Nd:LMA自锁模飞秒激光器腔结构最佳化设计

Nd:LMA晶体相对低的非线性折射率n₂,使得针对Ti:Sapphire自锁模激光器设计的腔,对Nd:LMA介质而言并没有提供足够的自幅度调制(SAM).为了在Nd:LMA激光器上实现自锁模,在腔内引入一个具有大的非线性系数n₂、快的响应时间的SF57介质,这样就使得激光腔的结构比较特殊.在这篇文章里利用ABCD传输矩阵对该激光器的结构从理论上进行了分析和计算,使腔结构达到了最佳化设计.     

Generalised ABCD matrix treatment for laser resonators and beam propagation

A generalised ABCD matrix treatment for laser resonators and beam propagation is developed for computer programming. In this treatment, imaginary parts are introduced into the matrixes for all optical elements, and the beam quality factor M² and the index of the medium are also taken into account. When the imaginary parts of the complex matrix are zero and the beam quality factor M² and medium index are unity, the stated method is transformed back into the more classic ABCD matrix format in which the fundamental-mode Gaussian beam transmits through real elements in the vacuum. Based on this method, laser resonator software is realised by the VB programme language. The software can be used to analyse and design stable/unstable standing/travelling cavities, phase-conjugate cavities and beam transformations.     

Propagation properties of the beam generated by Gaussian mirror resonator passing through a paraxial ABCD optical system

By expanding a hard aperture function into a finite sum of complex Gaussian functions, approximate propagation formula is derived in the situation that the beam generated by Gaussian mirror resonator passes through a paraxial ABCD optical system with an annular aperture. The corresponding forms for a circular aperture and a circular black screen are also given. Some numerical simulations are shown to illustrate propagation properties and focusing properties of the beam passing through a paraxial ABCD optical system with the three different kinds of aperture.     

CW mode locked Nd:YVO4 laser pumped by 20-W laser diode bar

The efficient cw mode locking (cw-ML) regime was demonstrated in Nd:YVO₄ laser by means of saturable absorber mirror (SAM). The 0.3-at.% Nd³⁺ doped 10-mm-long YVO₄ crystal end pumped by 20-W diode module with a beam shaper was applied as a gain medium located in the close vicinity to the rear flat mirror of the first arm of Z-type resonator of 316 cm total length with two curved mirrors of 100-cm curvature radii. The SAM of 2%-saturable absorptance and saturation fluence of 50 μJ/cm² was mounted at the opposite end of a resonator. The developed “dynamically stable” cavity design mitigates detrimental role of thermal aberration in gain medium, enforcing clean perfect mode locking even for the highest pump densities. The cw-ML pulses with 47.5 MHz repetition rate and pulse durations in the range of 15–20 ps were observed for a wide range of pump powers and output coupler losses. In the best case, for 32% of output coupler transmission, up to 6.2 W of average power with near 35% slope efficiency was achieved. The thresholds for Q-switched ML, cw-ML regimes were 2.67 W and 6.13 W of pump power, respectively. For the maximum pump power of 20 W we obtained 133 nJ of pulse energy with 16-ps pulse duration, resulting in a peak power higher than 8 kW. The threshold energy density at SAM giving the QML regime was estimated to be about 30 μJ/cm², threshold of cw-ML regime was 220 μJ/cm².     

Yb3+-doped YVO4 crystal for efficient Kerr-lens mode locking in solid-state lasers

We report the first demonstration, to our knowledge, of soft-aperture Kerr-lens mode locking in a diode-pumped femtosecond Yb3+:YVO4 laser. Near-transform-limited pulses as short as 61 fs are generated around a center wavelength of 1050 nm with an output power of 54 mW and a pulse repetition frequency of 104.5 MHz. This is, to our knowledge, the shortest pulse generated directly from an Yb laser having a crystalline host material. The femtosecond operation has a mode-locking threshold at an absorbed pump power of 190 mW. The nonlinear refractive indexes of the Yb3+:YVO4 crystal have been measured to be 19 x 10(-16) cm2/W and 15 x 10(-16) cm2/W for the sigma and pi polarizations, respectively, at 1080 nm.     

