Split-step Beam Propagation Method Research Articles

Background-oriented schlieren experimental and simulation study of the effect of sidewalls on optical turbulence anisotropy in a thermal convection turbulence simulator

The anisotropic optical structure characteristics within the Rayleigh-Bénard (RB) turbulence simulator have been investigated through experiment and simulation. We collected data using background-oriented schlieren technology and extracted displacements using optical flow algorithms. The wavefront phase can be recovered using the conjugate gradient iterative algorithm. We analyzed the optical structural characteristics within the turbulence simulator under different temperature differences. It is observed that with an increased temperature difference, the spatial correlation of the refractive index field decreases at an accelerated rate as the separation distance increases, and the degree of anisotropy characterized by the phase structure function increases. Large eddy simulation and split-step beam propagation method complement experimental results. Numerical simulations demonstrate consistency with experimental data. The results show that the von Karman power spectral density fits the simulation and experimental phase structure function very well relative to the Kolmogorov power spectral density, which demonstrates that the simulation system combining dynamics simulation and optical simulation can achieve power spectral characteristics that are consistent with reality. The experimental and simulation results are crucial for understanding the flow field's optical structural properties inside the turbulence simulator. They will help to construct a suitable turbulence simulator for optical engineering applications and establish a rational scheme for analyzing its internal optical structural properties.

Does the degree of polarization of vector beams remain unchanged on atmospheric propagation?

All roads lead to Rome. In this article we propose a novel theoretical framework to demonstrate vector beams whose degree of polarization does not change on atmospheric propagation. Inspired by the Fresnel equations, we derive the reflected and refracted field of vector beams propagating through a phase screen by employing the continuity of electromagnetic field. We generalize the conventional split-step beam propagation method by considering the vectorial properties in the vacuum diffraction and the refractive properties of a single phase screen. Based on this vectorial propagation model, we extensively calculate the change of degree of polarization (DOP) of vector beams under different beam parameters and turbulence parameters both in free-space and satellite-mediated links. Our result is that whatever in the free-space or satellite-mediated regime, the change of DOP mainly fluctuates around the order of 10-13 to 10-6, which is almost negligible.

Limitations of homogeneous and segmented single-crystal compact TEM00-mode Ho3+:YAG laser resonators

We present investigations on output power limitations of near-diffraction-limited compact hbox {Ho}^{3+}:YAG laser resonators employing a homogeneous or segmented laser crystal. An approach for designing a segmented crystal is presented. Maximum output powers of {57.6},hbox {W} and {51.9},hbox {W} are reached with the homogeneous and segmented crystal, respectively, resulting in pulse energies of {1.14},hbox {mJ} and {1.04},hbox {mJ} at a repetition rate of {50},hbox {kHz} in Q-switched operation. Interferometric experiments are conducted to derive the radial temperature profile for both crystals. Simulations based on a split-step beam propagation method are used to model the longitudinal temperature gradient in both crystals.

Continuous-wave terahertz holographic diffraction tomography based on multiple scattering forward model

Terahertz waves exhibit the great potential applications due to their particular properties. Also, diffraction tomography (DT) is an effective way to detect the internal three-dimensional structure of the sample. The terahertz DT (THz-DT) is an attractive and growing technique. However, when the samples to be measured are with complex internal structures or high refractive index contrast, the multiple scattering effect is a challenging problem, and even becomes more serious because of the large wavelength of the THz wave, which makes the first-order Born/Rytov approximation hardly to be satisfied and brings severe distortions to the reconstruction results. In this paper, the continuous-wave THz-DT based on digital holography is presented including the multiple scattering effect, which can further improve the reconstruction fidelity. The multiple scattering forward model is introduced to describe the propagation process of the THz wave inside the sample, and the modified split-step beam propagation method is presented, which evolves from its original version as one of the main forward models for the visible optical DT. Both the reconstruction accuracy and computation efficiency are achieved higher. Then, the simulations and experiments of THz-DT are performed, and three-dimensional reconstruction results with more fine structures are obtained by applying the correspondingly developed optimization-based nonlinear inverse scattering algorithm. To our best knowledge, it is the first time to apply the multiple scattering forward model into THz DT to improve the reconstruction fidelity. Besides, the capacity of the proposed method to enable sparse-angle reconstruction is also demonstrated by reducing the number of rotation angles, which is particularly important for real-time measurement applications.

