Truncated Gaussian Beam Research Articles

Broadband third-order nonlinear optical responses of black phosphorus nanosheets via spatial self-phase modulation using truncated Gaussian beams

Spatial self-phase modulation (SSPM) effect is widely used in the characterization of 2D nanomaterials because of its terse principle and intuitive experimental setup, although the understanding of its physical mechanism is still incomplete so far. Herein, we use the truncated Gaussian beam (TGB) along the horizontal direction to investigate the SSPM effect in solution dispersions of black phosphorus (BP) nanosheets. By means of the gravity-promoted thermal convection effect, we observe the spatial separation of the circular symmetric narrow rings and the distorted wide rings in the upper half of the SSPM patterns at 532 nm and 1064 nm when the laser beam propagates horizontally through the vertically placed sample. Based on the theory of local and nonlocal optical nonlinearities and the experiment of the TGB SSPM, we confirm that the narrow and wide rings originate from the optical Kerr nonlinearity (OKN) and the thermally induced nonlinearity (TIN), respectively. This work provides an efficient approach to distinguish the OKN from the TIN in 2D nanomaterial dispersions, to separate their contributions, and to extract nonlinear optical parameters.

Analysis of the caustics of partially coherently combined truncated Gaussian beams.

A theoretical model of the caustics of partially coherently combined beams (CBCs) was derived to model CBCs in the case of truncated Gaussian beams. The model enables the analysis of different lattice structures and partial coherence effects, including incoherent beam combining. The impacts of the coherence state, geometry of the array and truncation losses were analyzed. An optimal truncation level of about 6% was found and a maximum CBC efficiency of 62% was determined for this CBC system. With an increase in the beam numbers in the lattice, the relative caustics length decreases, which is consistent with the general rules of diffraction theory.

Characteristics of the partially reflected terahertz wave: truncated beam propagation

We investigate a free-space propagation of a truncated terahertz beam created as a result of a partial reflection from a small-sized metallic reflector. By using terahertz time-domain spectroscopy, we obtain both magnitude and phase spectra of a clipped terahertz beam after its propagation by about 15 cm. Compared to a fully reflected terahertz beam, the partially reflected beam exhibits large variations in both magnitude and phase spectra in the entire spectral range investigated, i.e., from 0.2 to 1.3 terahertz. To model the free-space propagation of the truncated Gaussian beam, we decompose it using a super-Gaussian beam and several other Gaussian beams with phase factors separately assigned to each component. From this, we could successfully reproduce the experimental results, i.e., the modulations observed in both magnitude and phase spectra.

Propagation of truncated Gaussian beams and their application in modeling sharp-edge diffraction.

The two new kinds of truncated Gaussian beams, known as the half and quarter Gaussian beams, are defined as the product of the fundamental Gaussian beam with the Heaviside unit step function. Using the generalized Collins integral, the exact analytical propagation formulas are derived for the truncated Gaussian beams through paraxial optical systems. Combined with the Gaussian beam decomposition method, the truncated Gaussian beams are used to represent the sharp edges of a field after a hard aperture. The modified Gaussian beam decomposition method presented in this work enables the calculation of the diffraction of a given field from a hard aperture with an arbitrary shape. This solves one of the limitations of the conventional Gaussian beam decomposition method, which is its inability to accurately model the diffraction of fields with sharp edges, especially in the near field. The validity and accuracy of the proposed method are demonstrated using a few exemplary diffraction calculations.

