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We propose and evaluate numerically the self compression for Yb-doped solid-state laser pulses in a KDP crystal by the combination of the group-velocity dispersion and the self-phase modulation. The self compression is achievable as the group-velocity-dispersion coefficient of KDP crystals is negative around a 1-μm wavelength. Numerical results showed that the laser pulse in the range 50–200 fs can be compressed as short as 12.8 fs. This self-compression method is simple and low cost, which is possible to be applied to an Yb-doped solid-state laser system with a large-scale beam and an ultrahigh intensity in the regime of tens of femotoseconds.
©2007 Optical Society of America
1. Introduction
Ytterbium (Yb) doped laser materials have allowed significant breakthroughs in terms of efficiency, compactness and reliability. Yb-doped solid-state lasers have broad emission bandwidths at a wavelength around 1-μm and a possible pulse duration near 50 fs [1–2]. Using technique of optical parametric chirped amplification, a high-intensity Yb:LiYF4 laser has been demonstrated [3]. To generate laser pulses shorter than 50 fs with the Yb-doped laser materials, further broadening in the pulse spectrum is necessary. This can be achieved by the compression based on nonlinear optical effects in photonic crystal fibers [4], hollow fiber filled with noble gases compression [5], third-order nonlinearity in bulk materials [6], and cascading effect in nonlinear optical crystals [7].
As the Yb-doped solid-state laser pulses become possible in the femtosecond regime, the group-velocity dispersion (GVD) can play an important role in the pulse propagation in optical materials. In addition, as the high-intensity laser pulses have been possibly over tens of GW/cm2, the self-phase modulation (SPM) becomes apparent even in conventional optical materials, and subsequently the pulse spectrum is possibly broadened. Therefore, the combination of SPM and GVD provides a possible way to further compress the duration of the Yb-doped solid-state laser pulse in bulk optical materials.
In this paper, we propose a novel method to further compress the temporal duration of the Yb-doped solid-state laser pulse around a 1-μm wavelength by the combination of SPM and GVD effects in Potassium dihydrogen phosphate (KDP) crystal, which has merits of the direct use of an ultrahigh input intensity and a large beam size at low cost. We show numerically that the self compression in the Yb-doped solid-state laser pulse in KDP can be achieved to produce a laser pulse as short as 12.8 fs.
2. Combination of SPM and GVD in KDP
When a chirped laser pulse propagates in a dispersive medium, a dispersion-induced chirp is imposed on the pulse during its propagation. The output laser pulse is then compressed if the dispersion-induced chirp is in the opposite of the initial chirp. The laser of the positively chirped requires a medium of an anomalous GVD. A grating pair or a prism pair is generally used to provide such anomalous GVD.
If the input laser pulse is chirp-free, the SPM effect, in which the time-varying index of refraction produces a time dependent phase modulation of the pulse, contributes to spectral broadening of the pulse. The instantaneous frequency goes as the negative of the derivative of the intensity profile with respect to time. Assuming the nonlinear index is positive, this leads to a lowering of frequencies on the leading edge of the laser pulse and an increasing in frequencies at the trailing edge of the pulse. A pulse with a Gaussian envelope will thus acquire a roughly linear frequency chirp across the central portion of the pulse. Such positively chirped laser pulse based on SPM is similar as the pulse with an initial chirp, and can be simultaneously compressed in the anomalous-GVD regime of bulk optical materials through the interplay between SPM and GVD. For the purpose of achieving such soliton-effect pulse compression, the bulk material with an anomalous GVD is also required.
The GVD coefficient g of KDP crystals is shown in Fig. 1 for ordinary (solid line) and extraordinary (dashed line) polarization directions, respectively. The zero-dispersion wavelength for the ordinary polarization direction is approximately 984 nm. KDP exhibits anomalous dispersion (g<0) for wavelengths larger than the zero-dispersion wavelength. The value of g with respect to the central wavelength near 1040 nm of the Yb-doped solid-state laser is -9.24754×10-27 s2/m. Therefore, the self compression through SPM and GVD in KDP is possible for wavelengths around 1 μm. As the GVD coefficient for the extraordinary polarization direction is in the regime of normal dispersion (g>0) around 1 μm, the self compression by the combination of SPM and GVD is impossible in the extraordinary polarization direction of KDP crystals. In addition, for conventional optical materials such as BK7 glass and fused silica, their GVD coefficients are generally in the normal dispersion regime for the wavelength range of 1010–1100 nm. The GVD coefficients of other nonlinear optical and laser crystals such as Beta-barium borate (BBO), Lithium triborate (LBO), Potassium titanyl phosphate (KTP) [8], and Neodymium doped Yttrium Lithium Fluoride Crystal (Nd:YLF) [9] are also in the normal dispersion regime around the wavelength of 1 μm.
