Sphere In Free Space Research Articles

Free-carrier detection in a silicon slab via absorption measurement in 2D integrating cells.

2D integrating cells provide long optical path lengths on a chip by multiple reflections at highly reflective mirrors similar to integrating spheres in free space. Therefore, they build a promising platform for integrated optical absorption sensing. Here, we present first absorption measurements of free carriers generated by a modulated pump laser inside a 2D integrating cell in a silicon slab. The results can be used to evaluate the lifetimes of free carriers in silicon slabs for integrated optics. Employing a silicon-on-insulator platform with a silicon thickness of 220nm, we demonstrate measurements of the access free-carrier concentration on the order of 1-8·1015 cm-3 with lifetimes on the order of 0.1-1μs governed by surface recombination at the silicon interfaces. The measured lifetimes are dependent on free-carrier concentration, which confirms previous observations. The presented free-carrier absorption experiment verifies the sensitivity of 2D integrating cells to changes in the absorption coefficient and thus demonstrates the potential of 2D integrating cells for absorption sensing.

Von Laue’s theorem and its applications

von Laue’s theorem, as well as its generalized form, is strictly proved in detail for its sufficient and necessary condition (SNC). This SNC version of Laue’s theorem is used to analyze the infinitely extended electrostatic field produced by a charged metal sphere in free space, and the static field confined in a finite region of space. It is shown in general that the total (Abraham=Minkowski) electromagnetic momentum and energy for the electrostatic field cannot constitute a Lorentz four-vector. A derivative von Laue’s theorem, which provides a criterion for a Lorentz invariant, is also presented.

New developments in AEM discrete conductor modelling and inversion

Discrete conductor models like sphere and plate were introduced in the 1950s as modelling tools in airborne electromagnetic (AEM) survey interpretation. In the last 20 years, with the development of inversion techniques, they have been integrated into parametric inversion programs. The recent advent of powerful workstations makes them useful tools for interactive AEM interpretation. Different problems have been encountered in the implementation and application of discrete objects as modelling and inversion tools. The sphere response is modelled using a sum of spherical functions. Assuming that the radius of the sphere is small compared to the distance between the transmitter and receiver to the centre of the sphere, the response can be approximated by using only the first term of the solution. This approach is reviewed for modelling the response of a conductive sphere in free space or buried in a layered earth. Plate modelling is based on spectral methods or the integral equation method, which provide different techniques for estimating the response of a plate in free space. A comparison of the results of these techniques show differences attributed to the different discretisation methods. A case history from Abitibi, Canada, shows that plate inversion using two different inversion methods provides useful information when the target is a plate-like conductor in a resistive environment.

Time Domain Scattering of Scalar Waves by Two Spheres in Free-Space

The importance of the T-matrix method is well known when frequency domain scattering problems are of interest. With the relatively recent and wide-spread interest in time domain scattering problems, similar applications of the T-matrix method are expected to be useful in the time domain. In this paper, the time domain spherical scalar wave functions are introduced, translational addition theorems for the time domain spherical scalar wave functions necessary for the solution of multiple scattering problems are given, and the formulation of time domain scattering of scalar waves by two scatterers of arbitrary shape is presented. The whole analysis is performed in the time domain requiring no inverse Fourier integrals to be evaluated. Scattering by two soft spheres in free-space is studied as an application to check the numerical accuracy and demonstrate the utility of the expressions.

