Two-frequency Mutual Coherence Function Research Articles

Time-domain analysis of multiple scattering effects on the radar cross section (RCS) of objects in a random medium

ABSTRACT This paper presents a theory of the time-domain radar cross section (RCS) of objects in discrete random media. The time-domain formula is obtained by applying the inverse Fourier transform of the two-frequency mutual coherence function (MCF) which is derived from both the second-order Rytov approximation and the strong fluctuation theory. This theory includes the backscattering enhancement and the time-domain shower curtain effect that are not normally considered in the conventional theory. Numerical examples of the time-domain RCS of a conducting square plate in a discrete random medium characterized by the Gaussian phase function are shown to highlight the random media effects on the time-domain waveforms including time delay and pulse broadening in terms of optical depth and random medium location. Numerical results show that both pulse arrival time and pulse broadening increase significantly when the random media is placed far from the target. This degradation of the image quality, known as the shower curtain effect, can be explained by the characteristics of the incoherent component.

Temporal pulse broadening due to dispersion and weak turbulence in FSO communications

In this paper, a novel analytical expression for the temporal pulse broadening caused by atmospheric dispersion and weak turbulence is derived. We developed an on-axis two-frequency mutual coherence function (MCF) for dispersive pulse propagating in weak atmospheric turbulence. Based on the derived temporal pulse broadening expression we formulated the signal-to-noise ratio (SNR) for Gaussian pulse propagating along a horizontal path, and we deducted an analytical expression for the bit error rate (BER) caused by dispersion and weak atmospheric turbulence effect on terrestrial free-space optical transmission system. Specific examples are presented for the pulse broadening due to weak turbulence and atmospheric dispersion. The results show that the temporal pulse broadening increases with propagation distance and decreases with low outer scale turbulence. The analytical BER results validated by using Monte Carlo simulations for on-off keying (OOK) intensity modulation/direct detection (IM/DD) show that BER due to pulse broadening is very significant when the propagation distance is increased. However, longer propagation distance with minimum bit error rate can be achieved by increasing the input power.

Modified biological spectrum and SNR of Laguerre-Gaussian pulsed beams with orbital angular momentum in turbulent tissue.

We propose a modified biological spectrum that contains both short length-scale and long length-scale to study light propagation through turbulent biological tissue. Based on the two-scale modified biological spectrum, we derive an analytic expression of the two-frequency mutual coherence function of Laguerre-Gaussian pulsed beam and establish a model of the signal-to-noise ratio (SNR) of Laguerre-Gaussian pulsed beam carrying orbital angular momentum in turbulent biological tissue. The results show that the modified biological spectrum agrees well with experimental results. In addition, the structural length-scale of biological tissue has a significant influence on the bandwidths and SNR of orbital angular momentum states. This work provides theoretical preparation for more accurately medical diagnosis and optical imaging.

Study of Pulse Scattering from Two-Dimensional Non-Gaussian Rough Ocean Surfaces

In this paper the analytical expression of two-frequency mutual coherence function of the scattered wave from two-dimensional non-Gaussian rough ocean surfaces is derived first by the Kirchhoff approximation. The Gram-Charlier distribution instead of the conventional Gaussian distribution is utilized in carrying out the ensemble average operation in order to represent the asymmetric characteristics of ocean surfaces. Then the scattered pulse for arbitrary incident pulse is derived utilizing the two-frequency mutual coherence function for wide-sense stationary uncorrelated scattering channel. Finally the two-frequency mutual coherence function and the scattered pulse with a Gaussian modulated or rectangular incident pulse from two-dimensional non-Gaussian rough ocean surfaces for different wind speed are numerically simulated, and the pulse broadening phenomenon is discussed.

Study of Two-Frequency Mutual Coherence Function From Two-Dimensional Rough Surfaces

Study of pulse scattering from two-dimensional rough surfaces is of great theoretical and engineering significance. In this paper the analytical expression of two-frequency mutual coherence function of the scattered wave from two-dimensional rough surfaces is derived with the integral equation method, which includes both the single scattering and the multiple scattering and therefore is applicable to surfaces with small or large-scale roughness. Then the numerical simulation of the two-frequency mutual coherence function from two-dimensional surfaces with Gaussian distribution is performed, and the results are compared with those with Kirchhoff approximation.

