temporally delayed by another KDP crystal, resulting in a proper time predelay between the two fundamental laser pulses for the purpose of the self compression. The time predelay is primarily relating to the duration and peak intensity of the fundamental pulse, and nonlinear properties of the crystal. In general, a too short predelay cannot ensure enough interacting time while a too long predelay requires a thick crystal in which the pulses are subsequently influenced by GVD for femtosecond pulses. The optimum time predelay is generally 1~2τ_{1o}(τ_{1e}).

Fundamental pulses with the initial chirp should be generally compressed by a simple linear compressor before SHG. However, a frequency-chirped SH pulse may be required for some special applications such as degeneracy optical parametric amplification [2]. SHG properties as functions of the chirp-rate coefficient Φ are shown in Fig. 1 for several values of fundamental intensity at the 1-ps pulse duration and 2-ps time predelay. The chirp rate *b* is 5.36909×10^{24}
*m*/*s*
^{2} for Φ=4, and the corresponding bandwidth is therefore reduced to 6.1 nm. The optimized thickness of the type II KDP crystal [Fig. 1(a)] is determined by the maximum value of the product of the peak-intensity and energy conversion efficiencies under the phase-matching condition (Δ*θ*
_{0}=0). The energy conversion efficiency [Fig. 1(b)] and the SH duration τ_{2e} [Fig. 1(c)] are plotted based on the thickness shown in Fig. 1(a) at the same Φ. The SH duration is generally self-compressed for Φ=1 and increased with Φ. At a fundamental intensity of 5 GW/cm^{2}, the SH pulse is compressed to 262 fs with a 22.3-mm-thick KDP crystal and the corresponding energy conversion efficiency is 66%. However, there is no obvious compression in the SH pulse for Φ=4 at the same intensity. By further increasing the fundamental intensity, the self compression is also possible even for the chirped fundamental laser pulse. In general, high fundamental intensity and long time predelay generally cause some sub- or satellite pulses together with the main pulse. In addition, the self compression around Φ=1.5 is obvious than other Φ (Φ>1) for fundamental intensities of greater than 20 GW/cm^{2}, in which the initial frequency chirp of the fundamental pulse is partially compensated for by SPM and XPM due to the third-order nonlinear optical process.

Figure 2 shows SHG properties versus the fundamental duration of 0.1–1.1 ps without initial frequency chirp. The optimized thickness [Fig. 2(a)] is obtained with the same way used in Fig. 1(a), and the energy conversion efficiency [Fig. 2(b)] as well as the SH duration [Fig. 2(c)] are obtained under the optimized thickness at the given fundamental duration. The dependence on the chirp-free fundamental duration can be approximately divided into compression and uncompression regimes sensitive to the fundamental intensity. The SH duration with respect to the optimized thickness is linearly proportional to the fundamental duration in the uncompression regime, and inversely proportional to the fundamental duration in the compression regime. GVD will take an obvious role for pulses in the sub-picosecond regime, and generally stretch the pulse even for the chirp-free fundamental pulse. This means the self compression arising from GVM may be prevented by GVD. As shown in Fig. 2(c), the compression regime becomes broader with the increase in the fundamental intensity, and the GVD effect can be simultaneously suppressed because of strong energy exchange.

The evaluated results for the self-compression possibility with fundamental pulses of 50–125 fs are shown in Fig. 3. There is no obvious self compression in the SH pulse at 50 fs for the peak intensity in the range of 100–600 GW/cm2 since the fundamental duration is too short so that the overlapping and interacting time between the fundamental and SH pulses is not enough to ensure effectively energy exchange. As shown in Fig. 3(c), the SH durations for the fundamental laser pulses of 75 and 125 fs are respectively 38.4 and 25.9 fs at 500 GW/cm^{2}, and are probably compressed by using the fundamental pulse of greater than 500 GW/cm2. The shortest SH pulse duration is 21.8 fs at 600 GW/cm^{2} in a 1.9-mm-thick KDP crystal. However, the corresponding energy conversion efficiency is only 30%.

The damage threshold of the KDP crystal has been demonstrated to be over 400 GW/cm^{2} at the picosecond regime, and may be further higher for the fundamental pulse in the femtosecond regime [14, 15, 16]. If the KDP crystal can be strong enough to endure the ultrahigh fundamental intensity as high as 1 TW/cm^{2}, the SH pulse may be directly compressed to 27 fs in a 1.4-mm-thick KDP crystal with 50-fs fundamental pulses and 150-fs time predelay. The corresponding energy and peak-intensity conversion efficiencies are 60 % and 112 %, respectively. Meanwhile, sub-pulse or pedestal portions, which are generally parasitic in the compressed SH pulse at the picosecond regime, are almost nonexistent because of the limited energy exchange.

When the ultrahigh-intensity SH pulse propagates through the KDP crystal, the SPM and XPM effects due to the third-order nonlinear optical process result in an additional phase shift that is equivalent to the initial phase arising from the phase mismatching. These phase modulations become strong with intensities of the interacting pulses, and the conversion efficiency is subsequently reduced in general. An initial phase shift can partially compensate for the phase modulations to obtain high conversion efficiency by introducing an initial phase-mismatching angle Δ*θ*
_{0} [17]. As shown in Fig. 4, the maximum conversion efficiency as high as 77% is possible for the 1-TW/cm^{2} fundamental intensity at Δ*θ*
_{0}=-8.5 mrad. The other optimized angles Δ*θ*
_{0} for 0.75 and 0.5 TW/cm^{2} are -6.5 and -4 mrad, respectively. Conversely, the SH duration becomes shorter at Δ*θ*
_{0} with opposite signs. The self focusing or defocusing as well as the walk off can be ignored for the thickness shown in Fig. 4(a).