Broadband scattering from spherical shells in a waveguide: modeling and classification

In this presentation, the exact expression for scattering from a sphere in a Pekeris waveguide (e.g., Sammelmann and Hackman, J. Acoust. Soc. Am., 𝟠𝟚, 1987) is discussed. The importance of the sphere/interface rescattering terms is considered. A computationally faster multipath expansion approach is derived and its accuracy compared with the exact approach. For sufficiently high frequencies and in the case where the rescattering terms can be ignored, the multipath approach yields accurate predictions. The broadband scattering from an elastic-shelled sphere in a Pekeris waveguide is considered as a function of the frequency (or time for a pulse) and the sphere's range and depth in the waveguide. The classification problem is also discussed. The echos (time series or spectra) from a large set of spheres with varying parameters are generated and grouped into six classes corresponding to the various shell thicknesses and materials. Simple classifiers based upon temporal or spectral representations are considered for the spheres in free space and in a waveguide

Novel BI-FDTD Approach for the Analysis of Chiral Cylinders and Spheres

A versatile time-domain technique, known as bi-isotropic finite difference time domain (BI-FDTD), has recently been introduced for the numerical analysis of electromagnetic wave interactions with complex bi-isotropic media. However, to date only one-dimensional BI-FDTD schemes have been successfully implemented. This paper presents novel two-dimensional (2-D) and three-dimensional (3-D) dispersive BI-FDTD formulations for the first time. The update equations for these new 2-D and 3-D BI-FDTD approaches are developed and applied to the analysis of electromagnetic wave scattering by chiral cylinders and spheres in free space. The distinctive feature of this technique is the use of two independent sets of wavefields representing the left- and right-polarized waves in the chiral medium. This wavefield decomposition approach allows dispersive models for the chirality parameter as well as the permittivity and permeability of the medium to be readily incorporated into an FDTD scheme. The 2-D and 3-D BI-FDTD simulation results are compared with available analytical solutions for the scattering from a circular chiral cylinder and a chiral sphere respectively

The Impulse-Response Moments of a Conductive Sphere in a Uniform Field, a Versatile and Efficient Electromagnetic Model

The impulse response moments of a conductive sphere in free space excited by a uniform magnetic field can be used to approximate the moments of a sphere in a dipolar field. The numerical computations are straightforward and the approximation is especially good for higher-order moments. The greatest discrepancy is seen on the zeroth-order moment when the radius of the sphere is large. It is possible to improve the accuracy for the zeroth-order moment by modelling the large-radius sphere (in a dipole field) as the combined response of multiple small-radius spheres (each in a locally uniform field). The small spheres are closely packed inside the larger sphere. The discrepancy can be reduced to less than 15 % in this manner.The sphere in a uniform field can also be used to approximate the response of a body that has its currents constrained to flow in a plane with a specific orientation. This means that plate-like bodies or anisotropic spheres can also be modelled.The third-order moment has been calculated from data acquired during a MEGATEM airborne electromagnetic survey of the Reid-Mahaffy test site. There is an anomalous response in the third-order moment that can be modelled by a sphere at 170 m depth with a conductivity of 15 S/m and a radius of 40 m. The currents flowing in the sphere are constrained to flow in a vertical plane. This model is consistent with the geology of the area and a hole drilled to test the anomalous zone.

The effect of magnetic anomalies on transient electromagnetic data

The effect of magnetically polarisable material on transientelectromagnetic (TEM) responses is discussed using data calculated for a layered-Earth model and a sphere in free-space. The layered-Earth models are characterised by a resistive surface layer of elevated magnetic permeability to represent lateritic duricrust and ironstone. The corresponding results indicate that data measured by TEM systems such as SIROTEM, UTEM, QUESTEM and TEMPEST will only be marginally affected by the magnetic permeability of the surface layer. The sphere model is characterised by high conductivity and high magnetic permeability to represent a pyrrhotite-rich mineralisation. The corresponding TEMPEST responses as well as results from singular value decomposition analysis indicate TEMPEST data to be sensitive to magnetic permeability variations exceeding 0.2 μ0 , but only for conductive and sizeable structures.TEMPEST data acquired crossing a banded iron formationrelated magnetic anomaly of 8000 nT were found to be devoid of any indication of magnetic induction. The anomaly was modelled with a dike-shaped polygon suggesting a magnetic permeability of 2.0 μ0 with the corresponding EM data indicating a zone of low conductivity. The EM responses caused by dragging the receiver coils through the associated magnetic gradient is removed by the stacking process, but is retrievable from the streamed data.

