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Hansen T.B.,Seknion Inc.
IEEE Transactions on Antennas and Propagation | Year: 2014

A diagonal Gaussian translation operator for the time-harmonic fast multipole method (FMM) in two dimensions is examined numerically. The Gaussian translation operator depends on a beam parameter that determines its sharpness. When the beam parameter is set to zero, the Gaussian translation operator reduces to the standard FMM translation operator. The sampling rate can be determined straightforwardly to achieve any desired accuracy. The directionality of the Gaussian translation operator makes it possible to reduce the number of plane waves required to achieve a given accuracy. The required number of plane waves depends strongly on the actual source-receiver locations, not just on the diameter of the source and receiver regions. © 1963-2012 IEEE.


Hansen T.B.,Seknion Inc.
International Journal of Antennas and Propagation | Year: 2012

The system-matrix method for higher-order probe correction in spherical near-field scanning is based on a renormalized least-squares approach in which the normal matrix closely resembles the identity matrix when most of the energy of the probe pattern resides in the first-order modes. This method will be stressed-tested in the present paper by employing probes for which up to 49% of the pattern energy resides in the higher-order modes. The condition number of the resulting normal matrix will be computed, and its distance from the identity matrix displayed. It is also shown how the condition number of the normal matrix can be further reduced. Copyright © 2012 Thorkild B. Hansen.


Hansen T.B.,Seknion Inc.
IEEE Transactions on Antennas and Propagation | Year: 2014

A new exact plane-wave expansion is presented for general time-harmonic electromagnetic fields radiated by an arbitrary finite source region in three dimensions. The plane-wave expansion employs a directional spectrum that is proportional to the far-field pattern of the source. The directionality is achieved through an exact representation that involves a beam parameter, which determines the sharpness of the spectrum. When the beam parameter is set to zero, the new expansion becomes identical to the plane-wave expansion employed in the fast multipole method. The expansion is exact for observations points all the way up to the source region and includes evanescent waves. An antenna-antenna transmission formula follows straightforwardly. Numerical examples demonstrate that the new plane-wave expansion requires fewer plane waves than previously derived plane-wave expansions. © 2014 IEEE.


Analytic continuation of Gegenbauer's addition theorems produces a diagonal Gaussian translation operator for the fast multipole method (FMM) in three dimensions. The Gaussian beams affect only the translation operator, and as usual the field computation is performed with plane waves. Sampling theorems determine the plane-wave sampling rate required by the Gaussian translation operator. The formulation is based on an exact identity, so arbitrarily high accuracy can be achieved. The required sampling rate depends not only on the diameter of the source and receiver regions but also on the actual locations of the sources and receivers within those regions. The directionality of the Gaussian translation operator makes it possible to disregard a large fraction of the plane-wave translations. Numerical simulations reveal that for general source-receiver geometries the required number of plane-wave translations grows linearly with the diameter of the source-receiver groups. © 2013 Elsevier B.V.


An exact complex-space extension of the plane-wave Gegenbauer formula leads to a diagonal Gaussian translation operator for the fast multipole method (FMM) in two dimensions. The Gaussian beams are confined to the translation operator, and the fields are transmitted through plane waves as usual. The regions where the real source and receiver points can reside depend on the beam sharpness. As the beam gets sharper, the transverse dimensions of these regions get smaller. An arbitrarily high accuracy can be obtained with the Gaussian translation operator. The Gaussian translation operator makes it possible to disregard a large fraction of the plane-wave translations. The required sampling rate depends not only on the diameter of the source and receiver regions but also on the actual locations of the sources and receivers within those regions. For some common source-receiver geometries, the required sampling rate is below that of the standard translation operator. For other source-receiver geometries, the required sampling rate is greater than that of the standard translation operator. The theory is validated through numerical examples. © 2013 Elsevier B.V.


