geometrical theory of diffraction (gtd) uniform

$: Geometrical Theory of Diffraction (GTD) Uniform$
1/52

Geometrical Theory of Diffraction (GTD)

Uniform geometrical Theory ofDiffraction (UTD)

Sangeun Jang, Jaesun Park, Minseok Park

Applied Electromagnetic Technology Laboratory

Department of Electronics and Computer EngineeringHanyang University, Seoul, Korea

2/52

Outline

❑ Geometrical Optics

❑ Amplitude, Phase, and Polarization Relations

❑ Straight Edge Diffraction : Normal Incidence

❑ Oblique Incidence

▪ Straight Edge Diffraction

▪ Curved Edge Diffraction

❑ Uniform Theory of Diffraction

3/52

Geometrical Optics

▪ We shall describe an extension of geometrical optics which does account for these phenomena.

▪ Geometrical optics, the oldest and most widely used theory of light propagation, fails to account for certain

optical phenomena called diffraction.

▪ It is called the geometrical theory of diffraction.

4/52

Amplitude, Phase, and Polarization Relations

▪ When an electromagnetic wave impinges on this curved edge, diffracted rays emanate from the edge whose

leading term of the high-frequency solution for the electric field takes the form

5/52


6/52


▪ The diffracted field of (13-32) should be independent of the location of the reference point 0`, including 𝜌𝑐′ = 0.

7/52


▪ Using (13-33a), the diffracted field of (13-32) reduces to

𝜌𝑐 : distance between the reference point 𝑄𝐷(𝑠 = 0) at the edge and the second caustic of the diffracted rays.

8/52


▪ In general,

1. Wave front curvature of the incident field.

2. Angles of incidence and diffraction, relative to unit vector normal to edge at the point 𝑄𝐷 of diffraction.

3. Radius of curvature of diffracting edge at point 𝑄𝐷 of diffraction.

9/52


▪ When the edge is straight, the source is located a distance 𝑠′ from the point of diffraction, and the observations

are made at a distance s from the diffracted field can be written as

10/52

Straight Edge Diffraction : Normal Incidence

▪ If observations are made on a circle of constant radius 𝜌 from the edge of the wedge, it is quite clear that, in

addition to the direct ray, there are rays that are reflected from the side of the wedge, which contribute to the

intensity at point P.

▪ In order to examine the manner in which fields are diffracted by edges, it is necessary to have a diffraction

coefficient available.

▪ These rays obey Fermat’s principle, that is, they minimize the path between points O and P by including points

on the side of the wedge, and deduce Snell’s law of reflection.

▪ It would then seem appropriate to extend the class of such points to include in the trajectory rays that pass

through the edge of the wedge, leading to the generalized Fermat’s principle.

11/52


▪ By considering rays that obey only geometrical optics radiation mechanisms, we can separate the space

surrounding the wedge into three different field regions. Using the geometrical coordinates of Figure 13-13b, the

following geometrical optics fields will contribute to the corresponding regions :

12/52


▪ With these fields, it is evident that the following will occur :

1. Discontinuities in the field will be formed along the RSB boundary separating regions I and II (reflected

shadow boundary, 𝜙 = 𝜋 − 𝜙′), and along the ISB boundary separating regions II and III (incident

shadow boundary, 𝜙 = 𝜋 + 𝜙′).

2. No field will be present in region III (shadow region).

▪ Since neither of the preceding results should be present in a physically realizable field, modifications and/or

additions need to be made.

13/52


▪ To remove the discontinuities along the boundaries and to modify the fields in all three regions, diffracted fields

must be included.

▪ The fields of an electric line source satisfy the homogeneous Dirichlet boundary conditions Ez = 0 on both faces

of the wedge and the fields of the magnetic line source satisfy the homogeneous Neumann boundary condition𝜕𝐸𝑧

𝜕∅= 0 or Hz = 0 on both faces of the wedge.

▪ The infinitely long line source is parallel to the edge of the wedge, and its position is described by the coordinate

(𝜌′, ∅′).

