Uniform Diffracted Fields from a Perfectly Conducting Cylindrical Reflector with Modified Theory of Physical Optics

Abstract

The uniform diffracted fields are calculated on PEC cylindrical reflector by Modified Theory of Physical Optics (MTPO). It is aimed to convert the noncontinuous solution to a continuous solution by finding a uniform equation which does not contain any expression converging to 0 in the denominator part. Three axioms of MTPO theory are used to construct the integral equations for the perfectly electrically conducting surface application. The “edge-point” technique is used to find the diffracted field, and uniform solution is to be found via “detour parameter(s).” Finally, the obtained results are to be compared with the nonuniform ones, numerically.

1. Introduction

Physical Optics (PO) is a high frequency method to calculate the reflected and diffracted fields via induced current on a scatterer surface [1, 2]. The incident magnetic field is used to obtain the induced current [3]. However, there exists deficiency while obtaining edge-diffracted field [4]. The PO application of the black half plane surface is a typical example of such a deficiency [5].

The two following points can be defined as the cause of this lack. Firstly, the induced current is assumed on the reflection surface, whereas the aperture surface is not taken into account. Thus, the incident diffracted fields cannot be obtained due to this neglecting. Secondly, the edges are considered as continuous which cause to suppose the equality of reflection and incidence angle in the edge point.

Consequently, the innovative method of Modified Theory of Physical Optics (MTPO) has been derived in order to find the exact solution analytically on edge diffraction and to overcome the deficiencies of PO for edge currents [6]. The result is coherent with the solution of the exact solution of the Helmholtz equation [3]. MTPO was derived by Umul in 2004 via the paper about the scattering of the field from a “perfectly conducting” (PEC) half-plane surface [7]. Afterwards, the studies for the black half plane [8] and for the impedance half plane [9] were carried out by Umul.

Moreover, the study for “perfectly conducting cylindrical reflector” was fulfilled by Yalçın in 2007 [10], and the study for the scattering from “cylindrical parabolic impedance surface” was studied by Umul in 2008 [11]. Lastly, a brief explanation of MTPO and the synthesis of the studies have been presented [12].

There are three axioms of MTPO which are used to evaluate the asymptotic evaluation of scattering integral. First axiom is related to the induced current on a scatterer surface by which the reflected and reflected diffracted fields are obtained. However the incident diffracted field cannot be found. Hence, a second surface is assumed which is called an aperture surface. The induced currents on this surface are defined by the aid of Equivalent Source Theory, and so the incident and incident diffracted fields are calculated. Second axiom is the reflection/transmission angle which depends on the real and aperture surface coordinates and which is a function of integral variables. Last axiom is the definition of the vectors—different from PO—which divide the angle between the incident and reflected fields and which are the factors used to evaluate the induced current [13]. Additionally in MTPO integral, the phase function is expressed as a new form [14].

In this study, the uniform diffracted fields are calculated on PEC cylindrical reflector. Previously, on this surface, diffracted fields have been obtained as a nonuniform solution where the field equation diverges to infinity and which lead to discontinuity in the analysis [10]. Thus, in this work it is aimed to construct a continuous solution by finding a uniform equation and also to compare numerically the diffracted fields on PEC cylindrical reflector for the noncontinuous and continuous solutions of MTPO method.

A time factor 𝑒^𝑗𝜔𝑡 is assumed and suppressed throughout the paper.

2. MTPO Theory

For MTPO, induced currents on the real surface are defined as𝐽_es=𝑛₁×𝐻_𝑡𝑆1, (1a) where 𝐻_𝑡 is the total magnetic field on the real surface (𝑆₁). The induced currents on the aperture surface (𝑆₂) are also calculated via the following formulas: 𝐽_es=(𝑛₂×𝐻_𝑖)_𝑆2,𝐽_ms=−𝑛₂×𝐸_𝑖𝑆2, (1b)(1c)

where 𝐻_𝑖 denotes the incident magnetic field and 𝐸_𝑖 denotes the incident electrical field [7]. The vectors 𝑛₁ and 𝑛₂ are the ones mentioned in the third axiom of MTPO, and they are defined on the real (𝑆₁) and aperture (𝑆₂) surface, respectively. They can be formulized as𝑛₁()⃗()𝑛=sin𝑢−𝛼𝑡−cos𝑢−𝛼𝑛,₂=−cos(𝑣+𝛼)𝑡⃗+sin(𝑣+𝛼)𝑛, (2a)(2b)

where 𝛼 denotes the incidence angle and 𝑡⃗ and 𝑛 are the tangential and normal unit vectors of the surfaces, respectively, as they are shown in Figure 1. Moreover, 𝑢 is equal to (𝛼+𝛽)/2, and 𝑣 is equal to 𝜋/2−(𝛼−𝛽)/2.

