rcs reduction of open-ended circular waveguide cavity with...
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Progress In Electromagnetics Research, Vol. 146, 57–69, 2014
RCS Reduction of Open-Ended Circular Waveguide Cavity withCorrugations Using Mode Matching and Scattering Matrix Analysis
Young Dam Kim*, Ho Lim, Jung Hoon Han, Won Young Song, and Noh Hoon Myung
Abstract—A scattered radar signal by a jet engine inlet involves engine information of the target. Thejet engine RCS due to the engine inlet is therefore significant in radar target recognition. Accordingly,several accurate analyses of the jet engine RCS have been reported. Reduction of the jet engine RCSis also required for stealth technology. We suggest a new approach for jet engine RCS reduction thatinvolves the insertion of many small corrugations into the jet engine inlet. The proposed jet enginestructure is assumed to be a circular waveguide cavity with many corrugations. The structure is analyzedby combining a mode matching technique with a scattering matrix analysis. The proposed closed-formRCS solutions of the corrugated engine structure are validated with a MoM simulation using FEKO. Theproposed closed-form solution has numerical efficiency and rapid convergence. Using the characteristicsof the solution, we simply apply a genetic algorithm (GA) to optimize the structure of the corrugationsin terms of minimizing the RCS.
1. INTRODUCTION
The radar cross section (RCS) of a jet engine is important in that it plays a key role in radar targetrecognition. If a transmitted radar signal is scattered from the interior of the jet engine, the radar signalis modulated by rotating jet engine blades. The modulated signal referred to jet engine modulation(JEM) provides significant information, such as the spool rate, blade length and blade number. Byinterpreting the JEM signal, enemy troops are able to identify the fighter type [1]. Reduction of the jetengine RCS is therefore important for stealth technology. For this reason, the jet engine RCS has beenextensively studied in many electromagnetic problems.
Three remarkable methods to analyze jet engine RCS have been reported. The first is an exactsolution using a mode matching method with a scattering matrix analysis [2–6]. In this method, the jetengine is treated as a complex open-ended waveguide cavity structure. In order to analyze a structurethat has some obstacles, the scattering matrix method applies a modal analysis to each simple distinctsection composing the complex waveguide cavity structure. A scattering property of the entire structure,represented in matrix form, is attained by cascading the modal coefficients of all sections. The secondis an asymptotic solution using high-frequency methods such as shooting and bouncing rays (SBR) [7],geometrical theory of diffraction (GTD) [8], physical optics (PO) [9] and generalized ray expansion(GRE) [10]. Because the size of jet engine is typically much larger than the wavelength and thesemethods rely on the ray tracing method, they have good applicability to arbitrary and large structures.The third is a hybrid solution combining an integral or differential of Maxwell equations with the twomethods described above [11–14]. The mode matching method or high-frequency method is utilized forthe front part of the jet engine due to its simple structure. Low-frequency methods using Maxwell’sintegral equations or differential equations are applied to the end part of the jet engine which has
Received 28 February 2014, Accepted 1 April 2014, Scheduled 23 April 2014* Corresponding author: Young Dam Kim ([email protected]).The authors are with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 291Daehak-ro, Yuseong-gu, Daejon 305-701, Republic of Korea.
58 Kim et al.
complex jet engine blades. By combining two methods, both computational efficiency in the front partand solution accuracy in the end part are attainable.
Reduction of the jet engine RCS has been researched via two methods. The first is geometricalmodification of the jet engine structure, which backscatters incident radar waves to the opposite directionof the incidence. For reduction of the jet engine RCS, the straight jet engine structure is commonly bentinto an S-shape [6, 7]. This method is quite effective but the whole structure of the jet engine requireslarge modification relative to the straight jet engine. The second is using radar absorbing material(RAM) coating on the surface of the jet engine [8]. Although the RAM has a simple property of absorbingelectromagnetic wave in the molecular level, large RCS reduction is attained by coating the RAM. Inorder to obtain a better effect, many RAMs that have different permittivity and permeability shouldbe accumulated in a multilayered coating [15]. In the design of the multilayered coating, GA (GeneticAlgorithm) optimization of the multilayered RAM brings a significant reduction of the RCS [16, 17].However, this method is quite expensive with respect to material cost and maintenance.