Extended phase matching of high harmonic generation by plasma-induced defocusing

High-harmonic generation (HHG) of femtosecond lasers produces unique short-wavelength light pulses with femtosecond to attosecond duration. However, free electrons in the partially ionized gas medium are known to prevent phase matching and limit upconversion to low-energy harmonics only. We have demonstrated experimentally and theoretically an unconventional phase-matching scheme: extending the HHG phase matching cutoff by controlling the rapid self-defocusing effect of the driving laser. This method takes advantage of the additional intrinsic atomic dipole phase mismatch introduced by the rapid laser defocusing. This phase can be precisely controlled by adjusting the aperture of a simple iris, which truncates the input beam, to correctly compensate free-electron dispersion, resulting in tunable harmonic energy and phase matching cutoff extension. Based on this approach, we report experimental observation of extending the harmonic cutoff energy in Ar to ∼65 eV with a 400-fold increase in conversion efficiency using a tightly focused truncated Gaussian beam in a short high-pressure gas cell. This new defocusing-assisted phase matching in a highly ionized gas medium can be applied to different targets, laser wavelengths, and pulse durations to extend harmonic upconversion to higher cutoff energy with higher efficiency.

Double acousto-optic deflector system for increased scanning range of laser beams

A new laser scanning system is presented based on two wide-band acousto-optic deflectors. The interaction medium is tellurium dioxide. Anisotropic interactions take place under two different tangential phase matching configurations in such a way that the acousto-optic bandwidths add up. We demonstrate the feasibility of such a cascade deflection system for the wavelength of λ=514nm. The total frequency bandwidth is Δf=100MHz, equally distributed between the two acousto-optic deflectors. The total angular scan at the output is Δθ=4.4° leading to 125 resolvable spots for a 1mm truncated Gaussian beam.

The beam quality of a truncated Gaussian beam with aberrations

We present a method for the calculation of the beam quality factor of an aberrated laser beam. The closed-form equations are proved, and it is shown that the beam quality factor of an aberrated Gaussian beam depends directly on the Hermite coefficient of the wavefront, as well as the truncation ratio. Further numerical calculation proves that the proposed model leads to results matching those in published work.

Beam quality of truncated laser beams with amplitude modulations and phase fluctuations in turbulence

The closed-form expressions for the Rayleigh range zR and the M2-factor of truncated laser beams with amplitude modulations (AMs) and phase fluctuations (PFs) in turbulence are derived, and the beam quality is studied by taking the zR and the M2-factor as the characteristic parameters of beam quality. The M2-factor of truncated laser beams with AMs and PFs is always larger than that of truncated Gaussian beams both in free space and in turbulence. However, in turbulence the beam quality of truncated laser beams with AMs and PFs may be better than that of truncated Gaussian beams if the zR is taken as the characteristic parameter of beam quality. For laser beams with AMs and PFs in turbulence, the beam quality expressed in terms of zR is consistent with that in terms of the M2-factor versus the phase fluctuation parameter α, but not versus the intensity modulation parameter σA. The beam quality of truncated laser beams with AMs and PFs is less sensitive to turbulence than that of truncated Gaussian beams. The beam quality of laser beams with smaller α and larger σA is less affected by turbulence than those with larger α and smaller σA.

Study of the optimal truncation of a Gaussian laser for far-field propagation through atmospheric turbulence

Based on the generalized Huygens–Fresnel principle, a far-field propagation model of a truncated Gaussian beam through Kolmogorov atmospheric turbulence is proposed. Two parameters, the far-field focal peak intensity and the corresponding mean power density, are used to characterize the energy of the laser beam's long-distance transmission. Furthermore, an optimal truncation of the launch window to maximize the system performance has been investigated by numerical simulation. Several new formulae showing the relationship between the coefficient of the optimal launch window truncation and the turbulence strength are presented, which agree well with the real situation over a considerable turbulence regime.

Effects of truncated Gaussian beam on the performance of fiber optical synthetic aperture system

In the fiber optical synthetic aperture (FOSA) system, the diffraction of the Gaussian beam limited by the aperture in exit pupil plane of fiber collimator is studied theoretically, and the axial and transverse irradiance distributions are obtained. The point spread function (PSF) and modulation transfer function (MTF) of the truncated Gaussian beam array are computed numerically with different truncation factors. The results show that the diffraction of the truncated Gaussian beam array agrees with the uniform-beam Rayleigh diffraction when the truncation factor is less than 0.5, but little power is transmitted. The PSF and MTF are degraded, but more power can be contained when the truncation factor is larger. The selection of the truncation factor is a trade-off between the loss of transmission and the qualities of PSF and MTF in practical application.