Numerical estimates—based on ray methods—for the scattered acoustic fields of pairs of targets

The sounds fields scattered by a multiplicity of spherical bodies can be predicted in an exact analytic fashion. This was illustrated in a previous presentation (cf., the paper entitled ‘‘Exact Analysis of the Acoustic Scattering Produced by Arbitrarily Insonified Bodies Near Boundaries,’’ by Huang and Gaunaurd (in this session), which also included the exact analysis for two spheres in free space, at arbitrary frequencies, and accounting for multiple scattering interactions. That work was mainly concerned with the low- and mid-frequency regimes. However, at the high frequencies for which certain asymptotic approximations begin to hold true, one can construct relatively simple-looking expressions for the sound fields scattered by pairs of spherical bodies. These expressions become even simpler looking for bodies separated by distances larger than their sizes. Even when this is not the case, the asymptotic results seem to hold just as well. The final estimates are: (a) evaluated numerically, (b) compared to (now available) exact analytic solutions, (c) graphically displayed, and (d) physically interpreted. These estimates are the acoustical counterparts of analogous ones in the radar literature and they are useful to determine the minimal separation at which remote sensors can actually discriminate two (or more) interacting neighboring targets. [Work partially supported by the NSWC IR Program.]

Electrochemical simulations of mass transfer from isolated wet spots and droplets on realistic fluttering leaves

This paper reports on forced-convection mass transfer from isolated discs on rectangular plates as well as hemispheres on realistic fluttering leaves. An electrochemical method was used where the convective transfer of ions to the test electrode (the droplet or the wet spot) in an electrolytic flow system was measured as a function of flow rates, sizes of discs and hemispheres. Measurements showed that the local transfer coefficient for uniformly transferring plates varied as expected while the transfer from isolated discs on plates was much less a function of the distance from the leading edge. An expression to describe the transfer coefficient for an isolated disc as a function of distance from the leading edge was determined. An expression describing the transfer from hemispherical drops on fluttering leaves was derived and compared with the predictions from transfer theory for a sphere in free space.

Interpretation of Crone pulse electromagnetic data

Time‐domain electromagnetic (TEM) prospecting systems, including the Crone pulse electromagnetic (PEM) system, are becoming more widely used in the search for massive sulfides. We have developed interpretation aids for the moving‐source configuration of the Crone PEM system using the PLATE and SPHERE programs developed by the University of Toronto. PLATE models a thin rectangular sheet in free space and SPHERE models a two‐layer sphere in free space. The effects of varying the model parameters, particularly the dip, depth, and conductance for large plates, are shown. As the dip of a plate goes from 0 to 90 degrees, the negative side lobe over the conductor becomes less pronounced. The depth can be estimated from the amplitude of the positive peak over the edge of a conductor. The depth of exploration for the Crone PEM system is approximately twice the coil separation for shallow dipping plates, and one coil separation for steeply dipping plates when there is not a significant half‐space or overburden response. The ratio of the on‐conductor and off‐conductor side‐lobe areas gives an estimate of the dip if the depth of the target has been determined. The resolution of dip for plates dipping more than 60 degrees is poor. The conductance of a plate can be determined from either channel‐ratios or the estimated time constant. Channel‐ratios sometimes provide better estimates, because they have less dependence on the size of the conductor at earlier times. At early times, the target response can be obscured by conductive overburden. Only at late times would interpretation aids be useful in evaluating the target parameters. A dip‐depth nomogram and channel‐ratios gave good first estimates of the target parameters of three field examples interpreted by computer modeling.

