Bragg Wavenumber Research Articles

A theory on the distribution function of backscatter radar cross section from ocean waves of individual wavelength

A new and simple method of interpreting the of the backscatter radar cross section (RCS) from ocean waves of individual wavelength is presented. Using the Kirchhoff scattering (Physical Optics) theory, the RCS from the ambient waveheight spectrum is first computed as a function of the wavenumber. Differentiating this cumulative RCS yields the distribution function of the RCS from ocean waves of different wavenumbers. The present study shows, as a general feature, that the maximum RCS contribution tends to come from the ocean waves of wavenumbers closer to the Bragg wavenumber as the radar incidence angle increases. Under low wind speeds (/spl sim/2 m/s) and at incidence angles between 200 and 45/spl deg/, the dominant RCS contribution at both C-band and L-band comes from ocean waves of wavenumbers close to the Bragg wavenumber. Under intermediate to high wind speeds (/spl sim/10-15 m/s) and at the small incidence angle of 200, the radar backscatter is dominated by long ocean waves with little contribution from the Bragg waves at both bands. At the intermediate incidence angles (/spl sim/35/spl deg/ to 45/spl deg/), the RCS distribution is centered at wavenumbers near the Bragg wavenumber at all wind speeds from 2 m/s to 15 m/s. These features are more pronounced at C-band than at L-band. The present theory is based on the Kirchhoff scattering model, and as such, its validity may also be limited to a range of small to intermediate incidence angles where the polarization effect is not very significant. The RCS distribution is evaluated for the ambient sea surface and the surface perturbed by varying currents induced by the interaction between the current and bottom topography.

High-frequency acoustic scattering from medium variability

Laboratory results of high-frequency acoustic scattering measurements from thermal and saline driven plumes are reported. These experiments used broadband signals in a multi-static configuration. Far-field weak scattering theory describes the acoustic scatter from the medium variability. Expressions relating the acoustic scatter to the scattering field are given in terms of deviations from ambient of temperature and salinity or alternatively, sound speed and density. The angular dependence of the coefficient for the salinity difference has a stronger angular dependence than does the coefficient for the temperature difference. By using the common Bragg wave number comparisons [J. Oeschger, Proc. of 16th ICA and 135th ASA 2, 1335–1336 (1998)], multi-static measurements can be used to determine the source of scattering, either salinity or thermal in nature, without resorting to a priori information. Each angular dependent source-term normalization factor is applied in turn to the data and the one that yields the best agreement predicts the scattering source correctly. Results are also presented for medium mixing where the relative density fluctuations dominate the scattering mechanism. Again the common Bragg wave number comparisons using the multi-static data are able to determine correctly the source of scattering as predicted by supporting environmental measurements.

Radar signatures of marine mineral oil spills measured by an airborne multi-frequency radar

Radar signatures of mineral oil spills consisting of heavy and light fuel were measured by an airborne five-frequency ( L - S - C - X - and K -band) multi u polarization microwave scatterometer flown on a helicopter during a controlled oil spill experiment in the North Sea. The damping ratio, defined as the ratio of the backscattered radar power from an oil-free and an oil-covered sea surface, was measured at different radar frequencies and incidence angles such that the Bragg wavenumbers, k, between 20 radm 1 and 500 radm 1 were covered. The B following results were obtained: for the five oil spills deployed in the experiment the damping ratio, in general, increases monotonically from k 20 radm 1 to B k 500 radm 1 . At S - C - X - and K -band, the damping ratio is larger for heavy B u fuel than for light fuel spills, while at L- band it is almost the same. For heavy fuel, the damping ratio increases with increasing thickness of the oil layer. Furthermore, for wind speeds between 6 m s 1 and 10 m s 1 the damping of the backscattered radar power is independent of the polarization of the radar and its look direction relative to the wind. The results obtained in the present investigation agree well with those obtained from similar measurements with heavy fuel under high wind conditions.

