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.     

Propagation of Airy beams from right-handed material to left-handed material

Based on the ABCD matrix formalism,the propagation property of an Airy beam from right-handed material(RHM) to left-handed material(LHM) is investigated.The result shows that when the Airy beam propagates in the LHM,the intensity self-bending due to its propagation in the RHM can be compensated.In particular,if the propagation distance in the RHM is equal to that in the LHM and the refractive index of the LHM is n L =-1,the transverse intensity distribution of the Airy beam can return to its original state.     

Propagation of cylindrical vector beams in a turbulent atmosphere

<正>Based on the extended Huygens-Presnel principle,the propagation of cylindrical vector beams in a turbulent atmosphere is investigated.The intensity distribution and the polarization degree of beams on propagation are studied. It is found that the beam profile has a Gaussian shape under the influence of the atmospheric turbulence,and the polarization distribution shows a dip in the cross section as the beam propagates in the turbulent atmosphere.It is also found that the beam profile and the polarization distribution are closely related to beam parameter and atmospheric turbulence.     

Far-field vector-diffraction of off-axis parabolic mirror under oblique incidence

Based on a full vector-diffraction theory, a detailed theoretical study is carried out, aiming at providing a clear insight into the effects of different focusing and off-axis parabola parameters on far-field vector-diffraction properties of an offaxis parabolic mirror in the presence of misalignments of the incoming beam. The physical origin of these effects is also explored. The results show that the far-field intensity profile is altered by the distortion-, coma-, and astigmatism-like aberrations, which are caused by oblique incidence rather than inherent aberrations for the off-axis configuration. The radius of 90% encircled energy also increases but does not change monotonically with incident beam size increasing, or rather,it first decreases and then increases. The focal shift strongly depends on the effective focal length and oblique incidence angle, but it is almost independent of the beam size, which affects the focusing spot patterns. The intensity distribution produces a higher astigmatic image with off-axis angle increasing. Coma-like aberration starts to become dominant with beam size increasing and results in larger curved propagation trajectory. The incident polarization also affects the intensity distribution. The variation in the Strehl ratio with oblique incidence angle strongly depends on the misalignment direction and beam size as well. In addition, we find that the difference in locus between the catacaustic and the diffraction focus in the meridian is small. But the locus of the sagittal foci is obviously different from the locus of the meridian foci and the catacaustic focus. Moreover, the peak intensity of the sagittal focus is maximum, and the ratio of the peak intensity to that in the meridian plane is approximately 1.5. Understanding these effects is valuable for assessing a practical focused intensity and describing the motion of charged particles under a strong electric field in ultraintense laser–matter interaction.     

Characteristics of paraxial propagation of a super Lorentz—Gauss SLG01 mode in uniaxial crystal orthogonal to the optical axis

Analytical propagation expression of a super Lorentz-Gauss（SLG） 01 mode in uniaxial crystal orthogonal to the optical axis is derived.The SLG 01 mode propagating in uniaxial crystal orthogonal to the optical axis mainly depends on the ratio of the extraordinary refractive index to the ordinary refractive index.The SLG 01 mode propagating in uniaxial crystals becomes an astigmatic beam.The beam spot of the SLG 01 mode in the uniaxial crystal is elongated in the x-or y-direction,which is determined by the ratio of the extraordinary refractive index to the ordinary refractive index.With the increase of the deviation of the ratio of the extraordinary refractive index to the ordinary refractive index from unity,the elongation of the beam spot also augments.In different observation planes,the phase distribution of an SLG 01 mode in the uniaxial crystal takes on different shapes.With the variation of the ratio of the extraordinary refractive index to the ordinary refractive index,the phase distribution is elongated in one transversal direction and is contracted in the other perpendicular direction.This research is beneficial to the practical applications of an SLG mode.     