The role of simple computer models in interpretations of wide‐band, drill‐hole electromagnetic surveys in mineral exploration

Drill‐hole geophysical surveys are a means of extending the search for massive‐sulfide deposits to depths which are inaccessible to conventional surface techniques. The present investigation combines field and model studies of an electromagnetic (EM) prospecting method which utilizes a large, fixed transmitter loop with a downhole, axial‐component magnetic field sensor (solenoid). The system is shown to be well‐suited for detection of deeply buried massive‐sulfide conductors located in resistive host rock at appreciable distances from the drill hole. We propose that drill‐hole survey data collected with a wide‐band large‐loop EM system can be routinely used for estimating target parameters by forward modeling with two simple conductor shapes: a plate and a sphere in free space. Analysis of confined conductors is facilitated by “eigencurrent decomposition” of the induced current vortex into a set of noninteracting loops with simple RL‐circuit behavior. Solutions have been implemented in interactive computer programs which are fast, inexpensive, and sufficiently versatile to accommodate configurations and waveforms used in many practical EM systems. Both stationary and dynamic aspects of the induction process are exploited for diagnosis of three‐dimensional targets. Field studies at test sites in Sudbury and Noranda base‐metal mining areas with a commercial pulse EM system indicate that many important effects predicted by the model studies are, indeed, observable in survey data.

TRANSIENT ELECTROMAGNETIC RESPONSE OF A SPHERE IN A LAYERED MEDIUM*

AbstractThe Galerkin method of solving integral equations is well suited to the solution of the integral equations describing the transient response of a sphere embedded in a layered medium, which is excited by a large co‐axial loop.The transient response is calculated by transforming the steady state solutions obtained in the frequency domain.The analysis shows that the scattering matrix is extremely diagonally dominant and the maximum number of modes required to obtain convergence does not rapidly increase with frequency. The number of modes required is about eight. This type of scattering matrix can be taken to be an expression of the principle of elementary superposition. This principle is reflected in the decay curves. These show that the early part of the decay curves asymptotically approach the decay curves to be expected for a layered structure without the sphere. The slope of the latter stages of the decay curve gives a decay constant that is the same as was obtained for spheres in free space excited by planar or dipolar sources.The point of departure in time of these curves from the layered ground curves is delayed either by placing the sphere at a greater depth or by placing a more conductive overburden above the sphere.

'Twixt Earth and Sky with Rod and Tube; the Mobility and Classical Impedance Analogies

The techniques which comprise the mobility analogy have been introduced into the classical impedance analogy. The mobility and impedance analogies are set forth in form for practical use. The completely dual character of these two analogies is rendered evident by placing the elements of the mobility analogy on a series of left-hand pages and the dual elements of the impedance analogy on the facing right-hand pages. In each analogy a set of mechanical and acoustical symbols is provided so that mechanical and acoustical schematic diagrams can be drawn without the necessity of drawing either of the analogous electrical schematic diagrams. These mechanical and acoustical symbols resemble somewhat their analogous electrical symbols so as to indicate to anyone familiar with circuit theory the algebraic operations which are to be performed in the analysis. Whereas in an electrical schematic straight line represents an ideal wire, in a mobility schematic a straight line represents an ideal incompressible massless rod; in an impedance schematic a line represents a hydraulic tube filled with ideal massless incompressible liquid which can sustain either tension or compression. Correct mobility schematics are obtained by inspection through considering each mass as having one terminal fixed in a frame of reference called the earth which has zero mobility and infinite mass and is analogous to electrical ground; correct impedance schematics are obtained by inspection through considering each spring as having one terminal fixed in a force of reference (P0, etc.) called the sky which has zero impedance and infinite compliance (in both cases the analogous electrical element is a conducting sphere in free space, not a condenser). Although it is recommended that analogous electrical circuits not be used in routine analysis, they can be copied directly from a mobility schematic by considering mechanical earth analogous to electrical ground, or copied directly from an impedance schematic by considering mechanical sky analogous to electrical ground.