Figure 1 

The 𝑥𝑦-plane cross-section of the source and PEC cylinder.

The total electrical field can be written as 𝐸_𝑡=𝐸_is+𝐸_rs, (3) where incident scattered electrical field is𝐸_is=−𝑗𝜔𝜇₀4𝜋_𝑆𝑤𝑛₂×𝐻_𝑖𝑆2𝑒^{−𝑗𝑘𝑅2}𝑅₂𝑑𝑆^′+_𝑆2∇×𝑛₂×𝐸_𝑖𝑆2𝑒^{−𝑗𝑘𝑅2}𝑅₂𝑑𝑆^′, (4a) and reflecting scattered electrical field is 𝐸_rs=−𝑗𝜔𝜇₀4𝜋_𝑆1𝑛₁×𝐻_𝑡𝑆1𝑒^{−𝑗𝑘𝑅1}𝑅₁𝑑𝑆^′. (4b)

Then the related magnetic fields can also be formulized as 𝐻_is=14𝜋_𝑆𝑤∇×𝑛₂×𝐻_𝑖𝑆2𝑒^{−𝑗𝑘𝑅2}𝑅₂𝑑𝑆^′+𝑗𝜔𝜀4𝜋_𝑆2𝑛₂×𝐸_𝑖𝑆2𝑒^{−𝑗𝑘𝑅2}𝑅₂𝑑𝑆^′,𝐻_rs=14𝜋_𝑆1∇×𝑛₁×𝐻_𝑡𝑆1𝑒^{−𝑗𝑘𝑅1}𝑅₁𝑑𝑆^′. (5a)(5b)

3. Application

The incident electrical and magnetic fields from a line source are indicated as 𝐸_𝑖=−𝜔𝜇₀𝐼4𝐻₀⁽²⁾𝑘𝜌₁𝑒_𝑧⃗,𝐻_𝑖=𝐻_𝑖sin𝛼𝑒_𝜙⃗+cos𝛼𝑒_𝜌⃗, (6a)(6b)

respectively. 𝐻₀⁽²⁾(𝑘𝜌₁) is the Hankel function of the second kind where 𝑘 is the number of wavelength and 𝜌₁ is the distance between the source and the reflector. Therefore the reflected magnetic field is 𝐻_𝑟=𝐻_𝑖(−sin𝛽𝑒_𝜌⃗+cos𝛽𝑒_𝜙⃗). The total magnetic field which is used in the integral equation is obtained by 𝐻_𝑡=𝐻_𝑟+𝐻_𝑖=𝐻_𝑖sin𝛼𝑒_𝜙⃗+cos𝛼𝑒_𝜌⃗−sin𝛽𝑒_𝜌⃗+cos𝛽𝑒_𝜙⃗. (7)

The magnitude of the incident ray is equal to 𝜌₁=[𝜌^′2+𝜌²₀−2𝜌^′𝜌₀cos(𝜙^′−𝜙_𝑠)]^1/2 according to Figure 1. The surface current induced on 𝑆₁ is found as 𝐽_es=−𝑒_𝑧⃗2𝐻_𝑖cos𝛼+𝛽2. (8) The other surface currents which are induced on aperture surface are also calculated as𝐽_es=𝑒_𝑧⃗𝐻_𝑖𝐽cos𝑣,_ms=−𝐸_𝑖𝑒_𝜌⃗sin(𝑣+𝛼)−𝑒_𝜙⃗cos(𝑣+𝛼). (9a)(9b)

