On relating the perturbation theory and random cylinder generation to study scattered field

Physical Communication 39 (2020) 101003

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On relating the perturbation theory and random cylinder generation to study scattered field ∗

Fahad Masood, Usama Amin, Muhammad Arshad Fiaz , Muhammad Aqueel Ashraf Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan

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Article history: Received 26 September 2019 Accepted 3 January 2020 Available online 13 January 2020 Keywords: Scattering Radar cross section Perturbation theory Random media

a b s t r a c t Perturbation theory (PT) is used to calculate the scattered field from a slightly perturbed cylinder for both transverse magnetic (TM) and transverse electric (TE) polarized incidence fields. A relation between the PT and the model for random cylinder (RC) generation is developed which is important for the PT to remain valid during the numerical simulations. The scattered field expressions obtained using the PT are verified using the result obtained by method of moments (MoM). The solution obtained by using the PT is more efficient in terms of simplicity and computational time. The scattering cross section is analyzed by varying the perturbation parameter B and the parameter describing the RC generation N. It is noted that both B and N controls the roughness of the cylinder and hence affect the scattering pattern. The scattered field for TE polarized incident field is more sensitive to the change in B and N than TM case. © 2020 Elsevier B.V. All rights reserved.

1. Introduction Scattering from random objects is an area of special interest which can be applied in tropospheric wave propagation, wireless communication and radar technology etc. Scattering from the random media had been discussed as early as 1950 by Rice and Cartwright [1,2]. Their main focus was on planar random surfaces. Liang et al. reported the scattering by rough surfaces with large heights and slopes [3]. Further, scattering from a three-dimensional arbitrary layered media having periodic rough interfaces was presented in [4,5]. In literature, small amount of effort had been made to study the scattered field from cylindrical objects with random shape and radius as compared with the random planar surface scattering. Scattering from the deterministic cylindrical structures has widely been reported in the literature [6,7]. Scattering from a perfect electric conductor (PEC) cylinder with random radius placed under a slightly rough surface was studied by Fiaz et al. [8]. This work has been done analytically using the PT for rough surface scattering. Recently, the average cross polarized scattering from a random PEMC cylinder had been reported in [9]. Scattered field from an object of arbitrary shaped cylindrical structure had been discussed by Lawrence and Sarabandi [10,11]. Ashraf and Rizvi calculated the scattering from ∗ Corresponding author. E-mail addresses: [email protected] (F. Masood), [email protected] (U. Amin), [email protected] (M.A. Fiaz), [email protected] (M.A. Ashraf). https://doi.org/10.1016/j.phycom.2020.101003 1874-4907/© 2020 Elsevier B.V. All rights reserved.

a random cylinder using the MoM [12]. The PT requires small surface variations as compared to wavelength of incident field and it is a computationally efficient method while the MoM is shape dependent and the computational time increases manifolds if the scatterer size is large. In this paper, scattering from a slightly perturbed cylinder having shape perturbations is discussed. Both the TM and TE polarized incident fields are assumed. The solution is verified by implementing the well known MOM. Different models are reported in literature which can be used to generate the random cylinder [12–14]. In this research, a random cylinder has been generated using the model reported in [12]. This model is not limited by the choice of correlation function as in [13,14]. It gives a better control on the shape of the resultant cylinder and the cylinder will have no negative radius at any angle. An effort has been made to relate the PT conditions with the RC generation. This is very important as the conditions for the PT to be valid must not be violated in the generation of the random cylinder. Problem formulation is given in Section 2 and in Section 3, the numerical results are presented. The time dependency is taken as e jω t . 2. Problem formulation A random PEC cylinder with an arbitrary cross section is assumed as shown in Fig. 1. A plane wave is made incident to illuminate the scatterer. In polar coordinates, the radius of the

2

F. Masood, U. Amin, M.A. Fiaz et al. / Physical Communication 39 (2020) 101003

(cn0 + cn1 kB + cn2 (kB)2 + · · ·)ejnθ = 0

(7)

Comparing the coefficients of the zeroth order terms the following expression is obtained for cn0 cn0 = −

Jn (ka)

(8)

(2)

Hn (ka)

For B = 0, there is no perturbation and the resultant scattered field will be that of an unperturbed circular cylinder given by [15]

Vs = − Fig. 1. A circular random PEC cylinder illuminated by TM/TE polarized incident field.

random cylinder in terms of unperturbed cylinder radius a and perturbation constant B are expressed as

ρ ′ = a + Bp(θ )

(1)

where B satisfy the relation B ≪ λ. The TM or TE polarized incident field is denoted by Vi which represents electric or magnetic field component respectively. It is given by +∞ ∑

Vi =

(2)

n=−∞

Vs =

n=−∞

+∞ ∑

(2)

