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Bianisotropic multilayer analysis software formulation and user guide

1. W H Press S A Teukolsky W T Vetterling B P Flannery Numerical recipes in Fortran 77 Cambridge university press 1997 S Pancharatnam Achromatic combinations of birefringent plates memoir no 71 of the Raman Research Institute Bangalore pp130 136 March 1955 L D Landau E M Lifshitz Statistical Physics part 1 Elsevier Press 1980 L D Landau E M Lifshitz L P Pitaevski Electrodynamics of continuous media Elsevier Press 1984 CST Computer simulation technology See web site www cst com HFSS 8D Electromagnetic field simulation See web site www ansoft com E Anderson et al LAPACK Users Guide third edition SIAM publishers See also we site http www netlib org lapack lug Q par AJM anlay 1 v2 0 Page 68 of 68
2. and R is the real symmetric rotation matrix assumed independent of frequency R 3 naa 2 68 sinv cosr for real rotation angle v o and ol are generally complex frequency dependent conductivities expressed in Ohms Realisable forms for these may be given using equivalent circuit models as described in section 2 7 These relations are formally exact but only if o and c are known However they can often be estimated using approximate techniques e g using approximate boundary conditions and or low frequency methods and used to model the behaviour of a real frequency selective surfaces embedded in real materials The true structures they represent might be formed using periodic patterns of thin regions of conductor e g copper and or resistive materials e g carbon or indium tin oxide with without lumped passive or possibly semiconductor materials One example employed later of a simple anisotropic surface is a polarising grid with strip width and inter strip spacings small but not very small compared to a wavelength In this case the angle v defines the grid angle c 0 and all oo A more accurate model requires capacitive and inductive models o jwC and al 1 jwL where wC and wL are small Combining 2 64 and 2 65 and using a subscript i to refer to the values for the i interface we obtain E 2 EU e NG CE Aa 2 69 ml Hil JE X md Hz Gaby where I 0 3 mR X R I Sal
3. I is the 2 x 2 identity matrix The inverse SEU eum A fri 1toN 1 2 71 za oft 24i fti 7The accurate determination of the equivalent conductivities of FSS embedded in a general bianisotropic composite is a task requiring advanced numerical techniques outside of the scope of this formulation Commercial software packages such as CST 10 or HFSS 11 may be used for this purpose Q par AJM anlay 1 v2 0 Page 18 of 68 It is possible to employ this relation directly within the previously described method 1 concatenation replacing 2 27 by the concatenation 2 Oy S zw 2 exp jko zu 2 ZN 4 S7 2 exp jko z2 1 Mo S 2 PO 2na S wi exp jko zw 1 zw a S M v 1 3 M N 2 72 This method was adopted within an interim version version 00 02 00 of the software but can be seriously ill conditioned when any all of the X matrices contain both very small and very large terms The reason for this is that such structures are opaque to one polarisation and thus fail for the reasons outlined previously We will not consider this concatenation method any further except by way of a later example illustrating the problem 2 42 Partial waves revisited Although for the reasons outlined previously we are not able to provide a precise and uniformly valid partial wave analysis for general bianisotropic materials we are able to provide a similar method based on a singular value decomposition SVD of the
4. List of contents List of figures 1 Introduction 1 1 Scope of problem 1 2 Polarisation definitions and wave conventions 2 The formulation 2 1 The tensor constitutive relationships 2 2 Some dyadic and matrix notation for describing tensors 2 3 The transverse formulation method 1 2 4 The transverse formulation method 2 2 5 Energy measures and subsidiary quantities 2 6 Special types of materials 2 7 Surface impedance models 3 Software user guide 3 1 Introduction 2 Key word descriptions 3 3 Output file formats 3 4 Example 1 An anisotropic material for polarisation conversion in free space 3 5 Example 2 An anisotropic radome composite in free space 3 6 Example 3 A bianisotropic material in free space 3 7 Example 4 An anisotropic RAM 3 8 Example 5 A new kind of reflection polariser 3 9 Example 6 A sheet polariser showing the problem with formulation 1 3 10 Example 7 Optimised 4 sheet polarisers using formulation 2 4 References Q par AJM anlay 1 v2 0 iv vi vii N e o Ot n A 17 33 35 38 40 40 41 46 48 51 54 57 60 62 64 68 vi List of figures 1 1 2 1 2 2 2 3 2 4 2 5 2 6 2 7 3 1 3 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 3 10 3 12 3 13 3 14 3 15 Plane wave excitation of a multiple layer structure A structure consisting of N 1 layers shown as a free structure A structure consisting of N layers with impedance surfaces between layers The general canoni
5. 2 3 5 Determination of the reflectance and transmittance matrices The physical quantities of interest are the reflectance and transmittance matrices R and T which express the reflection and transmission coefficients in dB and phase for a wave incident on the material In order to obtain R and 7 we transform back into a TM TE coordinate base We therefore define 9 Tg 3 UE BO 2 59 and Ps ty Eya utu 2 60 ti tri The order of entries is merely for historical reasons and matches the order of entries in the computer software Q par AJM anlay 1 v2 0 Page 15 of 68 where abg COS Pin sin Pin a SIN Pin COS Pin 2 61 so that we may finally identify see 2 for details of the derivation of full wave from tangential coefficients co Pits Mua e e EE t 1 C08 Bin ia Tu Ty g t 1 COS Bin tl alae and cos R Ry E E 7 1 COS Gin Ray Ry i rijcos0s e Q par AJM anlay 1 v2 0 Page 16 of 68 2 4 The transverse formulation method 2 2 4 Introduction extensions to method 1 We wish to extend the method to permit an impedance surface of general electric anisotropic form to lie at the interface of any or all of the layers in the multi layer structure This is illustrated is figure 2 2 Note the change in the numbering convention compared with figure 2 1 and the use of i 0 andi N 1 to refer to the left hand and right hand free space semi infinite material
6. TENSOR munamei CONSTANT OVERGEN 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 TENSOR xinamei CONSTANT OVERGEN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 TENSOR zetanamei CONSTANT OVERGEN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Q par AJM anlay 1 v2 0 Page 48 of 68 where the character is not a program input this is shown to represent an extended line in the report The program generates sets of data for 100 frequencies so only the first two output blocks of the file output1 dat are shown These are theta deg 0 0000 phi deg 0 0000 frequency GHz Transmission and Reflection S parameters Index base TE inc TE out TE inc TM out TM inc TE out TM inc TM out 5 0000 9283 dB 7514 dB 7090 dB 9675 dB 5287 dB 9227 dB 4242 dB 0499 dB 3282 7854 2641 T547 T 1 1 1 8248 dB 55 2175 deg T 1 2 6 T 2 1 7 4916 dB 150 0044 deg T 2 2 R 1 1 9 4555 dB 104 9085 deg R 1 2 15 R 2 1 15 7090 dB 120 5267 deg R 2 2 8 TE Transmission Tilt angle degrees 82 5445 Axial ratio TM Transmission Tilt angle degrees 3 1228 Axial ratio TE Reflection Tilt angle degrees 69 0810 Axial ratio TM Reflection Tilt angle degrees 1 4923 Axial ratio input TE perpendicular polarisation balance 1 0000000 input TM parallel polarisation balance 1 0000000 theta deg 0 0000 phi deg 0 000
7. 2 00 0 00 1 00 0 00 There must be at least three rows All frequencies must be specified in sequentially increasing order The minimum and maximum frequencies must lie outside the requested range of frequencies defined by the FREQS tag 3 3 Output file formats 3 3 Format for lt filenamel gt The first output file specified in the FILENAME definition outputs much of the data shown in the log file This contains a list of the components of Z and R calculated at each angle and frequency The tilt angles axial ratios and power balances Prg and Pry are also given a typical output for a lossless material at two frequencies might look like Q par AJM anlay 1 v2 0 Page 46 of 68 theta deg 75 0000 phi deg 30 0000 Transmission and Reflection S parameters Index base TE inc TE out TE inc TM out TM inc TE out TM inc TM out frequency GHz 2176 1096 6687 4963 7080 8038 8725 2742 dB dB dB dB 19 44 25 26 dB dB dB dB 69 13 0428 5360 3351 0222 8358 21 14 1651 9551 2686 T 1 1 8 0023 dB 105 5079 deg T 1 2 25 T 2 1 43 0155 dB 177 5671 deg T 2 2 1 R 1 1 0 7999 dB 159 3768 deg R 1 2 21 R 2 1 28 6721 dB 131 8570 deg R 2 2 6 TE Transmission Tilt angle degrees 85 3825 Axial ratio TM Transmission Tilt angle degrees 0 3101 Axial ratio TE Reflection Tilt angle degrees 85 8520 Axial ratio TM Reflection Tilt angle degre
8. A and As respectively o 8 and y are the real Euler angles in degrees These are defined in section 2 6 3 TAB ORTHOROT refers to a tabulated orthotropic tensor with principal components that are frequency dependent In this case the list of parameters which follow are tabfile name lt a gt 0 vy Q par AJM anlay 1 v2 0 Page 43 of 68 where a 8 and y are the real Euler angles in degrees lt tabfile name gt is a character string specifying the name of an input file supplying the tabulated data see below This is also described in section 2 6 3 CONSTANT OVERGEN refers to a general tensor specified by its Cartesian components The list of parameters which follow are cA gt lt S Are gt lt R Ary gt lt S Ary gt TMA gt lt NA gt RD gt lt S Ayr gt lt R Ay gt lt NAg gt eA gt lt A lt R Azr gt BAG gt lt o4 gt lt SU gt ROCA gt NA Here the character is not a program file input it only has meaning above to signify a line continuation In the program input file all this data is entered on one line and the real and imaginary parts of A are specified in the order shown This is a total of 18 real numbers The definition of quantities is given in section 2 6 4 3 2 8 SURFACE Syntax is SURFACE surface number rotation angle lt name_l gt name 25 This is optional If present SIGMATYPE key words must also be present which define the p
9. Aso Aag 21 then we have mo d u An O 2 Aj 90 An An 0 3 a 2 9 0 0 0 the vectors AO Ag Ag 0 AD 0 2 10 A13 2 AY Ag Ai 2 11 0 0 and the scalar Azz A33 2 12 This decomposition is applied to jj and Q par AJM anlay 1 v2 0 Page 6 of 68 We also follow Rikte et al 1 in defining the 3 x 3 and 2 x 2 unit matrices as 1 0 0 I3 1 9 0 1 0 2 13 0 0 1 1 0 I 1 1 01 2 14 and the rotation matrix JeleL L 3 2 15 with the various notations depending on context Note that we will use bold type face to represent 2 x 2 matrices rather than the dyadic double underline notation when there is little chance of confusion Q par AJM anlay 1 v2 0 Page 7 of 68 2 3 The transverse formulation method 1 2 3 1 Introduction to method 1 In this section we provide a derivation with expanded comments of Rikte s method 1 This forms the basis of out first software implementation up to and including software version 00 01 005 However as we point out there are certain numerical conditioning problems associated with this method These are not usually serious if we do not generalise the method to include thin sheets such as polarising grids or certain types of frequency selective surfaces FSS between layers However we will wish to generalise the method to include such thin sheets and also make the method more stable when employing absorbent materials To this end w
10. degrees 90 0000 Axial ratio TM Transmission Tilt angle degrees 0 0000 Axial ratio TE Reflection Tilt angle degrees 90 0000 Axial ratio TM Reflection Tilt angle degrees 0 0000 Axial ratio input TE perpendicular polarisation balance 0 9983561 input TM parallel polarisation balance 0 9983561 theta deg 0 0000 phi deg 0 0000 frequency GHz Transmission and Reflection S parameters Index base TE inc TE out TE inc TM out TM inc TE out TM inc TM out T 1 1 0 0316 dB 13 2706 deg T 1 2 300 T 2 1 300 0000 dB 0 0000 deg T 2 2 0 R 1 1 23 9798 dB 104 7306 deg R 1 2 300 R 2 1 300 0000 dB 0 0000 deg R 2 2 23 TE Transmission Tilt angle degrees 90 0000 Axial ratio TM Transmission Tilt angle degrees 0 0000 Axial ratio TE Reflection Tilt angle degrees 90 0000 Axial ratio TM Reflection Tilt angle degrees 0 0000 Axial ratio input TE perpendicular polarisation balance 0 9967513 input TM parallel polarisation balance 0 9967513 Q par AJM anlay 1 v2 0 dB dB dB dB 13 75 dB dB dB dB 0000 6453 0000 8882 0000 2706 0000 2694 Page 52 of 68 deg deg deg deg deg deg deg deg Transmission coefficient in dB Co polar TE polarisation 1 Cc 10 15 Co polar TE Transmission Power dB m o 25 1 N 1 ji 1 N 0 5 10 15 20 25 30 35 40 Frequency in GHz Figure 3 2 Co polar Transmission for incident T