Simulation of spatial coherence of laser echo light field from random rough surface in turbulent atmosphere

According to the generalized Huygens-Fresnel principle, we derive the analytical formula for the complex degree of coherence of the echo light field under the von Karman atmospheric turbulence spectrum condition. Based on split-step beam propagation method of the turbulent phase screen and the target surface model, the fold pass propagation simulation of the laser in the turbulent atmosphere is realized. The dynamic speckle characteristics on the image plane are consistent with the experimental phenomenon. Firstly, the simulation results of the complex degree of coherence and phase structure function of the mirrored reflection echo light field are compared with the theoretical values, which verifies the correctness of the algorithm. Based on this, the complex degree of coherence of the echo light field reflected by the optical rough surface is calculated and analyzed. The results show that on a double-path turbulent flow path of 1.1 km, in other words, it transmits 2.2 km in unfolded mode, the spatial coherence of the echo light field is very sensitive to the root mean square value of height. When the root-mean-square value of height is close to the wavelength, the coherence is seriously degraded. When the correlation length of the target surface is much larger than the atmospheric coherence length, the coherence length of the echo light field is relatively close to the set spatial coherence length. When the correlation length of the target surface is close to the atmospheric coherence length, the influence of the rough surface of the target on the beam coherence cannot be ignored. When the correlation length of the target surface is much smaller than the atmospheric coherence length, the target surface characteristics have a dominant influence on the echo coherence, the spatial coherence of the light field is seriously degraded, and the echo is close to incoherent light. Considering the smooth target reflection surface, the greater the strength of turbulence, the faster the complex coherence decreases with space. The atmospheric coherence diameter <inline-formula><tex-math id="M1">\begin{document}${r_0}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212420_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212420_M1.png"/></alternatives></inline-formula> can be calculated further according to the complex degree of coherence. For the Pearson correlation coefficient the simulation value and theoretical value are both 0.998, which indicates that the atmospheric coherence diameter calculated by the complex degree of coherence has a high correlation with the theoretical value. This research provides a theoretical basis for the coherent detection scheme of echoes from rough surfaces in the turbulent atmosphere. The simulation algorithm extracts the target surface features by analyzing the variation of the complex coherence of laser echo signals in the turbulent atmosphere with the spatial distance, and also provides a method of using the known target surface to obtain path turbulence information.

Spatial evolution leading to multiple filament formation of higher order super Gaussian beam in bulk medium at input power P >>Pcr

Multiple filamentation in higher order super Gaussian laser beam propagating in chalcogenide glass is investigated by numerically solving nonlinear Schrödinger equation with split step beam propagation method at input power range of 40 – 160 Pcr and compared to the process in super Gaussian beam. The effect of raising the beam order on the deterministic nature of multiple filaments, threshold power of filamentation, length of filamentation and the number of filaments formed are analysed. The results indicate that within a range of beam order and input power, the process of multiple filamentation in super Gaussian beam may be controlled through beam order. The study further exemplifies the importance of deterministic filaments and the ‘role of beam order as a tool’ in control of self-guided laser beam properties in filamentation process.

Pilot beam-assisted adaptive compensation for atmospheric turbulence in free-space optical transmission of beams carrying orbital angular momentum

One of the major challenges for free-space optical (FSO) communication through atmosphere is the turbulence caused by the random variations in temperature and convective motion of air. It induces fluctuation in intensity and phase of the propagating optical signal and impairs the transmission quality. The FSO communication using beams carrying orbital angular momentum would be vulnerable to such fluctuation since they have a unique spatial phase profile of the helical phase front. In order to overcome the issue, we propose a pilot beam-assisted adaptive compensation method for the turbulence-induced phase fluctuation by using spatial light modulators and a wavefront sensor. The proposed method is a potential candidate for real-time compensation due to its fast operation without algorithmic converging process. The propagation of Laguerre–Gaussian beam through atmospheric turbulence is numerically analyzed based on the modified von Karman model and split-step beam propagation method. The simulation results reveal that the compensation performances are considerably improved by using the pilot beam which has the same azimuthal index as that of the data beam.

Multiple filamentation and control of properties of self-guided elliptical Gaussian laser beam

The spatial evolution of an elliptical Gaussian beam in fluoride glass (Kerr medium) is studied using nonlinear Schrodinger equation solved by split-step beam propagation method. Solutions obtained by numerical simulation are analysed to study the effect of variation of input beam ellipticity on the process of multiple filamentation in elliptical Gaussian beam, where emphasis is on the properties of the self-guided Gaussian laser beam by externally induced input beam astigmatism. Controlling the properties of self-guided laser beam is a necessity in generation of tailored filaments.

