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Sevgi, Levent (Dogus Author) -- Conference full title: International Symposium on Complex Computing-Networks, Dogus University: Istanbul, Turkey, 13-14, 2005. Novel, generic double-arm microstrip structures which can be used as both electromagnetic bandstop and bandpass filters are introduced. The design steps, numerical simulations, practical realization, and experimentations are briefly discussed. Examples of microstrip electromagnetic bandstop and bandpass filters are presented.

This paper introduces Matlab-based two dimensional (2D) virtual propagation tools (VT) which can be used to investigate EM propagation over user-specified nonflat terrain through inhomogeneous atmosphere. The VTs can be used for both engineering (GSM coverage planning, digital site survey, etc.) and educational purposes (e.g., in EM Theory, Wireless Communication, Antennas and Propagation lectures).

This paper presents the design, simulation, and experimentation of a novel low-cost small-size high-performance microstrip (MS) electromagnetic bandgap (EBG) structure for MIMIC applications. The simulations and measurements confirm that the stopband characteristics can be easily controlled by changing the stub lengths and the interstub gap. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 46: 399–401, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20999

Sevgi, Levent (Dogus Author) A novel generic microstrip structure that can be used as both bandstop and bandpass filters is introduced in this paper. The design steps, numerical simulations and experimentations are briefly discussed. Examples of microstrip electromagnetic bandgap (EBG) and bandpass filters are presented. Eur. Office of Aerospace Res. and Dev. of the USAF,IEEE Region 8 Office,SRI on Radar Syst. "Kvant-Radiolokatsiya", Kyiv, Ukraine,Joint Stock Co. "TPOAZ", Donetsk, Ukraine,Scientific-Prod. Company "TERA", Kyiv, Ukraine

Sevgi, Levent (Dogus Author) Modeling and numerical simulation in electromagnetics is discussed in this paper with emphasize placed on validation, verification and calibration. Canonical examples from antennas, propagation to microstrip circuits are presented. Tests against measurements, when available, are also presented. IEEE,IEEE Electromagnetic Compatibility Society,International Union of Radio Science, URSI,Research-and-production Enterprise "Proryv"

Applications in science and technology rely increasingly on electromagnetic (EM) field computations in either man-made or natural complex structures or environments. Also, majority of these applications are based on digital technology supported with intelligent software. They include (but not limited with) Radar and Satellite systems, Radio/TV broadcasting, Wireless communication, Remote sensing, Antenna design and analysis, Radar cross section (RCS) prediction and Stealth target design, Microwave networks, Multi-sensor integrated surveillance systems, Subsurface imaging, Electromagnetic compatibility (EMC) and Bio-Electromagnetics (BEM) [1].

Modeling and simulation strategies for the Surface Wave High Frequency Radars (SWHFR) are discussed in this tutorial paper. Their potential application areas are summarized, together with problems related to ground wave propagation, radar cross section (RCS) prediction, clutter elimination, noise and interference cancellation, etc. Challenges in SWHFR system design, antenna requirements, signal processing techniques as well as detection and tracking approaches are reviewed. CSSIP; IEEE, S Australia Sect

The objective of this special session is to gather international esteemed experts and scientists in a special session to discuss some current topics in EMC engineering and to emphasize the importance of signal processing in electromagnetics.

Sevgi, Levent (Dogus Author) -- Conference full title: 2003 IEEE International Symposium on Electromagnetic Compatibility, EMC 2003; Istanbul; Turkey; 11 May 2003 through 16 May 2003. Transmission line matrix (TLM) method algorithm is validated against exact analytical solutions for a typical EMC problem. The two test problems chosen for this purpose are the determination of fields inside a PEC resonator excited by a pulsed source distribution and the determination of the radiated fields from an aperture on the wall of the resonator. Almost perfect agreement was obtained between the TLM solutions and corresponding analytical results both in time and frequency domains. Results show that TLM can be used in an efficient way in addressing SE calculations and can also be applied to the calculation of EM fields in similar complex EM problem environments.

Sevgi, Levent (Dogus Author) -- Conference full title: 2003 IEEE International Symposium on Electromagnetic Compatibility, EMC 2003; Istanbul; Turkey; 11 May 2003 through 16 May 2003. A semi-symmetrical groove guide resonator at millimeter waves is investigated both experimentally and numerically. The structure is made of duraluminium and fabricated with a maximum relative tolerance of less than 0.5%. Its resonance frequencies are measured. The simulations are performed via the powerful time domain simulator FDTD (Finite Difference Time Domain). The comparisons will be presented and the results will be discussed.