We study the GVD characteristic of KDP crystals and try to apply it to the pulse compression of Yb-doped solid-state lasers in the femtosecond regime. KDP is a candidate material with an anomalous GVD for the self compression around 1 μm. As KDP can grow in large size over 0.5 m [10], the self compression is also applicable for ultrahigh power lasers with large-scale beam.
3. Theoretical model
We carried out simulations based on nonlinear wave equation to demonstrate numerically the possibility of the pulse self compression in KDP through the combination of SPM and GVD. The electric fields E of the laser pulse used in the propagation in KDP are expressed as
where z and (x, y) are the propagation and transverse axes, respectively, t is the time, A is the amplitude of the electric field, ω0 is the carrier frequency, k is the wave number, and j is the imaginary unit. By applying the slowly varying envelope approximation in the propagation, the coupled amplitude equations for A is then approximately given by [11–14],
where v g is the group velocity, g = ∂2 k/∂ω 2 is the GVD coefficient with respect to the carrier frequency, β = ∂3 k/∂ω 3 is the third-order dispersion coefficient at the carrier frequency, α is the linear absorption coefficient, T R is related to the slope of the Raman scattering determined by experiments, γ 0 is the nonlinear parameter related to the third-order susceptibility, which is connected to SPM and self focusing in KDP. The intensity I with respect to the amplitude E is given by I = cε 0 n|A|2/2 , where c is the vacuum speed of the light, n is the index of the material, and ε 0 is the vacuum permittivity.
Equation (1) involves GVD, third order dispersion, absorption, diffraction, and nonlinear effect. In addition, self-steepening effect and delayed Raman response are also included in the equation. Therefore, the pulse compression or expansion in the time domain and the beam focusing or defocusing effect in the spatial domain are possibly evaluated. The integration for the right-hand components of Eq. (1) is carried out in the spatiotemporal domain, and the integration for the other component is carried out in the frequency domain. Using a standard split-step beam-propagation algorithm with a fourth-order Runge-Kutta nonlinear integration [14], we studied the evolution of input laser wave with a Gaussian or a super-Gaussian transverse-mode profile and a Gaussian pulse shape as
where A 0 is the peak amplitude of the input laser pulse, G is 1 for the Gaussian transverse profile or is the positive integer (>1) controlling the degree of the edge sharpness of the super-Gaussian function, t 0 is the initial time of the peak position, D is the beam diameter, τ is the pulse duration, and b is the initial chirp. We define z=0 as the vacuum-crystal boundary. The central position x 0 and y0 are assumed to be zero. For the unchirped laser beam (b=0), the pulse duration is corresponding to the transform limit, and the spectrum of the input laser pulse is subsequently broadened with the increase in the initial chirp (b ≠ 0). The chirp b can be calculated from the derivative of the instantaneous frequency with respect to time.
4. Numerical results
Since only the ordinary polarization direction of KDP provides a negative GVD around the 1-μm wavelength, there are no limitations on the azimuth and phase-matching angles of KDP except for the optimum of the crystal thickness. The Yb-doped solid-state laser used in the calculation has a 50-fs temporal duration and a 32-nm bandwidth with a 1040-nm central wavelength, which meets the transform limit of Gaussian profile. The laser pulse enters KDP in the ordinary polarization direction. The integer G for the Gaussian function is 1 and the beam diameter D is 1 mm. The peak intensity of the input laser pulse is in the range 50–500 GW/cm2. If the input laser pulse is frequency-chirped, the corresponding pulse duration is thus larger than that of the transform limit. We assume that the input pulse with an initial chirp can reach its longest duration up to 200 fs. In addition, the nonlinear coefficient γ0 is 2.3 ×10-16 cm2/W at 1 μm. The parameter T R is assumed to be 3 fs.
