Consider an elliptic cylinder aligned along the z-axis as shown in Figure 1.

The elliptic coordinates are related to Cartesian coordinates by the relations

Eq. 1

where on the surface of the elliptic cylinder (e.g. )

Eq. 2

The unit vectors on the surface of the elliptic cylinder are given by

Eq. 3

where and an infinitesimal surface patch is given by .

A plane wave incident from is given by

Eq. 4

when this field is evaluated on the surface of the elliptic cylinder and denotes the orientation of the electric field. The magnetic field associated with Eq. 4 is given by

Eq. 5

where Y is the admittance and k is the wavenumber of the media through which the plane wave is propagating. As the source of the plane wave recedes to infinity (e.g. ), the curl may be approximated as and accordingly Eq. 5 is approximated by

Eq. 6

or dropping the ~, we have

Eq. 7

where the magnetic field is oriented in the direction .

The physical optics (PO) integral is given by

Eq. 8

where the primed coordinates indicate integration variables and the surface of integration corresponds to the lit surface of the cylinder. Other useful formulae relate the electric field to the magnetic field when the observer is far from the source

Eq. 9

We now complete the derivation for e-pol and h-pol separately.

For e-pol, we find that

Eq. 10

and

Eq. 11

where

Eq. 12

The resulting electric field obtained by combining Eq. 8 through Eq. 12

Eq. 13

where the following representations are made

Eq. 14

and L is the length of the cylinder. In Eq. 13, the lit region of the elliptic cylinder surface is shown to be of the incident azimuth angle, , and the axial dependency is represented by . This function will be defined in a later section.

For h-polarization, the equivalent to Eq. 10 is given by

Eq. 15

and

Eq. 16

where

Eq. 17

The far-zone PO fields are then given by

Eq. 18

and

Eq. 19

The integrals in Eq. 13, Eq. 18, and Eq. 19 may be evaluated using either numerical integration or the stationary phase method.

Consider the integral

Eq. 20

For the elliptic cylinder, is , , or as appropriate and the phase function and its derivatives are given by

Eq. 21

The stationary phase point, , is given by

Eq. 22

Additional useful formulae are

Eq. 23

Since , then the argument of the square root in Eq. 23 is greater than zero and hence the sign of is positive. Accordingly, Eq. 20 reduces to

Eq. 24

Eq. 23 and Eq. 24 can now be used to evaluate Eq. 13, Eq. 18, and Eq. 19.

For E-polarization, we have

Eq. 25

whereas for H-polarization, we have

Eq. 26

and

Eq. 27

The remaining information required to evaluate the PO integrals is the axial function. This is the integration of the axial phase function over the length of the cylinder

Eq. 28

where the function denotes a weight function over the length of the cylinder. Two weight functions are typically used: uniform and Gaussian. For a uniform window, the axial function reduces to

Eq. 29

where . For a Gaussian window

Eq. 30

where is the error function and the axial function becomes

Eq. 31

In Eq. 30, the constant is chosen so that the weight function is nearly zero at the endpoints . For example, choosing results in and this value minimized integral truncation effects.