11. u is the relative permeability tensor and f and are respectively the xi and zeta chirality tensors These tensors are dimensionless and may be expressed as 3 x 3 matrices eg is the permittivity of free space 8 854 x 10 2 F m in SI units Ng y Ho o is the impedance of free space 376 7 ohms and ug is the permeability of free space 4v x 1077 H m in SI units We will make no assumptions about the realisability of permitted dispersion relationships i e the nature of the frequency dependence of each of these tensors for physically realisable materials This is still a research topic and only limited results are available For most applications the full generality expressed in 2 2 is not required and some special cases are of use and are employed in describing inputs to the software implementation These are described in section 2 6 below However before proceeding further some slightly non standard notation is required 2 2 Some dyadic and matrix notation for describing tensors Rikte et al 1 employ dyadic notations for tensor algebra We have expanded on this slightly to avoid confusion on the dimension of various quantities Firstly all second order tensors can be expressed by square matrices in a specified coordinate base with certain transformation properties Depending on context these matrices and the matrices that operate on them may be either 2 x 2 or 3 x 3 depending on whether they are employed to represent
12. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The predicted reflection coefficients co and cross polar and the axial ratio are shown in figure 3 9 below The axial ratio is less than 1dB over the octave theta 0 phi 0 CP polarising structure Linear polarised input y polarised 15 T T T 10 Reflection Power and Axial ratio dB R 11 y component out R_12 x component out lt X axial ratio reflection x 1 1 1 5 10 15 20 Frequency in GHz 10 Figure 3 9 Output powers and axial ratio for a y polarised incident wave Q par AJM anlay 1 v2 0 Page 61 of 68 3 9 Example 6 A sheet polariser showing the problem with formulation 1 In this example we illustrate why there is a need to employ the more complicated second formulation of section 2 4 rather than the first formulation of section 2 3 when the structure comprises five low resistance five ohm thin polarising grids in free space The input file is given by STRUCTURE 5 FREE 123 4 5 FILENAME outputi dat output2 dat ANGLES 00 0 00 0 1 00 0 0 0 1 FREQS 500 0 500 0 100 MATERIAL 3 0 0030 epsname0 muname0 xinamel zetanamel MATERIAL 4 0 0030 epsname0 muname0 xinamel zetanamel MATERIAL 5 0 0030 epsname0 muname0 xinamel zetanamel MATERIAL 1 0 0030 epsname0 muname0 xinamel zetanamel MATERIAL 2 0 0030 epsnamel munamel xinamel zetanamel TENSOR epsname0 CONSTANT_ORTHOROT 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 TENSOR epsnamei CONSTANT_ORTHOROT 1 0 0
13. 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 TENSOR muname0 CONSTANT_ORTHOROT 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 TENSOR munamei CONSTANT_ORTHOROT 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 TENSOR xinamei CONSTANT ORTHOROT 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 TENSOR zetanamei CONSTANT ORTHOROT 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SURFACE 1 04 0 sigmal sigma2 SURFACE 2 10 0 sigmal sigma2 SURFACE 3 22 5 sigmal sigma2 SURFACE 4 35 0 sigmal sigma2 SURFACE 5 41 0 sigmal sigma2 SURFACE 6 45 0 sigmal sigma2 SIGMATYPE sigmal 1 1 0e 08 0 0 SIGMATYPE sigma2 1 5 0 0 00 Figures 3 10 and 3 11 show the predicted transmission coefficients using the first and second formulations The ill conditioning leads to incorrect predictions with formulation 1 as is clear from the figures The ill conditioning becomes worse the smaller the resistance value defined for sigma2 in the first formulation The second formulation remains stable for very small values currently limited in the surface impedance definition to 0 001 Ohms Q par AJM anlay 1 v2 0 Page 62 of 68 Transmission 0 7371 T T T Ox 2L ki a Li 9 T o 4f T 3 l o K S i 8T AA i iD i a A2 ee i 2 uj Bot sako 5 T2 Be i S pa FH 8 H i A 0 10 20 30 40 50 Frequency in GHz Figure 3 10 Transmission coefficients for a 45 degree polariser using software version 00 02 00 and ill conditioning formulation 1 Transmission
14. 1 Introduction 1 1 Scope of problem There is an increasing use of artificial dielectrics and metamaterials intended for multiple applications in antenna engineering Any material which is uniform on one scale but shows structure on a smaller typically but not always sub wavelength scale can be regarded as such These can often be modelled as linear and passive but where the constitutive relations relating the magnetic B and H fields and electric D and E fields are tensors Many materials exist in nature which require anisotropic tensor relations especially at op tical frequencies but many have been synthesised artificially which have no known natural analogues It is of importance to be able to analyse these materials and to determine the transmission and reflection characteristics Applications include the design of filters polaris ers linear to circular linear to linear etc and as backings for antenna arrays This analysis assumes a layered material with an arbitrary number of layers each composed of a general bianisotropic material Each layer is assumed infinite in lateral extent with finite specified thickness The material is assumed to be impinged upon by a plane wave of arbitrary polarisation and angle of incidence The structure may be open with free space on either side or backed by a perfect conductor In this case its electromagnetic properties are specified by a single reflection and transmission matrix from which other au
15. CONSTANT ORTHOROT or where the A values are frequency dependent and specified by a user supplied data file of tabulated data from which the program interpolates In this option it is assumed that the Euler angles remain independent of frequency The keyword used is TAB ORTHOROT Under TAB ORTHOROT the real and the imaginary parts of A are interpolated indepen dently under cubic spline interpolation using the spline and splint subroutines defined in section 3 3 of 6 A minimum of three data points are required with an interpolation range that covers the requested frequency range Natural splines are assumed with zero second derivative on the boundaries of the data set 89 General realisable dispersion relations for tensors are not known to us so this option must be used with care We have no knowledge of rules concerning realisable frequency dependent changes in the Euler angles associated with more general tensors e g associated with triclinic crystal forms Q par AJM anlay 1 v2 0 Page 36 of 68 2 64 general tensors Most generally tensors may be entered in Cartesian x y z form where u or take the form Aag Ary Arz A Age Ay Ags 2 150 i Box Asy Azz It is assumed that A are defined independent of frequency but may be arbitrary complex numbers General conditions for realisability of materials are not known especially when non zero and are present Note that it is not necessary for a material to be recipro
16. ELA EE W Az Z Ri 1 i jwla4 In all cases we must limit the allowed conductivity to some finite value Too large a value will result in numerical ill conditioning in the determination of the X7 terms of the canonical layer structures We limit values so that the minimum value of Z is 6 0 001 Ohms i e if Z then define Z 6 2 156 Q par AJM anlay 1 v2 0 Page 38 of 68 e 2 oN AR 9 g o m Ar AAAS Model 1 Model 2 Ly ALLL C R4 Ly C S R4 9 S NAA Sb Ao Model 3 Model 4 Figure 2 7 Equivalent circuit models for the principal surface admittances Q par AJM anlay 1 v2 0 Page 39 of 68 3 Software user guide 3 1 Introduction The software is written in extended Fortran 77 which may be compiled under the Linux operating system using the gnu g77 or gfortran Fortran compiler At the time of writing the software version is at version 00 03 002 It is operated by the name of the executable anlay followed by the name of the input file anlay lt input file name gt The input file is a standard ASCII file using key words which are case insensitive Each line contains one or more key words and numerical parameters Lines can generally be entered in any order with the main purpose being defined by the first primary key word of each line If a primary key word is not recognised it will be ignored A list of primary key words are in no particular order FILENAME STRUCTURE ANGLES FREQS MATE
17. FREQS lt f gt lt fs gt ng This specifies the requested range of frequencies defined by f fer holds for all integers 1 lt k n fs and f are real numbers specified in MHz m is a positive integer greater than or equal f to one There can be one and only one FREQS specifier in an input file 3 2 6 MATERIAL Syntax is MATERIAL lt itype gt lt thickness gt lt epsilon name gt lt mu name gt lt xi name gt lt zeta name gt lt itype gt is a positive integer specifier used in the STRUCTURE definition lt itype gt must be unique to each MATERIAL lt thickness gt is a real number specifying the thickness of the layer of material in meters lt epsilon_name gt is a character string defining the name of the relative permittivity tensor defined by a TENSOR key word mu name is a character string defining the name of the relative permeability tensor ji defined by a TENSOR key word E xi name is a character string defining the name of the chirality tensor defined by a TENSOR key word Q par AJM anlay 1 v2 0 Page 42 of 68 lt zeta_name gt is a character string defining the name of the chirality tensor defined by a TENSOR key word i There can be any number of MATERIAL specifiers more than are necessary to define the material However there must be sufficient MATERIAL specifiers to do so If a STRUC TURE definition refers to a MATERIAL specifier lt itype gt that does not exis
18. Given t and r calculation of the R and T matrices follows exactly as before as given in section 2 3 5 Q par AJM anlay 1 v2 0 Page 32 of 68 2 5 Energy measures and subsidiary quantities Conservation of energy implies that for a passive realisable material the total fraction of absorbed incident power converted to heat is given by Pre Ti Ba Ral Ry lt 1 2 131 for an incident TE wave and Prum Hu P Rog Ry KI 2 132 for an incident TM wave If the material is lossless then there is equality in both cases Both Pre and Pry are software outputs and provide a useful diagnostic for program functionality They are also useful for high power applications where in conjunction with other factors they may be used to estimate temperature rise due to heating In general the 7 and R coefficients are complex numbers and express both the phase and amplitude referenced to the first interface z 21 The software expresses these quantities with square magnitudes in dB and phase in degrees Txap 10 log o 7 Rasan 10 logio Ral 2 133 and ephase IB0 m arctana S 2 RC Raxphase 180 m parebans SUR ROUES 2 134 where the subscript refers to any entry or L and the arctan function is the angle unambiguous arc tangent defined by arctano y x arg x jy for real x and y Since T and R transform a linearly polarised wave of either polarisation state into one of arbitrary polarisation we m
19. coincide with the principal axes of the crystal The orthorhombic structure is described by three perpendicular dyads and is of practical use since artificial structures are relatively easy to make on such a lattice The tetragonal and cubic crystal forms are special cases of the orthorhombic and the uniaxial and isotropic tensors are special cases of the orthotropic tensor As above usually employed to describe the relative permittivity tensor or relative perme ability tensor the program is structured in such a way that any constitutive tensor of any material may take this form If a tensor u or is expressed by a generic tensor A then the special orthotropic form is given by A 0 0 AU 0 0 U 2 148 0 0 As where A1 A and 3 are complex numbers and U is a real valued unitary rotation matrix of the form cosy siny 0 1 0 0 cosa sina 0 U siny cosy 0 0 cos sin sina cosa 0 2 149 0 0 1 0 sinG cosZ 0 0 1 where a 9 and y are the real valued Euler angles A general orthotropic tensor is thus specified by three complex and three real independent numbers 9 degrees of freedom When referring to the permittivity or permeability tensors 44 A and A3 should have zero or negative imaginary parts for passive realisable materials The program allows two options for these tensors Either a specification of this tensor which is independent of frequency over the designated range of frequencies using the keyword
