Investigation in the far field characteristics of TM polarized Gaussian beam from the vectorial structure

Based on the vectorial structure of electromagnetic beam and the method of stationary phase, the analytical TE and TM terms of TM polarized Gaussian beam, the rigorous solution of Maxwell's equations for a confocal resonator, have been presented in the far field. Then in terms of the vectorial structure, TM polarized Gaussian beam is compared with Gaussian TEM₀₀ mode. The TE term is located at the y-axis, and the TM term the x-axis. At the non-paraxial case, the whole beam spot is elliptical, and the long axis is located at the y-axis. Moreover, the whole beam spot of TM polarized Gaussian beam is smaller than that of Gaussian TEM₀₀ mode. At the paraxial case, the whole beam spot is circular, and TM polarized Gaussian beam reduces to be Gaussian TEM₀₀ mode.     

Tailored TEMp0 beams for large size 3-D laser prototyping

The Laser-Lithography technique allows the fabrication of complex objects having microsizes by selectively solidifying polymeric materials layer by layer upon exposure to a focused Gaussian laser beam having a beam propagation factor M²=1. We can expect that extension of this technique to large sizes 3-D prototyping comes up against a large increase in the design time. A possible solution is the increase in the focused spot size, but unfortunately at the price of a great reduction in the longitudinal resolution due to the resulting increase in the depth of focus. To overcome these difficulties, we propose the use of a rectified TEM_p0 beam allowing the obtaining of a Gaussian beam intensity profile in the focus plane of a lens. The reshaped TEM_p0 beam has a beam propagation factor M²≈(2p+1), and this yields to a relative improvement of the longitudinal resolution although the spot size is increased for reducing the processing time.     

Measurement of the diameter of a laser beam

A new and simple technique for measuring the effective diameter of a laser beam used in material processing is described. The time for the temperature of a spot heated by the laser beam to rise to 90% of equilibrium is compared with that predicted theoretically for a Gaussian TEM₀₀ laser beam. A Gaussian beam diameter equivalent is thus deduced. This calculated diameter is of particular relevance to applications where the laser is used as a heat source.     

Diffraction properties of opaque disks outside and inside a laser cavity

Diffraction of symmetrical Laguerre-Gauss TEM_p0 beams incident on an opaque disk known as a stop is considered. The near- and far-field patterns are studied. Thanks to zero-field occluding, conversion from TEM₁₀ beam to dark hollow beam can be achieved with better efficiency than from a TEM₀₀ beam. It is shown that the fundamental mode of a laser cavity including a diaphragm and a stop can be TEM₀₀- or TEM₁₀-like in shape depending on their size. This result is interpreted from the new divergence hierarchy, which characterises the diffracted TEM_p0 beams emerging from the stop.     

Amplitude and phase clipping: Strehl ratio versus divergence

We consider the transverse characteristics of a Gaussian laser beam subject to a phase or amplitude clipping due to a pupil which is a π-plate or an opaque disc (stop). In particular, we consider the correlation between two features, the Strehl ratio and divergence angle, usually used for characterising the focusability of a diffracted beam. It is demonstrated that the Strehl ratio does not give systematically a global view, from a divergence point of view, on the transverse properties of a Gaussian beam suffering amplitude or phase diffraction. In addition, we consider the case of self-diffraction of a Gaussian beam upon a Gaussian phase aberration of same width, and it is found that the on-axis intensity describes correctly the whole diffracted beam cross-section, from a divergence point of view, only if the central phase shift is smaller than π. Another example showing that the focusability of a pure high-order Laguerre–Gauss TEM_p0 beam, free from any clipping, cannot be correctly described by Strehl ratio is also considered.     

Intensity fluctuations in J-Bessel–Gaussian beams of all orders propagating in turbulent atmosphere

The scintillation index of a J _n -Bessel–Gaussian beam of any order propagating in turbulent atmosphere is derived and numerically evaluated at transverse cross-sections with the aid of a specially designed triple integral routine. The graphical outputs indicate that, just like the previously investigated J ₀-Bessel–Gaussian beam, higher-order members of the family also offer favorable scintillation characteristics at large source sizes. This advantage is maintained against rising beam orders. Viewed along the propagation axis, beams with lower orders and smaller widths exhibit smaller values of the scintillation index at shorter propagation distances and large values at longer propagation distances. Further, it is shown that the scintillation index of the J _n -Bessel–Gaussian beams (n>0) is larger than that of the fundamental Gaussian and the J ₀-Bessel–Gaussian beams only near the on-axis points, while remaining smaller towards the edges of the beam.     