(−j)n cn1 Hn(2) (ka)ejnθ =

n=−∞

Hn(2) (kρ )ejn(φ−φi )

(9)

+∞ 2jp(θ ) ∑

π ka

ejnθ

(−j)n

n=−∞

(2)

(10)

Hn (ka)

The higher terms of (kB)2 can be neglected as kB ≪ 1. The Fourier series representation of p(θ ) is +∞ [ ∑

Ym ej(mθ +ψm )

p(θ ) = Re

]

(11)

Putting the above equation in Eq. (10) we get the solution for cn1 ,

cn1 =

+∞ ∑

2jn+1

(−j)q

Yn−q ejψn−q

π kaHn(2) (ka) q=−∞

(2)

The scattered field up to first order is given by

cn1

Vs =

where etc. are the unknown coefficients and can be found by imposing the boundary conditions at the perturbed cylinder surface. 2.1. Solution for TM polarization

+∞ ∑

[

V0 (−j)n −

n=−∞

+∞ ∑

(−j)q

To find the unknown scattering coefficients, the following boundary conditions will be applied at the surface of the cylinder

Jn (ka) (2) Hn (ka)

Yn−q ejψn−q (2)

Hq (ka)

q=−∞

(12)

Hq (ka)

V0 (−j)n (cn0 + cn1 kB + cn2 (kB)2 + · · ·)Hn(2) (kρ )ejn(φ−φi ) (3)

n=−∞

cn0 ,

Jn (ka) Hn (ka)

The first order scattering coefficient cn1 can be calculated by comparing the first order terms as

E

where V0TE = ωµ0 and V0TM = k0 . By writing perturbation series, 0 0 the scattered field can be represented as [11] +∞ ∑

V0 (−j)n

m=−∞

V0 (−j)n Jn (kρ )ejn(φ−φi ) E

+∞ ∑

+

2jn+1

π kaHn(2) (ka)

]

kB Hn(2) (kρ )ejn(φ−φi )

(13)

In the next section, the unknown coefficients for the TE polarization will be derived. 2.2. Solution for TE polarization

Vzi + Vzs = 0

(4)

After putting the expressions for the incident and scattered fields into boundary condition, we get +∞ ∑

(−j)n Jn (ka)ejn(φ−φi )

In the TM case, it is seen that the electric field Ez is always tangent to the surface of the cylinder while for the TE case, Eφ is always tangent to the cylinder boundary. So, the tangential electric field component will be found using the unit tangent vector defined by [11]

n=−∞

+

+∞ ∑

(−j)

n

(cn0

+

cn1 kB

+

cn2 (kB)2

)Hn(2) (ka)ejn(φ−φi )

+ ···

=0

(5)

n=−∞

{

}

Fn (kρ ′ ) = Fn k(a) + kBp(θ ) = Fn (ka) + Fn′ (ka)kBp(θ ) + · · · . Putting the above equation in Eq. (5), we get

+∞ ∑ n=−∞

1+ζ

2

+√

φˆ

(14)

1 + ζ2 ∂ p(θ )

(6)

where ζ = a+Bp(θ ) and ∂ p(θ ) = ∂θ . Using the tangential components of the electric field, the boundary condition is given by i s Vtan + Vtan =0

(−j)n [Jn (ka) + Jn′ (ka)kBp(θ ) + · · ·]ejnθ i Vtan =

[

(−j)n Hn(2) (ka) + Hn′(2) (ka)kBp(θ ) + · · ·

(15)

The tangent electric field can be calculated using the Maxwell’s equation as [11]

n=−∞

+

ζ ρˆ

B∂ p(θ )

The Taylor series expansion of Bessel and Hankel functions about ka can be written as

+∞ ∑

tˆ = √

]

1

+∞ ∑

1 j−n √ jωϵ 1 + ζ2 n=−∞

[

] ζ jn ′ ′ ′ J (k ρ ) − kJ (k ρ ) ejn(φ−φi ) n n ρ′ (16)

F. Masood, U. Amin, M.A. Fiaz et al. / Physical Communication 39 (2020) 101003

given by

and s = Vtan

1

+∞ ∑

jωϵ

n=−∞

[

1

√

1 + ζ2

|kp(θ )| ≪ 1

j−n (cn0 + cn1 kB + cn2 (kB)2 + · · ·)

] ζ jn (2) ′ ′ ′(2) (k ρ ) ejn(φ−φi ) (k ρ ) − kH H n ρ′ n

(17)

Substituting the expressions for the tangential incident and scattered fields in Eq. (15), we obtain

−

+∞ ∑

1 jωϵ

j−n √

n=−∞

[ ζ jn

1 1+ζ

ρ′

2

{Jn (ka) + Jn′ (ka)kBp(θ ) + · · ·}

+∞ ∑

j−n (cn0 + cn1 kB + cn2 (kB)2 + · · ·) √

1

(18)