20. layer radar absorbing material RAM using the PEC termination option It is taken to include some anisotropy and a small non unity relative permeability but is not intended to represent any particular synthesised material Also note that a constant valued lossy relative permittivity or permeability is not realisable though it is often used as an approximation over a limited bandwidth The input file is given as STRUCTURE 3 PEC 1 2 3 FILENAME outputi dat output2 dat ANGLES 00 0 15 0 5 00 0 0 0 1 FREQS 200 0 200 0 130 MATERIAL 1 0 0100 epsnamel munamel xinamel zetanamel MATERIAL 2 0 0100 epsname2 munamel xinamel zetanamel MATERIAL 3 0 0100 epsname3 munamel xinamel zetanamel TENSOR epsnamei CONSTANT ORTHOROT 1 1 0 3 1 1 0 3 1 2 TENSOR epsname2 CONSTANT ORTHOROT 1 3 0 4 1 3 0 4 1 5 TENSOR epsname3 CONSTANT ORTHOROT 1 5 0 6 1 5 0 6 1 8 TENSOR munamei CONSTANT OVERGEN 1 3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 95 0 1 TENSOR xinamei CONSTANT OVERGEN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 TENSOR zetanamei CONSTANT OVERGEN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 As in previous ezamples the V symbol is used to indicate line continuation A section of data from the file output1 dat is given here theta deg 60 0000 phi deg 0 0000 frequency GHz Transmission and Reflection S parameters Index base TE inc TE out
21. same as the eigenvectors and hence independent of z except possibly in degen erate cases However as previously mentioned P is not a normal matrix Consequently the singular vectors are dependent on layer thickness and the singular values have magni tudes that vary with z For example the singular values do not have unit magnitude for a lossless isotropic material and oscillate with z between defined bounds Also they assume four different values at non normal incidence With the SVD we have DO Fed x No ma U fori 1 N 2 77 1 ooo Nan oy Do w DD O Q par AJM anlay 1 v2 0 Page 21 of 68 where V and U are unitary matrices such that the inverses are equal to the complex conju gate transposes U Ul and V V respectively containing the left and right singular vectors entered as columns and p are the defined positive real singular values The A are ordered such that 1 gt Ag gt 332 A4 The decomposition is well conditioned for any matrix P though as given below we need specific constructions for V and U when there is high level degeneracy for which all singular values are equal Note that V and U are both defined in terms of the electrical properties of layer i 1 as illustrated in figure 2 2 With this ordering of A we define nem fori 1 N 2 78 P3 ziii i not Hay zi pin where the superfix designates the field vector just to the left of the interface at the designa
22. these are known and free of the degeneracy problems associated with generic bianisotropic media as previously discussed There is no SVD em ployed and it is important to realise that except at normal incidence both V and U U yy are non unitary in contrast to the pseudo partial waves defined in the layers This is why we maintain the distinction of notating the inverses of V and U as V and U rather than their complex conjugate transposes when we consider an algorithm valid for all i 0 to N 1 Using 2 44 2 45 2 46 2 48 and 2 49 we may define p 21 pi a V2F a 95 2N uq V2 F 241 pi zw V2F 2ng 2 89 in which case 1 I I i UNT x 2 wt _w 2 90 and a pra kh a we MS 5 Ka 2 91 with the normalisation constant v2 in 2 89 chosen only to ensure that 2 90 and 2 91 are of similar form Note that U ij and LE are not employed in the algorithm and need not be defined Q par AJM anlay 1 v2 0 Page 24 of 68 2 4 5 Canonical structures We now choose to concatenate the layered structure one layer at a time in a manner similar to that employed in 2 for FSS structures or in appendix D of 1 One general canonical structure and one special case canonical structure are required and one optional special case The general structure comprises a single layer terminated on its right hand side by a surface impedance sheet of the kind described in section 2 4 1 The mandatory special
23. where the 2 x 2 block matrix elements are given by Mir 46 0 a ky ko 4 6 9 nue P 6 Bal bo E226 aT eue E22 ky ko eG 2 19 Mia Jp P a ky ko TC Wake Ko hag O I Exe J a J u 9 Gk ko Gg Gas OT 2 20 M2 e O EE ae O y c 554 Kk ko F ae oa J K ko E 9 Cue ce Jk ko eC 0 2 21 M5 Sud ae Liz k ko MENI Eai PI a J K ko E D C k ko S Qu exi NT 2 22 where i a 2 23 Ezz zz Pontus The case where a oo is a theoretical possibility within the formulation for which we take no special care Currently the software will terminate with an appropriate warning if a is excessively large This is one of several special cases where the algorithm is currently ill conditioned but where there is unlikely to be a problem for realistic materials The expressions clearly take simpler forms under the special cases where the materials are isotropic bi isotropic or anisotropic Expressions in these special cases are given in 1 and have been implemented for diagnostic purposes within the software Solution of 2 17 within a single layer is given by n PO oa es 2 24 where P 4 is the matrix exponential PO z 2 exp jko z 1 M 2 25 it is not required to use the special cases within the software except to make diagnostic checks so the routines for special purpose constr
24. with an incident TE L wave and the axial ratio associated with the incident TE wave are illustrated in figure 3 1 Since this is normal incidence the wave is also TM The component shown is for an incident wave polarised in the direction which for o 0 is y polarised theta 0 phi 0 CP polarising structure Linear polarised input y polarised 10 T T T Transmission Power and Axial ratio dB 11 y component out T 12 x component out axial ratio 10 L 1 1 5 10 15 20 25 Frequency in GHz Figure 3 1 Output powers and axial ratio for a y polarised incident wave Q par AJM anlay 1 v2 0 Page 50 of 68 3 5 Example 2 An anisotropic radome composite in free space In this example we replicate Rikte et al s example 8 2 of 1 This represents an anisotropic 3 layer radome structure comprising an ABA structure with Epoxy E glass skins over a low density Rohacell The input file is given by STRUCTURE 3 FREE 1 2 3 FILENAME output2a dat output2b dat ANGLES 00 0 15 0 6 0 0 0 0 1 FREQS 500 0 500 0 81 MATERIAL 1 0 0008 epoxy eglass munamel xinamel zetanamel MATERIAL 2 0 0064 rohacell munamel xinamel zetanamel MATERIAL 3 0 0008 epoxy eglass munamel xinamel zetanamel TENSOR epoxy eglass CONSTANT_OVERGEN 4 444 0 096792 0 0 0 0 0 0 0 0 0 000 0 0 4 444 0 096792 0 0 0 0 0 000 0 0 0 0 0 0 4 23 0 104904 TENSOR rohacell CONSTANT_OVERGEN 1 10 0 00044 0 0 0 0 0 0 0 0 0 0
25. with the special case canonical structure defined above We then use the general case canonical structure to represent the effect of each successive layer until we have completed the composite Suppose we know the pseudo partial R and T waves p 21 and p zi given the pseudo partial incident A and W waves p7 2 and p 21 for that part of the composite composed of the left hand semi infinite medium and layers 1 to i 1 with interfaces 1 to 7 and we wish to add the canonical structure comprising layer i with interface i 1 for i 1 to N This is illustrated in figure 2 6 Let the connection in the partial composite layers 1 to i 1 plus the first interface be defined by the matrix X with sub matrices Xj De B x o a fori 1ltoN 2 121 Q par AJM anlay 1 v2 0 Page 29 of 68 Partial composite Layer i comprising first interface and plus interface i 1 layers 1 to i 1 p zi p 2 x p 2 p i Zi Z Figure 2 6 Concatenation of partial structure with the next layer Ifi N then this refers to the complete composite and we need consider no further concate nation For i lt N we suppose that for the layer plus interface for i gt 1 we employ 2 96 to 2 99 The p pseudo partial waves at the centre of the structure must be well defined in terms of the pseudo partial waves entering the structure at the left of the partial composite and at the right of the i layer We find that Q p
26. x H 2 28 Expressing the vector fields as the sum of transverse and z directed components E E E H H H k ke k 8 ko 2 29 together with the other properties of a plane wave k H 0 k E 2 30 2 31 we obtain ELE NG k H Et 2 5 kf k2 H 2 32 g 53 but eso x ar 5E 2 32 from which we obtain the expression relating tangential components b Hr E P kzxH 2 king 2 33 ko kz If the plane wave is propagating in a forward direction away from the origin with angle 0 to the z axis then k k k kosin k kocos 2 34 Note that according to our definitions the incident and transmitted waves point towards the origin so for these waves 0 Oin T bix whereas for the reflected wave which points away 0 Ore Now let us write H Hy H p in the TE TM coordinate base 2 35 Q par AJM anlay 1 v2 0 Page 12 of 68 E Ej E in the TE TM coordinate base 2 36 for arbitrary coefficients Hj H E and E Substituting into 2 33 we obtain the matrix form in the TE TM coordinate base a No end NE ah a m 2 37 We require in a Cartesian coordinate system such that H A i y 2 38 E TH E 2 39 Transforming between coordinate systems define D ae sing cos 2 40 cosp sing ais E 1 cos0 0 H nM cos 2 2o Post 0 as b Gan since J v vJ we may w
27. 0 0 0 1 10 0 00044 0 0 0 0 0 00 0 0 0 0 0 0 1 10 0 00044 TENSOR munamei CONSTANT_OVERGEN 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 N 0 0 0 0 0 0 0 0 1 0 0 0 TENSOR xinamei CONSTANT OVERGEN 0 0 0 0 0 0 0 0 0 0 0 0 N 0 0 0 0 0 0 0 0 0 0 0 0 N 0 0 0 0 0 0 0 0 0 0 0 0 TENSOR zetaname1 CONSTANT OVERGEN 0 0 0 0 0 0 0 0 0 0 0 0 N 0 0 0 0 0 0 0 0 0 0 0 0 N 0 0 0 0 0 0 0 0 0 0 0 0 where as above the symbol just signifies a line continuation for illustration purposes and is not present in the actual input file where each continued data set appears on the same line The first two blocks of the output file output2a dat are given below The TE and TM plotted data are illustrated in figures 3 2 and 3 3 below using the same scales as in 1 As far as we can tell the results appear to be the same Q par AJM anlay 1 v2 0 Page 51 of 68 theta deg 0 0000 phi deg 0 0000 frequency GHz Transmission and Reflection S parameters Index base TE inc TE out TE inc TM out TM inc TE out TM inc TM out 0000 dB 0116 dB 0000 dB 8785 dB 418 418 370 370 0000 dB 0316 dB 0000 dB 9798 dB 412 412 376 376 7200 7200 2088 2088 7509 7509 3105 3105 T 1 1 0 0116 dB 6 6453 deg T 1 2 300 T 2 1 300 0000 dB 0 0000 deg T 2 2 0 R 1 1 29 8785 dB 98 1118 deg R 1 2 300 R 2 1 300 0000 dB 0 0000 deg R 2 2 29 TE Transmission Tilt angle
28. 0 T T T T 2 L m B I x 9 x S9 r k 8 j o K x c i TA i 2 T 12 x i 2 eL T21 43k i E i T 22 n o Ed i S I i i sk 10 1 1 1 i 0 10 20 30 40 50 Frequency in GHz Figure 3 11 Transmission coefficients for a 45 degree polariser using software version 00 03 02 and no ill conditioning formulation 2 Q par AJM anlay 1 v2 0 Page 63 of 68 3 10 Example 7 Optimised 4 sheet polarisers using formulation 2 In this example we use the second formulation within a simplex optimiser where a certain degree of robustness is required The structure pre supposes three layers of low dielectric constant foam e 1 1 with four idealised polarising grids angled differently on each interface The layers are each 3 2 mm thick with a frequency range nominally between 2 and 42 GHz Analysis is conducted at normal incidence The angle v for the fourth interface is fixed at 45 degrees while the other three values of V are controlled by the optimiser Two examples 7a and 7b are given using different cost functions The input files after optimisation are presented where the frequency range is used to control the cost function Once optimised the frequency range in the input files is extended to present results between 0 2 and 45 GHz 3 10 1 Example 7a In this case an optimiser cost function is defined as the minimum total transmitted power 751 T22 expressed as a power not in dB over the defined frequency interval an
29. 0 frequency GHz Transmission and Reflection S parameters Index base TE inc TE out TE inc TM out TM inc TE out TM inc TM out T 1 1 2 0506 dB 71 1225 deg T 1 2 6 T 2 1 7 3768 dB 132 0308 deg T 2 2 sl R 1 1 8 6690 dB 128 4056 deg R 1 2 17 R 2 1 17 4242 dB 92 6735 deg R 2 2 8 TE Transmission Tilt angle degrees 81 3639 Axial ratio TM Transmission Tilt angle degrees 2 9624 Axial ratio TE Reflection Tilt angle degrees 73 7973 Axial ratio TM Reflection Tilt angle degrees 5 1166 Axial ratio input TE perpendicular polarisation balance 1 0000000 input TM parallel polarisation balance 1 0000000 Q par AJM anlay 1 v2 0 4 4918 0706 6827 x 5 ds 9 7327 135 115 4733 9675 b9 27 dB dB dB dB 151 133 3265 2605 87 16 dB dB dB dB 6844 5389 4798 9821 Page 49 of 68 deg deg deg deg deg deg deg The first 8 columns of the file output2 dat are shown below freq GHz theta deg phi deg t ii db t i12 db t 21 db t 22 db t_11 deg 5 00000 0 00000 0 00000 1 8248 6 9283 7 4916 1 7514 55 2175 5 20000 0 00000 0 00000 2 0506 6 5287 7 3768 1 9227 71 1225 5 40000 0 00000 0 00000 2 2404 6 1519 7 1477 2 0157 86 4358 5 60000 0 00000 0 00000 2 3501 5 8632 6 7908 2 0322 101 3389 When plotted by third party software the 7j Tj linear co and cross polar transmitted power coefficients associated