Complex Gaussian functions expansion method applied to truncated Gaussian beams

An analytical expression for the beam propagation factor (M 2 factor) of truncated Gaussian beams was derived by using the complex Gaussian functions expansion method. The reasonability of the approximation of complex Gaussian functions expansion method is studied, and a comparison of this method with the generalised truncated second-order moments method and the asymptotic analysis method is also made. In general, an easy analytical expression for the M 2 factor of truncated laser beams can be derived by using the complex Gaussian functions expansion method. The M 2 factor obtained by using the complex Gaussian functions expansion method is more consistent with that in practice than that obtained by using two other methods. The analytical results obtained by using the complex Gaussian functions expansion method can be reduced to that for the non-truncated case when the truncation parameter is sufficiently large. Therefore, the complex Gaussian functions expansion method is a suitable approximation method for studying the M 2 factor of truncated laser beams.

Generalized M2G factor of truncated partially coherent Hermite-Gaussian beam

Based on the method of generalized truncated second-order moment, the analytical formula of the generalized beam propagation factor (M2G factor) of truncated partially coherent Hermite-Gaussian(H-G) beams is derived. The analytical results obtained from truncated fully coherent H-G beam, truncated Gaussian Shell Model (GSM) beam and truncated Gaussian beam are given as particular examples in this paper. It is shown that the M2G factor of truncated partially coherent H-G beam depends on truncated parameter δ, beam order m, and beam coherence parameter α. When the value of δ is very small, the odd-even groups happen to the M2G factor, i.e., the values of M2G factor with different values of the odd m are nearly the same, and so are they for the even m case. However, this phenomenon disappears as δ increases. For the truncated GSM beams with different values of δ, there exist the cross points between the curves of the M2G factor versus α. However, this phenomenon may disappear for truncated partially coherent H-G beams. In addition, the larger the m is, the more the M2G factor is affected by the δ, and the effect of aperture on M2G factor may be neglected when the δ is larger.

Irradiation of amorphous Ta42Si13N45 film with a femtosecond laser pulse

Films of 260 nm thickness, with atomic composition Ta_(42)Si_(13)N_(45), on 4 silicon wafers, have been irradiated in air with single laser pulses of 200 femtoseconds duration and 800 nm wave length. As sputter-deposited, the films are structurally amorphous. A laterally truncated Gaussian beam with a near-uniform fluence of ~0.6 J/cm^2 incident normally on such a film ablates 23 nm of the film. Cross-sectional transmission electron micrographs show that the surface of the remaining film is smooth and flat on a longrange scale, but contains densely distributed sharp nanoprotrusions that sometimes surpass the height of the original surface. Dark field micrographs of the remaining material show no nanograins. Neither does glancing angle X-ray diffraction with a beam illuminating many diffraction spots. By all evidence, the remaining film remains amorphous after the pulsed femtosecond irradiation. The same single pulse, but with an enhanced and slightly peaked fluence profile, creates a spot with flat peripheral terraces whose lateral extents shrink with depth, as scanning electron and atomic force micrographs revealed. Comparison of the various figures suggests that the sharp nanoprotrusions result from an ejection of material by brittle fraction and spallation, not from ablation by direct beam–solid interaction. Conditions under which spallation should dominate over ablation are discussed.