Jn′ (ka)

(19)

′(2)

Hn (ka)

The resultant scattered field for the unperturbed circular cylinder can be written as +∞ ∑

Jn′ (ka)

[

j−n −

]

′(2)

Hn (ka)

n=−∞

Hn(2) (kρ )ejn(φ−φi )

(20)

To calculate cn1 , we equate the first order terms of Eq. (18) +∞ ∑

j−n Hn′(2) (ka)cn1 ejnθ =

n=−∞

+∞ ∑

j− n

[

p(θ )Tn

−

′(2)

Hn (ka)

n=−∞

p′ (θ )2n

]

π (ka)3 Hn′(2) (ka)

ejnθ

(21) ′(2)

′′

′′(2)

′

where Tn = Jn (ka)Hn (ka) − Jn (ka)Hn the p′ (θ ) is given by +∞ [ ∑

jmYm ej(mθ +ψm )

p′ (θ ) = Re

(k1 a). The expression for

]

(22)

m=−∞

Substituting the above equation in Eq. (21), cn1 can be written as cn1

=

+∞ ∑

jn

jψn−q

Yn−q e j−q ′(2)

′(2)

Hn (ka)

Hq (ka)

q=−∞

[ Tq −

j2(n − q)q

] (23)

π (ka)3

Vs =

{Tq −

1 jωϵ

[

j−n −

π (ka)3

+

′(2)

Hn (ka)

n=−∞

j2(n − q)q

Jn′ (ka)

]

}kB

′(2)

Hn (ka) jn(φ−φi )

ρ )e

Hn(2) (k

jn

+∞ ∑ q=−∞

j−q

(26)

In [12], the finite term Fourier series is used to generate a random cylinder and it is given by N ∑

Yn cos(nθ + ψn )

(27)

(28)

The derivative of p(θ ) can be expressed as N

∂ p(θ ) ∑ = nYn sin(nθ + ψn ) ∂θ

The maximum value of the sum of the above series is N(N + 1)B S= (30) 2 From Eqs. (29), (30) and the second condition given in Eq. (26), we get

⏐ N(N + 1)B ⏐ ⏐ ⏐ ⏐ ⏐≪1

(31) 2 From Eqs. (28) and (31), the conditions for RC generation can be written as ⏐ N(N + 1)B ⏐ ⏐ ⏐ |kNB| ≪ 1, ⏐ (32) ⏐≪1 2 For N < (4π/λ − 1), the first condition dominates while the second condition dominates when N > (4π/λ − 1). It means that the perturbations must be small and the condition for gentle slope will be automatically satisfied for smaller values of N. For our simulations, N is a small number and only the condition given in Eq. (28) must satisfy. Fig. 2 shows the random cylinder generated by using Eq. (27). In Fig. 2a, N is kept constant while B is kept constant in Fig. 2b. It can be noticed that the randomness of the cylinder varies as the value of B or N is changed. Fig. 3 shows the scattering width σ of a random PEC cylinder obtained using the PT and MoM, where ∞

Yn−q ejψn−q

(29)

n=1

2π ⏐ ∑

σ2D =

The resultant scattered field is given by +∞ ∑

|≪1

|kNB| ≪ 1

In the above equation, the Taylor series expansion of the Hankel functions is used. Balancing the zeroth order terms of Eq. (18), the unknown scattering coefficient cn0 can be obtained as

Vs =

∂θ

It is assumed that the random amplitude Yn is independent and identically distributed between [−B, B] and also the phase ψn between [−π, π]. In this model, p(θ ) is zero mean and its variance is NB2 /6 where N denotes the number of terms. The maximum value of the sum of the series is |NB|. Putting the maximum value of p(θ ) in the first condition defined in Eq. (25), we get

jωϵ 1 + ζ2 n=−∞ [ ζ jn {H (2) (ka) + Hn′(2) (ka)kBp(θ ) + · · ·} ρ′ n ] − k{Hn′(2) (ka) + Hn′′(2) (ka)kBp(θ ) + · · ·} ejnθ = 0

cn0 = −

|

(25)

n=1

] − k{Jn (ka) + Jn (ka)kBp(θ ) + · · ·} ejnθ 1

and ∂ p(θ )

p(θ ) =

′′

′

−

3

λ

⏐ ⏐

⏐2 ⏐ ξn (cn0 + cn1 kB + cn2 (kB)2 + · · ·) cos(nθ )⏐

(33)

n=0

and

′(2)

Hq (ka)

ξn = (24)

From Eq. (23), it is seen that the function p′ (θ ) is not required, only coefficients are required. 3. RC generation and results In this section, the RC generation is discussed and relationship to the PT is developed. For the PT to be valid, the conditions are

{

1 n=0 2 n ̸= 0

(34)

Result obtained using the MoM compared well with that obtained by the PT for a slightly perturbed cylinder. The values of radius a = 0.5λ, B = 0.005 and N = 5. Figs. 4 and 5 show the scattered field from a perturbed PEC cylinder where a = 0.5λ and N = 5. For reference, result for a circular cylinder having a = 0.5λ is also shown. For a circular cylinder B = 0 and for a random cylinder B = 0.005. It is noted that for the small value of the perturbation parameter B,

4

F. Masood, U. Amin, M.A. Fiaz et al. / Physical Communication 39 (2020) 101003

Fig. 2. Generation of the random cylinder with a = 0.5λ.