30. 12 and 3 13 These angles are principally controlled by the requirement for best performance at the top and bottom of the frequency interval 3 10 2 Example 7b In this case an optimiser cost function is defined as the maximum total reflected power orthogonal to the incident principal polarisation R22 expressed as a power not in dB over the defined frequency interval and we wish to minimise this quantity as a function of the angles r4 v and v3 The frequency interval is from 4 0 GHz to 40 6 GHz The final input file after optimisation is given by STRUCTURE 3 FREE 1 2 3 FILENAME outputi dat output2 dat ANGLES 00 0 0 0 1 00 0 0 0 1 FREQS 4000 0 200 0 184 MATERIAL 1 0 0032 epsnamel munamel xinamel zetanamel MATERIAL 2 0 0032 epsname2 munamel xinamel zetanamel MATERIAL 3 0 0032 epsname3 munamel xinamel zetanamel TENSOR epsnamel CONSTANT ORTHOROT 1 1 0 0 1 1 0 0 1 1 0 0 00 0 0 0 0 0 TENSOR epsname2 CONSTANT ORTHOROT 1 1 0 0 1 1 0 0 1 1 0 0 00 0 0 0 0 0 TENSOR epsname3 CONSTANT ORTHOROT 1 1 0 0 1 1 0 0 1 1 0 0 00 0 0 0 0 0 TENSOR munamei CONSTANT OVERGEN 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 N 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 TENSOR xinamei CONSTANT OVERGEN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 TENSOR zetanamei CONSTANT OVERGEN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SURFACE 0 150250E 02 sigmal sigma2 SURFACE 0 252412E 02 sigmal sigma2 1 2 SURFACE 3 0 33
31. 2 lp sing jcosp and we will abbreviate Ej Fin Gin 1 3 because it turns out that we only need these vectors evaluated when Qin The unit vectors 1 3 are the unit projections of 0 and o where A Oin cos Fine sin fin 1 4 We may similarly define unit vectors associated with the transmitted and reflected waves Din m by 1 5 and 0 C08 dine sin Oin O 1 6 an Note the negative sign on the Ey term for due to the angle change in 1 1 If the projection of a wave vector onto the x y plane lies in the Ej direction it is a directed transverse magnetic TM wave If the projection of a wave vector onto the x y plane lies in the direction it is a p directed transverse electric TE wave When fin 0 a wave is both TE and TM or TEM This needs no special case however since both j and are well defined here An incident wave pointing towards the origin has a vector wave number kin k k 1 7 The parallel perpendicular nomenclature can be confusing since some authors refer to the parallel wave as one in which the E field lies parallel in the plane of the material i e in the direction which is the reverse of the definition used here Q par AJM anlay 1 v2 0 Page 3 of 68 where the transverse component k ko sin bin ej k ko cos in 1 8 where kg is the free space wave number kg 2r Ag w co wi
32. 2320E 02 sigmal sigma2 SURFACE 4 45 0 sigmal sigma2 SIGMATYPE sigmal 1 1 0e 08 0 0 SIGMATYPE sigma2 1 0 00 0 0 Note that the optimum grid angles are approximately 15 0 25 2 33 3 and the pre defined 45 Plots showing the reflectivities and tranmissivities in dB are given in figures 3 14 and 3 15 The frequency interval was chosen so that the end points of the interval did not strongly control the performance and the results show an approximately equi ripple R22 as desired Q par AJM anlay 1 v2 0 Page 65 of 68 Reflection Reflection coefficient dB 30 1 J 1 L l l 1 1 0 5 10 15 20 25 30 35 40 45 Frequency in GHz Figure 3 12 Reflection coefficients R41 R45 R21 Rs in dB for example 7a Transmission 0 T T T T T Transmission coefficient dB Ad mE H 12 T21 8r 722 8 10 l L l L i i 1 l 0 5 10 15 20 25 30 35 40 Frequency in GHz Figure 3 13 Transmission coefficients T 715 T21 75 in dB for example 7a Q par AJM anlay 1 v2 0 Page 66 of 68 Reflection oO Ira B m tt ke a n 2 ni z Pox 8 151 i R11 5 S R12 x 1 R1 rea E 2 B R 22 8 Ko d a c 20 f ni T 61 E a o p 25 t 6 a B ti ni ni a tt 30 J p 1 1 1 1 1 H id 0 5 10 15 20 25 30 35 40 45 Frequency in GHz Figure 3 14 Reflection coefficients R11 R45 R21 R22 in dB for example 7b Transmission 0 T T T T
33. 44 lt z lt zi takes the form 2 25 it follows that provided jko z 2 M can be diagonalised by jko z z m 0 0 0 TU g 0 jko z zi M2 0 0 Mapa 0 0 jko z zi M3 0 with a matrix 8 whose inverse ee exists then matrix exponential theory implies that B a 2 75 and the complex eigenvalues are related by lj exp jko z z my for k 1 2 3 4 2 76 and hence the matrix of eigenmodes a is independent of z because 0 is independent of z Because the matrix M is not Hermitian indeed it is not even a normal matrix we can say little in general about the nature of the eigenvalues mg However for lossless isotropic materials m are real and hence l have unit magnitude This also appears to be true for several classes of lossless anisotropic materials Note that if 9 does not exist and this occurs for rank deficient matrices then 2 75 does not generally hold true and a may become dependent on z Rank deficiency is known to occur for certain angles of incidence in certain special kinds of anisotropic materials If one or more of the z take the same value then a is not unique There are differing physical interpretations depending on the nature of the degeneracy For example in an isotropic medium there are generally two pairs of distinct eigenvalues since there is no difference in wave propagation when the direction of the electric field is rotated about the direction of propagation In
34. 9 Provided the pseudo partial waves are correctly identified pj z 1 and p7 zi should have finite magnitudes for any choice of p3 zi and Py 241 with finite magnitudes This implies that X7 must be bounded and hence 4 must exist provided neither 43 nor A are zero The existence of 5 is thus assured if there is a unique ordering of the singular value pairs and if neither A3 nor A4 are zero The ordering problem for all equal singular values is addressed and solved in section 2 4 3 So far we have not come across a problem when A3 and or A4 is close to zero which we expect for very lossy materials but it should be noted that we have no mathematical proof that 4 always exists under these conditions Special case 1 required We now consider the special structure required when the surface is the first interface of the composite This is illustrated in figure 2 4 and we need to determine the X matrices defined with a 7 2 P 21 Xu Xy ka Al 2 100 pe la xi X5 pia 2199 We now define po 11 12 de 2 101 2o E S22 U 5 Ys Kal such that u g p5 21 V p a Q par AJM anlay 1 v2 0 Page 26 of 68 Semi infinite region of free space region 0 p Z4 Figure 2 4 The special canonical structure associated with the first interface The inverse matrix is well defined and given by Es nde am 0 21 S22 m cz oh md zy 2 103 Solution now takes the
35. Bianisotropic multi layer analysis software formulation and user guide Q par AJM anlay 1 v2 0 Q par Angus Ltd F IDEAS ENGINEERED Cover vii 68 pages May 2011 Barons Cross Laboratories Leominster Herefordshire HR6 8RS UK Tel 44 0 1568 612138 Fax 44 0 1568 616373 Web www q par com E mail sales q par com Report This document has been prepared by Q par Angus Ltd and may not be used or copied without proper authorisation Copyright 2011 Q par Angus Ltd U K Q par AJM anlay 1 v2 0 Author Date Issued by Q par AJM anlay 1 v2 0 Q par Angus Ltd May 2011 Q par Angus Ltd Barons Cross Laboratories Leominster Herefordshire HR6 HRS UK Document changes record Issue Date Change summary Issue 1 0 September 2010 First version Issue 1 1 October 2010 Added remarks on general tensor forms Issue 2 0 May 2011 Revised formulation and impedance surfaces Q par AJM anlay 1 v2 0 Abstract This document describes the formulation of theory for the plane wave excitation of multi layered structures containing bianisotropic materials and surface impedance sheets It also provides a user guide for the software constructed around the method and examples showing its use One example shows the design of a novel octave bandwidth linear to circular reflector polariser Q par AJM anlay 1 v2 0 List of contents Document changes record Abstract
36. E waves Transmission coefficient in dB Co polar TM polarisation 0 t4 FH m 2 5 2f o n 5 8 E o c S E er 0deg 15 deg S 30deg 45 deg 5 F 60deg O O 75 deg 6 FH i fi fi fi fi fi 1 fi 0 5 10 15 20 25 30 35 40 Frequency in GHz Figure 3 3 Co polar Transmission for incident TM waves Q par AJM anlay 1 v2 0 Page 53 of 68 3 6 Example 3 A bianisotropic material in free space In this example we replicate Rikte et al s example 8 3 of 1 This represents a bianisotropic single layer structure employed as a model of an N material STRUCTURE 1 FREE 1 FILENAME output3a dat output3b dat ANGLES 00 0 2 0 45 0 0 2 0 46 FREQS 10000 0 00 0 1 MATERIAL 1 0 030 epsnamel munamel xinamel zetanamel TENSOR epsname1 CONSTANT OVERGEN 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 TENSOR munamei CONSTANT OVERGEN 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 TENSOR xinamei CONSTANT OVERGEN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 TENSOR zetanamei CONSTANT OVERGEN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 As above the syntax is used just to identify a line continuation Also note that the non zero entries in the and tensors are imaginary and are thus of opposite sign to that employed in example 8 3 of 1 due to the a
37. P matrices We will call this a quasi partial wave analysis In what follows we begin with a description of a proper eigenmode analysis and describe some of the problems involved We then proceed with the SVD showing some relationships and some important differences between the two approaches A full partial wave analysis decomposes the total tangential electric and magnetic fields on either side of an interface in terms of the tangential components of forwards and backwards wave components that correspond to the eigenmodes within the medium the given side of the interface These wave components represent a set of linear combinations of the E and nod Hay vector fields dependent on the constitutive properties of the layer but independent of the position within the layer Within the layer we may write this set of linear combinations by the 4 x 4 matrix a such that h z 0 0 0 E 2 0 b z 0 0 E 2 Ty ps un l0 0 bo BA tnd a A 0 0 0 LC 2 73 and formally we may identify L 000 al koO0 0 P 2 41 zi a 0 0 ls 0 a 2 74 0004 Q par AJM anlay 1 v2 0 Page 19 of 68 Such a diagonalisation is obtained if the rows of a are the eigenmodes of the medium expressed in the E noJ gel base and the z represent the complex modal propagation factors whose magnitude and phase are generally both functions of z Suppose M is that matrix M independent of z corresponding to the ij layer Because PO zzi for 2
38. RIAL TENSOR SURFACE SIGMATYPE Used to designate output file names generated by the software Top level description of the structure in terms of materials specified by the MATERIAL type Used to define the range of angles of incidence required Used to define the range of frequencies required Referenced by the STRUCTURE key word Designates a material by its thick ness and TENSOR names used to represent e y and Referenced by the MATERIAL key word Defines a tensor by tensor case name and material parameters Optional word which if present defines a surface impedance sheet associated with the 7 interface specified by a rotation angle v and two principal admit tance pointers Referenced by the SURFACE key word Defines the principal admittance specified by a model number and a list of equivalent circuit parameters Q par AJM anlay 1 v2 0 Page 40 of 68 3 2 Key word descriptions 3 2 4 General comments All key words and numbers are separated by either spaces or commas Either may be used Each primary key word is associated to a single line 3 2 2 FILENAME Syntax is FILENAME lt filenamel gt lt filename2 gt lt filenamel gt and lt filename2 gt are character strings They are the names of output files lt filenamel gt displays data in the same manner as the log file lt filename2 gt outputs data in column format see example of output files There can be one and only one FILENAME specifier in an
39. T T T T 2r ea 9 5 a 4f g oO n o i o L4 c R 5 7 i 6 H 7 i i iS i H Eo H i k udi AA i l Bi i T 21 ea i SEE TOD weld i 10 i 1 1 1 L l L L 0 5 10 15 20 25 30 35 40 45 Frequency in GHz Figure 3 15 Transmission coefficients 711 715 T21 759 in dB for example 7b Q par AJM anlay 1 v2 0 Page 67 of 68 10 11 12 References Sten Rikte Gerhard Kristensson Michael Andersson Propagation in bian isotropic media reflection and transmission Report CODEN LUTEDX TEAT 7067 1 32 1998 Lund Institute of Technology Department of Electroscience Sweden Available on web site http www es lth se teorel Publications TEAT 7000 TEAT 7067 pdf Andrew Mackay The mathematical formulation of QDAS Q par Dichroic Array Software Report Q par FSS TRFSS2 1 1 Q par Angus Ltd UK April 2003 Available on web site http www q par com Cleve Moler Charles V Loan Nineteen dubious ways to compute the exponential of a matrix twenty five years later SIAM review Vol 45 No 1 pp1 46 February 2003 Available on web site http www cs cornell edu cv ResearchPDF 19ways pdf Roger B Sidge EXPOKIT Software package for computing matrix exponentials Transactions on Mathematical Software 1998 Documentation and source code available on web site http www maths uq edu au expokit W V T Rusch P D Potter Analysis of reflector antennas Academic Press 1970