Beam spreading of truncated Gaussian beams in Kolmogorov turbulence

Based on the extended Huygens–Fresnel principle and the hard-edge aperture function expanded as the sum of finite-term complex Gaussian function, analytical expression for average intensity of truncated Gaussian beam in Kolmogorov turbulence is derived, and some limiting cases are discussed. The influences of many factors, such as Fresnel number, wavelength, truncation parameter and structure constant on beam spreading are studied with the help of the average intensity formula. We found that the peak value of average intensity decreases and the beam spot spreads with the decrease of Fresnel number. The change of peak intensity against Fresnel number is slower with large aperture than that with small aperture. When Fresnel number is not small enough the influence of turbulence on intensity profiles is so small that can be neglected if structure constant is small.

Propagation of phase-locked truncated Gaussian beam array in turbulent atmosphere

Truncation manipulation is a simple but effective way to improve the intensity distribution properties of the phase-locked Gaussian beam array at the receiving plane. In this paper, the analytical expression for the propagation of the phase-locked truncated Gaussian beam array in a turbulent atmosphere is obtained based on the extended Huygens–Fresnel principle. Power in the diffraction-limited bucket is introduced as the beam quality factor to evaluate the influence of different truncation parameters. The dependence of optimal truncation ratio on the number of beamlets, the intensity of turbulence, propagation distance and laser wavelength is calculated and discussed. It is revealed that the optimal truncation ratio is larger for the laser array that contains more lasers, and the optimal truncation ratio will shift to a larger value with an increase in propagation distance and decrease in intensity of atmosphere turbulence. The optimal truncation ratio is independent of laser wavelength.

Impact of ablation efficiency reduction on post-surgery corneal asphericity: simulation of the laser refractive surgery with a flying spot laser beam

We developed a rigorous simulation model to evaluate ablation algorithms and surgery outcomes in laser refractive surgery. The model (CASIM: Corneal Ablation SIMulator) simulates an entire surgical process, which includes calculating an ablation profile from measured wavefront errors, generating a shot pattern for a flying spot laser beam, simulation of the shot-by-shot ablation process based on a measured or modeled beam profile, and healing of the cornea after surgery. Using simulated post-surgery corneal shapes for various ablation parameters and beam fluences, we calculated angular dependence of ablation efficiency and the amount of increase in corneal asphericity. Without considering the effect of corneal healing, our result shows the following; 1) ablation efficiency reduction in the periphery depends on the peak fluence of the laser beam, 2) corneal asphericity increases even in the surgery using an ablation profile based on the exact Munnerlyn formula, contrary to previous reports, and 3) post-surgery corneal asphericity increases by a smaller amount in high fluence small Gaussian beam surgery than in low fluence truncated Gaussian beam. Our model can provide improved ablation profiles that compensate for the change of corneal asphericity and induction of spherical aberration in a flying spot laser system, resulting in better surgery outcomes in laser refractive surgeries.

Short access time acousto-optic deflector based on two cascaded Paratellurite devices

Acousto-optic deflectors (AODs) are used in many applications requiring optical scanning devices with various specifications regarding access time, resolution and wavelength. High time-bandwidth products can be achieved with AODs made in a tellurium dioxide (TeO2) crystal. As paratellurite crystals are available with relatively large dimensions, the time-bandwidth product can be increased by enlarging the optical aperture. However, the access time is increased which is not suitable for random access scanning mode. A new laser scanning system is then presented based on two wide-band paratellurite acousto-optic deflectors. Anisotropic interactions take place under two different tangential phase matching (TPM) configurations in such a way that the acousto-optic bandwidths add up. The optical arrangement of the two cascaded AODs is detailed. The feasibility of such a cascade deflection system has been demonstrated for the green wavelength λ= 514 nm of an argon laser. The total frequency bandwidth is Δft = 100 MHz, equally distributed between the two acousto-optic deflectors. The total angular scan at the output is Δθt = 4.5 ° leading to more than 120 resolvable spots for a 1 mm truncated Gaussian beam and a short access time (as low as 1.5 μs).