Our numerical calculations are carried out only for the KDP crystal as its transverse size can be larger than 0.5 m and applied to the large-scale laser system for the laser fusion. In fact, some other crystals such as lithium formate (LBO) and ammonium dihydrogen phosphate (ADP) are also usable because their group velocities satisfy approximately the requirement of the self compression.

## 4. Conclusions

We described numerical simulations on SHG in a type II KDP crystal with a time predelay for picosecond and/or femtosecond Yb-doped solid-state lasers, and clarified the dependence of the self compression in the SH laser pulse on the initially frequency chirp, fundamental duration and intensity, and phase-mismatching angle. The self compression in the SH laser pulse is achievable even with the chirped fundamental laser pulse if the fundamental intensity is high enough to improve the energy exchange. For fundamental laser pulses ranging from 50 fs to 1.1 ps, the SH conversion process is approximately divided into compression and uncompression regimes which are controllable by the fundamental intensity. High conversion efficiency or short SH duration can be achieved by properly introducing an initial phase-mismatching angle. Our numerical results demonstrated that a self-compressed SH laser pulse near 20 fs is possibly generated with the fundamental pulses from Yb-doped solid-state lasers.

## References and links

**1. **Y. Zaouter, J. Diderjean, F. Balembois, G. Lucas-Leclin, F. Druon, P. Georges, J. Petit, P. Golner, and B. Viana, “47-fs diode pumped Yb^{3+}:CaGdAlO_{4} laser,” Opt. Lett. **31**, 119–121 (2006). [CrossRef] [PubMed]

**2. **K. Yamakawa, M. Aoyama, Y. Akahane, K. Ogawa, K. Tsuji, A. Sugiyama, T. Harimoto, J. Kawanaka, H. Nishioka, and M. Fujita, “Ultra-broadband optical parametric chirped-pulse amplification using an Yb:LiYF_{4} chirped-pulse amplification pump laser,” Opt. Express **15**, 5018–5023 (2007). [CrossRef] [PubMed]

**3. **A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. **86**, 301–306 (1991). [CrossRef]

**4. **Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A **41**, 5645–5649 (1990) [CrossRef] [PubMed]

**5. **Y. Wang and B. Luther-Davies, “Frequency-doubling pulse compressor for picosecond high-power neodymium laser pulses,” Opt. Lett. **17**, 1459–1461 (1992). [CrossRef] [PubMed]

**6. **R. Danielius, A. Dubietis, G. Valiulis, and A. Piskaraskas, “Femotosecond high-contrast pulses from a parametric generator pumped by the self-compressed second harmonic of a Nd:glass laser,” Opt. Lett. **20**, 2225–2227 (1995). [CrossRef] [PubMed]

**7. **R. Danielius, A. Dubietis, A. Piskaraskas, G. Valiulis, and A. Varanavicius, “Generation of compressed 600–720-nm tunable femotosecond pulses by transient frequency mixing in a β-barium borate crystal,” Opt. Lett. **21**, 216–218 (1996). [CrossRef] [PubMed]

**8. **T. Zhang, Y. Kato, K. Yamakawa, H. Daido, and Y. Izawa
: “Peak intensity enhancement and pulse compression of a picosecond laser pulse by frequency doubling with a predelay,” Jpn. J. Appl. Phys. **34**, 3552–3561 (1995). [CrossRef]

**9. **T. Zhang, M. Yonemura, M. Aoyama, and K. Yamakawa
: “A simulation code for tempo-spatial analysis of three-wave interaction with ultra-short and ultra-high intensity laser pulses,” Jpn. J. Appl. Phys. **40**, 6455–6456 (2001). [CrossRef]

**10. **C. Y. Chien, G. Korn, J. S. Coe, J. Squier, G. Mourou, and R. S. Craxton, “Highly efficient second-harmonic generation of ultraintense Nd:glass laser pulses,” Opt. Lett. **20**, 353–355 (1995). [CrossRef] [PubMed]

**11. **L. Zheng and D. D. Meyerhofer, “Self- and cross-phase-modulation coefficients in KDP crystals measured by a Z-scan technique,” LLE Review **74**, 125–130 (1998).

**12. **R. A. Ganeev, I. A. Kulagin, A. I. Ryasnyansky, R. I. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO_{3} and BBO crystals,” Opt. Commun. **229**, 403–412 (2004). [CrossRef]

**13. **V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, *Handbook of nonlinear optical crystals*, (Springer-Verlag, 1991).

**14. **G. Duchateau and A. Dyan, “Coupling statistics and heat transfer to study laser-induced crystal damage by nanosecond pulses,” Opt. Express **15**, 4557–4576 (2007). [CrossRef] [PubMed]

**15. **P. DeMange, C. W. Carr, R. A. Negres, H. B. Radousky, and S. G. Demos “Multiwavelength investigation of laser-damage performance in potassium dihydrogen phosphate after laser annealing,” Opt. Lett. **30**, 221–223 (2005). [CrossRef] [PubMed]

**16. **H. Yoshida, H. Fujita, M. Nakatsuka, M. Yoshimura, T. Sasaki, T. Kamimura, and K. Yoshida, “Dependences of laser-induced bulk damage threshold and crack patterns in several nonlinear crystals on irradiation direction,” Jpn. J. Appl. Phys. **45**, 766–769 (2006). [CrossRef]

**17. **T. Zhang, M. Aoyama, and K. Yamakawa: “Noncollinear chirp-compensated second-harmonic generation with subpicosecond laser pulses,” Jpn. J. Appl. Phys. **39**, 1146–1150 (2000). [CrossRef]