Characteristics of paraxial propagation of a super Lorentz-Gauss SLG₀₁ mode in uniaxial crystal orthogonal to the optical axis

Analytical propagation expression of a super Lorentz-Gauss(SLG) 01 mode in uniaxial crystal orthogonal to the optical axis is derived.The SLG 01 mode propagating in uniaxial crystal orthogonal to the optical axis mainly depends on the ratio of the extraordinary refractive index to the ordinary refractive index.The SLG 01 mode propagating in uniaxial crystals becomes an astigmatic beam.The beam spot of the SLG 01 mode in the uniaxial crystal is elongated in the x-or y-direction,which is determined by the ratio of the extraordinary refractive index to the ordinary refractive index.With the increase of the deviation of the ratio of the extraordinary refractive index to the ordinary refractive index from unity,the elongation of the beam spot also augments.In different observation planes,the phase distribution of an SLG 01 mode in the uniaxial crystal takes on different shapes.With the variation of the ratio of the extraordinary refractive index to the ordinary refractive index,the phase distribution is elongated in one transversal direction and is contracted in the other perpendicular direction.This research is beneficial to the practical applications of an SLG mode.     

Axial distribution of Gaussian beam limited by a hard-edged aperture

In this letter, the axial distribution of Gaussian beam limited by a hard-edged aperture is studied. We theoretically analyze the axial diffraction of Gaussian beam limited by a hard-edged aperture, and give the simpler formulas of the axial diffraction intensities of Gaussian beam in Fresnel diffraction field and Fraun-hofer diffraction field. The corresponding numerical calculation of axial diffraction intensity distribution of Gaussian beam with different wave waist is provided and the evolution of the diffraction distribution with the wave waist of Gaussian beam is explained. As the especial cases of the truncated Gaussian beam, the Gaussian beam in free space and the parallel light limited by the aperture are discussed too, and the system parameters of the truncated Gaussian beam which can cause it to equal to these cases are given. The theoretical results conform to the numerical analysis.     

Double-distance propagation of Gaussian beams passing through a tilted cat-eye optical lens in a turbulent atmosphere

By using the extended Huygens-Fresnel diffraction integral and the method of expanding the aperture function into a finite sum of complex Gaussian functions, an approximate analytical formula of the double-distance propagation for Gaussian beam passing through a tilted cat-eye optical lens and going back along the entrance way in a turbulent atmosphere has been derived. Through numerical calculation, the effects of incidence angle, propagation distance, and structure constant on the propagation properties of a Gaussian beam in a turbulent atmosphere are studied. It is found that the incidence angle creates an unsymmetrical average intensity distribution pattern, while the propagation distance and the structure constant can each create a smooth and symmetrical average intensity distribution pattern. The average intensity peak gradually deviates from the centre, and the central average intensity value decreases quickly with the increase in incidence angle, while a larger structure constant can bring the average intensity peak back to the centre.     

Transverse Inhomogeneous Carrier-Envelope Phase Distribution of Idler Generated through Difference-Frequency-Generation

The transverse inhomogeneous carrier-envelope phase （CEP） distribution of idler generated through difference-frequency-generation （DFG） in quadratic nonlinear crystals is theoretically studied. In practical CEP stabilized DFG setups, the pump and the signal are usually Gaussian beams with non-uniform intensity distribution. Since the idler CEP is dependent on gain, this non-uniform intensity distribution leads to inhomogeneous gain across the aperture of the idler beam, resulting in a varying transverse idler CEP. Simulation results show that in practical settings, in the high-gain regime, transverse inhomogeneous CEP can be much smaller compared with π/2. However, when gain on the propagation axis reaches saturation, CEP difference can well exceed π/2.     

Propagation of Gauss--Bessel beams in turbulent atmosphere

This paper studies the propagation properties of Gauss--Bessel beams in a turbulent atmosphere. Based on the extended Huygens--Fresnel principle, it derives the intensity distribution expression for such beams propagating in a turbulent atmosphere. Then the influence of turbulence and source beam parameters on the beam propagation is studied in great detail. It finds that the intensity distribution of Gauss--Bessel beams will change into Gaussian profile in a turbulent atmosphere, and that stronger turbulence and smaller topological charges will lead to a faster changing.     