Telsiz iletişiminin büyük önem kazandığı günümüzde, özellikle atmosferdeki elektromanyetik dalga yayılımının çok iyi anlaşılması gerekmektedir. Yüzyılı aşan bir süredir ayrıntılı olarak incelenmesine rağmen hala güncelliğini korumakta olan elektromanyetik dalga yayılımı analitik tam çözümü bulunmayan karmaşık bir problemdir, Elektromanyetik problemlerin çözümünde, diğer mühendislik dallarında olduğu gibi, yalnızca basitleştirilmiş ve idealleştirilmiş yapılar için analitik yöntemler elde edilebilmektedir. Ancak bu yöntemlerden elde edilen fiziksel bilgi sayesinde karmaşık yapılarda ve gerçeğe yalan koşullarda analitik yaklaşık yada salt sayısal yöntemlerin güvenli olarak kullanılması mümkün olabilmektedir. Yarı analitik sayısal bir yöntem olan iki boyutlu (2D) Parabolik Denklem yöntemi, ki HFD (Hızlı Fourier Dönüşümü) ile çözüldüğünde SSPE (Split Step Parabolik Equation-Adım Adım Parabolik Denklem) adım almaktadır, elektromanyetik dalga yayılımı problemlerinde yaklaşık 20 yıldır kullanılan güçlü bir yöntemdir. Bu çalışmada karmaşık dalga yayılımı problemlerinin parabolik denklem yöntemiyle incelenmesi amaçlanmıştır. Çalışma içinde ilk olarak problemin yapısı hakkında genel bir bilgi yer almaktadır. Parabolik denklem yönteminin ayrıntıları, uygulanması esnasında gözönünde bulundurulması gereken noktalar, zayıf kaldığı yerler üzerinde ayrıntılı olarak durulmaktadır. Parabolik denklem yönteminin farklı senaryolara uygulanması ile elde edilen sonuçlar grafikleri ile birlikte verilmektedir. ANAHTAR KELİMELER : Parabolik denklem, dalga yayılımı, ayrık Fourier dönüşümü, arazi modelleme, atmosferik kırılma indisi, yol kaybı, propagasyon faktörü. The development of today's wireless communication systems requires realistic modeling of electromagnetic wave propagation through atmosphere. The electromagnetic wave propagation is a complex problem which is non-tractable via exact analytical methods. An analytical solution can only be obtained for simple, canonical structures and/or under various limitations. Therefore, an analytical approximate or pure numerical solutions are of interest in most of the cases. One of the well-known numerical solution techniques is the parabolic equation (PE) model. It has both closed and open form solutions. The open form solution, which is called Split Step Parabolic Equation (SSPE) and based on Fast Fourier Transformation (FFT), has been in use for decades, first in underwater acoustics and then in electromagnetics. It is a one-way propagator and neglects back scatter effects. In this study, the SSPE is taken into account. First a brief explanation on electromagnetic wave propagation over Earth's surface is given. Then the PE method presented in detail. Also, wave propagation over smooth ground as well as over non-fiat terrain is investigated in detail. Complex wave propagation scenarios are prepared and propagation characteristics are obtained via the SSPE program. Finally, the conclusions are presented. KEY WORDS : Parabolic equation, wave propagation, discrete Fourier transform, terrain modeling, refractivity. 71