Fig. 3. Comparison of scattered field from a perturbed PEC cylinder using the PT with that obtained using the MoM where a = 0.5λ, B = 0.005 and N = 5.

Fig. 4. Scattering cross section of a perturbed PEC cylinder for TM polarization where a = 0.5λ and B = 0.005.

the scattered field from the random cylinder is almost equal to that of the circular cylinder. Figs. 6 and 7 show the scattering cross section of a perturbed PEC cylinder with different value of the perturbation parameter where N = 5. When B is kept small, a little variation in the scattered field is observed while the large variations can be found when B is increased. Figs. 8 and 9 show the scattering cross section from a perturbed cylinder with different values of N where B = 0.005. Small variations in the scattered field is observed when N is kept small while the large variations can be seen when N is increased. Scattering for TE polarized field is more sensitive to the change in B and N than TM case.

4. Conclusion In this study, electromagnetic scattering from a random PEC cylinder is studied using the relationship between the PT and RC generation model. The numerical results obtained using the PT are compared well with the MoM. It is noticed that both N and B control the amount of roughness and affect the scattered field from the random cylinder. The variations are small for small values of B and N while an increase in the value of N and B results in large variations in the scattered field. Scattered field for TE

Fig. 5. Same as Fig. 4 except that TE polarization is assumed.

polarized incidence is more sensitive to B and N as compared to TM polarized incidence.

F. Masood, U. Amin, M.A. Fiaz et al. / Physical Communication 39 (2020) 101003

Fig. 6. Scattering cross section of a perturbed PEC cylinder for different values of B where a = 0.5λ and TM polarized incident field is assumed.

5

Fig. 9. Same as Fig. 8 except that TE polarized incident field is assumed.

Declaration of competing interest No potential conflict of interest was reported by the authors. Acknowledgments This work was supported by Quaid-i-Azam university research fund grant No. FNS/17-1789. References

Fig. 7. Same as Fig. 6 except that TE polarized incident field is assumed.

Fig. 8. Scattering cross section of a perturbed PEC cylinder for different values of N where a = 0.5λ and TM polarized incident field is assumed.

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[13] C. Eftimiu, Electromagnetic scattering by rough conducting circular cylinders I: Angular corrugation; II: Axial corrugation, IEEE Trans. Antennas Propag. 36 (5) (1988) 651–658, 659–663. [14] H. Ogura, H Nakayama, Scattering of waves from a random cylindrical surface, Math. Phys. 29 (4) (1988) 851–860. [15] C.A. Balanis, Advanced Engineering Electromagnetics, Wiley, New York, 2012. Fahad Masood received his master’s degree in electronics from university of Peshawar Pakistan in 2009. He is currently pursuing his Ph.D. degree in electronics from Quaid-i-Azam university Islamabad Pakistan. his main research interest includes computational and numerical techniques for electromagnetic scattering from random objects in remote sensing.

Usama Amin received his Bachelor’s degree in Electronics from Comsats University, Islamabad Pakistan. He has also received his MPhil degree in Electronics from Quaid-i-Azam University Islamabad, Pakistan in 2018. His main research work was scattering from perfect electric conductor cylinder of random shape.

Muhammad Arshad Fiaz obtained the degree of master of science and philosophy in the subject of electronics from Quaid-i-Azam university Islamabad, Pakistan in 2006 and 2008, respectively. In 2008, he attended the Doctoral School in Biomedical Engineering, Electromagnetism, and Telecommunications at department of Engineering Roma Tre university Rome, Italy. He received the Ph.D. degree in March 2012. Since May 2012, he has been working as assistant professor at department of Electronics, Quaid-i-Azam university Islamabad, Pakistan. His research interests include electromagnetic scattering from rough surface, detection and shape reconstruction of buried objects, and metamaterials.

Dr. Muhammad Aqueel Ashraf received the Ph.D. degree in Electronics from Quaid-i-Azam University, Islamabad Pakistan in 2009. He is currently serving as Associate Professor at Department of Electronics, Quaid-i-Azam University Islamabad. He is an active member of International Society of Optics and Photonics (SPIE). His main research interest includes electromagnetic scattering, computational and numerical techniques in electromagnetics, wireless communication.