Hollow Gaussian beams in strongly nonlocal nonlinear media

The propagation of hollow Gaussian beams in strongly nonlocal nonlinear media is studied in detail.Two analytical expressions are derived.For hollow Gaussian beams,the intensity distribution always evolves periodically.However the second-order moment beam width can keep invariant during propagation if the input power is equal to the critical power.The interaction of two hollow Gaussian beams and the vortical hollow Gaussian beams are also discussed.The vortical hollow Gaussian beams with an appropriate topological charge can keep their shapes invariant during propagation.     

Paraxial propagation of cosh-Airy vortex beams in chiral medium

Propagation dynamics of the cosh-Airy vortex(CAiV) beams in a chiral medium is investigated analytically with Huygens–Fresnel diffraction integral formula. The results show that the CAiV beams are split into the left circularly polarized vortex(LCPV) beams and the right circularly polarized vortex(RCPV) beams with different propagation trajectories in the chiral medium. We mainly investigate the effect of the cosh parameter on the propagation process of the CAiV beams.The propagation characteristics, including intensity distribution, propagation trajectory, peak intensity, main lobe's intensity, Poynting vector, and angular momentum are discussed in detail. We find that the cosh parameter affects the intensity distribution of the CAiV beams but not its propagation trajectory. As the cosh parameter increases, the distribution areas of the LCPV and RCPV beams become wider, and the side lobe's intensity and peak intensity become larger. Besides, the main lobe's intensity of the LCPV and RCPV beams increase with the increase of the cosh parameter at a farther propagation distance, which is confirmed by the variation trend of the Poynting vector. It is significant that we can vary the cosh parameter to control the intensity distribution, main lobe's intensity, and peak intensity of the CAiV beams without changing the propagation trajectory. Our results may provide some support for applications of the CAiV beams in optical micromanipulation.     

Spectral broadening induced by intense ultra-short pulse in 4H–SiC crystals

We report the observation of spectral broadening induced by 200 femtosecond laser pulses with the repetition rate of 1 kHz at the wavelength of 532 nm in semi-insulating 4H–SiC single crystals.It is demonstrated that the full width at half maximum of output spectrum increases linearly with the light propagation length and the peak power density,reaching a maximum 870 cm~(-1)on a crystal of 19 mm long under an incident laser with a peak power density of 60.1 GW/cm~2.Such spectral broadening can be well explained by the self-phase modulation model which correlates time-dependent phase change of pulses to intensity-dependent refractive index.The nonlinear refractive index n_2 is estimated to be1.88×10~(-15)cm~2/W.The intensity-dependent refractive index is probably due to both the nonlinear optical polarizability of the bound electrons and the increase of free electrons induced by the two-photon absorption process.Super continuum spectra could arise as crystals are long enough to induce the self-focusing effect.The results show that SiC crystals may find applications in spectral broadening of high power lasers.     

Z-scan analysis of high-order nonlinear refraction effect induced by using elliptic Gaussian beam

The irradiance of an elliptic Gaussian beam that is high enough to excite high-order nonlinear refraction effect is used to calculate the normalized on-axis transmittance function in the z-scan technique by introducing complex beam parameters which make the calculation simpler. The transmittance formula is applied to the first-, first two-, and first three-order nonlinearities. Numerical evaluation shows that the symmetry no longer holds when using an elliptic Gaussian beam instead of a circular Gaussian beam. A distortion is observed in the central part of the curve, which decreases as ellipticity increases. Moreover, the variation of the normalized peak-valley difference decreases as ellipticity decreases.     

Nonparaxial Dark-Hollow Gaussian Beams

The concept of nonparaxial dark-hollow Gaussian beams （DHGBs） is introduced. By using the Rayleigh-Sommerfeld diffraction integral, the analytical propagation equation of DHGBs in free space is derived. The on-axis intensity, far-field equation and, in particular, paraxial expressions are given and treated as special cases of our result. It is shown that the parameter f = 1/κω0 with κ being the wave number and ω0 being the waist width determines the nonparaxiality of DHGBs. However, the parameter range, within which the paraxial approach is valid, depends on the propagation distance. The beam order affects the beam profile and position of maximum on-axis intensity.     

