PO method for diffraction problems by impedance wedges

Optics Communications 284 (2011) 4289–4294

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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

A hybrid Maliuzhinets/PO method for diffraction problems by impedance wedges Yusuf Ziya Umul ⁎ Electronic and Communication Department, Cankaya University, Öğretmenler Cad, No: 14, Yüzüncü Yıl, Balgat, Ankara 06530, Turkey

a r t i c l e

i n f o

Article history: Received 11 March 2011 Received in revised form 21 June 2011 Accepted 23 June 2011 Available online 5 July 2011

a b s t r a c t The solution of Maliuzhinets of the diffraction problem of waves by an impedance wedge is transformed into a physical optics integral. The resultant expression is suitable for the investigation of various diffraction problems having impedance wedges. The method is applied to the scattering of waves by an impedance spherical reflector with wedge structure at its discontinuity. The results are examined numerically. © 2011 Elsevier B.V. All rights reserved.

Keywords: Physical optics Diffraction theory Electromagnetic scattering

1. Introduction The method of physical optics (PO) was first introduced by Macdonald for the investigation of the interaction of the electric waves by an obstacle on their path of propagation [1]. The technique is widely used for the examination of microwave reflector antennas [2,3] and scattering problems of waves by various shapes [4,5]. The main defect of PO is its edge point contributions which lead to the incorrect diffracted waves. In 2004, we showed that it is possible to obtain the exact solution of the scattering problem of the electromagnetic waves by a perfectly conducting half-pane, which was not possible before [6]. Our approach was based on three axioms, the mathematical and physical basis of which were explained in following papers [7,8]. We also applied the modified theory of physical optics (MTPO) to the diffraction problems by conducting wedge [9], impedance wedge [10] and impedance half-plane [11]. It is important to note that the MTPO scattering integrals satisfy the Helmholtz equation [12]. The improved PO technique was also used by Ando et al. for the study of RCS of some canonical shapes [13,14]. The aim of this paper is to introduce a hybrid method of solution for the scattering problems of waves by impedance wedges. With this purpose, we will take into consideration the solution of Maliuzhinets [15] and transform the diffracted field into a PO type integral by using the philosophy of MTPO. The resultant scattering integral will be applied to the diffraction of waves by an impedance spherical reflector with wedge discontinuity. The results will be plotted and examined numerically. Our motivation is to obtain a general PO integral that can be used for the investigation of scatterers with various geometries, having wedge discontinuities.

⁎ Tel.: + 90 312 2844500; fax: + 90 312 2848043. E-mail address: [email protected]. 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.06.045

It is important to note that there is also an alternative solution of this problem, which was put forward by Senior [16]. However, we recently showed that these two solutions do not satisfy the impedance boundary condition on the surfaces of the wedge or half-plane [17,18]. Furthermore, the solution of Senior does not compensate the discontinuity of the geometrical optics (GO) fields at the transition regions [19]. The scattered field expression of Maliuzhinets is continuous everywhere. For this reason, we chose to transform the diffracted field of Maliuzhinets to the PO integral. A time factor of exp(jωt) is suppressed throughout the paper. ω is the angular frequency.

2. Theory We take into account an impedance wedge, illuminated by the incident plane wave of ui = u0 exp ½ jk ðx cos ϕ0 + y sin ϕ0 Þ

ð1Þ

for u0 is the complex amplitude. ϕ0 is the angle of incidence and k wavenumber. u represents the z component of the electric field. The cylindrical coordinates are defined by (ρ,ϕ,z). The geometry of the problem is given in Fig. 1. P and Q express the points of observation and scattering (integration). The outer angle of the wedge is equal to 2φ. Z1 and Z2 are the impedances of the wedge's upper and lower surfaces. β is the angle of scattering [6] and R the distance between the observation and integration points. The scattered fields by the wedge can be written as us ðP Þ = ui ðP Þ + uPO ðP Þ

ð2Þ

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diffracted field reads ud = −

u0 expð−jπ = 4Þ f ðπ−ϕÞ expð−jkρÞ pffiffiffiffiffiffi pffiffiffiffiffiffi ; cosϕ + cosϕ0 2π kρ

ð8Þ

since β is π − ϕ at the edge point. We will determine the value of f (β) by considering the diffracted field expression of Maliuzhinets, which can be written as udM =

u0 expð−jπ = 4Þ Π ðϕ; ϕ0 Þ expð−jkρÞ pffiffiffiffiffiffi pffiffiffiffiffiffi cosϕ + cosϕ0 2π kρ