40. TE inc TM out TM inc TE out TM inc TM out T 1 1 300 0000 dB 0 0000 deg TG T 2 1 300 0000 dB 0 0000 deg T 2 2 R 1 1 5 5878 dB 79 1200 deg R 1 2 R 2 1 300 0000 dB 0 0000 deg R 2 2 TE Transmission Tilt angle degrees 45 0000 Axial TM Transmission Tilt angle degrees 45 0000 Axial TE Reflection Tilt angle degrees 90 0000 Axial TM Reflection Tilt angle degrees 0 0000 Axial input TE perpendicular polarisation balance 0 276200 input TM parallel polarisation balance 0 389305 300 300 300 A ratio ratio ratio ratio 2 9 2 0000 0000 dB 0 0000 deg 0000 dB 0 0000 deg 0000 dB 0 0000 deg 0971 dB 163 9202 deg 324 2607 dB 324 2607 dB 394 5698 dB 407 0541 dB Q par AJM anlay 1 v2 0 Page 57 of 68 The transmission coefficient for a PEC backed structure is defined as zero which in dB terms is assigned a finite number 300 dB The axial ratios and tilt angles attributed to transmission are similarly ill defined in this instance but are assigned numerical values without causing software faults Figures 3 6 and 3 7 show the co polar TE and TM reflection coefficients as a function of frequency and incidence angle This example predicts that deliberate use of anisotropy has the potential to equalise the TE and TM reflection coefficients as the angle of incidence is changed Often and this is the case here if the materials are isotropic there is an increasing diverge
41. ar AJM anlay 1 v2 0 Page 30 of 68 P3 zi I X Xa Xi p 21 pues XXa X 2X d0 VF Gi 2 122 Py z X53 XXa X p 21 T xaa x XXa Xh T I X pt zim Substituting these expressions into the right hand sides of 2 121 and 2 92 we obtain the new matrix terms for the concatenated structure defined by the matrix Y with 2 x 2 bock matrices Y where Dy 2 41 Yu Yn p 41 en 2 123 p 21 Ya Y Py 241 we obtain Yi LXi 2 124 Yi Li X oX Xj 2 125 You X5 X5 LoX 2 126 Y XS LX I X7 2 127 where L Xt I XX 2 128 L XA E XX 2 129 Because we are employing pseudo partial waves the fields at the centre of the structure are guaranteed to exist given the waves entering the structure from the left and right and hence the matrix inverse appearing in the definition for L and L must exist Q par AJM anlay 1 v2 0 Page 31 of 68 The algorithm for determining Y for the entire structure for which i N proceeds as Step 1 Let X X for the special case first interface i 0 Step 2 Determine Y for i 1 layer 1 Step 3 Let X Y Step 4 Determine Y for next value of i and continue loop until 7 N in which case X is evaluated with i N X X 2 4 7 Determination of the tangential transmission and reflection matrices t and r Determination of the t and r matrices is trivial in this formulation In this case we simply have t Xi wheni N
42. ay also define the axial ratio and tilt angle of a transmitted and reflected wave for such an assumed linearly polarised wave We write T directed TE incident wave V Vilet ot i 2 135 i Male Ti directed TM incident wave T directed TE incident wave V Valei i cl T 2 136 2 Vale Tjj directed TM incident wave and similarly for the reflection coefficients with 7 replaced by R Using definitions from 5 let a P2 Pi 2 138 2 137 Q par AJM anlay 1 v2 0 Page 33 of 68 then the tilt angle is define by 1 AG arctan 2 V V2 cos V V3 and the axial ratio x given by v2 pae 2 where Vil Val 2 cos 6 Val Val sin Again x is expressed in dB by XaB 10 logio x since X gt 1 is real and positive Q par AJM anlay 1 v2 0 2 139 2 140 2 141 2 142 Page 34 of 68 2 6 Special types of materials 2 6 1 Isotropic tensors An isotropic material is defined by one for which 0 2 143 r13 2 144 and L ula 2 145 No special data entry is given for such materials but may be defined using any of the more general input options defined below For a realisable passive material e and u are complex numbers with zero or negative imaginary parts 2 6 2 Uniaxial tensors Usually employed to describe the relative permittivity tensor but also useful to describe the relative permeability tensor A u
43. cal The keyword employed for such tensors is CONSTANT OVERGEN Some general conditions for tensors are known to us some of which are listed below and may be useful in investigations of physically realisable materials Concerning the relative permittivity and permeability tensors For general lossless anisotropic e g triclinic structures in the absence of magnetic struc tures or strong external magnetic fields e and pu are real symmetric This implies that the principal directions of the tensors are orthogonal and real they correspond with orthogonal directions in real space Note however that for triclinic magnetic crystals the principal directions of e and u may vary with frequency and will not in general coincide A proof of this is given in 9 section 96 combined with 8 section 125 though it hinges on some very abstract physics depending on time reversibility which is violated in the presence of large i e a lot larger than the wave field perturbations magnetic fields and in the presence of magnetic structures or for certain classes of non linear effects Unfortunately what counts as a suitably violating magnetic structure is not defined In the presence of strong magnetic fields and presumably other kinds of time symmetry breaking effects a lossless material has tensors which are complex Hermitian rather than real symmetric In this case the tensor directions are orthogonal but complex and hence do not
44. cal structure associated with layer 7 The special canonical structure associated with the first interface The special canonical structure referring to layer terminated by a perfect conductor Concatenation of partial structure with the next layer Equivalent circuit models for the principal surface admittances Output powers and axial ratio for a y polarised incident wave Co polar Transmission for incident TE waves Co polar Transmission for incident TM waves Transmittance Trp for incident TE waves linear scale Transmittance T for incident TE waves linear scale Co polar Reflection for incident TE waves Co polar Reflection for incident TM waves An ABA 2 PEC terminated structure for the design of a reflector polariser linear to circular Output powers and axial ratio for a y polarised incident wave Transmission coefficients for a 45 degree polariser using software version 00 02 00 and ill conditioning formulation 1 Transmission coefficients for a 45 degree polariser using software version 00 03 02 and no ill conditioning formulation 2 Reflection coefficients 7941 A45 R21 Rs in dB for example 7a Transmission coefficients 711 715 721 72 in AB for example 7a Reflection coefficients 7941 45 31 Rs in dB for example 7b Transmission coefficients 711 715 721 72 in dB for example 7b Q par AJM anlay 1 v2 0 10 17 25 2T 28 30 39 50 53 53 56 56 59 59 60 61 63 63 66 66 67 67 vii
45. coincide with directions in real space For lossy materials in the absence of time reversal breaking effects e and u are complex symmetric not Hermitian Both the real and imaginary parts of the tensors are symmetric with real orthogonal principal directions that do not generally coincide For lossy materials in the presence of time reversal breaking effects the tensors can be neither complex symmetric nor complex Hermitian For a passive material the principal values eigenvalues must be complex with zero or negative imaginary part What constraints this places on the matrix terms or on the principal directions is not known to me Q par AJM anlay 1 v2 0 Page 37 of 68 2 7 Surface impedance models Any models for determining the principal surface admittances c and ol associated with a surface impedance interface should be physically realisable and obey causality A convenient way to guarantee this is to model frequency dependence using equivalent circuits Any number are possible but currently as of version 00 03 00 of the software we have four forms which are useful for the modelling of simple resonant and non resonant structures In what follows we define 1 1 PAGE a Or Zu gli 2 151 The equivalent circuits are shown in figure 2 7 with Z given as follows Model 1 Z Ry jw Ly 2 152 Model 2 i 1 jo C 2 153 2 K IWC Model 3 1 Z R jw L 2 154 i UG jwLA Model 4 l jwCi 2 155
46. d we wish to maximise this quantity as a function of the angles r4 v and v3 The frequency interval is from 1 6 GHz to 41 6 GHz The final input file after optimisation is given by STRUCTURE 3 FREE 1 2 3 FILENAME outputi dat output2 dat ANGLES 00 0 0 0 1 00 0 0 0 1 FREQS 1600 0 200 0 201 MATERIAL 1 0 0032 epsnamel munamel xinamel zetanamel MATERIAL 2 0 0032 epsname2 munamel xinamel zetanamel MATERIAL 3 0 0032 epsname3 munamel xinamel zetanamel TENSOR epsname1 CONSTANT ORTHOROT 1 1 0 0 1 1 0 0 1 1 0 0 00 0 0 0 0 0 TENSOR epsname2 CONSTANT_ORTHOROT 1 1 0 0 1 1 0 0 1 1 0 0 00 0 0 0 0 0 TENSOR epsname3 CONSTANT_ORTHOROT 1 1 0 0 1 1 0 0 1 1 0 0 00 0 0 0 0 0 TENSOR munamei CONSTANT_OVERGEN 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 N 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 TENSOR xinamel CONSTANT OVERGEN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 TENSOR zetanamei CONSTANT OVERGEN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SURFACE 0 202284E 02 sigmal sigma2 SURFACE 0 285053E 02 sigmal sigma2 1 2 SURFACE 3 0 366852E 02 sigmal sigma2 SURFACE 4 45 0 sigmal sigma2 SIGMATYPE sigmal 1 1 0e 08 0 0 SIGMATYPE sigma2 1 0 00 0 0 9 Using in house software optimiz Q par AJM anlay 1 v2 0 Page 64 of 68 Note that the optimum grid angles are approximately 20 2 28 5 36 7 and the pre defined 45 Plots showing the reflectivities and tranmissivities in dB are given in figures 3
47. e directions in the material An example of this is the conical refraction effect where there is a cone of permitted wave directions in the material for a given incident wave For many examples where materials are not very lossy or as is the case in this section we are not coupling FSS interfaces with strong evanescent fields through such media the numerical ill conditioning appears to be rarely problematic and was not an issue in the examples presented in 1 and duplicated later in this report However if software such as formulated for QDAS 2 were to employ such materials it is necessary to solve this problem One but not the only consequence of such ill conditioning is for the appearance of large numbers in P 9 z N 1 21 Some tests and diagnostics are included within the software to check for such problems but ill conditioning can also manifest by a successive degradation in accuracy resulting in poor condition numbers for certain matrix inverses required in the next section Q par AJM anlay 1 v2 0 Page 11 of 68 2 3 4 Determination of the tangential reflection and transmission matrices In this section we employ the same formulation as given 1 but derive the results is a different manner Other than to provide confirmation we believe it provides a slightly more intuitive picture in terms of TE and TM plane wave components Firstly any plane wave in free space satisfies the relation between E and H fields given by E nok
48. e will also detail a second novel formulation in section 2 4 which will form the basis of software version 00 03 00 The first subsection section 2 3 2 describes formulation elements which are common to all our transverse methods Section 2 3 3 outlines Rikte s basic concatenation method with no thin sheets between layers Section 2 3 4 describes the derivation of the reflection and transmission coefficient matrices employing the above concatenation method Some of the terms defined here will be employed in later formulations but it is essentially descriptive of method 1 Section 2 3 5 derives the reflectance and transmittance matrices given the tangential reflection and transmission matrices This section is common to all methods 2 3 0 Common formulation elements Within a homogeneous region i e within one of the layers of a layered material Maxwell s equations can be written in terms only of the transverse field components If we write the x and y components of the field as the 2 component column vectors Eu EO Es Ey y H oy HY Hz Hy 2 16 then it can be shown 1 that all SA jkyM 0 ty 2 1 dz Nod H y 2 Ha TZ H y 2 7 where M M 9 is a 4 dimensional matrix containing terms of the relative permittivity permeability and chirality tensors as defined in appendix C of 1 For a general bianisotropic medium Mir Mie 2 18 Mar Mo 2 18 IS Q par AJM anlay 1 v2 0 Page 8 of 68