Probability distribution of the orbital angular momentum mode of the ultrashort Laguerre-Gaussian pulsed beam propagation in oceanic turbulence

Abstract By introducing the orbital angular momentum (OAM) eigenfunctions of the pulsed beam and two-frequency mutual coherence function in oceanic turbulence, we establish the received probability distribution of OAM modes of the ultrashort Laguerre-Gaussian pulsed beam propagating through oceanic turbulence. Our results show that the higher rate of dissipation of kinetic energy per unit mass of fluid, larger inner scale, smaller ratio of temperature and salinity contributions to the refractive index spectrum, or lower dissipation rate of the mean-squared temperature will cause the smaller pulse duration by its full-width half-maximum. The pulse duration has a weak dependence on the OAM quantum number. In addition, the smaller OAM quantum number and the larger transmission distance cause the smaller pulse delay. Our results will be useful for vortex pulsed beam in the optical ocean communication and remote sensing.

Two-Frequency Coherence Function for the Field of a Wave Propagating Through a Multiscale Randomly Inhomogeneous Medium

We study the possibilities of integral representation for the two-frequency mutual coherence function of the wave field in a randomly inhomogeneous ionosphere. The integral representation was obtained using the Double Weighted Fourier Transform (DWFT) method. We point out that the conditions of validity of the geometrical-optics approximation for frequency coherence are weaker than the same condition for individual samples. Examples of calculation of the frequency coherence functions for waves in the ionospheric plasma with the irregularities described by the Gaussian spectrum and Shkarofsky’s model are given. Simulation results show that diffraction effects reduce the width of the frequency coherence function. The capabilities of the methods for spatial processing of the wave field and its two-frequency mutual coherence function, which eliminate these effects through the Fresnel and DWFT inversions, are examined.

Influence of oceanic turbulence on propagation of the Gaussian pulsed X wave carrying orbital angular momentum

Abstract By developing the two-frequency mutual coherence function of the vortex wave, we derive the analytical expression of the coherence function of the Gaussian pulsed X wave carrying orbital angular momentum propagating in the anisotropic turbulence of ocean. Employing this coherence function, we establish the received probability density of orbital angular momentum modes including the time factor. Applying this probability density, we investigate the influences of turbulence and beam parameters on the received signal probability. Our results show that the anisotropic oceanic turbulence with a smaller ratio of temperature and salinity contributions to the refractive index spectrum, higher rate of dissipation of kinetic energy per unit mass of fluid, larger inner scale, or lower dissipation rate of the mean-squared temperature causes the larger signal probability density; the Gaussian pulsed X wave with large sources transverse size, small Bessel cone angle, low orbital angular momentum quantum number or broad initial half-pulse width has stronger resistance to oceanic turbulent interference. This work provides theoretical preparation for the implementation of the spatiotemporal communication with ultrafast vortex pulse.

Cross-frequency coherence and pulse propagation in a turbulent atmosphere.

Cross-frequency coherence of acoustic signals in a turbulent atmosphere is an important consideration for source localization with acoustic sensor arrays and for remote sensing of the atmosphere with sodars and tomography techniques. This paper takes as a starting point recently derived, closed-form equations for the spatial-temporal correlation function of a broadband acoustic signal propagating in a turbulent atmosphere with coupled spatial-temporal fluctuations in temperature and wind velocity. This theory is employed to calculate, based on the Rytov approximation, the two-point, two-time, two-frequency mutual coherence function of plane and spherical waves in the weak scattering regime. The cross-frequency coherence for these waveforms is then obtained and compared with that in the geometrical acoustics approximation. The coherence bandwidth is calculated and analyzed for typical meteorological regimes of the atmospheric surface layer and parameters of sound propagation. The results obtained are compared with available experimental data. The cross-frequency coherence is also used to study the effect of atmospheric turbulence on the mean intensity of an acoustic pulse propagating in a turbulent atmosphere.

Broadening of ultra-short pulses propagating through weak-to-strong oceanic turbulence

In this paper, the new approach of correlation function of the complex phase perturbed by oceanic turbulence is shown. Based on this new approach, the general formula of the two-frequency mutual coherence function (MCF) of ultra-short pulses in oceanic turbulence is derived. Using a temporal moments approach and combining with this new formula for the MCF, the analytical expression for the pulse width is deduced. Besides, the quantity of Rytov variance σR2 in oceanic turbulence is obtained, which is widely used as a measure of the strength of turbulence. In particular, the on-axis relative pulse broadening and turbulent effective coefficient of ultra-short pulses (i.e., femtosecond–picosecond regime) propagating through oceanic turbulence are investigated.