In this paper, we suggest a new method for engine RCS reduction where a corrugated open-endedcircular waveguide cavity is treated as the jet engine analysis model. Many small-sized corrugationscompared to the size of the jet engine are inserted into the waveguide cavity. The inserted corrugationsserve as a waveguide filter which suppresses surface current in the waveguide cavity, and the RCS ofthe jet engine is consequentially reduced. This method can feasibly be realized without modificationof the whole engine structure, and it is also competitive in terms of cost and maintenance. In orderto analyze the structure, we used a mode matching technique combined with a scattering matrix toobtain an analytic solution. To analyze the open-end of the structure, the mode-matching techniquewas reciprocally able to convert a closed form equation of both incident and scattering waves to matricesof the excited modal coefficients at the aperture [18]. For the complicated corrugations in the circularwaveguide [19], many unit scattering matrices matching the unit circular waveguide as part of thewhole corrugated waveguide were cascaded step by step [20]. Combining the mode matching and thescattering matrix solutions, a closed-form RCS solution was finally obtained. Since the final solution hasthe characteristics of a modal analysis, the solution rapidly converges and is numerically efficient. Usingthese two characteristics, we suggested a simple cost function for optimizing the corrugation structure.Applying the GA algorithm, we found an optimized corrugation design for minimizing RCS.
2. RCS ANALYSIS OF OPEN-ENDED CIRCULAR WAVEGUIDE CAVITY WITHCORRUGATIONS
2.1. Analysis Structure and Methods
The suggested model for the RCS reduction is illustrated in Figure 1. We assume the jet engine model asa large open-ended circular waveguide cavity. In order to reduce the RCS of the jet engine, small-sizedcorrugations are inserted into the waveguide.
Figure 2 shows divided regions of the open-ended circular waveguide with corrugations and describes
Figure 1. Open-ended circular waveguide cavity with corrugation.
Progress In Electromagnetics Research, Vol. 146, 2014 59
Figure 2. Divided structure with three regions corresponding to each analysis method.
the analysis method according to each region. The aperture of the analysis structure is orthogonal tothe z axis, and the analysis structure is divided into three regions with respect to the analysis method,as shown in Figure 2. First, we used mode-matching at the open-ended aperture, and the obtainedequations were transformed to matrix equations for connection with the scattering-matrix method.Second, the corrugations were divided into many simple sections, and the scattering matrix methodwas applied to each section. Cascading all sections, a scattering characteristic was obtained for allcorrugations. Third, a boundary condition by perfect electric conductor (PEC) termination was simplyapplied. With the three conditions, the unknown coefficients of Fourier series and transform AI, BI,AII, BII and BIII were obtained. Finally, the RCS of the analysis structure was evaluated.
2.2. Field Representation
We utilize normalized field equations for simplicity of the field expression in [19]. This normalization isequal to assign both ε and µ to unity. With e−iωt time convention, the field is given by
E =√
µE′ H =√
µH ′ t =√
µεt′ (1)Maxwell’s equations for the normalized field expressions are as follows.