49. es 3 4146 Axial ratio input TE perpendicular polarisation balance 1 0000000 input TM parallel polarisation balance 1 0000000 theta deg 75 0000 phi deg 30 0000 frequency GHz Transmission and Reflection S parameters Index base TE inc TE out TE inc TM out TM inc TE out TM inc TM out T 1 1 3 1185 dB 131 2179 deg T 1 2 19 T 2 1 21 9522 dB 68 4933 deg T 2 2 0 R 1 1 3 1780 dB 137 3807 deg R 1 2 16 R 2 1 18 6051 dB 104 7379 deg R 2 2 8 TE Transmission Tilt angle degrees 81 5764 Axial ratio TM Transmission Tilt angle degrees 0 3104 Axial ratio TE Reflection Tilt angle degrees 84 0736 Axial ratio TM Reflection Tilt angle degrees 12 0733 Axial ratio input TE perpendicular polarisation balance 1 0000000 input TM parallel polarisation balance 1 0000000 Q par AJM anlay 1 v2 0 129 134 5728 9853 b7 171 dB dB dB dB 48 162 103 8164 152 dB dB dB dB 1274 8070 6547 0109 5373 Page 47 of 68 deg deg deg deg deg deg deg deg 3 3 2 Format for lt filename2 gt The second output file contains data in column format column 1 column 2 column 3 columns 4 5 6 7 columns 8 9 10 11 columns 12 13 14 15 columns 16 17 18 19 column 20 21 column 22 23 frequency in GHz theta 0 in degrees phi Pin in degrees The square magnitude of the 7 entries T 1 1 T 1 2 T 2 1 and T 2 2 respectively
50. f l values at least one of whose magnitude becomes exponentially large as the layer thickness and or the loss factor becomes large Similarly the second pair must feature at least one value whose magnitude become exponentially small Somehow we need to regroup the fields at z and z so that something similar to a re flected backwards propagating wave at z and a transmitted forwards propagating wave at 2441 are determined as a function of the backwards incoming wave at z and the forwards incoming wave at 2 41 This interpretation would be precise if we had a true partial wave decomposition However all that is required is a similarity such that the exit fields contain no exponentially large terms in the dependence on the entry fields One possibility is to use the eigenvalue magnitudes in order to re group the fields eigenvectors when they are unique as exit entry pairs Motivated by this philosophy instead of performing the eigenvalue decomposition of PW we perform its singular value decomposition SVD This has various advantages Firstly the SVD is inherently better conditioned that the eigenvalue decomposition Secondly we are only interested in the magnitudes of the eigenvalues There are important differences though If P is a normal matrix the singular values of the SVD and the magnitudes of the eigen values take the same values Also the singular vectors would be up to a constant phase factor the
51. f version 00 03 02 of the software there are four model types 1 2 3 or 4 defined Parameters list of parameters are a list of real numbers which specify the equivalent circuit parameters associated with a given model number These comprise resistances in Ohms capacitances in pF and inductances in nH For model number 1 parameter 1 is R and parameter 2 is L For model number 2 parameter 1 is Ri and parameter 2 is C1 For model number 3 parameter 1 is Rj parameter 2 is Lj and parameter 3 is C1 For model number 4 parameter 1 is Rj parameter 2 is Lj and parameter 3 is C1 Q par AJM anlay 1 v2 0 Page 45 of 68 3 2 10 Data format for tabulated input file lt tabfile_ name gt This file is required if the TAB ORTHOROT option is used on the TENSOR definition The input file is Fortran free format consisting of a column of frequencies and three columns of complex numbers Entry on each row is of the form lt fs gt lt Aqu BONG gt lt Nga gt where f is a frequency specified in MHz and Aj A2 and As are those values of A1 Ag and A3 defined at the frequencies f f is a real number v As and A3 are complex numbers For example a typical input file specifying A at 7 9 10 12 and 15 GHz might take the form 7000 0 3 00 0 00 2 00 0 00 1 00 0 00 9000 0 4 00 0 00 3 00 0 00 2 00 0 00 10000 0 7 00 0 00 4 00 0 00 2 00 0 00 12000 0 4 00 0 00 3 00 0 00 2 00 0 00 15000 0 3 00 0 00
52. full component or transverse component fields Except where specifically noted we employ a Cartesian x y z coordinate base Similarly vectors may be either of dimension 3 or dimension 2 Where there is possible confusion we will use the superscript n to represent an n dimensional quantity In this report n may be 2 3 or 4 Q par AJM anlay 1 v2 0 Page 5 of 68 There are two basic tensor operations an outer and an inner product The outer product of two 3 dimensional vectors is the matrix defined by ay a b Q4 02 a163 ab a p a2 bj by b3 ab agb b3 2 3 a3 a3b a309 a3b3 The inner product of two 3 dimensional vectors is the usual dot product a b a b a24 a3b3 2 4 We do not require the outer product of matrices or the outer product of a matrix with a vector so these are not defined here The inner product of two matrices is the standard matrix matrix multiplication and will be given no special notation The inner product of a matrix with a vector depends on the order of operation as An A12 A13 by A A22 A3 b 2 5 A31 A32 A33 b3 Im IIl gt T I or An A1 A13 b A bTA ATb T b bo b3 As A9 A3 2 6 A31 A32 A33 Using this notation we have a tensor decomposition A A 843A94 AP 34 4 22 2 7 In a Cartesian coordinate base such that the matrix A transforms between the Cartesian vectors x1 Y1 21 and x2 Y2 22 To Ay Arn A13 T Y2 x Aoi Ag Aog Yi 2 8 22 Az
53. ilt angle degrees 85 8374 Axial ratio 41 5085 dB TM Transmission Tilt angle degrees 3 7623 Axial ratio 43 2836 dB TE Reflection Tilt angle degrees 89 5623 Axial ratio 44 3287 dB TM Reflection Tilt angle degrees 0 4394 Axial ratio 40 4651 dB input TE perpendicular polarisation balance 1 0000000 input TM parallel polarisation balance 1 0000000 Excitation is with a plane wave at a single frequency where the material is exactly one wavelength thick kod 27 Three dimensional plots are given of the total transmitted power as a function of both in and Qin for the TE and TM polarisations These are defined in our notation by Trp Tru Tl Pal Te 3 1 These are shown in figures 3 4 and 3 5 and appear to agree well with the figure 8 predictions of 1 The results actually differ little from results not shown when 0 except for angles approaching grazing incidence Q par AJM anlay 1 v2 0 Page 55 of 68 deg deg deg deg deg deg deg deg TE transmission total power coefficient linear power scale ae 0 4 0 2 50 theta in degrees Figure 3 4 Transmittance Trp for incident TE waves linear scale TM transmission total power coefficient linear power scale at Figure 3 5 Transmittance Trym for incident TE waves linear scale Q par AJM anlay 1 v2 0 Page 56 of 68 3 7 8 Example 4 An anisotropic RAM In this example we consider a three
54. input file 3 2 5 STRUCTURE Syntax is STRUCTURE lt n gt type lt iji ip gt lt n gt is an integer defining the number of layers lt type gt is a character string which must be either FREE or PEC If the former the structure is assumed to lie in free space with air vacuum as the left and right half spaces If the latter it is assumed that the structure is terminated by a perfect conductor on its right hand side The integers 7 to in are positive integers There must be n of them but they need not be unique Each number refers to a MATERIAL type defined below The materials to which they refer must be defined but materials can be defined which are not used in the structure The numbers reference in sequence starting from the left most material on which the incident wave impinges to the right most material the materials employed in the multi layer structure There can be one and only one STRUCTURE specifier in an input file 3 2 4 ANGLES Syntax is ANGLES lt b gt lt bs gt no lt Qs gt ds no gt Q par AJM anlay 1 v2 0 Page 41 of 68 This specifies the requested range of incident angles defined by Pin bs j 1 0s for all integers 1 lt i ng and 1 j lt ng 0 05 Ps and s are real numbers specified in degrees ng and ng are positive integers greater than or equal to one There can be one and only one ANGLES specifier in an input file 3 2 5 FREQS Syntax is
55. lar vectors may leave sub blocks of e3 U mO containing rows or columns of zeros and hence leave the sub blocks non invertible To avoid the problem we adopt a special procedure if the ratio A4 X gt 10 e 2 86 where e is some small non zero number currently we use a value e 10 If 2 86 is true then we define lala gt 5 2 87 Q par AJM anlay 1 v2 0 Page 23 of 68 where K is chosen as a unitary matrix whose 2 x 2 block sub matrices are invertible A real symmetric matrix satisfying this requirement is the Klein matrix E gt ud 1 al 1 1 1 1 1 ep c 1 K 5 5 2 88 1 1 1 1 The special form 2 87 satisfies the SVD when all singular values are equal in such a manner that all 2 x 2 sub blocks of U and V are invertible this is not the default equal singular value definition of U and V under the ZGESVD algorithm It may be adopted as a default condition for any SVD calculated in this report and employed as part of a wrapper for the ZGESVD algorithm 2 4 4 Special U and V for the left and right semi infinite media In section 2 4 2 we defined U and V for i 1 to N However we also require a partial or pseudo partial wave description i in the semi infinite left hand and right hand half planes In particular Ve and U ed and their inverses must be defined We are free to do this in various ways but it is convenient to define them in terms of the true partial waves in free space since
56. lternative i j harmonic sign convention as described in the introduction of this report The first two blocks of the output file output3a dat are given below Q par AJM anlay 1 v2 0 Page 54 of 68 theta deg 0 0000 phi deg 0 0000 frequency GHz 10 0000 Transmission and Reflection S parameters Index base TE inc TE out TE inc TM out TM inc TE out TM inc TM out T 1 1 2 1270 dB 72 2058 deg T 1 2 300 0000 dB 0 0000 T 2 1 300 0000 dB 0 0000 deg T 2 2 1 2374 dB 95 2271 R 1 1 4 1203 dB 162 2058 deg R 1 2 300 0000 dB 0 0000 R 2 1 300 0000 dB 0 0000 deg R 2 2 6 0568 dB 5 2271 TE Transmission Tilt angle degrees 90 0000 Axial ratio 398 2988 dB TM Transmission Tilt angle degrees 0 0000 Axial ratio 398 7988 dB TE Reflection Tilt angle degrees 90 0000 Axial ratio 406 1766 dB TM Reflection Tilt angle degrees 0 0000 Axial ratio 414 7526 dB input TE perpendicular polarisation balance 1 0000000 input TM parallel polarisation balance 1 0000000 theta deg 0 0000 phi deg 2 0000 frequency GHz 10 0000 Transmission and Reflection S parameters Index base TE inc TE out TE inc TM out TM inc TE out TM inc TM out T 1 1 2 1491 dB 72 1889 deg T 1 2 24 8513 dB 101 1878 T 2 1 24 8513 dB 101 1878 deg T 2 2 1 2573 dB 95 2133 R 1 1 4 1226 dB 162 2179 deg R 1 2 44 3328 dB 123 7232 R 2 1 44 3328 dB 56 2768 deg R 2 2 6 0545 dB 5 2460 TE Transmission T
57. nce between TE and TM results as 0 is increased Q par AJM anlay 1 v2 0 Page 58 of 68 Reflection coefficient in dB A three layer radar absorbent material R 11 co polar TE theta 00 deg R 11 co polar TE theta 15 deg R 11 co polar TE theta 30 deg R 11 co polar TE theta 45 deg R 11 co polar TE theta 60 deg Frequency in GHz Figure 3 6 Co polar Reflection for incident TE waves Reflection coefficient in dB A three layer radar absorbent material 25 R 22 co polar TM theta 00 deg R 22 co polar TM theta 15 deg R 22 co polar TM theta 30 deg R2 R2 2 co polar TM theta 45 deg 2 co polar TM theta 60 deg Frequency in GHz Figure 3 7 Co polar Reflection for incident TM waves Q par AJM anlay 1 v2 0 Page 59 of 68 3 8 Example 5 A new kind of reflection polariser In this example we have used the software to explore a possibility for designing a linear to circular polariser that operates in reflection over an octave bandwidth To our knowledge there are currently no analytical methods unlike the transmission mode device 7 for the design of wide band polarisers of this type but we expect a number of possible applications for high power reflector antennas It would appear that an A B A 2 structure backed by a perfect conductor shows good performance This is a structure as sketched in figure 3 8
58. niaxial tensor is associated with a tetragonal or hexagonal crystal form Isotropic tensors are a special case of the uniaxial The program is structured in such a way that any constitutive tensor of any material may take this form There is no requirement that tensors share a common set of principal axes If a tensor u or is expressed by a generic tensor A then the special uniaxial form is given by A ail UU Agtt 2 146 where amp is a real unit vector of arbitrary direction and a and a are arbitrary complex numbers When referring to the relative permittivity or relative permeability a and a must have zero or negative imaginary part for a realisable material In the software t is defined by un normalised component inputs u1 us u3 such that ui us ua AJ u ui us 2 147 A general uniaxial tensor is thus specified by two complex and two real independent num bers 6 degrees of freedom The program assumes a specification of this tensor which is independent of frequency over the designated range of frequencies using the keyword CON STANT_UNIAX see later for software use Q par AJM anlay 1 v2 0 Page 35 of 68 2 6 3 Orthotropic tensors An orthotropic tensor is defined as one with three mutually orthogonal real valued eigenvec tors whose directions are fixed in space Such are associated with an orthotropic material or orthorhombic crystal form where the principal axes of the tensor