The nonparaxial propagation of the TEM0l doughnut beam with the orbital angular momentum in the far field

The analytical expression for nonparaxial electric field amplitude of the TEM_0l^? doughnut beam with the orbital angular momentum quantum number l is derived in the far field by means of the angular spectrum representation and the stationary phase method. It is shown that the divergent angle of the far field of TEM_0l^? doughnut beam will be smaller with the decreasing of the orbital angular momentum quantum number l or the increasing of the beam waist width w of the initial beam. And the maximal radial intensity of the beam is decreased with its propagation at different rates for different l and w.     

Experimental explorations of the high-order Gaussian mode transformation based on blind-optimization adaptive optics

We present research on high-order Gaussian laser beam transformation by using adaptive optics (AO) technique. Numerical simulation for a mode transformation system based on stochastic parallel gradient descent (SPGD) algorithm was accomplished, along with experimental results obtained for a transformation of TEM₂₀ mode with 37 control channels. The far-field energy centrality of TEM₂₀ mode was advanced and the evaluation function increased more than a factor of 6 after transformation. The experiments results indicate the feasibility of blind-optimization AO in mode transformation system.     

Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere

The propagation properties of flattened Gaussian beam with aperture in turbulent atmosphere have been studied by using the extended Huygens-Fresnel principle. From the study and numerical calculation, the effects of aperture on the propagation of flattened Gaussian beams in turbulent atmosphere have been illuminated. It shows that when the value of the truncation parameter δ is bigger, for example δ?2, the effects of aperture on the propagation properties are too small to be neglected. But when the truncation parameter δ is smaller, for example δ<2, the effects of aperture are complex. The peak value of the average intensity descends more rapidly and the beam spot spreads quicker with aperture than that without aperture when the propagation distance increases. Meanwhile, with the propagation distance increasing, the average intensity profiles of flattened Gaussian beams gradually convert into Gaussian average intensity profiles. In addition, some limiting cases are also discussed. It agrees with the existing results.     

Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd: YAG rod

The effects produced by a birefringent rod on the beam quality factor,M _infr ^sup2 , of a Gaussian TEM₀₀ laser beam are studied. Analytical expressions for the beam parameters at the output plane of a rod amplifier are derived. The theory predicts that the beam quality degrades rapidly with the heat power dissipated in the rod.     

Focus shaping of tightly focused TEM11 mode cylindrically polarized Laguerre Gaussian beam by diffractive optical element

We study the focus shaping of tightly focused TEM₁₁ mode cylindrically polarized Laguerre Gaussian beam with high numerical aperture lens axicon system is investigated theoretically by vector diffraction theory. The intensity pattern at the focus can be tailored by appropriately adjusting the rotation angle. We show that the high NA lens axicon system can generates a sub wavelength focal spot, focal hole, focal splitting and flat-topped focal shapes with extended depth of focus.     

High-power Q-switched laser with high-order Laguerre–Gaussian modes: application for extra-cavity harmonic generations

We present a compact efficient scheme for generation of high-order Laguerre–Gaussian TEM _p,0 modes in a diode-pumped actively Q-switched Nd:YVO₄ laser. The resonator is composed of two plane mirrors with an intracavity concave lens to expand the cavity mode size for generating a high-power Laguerre–Gaussian TEM_5,0 mode. We further confirm that the TEM_5,0 mode is noticeably superior to the TEM_0,0 mode in the processes of second and third harmonic generations.     

Thermo-Mechanical Analysis of Single-Disc Edge-Cooled Silicon Nitride Millimeter Wave Window for 200 kW CW Gyrotrons

This paper investigates the possibility of using Silicon Nitride Composite (Kyocera SN-287) as single-disc, edge-cooled window for gyrotrons operating below 200 kW CW in the frequency range 28-42 GHz. Rotationally symmetric TE_0n, and TEM₀₀ Gaussian modes of rf transmission through the window have been considered. Analysis performed using a one dimensional (1D) finite difference (FD) code reveals that thermal stresses developed due to non-uniform temperature distribution on the disc surface are well within manageable limits for a 200 kW 42 GHz gyrotron proposed for some ECRH applications. For industrial gyrotrons, for microwave material processing and operating at a maximum power level of 100 kW CW, Si₃N₄ windows may be a cost effective replacement for sapphire windows. It is found that a TE₀₂ profile results in lower peak temperature at the window disc in comparison to a Gaussian beam profile and allows the use of smaller discs.     