ÖZET Bu çalışmada, kısaca FDTD olarak tanımlanan Zamanda Sonlu Farklar (Finite- Difference Time-Domain) yöntemi ayrıntılı olarak incelenmiştir. FDTD yöntemi FD olarak bilinen sonlu farklar yönteminin 1966 yılında Yee tarafından Maxwell denklemlerine uyacak şekilde zaman domeni için genişletilmesiyle birlikte ortaya atılmış, özellikle 1980'lerin ortalarında bilgisayarların hız ve kapasitelerindeki hızlı artışla birlikte elektromagnetik problemler için en çok kullanılan yöntemlerden biri haline gelmiştir. Çalışma sırasında öncelikle FDTD yönteminde ortam modelleme, sayısal dispersiyon, zamanda ve konumda ayrıklaştırma, hata analizi gibi konular incelenmiş, daha sonra da konumda sınırlı sayıda hücre kullanılması nedeniyle oluşan ya pay sınırlardaki yansımaların giderilmesi için uygulanan sınır koşulları üzerinde durulmuştur. Sonsuza giden sınırları simüle eden birinci-derece ve ikinci-derece Mur, Higdon, DBC (Dağıtıcı Sınır Koşulu), Liao ekstrapolasyonu gibi açık sınır koşullan teorik olarak açıklandıktan sonra, 1994 yılında Berenger tarafından ortaya atılan PML (Mükemmel uyumlu Tabaka) yöntemi anlatılmıştır. Uygulamalar sırasında mikroşerit hatlardaki karakteristik empedans, efektif dielektrik sabiti ve yansıma katsayısı FDTD yöntemi ile sayısal olarak bu lunmuş ve bu değerler yardımıyla açık sınır koşullarının etkileri birbiri ile karşılaştırılmıştır. Bu işlemlerin sonucunda gerekli optimizasyonlar yapıldığı taktirde PML yönteminin en güvenilir sonuçları verdiği görülmüştür. Ancak PML yöntemi FDTD algoritmasında ihtiyaç duyulan hafıza miktarını arttırmakta ve programın çalışmasını da yavaşlatmaktadır. Bu yüzden, çok hassas analizlere gerek duyulmuyorsa, incelenen yapıya ve probleme uygun olan başka bir açık sınır koşulu kullanılabilir. En son olarak FDTD yönteminin farklı problemlere ve yapılara ne şekilde uygulanacağına dair örnek olması için alçak geçiren filtre, çeyrek dalga empedans transformatörü, mikroşerit küple devresi gibi düzlemsel mikroşerit yapılar FDTD ile incelenmiştir. v iii SUMMARY THE FINITE-DIFFERENCE TIME-DOMAIN METHOD AND ABSORBING BOUNDARY SIMULATIONS In this study the Finite-Difference Time-Domain (FDTD) method is investigated. FDTD is widely regarded as one of the most popular full-wave computational electromagnetics (EM) algorithm. It was first investigated in 1966 by Yee [1] and only a few studies have appeared until the end of 1970s because;. Large computer storage and high speed requirements have not been supplied until the mid of 1980s.. FDTD, as introduced by Yee was not suitable for most of the EM problems such as antenna simulations and RCS calculations [27] etc.. Source implementations and open boundary simulations have not been han dled in early FDTD, as introduced by Yee. Yee [1] cliscretized Maxwell's two curl equations directly in time and spatial do mains and put them in iterative form. In Yee formulation the physical volume of interest is divided into cubic reference cells where characreristics of the medium are defined by three parameters; permittivity (e), conductivity (cr) and permeabil ity (fi). Within the reference cell three electric and three magnetic field compo nents are located at different locations in such a way to minimize the computation duty after the discretization of two curl equations by using central-difference ap proach (or by taking up to the second order terms in their Taylor's expansion). Besides the differences in the locations of six field components, there is also a half time step difference between electric and magnetic field components. This is called a leap-frog computation. During FDTD simulation electric field com ponents are calculated at each cell at time instants t = 0,At,2At,3At,... etc., but magnetic field components are calculated at t = At/2,3At/2, 5At/2,... etc., having a At/2 time difference between electric, and magnetic field components. Therefore, in electromagnetic analysis, synchronization in both time and spatial domains are needed for E and H field or V and / calculations at a fixed point in FDTD volume. This is fortunately accomplished by simple cell and time averag ing processes. Since 1980, hundreds of publications about FDTD have appeared related to the algorithmic improvements as well as its applications to broad range of complex electromagnetic problems. Major improvements may be groupped in: ix. Narrow and broad band source simulations and their injections in given FDTD cells.. Open boundary simulations which extend the ability of FDTD algorithms to