∇× E′ = in∂H ′
∂t′∇× H ′ = in
∂E′
∂t′(2)
where n is the refractive index of the medium. The electric and magnetic fields are expressed withcombination of electric and magnetic Hertz potentials such that
E′ = ∇×∇× Πm + ω′n∇× Πe (3)H ′ = ∇×∇× Πe + ω′n∇× Πm (4)
where Πm and Πe are the magnetic Hertz potential and the electric Hertz potential, respectively. Inregions I and II (the corrugated circular waveguide), the tangential fields with respect to the waveguideaperture are represented as
EI =∞∑
v=−∞
2NI∑
u=1
[A(v)
I (u)eikIvuz + B(v)
I (u)e−ikIvuz
]e(v)I (u) (5)
HI =∞∑
v=−∞
2NI∑
u=1
[A(v)
I (u)eikIvuz −B(v)
I (u)e−ikIvuz
]h(v)
I (u) (6)
EII =∞∑
v=−∞
2NII∑
u=1
[A(v)
II (u)e−ikIIvuz + B(v)
II (u)eikIIvuz
]e(v)II (u) (7)
60 Kim et al.
HII =∞∑
v=−∞
2NII∑
u=1
[−A(v)
II (u)e−ikIIvuz + B(v)
II (u)eikIIvuz
]h(v)
II (u) (8)
where the unknown modal coefficients of the Fourier series are defined as
A(v)I,II =
[AI,II(v)
m (1) , . . . , AI,II(v)m (NI,II) , AI,II(v)
e (1) , . . . , AI,II(v)e (NI,II)
]T(9)
B(v)I,II =
[BI,II(v)
m (1) , . . . , BI,II(v)m (NI,II) , BI,II(v)
e (1) , . . . , BI,II(v)e (NI,II)
]T(10)
and
e(v)I,II =
[eI,II(v)m (1) , . . . , eI,II(v)
m (NI,II) , eI,II(v)e (1) . . . , eI,II(v)
e (NI,II)]T
(11)
h(v)I,II =
[hI,II(v)
m (1) , . . . , hI,II(v)m (NI,II) , hI,II(v)
e (1) , . . . , hI,II(v)e (NI,II)
]T(12)
eI,II(v)m = iβI,II
vn ∇ΠI,IIm , eI,II(v)
e = ωnI,II∇ΠI,IIe × z (13)
hI,II(v)m = ωnI,II∇ΠI
m × z, hI,II(v)e = iβ′I,IIvn ∇ΠI,II
e (14)
ΠI,II(v)m = eivφJv
(χvn
aI,IIr
), ΠI,II(v)
e = eivφJv
(χ′vn
aI,IIr
)(15)
eI,II(v)m (u) = iβI,II
vu
[reiνφ χνu
aI,IIJ ′ν
(χνu
aI,IIr
)+ φ
1riνeiνφJν
(χνu
aI,IIr
)](16)
eI,II(v)e (u) = ωnII
[r1riνeiνφJν
(χ′νu
aI,IIr
)− φeiνφ χ′νu
aI,IIJ ′ν
(χ′νu
aI,IIr
)](17)
hI,II(v)m (u) = ωnII
[r1riνeiνφJν
(χνu
aI,IIr
)− φeiνφ χνu
aI,IIJ ′ν
(χνu
aI,IIr
)](18)
hI,II(v)e (u) = iβ′I,IIvu
[reiνφ χ′νu
aI,IIJ ′ν
(χ′νu
aI,IIr
)+ φiνeiνφJν
(χ′νu
aI,IIr
)]a. (19)
Note that Jv(·) is the Bessel function of the first kind and order v, and J ′v(·) denotes differentiationwith respect to the argument. χvu is defined as the uth root of the v-order Bessel function Jv(·) andχ′vu is defined as the uth root of the derivative of the v-order Bessel function J ′v(·). a is the radius ofthe aperture in the relevant region. r and φ denote unit vectors in cylindrical coordinates. Propagationconstant for the transverse magnetic (TM) mode and transverse electric (TE) mode (magnetic Hertzpotential and electric Hertz potential) are respectively represented as
kI,IIvu =
βI,IIvu =
√(ωnI,II)
2 −(
χvu
aI,II
)2
β′I,IIvu =
√(ωnI,II)
2 −(
χ′vuaI,II
)2. (20)
In the region III (the open-ended region) the outer scattered fields by the inlet of the waveguideare represented as
EIII =∞∑
v=−∞