On the Reduction of the Radar Backscatter by Oceanic Surface Films: Scatterometer Measurements and Their Theoretical Interpretation

During the two SIR-C/X-SAR missions in 1994, surface film experiments were performed in the North Sea with a 5-frequency/multipolarization scatterometer flown on a helicopter, in order to investigate the reduction of the radar backscatter in the presence of quasibiogenic and anthropogenic sea surface films, particularly, at different wind speeds. Under all wind conditions encountered in this study, the measured damping ratio (i.e., the ratio of the radar backscatter from a slick-free and a slick-covered water surface) increases with increasing Bragg wavenumber. It is shown that not only Marangoni damping theory, but also wind-induced effects, primarily the energy input by the wind into the wave spectrum, also have to be taken into account. The reductions measured at low to moderate wind speeds (3.5–4 m/s and 5 m/s) are qualitatively explained by means of a comparison of the different source terms of the action balance equation. For the case of high wind speed (12 m/s) a theoretical model for the damping ratios is developed. Using this model, the experimental data can well be reproduced, and the absence of the Marangoni damping maximum at intermediate Bragg wavenumbers (approximately 100 rad/m) can be interpreted. Furthermore, the model can explain the similarities between the radar backscatter reductions measured over quasibiogenic and anthropogenic surface films under high wind conditions.

High-frequency acoustic scattering from thermally generated turbulence

The results of a laboratory experiment are presented for the case of high-frequency broadband multistatic acoustic scatter from a thermally generated buoyant plume. The dominant scattering mechanism is changes in the relative compressibility, which is related to the temperature difference from ambient. Far-field weak scattering theory is used to describe the scattering process. This results in a simple relationship between the complex acoustic scatter and the one-component, three-dimensional Fourier transform of the temperature field measured at the scattering producing Fourier component K, defined as the Bragg wave number. This wave number is the magnitude of the difference wave vector between the scattered and incident wave vectors. Theory predicts for measurements made in the same scattering direction that spectral estimates for overlapping Bragg wave numbers be identical. Utilizing data collected from simultaneous multiple bistatic configurations with parallel Bragg wave vectors yields a more fully resolved spectral estimate of the scattering producing temperature field. A time series of the spectral estimate of the turbulent temperature field is shown for a three-channel bistatic common Bragg direction measurement. The results indicate agreement between the spectral estimates in the regions of overlapping Bragg wave numbers, validating the theoretical description. The time series also shows strong, yet highly variable nulls in the wave-number spectrum of the turbulence. [Work supported by ONR.]

Theory of double-chirped mirrors

A theory of double-chirped mirrors (DCMs) for dispersion compensation in ultrashort pulse laser sources is presented. We describe the multilayer interference coating by exact coupled-mode equations. They show that the analysis and synthesis of a coating with a slowly varying chirp in the layer thicknesses can be mapped onto a weakly inhomogeneous transmission line problem. Solutions of the transmission line equations are given using the WKB-method. Analytic expressions for reflectivity and group delay are derived. The solutions show that the main problem in chirped mirror design is the avoidance of spurious reflections, that lead to Gires-Tournois-like interference effects responsible for the oscillations in the group delay. These oscillations are due to an impedance matching problem of the equivalent transmission line. The impedance matching can be achieved by simultaneously chirping the strength of the coupling coefficient and the Bragg wavenumber of the mirror. An adiabatic increase in the coupling coefficient removes the typical oscillations in the group delay and results in broad-band mirrors with a controlled dispersion. Finally, the mirror is matched to air with a broadband antireflection coating. We discuss a complete design of a laser mirror with a reflectivity larger than 99.8% and a controlled dispersion over 300-nm bandwidth. Using such mirrors in a Ti:sapphire laser, we have demonstrated /spl ap/30-fs pulses, tunable over 300 nm, as well as 8-fs pulses from the same setup. A different design resulted in 6.5-fs pulses.

Threshold and above-threshold analysis of corrugation-pitch-modulated DFB lasers with inhomogeneous coupling coefficient

DFB lasers with longitudinal variations in their grating pitch and, consequently, their Bragg wavenumber (corrugation-pitch-modulated or CPM-DFB) can achieve single-mode oscillation with good threshold gain margins and flatter longitudinal photon-density distributions than conventional quarter-wave-shift (QWS) DFB lasers. Moreover, substantial improvements of these two characteristics are provided by adequate variation of the coupling coefficient κ along the cavity In the paper, a three-section corrugation-pitch-modulated DFB laser with distributed-coupling-coefficient (DCC-CPM-DFB) is studied extensively. Threshold and above-threshold numerical simulations show that these structures can achieve very good single-mode operation by correct optimisation of the Bragg detuning and coupling strength profile at threshold and can maintain these characteristics up to high injection currents.