Spatio-temporal modulation instability of surface plasmon polaritons in graphene-dielectric heterostructure

Using the Jacobi Elliptic Functions, an analytical solution is developed for the nonlinear amplitude equation of Surface Plasmon Polaritons (SPPs) in a graphene-dielectric waveguide. It is shown that the field localization of SPPs coupled with TM polarized terahertz light can be enhanced if the nonlinearity is increased. On the side, a numerical solution based on Split Step Beam Propagation Method (SSBPM) suggests that spatial Modulation Instability (MI) can be dominant. Accordingly, larger nonlinearity leads to the generation of discrete plasmon solitons rather than the diffracted profile resulted for the modest nonlinearity. Adding then the temporal variations to the nonlinear amplitude equation and solving numerically by predictor –corrector method, it is revealed that temporal MI appears as ultrashort pulse trains with multi-periodic behavior. Evoking the similarity with a laser cavity, the waveguide can be assumed as a spasing system - assuming the large nonlinearity regime-in which if the coupling depth is raised, the character of MI will be changed from the convective to absolute and the amplitude of SPPs will grow fast. The procedure can be even chaotic and unpredictable. This spasing system is suitable for applications of the electro-optical circuitry, optical amplification, optical communication and biomedical sensing through which the low power consumption and non-destructivity are of important traits.

Experimental and numerical study of underwater beam propagation in a Rayleigh-Bénard turbulence tank.

The propagation of a laser beam through Rayleigh-Bénard (RB) turbulence is investigated experimentally and by way of numerical simulation. For the experimental part, a focused laser beam transversed a 5 m×0.5 m×0.5 m water filled tank lengthwise. The tank is heated from the bottom and cooled from the top to produce convective RB turbulence. The effect of the turbulence on the beam is recorded on the exit of the beam from the tank. From the centroid motion of the beam, the index of refraction structure constant Cn2 is determined. For the numerical efforts RB turbulence is simulated for a tank of the same geometry. The simulated temperature fields are converted to the index of refraction distributions, and Cn2 is extracted from the index of refraction structure functions, as well as from the simulated beam wander. To model the effect on beam propagation, the simulated index of refraction fields are converted to discrete index of refraction phase screens. These phase screens are then used in a split-step beam propagation method to investigate the effect of the turbulence on a laser beam. The beam wander as well as the index of refraction structure parameter Cn2 determined from the experiment and simulation are compared and found to be in good agreement.

Multiple filamentation and control of properties of self-guided super Gaussian laser beam

Spatial evolution of super Gaussian beam in Chalcogenide glass (Kerr medium) is investigated for input power in the range 15Pcr to 180Pcr using Nonlinear Schrödinger Equation. Solutions are obtained by numerical simulation with split step beam propagation method and are analysed to study the effect of input noise level on the process of multiple filamentation in super Gaussian beam, where emphasis is on the control of properties of the self-guided super Gaussian laser beam by externally induced input noise. Control of properties of self–guided laser beam is a necessity in generation of tailored filaments.

Pulse Propagation with Self-Phase Modulation in Nonlinear Chiral Fiber and Its Applications**Supported by the National Natural Science Foundation of China under Grant No 60977032, and the Program for Innovation Research of Science of Harbin Institute of Technology (PIRS-HIT) under Grant No T201407.

From Maxwell's equations and Post's formalism, a generalized chiral nonlinear Schrödinger equation (CNLSE) is obtained for the nonlinear chiral fiber. This equation governs light transmission through a dispersive nonlinear chiral fiber with joint action of chirality in linear and nonlinear ways. The generalized CNLSE shows a modulation of chirality to the effect of attenuation and nonlinearity compared with the case for a conventional fiber. Simulations based on the split-step beam propagation method reveal the role of nonlinearity with cooperation to chirality playing in the pulse evolution. By adjusting its strength the role of chirality in forming solitons is demonstrated for a given circularly polarized component. The application of nonlinear optical rotation is also discussed in an all-optical switch.

Optimal distribution of the optical intensity within a laser beam for optical wireless communications

This study is focused on analysing the optimal distribution of optical intensity within a radiated laser beam at a transmitter plane which is propagated through free space as well as through a turbulent atmosphere. To analyse the propagation of an optical wave through atmospheric turbulence, the simulation, based on the split-step beam propagation method, was utilised. The aim of this study is to determine the optimal parameters for a flattened Gaussian beam at the transmitter plane. As a result, the scintillation index should be reduced. There are a number of theoretical studies aimed at propagating the flattened beam despite the lack of experimental work in this area. Therefore the techniques for generating the flattened Gaussian beam will also be mentioned.

Beam propagation analysis on thickness measurements in quantitative phase microscopy

The two-dimensional thickness profile of a phase object can be measured by phase microscopy by assuming that the light passes straight through the sample such that the measured phase profile is proportional to the thickness of the sample. However, any non-uniform index structure in a sample bends the straight light path by refraction and diffracts the non-uniform transverse phase structure of the wavefront along the propagation path within a sample. We investigated the consequence of these two effects within a phase object using a split-step beam propagation method that considers beam paths through a 3-μm-diameter bead sample. Our simulation results show that the phase profile of light just after passing through a sample differs significantly from an ideal phase profile. We verified these simulation results by comparing them with experimental data obtained with a Mach–Zehnder interferometer.