∫ ∞
0B(v)
III(ζ)eiκze(v)III(ζ)dζ (21)
HIII =∞∑
v=−∞
∫ ∞
0B(v)
III(ζ)eiκzh(v)III(ζ)dζ (22)
where B(v)III(ζ) is the unknown spectral coefficient of the Fourier transform and
e(v)III(ζ) =
[eIII(v)m (ζ) eIII(v)
e (ζ)]T
(23)
h(v)III(ζ) =
[hIII(v)
m (ζ) hIII(v)e (ζ)
]T(24)
Progress In Electromagnetics Research, Vol. 146, 2014 61
eIII(v)m = iκ∇ΠIII
m , eIII(v)e = ωnIII∇ΠIII
e × z (25)
hIII(v)m = ωnIII∇ΠIII
m × z, hIII(v)e = iκ∇ΠIII
e (26)
ΠIII(v)m = ΠIII(v)
e = eiυφJυ (ζr) (27)
κ =√
(ωnIII)2 − ζ2 (28)
eIII(v)m (ζ) = iκ
[reiνφζJ ′ν (ζr) + φ
1riνeiνφJν (ζr)
](29)
eIII(v)e (ζ) = ωnIII
[r1riνeiνφJν (ζr)− φeiνφζJ ′ν (ζr)
](30)
hIII(v)m (ζ) = ωnIII
[r1riνeiνφJν (ζr)− φeiνφζJ ′ν (ζr)
](31)
hIII(v)e (ζ) = iκ
[reiνφζJ ′ν (ζr) + φ
1riνeiνφJν (ζr)
]. (32)
We assume that the incidence plane wave impinges with the angle θi, as shown in Figure 2. Forconvenience of offsetting in the calculation, we set arbitrary fields reflected by a virtual PEC plate, andthese fields are satisfied with the law of reflection. Although the offsetting only provides a valid θi rangefrom 0 to 50, the range is sufficient to observe scattering effect by the jet engine inlet because thesignificant JEM emerges around the normal axis of the jet engine aperture (θi = 0). The two pairs ofthe incidence and reflected plane wave according to the TM and TE polarization are represented as
(H i
)TM
= yeikx sin θi+ikz cos θi = ye−ikz cos θi
∞∑v=−∞
ivJv (kr sin θi) eivφ (33)
(Hr
)TM
= yeikx sin θi−ikz cos θi = ye−ikz cos θi
∞∑v=−∞
ivJv (kr sin θi) eivφ (34)
(Ei
)TE
= yeikx sin θi−ikz cos θi = ye−ikz cos θi
∞∑v=−∞
ivJv (kr sin θi) eivφ (35)
(Er
)TE
= −yeikx sin θi+ikz cos θi = −yeikz cos θi
∞∑v=−∞
ivJv (kr sin θi) eivφ (36)
2.3. Mode Matching and Scattering Matrix Analysis
The incidence plane wave is represented as the sum of the Bessel function of the first kind, representedas v. The entire fields are also represented in the same manner. Boundary conditions for the entirefields are then individually valid with respect to the arbitrary v. Therefore, the final solution can beobtained by superposition in terms of v. For an arbitrary v, tangential electric field continuity at theaperture requires
E(v)II = E
(v)III +
(Ei + Er
)(v)
TM,TE(37)
For convenience, we apply power orthogonality described in [21] to (37), and the equations arerepresented as (A1). In order to connect mode-matching equations with scattering matrix, the equationsare transformed into a matrix form such as
B(v)III(ζ) =
(Q(v)
o
)−1P (v)
o
(A(v)
II +B(v)II
)(38)
where Q(v)o and P
(v)o are represented as (A3) and (A4), respectively. In the same manner, the tangential
magnetic field continuity at the aperture requires
H(v)II = H
(v)III +
(H i + Hr
)(v)
TM,TE(39)