Small-angle light scattering studies of dense AOT-water-decane microemulsions

We have performed extensive studies of a three-component microemulsion system composed of AOT-water-decane (AOT=sodium-bis-ethylhexyl-sulfosuccinate is an ionic surfactant) using small-angle light scattering (SALS). The small-angle scattering intensities are measured in the angular interval 0.001–0.1 radians, corresponding to a Bragg wave number range of 0.14 μm−1<Q<<1.4 μm−1. The measurements were made by changing temperature and volume fraction ϕ of the dispersed phase (water + AOT) in the range 0.05<ϕ<0.75. All samples have a fixed water-to-AOT molar ratio,w=[water]/[AOT]=40.8, in order to keep the same average droplet size in the stable one-phase region. With the SALS technique, we have been able to observe all the phase boundaries of a very complex phase diagram with a percolation line and many structural organizations within it. We observe at the percolation transition threshold, a scaling behavior of the intensity data. This behavior is a consequence of a clustering among microemulsion droplets near the percolation threshold. In addition, we describe in detail a structural transition from a droplet microemulsion to a bicontinuous one as suggested by a recent small-angle neutron scattering experiment. The loci of this transition are located several degrees above the percolation temperatures and are coincident with the maxima previously observed in shear viscosity. From the data analysis, we show that both the percolation phenomenon and this novel structural transition are derived from a large-scale aggregation between microemulsion droplets.

Distributed feedback lasers with an S-bent waveguide for high-power single-mode operation

DFB lasers with variations in the Bragg wavenumber along the cavity (corrugation-pitch modulation) can be designed to operate in a single mode up to high output powers. Practical implementation of various pitch-modulated DFB cavities is facilitated by bending the laser waveguide over a uniform grating, so that the Bragg wavenumber is varied along the cavity. In this paper, above-threshold behavior of such lasers is studied, and design rules for high power single-mode operation are developed. S-bent waveguide DFB lasers, fabricated according to our design criteria, exhibit single-mode operation with 40 dB side mode suppression ratio from threshold to 10/spl times/I/sub th/.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the theory of Δk radar observations of ocean surface waves

We consider the theory of waves scattered from a moving, rough, and dispersive surface in the small perturbation limit. The first-order scattered field for a time-dependent surface is obtained in the far zone of scattering in terms of the two-dimensional spectral amplitude of the surface and its dispersion relation. We develop a rigorous /spl Delta/k radar theory and show that the nonzero output of a /spl Delta/k radar occurs only when the Bragg condition for each signal component is satisfied separately. The frequency correlation function of the scattered field is then proportional to the mean value of the product of the spectral amplitudes of the surface at the corresponding Bragg wavenumbers. The mean value of this product is nonzero only for surfaces that have a locally varying spectrum and is proportional to the Fourier transform (with the argument /spl Delta/k) of the variation of the local spectrum with respect to the pattern position. Such variations may be caused by either amplitude or phase modulation of the surface structure. In the former case, our results are similar to the results of existing theory. The latter case of phase modulation of the surface (for example, internal waves interacting with capillary waves) cannot be explained by previous theory.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

In situ measurements of capillary‐gravity wave spectra using a scanning laser slope gauge and microwave radars