14/52


A. Modal Solution

where

▪ In acoustic terminology, the Neumann boundary condition is referred to as hard polarization and the Dirichlet

boundary condition is referred to as the soft polarization.

▪ This series is an exact solution to the time-harmonic, inhomogeneous wave equation of a radiating line source

and wedge embedded in a linear, isotropic homogeneous, lossless medium.

15/52


B. High-Frequency Asymptotic Solution

▪ Many times it is necessary to determine the total radiation field when the line source is far removed from the

vertex of the wedge.

where

16/52


▪ For example, if 𝛽𝜌 is less than 1, less than 15 terms are required to achieve a five-significant-figure accuracy.

▪ The infinite series of (13-40a) converges rapidly for small values of 𝛽𝜌.

▪ However, at least 40 terms should be included when 𝛽𝜌 is 10 to achieve the five-significant-figure accuracy.

▪ To demonstrate the variations of (13-40a), the patterns of a unit amplitude plane wave of hard polarization

incident upon a half plane (n = 2) and 90° wedge (n = 3/2), computed using (13-40a), are shown in Fig 13-15.

Computed patterns for the soft polarization are shown in Fig 13-16.

17/52


▪ Whereas (13-40a) represents the normalized total field, in the space around a wedge of included angle WA =(2 − 𝑛)𝜋, of a unity amplitude plane wave incident upon the wedge, the field of a unit amplitude cylindrical

wave incident upon the same wedge and observations made at very large distances can be obtained by the

reciprocity principle.

18/52


▪ In order to derive an asymptotic expression for 𝐹(𝛽𝜌) by the conventional method of steepest descent for

isolated poles and saddle points, it must first be transformed into an integral or integrals of the form

▪ A high-frequency asymptotic expansion for 𝐹(𝛽𝜌) in inverse powers of 𝛽𝜌 is very useful for computational

purposes, because of the slow convergence of (13-40a) for large values of 𝛽𝜌.

where

and the Bessel functions are replaced by contour integrals in the complex z plane, of the form

or

where the paths C and C` are shown in Figure 13-18a.

19/52


20/52


▪ Doing this and interchanging the order of integration and summation, it can be shown that (13-40a) can be

written as

with 𝜉∓ = ∅ ∓ ∅′.

where

▪ Using the series of expansions of

and that

21/52


where the negative sign before C indicates that this integration path is to be traversed in the direction opposite to

that shown in Fig 13-18a.

▪ It is now clear that 𝐹(𝛽𝜌) has been written in an integral of the form of (13-41), which can be evaluated

asymptotically by contour integration and by the method of steepest descent.

22/52


C. Method of Steepest Descent

▪ Using the complex z-plane closed contour CT,

23/52


where SDP±π is used to represent the steepest descent paths passing through the saddle points ±π. We can

rewrite (13-49) as

▪ The closed contour CT is equal to the sum of

▪ The closed contour of the first term on the right side of (13-50) is evaluated using residue calculus and the other

two terms are evaluated using the method of steepest descent.

▪ When evaluated, it will be shown that the first term on the right side of (13-50) will represent the incident

geometrical optics and the other two terms will represent what will be referred to as the incident diffracted field.

▪ A similar interpretation will be given when (13-48b) is evaluated; it represents the reflected geometrical optics

and reflected diffracted fields.

24/52


▪ Using residue calculus, we can write the first term on the right side of (13-50)

▪ To evaluate (13-51),

where

25/52


▪ Equation 13-52 has simple poles that occur when

or

where

▪ Using residue calculus,

provided that

26/52


▪ Thus, (13-54) and (13-51) can be written, respectively, as

▪ The 𝑈(𝑡 − 𝑡0) function in (13-55b) is a unit step function defined as

▪ The unit step function is introduced in (13-55b) so that (13-53b) is satisfied. When 𝑧 = ±𝜋, (13-53b) is

expressed as

for 𝑧𝑝 = +𝜋 and

for 𝑧𝑝 = −𝜋.

27/52


▪ For the principal value of 𝑁(𝑁± = 0), (13-55b) reduces to

▪ This is referred to as the incident geometrical optics field, and it exists provided ∅ − ∅′ < 𝜋 .