Nonlinear change of on-axis pressure and intensity maxima positions and its relation with the linear focal shift effect

A comprehensive experimental, analytical and numerical study of the true focal region drift relative to the geometrical focus (focal shift effect) in acoustic focused beams and its nonlinear evolution is presented. For this aim, the concept of Fresnel number, proportional to the linear gain, is introduced as a convenient parameter for characterizing focused sources. It is shown that the magnitude of the shift is strongly dependent on the Fresnel number of the source, being larger for weakly focused systems where a large initial shift occurs. Analytical expressions for axial pressure distributions in linear regime are presented for the general case of truncated Gaussian beams. The main new contribution of this work is the examination of the connection between the linear and nonlinear stages of the focal shift effect, and its use for the estimation of the more complicated nonlinear stage. Experiments were carried out using a continuous-wave ultrasonic beam in water, radiated by a focused source with nominal frequency f = 1 MHz, aperture radius a = 1.5 cm and geometrical focal distance R = 11.7 cm, corresponding to a Fresnel number N F = 1.28. The maximum measured shifts for peak pressure and intensity were 4.4 and 1.1 cm, respectively. The evolution of the different maxima with the source amplitude, and the disparity in their axial positions, is interpreted in terms of the dynamics of the nonlinear distortion process. Analytical results for the particular case of a sound beam with initial Gaussian distribution are also presented, demonstrating that the motion of peak pressure and peak intensity may occur in opposite directions.

An alternative analytical description of truncated Gaussian beams

The paper presents a simple analytical method for the diffraction and propagation of truncated or apertured Gaussian beams. The Bessel function, which appears in the Fresnel and Fraunhofer field integrals, is expanded into an approximate sum of a set of Gaussian functions. The field integrals for a truncated Gaussian beam are then analytically performed in terms of the elementary exponential or Gaussian functions only. Analytical results show that there is a good agreement between the present method and direct numerical integration. Some possible extensions of this method to other apertured beams are also suggested.

Calcium fluoride windows for high-energy chemical lasers

The development of high-energy lasers requires optical windows capable of handling megajoule beam energies without compromising the system’s performance. Calcium fluoride (CaF2) has been identified as a prime candidate for windows operating at chemical laser wavelengths due to very low bulk absorption and exceptionally small thermal lensing coefficients; it is, however, vulnerable to structural failure owing to poor mechanical strength characteristics and a large thermal stress factor. It is, therefore, essential to properly assess the ultimate potential of this material, which we attempt to do here in the following manner: (a) We assemble reliable numbers for all pertinent properties of (111)-oriented CaF2 single crystals and polycrystalline isotropic aggregates (PIAs), such as fusion-cast CaF2, which requires addressing issues relating to the elastic properties, the stress-optic coefficients, and the flexural strength. (b) We provide correct analytical expressions for evaluating the impact of pressure- and beam-induced effects on wave-front phase distortions and mechanical failure modes, taking advantage of a previous investigation [J. Appl. Phys. 98, 043103 (2005)]. (c) We perform detailed calculations on “model” windows made of either (111)CaF2 or (PIA)CaF2 that transmit optimally truncated Gaussian beams at wavelengths of 1.15 and 3.39μm, for run times such that lateral heat conduction and surface cooling can be ignored. Our main conlusions are as follows: (a) With CaF2 windows thermal lensing, as measured in terms of the Strehl ratio and on assuming coating absorptances of no more than 3×10−5, is of no consequence in the sense that catastrophic failure may occur at fluence levels way below the threshold for optical distortion. (b) Evidence of a poor Weibull shape factor (m≃3.5) degrades the design safety margins, which requires operating at peak intensities of no more than 100kW∕cm2 to achieve optimum on-target fluences. (c) Regarding the issue of (111)CaF2 vs (PIA)CaF2, we note that fusion-cast material outperforms single crystals based on the figure of merit for distortion, as well as fracture and yield strengths, but contrary to (111)-oriented material, it exhibits birefringence that may rule out its use if depolarization is of concern.