Pulse broadening and beam spread of polarized laser pulse beam on slant path in turbulence atmospheric

This paper is based on the extended Huygens – Fresnel principle, the two-frequency mutual coherence function and the Rytov approximation method. We have derived the novel expressions for the pulse broadening and beam effective spread radius of the elliptically polarized Gaussian pulse beam propagating through atmospheric turbulence along a slant path. Meanwhile the effect of weak optical turbulence on the pulse broadening and the beam spread in a slant path are analyzed. The results show that the pulse broadening of the elliptically polarized Gaussian pulse beam on slant path turbulence atmospheric depends on the zenith angle η and receiver height H. The inner scale factor β appears to have an insignificant effect on the pulse broadening. Beam spread is mainly depends on the initial waist radius w0, wavelength λ and polarization angle φ. With the increasing of propagation distance L, beam effective spread radius w(L) decreases as w0 increases, and it increases with λ, in addition, w(L) increases as φ increases.

Two-frequency mutual coherence function for Gaussian-beam pulses propagating along a horizontal path in weak anisotropic atmospheric turbulence.

A theoretical formulation of the spherical-wave two-frequency mutual coherence function (MCF) for a propagation path characterized by a complex ABCD matrix with anisotropic atmospheric turbulence existing somewhere is developed. A specialization of this formulation leads to an expression for the two-frequency MCF of an equivalent pulsed Gaussian beam propagating in weak anisotropic atmospheric turbulence along a horizontal line-of-sight path; relevant closed-form analytical solutions under both near- and far-field conditions are obtained. The small- and large-scale solutions for both the plane- and spherical-wave spatial-coherence radii in either horizontal or vertical direction are derived. Analysis shows that the formula for the on-axis two-frequency MCF of a pulsed Gaussian beam under the weak-turbulence condition in both the near- and far-field regions is distinguished from that applicable in the strong-turbulence limit only by whether the turbulence-induced beam broadening can be thought of as negligible. Under both the near- and far-field conditions, the turbulence-induced increment of the mean-square temporal-pulse half-width is proportional to the effective anisotropy factor of turbulence. The MCF becomes statistically anisotropic due to the anisotropy of turbulence. For the spatial coherence radius of either a plane or spherical wave propagating along a horizontal line-of-sight path in anisotropic atmospheric turbulence, the corresponding small-scale solution is proportional to that for the plane-wave spatial-coherence radius in the isotropic-turbulence case with a proportionality coefficient depending only on the effective anisotropy factor of turbulence. The corresponding large-scale solution is proportional to that for the plane-wave spatial-coherence radius in the isotropic-turbulence case with a proportionality coefficient that depends on both the effective anisotropy factor and spectral index of turbulence.

Characterization of temporal pulse broadening for horizontal propagation in strong anisotropic atmospheric turbulence.

The on-axis two-frequency mutual coherence function (MCF) for beam waves propagating along a horizontal path in strong anisotropic atmospheric turbulence is theoretically formulated by making use of the extended Huygens-Fresnel principle. Based on this formulation, a new closed-form expression for the mean square temporal width of Gaussian-beam-wave pulses passing horizontally through strong anisotropic atmospheric turbulence is developed. With the help of this expression, the increments of mean square temporal pulse width due to strong anisotropic atmospheric turbulence under various conditions are further calculated. Results show that the increment of mean square temporal pulse width due to strong anisotropic atmospheric turbulence is basically proportional to the effective anisotropic factor in most situations of interest, with the possible exception of cases in which both the Fresnel ratio and spectral index become relatively small; increasing the effective anisotropic factor can reduce the number of the said exceptions; the turbulence-induced increment of mean square temporal pulse width enlarges as the spectral index increases with a fixed value of the nondimensional turbulence-strength parameter. It is also illustrated that a significant enlargement in the turbulence-induced increment of mean square temporal pulse width occurs by changing the Fresnel ratio from a large to a tiny value if both the effective anisotropic factor and spectral index are relatively small.