▪ In evaluating the last two terms on the right side of (13-50), the contributions from all the saddle points along the

steepest descent paths must be accounted for.

▪ In this situation, however, only saddle points at 𝑧 = 𝑧𝑠 = ±𝜋 occur. These are found by taking the derivative

of (13-48e) and setting it equal to zero. Doing this leads to

▪ The form used to evaluate the last two terms on the right side of (13-50) depends on whether the poles of (13-

53a), which contribute to the geometrical optics field, are near or far removed from the saddle points at 𝑧 =𝑧𝑠 = ±𝜋 .

28/52


▪ When the poles of (13-53a) are far removed from the saddle points of (13-59), then the last two terms on the

right side of (13-50) are evaluated using the conventional steepest descent method for isolated poles and

saddle points.

▪ Doing this, we can write the last two terms on the right side of (13-50), using the saddle points of (13-59)

▪ Combining (13-60a) and (13-60b), it can be shown that the sum of the two can be written as

29/52


▪ Following a similar procedure for the evaluation of (13-48b), it can be shown that its contributions along CT and

the saddle points can be written in forms corresponding to (13-58) and (13-61) for the evaluation of (13-48a).

▪ Equation 13-62b is valid provided the observations are not made at or near the reflection shadow boundary of

Figure 13-13. When the observations are made at the reflection shadow boundary∅ + ∅′ = 𝜋, (13-62b)

becomes infinite.

30/52


▪ If the poles of (13-53a) are near the saddle points of (13-59), then the conventional steepest descent method of

(13-60a) and (13-60b) cannot be used for the evaluation of the last two terms of (13-50) for 𝐹1(𝛽𝜌), and

similarly for 𝐹2(𝛽𝜌) of (13-48b).

▪ One method that can be used for such cases is the so-called Pauli-Clemmow modified method of steepest

descent.

▪ The main difference between the two methods is that the solution provided by the Pauli-Clemmow modified

method of steepest descent has an additional discontinuous function that compensates for the singularity along

the corresponding shadow boundaries introduced by the conventional steepest-descent method for isolated

poles and saddle points.

▪ This factor is usually referred to as the transition function, and it is proportional to a Fresnel integral.

31/52


where

32/52


D. Geometrical Optics and Diffracted fields

▪ After the contour integration and the method of steepest descent have been applied in the evaluation of (13-47),

as discussed in the previous section, it can be shown that for large values of 𝛽𝜌, (13-47) is separated into

where 𝐹𝐺(𝛽𝜌) and 𝐹𝐷(𝛽𝜌) represent, respectively, the total geometrical optics and total diffracted fields

created by the incidence of a unit amplitude plane wave upon a two-dimensional wedge.

33/52


▪ In summary, then, the geometrical optics fields 𝐹𝐺 and the diffracted fields 𝐹𝐷 are represented, respectively, by

34/52


▪ If observations are made away from each of the shadow boundaries so that 𝛽𝜌𝑔± ≫ 1, (13-66a) and (13-66b)

reduce, according to (13-61) and (13-62b), to

which is an expression of much simpler form, even for computational purposes.

35/52


E. Diffraction Coefficients

▪ The incident diffraction function 𝑉𝐵𝑖 of (13-66) can also be written, using (13-66a) through (13-66f), as

where

▪ In a similar manner, the reflected diffraction function 𝑉𝐵𝑟 of (13-66) can also be written, using (13-66a) through

(13-66f), as

where

36/52


▪ With the aid of (13-68) and (13-69), the total diffraction function 𝑉𝐵 𝜌, ∅ ∓ ∅′, 𝑛 = 𝑉𝐵𝑖 𝜌, ∅ − ∅′, 𝑛 ∓

𝑉𝐵𝑟 𝜌, ∅ + ∅′, 𝑛 of (13-66) can now be written as

or

37/52


▪ The soft Ds and hard Dh diffraction coefficients can ultimately be written, respectively, as

38/52


▪ The soft Ds and hard Dh diffraction coefficients can ultimately be written, respectively, as

▪ The formulations are part of the often referred to uniform theory of diffraction (UTD), which are extended to

include oblique incidence and curved edge diffraction.