The surface element can be shown as 𝑑𝑆^′=𝑎𝑑𝜙^′𝑑𝑧^′, where the radius of the cylinder is 𝜌^′=𝑎. By the Green function, which is equal to 𝐺(𝑟,⃗𝑟^′⃗)=𝑒^{−𝑗𝑘𝑅}/𝑅, the equation can be written to solve the integral ^∞_𝑧=−∞𝑒^{−𝑗𝑘𝑅1}𝑅₁𝑑𝑧^′=^∞_{𝑧′=−∞}12𝑗^∞_𝜁=−∞𝐻₀⁽²⁾𝑘²−𝜁²𝜌₁𝑒^{−𝑗𝜁(𝑧−𝑧′)}𝑑𝜁𝑑𝑧^′. (10) By the property ∫^∞_{𝑧′=−∞}𝑒^{𝑗𝜁𝑧′}𝑑𝑧^′=2𝜋𝛿(𝜁), the Green function can be simplified to ^∞_𝑧=−∞𝑒^{−𝑗𝑘𝑅1}𝑅₁𝑑𝑧^′=𝜋𝑗𝐻₀⁽²⁾𝑘𝜌₁. (11) By the utilization of (4a) and (4b) the total scattered field can be obtained as 𝐸_𝑠=𝐸_is+𝐸_rs=−𝑒_𝑧⃗𝑘²𝑍₀𝐼𝑎8×^𝜙0_{𝜙′=−𝜙0}cos𝛼+𝛽2𝐻₀⁽²⁾𝑘𝜌₁𝐻₀⁽²⁾𝑘𝜌₂𝑑𝜙^′+^−𝜙0_{𝜙′=𝜙0}sin𝛼−𝛽2𝐻₀⁽²⁾𝑘𝜌₁𝐻₀⁽²⁾𝑘𝜌₂𝑑𝜙^′. (12) By the Debye’s asymptotic expansion of the Hankel function 𝐻₀⁽²⁾(𝑘𝜐)≈2𝜋𝑒^{−𝑗𝑘𝜐+𝑗(𝜋/4)}√,𝑘𝜐 (13) the total scattered field equation is converted to 𝐸_𝑠=−𝑒_𝑧⃗𝑘𝑍₀𝐼𝑎×4𝜋_{𝜙0𝜙′=−𝜙0}cos𝛼+𝛽2𝑒^{−𝑗𝑘(𝜌1+𝜌2)}√𝜌₁𝜌₂𝑑𝜙^′+^−𝜙0_{𝜙′=𝜙0}sin𝛼−𝛽2𝑒^{−𝑗𝑘(𝜌1+𝜌2)}√𝜌₁𝜌₂𝑑𝜙^′. (14) Thus, the scattering integral equations, which are obtained by MTPO, are in a form of 𝐼_𝑆=^∞_{𝑥′=−∞}𝑓𝑥^′𝑒^{−𝑗𝑘𝜓(𝑥′)}𝑑𝑥^′, (15) where 𝑓(𝑥^′) is represented by the sinusoidal term and 𝜓(𝑥^′) is represented by (𝜌₁+𝜌₂). The phase function has been found as 𝜓𝜙^′=𝜌₁+𝜌₂=𝜌₀cos𝜎+𝜌^′cos𝛼+𝜌cos𝛾+𝜌^′cos𝛽 (16) by utilizing the angles and the distances shown Figure 1. If the phase function is derived with respect to 𝜙^′, the equation 𝜓^′𝜙^′=−𝜌₀sin𝜎𝑑𝜎𝑑𝜙^′−𝜌^′sin𝛼𝑑𝛼𝑑𝜙^′−𝜌sin𝛾𝑑𝛾𝑑𝜙^′−𝜌^′sin𝛽𝑑𝛽𝑑𝜙^′ (17) can be obtained. The angles can be rewritten by Figure 1 as follows:𝜎=𝜋±𝜙_𝑠∓𝜙^′−𝛼,𝛾=𝜋∓𝜙_𝑠±𝜙^′−𝛽. (18a)(18b)

By the sine relations, the following equations are obtained: 𝜌₀=𝑎sin𝛼,𝜌sin𝜎=𝑎sin𝛽.sin𝛾 (19) If the angle equations are written in terms of the derivation of phase function, the derivative of the phase equation becomes as follows: 𝜓𝜙^′=𝑎sin𝛼−𝑎sin𝛽. (20) This equation is equal to 0, as long as 𝛼=𝛽 which eliminates the second term of (12).