Two-Frequency Mutual Coherence Function of a Gaussian Beam Pulse Propagated Through Non-Kolmogorov Turbulent Atmosphere

Over the past half century, satellite-laser-communication technology has been the focus in many countries and areas due to its higher data transmission rates, smaller terminals, low probability of intercept, etc. Much manpower and material and financial resources have been put into the development of this technology. Usually, longer pulses can be modeled as if they were continuous-wave pulses during the time when the pulse is transmitted. However, this is not the case for ultrashort pulses that are usually used as a carrier in satellite-laser communication. For satellite-to-ground or ground-to-satellite links, the atmospheric effects are more deleterious to the laser pulse and their analysis is also more complicated. For a long time, laser pulse propagation in the turbulent atmosphere were studied based on the Kolmogorov model. However, increasing experimental evidence and theoretical investigations have shown that the Kolmogorov model is not the only possible turbulent one in the atmosphere, and there exists another turbulence, namely, the non-Kolmogorov turbulence. So it is necessary to perform the analysis of the laser-pulse propagation in the non-Kolmogorov turbulence. The two-frequency mutual coherence function (MCF) plays an important pole in determining the basic characteristics associated with the laser-pulse propagating in the atmospheric turbulence. In this paper, we consider a theoretical power spectrum of the refractive-index fluctuations with a generalized power law, derive the two-frequency MCFs of a Gaussian-beam pulse in weak turbulence for a horizontal path, and analyze the influence of spectral power-law variations on the two-frequency MCFs. We present an analytical expression for a collimated-beam pulse.

Scattering and propagation of UV pulses in soot aerosols

Scattering and propagation of a UV pulse in soot aerosols are studied using generalized multi-sphere Mie theory (GMM) and a two-frequency mutual coherence function. Soot aerosols are obtained by the diffusion-limited aggregation (DLA) model. Scattering characteristics of aggregate structures in soot aerosols are analyzed by GMM theory in detail. Scattering intensities versus scattering angles are given and discussed. The effects of different-positions of the aggregate on the scattering intensities, scattering cross section, extinction cross section, absorption cross section and asymmetry factor are computed and compared. The two-frequency mutual coherence functions of UV pulses in soot aerosols are simulated, and the effects of optical distance, frequency difference are analyzed.

Scattering and propagation of terahertz pulses in random soot aggregate systems

Scattering and propagation of terahertz pulses in random soot aggregate systems are studied by using the generalized multi-particle Mie-solution (GMM) and the pulse propagation theory. Soot aggregates are obtained by the diffusion-limited aggregation (DLA) model. For a soot aggregate in soot aggregate systems, scattering characteristics are analyzed by using the GMM. Scattering intensities versus scattering angles are given. The effects of different positions of the aggregate on the scattering intensities, scattering cross sections, extinction cross sections, and absorption cross sections are computed and compared. Based on pulse propagation in random media, the transmission of terahertz pulses in random soot aggregate systems is determined by the two-frequency mutual coherence function. Numerical simulations and analysis are given for terahertz pulses (0.7956 THz).

Envelope synthesis of a cylindrical outgoing wavelet in layered random elastic media based on the Markov approximation

SUMMARY In the heterogeneous earth medium, short period seismograms of an earthquake are well characterized by their smooth envelopes with random phases. The Markov approximation has often been used for the practical synthesis of their envelopes for a given frequency band. It is a stochastic extension of the phase screen method to synthesize wave envelopes in media with random fluctuations under the condition that the wavelength is shorter than the correlation distance of the fluctuation. We propose an extension of the Markov approximation for the envelope synthesis to the case that an isotropically outgoing wavelet is radiated from a point source in horizontal layered random elastic media, where different layers have different randomness and different background velocities. In each layer, we solve the master equation for the two frequency mutual coherence function (TFMCF) which contains the information of the intensity in the frequency domain. Just below each layer boundary, we calculate the angular spectrum which is the expression of the TFMCF in the transverse wavenumber domain for up-going wavelets. The angular spectrum shows the ray angle distribution of intensities of scattered waves. Multiplying it by the square of transmission coefficients calculated from the background velocity contrast at the boundary, we evaluate the angular spectrum just above it. We neglect P to S (S to P) conversion scattering inside of each layer; however, we take into account the mode conversion at the layer boundary. Different from the vertical incidence of a plane wavelet, the wavefront expands with time and its curvature is modified at the layer boundary due to the Snell’s law. Approximating the wavefront in the second layer by a circle for a small incidence angle, we may shift the real origin to the pseudo-origin of the wavefront circle, which leads to the change in geometrical spreading factor. Finally, we calculate the mean square envelope from the TFMCF by using an FFT. By multiplying the angular spectrum by conversion or reflection coefficients and calculate the TFMCF for the converted or reflected wavelets at layer boundaries, we can calculate any phase generated due to velocity discontinuities. For the reflected wavelets, we solve the master equation of the TFMCF downward. To confirm the validity of the method, we directly synthesize mean square envelopes in 2-D two-layered random elastic media and compare them with the averaged envelopes calculated by finite difference (FD) simulations of the wave propagation in random elastic media. We find that the Markov envelopes well agree with the FD envelopes not only for a transmitted wavelet but also for a P to S converted wavelet and a reflected wavelet at a layer boundary.