62 Kim et al.
The equation applied power orthogonality to (39) is represented as (A2), and the matrix form equationyields
R(v)o
(B(v)
II −A(v)II
)= −
∫ ∞
0
(P (v)
o
)HB
(v)III(ζ)dζ + V(v)
TE,TM (40)
where R(v)o and V(v)
TE,TM are represented as (A5) and (A6) respectively. Substituting (38) into (40), weobtain
R(v)o
(B(v)
II −A(v)II
)= −X(v)
(A(v)
II +B(v)II
)+ V(v)
TE,TM (41)
(a) (b)
(c)
Figure 3. Scattering matrix method (a) continuous structure, and (b) discontinuous structure, and(c) cascading method between two structures.
Progress In Electromagnetics Research, Vol. 146, 2014 63
whereX(v) =
∫ ∞
0
(P (v)
o
)H (Q(v)
o
)−1P (v)
o dζ. (42)
X(v) is represented as (A7), and the integrals in the X(v) are numerically evaluated by the adaptiveGauss-Kronrod quadrature method in [22].
It is necessary to obtain another condition by using the scattering matrix of the corrugations withthe boundary condition of the PEC-termination. The PEC boundary condition is simply expressed asB(v)
I = −A(v)I . We then obtain
B(v)II =
(S
(v)22 − S
(v)21
(I + S
(v)11
)−1S
(v)11
)A(v)
II (43)
where S(v)11 , S
(v)12 , S
(v)21 and S
(v)22 are evaluated by using Figure 3 describing the scattering matrix method
case by case. The three cases are a continuous structure, a discontinuous structure and a cascadingmethod between two structures AI, AII, BI and BII are modal coefficient vectors of Fourier seriesrepresenting electromagnetic fields in the waveguide. These vectors have relation as
[BI
BII
]=
[S11 S12
S21 S22
] [AI
AII
]. (44)
The P , Q and R matrices for calculating the scattering matrix are represented as (B1), (B2) and (B3)respectively.
Using both (41) and (43), the unknown modal coefficients at the aperture A(v)II + B(v)
II can beevaluated as
A(v)II +B(v)
II =(Ψ(v) + I
)(R(v)
o
(Ψ(v) − I
)+ X(v)
(Ψ(v) + I
))−1V(v)
TE,TM (45)
where
Ψ(v) =(
S(v)22 − S
(v)21
(I + S
(v)11
)−1S
(v)11
). (46)
Defining Ω(v)TM,TE = A(v)
II + B(v)II and Ω(v)
TM,TE = [Ω(v)m Ω(v)
e ]TTM,TE, the unknown spectral coefficient of
the scattering field B(v)III(ζ) is represented as
B(v)III(ζ) =
(Q(v)
o
)−1P (v)
o Ω(v)TM,TE (47)
2.4. Closed-Form RCS Solution
The scattered electric and magnetic fields are obtained by substituting (47) into (21) and (22),respectively. In the far field observation, the stationary phase method can be applied to the semi-infinite bounded integrals about ζ. The obtained fields with respect to the incidence polarization arethen substituted into the definition of the RCS, represented as
σTM = limrs→∞
(4πr2
s
|Eθ|2 + |Eφ|2|Ei|2
), σTE = lim
rs→∞
(4πr2
s
|Hθ|2 + |Hφ|2|Hi|2
). (48)
Finally we can express the closed-form RCS solution of the analysis structure as
σTM=4π
∣∣∣∣∣∣∣
∞∑v=−∞
NII∑u=1
eivφe−i( v2 )π
Ω(v)m (u) iβvuχvu
k2 sin θi cos 2θiJ′v(χvu )[
(k sin θi)2−
(χvuaII
)2]
+Ω(v)e (u) ωnII (iv) (cot θi − sin θi) Jv(χ′vu)
Jv (kaII sin θi)
∣∣∣∣∣∣∣
2
+
∣∣∣∣∣∣∞∑
v=−∞
NII∑u=1
eivφe−i( v2 )π
Ω(v)
e (n)ω nIInIII
χ′2vuaII
(k cos θi)Jv(χ′vu )[(k sin θi)
2−(
χ′vuaII
)2]
J ′v (kaII sin θi)
∣∣∣∣∣∣
2
(49)