Kurtosis parameters of super Lorentzben Gauss beams through a paraxial and real ABCD optical system

Based on the propagation equation of higher-order intensity moments, analytical propagation expressions for the kurtosis parameters of a super Lorentz-Gauss (SLG) SLG₀₁ beam through a paraxial and real ABCD optical system are derived. By replacing the parameters in the expressions of the kurtosis parameters of the SLG₀₁ beam, the kurtosis parameters of the SLG₁₀ and SLG₁₁ beams through a paraxial and real ABCD optical system can be easily obtained. The kurtosis parameters of an SLG₀₁ beam through a paraxial and real ABCD optical system depend on two ratios. One is the ratio of the transfer matrix element B to the product of the transfer matrix element A and the diffraction-free range of the super-Lorentzian part. The other is the ratio of the width parameter of the super-Lorentzian part to the waist of the Gaussian part. As a numerical example, the properties of the kurtosis parameters of an SLG₀₁ beam propagating in free space are illustrated. The influences of different parameters on the kurtosis parameters of an SLG₀₁ beam are analysed in detail.     

Hybrid optical resonator type laser proposed for direct coupling into optical fiber

Combining one stable resonator with a concave mirror and a coupling-hole plane mirror, and another unstable resonator having a concave mirror and a convex mirror, a new hybrid optical resonator type laser has been proposed by overlapping the concave mirror as a common one and fitting the convex mirror exactly to the coupling-hole. Using theABCD law, the so-called g-parameters are first found for beam confinement. Then, based on the eigen-equation, the field distribution on the plano-convex mirror is obtained with the g-parameters above. Knowing this field distribution results in an effective transmittanceG _{e e}. Finally, the ratio of the inner convex to outer plane mirror radii can be determined for an optimum coupling in both the limiting cases of homogeneously and inhomogeneously broadened gain media. Some numerical examples will also be given for a CO₂ laser having the proposed hybrid resonator.     

Propagation of flat-topped beams

The propagation of flat-topped beams passing through paraxial ABCD optical system is investigated based on the propagation formulas of Gaussian beam. The focal shift of focused coherent flat-topped beam is also studied in detail. Analytical expressions of the M² factor and the far-field intensity distribution for flat-topped beams are derived on the basis of second-order moments.     

Third- and fourth-order parametric characterization of partially coherent beams propagating throughABCD optical systems

Within the formalism of the Wigner distribution function, third- and fourth-order measurable beam parameters are proposed and their physical meaning is investigated. The general propagation law of such characteristic quantities inABCD optical systems is derived and a number of paramters are shown to be invariant under propagation inABCD devices.     

Propagation characteristics of the two-dimensional off-axial Hermite-cosh-Gaussian beams through rectangular apertured and misaligned optical systems

By introducing a hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression for the two-dimensional off-axial Hermite-cosh-Gaussian beams passing through a rectangular apertured and misaligned paraxially ABCD optical system has been derived. The results provide more convenience for studying their propagation and transformation than the usual way by using diffraction integral directly. Some numerical simulations are also illustrated for the propagation characteristics of a two-dimensional off-axial Hermite-cosh-Gaussian beam through a rectangular apertured ABCD optical system.     

Nonlinear refractive index of RECOB (RE = Gd and La) crystals

The Z-scan technique is employed to obtain the nonlinear refractive index (n ₂) of the Ca₄REO(BO₃)₃ (RECOB, where RE = Gd and La) single crystals using 30 fs laser pulses centered at 780 nm for the two orthogonal orientations determined by the optical axes (X and Z) relative to the direction of propagation of the laser beam (k//Y// crystallographic b-axis). The large values of n ₂ indicate that both GdCOB and LaCOB are potential hosts for Yb:RECOB lasers operating in the Kerr-lens mode locking (KLM) regime.     

Thermal lens determination of LD end-pumped solid-state laser with stable resonator by slit scanning method

A simple method to determine the thermal focal length of LD end-pumped solid-state laser with stable resonator is presented. The M² factor describing the quality of the beam can be obtained by scanning a slit through the multi-mode Gaussian beam field. The waist width of the beam and the corresponding TEM₀₀ under the same parameters of laser are then deduced through the law of multi-mode Gaussian beam propagation. Based on the standard matrix theory of stable resonator, the thermal focal length of the gain medium can be easily achieved. To show the application of this approach, the thermal focal length of an LD pumped Nd:YVO₄ laser is measured and the experimental results are in agreement with the theoretical prediction.     