handle antenna radiation simulations and radar cross-section (RCS) cal culations.. Both frequency and time domain near- to- far field transformations based on Huygen's equivalent source principle. where effords are still needed for further modifications. Some of the applications may also be groupped as;. The analysis of planar microstrip structures,. Specific Absorption Rate (SAR) calculations in human tissues near electro magnetic sources,. Mutual effects of hand-held receivers and human head,. Antenna simulations and RCS calculations,. Analysis of waveguiding structures,. The simulation of ground-penetrating radars,. The simulation of microwave ovens. In this study, PML which is the most, efficient and available ABC, is implemented numerically for 3D rectangular FDTD volume. Although PML is effective in absorbing guided waves of all angular distribution, its implementation is quite difficult. Only a few studies related to the application of PML to 3D-FDTD [18], [17], [19], [22] have appeared in the literature up to day. For this reason, this study aims to focus on not only FDTD analysis of complex structures, but also 3D PML implementations. Three dimensional (3D) FDTD mesh suggested by Yee [1] is shown in Section 2.1 (see figure 2.1). In a lossy source-free medium, Maxwell's curl equations are given as 9H - - ^o-jT = -V xE-crE (la) BE.^. = Vx// (lb)where £q, (iç,, a are permittivity, permeability and conductivity respectively, and they are discretized as A/ H?(iJ,k) = H^iiJ.k) - -- [E*(i,j,k) - E^(i,j,k- 1)] At [E:(i,j,k)-E?(i,j-l,k)} (2a) Hj{i%j,k) = H*-/i,j,k) - -£L [E,*) - E:(i,j,k- 1)] (2b) ff?(», j,*) = Hr/i,j,k) - -^- [££(*,**) - ^(i, j - 1,*)] A* [JEJ(i,i,Ar)-^(i-l,j,*)] (2c) /z0A:r 2e-jrAj 2A* Wj'-*> = 2^ME``{i'j'k) 2A/ 2Aİ [H?(i,j,k)-H:(i-lJ,k)] {2e + aAt)Ax 2At + WT^Wzm''hk)-H:i''i'k-l)] (2e) In these equations h = n + 1/2 and e = £o£r- Because of the 3D-FDTD mesh structure, electric fields are calculated at the integer multiplicants of time and magnetic fields are calculated at the fractional multiplicants of time. As it is common in all numerical methods there exist unexpected reflections from the boundaries of the FDTD computation space unless the boundaries are cov ered with appropriate boundary conditions. In order to remove the effects of these reflections the methods listed below are used in FDTD applications: xi. Boundaries are covered with PEC (perfectly electric conductor) by assigning zero value to the tangential electic field components during time simulations.. Boundaries are covered with PMC (perfectly magnetic conductor) by as signing zero value to the tangential magnetic fields components during time simulations.. Field values of the cells which are adjacent to the boundary inside FDTD are assigned to field values of the cells which are adjacent to the boundary outside (symmetry condition).. Radiation condition called as absorbing boundary condition (ABC) is ap plied at the boundary planes. In this thesis, different ABCs are theoretically explained and numerically com pared with each other according to their efficiencies and accuracies in FDTD applications. First-order MUR ABC: With the help of first-order approximation of the three dimensional wave equation first-order MUR ABC [2] at x - 0 boundary is given as (dr - %ldt)Et = 0 (3a) where E% represents the tangential electric field component relative to the bound ary wall. The discretized form of Eq.(3a) will then be £,`+1(0,j,A-) = £r(U,Â0 + ^^(İ^HW,*) - £T(0,i,*)) (3b) For example, the tangential electric field components at x - 0 boundary are Ey and Ez. Eq.(3b) tells that, Ey(i,j, k) and Ez(i,j, k) for all (j, k) will be related to the current (n) and one past (n - 1) time instant values of themselves and to the nodes one inside. By doing this, first-order Mur condition is satisfied. Second- order MUR ABC: With the help of second-order approximation of the three dimensional wave equation second-order MUR ABC [2] at x = 0 boundary is given as ( PML(o`.o,'.o.o) y f! Xi),o) PML«j`,o,;,o`.%) PML(o.ao>0'`) PML(oa.o»a`o,) Figure 1. 2D FDTD and PML structure yf PML J-JL ¦+-E, O H, H` © H` Figure 2. Right-upper part of 2D FDTD+PML structure PML is applied to two-dimensional FDTD as shown in Figure 1. FDTD equations are applied in the inner part and PML equations are used in the PML region. For the right-upper part of the Figure 1 (see Figure 2) in PML region (/ > IL and J > JL in Fig.3) the field equations are gj(i)Ai gar(l)At Eny+i{i,j) = e~ i) (l- e tr*(i+l/2)At CO ')