Temporal broadening of optical pulses propagating through non-Kolmogorov turbulence

General formulations of the temporal averaged pulse intensity for optical pulses propagating through either non-Kolmogorov or Kolmogorov turbulence are deduced under the strong fluctuation conditions and the narrow-band assumption. Based on these formulations, an analytical formula for the turbulence-induced temporal half-width of spherical-wave Gaussian (SWG) pulses is derived, and the single-point, two-frequency mutual coherence function (MCF) of collimated Gaussian-beam waves in atmospheric turbulence is formulated analytically, by which the temporal averaged pulse intensity of collimated space-time Gaussian (CSTG) pulses can be calculated numerically. Calculation results show that the temporal broadening of both SWG and CSTG pulses in atmospheric turbulence depends heavily on the general spectral index of the spatial power spectrum of refractive-index fluctuations, and the temporal broadening of SWG pulses can be used to approximate that of CSTG pulses on the axis with the same turbulence parameters and propagation distances. It is also illustrated by numerical calculations that the variation in the turbulence-induced temporal half-width of CSTG pulses with the radial distance is really tiny.

Synthesis and applicable condition of vector wave envelopes in layered random elastic media with anisotropic autocorrelation function based on the Markov approximation

SUMMARY The Markov approximation is a powerful stochastic method for the direct synthesis of wave envelopes in random velocity fluctuated media when the wavelength is shorter than their correlation distance. To apply the Markov approximation to realistic cases, we consider horizontal layered random media characterized by an anisotropic autocorrelation function (ACF), where different layers have different randomnesses and different background velocities having step-like changes. Solving the parabolic master equation for the two frequency mutual coherence function (TFMCF) in random elastic media for the vertical incidence of a plane wavelet from the bottom, we calculate the angular spectrum just before the first velocity boundary. Multiplying transmission or PS conversion coefficients of the boundary by the angular spectrum, we calculate the angular spectrum and TFMCF on the other side of the boundary. Then we solve the master equation for the forward propagating wavelet in the second layer. Taking the same procedure for each layer boundary, we finally obtain the mean square (MS) vector envelopes on the free surface on the top layer. For the practical simulation, we use 2-D random media characterized by a Gaussian ACF. We numerically confirm the validity of the envelope synthesis for a specific case of layered random media with anisotropic ACF by comparing with finite difference (FD) simulations of elastic waves. Considering the Earth structure, the horizontal correlation distance is larger than the vertical one and the velocity fractional fluctuation becomes weak as depth increases, the Markov approximation is good for modelling the primary wavelet and also the converted wavelet for the vertical incident wavelet. We derive an applicable range of the Markov approximation for random media with anisotropic ACF by comparing with FD simulations. The results show that the Markov approximation is accurate when the wavelength is comparable or shorter than the both of vertical and horizontal correlation distances and the MS fractional fluctuation is much smaller than the ratio of squared horizontal correlation distance to the product of the vertical correlation distance and propagation distance.

Two-frequency mutual coherence function of scattering from arbitrarily shaped rough objects

Based on the Physical optics approximation, the scattering field in the far zone by arbitrarily shaped objects with slightly rough surface which obeys Gaussian distribution and its two-frequency mutual coherence function are derived theoretically, and the numerical results for rough spheres and rough cylinders are given and analyzed. The results show that the function has closely relationship with the roughness and the dimension of the rough objects. The roughness and the curvature of the object influence both the amplitude and the profile of the two-frequency mutual coherence function. Also, the smaller the radius of the object, the larger the coherent bandwidth. The two-frequency mutual coherence function can be used to investigate the laser pulse scattering characteristics of arbitrarily shaped rough objects, provide theoretical basis for target recognition.