High-Frequency Methods

• Full wave methods, such as MoM, FEM, FDTD, and etc., require

large amount of memory.

• When the structure is much large comparing to wavelength, optics

approximation can be used, i. e., high-frequency methods.

Geometrical Optics

• Only consider propagation of energy and amplitude without

considering polarization.

• Ray: direction of energy propagation. Straight line in a

homogeneous media.

• Ray tube: a bundle of ray containing energy.

• Eikonal: Surface normal to ray, wavefront. Equivalently, the equal

phase surface of an electromagnetic waves.

Energy conservation

Within a ray tube, energy must conserve.

or

where

: power density at point .

: ray tube area at point .

: power density at point .

: ray tube area at point .

Considering radii of curvature, we have

Example:

Plane wave: , .

Cylindrical wave: , .

Spherical wave; , .

Assume at point 0, the phase is , then,

Two dimensional Case: Reflection from a curve surface

: radius of curvature of reflected ray tube.

: radius of curvature of incident ray tube.

: radius of curvature of the surface.

: incident angle.

Example: Plane wave incident on a sphere

Plane wave:

Sphere:

Normal incidence:

Therefore, . Consider 3-D case,

where is the electric field amplitude at the incident

point.

Radar cross-section:

We have

Wedge Diffraction Theory• From previous sphere scattering example, geometry optics can not

predict the fields in the shadow region.

• Must consider diffracted rays at edges, tips or curve surfaces.

Diffraction of an Infinite Half-plane

Assume plane wave incidence.

Using Huygen’s principle

Let . If for , , then

which means contribution from the field

near the edge is dominate.

Let , for small , then

Extending the limit to infinity since the contribution from to infinity

is small, we have

The integral is Fresnel integral and can be found.

Shadow Boundary

Assume plane wave incident at an angle

.

Region I: incident and reflected waves

plus diffracted waves.

Region II: incident waves plus diffracted

waves.

Region III: diffracted waves.

Then, the total field can be consistent of

four components:

where

: incident wave in region I and II.

: reflected wave in region I.

: diffracted wave in all regions due to the incident waves.

: diffracted wave in all regions due to the reflected waves.

Example: , shadow boundary at and .

Calculation of Diffracted Fields

1. Infinite half plan formula:

where , .

2. Asymptotic wedge diffraction formula

where interior wedge angle is .

Accurate only outside shadow boundary and for

Property of diffracted wave: cylindrical wave caused by a line current.

amplitude , phase dependancy .

In the above formula, results are inaccurate near shadow boundary and

when . Uniform Theory of Diffraction (UTD) are required, which

is a more complicated formula improved upon previous formula using

ray optics.

E-Plane Analysis of Horn Antennas

• 2-dimensional

• magnetic line source at apex producing cylindrical wave.

• two diffracted waves generated by the two horn edges.

• only consider single or double diffraction.

• Total field = direct field + diffracted fields.

• Direct field calculated by cylindrical wave assumption and only

exist in the region bounded by the extension line of the horn.

• Diffracted field calculated by UTD in all region.

Cylindrical Parabolic Reflector Antenna

• 2-dimensional

• line source at focus with feed pattern

• two diffracted waves generated by the two edges.

• only consider single diffraction.

• Total field = direct field + diffracted fields.

• Direct field calculated by aperture integration.

• Diffracted field calculated by UTD in all region.

Radiation by a Monopole on a Finite Ground Plane

• Direct field: dipole radiation pattern.

• Diffracted field: edges.