ð9Þ

where the function Π(ϕ, ϕ0) can be introduced as Π ðϕ; ϕ0 Þ =

0Þ π cos πðφ−ϕ 2ϕ

2φM ðφ−ϕ0 Þ

½qðφ−ϕ−πÞ−qðφ−ϕ + π Þð cosϕ + cosϕ0 Þ ð10Þ

for M(x) is defined by Fig. 1. Diffraction geometry of waves by a wedge with different face impedances.

where uPO is the PO field, which can be introduced as uPO ðP Þ =

jk expð−jkR1 Þ 0 ∬ J ðQ Þ dS ; 4π S PO R1

ð3Þ

for S is the illuminated surface of the scatterer [20]. R1 is equal to [(x − x′) 2 + y 2 + (z − z′) 2] 1/2 for our problem. JPO is the PO current that can be defined by JPO ðQ Þ = 2ui ðQ Þf ðβÞ

ð4Þ

where f(β) is a function to be determined. This function gives an angular relation between the incident and scattered fields on the surface and β is the angle of scattering as shown in Fig. 1. In the classical PO theory, f(β) is constant and only depends on the angle of incidence. ui(Q) is equal to u0exp(jkx′cosϕ0) for the incident wave, given in Eq. (1). The integral, in Eq. (3), can be rewritten as ∞

uPO ðP Þ =

∞

jk expð−jkR1 Þ 0 0 ∫ ∫ u ðQ Þf ðβÞ dx dz 2π z0 = −∞ x0 = 0 i R1

ð5Þ

when the geometry, in Fig. 1, is taken into account. If there is symmetry according to z′, this part of the integral can be evaluated directly and gives a Hankel function of second kind zero order [6]. The PO integral reads

    π π MðxÞ = Mφ x + φ + −η1 Mφ x + φ− + η1 Mφ 2 2     π π x−φ + −η2 Mφ x−φ− + η2 2 2

ð11Þ

in terms of the Maliuzhinets function, Mφ. [15]. η1 and η2 are sin− 1(Z0/Z1) and sin− 1(Z0/Z2) respectively. Z0 is the impedance of the free space. The Maliuzhinets function is introduced by ( ) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi      ∞ πx 1 πx πv dv exp ∫ ln 1 + j tan tanh cos Mφ ðxÞ = : 4φ 2π −∞ 4φ 4φ chv ð12Þ q(x) has the expression of qðxÞ =

MðxÞ πx 0Þ sin 2φ − sin πðφ−ϕ 2φ

:

ð13Þ

f(π− ϕ) is found to be f ðπ−ϕÞ = −Π ðϕ; ϕ0 Þ

ð14Þ

when Eq. (8) is equated to Eq. (9). Thus f(β) can be determined as f ðβÞ = −Π ðπ−β; ϕ0 Þ

ð15Þ

by taking into account the relation of ϕ = π− β. The PO integral of the scattered waves by a wedge with different face impedances becomes uPO ðP Þ = −

k expðjπ = 4Þ ∞ expð−jkRÞ 0 pffiffiffiffiffiffi pffiffiffiffiffiffi ∫ ui ðQ ÞΠ ðπ−β; ϕ0 Þ dx 2π kR 0

ð16Þ

when Eq. (15) is used in Eq. (6). The PO integral can be written as uPO ðP Þ =

∞

k exp ðjπ = 4Þ expð−jkRÞ 0 pffiffiffiffiffiffi pffiffiffiffiffiffi ∫ ui ðQ Þf ðβÞ dx 2π kR 0

ð6Þ

when the Debye asymptotic expansion of the Hankel function is used. R has the expression of [(x − x′) 2 + y 2] 1/2. The diffracted field can be evaluated with the method of edge point at x′ = 0. The formulation of the edge point technique can be given by ∞

∫ hðxÞ exp½jkgðxÞdx≈− xe

1 hðxe Þ exp½jkgðxe Þ jk g 0 ðxe Þ

ð7Þ

for xe is the edge point of the integral [21]. h(x) and g(x) are the amplitude and phase functions of the integral. As a result, the wedge

uPO ðP Þ = −

jk expð−jkR1 Þ 0 ∬ u ðQ ÞΠ ðπ−β; ϕ0 Þ dS R1 2π S i

ð17Þ

for the general case of arbitrary surface with wedge discontinuity. The scattered field reads us ðP Þ = ui ðP Þ−

jk expð−jkR1 Þ 0 ∬ u ðQ ÞΠ ðπ−β; ϕ0 Þ dS 2π S i R1

ð18Þ

according to Eq. (2). The PO surface current can be defined as JPO ðQ Þ = −2ui ðQ ÞΠ ðπ−β; ϕ0 Þ for the scatterers with wedge discontinuities.