39/52

Straight Edge Diffraction : Oblique Incidence

▪ The normal incidence and diffraction formulations of the previous section are convenient to analyze radiation

characteristics of antennas and structures primarily in principal planes.

▪ However, a complete analysis of an antenna or scatterer requires examination not only in principal planes but also

in nonprincipal planes, as shown in Figure 13-30 for an aperture and a horn antenna each mounted on a finite size

ground plane.

40/52


▪ The diffraction of an obliquely incident wave by a two-

dimensional wedge can be derived using the geometry

of Figure 13-31.

▪ The diffracted field, in a general form

where : the dyadic edge diffraction coefficient

for illumination of the wedge by plane, cylindrical, conical, or

spherical waves.

▪ The radial unit vectors of incidence and diffraction

▪ The dyadic diffraction coefficient

where Ƹ𝑠′ points toward the point of diffraction

where 𝐷𝑠 : the scalar diffraction coefficients for soft polarizations

𝐷ℎ : the scalar diffraction coefficients for hard polarizations

41/52


▪ We can write in matrix form the diffracted E-field

components that are parallel (𝐸𝛽0𝑑 ) and perpendicular

(𝐸𝜙𝑑) to the plane of diffraction

where

= component of the incident E field parallel to

the plane of incidence at the point of diffraction

QD

= component of the incident E field

perpendicular to the plane of incidence at the

point of diffraction QD

▪ 𝐷𝑠 and 𝐷ℎ are the scalar diffraction coefficients

where

42/52


where 𝜌1𝑖 , 𝜌2

𝑖 = radii of curvature of the incident wave front at 𝑄𝐷𝜌𝑒𝑖 = radius of curvature of incident wave front in the edge-fixed plane of incidence

▪ In general, L is a distance parameter that can be found by satisfying the condition that the total field (the sum of the

geometrical optics and the diffracted fields) must be continuous along the incident and reflection shadow

boundaries.

▪ A general form of L

▪ For oblique incidence upon a wedge, the distance parameter can be expressed in the ray-fixed coordinate system

▪ The spatial attenuation factor A(s’, s) which describes how the field intensity varies along the diffracted ray

43/52


▪ If the observations are made in the far field (𝑠 ≫ 𝑠′ or 𝜌 ≫ 𝜌′), the distance parameter L and spatial attenuation

factor 𝐴(𝑠′, 𝑠) reduce, respectively, to

▪ For normal incidence, 𝛽0 = 𝛽0′ = 𝜋/2.

44/52

Curved Edge Diffraction : Oblique Incidence

▪ The edges of many practical antenna or scattering structures are not straight, as demonstrated in Figure 13-34 by

the edges of a circular ground plane, a paraboloidal reflector, and a conical horn.

▪ In order to account for the diffraction phenomenon from the edges of these structures, even in their principal planes,

curved edge diffraction must be utilized.

45/52


▪ Curved edge diffraction can be derived by assuming an

oblique wave incidence on a curved edge where the

surfaces forming the curved edge in general may be

convex, concave, or plane.

▪ Since diffraction is a local phenomenon, the curved edge

geometry can be approximated at the point of diffraction

𝑄𝐷 by a wedge whose straight edge is tangent to the

curved edge at that point and whose plane surfaces are

tangent to the curved surfaces forming the curved edge.

▪ This allows wedge diffraction theory to be applied directly

to curved edge diffraction by simply representing the

curved edge by an equivalent wedge.

▪ Analytically, this is accomplished simply by generalizing

the expressions for the distance parameter L that appear

in the arguments of the transition functions.

46/52


▪ The general form of oblique incidence curved edge diffraction can be expressed in matrix form .

▪ However, the diffraction coefficients, distance parameters, and spatial spreading factor must be modified to

account for the curvature of the edge and its curved surfaces.