59. put quantities The reader is strongly advised to read 1 for more detailed explanation of certain points z z incident plane wave reflected plane wave I I l I T I l I transmitted wave x Figure 1 1 Plane wave excitation of a multiple layer structure 1 2 Polarisation definitions and wave conventions We assume throughout this report a harmonic time dependence e where w 27 f is the angular frequency and f is the frequency in Hz This implies for an isotropic material with scalar relative permittivity and scalar relative permeability e and ur that e and u have zero or negative imaginary parts is the material is passive and lossy Similar requirements suitably generalised may be made of the relative permittivity and permeability tensors for an anisotropic material see later In Rikte et al 1 the alternative e7 time convention is employed Thus their i should be replaced whenever it occurs by j in our formulation We employ the 7 notation for consistency with 2 as is common in electrical engineering rather than in physics and mathematics Q par AJM anlay 1 v2 0 Page 2 of 68 A plane wave is defined in two transverse coordinate systems in terms of the unit vectors and 9 and in terms of the unit vectors parallel and 6 perpendicular at an angle to the x axis in the x y plane This follows Rikte et al notation with l cos gsing 1
60. rge as well as exponentially small terms under certain realisable conditions This includes where one or many of the layers are very lossy When relating the transmitted and reflected waves to the tangential fields at the interfaces z and zw 4 these large terms are al gebraically removed as may be seen below but the formulation can be ill conditioned when cancellations are required in the differences of large numbers The problem is mentioned in 1 but not solved The solution would appear to be difficult since it involves wave splitting within the general bianisotropic material of each layer in terms of forward and backward propagating wave modes While formally this is possible and the method for doing this is given in appendix D of 1 in practice it is hard Such a method requires the eigen decomposition of each of the matrices M for each layer prior to matrix exponentiation The determination of eigenvectors and eigenvalues is not a problem and actually provides a simple means of matrix exponentiation but only when the eigenvalues are distinct and the matrices M are normal matrices The problem occurs when this is not or nearly not the case when we need to treat all possible special cases of matrix exponentiation in a well conditioned manner The mathematical problems are addressed in 3 Physically there are certain conditions for which it is known that a non uniaxial biaxial lossless material can give rise to degenerate wav
61. rincipal admittance names 1 and 2 If not present it is assumed there is no admittance surface Parameter surface number is an integer 1 lt i lt N 1 describing the interface number for which the surface is defined where N is the total number of layers within the compos ite Integer 1 specifies the left most interface with interface numbers defined sequentially thereafter Parameter rotation angle is a real number specifying the rotation angle v in degrees This defines the alignment angle associated with the principal admittances Parameter lt name_l gt is a character string specifying the name of the first principal admit tance c defined by the SIGMATYPE key word Parameter lt name_2 gt is a character string specifying the name of the second principal admittance ol defined by the SIGMATYPE key word 3 2 9 SIGMATYPE Syntax is SIGMATYPE lt name gt lt model_number gt list of parameters Q par AJM anlay 1 v2 0 Page 44 of 68 This key word is mandatory if it is required to define a lt name gt specified by the SURFACE key word If no SURFACE key words are present no SIGMATYPE key words are required Parameter lt name gt is a character string which must match lt name_l gt or name 2 gt in the SURFACE definition The SIGMATYPE then defines the referenced admittance Parameter lt model_number gt is an integer that refers to the equivalent circuit model number employed Currently as o
62. rite 8 wenos g 2 42 where j m we voy S 2 Jolo 2 43 Now for consistency with 1 let us define W 6 evaluated when 0 in and Qin These in our notation are the required values for a reflected wave Thus define W W Oin Pin in which case cos din COS Pin Sin din 1 sin din COS Pin SIN Pin W cos bin a 2 COS Qin SIN Pin sin din cos Oin V COS Pin SIN Pin COS Pin 2 44 and its inverse zi 1 COS Pin COS Pin SIN Pin sin din COS Pin SIN Pin W ae cos On 2 COS COS Pin SIN Pin sin Pin COS Pin SIN Qin COS Pin 2 45 In order to relate the tangential fields at either side of the structure to R and T it is necessary to decompose the free space fields into incident reflected and transmitted waves This is the so called wave splitting method which is straightforward to do in an isotropic Q par AJM anlay 1 v2 0 Page 13 of 68 material Equations 1 9 decompose the full wave into incident reflected and transmitted parts Associated with each we may consider the tangential components In the left hand half plane there is an incident and reflected wave and in the right hand half plane there is a transmitted and a possible incoming wave which is assumed not present for a complete free structure In Cartesian coordinates we write Ba BO E C oora E Ue Ooo 2 46 and IL EG Oonan G Glen 2 47 where the superscript refers to a wave p
63. ropagating from the left to the right half plane and the refers to a wave propagating from the right to the left half plane To simplify notation and in keeping with 1 define F z F z bin F z mE I z Oin G z G z Gin G7 z2 eG z Oin 2 48 Since W 6in Pin W x fin Pin equation 2 42 implies nod G z FW E 2 2 49 which is Rikte et al s result We now substitute 2 49 into 2 26 to obtain 1 Ft zn 1 pe Tu Ti Ft z 2 50 PF zw 1 Ta To NE 2 where IP Pa P W W P W P W 2T 12 Pai PW W P W P W7 2T4 Pau P W t W Pa W P W 2T Pu Pi W W Pa W P W 2 51 where the 2 x 2 matrices P are the submatrices of PO zy 4 21 P P Bisa Em pe 2 52 Q par AJM anlay 1 v2 0 Page 14 of 68 At this point we define the tangential reflection and transmission coefficient matrices r and t such that F a rE u Eis tE z2 2 53 with components E Tee Tay 2 54 t ber tay 2 55 and These matrices are given by 1 t Tu Tor for a free structure 2 56 for a free structure In the special case where the structure is terminated by a perfect conductor these matrices are not properly defined In this case E zw 1 0 and the first row of 2 26 is given by 0 Pi Pi RE d 2n 2 57 Sane e 1 1 I c Z 0 aid a Pa Bok a for a PEC terminated structure 2 58
64. s to the left and right of the structure Such an impedance surface is defined as thin i e of essentially zero thickness anisotropic with two orthogonal principal axes and realisable For it to be realisable we preclude the possibility of magnetic currents lacking the existence of magnetic monopoles and thus the tangential electric field is assumed to be continuous across the interface E ua ey Impedance surfaces 1 2 3 N N 1 I incident wave direction Figure 2 2 A structure consisting of N layers with impedance surfaces between layers To describe the electrical behaviour of such an interface let the field just to the left of the interface be designated by a superfix and just to the right by a superfix The vector field pair Bi a nod Hoy 2 designates the tangential electric and rotated magnetic field just to the left of the interface at z 2 and the vector field pair Ens Nod Hoy 2i designates the tangential electric and rotated magnetic field just to the right of the interface at z zi The boundary conditions applied to the interface are given by E gt z Eia 2 64 TY y Q par AJM anlay 1 v2 0 Page 17 of 68 and nod Hoy 2 Hey noo Ez i 2 65 Try Try where is the 2 x 2 conductivity tensor with units in Ohms defined by c R XR 2 66 where X is the diagonal matrix defining the principal value conductivities X a i 2 67
65. same form as the general case with the absence of the diagonal A 5 matrices Xt Ha T X 92 3 Xa s 3 1 X Su S12 S92 Sh Special case 2 optional We now consider the special structure consisting of a single final layer layer i N termi nated by a perfect conductor PEC This is optional and provides an alternative method of specifying that the final interface is a perfect conductor The alternative is to specify a general interface with principal conductivities that are both infinite In the software the use of the PEC option in the STRUCTURE definition activates this case in which case any subsequent definition of an impedance surface at interface N 1 is ignored The structure is illustrated in figure 2 5 where we illustrate the case at an arbitrary value of i and we need to determine the X7 matrices defined with CAS GR 38 GE eus Q par AJM anlay 1 v2 0 Page 27 of 68 PEC interface Figure 2 5 The special canonical structure referring to layer terminated by a perfect conductor where p 2 41 0 and p7 z is independent of pt 2 41 At the perfect conducting inter face z 241 the total tangential electric field is zero i e E zi 1 EG 0 2 109 These respectively imply that Dia rug 2 110 i 1 ry Nod Hz 1 V p zi i 2 111 i _ 5 z Pay a not sully ee Let us write Uu Un Uam U Us 2 112 Vu Vie V 2 113 Va Vo
66. so we have Ui Py iai U4 p 24x1 Vip zis Views na 0 2 114 hence Py 2 41 Ui Ui Py ii 2 115 Q par AJM anlay 1 v2 0 Page 28 of 68 and 1 A 0 E N3 0 pr zi af 1 x Va Vo ng x ete 2 116 where the A are those referring to the iy layer Consequently X 0 2 117 X5 UpUn 2 118 ea mo 1 2 0 SA Ng 0 Xa U NM UL oM and x 0 2 120 We might implement this special case anywhere within the composite but it is required only when i N Since the p vectors represent pseudo partial waves all the X7 entries must exist Thus provided U and V1 contain no zero eigenvalues and hence are invertible then the inverses Uj and Vj also exist There is no problem showing existence for the U matrices when i N since these refer to the free space values 2 90 For the V matrix or for i Z N we rely on the fact that both Uy and V y are unique up to a constant phase factor provided there is no singular value degeneracy discussed previously We therefore expect existence of the inverses since if they did not then the layer problem would be inherently badly conditioned which makes no physical sense As with the general canonical structure we do not however have a mathematical proof of this 2 4 6 Concatenation with the canonical structures In order to consider the effect of all layers in the composite we now proceed with a con catenation where we build up the composite a layer at a time starting
67. specified in dB The phase of the 7 entries T 1 1 T 1 2 T 2 1 and T 2 2 respectively specified in degrees The square magnitude of the R entries R 1 1 R 1 2 R 2 1 and R 2 2 respectively specified in dB The phase of the R entries R 1 1 R 1 2 R 2 1 and R 2 2 respectively specified in degrees The axial ratio associated with TE transmission and TM trans mission respectively The axial ratio associated with TE reflection and TM reflection respectively 3 4 Example 1 An anisotropic material for polarisation conversion in free space This is an example showing a three layer material which might be used to convert linear to cir cular polarisation It is an un optimised half wavelength half wavelength quarter wavelength design based on the method of Pancharatnam 7 Note the relative permittivity tensors of the first three layers are rotated in the x y plane by angles a 7 34 and 100 respectively The input file is given by STRUCTURE 3 FREE 1 2 3 FILENAME outputi dat output2 dat ANGLES 00 0 0 0 1 00 0 0 0 1 FREQS 5000 0 200 0 100 MATERIAL 1 0 0200 epsnamel munamel xinamel zetanamel MATERIAL 2 0 0200 epsname2 munamel xinamel zetanamel MATERIAL 3 0 0100 epsname3 munamel xinamel zetanamel TENSOR epsnamei CONSTANT ORTHOROT 3 0 0 0 1 5 0 0 3 0 0 0 07 0 0 0 0 0 TENSOR epsname2 CONSTANT ORTHOROT 3 0 0 0 1 5 0 0 3 0 0 0 34 0 0 0 0 0 TENSOR epsname3 CONSTANT_ORTHOROT 3 0 0 0 1 5 0 0 3 0 0 0 100 0 0 0 0 0
68. structure comprises the first interface of the multi layer structure fronted by a semi infinite region of free space The optional special case is that where the final layer is terminated by a perfect conductor General case Figure 2 3 shows a single layer layer i for i 1 N terminated by the surface impedance sheet and the associated p vectors We now require to find the 2 x 2 matrix entries A X X and X7 evaluated on the ij layer i e for i 1 to N but to avoid notation complexity the 7 dependence of the X terms is suppressed defined by o je xs fori 1to N 2 92 pi zi X5 X Py 241 Layer i Zi Z Figure 2 3 The general canonical structure associated with layer i Using equations 2 78 and 2 80 together with 2 69 and 2 70 we first relate the p vectors either side of the interface at 2 1 This is defined by the matrix _ composed of 2 x 2 sub matrices 11 S12 S21 and S22 i 21 S22 i4 1Si4 1 i amp uisir for i 1to N 2 93 Q par AJM anlay 1 v2 0 Page 25 of 68 such that PI std 262 aminin tasa P The inverse matrix is well defined and given by Si i ciz ia SH SU fori 1toN 2 95 Dog Us UK v Me si 51 22 i 2141231 Together with 2 84 and 2 85 this implies 1 Joy A3 0 Xi S2 i NG 2 96 X 6 Toh 2 97 j 1 Ai 0 E s 0 X5 Ng 1 2 S12 S22 o Ng 2 98 7 1 0 X5 ni L ds si S12 S22 a 2 9