ð19Þ

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3. Application: scattering of waves by an spherical impedance reflector with wedge discontinuity We take into account a spherical reflector, illuminated by the plane wave, which has the expression of ui ðP Þ = u0 expðjkzÞ:

ð20Þ

The geometry of scattering is given in Fig. 2. QE is the edge point of the reflector. The spherical coordinates are defined by (r,θ,ϕ). →n is the unit normal vector of the spherical surface. The radius of the sphere is equal to a. The geometry is symmetric according to ϕ. We take into account the relations of π β→ −β 2

ð21Þ

and π 0 ϕ0 → −θ 2

ð22Þ

in Eq. (18) according to Fig. 2. The PO integral of the problem reads 2

uPO ðP Þ = −

jka 2π

2π

θ

∫

∫

φ0 = 0

θ0 = 0

0

ui ðQ ÞΠ

Fig. 3. Geometry of the reflected GO waves by the spherical reflector.

π  expð−jkR Þ π 0 0 0 0 1 + β; −θ sinθ dθ dϕ 2 2 R1 ð23Þ

for R1 is r 2 + a 2 − 2ra[sinθsinθ′cos(ϕ − ϕ′) + cosθcosθ′]} 1/2. ui(Q) is exp(jkacosθ′). Since the problem is symmetric according to ϕ′ and varies from 0 to 2π, we can evaluate this part of the scattering integral by the method of stationary phase. The first derivative of R1 according to ϕ′ gives

ra sinθ sinθ sin ϕ−ϕ0 ∂R1 = − ; R1 ∂ϕ0 0

ð24Þ

points are the stationary phase point that can be obtained by equating the first derivative of the phase function to zero. The third critical point is the edge point, at θ′ = θ0. The contribution of this point gives the edge diffracted waves and can be evaluated by the method of edge point. 4. Asymptotic evaluation of the scattering integral The integral, in Eq. (25), is taken into consideration. First of all, we will evaluate the stationary phase point contributions of the PO integral. These contributions lead to the GO waves. The phase function of the integral can be written as

which leads to the stationary phase point of ϕs = ϕ when equated to zero. The PO integral reads

0

0 g θ = a cosθ −R;

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi θ0 π  expð−jkRÞ a sinθ0 ka exp j π4 π 0 0 pffiffiffiffiffiffi pffiffiffiffiffiffi uPO ðP Þ = − + β; −θ dθ ∫ ui ðQ ÞΠ 2 2 r sinθ 2π θ0 = 0 kR

the first derivative of which gives

ð25Þ after the evaluation of the ϕ′ part. The details of the stationary phase method can be found in Ref. [6]. R is [r 2 + a 2 − 2racos(θ − θ′)] 1/2. The integral, in Eq. (25), has three critical points. The first two critical



ra sin θ−θ0 dg 0 : 0 = −a sinθ + dθ R

ð26Þ

ð27Þ

Eq. (27) can be arranged as

dg 0 = −a sinθ − sinβ dθ0

ð28Þ

when the geometry, in Fig. 2, is taken into account. Thus we can find two stationary points at βs1 = θs and βs2 = π − θs. θs is the stationary phase value of θ′ and represents the value of θ′ at Q where the incident ray hits. The first stationary phase point gives the reflected GO waves by the surface of the reflector. The second one expresses the transmitted rays through the spherical surface as if there is not any obstacle at point Q. The second derivative of the phase function reads d2 g =a dθ02

 cosβ

 dβ 0 0 − cosθ ; dθ

ð29Þ

which yields the equation of

Fig. 2. Scattering geometry of waves by a spherical reflector with wedge discontinuity.

0

2 R cosβ + cosθ + a cos β d2 g = −a R dθ02

ð30Þ

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90

Scattered wave

90

1.5

120

1.5

120

Diffracted wave

60

GO wave

60

1

1

150

30 0.5

30

150 0.5

180 180

0

0

210 210

330

330 300

240 240

270

300 270

Fig. 6. Total scattered wave.

Fig. 4. Incident scattered wave.