64 Kim et al.
σTE=4π
∣∣∣∣∣∣∞∑
v=−∞
NII∑u=1
eivφe−i( v2 )πΩv
e (u) i nIInIII
χ′2vuaII
Jv(χ′vu)(k2 cos θi)J ′v(kaII sin θi)[(k sin θi)
2−(
χ′vuaII
)2]
∣∣∣∣∣∣
2
+
∣∣∣∣∣∣∣
∞∑v=−∞
NII∑u=1
eivφe−i( v2 )π
Ωvm (u) ωβvuχvuJ ′v (χvu) (k sin θi)[
(k sin θi)2−
(χvuaII
)2]
+Ωve (u) ω2nIIvJv(χ′vu) 1
(k sin θi)
Jv (kaII sin θi)
∣∣∣∣∣∣∣
2
(50)
3. NUMERICAL COMPUTATIONS AND GA OPTIMIZATION
For numerical calculation from the point of convergence, the limitation of v must be determined as afinite number, not an infinite number. The limit is proportional to the aperture size of the cylinder, andv having a finite range from −11 to 11 is appropriate for the case aII = 5λ. When v is determined, themaximum modal order at the aperture NII is automatically determined by (20). Because of the rapidlyconvergent series in the equations, the calculation is numerically efficient.
Figure 4 shows the specific structure employed for verification. The monostatic RCS calculationby (49) and (50) for the structure in Figure 4 was evaluated and verified through comparison with aFEKO simulation by the method of moments (MoM). Figure 5 and Figure 6 show the RCS comparisonin terms of an incident plane wave, respectively. The results of the proposed solution agreed well withthe MoM simulated results for θi range from 0 to 35, while a slight error occurred with the proposedsolution for θi range more than 35. The reason of the error was using a pair of an incidence and reflectedplane wave, as discussed above. The reflected plane wave by the virtual PEC plate were assumed toapply the mode-matching method to the aperture. The assumed PEC plate improperly converted a rimedge into a 90 half-plane wedge, and resulted in an error which was proportional to the θi. For a smallθi, the scattered field by the inside of the circular waveguide cavity is dominant in comparison with anedge diffracted field, therefore the error due to the edge is negligible value. In contrast, the effect of theedge diffraction is noticeable for θi more than 35, thus an accuracy of the proposed solution graduallydecrease with increasing θi. The proposed solution provides correct analysis of a RCS for an strongJEM effect occur with θi range from 0 to 30. In order to analyze the RCS for an weak JEM effectoccur with θi more than 30, the rim edge diffraction must be supplementary considered for accuracy.
For improved reduction of the RCS, the structure of the corrugations requires optimization. Weapplied GA optimization to the number and unit cell shape of the corrugations. A cost function wasdefined as
f =∣∣∣∣sT
[(R(1)
o
(Ψ(1) − I
)+ X(1)
(Ψ(1) + I
))−1]s∣∣∣∣ (51)
where s = [1, 1, . . . , 1]T .
Figure 4. Verification design of the open-ended circular waveguide cavity with corrugations.
Progress In Electromagnetics Research, Vol. 146, 2014 65
(a)(b)
Figure 5. θ-θ pol. RCS of the open-ended corrugated circular waveguide in Figure 4.(a) RCS calculated by the mode-matching andscattering matrix. (b) Simulated result by theMoM.
(a)(b)
Figure 6. φ-φ pol. RCS of the open-ended corrugated circular waveguide in Figure 4.(a) RCS calculated by the mode-matching andscattering matrix. (b) Simulated result by theMoM (FEKO).
Figure 7. Optimized design of the circular waveguide cavity with corrugations.