Sevgi, Levent (Dogus Author) This special issue entitled ‘‘Modeling and Simulation Challenges in Microwave Engineering’’ contains invited/ selected articles from The 12th Mediterranean Microwave Symposium (MMS 2012, http://mms.dogus.edu.tr), which was held at Dogus_ University during September 2–5, 2012 at the same time together with The 6th URSI-Turkey National Assembly and Congress (URSI-Turkey 2012, http://ursi.dogus.edu.tr). Technical programs of MMS 2012 covered a vast array of electromagnetic and microwave topics, including microwave circuit design and applications, the latest breakthroughs in antenna design, wireless communications, exotic materials, biomedical applications, numerical methods, etc.

ÖZET Bu çalışmada, kırılma indisinin enine ve boyuna iki iki boyutlu değiştiği ortamlarda dalga iletimi problemi Parabolik Denklem (PD) yöntemi ile ince lenmiştir. Öncelikle yüksek ve çok yüksek frekanslarda küresel bir zemin üzerine düşey olarak yerleştirilen kısa bir dipolden çıkan elektromagnetik dalgaların ya yılma özellikleri ve davranışları incelenmiştir. Ortam parametreleri kontuna bağlı olduğundan problemler basit analitik bi çimde ortaya konamazlar. Bu nedenle yaklaşık ya da salt sayısal yöntemlerle çözülmeye çalışılırlar. Dalga iletim problemlerinde, analitik - sayısal yaklaşık çözümlerden en yaygın olarak kullanılanlarından biri PD yöntemidir. Burada, öncelikle yöntem tüm dikkat edilmesi gereken noktalarının üzerinde durularak incelenmiş, daha sonra tam çözümleri bilinen kanonik bir yapı ele alınarak bu tam çözümlerle PD çözümleri karşılaştırılmıştır. Son olarak da, tam çözümleri bilinmeyen problemlere PD uygulanarak elde edilen sonuçlar yorumlanmıştır. Bu amaçlarla, çalışmanın bütününde olabildiğince çok değişik parametre takımları ile incelemeler yapılmış ve elde edilen tüm sonuçlar grafiklerle ve açıklamalarla verilmiştir. iv SUMMARY PARABOLIC EQUATION (PE) METHOD IN GROUND WAVE PROPAGATION In this thesis, the propagation phenomena in 2D complex environments with arbitrary transverse and slow longitudinal variations is described in terms of Parabolic Equation (PE) method. The model presented here represents propa gation of electromagnetic waves over a spherical, finitely conducting earth and allows specification of frequency, polarization, antenna pattern, antenna altitude and tilt angle. The well-known solution method of the parabolic form of the wave equation, split-step PE (SSPE) is compared with the reference solutions obtained from Helmholtz wave equation where analytic solutions exist and also with each other through surface duct-to-elevated duct transition to show the transformation of the surface trapped modes into the elevated beams. Propagation in complex waveguide environments which may involve bound aries and/or transversely confining refractive index variations are becoming more and more interesting for the electromagnetics and microwave society. Such prob lems can be solved with analytical methods which are mostly based on mode and/or ray summations or their hybrid combination. Analytical solutions are very instructive since they give a good insight into the physics associated with non-separable problems. However, they exist only for a small amount of ideal problems which are mostly non-physical, for this reason, a lot of effort have been given on finding approximate mode-like wave objects to built fields in non- separable environments. PE method combines the physical insight of the analytical solutions and the generality of numerical solutions in a computationally efficient scheme[8,9]. The PE method can be used to solve wave propagation problems everywhere, where in the variation of the geometry and/or medium parameters can be arbitrary in the transverse domain but is slow in the longitudinal direction so as to allow approximate reduction of the wave equation to the parabolic form. The parabolic form of the wave equation is very suitable for the numerical computations. Once the initial transverse field profile is given, SSPE algorithm, which is built to satisfy the transverse boundary conditions automatically, computes the transverse field distributions at any longitudinal range value proceeding step by step. At each range step the fast Fourier transform (FFT) routine is used to carry out thetransformations between the spatial transverse coordinate and transverse wave number coordinate. SSPE method has been applied to many important and interesting wave problems also including the longitudinally varying guiding media. In this study, first the complete derivation of the parabolic wave equation from the vector wave equation is presented[7,10]. The assumed spherical earth geometry is shown in figure 1. We are concerned with describing the propagation from a source located at 9 = 0 and r = rs in the region r > a (where a is the earth's radius) and in the far field of the source antenna. The emphasis in the following discussion is on the case where the source is a vertical electric dipole (VED), but the corresponding results for the horizontal electric dipole (HED) may be provided at various point. Source Location short dipole e0,a= 0 >M0 P(r,e>0 (ll.b) U(x,z)/z^±oo^0 (ll.c) The location of the boundaries and the transverse refractive index variation of the medium permits the seperation of the 2D wave equation into transverse and longitudinal components which are ID differantial equations. The longitudinal equation gives a phase variation of exp(- ifiz) with exp(iu}t)time dependence, while transverse equation reduces to * + k2n2(x)-Ş2 u(x) = 0 u(x, z) = ü{x)t-lfiz { 12) _dx* Defining new transverse variables as k2n2{x) - f = Ax + B; p = -A~2/:i{Ax + B) ( 13) A = -a0k2 B = k2Q- &2, yields d2 - -P dp ü(p) = 0 (14) IXdifferent ranges and/or altitudes. The PD program uses FFT and inverse FFT modules at each longitudinal computation step. Extensive numerical tests have been performed for convergence, and for minimization of the anti-alising effects and truncation errors. Here, examples are presented wherein SSPE solutions on a surface duct and surface-to-elevated duct transition problems are compared with reference solu tions. The better critical transition is characterized by the transverse and lon gitudinal refractivity variations of the environment. The calculations and com parisons can be made at all radiowave frequency ranges but the results are given for the lower half (i.e., typically 3 - 15 MHz.) of the HF band. These results are applicable to surface wave HF communication especially for the surface wave HF radar applications. In the computations the coverage of the altitude and range extends to 3km. and 200km., respectively, which is the typical region of interest in surface wave HF communication systems. The initial refractivity gradient for the model is chosen as dN/dh = iOONunits/km which is associated with at mospheric ducts ör trapping layers. The positive constant ao which controls the surface duct height will then be 4. 10-7. n(x,z) i Z = -100 z = 0 z = 100 Range (km) Figure 2. The structure of the problem and the refractive index variations at different observation ranges SSPE algorithm is first tested on longitudinally invariant surface duct defined by the first profile in fig. 2. Assuming that the initial 400 N units /km. transverse refractivity gradient remains invariant for all observation ranges, yields an exact solution in terms of Normal Modes (NM) [20]. The first comparison is carried out between the NM(exact solution) and PE computations. Fig. 3. shows various altitude profiles at different observation ranges. The initial profile is built with the superposition of the equally excited first ten trapped modes and then fed into the PD.FOR algorithm. The truncation of the spectrum in kx domain is chosen to cover the transverse wave number of the highest NM considered in the altitude field distribution for the SSPE computations. The loss factor of a is introduced XIbetween Xmax/2 and Xmax. A very good agreement between the NM and SSPE results is clearly seen from the figure for the altitudes extending from the surface to five kilometers. 10000 7600 - ^-»., 2500- Figure 3. The comparisons of the exact analytical results(NM) with the SSPE compu tations at different ranges (a: loss factor) In the PD.FOR algorithm, the initial transverse profile can be constructed to correspond to the radiation field of specific antenna isotropic in the horizontal plane. In order to compare the SSPE method with NM results, the initial trans verse profile of the antenna pattern is simulated with the superposition of the NM with the suitably chosen excitation coefficients. The radiation pattern in the elevation plane simulating the Log-Periodic dipol antenna for 3MHz. is apporxi- mated with the superposition of suitably excited the first ten NM assumed which are then fed into SSPE algorithm. The results of the NM and SSPE computations for the transverse field profiles at different observation ranges are given in fig. 4. An excellent agreement between the two methods is clearly seen in the figure for the Dirichlet type boundary condition. Finally, the surface duct-to-elevated duct transition is examined. There is no reference solution for the surface duct-to-elevated duct transition. Therefore, only the SSPE results corresponding to this transition are given Fig. 5. The initial transverse field distribution at the begining of the transition region (c = Z/) is again calculated via LINEER.FOR as a modal superposition over the modes of the homogeneous surface duct. In fig. 5., the first two modes are considered in order to be able to demonstrate the transformation of energy from a trapped mode to the radiating beam. This transformation is clearly seen from the figure. In this study, a powerful tool for one way propagation problems is examined on a canonical surface duct as well as on a complex surface duct-to-etevated duct transition problems. The SSPE algorithm in this form may be used for all kinds of refractive index profiles under slow longitudinal variations or in the paraxial regions and for any transverse boundary variations [18,19]. xn(a) 6000 4000- 2000- (b) Figure 4. The comparisons of the analytical solution based on modal summation and SSPE algorithm. Figure 5. The transformation of the surface trapped modes into elevated beams for the lowest two modes. xiu 72