▪ Where 𝐷𝑖 and 𝐷𝑟 can be found by imposing the continuity conditions on the total field across the incident and

reflection shadow boundaries.

▪ 𝐷𝑖 and 𝐷𝑟

where

where 𝜌1𝑖 , 𝜌2

𝑖 = radii of curvature of the incident wave front

at 𝑄𝐷𝜌𝑒𝑖 = radius of curvature of incident wave front in

the edge-fixed plane of incidence

𝜌1𝑟, 𝜌2

𝑟 = principal radii of curvature of the reflected

wave front at 𝑄𝐷𝜌𝑒𝑖 = radius of curvature of the reflected wave

front in the plane containing the diffracted

ray and edge

47/52


▪ For far-field observation, where 𝑠 ≫ 𝜌𝑒𝑖 , 𝜌1

𝑖 , 𝜌2𝑖 , 𝜌𝑒

𝑟 , 𝜌1𝑟 , 𝜌2

𝑟,

▪ If the intersecting curved surfaces forming the curved edge are plane surfaces that form an ordinary wedge, then

the distance parameters are equal.

48/52


▪ The spatial spreading factor 𝐴(𝜌𝑐 , 𝑠) for the curved edge diffraction takes the form

where 𝜌𝑐 = distance between caustic at edge and second caustic of diffracted ray

𝜌𝑒 = radius of curvature of incidence wave front in the edge-fixed plane of

incidence which contains unit vectors Ƹ𝑠′ and Ƹ𝑒𝜌𝑔 = radius of curvature of the edge at the diffraction point

ො𝑛𝑒 = unit vector normal to the edge at 𝑄𝐷 and directed away from the

center of curvature

Ƹ𝑠′ = unit vector in the direction of incidence

Ƹ𝑠 = unit vector in the direction of diffraction

𝛽0′= angle between Ƹ𝑠′ and tangent to the edge at the point of diffraction

Ƹ𝑒 = unit vector tangent to the edge at the point of diffraction

49/52

Uniform geometrical theory of diffraction (UTD)

▪ In numerical analysis, the uniform geometrical theory of diffraction (UTD) is a high-frequency method for

solving electromagnetic scattering problems from electrically small discontinuities or discontinuities in more

than one dimension at the same point.

▪ UTD is an extension of Joseph Keller’s geometrical theory of diffraction (GTD).

▪ The uniform theory of diffraction approximates near field electromagnetic fields as quasi optical and uses ray

diffraction to determine diffraction coefficients for each diffracting object-source combination.

▪ These coefficients are then used to calculate the field strength and phase for each direction away from the

diffracting point.

▪ These fields are then added to the incident fields and reflected fields to obtain a total solution.

50/52


ObstacleIncident Wave

Shadow region

<GTD>

ObstacleIncident Wave

Shadow region

<UTD>

In GTD method, shadow

region cannot be

calculated!!

In UTD method, shadow

region can be

calculated!!

51/52


✓ QE : The point of diffraction.

✓ ed : Edge diffracted ray caused by incident ray at QE.

✓ sr : Surface diffracted ray.

✓ sd : In the case of convex surfaces, the surface ray sheds

a surface diffracted ray sd from each point Q on its path.

✓ ES : The boundary between ed and sd.

✓ SB : The shadow boundary of the incident field.

✓ RB : Simply as the reflection boundary.

52/52


▪ UTD process

1. Close the distance between QE and source point to ‘0’.

2. 𝜌1𝑟 and 𝜌2

𝑟 become ‘0’ and Lr also becomes ‘0’.

3. Equation (55) no longer needs to be calculated.

4. Through the above process to position the ES and SB in the same position.

5. The shadow region can be calculated due to differences in diffraction coefficient of Keller’s.

6. Consequentially, ES and SB are placed in the same line, enabling interpretation of shadow region.

( )( ) ( )

( )

( )

( )( )

' '

'

, 0 ' '0

exp 4, ; 55

2 2 sin cos 2 cos 2

i r

s h

F kLa F kL ajD

k

− + − − = − +

geometrical theory of diffraction (gtd) uniform

Documents