Capillary‐gravity wave spectra are measured using a scanning laser slope gauge (SLSG), and simultaneously by X and K band Doppler radars off the Chemotaxis Dock at the Quissett campus of the Woods Hole Oceanographic Institution at Woods Hole, Massachusetts. Wave spectral densities estimated from the radar measurements using the Bragg theory agree with those measured using the SLSG at the Bragg wavenumber to within a few decibels, suggesting that Bragg scattering theory is valid for the conditions of this experiment. The observed degree of saturation of capillary‐gravity waves is in reasonable agreement with measurements by Jähne and Riemer (1990) obtained from measurements in a large wind‐wave flume at intermediate wind speeds, but our data indicate a higher degree of saturation at very low wind speeds. The rate at which the slope‐frequency spectrum falls off, however, is much lower in the field than in laboratories, even at moderate winds, suggesting long waves are responsible for a large Doppler shift of capillary‐gravity waves. Close examination of combined wavenumber‐frequency slope spectra also reveals significant smearing of the spectra in the frequency domain due to long waves. These observations confirm that spatial measurements (wavenumber spectra measurements) are essential for characterizing short capillary‐gravity waves, since this strong Doppler shift will dramatically change apparent frequency spectra.

Hydrodynamic modulation of short wind‐wave spectra by long waves and its measurement using microwave backscatter

In this paper we use results of microwave backscattering experiments over the past decade to attempt to present a coherent picture of the ocean wave‐radar modulation transfer function (MTF) based on composite surface theory, short‐wave modulation, and modulated wind stress. A simplified relaxation model is proposed for the modulation of the gravity‐capillary wavenumber spectrum by long waves. The model is based on the relaxation rate and the equilibrium gravity‐capillary wavenumber spectrum. It differs from previous models by including all airflow modulation effects in the response of the equilibrium spectrum to changes in the airflow. Thus the explicit modulation of individual source functions such as wind input, short‐wave dissipation, and nonlinear interactions need not be known in order to calculate the hydrodynainic MTF. By combining this new model of the hydrodynamic MTF with microwave measurements, we attempt to determine wind shear stress modulation caused by the long waves. In order to extract the hydrodynamic MTF from the microwave data, we remove tilt and range change effects from the measured MTFs using the published analytical forms for these effects. Our results show that the inferred hydrodynamic MTF is higher for II polarization scattering than for V polarization. Since this is impossible if we have obtained the true hydrodynamic MTF, these results strongly indicate a problem with composite scattering theory as it has been traditionally applied. One explanation for this result may be the effects of intermediate‐scale waves suggested by Romeiser et al. (1993). Since these effects are much stronger for H polarization than for V polarization, they may explain our observed discrepancy and, if so, imply that V polarization return should yield an acceptable upper limit for the true hydrodynamic MTF. Thus we incorporate our V polarization results into the proposed model to estimate an upper limit for the wind shear stress modulation along the long‐wave profile. We infer that the primary source of modulation of Bragg resonant waves depends strongly on Bragg wavenumber and windspeed. For low values of these quantities, straining by long‐wave orbital velocities dominates the modulation process, while for higher values modulated wind stress becomes increasingly important. Our calculations indicate that wind stress modulation dominates the process for 3 cm Bragg waves at moderate to high wind speeds.

Estimates of the scattering cross section of equatorial electrojet irregularities over the 3‐ to 7‐MHz frequency band

Esq echo intensities recorded by a dynasonde system operated at Huancayo, Peru, are used to estimate the scattering cross section of equatorial electrojet plasma irregularities in the 3 to 7 MHz frequency band. The scattering radar cross section of vertically propagating ∼20‐m scale electrojet irregularities is estimated as ∼3×10−7 m−1 over a period exhibiting ƒ0E values of 2.7 to 2.9 MHz. The cross‐section variation with Bragg wavenumber k is found to be characterized with a power law index of approximately −3 over the HF band.

Comparison of UMPE/PEREV bistatic reverberation predictions with observations in the ARSRP Natural Lab

During the summer of 1993 acoustic experiment sponsored by the ONR Acoustic Reverberation Special Research Program, bistatic bottom reverberation measurements were made within the ARSRP Natural Laboratory near the mid‐Atlantic ridge. Employing a version of the University of Miami Parabolic Equation (UMPE) model [MPL Tech. Memo, 432 (1993)] incorporating Tappert’s PE Reverberation (PEREV) model, calculations of monostatic and bistatic reverberation within the Natural Laboratory have been made. The UMPE/PEREV model predicts a strong correlation between areas of high reverberation and high bottom ensonification. Furthermore, the wave scattering strength predicted by the PEREV model is shown to depend on the bottom roughness spectrum evaluated at the Bragg wave number 2k0 sin(φs/2) where φs is the azimuthal scattering angle. Due to the typical power‐law form of the roughness spectrum, this suggests that the wave scattering strength is enhanced in the direction of forward scatter. These predictions will be com...

