1. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 1 /19 Near-to-Far-Field Transformation (1 Session) (1 Task)

2. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 2 /19 Near-to-Far-Field Transformation  Plane Wave generation by using a physical reflector:

3. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 3 /19 Near-to-Far-Field Transformation  Plane Wave generation by using a non-physical system in FDTD:  TF/SF region in zoned FDTD space lattice, permits a systematic near-to-far-field (NTFF) transformation.  Using near-field data obtained in a single FDTD modeling run, NTFF efficiently and accurately calculates complete far-field bi-static scattering or radiation pattern of an antenna.  There is no need to extend FDTD space lattice to far field.  Using EM Theorem of Surface Equivalent Theory (SET) and using Green's theorem, scattered or radiated fields that are tangential to a closed virtual surface can be integrated to provide far-field response.  Virtual surface is independent of nature of structure being modeled, and can have a fixed rectangular shape to conform with a Cartesian FDTD grid.  This yields powerful SET in two dimensions, which is subsequently extended to general 3D-case.  Resulting analytical NTFF expressions have been implemented in many FDTD codes.  Next, time-domain NTFF transformation, which permits a direct computation of scattered or radiated field-versus-time waveforms, is generated.

4. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 4 /19 Near-to-Far-Field Transformation It should be emphasized that C a is not a physical surface It is a virtual surface.

5. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 5 /19 Near-to-Far-Field Transformation  Application of Green's Theorem (cont.):  Substituting into S integral:  Therefore the first equation is simplified as:  Far-Field Limit:  Consider analytical form of Green function which in 2D, G(r|r') is given by Hankel function as:  Consider an observation point P =r in far field as:  K|r-r‘| is very large and then limiting expression G(r|r') is:  Using:  Taking square root, and then expanding right side by binomial expansion:  Repeating this square-root procedure:

6. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 6 /19 Near-to-Far-Field Transformation  Application of Green's Theorem (cont.):  Substituting the results:  By using a similar way:  Substituting results in following expression for far-field E z :  Reduction to Standard Form:  in Cartesian coordinates:  Maxwell's equations in 2D let us replace x and y partial derivatives of E z with corresponding magnetic field quantities:  Then, it follows that:  Exploiting a vector identity, we can write:

7. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 7 /19 Near-to-Far-Field Transformation  Substituting into E z (r) yields NTFF transformation in standard form:  Where:  If we now identify a complex-valued pattern function F(φ):  Bistatic radar cross section (RCS) in is defined as:  Because C a can have an arbitrary shape, we can conveniently assign it to lie along a rectangle in scattered-field zone of FDTD grid.  If this rectangle is populated with time-domain E z components, complex phasor values E z of these components are first obtained via a discrete Fourier transform (DFT) algorithm.  Then, phasor E-field data can be used directly in numerical calculation of recent integrals. are tangential equivalent currents observed at C a defined by SET

8. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 8 /19 Near-to-Far-Field Transformation  Obtaining Phasor Quantities Via DFT:  Field data used in NTFF transformation are phasor quantities.  At each field point on virtual surface C a these data can be efficiently and concurrently obtained for multiple frequencies with only one FDTD run.  We need only provide an impulsive wideband electromagnetic excitation of structure of interest, and perform a recursive DFT "on the fly“ (i.e., concurrently with FDTD time-stepping) for each frequency.  Computer storage is quite reasonable, with only two numbers (i.e., the field magnitude and phase) required to store DFT results for each frequency at each field point on virtual surface.  Therefore, a single FDTD run can generate complete far-field distribution of a structure (i.e., its bistatic RCS pattern or its radiation pattern) at many frequencies.  Refer to DFT-code by Fortran in the book as:

9. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 9 /19 Surface Equivalence Theorem  Surface Equivalence Theorem:  knowledge of equivalent currents tangential to any closed contour surrounding a EM wave interaction structure is sufficient to obtain far field via a simple integration.  In fact, we can think of region within observation contour as being source-free and field-free.  This idea, forms basis of surface equivalence theorem (SET) a powerful concept in EM theory.  Degree of accuracy depends on knowledge of tangential components of fields over closed surface.  Final equivalent: This is required to satisfy boundary conditions

10. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 10 /19 Extension to 3D-Phasor Domain  Extension to 3D-Phasor Domain:  Using SET, NTFF transformation can be readily extended to FDTD modeling.  Virtual surface is a six-sided rectangular that encloses structure of interest in scattered-field zone.  Along each side, equivalent J s and M s are calculated using DFTs applied to FDTD-computed H t and E t  Then, these currents are integrated with free-space Green function weighting to obtain far-fields:  where:  Using:  Neglecting radial field components of negligible amplitude compared to θ & φ components:.

11. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 11 /19 Extension to 3D-Phasor Domain  Given Cartesian of FDTD lattice and its NTFF transformation, it is convenient to first calculate N & L in rectangular coordinates, and then transform to spherical coordinates:  For each face of integration surface S:  The two faces of S located at:  Nonzero components:  Exponential phase term:  Integration limits:

12. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 12 /19 Extension to 3D-Phasor Domain  The two faces of S located at:  Nonzero components:  Exponential phase term:  Integration limits:  The two faces of S located at:  Nonzero components:  Exponential phase term:  Integration limits:

13. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 13 /19 Time Domain Near-to-far-field Transformation (TD-NTFF)  Time Domain Near-to-far-field Transformation (TD-NTFF):  An efficient time-domain NTFF transformation, that calculates time waveforms in far field, is proposed by “Luebbers”.  These calculations are performed concurrently with normal FDTD time-stepping.  Luebbers' method involves setting up time-dimensioned arrays for far-field vector potentials.  Each array element is determined by conducting a recursive sum of contributions from time-domain electric and magnetic current sources just computed via FDTD on S.  These contributions are delayed in time according to propagation delay between a source element on S and far-field observation point.  If far-field bistatic RCS pattern or antenna radiation pattern is required at specific frequencies, field- versus-time waveform obtained in this manner can be post processed via an FFT.  By starting:  By applying inverse Fourier transformation to each term of W & U: [8] Luebbers, R. J., K. S. Kunz, M. Schneider, and F. Hunsberger, "A finite-difference time-domain near zone to far zone transformation," IEEE Trans. Antennas Propagat.,Vol. 39, 1991, pp.429-433. (1) (2) (3) (4) is source-to observation point time delay.

14. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 14 /19 Time Domain Near-to-far-field Transformation (TD-NTFF) (5) virtual surface S

15. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 15 /19  Following rectangular-to-spherical vector component conversion is performed:  These time waveforms of vector potentials can be inserted into:  Note that amplitudes of these waveforms diminish as 1/ r, so that it is possible to obtain a normalized far-field response that is independent of distance from origin simply by multiplying by r.  [8] reported a post processing FFT of its normalized far-field waveforms to yield wideband RCS response of generic PEC flat plates.  Very good agreement with frequency-domain MOM data was reported. Time Domain Near-to-far-field Transformation (TD-NTFF) are desired time waveforms of E at observation point r in far zone.

16. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 16 /19  Modified NTFF is used to more accurately calculate backscattering from strongly forward-scattering.  Scattering of biological tissues (having in 1-10λ size range) are generally characterized by strong forward-scattering lobes. (CH16).  Unfortunately, basic NTFF, it has been found to be difficult to apply FDTD to accurately calculate backscattering spectra of strongly forward-scattering objects.  Instead of trying to improve the FDTD near-field calculation, [16] proposes a simple alternative: just omit forward plane of NTFF surface when performing integration to calculate far-field backscattering.  Example: Backscattering of a single PWS, dielectric sphere a=3um of relative permittivity ε r =1.21 Modified NTFF Procedure Sphere is mapped with a staircased surface into a uniform, 3D-FDTD space lattice of cubic cell size Δ=25nm. Two FDTD backscattering spectra are shown for this sphere; One calculated with full NTFF surface integration, Other calculated with a partial surface integration that omits NTFF plane in forward scattering region.

17. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 17 /19  Backscattering spectrum of a PWS to triplet of dielectric spheres of relative permittivity ε r =1.21  Larger sphere is 2um in diameter, and two smaller spheres are 1um in diameter.  Spheres are mapped with staircased surfaces into a uniform 3D-FDTD space lattice of cubic cell size Δ=25nm. Source: Li et all, IEEE Antennas and Wireless Propagation Lett., 2005, pp. 35-38, IEEE.  LN8T1: (P7.4) – 4/24  Implement phasor-domain NTFF transformation in a 2D-TM z FDTD code. Using 2-Methods  Write a DFT subroutine to obtain necessary phasor data at a rectangular virtual surface surrounding structure of interest.  Conduct numerical experiments modeling illumination of a flat PEC plate or square PEC cylinder to obtain complete bistatic RCS pattern.  Compare results with data in literature, such as:  Umashankar, K. R., and A. Taflove, "A novel method to analyze electromagnetic scattering of complex objects," IEEE Trans. Electromagnetic Compatibility, Vol. 24, 1982, pp. 397-405. Modified NTFF Procedure

18. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 18 /19  Summary & introduced topics: 1.Basis of frequency and time domain NTFF transformations for FDTD simulations is introduced. 2.Using phasor-domain Green's theorem to prove that scattered or radiated fields tangential to a virtual surface enclosing structure being modeled can be integrated to provide complete far-field pattern. 3.virtual surface is independent of shape or composition of structure being modeled. 4.Phasor data needed for this calculation can be efficiently obtained during FDTD run by running a concurrent DFT on time-stepped field components tangential to designated virtual surface in lattice. 5.These discussions motivated a review of a powerful phasor-domain surface equivalence theorem. 6.A numerical implementation is used for a time-domain NTFF transformation which permits direct computation of scattered or radiated pulses at a set of angles in far field.  This procedure can be used instead of phasor-domain transformation discussed earlier, if required data are time waveforms, and there are relatively few far-field observation angles of interest. 7.A recent simple modification of NTFF procedure that greatly improves accuracy in calculating backscatter from strongly forward-scattering objects, such as biological cells illuminated by light and certain types of low-observable vehicles. 8.Modification involves simply omitting forward plane of NTFF surface when performing integration to calculate far-field backscattering. 9.This avoids "subtraction noise“ problem posed by requirement for near cancellation of relatively large field values collected on this plane when integrated into far backscattered field. Near-to-Far-Field Transformation

19. Numerical ElectromagneticsLN08_NTFF zakeri@nit.ac.ir 19 /19  Summary on NTFF:  There are two alternatives performing NTFF transformation: 1.On-the-fly (real-time) time-domain transformation that calculates time waveforms of scattered or radiated E-and H-fields at previously specified angular positions in far field. These calculations are performed simultaneously with normal FDTD time-stepping. 2.Frequency-domain transformation that computer transformation of fields in whole far field region at a selected frequency point. First Discrete Fourier Transform (DFT) has to be performed on collected equivalent current densities over virtual surface within FDTD iteration loop. Then based on results near-to-far-field transformation is conducted on specified frequency point.  NTFF transformation alternatives in CADs:  FD near-to-far-field transform: This transform is useful if the user wants the far-field information at a number of prescribed frequencies. A compensation procedure has been implemented for significantly reduce the dispersion error of the incoming wave.  CW near-to-far-field transform: For problems where a continuous wave source with a single frequency is used and an efficient continuous wave near-to-far-field transformation can be utilized.  TD near-to-far-field transform: This transform directly computes the scattered or radiated field versus time during the FDTD time stepping. The frequency domain fields can then be obtained by FFT post-processing. Near-to-Far-Field Transformation