High efficiency TEM00 mode, diode-pumped, conduction-cooled, Nd:YAG zig-zag slab laser

We report a compact, conduction-cooled, highly efficient, continuous wave (CW) Nd:YAG slab laser in diode-side-pumped geometry. To achieve high efficiency, a novel laser head for Nd:YAG slab has been developed. For an absorbed pump power of 27.6 W, maximum output power of 10.4 W in multimode and 8.2 W in near-diffraction-limited beam quality has been obtained. Slope and optical-to-optical conversion efficiencies are 45.3% and 37.7% in multimode with beam quality factors (M²) in x and y directions equal to 32 and 8, respectively. TEM₀₀ mode operation was achieved in a hybrid resonator with slope and optical-to-optical conversion efficiencies of 43.2% and 29.7%, respectively. Beam quality factors in x and y directions are ?1.5 and ?1.6 for the whole output power range. The laser radiation was linearly polarized and polarization contrast ratios are >1200:1 in the multimode and 1800:1 in the TEM₀₀ mode operation. In passive Q-switching with Cr⁴⁺:YAG crystal of 68% initial transmission, 18 ns pulsewidth has been achieved with an average power of 2 W at a repetition rate of 16 kHz.     

Anomalous spectral behavior of vectorial apertured ultrashort pulsed beams beyond the paraxial approximation

Based on the vectorial Rayleigh diffraction integrals, the analytical expression for the spectral intensity of a vectorial nonparaxial ultrashort pulsed Gaussian beam diffracted at a circular hard-aperture is derived. The effect of f-parameter (f = 1/k₀w₀) on the spectral anomalies near phase singularities of the vectorial nonparaxial ultrashort pulsed beams is studied. It is shown that the spectral switch near the phase singularity of diffracted vectorial nonparaxial ultrashort pulsed beam still exists beyond paraxial regime, but disappears when the f-parameter is larger than a certain value.     

Determination of the ten second order intensity moments

The general astigmatic beam can be characterized by its ten second order moments in first order approach. Most intensity moments, except the beam twist, can be determined by measuring the intensity in a reasonable number of positions around the waist of the beam. The beam twist is determined by applying a rotated cylindrical lens. The ten intensity moments of two kinds of astigmatic beam were determined: a simple astigmatic TEM_8,0 Hermite–Gaussian beam and a twisted beam generated from the TEM_8,0 mode. The experimental results were compared with the theoretical calculations and demonstrate that the ten second order moments of a beam can be determined in a rather simple way.     

大气湍流对径向分布高斯列阵光束扩展和方向性的影响

采用积分变换的技巧,推导出了径向分布高斯列阵光束通过湍流大气传输的二阶矩束宽和远场发散角的解析公式,并详细研究了大气湍流对光束扩展和方向性的影响.研究表明,相干合成情况下,子光束数N越小、径向分布半径r₀越大,列阵光束扩展受湍流影响越小.相干较非相干合成时列阵光束的扩展小,但非相干合成时列阵光束扩展受湍流的影响比相干合成时的小.特别地,N足够小或r₀足够大时,相干与非相干合成列阵光束的远场束宽相等.另一方面,还给出了相干和非相干合成径向分布高斯列阵 关键词： 径向分布高斯列阵光束 大气湍流 相干和非相干合成 二阶矩束宽     

单口径相干合成系统激光光束的M2因子研究

单口径相干合成系统激光光束的光束质量是一个亟待解决的重要问题.基于二阶矩定义, 文中给出了单口径TEM₀₀, TEM₀₁及TEM₁₀两两相干光束M²因子的解析表达式, 并比较分析了束腰宽度、传输距离、振幅之比,以及源场位置矢量对相干光束M²因子的影响, 得到了诸如源场位置参量d₁<100λ时,各相干光束M²因子恒定,反之, 其随位置参量d₁的增大而增大等一些结论.最后,文章对两TEM₀₀模相干光束M²因子的 部分理论进行了实验验证.     

Cross focusing of two coaxial spatial modes of a single frequency of a laser beam in a nonlinear dielectric

The cross-focusing of two co-axial transverse modes of a laser beam viz. TEM₀₀ and TEM₀₁ modes of the same frequency in a non-linear dielectric has been studied. When the TEM₀₀ mode is stronger than the TEM₀₁ mode, both modes focus on the axis. In the opposite case, defocusing occurs at the axis and energy concentrates in a ring.     

Beam wander of J 0- and I 0-Bessel Gaussian beams propagating in turbulent atmosphere

Root mean square (rms) beam wander of J ₀-Bessel Gaussian and I ₀-Bessel Gaussian beams, normalized by the rms beam wander of the fundamental Gaussian beam, is evaluated in atmospheric turbulence. Our formulation is based on the first and the second statistical moments obtained from the Rytov series. It is found that after propagating in atmospheric turbulence, the collimated J ₀-Bessel Gaussian and the I ₀-Bessel Gaussian beams have smaller rms beam wander than that of the Gaussian beam, regardless of the choice of Bessel width parameter. However, the extent of such an advantage depends on the chosen width parameter, Gaussian source size, propagation distance and the wavelength. Focusing at finite distances of the considered beams causes the rms beam wander to decrease sharply at the propagation distances equal to the focusing parameter.