(a) (b)
Figure 8. RCS comparison among the three structures. (a) θ-θ pol. (b) φ-φ pol.
66 Kim et al.
The cost function is proportional to the sum of the modal coefficients (45) at the aperture, andonly the dominant order of v = 1 is used for convenience. The reduction of the cost function leads toreduction of the scattered wave intensity. Utilizing the GA to minimize the cost function, we obtainedthe optimized design of the corrugations for the RCS reduction, as shown in Figure 7.
Figure 8 shows the RCS comparison results among the three structures. The first is the non-corrugated circular waveguide cavity with aII = 5λ. The second is the circular waveguide cavity withnon-optimized corrugations in Figure 4. The third is the circular waveguide cavity with optimizedcorrugations using GA in Figure 7. The RCS results of all structure were evaluated by the proposedsolution (49) and (50). Finally, we confirmed that the RCS is reduced by adding the corrugations tothe jet engine inlet.
4. CONCLUSION
For the first time, we analyzed the RCS of the open-ended circular waveguide cavity with corrugationsby using mode-matching and scattering matrix methods. We validated the suggested RCS solutionwith a MoM simulation. The suggested solution with a scattering matrix analysis can be applied toan arbitrary corrugated structure. The suggested modal solution with rapidly converging series alsoprovides computational efficiency. Using these two characteristics, we applied the GA to reconfigurethe shape of the corrugations to reduce the RCS. Finally, we obtained the optimized design of thecorrugations. The optimized design led to improved RCS reduction. In future work, we will select thebest optimization algorithm, variables and cost function for advanced reduction of the jet engine RCS.
APPENDIX A. APPLYING POWER ORTHOGONALITY TO THE BOUNDARYCONDITIONS
NII∑
u=1
[A(v)
II (u) + B(v)II (u)
] ∫ aII
0
(eII(v)m (u)× h
III(v)∗m (ζ) + e
II(v)e (u)× h
III(v)∗m (ζ)
+eII(v)m (u)× h
III(v)∗e (ζ) + e
II(v)e (u)× h
III(v)∗e (ζ)
)· ds
=∫ ∞
0B
(v)III (ζ)
∫ aII
0
(eIII(v)m (ζ)× h
III(v)∗m (ζ) + e
III(v)e (ζ)× h
III(v)∗m (ζ)
+eIII(v)m (ζ)× h
III(v)∗e (ζ) + e
III(v)e (ζ)× h
III(v)∗e (ζ)
)· dsdζ
+∫ ∞
0
∫ aII
0
((Ei + Er
)(v)
TM,TE× h
III(v)∗m