Theoretical and experimental study on self-accelerating Bessel-like Hermite-Gaussian beams

Phase modulation is an important method of designing accelerating optical beams. In this paper, we present new self-accelerating non-diffracting Bessel-like Hermite-Gaussian beams based on our previous research on Bessel-like beams. The evolutions of the beams along different trajectories are studied numerically and experimentally. These beams are designed by modulating the phase of the initial Hermite-Gaussian beams. With the split-step beam propagation method, we show numerically that (0,1), (1,0), (1,1) and (1,2) modes of the Bessel-like Hermite-Gaussian beams can propagate along different predesigned trajectories. With the computer-generated hologram and spatial light modulator, we observe the propagating behaviors of the Bessel-like Hermite-Gaussian beams along different trajectories, including parabolic, hyperbolic, hyperbolic secant and 3D trajectories. Experimental results show good agreement with the theoretical prediction. It is also demonstrated that the peculiar profile of Bessel-like Hermite-Gaussian beams exhibits nondiffracting and self-healing properties propagating along the tunable trajectories. These beams generalize the concept of Bessel-like beams, suggesting that more exotic optical beams can be obtained.

Generation of autofocusing beams with multi-Airy beams

Thus far, focusing a single beam and combining multi-optical Gaussian beams have become two major methods of locally enhancing output laser intensity. In this paper, we propose a method of generating free space auto-focusing beams by combining multiple one-dimensional Airy beams. By employing split-step beam propagation method, auto-focusing beams generated by four and eight one-dimensional Airy beams are numerically simulated, and their intensity distributions are displayed from both transverse plane and side-view propagation. By utilizing a spatial light modulator and computer generated phase masks, these auto-focusing beams are experimentally demonstrated. Our experimental results accord well with the numerical simulations.

Dynamical stable cavity of side-pumped thin-disk laser with slanted sides

In this article, we use the results of finite-element analysis (FEA) of temperature distribution, deformation, and stress to provide a full three-dimensional simulation of a dynamically stable cavity by propagating wavefront into hot, thermally deformed Yb:YAG/YAG thin-disk laser, using the fast Fourier transform (FFT) split-step beam propagation method (BPM). The wave optics computation therefore delivers realistic results for important features of a laser-like intensity, phase profile, and resonator eigenvalues and higher order-eigenmodes of the laser beam.

Iterative nonlinear beam propagation using Hamiltonian ray tracing and Wigner distribution function

We present an iterative method for simulating beam propagation in nonlinear media using Hamiltonian ray tracing. The Wigner distribution function of the input beam is computed at the entrance plane and is used as the initial condition for solving the Hamiltonian equations. Examples are given for the study of periodic self-focusing, spatial solitons, and Gaussian-Schell model in Kerr-effect media. Simulation results show good agreement with the split-step beam propagation method. The main advantage of ray tracing, even in the nonlinear case, is that ray diagrams are intuitive and easy to interpret in terms of traditional optical engineering terms, such as aberrations, ray-intercept plots, etc.

Three-dimensional finite difference split-step nonparaxial beam propagation method: new method for splitting of operators

We present a new method for splitting of operators in the three-dimensional finite difference split-step nonparaxial beam propagation method. The method increases the accuracy and the efficiency in terms of speed and memory requirements for three-dimensional wide-angle beam propagation. It also makes the application of the perfectly matched layer (PML) boundary condition very simple.

Solving Soliton Perturbation Problems by Introducing Rayleigh's Dissipation Function

We solve soliton perturbation problem in nonlinear optical system by introducing Rayleigh's dissipation function in the framework of variational approach. The adopted process facilitates variational approach to be applied on dissipative system where the Lagrangian and Hamiltonian are difficult to form. Exploiting the idea, loss and filtering problems are evaluated with convincing results. Considering other perturbing terms like two soliton interactions, intrapulse Raman scattering, self-steepening, and two-photon absorption in extended nonlinear Schrodinger equation, Rayleigh's dissipation function is configured intuitively so that the generalized Euler-Lagrange equation converges to the related governing equation of the pulse propagation. The process evolves a set of differential equations exploiting the dynamics of different pulse parameters under the influence of perturbations. The obtained analytical results are verified with generalized Kantorovich approach and compared with previous reported results. Numerical simulations based on the split-step beam propagation method are employed to calculate the pulse evolution parameters and the derived results are found to be corroborated well with the analytical predictions.