The edge diffracted fields can be written by the aid of the quantities 𝛼=𝛼_𝑒,𝛽=𝛽_𝑒,𝜙^′_𝑒=𝜙₀. (21)

In order to calculate diffraction field, the equation 𝐸_𝑑≈∓𝑒_𝑑⃗1𝑓𝑥𝑗𝑘𝜓^′𝑥^′′𝑒^{−𝑗𝑘𝜓(𝑥′)} (22) is used [4, 15]. In this equation𝑓(𝑥^′) is expressed as magnitude, 𝜓(𝑥^′) is expressed as phase function and 𝑒_𝑑⃗ is a unit vector of diffracted field. The minus and plus signs represent the upper and lower limits, respectively. In fact, this equation is the derived version of edge-point technique. The factors in this equation can be found as𝑓_𝑟𝜙₀=−𝑗𝑘𝑍₀𝐼𝑎𝛼4𝜋cos_𝑒+𝛽_𝑒/2𝑙₀𝑙₁,𝑓_𝑖𝜙₀=−𝑗𝑘𝑍₀𝐼𝑎𝛼4𝜋sin_𝑒−𝛽_𝑒/2𝑙₀𝑙₁ (23a)(23b)

for reflected diffracted term and for incident diffracted term, respectively. The phase function and the derivative of the phase function can be also shown as 𝜓𝜙₀=𝑙₀+𝑙₁,𝜓^′𝜙₀=𝑎sin𝛼_𝑒−sin𝛽_𝑒, (24a)(24b)

where 𝑙₀=𝜌₀cos𝜎+acos𝛼_𝑒,𝑙₁=𝜌cos𝛾+𝑎cos𝛼_𝑒. (25a)(25b)

The total edge diffraction is calculated as 𝐸_𝑑=𝐸_id+𝐸_rd=𝑒_𝑧⃗𝑍₀𝐼18𝜋sin𝛼_𝑒−𝛽_𝑒−1/2cos𝛼_𝑒+𝛽_𝑒×𝑒/2_{−𝑗𝑘(𝑙0+𝑙1)}𝑙₀𝑙₁. (26) According to the calculation, the edge diffraction field result contains sinusoidal terms at the denominator [10]. It can be easily seen that diffracted field value diverges to infinity as 𝛼sin^𝑒−𝛽_𝑒2𝛼⟶0,cos^𝑒+𝛽_𝑒2⟶0. (27)

The discontinuity can be detected on the values of transition regions. 𝛼_𝑒=𝛽_𝑒 is the reflection boundary, whereas 𝛼_𝑒=𝜋−𝛽_𝑒 is the shadow boundary. Thus, the “detour parameter” method is to be used to eliminate the value equalizing the denominator to zero [16–18]. The formula of detour parameter can be shown as 𝜉_𝑖,𝑟=−𝜓_𝑖,𝑟−𝜓_𝑑. (28)

The “detour parameter” is defined for the Fresnel function equation, and the Fresnel function can be written in terms of “detour parameter” as [18] 𝐹𝜉̂_𝑖=𝐹𝜉_𝑖sgn𝜉_𝑖=𝑒^{−𝑗𝜉2𝑖−𝑗𝜋/4}2√𝜋𝜉_𝑖,𝐹𝜉̂_𝑟=𝐹𝜉_𝑟sgn𝜉_𝑟=𝑒^{−𝑗𝜉2𝑟−𝑗𝜋/4}2√𝜋𝜉_𝑟, (29a)(29b)

where 𝐹𝜉_𝑖,𝑟=𝑒^𝑗𝜋/4√𝜋^∞_{𝜉,𝑖,𝑟}𝑒^−𝑗𝑡2𝑑𝑡. (30)