Accuracy of scattering approximations for acoustical remote sensing.

Acoustical remote sensing of interface roughness usually involves an interpretation of the scattered signal assuming the forward problem is understood. The accuracy of the standard scattering approximations for power law roughness spectral densities typical of the seafloor will be reviewed. The approximations include the Kirchhoff approximation and the lowest-order perturbation approximation. The effects of roughness at wave numbers higher than the Bragg wave number will also be considered. Exact integral equation results for 1-D surfaces are used for ground truth. [Work supported by ONR.]

Radar Sea Returns—Ocean-Ripple Spectrum and Breaking-Wave Influence

Radar cross sections reported by Guinard et al. are normalized with respect to their mean value obtained with different bands but at the same wind velocity. The results confirm that there are significant differences between returns with VV and HH polarizations. The VV-polarized returns, being associated more closely with distributed ocean ripples, support the equilibrium spectrum of ocean waves k−3.5 with the wavenumber k < 0.5 rad cm−1, and the saturated spectrum k−4 at higher wavenumbers. The HH-polarized returns at low wavenumbers have the same trend as those of the VV-returns, but are smaller in magnitude. At high wavenumbers (k > 0.5 rad cm−1), the HH-returns retain the trend of increasing continuously with the Bragg wavenumber; this is believed to be caused by the surface roughness produced by individual breaking waves. Contributions of returns produced by each mechanism are then discussed for radars having various bands. Functional, power-law type, dependence of returns on the wind-friction velocity is also found to vary systematically with the Bragg wavenumber; the exponent of power law increases with the Bragg wavenumber, following k⅓.

Calculation of radar backscatter modulations from internal waves

Calculations of microwave backscatter from the ocean surface using the small‐perturbation method (first‐order Bragg approximation), a modified Kirchhoff approximation, and a two‐scale composite model derived from the modified Kirchhoff expression are described, and predictions of the modulation in the cross section induced by internal wave features are presented. The calculations demonstrate that the modified Kirchhoff approximation and the composite model yield equivalent cross sections if the cutoff wave number separating the long‐ and short‐wavelength regions in the composite model is chosen to be roughly one‐third the Bragg wave number. The interaction of internal waves and surface waves, assumed to modulate an equilibrium wind wave spectrum over wavelengths extending from about 0.1 to 10 m, results in significant perturbations to the L band (24 cm) and X band (3 cm) cross sections at incidence angles from 23° to 50°. These calculations indicate that the large modulations observed in X band synthetic aperture radar images of internal waves are due, at least in part, to the tilting of the X band Bragg scatterers by the internal‐wave‐modulated long‐wavelength surface waves.

Resonant reflection of water waves in a long channel with corrugated boundaries

A one-dimensional wave equation is derived for water-wave propagations in a long channel with corrugated boundaries. The amplitude and the wavelength of boundary undulations are assumed to be smaller than and in the same order of magnitude as the incident wavelength, respectively. When the Bragg reflection condition (i.e. the wavenumber of the boundary undulations is twice that of the incident wavenumber) is nearly satisfied, significant wave reflection could occur. Coupled equations for transmitted and reflected wave fields are derived for the near resonant coupling. The detuning mechanism is attributed to the slight deviation in the wavenumber of the corrugated boundaries from the Bragg wavenumber. Analytical solutions are obtained for the cases where the boundary undulations are within a finite region. The application of the present theory to the design of a harbour resonator is discussed.

Doppler Spectra in Microwave Scattering from Wind Waves

Doppler spectra in microwave scattering from wind waves have been measured at X band (3.2 cm) and K band (1.25 cm) in a short fetch wind wave tank. The scattering cross sections, mean frequency shifts, and bandwidths obtained are compared with results of measurements of the directional slope spectra, mean drift, and particle velocities of the wind waves. The scattering is attributed to low order Bragg scattering, the phase speeds of the wind waves are deduced and the dependence of the Doppler bandwidth on wind speed and Bragg wavenumber is given.