Fig. 2. Distribution of (a) pulse shapes and (b) spectra of the laser pulse over a 50-mm-thick KDP. d is the propagating distance in the crystal. The intensity is normalized by the maximum value. The frequency axis is the product of the frequency ω and the pulse duration τ.
The temporal and spectral evolutions of the laser pulse propagating in KDP are shown in Fig. 2 at the input intensity of 50 GW/cm2 and the pulse duration of 50 fs. The pulse becomes short with the crystal distance d, and the corresponding spectrum broadens simultaneously. At the distance of 39.5 mm, the pulse duration is shortest with a FWHM value of 13.8 fs. The trailing portion of the laser pulse becomes steeper due to the self-steepening effect and the asymmetry appears obviously in the spectral distribution. In addition, a shift of the pulse spectrum to the red side is primarily resulted from the intrapulse Raman scattering. The laser pulse with respect to the shortest duration seen in Fig. 3 is asymmetric and is superimposed on a pedestal. When the pulse propagates over 39.5 mm, both pulse shape and spectrum split into several peaks, resulting in broadening of the temporal and spectral distributions.
Fig. 3. (a) Temporal and (b) spectral profiles of the compressed laser pulse at the crystal thickness of 39.5 mm for the input intensity of 50 GW/cm2 and the input pulse duration of 50 fs.
Figure 4 shows the optimized crystal thickness and the compressed pulse duration (FWHM) as functions of the input intensity for the pulse duration of 50 fs, in which the solid line is for the nonlinear coefficient γ0, and the dashed line is for 1.1γ0 because values of γ0 are slightly different with references [15–16]. The optimized thickness of KDP is inversely proportional to the input intensity. For low intensity pulses, the negative GVD effect dominates over SPM, and longer thickness is required to increase the SPM effect. By optimizing the thickness under the given input intensity and pulse duration, SPM drives the pulse to compress, and SPM and negative GVD compensate for each other resulting in chirp-free-compressed pulse. A high intensity leads to residual SPM and the corresponding pulse duration is slightly broadened. The optimized thickness and pulse duration for the intensity of 50 GW/cm2 are 39.5 mm and 13.8 fs, respectively, and further reduced to 11.8 mm and 12.8 fs for the intensity of 250 GW/cm2. As the increase in the nonlinear coefficient γ0, the optimized crystals get thinner, but the condition for the shortest pulse duration has no obvious variation.
Fig. 4. (a) Optimized results for crystal thickness and (b) compressed pulse duration as functions of the input intensity at the input pulse duration of 50 fs.
Fig. 5. (a) Optimized results for crystal thickness and (b) compressed pulse duration as functions of the input pulse duration at the input intensity of 50 GW/cm2.
Figure 5 shows the optimized thickness and compressed duration as functions of the input pulse duration at the input intensity of 50 GW/cm2. Except for the transform-limit duration of 50 fs, other input laser pulses are frequency-chirped, and the corresponding compressed durations are larger than that of the transform limit.
Figure 6 is the surface plot of the laser beam, which is obtained by temporally integrating the intensity distribution after propagating in an 11.8-mm-thickness KDP at the input pulse duration of 50 fs and the input intensity of 250 GW/cm2. In comparison with that of the input Gaussian distribution of the laser beam, a minute self-focusing effect appears around the beam center where the intensity is highest. This is similar to the pulse compression in the time domain due to the compensation of GVD for SPM. Since the optimized thickness and the compressed pulse duration are sensitive to the input intensity as shown in Fig. 7, the input beam with uniform spatial distribution is necessary to obtain the same compressed duration. This means that the integer G in Eq. (2) should be larger than 1.
Fig. 7. Spatiotemporal distribution (Y-t) of the compressed laser pulse with the same conditions used in Fig. 6.
5. Conclusions
We proposed and evaluated numerically the self compression of the Yb-doped solid-state laser pulse in a KDP crystal by the combination of GVD and SPM. This is achievable as the GVD coefficient of KDP is negative around the 1-μm wavelength. The GVD and SPM effects in KDP can compensate for each other for input laser pulses with initial intensities over tens of GW/cm2 in the femtosecond regime. Numerical results showed that the pulse duration can be compressed as short as 12.8 fs from the input laser pulse of 50 fs. This self-compression scheme is simple and low cost, and is applicable to Yb-doped solid-state laser systems with large-scale beam and ultrahigh intensity.
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