U(x) is the unit step function, which is equal to one for x N 0 and zero otherwise. l is the stationary phase value of R and equal to [r 2 + a 22racos(θ − θs)] 1/2. θ1 can be defined by the relation of

when the relation of dβ R + a cosβ =− dθ0 R

ð31Þ

is taken into account according to Fig. 2. The reflected GO wave is found to be GO ur

= −u0

Π

π

2

+ θs ; π2 −θs cosθs

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kla cosθs a sinθs expð−jklÞ pffiffiffiffi ð32Þ 2l + a cosθs r sinθ kl

expðjka cosθs ÞU ðθ1 −θÞ after the phase point evaluation of the integral, in Eq. (25), at βs = θs. θs can be determined from the equation of −1

2θs −θ = sin

a sinθs : r

ð33Þ

−1

θ1 = 2θ0 − sin

a sinθ0 r

ð34Þ

and shows the location of the reflection boundary. θ0 is the coordinate of the edge point QE.The geometry of reflection is given in Fig. 3. Qs is the point of reflection. The last reflection point of the GO wave is located at θs = θ0. After this point, the reflected GO waves will vanish. Þ sinðϕ−α Þ− sinη1 For this reason the unit step f MðM2ϕ−α = − sin ðα Þ ðϕ−α Þ + sinη1 unction, in Eq. (32), exists. The relation of Π

π

2

+ θs ; π2 −θs cos φ−ðπ =n2Þ + θs sinθs − sinη1 =− s cosθs sinθs + sinη1 sin π−2θ 2n

ð35Þ

according to the Equation of Mð2φ−α Þ sinðφ−α Þ− sinη1 =− M ðα Þ sinðφ−α Þ + sinη1

ð36Þ

for n is equal to 2φ/π [22]. Thus the reflected GO wave can be written as Scattered wave

90

0.2

120

Diffracted wave

60

GO

ur

= u0

GO wave

cos φ−ðπ =n2Þ + θs sinθs − sinη1 sinθs + sinη1 sin π−2θs 2n

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kla cosθs a sinθs expð−jklÞ pffiffiffiffi expðjka cosθs ÞU ðθ1 −θÞ 2l + a cosθs r sinθ kl

ð37Þ

0.15 0.1

150

30

0.05

180

0

when Eq. (35) is used in Eq. (32). The stationary phase point, at βs2 = π − θs, leads to the transmitted GO waves through the reflector surface. However this component is fictitious and vanishes when ui is added to uPO. The incident scattered wave reads GO

ui 330

210

300

240 270

Fig. 5. Reflected scattered wave.

= u0 expðjkzÞ½1−U ðθ−θ2 Þ

ð38Þ

where θ2 represents the location of the shadow boundary and can be evaluated as θ2 = sin

−1

a sinθ0 r

according to Fig. 3.

ð39Þ

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The diffracted field can be evaluated by using the edge point method at θ′ = θ0. The formula is given in Eq. (7). The wedge diffracted field reads

respectively. The value of the diffraction angle βe can be determined by the equation of





rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp −j π4 Π π2 + βe ; π2 −θ0 a sinθ0 expð−jkRe Þ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi ud = expðjka cosθ0 Þ θ0 0 r sinθ kRe 2 2π sin βe −θ cos βe + 2 2

βe = cos

ð40Þ for βe is the edge value of β. Re is [r 2 + a 2-2racos(θ − θ0)] 1/2. Eq. (40) can be arranged as ð41Þ

ud = udi + udr

where udi and udr are the incident and reflected diffracted waves that can be defined by udi =

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi



exp −j π4 Π π2 + βe ; π2 −θ0 β + θ0 a sinθ0 1 pffiffiffiffiffiffi sin e cosθ0 2 r sinθ cos βe 2 2π

+ θ0 2

expðjka cosθ0 Þ

expð−jkRe Þ pffiffiffiffiffiffiffiffi kRe

−1

r cosðθ−θ0 Þ−a : Re

The scattered wave is the sum of the GO and diffracted fields. 5. Numerical results In this section, we will analyze the scattered, diffracted and GO waves numerically. The observation distance (r) is taken as 6λ for λ is the wavelength. The radius of the sphere (a) is equal to 2λ. The outer angle of the wedge discontinuity is 3π/2. In this case [24], the Maliuzhinets function becomes M3 = 4 ðxÞ =

x + cos x−π 6 cos 6

ð42Þ and udr =

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi



exp −j π4 Π π2 + βe ; π2 −θ0 β −θ0 a sinθ0 1 expð−jkRe Þ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi cos e expðjka cosθ0 Þ 2 r sinθ sin βe −θ0 cosθ0 kRe 2 2π 2