+(Ei + Er
)(v)
TM,TE× h
III(v)∗e
)· dsdζ (A1)
NII∑
u=1
[B(v)
II (u)−A(v)II (u)
] ∫ aII
0
(hII(v)m (u)× e
II(v)∗m (u) + h
II(v)e (u)× e
II(v)∗m (u)
+hII(v)m (u)× e
II(v)∗e (u) + h
II(v)e (u)× e
II(v)∗e (u)
)· ds
=∫ ∞
0B
(v)III (ζ)
∫ aII
0
(hIII(v)m (ζ)× e
II(v)∗m (u) + h
III(v)e (ζ)× e
II(v)∗m (u)
+hIII(v)m (ζ)× e
II(v)∗e (u) + h
III(v)e (ζ)× e
II(v)∗e (u)
)· dsdζ
+∫ ∞
0
∫ aII
0
((H i + Hr
)(v)
TM,TE× e
II(v)∗m(
H i + Hr)(v)
TM,TE× e
II(v)∗e
)· dsdζ (A2)
P (v)o =
−iωnIIIβvuχvuJ ′v(χvu) ζ2Jv(ζb)
ζ2−(χvub )2 −ω2nIInIII(iv)Jv(χ′vu)Jv(ζb)
0 iωnIIχ′2vu
b Jv(χ′vu) κ∗IIIζJ ′v(ζb)
ζ2−(
χ′vub
)2
(A3)
Q(v)o =
(−iωnIIIκ3ζ 00 −iωnIIIκ
∗IIIζ
)δ(ζ − ζ ′) (A4)
R(v)o =
(− i
2ωnIIβ∗vuχ2
vu [J ′v(χvu)]2 00 − i
2ωnIIβ′vu
(χ′2vu − v2
)[Jv(χ′vu)]2
)(A5)
Progress In Electromagnetics Research, Vol. 146, 2014 67
V(v)TE=
0−2iv cos θiωnII
χ′2vuaII
Jv(χ′vu) J ′v(kIIIaII sin θi)
(kIII sin θi)2−
(χ′vuaII
)2
(A6a)
V(v)TM=
(2ivβvuχvuk3 sin θiJ
′v(χvu) J ′v(kIIIaII sin θi)
(kIII sin θi)2−(χvu
a )2
2iv ωnIIvkIII sin θi
Jv(χ′vu)Jv(kIIIaII sin θi)
)(A6b)
X(v) =∫ ∞
0
(P (v)
o
)H (Q(v)
o
)−1P (v)
o dζ =(
X11 X12
X21 X22
)(A7a)
X11mn =
∫ ∞
0
iωnIIIβ∗vnβvmχvmχvnJ ′v(χvm)J ′v(χvn)ζ3 [Jv (ζb)]2
κIII
[ζ2 − (χvm
b
)2] [
ζ2 − (χvn
b
)2] dζ (A7b)
X12mn =
∫ ∞
0
iω2nIInIIIvβ∗vnχvnJv(χ′vm)J ′v(χvn)ζ [Jv (ζb)]2
κIII
[ζ2 − (χvn
b
)2] dζ (A7c)
X21mn =
∫ ∞
0
iω2nIInIIIvβvmχvmJv(χ′vn)J ′v(χvm)ζ [Jv (ζb)]2
κIII
[ζ2 − (χvm
b
)2] dζ (A7d)
X22mn =
∫ ∞
0
iω3n2IInIIIv
2Jv(χ′vm)Jv(χ′vn) [Jv(ζaII)]2
κIIIζdζ
+∫ ∞
0
iωn2II
nIII
(χ′vmχ′vn
aII
)2Jv(χ′vm)Jv(χ′vn)κIIIζ [J ′v (ζaII)]
2
[ζ2 −
(χ′vmaII
)2] [
ζ2 −(
χ′vnaII
)2] dζ (A7e)
APPENDIX B. P , Q, R MATRICES FOR SCATTERING MATRIX ANALYSIS
P (v) =(
P 11 P 12
0 P 22
)(B1a)
P 11mn = iωnIIβvm
χvm
a
χvn
b
ab χvnJ ′v (χvm) Jv
(ab χvn
)(χvm
a
)2 − (χvn
b
)2 (B1b)
P 12mn = −ω2nInII(iv)Jv(χ′vm)Jv
(a
bχvn
)(B1c)
P 22mn = −iωnIβ
′∗vn
χ′vmχ′vn
ab
χ′vmJv (χ′vm) J ′v(
ab χ′vn
)(
χ′vma
)2−
(χ′vn
b
)2 (B1d)
Q(v) =(
Q11 00 Q22
)(B2a)
Q11 = diag[− i
2ωnIIβvnχ2
vn
[J ′v(χvn)
]2]
(B2b)
Q22 = diag[− i
2ωnIIβ
′∗vn
(χ′2vn − v2
) [Jv(χ′vn)
]2]
(B2c)
R(v) =(
R11 00 R22
)(B3a)
R11 = diag[− i
2ωn1β
∗vmχ2
vm
[J ′v(χvm)
]2]
(B3b)
R22 = diag[− i
2ωnIβ
′vm
(χ′2vm − v2
) [Jv(χ′vm)
]2]
(B3c)
68 Kim et al.
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