Properties of multi-Gaussian Schell-model beams carrying an edge dislocation propagating in oceanic turbulence

Based on the theory of coherence, the model of multi-Gaussian Schell-model(MGSM) beams carrying an edge dislocation generated by the MGSM source is introduced. The analytical cross-spectral density of MGSM beams carrying an edge dislocation propagating in oceanic turbulence is derived, and used to study the evolution properties of the MGSM beams carrying an edge dislocation. The results indicate that the MGSM beam carrying an edge dislocation propagating in oceanic turbulence will evolve from the profile with two intensity peaks into a flat-topped beam caused by the MGSM source, and the beam will evolve into the Gaussian-like beam due to the influences of oceanic turbulence in the far field.As the propagation distance increases, the MGSM beam carrying an edge dislocation propagating in oceanic turbulence with the larger rate of dissipation of mean-squared temperature(χT) and ratio of temperature to salinity contribution to the refractive index spectrum(?) or the smaller rate of dissipation of kinetic energy per unit mass of fluid(ε) evolves into the flat-topped beam or a Gaussian beam faster.     

Phase Control of Few-Cycle Coherent Terahertz Radiation Pulses

We report a method to control the carrier-envelope phase shift in nearly single-cycle coherent terahertz (THz) radiation pulses, in which we used Kerr effect and two-photon absorption in the generated crystals due to the pump beam peak intensity. Dependence of relative carrier-envelope phase shift of THz pulses on the pump beam intensity in the ZnTe crystal was measured experimentally. Under our experimental condition, we measured that maximum relative phase shift is 0.29 radians.     

Beam divergence effects on high power optical parametric oscillation

The beam divergence effects of the input pump laser on a high power nanosecond optical parametric oscillator （OPO） have been numerically simulated. The OPO conversion efficiency is affected due to the angular deviation of real laser beams from ideal phase matching conditions. Our theoretical model is based on the decomposition of the Gaussian beam and assumes each component has a single deviation angle and thus a particular wave vector mismatch. We take into account the variable intensity profile in the spatial and temporal domains of the Gaussian beam, the pump depletion effects for large-signal processes as well as the oscillatory effects of the three waves. Two nonlinear crystals β-BaB2O4 （BBO） and LiB305 （LBO） have been investigated in detail. The results indicate that the degree of beam divergence strongly influences the maximum pump intensity, optimum crystal length and OPO conversion efficiency. The impact of beam divergence is much more severe in the case of critical phase-matching for BBO than in the case of non-critical phase-matching for LBO. The results provide a way to choose the optimum parameters for a high power ns OPO such as the nonlinear material, the crystal length and the pump intensity, etc. Good agreement is obtained with our experimental results.     

Off-Axis Astigmatic Gaussian Beam Combination Beyond the Paraxial Approximation

Taking the off-axis astigmatic Gaussian beam combination as an example, the beam-combination concept is extended to the nonparaxial regime. The closed-form propagation expressions for coherent and incoherent combinations of nonparaxial off-axis astigmatic Gaussian beams with rectangular geometry are derived and illustrated with numerical examples. It is shown that the intensity distributions of the resulting beam depend on the combination scheme and beam parameters in general, and in the paraxial approximation （i.e., for the small f-parameter） our results reduce to the paraxial ones.     

Propagation of the off-axis superposition of partially coherent beams through atmospheric turbulence

The propagation properties of the off-axis superposition of partially coherent beams through atmospheric turbulence and their beam quality in terms of the mean-squared beam width w(z) and the power in the bucket (PIB) are studied in detail, where the effects of partial coherence, off-axis beam superposition and atmospheric turbulence are considered. The analytical expressions for the intensity, the beam width and the PIB are derived, and illustrative examples are given numerically. It is shown that the maximum intensity I_max and the PIB decrease and w(z) increases as the refraction index structure constant C_n² increases. Therefore, the turbulence results in a degradation of the beam quality. However, the resulting partially coherent beam with a smaller value of spatial correlation parameter γ and larger values of separate distance x_d and beam number M is less affected by the turbulence than that with a larger value of γ and smaller values of x_d and M. The main results obtained in this paper are explained physically.