Instabilities in coupled photorefractive ring cavities and self-pumped phase conjugators

We investigate the temporal instabilities of mode intensities in two coupled unidirectional photorefractive ring resonators. The first resonator is driven by an external laser beam via photorefractive two-wave mixing. The internal oscillating beam is then employed to drive the second ring resonator. The second ring resonator provides a nonlinear loss mechanism for the coupled system. Complete spatial-temporal equations for describing the coupled system are derived and mean-field approximation is employed to simplify the transient analysis. The results of linear stability analysis indicate that the coupled system exhibits instability in the off-state and steady-state operation. The instability is explained in terms of competition between nonlinear gain and loss. The results are presented and discussed.Part II on Numerical Results will be published in a forthcoming issue of Applied Physics B     

Finite thickness lens model for self-focusing (defocusing) in Kerr medium

A “finite thickness lens” model for self-focusing (defocusing) in Kerr medium is presented. An on-axis normalization transmittance formula is presented for arbitrary nonlinear phase shift for the finite thickness Kerr medium by introducing a nonlinear ABCD-matrix for the transition of a Gaussian beam from linear to non-linear medium, without complex calculation for the beam radius at the far field aperture. The variation of the peak and valley transmittance difference is found to enhance linearly as the phase shift at the focus increases by increasing the thickness of the medium. If the ratio of the Rayleigh istance divided by the thickness of the medium (d/z0) is constant and small enough, the peak and valley transmittance difference stays constant. Finally, a qualitative formula is presented to express the relationship between the system parameters and the on-axis phase shift at the focus.      

Beam propagation in a linear or nonlinear lens-like medium usingABCD ray matrices: the method of moments

We derive exact expressions for the evolution of the second order moment of the intensity distribution of an arbitrary beam propagating in a nonlinear Kerr medium with a quadratic index profile. The results can be recast in terms of theABCD matrix formalism after introducing a generalized complex radius of curvature,Q(z). Various definitions of the beam quality factor are introduced. Numerical simulations reveal the interest of this approach.An Invited Paper     

轴对称光束的克尔透镜矩阵及其在克尔透镜锁模中的应用

在非线性薛定谔方程的基础上较严格简明地推导了克尔介质的轴对称光束及高斯光束的ＡＢＣＤ传播矩阵，该矩阵可用于设计克尔透镜锁模，以腔二端为平面镜的任意光学系统为例，获得了简单的普遍公式。     

束旋转90°对束变换环孔腔输出性能的影响

 使用积分方法和差分方法结合的快速计算方法，在实际燃烧驱动氟化氢化学激光器增益介质沿径向分布不均匀的情况下，考虑在光束变换环形孔径激光谐振腔的紧束段引入光束旋转90°对其输出性能的影响。计算结果显示：在无腔失调的情况下，在紧束段引入光束旋转对束变换环孔激光谐振腔输出功率及光束质量的改善并不显著；在存在腔失调的情况下，紧束段光束旋转对于输出功率和光束质量的改善有显著效果。     

Influence of spherical aberrations on the performance of dynamically stable resonators

Spherical aberrations of thermal lens of the active media are severe when the solid state lasers are strongly pumped. The aberrations influence the mode profile, the diffraction loss, the output power and other laser beam properties, especially for a dynamically stable resonator with large TEM₀₀-mode volume. Self-consistent modes in a resonator with aberrations are calculated using Fox-Li diffraction iterative algorithm. There are side lobes with the fundamental mode when the spherical aberration is introduced, hence the beam quality is deteriorated. The calculated results agree well with the experimental data. The beam filling factor plays an important role in the design of a laser system. The optimum value of the beam filling factor should be determined by the experimental results according to the spherical aberration effect.     

Efficient design considerations for end-pumped solid-state-lasers

The temperature distribution in the gain medium and key design parameters for an ideal-four level end-pumped solid-state laser have been analyzed depending on the crystal's length, absorption coefficient, and pump beam M² factor. The optimum key design parameters and thermal focal lens are obtained by minimizing the root mean square of pump beam radius in the laser crystal. It is found that the focal thermal lens and key design parameters are dependent on the gain medium characteristics and pump beam properties as well. By considering the Poisson equation in cylindrical coordinate and Top-Hat pumping profile, an analytical formula has been derived to introduce the thermal focal length in the end-pumped lasers. A formula is also presented to relate the requirements of pump source to the gain medium properties for working laser at the design point.     

利用特征向量法分析有源失调腔的2维模场

 利用一种用于光腔模式及光束传输模拟计算的特征向量法计算有源腔的2维失调,通过对谐振腔几何关系的求解,以及对增益进行薄层模型处理,最后利用特征向量法计算有源腔失调后的模式分布,发现失调造成了模式强度分布不均匀,并带来模式阶数的改变。计算了相应的光束质量因子,结果表明:在一定失调范围内,失调可能会带入新的模式竞争;谐振腔在一定范围的失调情况下,各阶模式的光束质量可能变好,也有可能变差,随着模式阶数的增加光束质量整体趋势是越来越差;原来简并的模式可能不再简并。该方法可以为计算失调后谐振腔的模场带来方便。