69. t the program will close with an appropriate error message 3 2 7 TENSOR Syntax is TENSOR lt materialname gt tensor type list of parameters material name gt is the name referenced by the MATERIAL definition and must match one of the material names so defined Each TENSOR definition must have a unique material name or else the program will terminate with a suitable warning lt tensor_type gt defines the type special case or method of description of tensor Cur rently permitted key words are CONSTANT OVERGEN CONSTANT UNIAX CON STANT ORTHOROT and TAB ORTHOROT These are defined in section 2 6 The list of parameters which follow depend on this key word CONSTANT UNIAX refers to a uniaxial tensor The list of parameters which follow are lt a1 gt lt S a1 gt lt Raz gt S ag gt uu lt Uz gt lt uz gt where R a and Y a are real numbers specifying the real and imaginary parts of a R a2 and 9 a2 are real numbers specifying the real and imaginary parts of a and u1 us and us are real numbers defining the uniaxial direction vector These are defined in section 2 6 2 CONSTANT_ORTHOROT refers to an orthotropic tensor The list of parameters which follow are IRA gt lt SO gt lt KROQ gt lt SA2 gt lt N33 gt lt SA3 gt lt a gt lt Bro lt y gt where A41 S A1 R A2 S A3 and A3 and S A3 are real numbers specifying the real and imaginary parts of A
70. ted value of z and VO V fori 1 N 2 79 i4 1 pt a mu m for i 1 N 2 80 d P zi O29 7ay where the superfix designates the field vector just to the right of the interface at the designated value of z and Similarly U eU fori 1 N 2 81 i 1 i 1 Across a layer between interfaces we therefore have 0 0 0 CEA MS To E Gi d TO See We may therefore define the vectors pr zi 1 P3 2 41 Pj zi and pj zi as pseudo partial waves with the property that the vector magnitudes p 253 lt KIp iz Ip zi 1 lt Kp 2 py z lt Klor a pi z lt Klp C 2 83 for any physical lossy or lossless structure where K is some constant independent of the layer thickness 2 1 zi We may therefore identify P3 261 as taking the role of an outgoing wave travelling left to right transmitted T field Q par AJM anlay 1 v2 0 Page 22 of 68 p 243 as taking the role of an incoming travelling wave moving right to left incident W field P3 zi as taking the role of an incoming wave travelling left to right incident A field P3 24 as taking the role of an outgoing wave travelling right to left reflected R field These p vectors are uniquely defined up to a common phase factor e provided the singular values are distinct non degenerate but are not unique otherwise However since they always satisfy the above bounds this lack of uniq
71. th Ag as the free space wavelength and co the speed of light in vacuum 9 and k k form the with the wave orthonormal coordinate system for the incident wave The electric field vector amplitudes associated with the incident transmitted and reflected waves may be defined by En 05 Oin Din Bay ai gt T ae bre Epa SA st aO 1 9 where the a coefficients are in general complex scalars The principal required outputs from the formulation are the reflectance and transmittance matrices that relate the a coefficients We define these by T T Ti Tiz T l 1 10 2 Tii o 2 ae and R Ry Rit Rig R 1 11 Ex Ry D m ic Pus a x Qin alo TE o 1 12 tx in and T p Orr Qin NO R NO 1 13 where the T superscript represents the matrix transpose The matrix transpose is employed to permit the same definitions of T and R as are used in 2 3Rikte et al 1 do not explicitly employ these matrices in their formulation Q par AJM anlay 1 v2 0 Page 4 of 68 2 The formulation 2 1 The tensor constitutive relationships At a given frequency the Maxwell equations relating D E B and H fields are V x E jkocoB V x noH jkoconoD 2 1 If materials are assumed to be linear then the most general constitutive relationship between the fields is given by D leE mg H B HCE4mp H 2 2 where is the relative permittivity tensor
72. this case there are only two eigenmodes corresponding to the forwards and backwards travelling waves Another situation arises when the thickness of the layer approaches zero In this case P I the identity matrix and it becomes impossible to determine the eigenmodes from the matrix P A similar situation arises in an isotropic medium at normal incidence when e Hr These examples serve to show problems that may be encountered in trying to uniquely determine the partial wave modes from a knowledge only of PO zx 2 41 when a material is almost isotropic or when the material is electrically thin even when 67 exists Q par AJM anlay 1 v2 0 Page 20 of 68 kes However while determining the partial wave modes may be desirable it is not entirely necessary The ill conditioning in determination of the transmission and reflection coefficients arises principally because of the existence of exponentially growing terms in P that occur when loss is present or when some other loss of information occurs when we introduce the sheet interface media described earlier When loss is present the z contain terms that grow or decay exponentially with z Since P defines the fields at 2 44 gt z given the fields at z a forwards propagating wave contains terms which decay exponentially with z and a backwards propagating wave terms which grow exponentially with z Because there are two forwards and two backward waves there must be a pair o
73. uction are generally commented out in the source code Q par AJM anlay 1 v2 0 Page 9 of 68 where z and z lie in the same homogeneous region Accurate computation of general matrix exponentials is surprisingly difficult and is still a research topic in mathematics Generally there should be no problem if the matrix to be exponentiated is well conditioned in the sense of a normal matrix but this may not always be the case for certain kinds of realisable materials In particular the Cayley Hamilton method described and presumably employed in 1 will fail when the eigenvalues of M coalesce This will occur for nearly isotropic materials which are normal matrix well conditioned and for Tellegen materials which may not be normal matrix well conditioned As of the year 2003 the state of the art concerning matrix exponentiation is given by Moler and Loan 3 For small order dense general complex matrices such as M it would seem that the best algorithm is one employed and with freely available source code in the Expokit software suite 4 We employ the routine zgpadm which uses a variant of the scaling and squaring method 2 3 3 Basic formulation with no surface discontinuities If there are no surface discontinuities i e no thin material sheets or frequency selective surfaces between layers then the fields E and H are both continuous across the boundary between layers and the general solution to the fields across a multiple la
74. ueness is not generally a problem except for certain special cases where further information is required to construct suitable U and V matrices as shown later z The general rule is now to express outgoing pseudo partial waves only in terms of incoming ones at each layer of the structure Re ordering 2 82 we have pt zi Ya na Jar n 2 84 mena MANG PIO 2 85 which are both well conditioned in the sense that 1 1 1 15 A3 and A4 always exist and are well behaved 2 4 3 Computation of the SVD and degenerate singular values As given in equation 2 77 the SVD is well conditioned and we employ the LAPACK 12 routine ZGESVD for this task However while well conditioned the U and V matrices are non unique not simply up to a phase factor when there is singular value degeneracy This is not a problem when we have two pairs of equal singular values since the relative ordering of singular vectors associated with A and As or between A3 and A4 is unimportant for our purposes However there is a potential problem when all singular values are equal or sufficiently close that the singular vectors incorrectly mix incoming and outgoing pseudo partial waves The problem arises because if the singular vectors are incorrectly identified the method of calculation of the matrix elements X7 described later in section 2 4 5 may fail In particular the matrix 5 may not exist when 0 This is because an inappropriate ordering of the singu
75. where each layer is anisotropic in the xy plane co aligned and rotated 45 to the incident linearly polarised wave material type A B A linear in mo perfect i conductor circular out gt _ gt Qu QUE gt h h h2 thicknesses Figure 3 8 An ABA 2 PEC terminated structure for the design of a reflector polariser linear to circular The input file shown below represents a material with h 22 mm Material A has Exe 2 6 and ey 1 5 Material B has css 3 0 and ey 1 5 The materials are then rotated by 45 about the z axis Input file is STRUCTURE 3 PEC 1 2 3 FILENAME outputi dat output2 dat ANGLES 00 0 0 0 1 00 0 0 0 1 FREQS 5000 0 200 0 100 MATERIAL 1 0 0022 epsnamel munamel xinamel zetanamel MATERIAL 2 0 0022 epsname2 munamel xinamel zetanamel MATERIAL 3 0 0011 epsname3 munamel xinamel zetanamel TENSOR epsnamei CONSTANT ORTHOROT 2 6 0 0 1 5 0 0 2 6 0 0 45 0 0 0 0 0 TENSOR epsname2 CONSTANT ORTHOROT 3 0 0 0 1 5 0 0 3 0 0 0 45 0 0 0 0 0 TENSOR epsname3 CONSTANT ORTHOROT 2 6 0 0 1 5 0 0 2 6 0 0 45 0 0 0 0 0 Q par AJM anlay 1 v2 0 Page 60 of 68 TENSOR munamei CONSTANT_OVERGEN 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 TENSOR xinamei CONSTANT OVERGEN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 TENSOR zetanamei CONSTANT OVERGEN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
76. xiliary parameters such as absorbed energy degree of depolarisation axial ratio etc can be determined We update the original Rikte formulation to provide a method which is substantially well conditioned for arbitrary materials of arbitrary thickness and also allows the option of placing thin impedance sheets e g frequency selective surfaces or polarising grids between layers We will assume that each layer is infinite in the x y plane and transverse to the z direction An incident plane wave is specified with incidence angles 9 in and the transmitted and reflected waves by the angles Oix diz and Ore Ore respectively Figure 1 1 indicates such a structure and the coordinate system associated with it It is important to note that the Cartesian x y z coordinate system employed is a global one Under this and using Snell s law we have Bis Oin Qta Pin Ora Tm Oin Pra Qin 1 1 assuming the wave is always described pointing in the same direction towards the origin with respect to the coordinate system We term this the with the wave coordinate system and its use has some subtle consequences that will be described later This coordinate system is also the one employed by us in 2 Q par AJM anlay 1 v2 0 Page 1 of 68 The main analysis in this report is taken from Rikte et al 1 with some modifications see later principally concerned with evaluation of matrix exponentials and in the definition of out
77. yer structure can be concatenated as the product of the solution to a single layer as presented in 1 Suppose a structure consisting of N 1 layers N interfaces with a wave incident in medium 1 and transmitted for a free structure in material N as shown in figure 2 1 The tangential incident wave direction z z 1 Z Z Figure 2 1 A structure consisting of N 1 layers shown as a free structure fields at the first interface where z 21 are related using 2 24 to the tangential fields at the final interface where z zy_1 by dpi M BN aun UNS n Software may now exist with better more accurate more robust and or more efficient algorithms especially given the seven years since the publication of 3 but we have been unable to find any such non commercial software Experiments to date have found zgpadm to be quite sufficient for our application Q par AJM anlay 1 v2 0 Page 10 of 68 where Paa z1 exp jko zn 1 ZN 2 M y ud exp jko zw 2 zu 3 M w 2 9 exp jko z2 21 M 5 2 27 Note that equation 2 27 involves the product of matrix exponentials In general this cannot be written as the matrix exponential of the sum of the arguments since for non normal matrices exp A exp B exp A B This formulation as used by 1 is relatively straight forward but is not always well con ditioned The problem lies with the fact that in general P an1 21 involves both expo nentially la

Bianisotropic multilayer analysis software formulation and user guide

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