Hansen T.B.,Seknion Inc.
IEEE Transactions on Antennas and Propagation | Year: 2011

A general method for higher-order probe correction in spherical scanning is obtained from a renormalized least-squares approach. The renormalization causes the normal matrix of the least-squares problem to closely resemble the identity matrix when most of the energy of the probe pattern resides in the first-order modes. The normal equation can be solved either with a linear iterative solver (leading to an iterative scheme), or with a Neumann series (leading to a direct scheme). The computation scheme can handle non-symmetric probes, requires only the output of two independent ports of a dual-polarized probe, and works for both φ and θ scans. The probe can be characterized either by a complex dipole model or by a standard spherical-wave representation. The theory is validated with experimental data. © 2011 IEEE.


Hansen T.B.,Seknion Inc. | Kaiser G.,Center for Signals and Waves
IEEE Transactions on Antennas and Propagation | Year: 2011

Huygens' relations that express wave fields of primary sources in terms of Huygens' sources on a spherical surface remain valid when the sphere radius is a complex number in a certain bounded domain. Such Huygens relations are derived for frequency and time-domain acoustic and electromagnetic fields. Interior and exterior source configurations are combined to obtain dual-sphere Huygens' relations, in which the primary sources are enclosed by one sphere and the observation points by another. The Huygens sources are complex point sources that exhibit directivity, which for certain parameter ranges makes it possible to achieve high accuracy with only a small fraction of the Huygens surface included in the integration. In general, for a dual-sphere configuration with fixed physical dimensions, the fraction of the spheres required to achieve a given accuracy diminishes with frequency, up to a certain frequency limit. Beyond this upper frequency limit the size of the required regions on the two spheres remain roughly constant. The upper frequency limit is increased when the imaginary part of the complex sphere radius is increased. A similar result holds in the time domain with respect to diminishing pulse width. © 2011 IEEE.


Hansen T.B.,Seknion Inc.
Wave Motion | Year: 2015

An array is constructed to radiate the far-field pattern of a single complex point source. For any nonzero error tolerance, the physical dimension of the array is smaller (sometimes much smaller) than the diameter of the branch-cut disk of the complex point source. The inverse source problem is formulated for non-resonant arrays with reactive zones that do not extend significantly beyond the physical dimensions of the array. Both time-harmonic and pulsed beams are considered. In numerical examples, each array element consists of real point sources in an end-fire configuration. © 2015 Elsevier B.V.


Hansen T.B.,Seknion Inc.
IEEE Transactions on Antennas and Propagation | Year: 2012

Outgoing spherical vector-wave functions are expressed in terms of Gaussian beams (also called complex source-point beams) radiating in all directions. By use of a vector-wave expansion, the electromagnetic field of an arbitrary source of finite extent is thus expressed in terms of Gaussian beams with weights determined directly from the spherical expansion coefficients of the source. These outgoing-wave formulas allow the field of any transmitting antenna to be expressed in terms of Gaussian beams. Elementary Gaussian-beam receivers are introduced as the electromagnetic field at complex points in space. The outputs of elementary Gaussian-beam receivers pointing in all directions determine the spherical expansion coefficients in a standing-wave expansion. These standing-wave formulas allow the output of any receiving antenna to be expressed in terms of the outputs of elementary Gaussian-beam receivers. Combining the formulas for outgoing and standing waves produces a new antenna-antenna transmission formula based solely on Gaussian beams. The theory is exact and validated through numerical examples. © 2006 IEEE.


An exact representation is presented for the field inside a sphere (the observation sphere) due to primary sources enclosed by a second sphere (the source sphere). The regions bounded by the two spheres have no common points. The field of the primary sources is expressed in terms of Gaussian beams whose branch-cut disks are all centered at the origin of the source sphere. The expansion coefficients for the standing spherical waves in the observation sphere are expressed in terms of the output of Gaussian-beam receivers, whose branch-cut disks are all centered at the origin of the observation sphere. In this configuration the patterns of the transmitting and receiving beams "multiply" to produce a higher directivity than is usually seen with Gaussian beams. The areas on the unit sphere, which must be covered by the transmitting and receiving disk normals to achieve a given accuracy, diminish as 1/(ka) for ka→∞, where a is the disk radius and k is the wavenumber. This 1/(ka) behavior leads to a single-level method with O(N 3/2) complexity for computing matrix-vector multiplications in scattering calculations (N is the number of unknowns). © 2010 Elsevier B.V.

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