ð43Þ respectively. udi approaches to infinity at βe = π − θ0, which is the location of the shadow boundary. In a similar way, udr is not finite at the reflection boundary that is located at βe = θ0. For this reason, the diffracted field expressions, in Eqs. (42) and (43), are not uniform. The uniform representations of the waves can be obtained by using the asymptotic relation of  

2 exp −j π4 exp −jx pffiffiffi signðxÞF ½x≈ x 2 π

ð44Þ

for xN N 1 [23]. sign(x) is the signum function, which is equal to 1 for x N 0 and − 1 for x b 0. F[x] is the Fresnel integral that can be defined by ∞ exp j π4 pffiffiffi ∫ π x

F ½x  =

  2 exp −jt dt:

ð45Þ

We define the detour parameters of qffiffiffiffiffiffiffiffiffiffi β + θ0 ξi = − 2kRe cos e 2 and ð47Þ

for the incident and reflected diffracted waves. The detour parameter represents the difference of the path ways of the GO and diffracted waves. It is an important parameter, because the change in the sign of the diffracted field is determined by this expression. The uniform incident and reflected diffracted fields can be written as udi = −

Π

π

2

+ βe ; π2 −θ0 cosθ0



β + θ0 sin e 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a sinθ0 signðξi ÞF ½ξi  r sinθ

ð48Þ

expðjka cosθ0 Þ exp½jkRe cosðβe + ϕ0 Þ and udr = −

Π

π

2

+ βe ; π2 −θ0 cosθ0



β −θ0 cos e 2

cos2 π6 cos 6x

π

:

ð51Þ

The observation angle (θ) varies in the interval of [0, π]. sin(η1) and sin(η2) have the values of 2 and 4 respectively. θ0 is taken as π/4. Fig. 4 shows the variation of the incident scattered, diffracted and GO waves versus the observation angle. The GO field is given by Eq. (38). The spherical surface obstructs the incident wave and a geometrical shadow region forms behind the reflector at θ∈[π − θ1, π]. The incident diffracted field, given in Eq. (48), has an amplitude, which is the half of the amplitude value of the GO field at the shadow boundary. The diffracted field has also a phase shift of π in this boundary. Thus it compensates the discontinuity of the GO wave and the scattered field, which is the sum of the GO and diffracted waves, is continuous everywhere. Fig. 5 depicts the variation of the reflected scattered, diffracted and GO fields with respect to the observation angle. The GO wave is discontinuous at the reflection boundary and the diffracted field compensates this discontinuity. As a result, the scattered wave is continuous. The variation of the total scattered wave, which is the sum of the incident and reflected scattered fields versus the observation angle, is plotted in Fig. 6. The depth of variation of the interference fringes is more intensive for θ b θ2, which is the illuminated region. Since two GO fields interfere with each other in this region. 6. Conclusion

ð46Þ

qffiffiffiffiffiffiffiffiffiffi β −θ0 ξr = − 2kRe sin e 2

ð50Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a sinθ0 signðξr ÞF ½ξr  expðjka cosθ0 Þ exp½−jkRe cosðβe −ϕ0 Þ r sinθ

ð49Þ

In this paper, we transformed the diffracted field expression, obtained by Maliuzhinets for the scattering problem of waves by a wedge with different face impedances, into a PO type integral. The new integral is applied to the diffracted problem of waves by a spherical convex reflector with wedge discontinuity. The scattering integral is evaluated asymptotically and the scattered, diffracted and GO waves are plotted numerically. The importance of this study is the introduction of a general scattering PO integral, given by Eq. (18), that can be applied to the diffraction problem of waves by various impedance surfaces which have wedge discontinuities. The solution of these problems was not possible before by the method of PO, since the surface current is defined in the direct illuminated portion of the scatterer and the PO integral sees the wedge geometry as a half-plane. The second important aspect of the paper is the consideration of wedge with different face impedances. Thus the method of PO is improved for these types of problems by using the axioms of MTPO. References [1] H.M. Macdonald, Philos. Trans. R. Soc. Lond. A 212 (1913) 299. [2] S. Silver, Microwave Antenna Theory and Design, IEE, London, 1997. [3] S. Silver, J. Opt. Soc. Am. 52 (1962) 131.

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