If (26) is rewritten as 𝐸_𝑑=𝐸_id+𝐸_rd=𝑒_𝑧⃗𝑘𝑍₀𝐼18𝜋sin𝛼_𝑒−𝛽_𝑒−1/2cos𝛼_𝑒+𝛽_𝑒×𝑒/2_{−𝑗𝑘𝑙0}𝑒^{−𝑗𝑘𝑙1}𝑘𝑙₀𝑘𝑙₁, (31) the conversion to the Fresnel function (𝜉_𝑖,𝑟) can be satisfied by expressing in terms of (29a) and (29b) as follows: 𝐸_𝑑=𝑒_𝑧⃗𝑘𝑍₀𝐼4√𝑒2𝜋2𝑘𝑙₀sin𝛼_𝑒−𝛽^{−𝑗𝑘𝑙0(2sin2((𝛼𝑒−𝛽𝑒)/2)+cos(𝛼𝑒−𝛽𝑒))}_𝑒−𝑒/2_{−𝑗𝑘𝑙0(2cos2((𝛼𝑒+𝛽𝑒)/2)−cos(𝛼𝑒+𝛽𝑒))}2𝑘𝑙₀cos𝛼_𝑒+𝛽_𝑒𝑒/2𝑘𝑙^{−𝑗𝑘𝑙1}₁. (32) Hence the arguments of the Fresnel function can be obtained as 𝜉_𝑖=2𝑘𝑙₁𝛼sin^𝑒−𝛽_𝑒2,𝜉_𝑟=2𝑘𝑙₀𝛼cos^𝑒+𝛽_𝑒2. (33)

As a result of these operations, explained previously, 𝐸_𝑑=𝐸_id+𝐸_rd=𝑒_𝑧⃗𝑘𝑍₀𝐼𝑒^𝑗(𝜋/4)2√𝑒2𝜋𝑘𝑙^{−𝑗𝑘𝑙0}₀×𝑒^{−𝑗𝑘𝑙1cos(𝛼𝑒−𝛽𝑒)}𝐹𝜉_𝑖−𝑒^{𝑗𝑘𝑙1cos(𝛼𝑒+𝛽𝑒)}𝐹𝜉_𝑟 (34) uniform equations are obtained.

4. Numerical Analysis

The obtained results are to be compared numerically with the results of the nonuniform solution found by MTPO method. All the analyses are to be done with the value 𝑘𝜌₁=100. Firstly the nonuniform field is analyzed for 𝜙₀=𝜋/7. As it can be seen in Figure 2, the magnitude of the diffracted fields diverges to infinity at the value 𝜙=15𝜋/7, since sin((𝛼_𝑒−𝛽_𝑒)/2) converges to 0 at that value. Also for 𝜙=13𝜋/7, the denominator of the second term of the nonuniform solution converges to 0, as cos((𝛼_𝑒+𝛽_𝑒)/2) converges to 0, and the magnitude diverges to infinity.

Figure 2 

The nonuniform solution at 𝜙0=𝜋/7.

When the “detour parameter” is applied, the uniformization of the solution leads the results to have a finite output for all values and that can be visualized in the Figure 3.

Figure 3 

The uniform solution at 𝜙0=𝜋/7.

As uniform and nonuniform solutions are checked together, the result in Figure 4 can be obtained.

Figure 4 

The nonuniform and uniform solution at 𝜙0=𝜋/7.

Also these two types of solutions can be also analyzed for 𝜙₀=𝜋/4. In this analysis, for nonuniform solution at the values 𝜙=7𝜋/4 and 𝜙=9𝜋/4 the terms cos((𝛼_𝑒+𝛽_𝑒)/2) and sin((𝛼_𝑒−𝛽_𝑒)/2) converge to 0, respectively, as indicated in Figure 5. The solution is continuous for the uniform solution.

Figure 5 

The nonuniform and uniform solution at 𝜙0=𝜋/4.

5. Conclusion

In this study, the uniform diffracted fields are calculated on PEC cylindrical reflector with Modified Theory of Physical Optics (MTPO). It is aimed to construct a continuous solution by finding a uniform equation. MTPO theory has been explained with three axioms to construct the integral equations. The application of this theory on cylindrical electrically conducting surface has been carried out. For the diffraction the “edge-point” technique has been used, and uniform solution has been found out via “detour parameter(s).” Furthermore, the obtained results have been compared with the nonuniform ones numerically, and uniformization of the solution has been detected. Finally, by this innovative method, which has been derived in 2004, some other studies on parabolic electrically conducting surface can also be achieved in the near future.

Acknowledgment

This study is a part of M.S. thesis of Mücahit Sarnık.