'Twixt Earth and Sky with Rod and Tube

The techniques which comprise the mobility method of vibration computation are introduced into the classical impedance analogy; instead of drawing an analogous classical electrical schematic, a new set of mechanical and acoustical symbols is introduced which permits the drawing of a (classical) hydraulic schematic directly by cautious inspection of the structure to identify the elements and their terminals. In a mobility schematic, a straight line represents an ideal rigid massless rod; in a hydraulic schematic, a line represents a tube of unit cross section filled with ideal massless incompressible liquid which can sustain either tension or compression. Revised mobility schematic symbols are provided for mechanical and acoustical elements, and hydraulic schematic symbols for the same elements have been visualized as tubes filled with mercury or molasses, or chambers filled with methane (or air); all of these symbols resemble their analogous electrical symbols. Rigid and hydraulic junctions are distinguished by different symbols. Correct mobility schematics are obtained by inspection through considering each mass as having one terminal fixed in a frame of reference called the ground, which has zero mobility and infinite mass; correct hydraulic schematics are obtained by inspection through considering each spring as having one terminal fixed in a force of reference (P0 etc.) called the sky, which has zero impedance and infinite compliance (in both cases the analogous electrical element is a conducting sphere in free space, not a condenser). While it is recommended that analogous electrical circuits not be used, they can be copied directly from a mobility schematic by considering mechanical ground analogous to electrical ground, or copied directly from a hydraulic schematic by considering mechanical sky analogous to electrical ground.

THE PHASE DIFFERENCE AND AMPLITUDE RATIO AT THE EARS DUE TO A SOURCE OF PURE TONE

Measurements of phase difference and amplitude ratio of the sound entering the ears of a man-shaped wax dummy were taken with the source at varying azimuths around the head; at 256, 1024, and 1944 cycles per second; and at distances of 20, 50, 100, and 400 cm. from the head. The measurements at the two lower frequencies agree with the values published by Hartley and Fry computed on the assumption that the head is a rigid sphere in free space, except that the observed values of amplitude ratio are about 13 percent less than the computed values on account of the interference of the neck. The observed values at 1944 cycles are quite different from the computed values.

The Binaural Localization of Pure Tones

Measurements of phase difference and amplitude ratio of the sound entering the ears of a man-shaped wax dummy were taken with the source at varying azimuths around the head; at 256, 1024 and 1944 cycles per second; and at distances of 20, 50, 100 and 400 cm. from the head. The measurements at the two frequencies agree with the values published by Hartley and Fry computed on the assumption that the head is a rigid sphere in free space, except that the observed values of amplitude ratio are about 13 per cent. less than the computed values on account of the interference of the neck. The observed values at 1944 cycles are quite different from the computed values. As a converse experiment, pure tones of frequency 256 cycles were presented binaurally to eight different observers, the phase difference and amplitude ratio of the acoustical pressures at the two ears being known and adjustable. The observers were asked to state the direction and distance of the apparent source of sound. Only three of the eight observers were influenced in their judgment of direction by the phase difference and even they were rather inaccurate. All were influenced more or less in judging direction by the amplitude ratio. None was able to tell the distance of the source by the amplitude ratio.

On the electric resistance to the motion of a charged conducting sphere in free space or in a field of force

In his investigations on the mode of decay of vibratory motion, Prof. Love has demonstrated the great importance of considering the effect of the medium. In the particular case of a pendulum vibrating in air, it appears that the customary interpretation of the reaction of the medium on the pendulum, as an addition to the effective inertia along with a viscous term, is a good approximation under certain conditions. But in order to get an exact idea of what goes on, and justify the usual interpretation, it is necessary to examine in detail the motion of the medium, for it is only by this means that we can prescribe the conditions under which the usual interpretation is valid, and determine when this interpretation fails. Turning to the case of electrical vibrations, it appears to me that Prof. Love’s results have a very important bearing on the questions of electrical inertia and electrical damping. The present paper is an attempt to apply his method to some questions connected with the motion of an electrified spherical conductor.

Societies and Academies

LONDON. Royal Society, March 2, 1905.—“On the Electr Resistance to the Motion of a Charged Conducting Sphere in Free Space or in a Field of Force.” By G. W. Walker. Communicated by Prof. A. E. H. Love, F.R.S.